FRANKLIN INSTITUTE LIBRARY 
 
 PHILADELPHIA, PA. 
 
 REFERENCE 
 
CONCRETE 
 ENGINEERS' HANDBOOK 
 
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CONCRETE 
 ENGINEERS' HANDBOOK 
 
 DATA FOR 
 
 THE DESIGN AND CONSTRUCTION OF PLAIN 
 AND REINFORCED CONCRETE STRUCTURES 
 
 BY 
 
 GEORGE A. HOOL, S. B. 
 
 PROFESSOR OF STRUCTURAL ENGINEERING, THE UNIVERSITY 
 OP WISCONSIN 
 
 AND 
 
 NATHAN C. JOHNSON, M. M. E. 
 
 CONSULTING CONCRETE ENGINEER, NEW YORK CITY 
 
 ASSISTED BY 
 
 S. C. HOLLISTER, B. S. 
 
 RESEARCH ENGINEER, CORRUGATED BAR CO. 
 
 with chapters by 
 Harvey Whipple, Adelbert K LIjiLls,., - , « ^. „ 
 Walter S. Edge, A. G. Hillberg ' ' 
 
 AND Leslie H. ALLfi;<j ^ i.^; ;^ = > 
 
 First Edition 
 Third Impression 
 
 McGRAW-HILL BOOK COMPANY, Inc. 
 239 WEST 39TH STREET. NEW YORK 
 
 LONDON: HILL PUBLISHING CO., Ltd. 
 
 6 & 8 BOUVERIE ST., E. C. 
 1918 
 
COMS 
 
 TA 
 
 IfiS 
 
 Copyright, 1918, by the 
 McGraw-Hill Book Company, Inc. 
 
 First printing^ May, 1918 
 
 Second printing, November, 1918 
 Third printing, April, 1919 
 
 Total Issue, 11,000 
 
 X 11 K MAPIiE PKESS TT O R K F A. 
 
PREFACE 
 
 This handbook has been prepared to make available in concise form the best of present day 
 knowledge concerning concrete and reinforced concrete and to present complete data and 
 details, as well as numerous tables and diagrams, for the design and construction of the principal 
 types of concrete structures. Although intended as a working manual for the engineer, the 
 first few sections of the book may be read with profit by any one engaged in concrete work. 
 In these sections an effort has been made to present the latest authoritative knowledge in 
 regard to the making and placing of concrete in such form that it may be applied in the field 
 to the betterment of construction. 
 
 In preparing this book the authors have been ably assisted by Mr. S. C. Hollister, Research 
 Engineer of the Corrugated Bar Company. Special credit is due Mr. Hollister for his applica- 
 tion of the slope-deflection method to the development of formulas for rigid frame structures; 
 also for material relating to flexure of annular sections and to restraint of standpipe sides 
 by connection with the base, this material being published here through the courtesy of 
 the Corrugated Bar Company. 
 
 The authors also are greatly indebted to Messrs. Harvey Whipple, Adelbert P. Mills, 
 Walter S. Edge, A. G. Hillberg, and Leslie H. Allen for the important chapters which they have 
 prepared. These men are specialists; and their contributions will prove of great value to the 
 engineering profession. 
 
 The chapter on dams, written by Mr. A. G. Hillberg, has been made brief, but a sufficiently 
 extensive presentation is given to enable the reader to decide intelligently what type of dam 
 to adopt. It is desired to call attention to the paragraphs dealing with siphonic spillways 
 as there is practically no text-book information on that subject. 
 
 In writing this book the authors have drawn from the three volumes of Hool's "Reinforced 
 Concrete Construction" only where the preparation of new material would have been sub- 
 stantial duplication. 
 
 The authors are under obligation to Messrs. Wm. J. Fuller and Frank C. Thiessen for many 
 helpful suggestions and for assistance in making calculations for some of the tables and diagrams. 
 Acknowledgments are also due to Mr. C. M. Chapman and others for suggestions and criticisms 
 of manuscript; and to a large number of engineers who have supplied data and details, and have 
 generously given their views in regard to both theory and practice. 
 
 The authors are indebted to Mr. Clifford E. Ives for his excellent work in preparing all 
 drawings made expressly for this handbook. 
 
 G. A. H. 
 
 April, 1918. N. C. J. 
 
 V 
 
Digitized by the Internet Archive 
 in 2015 
 
 https://archive.org/details/concreteengineerOOhool 
 
STANDARD NOTATION 
 USED THROUGHOUT THIS VOLUME 
 SEE APPENDIX D 
 
TABLE OF CONTENTS 
 
 Section 1. Materials 
 
 Cement 
 
 Art. Page 
 
 1. Classification, composition, and uses 
 
 of the principal cementing 
 
 materials 1 
 
 a. Gypsum plasters 1 
 
 h. Common lime 1 
 
 c. Hydraulic lime 2 
 
 d. Puzzolan or slag cement. 2 
 
 e. Natural cement 3 
 
 /. Portland cement 4 
 
 2. Portland and natural cements com- 
 
 pared 4 
 
 3. Constitution of Portland cement 5 
 
 4. Setting and hardening of Portland 
 
 cement 5 
 
 5. Manufacture of Portland cement .... 6 
 
 a. Raw materials 6 
 
 h. Proportioning the raw materials . 7 
 
 c. Grinding and mixing 7 
 
 d. Burning the cement mixture 7 
 
 e. Treatment of the clinker 7 
 
 6. Manufacture of natural cement 8 
 
 a. Raw material 8 
 
 h. Process of manufacture 8 
 
 7. Testing of cement 8 
 
 a. Sampling 8 
 
 6. Uniformity in cement testing .... 8 
 
 c. The personal factor 8 
 
 d. Kinds of tests 8 
 
 e. Fineness 8 
 
 /. Normal consistency 9 
 
 g. Time of setting 9 
 
 h. Tensile strength 10 
 
 i. Relation between tensile and 
 
 compressive strength 10 
 
 j. Compressive strength 10 
 
 k. Soundness 10 
 
 I. Specific gravity 11 
 
 m. Chemical analysis 11 
 
 8. Specifications for cement 11 
 
 9. Containers for cement 11 
 
 10. Storing of cement 11 
 
 11. Seasoning of cement 12 
 
 Art. Page 
 
 12. Use of bulk cement 12 
 
 13. Weight of cement 12 
 
 Aggregates 
 
 14. Definitions 12 
 
 15. General requirements 12 
 
 16. Classification — Coarse and fine 12 
 
 17. Qualities of fine aggregates — General 13 
 
 18. Qualities of coarse aggregates — Gen- 
 
 eral 13 
 
 19. Materials suitable for coarse aggre- 
 
 gates 13 
 
 20. Igneous rocks 13 
 
 a. Granite 13 
 
 h. Trap-rock of diabase 14 
 
 21. Sedimentary rocks 14 
 
 a. Sandstone 15 
 
 h. Limestone 15 
 
 22. Metamorphic rocks 16 
 
 23. Gravel 16 
 
 24. Blast-furnace slag 17 
 
 25. Cinders 17 
 
 26. Materials suitable for fine aggregates 17 
 a. Special characteristics of sand. . . 18 
 6. Crushed stone and screenings. ... 19 
 
 c. Sea sand 19 
 
 d. Standard sand 19 
 
 27. Requirements of fine aggregate as to 
 
 shape and size of particles 19 
 
 28. The selection of sand 20 
 
 29. Requirernents of coarse aggregate as 
 
 to shape and size of particles 20 
 
 30. Impurities in aggregates 21 
 
 31. Size and gradation of aggregate par- 
 
 ticles 22 
 
 a. Grading of mixtures 22 
 
 h. Grading, density, and strength. . . 22 
 
 c. Money value of grading 23 
 
 32. Mechanical analysis of aggregates ... 23 
 
 33. Specific gravity of aggregates 25 
 
 34. Voids in aggregates 25 
 
 a. Percentage of voids 25 
 
CONTENTS 
 
 Art. Page 
 
 h. General laws 25 
 
 c. Effect of moisture on voids in sand 
 
 and screenings 26 
 
 d. Percentage of voids determined 
 
 by weight 26 
 
 35. Tests of aggregates 27 
 
 36. Notes on the selection and testing of 
 
 aggregates 30 
 
 37. Specifications for aggregates 30 
 
 Water 
 
 38. General requirements 31 
 
 39. Examination of water 31 
 
 40. Functions of water 31 
 
 41. Influence of quantity of water on 
 
 strength of concrete 32 
 
 42. Influence of quantity of water on flux- 
 
 ing of cement 32 
 
 43. Influence of quantity of water on 
 
 lubrication of concrete mixture. . 33 
 
 44. Influence of quantity of water on 
 
 space occupied in resulting con- 
 crete 33 
 
 45. Harmful effects of voids caused by 
 
 excess water 34 
 
 46. Excess water the cause of ''day's 
 
 work planes" 34 
 
 47. Excess water the cause of large lait- 
 
 ance deposits 34 
 
 48. Excess water and waterproof concrete 35 
 
 49. Excess water causes unsatisfactory 
 
 concrete floor surfaces 35 
 
 50. Excess water prevents bonding new 
 
 concrete to old 35 
 
 51. Excess water and concreting in cold 
 
 weather 35 
 
 52. Suggested procedures to guard 
 
 against use of excess water 36 
 
 Reinforcement 
 
 53. Types of reinforcement 36 
 
 54. Surface of reinforcement 37 
 
 55. Quality of steel 37 
 
 56. Working stresses 37 
 
 57. Coefficient of expansion 37 
 
 Proportioning Concrete 
 1. Properties of concrete, dependent 
 upon properties and relative pro- 
 portions of constituent materials 63 
 
 Art. Page 
 
 58. Modulus of elasticity 37 
 
 59. Steel specifications 37 
 
 a. For bars rolled from billets 37 
 
 h. For rerolled bars 40 
 
 60. Factors affecting cost of reinforcing 
 
 bars 42 
 
 61. Deformed bars 42 
 
 a. Diamond bar 42 
 
 h. Corrugated bars 43 
 
 c. Havermeyer bars 43 
 
 (i. Rib bar 44 
 
 e. Inland bar 44 
 
 /. American bars 44 
 
 62. Wire fabric 45 
 
 a. Welded wire fabric 46 
 
 h. Triangle-mesh wire fabric 47 
 
 c. Unit wire fabric 48 
 
 d. Lock-woven steel fabric 49 
 
 e. Wisco reinforcing mesh 50 
 
 63. Expanded metal 50 
 
 d. Streelcrete 51 
 
 h. Kahn mesh 52 
 
 c. Corr-X-metal 52 
 
 d. Econo 53 
 
 e. G. F. expanded metal 53 
 
 64. Rib metal 55 
 
 65. Self-centering fabrics 55 
 
 a. Hy-rib 56 
 
 h. Corr-mesh 56 
 
 c. Self-centering 56 
 
 d. Chanelath 56 
 
 e. Ribplex 57 
 
 /. Dovetailed corrugated sheets .... 57 
 
 66. Reinforcing systems for beams, gird- 
 
 ers and columns 57 
 
 a. Kahn system 57 
 
 h. Cummings system . 58 
 
 c. Unit system 58 
 
 d. Corr system 60 
 
 c. Hennebique system 60 
 
 /. Pin-connected system 60 
 
 g. Luten truss 60 
 
 h. Xpantruss system 60 
 
 i. Shop fabricated reinforcement 
 
 system 60 
 
 2. Theory of proportioning 63 . 
 
 3. The strength elements of concrete. ... 64 
 
 4. Proportioning for high strength con- 
 
 cretes 64 
 
 Section 2. General Methods of Construction 
 
CONTENTS 
 
 Art. Page 
 
 5. Weakness due to poor proportioning . 64 
 
 6. Unit of proportioning 65 
 
 7. Arbitrary proportions 65 
 
 8. Proportioning by void determinations 65 
 
 9. Proportioning by mechanical analy- 
 
 sis 68 
 
 10. Proportioning by maximum density 
 
 tests 68 
 
 1 1 . Checking materials on the j ob 69 
 
 12. Proportions and the measurement of 
 
 materials 70 
 
 13. Proportioning bank-run gravel 70 
 
 14. Proportioning crusher-run stone 71 
 
 15. Proportioning blast-furnace slag and 
 
 cinders 71 
 
 16. Proportioning water 71 
 
 . 17. Success in proportioning 72 
 
 Mixing, Transporting and Placing Con- 
 crete 
 
 18. Mixing concrete 72 
 
 19. Amount of water to be used in mixing 
 
 concrete 72 
 
 20. Transporting concrete 72 
 
 21. Depositing concrete in forms 73 
 
 22. Continuous and even depositing in 
 
 forms 73 
 
 23. Continuous depositing to avoid stop- 
 
 page planes 74 
 
 24. Bonding set and new concrete 75 
 
 25. Removal of entrained air 75 
 
 26. Spading, puddling and tamping 75 
 
 27. Depositing concrete through water. . 76 
 
 28. Remixed and retempered concrete, . . 76 
 
 29. Concreting in hot weather and in 
 
 cold weather 76 
 
 a. Pre-heating aggregates and water 77 
 
 h. Means for heating aggregates. ... 77 
 
 c. Enclosure and heating of forms. . 77 
 
 d. Protection against frost 78 
 
 e. Freezing of concrete 78 
 
 /. Use of anti-freezing mixtures .... 78 
 
 g. Protection against heat 78 
 
 Field Tests of Concrete 
 
 30. Object of field tests 78 
 
 31. Limitations inherent in field tests. . . 78 
 
 32. Comparative tests on field-molded 
 
 and structural specimens 79 
 
 xi 
 
 Art, Page 
 
 33. Value of tests on field-molded test 
 
 specimens 79 
 
 34. Transverse tests on beam specimens. . 79 
 
 35. Core drill test specimens from actual 
 
 structures 79 
 
 36. Suggested methods for making and 
 
 testing field specimens of con- 
 crete 79 
 
 37. Pre-use tests of materials 82 
 
 Waterproofing Concrete 
 
 38. Meaning of "waterproof " 82 
 
 39. Resistance of concretes to water ac- 
 
 tion 82 
 
 40. Resistance of concretes to water pene- 
 
 tration 83 
 
 41. Degree of impermeability attainable. 83 
 
 42. Porosity of commercial concretes . . . , 83 
 
 43. Excess water as a cause of porosity. . 84 
 
 44. Shrinkage cracks 84 
 
 a. Types of shrinkage cracks 84 
 
 h. Shrinkage cracks and porosity. . . 85 
 
 c. Prevention of shrinkage cracks. . . 85 
 
 45. Pervious concretes and laitance 86 
 
 46. Effect of temperature and atmos- 
 
 pheric effects on water-tightness 86 
 
 47. Integral waterproofing compounds . . 86 
 
 a. Integral waterproofing classifica- 
 
 tion 87 
 
 b. Value of integral waterproofing 
 
 compounds 87 
 
 c. Rendering defective structures 
 
 impervious 87 
 
 48. Waterproofing by cement grouting. . 88 
 
 49. Membranous waterproofings 88 
 
 a. Application of membranous 
 
 waterproofing 88 
 
 b. Continuity of membrane 89 
 
 c. Protection of waterproofing 89 
 
 50. Rules for making concrete impervious 89 
 
 Finishing Concrete Surfaces 
 
 51. Character of surface finish desired. . . 90 
 
 52. Removing form marks 90 
 
 a. Tooling 90 
 
 b. Rubbing 91 
 
 c. Brushing 92 
 
 d. Sand-blasting 92 
 
 53. Use of colored aggregates 92 
 
xii 
 
 CONTENTS 
 
 Art. Page 
 
 54. Addition of colors to concrete 93 
 
 55. Use of white cement 93 
 
 56. Plaster finishes 93 
 
 57. Surfacing concrete floors 93 
 
 58. Specification for the production of a 
 
 rubbed surface 93 
 
 Forms 
 
 59. General requirements 93 
 
 60. Economical considerations 94 
 
 61. Lumber for forms 94 
 
 62. Removal of forms 95 
 
 63. Number of sets of forms in building 
 
 work 96 
 
 64. Examples of form design 96 
 
 a. Column forms ; 96 
 
 h. Beam and girder forms 103 
 
 c. Slab forms 104 
 
 d. Column heads 108 
 
 e. Wall and pier forms Ill 
 
 65. Design of forms 120 
 
 a. Values to use in design 121 
 
 h. Drafting room methods 123 
 
 66. Tables and diagrams for designing 
 
 forms 124 
 
 a. Notation. 124 
 
 h. Fiber stresses allowed 124 
 
 c. Formulas used 125 
 
 67. Systematizing form work for build- 
 
 ings 131 
 
 a. Saw mill and yard 131 
 
 6. Shop procedure 131 
 
 c. Cleat spacing 134 
 
 d. Planning of field work 134 
 
 e. Stripping of forms 134 
 
 68. Steel forms 135 
 
 69. Construction notes 138 
 
 Bending and Placing Reinforcement 
 
 70. Checking, assorting and storing steel 139 
 
 71. Bending of reinforcement 139 
 
 o. Types of bends 139 
 
 h. Hand devices 139 
 
 c. Power-operated benders 141 
 
 d. Care to be exercised in bending. . 141 
 
 e. Bending of slab reinforcement. . . . 142 
 
 72. Placing of reinforcement 142 
 
 73. Devices for supporting reinforcing 
 
 bars 143 
 
 Manufacture and Use op Concrete 
 
 Stone, Block and Brick 
 Art. Page 
 
 74. Development of the industry 146 
 
 75. Two main lines of work 147 
 
 76. Methods of manufacture 147 
 
 a. Dry tamp method 147 
 
 h. Pressure method 148 
 
 c. Wet-cast method 148 
 
 77. Consistency 148 
 
 78. Commercial molds 149 
 
 79. Operation of machines 150 
 
 a. Tamping 150 
 
 80. Gang molds for wet-cast products .. . 151 
 
 81. Materials 151 
 
 a. Cement (storage and conveying) 151 
 
 h. Aggregates (kind and quality) . . . 152 
 
 82. Mixing >. 152 . 
 
 a. Mixers (general type) 153 
 
 h. Mixing dry and mixing wet 153 
 
 c. Agitation subsequent to mixing 
 
 in wet-cast work 153 
 
 d. Mixing facing materials 153 
 
 83. Placing 154 
 
 a. Buckets and hoppers 154 
 
 h. Wheelbarrows 155 
 
 c. Pallets 155 
 
 d. Bankers 156 
 
 84. Curing 156 
 
 a. Natural curing 156 
 
 h. Steam curing 157 
 
 85. Special molds 159 
 
 a. Wood molds 159 
 
 h. Plaster molds 159 
 
 c. Glue molds 160 
 
 d. Combination molds 161 
 
 e. Waste molds 161 
 
 86. Sand molds and casting in sand 161 
 
 87. Surfaces 162 
 
 a. Face design in standard units. ... 162 
 
 h. Facing materials 163 
 
 c. Colors 164 
 
 d. Spraying 165 
 
 e. Brushing 165 
 
 /. Rubbing 166 
 
 g. Tooling 166 
 
 h. Mosaics 167 
 
 i. EflSorescence 167 ■ 
 
 j. Air bubbles 167 
 
 k. Crazing 167 
 
 88. Specifications of the American Con- 
 
 crete Institute 168 
 
CONTENTS 
 Section 3. Construction Plant 
 
 Preparation of Concjiete Aggregates 
 Art. Page 
 
 1. Preparation of crushed-stone aggre- 
 
 gate 171 
 
 a. Preparation of site for quarrying. 171 
 h. Quarrying 171 
 
 c. Drills 171 
 
 d. Stone crushers 172 
 
 e. Screening and grading of crushed 
 
 stone 172 
 
 /. Washing crushed stone 172 
 
 g. Crushed limestone 172 
 
 2. Screening of sand and gravel 172 
 
 3. Washing of sand and gravel 173 
 
 Handling and Storage of Materials 
 
 4. 
 
 General considerations 
 
 173 
 
 5. 
 
 
 174 
 
 6. 
 
 Shoveling materials directly from cars 
 
 
 
 
 174 
 
 7. 
 
 Storage and care of sand 
 
 174 
 
 8. 
 
 
 174 
 
 9. 
 
 Unloading economies 
 
 175 
 
 10. 
 
 Proper size and type of shovel 
 
 175 
 
 11. 
 
 
 176 
 
 12. 
 
 Bucket unloaders and conveyors. . . . 
 
 177 
 
 13. 
 
 Belt conveyors 
 
 177 
 
 14. 
 
 Storage and handling of sack cement 
 
 177 
 
 15. 
 
 Bundling and storage of empty 
 
 
 
 cement sacks 
 
 178 
 
 16. 
 
 Storage and handling of water 
 
 179 
 
 17. 
 
 A typical installation 
 
 179 
 
 
 Concreting Plant 
 
 
 18. 
 
 Plant economics 
 
 181 
 
 
 a. First cost 
 
 181 
 
 
 6. Cost of installation 
 
 181 
 
 Concrete Floors and Floor Surfaces 
 
 1. 
 
 The concrete floor problem 
 
 205 
 
 2. 
 
 "Dusting" of concrete floors 
 
 206 
 
 3. 
 
 Making good concrete floors and floor 
 
 
 
 surfaces 
 
 206 
 
 4. 
 
 Special surface finishes 
 
 206 
 
 
 a. Surface grinding 
 
 206 
 
 
 
 206 
 
 Art. Page 
 
 c. Cost of operation 181 
 
 d. Cost of maintenance 181 
 
 e. Cost of removal 181 
 
 /. Salvage 181 
 
 19. Balancing the plant 181 
 
 20. Typical plants 181 
 
 21. Machine vs. hand mixing 186 
 
 22. Types of mixers 187 
 
 a. Drum mixers 187 
 
 6. Trough mixers 187 
 
 c. Gravity mixers 188 
 
 d. Pneumatic mixers 188 
 
 23. Machine mixing 189 
 
 a. Time of mixer operations 189 
 
 6. Time of mixing 189 
 
 c. Drum speeds 190 
 
 d. Loading the mixer 190 
 
 Charging hoppers 190 
 
 Power loaders 191 
 
 Low charging mixers 191 
 
 e. Measuring materials 192 
 
 /. Discharge of the mixer 192 
 
 24. Transporting and placing concrete . . 193 
 
 a. Barrows 193 
 
 6. Concrete carts 193 
 
 c. Buckets 194 
 
 d. Cableways and buckets 195 
 
 e. Spouts or chutes 195 
 
 /. Sections used in spouting 196 
 
 g. Hoists 200 
 
 25. Spouting plants . . 203 
 
 a. Boom plants 203 
 
 h. Guy line plants 203 
 
 c. Tower plants 203 
 
 d. Combinations of spouting systems 203 
 
 e. Regulating flow of concrete in 
 
 spouting plants 203 
 
 c. Finish produced by removal of 
 
 water from surface 207 
 
 d. Integral hardeners and surface 
 
 compounds 207 
 
 5. Causes of common defects in concrete 
 
 floors 207 
 
 6. Remedial measures 209 
 
 a. Retopping 209 
 
 h. Chemical hardeners 210 
 
 Section 4. Concrete Floors and Floor Surfaces, Sidewalks, and Roadways 
 
xiv 
 
 CONTENTS 
 
 Abt. Page 
 
 c. Use of oils 210 
 
 d. Floor coatings and paints 210 
 
 Concrete Sidewalks 
 
 7. Structural functions 210 
 
 8. Essential qualities 210 
 
 9. The making of concrete sidewalks. . . 211 
 
 a. Porous subbase 211 
 
 h. Concrete base 211 
 
 c. Top of wearing surface 211 
 
 d. Surface finishing 211 
 
 e. Surface protection and curing.. . . 211 
 /. Protecting sidewalks in hot weather 212 
 g. Special surface finishes 212 
 
 10. Vault light pavements 212 
 
 Strength of Cement Mortar and Plain 
 Concrete 
 
 1. Strength in general 215 
 
 2. Laboratory tests, their use and signifi- 
 
 cance 215 
 
 3. Neat, mortar, and concrete strength 
 
 compared 217 
 
 4. Aggregates of mortar and concrete. . . 218 
 
 5. Effect of mineral character of aggre- 
 
 gates 219 
 
 6. Effect of shapes and size of aggre- 
 
 gates .• 221 
 
 7. Relation between density and 
 
 strength 222 
 
 8. Effect of mica, clay, and loam in ag- 
 
 gregates 224 
 
 9. Effect of consistency 225 
 
 10. Compressive and tensile strengths 
 
 compared 227 
 
 1 1 . Strength of plain concrete columns . . 229 
 
 12. Effect of method of mixing 230 
 
 13. Effect of method of placing 231 
 
 14. Effect of regaging 232 
 
 15. Effect of curing conditions 234 
 
 16. Effect of freezing 236 
 
 17. Effect of salts 236 
 
 18. Effect of hydrated lime and water- 
 
 proofing compounds 238 
 
 19. Effect of sea water used. in gaging. . . 238 
 
 20. Effect of oils used in gaging 240 
 
 21. Effect of laitance 241 
 
 22. Rate of increase in mortar strength, 
 
 retrogression 241 
 
 Abt. Page 
 
 11. Concrete curbing 212 
 
 12. Summary 212 
 
 Concrete Roadways 
 
 13. Structural functions 213 
 
 14. Essential qualities 213 
 
 15. One-course and two-course pave- 
 
 ments 213 
 
 16. The making of concrete roadways. . . 213 
 
 a. Porous subbase 213 
 
 h. Proportioning and selecting of 
 
 materials 213 
 
 c. Joints 214 
 
 (/. Curing 214 
 
 e. Consistency 214 
 
 23. Transverse strength 242 
 
 24. Shearing strength 243 
 
 25. Adhesive strength 246 
 
 a. Adhesion to concrete previously 
 
 placed 246 
 
 h. Adhesion or bond to steel (see 
 
 Art 2, Sect. 6) 247 
 
 26. Strength of natural cement mortar 
 
 and concrete 247 
 
 27. Strength of cinder concrete 248 
 
 28. Working stresses (see Appendix B) . . 250 
 
 Elastic Properties of Cement Mortar 
 AND Concrete 
 
 29. Stress-strain curves for mortars and 
 
 concretes 250 
 
 30. Yield point 251 
 
 31. Modulus of elasticity 251 
 
 Contraction and Expansion of Cement 
 Mortar and Concrete 
 
 32. Coefficient of expansion 252 
 
 33. Moisture changes 253 
 
 Durability of Cement Mortar and Con- 
 crete 
 
 34. Fire-resistant properties 254 
 
 35. Weathering qualities 255 
 
 36. Abrasive resistance 255 
 
 37. Action of sea water 256 
 
 38. Action of alkali 257 
 
 39. Action of acids, oils, and sewage .... 257 
 
 40. Electrolysis in concrete 258 
 
 Section 5. Properties of Cement Mortar and Plain Concrete 
 
CONTENTS 
 
 XV 
 
 Art. Page 
 
 41. Effect of manure 259 
 
 Miscellaneous Properties of Cement 
 Mortar and Concrete 
 
 42. Rise of temperature in setting 259 
 
 1. Advantages of combining concrete 
 
 and steel 265 
 
 2. Bond between concrete and steel 265 
 
 Pull-out tests 266 
 
 Relation of bond stress to slip of 
 bar as load increases 266 
 
 Bond resistance in terms of com- 
 pressive strength of concrete . . . 267 
 
 Distribution of bond stress along 
 a bar 267 
 
 Variation of bond resistance with 
 size, shape, and condition of sur- 
 face of bar 267 
 
 Anchoring of reinforcing bars 268 
 
 Influence of method of curing 
 concrete 268 
 
 Rectangular Beams and Slabs 
 
 1. Forces to be resisted 273 
 
 2. Distribution of stress in homogeneous 
 
 beams 273 
 
 3. Assumptions in theory of flexure for 
 
 homogeneous beams . 275 
 
 4. Plain concrete beams 275 
 
 5. Purpose and location of steel rein- 
 
 forcement 275 
 
 6. Tensile stress lines in reinforced con- 
 
 crete beams 275 
 
 7. Flexure formulas for reinforced con- 
 
 crete beams 276 
 
 8. Assumptions in flexure calculations . . 276 
 
 9. Flexure formulas for working loads — 
 
 straight line theory 276 
 
 10. Flexure formulas for ultimate loads . . 279 
 
 11. Flexure formulas for working loads 
 
 and for ultimate loads compared 280 
 
 12. Lengths of simply supported beams. . 280 
 
 13. Shearing stresses 280 
 
 14. Methods of strengthening beams 
 
 againstf ailurein diagonal tension 281 
 
 15. Moment and diagonal-tension tests — 
 
 General 281 
 
 Art. Page 
 
 43. Porosity 261 
 
 44. Permeability and absorptive proper- 
 
 ties 262 
 
 45. Protection of embedded steel from 
 
 corrosion 262 
 
 46. Weight of mortar and concrete 263 
 
 Influence of freezing of concrete. . . 268 
 
 Influence of age and mix of con- 
 crete 268 
 
 Effect of continued and repeated 
 load 268 
 
 Effect of concrete setting under 
 
 pressure 269 
 
 Beam tests 269 
 
 3. Length of embedment of reinforcing 
 
 bars to provide for bond 270 
 
 4. Ratio of the moduli of elasticity 270 
 
 5. Behavior of reinforced concrete under 
 
 tension 271 
 
 6. Shrinkage and temperature stresses . . 271 
 
 7. Weight of reinforced concrete 272 
 
 16. Bond stress 284 
 
 17. Web reinforcement in general 285 
 
 18. Region where no web reinforcement 
 
 is required 286 
 
 19. Vertical stirrups 289 
 
 20. Method of placing stirrups from the 
 
 moment diagram 291 
 
 21. Bent rods and vertical stirrups for 
 
 web reinforcement 296 
 
 22. Points where horizontal reinforce- 
 
 ment may be bent 297 
 
 23. Transverse spacing of reinforcement 300 
 
 24. Depth of concrete below rods 300 
 
 25. Ratio of length to depth of beam for 
 
 equal strength in moment and 
 shear 301 
 
 26. Economical proportions of rectangu- 
 
 lar beams 302 
 
 27. Rectangular beams with steel in top 
 
 and bottom 302 
 
 a. Formulas for determining per- 
 centages of steel in double-rein- 
 forced rectangular beams 304 
 
 28. Deflection of rectangular beams 304 
 
 a. Maney's method 305 
 
 Section 6. General Properties of Reinforced Concrete 
 
 Section 7. Beams and Slabs 
 
CONTENTS 
 
 Art. Page 
 
 h. Turneaure and Maurer's method 305 
 
 29. Slabs 306 
 
 a. Moments in continuous slabs 306 
 
 h. Provision for negative moment in 
 
 continuous slabs 306 
 
 c. Floor slabs supported along four 
 
 sides 307 
 
 d. Cross reinforcement in slabs 307 
 
 T-Beams 
 
 30. T-beams in floor construction 307 
 
 31. Tests of T-beams 307 
 
 32. Flange width 308 
 
 33. Bonding of web and flange 308 
 
 34. Flexure formulas 308 
 
 a. Formulas for determining dimen- 
 sions and steel ratio for given 
 working stresses 309 
 
 35. Designing for shear 310 
 
 36. General proportions of T-beams 310 
 
 37. Economical considerations 310 
 
 38. Conditions met with in design of T- 
 
 beams 310 
 
 39. Design of a continuous T-beam at the 
 
 supports 311 
 
 40. T-beams with steel in top and bottom 313 
 a. Formulas for determining per- 
 centages of steel in double-rein- 
 forced T-beams 313 
 
 41. Deflection of T-beams 313 
 
 Special Beams 
 
 42. Wedge-shaped beams 314 
 
 1. Column types 371 
 
 2. Plain concrete columns or piers 371 
 
 3. Columns with longitudinal reinforce- 
 
 ment 371 
 
 4. Columns with hooped and longitudi- 
 
 nal reinforcement 372 
 
 5. Columns reinforced with structural- 
 
 steel shapes 372 
 
 1. Theory in general 385 
 
 2. Analytical determination of stresses in 
 
 rectangular sections 387 
 
 a. Case I. — Compression over the 
 
 whole section — steel top and 
 
 bottom 387 
 
 h. Case II. — Tension over part of 
 
 section — steel top and bottom . . 394 
 
 Art. Page 
 
 43. Beams of any complex or irregular 
 
 section 314 
 
 a. Analytical method 314 
 
 h. Graphical method 316 
 
 Shear and Moment in Restrained and 
 Continuous Beams 
 
 44. Span length for beams and slabs 318 
 
 45. Recommendations of Joint Com- 
 
 mittee as to positive and negative 
 
 moments 318 
 
 46. Theorem of three moments 318 
 
 47. Uniform load over all spans 323 
 
 48. Fixed and moving concentrated 
 
 loads 324 
 
 a. Influence lines 324 
 
 49. Moving uniform loads 329 
 
 50. Maximum moments from uniform 
 
 loads 330 
 
 51. Beam concentrations 331 
 
 52. Negative moment at the ends of con- 
 
 tinuous beams 334 
 
 53. Bending up of bars and provision for 
 
 negative moment 335 
 
 54. Continuous beams with varying 
 
 moment of inertia 339 
 
 Designing Tables and Diagrams for 
 Beams and Slabs 
 
 55. Illustrative problems (12 in all) 341 
 
 56. Leffler's comprehensive beam chart. 348 
 
 57. Beard and Schuler's comprehensive 
 
 charts 350 
 
 6. Working stresses 373 
 
 7. Recommendations of the Joint Com- 
 
 mittee 373 
 
 8. Tables and diagrams 374 
 
 9. Reduction formula for long columns. . 381 
 10. Columns supporting bracket loads .. . 381 
 
 c. Case III. — Tension over part of 
 
 section — steel in tension face only 403 
 
 3. Graphical determination of stresses . . 406 
 
 o. Rectangular sections 406 
 
 h. Hollow-circular sections 407 
 
 c. Solid circular and other sections . . . 409 
 
 Section 8. Columns 
 
 Section 9. Bending and Direct Stress 
 
CONTENTS 
 
 Section 10. Moments 
 
 Art. Page 
 
 1. Importance of the subject 411 
 
 2. Method of analysis 411 
 
 3. AppUcation of method of analysis to 
 
 simple cases 414 
 
 4. Conception of rigidity of building 
 
 frames 415 
 
 5. Moments at interior columns in 
 
 beam-and-girder construction. . 416 
 
 a. All terminals hinged 416 
 
 b. All terminals fixed 417 
 
 c. Columns hinged. Outer girders 
 
 with constant moment 418 
 
 (I Columns fixed. Outer girders 
 
 with constant moment 418 
 
 e. Point of inflection at center of 
 
 columns. Outer girders hinged . 418 
 /. Point of inflection at center of 
 
 columns. Outer girders fixed. . . 418 
 g. Point of inflection at center of 
 
 columns. Outer girders with 
 
 constant moment 419 
 
 in Rigid Building Frames 
 
 Art. Page 
 
 h. Point of inflection at center of 
 
 upper columns. Lower columns 
 fixed. Outer girders with con- 
 stant moment 419 
 
 i. Point of inflection at center of 
 
 upper columns. Lower columns 
 fixed. Outer girders hinged. . . 419 
 j. Point of inflection at center of 
 upper columns. Lower columns 
 fixed. Outer girders fixed 420 
 
 6. Moments at exterior columns in 
 
 beam-and-girder construction. . . 420 
 
 a. Inner end of girder hinged 425 
 
 b. Inner end of girder fixed 425 
 
 7. Moments in columns in flat-slab 
 
 construction 425 
 
 8. Criteria for maximum combined 
 
 stresses in columns 426 
 
 a. Interior columns 426 
 
 b. Exterior columns 426 
 
 9. Wind stresses in building frames .... 427 
 
 10. Roof frames 428 
 
 11. L-frames 429 
 
 Section 11. Buildings 
 
 Floors — General Data 
 
 1. General types of concrete floors. . . . 431 
 
 2. Floor loads 431 
 
 3. Economic considerations 432 
 
 4. Floor surfaces 432 
 
 5. Small floor openings 433 
 
 6. Provision for the attachment of 
 
 shaft-hangers and springer pipes 434 
 
 7. Bedding machinery 437 
 
 8. Waterproof floors 438 
 
 9. Tests 438 
 
 10. Basement floors 438 
 
 Monolithic Beam and Girder Con- 
 struction 
 
 11. Ordinary type of beam and girder 
 
 construction 439 
 
 12. Hollow tile construction 447 
 
 Flat Slab Construction 
 
 13. General description 457 
 
 14. Advantages over the beam-and-girder 
 
 type 457 
 
 15. Classes of buildings to which adapted 458 
 
 16. Remarks regarding design 459 
 
 17. Systems 461 
 
 a. Barton Spider Web system 461 
 
 b. Cantilever Flat-slab construction 463 
 
 c. Simplex system 465 
 
 d. Mushroom system 465 
 
 e. Watson system 466 
 
 /. Akme system 467 
 
 g. Corr-plate floors. 470 
 
 h. S-M-I system 471 
 
 i. Three-way system 476 
 
 18. Patents 479 
 
 19. Loading tests 480 
 
 20. Methods of design and problems 487 
 
 a. Computations for Akme system.. 488 
 
 b. Computations for Corr-plate 
 
 floors 489 
 
 c. Computations based on Pitts- 
 
 burgh Ruling 492 
 
 d. Computations based on Chicago 
 
 Ruling 494 
 
 e. Computations based on Ruling of 
 
 American Concrete Institute . . . 495 
 
xviii 
 
 CONTENTS 
 
 Art. Page 
 
 /. Roof design 496 
 
 g. Beams in flat-slab floors 497 
 
 h. Columns 497 
 
 i. Brick exterior wall supports 498 
 
 21. Tables 498 
 
 22. Construction methods and safeguards 506 
 
 Unit Construction 
 
 23. Method of construction in general. . . . 508 
 
 24. Advantages of the unit method 508 
 
 25. " Unit-bilt" system 508 
 
 26. Ransome unit system 509 
 
 Steel Frame Construction with Con- 
 crete Slabs 
 
 27. Types of construction 511 
 
 28. Wrapping of I-beams 511 
 
 29. Types illustrated 511 
 
 Roofs 
 
 30. Structural design 512 
 
 31. Loading 512 
 
 32. Prevention of condensation on con- 
 
 crete roof slabs 513 
 
 33. Concrete roof surfaces 516 
 
 34. Separate roof coverings 517 
 
 35. Drainage '. 518 
 
 36. Parapet walls 521 
 
 37. Saw-tooth construction 526 
 
 Art. Page 
 
 38. Trainshedof C/m7-5z7f construction . . . 527 
 
 Columns 
 
 39. Details of design 527 
 
 40. Loading 530 
 
 41. Column brackets 530 
 
 Walls and Partitions 
 
 42. Bearing walls 532 
 
 43. Curtain walls 533 
 
 44. Brick and other veneer 539 
 
 45. Window openings 540 
 
 46. Door openings 543 
 
 47. Basement walls 545 
 
 48. Partitions 545 
 
 Stairs 
 
 49. General design 549 
 
 50. Methods of supporting stairs 550 
 
 51. Stair details 551 
 
 Elevator Shafts 
 
 52. Elevator-shaft pits 552 
 
 53. Pent houses 553 
 
 Provisions for Contraction and Expan- 
 sion 
 
 54. Methods employed 554 
 
 Section 12. Foundations 
 
 1. Bearing capacity of soils 557 
 
 2. Pressure on the soil 558 
 
 3. Plain concrete footings 558 
 
 4. Advantages in using reinforced con- 
 
 crete for foundations 558 
 
 5. Wall footings 559 
 
 6. Types of column footings 559 
 
 7. Single column footings 559 
 
 8. Combined column footings 565 
 
 9. Cantilever footings 568 
 
 10. Raft foundations 570 
 
 11. Examples of column footings 571 
 
 12. Concrete piles 572 
 
 a. Piles molded in place 572 
 
 b. Piles molded before driving 574 
 
 Section 13. Retaining Walls 
 
 1. Earth pressure 575 
 
 a. Rankine's formula for resultant 
 
 active earth pressure 576 
 
 h. Coulomb's wedge of maximum 
 
 pressure 577 
 
 c. Comparison of Coulomb and Ran- 
 
 kine results 580 
 
 d. Useful interpretation of results of 
 
 earth pressure theories 580 
 
 2. Live load on top of fill — Equivalent 
 
 surcharge 580 
 
 3. Live load on top of fill — Pressure dis- 
 
 tribution 58 1 
 
 4. Stability of a retaining wall 581 
 
CONTENTS 
 
 xix 
 
 Akt. Page 
 
 a. So-called "Factor of Safety" 584 
 
 b. Factor of limitation 584 
 
 5. Types of retaining walls 584 
 
 6. Design of plain concrete walls 584 
 
 a. Formulas and diagrams for the 
 
 two principal types 584 
 
 h. Trautwine's table 586 
 
 c Selection of preliminary section. . . 587 
 
 7. Design of cantilever or T-walls of 
 
 reinforced concrete 587 
 
 a. To determine approximate base 
 
 width 587 
 
 b. Stem 589 
 
 c. Base slab 590 
 
 Art. Page 
 
 d. Expansion joints 592 
 
 8. Design of counterforted walls 592 
 
 a. Thickness and spacing of counter- 
 
 forts 592 
 
 b. Vertical or face wall 593 
 
 c. Back floor slab 593 
 
 d. Cantilever toe slab 596 
 
 e. Methods of reinforcing counter- 
 
 forted wall 596 
 
 9. Special types of reinforced-concrete 
 
 walls.. 600 
 
 10. Construction of retaining walls 601 
 
 a. Back-filling and drainage 601 
 
 6. Forms 602 
 
 Section 14. Slab and Girder Bridges 
 
 Slab Bridges 
 
 1. Slabs under concentrated loading. . . 603 
 
 a. Illinois tests 603 
 
 b. Ohio tests 604 
 
 c. Tests by Goldbeck 605 
 
 2. Slab bridges of single span 605 
 
 3. Slab bridges of multiple spans 607 
 
 a. Concrete pile trestles 607 
 
 b. Pier trestles 610 
 
 c. Trestles with framed bents 610 
 
 d. Cantilever flat-slab construction.. 612 
 
 Simple Girder Bridges 
 
 4. Deck girders 613 
 
 5. Through girders 617 
 
 Continuous Girder Bridges 
 (Monolithic construction) 
 
 6. Expansion joints 622 
 
 7. Examples of typical bridges of the 
 
 continuous girder type 623 
 
 8. Analysis of stresses in rigid viaduct 
 
 structures 625 
 
 a. Viaduct frames 628 
 
 b. Four-span viaduct frame with 
 
 rigidly connected column tie ... . 629 
 
 c. Three-span viaduct frame with 
 
 rigidly connected column tie. ... 631 
 
 d. Two-span viaduct frame with 
 
 rigidly connected column tie . . . 632 
 
 e. One-span viaduct frame with 
 
 rigidly connected column tie . . . 633 
 /. Four-span viaduct frame 633 
 
 g. Three-span viaduct frame 634 
 
 h. Two-span viaduct frame 634 
 
 i. One-span frame, unequal columns 636 
 
 j. Temperature stresses 636 
 
 k. Effect of fixed bases 638 
 
 I Viaduct bent 638 
 
 Cantilever Bridges 
 
 9. Theory of design 639 
 
 10. Examples of cantilever bridges 639 
 
 Section 15. Concrete Floors and Abutments for Steel Bridges 
 
 1. Concrete floors on steel bridges 643 
 
 2. Abutments for steel bridges 645 
 
 3. Types of abutments 645 
 
 4. Pier abutments of plain concrete. . . . 645 
 
 5. Pier abutments of reinforced concrete 646 
 
 6. Wing abutments 646 
 
 7. Cellular abutments 647 
 
 8. U-abutments 647 
 
 9. T-abutments 647 
 
 10. Buried-pier abutments 648 
 
 11. Skeleton and arched abutments 648 
 
 12. Care in constructing abutments 649 
 
XX 
 
 CONTENTS 
 
 Section 16. Arches 
 
 General Data 
 Art. Page 
 
 1. Definitions 651 
 
 2. Curve of the in trades 651 
 
 a. Three-centered curve 652 
 
 b. Semi-elUpse 652 
 
 c. Parabola 653 
 
 3. Arrangement of spandrels 653 
 
 4. Piers and abutments 653 
 
 5. Depth of filling at crown 654 
 
 6. Loads 654 
 
 7. Empirical rules for thickness of arch 
 
 ring 656 
 
 8. Approximate formula for best shape 
 
 of arch axis 658 
 
 9. Proper thickness of arch ring in the 
 
 haunch for given thicknesses at 
 crown and springing 658 
 
 10. Dead loads and their action lines 658 
 
 11. Approximate method of testing trial 
 
 arch 658 
 
 12. Use of temporary hinges in arch 
 
 erection 659 
 
 13. Use of reinforcement in concrete 
 
 arches 660 
 
 14. Classification of arch rings 660 
 
 Analysis of the Arch by the Elastic 
 Theory 
 
 15. Deflection of curved beams 660 
 
 16. General procedure in arch analysis . . 661 
 
 17. Notation 662 
 
 18. Formulas for thrust, shear, and 
 
 moment 663 
 
 19. Division of arch ring for constant j . . 663 
 
 20. Loadings to use in computatioi\s 664 
 
 21. Use of influence lines 664 
 
 22. Internal temperature investigations. 664 
 
 23. Shrinkage stresses 664 
 
 24. Deflection at any point 664 
 
 25. Method of procedure in arch-ring 
 
 design 665 
 
 26. Uncertainty as to fixedness of ends of 
 
 arch 665 
 
 27. Skew arches 665 
 
 28. Unsymmetrical arches 665 
 
 Art. Page 
 
 a. Origin of coordinates between 
 
 divisional lengths 665 
 
 b. Origin of coordinates at crown. . . 666 
 
 c. Origin of coordinates at left 
 
 springing 667 
 
 29. Arch structure of two spans with 
 
 elastic pier 667 
 
 Cochrane's Formulas and Diagrams for 
 
 Use in the Design of Symmetrical 
 Arches in Accordance with 
 the Elastic Theory 
 
 30. Accuracy of formulas and diagrams . . 669 
 
 3 1 . Difficulties and uncertainties involved 
 
 in applying the elastic theory . . 669 
 
 32. Best shape of arch axis 669 
 
 33. Variation in thickness of arch ribs. . . . 670 
 
 34. Influence-line diagrams 673 
 
 35. Diagrams for moments, thrusts, and 
 
 average stresses 681 
 
 36. Approximate method of correcting 
 
 maximum moments when actual 
 arch axis deviates from assumed 
 axis 686 
 
 Details of Arch Bridges 
 
 37. Spandrel details in earth-filled 
 
 bridges 691 
 
 38. Spandrel details in open-spandrel 
 
 bridges 694 
 
 39. Piers and abutments 702 
 
 40. Railing and ornamental details 702 
 
 Construction of Arches 
 
 41. Arch-ring construction 702 
 
 42. Centering 704 
 
 a. Timber centers 704 
 
 b. Steel centers 712 
 
 Three-hinged Arches 
 
 43. General discussion 715 
 
 44. Methods of analysis 715 
 
 45. Common type of hinges 717 
 
 46. Methods of construction 717 
 
 47. Details of design. 719 
 
CONTENTS 
 Section 17. Hydraulic Structures 
 
 Dams 
 
 Art. Page 
 
 1. Preliminary studies 723 
 
 a. Locating 723 
 
 h. Geological investigations 723 
 
 c. Selecting a suitable type of dam., 724 
 
 d. Height of structure 724 
 
 e. Hydrographic investigations 725 
 
 /. Capacity of reservoir 726 
 
 2. Design of foundation 728 
 
 a. Grouting 728 
 
 6. Cut-off walls " 728 
 
 c. Caissons 728 
 
 d. Pilings 728 
 
 6. Sheet piling 728 
 
 3. Design of dams of gravity section . . . 729 
 
 a. Hydrostatic pressure 729 
 
 h. Profiles of dams 729 
 
 c. Uplift 731 
 
 d. Wind pressure 732 
 
 e. Ice thrust 732 
 
 /. Initial stress 732 
 
 g. Temperature stresses. 732 
 
 h. Stresses in masonry and on foun- 
 
 dation 733 
 
 i. Shearing stresses 734 
 
 j. Final calculation 734 
 
 4. Design of arched dams 736 
 
 a. Constant radius dams 736 
 
 h. Constant angle dams 738 
 
 5. Design of reinforced-concrete dams. . 739 
 
 a. Cut-off walls 740 
 
 h. Foundation mattress 740 
 
 c. Buttresses 740 
 
 d. Bracing 741 
 
 e. Apron 741 
 
 /. Multiple-arch dams 743 
 
 6. Earthern dams with concrete core 
 
 wall 745 
 
 7. Passing the discharge 748 
 
 a. Form of spillway. 748 
 
 h. Discharge capacity 749 
 
 c. Profiles of spillways 749 
 
 d. Overflow dams 753 
 
 e. Sluices 754 
 
 /. Siphonic spillways 755 
 
 8. Movable dams 758 
 
 a. Requiring operating machinery. . 759 
 h. Operating under hydrostatic pres- 
 sure differences 759 
 
 Art. Page 
 
 c. Automatically operating 759 
 
 9. Fish ladders 759 
 
 Reservoirs 
 
 10. General types 760 
 
 11. Quality of concrete for reservoir 
 
 masonry 760 
 
 12. Open basins with embankment walls,. 761 
 
 13. Concrete floors for reservoirs 761 
 
 14. Groined and flat floors 762 
 
 15. Concrete walls for open reservoirs. . . . 763 
 
 16. Partition and outside walls 764 
 
 17. Provision for ice 764 
 
 18. Covers or roofs for reservoirs and 
 
 basins 764 
 
 19. Groined arch construction 764 
 
 20. Construction details of columns and 
 
 roof 764 
 
 Standpipes and Small Tanks 
 
 21. Analysis of stresses in standpipes . . . . 765 
 
 22. Restraint at base 765 
 
 23. Shear at base 767 
 
 24. Small tanks 768 
 
 25. Construction details of tanks and 
 
 standpipes 769 
 
 26. Precautions in construction 771 
 
 Elevated Tanks 
 
 27. Analysis of stresses 771 
 
 28. Supporting tower 774 
 
 Culverts 
 
 29. General considerations 775 
 
 30. Factors in culvert design 776 
 
 a. Culvert efficiency 776 
 
 h. Waterway required 776 
 
 c. Length of culverts 776 
 
 d. Design of ends 777 
 
 31. Pipe culverts 777 
 
 a. Pressure in trenches 778 
 
 h. Strength of pipe 781 
 
 c. Circular culverts cast in place,,. . 782 
 
 32. Box culverts 783 
 
 a. Forms of box culverts 785 
 
CONTENTS 
 
 Art. Page 
 h. Loading 785 
 
 c. Design of cross-section 785 
 
 Type I. The closed frame. ... 787 
 Type II. Open frame with fixed 
 
 walls 788 
 
 Type III. Open frame hinged 
 
 at the base 789 
 
 Diagrams 794 
 
 d. Construction 794 
 
 33. Arch culverts 796 
 
 a. Design of cross-section 796 
 
 Art. Page 
 
 h. Forms 797 
 
 Conduits and Sewers 
 
 34. Stresses due to internal pressure 799 
 
 35. External earth pressure on circular 
 
 pipe 799 
 
 36. Large conduits and sewers not circu- 
 
 lar 799 
 
 37. Construction 802 
 
 38. Longitudinal reinforcement 802 
 
 39. Examples of reinforced-concrete pipe. 803 
 
 40. Forms for sewers 803 
 
 Section 18. Miscellaneous Structures 
 
 Deep Grain Bins or Silos 
 
 L Action of grain in deep bins 805 
 
 2. Janssen's formulas for pressure in 
 
 deep bins 805 
 
 3. Conclusions from tests 806 
 
 4. Design of walls 808 
 
 a. Vertical load carried by walls. . 808 
 h. Wind stresses on a horizontal sec- 
 tion 808 
 
 c. Thickness of walls 808 
 
 d. Horizontal reinforcement — circu- 
 
 lar sections 808 
 
 e. Rectangular sections 808 
 
 /. Hexagonal bins 809 
 
 5. Construction 809 
 
 Shallow Bins 
 
 6. Sloped sides — level full (Case I) 811 
 
 7. Partly vertical sides — level full (Case 
 
 II) 811 
 
 8. Sloped sides — fill heaped to angle of 
 
 repose (Case III) 811 
 
 9. Partly vertical sides — fill heaped 
 
 (Case IV) 812 
 
 10. Thrust due to P4 812 
 
 11. Data for bin design 812 
 
 12. Submerged storage for coal 813 
 
 13. Dock pockets 815 
 
 Chimneys 
 
 14. Dead load stresses 816 
 
 15. Stresses on annular sections in flexure 816 
 
 16. Wind stresses in chimneys of rein- 
 
 forced concrete 818 
 
 17. Chimney with no vertical reinforce- 
 
 ment 818 
 
 18. Longitudinal shear in chimneys 819 
 
 19. Temperature stresses in chimneys. . . . 819 
 
 20. Chimney construction 820 
 
 21. Bases for chimneys 820 
 
 Section 19. 
 
 Estimating Unit Costs 
 
 1. Division of the work 823 
 
 2. Estimating unit cost of concrete 823 
 
 a. Materials 823 
 
 h. Labor 824 
 
 c. Plant 825 
 
 d. Summary 825 
 
 3. Estimating unit cost of forms 826 
 
 a. Considerations involved 826 
 
 h. Materials 827 
 
 c. Labor 827 
 
 Estimating 
 
 d. Summary 828 
 
 4. Estimating unit cost of steel rein- 
 
 forcement 828 
 
 5. Estimating unit cost of surface finish 829 
 
 Estimating Quantities 
 
 6. Systematic procedure advisable 830 
 
 7. Rules for measurement of concrete 
 
 work 830 
 
 8. Estimating amount of form work.. . . 831 
 
 9. Estimating amount of steel 832 
 
 10. Estimating amount of surface finish . . 832 
 
CONTENTS 
 
 XXIU 
 
 Appendix A 
 
 Page 
 
 Standard specifications and tests for Port- 
 land cement 833 
 
 Appendix B 
 
 Working stresses 845 
 
 Appendix C 
 
 Rulings Pertaining to Flat-Slab Design 
 
 Ruling on the design of cantilever flat- 
 slab construction in the city of 
 Pittsburgh 847 
 
 Ruling covering design of flat-slab con- 
 struction in the city of Chicago. . . 849 
 
 Chicago reinforced-concrete flat-slab rul- 
 ing amended 851 
 
 Page 
 
 Final report of special committee of the 
 American Society of Civil Engi- 
 neers — part pertaining to flat-slab 
 design 854 
 
 Standard building regulations for the use 
 of reinforced concrete. American 
 Concrete Institute, 1917 — part 
 pertaining to flat-slab floors 858 
 
 Appendix D 
 
 Standard notation 
 
 861 
 
 Appendix E 
 
 Index 
 
 Concrete barges and ships — extract from 
 report of the Joint Committee of 
 the American Concrete Institute 
 and Portland Cement Association 863 
 
 867 
 
CONCRETE ENGINEERS^ 
 HANDBOOK 
 
 SECTION 1 
 
 MATERIALS 
 CEMENT 
 
 1. Classification, Composition, and Uses of the Principal Cementing Materials. — Cement- 
 ing materials used in structural work may be divided into two main classes — non-hydraulic 
 and hydraulic. Non-hydraulic cements, as the name implies, will not set and harden under 
 water; while hydraulic cements will harden in either water or air. Following is a list of the 
 •structural cements of commercial importance: 
 
 Non-hydraulic 
 
 Hydraulic 
 
 Gypsum plasters 
 Common lime 
 
 Hydraulic lime 
 
 {Grappier cement, a by-product) 
 Puzzolan cement 
 Natural cement 
 Portland cement 
 
 (Adulterated or modified Portland cement) 
 
 la. Gypsum Plasters. — Gypsum plasters are made by partial or complete de- 
 hydration of relatively pure or impure natural gypsum. The setting of these plasters is a 
 recrystallization from a solution formed by admixture of the partially or totally dehydrated 
 material with water, reforming the original substance. [Pure gypsum is a hydrous crystalline 
 calcium sulphate (CaS04 + 2H2O) ; and in its raw uncalcined state is used as an adulterant 
 to retard the setting of Portland and natural cements. Plaster of Paris (CaS04 + 3'^H20) 
 is also used for the same purpose and is a refined plaster made from pure gypsum by 
 dehydration.] 
 
 Gypsum plasters, of one variety or another, are used principally on interior walls and floors. 
 They are also used in the form of molded hollow blocks and tiles for fireproof interior partition 
 walls, and as one variety of "stucco" for the architectural adornment of buildings. 
 
 16. Common Lime. — Common lime is made by burning limestone (CaCOs) 
 at a temperature of about 900°C. until its carbon dioxide (CO2) is driven off as gas. The residue 
 is common lime (CaO), known commercially as ''quicklime." On addition of water this 
 product slakes with evolution of heat and much increase of volume, forming a paste of lime 
 hydrate, or calcium hydroxide (Ca[0H]2) known as "Hme putty" or, on dilution with water, as 
 "cream of lime." 
 
 Even in the purest limestone to be found in nature some impurities are present. Generally 
 a part of the lime (CaO) is found replaced by a certain percentage of magnesia (MgO), and clay 
 
 1 
 
2 
 
 CONCRETE ENGINEERS' HANDBOOK 
 
 [Sec. 1-lc 
 
 is also present to some extent. [Clay is composed chiefly of silica (Si02) and alumina (AI2O3), 
 and usually contains some iron oxide (Fe203).] In the manufacture of quicklime, magnesia 
 acts in much the same manner and may be considered the equivalent of lime, which makes it 
 possible to use limestone which is high in magnesia. Quicklimes are divided into four main 
 types according to the relative content of calcium oxide (CaO) and magnesium oxide (MgO). 
 These are: 
 
 1. High-calcium; quicklime containing 90% or over of calcium oxide. 
 
 2. Calcium; quicklime containing not less than 85% and not more than 90% of calcium 
 oxide. 
 
 3. Magnesian; quicklime containing between 10 and 25% of magnesium oxide. 
 
 4. Dolomitic; quicklime containing over 25% of magnesium oxide. 
 
 Following is an average analysis of ten high-calcium quicklimes and two dolomitic quick- 
 limes : 
 
 
 Si02 
 
 AI2O3 
 
 FezOs 
 
 CaO 
 
 MgO 
 
 
 % 
 
 % 
 
 % 
 
 % 
 
 % 
 
 High-calcium . . 
 
 0.81 
 
 0 22 
 
 0.23 
 
 94.98 
 
 1.39 
 
 Dolomitic 
 
 0.87 
 
 0.32 
 
 0.29 
 
 60.13 
 
 36.12 
 
 [Analysis also shows carbon dioxide (CO2) and water (H2O) to be present in small amounts.] 
 Practically all lime used in construction is made into mortar by adding sand to the paste 
 of lime hydrate, as sand is not only cheaper than lime but diminishes the great shrinkage which 
 accompanies the setting and hardening of lime putty. This hardening is due mainly to crys- 
 tallization, but in addition some of the water in the hydroxide is gradually replaced by carbon 
 dioxide from the atmosphere, causing a small part of the hydroxide to revert to the original cal- 
 cium carbonate (CaCOs). 
 
 Although common lime is used chiefly in combination with sand as a mortar in laying 
 ordinary brick and stone masonry, it is also used extensively as an interior wall plaster and 
 for gaging hydraulic cement mortars, either to make them easier to work or to reduce their 
 permeability. 
 
 Hydrated lime is quicklime slaked at the place of manufacture. Its market form is that of 
 a dry powder, and as such it can be mixed with sand more easily than can lime paste made by 
 slaking ordinary quicklime on the work. 
 
 Ic. Hydraulic Lime. — Hydraulic lime is made by burning argillaceous or silicious 
 limestone at a temperature not less than 1000°C. When showered with water the product 
 slakes completely or partially without sensibly increasing in volume, and possesses hydraulic 
 properties due to the combination of calcium with silica contained in the limestone as an 
 impurity, forming calcium silicate. It is the universal practice to slake the lime at the place 
 of manufacture on account of the better results obtained. 
 
 Grappier cement is a by-product in the manufacture of hydraulic lime, produced by grind- 
 ing the lumps of underburned and overburned material which do not slake. As might be 
 inferred, grappier cement possesses properties similar to those of hydraulic lime. 
 
 Hydraulic lime is not manufactured in the United States on account of the abundance 
 of raw materials suitable for the manufacture of Portland cement, with which hydraulic lime 
 cannot compete as a structural material. A number of hydraulic limes and grappier cements 
 are marketed as "non-staining cements" — that is, they do not stain masonry For this reason 
 a considerable amount of this cementing material is annually imported from Europe for purposes 
 of architectural decoration. 
 
 Id. Puzzolan or Slag Cement. — Puzzolan cement is made by incorporating 
 hydrated lime with a silicious material, such as granulated blast-furnace slag, of suitable fine- 
 ness and chemical composition. In Europe a natural puzzolanic material, such as volcanic ash. 
 
Sec. 
 
 MATERIALS 
 
 3 
 
 is used at some plants in place of the blast-furnace slag. Silica, when finely enough divided, is 
 soluble in water and chemically active. For this reason the materials are finely pulverized and 
 intimately mixed by grindmg, but are not calcined, the formation of calcium silicate taking 
 place slowly and at ordinary temperatures. 
 
 Although this type of cement possesses hydraulic properties, it should not be confused with 
 slag Portland cement (sometimes called steel Portland cejnent) which is produced by calcining 
 finely divided slag and lime in a kiln and pulverizing the resulting clinker. Analysis of the small 
 number of puzzolan or slag cements manufactured in this country shows approximately the 
 following range in composition: 
 
 Si02 
 % 
 
 AI2O3 + FeaOa 
 + FeO 
 
 % 
 
 CaO 
 
 % 
 
 MgO 
 
 S 
 
 % 
 
 CO2 + H2O 
 
 % 
 
 27.2 to 31.0 
 
 11.1 to 14.2 
 
 50.3 to 51.8 
 
 1,4 to 3.4 
 
 0.15 to 1.42 
 
 2 . 6 to 5 . 3 
 
 They are normally slower in setting than Portland cements and on this account are usually 
 treated with materials which will hasten the set — such as burned clay, high-alumina slags, caus- 
 tic soda, sodium chloride, or potash. Puzzolan cements made from slag may be distinguished 
 by their light lilac color, absence of grit, and low specific gravity (2.60 to 2.85). They also are 
 high in sulphides, which render them liable to disintegration in air, nor are they suited for use in 
 sea water, where there is always an excess of sulphates. 
 
 Puzzolan cement is not as strong or reliable as either natural or Portland cement and should 
 be used only in unimportant structures or in unexposed work, such as foundations, where weight 
 and bulk are more important than strength. 
 
 le. Natural Cement. — Natural cement, as its name implies, is made from rock as 
 it occurs in nature. This rock is an argillaceous (clayey) hmestone, or other suitable natural 
 rock, and it is burned at a temperature of from 900° to 1300°C., the clinker being then finely 
 pulverized. The product does not slake, but possesses strong hydraulic properties, calcium 
 silicate being formed and acquiring strength and rigidity through crystallization. 
 
 Unfortunately, the composition and characteristics of natural cement are subject to consider- 
 able variation. This is to be expected, since the composition of the rock from which it is made 
 not only varies in different localities but is further subject to variation to some extent at least, 
 even in the same deposit. Portland cement, on the contrary, is an artificial mixture, subject 
 to control. This fact, together with its slow setting, is mainly responsible for the decrease in the 
 use of natural cement and the adoption of Portland cement in all important structures. 
 
 In spite of these disadvantages, however, it is a significant fact that natural cements do not 
 show disintegration with passage of time, while Portland cements frequently are most erratic 
 in behavior. On comparing analyses of typical natural and Portland cements, it is at once 
 noticed that natural cements have a higher percentage of silica, about the same percentage of 
 alumina and a lower percentage of lime than have Portland cements. Excess hme, so generally 
 prevalent in hydrated Portland cements, and not necessarily resultant on "free lime," is fre- 
 quently the cause of much trouble. There is a distinct field of usefulness for natural cement 
 which is largely overlooked by engineers at the present time. In many cases, perplexing prob- 
 lems could be effectively solved by its employment. 
 
 The following summary shows the range in composition of an average analysis of six well- 
 known American natural cements: 
 
 Si02 
 
 AI2O3 
 
 Fe203 
 
 CaO 
 
 IMgO 
 
 % 
 
 % 
 
 % 
 
 % 
 
 % 
 
 22.3 to 29.0 
 
 5 . 2 to 8 . 8 
 
 1 . 4 to 3 . 2 
 
 31.0 to 57.6 
 
 1.4 to 21.5 
 
4 
 
 CONCRETE ENGINEERS' HANDBOOK 
 
 [Sec. 1-1/ 
 
 [Analysis also shows varying small amounts of alkalies (K2O and Na20), anhydrous sulphuric 
 acid or sulphur trioxide (SO3), carbon dioxide (CO2), and water (H2O). Magnesia (MgO) is 
 usually regarded as equivalent to lime in its action.] The specific gravity of natural cements 
 range from 2.7 to 3.1, with an average of 2.85. 
 
 Natural cement is adapted to many uses, but its relatively low strength and slow hardening 
 limit its field to structures where high stresses will not be imposed for several months after plac- 
 ing the concrete, as in large or massive structures where weight and mass are more essential than 
 early strength — that is, in such structures as dams, abutments, foundations, and many under- 
 ground structures. Mortar made with natural cement (either alone or mixed with lime mor- 
 tar) is excellent for laying ordinary brick and stone masonry. 
 
 If, Portland Cement. — Portland cement is made by finely pulverizing the clinker 
 produced by burning a definite artificial mixture of silicious (containing silica), argillaceous 
 (containing alumina), and calcareous (containing lime) materials to a point somewhat beyond 
 where they begin to fuse or melt. The product is one that does not slake and possesses strong 
 hydraulic properties. The essential components of Portland cement — namely: silica, alumina, 
 and lime — are obtained from many different sources, but the proportions used of the raw mate- 
 rials are always such that the chemical composition of the different Portland cements is constant 
 within narrow limits. The percentages of the principal components range about as follows: 
 
 Si02 
 
 AI2O3 
 
 Fe203 
 
 CaO 
 
 MgO 
 
 % 
 
 % 
 
 % 
 
 % 
 
 % 
 
 19 to 25 
 
 5 to 9 
 
 2 to 4 
 
 60 to 64 
 
 1 . 0 to 2 . 5 
 
 [Small amounts of alkalies (K2O and Na20) and sulphur trioxide (SO 3) are also present. Mag- 
 nesia (MgO) is considered by some as an impurity, while other investigators claim it is equiva- 
 lent to lime (CaO) in its action. Alumina (AI2O3) and iron oxide (Fe203) do not act entirely 
 alike but are usually considered to have the same functions.] The specific gravity of Portland 
 cements range from 3.1 to 3.20, with an average of 3.15. 
 
 Portland cement is by far the most important cementing material used in modern engineer- 
 ing construction. It is adapted for use in concrete and mortar for all types of structures where 
 strength is of special importance, or in structures exposed to wear or to the elements. It 
 should invariably be employed in reinforced-concrete construction because of its high early 
 strength and generally uniform quality. 
 
 A number of special cements employing Portland cement as a base are made by grinding in 
 adulterating materials after calcination. These adulterants include clay, slaked lime, sand, 
 slag, natural cement, limestones, and natural puzzolanic material or tufa. The action of these 
 materials is essentially to promote combination between lime from the cement and silica from 
 the adulterant, with formation of silicate of lime. In some cases these silicious adulterants 
 improve the quality of concrete made from such cements, but this result cannot be expected 
 from all forms of adulteration. 
 
 Sand and puzzolanic material have perhaps been used the most extensively and successfully 
 of any of the adulterants, producing products known as sand cement and tnfa cement respectively. 
 These cements have been used principally on large work where freight rates are high and long 
 wagon hauls combine to make the cost of undiluted Portland cement excessive. Cement 
 specifications in common use are of a character to exclude any grinding in of materials after 
 calcination, presumably on the ground that specifications permitting any adulteration would 
 be subject to abuse so that the results obtained would be uncertain. 
 
 2. Portland and Natural Cements Compared. — The distinguishing properties of natural and 
 Portland cements and the chief differences in manufacture may be summarized as follows: 
 
Sec. 1-3] 
 
 MATERIALS 
 
 5 
 
 Natural cement 
 
 Portland cement 
 
 Calcination temperature 
 Chemical composition. . . 
 
 Raw material .... 
 Type of kiln used 
 
 Natural rock 
 
 Mostly the vertical stationary 
 type 
 
 Low, but variable 
 
 Variable, not under control 
 
 Artificial mixture 
 
 Slanting cylindrical revolving 
 
 type 
 Relatively high 
 Controllable within narrow 
 
 limits 
 
 Bluish or steel gray 
 
 Specific gravity 
 
 Rate of setting 
 
 Strength 
 
 Degree of grinding 
 Soundness 
 
 Color 
 
 Yellow to brown 
 2.7 to 3.1 
 Relatively rapid 
 Low, especially at early age 
 Usually rather coarse 
 Will not usually stand steam 
 test 
 
 3 . 1 to 3 . 2 
 
 Relatively slow 
 Relatively high 
 Relatively fine 
 
 Required to stand steam test 
 
 In structures where either natural or Portland cement may be used, and where economy 
 is the governing consideration, the choice of cement should be based on a comparison of the 
 costs per cubic yard of the required mortar or concrete mixtures. Decisions usually are 
 desired either between a 1:2 natural cement mortar and a 1:3 Portland cement mortar, or 
 between a 1:2:4 natural cement concrete and a 1:4:8 Portland cement concrete. 
 
 3. Constitution of Portland Cement.^ — The latest optical and microscopical examinations 
 of Portland-cement clinker, and of all the substances which were formally considered as likely 
 to be formed in manufacture, show Portland cement to be made up largely of the three com- 
 pounds 3CaO-Si02, 2CaO-Si02, and 3CaO-Al203. The tri-calcium silicate appears the best 
 cementing compound and it is probable that the higher its percentage, the better the cement. 
 The small amounts present of Fe203, MgO, alkalies, etc. have but little effect on the three 
 major compounds but their presence aids materially in manufacture by promoting the combina- 
 tion of CaO with AI2O3 and Si02. 
 
 A perfectly burned cement clinker consists of about 36% of tri-calcium silicate, 3CaO-Si02; 
 33% of di-calcium silicate, 2CaO-Si02; 21% of tri-calcium aluminate, 3CaO-Al203; and 10% 
 of minor constituents. The principal cementing compound, tri-calcium silicate, 3CaO-Si02, 
 is the last constituent to form completely in Portland-cement manufacture; and this compound 
 is formed by the combination of CaO with 2CaO-Si02. When cement clinker is not perfectly 
 burned there is evidently less 3CaO-Si02. formed and more 2CaO-Si02. There is also a certain 
 percentage of free lime (CaO) present, the amount depending upon the degree of burning. 
 
 4. Setting and Hardening of Portland Cement. ^ — The setting and hardening of Portland 
 cement is caused principally by hydration in the order named of the three major constituents — 
 3CaO Al203, 3CaO Si02, and 2CaO Si02. When water is added to Portland cement, these 
 constituents form first amorphous and later both crystalline and amorphous hydrated materials 
 which act much as does ordinary glue, except that since they are of mineral origin and largely 
 insoluble, hardening progresses even under water. 
 
 Of these hydration products, the compound tri-calcium aluminate (3CaO-Al203) when 
 mixed with water sets and hardens very quickly; tri-calcium silicate (3CaO-Si02) sets and 
 hardens somewhat less rapidly; and di-calcium silicate (2CaO-Si02) reacts slowly. Hardening 
 occurs only after the lapse of a long period of time. The initial set of cement is due undoubtedly 
 to the hydration of 3CaO-Al203; the early hardness and cohesive strength is due to this hydration 
 
 1 From paper before Am. Cone. Inst., Feb., 1916, by G. A. Rankin, Geophysical Laboratory, Carnegie Inst, 
 of Wash. 
 
 2 See Klein and Phillips: Tech. Paper, 43, U. S. Bureau of Standards. 
 
 Bates and Klein: Tech. Paper, 78, U. S. Bureau of Standards. 
 
6 
 
 CONCRETE ENGINEERS' HANDBOOK 
 
 [Sec. 1-5 
 
 and to that of the 3CaO Si02; while the gradual increase in strength is due to the further hydra- 
 tion of these two compounds together with the hydration of the 2CaO-Si02. 
 
 The compound 3CaO-Si02 appears to be the best cementing constituent of this group, as 
 it is the only one of the three which when mixed with water will set and harden within a reason- 
 able time to form a mass which is comparable in hardness and strength to Portland cement. 
 Although 3CaO-Al203 sets and hardens rapidly, it is rather soluble in water and is not particu- 
 larly durable or strong. The compound 2CaO-Si02, however, requires too long a time to harden 
 to be in itself a valuable cementing material. 
 
 5. Manufacture of Portland Cement. 
 
 5a. Raw Materials. — Silica, alumina, and lime — the essential components of 
 Portland cement — occur as ingredients in a large number of natural materials of widely vary- 
 ing character. In none of these, however, do the three components occur in the exact pro- 
 portions required in Portland-cement manufacture so that an artificial mixture of several 
 materials has to be resorted to. The following combinations of raw materials are used in differ- 
 ent cement plants in this country: 
 
 1. Cement rock and limestone. 
 
 2. Marl and clay (or shale). 
 
 3. Limestone and clay (or shale). 
 
 4. Blast-furnace slag and limestone. 
 
 5. Chalk and clay. 
 
 Cement rock is an argillaceous limestone containing about 68 or 72% of lime carbonate, 
 18 to 27% of clayey matter, and not over 5% of magnesium carbonate. It is a dark slatey 
 limestone, rather soft in texture, and is almost ideal for cement making due to the fact that 
 it is easy to quarry and grind, and is usually so well balanced in composition that but a small 
 amount of comparatively pure limestone needs to be added. This rock is found in many parts 
 of the country but so far has been used in the manufacture of Portland cement only in the 
 Lehigh district of eastern Pennsylvania and western New 'Jersey, a district producing nearly 
 one-third the entire output of the United States. 
 
 Limestone suitable for cement manufacture is composed principally of calcium carbonate 
 together with more or less impurities. The following shows the approximate range of com- 
 position of such limestones : 
 
 CaCOa 
 
 % 
 
 Si02 
 
 % 
 
 AI2O3 + Fe203 
 % 
 
 MgCOa 
 
 % 
 
 88.0 to 98.0 
 
 0 . 3 to 8 . 0 
 
 0.2 to 2.1 
 
 0.2 to 4.2 
 
 Sulphur as SO3 and various alkalies may also be present in small percentages. 
 
 Marl is almost pure calcium carbonate. It is a soft, wet earth found in the basins of dried- 
 up lakes and in swamp regions, deposited either by chemical agencies or through the phys- 
 ico-chemical agencies of certain forms of vegetable and animal life. 
 
 Clays and shales are of the same general composition, differing only in degree of solidifi- 
 cation. Clays result from the decay of shales, and like their parent rock, are composed chiefly of 
 sihca (Si02) and alumina (AI2O3), and usually iron oxide (Fe203). The proportion of silica in 
 clay suitable for cement manufacture should not be less than 55 to 65%; and the combined 
 amount of alumina and iron oxide should be between one-third and one-half the amount of 
 silica. A clay with these proportions of principal constituents is highly siliceous, producing 
 a cement clinker which is comparatively easy to grind. Clay containing a greater portion of 
 alumina produces a hard clinker and a quick-setting cement which is more severely attacked by 
 sea water. 
 
 Blast-furnace slag is a compound formed from impurities in the iron ore and the limestone 
 
Sec. 1-56] 
 
 MATERIALS 
 
 7 
 
 used as a flux in the blast furnace. Following is a typical analysis of slag used in the manu- 
 facture of cements: 
 
 Si02 
 
 % 
 
 AI2O3 + Fe203 
 
 % 
 
 CaO 
 
 % 
 
 MgO 
 
 % 
 
 33.10 
 
 12.60 
 
 49.98 
 
 2.45 
 
 Slag, when allowed to cool quickly, becomes a hard glassy mass, very tough and durable. In 
 order to use slag economically in cement manufacture the molten slag is run into large cisterns, 
 and there converted into a granular substance by directing against it innumerable little streams 
 of air and water. This disintegrated slag has the appearance of coarse brown sugar; and in this 
 form it is used as a raw material for Portland cement. 
 
 Chalk is a soft earthy variety of calcium carbonate, formed from the remains of minute 
 organisms. It also sometimes contains small amounts of silica, alumina, and magnesia. 
 Its use as a material for cement manufacture is limited. 
 
 5b. Proportioning the Raw Materials. — Two rules are in use for proportioning 
 the raw materials used in the manufacture of Portland cement. Newberry's rule is as follows: 
 
 Max. lime = 2.8 (%Si02) + 1.1 (roAlgOa) 
 
 Eckel's rule (called the ''cementation index"), which is really a modification of the above rule, 
 takes the magnesia and iron oxide into account : 
 
 2.8 (%Si02) + 1-1 (%Al203) + 0.7 (%Fe203) ^ 
 %CaO + 1.4 (%MgO) 
 
 A value of the cementation index below 1 means an excess of lime or magnesia in the cement 
 which will cause expansion, or unsoundness. In practice it is customary to reduce by about 
 10% the proportion of lime found by the above rule to avoid any chance of obtaining an unsound 
 cement. Although the above rules are not based on the most recent investigations of the 
 constitution of Portland cement there is no immediate prospect of any change being made in 
 practice in the methods of proportioning because the present rules are known by experience to 
 produce excellent results. 
 
 5c. Grinding and Mixing. — The admixture and grinding of the raw materials 
 before calcination is accomplished by either a wet or ai dry process. In the wet process, prin- 
 cipally for plants using marl, the raw materials are ground and fed into rotary kilns in the 
 form of a slurry containing sufficient water to make it of a fluid consistency. In the dry process 
 raw materials are ground and mixed in the dry state. The larger portion of Portland cement 
 manufactured in the United States at the present time is made by plants using the dry process, 
 
 5d. Burning the Cement Mixture. — Rotary kilns are used in almost all American 
 Portland-cement plants. These kilns are slightly inclined to the horizontal and revolve at 
 about the rate of one revolution per minute. The ground raw materials are fed in at the upper 
 end and are carried forward and tumbled over and over by the slant and revolution of the kiln. 
 As the materials advance they are reduced by the hot gases from the burning fuel which is fed 
 in at the lower end. The clinkers formed vary in size from in. up to about 1}4 in. in 
 diameter. It takes about an hour for a particle of raw material to traverse the entire dis- 
 tance from the feed to the outlet. 
 
 5e. Treatment of the Clinker. — After cement clinker is cooled it is crushed and 
 passed through preliminary grinding mills. Then gypsum is added and the clinker ground to 
 a fine powder. If the clinker was used without the addition of gypsum, it would take an 
 almost immediate set. Approximately 2 lb. of gypsum (CaS04) (or plaster of Paris) is used to 
 every 100 lb. of clinker. 
 
CONCRETE ENGINEERS' HANDBOOK 
 
 [Sec. 1-6 
 
 6. Manufacture of Natural Cement. 
 
 6a. Raw Material. — The raw material used in the manufacture of natural cement 
 is a natural argillaceous limestone containing from 13 to 35% of clayey material. About 15% 
 of the clayey material is silica, the balance being alumina and iron oxide. The kind of limestone 
 generally used contains a considerable proportion of magnesium carbonate in place of calcium 
 carbonate. When the rock varies greatly in composition, materials from different strata are 
 mixed together to give as uniform a product as possible. The wide range allowed in the com- 
 position of natural cement, however, does not warrant great refinement in the analysis of the rock, 
 
 66. Process of Manufacture. — Natural cement is usually manufactured in ver- 
 tical kilns lined with firebrick. These kilns are of the mixed-feed type, the rock (not crushed) 
 and fuel being charged in alternate layers. The temperature required in burning is considerably 
 below that required in Portland-cement manufacture because temperatures higher than 1300°C. 
 would fuse the material to a slag having no hydraulic properties. The temperature employed, 
 however, is sufficient to cause the formation of silica compounds with the lime and magnesia. 
 The clinker is taken out at the bottom of the kiln as it is burned; and then it is crushed and 
 ground. Grinding of the clinker is not usually carried as far as that of Portland although some 
 of the newer mills use grinding machinery similar to that in Portland-cement plants. 
 
 7. Testing of Cement. — For standard methods of cement testing, see Appendix A. 
 
 7a. Sampling. — Tests should be conducted only on representative samples. 
 For method of sampling, see Appendix A, page 834. 
 
 76. Uniformity in Cement Testing. — In order to obtain results in cement testing 
 which will be of the greatest value, definite and uniform methods should be used. Results 
 depend not only on the quality of the cement but also on the temperature and percentage of 
 water used in mixing, the method of mixing and molding test specimens, the temperature and 
 humidity of the air, the character of the sand used, and the type of apparatus employed. 
 
 7c. The Personal Factor. — The personal factor has considerable effect on results 
 obtained in cement testing and, on this account, only experienced, well-qualified men should be 
 employed in making tests. Results by untrained or careless operators are really worse than 
 nothing and may be positively misleading. The comparative results, however, by any one 
 experienced observer are generally consistent and are of value. It is usually advisable to have 
 the testing done at some well-established and properly equipped cement-testing laboratory. 
 
 7d. Kinds of Tests. — The following cement tests made regularly are recom- 
 mended for construction work of importance and also in all cases where the cement to be used 
 does not work satisfactorily : 
 Fineness. 
 Time of setting. 
 
 Tensile strength of standard mortar. (Compressive strength of standard mortar the 
 
 best criterion.) 
 Soundness. 
 
 On unimportant construction it is generally safe to use a well-known brand of Portland cement 
 without testing, or to make simply the test for soundness. 
 
 7e. Fineness. — Fine grinding has a great influence on the properties of cement. 
 It increases the ability of the cement to react readily with water and enables the cement 
 particles to coat the sand grains more thoroughly. In other words, the finer the cement, all 
 other conditions being the same, the stronger will be the mortar produced with a given sand. 
 
 The fineness of cement is measured by determining the percentage by weight which will 
 be retained on a standard 200-mesh sieve. Standard specifications^ require that the residue 
 shall not exceed 22%. Most mills are now equipped to grind cement to such a fineness that 
 even less than 10% is retained. 
 
 It has long been generally recognized that the coarser particles in cement are practically 
 
 1 For description of standard sieve, see Appendix A, p. 836. 
 
 2 For standard specifications, see Appendix A, p. 833. 
 
Sec. 1-7/] 
 
 MATERIALS 
 
 9 
 
 inert and that at least the earlier cementing value is due chiefly to the grains that will pass the 
 No. 200 sieve. Because of this fact the present standard method of testing for fineness is unsat- 
 isfactory, as no attempt is made therein to determine the further fineness of the greater and 
 more valuable part of the cement. To remedy this defect an air-analyzer^ has recently been 
 perfected at the Bureau of Standards which makes it possible to further divide a cement which 
 passes a 200-mesh sieve into four definite sizes. This apparatus had been standardized and will 
 undoubtedly come into extensive use. 
 
 Increased fineness has the effect of making a cement quicker in setting and hardening, 
 the high-alumina cements being the most affected. Fine grinding also affords additional 
 opportunity for seasoning and thus indirectly improves the soundness of cement. 
 
 7/. Normal Consistency. — Tests for time of setting, strength, and soundness 
 are greatly influenced by the quantity of water used in mixing. In order to have all results 
 comparable with one another, a determination is made in each case of the quantity of water 
 necessary to be added to a given weight of cement to give a standard or normal consistency. 
 
 A simple method of finding normal consistency is to mix a quantity of cement paste and 
 make up from the paste a ball about 2 in. in diameter. The ball is then dropped upon the 
 testing table from a height of 2 ft. The paste is of normal consistency when the ball does not 
 crack and does not flatten more than one-half of its original diameter. The finer the cement, 
 the more water is required for normal consistency. For this test the room and the mixing 
 water should be kept at standard temperature. 
 
 Another method of finding normal consistency which is more commonly used and gives 
 more concordant results is by the use of the Vicat needle apparatus (see Fig. 2, Appendix A, 
 page 837). The manner of making this test is explained in Appendix A, page 838. 
 
 Ig. Time of Setting. — The time of setting of a cement may vary within wide 
 limits and is no certain criterion of quality, but it is important in that it indicates whether or 
 not the cement can be used advantageously in ordinary construction. A cement may set so 
 quickly that it is worthless for use as a building material (since handling cement after it com- 
 mences to set weakens it and causes it to disintegrate), or it may set so slowly that it will 
 greatly delay the progress of the work. 
 
 Age of cement has a great effect upon the setting time, and tests should preferably be made 
 after delivery of the cement on the work. Most cements absorb moisture from the air and lose 
 some of their hydraulic property on storage. It also occasionally happens that the gypsum 
 added in manufacture loses its effectiveness in a short time, and in consequence the cement 
 becomes quick setting. The cause of this loss of effectiveness of the gypsum is due usually to 
 the composition of the cement and may be remedied by increasing the lime content. 
 
 Aside from the consideration of age, the conditions which accelerate setting are: finely 
 ground and lightly burned material; dry atmosphere; small amount of water used in gaging; 
 and high temperature of both water and air. Since the time of set is influenced by so many 
 factors, tests should alwaj'-s be made with extreme care under standardized conditions. 
 
 There are two distinct stages in setting: (1) the initial set; and (2) the hard or final set. 
 The best cements should be slow in taking the initial set but after that should harden rapidly. 
 Portland cement should acquire the initial set in not less than 45 min. when the Vicat needle 
 is used (see Appendix A, page 837), and hard set in not more than 10 hr.2 The time of initial 
 set is controlled largely by the amount of sulphate (gypsum or plaster of Paris) which is added 
 in making the cement. 
 
 A cement has taken its initial set when it will not thoroughly reunite along the nurf aces of 
 a break. It has taken its final set when it begins to have appreciable strength and hardness. 
 
 There are two methods in common use for finding the time of setting. The method usually 
 preferred is by the use of the Vicat needle apparatus explained in Appendix A, page 838. The 
 
 1 Copies of Tech. Paper 48, the publication upon this subject, may be obtained, free of charge, upon appli- 
 pation to the Bureau of Standards, Washington, D. C. 
 ? 3ee standard specifications in Appendix A, p. 833, 
 
10 
 
 CONCRETE ENGINEERS' HANDBOOK 
 
 [Sec. l-7h 
 
 other method is by the use of the standardized Gillmore needles described in Appendix A, 
 page 841. 
 
 7h. Tensile Strength. — The testing of cement in tension is to obtain some meas- 
 ure of the strength of the material in actual construction. In other words, tests of tensile 
 strength are made primarily to determine whether the cement will be likely to have a continued 
 and uniform hardening in the work, and whether it will have such strength when placed in 
 mortar or concrete that it can be depended upon to withstand the strain placed upon it. 
 
 The small shapes made for testing are called briquettes (see details of standard test piece in 
 Appendix A, page 842) and have a minimum cross-sectional area of 1 sq. in. — that is, at the 
 place where they will break when tested. Standard mortar used in -testing is composed of 1 
 part cement to 3 parts of standard sand^ from Ottawa, 111. 
 
 It is customary to store the briquettes, immediately after making, in a damp atmosphere 
 for 24 hr. They are then immersed in water until they are tested. This is done to secure 
 uniformity of setting, and to prevent the drying out too quickly of the cement, thereby prevent- 
 ing shrinkage cracks which greatly reduce the strength. 
 
 Specifications for tensile strength of cement usually stipulate that the material must pass a 
 minimum strength requirement at 7 and 28 days. This is required in order to determine the 
 gain in strength between different dates of testing so that some idea may be obtained of the 
 ultimate strength which the cement will attain. A first-class cement, when tested, should 
 give the values for tensile strength stated in the standard specifications (see Appendix A, 
 page 833). 
 
 7^. Relation between Tensile and Compressive Strength. — Since cements are 
 rarely depended upon to withstand tensile stresses, the test for tensile strength has undoubtedly 
 become standard on account of the popular belief that there exists a more or less definite and 
 constant relation between the tensile and compressive strengths. It can be shown, however, 
 that the ratio of compressive to tensile strength of cement mixtures is by no means constant 
 at all ages and varies greatly with different cements and with different mixtures. Thus the 
 tensile strength cannot usually be regarded as any more than a very approximate indication 
 of the probable compressive strength of the same cement. 
 
 7j. Compressive Strength. — Compressive strength of cement mortar is undoubt- 
 edly a better criterion by which to judge the suitability of a cement for use in construction. 
 The American Society for Testing Materials has tentative specifications and methods of tests 
 for compressive strength of Portland-cement mortar ^ which, when adopted as standard by the 
 Society, will be inserted in and made a part of the American Specifications and Methods of 
 Tests for Portland Cement. A foreign standard specification is as follows: 
 
 "Slowly setting Portland cement shall show a compressive strength of at least 120 kg. per 
 sq. cm. (1710 lb. per sq. in.) when tested with 3 parts by weight of standard sand, after 7 days' 
 hardening, 1 day in moist air and 6 days under water; after further hardening of 21 days in the 
 air at room temperature (15° to 20°C.) the compressive strength shall be at least 250 kg. per 
 sq. cm. (3570 lb. per sq. in.). In cases of controversies, only the test after 28 days is decisive. 
 
 ''Portland cement which is intended for use under water shall show a compressive strength 
 of at least 200 kg. per sq. cm. (2850 lb. per sq. in.) after 28 days' hardening, 1 day in moist air 
 and 27 days in water." 
 
 7k. Soundness. — A cement to be of value must be perfectly sound; that is, 
 it must remain constant in volume and not swell, disintegrate, or crumble. Excess of either 
 lime, magnesia, or sulphates may cause unsoundness. The usual method of testing is to form 
 a small pat of neat cement about 3 in. in diameter, }i in. thick at the center, and tapering to a 
 thin edge. This pat should remain 24 hr. in moist air and 5 hr. in an atmosphere of steam at a 
 temperature between 98 and 100°C. upon a suitable support 1 in. above boiling water. To 
 pass the soundness test satisfactorily, the pat should remain firm and hard, and show no signs 
 
 1 See Appendix A, p. 841. 
 
 2 See Proc. of the Society, vol. xvi (1916), part I (pp. 590-593). 
 
Sec. 1-7/] 
 
 MATERIALS 
 
 11 
 
 of cracking, distortion, checking or disintegration. The steam test is what is called an acceler- 
 ated test and is for the purpose of developing in a short time (5 hr.) those qualities which tend to 
 destroy the strength and durability of a cement.^ 
 
 7/. Specific Gravity. — A test for finding the specific gravity of Portland cement 
 was originally considered to be of value in detecting adulteration and underburning, but is no 
 longer thought to be of much importance in view of the fact that other tests lead to niore definite 
 conclusions. One trouble has been that specific gravity is not alone lowered by the above 
 causes. Seasoning of either cement or cement clinker, for instance, although known to be 
 desirable and in some cases absolutely necessary, lowers the specific gravity materially. On the 
 other hand, many underburned cements show a specific gravity much higher than that set by 
 standard specifications. These considerations, together with the fact that the principal adul- 
 terants have a specific gravity very near that of Portland cement, make it difficult in the specific 
 gravity test to obtain results from which accurate conclusions can be drawn. The test in any 
 case is without value unless every precaution is taken to have accurate results, as otherwise 
 only very large amounts of adulterated material could be discovered. When the specific 
 gravity of a cement falls below 3.10, standard specifications^ allow a second test to be made 
 upon an ignited sample — the idea being that ignition will lower the specific gravity of adulter- 
 ated cement. This second test, however, is usually of little value as the ignition loss of most 
 adulterants is low and as the specific gravity of an ignited sample of cement is invariably higher 
 than that of the original sample. 
 
 7m. Chemical Analysis. ^ — If the tests of a cement for time of setting, strength, 
 and soundness seem to indicate adulteration, resort may be had to chemical analysis. Such 
 analysis is not usually made in routine commercial testing. Chemical analysis not only serves 
 as a valuable means of detecting adulteration but shows the amounts of magnesia (MgO) and 
 sulphuric anhydride (SO3) contained in the cement. Specifications usually limit the amount of 
 MgO to about 5% and SO3 to about 2% because of fear that more of these materials may make 
 the cement unsound. 
 
 8. Specifications for Cement. — Standard specifications are given in Appendix A. 
 
 9. Containers for Cement. — Cement may be obtained in cloth or paper bags, in bulk, and 
 in barrels. 
 
 Cloth bags are the containers most generally used since manufacturers will refund the 
 extra charge for the bags when returned in good condition. The consumer, however, must 
 prepay the freight when returning the empty bags to the mill. The cloth bag will stand trans- 
 portation, and its size and shape make it convenient to handle. If properly cared for, it may be 
 used over and over again. Paper bags are more delicate and have no return value. Wooden 
 barrels are advisable when the work is in a damp location, as in marine construction. Bulk 
 cement requires special preparations for handling and storage. 
 
 10. Storing of Cement. — Cement either in containers or in bulk should be stored within 
 a tight, weather-proof building, at least 8 in. away from the ground and an equal distance from 
 any wall, so that free circulation of air may be obtained. In case the floor of a storage building 
 is laid directly above the ground, it would be well to give the cement an additional 8-in. eleva- 
 tion by means of a false floor, so as to insure ventilation underneath. The cement should 
 further be stored in such a manner as to permit easy access for proper inspection and identifi- 
 cation or removal of each shipment. When cement is not mill-tested, a proper period before 
 cement is needed should be allowed by the contractor for inspection and tests, this period being 
 determined by the provisions of the specifications governing his contract. 
 
 Where cement in bags is stored in high piles for long periods, there is often a shght tendency 
 in the lower layers to harden, caused by the pressure above; this is known as warehouse set. 
 
 1 The Lackawanna Railroad Co. requires that Portland cement used in its structures shall remain sound after 
 being subjected to boiling under a 20-atmosphere pressure. This is called the Autoclave Test. 
 
 2 See Appendix A, p. 833. 
 
 3 For method to be followed in making a chemical analysis, see Appendix A, p. 834. 
 
12 
 
 CONCRETE ENGINEERS' HANDBOOK 
 
 [Sec. 1-11 
 
 Cement in this condition is in every way fit for service and can be reconditioned by letting 
 each sack drop on a sohd surface before using the cement contained. 
 
 11. Seasoning of Cement. — A moderate amount of seasoning in weather-tight sheds often 
 improves the quahty of the cement. Fresh cement contains small amounts of free or loosely 
 combined lime which does not slake freely and causes expansion after the mass has set, endan- 
 gering the structure in which it is used. During the time of seasoning such free lime is changed 
 first to hydrate and then to carbonate of lime which does not swell on wetting. Usually cement 
 is seasoned at the mills before shipping, but, with the best mills, the stock house may run so 
 low in periods of rush that a chance will be taken on fresh material. Well-seasoned cement, 
 therefore, may be lumpy, but the lumps are easily broken up. If, however, the cement has 
 been subjected to excessive dampness, or has been wet, lumps will be formed which are hard 
 and difficult to crush. A distinction should be made, so that the latter will not be used with- 
 out sifting and rejection of hardened portions. 
 
 12. Use of Bulk Cement. — Within the past few years considerable cement has been shipped 
 in bulk to cement-product factories and to construction jobs adjacent to railroad tracks. Econ- 
 omy has, in these instances, resulted from the saving in labor, and from the elimination of pack- 
 age losses and expense. There seems to be no difficulty in shipping bulk cement in tight box cars. 
 
 13. Weight of Cement. — A barrel of Portland cement weighs 376 lb., not including the 
 barrel, and a bag of Portland cement weighs 94 lb.; in other words there are 4 bags to a barrel. 
 
 A barrel of natural cement varies in weight according to the locality in which it is manu- 
 factured. A barrel of Western cement usually weighs 265 lb. and a barrel of Eastern cement 
 300 lb. A bag of natural cement is usually one-third of a barrel. 
 
 A barrel of puzzolan cement is usually assumed to contain 330 lb. net, and there are 4 
 bags to the barrel. 
 
 A cement barrel weighs about 20 lb. on an average. 
 
 AGGREGATES 
 
 14. Definitions. — "Aggregates" is a general classifying term applied to those inert {i.e., 
 chemicall}'' inactive) materials, both fine and coarse, which, when bound together by cement, 
 form the substance known as concrete. Fine aggregates are materials such as natural sand or 
 rock screenings. Coarse, or large aggregates, or ballast are materials such as natural gravel, 
 crushed rock, or by-product materials such as cinders or crushed blast-furnace slag. 
 
 15. General Requirements. — Aggregates, fine and coarse, compose approximately 90% 
 or more of the substance of concrete. From this it follows that the properties of aggregates 
 must correspond and be at least equal to the properties desired in the concrete. 
 
 The usual service requirements are that aggregate shall be dense, hard, durable, structur- 
 ally strong and, for aggregates in concretes exposed to water action, insoluble. Further, since 
 concrete is formed by bonding of aggregates with cement, they must permit by their physical 
 characteristics (such as roughness) the adhesion of cement; and always all particles must be 
 clean, so that a surface coat of one kind or another may not prevent physical contact with ce- 
 ment, or destroy its properties through chemical action. 
 
 16. Classification of Aggregates. — The usual classification of aggregates is into two divi- 
 sions, based upon size. 
 
 Coarse aggregates are all particles of gravel, crushed stone, or other materials 
 above 3^:4 in. diameter. 
 
 Fine aggregates are all particles below ^ in. in size. Particles of such size are 
 further divided by defining "sand" as all mineral particles from 2 mm. to 0.5 mm. in 
 diameter; "silt," all particles from 0.5 mm. to 0.005 mm. in diameter; "clay," all particles 
 having a diameter less than 0.005 mm.; and "loam" as a mixture of any of the above finer 
 varieties with organic matter — i.e., of vegetable or animal origin. It is particularly such organic 
 matter rather than size of particle which renders loam unfit for concrete work, as through some 
 
Sec. 1-17] 
 
 MATERIALS 
 
 13 
 
 chemical action not yet fully understood, possibly through formation of an organic acid, it injuros 
 or inhibits the proper action of cement. 
 
 17. Qualities of Fine Aggregates. — General. — When it is remembered that the finer natural 
 materials are derived from rocks by disintegration and by "weathering," or breaking down 
 through frost action, water and wind erosion, or kindred agencies, the differences in quality 
 so often found in sand deposits, with possibly the presence of foreign materials, are not surprising. 
 Further, sands necessarily partake of the qualities of the rock from which they are derived. 
 Silicious quartz sands are best for concrete work, but crushed sands from any durable rock will 
 answer, if natural sand of proper quality cannot be obtained. 
 
 18. Qualities of Coarse Aggregates. — General. — For coarse aggregates, any crushed rock of 
 durable character, or any clean, hard, natural gravel not subject to ready disintegration may 
 properly be used. In general, the better the stone or gravel, the better the resulting concrete. 
 For this reason, granite, trap, or hard limestone are preferred for large aggregates, but any rock 
 will serve which is sound, which has adequate strength and does not contain objectionable 
 mineral inclusions liable to decompose, such as 
 
 iron pyrites, FeS2, which may form sulphuric 
 acid by oxidation (see Fig. 1). 
 
 Since the properties of any concrete are so 
 closely related to the properties of its compo- 
 nents, it is essential to an understanding of the 
 value of any stone as an aggregate that something 
 be known of the origin, nature, and properties 
 of the varieties in common use. 
 
 19. Materials Suitable for Coarse Aggre- 
 gates. — Roughly, rocks suitable for use as aggre- 
 gate fall into three groups. These are: (1) Gran- 
 ite and other igneous rocks; (2) sandstones and 
 other sedimentary rocks; (3) limestones and re- 
 lated rocks. A fourth division comprises slates 
 and shales, but as these weather rapidly with for- 
 mation of clay, they are unsuited for use in con- 
 crete. 
 
 The physical character of a rock depends 
 upon two things — its mineral constituents and 
 its structure. If the mineral constituents are themselves durable, but massed together in a 
 manner structurally weak, rapid weathering, with formation of sand through liberation of 
 mineral grains, is to be expected. Such a rock would make a poor concrete. On the other 
 hand, a dense structure with like mineral constituents would make an excellent aggregate. 
 A dense structure and weak mineral constituents are sometimes associated, but Nature has 
 generally cared for such rocks by bringing about their decomposition, so that they exist only as 
 sand. 
 
 20. Igneous Rocks. — Igneous rock is a general term descriptive of all rocks formed from 
 molten matter which has consolidated either into mineral, or glass, or both. Among such 
 rocks are granite and trap rock. Many classifications of a more-or-less satisfactory nature 
 have been devised; but all sorts of gradations exist between the various types, rendering their 
 descriptive identification difficult. 
 
 20a. Granite. — Granite is well-known by its characteristic appearance (sec 
 Fig. 2). In structure, it is a blend of quartz (crystallized silica dioxide), orthoclase, and mica, 
 though this latter may be replaced by hornblende. It is exceedingly dense, hard, and durable, 
 consisting entirely of minerals with no glass or uncrystallized material between its constituent 
 grains. 
 
 Granites possess the strength and durability desirable in an aggregate, but they are of low 
 
 Fig. 1. — Iron pyrites (FeSj) in concrete aggregate 
 oxidizing and destroying adjacent matrix. (Magni- 
 fied 40 diams.) 
 
14 
 
 CONCRETE ENGINEERS' HANDBOOK 
 
 [Sec. 1-206 
 
 toughness. In addition, if used in concretes exposed to more than ordinary heat, as in chimneys, 
 there is a decided tendency to disintegrate, due to unequal mineral grain expansions. Granites 
 are not often used as aggregate, their ornamental value precluding less profitable use. 
 
 206. Trap Rock or Diabase. — Trap rock (Fig. 3) and fine-grained basic and vol- 
 canic rocks are generally hard, of high abrasive value adhering well to cement. These rocks 
 
 Surface appearance. Internal structure. 
 
 (Natural size.) (Magnified 60 diams.) 
 
 Fig. 2. — Granite. 
 
 have a closely interlaced mineral structure and generally good resistance to stress. Care should 
 be taken not to choose a trap rock having a considerable percentage of iron present in low oxide 
 form, as this may absorb oxygen, forming a higher oxide, with expansion and probably rupture. 
 
 In general, trap rock (and rock of similar character, in which class are included many of the 
 ''green-stones") makes a very excellent aggregate although in some respects its excellence 
 
 Surface appearance. Internal structure. 
 
 (Natural size.) (Magnified 60 diams.) 
 
 Fig. 3.— Trap rock. ^ 
 
 has been exaggerated. It has, however, a very high compressive strength and, as this quality 
 is very desirable in concretes, its use has become widespread. It is not always procurable 
 without excessive cost but, where price is not prohibitive, its use is advantageous. 
 
 21. Sedimentary Rocks. — To the sedimentary series of rocks belong all those solidified 
 deposits which have accumulated at the bottom of bodies of water. Originally, these materials 
 were derived from the land surface and transported to the sea or lakes, either by mechanical 
 carriage, or by solution in water. Many of the minerals contained in sedimentary rocks were 
 
Sec. l-21a] 
 
 MATERIALS 
 
 15 
 
 derived directly from the decay of igneous or volcanic rocks, although additional chemical 
 changes supplementing this more-or-less complete decomposition of the original minerals may 
 have resulted in the formation of new minerals found in the sedimentary series. With passage 
 of time and the action of various chemical and mechanical agencies, these sedimentary deposits 
 solidified into the stratified rocks of one kind or another found throughout the entire surface 
 of the earth. 
 
 21a. Sandstone. — One of the most important of the sedimentary rocks is sand- 
 stone (Fig. 4). In structure, it is natural concrete, composed of finely divided mineral parti- 
 cles, cemented together in more-or-less close relation by iron or alumina or by calcium com- 
 pounds. The character of any sandstone depends, therefore, on the mineral character of its 
 component grains; on the size and shape of these grains; on their arrangement within the rock; 
 and on the nature of the material cementing them together. 
 
 Quartz particles form by far the greatest percentage and the most desirable constituent of 
 sandstone. Feldspar is also frequently present, and occasionally, hornblende, chlorite, garnet, 
 magnetite, and calcite. In the best sandstones, the grains are arranged uniformly through the 
 
 SurfiK c appearance. Internal structure. 
 
 (Natural size.) (Magnified 60 diams.) 
 
 Fig. 4.— Sandstone. 
 
 mass, although frequently coarser and finer particles are arranged in layers, giving a stratified 
 appearance to the stone. 
 
 So far as its use in concrete is concerned, the most important feature of a sandstone is the 
 nature of the cementing material combining its constituent grains. Argillaceous sandstones 
 in which the cementing material is lime (usually lime carbonate) may be crushed with compara- 
 tive ease, but they disintegrate rapidly on exposure to weathering agents, such as water or air. 
 Such stone may be readily identified by its effervescence when treated with a drop of hydro- 
 chloric acid. Sandstones cemented by oxide of iron are generally red in color, the shade being a 
 rough indication of the amount of iron present. Many of these sandstones disintegrate very 
 rapidly on exposure to the weather, forming the so-called ''rotten stones" so often found in 
 gravel. 
 
 Sandstones cemented solely by clay should never be used in concrete, as the simple pene- 
 tration of moisture is sufficient to disintegrate them, rendering them practically valueless as 
 aggregate. A good accelerated test is to boil }i-m. fragments of the stone in water. Rapid 
 disintegration indicates a weak stone, with a tendency to weather rapidly, and unsuited for use 
 as aggregate. 
 
 216. Limestone. — ^Limestone is carbonate of Ume deposited on the floors of 
 bodies of water and subsequently hardened into rock. This precipitation of lime may have 
 been effected from the water, or through the agency of animal or vegetable life. That is to say, 
 
16 CONCRETE ENGINEERS' HANDBOOK [Sec. 1-22 
 
 some limestones are chemical precipitates, while others are formed from the shells and other 
 hard parts of animals, as well as from hardened tissues of certain plants. Such plant and animal 
 forms fossilized are often seen in limestone fragments (see Fig. 10, page 19). 
 
 Compact limestone (Fig. 5) varies in texture from coarse to exceedingly fine. (It is prac- 
 tically impossible to obtain a good photograph of the surface of limestone on account of its 
 dark color and uniform texture.) It is only occasionally pure carbonate of lime, usually contain- 
 ing greater or less percentages of magnesia. Either magnesian limestone, or pure calcic lime- 
 stone is very well suited for use as a concrete aggregate. Any considerable percentage of clay 
 in limestone, however, is very undesirable, as it softens the rock and renders it very liable to 
 disintegration. Limestone is found in many colors. White, gray, yellow, blue, and green are 
 those of most frequent occurrence. 
 
 In general, limestone makes a very good coarse aggregate for concrete. When crushed to 
 the finer sizes, it has a flaky fracture which renders it somewhat unsuitable for use as sand unless 
 it is rerolled. Natural limestone sands are of infrequent occurrence, as limestone is soluble 
 
 to as high a percentage as 90%, so that the usual 
 weathering processes result in solution, rather than 
 fragmentary disintegration. 
 
 22. Metamorphic Rocks. — Rocks of either igneous 
 or sedimentary origin have often been subjected to 
 such severe treatment in the long course of geologic 
 history, that their ordinary character is much altered. 
 Crushing of the earth's crust, the weight of overlying 
 material, and contact with hot molten rock from the 
 interior are among the causes contributing to the 
 change. Such rocks are classed as ''metamorphic." 
 
 There are many metamorphic rocks, the whole 
 group constituting a very high percentage of the sur- 
 face of the earth's crust. Some of them are of value 
 as aggregates in concrete; while others, notably the 
 slates and shales, have a weak stratified structure 
 and weather so rapidly that their value in concrete is 
 almost nothing. 
 
 The above classification gives a general indica- 
 FiG. 5.— Internal structure of limestone. ^ion of those rocks which when crushed are of value 
 (Magnified 20 diams.) as Concrete aggregates. The first caution to be ob- 
 
 served in selecting them is to be sure that they do not 
 contain objectionable impurities in their substance; and the second important caution is to be 
 sure they are clean. 
 
 23. Gravel. — Gravel of good quality (Fig. 6) makes excellent concrete (see Tech. Paper 58, 
 Bureau of Standards, Washington, D. C). Gravel is nothing more nor less than natural rock, 
 broken away from parent ledges and worn round by the rolling of streams. Its natural proper- 
 ties, therefore, are identical with the rock of which it once formed a part. Provided it has not 
 decayed through being in relatively small masses, the properties natural to this parent rock 
 are to be expected of a gravel. The surface of gravel is usually very rough; and from consider- 
 ations of character of surface presented for adhesion of cement, it should produce as good as, and 
 even better, concrete than crushed stone. Certainly, there is no reason against its use, provided 
 it is clean and of good mineral quality. 
 
 It is desired to emphasize in this connection that not the least important of all the qualities 
 of stone or gravel is its cleanness. The percentage of concretes in which cement and aggregates 
 have little or no adhesion due to a coating of dirt (a coating of "matter out-of-place") is sur- 
 prisingly large; and the careless acquiescence of engineers in the use of such materials is result- 
 ing in a general inferiority of concrete structures. ''Dirt" in such cases may be visible (as 
 
Sec. 1-24] 
 
 MATERIALS 
 
 17 
 
 Fig. 6. — Enlarged surface of piece of 
 gravel, showing roughness. (Magnified 
 5 diams.) 
 
 when the coating is clay, or of tenacious dust due to crushing) or it may be quite invisible (such 
 as a coating of colloidal, transparent, organic matter) requiring chemical procedure for its 
 detection. A coating of any character is not to be disregarded, when first-quality concretes 
 are desired. At best it is a detriment and oftentimes proves a serious defect, greatly weakening 
 the concrete. 
 
 24. Blast-furnace Slag. — Slag from blast furnaces, crushed to proper size, has much to 
 recommend it for mass construction. Slag is a hard though very porous material, of high 
 compressive strength; and in certain localities is relatively 
 
 cheap as compared to stone of good qualities. Offering 
 a rough, pitted surface for the adhesion of cement, it 
 produces a very strong concrete, but care should be taken 
 that its sulphur content is low, else passage of time may 
 bring about disintegration of the concrete. Some steel 
 companies exercise great care in the preparation of slag 
 for aggregate, weathering it in thin layers for 2 or 3 ^''ears 
 before marketing, but the advisability of its use in con- 
 cretes exposed to dampness and especially in thin sec- 
 tions is yet in controversy (see Proc. Am. Soc. Test 
 Mat., 1913). 
 
 25. Cinders. — Furnace cinders as an aggregate are 
 used only in inferior grades of mass concrete, or for 
 fireproofing. Cinders have low structural strength, high 
 porosity, and oftentimes as an added objection, high sul- 
 phur content. In more than one instance, sulphuric acid 
 resulting from sulphur decomposition in cinder concrete 
 
 floors has eaten away conduits and piping, and has even attacked reinforcing and structural 
 steel. Cinder concrete is of value chiefly because of its cheapness and low^ specific gravity, 
 but discrimination is required in its use. 
 
 26. Materials Suitable for Fine Aggregates. — All fine aggregates are essentially rock 
 fragments, crushed to varying degrees of fineness, either by the natural processes of weathering, 
 disintegration, or glacial action, or by man with his machines. Sand deposits are masses of 
 
 weathered rock minerals, transported, collected, and 
 sorted by the age-long action of streams (see Fig. 7) . 
 
 From the earliest ages the formation of sand, 
 silt, and clay has been going on through the break- 
 ing down of rocks. The changes involved in these 
 processes are part physical and part chemical. All 
 changes produced at or near the surface by atmos- 
 pheric agents, which result in more or less complete 
 disintegration and decomposition, are classed under 
 the general term of ''weathering." The action of 
 physical agents alone, which results in the rock break- 
 ing down into smaller particles without destroying 
 its identity, is termed "disintegration" (see Fig. 8). 
 On the other hand, the action of chemical agents de- 
 stroys the identity of many of the minerals by the 
 formation of new compounds, and this latter process is known as "decomposition." Silt 
 and clay generally result from decomposition; and, as such chemical change has altered the 
 character of the material (usually to its detriment so far as concrete purposes are concerned), 
 that is one reason, but not the only reason, against permitting their presence in concrete sand. 
 
 Since coarse sands are of a size to retain and partake of the nature and properties of the 
 parent rock, the structure of at least larger particles should be identical with the structures of 
 1 From "Engineering Geology," by Ries and Watson. 
 2 
 
 Fig. 7. — Concretionary sandstone weathering 
 to form sand.i 
 
18 
 
 CONCRETE ENGINEERS' HANDBOOK 
 
 [Sec. l-26a 
 
 Fig. 8. — Shale deposit lying be- 
 tween hard sandstone ledges, disin- 
 tegrating with formation of clay. 
 
 such rocks; and their strength and fitness for use in concrete may be judged with more or less 
 accuracy from a consideration of the structure and strength of those rocks known to be suitable 
 for concrete work. There are few rocks which do not contain silica in greater or lesser 
 quantities. 
 
 Because of its hardness and resistance to chemical agents, quartz or silica is, therefore, the 
 commonest mineral in sand. Qther minerals such as feldspar, mica, etc., though originally 
 present, because of their lesser resistance, have been more readily decomposed by the action 
 of the elements; and by reason of their complete disintegration with resultant fine state of sub- 
 division, have been removed by wind and water. Quartz crys- 
 tals, therefore, remain as the most evident survivors of the 
 parent rock and their survival is evidence of their desirable 
 qualities for concrete. 
 
 26a. Special Characteristics of Sand. — A simple 
 and illustrative example of a rock from which quartz sand may 
 be derived is sandstone. This stone is built up almost wholly 
 of quartz grains, cemented together by iron oxide, calcium car- 
 bonate, or clay, and on the nature of the cementing material 
 depends the strength and hardness of the stone. The struc- 
 ture of a hard sandstone is shown in Fig. 9. In this stone the 
 cementing material is iron oxide. 
 Under certain conditions of use a fragment of sandstone, such as would be represented by a 
 large sand grain, might be unfit as a material for concrete. Such a condition would be repre- 
 sented by subjecting concrete containing such sand to extreme of heat, as in a fire. Under 
 like conditions it would be expected that a large fragment of the same stone would break up, or 
 spall," and it is actually found that some sand grains repeat in miniature the behavior of the 
 larger pieces of stone. Such a condition of heat is, of course, unusual and extreme, and would 
 not prejudice the use of such sand for most purposes. 
 
 If the cementing material of a sandstone sand grain be calcium carbonate, it may be dis- 
 solved by natural water, since such waters contain an ap- 
 preciable percentage of carbon dioxide, or carbonic-acid gas. 
 Such a sand, therefore, would be unsuited for use in a water- 
 storage reservoir, or in drainage tile, or in aqueducts of any 
 kind, or in other constructions which are designed to be im- 
 ipermeable to water. Fortunately, this calcium-carbonate 
 cement in sand is easy of detection by adding a drop of 
 muriatic acid and noting effervescence, or the lack of it. 
 
 Clay cement in a sandstone is quite undesirable. It 
 is frequently the case that a sand thought to be quite per- 
 fect for use in concrete, by reason of its whiteness and good 
 grading in size, is in reality quite dangerous. Clay is not a 
 strong cement, and a sand of which the particles are built 
 up with this cementing material is readily crushed. This 
 is especially true in concrete, for the clay readily absorbs 
 water and becomes a soft paste, leaving the component 
 
 sand grains loose and without contact with the Portland cement save at the outer surface of the 
 outside particles. This, of course, weakens the concrete seriously if all the sand is of this same 
 general character. 
 
 Not all sands, however, are composed of mineral grains, as are those previously mentioned. 
 It is not infrequently the case that the rock from which they came has been formed by the 
 fossilizing, or partial fossilizing, of minute prehistoric shells. A section of limestone, built up 
 in this way, is shown in Fig. 10. Other rocks contain like fossil materials in combination with 
 quartz grains and cementing material, as in fossiliferous sandstone. In many sands derived 
 
 Fig. 9. — Hard sandstone. Quartz 
 grains cemented by iron oxide. (Nat- 
 ural size.) 
 
Sec. 1-266] 
 
 MATERIALS 
 
 19 
 
 from such rocks the structure of the shell is so perfectly preserved that the fossils retain 
 their hollow struf ture and, further, are easily decomposed by agents which either reach them 
 before their incorporation in the concrete, or afterward, by dissolving out the softer portions. 
 The use of such a sand, therefore, is not advisable in concrete which is intended to be impervious 
 to water, or to possess a high strength. 
 
 26b. Crushed Stone and Screenings. — Crushed stone screenings, when free 
 from clay, usually make excellent sand. These screenings ordinarily give a stronger mortar 
 than natural sand but are likely to contain an undue amount of dust, especially when obtained 
 from soft stone, and should be screened and washed to get rid of the finest particles before being 
 used in mortar or concrete. 
 
 Crushed limestone makes a concrete of excellent early strength, provided the crushings are 
 rerolled, as limestone breaks with a flat, scaly fracture, giving particles that are structurally 
 weak and that are very hard to compact in the manner necessary to 
 give an impervious concrete. For work exposed to water this 
 point is of great importance. Furthermore, if there is porosity in 
 such concretes, the high solubility of the limestone fragments in 
 water is a further disadvantage. In such cases, percolation pro- 
 ceeds at an increasing rate with passage of time, due to bodily 
 removal of the fine aggregate by solution, leaving a honeycomb 
 structure behind. 
 
 26c. Sea Sand. — Sea sand is usually well suited 
 for use as fine aggregate for concrete, so far as structure, mineral 
 composition, and cleanness are concerned. It is, however, usually 
 of such fineness that its use is inadvisable if undiluted by coarser 
 
 J.' Hi, X' UBOii-uceinufi, lime- 
 
 particles. Saline deposits on the grains, when derived from pure stone. (Magnified 20 diams.) 
 sea water, should not be of a nature detrimental to concrete. It 
 
 is unwise to take such sands close to tide limits, as the newer sands close to water, teem 
 with minute organic life. 
 
 2Qd. Standard Sand. — The standard sand used in tests of mortars is a natural 
 sand obtained at Ottawa, 111., passing a screen having 20 meshes and retained on a screen having 
 30 meshes per lin. in., prepared and furnished by the Ottawa Silica Co. at a cost of 2 cts. per 
 lb., f.o.b. cars, Ottawa, 111. The grains of this sand are rounded and readily compacted, the 
 percentage of voids being about 37%. 
 
 It is to be noted that standard sand gives about the lowest value attainable with sand in 
 combination with cement, because of its uniform size of grain. Yet the present acceptance 
 tests for a commercial sand provide only that it shall in like combination with a like cement, 
 attain not less than 75% of the strength of the lowest value obtainable. This standard is 
 decidedly low and permits the use of almost any sand, even one of poor quality. 
 
 27. Requirements of Fine Aggregate as to Shape and Size of Particles. — It is exceedingly 
 difficult in choosing a fine aggregate for concrete work to balance all considerations. Time is a 
 factor of utmost importance in all construction operations. Therefore, where delays would 
 be entailed by the selection of one sand, the qualities of which are superior to those of another 
 sand that is more readily obtained, it is more than probable that considerations of superior 
 quahty will have little weight. It is unfortunately true that regardless of all that has become 
 known in regard to the importance of sands, their quality will be generally disregarded in favor 
 of cheapness or convenience until engineers and owners demand and insist upon concretes of 
 proper quality and refuse payment for those not coming up to standard. When inferior materials 
 at a less price are as readily marketable when incorporated in concrete as first-grade materials, 
 the contractor, as vendor, is not to be censured if he realizes every opportunity afforded him 
 to reahze the largest possible profit. The buyer and his agents receive only what they demand. 
 
 Usual specifications for concrete sands permit of little discrimination on the part of the 
 supervising engineer, however conscientious he may be. Provision that the sand shall be "clean^ 
 
20 
 
 CONCRETE ENGINEERS' HANDBOOK 
 
 [Sec. 1-28 
 
 sharp and coarse" means nothing, as no standards are defined as comparisons and the deter- 
 mination is left solely to the judgment of individuals oftentimes quite incompetent and unskilled. 
 
 Sharpness as a quality requirement for sand is archaic. It has little or no definite meaning; 
 and rarely are two individuals agreed as to how sharpness should be determined. To some it 
 defines the sound given off when sand is rubbed in the hand. To others, it is measured by abra- 
 sive quality, determined in the same way. To others, it indicates a certain angularity judged 
 solely by the eye. If it were but remembered that all natural sands are water-borne and water- 
 worn, with inevitable rounding of grains, the fallacy of "sharpness," whatever its interpretation, 
 as a standard of quality in natural sands, would be evident. 
 
 Cleanness in sands is most important, for reasons before given. Not all dirt coatings on sand 
 are detectable, short of laboratory procedures; and unfortunately, much sand is judged as to 
 cleanliness by rubbing in a hand that itself is usually none too clean, the fitness of the sand being 
 judged by the deposit it leaves behind. Judging a sand in this way without supplemental 
 tests betokens ignorance, or carelessness, or both. Cleanness is an imperative necessity, but 
 it should be judged by adequate tests, not by such haphazard methods as the foregoing. 
 
 Coarseness in sands, as opposed to excessive fineness, is a desirable quality, but coarseness 
 alone, without finer materials and especially when judged without standards, is no criterion 
 of fitness for use in concrete. As before pointed out, coarse sands have less surface area than 
 have fine sands, requiring less cement and being more readily coated. Such a, requirement, 
 if properly judged, is therefore advantageous. As an insurance against excessive clay or loam, 
 the requirement of coarseness may also be of benefit. 
 
 What really is needed is: (1) a general and thorough understanding by engineers and 
 contractors alike as to the fundamental relations existing between the various materials forming 
 concrete (see chapter on ''Proportioning" in Sect. 2); (2) an appreciation of the importance 
 of sands in the production of good concrete; (3) their selection on a basis of quality; and (4) to 
 make the foregoing of value, a rigid insistence upon conformity to standard by tests of each 
 shipment made to the job with ruthless rejections of inferior materials. A single test on a 
 sample which may or may not be representative of the bulk of material is of no value whatever, 
 unless it is supplemented by comparative tests on the materials actually delivered. 
 
 28. The Selection of Sand. — The only logical procedure in the selection of a sand for con- 
 crete is: 
 
 1. Determine its granulometric analysis by screening. 
 
 2. Determine its cleanness by washing, or by chemical tests. 
 
 3. Determine its actual strength value in concrete by test. 
 
 4. Check all shipments for cleanness and uniformity of grading. 
 
 29. Requirements of Coarse Aggregate as to Shape and Size of Particles. — Since stone is 
 one of the strongest, if not the strongest constituent of concrete, the greater the percentage of 
 stone {i.e., the nearer concrete actually approaches natural stone in strength and density) the 
 stronger is the concrete. It follows, then, other things being equal, that the larger the stone, 
 the stronger will be the concrete, since each piece of stone has greater mass density than would 
 its components unless compacted and united by Nature's unapproachable processes. 
 
 There are, however, certain limitations as to size of stone imposed by certain classes of work. 
 In reinforced work, the plastic concrete must fit itself closely around the reinforcing metal, so 
 that 1 to in. is the greatest diameter of particle that experience demonstrates is advisable 
 to use. 
 
 Concrete of this character obviously requires more cement than would concrete using 
 larger stone, since the stone surface to be coated is greater. In mass work, on the other hand, 
 crushed stone of 23-^ to 3 in. diameter may be advantageously employed, with less cement. 
 For these reasons, if for no others, richer mixtures are specified in reinforced work and leaner 
 mixtures in mass work. It should be borne in mind, however, that size of stone is not alone the 
 determining factor in this regard, but that grading of stone and size and grading of sand is of 
 
Sec. 1-30] 
 
 MATERIALS 
 
 21 
 
 even more importance as influencing the quantity of cement required, with a corresponding 
 effect on the quaUty of concrete. 
 
 Plums are large stone, 5 in. or more in least diameter, thrown into plastic mass concrete, 
 largely with the object of using them as cheap space-fillers. Their use is also to be commended 
 for the reasons before given, provided that the plums themselves are of the proper quality of 
 stone and that they are not of such size as to cut through the concrete section. A safe rule to 
 follow in the use of plums is that they shall be of a maximum size such that not less than 6 
 in. of concrete shall intervene between them and the forms at any point. In using very large 
 plums, this thickness of intervening concrete should be materially increased. 
 
 The shape of particle of large aggregates is of relatively little importance. Cleanliness, 
 grading, and character of rock have far greater influence on the concrete than has angularity or 
 roundness of particle. 
 
 30. Impurities in Aggregates. — In order that cement may adhere to sand grains and to 
 particles of coarse aggregate, each grain or particle must bear no coating such as would prevent 
 either proper chemical action between cement and water, or a proper bond between cement 
 and aggregates. 
 
 This requirement applies to both fine and coarse aggregates, but is of relatively more impor- 
 tance with respect to the former, as sand in its natural state and as used in ordinary concrete 
 construction is more likely to contain dirt in sufficient amount and of such kind to cause appreci- 
 able injury than is coarse aggregate. 
 
 Clay and silt are impurities of most frequent occurrence in sand and gravel. These mate- 
 rials are the result of decomposition of natural rock of various kinds and it is almost inevitable 
 that they should be associated with sand. 
 
 Each of these impurities causes injury to mortar or concrete not only when it exists as a 
 coating on the sand or gravel particles, but is equally undesirable when it occurs in such 
 amounts, or so unequally distributed, that its extremely fine grains ''ball up" and stick together 
 when wetted, so as to remain in lumps in the finished mortar or concrete. If, however, the par- 
 ticles of these impurities are distributed so that they do not bind together on the addition of 
 water; and if they are not contaminated by organic matter, experiments have shown that with 
 sand that is not too fine, no serious harm results in lean mortars and concretes from their pres- 
 ence to the extent of from 10 to 15%. In fact, either clay or silt are often found beneficial as 
 they increase the density by filling some of the voids, thus increasing the strength and water- 
 tightness besides making the mortar or concrete work smoothly. In rich mortars and con- 
 cretes the density and consequently the strength is lowered by even slight additions of clay or 
 silt as the cement furnishes all the fine material that is required. 
 
 A coating of organic matter on sand grains, such as loam, appears not only to prevent the 
 cement from adhering but also to affect it chemically. In some cases a quantity of organic 
 matter so small that it cannot be detected by the eye and is only sHghtly disclosed by chemical 
 tests has prevented the mortar or concrete from reaching any appreciable strength.'- Tannic 
 acid, colloidal sewage, manure, sugar, tobacco juice are instances of organic contamination 
 destructive to concrete. 
 
 Mica in sand or stone is objectionable because of its low mechanical strength and its lami- 
 nated scaly structure. Even a small amount of this impurity in sand may seriously reduce the 
 strength of a mortar or concrete. Mica is especially injurious in sands for concrete surface 
 work as the scaly flakes cause the surface to dust and peel. 
 
 Furthermore, it is to be noted that water does not wet the surface of mica. Necessarily, 
 this precludes attachment of cement, so that not only is the material weak of itself, but it lies 
 in the mass without attachment, inviting disintegration. 
 
 Mica schist is totally unfit for use as large aggregate, both because of the foregoing reasons 
 and also because of its rapid decomposition on exposure to air. 
 
 1 A new process for the detection of such coatings is being developed under the auspices of Committee C-9 
 of the A.S.T.M., at Lewis Institute, Chicago, 111. by Prof. Abrams and Dr. Harder. See footnote on p. 29, 
 
22 
 
 CONCRETE ENGINEERS' HANDBOOK 
 
 [Sec. 1-31 
 
 Mica above 1 % in concrete sands is very objectionable. 
 
 Iron 'pyrites or fooVs gold — a bright, yellow substance with metallic luster — is chemically 
 iron sulphide (Fe2S). This is a very common impurity in stone; and its undesirability lies in its 
 ready oxidation in the presence of water with formation of sulphuric acid (H2SO4), which latter 
 readily attacks the cement of concrete with disastrous consequences (see Fig. 1, page 13). 
 
 In large fragments of stone, as in coarse aggregate, the presence of a small amount of this 
 substance is not objectionable, but when fine aggregates are made from this same rock, the py- 
 rites previously isolated are exposed, with resultant increase in quantity and rate of oxidation 
 and with corresponding formation of acid. 
 
 Some sand deposits also contain unoxidized iron sulphide, though such deposits are an 
 exception. 
 
 Finely powdered dust present in crushed stone screenings causes approximately the same 
 effect upon the strength of mortar or concrete as does the presence of silt or clay in like quanti- 
 ties. It is essential for the best work that this dust be removed by screening and washing, 
 in the same manner that silt is removed from sand and gravel, though it may later be used 
 advantageously in known quantities by recombination. 
 
 31. Size and Gradation of Aggregate Particles. — The weakest and most changeable element 
 in any cement (plus water)-sand-stone combination is the cement matrix in which the aggre- 
 gates are embedded. The strongest and least changeable elements are the sand and stone. 
 It follows, therefore, that strong, enduring concrete should contain as large a percentage as 
 possible of aggregates consistent with proper embedment and cohesion, together with requisite 
 plasticity to permit ready placing in forms. This conclusion also is true from an economic 
 point of view, since sand and stone are much cheaper, bulk for bulk, than cement. 
 
 Further, tests of concrete show that, with the same percentage of cement to a unit volume 
 of concrete, that mixture which gives the smallest volume and has, therefore, the greatest den- 
 sity,^ usually produces the strongest and most impermeable concrete. This rule, it should be 
 said, does not strictly apply to water-tightness, as permeability is influenced by size of voids 
 as well as density. 
 
 31a. Grading of Mixtures. — Other things being equal, the best aggregates, fine 
 and coarse, for use in concrete are those which are so graded in sizes of particles that the per- 
 centage of voids, or hollow spaces, in the resulting concrete is reduced to a minimum. The same 
 law applies also to mortar mixtures, so that if concrete is considered as a quantity of relatively 
 large stone set in a pudding, or bedding of mortar, the best sand as to size is one which, if mixed 
 with the given cement in the required proportions to standard consistency, yviU produce the 
 smallest volume of mortar; and the best concrete will be one in which the particles of stone are 
 so graded as to permit in a given volume a maximum quantity of stone being bedded in such 
 a mortar. 2 
 
 316. Grading, Density, and Strength. — It has been found that the densest mix- 
 ture occurs with particles of graded sizes; and also that the least density occurs when the grains 
 are all of the same size. Coarse sands, or fine sands alone are thus inferior to graded sands for 
 concrete, but of the two extremes the coarse sand is preferable because its particles are more 
 readily coated with cement particles. Further, a coarse sand has a less total grain surface in 
 a unit volume than a fine sand, even when the sands considered contain the same proportion 
 of solid matter and voids. Less total grain surface means less cement and less water required 
 to coat the grains. Furthermore, these interactions are cumulative, for the additional amount 
 of cement and water required in the case of fine sand reduces the density of the resulting mortar 
 and likewise its strength as well as increasing its cost. 
 
 31c. Money Value of Grading. — One reason is here evident, both from an 
 engineering as well as from a purely doUars-and-cents standpoint, that care and attention be 
 
 1 The density of a mortar or concrete as here referred to is the ratio of the volume of the solid particles to 
 the total volume. 
 
 2 For the use of 6 to 8-in. aggregate, see Eng. Rec, May 1, 1915. 
 
Sec. 1-32] 
 
 MATERIALS 
 
 23 
 
 given to the grading of sands for concrete. If it were possible to compute the total saving in 
 the annual concrete production of the United States, both direct (by lessened quantity of cement 
 required) or indirect (by increased durability and usefulness and prevention of disintegration) 
 that would result from the use of proper sands, the amount would be almost beyond imagina- 
 tion. It is probable that from this neglect alone not less than 20% of the total annual expendi- 
 ture for concrete is unnecessary waste. It has been proven in England that on a strict 1:2:4 
 basis, using different aggregates, the cement required for a quantity of concrete varied from 
 100 to 130 bags — a difference of 30 % in direct cement cost, without counting the variation in 
 quality of the several concretes, with like variation in their durability and value. At the pres- 
 ent time, a comfortable ignorance is general among engineers and contractors alike on these 
 important matters, but in the not distant future, an awakened intelligence on the part of all 
 will demand reform. 
 
 32. Mechanical Analysis of Aggregates.— The value of an aggregate, sand or stone, with 
 reference to its size may be determined by means of a sieve analysis. This analysis consists of 
 sifting the material as supplied through several different sieves, and then plotting upon a dia- 
 gram the percentage by weight which is passed (or retained) by each sieve — abscissae (hori- 
 zontal) representing size of grain and ordinates (vertical) representing percentage of any size 
 passing each sieve. ^ Such a sieve analysis may appear of little use as regards the making and 
 placing of 100,000 yd. of concrete, but experiment has developed definite laws establishing the 
 relation of percentages and sizes of particles to maximum density and strength of concrete so 
 that such a sieve analysis may be directly translated into terms of commercial and engineering 
 economy. 
 
 A typical analysis of three natural sands — a fine, a medium, and a coarse sand is given 
 in Fig. 11.2 Uniform grading is indicated by an approach to a straight line and the variation 
 from the grading found to give best results in practice is observed without difficulty. 
 
 A mechanical or sieve analysis is also useful in studying the size of the particles of the coarse 
 aggregate. 3 Fig. 12 illustrates the analysis of a bank gravel and a crushed stone as it came from 
 the crusher, without screening. 
 
 The following mechanical test for sand has been used in the laboratory of the Board of 
 Water Supply of the City of New York: 
 
 Samples of the material proposed for use in mortar or concrete shall be prepared for testing by passing them 
 through a No. 4 sieve. Of the material passing this sieve not more than 9.5 % shall pass a No. 8 sieve, not more than 
 40 % a No. 50 sieve, and not more than 15 % a No. 100 sieve. 
 
 1 It is greatly to be regretted that there is no standard in the United States in matters of this kind. Such va- 
 riations make for confusion and waste. A proposed standard, which has much to recommend it, varies successive 
 screen openings by a constant ratio of \/2. The following sizes of sieves are desirable for analyzing sand, although 
 a very useful analysis may be made with fewer sizes: 
 
 Commercial No. 
 
 5 
 
 8 
 
 10 
 
 16 
 
 20 
 
 30 
 
 40 
 
 60 
 
 100 
 
 200 
 
 Approximate size of hole 
 in inches. 
 
 0.165 
 
 0.096 
 
 0.073 
 
 0.042 
 
 0.034 
 
 0.020 
 
 0.015 
 
 0.009 
 
 0.0055 
 
 0.0026 
 
 Sieves are given commercial numbers, which agree approximately with the number of meshes to the linear 
 inch. The actual size of hole, however, varies with the gage of wire used by different manufacturers and every 
 Bet of sieves should be separately calibrated. The screen with H-in. openings is generally used for separating out 
 large material from sand. The No. 4 sieve with four meshes per linear inch is practically its equivalent. 
 
 2 A new portable instrument for making mechanical analysis of sands quickly in the field is now manufactured 
 by Kolesch & Co., of New York (see Eng. Rec, June 26, 1915: also Fig. 2, Sect. 2, p. 69). 
 
 3 For coarse aggregate analysis, the following sizes of sheet brass sieves with round holes are desirable; 3 in., 
 2H in., 2 in., IM in., IM in., 1 in., H in., H in. and M in. As is also the case with a like analysis of sand, a 
 straight line on a mechanical-analysis diagram indicates a uniform grading. 
 
24 
 
 CONCRETE ENGINEERS' HANDBOOK 
 
 [Sec. 1-32 
 
 Material in which the percentage passing any one sieve or two sieves exceeds the specified percentage may be 
 used, provided there is a different percentage passing the other sieves or sieve under the limiting percentage equal 
 to at least twice the excess. 
 
 The sieves for testing shall be defined as follows: 
 
 No. 8 mesh holes, 0.0955 in. wide. No. 23 wire. 
 No. 50 mesh holes, 0.0110 in. wide. No. 35 wire. 
 No. 100 mesh holes, 0.0055 in. wide, No. 40 wire. 
 
 In the standard specifications for concrete pavement adopted by the American Concrete 
 Institute, the fine aggregate is required to pass, when dry, a screen having 3^^-in. openings. 
 Not more than 20% is allowed to pass a sieve having 50 meshes per linear inch, and not more 
 
 0 ^ 0.025 O.O50 0.075 0.100 0.IE5 0.150 0.175 0 EOO Q2E5 0.250 
 
 Diameter of Particle in (nches 
 Fig. 11. — Typical mechanical analyses of fine, medium, and coarse sands. 
 
 than 5% is allowed to pass a sieve having 100 meshes per linear inch. The coarse aggregate 
 is specified as such as will pass a iK-in. round opening and will be retained on a screen having 
 3^-in. openings. It is also required that natural mixed aggregate shall not be used as it comes 
 from deposits, but shall be screened and used as specified. 
 
 The Joint Committee recommends that not 
 more than 30% of sand by weight should pass a 
 sieve having 50 meshes per linear inch. 
 
 According to heretofore accepted theory, sand 
 for mortar or concrete should have its grains 
 uniformly graded. Recent tests, ^ however, by 
 Prof. McNeilly at Vanderbilt University seem to 
 indicate that there should be a jump in the grading 
 from the sieve No. 40 size particles to the sieve 
 No. 10 size particles. The following conclusions 
 were derived from these tests: 
 
 The best sizing of grains in a commercially-sieved aggre- 
 gate is about as follows: 53 % to be caught between the No. 4 
 and No. 10 sieves; 47 % fines to be passed by the No. 40 sieve 
 (this includes the cement). 
 
 There is reason to believe that the coarse aggregate in concrete could be beneficially sized on a basis similar 
 to the above; that is to say, there should be a jump in size from coarse to quite fine instead of the usually accepted 
 graded material. 
 
 For the method of proportioning by mechanical analysis, as developed by Wm. B, Fuller 
 and by the U. S. Bureau of Standards, see page 68. 
 
 1 See Eng. Rec, Ncv, 27, 1915. (Conclusions derived from these tests have not as yet received general 
 acceptance.) 
 
 Fig. 
 
 Q25 050 075 igo 1.25 150 
 
 Diameter of par+icle 'in .inches 
 
 12. — ^Typical mechanical analyses of bank 
 gravel and crushed stone. 
 
Sec. 1-33] 
 
 MATERIALS 
 
 25 
 
 33. Tests for Specific Gravity of Aggregates. — Ihe specific gravity of a substance is the 
 ratio of the weight of an absolutely solid unit volume of the substance to the weight of a unit 
 volume of water. This ratio for aggregates may be determined as follows: 
 
 1. By pouring a given weight of sand into a given volume of water and finding the increase 
 in volume of the liquid. (Enough material should be used to give sufficient accuracy.) 
 
 2. By suspending pieces of coarse aggregate by a thread from chemical scales and noting 
 weight in air and weight when hanging in water. (The difference in weight is the weight of 
 the water which the aggregate displaces.) 
 
 Finding specific gravity of particles of sand and stone is preliminary to one method of 
 determining the percentage of voids. The specific gravity of sand is practically a constant, 
 with a value of 2.65. The specific gravity of gravel is also quite uniform, the average being 
 2.66. Average values for stone varies with the kind and the locality, ranging from 2.4 (sand- 
 stone) to 2.9 (trap). Cinders have an average specific gravity of 1.5. 
 
 Before determining specific gravity, sand or fine stone should be dried in an oven at a 
 temperature as high as 212°F. until there is no further loss of weight. If the stone is of a 
 porous nature, it should be moistened sufficiently to fill its pores, and then the moisture on the 
 surface should be removed by means of blotting paper. Such a procedure, of course, does not 
 determine absolute specific gravity but gives a result that should be used in determining the 
 percentage of voids for proportioning concrete mixtures. 
 
 34. Voids in Aggregates. 
 
 34a. Percentage of Voids. — The percentage of voids in dry sand ranges from 
 28% for a coarse, well-graded natural sand to 40 to 45% for a very uniform natural or screened 
 sand. The range for coarse aggregates is from 25 to 55%. 
 
 The percentage of voids in aggregates may be determined by two methods : 
 
 1. By determining the specific gravity of the solid particles and then weighing a given 
 volume of the aggregate and computing therefrom the percentage of voids. (Pores in porous 
 stone should be filled with water. See Art. 33.) 
 
 Let S = specific gravity, W = weight per cubic foot of the dry aggregate, and P = per- 
 centage of voids. Then, since water weighs 62.5 lb. per cu. ft., 
 
 P = (i " WEs) 
 
 2. By finding the amount of water required to fill the voids in a given volume of aggregate. 
 Let V = volume of water required to fill the voids and T = total volume, or given volume 
 
 of the aggregate. Then 
 
 With sand or fine broken stone the percentage of voids by this method should be obtained 
 by dropping the aggregate into a vessel containing water. U K = volume displaced by the 
 aggregate and T = given volume of the aggregate, then 
 
 Pouring water into fine aggregate does not give reliable results because it is physically impossible 
 to drive out all the air. 
 
 The percentage of voids is considerably affected by the degree of compactness of the 
 aggregate. Moderate shaking of coarse aggregate, for example, will reduce the volume of 
 voids by as much as 10%. Loose measurement is usually considered preferable for the coarse 
 aggregate since sand and cement separate the stones to a considerable extent in the concrete 
 as placed. Voids in sand are usually determined with reference to the dry material well shaken. 
 
 346. General Laws.— 1. A mass of spheres of any uniform size if carefully 
 piled in the most compact manner would have 26% voids. If the same mass of spheres were 
 
26 
 
 CONCRETE ENGINEERS' HANDBOOK 
 
 [Sec. l-34c 
 
 poured into a receptacle and the spheres allowed to arrange themselves, it has been found by 
 experiment that 44% would be the smallest percentage of voids which could be obtained under 
 the best conditions. 
 
 2. A material having particles all of a uniform size and shape contains practically the same 
 percentage of voids as a material having particles of a corresponding similar shape but of a 
 different uniform size. 
 
 3. In any material the largest percentage of voids occurs with particles all of the same size 
 and the smallest percentage occurs with particles of such different sizes that the voids of each 
 size are filled with the largest particles which will enter them. Thus, an aggregate consisting 
 of a mixture of stones and sand has a less percentage of voids than sand alone. 
 
 4. Materials with round particles contain less voids than materials with angular particles 
 screened to the same size 
 
 34c. Effect of Moisture on Voids in Sand and Screenings. — The percentage 
 of voids in sand is greatly affected by moisture. The reason for this lies in the fact that when 
 water surrounds a particle of sand it occupies space and separates this particle from grains 
 adjacent to it. Since fine sand has a larger number of grains per unit volume, it is more 
 affected than is coarse sand. If either loose or tamped sand is mixed with a small percentage 
 of water and kept either loose or thoroughly tamped, it will be found to increase considerably 
 in volume and weigh less per cubic foot. A maximum volume will be obtained with the addition 
 of from 5 to 8% of water by weight. Greater percentages will give a less increase of volume 
 until finally when the sand is thoroughly saturated it will have a volume slightly less than the 
 original. A sand that has, say, 35% voids, may contain from 27 to 44% of voids depending 
 upon the degree of compactness and the percentage of water. Natural sand as it ordinarily 
 comes from the bank contains from 2 to 4% of moisture by weight. 
 
 34cZ. Percentage of Voids Determined by Weight. — The specific gravity of 
 gravel particles and of sand grains is usually nearly constant, varying between 2.6 and 2.7. 
 
 Percentages of Voids in Sand and Gravel Corresponding to Different Weights 
 
 PER Cubic Foot 
 
 (Based on an Average Specific Gravity of 2.65) 
 
 Percentages of 
 moisture by weight 
 
 Weight per cubic foot of sand or gravel 
 
 75 
 
 80 
 
 85 
 
 90 
 
 95 
 
 100 
 
 105 
 
 110 
 
 115 
 
 120 
 
 125 
 
 0 
 
 54.7 
 
 51 
 
 7 
 
 48.7 
 
 45.7 
 
 42.7 
 
 39.6 
 
 36.6 
 
 33.6 
 
 30.6 
 
 27.5 
 
 24.5 
 
 1 
 
 55.2 
 
 52 
 
 2 
 
 49.2 
 
 46.2 
 
 43.2 
 
 40.2 
 
 37.3 
 
 34.2 
 
 31.2 
 
 28.2 
 
 25.3 
 
 2 
 
 55.6 
 
 52 
 
 7 
 
 49.8 
 
 46.7 
 
 43.8 
 
 40.8 
 
 37.9 
 
 34.8 
 
 31.9 
 
 28.9 
 
 26.0 
 
 3 
 
 56.1 
 
 53 
 
 1 
 
 50.3 
 
 47.3 
 
 44.4 
 
 41.4 
 
 38.5 
 
 35.5 
 
 32.6 
 
 29.7 
 
 26.7 
 
 4 
 
 56.5 
 
 53 
 
 6 
 
 50.8 
 
 47.8 
 
 45.0 
 
 42.0 
 
 38.1 
 
 36.2 
 
 33.3 
 
 30.4 
 
 27.5 
 
 5 
 
 57.0 
 
 54 
 
 1 
 
 51.3 
 
 48.4 
 
 45.5 
 
 42.6 
 
 39.8 
 
 36.9 
 
 34.0 
 
 31.2 
 
 28.3 
 
 6 
 
 57.5 
 
 54 
 
 6 
 
 51.8 
 
 48.9 
 
 46.0 
 
 43.2 
 
 40.4 
 
 37.5 
 
 34.7 
 
 31.7 
 
 29.0 
 
 7^ 
 
 57.9 
 
 55 
 
 1 
 
 52.3 
 
 49.5 
 
 46.6 
 
 43.8 
 
 41.0 
 
 38.2 
 
 35.4 
 
 32.6 
 
 29.8 
 
 8 
 
 58.3 
 
 55 
 
 5 
 
 52.8 
 
 50.0 
 
 47.2 
 
 44.5 
 
 41.6 
 
 38.9 
 
 36.1 
 
 33.3 
 
 30.6 
 
 9 
 
 58.8 
 
 56 
 
 0 
 
 53.3 
 
 50.6 
 
 47.8 
 
 45.0 
 
 42.3 
 
 39.5 
 
 36.8 
 
 34.0 
 
 31.3 
 
 10 
 
 59.2 
 
 56 
 
 5 
 
 53.9 
 
 51.1 
 
 48.4 
 
 45.6 
 
 42.9 
 
 40.2 
 
 37.5 
 
 34.7 
 
 32.1 
 
Sec. 1-35] 
 
 MATERIALS 
 
 27 
 
 On account of this fact the percentage of voids in sand and gravel may be considered to vary 
 inversely as the weight per cubic foot of dry material. Knowing the weight per cubic foot and 
 assuming a specific gravity of 2.65, the percentage of voids in dry sand and gravel may be 
 readily found as explained under Method (1) in Art. 34a. The percentage of voids in moist 
 sand or gravel may be determined in the same manner as for the dry aggregate except that the 
 weight per cubic foot of the moist material should be considered as decreased by the weight of 
 moisture which the sand or gravel contains. The foregoing table gives percentages of 
 voids for sands and gravels of different weights per cubic foot and with different percentages 
 of moisture by weight. The table may be used for any aggregate with a specific gravity of 
 approximately 2.65. 
 
 35. Tests of Aggregates. ^ — Tests of an aggregate for use in mortar or concrete may be 
 divided into two general classes: 
 
 1. Tests to determine the general suitability of the aggregate. 
 
 2. Tests to determine those characteristics of the aggregate which have an influence on 
 its general suitability. 
 
 Tests of the first class comprise those for determining the quality of the mortar or concrete 
 that can be made from the given aggregate. These tests may be called Tests for Acceptance. 
 Tests of the second class include those which may be made to determine the cause of any 
 failure of an aggregate to pass the tests of the first class. These tests of the second class, which 
 may be called Tests for Quality, are useful not only to discover the cause of failure of an aggregate 
 to pass the tests for acceptance, but may be employed to determine the methods of improving 
 a given aggregate and of comparing different aggregates as to special characteristics. 
 
 No standards for acceptance tests of concrete aggregates have been established by the 
 American Society for Testing Materials, although the need of such standards is now fully 
 realized and will soon be satisfied. The most advanced practice in this direction is probably 
 that represented by the procedure of the Materials Testing Division of the New York Public 
 Service Commission whose methods have been given wide publicity. The standards quoted 
 in abridged form below are derived from this source (Eng. Rec, Jan. 8, 1916 and Eng. News, 
 Feb. 4, 1915). 
 
 Fine Aggregates. — Complete tests of a fine aggregate comprise: 
 
 1. Determination of % retained on No. 4 square-hole sieve. 
 
 2. Mechanical analysis of portion passing No. 4 square-hole sieve. 
 
 3. Determination of silt by washing on No. 100 sieve. 
 
 4. Determination of silt by decantation. 
 
 5. Compressive tests of 2-in. cubes. 
 
 6. Microscopical examination. 
 
 7. Weight per cubic foot. 
 
 8. Voids. 
 
 9. Specific gravity. 
 
 10. Reaction to litmus. 
 
 11. Quantitative test for organic matter as indicated by loss on ignition. ^ 
 
 12. - Density in mortar. 
 
 13. Determination of insoluble silica. 
 
 It is seldom necessary to make more than the first six of the above tests and frequently only one or two of them 
 are necessary. 
 
 1. The entire sample is screened on a No. 4 square-hole sieve and the % retained is computed upon the basis of 
 the original weight of the sample. No correction is made for moisture contained. 
 
 (A No. 4 sieve has clear openings of 0.20 in., while the H-in. sieve recommended by the Joint Committee of 
 the National Engineering Societies has 0.25-in. clear openings made by drilling round holes in a plate. The differ- 
 ence in results obtained with the two sieves is negligible and either is satisfactory.) 
 
 ^ See also the following articles on sand testing: 
 
 Cloyd M. Chapman and Nathan C. Johnson: "The Economic Side of Sand Testing," Eng. Rec, June 12, 19 
 and 26, 1915. 
 
 Cloyd M. Chapman: "The Testing of Sand for Use in Concrete," Eng. News, Feb. 5, 1914, and Mar. 12, 1914. 
 Ralph E. Goodwin: "Standard Practice Instructions for Concrete Testing," Eng. News, Feb. 4 and 11, 1915. 
 2 A new colorimetric test is being developed. See footnote on p. 29. 
 
28 
 
 CONCRETE ENGINEERS' HANDBOOK 
 
 [Sec. 1-36 
 
 2. The mechanical analysis of the portion of the sample which has passed the No. 4 sieve is made by use of 
 sieves Nos. 8, 16, 30, 50, and 100. About 150 grams of this material is separated by the method of quartering, and 
 110 grams of this is weighed and dried beneath a gas burner. One hundred grams of the dry material is placed on 
 the No. 8 sieve, the other-sieves being nested below the No. 8 in the order of increasing fineness, and the entire nest of 
 sieves is mechanically agitated. With the agitator used, the amount of sieving is standardized by always turning 
 the hand crank 200 revolutions. At the end of the operation the material retained on each sieve, and that which 
 has passed the No. 100 sieve, is weighed. The % of the whole sample which passes each sieve is now computed, 
 and the mechanical analysis curve is plotted (see Art. 32) . 
 
 3. Silt by washing on the No. 100 sieve is determined for a sample obtained by quartering which weighs about 
 220 grams in its natural moist condition. The sample is dried slowly at temperatures not greatly above ordinary 
 room temperature to avoid baking any clay or similar matter. Two hundred grams of the dry sample is now weighed 
 on the No. 100 sieve, soaked in water for a few moments to soften any lumps, washed under a gentle stream of water, 
 dried under a gas burner, and reweighed. The % of silt is the loss in weight multiplied by 100 and divided by 200. 
 
 (The washing test obtains the true silt value because it removes clay which adheres to the grains as a coating 
 which is not separated by sieving in a dry state.) 
 
 4. Determination of silt by decantation is a test for field use only. About 20 c.c. quartered from a carefully 
 selected sample is placed in a 100-c.c. graduated glass cylinder with about 30 c.c. of lukewarm water. The mixture 
 is stirred with a wire for 30 sec, allowed to settle for 30 sec, and the water decanted into a second 100-c.c. cylinder. 
 The sand left in the first cylinder is stirred up with a fresh portion of water and the process repeated. This is done 
 4 times. After 1 hr. the volume of silt in cylinder No. 2 and the volume of clean sand in cylinder No. 1 is noted 
 and recorded. The % of silt is 100 multipHed by the number of cubic centimeters of silt in cylinder No. 2 and 
 divided by the sum of the volumes of silt in cylinder No. 2 and clean sand in cylinder No. 1. 
 
 (The method of decantation is better than the method of allowing the silt to settle on top of the sand in one 
 cylinder but is not as satisfactory as the method of washing.) 
 
 5. Compressive tests of 2-in. cubes are made by the methods recommended by the Committee on Uniform 
 Tests of Cement, of the American Society of Civil Engineers {Trans. Am. Soc. C. E., vol. 75, p. 665). Fine aggre- 
 gate must not be dried, but the natural moisture is determined on a separate sample, and is counted as a part of the 
 water used for mixing, not as a part of the weight of sand. Proportions are 1 : 3 by weight. The consistency em- 
 ployed is 60% more water than that required to make standard Ottawa sand mortar of "normal consistency." 
 Ottawa sand specimens of the same consistencj' are made with each set of specimens from commercial sand. Tests 
 are made at ages of 3, 7, and 28 days. Specimens whose weights vary more than 3 % from the average are rejected. 
 Cubes are stored in water up to time of crushing, A spherical bearing block and two thicknesses of blotting paper 
 above and below are used in testing. Since the wet consistency used lowers the strength at early periods, the results 
 are permitted to fall below those for Ottawa sand specimens of standard consistency by the following amounts or 
 less: at 3 days — 10%; at 7 days — 5%; at 28 days — 1%. 
 
 (The wet consistency is used because it more nearly represents working conditions and because some sands fail 
 in wet consistencies although satisfactory in "normal consistency." No tensile tests are required because "it is 
 thought that compressive tests more nearly represent the conditions of the work and that modern practice is tending 
 toward compressive tests.") 
 
 6. Microscopical examination is made for the purpose of detecting the presence of a crust or film of organic 
 matter on the grains which cannot be detected by other means. 
 
 7. Weight per cubic foot is determined by using an 8 by 16-in. cylindrical concrete mold and a second cylinder 
 of smaller diameter but high enough to have a cubic capacity slightly greater than that of the 8 by 16-in. cylinder. 
 The smaller cylinder is placed within the larger one and filled with fine aggregate. It is now drawn out allowing 
 the aggregate to flow out at the bottom into the larger cylinder. The material is now struck off level with the top of 
 the measure and weighed. After weighing, the material is dried and again placed in the larger cylinder by use of 
 the smaller cylinder as before. The weight of the material dry is then determined, the volume being found by 
 measuring down from the top of the cylinder. 
 
 8. Voids are determined by using the sample whose apparent volume and dry weight have been determined in 
 test No. 6. It only remains to determine the absolute volume of solid matter. This can be computed from the 
 weight, assuming the specific gravity 2.65, or it can be found by placing the entire sample in a receptacle containing 
 water and weighing the whole, emptying out the aggregate, and reweighing the receptacle filled with water to the 
 previous level. The difference in weighings minus the weight of dry aggregate is the weight of a volume of water 
 equal to the absolute volume of the aggregate. This divided by the weight of water is the absolute volume of 
 aggregate. A hook gage clamped to the water receptacle is used to determine water levels accurately. Finally, 
 the % of voids in aggregate 
 
 100 (Apparent volume — Absolute volume) 
 Apparent volume 
 
 9. Specific gravity is determined by slowly and carefully introducing the aggregate into a specific gravity 
 flask or a graduated glass cylinder containing water and noting the volume of water displaced by a known weight 
 of material. The specific gravity of the aggregate 
 
 Weight of aggregate in grams 
 
 Displaced volume in cubic centimeters 
 
Sec. 1-35] 
 
 MATERIALS 
 
 29 
 
 10. The reaction to litmus demonstrates the presence of injurious alkalies. 
 
 11. Organic matter as indicated by the ignition loss is determined as follows:' 
 
 Two 25-gram samples are weighed from a 7o-gram sample containing natural moisture which has been pre- 
 viously selected by the method of quartering. These are placed in beakers "A" and " B" and each is covered with 
 60 c.c. of water at 100°F. The sand and water in beaker "A" is stirred briskly with a glass rod, allowed 30 sec. 
 to settle, and a part of the water decanted onto a previously dried and weighed filter paper. The remaining portion 
 is again stirred, allowed to settle and a further quantity of water decanted. The proceps is repeated until all the 
 water has been filtered. An additional 30 c.c. of water at 100°F. is added to the sand in the beaker and the process 
 is repeated again. The filtrate is allowed to drain, and the washed sand is dried under a gas burner. The same 
 procedure is followed with beaker " B," using a second filter paper. The filter papers with filtrate on them are now 
 dried in an oven at 100°C. to constant weight (a temperature of lOO^C. must not be exceeded) and weighed. The 
 excess in weight over the dry weight of the filter papers is the weight of silt. The filter papers are now folded care- 
 fully and ignited thoroughly in a platinum crucible. The residue is weighed, and the dry washed sand is also 
 weighed. Finally, % loss on ignition 
 
 100 (Weight of silt — Weight of crucible ash) 
 Weight of silt + Weight of dry washed sand 
 
 (This test is seldom necessary because more direct results are obtained by tests of 2-in. cubes.) 
 
 12. Density is determined by weighing the relative amounts of cement, sand, and water, according to the 
 proportions used, mixing, placing mix in a graduated cylinder and noting the final volume of the set mortar. The 
 net weight of this mortar is also determined and compared with the sum of the individual weights of cement, sand, 
 and water in the mix. The weight of material left adhering to mixing slab and tools is thus ascertained. This 
 loss is apportioned between the cement, sand, and water according to the relative weights of each as originally 
 combined, and the corrected amount by weight of each constituent in the set mortar is thus computed. The 
 corrected weights of cement and aggregate in the set mortar are now divided by their respective specific gravi- 
 ties to obtain absolute volumes and the sum of these absolute volumes divided by the total volume of set mortar 
 is the density, or solidity ratio. 
 
 13. The determination of insoluble silica is seldom called for but when required calls for the services of an 
 experienced analytical chemist. 
 
 Coarse Aggregates. — Complete tests of a coarse aggregate comprise: 
 
 1. Mechanical analysis. 4. Voids. 
 
 2. Cleanliness. 5. Specific gravity. 
 
 3. Weight per cubic foot. 6. Crushing strength of stone. 
 It is usually necessary to make only the first two of the above tests. 
 
 1. Mechanical analysis of coarse aggregate is made by mechanically agitating a nest of six sieves, the clear 
 openings in the wire meshes of which are 2 in., 1>^ in., 1 in., ?4 in., H in., and H in. respectively. A 25-lb. sample 
 obtained by quartering a larger sample is used. The material retained on each sieve after 100 bumps of the 
 rocker apparatus is weighed, and the percentage passing each sieve is computed. 
 
 2. Cleanliness is judged by inspection only. 
 
 3. Weight per cubic foot is determined by pouring the material slowly into an 8 by 16-in. cylinder mold from 
 a height of 2 ft. above the bottom, striking off the top and weighing. If the material is very wet it is previously 
 dried sufficiently to remove surplus water, but not enough to dry out the pores. 
 
 4. Voids are determined by the method used for fine aggregate. 
 
 5. Specific gravity is determined by weighing a number of representative particles of the stone after thorough 
 drying to remove all moisture in the pores, allowing the material to cool, filling the pores by boiling in water and 
 
 1 A colorimetric test for organic impurities in sands is being developed under the auspices of committee C-9 
 of the A.S.T.M. Sodium hydroxide (NaOH) is added to a sample of sand at ordinary temperature and the depth 
 of color resulting has been found to furnish a measure of the effect of the impurities on the strength of mortars 
 made from such sands. See Circular No. 1 of Structural Materials Research Laboratory, Lewis Institute, Chicago. 
 
 The method for field tests is described in the circular as follows: 
 
 "Fill a 12-oz. graduated prescription bottle to the 4M-oz. mark with the sand to be tested. Add a 3% solu- 
 tion of sodium hydroxide until the volume of the sand and solution, after shaking, amounts to 7 oz. Shake thor- 
 oughly and let stand over night. Obsers^e the color of the clear supernatant Uquid. 
 
 "In approximate field tests it is not necessary to make comparison with color standards. If the clear super- 
 natant liquid is colorless, or has a light yellow color, the sand may be considered satisfactory in so far as organic 
 impurities are concerned. On the other hand, if a dark-colored solution, ranging from dark reds to black is obtained 
 the sand should be rejected or used only after it has been subjected to the usual mortar strength tests. 
 
 " Field tests made in this way are not expected to give quantitative results, but will be found useful in: 
 
 1. Prospecting for sand supplies. 
 
 2. Checking the quality of sand received on the job. 
 
 3. Preliminary examination of sands in the laboratory. 
 
 "An approximate volumetric determination of the silt in sand can be made by measuring or estimating the 
 thickness of the layer of fine material which settles on top of the sand. The % of silt by volume has been found to 
 vary from 1 to 2 times the % by weight." 
 
30 
 
 CONCRETE ENGINEERS' HANDBOOK 
 
 [Sec. 1-36 
 
 allowing the material to stand in the water until cool, removing the surface water with a towel, weighing, placing 
 particles in a 300-c.c. graduate, pouring in a sufficient measured volume of water to cover the stone, noting the com- 
 bined volume of water and stone, computing the volume of stone alone, and computing the specific gravity by divid- 
 ing the original dry weight of the stone in grams by the displaced volume in cubic centimeters. Data are also 
 afforded by this test for determining the % of absorption of the stone based on dry weight. 
 
 6. Crushing strength of the stone has never been used as an acceptance test of aggregate. 
 
 36. Notes on the Selection and Testing of Aggregates. — Sand and stone for important or special 
 work should be tested in some well-equipped laboratory and, of course, the tests should be made 
 before the aggregates are purchased or the concrete mixed. The tests should be upon representative 
 samples, and materials should be checked for uniformity as delivered. In sampling natural sands 
 and gravels the greatest care must be used. 
 
 Sand should also be regularly tested as construction work progresses. Often a sand that is 
 found entirely suitable at the start will be found entirely different in later deliveries. Proportions 
 of materials should change whenever the mechanical analysis shows a decided change in the grada- 
 tion of the sand grains. 
 
 Mechanical analyses of sands, or volumetric tests of mortars or concretes made from the given 
 sands, sometimes show that a stronger mortar or concrete may be obtained by mixing two sizes of 
 sands from different portions of the same bank, making the single requirement that a definite pei- 
 centage is to be retained on a certain sieve. Mechanical analyses and volumetric tests are also 
 useful in studying two or more sands to determine the one most suited for the given work. Fre- 
 quently a properly proportioned mixture of sand and crushed stone screenings will produce a better 
 sand for mortor or concrete than either one used separately. 
 
 For the best results, sand for mortar requires more fine material than sand for concrete. 
 
 For maximum water-tightness a mortar or concrete may require a slightly larger proportion of 
 fine grains in the sand than for maximum density or strength. Gradel tends to produce a more water- 
 tight concrete than broken stone. 
 
 A high unit weight of material and a correspondingly low percentage of voids are indications of 
 coarseness and good grading of particles. However, the impossibility of establishing uniformity 
 of weight and measurement due to different percentages of moisture and different methods of 
 handling make these results merely general guides that seldom can be taken as positive indications 
 of true relative values. This is especially true of the fine aggregates in which percentages of voids 
 increase and weights decrease with the addition of moisture up to 5 to S%. 
 
 Aggregates that contain harmful impurities may sometimes be made satisfactory for concrete 
 work by washing. 
 
 Some sands which contain impurities have been found to prevent hardening with one brand of 
 cement and to give satisfactory results with another brand. 
 
 A chemical analysis of aggregates is desirable in many cases, and a microscopical examination 
 will often prove of value. 
 
 Cement adheres more readily to sand grains with rough, unpolished surfaces. 
 
 Usually an artificial sand or crushed stone will safely contain a greater percentage of fine 
 material than natural sand. 
 
 37. Specifications for Aggregates. — The specifications adopted by the Pubhc Service 
 Commission for quality of concrete aggregates used in New York City subway construction 
 are noted below (Eng. News, Feb. 11, 1916; Eng. Rec, Jan. 8, 1916). 
 
 Fine aggregates shall conform to the following requirements: 
 
 Mechanical Grading. — 
 
 Size of opening square 
 holes (inches) 
 
 Commercial number 
 of sieve 
 
 Limit of fineness (% 
 passing). Not more 
 may pass 
 
 Limit of coarseness (% 
 passing). Not less 
 must pass 
 
 0.200 
 
 4 
 
 100 
 
 95 
 
 0.100 
 
 8 
 
 95 
 
 85 
 
 0.042 
 
 16 
 
 75 
 
 40 
 
 0.021 
 
 30 
 
 50 
 
 20 
 
 0.011 
 
 50 
 
 30 
 
 2 
 
 0.006 
 
 100 
 
 6 
 
 
Sec. 1-38] 
 
 MATERIALS 
 
 31 
 
 Silt. — Not over 6 % by dry weight shall pass a No. 100 sieve when screened dry. Not over 10 % , dry weight, 
 shall pass a No. 100 sieve when washed on the sieve with a stream of water. 
 
 Both of the above tests shall be made, and neither limit shall be exceeded. 
 The following test is for field use only: 
 
 Not over 10% by volume shall be silt when the test is made by decanting from test tubes (method described 
 in Art. 35). 
 
 Strength in Mortar. — Fine aggregate shall be of such quality that mortar composed of 1 part Portland cement 
 and 3 parts fine aggregate by weight will show a tensile and compressive strength at least equal to the strength of 
 1 : 3 mortar of the same consistency made with the same cement and standard Ottawa sand. Fine aggregate shall 
 not be dried before being made into mortar, but shall contain natural moisture. 
 
 Organic Matter.^ — Loss on ignition shall not exceed 0.1 % of the total dry sand by weight, nor 10 % of the silt 
 obtained by decantation. 
 
 Coarse aggregate for concrete shall conform to the following requirements: 
 
 Mechanical Grading — 
 
 Size of opening 
 square holes 
 (inches) 
 
 Limit of fineness ( % 
 passing). Not more 
 may pass 
 
 T,imit of coarseness 
 ( % passing). Not 
 less must pass 
 
 2 
 
 
 100 
 
 IH 
 
 100 
 
 95 
 
 1 
 
 80 
 
 40 
 
 H 
 
 60 
 
 25 
 
 H 
 
 40 
 
 10 
 
 H 
 
 5 
 
 
 Cleanliness. — All broken stone aggregate must be so free from dust that samples caught as the material falls 
 from the conveyor belt at the plant will be within the limit of fineness. All gravel must be thoroughly washed at the 
 plant. 
 
 WATER 
 
 38. General Requirements. — The water used in mixing mortar or concrete should be free 
 from oil, acids, alkalies, or vegetable matter, and should be of a quality fit for drinking purposes. 
 The presence of oils is easily detected by the well-known iridescent surface film. Vegetable 
 matter can sometimes be detected by observing floating particles, or by turbidity. Chemical 
 determinations are better and more certain. 
 
 39. Examination of Water. — Tests of water for acidity or alkalinity may be made by means 
 of litmus paper, procured at any chemist's. " If blue litmus remains blue on immersing in the 
 water, then the property is either neutral or alkaline; if the color changes to red, then the prop- 
 erty is acidic. If there is a dangerous amount of acid present, the change in color will be very 
 rapid. Likewise, if red litmus changes very quickly to blue, the water will be found to contain 
 a dangerous amount of strong alkali. If the change of color is slow and faint in either test, 
 the indication may be disregarded. A solution of phenol-phthalein is a delicate test for 
 alkalinity. 
 
 Whenever a water does not appear satisfactory, its effect upon the strength and setting 
 quahties of a cement should be determined by direct test on mixtures. 
 
 40. Functions of Water. — The functions of water in concrete are: 
 
 1. Water reacts with cement to form a binding material which unites otherwise non- 
 cohering sand and stone. 
 
 2. Water operates to flux both dissolved and undissolved cementing substances over the 
 surfaces of sand grains and stone particles (or pieces of gravel), rendering possible extensive 
 and close adhesion by carrying these substances into the minute and multitudinous surface 
 irregularities of the particles, where they are absorbed as water is later absorbed or evaporated. 
 
 3. Water acts as a lubricant between sand particles and stone particles so that placement 
 of harsh and irregular materials in molds and forms is rendered easy. 
 
 1 For colorimetric test see footnote on p. 29. 
 
32 
 
 CONCRETE ENGINEERS' HANDBOOK 
 
 [Sec. 1-41 
 
 4. Water itself occupies space in the mass. 
 
 41. Influence of Quantity of Water on Strength of Concrete. — The first function of water 
 cited in the preceding article is basic and essential to the manufacture of concrete. If there 
 is insufficient water, obviously its reaction with cement will prematurely cease; and if there is 
 too much water, it is equally obvious that the cementing products may be too dilute to develop 
 proper strength since cement depends for its early strength and for a considerable part of its 
 later strength upon the hardening of amorphous or glue-like substances. Undue dilution of 
 these substances is readily possible, but it is accompanied by impairment of strength, just as 
 glue may be a valuable adhesive when of proper consistency, while the same glue, if too dilute, 
 may useless for like purposes although later evaporation may gradually restore its cementitious 
 
 3250 
 
 value. Cement depends further for its strength upon interlacing crystals; and crystallization 
 takes place only from saturated or supersaturated solutions. If, therefore, excess water is 
 added, such strength as these crystals may confer is further impaired. The influence of water 
 in greater or less quantities on the strength of concrete is shown in Fig. 13. ^ 
 
 42. Influence of Quantity of Water on Fluxing of Cement. — The second function of water 
 cited above — its action as a carrier (or flux) of cementing substances — is obvious. In bringing 
 sand, stone and cementing substances into intimate contact (Fig. 14), it acts physically in a 
 manner analogous to its earlier chemieal role. 
 
 Inevitably, however, this desirable function of water is closely dependent upon its fourth 
 
 ^ See L. N. Edwards: Proc. Am. Soc. Test. Mat., 1917. 
 
 D. A. Abrams: Concrete (C. M. Edition), July, 1917. 
 
Sec. 1-43] 
 
 MATERIALS 
 
 33 
 
 function — ^viz., its occupancy of space. It is readily seen that if the minute irregularities in 
 the surface of stone are first filled with water, and, because of initial excess of water, the cement- 
 ing solution is then too weak, a strong, intimate attachment of cement will be inhibited, both 
 because the irregularities are already filled and also because of excessive dilution. 
 
 Furthermore, water, particularly when charged with gelatinous aluminates from cement, 
 has ability to occlude a very high percentage of air. This air, as minute bubbles, firmly attaches 
 itself to the sand and stone. ^ It also remains between parti- 
 cles to such an extent as oftentimes to completely isolate a large 
 percentage of the materials. Given excess water , therefore, 
 and a proportionate amount of occluded air, detriment to con- 
 crete is sure to arise from the primary fault in an increasing 
 ratio (see Fig. 15). This explains to a large degree the lower 
 strengths with prolonged mixing in present-type machines 
 found by some investigators. 
 
 43. Influence of Quantity of Water on Lubrication of 
 Concrete Mixture. — The function of water as a lubricant of 
 concrete is very important, but its importance can be over- 
 estimated, particularly when balanced against the detrimental 
 effects which may and often do result. It is not necessary fig. 14— Cement particles 
 to add great quantities of water to concrete to make it easy- ^^i'^^ 9yer surfaces of sand grains. 
 
 ° ^ (.Magnined 20 cliams.) 
 
 flowing if the concrete is sufficiently mixed. The more con- 
 crete is mixed, the smoother working it becomes and the less water is superficially evident. 
 Cement is continually hydrating in the mixing action; and in process of hydration, large 
 quantities of hydrated lime are formed. This has a very pronounced effect in lubricating 
 the mass, and furthermore keeps it coherent. Excess water, on the other hand, promotes 
 separation of the constituent materials, offsetting the good effects of hydration and render- 
 ing the concrete extremely harsh in working and 
 difficult to handle. More mixing, therefore, or 
 more efficient mixing through improved mixing 
 devices, should -be relied upon for easy placing, 
 rather than excess water, 
 
 44. Influence of Quantity of Water on 
 Space Occupied in Resulting Concrete. — The 
 fourth function of water in concrete — that of 
 occupying space in the mass — is so important 
 and so varied in its manifestations that a large 
 treatise would be too small for adequate pre- 
 sentation. A few leading considerations may, 
 however, serve to stimulate individual thought 
 in this regard. 
 
 There are few substances so incompressible 
 as water. Beyond question, although water is 
 mobile, a given quantity occupies definite space. 
 Concrete in forms is essentially in a confined space. 
 In this form space are cement, sand, stone, and 
 water, each occupying its proportionate share of 
 the total volume of concrete. So long as the form remains tight, these substances must all 
 remain substantially in place. If the form leaks, which is contrary to practice, more or less 
 water, with a greater or less quantity of cement in solution, may escape. This escape may be 
 before, or after the mass has taken either initial or final set. In this latter case, a hollow space, 
 
 1 See Ost,wald: "Colloidal Chemistry," p. 70-118. 
 3 
 
 Fig. 15. — Water and air voids in concrete. 
 (Natural size.) 
 
34 
 
 CONCRETE ENGINEERS' HANDBOOK 
 
 [Sec. 1-45 
 
 Fig. 16. — Water voids in concrete. 
 (Magnified 18 diams.) 
 
 or void of greater or less volume remains in the concrete where once was water in greater or less 
 (luantity or a solution too dilute to solidify (see Fig. 16). But leakage need not necessarily 
 occur in this way. After forms are removed, any uncombined water or dilute solution will be 
 free to escape, either by gravity, or by capillary suction aided by evaporation, leaving behind as 
 
 a void the space it required in the mass. Such 
 action is evidenced in hundreds of concretes 
 examined. 
 
 45. Harmful Effects of Voids Caused by 
 Excess Water. — From the above it is evident 
 that the more uncombined water, the more voids 
 in the set concrete. Conversely, the more voids, 
 the less the closeness of compacting of sand and 
 stone; and the less this compacting, the less the 
 density, strength, durability and value of the 
 hardened mass. Furthermore, the proportions 
 of the concrete are seriously unbalanced (see Arts. 
 2 and 16, Sect. 2). 
 
 The space loss referred to is only a small 
 part of the ultimate damage. Physical stress 
 due to loading may be the least intense of the 
 stresses to which concrete is subjected. Physical, 
 or chemical and physico-chemical stresses set 
 up * in the mass after hardening, through dis- 
 ruptive freezing, or through percolating water 
 alone or carrying chemically active agents, are of far greater intensity. Each pore, or void, 
 is a potential aid to such destructive agents; and enlarged by initial attack, soon become 
 an active aid and abettor. First loss, therefore, may be of minor importance. Induced 
 weaknesses, augmenting primary deficiencies, must be reckoned with to an increasing degree. 
 
 46. Excess Water the Cause of Day's Work Planes. — Perhaps the commonest evidence, 
 to be found on every hand, as to the effects of excess water in concrete are "day's work planes." 
 In the early life of a structure such as a buttress wall, these planes are hidden either by a smooth 
 mortar surface at contact with forms, or by a later-applied wash or coat of cement plaster. 
 But as months pass and the structure is subjected to water action in greater or less degree, 
 from one source or another, these planes are made more and more evident by seepage along them. 
 When such seepage is in quantity, it may be detected as a film of water, or, with rapid evapora- 
 tion, by crystalline deposits. When seepage is less, it may be evidenced by a patch of efflores- 
 cence, but in each case the underlying cause of water passage is "laitance, " which is largely 
 caused by the use of excess water. 
 
 47. Excess Water the Cause of Large Laitance Deposits. — "Laitance," or "day's work 
 planes," may be of small bulk, relative to the total mass of concrete, yet in some instances, 
 laitance is found to an exaggerated and oftentimes to a dangerous extent. Whenever forms 
 are filled by dumping concretes continuously in one spot, with dependence upon hoeing-down, 
 or natural flow for distribution of heavier materials into lower parts of forms, it is inevitable 
 that water and the finer materials suspensible in water, including much of the cement, should 
 separate from the heavier materials; and that they should form, when solidified above the con- 
 crete a deposit or stratum of greater or less thickness and extent, which will be entirely composed 
 of muck or "laitance." This material is chalky and of low strength. It is very absorbent; 
 and when saturated is of little better value than so much wet, sandy clay. 
 
 Instances of the formation of "laitance " in quantities and in situations where it is dangerous 
 are foinid in columns poured in two or more sections. Although not approved by building codes, 
 contractors, for their own convenience or to save on forms, will sometimes pour half a column, 
 allowing it to set before continuing to the top. Inevitably, lighter materials rise in the form. 
 
Sec. 1-48] 
 
 MATERIALS 
 
 35 
 
 Necessarily the joint thus formed in the middle of the column is of inferior material; and of 
 a material which cannot bond with material subsequently poured. If this procedure is again 
 followed, the lighter part of this latter also rises so that a stratified column, with another 
 ''laitance" section at the column head will result. If after removal of forms this material 
 should become wet from any cause, crushing and sHding is to be expected with possibly collapse 
 of the column and its supported load. Columns should not only be poured in one section, but 
 they furthermore should be poured of concrete of such consistency that "laitance" will not 
 accumulate; and it would also be a desirable precaution to overflow the form to remove such 
 accumulations as may rise. It is better to waste a portion of material at the top, in order to 
 be sure that there may be no ''laitance" at the column head, rather than to have any question 
 as to strength or security. ^ 
 
 48. Excess Water and Waterproof Concrete. — It is difficult to find a truly waterproof 
 field concrete, largely because excess water is so generally used in mixing. The majority of 
 structures are of such size that they cannot be poured continuously. This necessarily means 
 stoppage of work for greater or less intervals. Stoppage of work with wet concretes always 
 means a layer of ''laitance;" and this inevitably prevents succeeding layers from bonding, 
 entailing a chain of consequences. A radical change in such field procedures is demanded, if 
 these difficulties are to be overcome. 
 
 49. Excess Water Causes Unsatisfactory Concrete Floor Surfaces. ^ — It is difficult to 
 insure that concrete floors shall be dustless. The functions of a concrete floor are to bear loads 
 as well as to withstand abrasion and impact, these latter being the severest service to which 
 it is subjected. It is unfortunate that the top of a concrete floor is the surface on which depend- 
 ence is placed, as in possibly nine cases out of ten, this surface coat, both by virtue of its initial 
 consistency and also because of water later brought to the surface through troweling, is partly 
 or wholly composed of ''laitance." 
 
 To remedy these defects, floor hardeners of one kind or another, are added to the concrete 
 in mixing. Few of these substances should have any real place in the concrete-floor industry. 
 Most are inferior to quartz sand in hardness and strength, but because of the prevalence of 
 unsatisfactory concrete floors and because of the human tendency to escape consequences by 
 purchasing immunities, such alleged remedies find ready sale. If, instead of buying integral 
 floor hardeners, less water were put into concrete floors and a good quality of graded sand with 
 Portland cement used in well-mixed and properly placed concrete, there would be less need of 
 tonics. 
 
 60. Excess Water Prevents Bonding New Concrete to Old. — One result that can be gua- 
 ranteed is the failure of effective bonding between new concrete and old. Various expedients 
 from time to time have been claimed to bring about effective results in this regard, but little 
 has as yet been unquestionably accomplished. Washing the surface of old concrete with hydro- 
 chloric acid is ineffective and wrong except so far as it may clean off surface dirt and carbonated 
 deposits. Picking the surface rarely goes deep enough or covers enough surface. The inher- 
 ent difficulty underlying all attempts at bonding, is the identical trouble that causes day's 
 work planes, or that makes the wearing surface of concrete floors unsatisfactory, i.e., the exist- 
 ence of a light, chalky, insecure material, substituted at the critical plane for a substance which 
 should be durable and secure. 
 
 51. Excess Water and Concreting in Cold Weather. — Concreting in cold weather is always 
 attended by some risk, even when forms remain in place until milder weather. Heating of 
 aggregates is seldom adequate, and the heat transmitted through wooden forms after pouring 
 is small in quantity. It should be remembered that at 40°F. the reaction between water and 
 cement and the production of cementing strength is only one-fourth as rapid as at 50°F. and 
 less than one-ninth as rapid as at 70°F. Dilution by excess water of such feeble solutions 
 increases the danger, as is evidenced by frequent winter failures. Furthermore, at 39°F. 
 
 ^See Gillmore: "On Limes, Hydraulic Cements and Mortars," 1872; p. 242. 
 2 See Sect. 4, "Concrete Floors and Floor Surfaces, Sidewalks, and Roadways." 
 
36 
 
 CONCRETE ENGINEERS' HANDBOOK 
 
 [Sec. 1-52 
 
 some subtle change occurs in water which decreases its chemical ability even before actual 
 freezing ;i and at this latter point occurs expansion of 8% by volume, with exertion of some 300 
 tons disruptive pressure per square inch of surface. 
 
 The greater the quantity of water in a cold-weather concrete, therefore, the greater the 
 liability to dilution, little strength, frost disruptions, and failures. The potency of excess 
 water in these respects is just beginning to receive due recognition. 
 
 52. Suggested Procedures to Guard Against Use of Excess Water. — Excess water in con- 
 crete should be rigidly guarded against. To insure the use of less water, specifications must 
 embody provisions giving the engineer authority for its regulation. To this end, the following 
 partial specification is suggested: 
 
 1. Concrete shall be an intimate mixture of sand, stone (or gravel), cement and water 
 of the several kinds and qualities herein specified and in proportions as specified, subject to 
 modification by the Engineer. 
 
 2. The proportions and quantities of all materials, including water, shall be as directed by 
 the engineer and shall be subject at all times to such change as his tests or judgment may 
 dictate as advisable. 
 
 3. All materials shall be accurately measured in measures of approved type and known 
 capacity. 
 
 Cement shall be measured by the standard sack or, if in bulk, by weight, 94 lb. being 
 taken as an equivalent of one sack. Loose measurement of cement is prohibited. 
 
 Sand and stone shall be measured in struck measures of a capacity and type approved by 
 the engineer. Measurement in wheelbarrows of a type which do not admit of a struck measure- 
 ment will not be permitted. 
 
 Water shall be measured at each mixer in containers adapted to ready adjustment and to 
 accurate delivery of variable quantities. Supplementing the delivery of such measuring con- 
 tainers by additions of water, because of slowness of discharge or for any other reason, will not 
 be permitted. 
 
 4. Concrete of a plastic consistency shall be required in all parts of the work, unless per- 
 mission be given by the engineer for the use of drier and stiffer mixtures. Sloppy and overwet 
 concretes are strictly prohibited. The quantity of water, therefore, will be subject to regulation 
 at all times by the engineer according to the requirements of the aggregates in use at that time. The 
 rejection and removal of overwet concretes either before or after placing in forms may, at 
 the engineer's discretion, be required of the contractor without compensation. 
 
 REINFORCEMENT 
 
 53. Types of Reinforcement. — The reinforcing steel in reinforced-concrete construction 
 is mostly in the form of rods, or bars, of round or square cross-section. These vary in size from 
 K to % in. for light floor slabs, up to iVi to 1}4 in. as a maximum size for heavy beams and 
 columns. Both plain and deformed bars are used. With plain bars the adhesion between steel 
 and concrete is depended upon to furnish the necessary bond strength. With deformed bars 
 the usual adhesion is supplemented by a mechanical bond, the amount of this bond in any 
 given case depending upon the shape of the bar. The adhesion of concrete to flat bars is less 
 than for round or square bars, but the flat deformed bar possesses advantages over other forms 
 when used as hooping for tanks, pipes, and sewers where the reduc ed thickness of the bar allows 
 the concrete section to have a greater effective depth for the same total thickness of concrete. 
 
 Wire fabric and expanded metal in various forms are used to a considerable extent in 
 slabs, pipes, and conduits. These types of reinforcement are easy to place and are especially 
 well adapted to resist temperature cracks and to prevent cracking of the concrete from impact 
 or shock. 
 
 1 See O. D. Van Engelen: Century Magazine, April, 1917. 
 
Sec. 1-54] 
 
 MATERIALS 
 
 37 
 
 A number of combinations of forms are employed to a greater or less extent. These 
 combinations are known as systems. 
 
 54. Surface of Reinforcement. — A rough surface on steel has a higher bond value for use 
 in concrete than a smooth surface, consequently a thin film of rust on reinforcement should not 
 cause its rejection. In fact in the case of cold-drawn wire which presents a very smooth surface, 
 a slight coating of rust is a decided advantage. Loose or scaly rust, however, should never be 
 allowed. Reinforcement in this state of corrosion may be used if first cleansed with a stiff 
 wire brush or given a bath of hydrochloric acid solution (consisting of 3 parts acid to 1 part 
 water) and then washed in clean running water. Oiling and painting of reinforcing steel 
 should not be permitted as its bonding value is greatly reduced thereby. 
 
 55. Quality of Steel. — Authorities differ as to the quality of steel to be used for reinforce- 
 ment. Mild steel is the ordinary structural steel occurring in all structural shapes. High steel 
 or steel of hard grade has a greater percentage of carbon than mild steel and is also known as 
 high-carbon or high elastic-limit steel. 
 
 Brittleness is to be feared in high steel, although this quality is not so dangerous when the 
 metal is used in heavy reinforced-concrete members — for example, in heavy beams or slabs — ■ 
 as the concrete to a large extent absorbs the shocks and pro- 
 tects the steel. All high steel should be carefully inspected 
 and tested in order to prevent any brittle or cracked material 
 from getting into the finished work. Steel of high elastic Fig. 17.— Cold-twisted square bar. 
 limit is seldom employed where plain bars are used. 
 
 Cold twisting increases the elastic limit and ultimate strength of mild-steel bars. The 
 increase, however, is not definite, varying greatly with slight variations in the grade of the 
 rolled steel. A square twisted bar is shown in Fig. 17. 
 
 56. Working Stresses. — The generally accepted working stress for mild steel is 16,000 
 lb. per sq. in. and 18,000 to 20,000 lb. per sq. in. for high steel and cold-twisted steel. A stress 
 not greater than 16,000 lb. per sq. in. is recommended by the Joint Committee for all grades of 
 steel. 
 
 57. Coefficient of Expansion. — The coefficient of expansion of steel is approximately 
 0.0000065 degree Fahreinheit. 
 
 58. Modulus of Elasticity. — The modulus of elasticity of all grades and kinds of steel is 
 about the same and is usually taken as 30,000,000 lb. per sq. in. in both tension and compression. 
 
 59. Steel Specifications. — The following specifications are those of the Association of 
 American Steel Manufacturers for concrete reinforcement bars rolled from billets, adopted 
 March 22, 1910 (revised 1912 and 1914) : 
 
 Manufacturers' Standard Specifications for Concrete Reinforcement Bars 
 
 Rolled from Billets 
 
 1. Manufacture. — Steel may be made by either the open-hearth or Bessemer process. Bars shall be rolled 
 from standard new billets. 
 
 2. Chemical and Physical Properties. — The chemical and physical properties shall conform to the limits as 
 shown in the table on the following page. 
 
 3. Chemical Determinations. — In order to determine if the material conforms to the chemical limitations 
 prescribed in paragraph 2 herein, analysis shall be made by the manufacturer from a test ingot taken at the time 
 of the pouring of each melt or blow of steel, and a correct copy of such analysis shall be furnished to the engineer 
 or his inspector. 
 
 4. Yield Point. — For the purposes of these specifications, the yield point shall be determined by careful ob- 
 servation of the drop of the beam of the testing machine, or by other equally accurate method. 
 
 5. Form of Specimens. — (a) Tensile and bending test specimens may be cut from the bars as rolled, but 
 tensile and bending test specimens of deformed bars may be planed or turned for a length of at least 9 in. if deemed 
 necessary by the manufacturer in order to obtain uniform cross-section. 
 
 (b) Tensile and bending test specimens of cold-twisted bars shall be cut from the bars after twisting, and shall 
 be tested in full size without further treatment, unless otherwise specified as in (c), in which case the conditions 
 thereon stipulated shall govern. 
 
 (c) If it is desired that the testing and acceptance for cold-twisted bars be made upon the hot-rolled bars 
 before being twisted, the hot-rolled bars shall meet the requirements of the structural-steel grade for plain bars 
 shown in this specification. 
 
38 
 
 CONCRETE ENGINEERS' HANDBOOK 
 
 [Sec. 1-59 
 
 
 Structural-steel 
 
 Intermediate 
 
 Hard errade 
 
 
 
 grade 
 
 grade 
 
 
 
 Cold- 
 
 Properties considered 
 
 Plain 
 bars 
 
 ue- 
 formed 
 bars 
 
 Plain 
 bars 
 
 Via 
 
 formed 
 bars 
 
 Plain 
 bars 
 
 De- 
 formed 
 bars 
 
 twisted 
 
 Phosphorus maximum: Bessemer. . .. 
 
 0.10 
 
 
 
 0.10 
 
 0.10 
 
 0.10 
 
 0.10 
 
 0.10 
 
 0.10 
 
 
 0.06 
 
 0.06 
 
 0.06 
 
 0.06 
 
 0.06 
 
 0.06 
 
 0 06 
 
 Ultimat/6 tensile strength^ lb. per sq. 
 
 
 
 
 Tn/s"! nnn 
 
 80,000 
 
 80,000 
 
 
 in. 
 
 
 
 
 
 min. 
 
 min. 
 
 only 
 
 Yield point, minimum lb. per sq. in. . 
 
 33,000 
 
 33.000 
 
 40,000 
 
 40,000 
 
 50,000 
 
 50,000 
 
 •^^ nnn 
 
 Elongation, % in 8-in. minimum. . . . 
 
 1,400,000 
 
 1,250,000 
 
 l,oOO,OUO 
 
 1,125,000 
 
 1,200,000 
 
 1,000,000 
 
 5% 
 
 Tens. str. 
 
 Tens. str. 
 
 Tens. str. 
 
 Tens. str. 
 
 Tens. str. 
 
 Tens. str. 
 
 Cold bend without fracture: Bars 
 
 180 deg. 
 
 180 deg. 
 
 180 deg. 
 
 180 deg. 
 
 180 deg. 
 
 180 deg. 
 
 180 deg. 
 
 under ?4-in. diameter or thickness. 
 
 d = It 
 
 d = 2t 
 
 d = 2t 
 
 d = 3t 
 
 d = St 
 
 d = 4< 
 
 d = 2t 
 
 Bars ?4-in. diameter or thickness and 
 
 180 deg. 
 
 180 deg. 
 
 90 deg. 
 
 90 deg. 
 
 90 deg. 
 
 90 deg. 
 
 180 deg. 
 
 over. 
 
 d It 
 
 d = 2t 
 
 d = 2t 
 
 d = 3t 
 
 d = 3t 
 
 d = 4t 
 
 d = 3t 
 
 The intermediate and hard gra 
 
 des will be 
 
 used only 
 
 when spe 
 
 cified. 
 
 
 
 
 6. Number of Tests. — (a) At least one tensile and one bending test shall be made from each melt of open- 
 hearth steel rolled, and from each blow or lot of 10 tons of Bessemer steel rolled. In case bars differing H in. and 
 more in diameter or thickness are rolled from one melt or blow, a test shall be made from the thickest and thinnest 
 material rolled. Should either of these test specimens develop flaws, or should the tensile test specimen break 
 outside of the middle third of its gaged strength, it may be discarded and another test specimen substituted there- 
 for. In case a tensile test specimen does not meet the specifications, an additional test may be made. 
 
 (6) The bending test may be made by pressure or by light blows. 
 
 7. Modifications in Elongation for Thin and Thick Material. — For bars less than Yie in. and more than % 
 in. nominal diameter or thickness, the following modifications shall be made in the requirements for elongation: 
 
 (o) For each increase of }^ in. in diameter or thickness above % in. a deduction of 1 shall be made from the 
 specified percentage of elongation. 
 
 {b) For each decrease of Me in. in diameter or thickness below Yie in. a deduction of 1 shall be made from the 
 specified percentage of elongation. 
 
 (,c) The above modifications in elongation shall not apply to cold-twisted bars. 
 
 8. Number of Twists. — Cold-twisted bars shall be twisted cold with one complete twist in a length equal to 
 not more than 12 times the thickness of the bar. 
 
 9. Finish. — Material must be free from injurious seams, flaws or cracks, and have a workmanlike finish. 
 
 10. Variation in Weight. — Bars for reinforcement are subject to rejection if the actual weight of any lot varies 
 more than 5 % over or under the theoretical weight of that lot. 
 
 The following specifications are those of the American Society for Testing Materials for 
 concrete reinforcement bars rolled from billets: 
 
 Standard Specifications for Billet-steel Concrete Reinforcement Bars 
 
 (American Society for Testing Materials) 
 
 1. (o) These specifications cover three classes of billet-steel concrete reinforcement bars, namely: plain, 
 deformed and cold-twisted. 
 
 (6) Plain and deformed bars are of three grades, namely: structural-steel, intermediate and hard. 
 
 2. (a) The structural-steel grade shall be used unless otherwise specified. 
 
 (fe) If desired, cold-twisted bars may be purchased on the basis of tests of the hot-rolled bars before twist- 
 ing, in which case such tests shall govern and shall conform to the requirements specified for plain bars of structural- 
 steel grade. 
 
 Manufacture. — 3. (a) The steel may be made by the Bessemer or open-hearth process. 
 (6) The bars shall be rolled from new billets. No reroUed material will be accepted. 
 
 4. Cold twisted bars shall be twisted cold with one complete twist in a length not over 12 times the thickness 
 of the bar. 
 
 Chemical Properties and Tests. — 5. The steel shall conform to the following requirements as to chemical 
 composition: 
 
Sec. 1-591 
 
 MATERIALS 
 
 39 
 
 Phosphorus, Bessemer not over 0.10% 
 
 Open-hearth not over 0 . 05 % 
 
 6. An analysis of each melt of steel shall be made by the manufacturer to determine the percentages of car- 
 bon, manganese, phosphorus and sulphur. This analysis shall be made from a test ingot taken during the pouring 
 of the melt. The chemical composition thus determined shall be reported to the purchaser or his representative, 
 and shall conform to the requirements specified in Sect. 5. 
 
 7. Analyses may be made by the purchaser from finished bars representing each melt of open-hearth steel, 
 and each melt, or lot of 10 tons, of Bessemer steel. The phosphorus content thus determined shall not exceed that 
 specified in Sect. 5 by more than 25%. 
 
 Physical Properties and Tests. — 8. (a) The bars shall conform to the following requirements as to tensile 
 properties: 
 
 
 Plain bars 
 
 Df;formed bars 
 
 Cold- 
 twisted 
 bars 
 
 Properties considered 
 
 Structural- 
 steel 
 grade 
 
 Inter- 
 mediate 
 grade 
 
 Hard 
 grade 
 
 Structural- 
 steel 
 grade 
 
 Inter- 
 mediate 
 grade 
 
 Hard 
 grade 
 
 Tensile strength, lb. per sq. 
 
 55,000 
 
 70,000 
 
 80,000 
 
 55,000 
 
 70,000 
 
 80,000 
 
 Recorded 
 
 in. 
 
 to 
 70,000 
 
 to 
 85,000 
 
 min. 
 
 to 
 70.000 
 
 to 
 85.000 
 
 min. 
 
 only 
 
 Yield point, min., lb. per sq. 
 in. 
 
 33,000 
 
 40,000 
 
 50,000 
 
 33,000 
 
 40,000 
 
 50,000 
 
 55,000 
 
 Elongation in 8 in. min. %' 
 
 1,400,000 
 
 1,300,000 
 
 1,200,000 
 
 1,250,000 
 
 1.125,000 
 
 1,000,000 
 
 5 
 
 Tens. str. 
 
 Tens. str. 
 
 Tens. str. 
 
 Tens. str. 
 
 Tens. str. 
 
 Tens. str. 
 
 1 See Sect. 9. 
 
 (b) The yield point shall be determined by the drop of the beam of the testing machine. 
 
 9. (a) For plain and deformed bars over in. in thickness or diameter, a deduction of 1 from the percent- 
 ages of elongation specified in Sect. 8(o) shall be made for each increase of }i in. in thickness or diameter above 
 Vi in. 
 
 (b) For plain and deformed bars under Yie in. in thickness or diameter, a deduction of 1 from the percentages 
 of elongation specified in Sect. 8(a) shall be made for each decrease of Mo in- in thickness or diameter below lie in. 
 
 10. The test specimen shall bend cold around a pin without cracking on the outside of the bent portion, as 
 follows : 
 
 Thickness or diameter of 
 bar 
 
 Plain bars 
 
 Deformed bars 
 
 Cold- 
 twisted 
 bars 
 
 Structural- 
 steel 
 grade 
 
 Inter- 
 mediate 
 grade 
 
 Hard 
 grade 
 
 Structural- 
 steel 
 grade 
 
 Inter- 
 mediate 
 grade 
 
 Hard 
 grade 
 
 
 180 deg. 
 
 d = t 
 180 deg 
 
 d = t 
 
 180 deg. 
 d = 2t 
 90 deg. 
 d = 2t 
 
 180 deg. 
 d = 3t 
 90 deg. 
 
 d = 3t 
 
 180 deg. 
 
 d = t 
 180 deg. 
 
 d = 2t 
 
 180 deg. 
 d = Zt 
 90 deg. 
 
 d = 3t 
 
 180 deg. 
 d = U 
 90 deg. 
 d = \t 
 
 180 deg. 
 d = 2t 
 ISO deg. 
 
 d = 3< 
 
 
 
 d = diameter of pin about which the specimen is bent. 
 t = thickness or diameter of specimen. 
 
 11. (a) Tension and bend test specimens for plain and deformed bars shall be taken from the finished bars, 
 and shall be of the full thickness or diameter of bars as rolled; except that the specimens for deformed bars may be 
 machined for a length of at least 9 in., if deemed necessary by the maiiufacturer to obtain uniform cross-section. 
 
 (6) Tension and bend test specimens for cold-twisted bars shall be taken from the finished bars, without 
 further treatment; except as specified in Sect. 2(6). 
 
 12. (a) One tension and one bend test shall be made from each melt of open-hearth steel, and from each melt, 
 or lot of 10 tons, of Bessemer steel; except that if material from one melt differs % in. or more in thickness or diam- 
 eter, one tension and one bend test shall be made from both the thickest and the thinnest material rolled. 
 
 (6) If any test specimen shows defective machining or develops flaws, it may be discarded and another soeci- 
 men substituted. 
 
40 
 
 CONCRETE ENGINEERS' HANDBOOK 
 
 [Sec. 1-59 
 
 (c) If the percentage of elongation of any tension test specimen is less than that specified in Sect. 8(a) and any 
 part of the fracture is outside the middle third of the gage length, as indicated by scribe scratches marked on the 
 specimen before testing, a retest shall be allowed. 
 
 Permissible Variations in Weight. — 13. The weight of any lot of bars shall not vary more than 5% from the 
 theoretical weight of that lot. 
 
 Finish. — 14. The finished bars shall be free from injurious defects and shall have a workmanlike finish. 
 
 Inspection and Rejection. — 15. The inspector representing the purchaser shall have free entry, at all times 
 while work on the contract of the purchaser is being performed, to all parts of the manufacturer's works which con- 
 cern the manufacture of the bars ordered. The manufacturer shall afford the inspector, free of cost, all reasonable 
 facilities to satisfy him that the bars are being furnished in accordance with these specifications. All tests (except 
 check analyses') and inspection shall be made at the place of manufacture prior to shipment, unless otherwise speci- 
 fied, and shall be so conducted as not to interfere unnecessarily with the operation of the works. 
 
 16. (a) Unless otherwise specified, any rejection based on tests made in accordance with Sect. 7 shall be re- 
 ported within 5 working days from the receipt of the samples. 
 
 (6) Bars which show injurious defects subsequent to their acceptance at the manufacturer's works will be 
 rejected, and the manufacturer shall be notified. 
 
 17. Samples tested in accordance with Sect. 7, which represent rejected bars, shall be preserved for 2 weeks 
 from the date of the test report. In case of dissatisfaction with the results of the tests, the manufacturer may make 
 claim for a rehearing within that time. 
 
 Reinforcing bars rolled from old rails are being used tc a considerable extent in reinforced- 
 concrete work and seem to be giving satisfaction, especially for unimportant work such as 
 footings, retaining walls, and possibly in slabs where the failure of one rod could not wreck 
 the structure. The specifications for rail-steel concrete reinforcement bars adopted by the 
 Association of American Steel Manufacturers April 20, 1912 (revised April 21, 1914) are as 
 follows : 
 
 Manufacturers' Standard Specifications for Rail-steel Concrete Reinforcement Bars 
 
 1. Manufacture. — All steel shall be rolled from standard section Tee rails. 
 
 2. Physical Properties. — The physical properties shall conform to the following limits: 
 
 3. Yield Point. — For the purposes of 
 these specifications, the yield point shall 
 be determined by careful observation of 
 the drop of the beam of the testing ma- 
 chine, or by other equally accurate method. 
 
 4. Form of Specimens. — (a) Tensile 
 and bending test specimens may be cut 
 from the bars as rolled, but tensile and 
 bending test specimens of deformed bars 
 may be planed or turned for a length of 
 at least 9 in. if deemed necessary by the 
 manufacturer in order to obtain uniform 
 cross-section. 
 
 (6) Tensile and bending test speci- 
 mens of hot-twisted bars shall be cut from 
 the bars after twisting, and shall be tested 
 in full size without further treatment, un- 
 less otherwise specified. 
 
 5. Number of Tests. — (a) One ten- 
 sile and one bending test shall be made 
 from each lot of 10 tons or less of each 
 
 size of bar rolled from rails varying not more than 10 lb. per yd. in nominal weight. Should a test specimen 
 develop flaws, or should the tensile test specimen break outside of the middle third of its gaged length, it may 
 be discarded and another test specimen substituted therefore. In case a tensile specimen does not meet the 
 specifications, an additional test may be made. 
 
 (b) The bending test may be made by pressure or by light blows. 
 
 6. Modtfications in Elongation for Thin and Thick Material. — For bars less than ^ie in. and more than ^4 
 in. nominal diameter or thickness, the following modifications shall be made in the requirements for elongation: 
 
 (a) For each increase of 3^^ in. in diameter or thickness above ?4 in., a deduction of 1 shall be made from 
 the specified percentage of elongation. 
 
 (6) For each decrease of Me in. in diameter or thickness below Yif, in., a deduction of 1 shall be made 
 from the specified percentage of elongation. 
 
 7. Number of Twists. — Hot-twisted bars of rail carbon steel shall be twisted with one complete twist in a 
 length equal to not more than 12 times the thickness of the bar. 
 
 
 Rail-steel grade 
 
 Properties considered 
 
 Plain 
 bars 
 
 Deformed 
 and hot- 
 twisted bars 
 
 Ultimate tensile strength, minimum, lb. 
 
 per sq. in. 
 Yield point, minimum, lb. per sq. in. . . . 
 
 Elongation, % in 8-in. minimum 
 
 80,000 
 
 50,000 
 1,200,000 
 
 80,000 
 
 50,000 
 1,000,000 
 
 
 Tens. str. 
 
 Tens. str. 
 
 Cold bend without fracture: Bars under 
 H in. diameter or thickness. 
 
 180 deg. 
 d = 3t 
 
 180 deg. 
 d = U 
 
 Bars H in. diameter or thickness and 
 over. 
 
 90 deg. 
 
 d = 3t 
 
 90 deg. 
 d = 'it 
 
Sec. 1-59] 
 
 MATERIALS 
 
 41 
 
 8. Finish. — Material must be free from injurious seams, flaws or cracks, and have a workmanlike finish. 
 
 9. Variation in Weight. — Bars for reinforcement are subject to rejection if the actual weight of any lot 
 varies more than 5% over or under the theoretical weight of that lot. 
 
 Properties considered 
 
 Plain bars 
 
 Deformed 
 and hot- 
 twisted bars 
 
 Tensile strength, lb. per sq. in 
 
 80.000 
 50,000 
 1.200,000 
 
 80.000 
 50.000 
 1.000,000 
 
 
 Tens. str. 
 
 Tens. str. 
 
 1 See Sect. 
 
 ReroUed bar specifications have also been adopted by the American Society for Testing 
 Materials after an extended series of tests. The specifications follow: 
 
 Standard Specifications for Rail-steel Concrete Reinforcement Bars 
 
 (American Society for Testing Materials') 
 
 1. The specifications cover three classes of rail-steel concrete reinforcement bars, namely: plain, deformed, 
 and hot-twisted. 
 
 Manufacture. — 2. The bars shall be rolled from standard section Tee rails. 
 
 3. Hot-twisted bars shall have one complete twist in a length not over 12 times the thickness of the bar. 
 
 Physical Properties and Tests. — 4. (a) The bars shall conform to the following minimum requirements as 
 to tensile properties: 
 
 {h) The yield point shall be deter- 
 mined by the drop of the beam of the test- 
 ing machine. 
 
 5. (a) For bars over Yi in. in thick- 
 ness or diameter, a deduction of 1 from the 
 percentages of elongation specified in Sect. 
 4(a) shall be made for each increase of 
 in. in thickness or diameter above ?4 in. 
 
 (6) For bars under Jle in. in thickness 
 or diameter, a deduction of 1 from the per- 
 centages of elongation specified in Sect. 4(a) 
 shall be made for each decrease of Me in. 
 in thickness or diameter below Ha in. 
 
 6. The test specimen shall bend cold around a pin without cracking on the outside of the bent portion, as 
 follows: 
 
 7. (a) Tension and bend test speci- 
 mens for plain and deformed bars shall be 
 taken from the finished bars, and shall be of 
 the full thickness or diameter of bars as 
 rolled; except that the specimens for de- 
 formed bars may be machined for a length 
 of at least 9 in., if deemed necessary by the 
 manufacturer to obtain uniform cross- 
 section. 
 
 (6) Tension and bend test speci- 
 mens for hot-twisted bars shall be taken 
 from the finished bars, without further 
 treatment. 
 
 8. (o) One tension and one bend test 
 shall be made from each lot of 10 tons or less of each size of bar rolled from rails varying not more than 101b. per 
 yd. in nominal weight. 
 
 (fe) If any test specimen shows defective machining or develops flaws, it may be discarded and another speci- 
 men substituted. 
 
 (c) If the percentage of elongation of any tension test specimen is less than that specified in Sect. 4 (a) and any 
 part of the fracture is outside the middle third of the gage length, as indicated by scribe scratches marked on the 
 specimen before testing, a retest shall be allowed. 
 
 Permissible Variations in Weight. — 9. The weight of any lot of bars shall not vary more than 5% from the 
 theoretical weight of that lot. 
 
 Finish. — 10. The finished bars shall be free from injurious defects and shall have a workmanlike finish. 
 
 Inspection and Rejection. — 11. The inspector representing the purchaser phall have free entry, at all times while 
 work on the contract of the purchaser is being performed, to all parts of the manufacturer's works which concern 
 the manufacture of the bars ordered. The manufacturer shall aff'ord the inspector, free of cost, all reasonable facili- 
 ties to satisfy him that the bars are being furnished in accordance with these specifications. All tests and inspec- 
 tion shall be made at the place of manufacture prior to shipment, unless otherwise specified, and shall be so con- 
 ducted as not to interfere unnecessarily with the operation of the works. 
 
 12. Bars which show injurious defects subsequent to their acceptance at the manufacturer's works will be 
 rejected, and the manufacturer shall be notified. 
 
 Thickness or diameter of bar 
 
 Plain bars 
 
 Deformed 
 and hot- 
 twisted bars 
 
 
 180 deg. 
 d = 3t 
 90 deg. 
 
 d = 3t 
 
 180 deg. 
 d ^ At 
 90 deg. 
 
 d = it 
 
 
 
 d = diameter of pin about which the specimen is bent. 
 t = thickness or diameter of the specimen. 
 
42 
 
 CONCRETE ENGINEERS' HANDBOOK 
 
 [Sec. 1-60 
 
 60. Factors Affecting Cost of Reinforcing Bars.— In order to insure minimum cost and 
 prompt delivery of steel reinforcing bars, the steel schedule for a reinforced-concrete structure 
 should call for bars of as few different sizes and lengths as possible. Bars of odd 16th sizes 
 are seldom to be found in stock (except the ^e-in. size which is frequently used for slab rein- 
 forcement) and shipments from the mill on such sizes are likely to be very slow. Designers 
 should always bear in mind this fact and arrange to use either round or square bars in 3^^-in. 
 sizes. Wherever possible, steel lengths that do not vary greatly on the schedule should all be 
 made equal since an order calling for only a few different lengths will be put through the mill 
 much faster than one calling for many different lengths. 
 
 The following size extras for bars less than ^^-in. are standard with all mills and are the 
 same for either round or square bars: 
 
 Size Extras for Rounds and Squares in Cents per 100 Lb. 
 
 ^-in. and larger Base 
 
 to 1 He-ill 5 cts. extra 
 
 M to ^le-in 10 cts. extra 
 
 ]^G-in 20 cts. extra 
 
 ^^-in 4 25 cts. extra 
 
 ^6 -in 35 cts. extra 
 
 3'^-in 50 cts. extra 
 
 It should be noticed that a higher size extra must be paid for an odd 16th size below ^:^-in. 
 than for the next larger J-^-in. size. This fact alone offsets any advantage in saving steel 
 by always calling for the nearest theoretical size whether odd or even. 
 
 Where the character of the work requires small bars a saving in cost is obtained by using 
 round bars owing to the difference in size extras between rounds and squares of equivalent area. 
 
 Lengths less than 5 ft. should be avoided, if possible, as they are subject to the following 
 cutting extras, whether sheared or hot-sawed: 
 
 Lengths over 24 in. and less than 60 in 5 cts. per 100 lb. 
 
 Lengths 12 in. to 24 in. inclusive 10 cts. per 100 lb. 
 
 Lengths under 12 in 15 cts. per 100 lb. 
 
 All orders calling for less than 2000 lb. of the same size and shape are subject to the follow- 
 ing extras: 
 
 Quantities less than 2000 lb. but not less than 1000 lb 15 cts. per 100 lb. 
 
 Quantities less than 1000 lb 35 cts. per 100 lb. 
 
 61. Deformed Bars. — The following deformed bars are in common use: 
 
 61a. Diamond Bar (Fig. 18). — Furnished by Concrete-Steel Engineering Co., 
 New York City. The standard sizes are as follows: 
 
 Diamond Bars 
 
 Size in inches 
 
 
 
 Vl6 
 
 
 5/8 
 
 H 
 
 
 1 
 
 VA 
 
 IH 
 
 Area in square inches. . . . 
 
 0.0625 
 
 0.1406 
 
 0.19 
 
 0.25 
 
 0.39 
 
 0.56 
 
 0.76 
 
 1.00 
 
 1.26 
 
 1.56 
 
 Weight per foot in pounds. 
 
 0.213 
 
 0.478 
 
 0.65 
 
 0.85 
 
 1.33 
 
 1.91 
 
 2.60 
 
 3.40 
 
 4.30 
 
 5.31 
 
 Fig. 18. — Diamond bar. 
 
 It should be noted that the weights and areas of Diamond bars are equal to those of plain 
 square bars of like denominations. 
 
Sec. 1-616] MATERIALS 43 
 
 616. Corrugated Bars (Fig. 19).— Furnished by Corrugated Bar Co., Buffalo, 
 N. Y. The standard sizes are as follows: 
 
 Corrugated Rounds 
 
 Size in inches 
 
 H 
 
 H 
 
 ri6 
 
 ^/^ 
 
 H 
 
 
 1 
 
 11^ 
 
 IH 
 
 Net area in square inches 
 
 Weight per foot in pounds 
 
 0.11 
 0.38 
 
 0.19 
 0.66 
 
 0.25 
 0.86 
 
 0.30 
 1.05 
 
 0.44 
 1.52 
 
 0.60 
 2.06 
 
 0.78 
 2.69 
 
 0.99 
 3.41 
 
 1.22 
 4.21 
 
 Corrugated Squares 
 
 Size in inches 
 
 Vi 
 
 H 
 
 H 
 
 H 
 
 H 
 
 
 1 
 
 IH 
 
 VA 
 
 Weight per foot in pounds 
 
 0.06 
 0.22 
 
 0.14 
 0.49 
 
 0.25 
 0.86 
 
 0.39 
 1.35 
 
 0.56 
 1.94 
 
 0.76 
 
 2.64 
 
 1.00 
 
 3.43 
 
 1.26 
 4.34 
 
 1 . 55 
 5.35 
 
 Fig. 19. — Corrugated bars. 
 
 61c. Havermeyer Bars (Fig. 20). — Furnished by Concrete Steel Co., New York 
 City. The following table gives the weights and areas of the standard Havermeyer bars: 
 
 Havermeyer Bars 
 
 
 Squares 
 
 Rounds 
 
 Flats 
 
 Size in inches 
 
 Area in 
 
 Weight 
 
 Area in 
 
 Weight 
 
 Size in 
 
 Area in 
 
 Weight 
 
 
 square 
 
 per foot in 
 
 square 
 
 per 
 
 foot in 
 
 square 
 
 per foot in 
 
 
 inches 
 
 pounds 
 
 inches 
 
 pounds 
 
 inches 
 
 inches 
 
 pounds 
 
 H 
 
 0 . 0625 
 
 0.212 
 
 0.0491 
 
 0 
 
 167 
 
 ixH 
 
 0.2500 
 
 0.850 
 
 He 
 
 0.9770 
 
 0.332 
 
 
 
 
 
 0.3750 
 
 1.280 
 
 H 
 
 0.1406 
 
 0.478 
 
 0.1104 
 
 0 
 
 375 
 
 \\ix% 
 
 0.4690 
 
 1.590 
 
 
 0.2500 
 
 0.850 
 
 0.1963 
 
 0 
 
 667 
 
 
 0 . 4688 
 
 1.590 
 
 H 
 
 0.3906 
 
 1.328 
 
 0.3068 
 
 1 
 
 .043 
 
 
 0.5625 
 
 1.913 
 
 % 
 
 0.5625 
 
 1.913 
 
 0.4418 
 
 1 
 
 502 
 
 WixM 
 
 0.7500 
 
 2.550 
 
 % 
 
 0.7656 
 
 2.603 
 
 0.6013 
 
 2 
 
 044 
 
 mxH 
 
 0 . 6563 
 
 2.230 
 
 1 
 
 1.0000 
 
 3.400 
 
 0.7854 
 
 2 
 
 670 
 
 mxviG 
 
 0.7656 
 
 2.600 
 
 
 1.2656 
 
 4.303 
 
 0.9940 
 
 3 
 
 379 
 
 ly^xM 
 
 0.8750 
 
 2.980 
 
 
 1.5625 
 
 5.312 
 
 1.2272 
 
 4 
 
 173 
 
 
 
 
44 
 
 CONCRETE ENGINEERS' HANDBOOK 
 
 [Sec. l-6ld 
 
 Special sizes of 1^^-in. and l3>2-in. square Havermeyer bars can be rolled by special arrange- 
 ment, but are not carried in stock. A size extra of 10 cts. applies against 1 by ^-^-in. and 13-2 
 by ^{Q-in. flats; all other sizes tabulated take the base price. 
 
 Qld. Rib Bar (Fig. 21). — Furnished by Trussed Concrete Steel Co., of Youngs- 
 town, Ohio and Detroit, Mich. The following sizes are standard: 
 
 Rib Bar 
 
 Size in 
 inches 
 
 Area in 
 square inches 
 
 Weight per 
 linear foot 
 in pounds 
 
 Size in 
 inches 
 
 Area in 
 square inchSs 
 
 Weight per 
 linear foot 
 in pounds 
 
 ys 
 
 0.1406 
 
 0.48 
 
 
 0.7656 
 
 2.65 
 
 
 0.2500 
 
 0.86 
 
 1 
 
 1.0000 
 
 3.46 
 
 % 
 
 0.3906 
 
 1.35 
 
 
 1.2656 
 
 4.38 
 
 
 0.5625 
 
 1.95 
 
 
 
 
 Fig. 21.— Rib bar. 
 
 60e. Inland Bar (Fig. 22).— Furnished by Inland Steel Co., Chicago. 
 Sizes in. to in. inclusive with single row of stars on each side. 
 Sizes in. to IJ^ in. inclusive with double row of stars on each side. 
 
 Lengths may be obtained up to 85 ft. Supplied in both open-hearth steel and rail carbon 
 steel. 
 
 Standard sizes are as follows: 
 
 Inland Bar 
 
 Size in inches 
 
 % 
 
 K2 
 
 % 
 
 
 
 1 
 
 
 IH 
 
 Area in square inches .... 
 Weight per foot in pounds. 
 
 0.140 
 0.485 
 
 0.250 
 0.862 
 
 0.390 
 1.341 
 
 0.562 
 1.932 
 
 0.765 
 2.630 
 
 1.000 
 3.434 
 
 1.265 
 4.349 
 
 1.562 
 5.365 
 
 Fig. 22.— Inland bar. 
 
 Rail carbon steel bars not rolled larger than 1 in. 
 
 61/. American Bars (Fig. 23). — Furnished by American System of Reinforcing, 
 Chicago. The following sizes are standard: 
 
Sec. 1-62] MATERIALS 45 
 
 American Bars 
 
 
 Squares 
 
 Rounds 
 
 Size in inches 
 
 Net area in square 
 inches 
 
 Weight per foot in 
 pounds 
 
 Net area in square 
 inches 
 
 Weight per foot in 
 pounds 
 
 H 
 
 0.141 
 
 0.48 
 
 0 110 
 
 U . 00 
 
 
 0.250 
 
 0.85 
 
 0.196 
 
 0.68 
 
 /8 
 
 0.391 
 
 1.33 
 
 0.307 
 
 1.05 
 
 
 0.563 
 
 1.92 
 
 0.442 
 
 1.51 
 
 
 0.766 
 
 2.61 
 
 0.602 
 
 2.05 
 
 1 
 
 1.000 
 
 3.40 
 
 0.786 
 
 2.68 
 
 
 1.270 
 
 4.31 
 
 0.994 
 
 3.38 
 
 
 1.560 
 
 5.32 
 
 1.230 
 
 4.19 
 
 Fig. 23. — American bars. 
 
 62. Wire Fabric. — This material is used to a considerable extent for floors, roofs, walls, 
 vaults, pavement, etc., and has been found to possess many valuable quaUties. Wire fabric 
 is made of steel wires crossing generally at right angles and secured at the intersections. The 
 heavier wires run lengthwise and are called carrying wires; the lighter ones cross these and are 
 called distributing or tie wires. One distinct advantage in the use of fabric is that it preserves 
 uniform spacing of the steel. 
 
 The steel wire gage adopted as standard for all steel wire upon recommendation of the 
 United States Bureau of Standards is given in the following table: 
 
 Steel Wire Gage 
 
 Diameter, 
 inches 
 
 Steel wire 
 gagei 
 
 Diameter, 
 inches 
 
 Area, square 
 inches 
 
 Pounds per 
 foot 
 
 Pounds per 
 mile 
 
 Feet per 
 pound 
 
 
 
 0 
 
 5000 
 
 0 
 
 19635 
 
 0 
 
 .6668 
 
 3,521.0 
 
 1 
 
 .500 
 
 
 7/0 . 
 
 0 
 
 4900 
 
 0 
 
 18857 
 
 0 
 
 6404 
 
 3,381.0 
 
 1 
 
 .562 
 
 
 
 0 
 
 46875 
 
 0 
 
 17257 
 
 0 
 
 5861 
 
 3,094.0 
 
 1 
 
 .706 
 
 
 6/0 
 
 0 
 
 4615 
 
 0 
 
 16728 
 
 0 
 
 5681 
 
 2,999.0 
 
 1 
 
 760 
 
 Ke 
 
 
 0 
 
 4375 
 
 0 
 
 15033 
 
 0 
 
 5105 
 
 2,696.0 
 
 1 
 
 959 
 
 
 5/0 
 
 0 
 
 4305 
 
 0 
 
 14556 
 
 0 
 
 4943 
 
 2,610.0 
 
 2 
 
 023 
 
 
 
 0 
 
 40625 
 
 0 
 
 12962 
 
 0 
 
 4402 
 
 2,324.0 
 
 2 
 
 272 
 
 
 4/0 
 
 0 
 
 3938 
 
 0 
 
 12180 
 
 0 
 
 4136 
 
 2,184.0 
 
 2 
 
 418 
 
 H 
 
 
 0 
 
 3750 
 
 0 
 
 11045 
 
 0 
 
 3751 
 
 1,980.0 
 
 2 
 
 666 
 
 
 3/0 
 
 0 
 
 3625 
 
 0 
 
 10321 
 
 0 
 
 3505 
 
 1,851.0 
 
 2 
 
 853 
 
 
 
 0 
 
 34375 
 
 0 
 
 092806 
 
 0 
 
 3152 
 
 1,664.0 
 
 3 
 
 173 
 
 
 2/0 
 
 0 
 
 3310 
 
 0 
 
 086049 
 
 0 
 
 2922 
 
 1,543.0 
 
 3 
 
 422 
 
 1 Formerly called the "American Steel & Wire Go's. Gage." 
 
46 CONCRETE ENGINEERS' HANDBOOK [Sec. l-62a 
 
 Steel Wire Gage. — {Continued.) 
 
 Diameter, 
 inches 
 
 Steel wire 
 gage 
 
 Diameter, 
 inches 
 
 Area, square 
 inches 
 
 Pounds per 
 foot 
 
 Pounds per 
 mile 
 
 Feet per 
 pound 
 
 Vi 
 
 
 0.3125 
 
 0.076699 
 
 0.2605 
 
 1,375.0 
 
 3.839 
 
 
 0 
 
 0.3065 
 
 0.073782 
 
 0.2506 
 
 1,323.0 
 
 3.991 
 
 
 1 
 
 X 
 
 n 2R'^n 
 
 0 062Q02 
 
 n 91 "^fi 
 
 1 128 0 
 
 
 %2 
 
 
 0.28125 
 
 0.062126 
 
 0.2110 
 
 1,114.0 
 
 4.74 
 
 
 2 
 
 0.2625 
 
 0.054119 
 
 0.1838 
 
 970.4 
 
 5.441 
 
 \4 
 74 
 
 
 0 2500 
 
 0 04Q087 
 
 0. 1667 
 
 880.2 
 
 5 QQQ 
 
 
 3 
 
 0.2437 
 
 0.046645 
 
 0.1584 
 
 836.4 
 
 6.313 
 
 
 4 
 
 0.2253 
 
 0.039867 
 
 0.1354 
 
 714.8 
 
 7.386 
 
 /3 2 
 
 
 0 21875 
 
 0 037583 
 
 0. 1276 
 
 673.9 
 
 7. 835 
 
 
 5 
 
 0.2070 
 
 0.033654 
 
 0.1143 
 
 603.4 
 
 8.750 
 
 
 6 
 
 0.1920 
 
 0.028953 
 
 0.09832 
 
 519.2 
 
 10.17 
 
 /16 
 
 
 0. 1875 
 
 0 027612 
 
 0 0Q377 
 
 495. 1 
 
 10 66 
 
 
 7 
 
 0.1770 
 
 0.024606 
 
 0.08356 
 
 441.2 
 
 11.97 
 
 
 8 
 
 0.1620 
 
 0.020612 
 
 0.07000 
 
 369.6 
 
 14.29 
 
 /3 2 
 
 
 0 15625 
 
 0 01 91 75 
 
 0 06512 
 
 343. 8 
 
 15. 36 
 
 
 9 
 
 0.1483 
 
 0.017273 
 
 0.05866 
 
 309.7 
 
 17.05 
 
 
 10 
 
 0.1350 
 
 0.014314 
 
 0.04861 
 
 256.7 
 
 20.57 
 
 
 
 0.125 
 
 0 012272 
 
 0 04168 
 
 220.0 
 
 24.00 
 
 
 
 0. 1205 
 
 0 01 1 404 
 
 0 03873 
 
 204.5 
 
 25. 82 
 
 
 12 
 
 
 0 0087417 
 
 0 02Q6Q 
 
 156. 7 
 
 33. 69 
 
 
 
 0.09375 
 
 0.0069029 
 
 0 . 02344 
 
 123.8 
 
 42.66 
 
 
 13 
 
 0.0915 
 
 0.0065755 
 
 0.02233 
 
 117.9 
 
 44.78 
 
 
 14 
 
 0.0800 
 
 0.0050266 
 
 0.01707 
 
 90.13 
 
 58.58 
 
 
 15 
 
 0.0720 
 
 0.0040715 
 
 0.01383 
 
 73.01 
 
 72.32 
 
 
 16 
 
 0.0625 
 
 0.0030680 
 
 0.01042 
 
 55.01 
 
 95.98 
 
 
 17 
 
 0.0540 
 
 0.0022902 
 
 0.007778 
 
 41.07 
 
 128.60 
 
 The manner of securing the intersections of wire fabric has given rise to a number of 
 different types, several of the principal ones of which are given 
 n f] below. 
 
 K ' ■ . ' ^ 62a. Welded Wire Fabric. — Welded wire fabric, 
 
 ■ Fig. 24, manufactured by the CHnton Wire Cloth Co., is a gal- 
 
 " ' vanized wire mesh made up of a series of parallel longitudinal 
 
 wires, spaced a certain distance apart and held at intervals by 
 
 ' ' ' ■ " means of transverse wires, arranged at right angles to the longi- 
 
 ^ — — — tudinal ones, and welded to them at the points of intersection 
 [J — — ^ ^ ^ patented electrical process. Longitudinal wires can be 
 
 Fig. 24.— Welded wire fabric, spaced on centers of 2 or more in., in steps of 3^ in. Transverse i 
 
 wires can be spaced on centers of 1 to 18 in, inclusive, in steps | 
 of 1 in. and on centers of 10 to 18 in. inclusive, in steps of 2 in. The following table shows 
 
Sec. 1-626] 
 
 MATERIALS 
 
 47 
 
 the sizes and areas of the wire used. Rolls kept in stock vary in length between 150 and 200 
 ft. and between 56 and 100 in. in width. The wire will develop an average ultimate strength 
 of 70,000 to 80,000 lb. per sq. in. 
 
 Welded Wire Fabric 
 
 
 
 
 
 
 
 
 Area per 
 
 foot of width 
 
 in longitudinal wires only 
 
 
 Diameter of 
 longitudinal 
 wires 
 (inches) 
 
 Area of one 
 longitudinal 
 wire (square 
 inches) 
 
 
 Spacing of 
 transverse 
 wires 
 (inches) 
 
 
 
 
 
 
 
 
 
 
 G&gG of 
 longitud- 
 inal wires 
 
 Gage of 
 transverse 
 wires 
 
 
 
 Spacing of longitudina 
 
 wires 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 2 in. 
 
 3 in. 
 
 4 
 
 in. 
 
 5 in. 
 
 6 in. 
 
 0000 
 
 0 
 
 394 
 
 0 
 
 122 
 
 3 
 
 16 
 
 0 
 
 735 
 
 0 
 
 490 
 
 0 
 
 367 
 
 0 
 
 294 
 
 0.245 
 
 r\r\r\ 
 OOU 
 
 0 
 
 363 
 
 0 
 
 103 
 
 
 1 A 
 ID 
 
 0 
 
 619 
 
 0 
 
 413 
 
 0 
 
 310 
 
 0 
 
 248 
 
 n 9nA 
 
 00 
 
 0 
 
 331 
 
 0 
 
 086 
 
 4 
 
 16 
 
 0 
 
 516 
 
 0 
 
 344 
 
 0 
 
 258 
 
 0 
 
 207 
 
 n 1 79 
 
 0 
 
 n 
 u 
 
 
 n 
 U 
 
 ri7A 
 \j( ^ 
 
 6 
 
 16 
 
 0 
 
 443 
 
 0 
 
 295 
 
 0 
 
 221 
 
 0 
 
 177 
 
 0.148 
 
 1 
 
 0 
 
 283 
 
 0 
 
 063 
 
 O 
 
 1 A 
 
 lo 
 
 0 
 
 377 
 
 0 
 
 252 
 
 0 
 
 189 
 
 0 
 
 151 
 
 n 1 0A 
 
 2 
 
 0 
 
 263 
 
 0 
 
 054 
 
 8 
 
 16 
 
 0 
 
 325 
 
 0 
 
 217 
 
 0 
 
 162 
 
 0 
 
 130 
 
 0.108 
 
 3 
 
 0 
 
 244 
 
 0 
 
 047 
 
 8 
 
 16 
 
 0 
 
 280 
 
 0 
 
 187 
 
 0 
 
 140 
 
 0 
 
 112 
 
 0.093 
 
 4 
 
 0 
 
 225 
 
 0 
 
 040 
 
 9 
 
 16 
 
 0 
 
 239 
 
 0 
 
 160 
 
 0 
 
 120 
 
 0 
 
 096 
 
 0.080 
 
 5 
 
 0 
 
 207 
 
 0 
 
 034 
 
 9 
 
 16 
 
 0 
 
 202 
 
 0 
 
 135 
 
 0 
 
 101 
 
 0 
 
 081 
 
 0.067 
 
 6 
 
 0 
 
 192 
 
 0 
 
 029 
 
 10 
 
 16 
 
 0 
 
 174 
 
 0 
 
 116 
 
 0 
 
 087 
 
 0 
 
 069 
 
 0.058 
 
 7 
 
 0 
 
 177 
 
 0 
 
 025 
 
 10 
 
 16 
 
 0 
 
 148 
 
 0 
 
 098 
 
 0 
 
 074 
 
 0 
 
 059 
 
 0.049 
 
 8 
 
 0 
 
 162 
 
 0 
 
 021 
 
 10 
 
 12 
 
 0 
 
 124 
 
 0 
 
 082 
 
 0 
 
 062 
 
 0 
 
 049 
 
 0.041 
 
 9 
 
 0 
 
 148 
 
 0 
 
 017 
 
 11 
 
 12 
 
 0 
 
 104 
 
 0 
 
 069 
 
 0 
 
 052 
 
 0 
 
 041 
 
 0.035 
 
 10 
 
 0 
 
 135 
 
 0 
 
 014 
 
 12 
 
 12 
 
 0 
 
 086 
 
 0 
 
 057 
 
 0 
 
 043 
 
 0 
 
 034 
 
 0.029 
 
 626. Triangle-mesh Wire Fabric. — Triangle-mesh steel-wire fabric, manu- 
 factured by the American Steel & Wire Co., is made with both single and stranded longitudinal 
 or tension members. That with the single wire longitudinal is made with one wire varying in 
 size from a No. 12 gage up to and including a 3'^-in. diameter, and that with the stranded longi- 
 
 FiG. 25. — Triangle-mesh wire fabric. 
 
 tudinal is composed of two or three wires varying from No. 12 gage up to and including No. 4 
 wires stranded or twisted together with a long lay. These longitudinals either solid or stranded 
 are invariably spaced 4-in. centers, the sizes being varied in order to obtain the desired cross- 
 sectional area of steel per foot of width (see Fig. 25). 
 
 The transverse or diagonal cross wires are so woven between the longitudinals that triangles 
 
48 
 
 CONCRETE ENGINEERS' HANDBOOK 
 
 [Sec. l-62c 
 
 are formed by their arrangement. These diagonal cross wires are woven either 2 or 4 in. apart, 
 as is desired. Triangle-mesh wire reinforcement is made in lengths of 150, 200, and 300 ft. 
 and in widths from 18 to 58 in. (4-in. steps). The table following shows the number and gage 
 of wires and the areas per foot width when the longitudinals and cross wires are spaced 4 in. 
 on centers. 
 
 Triangle-mesh Wire Fabric 
 
 Style 
 number 
 
 Number of 
 long. 
 
 Gage of 
 wire, G3<ch. 
 long. 
 
 Gage of 
 cross wires 
 
 Sectional 
 square inches 
 
 bectional 
 area, cross 
 wires, 
 square inches 
 
 Cross-sec- 
 tional area 
 
 per foot 
 
 width 
 
 Approximate 
 per 100 sq. ft. 
 
 A 1 
 
 4^ 
 
 
 a 
 D 
 
 1 A 
 
 14 
 
 0 . 087 
 
 A AO C 
 
 0 . Ozo 
 
 A 1 AO 
 
 0 . lOz 
 
 A 0 
 
 43 
 
 C 1 
 
 
 Q 
 O 
 
 1 A 
 
 14 
 
 U . ObZ 
 
 A AO K 
 
 0 . Ozo 
 
 A A'TT 
 
 0.0// 
 
 0 A 
 
 34 
 
 61 
 
 
 10 
 
 14 
 
 0.043 
 
 0.025 
 
 0.058 
 
 27 
 
 / ^ 
 
 
 
 1 A 
 
 14 
 
 U . yjZb 
 
 A AO K. 
 
 0 . Uzo 
 
 A nA 1 
 0 . 041 
 
 0 1 
 zl 
 
 
 
 
 1 O 1 / 
 
 0 . 14/ 
 
 A AOO 
 
 0. 038 
 
 A 1 TA 
 0 . 170 
 
 TO 
 
 7z 
 
 24 
 
 
 4 
 
 mi 
 
 0.119 
 
 0.038 
 
 0.142 
 
 62 
 
 
 
 o 
 
 
 U . iUi 
 
 A AQQ 
 
 A 1 OA 
 
 U . Iz4 
 
 00 
 
 
 
 D 
 
 101/ 
 
 n AC? 
 U . 08/ 
 
 A AOO 
 
 0 . 038 
 
 A 1 1 A 
 
 0. 110 
 
 K A 
 
 50 
 
 271 
 
 
 8 
 
 mi 
 
 0.062 
 
 0.038 
 
 0.085 
 
 41 
 
 OQ 1 
 
 
 10 
 
 1 O 1 / 
 
 12>^ 
 
 0 . 04d 
 
 A AOO 
 
 0 . 008 
 
 A A£!£! 
 0 . OOD 
 
 0 A 
 
 34 
 
 
 
 1 o 
 IZ 
 
 
 0 . UZb 
 
 A AOO 
 
 0 . 038 
 
 A r\ A Pi 
 
 0 . 04y 
 
 OQ 
 
 z8 
 
 311 
 
 2 
 
 4 
 
 mi 
 
 0.238 
 
 0.038 
 
 0.261 
 
 106 
 
 
 o 
 
 z 
 
 r 
 0 
 
 101/ 
 
 0 . ZOZ 
 
 A AOO 
 
 0 . 008 
 
 A 00 K 
 
 0 . zzo 
 
 00 
 9z 
 
 33 
 
 2 
 
 6 
 
 m'2 
 
 0.174 
 
 0.038 
 
 0.196 
 
 82 
 
 34 
 
 2 
 
 8 
 
 m^i 
 
 0.124 
 
 0.038 
 
 0.146 
 
 63 
 
 35 
 
 2 
 
 10 
 
 123^ 
 
 0.086 
 
 0.038 
 
 0.109 
 
 50 
 
 36 
 
 2 
 
 12 
 
 mi 
 
 0.052 
 
 0.038 
 
 0.075 
 
 37 
 
 38' 
 
 3 
 
 4 
 
 mi 
 
 0.358 
 
 0.038 
 
 0.380 
 
 151 
 
 39 
 
 3 
 
 5 
 
 123^ 
 
 0.303 
 
 0.038 
 
 0.325 
 
 130 
 
 401 
 
 3 
 
 6 
 
 123^ 
 
 0.260 
 
 0.038 
 
 0.283 
 
 114 
 
 41 
 
 3 
 
 8 
 
 123^ 
 
 0.185 
 
 0.038 
 
 0.208 
 
 87 
 
 421 
 
 3 
 
 10 
 
 mi 
 
 0.129 
 
 0.038 
 
 0.151 
 
 66 
 
 43 
 
 3 
 
 12 
 
 mi 
 
 0.078 
 
 0.038 
 
 0.101 
 
 47 
 
 Elastic limit of regular stock is from 50,000 to 60,000 lb. per sq. in. Ultimate strength is 
 85,000 lb. per sq. in. or over. Higher elastic limits and breaking strengths are furnished when 
 required. Material may be obtained either plain or galvanized. Unless otherwise specified, 
 shipments are made of material not galvanized. 
 
 62c. Unit Wire Fabric. — A rectangular-mesh staple-locked fabric (Fig. 26) is 
 furnished by the American System of Reinforcing. The wire used is of high tensile steel and 
 
 1 Styles usually carried in stock. 
 
Sec. 1-Q2d] 
 
 MATERIALS 
 
 49 
 
 is secured at the intersections by No. 14 wire. Standard sizes are shown in the following table. 
 The fabric is galvanized and comes in standard widths of 3, 4, and 5 ft., 200 lin. ft. in a roll. 
 
 Fig. 26. — Unit wire fabric. 
 
 Unit Wire Fabric 
 
 Gage of longitud- 
 inal wires 
 
 Gage of cross wires 
 
 Distance center t 
 Longitudinal wires 
 
 3 center in inches 
 Cross wires 
 
 Sectional area in sq. 
 in., foot width 
 
 11 
 
 11 
 
 6 
 
 6 
 
 0 
 
 023 
 
 10 
 
 10 
 
 6 
 
 6 
 
 0 
 
 028 
 
 9 
 
 
 6 
 
 6 
 
 0 
 
 035 
 
 9 
 
 
 4 
 
 12 
 
 0 
 
 05 
 
 9 
 
 
 3 
 
 12 
 
 0 
 
 07 
 
 8 
 
 
 4 
 
 12 
 
 0 
 
 062 
 
 7 
 
 
 4 
 
 12 
 
 0 
 
 074 
 
 6 
 
 
 4 
 
 12 
 
 0 
 
 087 
 
 5 
 
 
 4 
 
 12 
 
 0 
 
 10 
 
 4 
 
 
 4 
 
 12 
 
 0 
 
 12 
 
 3 
 
 
 4 
 
 12 
 
 0 
 
 14 
 
 Q2d. Lock-woven Steel Fabric. — Lock-woven steel fabric (Fig. 27) is also known 
 as Page Special Process fabric. It is manufactured by the Page Woven Wire Fence C'o., of 
 
 Fig. 27. — Lock-woven steel fabric. 
 
 Monessen, Pa. and is controlled by W. W. Wight & Co. of New York City. This fabric is 
 usually made 54 in. wide with special widths from 18 to 54 in. The longitudinal wires are made 
 by a special process which gives them an ultimate tensile strength of 180,000 lb. per sq. in. 
 with an elastic hmit of about 70% of the ultimate. The material is galvanized and is furnished 
 in rolls of 150, 300, 450 and 600 ft. in length. The table on page 50 gives the characteristics 
 of the different styles. 
 4 
 
50 CONCRETE ENGINEERS' HANDBOOK [Sec. l-62e 
 
 Lock-woven Steel Fabric 
 
 Style 
 
 Gage 
 
 Spacing 
 
 in inches 
 
 Sectional 
 area in sq. in. 
 per foot width 
 
 Ultimate 
 strength in 
 pounds per 
 foot width 
 
 Weight per 
 100 sq. ft. 
 
 Long. 
 
 Trans. 
 
 Long. 
 
 Trans. 
 
 14P 
 
 14 
 
 14 
 
 3 
 
 12 
 
 0.0201 
 
 3,621 ■ 
 
 11.04 
 
 13P 
 
 13 
 
 14 
 
 3 
 
 12 
 
 0 . 0265 
 
 4,790 
 
 12.91 
 
 12P 
 
 12 
 
 14 
 
 3 
 
 12 
 
 0 . 0350 
 
 6,300 
 
 15.85 
 
 IIP 
 
 11 
 
 14 
 
 3 
 
 12 
 
 0.0452 
 
 8,140 
 
 17.47" 
 
 9P 
 
 9 
 
 14 
 
 3 
 
 12 
 
 0 . 0680 
 
 12,390 
 
 28.62 
 
 8P 
 
 8 
 
 14 
 
 3 
 
 12 
 
 0.0824 
 
 14,280 
 
 34.82 
 
 7P 
 
 7 
 
 14 
 
 3 
 
 12 
 
 0.0984 
 
 17,720 
 
 39.48 
 
 14D 
 
 14 
 
 14 
 
 
 12 
 
 0 . 0402 
 
 7,242 
 
 22.08 
 
 13D 
 
 13 
 
 14 
 
 •IK 
 
 12 
 
 0.0532 
 
 9,580 
 
 25.82 
 
 12D 
 
 12 
 
 14 
 
 IK 
 
 12 
 
 0 . 0700 
 
 12,600 
 
 31.70 
 
 
 11^ 
 
 14 
 
 IK 
 
 12 
 
 0.0795 
 
 14,313 
 
 33.25 
 
 1 1 "n 
 
 11 
 
 14 
 
 1 1^ 
 
 1X2 
 
 1 9 
 
 0 0004. 
 
 1 a 900 
 
 o'k . 
 
 93^D 
 
 93^ 
 
 14 
 
 
 12 
 
 0.12498 
 
 22,450 
 
 53.43 
 
 9D 
 
 ■ 9 
 
 14 
 
 
 12 
 
 0.1376 
 
 24,780 
 
 57.20 
 
 8D 
 
 8 
 
 14 
 
 IK 
 
 12 
 
 0.1648 
 
 29,640 
 
 69.64 
 
 7D 
 
 7 
 
 14 
 
 IK 
 
 12 
 
 0.1968 
 
 35,440 
 
 78.96 
 
 62e. Wisco Reinforcing Mesh. — Wisco mesh is manufactured by the Witherow 
 Steel Co., Pittsburgh, Pa. It is made from the best grade of open-hearth steel and has a high 
 tensile strength. All longitudinals are spaced 3 in. c. to c. and cross wires 12 in. c. to c. Stand- 
 ard rolls are 150 and 300 ft. in length. Width of rolls are furnished in any multiple of 3 in. 
 from 18 to 54 in. Properties of the Wisco mesh are given in the following table: 
 
 Wisco Mesh 
 
 . Style 
 
 Sectional 
 area per 
 foot width 
 
 Weight per 
 square foot 
 
 Style 
 
 Sectional 
 area per 
 foot width 
 
 Weight per 
 square foot 
 
 style 
 
 Sectional 
 area per 
 foot width 
 
 Weight per 
 square foot 
 
 14 
 
 0.020 
 
 0.110 
 
 9K 
 
 0.062 
 
 0.277 
 
 6 
 
 0.116 
 
 0.465 
 
 12 
 
 0.035 
 
 0.158 
 
 9 
 
 0.069 
 
 0.286 
 
 29 
 
 0.138 
 
 0.556 
 
 11 
 
 0.046 
 
 0.175 
 
 8 
 
 0.083 
 
 0.341 
 
 27 
 
 0.197 
 
 0.775 
 
 10 
 
 0.058 
 
 0.223 
 
 7 
 
 0.098 
 
 0.395 
 
 26 
 
 0.230 
 
 1.036 
 
 panded metal (Fig. 28) is one of the oldest forms of sheet rein- 
 forcement. It is formed by slitting a sheet of soft steel and 
 then expanding the metal in a direction normal to the axis of 
 the sheet. The principal advantages claimed for this type of 
 reinforcement are the following: (!) An increased ultimate 
 strength and high elastic limit for low-carbon steel when the 
 diamond-shaped meshes are formed by cold drawing the metal; 
 (2) a mechanical bond with the surrounding concrete; (3) great 
 efficiency in the carrying of concentrated loads due to the ob- 
 
 63. Expanded Metal.—: 
 
 Fig. 28. — Expanded metal. 
 
Sec. l-63a] 
 
 MATERIALS 
 
 51 
 
 liquity of the strands; (4) an increased ductility because of the fact that the diamonds or quad- 
 rilaterals tend to close under severe loading; (5) a greater slab strength as the effect of closing 
 up of the diamonds is to introduce a compression into the concrete at the lower part of the 
 slab. Expanded metal and other sheet metal is made according to the U. S. Standard gago 
 which differs but slightly from the Steel Wire gage given on page 45. 
 
 63a. Steelcrete. — Manufactured by the ConsoUdated Expanded Metal Cos., 
 Rankin, Pa. The designation of the material gives the width of the diamond, the gage of the 
 plate and the cross-section per foot of width. Size 3-9-15 means that it is a 3-in. diamond, 
 made out of No. 9 plate, having a sectional area per foot of width of 0.15 sq. in. All standard 
 meshes have a diamond 3 by 8 in. The standard sizes and gages are given in the following 
 table: 
 
 "Steelcrete" Expanded Metal 
 
 Designation 
 of mesh 
 
 Width of 
 diamond 
 in incliGS 
 
 Size of mes] 
 
 Length of 
 diamond 
 in inches 
 
 1 
 
 Section in 
 sq. in. per 
 ft. of 
 width 
 
 wt. per 
 sq. ft. 
 
 in 
 pounds 
 
 No. of 
 
 sheets 
 
 in a 
 bundle 
 
 Size of standard 
 sheets 
 
 No. of 
 sq. ft. 
 
 in a 
 bundle 
 
 Wt. pet 
 bundle 
 in lb. 
 
 3-13-075 
 
 3 
 
 8 
 
 0.075 
 
 0.27 
 
 10 
 
 
 6'0" X 8'0" 
 
 480 
 
 129.6 
 
 
 
 
 
 
 
 
 6 0 X 12'0 
 
 720 
 
 194.4 
 
 3-13-10 
 
 3 
 
 8 
 
 0.10 
 
 0.37 
 
 7 
 
 
 6'9" X 8'0" 
 
 378 
 
 139.9 
 
 
 
 
 
 
 
 
 ^ 6'9" X 12'0" 
 
 567 
 
 209.8 
 
 3-13-125 
 
 3 
 
 8 
 
 0.125 
 
 0.46 
 
 7 
 
 
 5'3" X 8'0" 
 
 294 
 
 135.2 
 
 
 
 
 
 
 
 
 ^ 5'3" X 12'0" 
 
 441 
 
 202.9 
 
 3-9-15 
 
 3 
 
 8 
 
 0.15 
 
 0.55 
 
 5 
 
 
 ' 7'0" X 8'0" 
 
 280 
 
 154.0 
 
 
 
 
 
 
 
 
 ^ 7'0'' X. 12'0'' 
 
 420 
 
 231.0 
 
 3-9-20 
 
 3 
 
 8 
 
 0.20 
 
 0.73 
 
 5 
 
 
 ' 5'3" X 8'0" 
 
 210 
 
 153.3 
 
 
 
 
 
 
 
 
 5'3" X 12'0" 
 
 315 
 
 230.0 
 
 3-9-25 
 
 3 
 
 8 
 
 0.25 
 
 0.92 
 
 5 
 
 
 * 4'0" X 8'0" 
 
 160 
 
 147.2 
 
 
 
 
 
 
 
 
 4'0" X 12'0" 
 
 240 
 
 220.8 
 
 3-9-30 
 
 3 
 
 8 
 
 0.30 
 
 1.10 
 
 2 
 
 
 7'0" X 8'0" 
 
 112 
 
 123.2 
 
 
 
 
 
 
 
 
 ^ 7'0" X 12'0'' 
 
 168 
 
 184.8 
 
 3-9-35 
 
 3 
 
 8 
 
 0.35 
 
 1.28 
 
 2 
 
 
 ^ 6'0" X 8'0" 
 
 96 
 
 122.9 
 
 
 
 
 
 
 
 
 6'0" X 12'0" 
 
 144 
 
 184.3 
 
 3-6-40 
 
 3 
 
 8 
 
 0.40 
 
 1.46 
 
 2 
 
 
 ' 7'0" X 8'0'' 
 
 112 
 
 163.5 
 
 
 
 
 
 
 
 
 7'0" X 12'0'' 
 
 168 
 
 245.3 
 
 3-6-45 
 
 3 
 
 8 
 
 0.45 
 
 1.65 
 
 2 
 
 
 6'3" X 8'0" 
 
 100 
 
 165.0 
 
 
 
 
 
 
 
 
 6'3" X 12'0" 
 
 150 
 
 247.5 
 
 3-6-50 
 
 3 
 
 8 
 
 0.50 
 
 1.83 
 
 2 
 
 
 5'9" X 8'0'' 
 
 92 
 
 168.4 
 
 
 
 
 
 
 
 
 5'9" X 12'0" 
 
 138 
 
 252.5 
 
 3-6-55 
 
 3 
 
 8 
 
 0.55 
 
 2.01 
 
 2 
 
 
 5'3" X 8'0" 
 
 84 
 
 168.8 
 
 
 
 
 
 
 
 
 5'3" X 12'0" 
 
 126 
 
 253.3 
 
 3-6-60 
 
 3 
 
 8 
 
 0.60 
 
 2.19 
 
 2 
 
 j 
 
 4'9" X 8'0" 
 
 76 
 
 166.4 
 
 
 
 
 
 
 
 
 4'9" X 12'0" 
 
 114 
 
 249.7 
 
 The Consolidated Expanded Metal Cos. also make to order a 6-in. mesh, the size of the 
 diamond being 6 by 16 in. The gage of plate used is No. 4, or nearly K in. thick. Any cross- 
 sectional area desired up to and including 0.4 sq. in. can be obtained. The width of the sheets 
 depend on the sectional area. These companies also make a 4-in. mesh from No. 16 plate 
 which is unexpanded. Any length can be obtained up to 16 ft. The cross-sectional area per 
 
52 
 
 CONCRETE ENGINEERS' HANDBOOK 
 
 [Sec. 1-635 
 
 foot of width is 0.093 sq. in. Special meshes can be obtained having diamonds of in., 13^ 
 in., and 2 in. 
 
 636. Kahn Mesh. — Manufactured by the Trussed Concrete Steel Co., of 
 Youngstown, Ohio, and Detroit, Mich. The standard sizes and gages are the same as for 
 "Steelcrete." The Kahn Mesh may also be obtained with larger diamonds for reinforcing 
 concrete pavements. The sizes of the Kahn Road Mesh follow: 
 
 Kahn Road Mesh 
 
 
 Decimal 
 designation 
 
 Size of mesh 
 
 Sectional 
 
 Size No. 
 
 Width of diamond 
 in inches 
 
 Length of diamond 
 in inches 
 
 area 
 in square inches 
 
 15 
 
 6-13-042 
 
 6 
 
 12 
 
 0.042 
 
 20 
 
 6-13-053 
 
 6 
 
 12 
 
 0.053 
 
 22 
 
 6-13-058 
 
 6 
 
 12 
 
 0.058 
 
 25 
 
 6-13-066 
 
 6 
 
 12 
 
 0.066 
 
 28 
 
 6-13-074 
 
 6 
 
 12 
 
 0.074 
 
 30 
 
 6-9-079 
 
 6 
 
 12 
 
 0.079 
 
 32 
 
 6-9-085 
 
 6 
 
 12 
 
 0.085 
 
 No. of sheets in bundle, 10. Standard width of sheets, 5 ft. Standard lengths of sheets, 8 ft., 10 ft., 12 ft., or 
 any equal divisions of these lengths. 
 
 63c. Corr-X-Metal. — Furnished by the Corrugated Bar Co., Buffalo, N. Y. 
 The weights, sectional areas and standard sizes of sheets are given in the following table: 
 
 Corr-X-Metal 
 
 Style 
 
 Size of mesh, 
 short way 
 (inches) 
 
 Nominal thickness 
 of metal 
 (gage) 
 
 Approximate weight 
 per square foot, 
 (pounds) 
 
 Net sec. area per foot 
 of width 
 (square inches) 
 
 F 
 
 3 
 
 10 
 
 0.51 
 
 0.150 
 
 G 
 
 3 
 
 10 
 
 0.6 
 
 0.176 
 
 H 
 
 3 
 
 10 
 
 0.9 
 
 0.265 
 
 J 
 
 3 
 
 10 
 
 1.2 
 
 0.353 
 
 K 
 
 3 
 
 16 
 
 0.278 
 
 0.082 
 
 L 
 
 
 16 
 
 0.4 
 
 0.118 
 
 M 
 
 
 12 
 
 0.56 
 
 0.164 
 
 R 
 
 
 12 
 
 0.66 
 
 0.194 
 
 S 
 
 H 
 
 13 
 
 0.84 
 
 0.246 
 
Sec. l-63c^] MATERIALS 53 
 
 Standard Size Sheets 
 
 Style 
 
 Long way of diamond 
 
 Short way of diamond 
 
 F 
 
 6', 8', 9' and 10' 
 
 8" 
 
 3', 
 
 4', 5' and 6' 
 
 G 
 
 0,8,9 and 10 
 
 o// 
 
 8 
 
 3', 
 
 4', 5' and 6' 
 
 H 
 
 6', 8', 9' and 10' 
 
 8" 
 
 
 4' and 5' 4" 
 
 T 
 J 
 
 6', 8', 9' and 10' 
 
 8" 
 
 3', 
 
 4' and 6' 
 
 K 
 
 6' and 8' and 10' 
 
 8" 
 
 3', 
 
 4', 5' and 6' 
 
 L 
 
 6' and 8' and 10' 
 
 6" 
 
 3', 
 
 4', 5' and 6' 
 
 M 
 
 6' and 8' and 10' 
 
 6" 
 
 
 4' and 5' 4" 
 
 R 
 
 6' 
 
 
 3', 
 
 4', 5' and 6' 
 
 S 
 
 6' 
 
 
 3', 
 
 4', 5' and 6' 
 
 63c?. Econo. — Furnished by the North Western Expanded Metal Co., Chicago, 
 111. Standard sizes and weights are as follows: 
 
 Econo Expanded Metal 
 
 No. 
 
 Weight per 
 square foot, 
 pounds 
 
 Mesh and gage 
 
 Widths, feet 
 
 Lengths, feet 
 
 06-3 
 
 0.20 
 
 3"— 16 ga. 
 
 3, 4, 6 
 
 8 and 12 
 
 10-3 
 
 0.34 
 
 3" — 12 ga. 
 
 3, 4, 6 
 
 8 and 12 
 
 15-3 
 
 0.51 
 
 3"— 10 ga. 
 
 3, 4,6 
 
 8', 10' 6" and 12' 
 
 16-3 
 
 0.55 
 
 3"— 10 ga. 
 
 3, 4, 6 
 
 8', 10' 6" and 12' 
 
 20-3 
 
 0.68 
 
 3"— 10 ga. 
 
 3, 4, 6 
 
 8', 10' 6" and 12' 
 
 25-3 
 
 0.85 
 
 3"— 10 ga. 
 
 3, 4, 6 
 
 8', 10' 6" and 12' 
 
 30-3 
 
 1.02 
 
 3"— 10 ga. 
 
 3, 4, 6 
 
 8', 10' 6" and 12' 
 
 35-3 
 
 1.19 
 
 3"— 10 ga. 
 
 3, 4,6 
 
 8', 10' 6" and 12' 
 
 40-3 
 
 1.36 
 
 3"— 7 ga. 
 
 3' 6", 7' 0" 
 
 8 and 12 
 
 10-2M 
 
 0.34 
 
 2}i"—m ga. 
 
 3, 4, 6 
 
 8 and 12 
 
 15-2K 
 
 0.51 
 
 234"— 12 ga. 
 
 3, 4, 6 
 
 8 and 12 
 
 20-234 
 
 0.68 
 
 234"— 10 ga. 
 
 3, 4, 6 
 
 8 and 12 
 
 40-2M 
 
 1.36 
 
 234"— 7 ga. 
 
 3' 6", 7' 0" 
 
 8 and 12 
 
 10-1 H 
 
 JO. 34 
 \0.34 
 
 13^"— 18 ga. 
 11^"— 16 ga. 
 
 3, 4, 6 
 3, 4, 6 
 
 8 only 
 8 and 12 
 
 20-13^ 
 
 0.68 
 
 1>^"— 12 ga. 
 
 3, 4, 6 
 
 8 and 12 
 
 
 0.51 
 
 ^i"— 16 ga. 
 
 3, 4, 6 
 
 8 and 12 
 
 25-^ 
 
 0.85 
 
 M"— 12 ga. 
 
 3, 4, 6 
 
 8 and 12 
 
 20-1^ 
 
 0.68 
 
 3^"— 18 ga. 
 
 3, 4, 7 
 
 8 only 
 
 24-3^ 
 
 0.82 
 
 3^"— 16 ga. 
 
 2,4 
 
 8 only 
 
 The first two figures in the first column give the area of steel and the last figure gives the short 
 dimensions of mesh. Thus No. 30-3 has an area of 0.30 sq. in. per 12 in. of width and has a 
 mesh 3 in. wide. 
 
 63e. GF Expanded Metal— Manufactured by the General Fireproofing Co., 
 Youngstown, Ohio. Standard sizes are given in the table on page 54, 
 
54 CONCRETE ENGINEERS' HANDBOOK [Sec. 
 
 GF Expanded Metal 
 
 
 
 
 Standard size sheets 
 
 Style 
 
 Approx. weight per 
 sq. ft. in pounds 
 
 Deliveries 
 
 Lengths 
 
 Widths 
 
 
 
 
 Long w£ty of diRmonci 
 
 Short wsy of di^dnoncl 
 
 3-10-176 
 3-10-265 
 3-10-353 
 
 3-12-150 
 
 13^-12-194 
 
 M-12-246 
 
 0.60 
 0.90 
 1.20 
 
 0.51 
 0.66 
 0.84 
 
 Carried in stock 
 in standard sheets 
 
 6', 8', 9', 10'-8" 
 6', 8', 9', 10'-8" 
 6', 8', 9', 10'-8" 
 
 6', 8', 9', 10'-8" 
 6', 8' 
 6', 8' 
 
 3', 4', 5', 6' 
 4', 5'-4" 
 3', 4', 6' 
 
 3', 4', 6' 
 3', 4', 6' 
 3', 4', 6' 
 
 3-10-324 
 
 3-10-25 
 
 3-10-20 
 
 1.10 
 
 0.85 
 0.68 
 
 
 6', 8', 9', 10'-8" 
 6', 8', 9', 10-8" 
 6', 8', 9', 10'-8" 
 
 4'-4" 
 5-8" 
 5'-6" 
 
 3-10-162 
 
 3-10-15 
 
 3-12-125 
 
 0.55 
 0.51 
 0.425 
 
 
 6', 8', 9', 10'-8" 
 6', 8', 9', 10'-8" 
 6', 8', 9', 10'-8" 
 
 3', 4'-6" 
 3', 6' 
 4'-4", 6'-6" 
 
 3-12-10 
 
 3-16-082 
 
 3-16-059 
 
 0.34 
 
 0.278 
 
 0.20 
 
 Five days 
 
 6', 8', 9', 10'-8" 
 6', 8', 10'-8" 
 6', 8', 10'-8" 
 
 4', 6'-4" 
 3', 4', 5', 6' 
 3', 4', 6' 
 
 2^^-12-164 
 23^-16-118 
 23^-16-10 
 
 2-12-161 
 2-16-103 
 1>^-12-181 
 
 13-^-16-105 
 13^-18-088 
 1-12-234 
 
 0.56 
 0.40 
 0.34 
 
 0.547 
 0.351 
 0.61 
 
 0.36 
 
 0.308 
 
 0.796 
 
 to two 
 weeks 
 dependent 
 on size 
 order 
 and 
 unfilled 
 business on 
 books 
 
 6', 8', 10'-6" 
 6', 8', 10-6" 
 6', 8', 10'-6" 
 
 6', 8' 
 6', 8' 
 6', 8' 
 
 6', 8' 
 6', 8' 
 6', 8' 
 
 4', 5' 
 3', 4' 6' 
 4', 5' 
 
 4', 5' 
 4', 5' 
 4'-3" 
 
 4', 5' 
 3', 6' 
 4'-8" 
 
 1-16-175 
 1-18-125 
 ^^-1 6-1 54 
 
 0.597 
 0.425 
 0.525 
 
 
 6', 8' 
 6', 8' 
 6', 8' 
 
 3', 4', 6' 
 4'-4" 
 4'-4" 
 
 H-IS-U7 
 >^-l 8-220 
 
 0.50 
 0.75 
 
 
 6' 8' 
 6', 8' 
 
 3'-8" 
 4' 
 
 3-7-609 
 3-6-550 
 3-6-500 
 
 2.00 
 1.87 
 1.70 
 
 Mill 
 shipment 
 
 6', 8', 9', 10'-8' 
 6', 8', 9', i0'-8" 
 6', 8', 9', 10'-8" 
 
 5' 
 
 3', 4', 6' 
 4'-4" 
 
 3-6-450 
 3-7-400 
 
 1.53 
 1.36 
 
 only 
 
 6', 8', 9', 10'-8" 
 6', 8', 9', 10'-8" 
 
 4'-8" 
 5' 
 
 Note. — Interpret styles as follows: For example 3-10-176. 3 equals short dimension of 
 diamond in inches; 10 equals approximate gage; 176 equals 0.176 sq. in. sectional area per foot 
 of width. 
 
Sec. 1-64] 
 
 MATERIALS 
 
 55 
 
 64. Rib Metal. — Rib metal is manufactured by the Trussed Concrete Steel Co., and con- 
 sists of nine longitudinal ribs rigidly connected by light cross members. It is made from a 
 sheet of metal, fiat on one side and corrugated on the other. Strips of the metal adjacent to 
 the ribs are stamped out, and the sheet is drawn out into square meshes (Fig. 29). The standard 
 sheets are manufactured with meshes of from 2 to 8 in. and in all lengths up to 18 ft. The prop- 
 erties of rib metal are given in the table which follows: 
 
 Fig. 29. — Rib metal. 
 
 Rib Metal 
 
 Size 
 No. 
 
 Width of 
 standard sheet, 
 inches 
 
 Sq. ft. per 
 linear foot of 
 standard sheet 
 
 Area per ft. 
 width, sq. in. 
 
 Ult. tensile 
 strength per foot 
 of width 
 
 Safe tensile 
 strength per 
 foot of width, 
 pounds 
 
 2 
 
 16 
 
 1.33 
 
 0.54 
 
 38,880 
 
 9,720 
 
 3 
 
 24 
 
 2.00 
 
 0.36 
 
 25,920 
 
 6,480 
 
 4 
 
 32 
 
 2.67 
 
 0.27 
 
 19,440 
 
 4,860 
 
 5 
 
 40 
 
 3.33 
 
 0.216 
 
 15,552 
 
 3,888 
 
 6 
 
 48 
 
 4.00 
 
 0.18 
 
 12,960 
 
 3,240 
 
 7 
 
 56 
 
 4.67 
 
 0.154 
 
 11,088 
 
 2,772 
 
 8 
 
 64 
 
 5.33 
 
 0.135 
 
 9,720 
 
 2,430 
 
 Area of one rib = 0.09 sq. in. 
 Ultimate tensile strength = 6480 lb. 
 Safe tensile strength = 1620 lb. 
 
 65. Self-centering Fabrics. — Permanent centering fabrics (used mostly for reinforcement 
 in concrete floor slabs resting on steel beams) are stiffened by rigid, deep ribs which do away 
 with the use of slab forms. The mesh is made small enough to prevent ordinary concrete from 
 passing through. The centering fabric is laid over the supports, the concrete is poured on top 
 and the under side plastered. A simple brace along the middle of the slab span is sometimes 
 required to give sufficient strength to the ribs until the concrete has set. The permanent cen- 
 tering fabrics may be obtained either in flat or segmental form. 
 
 A serious disadvantage in this type of construction is the difficulty of providing efficient 
 fire-protection on the under side of the fabric. Bond with the concrete is also likely to be 
 insufficient. 
 
56 
 
 CONCRETE ENGINEERS' HANDBOOK 
 
 [Sec. l-65a 
 
 65a. Hy-Rib. — Hy-Rib (Fig. 30) is a steel sheathing, stiffened by deep ribs 
 formed from a single sheet of steel. It is controlled by the Trussed Concrete Steel Co. of 
 Youngstown, Ohio, and Detroit, Mich. 
 
 Fig. 30— Hy-rib. 
 
 Hy-Rib 
 
 Type of Hy-Rib 
 
 Formerly called 
 
 Height of 
 
 ribs 
 (inches) 
 
 Spacing 
 of ribs 
 (inches) 
 
 Width of 
 sheets 
 (inches) 
 
 Gage Nos. U. S. 
 Standard 
 
 l>^-in. Hy-Rib 
 
 Deep-Rib 
 
 m 
 
 7 
 
 14 
 
 22, 24, 26 
 
 i^l6-in. Hy-Rib 
 
 7-Rib 
 
 
 4 
 
 24 
 
 22, 24, 26, 28 
 
 i^e-in. Hy-Rib 
 
 3-Rib 
 
 
 8 
 
 16 
 
 24, 26, 28 
 
 ^^-in. Hy-Rib 
 
 6-Rib 
 
 Vs 
 
 4 
 
 20 
 
 24, 26, 28 
 
 Standard lengths, 6, 8, and 12 ft. 
 
 Other lengths are cut from standard lengths without charge except for waste. 
 1^-in. and ^^{Q-'m. Hy-Rib are shipped in bundles of eight sheets; '^^{Q-'m. and ^^-in. 
 Hy-Rib in bundles of sixteen sheets. 
 
 656. Corr-Mesh. — Corr-Mesh (Fig. 31) is furnished by the Corrugated Bar Co., 
 Buffalo, N. Y. It is a stiff-ribbed expanded metal, the ribs being spaced 3^^ 2 in. c. to c. The 
 
 Fig. 31. — Corr-mesh. 
 
 height of the ribs is in. and the width of the sheets is 12 % in. c. to c. of outside ribs. The 
 standard gages are No. 24, No. 26, and No. 28, although other gages can be obtained if required. 
 The standard lengths are 6, 8, 10 and 12 ft. The sheets are furnished either fiat or in various 
 types of curves. All metal is shipped painted unless specifically ordered otherwise. 
 
 65c. Self-Sentering. — Self-Sentering (Fig. 32) is manufactured by the General 
 Fireproofing Co., Youngstown, Ohio. It is made up of a series of heavy, cold-drawn ribs, 
 6 in. high, always spaced 3^^ in. c. to c, connected by a form of expanded metal — all cut 
 and drawn from one sheet of steel. Size of sheets — 29 in. wide by lengths of 4, 5, 6, 7, 8, 9, 
 10, 11 and 12 ft. Longer lengths up to 14 ft. furnished on special order. Self-Sentering is 
 made of No. 24, 26 and 28-gage metal. 
 
 Q5d. Chanelath.— Chanelath (Fig. 33), furnished by the North Western Ex- 
 panded Metal Co., Chicago, 111., is a type of expanded metal composed of a series of heavy 
 
Sec. l-65el 
 
 MATERIALS 
 
 57 
 
 cold-formed steel T-ribs connected together by a mesh known as "Kno-Burn" metal lath. 
 The T-ribs are }i in. high and spaced 4 in. c. to c. The flange of the T is 3^ in. wide. Ghane- 
 lath is manufactured and carried in stock ready for immediate shipment in the following sizes 
 of sheets: Lengths— 3, 4, 5, 6, 7, 8, 9, 10, 11 and 12 ft.; width?— 4, 8, 12, 16, 20, 24, 28, 32, 
 36, 40, 44 and 48 in. 
 
 65e. Ribplex. — Ribplex manufactured by the Berger Mfg. Co., Canton, Ohio, 
 is an expanded metal with ribs 4.8 in. on centers and in. high. Standard sheets are 24 in. 
 
 Fig. 32. — Self-sentering. 
 
 wide and are carried in Stock in 4, 5, 6, 7, 8, 9, 10, 11 and 12-ft. lengths. Sheets are made 
 in 28, 26 and 24 gages. 
 
 65/. Dovetailed Corrugated Sheets. — Sheets of thin steel corrugated so as to 
 form dovetailed grooves are manufactured by the Brown Hoisting Machinery Co., Cleveland, 
 Ohio, and by the Berger Mfg. Co., Canton, Ohio. The first-mentioned company manufacture 
 a plate known as Ferroinclave and the latter company furnish two types of plates known as 
 
 Fig. 33. — Chanelath. 
 
 Ferro-Lithic and Multiple Steel. The dovetailing in these plates serve to unite the plates to 
 the concrete. 
 
 66. Reinforcing Systems for Beams, Girders and Columns. 
 
 66a. Kahn System.— The Kahn trussed bar (Fig. 34), named for its inventor, 
 is rolled with flanges, which are bent up to resist the shear in the beam. For continuous beams, 
 inverted bars are placed over the supports in the upper part of the beam, extending over the 
 region of tension. Properties of Kahn trussed bars are shown in the following table: 
 
58 CONCRETE ENGINEERS' HANDBOOK [Sec. 1-666 
 
 Kahn Trussed Bars 
 
 Size in 
 inches 
 a Xb 
 
 Weight 
 in 
 
 pounds 
 per 
 foot 
 
 Area 
 
 Length of diagonals 
 in inches 
 
 Standard 
 
 Special 
 
 Square Section Bars 
 
 y2 X m 
 
 y4 X 23^6 
 
 1.4 
 2.7 
 
 0.41 
 0.79 
 
 12 
 
 12, 24, 36 
 
 6, 8, 18 
 8, 18, 30 
 
 New Section Bars 
 
 iy2 X 2H 
 
 m X 2H 
 2 X sy2 
 
 4.8 
 6.8 
 10.2 
 
 1.4.1 
 2.00 
 3.00 
 
 12, 24, 36 
 36 
 36 
 
 8, 18, 30 
 24, 30, 48 
 24, 30, 48 
 
 Note. — 8, 12, 18, 24, 30, 36, and 48-in. diagonals are sheared alternately. Six-in. diagonals 
 only are sheared opposite. 
 
 Fig. 34. — Kahn trussed bars. 
 
 What might be called the Kahn system is illustrated in Fig. 35. The collapsible column 
 hooping is shown more in detail in Fig. 36. The hooping is shipped in the form of flat, circular 
 coils of exact diameter and accurately spaced by means of special spacing bars. These coils 
 spring automatically into a complete hooped column on cutting the small fastening wires. Rib 
 bars (see Art. Qld) are ordinarily used as vertical reinforcement in conjunction with the hooping. 
 
 The collapsible column hooping is shipped complete with two spacing bars. Sizes of wire 
 for hooping: y, ^{q, %^ J-^e? and 3^-in. diameter. Diameter of coils: 9 to 30 in. Pitch: 
 ly to 12 in. Hooping, where desired, can also be obtained in bundles, coiled to the correct 
 diameter, and with separate spacing bars, ready for assembling in the field. 
 
 666. Cummings System. — The Cummings system is shown in Fig. 37. U- 
 shaped stirrups are used on the girder frame shown. They are shipped flat with the longitudinal 
 reinforcement, but are bent up to an inclined position on the work. The rods are held together 
 by means of a patented chair. In the Cummings hooped column, each hoop is securely attached 
 to the upright rods. ■ The hoops are made of flat steel, bent to a circle, with the ends riveted 
 or welded together in such a manner that the ends of the hoops protrude at right angles to 
 keep them the proper distance from the mold. Reinforcement of the Cummings system is 
 manufactured and sold by the Electric Welding Co., Pittsburg, Pa. 
 
 66c. Unit System. — Figs. 38 and 39 show the unit system of reinforcing con- 
 trolled by the American System of Reinforcing, Chicago, III. The girder frames are not stock 
 
Sec. l-66c] 
 
 MATERIALS 
 
 59 
 
60 
 
 CONCRETE ENGINEERS' HANDBOOK 
 
 [Sec. l-66d 
 
 frames but are built to meet the engineer's or architect's plans. Unit girder frames are pro- 
 vided with overlapping rods for continuous beams to reinforce against negative moment. 
 
 666?. Corr System. — Corr-bar girder frames (Fig. 40) and shop fabricated spirals 
 (Fig. 41) are furnished by the Corrugated Bar Co., Buffalo, N. Y. As with the unit system, 
 the girder frames are built to meet the engineer's or architect's plans. In the spiral reinforce- 
 ment the spacing bars consist of two or — in large columns — four spacers made of T-section 
 bars notched to receive the spiral. The spirals are made of cold-drawn wire and are furnished 
 in any length, in diameters of 10 to 36 in., pitch 1 to 4 in., and of the following sizes of wire: 
 
 Gage 
 
 Dia. of 
 wire 
 (inch) 
 
 Wt. of 
 wire (lb. 
 per foot) 
 
 Practical 
 equivalent 
 (inch) 
 
 Gage 
 
 Dia. of 
 wire 
 (inch) 
 
 wt. of 
 wire (lb. 
 per foot) 
 
 Practical 
 equivalent 
 (inch) 
 
 7/0 
 
 0.4900 
 
 0.6404 
 
 }/2 round 
 
 0 
 
 0.3065 
 
 0.2506 
 
 ^{q round 
 
 5/0 
 
 0.4305 
 
 0.4943 
 
 }{q round 
 
 3 
 
 0.2437 
 
 0.0466 
 
 round 
 
 3/0 
 
 0.3625 
 
 0.3505 
 
 % round 
 
 Fig. 38.— "Unit" frames. FtG. 39. — " Unit" spirals. 
 
 66e. Hennebique System. — One of the pioneers in concrete construction in 
 Europe is Mr. Hennebique, in France, and the system which still bears his name is shown in 
 Fig. 42. 
 
 66/. Pin-connected System. — Reinforcement in the pin-connected system con- 
 sists of bars made into a truss and ready for placing in the forms (see Fig. 43). 
 
 QQg. Luten Truss. — The Luten truss is shown in Fig. 44. The bars are rigidly 
 locked together to form the truss by a clamp, with a wedge that is self-locking when driven 
 home. The truss is especially adapted to highway culverts and bridges and is put out by the 
 National Concrete Co. 
 
 66/i. Xpantruss System. — The truss by this name is shown in Fig. 45, and is 
 applicable chiefly to beams, girders, and heavy slabs. This system is patented by The Con- 
 solidated Expanded Metal Cos. 
 
 66i. Shop Fabricated Reinforcement System. — In this system (Fig. 46) manu- 
 factured by the Lackawanna Steel Co., Lackawanna, N. Y., the standard bar is a troughed 
 
Sec. l-66^] 
 
 Fig. 40. — Corr-bar girder frame. 
 
 Fig. 41. — Corrugated Bar Co.'s spirals. 
 
 Fig. 42. — Hennebique system. 
 
 Fig. 43. — Pin-connected system. 
 
 Fig. 44. — Luten truss. 
 
62 
 
 CONCRETE ENGINEERS' HANDBOOK 
 
 [Sec. 1-m 
 
 section and the auxiliary reinforcing members, such as diagonal tension members, tie rods for 
 columns, walls, etc., are flat bars {}i by ^{q in.) with knobs on each edge. Fabrication is 
 
 Fig. 45. — Xpantruss system. 
 
 effected by placing a portion of the auxiliary flat, properly bent, within the trough and with a 
 bulldozer or other pressure machine squeezing the wings of the main bar and also gripping the 
 knobs of the flat. The upper or troughed part of the main bar is a constant. Increased area is 
 
 Fig. 46. — Shop fabricated reinforcement system. 
 
 developed by making the section deeper as required. Tests have shown that the rivet grip, as 
 it is called, is greater than the strength of the auxiliary member. 
 
SECTION 2 
 
 GENERAL METHODS OF CONSTRUCTION 
 PROPORTIONING CONCRETE 
 
 1. Properties of Concrete Dependent upon Properties and Relative Proportions of Con- 
 stituent Materials. — Despite prevalence of careless measurement of materials and arbitrary 
 specifying of proportions for concrete, it does not follow, as many infer, that such procedures 
 are right, either in practice or in theory, or that they should be continued. On the contrary, 
 such procedures are wasteful and inefficient and to meet their effects, the allowable stresses on con- 
 crete have been fixed at a standard so low that actual failure or disintegration can result only 
 from flagrant abuses of these lax and undesirable methods. All consideration of possible per- 
 fection aside, it is known beyond question that proper proportioning of selected materials is a 
 prerequisite to success, for concrete is wholly dependent for its properties upon the properties 
 and proportions of its constituent materials, both severally and in combination. 
 
 2. Theory of Proportioning. — Considering the composite nature of concrete as revealed 
 by a fractured section (Fig. 1), it will be seen at a glance that this substance is a pudding of 
 large stone particles set in a mass, or ''matrix," of other substances, which matrix is a mortar 
 of cement and sand. If this matrix be magnified, it will be seen to 
 be similar to the gross section, with sand grains as large par- 
 ticles held in a matrix of more-or-less hydrated cement. Theo- 
 retically, the large stone particles and also the sand grains should 
 lie as closely together as possible, so that if all fragments (sand 
 included) had come from crushing a solid cube of stone, their 
 reassembly would approach the cube's original volume, density, 
 and stress resistance. The aim in proportioning, therefore, is 
 efficient recombination. 
 
 It is impossible in practice to obtain the close rearrange- 
 ' ment and interlocking of the particles that is, on all counts, 
 
 desirable. There must and do remain between them, as is ^ x i^-^v^^^xw^-oi-v/iic 
 
 visually evident, inter-particle spaces or voids. To fill spaces or structure of concrete, 
 
 voids between large particles, finer materials of like nature and 
 
 origin are chosen ; and on theoretical grounds, it should be possible by measuring these inter- 
 particle spaces — as, for instance, by pouring measured quantities of water into a given con- 
 tainer filled with stone particles until the container can hold no more — to add a quantity of 
 sand to the stone of a volume equivalent to the added volume of water, so as to render the 
 measure almost solidly full of sand and stone particles, or rather, of stone particles since sand 
 is itself disintegrated rock. 
 
 It is to be expected, however, as is evidenced by visual magnification, that between the 
 sand grains must lie other and smaller inter-particle spaces in great number; and on the theo- 
 retical basis of proportioning, these should be filled by cement, so that the entire measure 
 might be solidly full of a composite which would closely approximate natural stone in texture, 
 density, and strength. But this ideal is not attained, partly because in this hypothesis the water 
 necessary for reaction with cement is not allotted space, and partly because no allowance is 
 made for the necessary surface coating of sand and stone by cement and water, with consequent 
 dispersion of particles. The water is assumed either to lie in inter-particle spaces not filled 
 
 63 
 
64 
 
 CONCRETE ENGINEERS' HANDBOOK 
 
 [Sec. 2-3 
 
 by cement or in inter-particle spaces in the cement itself, or else to be negligible in quantity and 
 volume. 
 
 The foregoing are the assumptions on which the void theory of proportioning is based. 
 There is hardly one of these assumptions that does not rely on false premises, so that the whole 
 void theory must be and is found at variance with practice, particularly when subjected to 
 comparison with results obtained by rough field procedure. Yet the idea behind the theory 
 is right, for it suggests by inference that density of natural stone is the ideal to be striven for 
 and that it may be obtained: (1) by using a maximum quantity of natural stone in fragments 
 large enough to possess unimpaired its inherent properties; and (2) by filling in the inter- 
 fragment spaces with maximum quantities of like mineral materials. The error lies not so 
 much in the standard thus chosen as in neglect to give proper consideration to the individual 
 and combinative properties of the several substances included in concrete, not the least impor- 
 tant of which is water, both as a space occupier as well as in its chemical and physical actions 
 with cement and with inert aggregate.^ It is always to be remembered that concrete is not 
 concrete without water; that its proportioning is equally important with the proportioning 
 of cement, stone, or sand; and that it affects by its quantity the proportions of the other 
 ingredients that may be placed in any given volume. 
 
 3. The Strength Elements of Concrete. — Any concrete will have as an upper limit of 
 stress resistance the properties of the most resistant of the materials entering into it. These 
 materials are usually stone and sand, with some strength preference in favor of sand, as most 
 sand particles are individual crystal units without cementing substance between them, while 
 natural stone is an aggregation of similar particles, built up in one way or another. Since 
 natural stone is formed under the most advantageous conditions as to arrangement of particles, 
 pressure, etc., and since in its composite nature, it is closely analagous to concrete, it may be 
 taken as the ultimate ideal in artificial stone made by the admixture of sand, stone, and water, 
 with Portland cement. Furthermore, in natural stone as in concrete, the weakest element 
 is the cementing substance which lies between the inert grains, for in each material this is 
 alterable by various agents and is at the same time of less inherent strength than the mineral 
 particles which it unites. 
 
 4. Proportioning for High-strength Concretes. — From the above, it follows that the 
 greater the proportion of mineral grains in any stone, brought about through compacting and 
 the exclusion of all but a film of cementing material (as in a very compact sandstone), the 
 higher will be its strength. In the same way, in artificial concretes, the greater the proportion 
 of natural stone (or of analogous mineral matter) and the less the quantity of cement, consistent 
 with proper coating of the inert materials, the greater will be the resulting strength, for Portland 
 cement in combination with water is the weakest element of this cement-sand-stone combina- 
 tion. In this connection it should be further remembered, that while ground Portland-cement 
 clinker (commercial powdered cement) is extremely hard and of great inherent strength, the 
 same substance in combination with water results in an entirely different product of different 
 chemical nature and composition, having relatively low strength. It is, therefore, actually 
 true that down to a certain limit, the less cement there is in any concrete the more enduring 
 and, at the same time, the cheaper will be that concrete. It becomes, then, of vital importance 
 not only to so select inert materials that they shall by their properties be able to endow 
 the concrete with high strength, but also to so choose their proportions that they shall be a 
 quantitative maximum in the mixture, the cement functioning in minimum quantities, as 
 an adhesive surface coverer and as a void filler. 
 
 6. Weakness Due to Poor Proportioning. — By reason of the remarkable properties of 
 Portland cement, carelessness in proportioning concrete has become tacitly accepted, if not 
 permitted and sanctioned practice. Fairly good results have been obtained in spite of gross 
 carelessness; and this has brought about a belief in the minds of perhaps a majority of con- 
 structors, that practically any materials in any proportions in combination with any cement 
 
 1 See chapter on "Water" in Sect. 1. 
 
Sec. 2-6] 
 
 GENERAL METHODS OF CONSTRUCTION 
 
 .65 
 
 will give the desired results. It is also unfortunately true, that in first results and appearance, 
 Portland-cement concrete, even when of inferior materials and in improper proportions, appears 
 equal to that properly made; but the rapidly increasing number of defective constructions 
 which are being brought to hght with the passage of time is bringing about an awakening 
 in regard to the causes of such disintegrations. Such apparently easy successes have given 
 rise to the so-called "arbitrary" proportions in widespread present use, but in probably a 
 majority of cases, not the least important of the causes which result in ultimate failure is the 
 use of these same arbitrary, "practical" proportions, which actually are not practical in one 
 case out of ten, so far as results achieved in making a good product are concerned. 
 I 6. Unit of Proportioning. — In specifying concrete or mortar mixtures a unit quantity of 
 1 cement is taken as a base. The cubic foot is the usual unit of quantity. On large work where 
 overhead bins and measuring-hoppers are provided, the cubic yard may be chosen. In general 
 the assumption is made that 1 sack of cement weighs 94 lb.; that it is 3^1 bbl.; and that it 
 I occupies 1 cu. ft.^ 
 
 7. Arbitrary Proportions. — Despite their inadvisabihty and incorrectness, arbitrary 
 proportions cannot be ignored. With the cubic foot of cement as a unit, these proportions 
 are commonly described as 1 : 2 : 4 or 1 : 3 : 6 or 1 : 4 : 8 or some other easily remembered ratio, 
 expressed in terms of volumes of sand and stone respectively, to the unit volume of cement. 
 
 Evil as is such arbitrary choosing of proportions, when such proportions are rendered 
 still further indefinite by inaccurate measurement of materials, the composition of the resulting 
 concrete is indefinite and uncertain to an extent that gives a new respect for the abilities of 
 Portland cement. It has been shown repeatedly on test, and confirmed by examination of 
 the resulting concrete, that measurement of materials in the usual manner by wheelbarrows or 
 shovelfuls, may bring about variations of from 100 to 200% in actual proportion of material 
 delivered to the concrete batch. This is particularly true with regard to sand. Since sand 
 occupies at least one-third of the volume of any concrete, and as the density and stress resistance 
 of the mixture is so largely dependent on the quantity of sand present, the variations introduced 
 through change of grading attendant on supplies from different localities, or through change 
 in moisture content, will be seen to be very serious. It is all too true that in commercial work, 
 a supposedly 1:2:4 concrete is quite as likely to be 1 : 4 : 8 or 1 : 5 : 9 or even worse, or on the 
 other hand it may be 1 : 1 : 2, or other combinations in indefinite number, with a corresponding 
 increase of unreliability and cement cost. 
 
 It should always be remembered that each of the inert materials of concrete has surfaces 
 and voids peculiar to itself, and the combination of any two is peculiar to them alone. A change 
 in either material must, therefore, result in new relations, with change in the burden imposed 
 on the cement, which (in combination with water) must function both as an adhesive, as a 
 surface cover, and as a void filler. Regardless of the apparent sanction generally given to the 
 use of arbitrary proportions in making concrete and to the apparent expediting of work by 
 slap-dash measurement, this practice should be prohibited on all work above that of the most 
 ordinary grade. 
 
 H. C. Johnson in "What is a 1:2:4 Concrete ?"2 states that with maintenance of strict 
 proportions with different materials, the cement demanded will range from 100 to 130 bags — 
 a variation of 4.5% in a definite quantity of concrete. The table on page 66 summarizes the 
 results of his experiments. 
 
 8. Proportioning by Void Determinations. — For reasons given in Art. 2 proportioning 
 materials by void determinations is obsolescent practice, not to be more countenanced than 
 that of arbitrary proportions, which it resembles. The determination of voids may give a rough 
 indication as to the inter-particle spaces existing in any fine or coarse aggregate, but it does not 
 
 1 A number of authorities have approved the adoption of 3.8 cu. ft. of cement to the barrel. This value is 
 more nearly exact and gives 100 lb. of cement to the cubic foot or 0.95 cu. ft. per sack. One standard sack, however, 
 may be and usually is considered as 1 cu. ft. 
 
 2 Concrete and Constructional Engineerimj (London), Feb., 1915. 
 5 
 
CONCRETE ENGINEERS' HANDBOOK 
 
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Sec. 2-81 GENERAL METHODS OF CONSTRUCTION 67 
 
 afford a basis on which to proportion the materials. If a cubic foot of broken stone contains 
 40% of voids, the void basis of proportioning is based on the assumption that cu. ft. of sand 
 should be added in order to fill these spaces, i.e., to bring the mass up to approximate solidity 
 and that if this quantity of sand in turn contains 35% of voids, that ^^qo X Viq cu. ft. of 
 cement should be added (plus a certain arbitrary percentage for coating sand and stone sur- 
 faces) to give to the mass actual solidity. 
 
 The inaccuracy of this method is partially due to the variation in voids in sand under differ- 
 ent conditions, this variation being sometimes sufficient to make a difference of 30% in the 
 amount of cement required. The following table shows variation in voids due to difference 
 in moisture alone. The figures in this table are for sand in a loose condition and the differences 
 would be still greater if the dry sand had been shaken and tamped. It is evident that the 
 method of proportioning by voids is valueless unless the sand is in the same state of compact- 
 ness when mixed in concrete as it is when the void test is made. 
 
 Physical Characteristics of Concrete Aggregates^ 
 
 (For aggregates in loose condition) 
 
 
 Voids (aggre- 
 gate contain- 
 ing natural 
 moisture) 
 
 Voids 
 (aggre- 
 gate 
 dry, 
 
 % 
 
 Weight (lb. per 
 cu. ft., aggregate 
 containing natural 
 moisture) 
 
 Weight (lb. per 
 cu. ft., aggre- 
 gate dry) 
 
 % moisture 
 in moist 
 aggregate 
 
 Specific 
 grav- 
 ity of 
 stone 
 
 Average of 4 good concrete 
 
 53 
 
 43 
 
 82 
 
 95 
 
 5 
 
 2.65 
 
 sands. 
 
 
 
 
 
 
 
 Long Island washed gravel 
 
 
 36 
 
 
 106 
 
 
 2.65 
 
 graded % in. to 1^^ in. 
 
 
 
 
 
 
 
 Commercial ^-in. lime- 
 
 
 44 
 
 
 97 
 
 
 2.80 
 
 stone. 
 
 
 
 
 
 
 
 Commercial l3^-in. lime- 
 
 
 46 
 
 
 95 
 
 
 2.80 
 
 stone. 
 
 
 
 
 
 
 
 Trap rock, graded % in. to 
 
 
 44 
 
 
 103 
 
 
 2.95 
 
 l^in. 
 
 
 
 
 
 
 
 In practice, therefore, the proportions obtained by void determinations do not hold. As 
 soon as water is added to sand, stone, and cement, very different physical relations are effective 
 from those that previously existed. The particles of cement and the finer particles of sand are 
 necessarily dispersed by the water which coats and lies between them; the larger particles of 
 sand are dispersed by this thin mortar of cement, fine sand, and water; and the finer and larger 
 stones in turn are dispersed in a similar manner by a like combination of the finer materials. 
 As pointed out in Art. 2, this fact is made very evident by the examination of the fractured sur- 
 face of any concrete. No matter whether the surface examined is in the gross, showing large 
 particles of stone, or whether it is magnified to make visible the very fine particles of sand 
 and cement, the same dispersion will be found to obtain, offering visual evidence, confirmed 
 by test, as to the necessary inaccuracy of void determinations as a basis of proportioning. 
 
 This may be made evident by a simple illustration. Assume a vessel containing a given 
 number of small spheres of varying sizes which may be considered as so many sand and cement 
 grains. It is evident that if these spheres fill the measure, a certain quantity of water may be 
 added without disturbing their inter-relations. That is to say, it is assumed that these spheres 
 will remain in surface contact one with another even after the addition of water. If, however, 
 the same spheres be put into a larger vessel and an additional quantity of water added, filling 
 
 1 R. E. Goodwin in Concrete, Nov., 1915. 
 
68 
 
 CONCRETE ENGINEERS' HANDBOOK 
 
 [Sec. 2-9 
 
 this vessel, then, if the mixture is uniform (which is the condition assumed to exist in mortar 
 and in concrete), these spheres must be dispersed and be out of surface contact one with another. 
 This represents in exaggerated illustration, but not in exaggerated degree the conditions as 
 they exist in commercial concretes, dispersion therein being progressive from the finest particles 
 to the largest, each successive grade assisting in the dispersion of the size next larger, with 
 resultant increased demand for cement and corresponding weakening of the mass. 
 
 Rather than proportion strictly on the basis of voids, therefore, a better way is first to 
 grade the aggregates, both coarse and fine, by sieve analyses. In this way, the voids are taken 
 cognizance of, though in a different way. A combination of materials may then be made such 
 as to give a mixture containing these materials in greatest quantities. The proportions of 
 aggregates in this mixture having been determined, the amount of cement required will then 
 depend very largely upon the strength needed or the degree of imperviousness required of the 
 concrete. It can be approximately estimated by determining the percentage of voids in the 
 mass, but on account of the errors introduced through the establishment of new conditions by 
 the introduction of water, this latter assumption is not to be recommended unless checked on 
 actual mixtures. 
 
 9. Proportioning by Mechanical Analysis. — Although the foregoing recognizes the exist- 
 ence of inter-particle spaces or voids, it properly should be termed ''proportioning by mechanical 
 analysis." Such proportioning is recombination after analyses are made by passing representa- 
 tive samples of the inert materials through successive sizes of standard-mesh screens; noting 
 the quantity passing and the quantity retained on each screen; and plotting these as a curve, 
 with sizes of screen openings as abscissae and percentages of material passing as ordinates 
 (see Art. 32, Sect. 1). By this procedure a more or less regular curve will be obtained for 
 the sand and for the stone; and its variation from a predetermined curve, such as that of Wil- 
 liam F. Fuller,^ or that advocated by the Bureau of Standards at Washington,^ may be deter- 
 mined. The deficiencies of one material, therefore, can be balanced against the advantages of 
 another; and by proper combination of the two, as determined from this curve, a determina- 
 tion may be had as to the proper proportions of the several materials. A few trials will give a 
 very close approximation; and if the qualities of the several materials are maintained to sample 
 throughout the work, these proportions may be safely followed. It is probable, however, that 
 the character of the materials will change more or less throughout the job, so that it is usually 
 necessary to continuously check the several shipments of materials as they go into the work; 
 and if they vary seriously from the established standard, to alter the proportions of the concrete 
 in accordance with variations noted by repetitions of the processes before noted. 
 
 10. Proportioning by Maximum Density Tests. — A direct test which reproduces actual 
 conditions is always preferable to an indirect test based on assumptions subject to variation. 
 The result desired in proportioning concrete is a mixture of maximum density, and the most 
 direct means to this end is the testing of trial mixtures. These are best made with concrete, 
 or the coarse aggregate may be omitted and the mortar alone used. The latter method, how- 
 ever, is not always representative, as in this case the voids in the coarse aggregate must be 
 determined and the concrete so proportioned that the mortar will fill the voids in the gravel 
 or the stone, with a certain arbitrary excess, thus introducing an element of error. As a factor 
 of safety, the amount of mortar should exceed the voids in the gravel or stone about 10%. 
 Nevertheless, this method has some justification as the percentage of voids in coarse aggregate 
 is less variable than in sand, and also because an error in determining them has less effect on the 
 quantitj^ of cement used. 
 
 In proportioning by trial mixtures, definite quantities of the materials in proportions first 
 determined by mechanical analysis are mixed with a requisite quantity of water and are put in 
 a metal cylinder about 1 ft. long by about 1}4 in. in diameter, and tamped. The volume they 
 occupy is then determined by measuring from the top of the cylinder. When this has been 
 
 1 See "Concrete, Plain and Reinforced," by Taylor and Thompson. 
 - See Bull. 58, Bureau of Standards, Washington, D. C. 
 
Sec. 2-11] 
 
 GENERAL METHODS OF CONSTRUCTION 
 
 determined the mixture is removed from the cyhnder, the latter cleaned and a new mixture made 
 and tried out in the same way, with a slight variation of the proportions of sand, stone and ce- 
 ment, but with the quantity of water constant. Very soon a mixture which will give the least 
 volume for any given quantity of materials will be found and this mixture will give the densest, 
 most impervious and strongest concrete with those materials. 
 
 This is known as a mixhire of maximum density, but it is apt to be inconvenient for use 
 in ordinary concrete work, inasmuch as it contains so large a proportion of stone that it is ex- 
 tremely harsh and difficult to work. To make it freer-working, more sand is generally added; 
 and, although something of strength and density is sacrificed by so doing, the advantages of 
 easy working and increased compactness in forms probably compensates for disadvantages 
 arising directly from any impropriety of proportions. 
 
 Proportioning by maximum density is very readily applied in the field, all that is necessary 
 being an iron pail and a pair of scales. Proportions can be determined for the concrete, without 
 the use of a laboratory apparatus or any unusual equipment, by weighing out the materials, 
 having due care that the sand is reasonably dry so that too great volumetric errors may not be 
 introduced, and then mixing them in the pail until a mixture of maximum density is obtained. 
 All tests are useless, however, unless the determined proportions apply to every batch mixed 
 and placed in forms. 
 
 11. Checking Materials on the Job J — When the materials used on the job are from the 
 same sources as those tested and from which tests the proportions to be used were determined, 
 it is a simple matter to check up their qualities. Sand and stone 
 from the same source do not vary much in quality, except in so far 
 as quality is influenced by size of particles. Having once established 
 by test the suitability of sand and stone for any grade of concrete 
 and having determined the proper proportions in which to use 
 them to attain a certain desired result, it is only necessary there- 
 after to see that the size, grading, and proportions of these ma- 
 terials are reasonably constant to insure uniform quality of con- 
 crete. Such a check on size and grading should be had on each 
 and every shipment of material and is easily obtained with a small 
 set of sieves, or in the case of sand, which is by far the more impor- 
 tant material, by means of a self-contained sand tester (see Fig. 2). 
 
 The regular and systematic testing of the size of the aggregates 
 gives data which will permit the engineer to tell without further 
 tests, whether the aggregates will produce a better or poorer con- 
 crete than that produced by the original or standard sample. This 
 fact is based on the well-established principle that, other things 
 being equal, the aggregate whose granulometric-analysis curve 
 most nearly approaches the line of maximum density will pro- 
 duce the best concrete. This makes it possible to determine with 
 
 reasonable certainty which of two sands of the same kind and from the same source, but 
 differing only in fineness, will make the better concrete. 
 
 To illustrate : Concrete is to be placed in a certain locali ty. There are to be heavy machin- 
 ery foundations and thick building foundation walls and footings below grade, with rein- 
 forced superstructure. The engineer in charge secures samples of the available concrete aggre- 
 gates, both fine and coarse, and sends them to the laboratory for test. The tests show that 
 although the best available sand has a strength in 1 : 3 mortar only 70% of that of standard 
 Ottawa sand, yet mixed in the proportions of 1 : 1^^ : 3^^ with the cement and coarse aggregates 
 to be used, the resulting concrete has a compressive strength of 2600 lb. per sq. in. at 28 days. 
 Other proportions give higher and lower strengths, depending on their richness, but as the 
 
 1 Chapman and Johnson: Eng. Rec, June 12, 19, 26, 1915. 
 2Kolesch & Co., Fulton St., New York. 
 
 Fig. 2. — Universal Sand 
 Tester — portable instrument for 
 making mechanical analyses of 
 sands.- 
 
70 
 
 CONCRETE ENGINEERS' HANDBOOK 
 
 [Sec. 2-12 
 
 design of the structure requires concrete having an ultimate strength of 2500 lb. per sq. in. 
 the 1:1^^: 3^^ proportion is used. For the foundations and footings, the designs being based 
 on an ultimate strength of 1500 lb. per sq. in. in the concrete, the proportion of 1 : 2}^ : A}^, 
 which gave in the test a compressive strength of 1550 lb. per sq. in., is chosen. Under the 
 present standard method of specifying sand, this particular sand could not have been used in 
 concrete. 
 
 Among the tests advantageously made in the field on a sand are granulometric-analysis 
 charts made with the sand tester (Fig. 2). This sand tester has five screens having 6, 10, 
 20, 35, and 65 meshes per in. Each screen in succession has openings one-half the width of 
 the openings in the preceding screen. The charts are averaged and a special guide chart is 
 prepared for the use of the inspector on the job. In making up this guide chart a permissible 
 variation of about 2.5% each way from the mean of the tests made on the sample, is allowed. 
 A copy of this chart is sent to the job and a copy kept in the office files. 
 
 As the sand arrives on the job the inspector, or some one designated by the superintendent, 
 makes tests with the sand tester and compares the resulting chart with the guide chart. If the 
 results show greater variation than is permissible — particularly if they show the sand to be 
 finer than shown on the guide chart — then the matter is taken up with the one who supplies 
 the sand. 
 
 By this method the quality of the aggregates is recognized and provided for in the selection 
 of proportions for the concrete, and enough cement is used to produce the desired quality. 
 In this way the uncertainty which is attendant upon separately testing each of the three 
 materials, and predicting therefrom the quality of the concrete resulting from their combina- 
 tion, is eliminated. The time required for testing the combination is no greater than that 
 required for testing any one of the materials. ^ 
 
 12. Proportions and the Measurement of Materials. — Proportioning always involves 
 measurement of materials. Even with the most exact determinations of proportions, if measure- 
 ment of materials in the field is inexact and variable, concrete so made will necessarily be a 
 substance of extremely uncertain value. Furthermore, so long as there is prevalent a tendency 
 to use excess water, even the most exact measurement of stone, sand, and cement may be 
 nullified. Those who seek the best results must use the utmost care not only in the initial 
 determination of the proper proportions, but also in the measurement of the quantities of 
 each material employed, and in checking the qualities of the materials that come on the 
 work. There will be a proportionate improvement in the general quality of concrete as at- 
 tention is more generally paid to these matters. 
 
 13. Proportioning Bank-run Gravel. — It is often questioned whether or not a natural 
 mixture of sand and gravel as taken from the bank is suitable for concrete work. Inherently 
 there should be no objection to this material, provided it is not contaminated by impurities, 
 but the proportions of the several grades of sand and gravel in any bank are extremely uncertain 
 and variable. Taking bank-run gravel and mixing it with cement in the proportions of 1 part 
 of cement to 6 of gravel is not in any sense equivalent to 1 part of cement, 2 of sand and 4 of 
 screened gravel, or to 1 of cement, l}i of sand or 4}^ of screened gravel, or any other equivalent 
 summation. 
 
 Gravel of itself, if of proper quality, makes a most excellent concrete, equal to that pro- 
 duced by the use of crushed stone. Sand in bank-run gravel is often of excellent qualities equal 
 in every way to that taken from large deposits of exclusively fine material. However, if bank- 
 run gravel is to be used, the relative proportions of sand and gravel must first be determined 
 by a series of tests on representative samples in sufficient number so that the average of the bank 
 may be determined with fair accuracy. These materials may then be combined with cement, 
 preferably by proportioning for maximum density. After proportions are determined in this 
 way, it may be found possible to use a bank-run gravel as it comes. On the other hand, it may 
 
 1 See C. M. Chapman: "Specifications for Concrete Aggregates." Procs. Am. See. Test. Mat., 1916. 
 
Sec. 2-14] 
 
 GENERAL METHODS OF CONSTRUCTION 
 
 71 
 
 be found necessary in some cases to screen out the finer materials from the coarse and recomhinc^ 
 them in proper proportions, the defining limits between gravel, sand, and other grades of 
 materials being as stated in Art. 16, Sect. 1. 
 
 This method of screening and recombination is always cumbersome and except on ex- 
 perimental or very large scales, impossible of putting into effective practice. In other cases, 
 after existing proportions of the several grades have been determined as above, a measured 
 quantity of sand, or of gravel, or of broken stone of a size or sizes lacking in the bank may be 
 added to the pit-run gravel, the quantities to be added having been determined by test. In 
 this way, the deficiencies of pit-run can be overcome by addition of other substances readily at 
 hand, or of certain of the screened-out portions of the bank itself. Only in these ways, how- 
 ever, can certainty as to proportions be secured; and it must further be borne in mind that 
 frequent tests should be made during progress of the work to insure uniformity. 
 
 Furthermore, great care should be exercised to make certain that silt in detrimental 
 quantities, or loam, are not present in bank-run gravel. Sand pits are less likely to contain 
 injurious quantities of silty materials than are gravel pits, by reason of the latter being the 
 bottom of an old stream bed or a like deposit, with all materials held therein, just as they chanced 
 to be when the waters receded. Furthermore, natural disintegration at the surface, with organic 
 additions, affects the quality of the material. Stripping away top layers is too often omitted. 
 
 14. Proportioning Crusher-run Stone. — The statements made with respect to pit-run 
 gravel apply in lesser degree to crusher-run stone. Different stones crush in different ways 
 with consequent variation in the character and quantity of the fine material incident to the 
 process. The softer stones give a larger yield of fine materials than the harder ones; and often 
 much of the very fine material is of a character unsuited to use in concrete. This applies es- 
 pecially to limestone, inasmuch as limestone has a flaky fracture in its fine particles, making the 
 particles very friable and rendering the adhesion of cement difficult, so that concrete made with 
 such material is pervious and of low strength. 
 
 Furthermore, where there are excessive fines in crushed rock materials, some of these fines 
 are merely impalpable dust. It is almost impossible for cement to properly coat particles of 
 this size, as the dust particles then approach in fineness the cement particles. With this very 
 fine material in large proportions, a considerable source of weakness is thus introduced into 
 concrete, inasmuch as these materials cannot be covered by cement. 
 
 The use of crusher-run materials of undetermined size and grading, therefore, introduces 
 an element of uncertainty in the making of concrete, which should not be permitted. The 
 course to pursue is similar to that indicated for the use of bank-run gravel — i.e., adequate 
 samples of the crushed materials should be taken; the proportions and size of fine and coarse 
 materials determined by mechanical analyses; the mixture of maximum density obtained; and 
 the proportions noted of each of the several materials required to produce this mixture. Then, 
 either by screening or by diluting the crusher run with screened materials or with extraneous 
 materials, concrete of proper quality can be more nearly assured. 
 
 15. Proportioning Blast-furnace Slag and Cinders. — It is difficult to proportion for 
 maximum density when blast-furnace slag or cinders are used as aggregate. This difficulty 
 arises largely because of the porosity of these two materials, cinders being especially absorptive 
 of water. A rough approximation as to proportions can be had with blast-furnace slag but with 
 cinders it is probably better not to attempt to secure accurate proportioning, inasmuch as the 
 use of cinder concretes is so restricted and their strength and impermeability are so low, as to 
 render any increase obtainable by refinement of methods of secondary importance. Further 
 reference as to the quality of these two materials will also be found in the chapter on "Aggre- 
 gates" in Sect. 1. 
 
 16. Proportioning Water.— Last but not least is the question of proportioning water in 
 concrete. This is often given so Httle thought as to make it considered of either minor or no 
 importance but it can be authoritatively stated that the strength of any concrete mixture is as 
 
72 
 
 CONCRETE ENGINEERS' HANDBOOK 
 
 [Sec. 2-17 
 
 dependent upon the proportion of water contained as it is upon the proportions of any or all of 
 the other materials. 
 
 Unfortunately, little is definitely known at the present time as to the proper proportions 
 of water. It is known, however, that the quantity depends both upon the demands of the 
 cement and also upon the character of aggregate employed, upon the surfaces to be covered, and 
 the voids to be filled. Research has been recently directed to these lines with highly important 
 results. 
 
 17. Success in Proportioning. — For success in proportioning, not only must the original 
 test determinations be right, and the specifications provide proper authority for their enforce- 
 ment, but these powers must be exercised and a rigid compliance compelled. Otherwise there 
 is no use in tests, and specifications are empty words. 
 
 MIXING, TRANSPORTING, AND PLACING CONCRETE 
 
 18. Mixing Concrete. — Although with careful superintendence hand-mixing will give good 
 results, machine-mixed concrete is usually of more uniform quality than that mixed by hand, 
 and is less expensive — except, of course, where the quantity of concrete is so small as to pro- 
 hibit the expense of purchasing or renting a mixer. The engineer should preferably reserve the 
 right to permit hand-mixing if practically unavoidable, but this method of mixing should be 
 resorted to only when machinery is unobtainable or where it is necessary to start work on a 
 large job before the machinery has arrived. 
 
 Some contractors mix the materials dry until a uniform color is secured and then add the 
 water. Others put the material and the water into the mixer at once. Either way can pro- 
 duce good results, except in hand-mixing, where the mixing of the cement and the sand in the 
 dry state is the general and better practice. 
 
 The strength of concrete is very largely dependent upon the thoroughness of mixing, and 
 much care is needed in this part of the work. No matter how suitable for the purpose the 
 materials and proportions of the same may be, insufficient mixing will result in inferior con- 
 crete. Time of mixing is treated in Art. 236, Sect. 3, and in Art. 12, Sect. 5. 
 
 The greatest care should also be exercised to make sure that the specified amounts of the 
 materials go into each batch of concrete. For measuring concrete aggregates, it is not good 
 practice to use the common form of contractor's wheelbarrow because the loads vary consider- 
 ably with the variation in the heaping of the barrow. Special barrows constructed with sides 
 nearly vertical can be obtained which will give the required amount when level full. The 
 proper measuring of materials is discussed in Art. 23e, Sect. 3. 
 
 19. Amount of Water to be Used in Mixing Concrete. — Sufficient water should be used 
 in mixing to obtain a concrete of sufficiently mushy consistency to be readily puddled. In re- 
 inforced work the amount of water should be such as to make the mixed concrete into a flowing 
 paste that will flow readily around the reinforcing steel and require only light tamping or pud- 
 dling to bring the mass to a homogeneous condition. A slight excess of water is preferable to 
 not enough, but there should not be any appreciable quantity of free water present. Concrete 
 is mixed with an excess of water if pools are immediately formed on top of the concrete when 
 deposited in the forms. Although the quantity of water needed in different batches will vary 
 occasionally because of the condition of the materials, the amount to use can be regulated best 
 by measurement. A tank with a float fastened to an indicator on the outside is easily con- 
 structed in connection with a concrete mixer. The effect of consistency on strength of concrete 
 is discussed in Art. 9, Sect. 5. For harmful effects from the use of excess water, see chapter on 
 "Water" in Sect. 1. 
 
 The general types of mixers are described in Art. 22, Sect. 3. 
 
 20. Transporting Concrete. — The transportation of concrete is not only an engineering 
 problem, often of first magnitude, but as a physical operation it is of prime importance in its 
 effect on the qualities of material in the manufacture of which transportation and transporta- 
 
Sec. 2-21] 
 
 GENERAL METHODS OF CONSTRUCTION 
 
 73 
 
 tion equipment are an incident. Briefly, the transportation system nmst be such: (1) that the 
 time interval elapsed between reception of concrete and its deUvery to forms will not cause it 
 ■ to dry, or to take initial set; (2) that the system shall be tight, so that more fluid portions may 
 not be lost in transit; (3) that the mode of transit shall not promote separation of ingredients; 
 (4) that the delivery shall be approximately continuous, so that mixtures of varying composition 
 may not be caused by stoppage and settling; (5) that it shall be efficient, rapid and economical. 
 In this summary of principles, the order of importance is such as to emphasize quality of pro- 
 duct delivered, as well as cheapness. 
 
 The varied and various appliances for the delivery of mixed concrete to forms are dis- 
 cussed and illustrated in Sect, 3 on ''Construction Plant." To each individual need must be 
 applied such means as careful analysis and study indicate, so correlated and systematized 
 that the ends desired will best be served. In the proper selection of transportation plant, per- 
 sonal experience and judgment enter as factors of such importance that, on large operations in 
 particular, profit or loss may depend wholly on them. In default of these, a safe rule is to study 
 methods and equipment used on operations of like character, either by first-hand inspection 
 or in printed reports; to supplement information so obtained by advice from those who have 
 had direct experience; and to so adjust and modify the ways and means indicated by the fore- 
 going as to suit them to the needs of a particular situation. 
 
 21. Depositing Concrete in Forms. — Responsibility for the character and quality of con- 
 crete does not end with its arrival at the forms. Depositing, or placement in forms is also an 
 operation of prime importance and its conduct is governed by elementary principles which are 
 similar to those that govern the transporting of concrete. These principles, directed toward 
 securing quality with economy, are: (1) that the concrete shall be continuously and evenly 
 placed in forms; (2) that it shall not be deposited continuously in one spot, with lateral flow 
 and (in wet concretes) gravity separation of lighter, more fluid portions from those that are 
 heavier; (3) that it shall not be deposited in forms in a manner tending to promote dissociation 
 or segregation of the component materials;^ (4) that so far as possible, forms shall be continu- 
 ously filled without stoppage, to prevent laitance, or stoppage planes; (5) that before new concrete 
 is deposited on concrete which has set, special precautions shall be taken to secure union between 
 the two; (6) that concrete shall be so deposited as to minimize the entraining of air; (7) that con- 
 crete shall be joggled in the forms or that forms shall be tapped on the outside after fiUing, suffi- 
 ciently to expel a considerable portion of entrained air; (8) that puddling and tamping shall 
 be done sufficient to bring about close filling of forms, and close contact with reinforcement; 
 (9) that larger aggregate shall be spaded away from forms at the concrete rises, permitting a 
 dense mortar coat and smooth finish at the exterior surface of the casting; (10) that uncombined 
 concrete shall not be deposited through water; (11) that concrete remixed or retempered after 
 initial set shall not be deposited in forms; and (12) that no concrete shall be deposited in cold 
 or very hot weather unless special and adequate precautions are taken. 
 
 22. Continuous and Even Depositing in Forms. — The temptation is very great to locahzc 
 delivery of concrete at one point, with reliance on gravity or hoeing for distribution to other 
 parts of a form. By such indulgence, one set-up of spout or barrow runways only is needed; 
 and by adding excess water, the form gets filled with less labor than where movement of the 
 spout, or movable or multiplication of runways are required. There are, of course, forms of 
 such section and dimensions that localized delivery is both permissible and advisable, but where 
 the form is long and high, localized delivery brings about a stratification or banding that not 
 only mars the appearance of the wall, but also provides fault planes along which seepage and 
 disintegration may proceed. 
 
 Furthermore, bearing in mind the harsh nature and heavy weight of concrete, the difficulty 
 of manual spreading in forms of concrete dumped on one spot creates a tendency to the use of 
 freer-flowing mixtures, and the ease with which a certain degree of flow may be brought about 
 
 1 See N. C. Johnson: Ena. Rec, Dec. 4, 11 :ind IS, lOlo. 
 
74 
 
 CONCRETE ENGINEERS' HANDBOOK 
 
 [Sec. 2-23 
 
 by the addition of water gives rise to the use of water in excess quantities in an attempt to 
 further accelerate the placing operations. (The consequences of such additions are treated at 
 length in chapter on "Water" in Sect. 1.) With such sloppy, or overwet concretes, but little 
 imagination is needed to conceive what actually happens in the forms. The more fluid por- 
 tions flowing off from the delivery mound carry with them much of the cement, together with 
 the lighter portions of sand, and fill the lower unoccupied parts of the form, there to solidify 
 
 Fig, 3. — "Soup" in forms. Fig. 4. — Loose stone section — the top of the 
 
 depositing heap. 
 
 in a chalky mass of "laitance" in which is embedded much cement needed by the stripped ag- 
 gregate left higher up. If materials are subsequently dumped into this "soup" before it sets, 
 segregations result by reason of the several materials settling through this fluid in the order of 
 their gravity. 
 
 Cause and effect are shown in Figs. 3 and 4. The cause in Fig. 3 is a knee-deep puddle of 
 these light materials. The effect, in Fig. 4, is a reservoir wall section almost devoid of cement 
 
 and sand — the top of the heap — while adjacent to it is a 
 lower section that can be chopped like chalk. ^ 
 
 Knowing the procedures to be avoided, substitute pro- 
 cedures suited to individual needs may be evolved. Local- 
 ized delivery brings a chain of evil consequences. Distributed 
 delivery avoids these, at an expense only slightly greater. 
 The gain in quality, endurance and value is worth the differ- 
 ence in first cost. 
 
 23. Continuous Depositing to Avoid Stoppage Planes. 
 — Even in concretes mixed only to a plastic consistency, 
 there is tendency for a scum of light, chalky material C'lait- 
 ance") to rise. The greater the quantity of water, the 
 thicker this deposit, which also is aggravated by silty sand 
 or dusty stone. Such a deposit at the top of a foundation 
 foundation slab. block is shown in Fig. 5. The scrolls were traced by a lath, 
 
 the depth of deposit being about in. and the thickness of 
 block about 4 ft. Concrete was subsequently deposited directly on this layer, as it is in thousands 
 of other instances daily, but in all of them, this will remain as a plane of weakness, ready to yield 
 when stress of proper character is imposed. Visual evidence of such yielding is furnished by 
 seepage of water and disintegrations starting at like planes in concretes on every hand. 
 
 1 See N. C. Johnson: Eng. Rec, Dec. 30, 1916. 
 
 D. A. Abrams: Concrete, April, 1917. Proc. Am. W. Wks. Assoc., 1916. 
 Carl Gaylor: Proc. Am. Soc. C. E., April, 1917, p. 669. 
 
Sec. 2-24] 
 
 GENERAL METHODS OF CONSTRUCTION 
 
 75 
 
 24. Bonding Set and New Concrete.— The foregoing is closely related to the problem of 
 bonding new and old concrete or, more properly, set concrete and concrete subsequently cast 
 upon it. Recognizing that at least on the majority of concretes, a top film or deposit of lait- 
 ance exists; that this deposit is loose in texture and non-coherent; and that a portion of it is 
 hydrolized cement, it is not to be expected that concrete subsequently placed in contact with it 
 shall adhere. It is known and recognized that a dust film on stone or gravel will prevent adhe- 
 sion of cement and it must no less be expected that a like film on solid concrete, often multi- 
 plied many times in thickness, will have like effect. Other and more complicated conditions 
 also affect the procurement of bond, but those above given are of themselves sufficient to ac- 
 
 ; count for the failure of many attempts (see Art. 50, Sect. 1). 
 
 ■ A first essential, therefore, in procuring bond is to remove this separating laitance film, 
 whether the set concrete is hours old, or years old. It is best to remove at least in. and pos- 
 sibly it may be necessary to remove several inches before clean, sound concrete and aggregates 
 are exposed. This surface should then be well washed and preferably soaked with clean water, 
 all loose material being removed. A wash of rich neat grout well scrubbed in with clean brushes 
 will provide a good bedment; and before this has set or dried, the new concrete should be depos- 
 ited, a first thin layer being tamped into place, followed by the full deposition. The foregoing 
 gives better guaranty of success than methods usually followed, but it should be borne in mind 
 that drying out of the fresh concrete surface, or drying or setting of the cement wash previous 
 to applying and ramming the first layer of concrete, or failure to deposit the remainder of the 
 concrete before this latter has taken set, will each be sufficient to cause failure to bond, as each 
 can and will duplicate in greater or less degree the separating film which first caused trouble. 
 
 Bonding fluids and compounds are marketed under various trade names, but these cannot 
 be successful unless conditions suitable for bond, as outlined above, are first established. 
 Hydrochloric acid is advocated by some as a wash preparatory to bonding, but the amount that 
 would be required if unassisted by picks or chisels in removing the usual laitance coat to a 
 sufficient depth makes its use prohibitive, both in cost and in time and labor required. Its use, 
 even when considerable effort is made to wash it away after use, is not to be recommended, as 
 concrete by its porosity, is capable of absorbing harmful quantities. 
 
 25. Removal of Entrained Air. — The customary mixing and depositing processes entrain 
 quantities of air. Even when the volumetric air content of a concrete appears low, a con- 
 siderable portion of the aggregate may actually be isolated by air, with little or no attachment 
 to cement.^ It is doubtful if the weaknesses produced in concrete by the occlusion of air are 
 appreciated. The evils of existing practices in this particular are to be deplored. In particular, 
 spouting unconfined from a height, or dumping from barrows in like manner probably do maxi- 
 mum damage in this particular. The present type of mixers work further evil in this regard. 
 But since many present fixed practices and equipment entail the occlusion of air, with no like- 
 lihood of an immediate change, the removal of as much as possible is logical progress. 
 
 To this end, vibrating rammers applied to the plastic concrete, or air hammers rapping the 
 outside of forms, or even sledge or maul blows^ have been used with good effect. In concrete- 
 products plants, vibrated molds have been used to obtain superior density; and in road work, 
 vibration by motor, appHed to mats on the fresh-laid concrete are said to produce superior 
 wearing qualities. ^ Certainly if the introduction of an objectionable impurity in a structural 
 material cannot be prevented, but its removal can be later effected, it is the part of constructive 
 engineering to overcome the undesirable effects while seeking to remove the cause. 
 
 26. Spading, Puddling and Tamping.— Forms should be closely filled, and, so far as pos- 
 sible, close contacting of form surfaces with smooth, plastic material should be brought about. 
 Since large aggregates tend to bridge over, or jam, leaving unsightly surface pockets, they should, 
 
 1 See N. C. Johnson: Eng. Rec, Jan. 23, 1915. 
 
 C. B. McCullough: Concrete, April, 1917. 
 
 2 See H. S. Carpenter; Eng. Rec, March 31, 1917. 
 
 8 The Vibrolithio Pavement of R. S. Stubbs Co., Austin, Tex. 
 
76 
 
 CONCRETE ENGINEERS' HANDBOOK 
 
 [Sec. 2-27 
 
 as the form is filled, be spaded back from form surfaces so that a dense, smooth 
 mortar may lie at exposed surfaces. Furthermore, since it is essential for structural 
 strength and for preservation of steel that the embedment of reinforcement be adequate with 
 close contacting of mortar, puddling of concrete should be progressively carried on as forms are 
 filled. 
 
 The temptation to use excessively wet concretes to lessen labor in the two foregoing opera- 
 tions is prevalent. For intricate reinforcement, a free-flowing concrete must be used, but it 
 is better to obtain the requisite flow by sufficient mixing, by the use of finer ballast and by pud- 
 dling than by indulging in excess water, which so generally defeats the intent of its use. 
 
 27. Depositing Concrete Through Water. — Care should be taken in depositing concrete 
 under water that it is not deposited through water, unless confined. 
 
 Underwater concretes are usually deposited by means of a tremie — a tube of about 1 ft. 
 diameter at the top, slightly flaring at the bottom and at the start of a length sufficient to reach 
 to the bottom. As deposition proceeds, the delivery end may be raised, but not out of the soft 
 deposited concrete, else water will enter, causing washing of concrete subsequently deposited. 
 The tremie must be kept full of concrete at all times; and deposition is assisted by moving the 
 bottom of the pipe slowly about, permitting gradual discharge. If the charge is lost, and the 
 tremie becomes filled with water, it is wise to add extra cement to the next charge, in order to 
 compensate for that which will be lost through washing away. Necessarily, a tremie is heavy, 
 so that scow, derrick or other handling arrangements must be provided. Care also must be 
 exercised in order that waves from passing boats may not lift the tremie as well as the scow, 
 causing loss of charge. 
 
 Underwater buckets, which are substantially boxes with bottom-dumping doors, have been 
 used in some underwater concreting, but their use is more costly than that of tremies and 
 possibly less satisfactory. Tilting buckets are not suited to underwater work, inasmuch as 
 their dumping subjects the concrete to washing. 
 
 Depositing in cloth bags of greater or less size to hold together the mixed concrete in pass- 
 ing through the water, has been successfully accomphshed.^ Paper bags are less successful 
 than are those of jute or burlap. The adhesion of successive bags is dependent upon trans- 
 fusion between and saturation of the bags with dissolved cementitious products, but in view of 
 the great mass in which the concrete is used in such operations, and its gravity functioning, lack 
 of strength at joining planes is of little moment. 
 
 28. Remixed and Retempered Concrete. — It is erring on the side of safety to reject all 
 concrete or mortar which has taken pronounced set, whether initial or final, or which requires 
 the addition of water and reworking to have requisite plasticity. The exact actions which take 
 place during initial set are not precisely known, but it is probable that in this process is begun an 
 interlacing crystallization which is later augmented by other crystallizations and depositions 
 of colloidal (amorphous or non-crystalline) material in the processes of final set and the sub- 
 sequent hardening. But whatever the exact process, it is known that retempering and re- 
 working of Portland-cement mixtures after initial set is decidedly disadvantageous at best, 
 resulting in a loose, unresistant product of inferior strength and coherence. This practice, 
 therefore, is to be avoided; and the operations of transporting and placing should never be of 
 such duration as to permit initial set, even in hot weather. 
 
 29. Concreting in Hot Weather and in Cold Weather. — The basis of all concrete is the 
 union of inert materials by substances produced through chemical reaction between Portland 
 cement and water. Any acceleration or retardation of this chemical process affects the quan- 
 tity and quality of binder resultant from this reaction; and any such alteration affects critically 
 the quality, strength, and endurance of concrete formed by admixture of this binding product 
 with sand and stone. 
 
 Temperature is known to control the rate of all chemical reactions. In general, heat 
 
 1 Proc. Am. Soo. C. E., vol. 39, p. 120; and vol. 47, p. 101. 
 
Sec. 2-29a] GENERAL METHODS OF CONSTRUCTION 
 
 77 
 
 accelerates and cold retards chemical union. ^ Furthermore, solution of some products, such as 
 gypsum (CaS04) contained in Portland cement, is active at relatively low temperatures and 
 inactive at higher temperatures, while solution of other products takes place in reverse order. 
 Relative evaporation speeds at different temperatures are also to be considered, with correlative 
 effect on the strength of concrete produced at any given time. It is reasonable to expect, 
 therefore, as is borne out in fact, that hot (weather) concretes are quick-setting and of early 
 strength and that cold (weather) concretes are slow-setting and of low strength; and on forget- 
 fulness of these obvious but inescapable facts rests responsibility for many a failure. 
 
 Particularly is this true of cold-weather concreting. At 40°F. concrete requires four 
 times as long a period to attain a given strength as the same concrete at 50°F.; and at 40°F. 
 about nine times as long as at 70°F. Below 40°F. the ratio still further increases. Many so- 
 called "mysterious" failures, in which the concrete is obviously not frozen, are to be explained 
 by delayed set and hardening, due to low temperatures alone. Below 40°F. the set is so 
 delayed down to and including 32°F. where rupture by ice formation occurs (requiring a 
 later extra period at elevated temperatures to induce reconsolidation in addition to that 
 normally required for setting at the average temperature prevailing) that computation must 
 be made for each instance to insure safety. 
 
 Using Portland cement of normal hardening rate, the following periods before removal of 
 forms in summer weather are suggested as representative of correct practice: 
 
 For concrete in mass work 24 to 48 hr. 
 
 For concrete in thin sections 48 to 60 hr. 
 
 For concrete columns 48 to 60 hr. 
 
 For concrete in beams and girders 12 to 21 days 
 
 For concrete in long span slabs 14 to 21 days 
 
 The period required in cold weather will be more or less protracted according to the 
 average temperature prevailing2 both prior to and during the setting period, inasmuch as tem- 
 peratures prior to mixing and placing will hold for the aggregates, even though in many cases 
 attempts at preheating have been made. 
 
 29a. Preheating Aggregates and Water. — Preheating sand, stone, and water 
 previous to admixture is an operation difficult adequately to perform. Each cubic yard of 
 materials will require approximately 1000 B.t.u. per degree rise in temperature. With an 
 indeterminate factor of heat transference, the fuel required on a day's operations may be com- 
 puted or, better still, such computation may be neglected and fuel added until the temperature 
 of the materials has been sufficiently raised. It is erring on the side of safety to have this 
 temperature judged by an unsensitive, calloused hand, rather than by a thermometer. Water 
 may more easily be made too hot, inducing flash set when mixed with cement. 
 
 296. Means for Heating Aggregates. — An old smokestack section, buried in 
 sand or stone, and fired with wood, is perhaps the best construction-job means of heating ag- 
 gregates. Steam jets are the least efficient. Water may be heated by either immersed steam 
 coils, or by steam jets, or by externally applied heat. A gasoline torch playing directly into the 
 mixer drum is sold as a concrete heater. 
 
 29c. Enclosure and Heating of Forms.— In cold-weather building operations 
 in particular, enclosure by canvas is desirable. Salamanders, or other heating units are kept 
 burning within to keep the temperatures somewhat elevated. It must be borne m mmd, 
 however, that at best the temperature of the enclosure is low; and that heat transference 
 through wooden forms to the concrete is slow. Such precautions, therefore, do not admit of 
 dispensing with preheating of aggregates and water, or of leaving forms in place for a requisite 
 time. 
 
 1 The speed of chemical reactions is approximately as the sixth power of the absolute temperature. 
 
 2 See A. B. McDaniel: Proc. Am. Con. Inst., 1915; also Art. 16, Sect. 5. 
 
78 
 
 CONCRETE ENGINEERS' HANDBOOK [Sec. 2-29d 
 
 29c?. Proteccion Against Frost. — The employment of manure in contact with 
 concrete is seriously objectionable. The heat of manure is derived from the decomposition of 
 its organic portions and in this process, compounds destructive of concrete are formed. Clean 
 straw, clean sawdust, or canvas will assist in protection against frost, but in addition, artificial 
 heat must be employed for temperatures below 35°F. if assurance of safety is desired. 
 
 29e. Freezing of Concrete. — If frozen before initial set, concrete will reconsoli- 
 date on later elevation of temperature with seemingly no impairment of strength. This holds 
 particularly for sections where there is sufficient hydrostatic head to recompact the mass as 
 the expansively disrupting ice is thawed, chemical reactions, in the interval, having been sus- 
 pended. It is better, however, to prevent freezing than to take chances. 
 
 29/. Use of Anti -freezing Mixtures. — Common salt (NaCl), or calcium chloride, 
 (CaCU), is the basis of most anti-freezing mixtures. Glycerine and alcohol also have been tried, 
 but both tend to lower the strength and there is also question as to the propriety of using gly- 
 cerine, because of possible organic decomposition and injury to the concrete. Calcium chlo- 
 ride or salt added to water will lower its freezing point, and in proportions of CaCl2 not to exceed 
 2% of the weight of cement or proportions of salt from 2 to 10% of the weight of water, have 
 been recommended and used, but the ill effects of salt so far outweighs its benefits— as for in- 
 stance, by promoting corrosion of steel — that it is better omitted. No anti-freezing compound 
 is better than salt; and none is equal to adequate heating of materials with proper maintenance 
 of temperature during the setting period. 
 
 29g. Protection Against Heat. — Aside from slab and thin wall construction, 
 protection of concrete against heat is rarely needed. For such purposes, protecting coverings 
 of straw, sawdust, sand, or canvas^ are usually sufficient. Evaporation must be guarded 
 against, as must also working after initial set, as in floating floor or sidewalk surfaces. Hot- 
 weather evils, however, are less troublesome than are those incident to cold-weather concreting 
 and are provided against with corresponding ease. 
 
 FIELD TESTS OF CONCRETE 
 
 30. Object of Field Tests. — The primary object of making field tests of concrete is either 
 to obtain information as to the strength of field concretes or assurance as to the strength and 
 integrity of a commercial structure. 
 
 31. Limitations Inherent in Field Tests. — Necessarily, field tests of concrete must be made 
 on specimens of such section — usually 6-in. cubes, or better, 6 by 12-in. or 8 by 16-in. cylinders 
 — that the maximum strength to be anticipated shall not exceed the capacity of testing appa- 
 ratus available. This limits the size of specimen to a considerable degree, which affects the 
 relationship between strength of test specimen and strength of a like section in the structure 
 according as aggregates of greater or less size are used. 
 
 Necessarily, also, the strength of test specimens has dependence upon the degree of compact- 
 ing and care of molding employed in their manufacture. In actual structures, quite dissimilar 
 internal conditions exist, with static head playing a more or less important part in consoli- 
 dation, this static head varying continually in each portion of the structure. It is therefore 
 difficult, if not impossible, to duplicate in test specimens pressure conditions obtaining in a 
 structure. 
 
 It is assumed, furthermore, that the materials incorporated in a small test specimen are 
 representative of, and in like quantities to, those making up concrete in the structure. It is a 
 regretable fact that a concrete mix is rarely of uniform composition in its several parts; and that 
 so great is this variation found to be that relatively small portions of any mix may or may 
 not represent in their constitution and properties when hardened a fair average of the con- 
 cretes in the structure. 
 
 Temperature conditions further increase discrepancies between test specimens and struc- 
 
 1 See Sect. 4 on "Concrete Floors and Floor Surfaces, Sidewalks, and Pavements." 
 
Sec. 2-32] 
 
 GENERAL METHODS OF CONSTRUCTION 
 
 79 
 
 tural concretes. When concrete is in considerable mass, temperature rise due to chemical re- 
 actions between cement and water are largely retained,^ and atmospheric variations exercise 
 less effect on the proper increase of strength. In small specimens, on the contrary, moicture 
 and temperature conditions are subject to abrupt change with consequent variation of proper- 
 ties in the hardened concrete. ^ The mode of applying stress is another factor tending to dis- 
 similarity and to misleading conclusions. 
 
 32. Comparative Tests on Field-molded and Structural Specimens. — Two notable series 
 of investigations are on record with respect to the value of field tests of concrete. Those of the 
 Public Service Commission of New York^ give comparative values between field-molded speci- 
 mens and specimens cut from the actual structure. Those of Kansas City, Mo.,^ were tests on 
 field-molded specimens alone, without comparative tests on specimens cut from the structure. 
 
 33. Value of Tests on Field-molded Test Specimens. — The indications of the tests above 
 mentioned are not favorable so far as consistency between laboratory and field is concerned, 
 and this is to be expected, as the practice of sampling concrete from a mixer; molding such 
 samples more or less inexpertly into small specimens; curing them under conditions dissimilar 
 to those structurally existing; and applying a breaking stress, must, obviously, give results of 
 doubtful value. Then again, by the time these test specimens are matured and broken, tons 
 upon tons of concrete have been piled on or around that portion of the structure of which they 
 might have been part, so that the removal of this concrete would be next to impossible, even 
 though test results should be adverse and indicate a low strength. The best that might be done 
 would be to so vary mixtures or procedures in subsequent parts of the work as to produce more 
 nearly the values desired. 
 
 34. Transverse Tests on Beam Specimens. — A variation in form of specimen and method of 
 testing introduced in the Welland Canal tests is of interest, though subject to all limitations 
 above set forth. In these tests^ the test specimen is a beam of rectangular section 4>^^ by 3}^ 
 in. and 3 ft. long, tested transversely. Such tests give little if any indication as to the ability of 
 a concrete to withstand applications of stress other than exactly similar to those applied in the 
 test. 
 
 35. Core Drill Test Specimens from Actual Structures. — In certain instances core borings 
 to secure test specimens have been made in completed structures. Such cores are more rep- 
 resentative of mass conditions than other specimens. In taking borings at the Municipal 
 Filter Plant, Cleveland, Ohio, both 6-in. and 4-in. cores were taken with a diamond drill bit, 
 the cores being subjected to examination and tests of various kinds. Somewhat similar work 
 but with a shot drill was done at the Ashokan Reservoir of the New York City water supply 
 system in 1915. 
 
 Either a shot or a diamond bit may be used in core borings. The shot bit is slower, cuts 
 reinforcing steel more readily, but gives a rough core. The diamond bit cuts rapidly, gives a 
 smooth, even core, but the diamond loss may be a serious item of expense, particularly where 
 tie-wires or reinforcing steel is encountered. Either bit is almost helpless where segregated 
 pockets of loose material are encountered. 
 
 36. Suggested Methods for Making and Testing Field Specimens of Concrete.— The fol- 
 lowing methods for making and testing field specimens of concrete are taken by permission 
 from the report of Committee C-9 of the Am. Soc. Test. Mat., June, 1917. 
 
 The following methods are presented not as final recommendations but as an outline of what in the opinion 
 of the committee represents the best practice at the present time. The necessity for greater attention to testing 
 concrete in construction, and for the adoption of a proper method for sampling the concrete to represent the 
 product of the various field operations, is recognized by engineers and contractors, eppecially in view of the tend- 
 
 1 Paul and Mayhew: Trans. Am. Soc. C. E., 1915, pp. 1225-1267. 
 
 2 A. B. McDaniel: Bull. 47, Univ. of 111., 1915. 
 Withey: Wise. Engr., Feb., 1915. 
 
 3 Eng. Rec, Sept. 4, 1915. 
 
 ^ Eng. News, Sept. 10, 1914. 
 
 6 Eng. Rec, July 24, 1915, p. 112. 
 
 i 
 
80 
 
 CONCRETE ENGINEERS' HANDBOOK 
 
 [Sec. 2-36 
 
 ency in many quarters to use a wet, sloppy concrete which may give a final strength much lower than that upon 
 which the design is based. 
 
 The tests are designed to provide an indication of the quality of the concrete which is placed in the structure 
 and character of workmanship in mixing. By providing damp sand storage for the test specimens, the variable 
 weather conditions are purposely disregarded although these sometimes greatly affect the final strength of the con- 
 crete. In comparing the results, the temperature and weather conditions must be taken into account. 
 
 Size and Shape of Specimen. — The test specimen should be of cylindrical form, with length twice the diameter. 
 When the coarse aggregate does not exceed l^-^ in. in diameter, a 6 by 12-in. cylinder may be used, although an 8 
 by 16-in. cylinder gives more concordant results. For larger-size aggregate a mold whose diameter is not less than 
 4 times the diameter of the largest size aggregate should be used. 
 
 Side view 
 
 Top view 
 Stack: 6?i"0.D. Cold-draivn seamfesf 
 steel tubing; ^e''¥valls. Make varrrw 
 slit along One element. May also 
 use 6" steel water pipe, machmed 
 mside. Slit along one element, so 
 that when closed will givg ^"insia9 
 ^,^iam^ter. , • 
 
 Top view 
 
 Side view 
 
 Fig. 6. 
 
 Fig. 7. 
 
 Molds and Apparatus. — Figs. G to 8 show types of molds which should be used in the field. Figs. G and 7 
 show types which are designed for repeated use; while the mold shown in Fig. 7 is destroyed in removing test piece, 
 or else has to be resoldered. While this latter mold is not adapted to continuous use and therefore is more expensive 
 in actual cost, it is quite convenient to use where but few tests are to be made and particularly advantageous 
 where specimens are to be shipped at early stages as the specimens can be left in the molds for shipment. "Various 
 modifications of each of these forms will suggest themselves, the only requirements being that they hold their 
 shape during the molding of the specimen and that the ends remain perpendicular to the side. 
 
 When bottomless forms are used, it will be necessary to provide a plane surface on which to mold the specimens. 
 Individual plates of glass M to H in. thick, or metal plates with plain surfaces about 2 in. larger than the diameter 
 
 of the mold, may be used, placing one under each test piece. 
 A piece of wax paper should be provided to place under each 
 test specimen to prevent the concrete from adhering to the 
 plate, or the plate may be oiled. 
 
 A central place should be selected for molding the 
 specimens, and a sand pile provided so that they may be 
 kept in damp sand as described below, to prevent undue 
 evaporation and obtain uniformity of storage conditions. 
 
 Sampling the Concrete. — Concrete for the test specimens 
 should be taken immediately after it has been placed in the 
 forms. Each sample should be taken from one place. A 
 sufficient number of samples — each large enough to make 
 one test specimen — should be taken at different points so 
 that the specimens made from them will give a fair average 
 of the work. The location from which each sample is taken 
 should be clearly noted for future reference. 
 
 In securing samples, the concrete is taken up irom the 
 mass by a shovel or similar implement and placed in a large 
 pail or in some other receptacle for transporting to the place 
 Care should be taken to see that each specimen represents the total mixture 
 
 Side view 
 
 Materia/: 
 h/oZOSavge 
 galvanized sfaef. 
 bj:^ lj- B ottom tightly soldered 
 
 Fig. 8. 
 
 where the specimens are molded, 
 of the concrete at that place. 
 
 Molding the Specimen. — The pails containing the samples of concrete should be taken to the place selected 
 for making the test pieces as quickly as possible. To offset segregation of materials during transportation, each 
 sample should then be dumped out of the pail into a non-absorbent water-tight receptacle, and without further 
 mixing immediately placed in the mold. Different samples should not be mixed together, but each sample should 
 make one specimen. 
 
Sec. 2-36] 
 
 GENERAL METHODS OF CONSTRUCTION 
 
 81 
 
 For working the concrete around the edges of the sides of the mold, a 3-in. round steel rod, 2 ft. long, should 
 be used. 
 
 Ramming should be avoided, but care should be taken to remove air pockets. The freshly made specimen 
 should be struck off and troweled level with the top of the form. The specimen should preferably be capped in the 
 field while it is in the mold so as to be ready for the testing machine. After the concrete has stiffened appreciably 
 and before the molds are removed, neat cement or a rather stiff 1 : 2 mortar may be used to fill the molds level full 
 A piece of plate glass or machined metal plate should then be worked around on the top of the mortar until it rests 
 on the form. This plate should be oiled or a piece of wax paper be placed between it and the concrete. If the forms 
 are carefully made, this will give top and bottom surfaces perpendicular to the sides of the specimens. To prevent 
 the specimen from drying out, it should be covered or otherwise protected. If desired, the mold itself may be 
 buried in sand while the specimen is being molded. 
 
 At the end of 48 hr. the specimens should be removed from the mold and buried in damp sand. In case the 
 molds shown in Fig. 8 are used, specimens may be buried in damp sand without the removal of the forms, thus 
 permitting shipment of the specimens in the molds. Test specimens made in the mold shown in Fig. 8 may be 
 removed by opening the soldered joint with a sharp tool. 
 
 Testing. — Ten days prior to the date of test, specimens should be well packed in damp sand or wet shavings 
 and shipped to the testing laboratory, where they should be stored either in a moist room or in damp sand until the 
 date of the test. It is assumed that ordinarily a 28-day test will be made, although tests at 7 and 14 days will give 
 some indications of the results to be expected at 28 days. In case 7-day tests are made, the test pieces should re- 
 main at the job as long as possible to harden, and should be shipped so as to arrive at the laboratory in time to 
 make the test on the required date. 
 
 The foregoing recommendations of the Am. Soc. Test. Mat. are subject to revision, it 
 being recognized that they are, by the nature and cost of equipment specified and require- 
 
 5/ofe of mold ■ 
 
 Fbrraffineorgwuf- 
 layer to level 
 bo^l-om--, 
 
 ments for curing conditions, more nearly allied to laboratory procedures than to testing in 
 the field. To overcome these limitations, the author has, with uniform success, used stock 
 cartons of paraffined paper, such as those shown in Fig. 8 A for field molds. As will be 
 seen, they are simply stout cartons with caps and they may 
 be had in quantities at prices ranging from 1^ cts. for 3)^ 
 by 7-in., to 6 cts. for 6 by 12-in. sizes. 
 
 Molding and puddling are accomplished in the usual 
 manner, the mold retaining its shape, and when full, capped, 
 with identifying data written directly on the cap. When the 
 specimen has matured, the mold is slit down the side with a 
 sharp knife, as in Fig. 8j5, and the shell removed. This leaves 
 a perfect specimen, as in Fig. 8C, whereon, it will be noted, 
 the top and bottom cardboards remain as cushions for the testing machine heads. 
 
 To insure even bottoms, it is well to set the empty cartons on loose sand during mold- 
 ing and until set. For such bottoms as are sprung, or out of true, a little melted paraffin, 
 or of cement grout (if time to set is permitted) poured in with the mold vertical, will ensure 
 a bedment so even as to make plaster preparation unnecessary (see Fig. 8D). 
 
 The advantages of this mold for field work are: 
 
 Fig. 
 
 5and bedmenl- 
 
 8D. — Leveling distorted 
 toms of waxed ends. 
 
 Disforfed^ 
 ..■pasretxxrrd 
 boffoni' 
 
 bot- 
 
82 
 
 CONCRETE ENGINEERS' HANDBOOK 
 
 [Sec. 2-37 
 
 (if 5000 
 
 ^ 4000 
 
 o 
 
 1. It is readily and cheaply procurable anywhere. 
 
 2. Temperature changes excepted, curing conditions for all specimens are always uniform 
 and alike, without bedding in sand or immersing in water, as the waxed carton retains all 
 water. 
 
 3. Shipment of specimens from job (where molded) to laboratory (for crushing) maybe 
 made in any manner convenient, curing proceeding uniformly throughout this period. 
 
 4. There are no molds to clean or to re-ship. 
 
 37. Pre-use Tests of Materials.— It is to be observed that recognition is accorded in the 
 above recommendations to the questionable commercial values of field tests of concrete. The 
 art is at present in a transition state. 
 
 Some field tests, however, are of value. One of these is as follows: It seems to hold true 
 that the strength of concrete is directly dependent upon the size-grading of its aggregate; that 
 
 this is particularly true as respects the fine ag- 
 gregate; and that of a selection of sands, con- 
 crete will be strongest when made with that 
 sand whose sum of percentage passing a given 
 series of screens is lowest. 
 
 In Fig. 9 is shown a series of curves pre- 
 pared by CM. Chapman illustrative of this 
 point. Percentages passing in these curves 
 are taken from record cards of the Universal 
 Sand Tester and illustrate the field practice 
 of Westinghouse Church Kerr & Co. in the 
 selection of sands. It is obvious from this 
 chart, that if the percentages passing is known 
 for any sand, the strength of a mortar made 
 from it in given proportions at any given age 
 (in this case, 28 days) may be read directly. 
 Although it has been held that the strength of 
 a mortar is not a measure of the strength of a 
 like mortar in concrete, the relationship is not 
 entirely misleading. On the contrary, it now 
 seems probable that pre-use tests of materials; 
 the establishment and maintenance of correct 
 proportions; and refinement of processes of 
 mixing and placing will afford the greatest de- 
 velopments in the concrete art.^ 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 \ 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 \ 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 \ 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 \ 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 V 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 N 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 N 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 \ 
 
 
 
 
 
 
 
 
 \ 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 \ 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 N 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 \ 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 N 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 K 
 
 ^- 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 \ 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 - 
 
 100 125 150 175 200 2S5 £50 Z15 300 325 350 375 400 
 
 Coefficienf of uniformity 
 Sum of''per cents passing' screens of Universal Sand Tester 
 
 Fia. 9. 
 
 WATERPROOFING CONCRETE 
 
 38. Meaning of "Waterproof." — ''Waterproof" as applied to concrete may, in its literal 
 sense, give rise to confusion and misunderstanding. "Water-resistant" to a specified degree, 
 or "impermeable" might more nearly define and delimit the abilities of concrete to withstand 
 attack from or permeation by water. ^ 
 
 39. Resistance of Concretes to Water Action. — Few concretes are free from one mani- 
 festation or another of water action. Except for minor surface attack, such action follows 
 water penetration, which latter may result from an actual hydraulic head, as in a dam, sewer, 
 aqueduct, or reservoir; or from a negative head induced by evaporation from an exposed 
 surface, as in a retaining wall, subaqueous tunnel, or sidewalk; or it may be caused by surface 
 wetting and mass absorption, as in a concrete building, or in stucco. Chemical attack by sol- 
 
 1 See R. E. Goodwin: Concrete, Nov., 1915. 
 
 * See M. O. Withey: "Permeability Test of Gravel Concrete." Proc. Western Soc. Engr's., 1914. 
 
Sec. 2-40] 
 
 GENERAL METHODS OF CONSTRUCTION 
 
 83 
 
 vents excepted (but inclusive of secondary frost action) the effects of penetrant water on con- 
 cretes are generally alike, though differing in degree. Water penetration is directly or indirectly 
 the cause of the majority of disintegrations in concrete and the degree to which water pene- 
 tration is permitted by the texture of any concrete is a direct measure of its strength and 
 endurance. 
 
 40. Resistance of Concretes to Water Penetration. — Penetration of water into concrete is 
 readiest by an actual physical passageway or passageways. Obviously, a given quantity in a given 
 time may enter by one large passageway, or by a multiplication of minute passageways. 
 Securing resistance to penetration is, therefore, to be ac- 
 complished by reduction of such passageways to a minimum, 
 both as to size and number, or by sealing them off. It 
 follows, therefore, that concretes of given materials arc 
 water-tight and water-resistant, as well as strong and en- 
 during, in proportion to their absolute densities. Con- 
 versely, concretes are weak, permeable, and of low endu- 
 rance in proportion to their porosities. ^ 
 
 41. Degree of Impermeability Attainable. — Absolute 
 freedom from water penetration is probably impossible of 
 attainment in the commercial manufacture of concrete. 
 Certainly, the average results of present practice warrant 
 that belief. An improvement in present-day work is not 
 only an imperative necessity, but, fortunately, practicable as well. 
 
 Illustrative of the difficulty of obtaining absolute impermeability in artificial concretes, 
 the structure of sandstone (Fig. 10) is worthy of study. This has before been cited^ as an ideal 
 concrete in structure, in that it has in combination silica (sand) particles closely compacted, 
 with a minimum of cementitious material between them. Yet sandstone of this grade is known 
 to be absorptive of water and to weather (disintegrate) rapidly. Arrows (a) and (6) indicate 
 the minute passageways in the cementing material between the silica particles through which water 
 entrance is secured and at which disintegrations center. And in further likeness to artificial 
 
 Fig. 10.— Medina sandstone show- 
 ing pores which render the stone ab- 
 sorbent. (Magnified 20 diams.) 
 
 Fig. 
 
 11. — Pinholes passageways in commercial 
 concrete. (Magnified 10 diams.) 
 
 concretes, the more cementing material in any sandstone, the higher its porosity and the lower 
 its strength and endurance. 
 
 42. Porosity of Commercial Concretes. — Necessarily, because of limitations imposed in 
 artificial concretes by inadequate compacting and consolidating processes, a density equal even 
 to sandstone cannot be obtained. Dispersion of aggregates in concrete, both coarse and fine, 
 with corresponding increase of cementing material between and around them, has been before 
 
 1 See chapter on "Proportioning Concrete," Sect. 2, and on "Properties of Plain Concrete," Sect. 5. 
 * See chapter on "Aggregates" in Sect. 1. 
 
84 
 
 CONCRETE ENGINEERS' HANDBOOK 
 
 [Sec. 2-43 
 
 noted. Isolation of aggregates by water^ and occlusion of air by mixing and placing methods 
 in current vogue^ have been pointed out at various times. The value of proper proportioning 
 as an aid to water-tightness has been the subject of frequent papers, discussions, and writings. 
 Field methods, however, provocative of undesirable conditions, have remained unchanged. 
 This argues either an apathy not creditable to the engineering profession and inimical to con- 
 crete, or else a confession of inability to remedy recognized evils. A present lack of adequate 
 presentation of the problem of impermeable concrete may be one reason for this state of 
 the art. 
 
 Search in commercial concretes for passageways capable of conveying water need not be 
 protracted to meet with reward. Such passageways vary in size from "pinholes," indicated 
 by the surface shown in Fig. 11, to those of finger size in Fig. 12. "A wall you could throw a 
 cat through" is verbatim repetition of a field characterization which is not infrequently ap- 
 plicable. It is often objected that "pinhole" passageways are not continuous, but no proof of 
 such assertion is offered; and while the converse is equally difficult of proof, the porosity of 
 pinholed concretes under test and the presence of like pinholes throughout any and every sec- 
 tion of such concretes gives warrant for belief that they are, by their multitude, of great im- 
 portance when their combined water-conveying abilities are considered. 
 
 43. Excess Water as a Cause of Porosity. — Aside from segregated pockets of stone (which 
 also are caused by excess water), water voids are, however, quantitatively more important than 
 
 are airholes as passageways for penetrant or percolating water. 
 Water voids are relatively massive, approaching segregations 
 even when not so classified; and inasmuch as the water once 
 lying in them has either flowed away, or been evaporated, con- 
 tinuity of passageways for subsequent water flow is strongly 
 indicated, if not proven by the fact of this loss. An example 
 of such water voids and flow passages in a commercial concrete, 
 intended to be of superior grade, is shown in Fig. 13. This is 
 typical of innumerable passageways of like character found 
 Fig. 13.— Water voids passage- trenerally in overwet concretes, 
 way in concrete. (Magnified . - ^t. • i i au • i i • ^ j: 
 
 diams.) 44. Shrinkage Cracks. — fehrmkage cracks m concrete are of 
 
 a general type and so universal as to be viewed by the majority 
 either with eyes unseeing, or regarded with the contempt that comes from familiarity. Their 
 importance both as a condition and as an indicator of internal processes is, however, of the 
 greatest importance. 
 
 Shrinkage cracks are of a general type, irregular in line and radiating from a common center, 
 usually a pore of greater or less size. This is to be expected, inasmuch as such a pore is at least 
 a possible point of egress for water from the mass immediately sur- 
 rounding. Further, flow is freest from such an open center, so that 
 under evaporation or other processes it soon becomes a point of 
 dryness; and inasmuch as it is already a point of weakness, relief 
 planes radiate from it- gradually as drying proceeds inward, until 
 shrinkage stresses are balanced by internal resistance. 
 
 44a. Types of Shrinkage Cracks. — Perhaps the com- 
 monest type of shrinkage crack is three-branched. This is to be 
 seen, on every hand in concretes and stuccos, both in the gross 
 (Fig. 14) and in microscopic sizes (Fig. 15). Where numerous 
 pore centers exist, complicated systems build up by the junction of a multitude of like radiat- 
 ing cracks from "crazing", with oftentimes, deeper and more serious disruptions, as in Fig. 
 16. In both Fig. 14 and Fig. 16 the centers have been outlined in circles. Like cracks in 
 
 1 See chapter on "Proportioning Concrete" in Sect. 2, and on "Water" in Sect. 1. 
 
 2 N. C. Johnson: Eng. Rec, Jan. 23, 1915; Dec. 30, 1916. 
 » N. C. Johnson: Eng. Rec, Deo. 4, 1915. 
 
Sec. 2-446] 
 
 GENERAL METHODS OF CONSTRUCTION 
 
 85 
 
 drying earth are everywhere to be observed, the three-branched crack permitting spherical 
 contraction and rehef with minimum disturbance. 
 
 446. Shrinkage Cracks and Porosity.— Necessarily, such shrinkage cracks in 
 concrete are open passageways for water. This is attested on every hand by the crystalline 
 filling of dissolved salts left behind in such shrinkage cracks. Fig. 17 is typical of such condi- 
 tions, which exist where the fluid supply is, or becomes, somewhat limited, so that super- 
 saturation and crystallization may be brought about. Moreover, as is evidenced by such 
 crystalline fillings, these cracks once conveyed fluid through the concrete. Cracks of like 
 formation are equally potent to convey other fluid; and, if the supply is ample, to remove 
 soluble portions of the concrete, with mechanical dislodgment and removal of inert particles 
 released by such solution. The original passageway is thus speedily enlarged, possibly to harm- 
 ful proportions and certainly to increased water-carrying capacity. Evidence as to such quanti- 
 tative removal of material is given on sheltered surfaces of concrete, such as inspection galleries 
 
 
 'if 
 
 
 
 
 nnil 
 
 Fig. 15. — Microscopic shrinkage crack. 
 (Magnified 150 diams.) 
 
 Fiu. 16. 
 
 P^a. 17. 
 
 of concrete dams, where deposits removed from the concrete and aggregating many tons are not 
 infrequently piled on floors and cling to walls. 
 
 44c. Prevention of Shrinkage Cracks. — Shrinkage cracks like the foregoing arc 
 difficult of prevention. They may be minimized in number and severity by: (1) use of 
 graded materials; (2) avoiding the use of excess water; (3) adequate mixing; (4) careful placing 
 to avoid segregation; and (5) curing (annealing) under proper conditions of moisture, so that 
 shrinkage stresses will be developed only at a rate commensurate with the slow increase of 
 strength in concrete. This, in connection with slow drying and hardening of colloids, largely 
 explains the high strength of concretes cured under water, and conversely, explains the easier 
 disintegration of concretes in which too rapid drying occurs. 
 
 But though the foregoing five principles, if made effective in practice under skilled direc- 
 tion, would result in better concretes as to water-tightness, with correlative strength and en- 
 durance, their field observance is so limited as to be negligible. Adequate mixing will not be 
 had, so long as engineers countenance and tacitly, if not openly, approve inadequate mixing in 
 the interest of quantity output. Graded sand and stone will not be used unless cheaper than 
 other available materials, so long as the same price per yard in forms is paid for one as for the 
 other. Excess water will be used until insistence is had for the use of lesser quantities. Careful, 
 uniform placing needs standardization and enforcement by engineers of such standards. 
 Moist curing, or annealing of large sections is often a physical impossibility, but often it can 
 be done if required. The practice of curing commercial concrete is now extending; a fact 
 which offers much encouragement. Though many problems yet remain unsolved, our present 
 knowledge is sufficient for great improvement, once engineering sentiment for right practice is 
 brought to the point of putting them into effect. The cause of pervious concretes lies not so 
 much in lack of knowledge as to how to make concretes that are impervious, as in neglect to 
 put into effect the knowledge at hand. 
 
86 
 
 CONCRETE ENGINEERS' HANDBOOK 
 
 [Sec. 2-45 
 
 45. Pervious Concretes and Laitance. — Laitance — the porous, chalky material which rises 
 during deposition to greater or less extent at the surface of concretes — is a chief foe of water- 
 tightness. The deposit is particularly deep with excess water, or too fine or dusty aggregates, 
 or both. Concrete subsequently placed on this laitance fails utterly to bond; and seepage 
 readily takes place along this construction or "day's work" joint, often followed by later dis- 
 integration. Laitance an inch or more thick, scrolled with a lath, on top of a newly poured 
 foundation block 4 ft. in depth, is shown in Fig. 5, page 74. In sections of greater height, as 
 in reservoir walls, especially where overwet concretes are indulged, the deposit will be of greater 
 depth, usually extending through the body of the wall to form a horizontal joint, open to per- 
 colating water. 
 
 Occasionally such laitance deposits are localized, forming pockets; and inasmuch as such 
 pockets are formed from the finer material assumedly lying distributed in inter-particle spaces 
 throughout the concrete mass, their isolation implies and often proves the existence of segre- 
 gated and open pockets of ballast at other points. This is easily to be understood when continu- 
 ous deposition in one part of overwet concretes is observed in field work with runoff of lighter 
 materials as the mound grows. 
 
 In all concrete work subject to water action in any degree, laitance planes may be sub- 
 stantially avoided by filling forms without permitting set of one portion before the portion next 
 above is deposited. Running off ''soupy" portions from the top of forms while the mass is 
 fluid is a palliative measure that may be used, or removal of laitance after setting may be at- 
 tempted. No measures yet devised are wholly adequate, in that none basically remove the 
 cause of complaint. 
 
 46. Effect of Temperature and Atmospheric Effects on Water-tightness. — Temperature 
 effects in finished structures, and those due to atmospheric changes, may result in opening 
 passageways capable of conveying water. The coefficient of expansion of concrete is approxi- 
 mately the same as that of steel (see Art. 32, Sect. 5). Adopting a linear unit, the movement 
 at any point may be found by multiplying the distance expressed in terms of this unit of this 
 point from a fixed point by the degrees change of temperature experienced or anticipated. In 
 massive masonry, the interior experiences little thermal change. Actual dimensional changes 
 in such structures are probably compensated for by internal flow.^ 
 
 Variations in moisture content, even after prolonged set, affect the volume of concrete 
 from 0.05 to 0.08% in the usual atmospheric range. This may be outwardly evidenced by 
 cracks, but is more generally taken up in internal stresses in the concrete and reinforcement. 
 Such changes are to be anticipated from our knowledge as to the colloid content of concretes 
 and the absorptive and dessicative properties of such materials. 
 
 If consideration of anticipated temperature and moisture changes indicates that a given 
 structure will be liable to cracking by stresses thus induced, expansion joints must be provided. 
 Their use is preferable to their omission in most cases, but they must be most carefully formed. 
 Copper or lead flashing, or asphalt or elaterite mastic and fabrics have been found efficacious 
 when properly applied, but all precautions must be observed as to obtaining density in the sur- 
 rounding concrete. With mastic compounds, dryness of the adjacent concrete must be secured 
 in order to obtain proper bond, else leakage at the joint will occur. 
 
 47. Integral Waterproofing Compounds. — The foregoing paragraphs have given an in- 
 sight into the causes of porous or leaking concretes. Where penetration and flow of water occur, 
 it has been pointed out that an actual, physical passage or passages exist, that these passage- 
 ways may be: (1) pinholes, or pores, resulting from occluded air; (2) water voids, or spaces left 
 by excess water; (3) shrinkage cracks radiating from a pore, or hole of greater or less size, with 
 joining of a myriad of like cracks into complicated systems of cracks in concretes that dry too 
 rapidly; (4) segregation due largely to excess water, with open pockets of stone; (5) laitance, in 
 pockets or strata, due largely to excess water; and (6) temperature or other cracks, due to at- 
 mospheric changes. 
 
 1 F. R. McMillan: Bull. University of Minnesota. 
 
Sec. 2-47a] 
 
 GENERAL METHODS OF CONSTRUCTION 
 
 87 
 
 It is to be expected that the customary violations of the natural laws governing concrete 
 which bring such passageways into existence should cause wide demand for something purchas- 
 able which would afford rehef from consequences. If any agent exists or is to be found, which, 
 when added to concrete made with lack of care, is capable either of preventing or of closing the 
 passageways through which water penetration or transmission is brought about, without detri- 
 ment to the strength or other properties of the concrete, it may be ranked as a noteworthy dis- 
 covery. There is some question, however, as to whether any substance, particularly in econom- 
 ical percentages, will accomplish this end to any save a minor degree. Consideration of the 
 open void volumes and segregated areas in many concrete structures reflect the magnitude of 
 the task. It is probable that proper practices and materials and they alone are adequate as 
 well as unsurpassed in securing water-tight and enduring concretes. 
 :i 47a. Integral Waterproofing Classification. — Integral waterproofings now on 
 
 I the market may be grouped under four heads: 
 
 (a) Special materials added to the mixing water. 
 
 (&) Special materials added dry to the cement at the job. 
 
 (c) Cement to which has been added the special materials during manufacture. 
 
 (d) Special materials and cement applied as a plaster, this being intended to so bond with 
 the concrete surface as to become integral with it. 
 
 The special materials employed in the foregoing are substantially as follows : 
 
 (a) Various forms of metallic salts, such as chloride of lime; oil emulsions; lime soaps, sus- 
 f: pehded in water; and like compounds. The actions of oil emulsions is to form soaps in combi- 
 nation with the lime of cement; that of soap solutions as lubricants and formers of insoluble fillers 
 by reaction with cement. Lime chloride has a catalytic action difficult properly to define, but 
 tending to hasten set rather than either to lubricate, or to form pore-filling compounds. 
 
 (b) Dry powders of floury consistency, formed of metallic stearates, such as lime soap, 
 often with alum and hydrated lime. Their properties are claimed to be void-filling and 
 lubricating. 
 
 (c) Like substances, or glycerides of limes, mixed with cement during manufacture. 
 
 (d) The same as (c), used as a surface plaster. 
 
 476. Value of Integral Waterproofing Compounds. — There is no general authori- 
 tative conclusion yet determined as to the value of integral compounds. Field testimony differs, 
 probably according as the methods and materials of one use have, through inherent excellence 
 or weakness, proven either adequate or inadequate to produce impervious concrete. The 
 most extensive work that has thus far been done is published in Tech. Paper 3, by R. J. Wig and 
 R. H. Bates of the U. S. Bureau of Standards. A majority of present commercial waterproofers 
 were tested in the course of the work therein detailed, but a subsequent series requested by 
 several manufacturers of tested compounds, with concretes to be made under commercial 
 conditions, has not yet matured. 
 
 The conclusion of the foregoing tests is that no additive, proprietary, or open, will of itself 
 overcome initial, serious deficiencies of material, or admit of defective practices; and no addi- 
 tive so far known is superior in results to an excess of cement and the use of graded sand of 
 proper quality with a little water as circumstances permit. 
 
 47c. Rendering Defective Structures Impervious. — It is often necessary to ren- 
 der an existing structure as nearly waterproof as possible. The end to be attained is, of course, 
 the closing of all water passageways. The proper method to use is dependent upon the size, 
 character, and origin of the pores or passageways in the concrete. If the pores are very small, 
 some inert filler such as clay or silt may be sufficient; or a soap and alum mixture, such as that 
 employed in the Sylvester process, may be applied. If the pores are of slightly larger size, paraf- 
 fine or a paraffine-carrying oil, or bitumen, or an asphaltic oil may be successfully used. Paraf- 
 fine may be applied either hot or cold. If applied cold, it is dissolved in a volatile carrier in 
 saturated solution. Applied to the surface of the concrete, it penetrates to a greater or less 
 depth according to the dryness and porosity of the concrete. Within a short time the volatile 
 
88 
 
 CONCRETE ENGINEERS' HANDBOOK 
 
 [Sec. 2-48 
 
 carrier is evaporated, leaving the paraffine in the holes. Paraffine may also be applied in a 
 molten condition and, to render successful its use, the concrete must first be rendered sufficiently 
 warm by artificial heat so that the melted paraffine may be thoroughly rubbed in. Hot paraffine 
 treatment is one of the most durable of waterproofing methods for work exposed to weather, but 
 it requires considerable experience to secure a successful result. 
 
 Bitumens of one grade or another are applied either in solution or hot, as in the paraffine 
 surface treatment. They also may be incorporated in paints which are applied to the surface. 
 
 Any bitumens employed must possess a high degree of elasticity and durability and must 
 have considerable bonding ability with the concrete. To this end all concrete surfaces to which 
 bitumens are applied should be thoroughly dried and preferably should be warm at the time of 
 applications. Material should be well rubbed into corners and recesses; and the waterproofing 
 film should be continuous throughout. 
 
 In applying bituminous paints and solutions it is a prerequisite to success that the coating 
 shall be applied on that side of the concrete against which the water pressure is exerted. If this 
 is done, the materials will be carried into the water passageways, but if this is neglected the 
 materials will be forced out so that their application is waste. 
 
 48. Waterproofing by Cement Grouting. — Neat cement grout has often been tried as a 
 waterproofing coating, applied either as a surface plaster or as a surface wash. It has also been 
 used as a crack filler, but inasmuch as it is virtually impossible to make a coating or filling of this 
 kind adhere to set concrete, its use is rarely, if ever, attended with success. In difficult situa- 
 tions attempts have been made to use cement grout under pressure as a waterproofer, but the 
 instances on record where this has been successfully done do not indicate generally satisfactory 
 results. It seems to be requisite that any waterproofing mixture shall be more or less plastic 
 and viscous and that it shall be so applied as to deform and closely fill passageways in the con- 
 crete under pressure of the water. 
 
 49. Membranous Waterproofings. — Membranous waterproofing is an elastic, continuous 
 sheet or membrane completely covering or surrounding a structure to be waterproofed (see Fig. 
 
 Tte cover I , 
 
 -s'-o-'-A 
 
 Fig. 1 
 
 •. o 
 
 '■-■■LBuH- laps 
 Nofe-'- Thickness of waterproofing exagger- 
 ated to distinguish the phes 
 
 Fig. 19. 
 
 18). This membrane is laid in several overlapping layers (Fig. 19), impregnated and fastened 
 down with some bituminous compound. The membranous system of waterproofing is adapted 
 principally to the waterproofing of structures in course of erection, such as subways, tunnels, 
 building foundations, retaining walls, arches, reservoirs, etc. 
 
 The bituminous materials employed as sealing compounds in the membrane method of 
 waterproofing are: (a) coal-tar pitch applied hot; (6) asphalts applied hot; (c) asphalt mastic 
 applied hot; {d) especially prepared asphaltic compounds sold under various trade names. 
 
 The membranes to be used with the above sealing compounds are: (a) tarred felt; (6) 
 asphalted felt; (c) burlap; id) burlap saturated with asphalt or tar; (e) combinations of canvas 
 and felt, or canvas and burlap, or felt and burlap. 
 
 49a. Application of Membranous Waterproofing. — Success of membranous 
 waterproofing depends largely upon the care with which the materials are applied. It is neces- 
 sary first to prepare the concrete surface. It must neither be too rough, nor too wet, nor cov- 
 ered with dirt or foreign substance; and it must not possess a glaze due to richness of cement 
 surface. It is, therefore, necessary: (1) that all dirt and foreign matter shall be removed before 
 
Sec. 2-496] 
 
 GENERAL METHODS OF CONSTRUCTION 
 
 SO 
 
 waterproofing is applied; (2) that when it is appHed the concrete shall be rendered dry, either by 
 drainage and evaporation or by the application of artificial heat; (3) that the concrete shall he 
 thoroughly set (as is indicated in the requirement for dryness) ; (4) that any glazed surfaces shall 
 be picked or rubbed down in order that the materials may adhere; (5) that form ties or other pro- 
 jections that might puncture the waterproofing shall be removed; and (6) that any metal sur- 
 faces encountered shall be dry, clean, and free from rust or dirt. 
 
 496. Continuity of Membrane. — Lack of continuity may be fatal to the success 
 of any waterproofing membrane. The waterproofing sheet must, therefore, be applied con- 
 \ tinuously over the whole surface to be treated, footings and foundations included. All joints 
 ■ in the membrane must be broken at least 4 in. on cross joints and 12 in. on longitudinal; and 
 at least 12 in. of lap must be left at corners to form good junctions with adjoining sections. 
 Where it is necessary to stop work, a lap of at least 12 in. shall be provided for joining on new 
 work. Each layer of bituminous or other material must com- 
 pletely cover the surface on which it is spread, without cracks or 
 blow-holes; and the fabric must be rolled out smoothly and pressed 
 over the cementing material so as to insure its sticking 
 thoroughly and evenly over the entire surface. 
 
 49c. Protection of Waterproofing.^ — After the 
 waterproofing has been put in place it must be properly pro- 
 tected from injury. Such injury may occur when backfilling 
 with earth; when depositing concrete against the waterproofing 
 (see Fig. 20); when laying brick or rubble, or from careless piling 
 of materials on the completed waterproofing work. Injury from 
 workmen's shoes is not infrequent. It should be remembered 
 that a completed membranous waterproofing is usually soft and 
 liable to injury and the chances of so doing should be reduced to a 
 
 minimum. A single point of entry for water, particularly if inaccessible when the work is 
 completed, may render ineffective all precautions against leakage. 
 
 The following table gives the numbers of ply of waterproofing required with various heads 
 of water: 
 
 50. Rules for Making Concrete Imper- 
 vious. — (a) To make concrete that shall be 
 impervious, the rules basically governing the 
 making of dense concrete apply with special 
 force. These are: 
 
 1. Use proper materials, i.e., clean and 
 preferably graded sand; hard, durable stone; 
 cement that conforms to standard; and clean 
 water. 
 
 2. Use proper proportions of proper ma- 
 terials, i.e., avoid arbitrary proportions; use 
 careful measurement for each batch; test 
 each shipment of sand for uniformity of 
 grading; if variation is found, properly com- 
 pensate by variation of proportions. 
 
 3. Properly and adequately mix the ma- 
 terials, i.e., not only stir together the several 
 ingredients, but prolong the operation suffi- 
 ciently to secure the needful consistency and 
 distribution, particularly of the cement. 
 
 Number of Ply of Waterproofing Required 
 FOR Varying Heads of Waters 
 
 
 Material 
 
 Head of 
 water 
 
 Coal tar 
 and 
 felt 
 
 Commer- 
 cial 
 asphalt 
 and felt 
 
 Special 
 felts and 
 
 com- 
 pounds 
 
 Asphalt 
 mastic, 
 thickness 
 in inches 
 
 0 
 
 2 
 
 2 
 
 1 
 
 h' 
 
 1 
 
 3 
 
 3 
 
 2 
 
 % 
 
 2 
 
 4 
 
 4 
 
 3 
 
 H 
 
 6 
 
 5 
 
 5 
 
 4 
 
 H 
 
 8 
 
 6 
 
 6 
 
 5 
 
 H 
 
 10 
 
 7 
 
 7 
 
 6 
 
 
 15 
 
 8 
 
 8 
 
 7 
 
 % 
 
 20 
 
 9 
 
 9 
 
 8 
 
 
 1 From 
 M. H. Lewis. 
 
 Modern Method of Waterproofing, 
 
90 
 
 CONCRETE ENGINEERS' HANDBOOK 
 
 [Sec. 2-51 
 
 4. Use a minimum of water that will permit adequate filling of forms and contact with 
 reinforcement. 
 
 5. Place carefully informs to avoid segregation or unequal distribution. 
 
 6. Expel as much as possible of occluded air by puddling, or vibrating, or jarring of forms as 
 filling proceeds. 
 
 7. Fill forms continuously to top, preferably overflowing, to avoid stoppage, or laitance planes. 
 
 8. Properly protect concrete against rapid evaporation and against unusual heat or cold during 
 the setting, hardening, and curing periods to avoid shrinkage, frost, or other crackings and 
 disruptions. 
 
 9. Remove visible segregations as soon as discovered, replacing with good concrete, well 
 rammed into place. Do not rely on surface plastering of defects. Such attempts are unwork- 
 man-like and ineffective. 
 
 10. Construct expansion or contraction joints with extreme care. Do not rely for water- 
 tightness on any supposed bond between abutting concrete sections. 
 
 (b) Integral waterproofings cannot be relied upon to avert the consequences of improper 
 manufacture. 
 
 (c) Membranous waterproofings are of service in closing water passageways in existing de- 
 fective structures when, and only when, carefully apphed. 
 
 FINISHING CONCRETE SURFACES 
 
 51. Character of Surface Finish Desired. — The character of finish desirable to produce on 
 concrete surfaces is determined by the ends to be served. The majority of requirements for 
 special finishes are architectural, varying according to the character of the structure and with 
 the location of the surface. 
 
 52. Removing Form Marks. — For all concrete work exposed to view, forms should be 
 exceedingly well constructed, producing plane surfaces and straight, sharp lines and true angles 
 
 in the finished concrete. In work of this 
 character, extra care is usually taken to ob- 
 tain even-textured, dense surfaces by using 
 a mixture of proper consistency and by 
 careful spading and puddling. Such sur- 
 faces necessarily reproduce all defects of the 
 mold, so that after-treatment is necessary to 
 remove the form marks, as well as to relieve 
 the "dead" color due to excess of cement at 
 the surface, with oftentimes efflorescence, 
 or other whitish deposits, indirectly occa- 
 sioned thereby. 
 
 It is elementary optics that blemishes 
 are least visible on a non-uniform light- 
 diffracting "matt" or stippled surface. 
 Form marks, therefore, are concealed by 
 producing such a surface. This may be 
 done by tooling, by rubbing, by brushing. 
 Fig. 21.-Rotary concrete surfacer being used. Note «^ ^y sand-blasting, and SUch treatments 
 
 contrast between finished and unfinished surface. have a further advantage of modifying the 
 
 dead color above referred to by exposing a 
 multitude of sand grains to fight, so that by reflection from their facets, the gray-green of 
 cement is relieved and brightened. 
 
 52a. Tooling. — Tooling concrete surfaces is more or less costly, depending upon 
 the length of time the concrete has set and upon its hardness. Bush-hammering, crandalling, 
 or axing may be done by hand, or a pneumatic or electric tool may be employed at considerable 
 
Sec. 2-526] 
 
 GENERAL METHODS OF CONSTRUCTION 
 
 91 
 
 advantage. One type of surfacing tool permitting of tooling or of grinding and the method of its 
 use is shown in Fig. 21. ^ Its capacity is rated at 60 to 70 sq. ft. of surface per hour on concrete 
 from 1 to 21 days old. Fig. 22 shows a hand bush-hammered surface of colored aggregates, and 
 Fig. 23 a picked surface. 
 
 Only small-sized aggregate should be used in facing material to be tooled, as it is hard to 
 dress and to obtain uniform results on surfaces where large angular stones are encountered. 
 The concrete should be thoroughly hardened before work is commenced, especially if sharp 
 clean surfaces are desired. The concrete should preferably be about 2 months old, although if 
 it is allowed to stand too long the labor involved will be unnecessarily great. 
 
 A variety of surface effects may be obtained by tooling, as the effect produced in any given 
 ■ case depends upon the kind of tool used. Some variation in the appearance of the finished sur- 
 face may also be obtained by the manner in which the tool is handled. By striking a perpen- 
 dicular blow no lines or marks are left in the surface, whereas with a glancing blow, tooth marks 
 are left which can be made parallel to each other or at various angles. Tooling cannot ordinarily 
 
 Fig. 22. Fig. 23. 
 
 Fig. 22. — Surface finish obtained by hand bush-hammering. The contrast of shades is produced by using dif- 
 ferent colored aggregates. The concrete in the dark portion was made with red sandstone, while the light por- 
 tion was made with trap rock. 
 Fig. 23. — Picked surface. 
 
 be performed satisfactorily on gravel concrete, as the pebbles will be dislodged before being 
 chipped. 
 
 526. Rubbing. — If a rubbed surface finish is desired, the coarse aggregate should 
 be well spaded back from the face of the work and the forms should be removed before the 
 concrete has set hard, preferably in a day or two after the concrete is poured. It is necessary 
 with green concrete to use care in removing forms to avoid spalling, as it is very difficult, if not 
 impossible, to adequately repair such spalls by patching with mortar. It is necessary, also, 
 to remove form wires or other projections before rubbing and to point with mortar any pockets 
 or open places in the surface. The process of rubbing consists in grinding down the surface 
 of the concrete sufficiently to remove all impressions of the timber or other irregularities, using 
 a brick of carborundum, emery, concrete, or soft natural stone. In connection with the rubbing 
 (which is accomplished with a circular motion), a thin grout composed of cement and sand 
 should be applied to the surface, well rubbed in, and the work afterward washed down with 
 clean water. The grout is used simply to fill surface imperfections and care must be taken not 
 to allow it to remain as a film on the surface. 
 
 This method of treatment produces a comparatively smooth surface of uniform color much 
 superior to that obtained by the all too prevalent method of painting with a grout, which almost 
 invariably crazes, cracks, and peels off. Rubbing is a very acceptable, cheap way of finishing 
 concrete surfaces. 
 
 1 The Berg Electric Rotary Surfacer, Elevator Supplies Co., N. Y. 
 
92 
 
 CONCRETE ENGINEERS' HANDBOOK 
 
 [Sec. ^-52c 
 
 Fig. 24. — Wire-brushed surface 
 treated with acid. 
 
 52c. Brushing. — A brushed finish is obtained by removing with a brush the 
 surface film of mortar which forms over the aggregate. This should be done while the concrete 
 surface is still green. The brushing should be started just as soon as it is possible to do so with- 
 out removing particles of aggregate. The time required for sufficient hardening can only be 
 determined by experimenting with the particular surface. The forms should be constructed so 
 they may be taken down in sections and give access to the face of the concrete without taking 
 away the standards which provide strength for the forms as a whole. 
 
 The process consists of scrubbing or brushing the concrete surfaces with ordinary fiber 
 (or wire) brushes and water until all the surface cement has been washed off, leaving the coarse 
 
 aggregate exposed. The ordinary fiber brush will not answer 
 if the surface is permitted to get too hard. A brush about 4 in. 
 wide, made by clamping together a sufficient number of sheets 
 of wire cloth, has been found to be very effective. 
 
 The appearance of a brushed finish can be improved by 
 washing with hydrochloric (muriatic) acid applied with a brush. 
 The acid (which should be diluted with 6 parts water) 
 thoroughly cleans the surface of the aggregate, thereby inten- 
 sifying the color and texture of same. The treated surface 
 can be made of any desired color by the proper selection of 
 colored aggregates, but care should be taken to make sure 
 that the materials used are not affected by the acid. The 
 surface should be thoroughly washed after the acid treatment 
 as otherwise it will have a mottled, streaky appearance. 
 
 A wire-brushed surface treated with acid is shown in 
 Fig. 24. 
 
 52d. Sand-blasting. — A sand-blast finish is very much the same in appearance 
 as that obtained by brushing the concrete surface while it is green. Sand-blasting produces a 
 granulated finish somewhat similar to sandstone but not so uniform, because the aggregates are 
 likely to be brought out irregularly. 
 
 The concrete should be thoroughly hardened before sand-blasting, especially if sharp 
 edges and surfaces of a fine, uniform texture are desired. All pronounced ridges should be 
 removed by tooling previous to sand-blasting since, in attempting to remove these ridges with 
 the sand-blast, the surface on either side is apt to be cut too deep and depressions in the surface 
 intensified or made more prominent instead of being erased. All strips nailed to the forms to 
 make moldings, joints, and courses should be left in place while the surface is being sand-blasted, 
 as otherwise the sharp angles and edges will be rounded off. 
 
 53. Use of Colored Aggregates. — The most satisfactory concrete surface of a given color 
 and texture can be obtained by properly finishing a surface faced with a mixture composed of 
 cement and an aggregate of the proper size and color. The color is obtained from the exposed 
 aggregate, and not by adding coloring matter to the mixture. The successful production of 
 the desired surface is also dependent upon the proper grading, proportioning, mixing, and plac- 
 ing of the materials and upon the method of finishing the surface. 
 
 Where special or expensive materials must be employed to obtain the desired finish, they 
 may be used only in the mixture applied as a facing to the exposed surface. Such a facing is 
 usually 1 to 1)^ in. thick. The best method of placing the facing material on vertical surfaces 
 is by the use of a metal facing form or mold, consisting of short lengths of iron plates 8 to 10 in. 
 wide and about 6 ft. long with three angle irons riveted to each plate. The upper end of the 
 long plate should be provided with handles and slightly flared to assist in depositing the material. 
 The metal form is placed against the wall form with the handles up and the outstanding legs of 
 the angles tight against the wooden form. The space between it and the back of the wall is 
 filled with ordinary concrete backing and the 1-in. or l}4-m. space between the metal form and 
 the face form is filled with facing material. The metal form should not be permitted to remain 
 
|3ec. 2-54] 
 
 GENERAL METHODS OF CONSTRUCTION 
 
 93 
 
 intil initial set takes place, but should be frequently raised to prevent the formation of seams 
 between the facing and the concrete backing. 
 
 : 64. Addition of Colors to Concrete. ^ — Pigments may be added to concrete to give colors 
 !)£ one shade or another. All such pigments should be chemically inert and of mineral origin. 
 
 lit is to be borne in mind that Portland cement is of itself a pigment, gray-green when unhy- 
 
 •Irated, white when hydrated; and that the brilliance of any pigment will be modified and its 
 •olor made somewhat uncertain due to varying percentages of hydrated or unhydrated cement 
 )resent at any portion of the surface. Cement stains of varying shades may be obtained in the 
 )pen market and, when used on dry concrete, give excellent results. ^ Inasmuch as in using 
 ;uch stains, the pigment is confined to the surface, they are more economical than are body 
 
 •;olors and there is less risk of detriment to the concrete through chemically active pigments. 
 
 I 65. Use of White Cement. — The uses of white Portland cement in decorative concretes 
 
 ire manifold. White Portland cement is necessarily a white pigment; and its use in connection 
 vith other pigments, or with colored aggregates, or stippled or mottled surfaces, should be 
 
 j'arefully tried out before the final mixture is determined and employed. 
 
 !' 66. Plaster Finishes. — Plasters of one kind or another rarely if ever bond effectively to 
 ;oncrete surfaces, without an intermediate agent. This phase of the art is in a transition stage 
 it present, but it is much to be desired that permanent bond may be produced between concrete 
 md plaster either applied directly, or on some intermediate agent of definite and proven value. 
 
 67. Surfacing Concrete Floors. ^ — Concrete floors are advantageously surfaced to prevent 
 I'lossible dusting as well as to produce a smooth finish. Decorative effects may be produced 
 is desired by the use of selected colored aggregates, with or without pigments or stains to color 
 :he mortar. The grinding operations for such floors are similar to the grinding of terazzo floors. 
 
 68. Specification for the Production of a Rubbed Surface. — -The following specification 
 is suggested : 
 
 Immediately upon stripping the forms from the concrete and while the material is green, patch all broken 
 lorners and such parts of the surface where the face has been removed with the removal of forms to bring 
 lame to the general level of the surrounding surface, using for this purpose a 1 : 1 mixture of cement and sand. 
 Ul wires, naills, etc., that may be projecting beyond the surface of the concrete shall be cut off at least H in. 
 Jack from the surface and the holes patched with a like mortar. 
 
 When all patches have well hardened and, in no case less than 48 hr. after patching, cut down with suitable 
 .ools all board makings, projections, swellings or lumps, so as to bring the ceiling or wall to a reasonably 
 straight surface. A reasonably straight surface shall be construed to be one which will show fair and 
 true when tested with a straight edge. 
 
 The rubbing may be done at any time convenient after the rough grinding has been completed. The 
 method of rubbing shall be as follows: 
 
 The entire surface is to be thoroughly wet down and kept moist. With carborundum or like abrasive 
 block, grind in a mix of 1:2 cement and white sand swabbed on the surface in small patches, the grinding-in 
 being done when the material is wet, water being added, if necessary, to keep the material sufficiently plastic 
 until the grinding operation has been completed. This material shall be ground into the surface until all voids, 
 lines, and wood markings have disappeared and until all surplus material not used for filling in the above- 
 mentioned voids, etc., has been removed from the surface. Should markings appear in spots where the plastic 
 material has been ground in in larger quantities, they shall be removed by rubbing lightly with a cork float. 
 
 No brush work or mixture of cement shall be applied on this surface after the grinding has been completed, 
 and where larger voids appear, same will be carefully pointed up with the ground-in material and with a cork 
 float. 
 
 The inside corners of all fillets, belt courses, and the intersecting planes of the ceiHng panels and beams, 
 shall be ground sharp. All outside edges shall be ground off either side and carefully smoothed off and straight- 
 ened with the cork float. 
 
 FORMS 
 
 59. General Requirements.— In order to obtain satisfactory concrete work the forms should 
 6e durable and rigid, and should be so well braced that bulging or twisting cannot occur. The 
 
 1 See Art. 4&, Sect. 4. 
 - See Art. 6d, Sect. 4. 
 
 * See chapter on "Concrete Floors and Floor Surfaces" in Sect. 4. 
 
94 
 
 CONCRETE ENGINEERS' HANDBOOK 
 
 [Sec. 2-60 
 
 joints, also, should be made tight enough to prevent any material leakage of the liquid mass, as 
 such leakage will mar the appearance of the finished work. 
 
 Forms should have sufficient strength to properly support the loads which they are called 
 upon to carry. Horizontal members, such as floor sheathing and supporting joists, should be 
 able to support the weight of the concrete and the construction load. Vertical members, such 
 as wall sheathing and supporting studs, should be designed to resist the hydrostatic pressure 
 of wet concrete which is about 145 lb. per sq. ft. for each vertical foot of height. 
 
 60. Economical Considerations. — The cost of forms constitutes a large item of expense in 
 the building of reinforced-concrete structures and this cost varies to a considerable extent with 
 the kind of form construction adopted. Form work, of course, should in every case leave the 
 finished concrete true to line and surface, but even with this accomplished a great deal can be 
 done in so designing forms and in so planning the detailed methods of their construction that 
 erection and removal will be greatly facilitated without undue waste of lumber. As a general 
 rule, the most important consideration is that of ease in form removal as great economy may be 
 effected by using form units over and over again with a minimum of repairs. 
 
 Simplicity and symmetry in form work should always be given consideration. In buildings, 
 uniform story heights should be selected whenever possible in order to prevent continual re- 
 making of column forms, and frequent changes in column sizes should be avoided (at least in 
 the case of light-floor construction), not only on account of the column forms themselves, but 
 on account of the beam and girder forms or slab forms which frame into them. Also, where it 
 is feasible, beam sizes should be so chosen that local standard widths of lumber may be employed 
 without splitting; and, at the same time, the sizes and spacing of the beams should be made so 
 uniform that the contractor may use his forms repeatedly, thus greatly reducing the expense for 
 lumber and eliminating the cost of making new forms. In many cases it will be found that a 
 slight excess of concrete will save many times its cost in carpenter work and lumber. 
 
 61. Lumber for Forms. — The kind of lumber to use for forms depends upon the character 
 of the work and the available supply in the local yards. Spruce seems to be the best all-round 
 material. It can readily be obtained in almost any locality and is undoubtedly an excellent 
 lumber to use for joists, studs, and posts. For sheathing, however, white pine is better than 
 spruce by reason of its smoothness and its resistance to warping, but this wood is generally too 
 expensive (except for cornice and ornamental work), and spruce makes a good substitute. 
 If white pine is to be used for sheathing, the fact should not be overlooked that this kind of 
 lumber has little durability on account of its extreme softness and would not give good results 
 if the forms were to be used many times. 
 
 Aside from spruce or white pine, Norway pine and southern pine are generally the most 
 available and give satisfaction. Hemlock is not usually desirable, especially for that part of 
 form, work which comes into contact with the concrete', but it is sometimes used for ledgers, 
 studs, and posts. This wood is too coarse-grained to be suitable for sheathing and is liable to 
 curl when exposed to the weather or to wet concrete. 
 
 It is safe to say that lumber which is only partially seasoned should be the kind employed 
 in form construction. Kiln-dried has a tendency to swell when soaked by the concrete, and 
 this swelling causes bulging and distortion of the forms. Green lumber, on the other hand, 
 dries out and shrinks if allowed to stand too long before the concrete is placed; fortunately, 
 though, this tendency of the green lumber to check and warp may be prevented to some extent 
 by keeping the boards thoroughly saturated with water. When using natural, well-seasoned 
 lumber care should be taken not to drive the work up too close, since forms should always be 
 left in a position to experience some slight swelling without any undesirable results. 
 
 Sheathing lumber should be dressed at least on one side and both edges, even for non-ex- 
 posed surfaces, as the removal and cleaning of the forms are greatly facilitated thereby. In face 
 work, where a smooth and true surface is quite important, the lumber employed should be 
 dressed on all four sides. Lumber which is dressed in this manner is easy to work up and place, 
 and this fact alone usually more than offsets the cost of dressing. 
 
Sec. 2-62] 
 
 GENERAL METHODS OF CONSTRUCTION 
 
 95 
 
 In floor and wall panels, tight joints between boards may be obtained by using either tongue- 
 and-grooved stock or stock cut with one edge beveled and the other square. Beveled-edge 
 stuff is especially good where very dry lumber is used, for the reason that buckling is prevented 
 by the edges crushing as the boards swell. Tongue-and-grooved stuff, however, gives the best 
 results under ordinary conditions. This stock suffers to some extent from breakage and costs 
 more than beveled-edge, but it gives smoother surfaces after repeated use. Joints in forms 
 for columns, beams, and girders are sometimes made tight simply by dressing the lumber true 
 to edge, forming what are called square joints or butt joints. 
 
 The thickness of lumber to use in form work depends to some extent upon the number of 
 times the forms are to be used and, in the case of floor and wall sheathing, upon whether the 
 boards are to be built into panels or nailed each time to the supporting joists or studs. Gener- 
 ally speaking, 1-in. stock dressed both sides to '^^{q in. (called J^-in. boards) is employed for 
 floor sheathing; 1, 13^, or l^^-m. stock (approximately l^^e-in., dressed) for column and wall 
 sheathing and for the sides of beams and girders; and 13^ or 2-in. stock (l^-in. dressed) for 
 beam and girder bottoms. The size and spacing of joists, studs, girts and posts depend upon 
 the strength and rigidity required (see Art. 65). 
 
 62. Removal of Forms. — The length of time that forms should remain in place depends 
 upon a number of factors, chief among which are: (1) the temperature and humidity of the air; 
 (2) the nature of stress to which the member is to be subjected; (3) the normal setting and hard- 
 ening properties of the concrete; and (4) the relation of the dead to the live load. 
 
 As is well known, weather has a great deal to do with the setting and hardening of concrete. 
 The warmer and drier the weather, the more rapid the hardening. Cold or wet weather, on 
 the other hand, retards hardening and the forms under such conditions must remain in place 
 a much greater length of time. If concrete freezes, it will not continue to harden until it has 
 thawed, and then it will acquire strength very slowly. ^ 
 
 The forms for members subjected to bending should remain in place longer than forms for 
 columns or walls. In other words, slabs, beams, and girders which depend upon the bonding 
 action between the steel and concrete for their ability to carry any material load should not be 
 left unsupported as early as vertical compression members which will bear their own weight 
 when the concrete is quite green. 
 
 The normal setting and hardening properties of the concrete vary with the quality of the 
 cement, the consistency and richness of the mixture, and the freedom of the small aggregate 
 from impurities. Some cements harden more rapidly than others and permit an earlier removal 
 of forms. Forms may also be removed sooner for rich mixtures than for lean. Then again, if 
 impurities are present in the sand, the concrete may set abnormally and failure will result if the 
 concrete is not given a sufficient time to harden. 
 
 The greater the proportion of dead to live load, the longer forms should remain. For 
 example, forms for an overhanging cornice should remain in place longer than forms for almost 
 any other member, since the dead weight is a very large percentage of the total weight which it 
 is designed to carry. By a similar reasoning, long-span beams or slabs should be supported a 
 longer time than short ones; roofs longer than floors; columns in the upper stories longer than 
 columns in the lower stories; and columns in general longer than footings. 
 
 It is becoming the practice on the best work to mold cubes of concrete from each day's 
 work and allow them to set on the job under the same conditions as the concrete in the struc- 
 ture. 2 When it is thought desirable to remove the forms, these cubes are tested for compression 
 in order to determine whether the concrete has sufficient strength to carry its load. Good judg- 
 ment should be exercised even under such conditions since the plan does not provide for the 
 possibility of an occasional poor batch of concrete. 
 
 Many builders have rules for the removal of forms under ordinary conditions. Such rules, 
 or course, can only be roughly approximate, and experience and judgment are necessary in 
 
 1 See Art. 29 and Art. 16, Sect. 5. 
 
 ^ See chapter on "Field Tests of Concrete." 
 
96 CONCRETE ENGINEERS' HANDBOOK [Sec. 2-63 
 
 using them. The following rules are recommended by the Illinois Department of Factory 
 Investigation :^ 
 
 Time Required Before Removing Forms 
 
 
 Above GO°F. 
 
 50° to 
 60°F. 
 
 40° to 50°F. 
 
 Less than 40°F. 
 
 Columns 
 
 Side forms for girders 
 and beams. 
 
 Bottom forms of slabs (6 
 
 ft. or less span). 
 Bottom forms of beams 
 and girders (less than 
 
 14-ft. span). 
 
 Form work in buildings should be so designed that the column forms may be removed 
 without in any way disturbing the supports of the beams and girders. This practice bares the 
 concrete of the column to the hardening action of the air and permits a defect to be detected 
 and remedied before any load is brought to bear upon the column. The beam and girder sides 
 are next taken down, but this is not done until the slabs are strong enough to stand up without 
 support. Sometimes, however, as a matter of safety, the slab supports (namely, the sheathing 
 and joists) are replaced with temporarj^ uprights bearing against a plank on the underside of the 
 slab. The beam and girder bottoms and the posts supporting them are left in place longer 
 than any of the other formwork, and should not be taken down until there is absolutely no 
 doubt as to the strength of the members supported. Walls are usually built separately from 
 the other parts of the building, and a wall form may be removed whenever the concrete in the 
 wall is hard enough to bear its own weight. 
 
 63. Number of Sets of Forms in Building Work. — The number of sets of forms which are 
 necessary in building work depends upon the weather, the variation in the shape of the members 
 from floor to floor, and the floor area. Under reasonably good summer conditions 13^^ sets of 
 forms (forms for 1)-^ stories) will serve the purpose and allow the work to progress at the usual 
 rate of a story in a week or 10 days. If the floor framing, however, is particularly complicated, 
 one set of forms may be the more economical if the speed of construction is not the leading con- 
 sideration. In such a case, of course, extra lumber must be used for beam and girder bottoms 
 and for supports which must be left in place. Where the floor area is large and the construc- 
 tion is fairly uniform throughout, even less than one set of forms with additional beam bottoms 
 and posts may be sufficient. 
 
 64. Examples of Form Design. 
 
 64a. Column Forms. — Fig. 25 shows a typical form for a rectangular column. 
 Two sides of the column are held together, as shown, by bolts, and the two opposite sides by 
 hardwood wedges between the bolt and the form. The sheathing or lagging boards run the 
 entire length of the column and are made up into panel units by means of the cleats, which serve 
 also as clamps. A somewhat similar column form is shown in Fig. 38, page 102. 
 
 A common type of rectangular column form where wooden wedges are used to do all the 
 clamping is illustrated in Fig. 26. The tightening is made possible by the use of stop blocks on 
 each clamping piece. When a reduction is made in the size of column, these blocks must either 
 be ripped off or additional blocks nailed on. The boards comprising each side of the column 
 form are usually battened or cleated together so as to form a panel unit. This practice allows 
 the clamps to be put on separately and thus permits stop blocks on the clamps to be easily 
 
 1 See also rules given in "Concrete, Plain and Reinforced," by Taylor and Thompson (1916 edition). 
 
 2 Add 1 day for each additional foot of span. 
 
 Within 3 days 
 
 5 
 
 days 
 
 Within 4 days 
 
 6 
 
 days 
 
 Within 4 days^ 
 
 8 
 
 days 
 
 Within 14 days^ 
 
 18 
 
 days 
 
 Not less than 10 days 
 Not less than 10 days 
 
 Not less than 14 days 
 
 Not less than 14 days 
 
 Not until tests 
 have been 
 made indicat- 
 ing that the 
 concrete is set. 
 
i Sec. 2-64a] 
 
 GENERAL METHODS OF CONSTRUCTION 
 
 97 
 
 changed when the column form is remade. It is a common practice to make up the sides of 
 , column forms with narrow strips of sheathing in order to facilitate the reduction in size of columns 
 If from floor to floor. A method of reduction sometimes employed by the Aberthaw Construction 
 
 Fig. 25. 
 
 Co. is shown in Fig. 27. The column form represented here is also illustrated in Plate III, 
 page 114. The use of narrow sheathing strips applies especially to interior column forms since 
 exterior or wall columns are usually of the same width from basement to roof and change but 
 little in thickness. 
 
 fa J Blocks nailed Hardwood 
 
 Fig. 26. 
 
 Fig. 28 and Plate II, page 113, show a method of bracing column forms by means of 4 
 by 4-in. "whalers" or posts. The use of Henderson clips should be noted. Sometimes the 
 bolting is done directly to the whalers. 
 7 
 
98 
 
 CONCRETE ENGINEERS' HANDBOOK 
 
 [Sec. 2-64a 
 
 la^gr/nj planed four 5/ 
 
 ■Clip 
 Ordinal 
 
 First. Reduction 
 
 \Pieces ' b'' removed, 
 side x" moved in, s'xA" 
 deal- cut doyvn and I'lcS" 
 set in as shown behind 
 boarding. 
 
 Fig. 27. 
 
 Second Reduction 
 Pieces "a" and b" removed, 
 side "x" moved in, 3x4''cuf 
 down and I x s" nailed^ 
 'behind cleats 
 
 n 
 
 
 n n - 
 
 1 
 
 
 . n 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 jj- 11 II • 
 
 1 
 
 
 Plan 
 
 Fie. 28.- 
 
 Detail Cff Column Section B-B 
 
 -Wall column forms employed in Morrill building, Portland. Me. 
 
Sec. 2-ma] 
 
 GENERAL METHODS OF CONSTRUCTION 
 
 99 
 
 Other types of rectangular column forms in common use are shown in Figs. 29, 30, and 31. 
 : The cleats which form part of the clamps are nailed to the sides when making. The clamp 
 [ shown in Fig. 31 is patented and known as the "New England Column Clamp," controlled 
 by the N. E. Column Clamp Co., Boston, Mass. It consists of angle irons punched at frequent 
 intervals with bolts for holding the form together wherever needed. Angles and bolts may be 
 I bought in the open market. The New England Column Clamp Co. sells only the rights to use. 
 Fig. 32 illustrates still another type of rectangular column form. The use of horizontal 
 sheathing should be noted. The sheathing is nailed to the vertical pieces before erection. 
 
 A patented steel clamp, known as the "Gemco Column Clamp" and manufactured by the 
 Gemco Mfg. Co., Milwaukee, Wis., is shown in Fig. 33. These clamps are described by the 
 manufacturers as follows: 
 
 Fig. 29. Fig. 30. Fig. 31.— New England column 
 
 clamp. 
 
 A Gemco Square-column Clamp is composed of four, straight, interchangeable, steel bars, each 2 in. wide 
 by YiQ in. thick. One end of each bar is provided with a hardened toothed locking dog pivoted between two plates 
 which are firmly riveted to the opposite sides of the bar. These plates fully protect the locking dog from damage, 
 but sufficient space is allowed to permit the bars to slide freely when the locking dog is not engaged. 
 
 The clamp is set by pressure on the tightening lever which slides over the free end of the bars, as shown in 
 Fig. 33. When the bars are drawn to position, the locking dog is set by pressure of the fingers or light tap of a 
 hammer and it positively locks the clamp. Tightening lever is detachable and can be used on any Gemco Square- 
 column Clamp. 
 
 A Gemco Clamp can be assembled, tightened and set in 1 min. It automatically squares a column and 
 can be tightened from all sides so as to positively close all cracks, thus retaining all the valuable part of the mixture 
 which would otherwise flow out with the water. Gemco Clamps will square a rectangular pier or column in the 
 same manner. 
 
 These clamps can be removed in the same time which it takes to adjust them. The tightening lever is 
 used to partially relieve the strain when the locking dog can be disengaged by the use of a claw hammer and a 
 nail or punch inserted in the hole at the corner of the dog. After two diagonally opposite corners are loosened the 
 entire clamp is free. By working two men on opposite sides of the column and at diagonal corners of the clamps, 
 the work of setting or wrecking can be completed in a very short time. 
 
 When knocked down these straight, interchangeable bars of steel are easily transferred from floor to floor 
 Or building to building. 
 
 To avoid unnecessary expense by using large clamps on small columns the Gemco Square-column Clamps 
 are manufactured in three stock sizes. Special sizes made only on specific order. 
 
100 
 
 CONCRETE ENGINEERS' HANDBOOK 
 
 [Sec. 2-64a 
 
 Stock No. 
 
 Column size, 
 inches 
 
 Weight, 
 pounds 
 
 
 24 
 
 291^ 
 
 C-11 
 
 30 
 
 35 
 
 C-12 
 
 36 
 
 
 C-13, Tightening lever 
 
 
 4-K 
 
 Another patented type of all-steel column clamp which requires no wood framing is shown 
 in Fig. 34 — manufactured by the K. & W. Clamp Co., Minneapolis, Minn. 
 
 All square or rectangular-column forms should have bevel strips in the corners. Sharp 
 corners in concrete work should always be avoided, not only because it is difficult to tamp 
 concrete so as to obtain perfect corners, but because the edges are likely to be chipped off when 
 the forms are removed or while the concrete is still green. The bevel strips are usually nailed 
 to two opposite panel units when the forms are being made up. 
 
 Fig. 32. 
 
 Octagonal and round-column forms with wooden clamps are not susceptible of ready 
 arrangement into units, and are clumsy and quite expensive to make. The problem seems to be 
 solved in present-day practice by the use of either iron bands or chains. Fig. 35 shows a 
 form clamp manufactured by the Sterling Wheelbarrow Co., Milwaukee, Wis., which can be 
 used on any type of column form. The band-iron placed around the form (view a) is passed 
 through the clamping head, and drawn tight. The manufacturers describe the operation as 
 follows: 
 
 The operator grasps the band with one hand and ratchets the lever several times. This ratcheting grips 
 and releases the band, drawing it perfectly tight no matter how kinked the band was before being put around the 
 form. When he sees that the band is tight enough, the operator presses the lever down close to the form, thus 
 crimping the band in such a way that slippage is impossible (view b), and locks the clamp by inserting the lever into 
 
Sec. 2-64a] 
 
 GENERAL METHODS OF CONSTRUCTION 
 
 101 
 
 one of the openings in the locking device (view c). When knocking down the form, it is necessary simply to release 
 the lever from the locking device, return the clamping head to position, and pull the band out. 
 
 C 
 
 Fig. 35. — Sterling form clamp. 
 
 A Gemco Clamp for round columns (Fig. 36) is described by the manufacturers in the 
 following manner: 
 
102 
 
 CONCRETE ENGINEERS' HANDBOOK 
 
 [Sec. 2-64a 
 
 A Gemco Round-column Clamp is made of a strip of m-in., 16-gage band iron, one end of which carries a 
 malleable-iron casting which holds the hardened tooth locking dog. When in use the free end of the band iron is 
 slipped under the locking dog and the clamp can be quickly and easily drawn to and set at any desired degree of 
 tension with a tightening lever. The lever is detachable and can be used on any size of Gemco Round-column 
 Clamp. 
 
 A clamp may be struck or wrecked in less time than required for setting, and in this operation the detachable 
 lever is used to relieve the strain. 
 
 Gemco Round Clamps are made in two stock sizes — for 24-in. and 36-in. columns. Special sizes made 
 to order. 
 
 Stock No. 
 
 Column size, inches 
 
 Weight, pounds 
 
 C-20 
 
 24 
 
 3 
 
 C-21 
 
 36 
 
 
 
 
 2K 
 
 Fig. 36. — Gemco round-column clamp. Fig. 37. — Universal round-column clamp. 
 
 Another patented clamp for circular columns is shown in Fig. 37, manufactured by the 
 Universal Form Clamp Co., Chicago, 111. These clamps are made in one size only, i.e., size 
 
 c C c < 
 
 Fig. 38. 
 
 C-1 for i^-in. rods, 5 in, long overall. A tightening wrench shown in Fig. 60, page 117, is 
 used with this clamp. 
 
Sec. 2-646] 
 
 GENERAL METHODS OF CONSTRUCTION 
 
 103 
 
 Wire and rod clamps described under the heading "Wall and Pier Forms" (page 111) are 
 used to some extent on forms for columns. 
 
 646. Beam and Girder Forms. — Fig. 38 is an isometric view of a typical floor. 
 Joists support the slab sheathing and are carried in turn by ledgers or joist bearers (see also 
 
 • Beam opening 
 
 Fig. 40. 
 
 Plates I and IV, pages 112 and 115 respectively). The method of clamping the beam and 
 girder forms is shown more in detail in Figs. 39 and 40. Separate blocks are sometimes used 
 instead of a joist bearer. Plate II, page 113, shows a method of 
 supporting slab forms by notching out the beam cleats to receive 
 the joists. 
 
 A number of different methods are employed to clamp beam 
 and girder forms and prevent them from spreading due to the 
 pressure of wet concrete. The method referred to above is per- 
 haps the most common, but bolts either above or below the beam ^^-3'' 
 bottom are sometimes employed. Clamping by means of ribbands 
 is well illustrated in Plates I to III inclusive, pages 112 to 114. 
 
 Typical girder-form construction is shown in Fig. 40. Some- Fig. 41. 
 
 times the girder sides are not erected complete, the portion between 
 
 the beams being erected in separate sections, as shown in Plate I, page 112. The lower part of 
 the girder sides are made of thick plank in order to provide a suitable support for the beams. 
 A common method of bracing the outer side of a wall beam is shown in Plate I. Another 
 
104 
 
 CONCRETE ENGINEERS' HANDBOOK 
 
 [Sec. 2-64c 
 
 method is by means of a cantilever brace. Methods shown in Fig. 28 and Plate II, page 113, 
 should also be noted. Fig. 41 shows a form of ordinary post or shore in common use. Haunch 
 design may be seen in Plates I and IV, pages 112 and 115 respectively. 
 
 64c. Slab Forms. — Two types of slab forms need to be considered for ordinary 
 construction where beams and girders are employed: (1) the panel type (the only type of form 
 construction so far treated) ; and (2) slab forms made up into box shape, comprising the joists 
 and the sides of the beams and girders. 
 
 In the usual type of construction, designated above as type 1, the beam and girder forms 
 are either erected complete and the posts set in place after the erection, or the beam and girder 
 bottoms are spiked in place to the post caps, and the sides are erected as separate units. After 
 this much is accomplished by either method, the joists are then put in position and the panel 
 
 Fig. 42. 
 
 units are placed on top of them without nailing. Notching out of the slab panels is usually 
 necessary at the columns. When the forms are to be used but once, the slab sheathing is not 
 made up in advance into panels, but is lightly nailed to the joists after they are in place. 
 
 Fig. 42 shows one method of making up slab forms into box shape — the form construction 
 being that used at the University of Wisconsin. Planks extend in two directions supporting 
 the cells. The steel and continuous stirrups are set for demonstration only. 
 
 A collapsible core box is used by at least one prominent builder (Fig. 43). A plank resting 
 on cleats on the sides of the cores forms the bottom of the beam mold. The main girders are 
 molded in similar spaces between the ends of the cores in one panel and those in the next panel. 
 The molds are made in two equal parts with a hinged joint through the longitudinal center line 
 of the upper surface, and are held open by transverse struts between the lower edges. When 
 the forms are to be removed, the transverse struts are knocked out and the boxes are collapsed. 
 The use of these molds presupposes a standardized layout. 
 
 Fig. 43. 
 
 Forms for flat-slab floors are much simpler than for the ordinary beam-and-girder construc- 
 tion — with the exception, of course, of the forms for the flaring column heads. Typical de- 
 signs of forms for flat-slab floors are shown in Fig. 44 and Plate V, page 116. Corrugated 
 iron has been used to a considerable extent for sheathing where the ceiling surface does not 
 need to be absolutely smooth. 
 
 A new and novel system of form work (Fig. 45) has been devised by Jesse E. Hodges, of 
 Cincinnati, which is applicable to both beam-and-girder and to fiat-slab constructions. 
 Edward O. Keator & Co. of Cincinnati, Ohio, are sole agents for this system. 
 
 The slab forms consist of very light metal sheathing held on a matting (usually wooden 
 strips 2 in. by 2}4 in. by 4 ft. 6 in.) which is supported by stringers placed about 4 ft. on centers. 
 
I 
 
 Sec. 2-64c] 
 
 GENERAL METHODS OF CONSTRUCTION 
 
 105 
 
 The matting is formed by connecting the small wooden strips by a light metal chain near each 
 end so as to allow an opening of about 2\i in. between the strips. The mats are made of short 
 lengths so that they can be rolled into bundles and not be too heavy for one or two men to carry. 
 Usually one stringer is used halfway between beams, and ledgers on the beam sides hold the 
 ends of the mat. The stringers do not have to be changed in length for different jobs as they 
 can be used in long lengths and lapped. 
 
 Fig. 44. 
 
 The adjustable shores (Fig. 46) are made from 8-ft. lumber, with a 4 by 4-in. post and two 
 2 by 4-in. side pieces, and are adjustable from 8 to 14 ft. This range is sufficient for ordinary 
 work but longer shores can, of course, be built for special work. In erecting one of these shores, 
 the clamps are loosened and the 4 by 4 drawn out to length designated by a measured pole 
 in the usual manner. Clamps are then tightened and the shores raised. The feature of this 
 
 Fig. 4o. — Hodges system of foriiiwork. 
 
 type of shore is a positive clamp which sets itself when the cam is in driving position and the 
 shore is under load. This shore also has a detachable head which may be left on, in setting 
 shores under a beam, or removed in using the shores for flat-slab floors. With the head removed 
 the form stringers rest in a fork made by the two upper pieces of the shore, which makes it un- 
 necessary for a man to climb up and nail cleats on each side of the joist after the shore is in place 
 
106 
 
 CONCRETE ENGINEERS' HANDBOOK 
 
 [Sec. 2-64c 
 
 Wedges are unnecessary with the Hodges shore as leveHng the centering for the floor may be 
 accompHshed with a kind of racket-jack arrangement shown in Fig. 47. To adjust the length 
 of shore a steel bar which is notched to form a rack is attached to one face of the 4 by 4. A fork- 
 
 preferably S'-O" 
 
 Top of 4"X4" beveled to pass 
 through battens 
 
 Section "A A" 
 Fig. 46. — Hodges adjustable shore. 
 
 ended lifting lever is then fitted over the timber, with its fulcrum bolt resting in one of the slots 
 of the bar and the ends of the fork engaging the lower ends of the 2 by 4-in. sticks. With the 
 
 Cam 
 
 Section X-Y 
 (Enlarged } 
 
 Fig. 47. 
 
 cams loosened, the upper portion of the shore is raised by bearing down on the lever handle, 
 and when in proper position is locked by the cams. 
 
 Another type of adjustable shore is shown in Fig. 48, manufactured by the H. W. Rocs 
 
jj Sec.2-64c] GENERAL METHODS OF CONSTRUCTION 107 
 
 Co., Cincinnati, Ohio, and known as the *'Roos Self-lock Adjustable Shore." It is made up 
 for average use with a 6-f t. piece of standard pipe and two pieces of 2 by 4's dressed side and edge, 
 8 ft. long. Only the necessary castings are sold by the company above mentioned. The opera- 
 tion of the shore consists in taking the extension member by the two legs and lifting it to the 
 
 Fig. 48. — Roos self-lock adjustable shore. Fig. 49. — Gemco adjustable steel shoring. 
 
 desired height. The slightest pressure on the two legs of the extension member toward the pipe 
 causes the yokes to grip and the shore is ready for the load. The greater the load the greater 
 the grip. For finer adjustment the shore can be raised or lowered with the jacking device 
 which can be attached to any point under the two legs of the shore by turning the handled screw 
 
 Fig. 50. 
 
 and the upper membar can be set at its final height with the adjusting lever. A turn of the 
 thumb-screw lock sets the yokes and prevents any possible release through outside jarring. 
 The jacking device is then removed for adjusting the next shore. 
 
 "Gemco Adjustable Steel Shoring" is shown in Fig. 49. The shores are adjustable in 
 
108 
 
 CONCRETE ENGINEERS' HANDBOOK 
 
 [Sec. 2-64d 
 
 height from 9 ft. to 12 ft. 6 in. For adjusting the height or raising the load, a detachable lever 
 is used. These shores are manufactured by the Gemco Mfg. Co., Milwaukee, Wis. 
 
 There are many other types of wooden floor forms but those described above illustrate the 
 main features in floor-form design. For steel floor forms see page 135. 
 
 Section A-B 
 
 Fig. 51. 
 
 64ci!. Column Heads. — The tops of column forms are sometimes made separate 
 from the column forms proper. Generally this is done either to avoid remaking the column 
 tops to fit varying sizes of beams and girders, or to facilitate the construction of special form 
 
 centering 
 
 Post" 
 
 /r'emoyable 
 \2'i<4'[ blocks 
 
 K 
 
 -'Framed to true 
 oct:7^on foJbe 2d'^p/us 
 c//ameter of co/umn 
 
 y/ood plank- centering 
 
 ■Jo/st 
 
 ^■Purlin 
 
 Fig. 52. 
 
 heads. Fig. 50 shows an assembly of column heads for regular interior columns of the 
 Larkin Go's, building, Buffalo, N. Y., designed by the Aberthaw Construction Co., the contrac- 
 tors. These heads are also shown, but not in detail, in Plate IV, page 115. 
 
 The problem of the column head in flat-slab construction has been met practically from 
 
Sec. 2-64dJ 
 
 GENERAL METHODS OF CONSTRUCTION 
 
 109 
 
 Fig. 54. — Steel column head to fit octagonal column. 
 
110 
 
 CONCRETE ENGINEERS' HANDBOOK 
 
 [Sec. 2-64(i 
 
 the first by the use of metal in some form. The metal mold used by the Aberthaw Construc- 
 tion Co. in the Massachusetts Cotton Mills at Lowell, Mass., is shown in Fig. 51. Note that 
 the head is very simple, a square where it leaves the column, with flaring beveled corners, which 
 finally form an octagon at the ceiling level. 
 
 Fig. 52 shows a type of column-head form used in constructing the Larkin warehouse at 
 Chicago, 111. It is one of the standard types of flaring column caps manufactured by the Des- 
 
 ^y- Apron Strip 
 
 ! <-Dia.Coh> 
 
 ijnih. 
 
 ■I2"R. 
 
 Top Diameter —X-Head 
 
 ''^ \^*30peninQ-Y extension 
 Column 
 
 rClamp 
 
 extension 
 
 Plan of Forming of 
 24-inch Column 
 
 Upper unit-'--'' 
 
 Plan of Joint Where 
 Units Telescope 
 
 Section of Complete 
 Column Form 
 
 Standard iS'Head 
 
 Wedffe--- 
 
 Fia. 55. — Adjustable steel column heads. Hydraulic Pressed Steel Co. 
 
 lauriers Column Mold Co., St. Paul, Minn. The columns themselves were molded by steel 
 forms and these forms were placed after the erection of the upper floor falsework. The column- 
 cap section was suspended from the floor forms and the remaining column sections bolted to it. 
 
 Figs. 53 and 54 show adjustable column heads as manufactured by the Blaw Steel Construc- 
 tion Co.; conical and octagonal heads only are illustrated here, but this company also makes 
 rectangular head molds and molds which spread from a rectangular section at the column to an 
 
Sec. 2-646] 
 
 GENERAL METHODS OF CONSTRUCTION 
 
 111 
 
 octagonal shape at the floor slab. The conical head is in two parts; the top portion remains the 
 same for all columns while the bottom is fitted to columns of various diameters. The octagonal 
 mold is also adjustable at the lower end and may be used in connection with either octagonal 
 and circular steel column forms or with wooden octagonal forms. The Hydraulic Pressed Steel 
 Co., Cleveland, Ohio, manufacture round steel column forms with both standard and adjust- 
 able column heads (see Figs. 55 and 56). 
 
 Sheef" mefal decking for depressed pane! 
 
 Cover opening with-' 
 quari-er strip 
 
 ^'Fa/se yyork block nailed 
 fo framing for support 
 of head form. 
 
 Frame fo gi/e from lj'fo2" 
 clearance for actual diameter 
 of head 
 
 •Bo/t together 
 
 Fig. 56. — Method of forming depressed panels. 
 
 64e. Wall and Pier Forms. — Forms for bridge piers and for walls of appreciable 
 height are usually constructed of either 1-in. or 2-in. plank nailed to studs and held by horizontal 
 waling pieces, with tie bolts extending across the pier or wall between opposite wales. The 
 wales, which often consist of two planks fastened together but separated by spacing blocks, 
 are set edgewise against the form studs and the tie bolts are carried through the openings which 
 occur in the waling pieces. Wire is sometimes used for bracing and is tightened either by wedges 
 or by twisting. The wire pulls against spreaders which are inserted between forms and which 
 are removed as- the concrete level rises. 
 
 Where bolts are employed in pier and wall construction, a number of different methods are 
 used for withdrawing the bolts. One method is to cover each bolt with old pipe cut somewhat 
 shorter than the inside dimensions of the forms, and to place a wood washer at each end of the 
 pipe. When the forms are taken down, the bolts are easily drawn out of the pipes, the wood 
 washers are then cut out of the face of the concrete, and the holes pointed up. Another method 
 is. to make the bolts in three pieces, with the middle piece occupying the same position between 
 the forms as the pipe above described. This middle section is connected with the end pieces by 
 means of ordinary unions. When the concrete has set sufficiently, one turn releases the end 
 sections and the holes left in the work are plugged with mortar. Still another method is to use 
 a patented casting with set screw, also a tightening wrench and rod puller. The tightening 
 wrench is a device for the purpose of exerting a pressure against the form to draw it in line or to 
 desired dimensions. The rod puller is for removing or pulling the tie rods from the concrete 
 after the concrete has set sufficiently to stand alone. 
 
 The second method described is shown in Figs. 57, 58 and 59. (In Fig. 59 wire is used in- 
 stead of the middle piece of bolt.) The bar couplings shown in Fig. 57 are manufactured by the 
 Marion Malleable Iron Works, Marion, Ind., and are stocked in seven sizes threaded to take 
 bolts H, H, 1, IH, and l}i in. "Universal Cone Nuts," shown in Fig. 58, are manu- 
 
8 
 
114 
 
 CONCRETE ENGINEERS' HANDBOOK 
 
 [Sec. 2-64e 
 
;ec.2-64el GENERAL METHODS OF CONSTRUCTION 115 
 
 Plate IV. 
 
 Section B-S 
 
 Assembly of forms, Larkin Co.'s building, Buffalo, N. Y. 
 
116 
 
 CONCRETE ENGINEERS' HANDBOOK 
 
 [Sec. 2-64c 
 
Sec. 2-64e] 
 
 GENERAL METHODS OF CONSTRUCTION 
 
 117 
 
 t/se of ty/re form tightener 
 
 One face of all to be finished Both faces of wall to be finished 
 
 Fig. 61. 
 
Fig. 64. — Typical pier forms used in bridges of Luten design. 
 
Sec. 2-64e] 
 
 GENERAL METHODS OF CONSTRUCTION 
 
 119 
 
 factured by the Universal Form Clamp Co., Chicago, 111. Cut shows washers used with cones 
 and countersunk into forms, but this is not necessary. Universal rod clamps (Fig. 60) can be 
 used on the outside of forms in place of nuts and washers. The device shown in Fig. 59, known 
 as "Tyscru," is being marketed by the Unit- Wall Construction Co., New York City. 
 
 A patented casting, rod tightener, and rod puller such as used in the third method above 
 described are shown in Fig. 60. These devices are manufactured by the Universal Form Clamp 
 Co., Chicago, 111. The rod clamps are made in six sizes, namely: No. 1 for 3'^-in. and ^ie-in. 
 rods; No. 2 for ^^-in. rods; No. 3 for }^-in. rods; No. 4 and No. 4 extra heavy for ^^-in. rods; and 
 
 ; No. 5 for ^^-in. rods. The No. 4 is recommended for column clamping and No. 4 extra heavy 
 
 ijifor large retaining walls. 
 
 Patented wire form clamps are illustrated in Figs. 61, 62 and 63. The wire form tightener, 
 shown in Fig. 61, is manufactured by the Marion Malleable Iron Works, Marion, Ind., and is 
 made in three sizes, threaded to take ^i, and 1-in. bolts. The tightener is simply attached 
 to a wire slipped through from one side of the forms and a bolt is screwed into it from the face 
 
 d4'^-l8'^ Jon^ivfaslen 
 
 Fig. 05. 
 
 side. By turning this bolt any degree of tension may be obtained in the wire. The " Universal 
 Wire Clamp" is illustrated in Fig. 62. The clamp is placed against wales or studding and wires 
 bent as shown in (a). The clamp should be placed in a position so as to allow the wires to enter 
 the slots at nearly right angles. For locking all that is necessary is to pull handle down so that 
 it takes a position as shown in (6) and (c). The ''American Wire Clamp" is shown in Fig. 63 
 and is similar in operation to the clamp just mentioned.. 
 
 Where especially good work is desired, forms are lined with galvanized iron. For high 
 piers or walls, the forms are constructed in large panels. After the concrete has been 
 constructed to a proper height and the last course has set several days, the panels are discon- 
 nected and hoisted to a higher position and then reassembled for concreting, and so on. 
 
 When the ends of bridge piers are rounding, special forms are necessary. In the construc- 
 tion of the Atherton Ave. bridge over the Pennsylvania R. R. tracks in Pittsburg, the forms 
 for the curved ends of the piers were built of 1-in. by 2-in. strips nailed to horizontal segmental 
 wales. These wales were nailed to the wales of the side forms. The rounding forms were kept 
 in place by wiring to dowels set in the foundation concrete. Before starting the erection of the 
 form work, a flexible panel was made by nailing galvanized-iron sheets to the 1-in. by 2-in. strips. 
 This panel was bent against the wales, which acted like hoops. Curved pier forms used in 
 bridges of Luten design is shown in Fig. 64. 
 
120 
 
 CONCRETE ENGINEERS' HANDBOOK 
 
 [Sec. 2-65 
 
 Fig. 65 illustrates a simple method employed by the Aberthaw Construction Co. for build- 
 ing a wall of considerable height by means of movable forms. A simple method is also shown 
 in Fig. 66. 
 
 Wall forms of small height may be braced by battered posts outside. Fig. 67 shows forms 
 for a concrete foundation wall for small buildings where no cellar is necessary. Such forms may 
 
 Fig. 66. Fig. 67. 
 
 either be constructed in sections or built in place. It should be noted that the forms are sus- 
 pended over the trench and are not allowed to rest upon the new concrete. Figs. 68 and 69 
 show other methods of form construction for low walls. No outside form is needed in the 
 lower part of such walls as shown in Fig. 68 when the earth is extremely hard and firm. 
 
 Fig. 70 shows a common type of form design for curtain walls below windows. Curtain 
 walls for complete wall panels are of similar construction. 
 
 Fig. 68. Fig. 69. 
 
 66. Design of Forms. — Although form design in most cases has been left entirely to the 
 discretion of the superintendent or foreman on the job, it has been found by a number of en- 
 gineering contractors that it pays on large or important work to have the forms designed in the 
 drafting room, provided such designing work is done under the direction of a man of practical 
 ability. This plan is not only more economical, but also eliminates the possibility of an error 
 being made in the proper size and spacing of form members. 
 
Sec. 2-65o] 
 
 GENERAL METHODS OF CONSTRUCTION 
 
 121 
 
 Many details of form design depend almost entirely upon judgment and experience; but 
 the size and spacing of supports for sheathing in slab, column, and wall forms and the spacing 
 of posts for slabs, beams, and girders are capable of being determined by scientific calculation 
 and such practice results in the use of a minimum quantity of lumber consistent with the deflec- 
 
 Forms -for Sill 
 fldte cast after Curfcrfn Wall ) 
 
 Elevation 
 
 tion allowed. In no case should the spacing of the supports be greater than a safe span for 
 the sheathing. Deflection should always be considered in order to give sufficient stiffness and 
 thus prevent partial rupture of the concrete. 
 
 65a. Values to Use in Design. — In calculating floor forms, we must add to the 
 weight of the concrete itself, a live load which 
 may be assumed as liable to come upon the 
 concrete while it is setting. This live load is 
 usually taken at 75 lb. per sq. ft. except in 
 cases where the floor is made a storage for 
 cement or sand. 
 
 The pressure of concrete against forms 
 depends principally upon the rate of filling 
 and the temperature. The accompanying 
 table^ is from tests made by Major Francis 
 R. Shunk, Corps of Engineers, U. S. A. 
 
 Tests on the pressure of wet concrete 
 
 iFrom Eng. Rec, Jan. 15, 1910. 
 
 Rate of filling 
 (vertical 
 feet per 
 hour) 
 
 Temperature 
 
 80° 
 
 70° 
 
 60° 
 
 50° 
 
 40° 
 
 2 
 
 530 
 
 560 
 
 600 
 
 680 
 
 790 
 
 3 
 
 690 
 
 720 
 
 810 
 
 920 
 
 1,080 
 
 4 
 
 820 
 
 870 
 
 980 
 
 1,130 
 
 1,340 
 
 5 
 
 930 
 
 990 
 
 1,120 
 
 1,310 
 
 1,570 
 
 6 
 
 1,020 
 
 1,090 
 
 1,250 
 
 1,480 
 
 1,780 
 
 7 
 
 1,090 
 
 1,170 
 
 1,350 
 
 1,620 
 
 1,970 
 
 8 
 
 1,130 
 
 1,240 
 
 1,440 
 
 1,740| 
 
122 
 
 CONCRETE ENGINEERS' HANDBOOK 
 
 [Sec. 2-65a 
 
 poured rapidly have been made under average conditions by the Aberthaw Construction Co., 
 Boston, Mass. The following table^ gives the results observed in the pouring of two columns 
 of small cross-sectional area. It should be noticed that the wet concrete exerted a hydrostatic 
 pressure equivalent to that of a liquid weighing from 140 to 150 lb. per cu. ft. or very nearly 
 that of a liquid having the same weight as the concrete. 
 
 Description of concrete 
 
 
 Head 
 (ft.) 
 
 Pressure 
 (lb. per 
 sq. ft.) 
 
 Hydraulic 
 equivalent 
 (lb, per cu. 
 
 ft.) 
 
 Time of pouring, 9 min 
 
 
 
 
 3.08 
 
 460 
 
 149 
 
 Mixture, 1: 1^:3 
 
 
 
 
 6.08 
 
 yuu 
 
 148 
 
 Stone, 1-in. run-of-crusher 
 
 
 No. 1 
 
 column 
 
 
 9.08 
 
 1,330 
 
 146 
 
 
 
 
 
 12.08 
 
 1,710 
 
 142 
 
 Wt. of mixture, 152 lb. per cu. ft j 
 
 
 
 
 15.08 
 
 2,110 
 
 140 
 
 Time of pouring, 14 min 
 
 
 
 
 2.75 
 
 407 
 
 148 
 
 
 
 
 
 5.75 
 
 840 
 
 146 
 
 Mixture, 1:1:1 
 
 
 No. 2 
 
 
 8.75 
 
 1,280 
 
 146 
 
 
 
 column 
 
 ■ 
 
 11.75 
 
 1,700 
 
 145 
 
 
 
 
 
 14.75 
 
 2,080 
 
 141 
 
 
 
 
 
 17.75 
 
 2,450 
 
 138 
 
 From the results of the tests just mentioned it would seem that 145 lb. per sq. ft. per foot 
 of height is a rational value to use for the lateral pressure of concrete in the design of forms. 
 This same value was also arrived at from laboratory and field tests made under the supervision 
 of Prof. A. B. McDaniel and N. B. Garver at the University of Illinois in 1913, 1914, and 1915. 
 
 Opinions differ as to the proper coefficient to use in the moment formula when making com- 
 putations for the strength of floor sheathing and joists. It is probable that the formula 
 
 M = may safely be used in all cases except for single-span joists where the concrete is conveyed 
 
 to place by small dump cars on a portable track. With dump cars the concentrations on a sin- 
 gle-span joist may be so heavy that the negative bending moments of the dead load will be 
 
 relatively very small as compared to the positive moments and the formula M = -g- should 
 
 undoubtedly be employed. When the bending moment is figured as ikf = the deflection 
 
 3 w"{l")'^ 
 
 should be determined by the deflection formula D = — — (see "Notation," page 124). 
 
 o84 iLl 
 
 This formula is the ordinary one for calculating deflection except that the coefficient is taken as 
 a mean between for a beam with fixed ends and ^^§4 for a beam with ends simply supported. 
 For lumber commonly used in form construction E may be assumed at 1,200,000 lb. per sq. in. 
 
 The fiber stresses allowed in form design may well be higher than those it would be desir- 
 able to use in more permanent construction. The following values are usually employed: 
 
 Maximum fiber stress in spruce or equal : 
 
 for timbers 1,200 lb. per sq. in. 
 
 for column yokes ; 1,800 lb. per sq. in. 
 
 Horizontal shear for spruce or equal • 200 lb. per sq. in. 
 
 Crushing perpendicular to grain in spruce or equal 400 lb. per sq. in. 
 
 1 From Concrete-Cement Age, Oct., 1913. 
 
Sec. 2-656] 
 
 GENERAL METHODS OF CONSTRUCTION 
 
 123 
 
 The crushing of form lumber perpendicular to the grain should not be overlooked. Although 
 a 3 by 4-in. or 4 by 4-in. post may be braced at least every 6 ft. in height and have apparently an 
 allowable compressive strength of about 800 lb. per sq. in., such a post should not be permitted 
 to carry more than one-half such a load due to crushing of the lumber perpendicular to the grain 
 in the crosspiece or girt over the post. Especially is this true where a granolithic finish is to be 
 cast with the slab, as settlement would ruin the slab surface. Large hardwood wedges should 
 be used to prevent any settlement due to crushing under the post. 
 
 The maximum deflection which may be allowed a form timber is not definitely known. 
 Some designers allow very small deflections for joists and use full live and dead load in design- 
 ing. Others reason that any serious deflection will be caused only by the dead weight of the 
 wet concrete and for that reason allow comparatively large deflections. Deflections allowed 
 in practice vary from an arbitrary maximum of }^ in. for all timbers to maximum deflections 
 of 3^6 0 of the span. 
 
 656. Drafting-room Methods. — The following article is taken by permission 
 from a paper presented at the Twelfth Annual Convention of the American Concrete 
 Institute by E,. A. Sherwin, Resident Engineer, Aberthaw Construction Co.: 
 
 The first study and drawings to be made form a general assembly. Usually several different combinations of 
 timbers and methods of assembling the panels are sketched up and compared as to cost. Other things being 
 equal, of course, the cheapest design is adopted. A record of costs of the various units which go to make up the 
 complete centering scheme is therefore necessary. 
 Points to be remembered in this study are: 
 
 1. Joists and girts should be in as few lengths as possible to save time in sorting on the job. 
 
 2. Use stock sizes and lengths of lumber. 
 
 3. Keep number of panels and pieces to a minimum. 
 
 4. Provide easy stripping. 
 
 5. Allow clearance enough for slight inaccuracies in making up and erecting, swelling of panels, etc. 
 
 6. Panels should be a whole number of boards in width, if possible, for ease in making up. 
 
 7. Units to be as big as can be handled and joists used as panel cleats where possible. 
 
 8. Provide for re-use of panels. 
 
 9. Beams to be handled as trough units when the job is regular and units can be re-used. 
 
 10. Consider use of floor domes or inverted boxes when beams are close together. In this way beam sides 
 and slab are erected, stripp^ed and moved as a unit. When either of the last two systems is used the beam sides 
 should be given a slope to prevent hard stripping. 
 
 11. Provide for reshoring if necessary. 
 
 12. Have bracing above the men's heads. 
 
 13. When four beam haunches occur at a column consider making haunches as a unit similar to a column 
 head in flat -slab construction. 
 
 14. Consideration of steel forms. 
 
 A general assembly shows how the various parts are to be put together, supported, tied, and braced to resist 
 the concrete pressure and the weights coming upon the members. This drawing in its final shape can be made on 
 tracing cloth with soft pencil. The dimension arrows, lettering and figures should be inked so that clear prints 
 can be made to avoid errors in the field. These assemblies should be filed permanently for reference on future work. 
 
 When it is necessary to strip the floor centering for re-use before the concrete has been in place long enough 
 to gain its full strength, provisions for proper support of the green slab must be made in the design of the floor forms. 
 The safest and cheapest method of doing this, in the writer's opinion, is to place boards between the floor panels 
 and wedge posts up to a bearing under these boards before the centering posts are knocked out. In this way the 
 slab is never left unsupported, as is the case when posts are replaced after all the centering is down. These boards 
 should be placed according to a plan and shall be located so as to shorten the spans of the main reinforcing bands. 
 
 After the general scheme for the forms has been decided upon the detail panel drawings are prepared. By 
 panel, in this paper, is meant several boards cleated together into a unit to be used as a form for some part of a 
 concrete member. Every different-sized panel is first sketched roughly on standard 6 by 9-in. sketching pads. 
 These pads have holes punched at the top, so that as sketches of the different kinds of panels are made they can be 
 bunched together and brass rivets put through the holes. These sketches are transferred in more detail onto thin 
 tracing paper. The standard sheet is 25 by 33 in., divided by a 1-in. space into two halves of five spaces each. 
 These details are used at the job mill for making up the panels. The blue-prints sent to the mill man are cut up 
 into 5 by 6-in. units, with one detail on each, and are given to the carpenters at the making-up benches. On these 
 detail sheets for each panel there is given the panel mark (which is stenciled onto the finished panel), the number 
 wanted, and the floor on which they are to be used. The sizes of the stock, dimensions and alterations, when panels 
 are to be re-used, should be clearly marked. 
 
 I 
 
124 
 
 CONCRETE ENGINEERS' HANDBOOK 
 
 [Sec. 2-66 
 
 A system of symbols as follows is easily learned by the workmen and should be kept standard: 
 B, beam side; BB, beam bottoms; F, floor slab panels; P, plinth forms; H, haunch forms; C, column sides; 
 W, wall panels. 
 
 Thus 518 means beam side number 18, and its location is shown on the key plan. 
 
 These detail sheets can be drawn up entirely with soft pencil on paper, as they are not valuable after the forms 
 are once made. Rubber stamps for a great deal of the lettering will save time. 
 
 Where time allows and plans are complete it is advantageous to get out one kind of panel at a time. In any 
 case the first panels to be sent to the making-up mill should be the typical floor panels. These can be used economic- 
 ally in building foundation walls, etc., and in this way the job can get an extra use out of them. 
 
 It is impossible to get a satisfactory cost unit for doing this form drawing. A price per square foot or per 
 detail varies widely according to the number of details required and the number of square feet that can be made 
 from one detail in different buildings. A big building where a large duplication of the different kinds of units is 
 possible will cost much less per square foot for this work than a small building which is badly cut up. The average 
 cost per sheet has been found to be about $4.50 complete. This includes time spent on studies, assemblies, key 
 plans, scheduling, and checking. The number of sheets required on buildings of different sizes of the same type 
 is fairly uniform according to the size. The number varies from 30 sheets on a small building up to 100 or more on 
 a large building with irregularities. By reducing the cost of the sheets to drafting-room hours, by dividing the 
 probable total cost by the average rate per hour, a fairly close estimate of the number of men required to turn out a 
 job in a specified time can be made. This is useful on rush work. 
 
 The key plan mentioned above is really a diagram of the floor plan upon which are shown the locations of 
 the various form panels. This plan is the only one which the workmen consult in erecting the formwork. It 
 must, therefore, be complete and clear. 
 
 This plan can best be made by tracing the floor plan on cloth from a blue-print, indicating the beams and holes 
 in the floor and the columns and walls in the story below. The lines should be heavily inked and the figures large 
 enough and spaced so that the tracing can be reduced and the photo reduced to convenient size for the foremen to 
 handle on the job. A big blue-print is inconvenient and fades in the strong sunlight. As a rule, to avoid mis- 
 understandings, a key plan should be prepared for each floor in the building. 
 
 The key plan may be supplemented by a letter to the job in which should be noted the assumptions made in 
 preparing the details. These notes might include such information as clearances allowed, grades at which forms are 
 to start, scheme for re-use, etc. 
 
 66. Tables and Diagrams for Designing Forms. 
 
 66a. Notation. — The following notation is used in the tables and diagrams: 
 b = breadth of member in inches. 
 d = depth of member in inches. 
 
 I = span of member in feet. " 
 
 I" 
 
 = span of member in inches. 
 
 
 w 
 
 = uniform load per linear foot. 
 
 
 w" 
 
 = uniform load per linear inch. 
 
 
 w' 
 
 = total load on floor in pounds per square foot = dead weight of slab per square foot 
 
 
 plus 75 lb. per sq. ft. live load. 
 
 
 h 
 
 = head in feet. 
 
 
 D 
 
 = deflection in inches. 
 
 
 E 
 
 = modulus of elasticity in pounds per square inch. 
 
 
 I 
 
 = moment of inertia in inches*. 
 
 
 f 
 
 = maximum fiber stress in pounds per square inch 
 
 
 M 
 
 = bending moment in foot-pounds. 
 
 
 Mr 
 
 s 
 
 = resisting moment in inch-pounds. 
 = spacing in inches. 
 
 
 n 
 
 = number of spaces. 
 
 
 n' 
 
 = number of spaces between column yokes. 
 
 
 k 
 
 = largest dimension of column in inches. 
 
 
 P 
 
 = concentrated load in pounds. 
 
 
 V 
 
 = total maximum vertical shear in pounds. 
 
 2\bdl 
 
 V 
 
 = maximum unit horizontal shear (pounds per square inch) 
 
 666. Fiber Stresses Allowed. — The fiber stresses allowed are those given in 
 
 Art. 65a. 
 
Sec. 2-66c] GENERAL METHODS OF CONSTRUCTION 125 
 
 66c. Formulas Used. — The spacing of joists was determined in Table I by the 
 
 formulas : 
 
 For flexure 
 
 For deflection 
 
 For horizontal shear 
 
 2000^2 
 
 D = 0.0225 
 
 a 
 
 When D = l-g in. 
 
 When D = 3^4 in. 
 
 1" 
 
 When D = 
 
 360 
 
 VsOOd 
 12 
 
 VlWOd 
 12 
 
 40 
 27' 
 
 (Depths for joists are taken li in. less than nominal sizes.) 
 
 In Table II, Part A: 
 For flexure 
 
 2000 
 
 In Table II, Part B : 
 For flexure 
 
 » = 1600^, 
 
 For D = }yi in. 
 
 s = 11,100^, 
 
 For horizontal shear 
 bd 
 
 For D = 
 
 r 
 
 360 
 
 s = 1780 
 
 bd^ 
 
 3200 
 
 For horizontal shear 
 bd 
 
 3200 
 
 w'l 
 
 Table III: 
 
 Deflection of in. governs. 
 
 590,000(145)d^ 
 
 In Table IV the concentrated loads from the joists were considered on a simple span in 
 calculating the bending moment. The worst case was assumed — that is, when one joist comes 
 it mid-span. Since a girt is usually continuous for at least two spans and the full live load never 
 ^caches it, the moment of resistance of the timbers was multiplied by 1.2. Nominal sizes of the 
 jirts were used in the computations. 
 
 For flexure 
 
 2406^2 + |-P(n2 + 2n) 
 
 3P(n + 1) 
 
 
 
 Horizontal shear is to be considered separately. From the above considerations it would seem 
 that an allowable horizontal shear of (200) (1.2) = 240 lb. per sq. in. may safely be u.sed. 
 
126 
 
 CONCRETE ENGINEERS' HANDBOOK 
 
 [Sec. 2-66C 
 
 Diagram I: 
 For flexure 
 
 For deflection 
 
 Diagram II: 
 For flexure 
 
 336,000d2 
 
 145/1 
 
 l,590,000d= 
 
 2380, 
 
 For deflection {D = 
 
 I" 
 
 'h{i"y 
 
 95,400d 
 
 •238 
 
 When d = 2 in. When = 4 in. When = 6 in. 
 
 I" = 28.3 in. I" = 40.0 in. V = 49.0 in. 
 
 These results are based on a value of /c = and the values given may at least 
 be increased to 30, 42, and 50 respectively. 
 For shear (y = 200 lb. per sq. in.) 
 
 I" =9d - {I" - k) 
 
 Table V: 
 
 19,060 
 
 h = Y' A/l9,060n' 
 
 Formulas to use with table: 
 
 ^. , ,„.095?)d2 
 
 Diameter of bolt 
 
 \2l" 
 
 Net area of washer 
 
 21" - k 
 
 Illustrative Problems. — 1. Required the proper spacing of 2 by 8-in. joists having a span of 7.5 ft. to 
 support forms for a 5-in. slab in beam-and-girder construction, assuming 1-in. sheathing. 
 
 Table I shows that 2 by 8-in. joists spaced 31 in. on centers will give sufficient strength if the bending moment 
 
 is assumed equal to j^- For this spacing the table indicates that the deflection is somewhat over }^in. but less 
 
 (7.5)2 
 
 than M in. and much less than Heo span. Accurately, D = 0.0225 -i^-y^ = 0.16 in. From Table III we find that 
 
 the spacing cannot be greater than 30 in. without the deflection of the sheathing exceeding H in., which is not 
 advisable. 
 
 wZ2 (7 5) 2 
 
 For M = -g-, the spacing would be (0.8) (31) = 25 in. from Table I and the deflection D = 0.030 -fJJ ^ 
 
 0.22 in. From Table II, the spacing would be 22 + H(8) = 24.7 in. 
 
 For M = — and the deflection limited to H in., Table II shows the proper spacing to be 21 + H(8) = 23.7 m. 
 
 2. Assume joists in the preceding problem to be supported midway between beams. Determine their 
 
 economical size and proper spacing, assuming M = Jq" and deflection limited to H in. 
 
 It will be sufficiently accurate to assume the span as 4 ft. From either' Table I or II, we find that 2 by 4-in. 
 joists may be employed spaced 25 in. on centers. 
 
 3. Determine size and span length of girt in the preceding problem to support the joists midway between 
 beams. 
 
 Load coming from each joist is (3.75) {^^12) (137.5) = 1075, or say 1000 lb., accurately enough. From Table 
 IV we find that a 3 by 4-in. girt with posts spaced 3.8 ft. c. to c. could be used, or a 3 by 6-in. girt with posts 5.6 
 ft. c. to c, or a 4 by 6-in. girt with posts 6.6 ft. c. to c. 
 
 Horizontal shear must be considered separately considering a joist to occur close to one support. Assuming 
 
 2260 ' ' ' ' 
 
 a 4 by ^-in. girt with posts 6.6 ft. on centers, V= 2260 lb. and v = ^*^4y(gy =" 142 lb. per sq. in., which is less than 
 the allowable value. 
 
GENERAL METHODS OF CONSTRUCTION 
 
 127 
 
128 
 
 CONCRETE ENGINEERS' HANDBOOK 
 
 [Sec. 2-66c 
 
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Sec. 2-66c] 
 
 GENERAL METHODS OF CONSTRUCTION 
 
 129 
 
 A 3 by 4-in. post could sustain (3) (4) (400) = 4800 lb. without injuring the fibers of the girt. It would only 
 be required to support 3450 lb., consequently this size of post is suitable. 
 
 4. Assuming the cross-section of beam below slab as 14 by 18 in. (23 in. total depth) determine the safe span 
 for the beam bottom to be made of 2-in. plank. 
 
 Live plus dead load on beam bottom is 75 + 2)'i2(150) = 3625 lb. per sq. ft. Diagram I shows the maximum 
 span to be 43 in. 
 
 5. Find the proper spacing of 3 by 4-in. posts to support the forms for 8 by 16-in. beams (cross-section given 
 below slab) spaced 6 ft. on centers with a 4-in. floor slab. Assume that no girt is placed at midspan of joists. 
 
 Total load on beam per linear foot is (125) (G) + = 883 lb. not considering the weight of 
 
 the forms, which may be neglected. 
 4800 
 
 of posts = -ggg- = 5.4 ft. 
 
 6. Determine the size and spacing of joists, girts, and posts to support an 11-in. flat slab floor. 
 Assuming 1-in. sheathing we find from Table III that the joists cannot be placed more than 27 in. on centers. 
 
 Table I shows that for 2 by 8-in. joists the spacing of girts may be made 6.5 ft. The load coming from each joist 
 is (6.5) (2JI2) (212.5) = 3110 lb., or accurately enough 3000 lb. From Table IV we find that for 4 by 6-in. girts the 
 
 posts may be placed 3.8 ft. on centers. Horizontal shear on girts must be considered separately, v = ^2. 't^tt^n = 273 
 
 144 
 
 Safe bearing of post on fibers of cap = (3) (4) (400) = 4800 lb. Safe spacing 
 
 lb. which is somewhat greater than the allowable value and a 6-ft. spacing of the girts is necessary. 
 
 Table III. — Safe Span for Floor Sheathing 
 
 (inches) 
 
 Based on M = — with deflection limited to H in. 
 lOj 
 
 (4) (6) 
 
 
 all 
 
 
 I 
 
 > 
 
 t 
 
 1 
 
 ^4 
 
 J 
 
 pounds per 
 square foot 
 (a ve plus dead) 
 
 
 
 
 
 3/n. 
 
 112.5 
 
 32 
 
 29 
 
 46 
 
 57 
 
 8 in. 
 
 1750 
 
 29 
 
 35 
 
 41 
 
 5/ 
 
 4 in. 
 
 125.0 
 
 31 
 
 3d 
 
 45 
 
 56 
 
 Sin. 
 
 187.5 
 
 28 
 
 35 
 
 41 
 
 50 
 
 5in 
 
 137.5 
 
 30 
 
 37 
 
 44 
 
 54 
 
 10/n 
 
 200.0 
 
 28 
 
 34 
 
 40 
 
 50 
 
 0 in 
 
 150.0 
 
 30 
 
 37 
 
 43 
 
 53 
 
 11 in 
 
 212.5 
 
 27 
 
 33 
 
 39 
 
 49 
 
 Tin. 
 
 162.5 
 
 29 
 
 36 
 
 42 
 
 52 
 
 12fn 
 
 225.0 
 
 27 
 
 33 
 
 39 
 
 48 
 
 7. What spacing of vertical studs is required for a wall form with l^-in. sheathing and a height of 12 ft. 
 Diagram I shows the spacing to be 18 in. 
 
 8. Assuming a 24 by 24-in. column with I" = 37 in., determine the spacing of the column yokes. 
 For 2 by 4-in. yokes placed on edge, the value of I" to be used in Table V should be 
 
 (37) (1.23^ 
 
 (2) (37) (24) - (24)2 
 
 372 
 
 42.5 in., say 42 in. 
 
 For 4 by 4-in. yokes, the value of 1" to be used should be (37) (0.87) (0.935) = 30 in. 
 
 For shear the actual value of I" must not be taken less than 9c? - (Z" - k) = (9) (4) - (37 - 24) = 23 in. 
 for either the 2 by 4-in. or the 4 by 4-in. yokes. Evidently the shear in the yokes will be less than the allowable. 
 
 Other things being equal. Diagram II shows that 1-in. sheathing would be more economical than VA-in. 
 when 2 by 4-in. yokes are used. The table shows that the same number of yokes would be used in the two cases. 
 
 The spacing center to center for the 2 by 4-in. yokes for a 10-ft. column with 1-in. sheathing would be as follows 
 in inches starting at the top: 30-20-14-11-10-9-8-7. 
 
 .095M2 
 
 vail 
 
 Where the strength of yokes governs their spacing, the bolts must have a diameter of 
 
 .^(0.095) (2) (16) 
 
 ues of k and I' 
 
 The net area of washer should be 
 
 Thus for the 2 by 4-in. yokes, diameter of the bolts must be at least 
 3bd^ (3) (2) (16) 
 
 50 
 
 using actual 
 
 0.25 in. 
 
 21" - k 
 
 9. Assuming a 12 by 12-in. column with I' 
 
 50 
 
 1.92 sq. in. 
 
 22 in., determine the spacing of 2 by 4-in. yokes placed on 
 
 edge. 
 
 The limiting value of I" for shear on yoke is (9) (4) - (10) = 26 in. Thus actual values oil" and k must 
 be considered as 26 in. and (26 - 10) = 16 in. respectively, which gives a value of I" to be used xn Table V of 
 1(2) (2 6) (16) - (16)2 ^/12 
 (26)2 \l6 
 
 (26) (1.23) 
 
 25.6 in., say 26 in. 
 
130 
 
 CONCRETE ENGINEERS' HANDBOOK 
 
 [Sec. 2-6Gc 
 
 1 Load coming from each Joisf In pounds j 
 
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 to 
 
 
 
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 Spacing 
 
 of 
 joists 
 
 
 
 
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 1 Load coming from each joist in pounds I 
 
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 VO 
 
 
Sec. 2-G7] 
 
 GENERAL METHODS OF CONSTRUCTION 
 
 67. Systematizing Formwork on Buildings.^ 
 
 67a. Sawmill and Yard. — The sawmill should be located after a study of the job site in a position 
 where there is plenty of space for piling the stock and finished panels. The sawmill shed and equipment should be 
 standard so that it can be erected and put in operation as soon as possible in order to reduce hand-sawing to the 
 minimum. A sawmill at its best is dangerous. For this reason every precaution should be taken to protect the 
 workmen by efficient saw, machine, and belt guards. The mill should be near the builcling so as to reduce the cost 
 of moving panels. This moving cost is an important item in obtaining low erection costs and will amount to a 
 
 Diagram I 
 
 Spacing of Vertical and Horizontal Studs (or Column Yokes) at Any Given Depth 
 
 Below Surface of Concrete 
 
 (Based on strength and deflection of sheathing) 
 
 If - 
 
 — 2Q Deflection limited to in. 
 
 0 
 / 
 
 z 
 
 3 
 4- 
 
 5 
 
 6 
 
 7 
 
 13 
 
 14 
 
 /6 
 
 17 
 18 
 /5 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 0^ 
 
 2^ 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 — o 
 —zoo 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 \ 
 
 
 'I 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 —400 
 
 — coo 
 
 — 800 Q 
 — 1000 ^ 
 
 — /eoo ^ 
 
 — ^ 
 
 — /400 , 
 
 — 1600 ^ 
 
 — /<300 ^ 
 
 — 2000 Yi 
 
 — I 
 
 — 2200 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 i 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 \* 
 
 7 
 
 
 .V 
 
 f 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 / 
 
 
 
 
 
 
 
 A 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 / 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 / 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 1 
 
 — 
 
 
 
 
 
 / 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 / 
 
 
 
 
 
 
 
 
 
 / 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 / 
 
 
 
 
 
 \ 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 t 
 
 
 
 I 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 1 
 
 
 
 
 
 
 
 
 
 1 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 \ 
 
 
 
 
 
 
 \\ 
 
 
 
 
 
 / 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 — 2400 
 
 — 2800 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 J 
 
 — 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 \ 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 10 15 20 25 30 35 40 45 50 
 
 3 " Spac/ngr In inches 
 
 considerable item when the mill is poorly located, either from lack of planning or from lack of space for the plant 
 around the building. 
 
 The mill yard should be so arranged that the stock will go in one direction from the lumber piles to the saws, 
 to the making-up benches and to the finished panel piles, which should be nearest the building. 
 
 The lumber when received is checked as to quality and quantity. The stock is then sorted and piled accord- 
 ing to size and length, each pile having its size plainly marked. 
 
 It is then an easy matter for the millman to make a lumber ledger of all the stock in his yard. Thus at all 
 times it is possible to know the stock available, for as the lumber is used deductions can be made and the running 
 total kept up to date. 
 
 676. Shop Procedure. — The millman divides the panel details into boards to make the required 
 width of the panel. The number of b6ards of each kind needed to make the .number of panels wanted are ordered 
 moved to the saw to be cut to proper length and thence to the make-up bench, where cleats of the proper size and 
 ^ By R. A. Sherwin, Resident Engineer, Aberthaw Construction Co. IVom paper presented at the Twelfth 
 Annual Convention, American Concrete Institute. 
 
132 
 
 CONCRETE ENGINEERS' HANDBOOK 
 
 [Sec. 2-676 
 
 Diagram II 
 
 Spacing of Column Yokes or Horizontal Studs at Any Given Depth Below Surface 
 
 OF Concrete 
 
 (Based on strength and deflection of the yokes or studs) 
 
 Based on M = 
 
 Deflection less than 3^ in. 
 
 /O 15 ^0 25 30 35 40 45 
 s = 3pac/n^ of column yokes or horizontal studs in inclnes 
 
 Directions for Using Diagram II and Table V 
 For 2X4-in. yokes or studs (flat) multiply actual I" by 1 .73 before using diagram or table. 
 For 2X4-in. yokes or studs (on edge) multiply actual I" by 1 .23 before using diagram or table. • 
 For 3X4-in. yokes or studs (on edge) multiply actual I" by 1 .00 before using diagram or table. 
 For 4X4-in. yokes or studs (on edge) multiply actual I" by 0.87 before using diagram or table. 
 For 3X6-in. yokes or studs (on edge) multiply actual I" by 0.67 before using diagram or table. 
 For 4X6-in. yokes or studs (on edge) multiply actual I" by 0.58 before using diagram or table. 
 For 6X6-in. yokes or studs (on edge) multiply actual I" by 0.47 before using diagram or table. 
 
 50 
 
 For 6X<i-in yokes or studs (on edge) multiply actual I" by 
 
 before using diagram or table. 
 
 Fo r col umns the value of I" to be used in diagram or table should be the value of I" as found above multiplied 
 
 'h - ^2 
 
 , in which expression the actual values of I" and h are to be substituted. 
 
 3 X 4-in. aiid 4 X 4-in. yokes, actual 
 
 3 X 6-in., 4 X 6-m., and 6X6-m. j ^ 
 
 In determining spacings from the diagram or table for 
 
 values of I" greater than |5o'o} ^"^^ ^^^^ ^ deflection of yokes greater than >8 in., and actual values of I" 
 
 greater than | 72 0 } ^ ^"^^ S^^® ^ deflection greater than 3^4 in. 
 
 In determining spacings from the diagram or table, actual values of I" must not be considered as less than deter- 
 mined by the formula 
 
 1" = 9d-{l" - k) 
 
 otherwise horizontal shear will be greater than 200 lb. per sq. in. The corresponding actual value of k (which will 
 be called k') should be determined by subtracting the value of a (see sketch"" from the value of I" found by the above 
 formula. The value of to use in diagram or table should then be found as explained above and finally multiplied 
 
Sec. 2-676] 
 
 GENERAL METHODS OF CONSTRUCTION 
 
 133 
 
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134 
 
 CONCRETE ENGINEERS' HANDBOOK 
 
 [Sec. 2-67c 
 
 length are already waiting, having been previously ordered through the mill. Another order to the bench car- 
 penters, which is clipped to the blue-print sketch of the panel, enables them to cleat together the boards into the 
 finished panel. The panel is then taken away by a laborer, stenciled with its location mark, oiled and piled until 
 ready for use in the building. 
 
 All this is done by orders written on standard order forms of three kinds, one for moving the stock, another 
 for sawing, and the last for making up. Duplicates of the orders are placed on the millman's progress board so 
 that he knows at all times how the work is progressing, for when an order is completed and returned to him its 
 duplicate is taken down. Each kind of order is of different color so as to be easily identified. 
 
 There are several points in connection with making up panels which may be considered here. 
 As much assembling as possible should be done at the bench. Rangers for wall beams can be attached and 
 ledgers to support joists can be nailed to the interior beam side cleats at the proper depth. When inserts need to 
 be placed in the sides of beams, the holes for the bolts which hold them can be located on the details and the holes 
 bored at the mill. The groove strip for steel sash and also the corner fillet can be put on the column sides and beam 
 bottoms. Bevel key strips for walls should also be nailed lightly to the panels when necessary. 
 
 Heavy cleats for big wall columns should be cut and bored, but not attached to the panels. The panel can be 
 cleated with IH-in. boards, and is thus made much lighter and easier to handle in erecting. Cleanouts at the 
 bottom, on two opposite sides of all columns, should be made at the mill. Reduction strips should be nailed at the 
 edge of all panels which reduce in size when reused. 
 
 All small pieces liabl • to get lost or used for other purposes can be dipped in red paint so that their small size 
 will be respected by the carpenters when looking for loose boards. 
 
 Much waste in making up li-m. panels can be prevented if the floor is laid out to make the majority of the 
 panels a whole number of boards wide. Roofers come 5?^ to bYz in. wide, and it is an easy matter to plan the 
 panels to come, say, seven or eight boards wide, which means no waste in ripping one board to make the width. 
 The lengths should be planned as near stock lengths as possible Panels made of spliced boards are expensive to 
 make and are easily broken. 
 
 New or Used Material? — It is not economical in labor to make up panels from lumber which has already been 
 in contact with concrete several times. If, however, panels are in good condition, they can be cleaned and repaired 
 at a considerable saving, both in labor and material. 
 
 67c. Cleat Spacing. — The cleat spacing for the various kinds of panels should be kept uniform so that 
 the strips on the benches, for spacing the cleats, will not need to be moved for every set of panels. 
 
 For beam sides 2 by 3-in. cleats, flat, can be used on panels up to 30 in. deep; 2 by 4-in., flat, up to 42 in.; and 
 3 by 4-in., on edge, above 42 in. deep. Nails should be specified as follows: 
 
 Wire nails should be used for formwork because 
 No. Size Width of board, inches ^ase in driving and drawing. The holding power is 
 
 , _ , also sufficient. Double-headed nails should be used 
 
 3 IQd 6H and 7¥i ■ . n u j u- u u x u i j u 
 
 in securing all boards which have to be loosened be- 
 
 2 lOd 2?4 to 5^4 f ... 
 
 t inj T *u o-!/ fore stripping. 
 
 1 10a Less than 2'?4 t»i • * t^- u ttt t a • 
 
 J 3 4 1 t Planmng of Field Work. — An im- 
 
 , , , ,. , ^ ^ f^^^ portant feature of the field work is a planning depart- 
 
 8d coated and clinch ends in %-in. boards. + u u • • i i • j 
 
 ' ment, whose business it is to plan the work in advance 
 
 so that at all times the several gangs will have definite 
 
 tasks to do and be supplied with suflBcient material with which to do the work. 
 
 The moving boss has an important position in this field work. He is responsible for getting the proper panels 
 to their correct location in the building and for having all other stock called for on the assembly plan on the job 
 ahead of the erection carpenters. When the forms are stripped he must move the panels, which are to be remade 
 before their next use, to the remaking benches, and thence to their next location. It is expensive to have high- 
 priced carpenters waiting for stock or using stuff not suited for the job they have to do. 
 
 The erection of the centering and the assembling of the panels, after the latter have been moved to their 
 proper location in the building, are done by carpenters. It is economical to employ good carpenters and to keep 
 them employed constantly. A gang of men that understand formwork on concrete buildings, the methods used, 
 and the grade of work desired, is a valuable asset for any firm specializing in reinforced-concrete work. 
 
 A competent foreman should be in charge of all carpenter labor. He should be consulted by the planning 
 department so that "team work" will prevail. 
 
 Quality in concrete building work is usually more desirable than low costs. Good lines and surfaces will be 
 remembered after the cost is forgotten. These results can be obtained only by the careful supervision of the erec- 
 tion of forms. 
 
 67e. Stripping of Forms. — When forms are to be reused, as is usually the case, stripping should be done 
 as carefully as possible. An intelligent foreman in charge of a stripping gang is a good investment. Any man can 
 wreck forms cheaply, but a man who can strip the forms and leave them in good condition for reuse is in the 
 end the cheapest stripper. He saves the time of the high-priced carpenters in remaking and fitting broken panels. 
 
 The question of when to strip is one for the building designer to decide, and his instructions should be care- 
 fully followed. It is cheap insurance to make enough forms so that no chances need be taken of weakening the 
 structure by stripping while the concrete is still green. 
 
 The panels which need to be altered before reuse should be remade from the blue-prints showing the neces- 
 sary changes. A bench for this purpose can be set up in the story where the forms are stripped. It is sometimes 
 
Sec. 2-68 J 
 
 GENERAL METHODS OF CONSTRUCTION 
 
 135 
 
 advisable in a high building to set up power saws in one of the upper stories for cutting off panels and getting out 
 stock for remaking and repairing. 
 
 A large gang of carpenters working overtime is sometimes necessary in order to deliver a building on time. 
 This is expensive, however. The workmen cannot give back in efficient labor the value of their high wages. Ten 
 hours is about all the average man can work without falling off greatly in efficiency. Night work on forms should 
 be avoided if possible. 
 
 When speed is not the essence of the contract, there is time to plan the erection work more carefully. Smaller 
 gangs can be used and their tasks and the material for them routed by slips similar to those used in making up panels. 
 
 An important item in the field work is the erection of economical safe stages. The stages should be carefully 
 designed as to the worst possible load to come upon them, and the design should be strictly followed in the field. 
 A man thoroughly familiar with lumber should be made inspector as to the quality of the stock used for stages. 
 A good stage makes the workmen more efficient and is a good insurance investment. 
 
 68. Steel Forms. — Steel forms have always been used more or less for sewers, curbs, and 
 sidewalks, but now the application of steel forms to floor and column construction is advancing 
 
 Fig. 71.— Blaw light wall forms on foundation wall for plant of Hubbard & Co., Pittsburgh, Pa. 
 
 at an exceedingly rapid rate. Steel forms are almost universally employed for circular columns 
 and for flaring column heads. Wall forms are" also much used and have given satisfaction, 
 especially in residences and other structures of the foundation-wall type. 
 
 Several types of floor forms are on the market and are used to some extent. The same is 
 true of forms for rectangular and octagonal columns. 
 
136 
 
 CONCRETE ENGINEERS' HANDBOOK 
 
 [Sec. 2-68 
 
 The adjustment in height of steel forms for circular columns is obtained by telescoping 
 the ends. Such forms are usually made up of a series of panels of thin galvanized steel held 
 rigidly in place, like staves in a barrel, by means of stiff steel bands. The panels are somewhat 
 flexible and are sprung in or out depending upon the size of the column. 
 
 Fig. 72.— Steel floretyles. 
 
 Fig. 53, page 109, and Plate V, page 116, show steel forms for circular columns, in place, 
 and ready for the pouring of the concrete. The construction around the column heads should 
 be noted. The form illustrated in Fig. 53 is manufactured by the Blaw Steel Construction Co. 
 and is so designed that all variations are taken care of with a single set of forms. Vertical ad- 
 
 FiG. 73. — iSteel floretyles with end caps. 
 
 justment is obtained by telescoping one form section inside of the section below it, an adjust- 
 ment of 18 in. being permitted between each two sections. Diameter adjustment is provided 
 for by the use of form panels of various standard widths. The edges of the panel sheets are 
 bent back to form flanges, and these flanges are slotted. The panels are kept to any desired 
 
 Fig. 74. — Steel floredomes. 
 
 curvature by segmental steel bands which slip through the slots in the panel flanges, and are 
 drawn tight by means of keys or wedges. The bands are not adjustable, and a complete set 
 must be provided for each diameter of column to be built. The form panels are made in three 
 standard lengths, 6 ft., 43^ ft., and 3 ft. The steel column forms are not used to support any 
 part of the floor, as is usually the case in wood-form construction, so that the floor forms may 
 be erected complete before the column molds are set up. 
 
Sec. 2-68] 
 
 GENERAL METHODS OF CONSTRUCTION 
 
 137 
 
 The ''Hydraulic " column forms shown in Fig. 55, page 110 (manufactured by the Hydrau- 
 lic Pressed Steel Co., Cleveland, Ohio) consist of galvanized-iron units held together at the 
 
 Fig. 75.— G. F. steel-tile. 
 
 Fig. 76. — Meyer steelforms. 
 
 joints with quick-acting clamps and shaped with steel ring. Any height of column or size of 
 head may be obtained. 
 
 Fig. 71 shows Blaw light wall forms for foundation-wall 
 work where wire ties are used. The standard wall-form panel 
 is 2 ft. square and provided with holes in the four flanging 
 angles for the passage of fasteners. Slots are also provided in 
 the plate to permit of the insertion of the wire ties. The hori- 
 zontal and vertical liners are used to keep the form straight and 
 to connect panels together so that they may be shifted in larger 
 units than single panels. This type of form is not limited to 
 two-course working, but may be used in pouring any desired 
 height of wall at one operation. 
 
 Wall and foundation forms manufactured by the Hydraulic Pressed Steel Co. consist of 
 uprights which are aligned and accurately spaced 3 ft. 3 in. c. to c. of pressed steel liners. 
 
 Between these uprights steel-faced plates are 
 clamped. 
 
 A type of permanent (non-removable) 
 steel form for floor construction which has a 
 similar function to hollow tile in terra-cotta 
 hollow tile floors, manufactured by the Trussed 
 Concrete Steel Co., is shown in Figs. 72 and 
 73. This type of floor form is also shown in 
 Fig. 35, page 59. " Steel Floredomes " shown 
 in Fig. 74 are manufactured by the same com- 
 pany for two-way construction in which the 
 loads are carried in two directions to the supports. The metal domes are deeply corrugated 
 to secure stiffness and are only open on the underside so that the joists extend on all sides of 
 the dome. The standard heights of ''Steel Floretyles:" 6, 8, 10, 12, and 14 in. Approxi- 
 
 FiG. 77. — Wiscoforms. 
 
138 
 
 CONCRETE ENGINEERS' HANDBOOK 
 
 [Sec. 2-69 
 
 mate width at base: 203^ in., exclusive of 1-in. flanges along bottom edges, which add 2 in. 
 to this dimension. Standard lengths (nominal) of all sizes: 4 ft. and 3 ft. — actual lengths are 
 4 ft. 1 in. and 3 ft. 1 in. to provide for end lap. ''Steel Floredomes" may be obtained in 
 depths of 6, 8, 10, and 12 in. and 21 by 21 in. base. Permanent steel floor forms similar to 
 "Steel Floredomes" are furnished by the General Fireproofing Co., Youngstown, Ohio (known 
 as G.F. Steel-Tile, Fig. 75). . Steel floor forms which are removable and may be re-used in succes- 
 sive floors of a building are furnished by the Concrete Engineering Co., Omaha, Neb. (known 
 as Meyer Steelforms, Fig. 76); and by the Witherow Steel Co., Pittsburg, Pa. (known as Wisco- 
 forms, Fig. 77). 
 
 69. Construction Notes. — Nails should be used sparingly in the construction of forms ex- 
 cept in those sections which are to be used over and over again without change. Unnecessary 
 nailing not only adds to the labor of wrecking but is liable to render the lumber unfit for con- 
 tinued use. Where nails must be used in the connection of form sections, the heads should be 
 left protruding so that they may be drawn without injury to the lumber. A special form of 
 double-headed nail is now on the market and gives satisfaction. 
 
 The location of column forms from floor to floor of a building should be determined by 
 means of the transit, and special care should be given to the erection of these forms in order to 
 make sure that they are set true to line and level. Column forms should be held in position by 
 diagonal braces in two directions nailed to adjustable or slotted blocks which are bolted to the 
 concrete slab — the bolts being placed when the floor is poured. Sometimes small pieces of 
 plank are employed instead of blocks, and these are nailed directly to the floor slab within 2 or 
 3 days after pouring and while the concrete is still green. 
 
 Trouble in the erection of floor forms may usually be traced to inaccuracies in form measure- 
 ments. If the column forms are of the proper widths and if the beam and girder forms are 
 cut to exact lengths, no trouble of great consequence can arise. A variation of more than }i 
 in. from the sizes shown on the drawings should not be permitted. 
 
 One method of erecting forms for rectangular columns is to nail three of the sides together 
 lightly before raising them to place, and then to set the remaining side afterward. This 
 enables the column reinforcement to be put in place before the form is set. Another method in 
 common use is to assemble the column form complete before raising, in which case the form 
 must be raised above the projecting reinforcement (belonging to the footing or column below) 
 and then lowered. 
 
 All forms for concrete require a coating of some lubricant to prevent the concrete from 
 adhering to the wood and making a rough, unpleasing appearance. Crude oil or petroline is 
 used to a considerable extent and preserves the forms against damage by alternate wetting and 
 drying. The forms should preferably be oiled before they are set in place. 
 
 Oil should not be used on forms against surfaces which are to be plastered, as oil prevents 
 the adhesion of the plaster. Wetting with water in such cases will be sufficient. 
 
 Beam and girder forms should be raised slightly higher at the center than at the ends in 
 order to prevent sagging. If this is done, deflection and compression of the supports will 
 finally leave the beams and girders in a level position. A deflection equal to % in. in every 10 
 ft. of length is usually provided for. 
 
 The sides of beam and girder forms should project over the edges of the bottom plank. 
 By so doing it becomes possible to leave the beam and girder bottoms in place after the sides 
 have been dropped. 
 
 Slab forms of the ordinary panel type should be made in sections (usually four to a floor 
 panel) in order to prevent binding and permit easy removal. A splice of in. is usually 
 allowed between adjacent sections, and this space is covered with a strip of sheet metal, thus 
 giving some leeway in fitting the sheathing panels into place without unnecessary cutting. 
 Sometimes it is a good plan to provide for a loose board between two panels near the center of 
 the span so that temporary uprights may be used to support the floor slab when the forms are 
 stripped. 
 
Sec. 2-70] 
 
 GENERAL METHODS OF CONSTRUCTION 
 
 139 
 
 All posts or shores should rest on large hardwood wedges, driven in pairs to an even bear- 
 ing. Hard driving of wedges should not be permitted as it is sure to injure the concrete which 
 is setting under them. In some instances wedges have been placed at the top of the posts 
 instead of at the bottom, as is the usual custom. The disadvantage of this method lies in the 
 difficulty of driving wedges while standing on a temporary support but, on the other hand, 
 by top-wedging, the shores or posts may be permanently braced before the form work is leveled. 
 
 Concrete, when poured under horizontal or inclined forms (such as in footings), will exert 
 an upward pressure, and such forms should be securely anchored. 
 
 When posts are placed on plank sills, great care should be exercised to avoid settlement be- 
 cause of the likelihood of hollows coming under the sills due to unevenness of concrete floors or 
 to thawing out of frozen ground. 
 
 Deflection should be carefully guarded against at a window head as a slight deflection at 
 this point will cause considerable trouble and expense in setting the window sash. 
 
 Forms which are to be used again should be cleaned as soon as they are taken down. 
 
 In removing forms the green concrete must not be disturbed by prying against it. 
 
 BENDING AND PLACING REINFORCEMENT 
 
 70. Checking, Assorting, and Storing Steel. — Steel should be checked, assorted, and stored 
 as soon as it is delivered at the site. It should be blocked up several inches from the ground 
 and should be stored in such a manner that those rods needed first may be easily reached. 
 
 71. Bending of Reinforcement. — Such a simple matter as bending of rods for concrete 
 reinforcement might seem to be almost too unimportant a subject to be worthy of very much 
 attention. However, when it comes to the proposition of making thousands of bends per day, 
 factors of time and expense in this work are most important. Of course, the structural design 
 should be such as to require the steel bends to be as few in number and kind as possible, but much 
 can be done to lessen expense by the manner of making these bends. In addition to economical 
 considerations, care should be taken to see that the bends are made true to line and plane, and 
 that the steel is not injured during the operation. 
 
 71a. Types of Bends. — In general, there are five different types of bends to be 
 made with reinforcing rods which may be indicated as follows: (1) bending of heavy beam and 
 girder rods; (2) bending of the vertical reinforcing rods of columns at or near floor level where 
 columns change size; (3) bending of stirrups and column hoops; (4) bending of slab reinforce- 
 ment, and (5) the coiling of rods or wire to form spiral column reinforcement. In making each 
 type of bend, the work should always be so arranged that all rods of the same size and shape 
 are bent at the same time. This avoids remeasuring and resetting of templets. 
 
 716. Hand Devices. — A simple method of bending heavy rods is shown in Fig. 
 78. Either steel bars or steel plugs 5 or 6 in. long are placed in holes in the bending table, these 
 holes being bored at the points where the 
 
 C/eafs on uncfers/de 
 
 ff 
 
 
 1 F 1 i . , 1 j 
 
 
 4 — 
 
 — l-i -y^^ h ^n: — '4"" 
 
 
 
 1 1 1 1 
 
 
 Fig. 78. 
 
 bends are to be made. A piece of plank 
 is nailed to the table, as shown, in order 
 to hold the rod in place while being bent. 
 The rod is then placed in the position GH 
 and bent around the plugs C and Z>, and 
 then around the plugs B and E until the 
 ends AB and EF are parallel to GH. 
 
 The manner in which rods are to be bent is generally left to the discretion of the steel 
 foreman. The general practice is to bend beam and girder reinforcement by using a heavy pipe 
 slipped on the rod and then making the bends as described, using either steel plugs or angles 
 bolted to the table. 
 
 In building two large reinforced-concrete buildings for the General Electric Co. at Schenec- 
 tady, N. Y., the Stone & Webster Engineering Corporation, of Boston, accomplished the bending 
 
140 
 
 CONCRETE ENGINEERS' HANDBOOK 
 
 [Sec. 2-716 
 
 of the heavy rods by means of two ^^-in. steel plates mounted one on top of the other on a bend- 
 ing table, and, in these, holes were drilled 3-in. on centers in both directions. Steel pegs were 
 dropped in the holes the proper distance and angle apart, and the rods were then bent by hand 
 using a 2-in. steam pipe. 
 
 R. C. Hardman, in Engineering Record, describes as follows a small home-made bar bender 
 for bending cold bars up to 1 ^-in. diameter, where not greater than 90-degree bends are required : 
 
 The apparatus consists essentially of a cast-iron plate containing two lugs between which the bar is placed, a 
 steel lever fastened to the plate by means of a steel pin about which it acts, and a set of fillers, as shown in Fig. 79. 
 
 The cast-iron plate can be cast of coarse metal in any foundry, the top of the plate with which the lever comes 
 into contact being machined. The bolt holes, certain ones of which must be countersunk to allow free action of the 
 
 lever, may be either cored or drilled. The space between the two 
 lugs should be slightly larger than the largest size bar to be bent. 
 The filler is a device made of strap iron by any blacksmith, which fits 
 around the lug opposite the lever, to insure a tight fit for bars 
 smaller than the maximum. A set of these to accommodate the 
 various commercial sizes of steel bars can be made at small cost. 
 To insure a good fit the edges of the lug around which the "filler" 
 is placed should be machined. The lever is made of 1 by 2-in. flat 
 steel forged to shape, with the face engaging the bar slightly upset 
 on the upper side. Its length is about 4 ft. 
 
 The operation of the apparatus can be readily seen in Fig. 79, 
 in which the lever, bar to be bent and a filler are shown dotted. 
 The apparatus is fastened to a bench by means of bolts, and counter- 
 sunk so that the top of the plate is flush with the top of the bench. 
 
 Two men can make cold bends under 1 in. in diameter. On 
 larger sizes three men are required unless recourse is had to a pipe 
 extension to the lever. 
 
 The cost of the apparatus should not exceed $12 to $15, and it will readily pay for itself on a small job which 
 will not admit of a more versatile bender. 
 
 Several hand and power devices are on the market for the bending of steel reinforcement and 
 for the making of spiral coils. These all have their merits and have given satisfaction. 
 
 Fig. 80 shows the Universal bar bender which may be fastened to any bench or plank. 
 It is a light, portable device weighing about 60 lb. and capable of bending all ordinary sizes of 
 reinforcing bars to any angle desired, without any adjustment being necessary. The top half 
 
 Fig. 79. — Small home-made bar bender. 
 
 Fig. 80. — Universal bar bender. 
 
 Fig. 81. — Wallace bar bender. 
 
 of the bender can be removed and used to bend bars after they are in place. The bar rolls 
 around the pin in bending, thus distributing the strain along the bar and reducing the chances 
 of fracture at the bend. 
 
 The bender is equipped with a 5-ft. crowbar for a handle, which may be removed and used 
 for other purposes. To bend large bars easily, the handle should be lengthened by using an 
 iron pipe over the crowbar. 
 
 A bar bender designed for heavy work and manufactured by the Wallace Supplies Mfg. Co., 
 Chicago, 111., is shown in Fig. 81. This machine has an auxiliary ratchet lever which operates 
 a pinion against a series of teeth in the frame at a large ratio, thus developing great power. The 
 
Sec. 2-71c] 
 
 GENERAL METHODS OF CONSTRUCTION 
 
 141 
 
 ratchet panel may be thrown out of engagement and machine operated with the reguhir lever 
 for light work. 
 
 A bender manufactured by the Waterloo Construction Co., Waterloo, Iowa, is shown in 
 Fig. 82. This bender bends reinforcing bars up to and including 13-^ in. The machine is fur- 
 nished with a detachable handle 7 ft. long for convenience in handling. 
 
 71c. Power-operated Benders. — Fig. 83 shows a power-operated truck-mounted 
 bar bender designed to bend any size of reinforcing rod that is likely to be used in building opera- 
 
 FiG. 82. — Waterloo bar bender. 
 
 tions. Any size of bar, from }4tol }i in., round, square or deformed, can be bent to any angle 
 desired; or spirals formed from 6-in. diameter to any required size. Weight, complete, ready 
 for shipment, 2700 lb. The machine is manufactured by Kardong Bros., Minneapolis, Minn. 
 
 lid. Care to be Exercised in Bending. — Bending reinforcement should be done 
 in such a manner that the bars will not break or crack at the bend; that is, the bends should not 
 be too sharp and the bending force should be applied gradually and not with a jerk. Reinfor- 
 cing bars should also be bent cold except for the unusual sizes of 1 ^2 in. and upward, when heat 
 
 Fig. 83. — Power-operated bar bender. 
 
 may be required. Warming up to a low cherry red should under all circumstances be the 
 highest heat permitted. Bars to be bent are generally not larger than IK in. and most hand- 
 bending machines on the market do not claim to bend bars of any greater diameter. 
 
 Structural grade bars will bend much more easily and with less danger of injury than either 
 the hard grade, rerolled steel, or cold-twisted bars. In fact, the three last-mentioned classes 
 of reinforcement will usually require heating in the larger sizes before bending can be safely 
 
142 
 
 CONCRETE ENGINEERS' HANDBOOK 
 
 [Sec. 2-71e 
 
 attempted. Square cold-twisted bars frequently give trouble in bending, due to the fact that 
 they arei apt to twist out of their proper plane. 
 
 lie. Bending of Slab Reinforcement. — Slab reinforcement is usually bent after 
 it is in place on the floor. A tool for this purpose is shown in Fig. 84. Sometimes, however, 
 
 slab rods are bent before being placed. This latter 
 
 ._ ^ method seems preferable since the bends can be more 
 
 lliil:' ■ accurately made. To keep the cost of such bending as 
 
 low as possible, the bending machine should be con- 
 ~> structed so that the bends may be made with great 
 
 rapidity. Such a machine is shoAvn in Fig. 85 and was 
 
 employed on the General Electric Co. 's buildings referred 
 
 to above. 
 
 This machine consists of two units which can be placed any distance apart and securely 
 bolted down to a table. Each part consists of a steel plate on which is mounted two smaller 
 plates having angle stops attached. Between the stops is located a casting, which is pivoted so 
 
 Pos/t/on of rod before bend/ncf 
 
 . Stop plafe 
 
 /l^g^^e Iron fastened 
 to sf/d/nj plate 
 
 Fig. 85. 
 
 that it can readily take any desired position between the angle stops. With the steel rod in 
 position, this casting can be turned by means of a long lever attached so as to move one of the 
 mounted plates. On the upper side of the casting are four lugs, and two of these lugs are con- 
 structed so that adjustable collars can be slipped over them, thus making a tight fit for the rods 
 which are to be bent. The amount and the angle of the offset 
 can be regulated by changing the distance through which the 
 lever is turned. This machine offsets a rod parallel to itself and 
 with one pull of the lever two bends can be made. 
 
 72. Placing of Reinforcement. — Steel should be thoroughly 
 cleaned before being placed in the forms in order to obtain a posi- 
 tive adhesion of the concrete to the steel. A slight film of red 
 rust is not objectionable, but no rod should be set in place on 
 which rust scales have formed (see Art. 54, Sect. 1). 
 
 All reinforcing metal should be securely fastened in correct 
 positions by wiring or otherwise before the placing of the con- 
 crete is begun. Particular attention should be given to loose-bar 
 reinforcement — that it is accurately and properly supported in 
 position and that it is not disturbed until the concrete is poured. 
 
 The advantages to be derived from placing beam-and-girder 
 reinforcement in frames has been considered in Art. 11, Sect. 11. 
 With steel in frames the erector has simply to line and level 
 
 them in the forms, place braces where necessary, and make end connections with abutting frames. 
 Column reinforcement should be made up into frames, the same as for beams and girders. 
 
 Slab and wall rods should be tied with wire at their intersections to prevent them from slip- 
 ping or getting out of place. The usual method has been to use a pair of pliers and to cut the 
 
 Fig. 86. — Wiring of reinforce- 
 ment using Curry Tyer. 
 
Sec. 2-73] 
 
 GENERAL METHODS OF CONSTRUCTION 
 
 143 
 
 wire into convenient lengths, A device known as the Curry Tyer has recently been placed on 
 the market for wire tying which has made a great record for itself in a very short time. Fig. 
 86 shows this to be a simple and practical device. The ties used for the binding are uniform in 
 length and the mechanical action of the tying tool gives the wire a uniform number of twists. 
 The tie is looped around the rods and the ends are placed on the hooks of the tying tool, then a 
 quick upward jerk of the wooden handle whirls the teeth and draws the wire up tightly. 
 
 Another method of tying slab and wall rods together at their intersections is shown in Fig. 
 87. These Bar-iys are manufactured by the Concrete Steel Co., in the three types illustrated. 
 
 Fig. 87. — Bar-tyi 
 
 The tys are quickly put in place and when once snapped on the bar will resist a tremendous 
 pressure. 
 
 73. Devices for Supporting Reinforcing Bars. — There are many excellent devices on the 
 market for supporting reinforcing bars at the proper distance from the forms. Some of these 
 devices not only support the rods but give them the proper spacing and lock them in position. 
 
 Beam spacers sold by the Universal Form Clamp Co., of Chicago, 111., are shown in Fig. 88. 
 They are made in three sizes — namely: 4 in., 5 in., and 6 in. — and space the bars accurately 
 
 both from the sides of form and center to center of bars. 
 Easy chairs sold by the same company are adaptable to 
 either a one-way or a two-way system of slab reinforce- 
 ment. Fig. 89 shows the chair before being placed in 
 position. The upright tying finger enables the chair to 
 be readily seized and put in position. Fig. 90 shows the 
 
 Fig. 88. — Beam spacers. 
 
 Fig. 89. — Easy chair. 
 
 Fig. 90. 
 
 chair applied to a one-way system of reinforcement. After the chair has been placed in posi- 
 tion the flexible tying fingers are bent over the bar from opposite sides by hand, firmly 
 securing the chair to the bar. This chair can readily be applied to a two-way system of re- 
 inforcement. Easy chairs are made in three sizes so as to provide for reinforcing bars varying 
 in diameter from to 1 in. 
 
 A device for supporting slab rods, known as the Securo locking spacer, is a light bar passmg 
 beneath the rods and having depending lugs which rest on the bottom of the form and so keep 
 the rods at the desired height. At the location of each rod the bar has two flexible clips which 
 
144 
 
 CONCRETE ENGINEERS' HANDBOOK 
 
 [Sec. 2-73 
 
 Fig. 97. — Hy-chair. 
 
Sec. 2-73] 
 
 GENERAL METHODS OF CONSTRUCTION 
 
 145 
 
 are bent around it, and these serve to insure the use of the proper number of rods. Three strips 
 of Securo slab bar spacers are used per panel for ordinary spans. Two additional upper spacers 
 are used where slab bars run in two directions. Upper spacers are also used in flat-slab con- 
 struction. The device is shown in Fig. 91 and is made by the Metal Building Materials Co. 
 of Chicago, 111. 
 
 Securo supporting and locking spacer for beam bars is shown in Fig. 92. Three spacers 
 are used per beam for ordinary spans. Upper beam spacers are employed when beam rods 
 
 Fig. 98. — Chair spacer. Fig. 99. — Continuous slab spacer. 
 
 are to be placed in two or more layers (Fig, 93). Securo beam spacers are furnished for any 
 widths of beams and for any number of bars in lower and upper layers. 
 
 Supporting and spacing devices manufactured by the Concrete Steel Co. are shown in 
 Figs. 94 to 100 inclusive. Ty-chairs, shown in Fig. 94, are made of spring steel wire for tying 
 any combination of reinforcing bars. These chairs are made in the following standard sizes: 
 
 No. 2 
 
 No. 3 
 
 No. 4 
 
 Combinations of cross bars. 
 
 Combinations of cross bars, 
 
 Combinations of cross bars. 
 
 inches 
 
 inches 
 
 inches 
 
 }yi and ^ 
 
 Yi and Y2 
 
 and 
 
 \i and 
 
 Y2 and 
 
 and % 
 
 M and % 
 
 Yi and M 
 
 Y/^ and 1 
 
 % and % 
 
 % and % 
 
 and 11^ 
 
 % and M 
 
 H and % 
 
 and y^ 
 
 
 
 J.'g and 1 
 
 The Easel-chairs shown in Fig. 95 are designed particularly for terra-cotta or steel tile and 
 joist construction. Single chairs supporting one bar are made 2 in. wide, and double chairs 
 supporting two bars are 4 in. wide and will fit any size bar. The standard Easel-chairs space 
 the underside of the bar 1 in. from the form. 
 
 Bar-chairs (Fig. 96) are used for supporting single bars. They are easily sprung into posi- 
 tion, locking on the bar with a strong tension grip. Bar-chairs are made for each size and shape 
 of reinforcing bars. Standard distance from underside of bar to forms is 1 in. 
 
 Hy-chairs are illustrated in Fig. 97. They are made from 2 by )^-in. flat steel with any 
 required height, and are used for supporting single reinforcing bars of any type or size. They 
 are particularly useful on fiat-slab buildings for rigidly holding the column bars. 
 
 Chair spacers and continuous slab spacers are shown in Figs. 98 and 99 respectively. 
 The "chair spacers" are made of spring steel wire. The "slab spacers" are made from J-^-in. 
 cold-rolled angle and the prongs are so flexible that they can be readily bent around the bars by 
 hand. In both types of spacers standard distance from underside of bar to form is 1 in. 
 
 Adjustable beam saddles (Fig. 100) are made from sheet steel and used in beams and girders 
 10 
 
146 
 
 CONCRETE ENGINEERS' HANDBOOK 
 
 [Sec. 2-74 
 
 for accurately spacing the bars and holding them the required distance from the forms. Stand- 
 ard distance from underside of bar to forms is 13-^ in. and 2 in. 
 
 The chair lock shown in Fig. 101 is manufactured by the Electric Welding Co., Pittsburg, 
 Pa. Stock sizes are as follows: ^^-in. round cross rod by 3^^, and J-^-in. round main 
 
 rods. Any size, however, can be furnished to fit any shape of rod. Chair pinchers are furnished 
 in two sizes for fastening the locks to cross rods. 
 
 Staple chairs shown in Fig. 102 are made from extra stiff sheet steel, cut and bent to develop 
 two pairs of pointed prongs projecting in opposite chairs. The chairs are driven into the 
 formwork as far as is desired and in the exact position the steel is to occupy. The bars are 
 placed and the upper prongs bent down over the bar with a quick blow of the hammer. The 
 driven-in points do not seem to make form removal difficult. 
 
 Fig. 100. — Adjustable bear 
 saddle. 
 
 Fio. 101.— Chair lock 
 
 Fig. 102. — Staple chairs. 
 
 THE MANUFACTURE AND USE OF CONCRETE STONE, BLOCK AND BRICK 
 
 By Harvey Whipple^ 
 
 74. Development of the Industry. — Although the development of the concrete products 
 industry, embracing, first of all, the manufacture of building units, began earlier than reinforced 
 concrete building construction, it has lagged behind it. This has been due, in general, to a lack 
 of trained management. And the trained management has been lacking, perhaps, because of 
 the apparent simplicity of operations involved in the manufacture of concrete building units.. 
 In the early days, the use of cheap machines of questionable value was common and many of 
 them were sold on such a basis as to tempt the more slovenly artisans. Men who had made 
 failures of other work sought easy money by making block beside a gravel bank and following 
 the inadequate directions which came with the machine. 
 
 Attracted by the seeming lack of any complexity in the essential operations of block 
 manufacture, by the cheapness of the machines with which the work was to be done, and by the 
 seeming lack of any necessity for any auxiliary equipment, it is not to be wondered at that there 
 were many enterprises in concrete products manufacture of woodshed and backyard magnitude. 
 The result was that many cheap buildings put up with this new product soon developed the 
 weaknesses which brought down upon an entire industry the condemnation which was earned 
 by its incompetent putterers. 
 
 Many of these early products were not even sound structurally. Many of them were 
 extremely porous. The appearance of a concrete block wall after a rain and the slowness with 
 which the moisture disappeared in subsequent sunshine, gave rise to the early belief that 
 concrete as a building material makes for dampness and is unfit for dwellings. The fact that 
 the early block had hollow spaces giving a cored wall equal to 25 to 50% of the cross-sectional 
 
 1 Managing Editor Concrete. Author "Concrete Stone Manufacture." 
 
Sec. 2-75] 
 
 GENERAL METHODS OF CONSTRUCTION 
 
 147 
 
 area, led to the extravagant claims not only that no moisture could get through the wall from the 
 outside, but that such a wall was sufficiently insulated so that there could he no possible danger 
 of condensation on the interior wall surfaces. The blockmakers had the architect against 
 them. The commonest type of block was an imitation of pitch face stone, which the architect 
 objected to, not so much because it was an imitation, as because it was a very poor imitation 
 of the real thing. Architects also objected to the early units because of their proportions in 
 height and length, commonly 8 in. high by 16 in. long. 
 
 The conditions which are gradually driving the incompetent out of the industry, and which 
 are closing down those plants which are inadequately equipped, poorly managed, and under- 
 capitalized, are bringing into the industry men who were first unattracted by what seemed to 
 be a "small fry" business. The possibilities of concrete stone manufacture have been de- 
 veloped very remarkably in the last 10 years, and much more rapidly in the last 4 or 5 years, so 
 that many of the leading architects are now specifying manufactured stone on an equal basis 
 with natural stone, and in some cases, in preference to natural stone in important building 
 enterprises. 
 
 i There are very few who doubt the value of well-made concrete building units and it remains 
 only for intelligent manufacturers to develop their business along lines entirely different from 
 those which characterized the industry's early efforts. 
 
 A most important thing for them to appreciate at the outset is the difference between 
 concrete units which are suitable for exposed walls and for the trim of first-class buildings, and 
 those other units which demand no architectural consideration and which are used to replace 
 
 ! common brick in foundation walls and such walls as are to be faced with some other material. 
 
 I 75. Two Main Lines of Work. — There are in general two types of concrete building units. 
 Their manufacture involves two distinct lines of work. One is in the production of standard 
 units in quantity; units which are structurally sound but have no special claim for use where 
 any architectural purpose is to be served. This is a simple bulk proposition, where the constant 
 flow of materials and the quantity of manufactured output are largely the determining factors 
 in the success of the enterprise. The other line of work is in the manufacture of trim stone or 
 standard units which are specially faced or otherwise surface treated to make them suitable for 
 exposed walls or trim in competition with natural stone, face brick, terra-cotta, and other 
 well-known building materials similarly used. 
 
 The success of one enterprise or the other depends quite as much upon the natural supply 
 of other building materials in the community as upon the enterprise and ability of the particular 
 manufacturer. A successful enterprise in the manufacture of trim stone and ornamental work 
 necessitates the employment of modellers, pattern makers, mold makers, workers in glue, 
 plaster, and wood; it involves the use of selected aggregates, more skilled workmen in the 
 molding department, and frequently the employment of stone cutters in the finishing 
 department. 
 
 76. Methods of Manufacture. — In the production of concrete building units there are the 
 wet process and the dry process. There is no well-understood definition which sharply distin- 
 guishes between a dry mixture and a wet mixture and the consistency of the concrete used varies 
 all along the line from that mixture which will just stick together when squeezed in the hand, to 
 the other 'extreme, a mixture which is of a soupy consistency. 
 
 There are three general classifications in manufacturing methods, each calling for the use of 
 different equipment. These three are the so-called dry-tamp method of manufacture; the so- 
 called pressure method; and the wet-cast method. 
 
 76a. Dry-tamp Method. — The dry-tamp method is the one most commonly 
 employed. It is the method whose products have in the main given concrete block its early 
 bad reputation; it is a method whose uses are quite satisfactory when in intelligent hands and 
 where there are methods of curing the products which contain so low a percentage of gaging 
 water; and it is the method whose abuses have resulted in many of the bad products that have 
 brought unmerited criticism in some cases of the entire output of the concrete products industry. 
 
148 
 
 CONCRETE ENGINEERS' HANDBOOK 
 
 [Sec. 2-766 
 
 The dry-tamp product may be made in an ordinary wooden box, but as commonly known it is 
 made in the mold boxes of simple machines, so familiar everywhere (see Fig. 103). The mix- 
 ture should have just as much water in it as will permit the quick removal of the product from 
 the mold in which it is made. The abuse of the method is in using too Httle water. 
 
 766. Pressure Method. — Pressure machines are not so common in the field as 
 they were at one time although their use is undoubtedly increasing at present. There are in use, 
 
 however, machines applying pressure hydraulically, 
 and others in which the pressure is exerted mechani- 
 cally by means of toggles operated by hand (see Fig. 
 104). It has been urged by some that the applica- 
 tion of pressure which is exerted evenly over one en- 
 tire face of a product does not result in so dense a unit 
 as is possible through tamping, the contention being 
 that this even pressure, allowing less free displace- 
 ment of individual particles than by tamping with a 
 small-headed hammer, induces an arching action, 
 particularly when crushed stone is used, this arching 
 action among the particles of stone resulting in voids. 
 There appears, however, to be little practical evi- 
 dence of the correctness of this belief. 
 
 76c. Wet-cast Method. — Wet -cast 
 concrete products are not easily defined nor are the 
 
 Fig. 103. — Hand tamp block machine. „ . , . , , • . ,i m i rm 
 
 factors which enter into them easily outlined, ihere 
 is more variation in the equipment used and in the methods pursued in wet-cast work than in 
 the other two methods. The quantity of water varies a great deal in wet-cast work, as be- 
 tween the consistency used in sand molds and that used in metal molds, for instance. With 
 sand molds, it is possible to use a very high percentage of water in the mix, providing the 
 mixture is constantly agitated before being deposited in the molds, because the excess moisture 
 
 Fig. 104. — Pressure block machine. 
 
 is, in large measure, taken up by the sand of the mold. The use of a very wet mixture in 
 metal gang molds (Fig. 105), or in a fairly tight mold of any kind, would result in a poor prod- 
 uct, due to the fact that much of the moisture could not readily escape and would be almost 
 sure to result in a porous product. The tendency to be overcome is the use of two much water. 
 
 77. Consistency. — The consideration of the various processes of manufacture has involved 
 some thought of consistency which, in a measure, defines those processes. It is now fairly well 
 
5ec. 2-78] 
 
 GENERAL METHODS OF CONSTRUCTION 
 
 149 
 
 istablished in the concrete field that the ideal consistency is in that mixture which will barely 
 •etain its shape when the forms are removed immediately after the concrete has been deposited 
 md pressed into shape. This is just a little wetter than can readily be used in dry-tamp ma- 
 ;hines. It is just a little wetter than is ordinarily used in pressure machines and it is a great 
 leal drier than the mixture which is ordinarily used in the wet-cast concrete. Since, however 
 various other conditions which contribute to the production of good concrete are more suscep- 
 ible of control in the case of factory work than in field work, the concrete products manufac- 
 ;urer has a somewhat wider latitude in the matter of consistency. 
 
 ; Concrete products manufacturers using dry-tamp equipment have constant difficulty with 
 
 employees in trying to get them to use a mixture of the wettest consistency which it is possible 
 
 ;o use in the mold boxes of their machines. 
 
 rhe drier the mix, granting that it is just wet 
 
 ;nough to stick together under tamping, the 
 
 ;asier it is to remove the product from the mold 
 
 vithout damage. The quantity of water which 
 
 ^ives the mixture an ideal consistency resulting 
 ;n water marks on the outside of the product 
 
 vhen it is removed from the mold will, it is gener- 
 
 illy contended in the field, cause the product to 
 
 stick to the face plates, resulting in damaged 
 products and in retarding the work. This stick- 
 : ng is undoubtedly due to a combination of the 
 
 tvater and fineness of the facing material in caus- 
 i ng a suction on the face plate which mars the 
 :"resh product. A wetter mixture with a slightly 
 ; coarser facing material is not so liable to cause 
 
 damage in removal from the mold. 
 
 The results of dr}^ mixtures are not nearly so Fig. 105. — Metal gang molds mounted on car. 
 
 Dad as might be expected. When the products 
 
 ire removed from the molding room promptly and put into a steam curing room where a warm 
 itmosphere saturated with moisture does not permit the evaporation of any of the moisture 
 ivhich has entered into the block in the first place, very good products can be obtained. 
 
 Mixtures so wet that they would give very unsatisfactory results under ordinary conditions 
 live high-class products when poured into sand molds. A recent tendency among sand-cast 
 stone manufacturers is toward less water and toward longer mixing, the additional mixing 
 serving in the place of so great an excess of water in obtaining a smoothly flowing mixture. 
 High crushing strengths and a high degree of density are obtained. Numerous architects show 
 a, preference for the manufactured product over the natural product because of less tendency 
 to discolor through the absorption of moisture and dirt. Where a mixture is poured, however, 
 into a rigid and non-absorptive mold, as in the use of steel gang molds, in block and brick manu- 
 facture, it is important that the moisture content be kept down just as low as is possible consist- 
 ent with a ready flow of the material into the forms and around the cores. 
 
 78. Commercial Molds. — Commercial molding equipment is, for the most part, very sim- 
 ple, the essential requirement being a mold box from which the product can be readily removed 
 when shaped. An idea which has been almost inseparable from concrete block from the incep- 
 tion of the molding machinery for its production, is that it shall provide a partially hollow 
 wall. This is very clearly shown in Fig. 106. The various sketches show how the designers 
 of different machines have varied the provisions for air space either in the unit itself or in the 
 wall as the block are laid up. The block shown at A, B, C, D and / are in one class, being 
 complete units in each case providing the entire wall thickness. At E are two separate thin- 
 «^all slabs which, when laid in the wall, are held by metal ties. The unit shown at H is for 
 light residence or other light wall construction, providing the plain outer wall surface on what 
 
150 
 
 CONCRETE ENGINEERS' HANDBOOK 
 
 [Sec. 2-79 
 
 is here shown as the upper side of the sketch. The projecting lugs provide a base for attaching 
 furring strips for lath and plaster. Blocks F and G each consists of two separate parts held by- 
 metal ties, cast in the blocks. The broad U-shape block J is designed for an interlocking 
 arrangement as laid in the wall, the straight faces of this block forming both interior and 
 exterior wall surfaces, giving a complete and continuous air space in the wall. The block 
 shown at K is for a similar construction, the lug on the block giving the desired bond between 
 the two sides of the wall. Still another type of block has three rows of vertical ducts, and two 
 horizontal. The center row is filled with slush concrete as laid up and provides for reinforcing 
 rods. 
 
 The hollow space serves to economize in material, to make a lighter, more easily-handled 
 building unit, to provide as laid up in the wall, either a series of vertical air ducts, as in the block 
 shown at A, C, D and /, or a more nearly continuous air space throughout the wall, as in 
 
 the block shown at E, F, G, H, J and 
 
 'r V' ° "V ? ^nn rS i%^L£^%^'p H A 'r\ r^°rs''r\^? ^' varied manner of provid- 
 
 , ^ , hi, j:uO:y:l^ /jDl^J^'l, |0;UQl);Q;^ ing air space in the wall lies the chief 
 A B c D difference between many of the ma- 
 
 3 , ' ' • ^««33>><^iP liA:/. " r ^^ i^ chines on the market. 
 
 J ^ Y . . ft-, c?^S-<S^ |j y 79. Operation of Machines.— In 
 
 the operation of the machines, con- 
 
 sidering more particularly the mold 
 boxes, the general type shown in 
 Fig. 103 is made either to make 
 
 Fig. 106.— Horizontal cross-sections of representative types of ^^^ck face down, face Up, Or with the 
 
 concrete block. face at the side. If block are to be 
 
 made with a special face design, as 
 for instance, the lamentable example of the rock-face block which is poorly conceived as an 
 imitation of pitch-face stone, then the face plate, which is to give this design, is usually at the 
 bottom. If the face of the block, however, is to be faced with a separate mixture of material for 
 a special texture or color, then the machine will be of either the face-down type or face-up 
 type. If block are to be produced solely for structural purposes, without face design, or with- 
 out a facing material used on the face side, the so-called stripper machines have given very 
 satisfactory results. In these machines, the block is produced upright in the mold just as it is 
 used in the v/all and the cores of the machine are introduced and removed by upward and down- 
 ward movements. The sides of the block are thus always perpendicular to the bed of the 
 machine, hence the block is stripped out of the mold with a troweling action. Other types 
 of machines tip the block over before it is removed on the pallet. 
 
 In some machines the cores are removed with a downward motion after the block has been 
 tipped over. The common way, however, is to withdraw the cores while the block is still face 
 down, leaving the hollow spaces lying in horizontal position. After the cores are removed a 
 block is turned over on the pallet. All this is done by the movement of two levers, one to 
 remove cores and one to tip over the mold box, or by automatic mechanism which is set in 
 motion by one lever movement. 
 
 One of the pressure machines has a sort of track of equal length on each side of the pressure 
 head (see Fig. 104). Mold boxes travel on this track, one box at each end. The box is filled 
 clear to the top, if it is to be a plain block, or if it is to be a faced block the backing material 
 is struck off at a depth of }i in. below the top and the facing material is put on and struck off. 
 The box is then rolled on the track to the center of the machine under the pressure head. 
 When the levers are released, the box is rolled back to its first position. A pallet is placed on 
 top and clamped in position. The box is then turned over and lowered so that the pallet rests 
 upon a stand placed to receive it. The block is thus released face downward on the pallet. 
 
 79a. Tamping. — In the use of machines in which block are compacted by means 
 of tamping, this tamping is done in three ways: (1) by hand solely; (2) by hand-operated pneu- 
 
Sec. 2-80] 
 
 GENERAL METHODS OF CONSTRUCTION 
 
 151 
 
 natic tampers; or (3) by means of machine tampers suspended above the mold box. Hand- 
 ^amping is most common, the common type of tamper (Fig. 107) being a double-ended instru- 
 ment with a center bar for a handle, and with one narrow spade-like head and one broad flat 
 lead, the narrow head being used beween cores and the broad head over the full block area 
 Defore and after the cores are introduced. In hand work, everything depends upon the oper- 
 itor and most manufacturers maintain that the operator relaxes his efforts to a considerable 
 extent toward the end of the day. In spite of this, a man who is accustomed to this work gives 
 3xcellent results with hand-tamping. 
 
 Machines for tamping have to be built to withstand severe jarring. They are supported 
 either by a framework from the floor of the factory or suspended from a framework above. 
 5uch tampers have several feet at the ends of plungers so 
 
 arranged as to fit the type of machine in use. These f f^ — I p r^ Tl 
 I plungers strike the concrete with equal force over the ^ 
 
 sntire open area of the mold. The plungers are so ad- Fig. io7. — Hand tamp. 
 
 :iusted as to be responsive to the depth of the material 
 
 |.n the mold box, so that the length of stroke varies as the mold box is filled, 
 j 80. Gang Molds for Wet-cast Products. — A type of equipment for making wet-cast 
 jproducts that is coming into more general use consists of gang molds. One kind is mounted 
 ipon cars (Fig. 105), which are run on tracks to receive the concrete at the mixer, and from there 
 io curing tunnels. Another kind is located in such a way throughout a stretch of floor as to 
 'equire the mixed concrete to.be brought to the molds. This process thus involves either the 
 ,jse of a very large number of individual molds in which the products must remain arranged 
 .jver a very large casting area for from 12 to 48 hr., or else it involves the use of molds set up 
 n such a way that they may be conveyed to a curing place when they are filled. When the 
 i products have become hard, the molds are simply taken down by a removal of the core pieces, 
 sides and division plates, and are oiled and set up again, either on platform or car. 
 81. Materials. 
 
 « 
 
 81a. Cement — Storage and Conveying. — As a rule concrete products manufac- 
 turers are satisfied to use any standard brand of Portland cement, which can usually be de- 
 pended upon to conform to the specifications of the American Society for Testing Materials. 
 There is, however, a disposition among some manufacturers, particularly those making a high 
 ^lass of trim stone and more particularly also where a rather wet mixture is used, to select their 
 brand of cement with some care. Some of these manufacturers believe that some brands have 
 1 tendency to cause crazing, which is one of the bugbears of the concrete stone manufacturer. 
 No manufacturer has been found, however, who can explain just exactly the reason for his pref- 
 9rence for one brand over another, except so far as his experience has seemed to show that the 
 use of one brand resulted in less crazing than another. There is a belief among some manu- 
 facturers that a cement which has been aged much longer than is ordinarily demanded is desir- 
 able in concrete stone manufacture and it is said that this older cement is less likely to give 
 hair-checking or crazing. In other respects, the quality of cement required in concrete products 
 manufacture scarcely differs from that in general concrete work. 
 
 In a small plant the cement is ordinarily stored on a platform as near as possible to the 
 level of the hopper which feeds the mixer. This may be, in some plants, at the second floor 
 level, chutes being used for cement as well as aggregates, or it may be at a level between the 
 first and second floor, determined by the level of the mixer itself. If stored in bags on the second 
 floor level, an elevator of some kind is provided unless the plant is so small as not to warrant 
 the use of equipment of this kind. It is possible,' sometimes, to have a railway siding on a 
 trestle and to use gravity conveyors with ball-bearing rollers to carry pallets bringing bags of 
 cement. With an arrangement of this kind, the cement is brought into the storage space 
 direct from the car with very little handling. In another plant the track may be slightly 
 above the level of the floor on which the mixer stands; or, even with it on the same level, it is 
 possible to build a platform at the level of the car floor and to pile bags of cement in such a 
 
152 
 
 CONCRETE ENGINEERS' HANDBOOK 
 
 [Sec. 2-816 
 
 way that they may be emptied into a cart filled with gravel from a bin along the side and dumped 
 into a mixer which stands just under the platform. 
 
 Bulk cement has not been used extensively in products manufacture but has been very 
 successfully used by a few. The cement may be scraped from the door of the car into a chute 
 feeding into the bottom of an elevator boot, the elevator lifting the cement into a bin in the 
 top of the plant from which it falls by gravity into a measuring box above the mixer hopper. 
 In another plant the cement has been loaded from the car to wheelbarrows, handled over a 
 runway to the hopper of a mixer. Where the output of the plant is large enough and it has 
 been possible to hold a car to use up its entire contents, this handling of the cement has been 
 very economical. When this has not been possible, the cement has been dumped from the wheel- 
 barrows into a bin close to the mixer from which it is shoveled into the mixer hopper. 
 
 816. Aggregates — Kind and Quality. — In the main, the aggregates used in the 
 general field of concreting, are suitable, except as to size, for concrete products manufacture. 
 In general, fine materials are used throughout the concrete products field. Better, cheaper 
 products can of course be made when larger aggregate can be used, the maximum size equal 
 to one-half the smallest dimension of the product. 
 
 Aside from quality as to cleanness, hardness, and so on, the concrete products manufacturer 
 has been most concerned with the consistency of the mixture with which the size and quality 
 of the aggregate have a great deal to do. A very dry mixture on one hand and a very wet mix- 
 ture on the other — both of them more common in the field of products manufacture than any 
 intermediate mixture — have both been an influence in favor of. rather fine materials. Most 
 manufacturers using dry-tamp equipment appear to be convinced that coarse materials cannot 
 be successfully used in a mixture containing little water because of a tendency of coarse mate- 
 rials to fall out of the product on removal from the molds and to cause a high percentage of 
 breakage. 
 
 Crushed limestone, especially when there is a rather high percentage of fine material, un- 
 doubtedly permits the use of more water than does sand. It is still a question whether or not 
 the excess of moisture used in a mixture of such material, becomes available for the hydration 
 of the cement in the curing period which follows the molding. 
 
 The prevalent belief is that a high percentage of fine materials should not be used, the 
 usual specifications being that a percentage no higher than 5 or 10% passing a 100-mesh screen 
 shall be used in the fine aggregate. 
 
 The most desirable qualities in concrete building units are, of course, strength and the 
 quality of resisting the attacks of the elements, to the end that they will not disintegrate the 
 concrete nor spoil its beauty through absorption of discolorative agents. 
 
 Bank-run and crusher-run materials should not be used. It is important that a concrete 
 products manufacturer's output be of even quality. To this end he should maintain a constant 
 supply of an aggregate of uniformly high quality, grading in size from fine to coarse. Any 
 amount of tamping or pressing, or care in puddling and pouring, or in placing the concrete, 
 is entirely unavailing if these operations have not been preceded by scrupulous care in the choice, 
 grading, and mixture of the materials, as essential to securing density in the product. It is 
 strongly recommended in connection with this chapter that reference be had to the chapter 
 on "Aggregates" in Sect. 1 and the chapter on ''Proportioning" in Sect. 2, as these chapters 
 treat in detail of the selection and grading of the materials in order to obtain the best results. 
 No manufacturer should determine his proportions arbitrarily but should first examine or have 
 examined samples of the material which he proposes to use and which he has reason to believe 
 will come to him in unvarying quality. To make sure that this quality is unvarying, frequent 
 tests should be made to determine the grading which should precede any decision as to the pro- 
 portions in the mixture. 
 
 82. Mixing. — No one in the concrete field has a wider latitude in selecting mixing equip- 
 ment than the concrete products manufacturer. His plant is stationary, the requirements of 
 
Sec. 2-82o] 
 
 GENERAL METHODS OF CONSTRUCTION 
 
 153 
 
 the plant are more or less regular; and he has no problems of getting about, here and there, 
 under varying conditions. 
 
 82a. Mixers — General Type. — The commonest types of mixers in concrete 
 products plants are those with the simple cylindrical drums in which a comparatively dry mix- 
 I ture of concrete ordinarily is turned out, and the continuous mixer, for a long time much despised 
 in the general field. In many concrete products factories where a wet mixture is being turned 
 ^ out in large quantities, types of mixers are used similar to those found in large construction 
 ' work in the field. In addition to these types, the small cylindrical mixer for handling facing 
 materials is common in almost every concrete products factory. There are frequently two or 
 three of these so that they may be used for facing mixtures of various kinds without change. 
 In connection with the mixers employed there is one thing which is perhaps unique in the prod- 
 ucts field and that is the adaptation of some of the best-known makes of continuous mixers to 
 the specific needs of the factory in which they are used. Local mechanical ability in each case 
 I has been able to set up these machines so as to give very satisfactory results. Not only has 
 ' the flow of dry materials been fixed under close control but the water is added in definite 
 quantity to give a continuous flow of a like mixture. It is almost invariable that the use of a 
 continuous mixer in a concrete products factory requires the erection of bins and hoppers over 
 those with which the machine is equipped. 
 
 j 826. Mixing Dry and Mixing Wet. — It is coming to be more general practice to 
 
 Imix the concrete materials dry in one mixer and to add the water in a second mixer. In a plant 
 where stone of a very high quality is manufactured, materials are stored on the second floor. 
 They are shoveled into a car in definite proportions with the cement on top. This car is shoved 
 along the track in front of the bins from which the materials are obtained and is dumped into 
 a cylindrical drum mixer where the materials are mixed dry. They are then dropped through 
 ' a chute to a continuous mixer on the first floor where the water is added, the quantity of water 
 being very carefully gaged to give a like consistency all the time. 
 
 Long and thorough mixing is particularly important in concrete products manufacture 
 where a homogeneous mixture of like color throughout is particularly desirable. Mixing the 
 materials dry and then adding water in another mixer is almost sure to make for greater 
 thoroughness in a combination of the materials. 
 
 82c. Agitation Subsequent to Mixing in Wet-cast Work. — Where materials are 
 mixed wet for casting in sand molds, it is highly important that the agitation of the mixture be 
 continued so as to prevent segregation of the materials. Even in mixtures where the size of 
 stone is little more than in., it is impossible in the wet mixture which is used, to prevent rhis 
 stone settling to the bottom of the receptacle as it is poured out from the mixer. For this reason 
 it is common practice to provide some means of keeping this mix agitated up to the time it 
 is placed in the molds. In the largest plants this is done in an auxiliary mixer travelling on an 
 overhead crane, driven by an electric motor — really a mixer in itself. It takes the materials 
 from the mixer proper, keeps them constantly agitated until the mix is deposited through a 
 pipe 3 or 4 in. in diameter to molds in a sand bed. In smaller plants where such equipment 
 seems unwarranted, it is common to use a large wooden cask either swung from a travelling 
 hoist or mounted on a truck moving about on tracks through the casting area. A workman 
 simply turns a crank operating paddles to keep the mixture agitated while it is being run off 
 through a spigot into the molds. 
 
 82(i. Mixing Facing Materials. — There are small cylindrical drum mixers specially 
 provided for mixing facing materials in small batches. Thorough mixing of the facing mixture is ■ 
 highly desirable so that there may be, in the case of special aggregates or the use of color in any 
 form, a thorough distribution of the material in order that it will not be spotted or in any way 
 uneven either in color or in texture. In the coarse-textured concrete stone, which is becoming 
 more popular, the mixture used is comparatively lean in cement and thorough mixing is, there- 
 fore, necessary to be sure of cementing the particles in place. These products are afterward 
 
154 
 
 CONCRETE ENGINEERS' HANDBOOK 
 
 [Sec. 2-83 
 
 brushed and it is important that the mixing be thorough so as to be sure of embedding the stone 
 particles. 
 
 83. Placing. — In a small concrete products plant where perhaps but two hand-operated 
 block machines are used, and where only two men may be employed at this work, it is not 
 uncommon to operate the mixer for a short period; pile up a batch of concrete in front of the 
 two machines and shovel it from the floor direct in the machines, each workman serving him- 
 self at this labor. This, however, is not the way of the modern concrete products plant which 
 does away as much as possible with wasteful hand methods, and by increasing the capacity 
 and output of the plant and increasing the quantity of machinery used, lowers the cost of the 
 product and in most cases improves its quality. 
 
 83a. Buckets and Hoppers. — In the majority of concrete products factories, 
 the mixed concrete is conveyed from the mixer by a bucket travelling by a trolley system to serve 
 the trim stone department and a row of block machines. From this travelling bucket the con- 
 crete is deposited in 4 to 5-cu. ft. batches in hoppers feeding to sloping tables just behind the 
 block machines (Figs. 108 and 109), the operators of the machines scraping the mixed concrete 
 into the mold boxes with very little lost effort. 
 
 Fia. 108. — Factory layout. (Block department in background.) 
 
 R = Curing rooms. 
 
 S = Second floor, or rather an intermediate floor above curing rooms where a battery of mixers (5) are located: 
 E = Elevator for boxes of mixed facing material. 
 
 B = Bucket (a part of an electric monorail system) for delivering mixed concrete to machines and at bankers. 
 H = Hoppers to receive mixed concrete. 
 C = Block car. 
 
 In a large factory, where there are several block machines, and a large-dimension stone 
 department, there is a battery of mixers at an intermediate level between the first floor and the 
 second floor, so arranged that the raw materials come in by conveyor belting at the level of the 
 hoppers over the various continuous mixers. The mixed concrete is fed into buckets which are 
 hooked to a monorail system operated electrically so that any workman anywhere in the large 
 molding room can have a box of mixed concrete delivered to him suited to his special work, j 
 by means of tracks and switches controlled from the starting point. The empty buckets are 
 then returned to be refilled. This is an elaborate system and an expensive installation. Few 
 existing factories probably have an output to warrant it. In a factor}'- making but one type 
 of units a conveyor belt brings the mixed material from mixer to machines. 
 
 The market offers machinery which couples the block machine with elevating equipment 
 and delivers concrete direct from the mixer to a hopper in the top of the machine and drops it as 
 required into the mold box under the tampers. Such equipment is usually coupled with machine 
 tampers. 
 
ec. 2-83ftJ 
 
 GENERAL METHODS OF OONSTRVCTlON 
 
 155 
 
 836. Wheelbarrows.— It is far more common in factories which put most of 
 heir effort on dimension stone, to find that concrete is handled chiefly in wheelbarrows. One 
 f the largest wet-cast stone factories in the East, and one of the largest factories making dry- 
 amp dimension stone in the Middle West, make use of wheelbarrows in transporting the 
 lixed concrete to the molds. In the case of the wet concrete, four pails are carried in a barrow, 
 'he situation is quite different, however, in various localities in respect to labor, and the situa- 
 ion is also affected by the fact that, in a factory making dimension stone which is to sell in 
 ompetition with natural stone, the actual labor in molding, tamping, and finishing is much 
 More per cubic foot of concrete than is the ordinary standard product made in machines. The 
 ulk of the concrete handled is, therefore, of less consequence than where standard units are the 
 hief products. 
 
 83c. Pallets.— The pallets used in standard block machines and brick machines 
 re commonly of two kinds— wood and iron. The pallets stand repeated changes from wet to 
 
 ':ry and are subject to severe wear. If iron pallets are neglected, they become coated with rust 
 nd concrete so as to be useless. It is recommended that to keep iron pallets in proper con- 
 
 jlition they should be kept coated with paraffine oil, or that they be dipped in a mixture of 
 
 Fig. 109. — Block machines with hoppers above are shown in the background and block cars in the foreground, 
 ^ote the two floor levels. The cars run on tracks into pits so that four decks can be loaded without too high a 
 each. The empty "decks" from the car at the left are placed on the car at the right as it is piled. 
 
 ierosene and axle grease, when the concrete can easily be wiped off at the end of each day's 
 rvork. The same treatment is recommended for the metal parts of the block machine. An- 
 )ther manufacturer suggests that iron pallets be dipped in a solution of 1 part lard oil and 1 
 3art kerosene, to keep the pallets clean and to prevent rusting. With wood pallets, the 
 hfiiculties are from swelling, splintering, warping and so on. It is necessary to use wood which 
 s as little subject to warping as possible, and have the pallets well treated in order to overcome 
 varp so far as it can be done. It is pointed out that pallets should not be made of one solid 
 jiece but of strips not more than 4 in. wide, with slightly open joints to allow for some expan- 
 lion. Some manufacturers recommend dipping wood pallets into hot linseed oil. Difficulty 
 rom slight swelling of the pallets is not important except in machines where pallets must fit 
 ^ery accurately as is not the case with most tamp machines. There are machines, however, 
 vhere the pallets must fit with great nicety and in such cases considerable difficulty has been 
 ixperienced in getting a pallet which will resist the severe treatment. One manufacturer 
 uffering such conditions finallj^ adopted a combination wood and metal pallet. 
 
 Most blocks are delivered on a pallet right side up, face perpendicular as they will be laid 
 
156 
 
 CONCRETE ENGINEERS' HANDBOOK 
 
 [Sec. 2-83d 
 
 in a wall. The condition of the pallet in this case is not so important. Where, however, the 
 block is turned over face down and delivered on a pallet in that position, it is very important, 
 particularly for face block, that the pallet be very true, and metal pallets are recommended 
 for such work where a true, smooth face is required. For rough-textured block, this is, of 
 course, not necessary, 
 
 ^Zd. Bankers. — Practically all dimension stone is made in special molds, these 
 molds resting on heavy plank pallets, supported in turn by bankers which may be of reinforced 
 concrete to minimize vibration. Such bankers are used in factories where dimension stone is 
 made by the dry-tamp process. For work of this kind, the placing of the concrete is much 
 slower than with wet-cast work, as the facing materials have to be built up vertically 2 or 3 in. at 
 a time on such faces of the stone as are to be exposed, besides placing on the bottom, and the 
 backing must be tamped in as the facing is brought up. Thus for the convenience of the 
 workmen the mold is placed on bankers at a height which is convenient for his work. 
 
 84. Curing. — In from 20 min. to 1 hr. after water has been added and the mixing of con- 
 crete completed, this mixture must be placed and it must not only be so handled subsequently 
 as not to disturb the hardening process but it must be kept in a condition which will aid that 
 process. Conditions for curing must be such that the product will not be rapidly dried, yet 
 as temperature influences the hardening — heat quickening it, cold retarding it, and freezing 
 interrupting the hardening process for the period in which the low temperature continues — it 
 is important that these things be considered in caring for products when they have been molded : 
 or cast. 
 
 The problems of curing do not present themselves as so serious a matter to the manufacturer 
 of wet-cast stone as to the manufacturer of dry-tamp products. In the wet-cast work where 
 there is already an excess of moisture, sufficient heat is practically the only essential, with con- 
 ditions which will prevent too-rapid drying. In tamp products, and for the most part in pressed 
 products where the moisture entering the mixture is only just about (or even a little less than) 
 that actually required for thorough hydration, it is very important that none of this moisture 
 be permitted to escape before complete hydration. 
 
 In curing wet-cast stone, block, and brick made in gang molds on cars, one method is to 
 move these cars on tracks to curing tunnels which are heated by steam. The tunnels are made 
 of concrete and just high enough to admit loaded cars. They are heated to give a rapid hard- 
 ening of the concrete. The cars are usually removed 24 hr. after they are placed in the tunnels, 
 the molds are taken down, and the products are carefully piled under sheds. The molds are 
 oiled, set up again on the cars and returned to the mixer to be refilled. In other wet-cast work, 
 where gang molds on cars are not employed, or where other molds are used with a wet mixture 
 on a large casting floor, it is common to leave these molds in place until the product is suffi- 
 ciently hard to be handled. The molds prevent a rapid escape of moisture in the early stages i 
 of hardening, and particularly is this true in sand cast work, where the sand is always damp 
 from having absorbed the excess moisture of the mixture. 
 
 84a. Natural Curing. — The recommendations in the old "Standard Practice" 
 of the American Concrete Institute with respect to natural curing are usually regarded as sound. 
 They are as follows: 
 
 Natural Curing. — The concrete products shall be protected from the sun and strong currents of air for a period 
 of at least 7 days. Throughout this period they shall be sprinkled at such intervals as is necessary to prevent dry- 
 ing, and maintained at a temperature of not less than 50°F. Such other precautions shall be taken a^ to enable the 
 hardening to take place under the most favorable conditions. Products must not be removed from the yard until 
 they are 21 days old. 
 
 Where products are cured in this, way, it is necessary that racks or cars be used so that 
 block on the pallets may be piled up in tiers. As standard practice requires that products be \ 
 sprinkled for 7 days, it is obvious that there must be curing shed space for 7 days' output and 
 a very large yard storage space in order to keep products until 21 days old. Building regula- 
 
I 
 
 5ec. 2-846] GENERAL METHODS OF CONSTRUCTION 157 
 
 ions in numerous cities require that products be at least 30 days old when cured in this way, 
 
 \ vithout regard to the crushing strength. 
 
 Inasmuch as this method of curing would require the use of a very large number of cars 
 
 , )n which to tier up the products, it is common, when natural curing is used, to apply the rack 
 ;ystem of storage. It will be obvious that this can be used only in rather small plants. The 
 •acks are usually built of 2 by 4's, each rack 16 to 20 ft. long. One rack can thus readily 
 iccommodate 26 blocks 8 by 8 by 16 in. These racks can be piled about four high. Four 
 •Qws of racks will thus accommodate about 400 blocks. 
 
 : With natural curing, the products are either moved on cars and placed upon racks, or are 
 ;arried on pallets direct from the machine to the racks. This latter method can only be used 
 n the smallest plants because the labor of carrying the block the distance required by racking 
 
 I n a large plant would make the cost excessive. 
 
 Sometimes products are made which are too large or too heavy or of too awkward shape 
 
 ; or removal to curing sheds. Until hardening has progressed to a considerable extent precau- 
 ion should be taken to see that these products are kept under suitable conditions to attain 
 
 ! strength. When left in the molding room, the products should be covered with wet cloths 
 md the cloths kept wet. This applies particularly to tamp products where the molds are 
 -emoved a day or so after manufacture. Sprinkling may be done systematically and thoroughly 
 
 \N\ih. a nozzle, which gives a fine, well-diffused spray. The nearer the spray approximates a 
 
 'ioating mist the more thoroughly it will do the work, reaching all the surface of products 
 stored in tiers on racks or cars. 
 
 ! Where the products are removed to sheds and cured in the natural way, it is obvious that 
 jonsiderable labor will be required to use the hose on the products and it would be difficult to 
 ase the hose in any effectual way. It is common, therefore, to install permanent sprinklers 
 m the curing sheds. 
 
 In an Eastern factory turning out high-class products from wood and plaster molds, the 
 time for the products to remain in the molds is from 5 hr. for small units up to 24 to 48 hr. for 
 the larger and more complicated pieces. Until the molds are removed it is not necessary to 
 ipply additional water to prevent the escape of the moisture contained in the product because 
 there is considerable water used in the mix. As soon as the molds are removed, sprinkling 
 oegins, using a fine spray. This factory, which is of old construction, has wooden columns 
 ibout 10 ft. apart in each direction. Pipes have been placed so that there is a water outlet at 
 3very column. A man is kept busy all day sprinkling the products and another man continues 
 the work at night. Great care is taken to keep the products moist until they dry out evenly. 
 
 In some ornamental work, manufacturers frequently make use of total immersion of prod- 
 acts to cure them. It is common in such cases to cover the product with wet cloths just as 
 soon as it has been removed from the mold and allow it to remain in this way until it has at- 
 tained sufficient strength to be handled and placed in the tank. Other manufacturers do not 
 use the immersion system but simply use the wet cloths. 
 
 846. Steam Curing. — General practice in steam curing makes use of a wet steam 
 and a low pressure to create a dense warm fog with all the moisture which can be introduced at a 
 temperature in the curing rooms between 100° and 130°F. While common practice in the field 
 does not warrant the use of high-pressure steam and while investigations with high-pressure 
 steam in curing have not gone far enough to suggest its adoption on a commercial basis, there 
 has been some investigation tending to show that steam under pressure up to 80 lb. can be 
 ased with success. This steam has to be employed in steam-tight compartments. The ordi- 
 aary curing rooms of the concrete products plant are not steam-tight. They are wide enough 
 to accommodate the cars which usually run on 24-in. track and high enough to permit four 
 decks of standard block to be piled on the cars, leaving some room for the accommodation of 
 special products to which it may be necessary to give steam-curing treatment. The curing 
 tunnels of the plant are usually about 60 to 90 ft. long, built of concrete block, with arched or 
 " A-shaped " ceiling— preferably a ceiling made by applying Portlant-cement plaster to a ribbed 
 
158 
 
 CONCRETE ENGINEERS' HANDBOOK 
 
 [Sec. 2-846 
 
 reinforcing mesh. The ceihng is built so as to carry the condensed moisture of the room 
 to the sides and away from the fresh products which dripping would damage. Curing 
 rooms are sometimes built wide enough to accommodate two or even three tracks, so laid out 
 that in the case of very wide products to be admitted to a curing room, only the middle track 
 can be used, allowing plenty of room for the projection of the products at the sides. At one 
 end the curing rooms usually open into the molding department as convenient as possible to 
 the machines supplying the greatest number of products to be cured, and the other end fre- 
 quently opens into a passageway connecting with the yard, or to the yard direct. 
 
 The construction of the doors has given concrete products manufacturers considerable 
 difficulty from time to time, due to the fact that metal doors rust, unless kept in perfect condi- 
 tion, wooden doors swell with the steam on one side while they remain dry and of their original 
 size on the other side, and canvas curtains are very short-lived. These curtains roll up and 
 are fastened at the sides, usually, with carriage buttons. In using them, allowance has to be 
 made for shrinkage. Galvanized sheet metal for doors lightly framed with wood, or small 
 angle irons, have given satisfaction. 
 
 The following recommendations of the American Concrete Institute with respect to steam 
 curing were made some time ago but are still considered as representing good practice. 
 
 The products shall be removed from the molds as soon as conditions will permit and shall be placed in a 
 steam-curing chamber containing an atmosphere of steam saturated with moisture for a period of at least 48 br. 
 The curing chamber shall be maintained at a temperature between 100° and 130°F. The products shall then be 
 removed and stored for at least 8 days. (This does not apply to high-pressure steam curing.) 
 
 From an excellent discussion of the proper use of steam in curing concrete products, the 
 following by W. M. Kinney in Concrete is quoted: 
 
 The principal object in curing concrete products with steam is to accelerate the hardening by means of heat 
 without endangering the concrete through loss of moisture by evaporation. Saturated steam will provide not only 
 heat but sufficient moisture to insure against injury from drying. 
 
 In the early history of concrete products manufacture it was customary to use exhaust steam for heating the 
 curing chamber, but as a sufficient quantity was not always to be procured, the natural resort was to use steam direct 
 from the boiler. The records show that in many cases large quantities of good concrete came to grief due to its 
 drying action, which was not at that time explained. 
 
 Especially difficult is the maintenance of a sufficient quantity of steam in a boiler at low pressure to heat 
 sufficiently any number of curing rooms. Coupled with this difficulty is the danger of the pressure rising con- 
 siderably above that necessary for proper curing. With this in view we have been recommending steam under 
 pressure, that is, around 30 to 45 lb., provided it be admitted through water. 
 
 The most satisfactory way of admitting steam in this manner is through a perforated pipe embedded in a 
 trough of water running through the center or along the sides of the curing chamber. The floor should be so sloped 
 that any water or condensation on the products or on the walls of the curing chamber will be returned to the trough 
 for re-evaporation. In this manner the trough is automatically kept full of water and we have yet to record a case 
 of trouble when this method of curing was employed. All of the heat which the steam contains on being emitted 
 is taken up immediately by the water and the result shown by evaporation. The temperature of the water is at 
 the boiling point due to the fact that steam is continually being forced through it and what heat is taken up by the 
 water is used in evaporating the water. This water evaporated at 212°F. is just as useful in warming the room as 
 is the steam at the same temperature. 
 
 To explain the drying action of steam under pressure when admitted to the curing chamber, let us assume that 
 we are taking steam from a boiler operating at 30-lb. gage pressure. The temperature of the steam in the boiler 
 is 252° atmospheric pressure. It is, of course, understood that steam under atmospheric pressure and having a 
 temperature of 252° is in an abnormal condition which we technically call superheated. This steam is in a similar 
 condition to that which would be obtained if water vapor at 212° were heated up to 252° away from contact with 
 water. Naturally the first thing that steam in this condition does is to avail itself of the first opportunity to reach 
 normal conditions and the most ready way is to absorb water from anything in its vicinity which is so possessed. 
 The result is a drying out of the concrete products which happen to be stored in the vicinity. 
 
 Another method of introducing steam into curing rooms is described by A. E. Cline as 
 follows : 
 
 The simplest way is to have a main pipe over the top of the curing-room doors, then from this lead a separate 
 IH-in. pipe to each room. Run this along one of the walls close to the floor but with slant enough to drain to the 
 farther end. Join each length with a T, having the third hole of the T reduced to }4 in. and turn this at right angles 
 
ec.2-85] GENERAL METHODS OF CONSTRUCTION 159 
 
 5 the wall. The 1 J^-in. pipe can be reduced to 1 in. at half the length of the room, and this on a supposition that 
 lie curing room is 80 ft. long. If using steam for power, have this main pipe connected with the exhaiLst during 
 he day and with the boiler at night. If the boiler pressure at night is greater than 10 lb. it is best to have a 
 sducing valve in the main line as high pressure is not wanted. 
 
 85. Special Molds. — Although the commercial market affords a wide range of machines for 
 (lolding various concrete units and a large number of special molds — not only for sills, lintels, 
 ,nd so on, used in building, but for various ornamental objects and special architectural pieces 
 -the progressive concrete products manufacturer no longer considers that his plant is fully 
 quipped until he has a department for making molds to meet the demands of architects in 
 urning oiit dimension stone according to special designs as may be required. 
 
 85a. Wood Molds. — ^The material most commonly used in making these molds 
 or dimension stone is wood and a large plant will have an extensive wood-working department 
 vith power saws, planers, etc., for cutting down the labor cost. By far the greater part of 
 limension-stone work in most factories will be made in wood molds — preferably white pine, 
 rhis will include sills, lintels, belt courses, cornices and so on. In general, for a plain piece of 
 v^ork the mold consists of side planks resting on a pallet with end pieces fitting inside (see 
 ng. 110). 
 
 All pieces have to be carefully finished, all small holes or cracks filled, ordinarily with 
 )laster, the entire work shellacked and then oiled before use. By clamping the side pieces 
 irmly, the end pieces are held in place on the outside against cleats. 
 vVhen the facing mixture has been placed on the bottom of the mold 
 md up the front side and part way on the ends, as the case requires, 
 he backing follows. When it is tamped all the way up, the work is 
 itruck off at the top and a plank very carefully bedded on top by means 
 )f a layer of bedding sand. This plank is then clamped in place, the 
 damps passing over the bottom pallet and the entire work is turned 
 )yer so that when the mold pieces are released the stone is right side up. 
 
 
 Clarr'p-'' 
 
 F 
 
 r 
 
 
 
 
 
 
 
 
 
 ion 
 
 laving been tamped in place in an up-side-down position. It is not yig. no.— Cross-sect 
 lecessary to use the bedding sand on very small products which will bed simple wood mold, 
 •eadily on a smooth plank surface. 
 
 In wet-cast work in which the molds cannot be removed immediately, it is common to 
 ise a large casting floor smoothly finished with concrete. This is shellacked and oiled as an 
 )rdinary mold surface would be treated and on it are set up side rails for plain work, with the 
 lecessary insert pieces and dividing partitions to produce the plain units in necessary lengths. 
 Sometimes for greater convenience the bottom of these molds is provided in a bench or table 
 rvith a concrete slab top. 
 
 Where long side rails are necessary channel irons of proper widths can be used to advan- 
 :age. Properly cleaned and oiled they will give long service and the principal thing to recom- 
 .-nend them is the fact that they do not warp as wood does frequently unless the grain is very 
 leavily filled and the surface shellacked and oiled, nor do they spring out of line from the weight 
 Df the concrete. This is something that has to be carefully considered in long work, where 
 3ven a slight wind due to the springing of the form frequently prevents the use of the stone on 
 aice fitting work. 
 
 The market affords a great many standard molds made of metal for various ornamental 
 pieces and standard architectural units. Manufacturers who are catering to discrimmatmg 
 architects will not depend upon standard units, however, but will be. prepared to meet the de- 
 signs of architects. 
 
 856. Plaster Molds.— Plaster molds are very extensively used in concrete stone 
 manufacture. In fact, plaster is used not only in making molds but in making models, and 
 Qot only in plaster molds themselves but in making molded inserts for wood and metal molds. 
 Fig. Ill shows how, by means of a template of thin metal, stiffened by a wood frame operated 
 on a smooth oiled surface, it is possible to make moldings for various purposes in mold manu- 
 
160 
 
 CONCRETE ENGINEERS' HANDBOOK 
 
 [Sec. 2-85c 
 
 facture. The template is guided by a straight-edge and moves on the table, pushing aside the 
 soft plaster, except in the desired section described by the template. Similar work requiring 
 curved outline is handled by mounting a template at the end of a pivoted arm at such a length 
 as to describe the required arc. The template then has a circular movement. 
 
 Plaster is readily used in making models from architects' drawings. When this is done 
 the plaster is cast in a large block from which the model can be carved with knives and suitable 
 chisels after the details have been outlined with a pencil on the various faces of the block. The 
 
 plaster model is then shellacked and oiled and the mold made 
 over this model, the mold also being made of plaster. 
 
 Flat panels in moderate relief without undercut may be 
 cast in draw molds. The model should be framed with wood 
 strips or clay ''fences" to control the plaster. It should then 
 be given two coats of shellac (orange) and when dry, should be 
 greased, using a mixture of 1 part stearine and 2 parts kerosene 
 (combined hot). If a clay model is being reproduced, grease 
 moiding. with lard oil. Sift plaster into a pan half full of water, until 
 
 plaster lies about an inch below the surface of the water. Stir 
 thoroughly. When it has thickened to a creamy consistency, apply to model, going over 
 the entire surface thinly at first and jarring the model which is best supported by a rigid 
 frame. The jarring eliminates bubbles and pinholes. Then pour in the plaster, reinforc- 
 ing as may be necessary with strips of burlap, excelsior or wood frame for heavy work. Jars, 
 urns, Capitols and similar objects must have piece molds, the model surface being divided 
 by clay fences against which the plaster is applied. When a section of plaster mold is hard, 
 the fence is removed and notches are cut in the plaster edge to key the adjoining mold sec- 
 
 FiG. 112. — Plaster molds. The dark modeled part is the plaster model. The parts A and B are the first two 
 
 pieces of a plaster mold. 
 
 tion (see Fig. 112). The edges are then shellacked and greased for ready separation of 
 the mold parts. When the parts are set up for casting the concrete, the mold previously shel- 
 lacked and greased is held by rope or by chain and turnbuckle, or by clamps, as the size, 
 weight, or shape of the work may necessitate (see Fig. 113). 
 
 85c. Glue Molds. — The use of gelatin or glue molds is necessary in all work 
 where there is intricate undercut in the model to be reproduced, it not being possible to remove 
 a rigid mold over these undercuts unless a plaster mold, for instance, is made in many pieces 
 to join at the undercut and thus pull away. When a glue mold is to be made, the model is 
 
Sec. 2-85rf] 
 
 GENERAL METHODS OF CONSTRUCTION 
 
 161 
 
 greased and covered, first with paper and then with modeUng clay to the thickness necessary 
 for the thickness of the glue mold. This clay covering is then greased and plaster is applied 
 over it, to form a shell with a hole or several holes at the top with air vents at various 
 heights. This is illustrated in Fig. 114, the lion head being the model to be reproduced. The 
 space immediately around it is first filled with clay and over this is the plaster shell with 
 a funnel at the top. When the plaster mold is hard it is removed and the clay and paper 
 
 Fig. 114.— Mak- 
 ing a glue mold. 
 
 Fig. 113. — Plaster mold with plank pallets clamped on top and bottom being turned over on banker after tamping 
 full, before release of mold parts from the fresh product. 
 
 cleaned from the model. The plaster shell and model are shellacked and oiled again and the 
 shell is then fitted in place over the model and the space between the model and the shell 
 is filled with the glue, the vent holes being stopped with clay as the shell is filled. For 
 ready handling, the plaster shell is usually divided in a number of pieces, together with the 
 glue mold, the division being made with a knife. Glue should be of the best quality and should 
 be melted in a double receptacle with very little water used in the vessel with 
 the glue. Over-heating takes the ' life" or elasticity out of the glue. 
 
 S5d. Combination Molds. — The manufacturer who has a well- 
 developed model and mold department will have workers in wood, in clay, and 
 in plaster, and men also familiar with the making and use of glue or gelatin 
 molds. As the work progresses, a resourcefulness in meeting special require- 
 ments will lead the manufacturer to make combinations of various mold- 
 making materials as, for instance, plaster inserts in wood molds, and small glue molds in con- 
 nection with plaster molds to take care of small areas of undercut in the model to be reproduced. 
 
 85e. Waste Molds. — Ornamental pieces, especially when there is to be consid- 
 erable duplication and rapid work is necessary, are sometimes made in so-called "waste" 
 molds of plaster. If the model is intricate, a glue mold is made first and in the glue mold a 
 glue model is cast. From the glue model as many duplicate molds are made of plaster as there 
 are pieces to be cast. When the concrete has become hard within the plaster mold, the plaster 
 is cut away and the concrete surface cleaned. Panels, without undercut, are reproduced in 
 a similar way without the necessity for first making a glue model. 
 
 86. Sand Molds and Casting in Sand. — Sand-molding of concrete has not been in exten- 
 sive use except by a few manufacturers particularly in the East, until recently. Patents 
 covering important features of sand-molding are just about to expire. 
 
 For ordinary work the sand, or the mixture of sand and stone dust with a little loam or 
 other ingredients to make the sand particles adhere, as in iron foundries, is used in large beds 
 on a big casting floor. The sand is packed around models; the models are so made that they 
 can be withdrawn at the top and the molds are filled with a very thoroughly mixed concrete 
 11 
 
162 
 
 CONCRETE ENGINEERS' HANDBOOK 
 
 [Sec. 2-87 
 
 at a flowing consistency in which there are usually no particles larger than 3^^ in. and most of 
 the aggregates much smaller than this. Less water is used than the consistency might suggest. 
 The ideal mix has no free water. The mixture is constantly agitated after leaving the mixer 
 proper so as to prevent segregation of materials. Mixing frequently is continued for from 10 
 to 15 min. which greatly facilitates the smoothness of the flow. Ordinarily, the concrete is 
 deposited through'a spout and some means adopted (as for instance holding a small board near 
 the bottom of the mold) to prevent injury to the sand mold by the heavy stream of concrete. 
 To preserve the edges at the back of this stone, which is usually the surface upward, and to 
 permit subsequent troweling of this upward surface, it is common to use wood strips placed in 
 the sand to give a more stable edge against which to work. It is customary to fill the sand 
 molds throughout one floor area; allow an hour for the very wet mixture to stiffen and settle 
 and then trowel the upper surfaces, filling in a little additional concrete on the backs of the stones 
 where it has settled. 
 
 When balusters, capitols, and similar pieces are to be made, liaving no large flat surface 
 to which the model parts will ''draw" on the 
 upper side of the sand bed, it is necessary to use 
 the flask method. A box for each of two or more 
 portions of a pattern supports the sand for a sec- 
 tion of the flask. The boxes are assembled to 
 
 Arch stone - T^vo or three piece pattern ^ 
 depending on direction or 'draw" 
 
 Nof-e:- Incasepaffem >i 
 IS drawn -Trorri sand as 
 shown by arrow 'A' wood 
 box IS not required and 
 pattern split in three 
 
 ^ parrs as shown 
 by solid lines 
 
 -<. • 
 
 V/ood 
 box. 
 
 1 
 
 Note - In case pattern is drawn as shown . . 
 
 by arrow 'B' if is split as shown by 
 dotted lines and wood box is usedi^' " 
 to take care of check ^^^-v 
 
 Nails removed before 
 pattern is complexly 
 moulded ■•. . i{ . 
 
 Fig. 115. — Splitting pattern for making sand mold. 
 
 Fig. 116. — Two ways of splitting a pattern. 
 
 complete a mold. The sand is sometimes mixed with a very small percentage of plaster to 
 give stiffness to the mold. 
 
 A great deal of the skill in successful stone manufacture with sand molds is in making the 
 models, around which the sand must be packed. Take for instance a stone like that illustrated 
 in lig. 115. The molded surface at the right is to be down and the model drawn from the sand 
 in the direction indicated by the arrow. It is necessary, therefore, to make a four-piece pattern 
 as indicated by the numbers 1, 2, 3, and 4. The parts are held lightly when first made by nails 
 as shown in Fig. 116, which shows two ways of splitting a pattern somewhat similar to that in 
 Fig. 115. The nails are loosened when the pattern is packed into the sand. Note the small 
 pattern part at 4 in Fig. 115. It is the custom among many manufacturers to eliminate the 
 undercut in such a small part in making a pattern, it being cheaper in such a case to let the stone 
 cutter put in the undercut after the concrete is hard. If a great number of these pieces were to 
 be made, it would, in most cases, be cheaper to make the pattern complete to avoid so much 
 stonecutting. 
 
 There is a tendency in practically^ all sand-cast concrete, for some of the sand in the sur- 
 rounding bed to be held by the cement and so leave lumps and uneven places on the casting. 
 Surface treatment of sand-cast concrete stone is, therefore, necessary, in most cases. 
 
 87. Surfaces. 
 
 87a. Face Design in Standard Units. — Design of units is a matter best left to 
 the judgment of architects who are specialists in such matters. Some early manufacturers of 
 block machinery and of concrete block were led astray by the ease with which the face of a 
 block could be cast to imitate anything. Face plates were supplied to imitate pitch-face stone, 
 cobble stone, bush-hammered stone, tooled stone and with all sorts of panels and borders. 
 Not satisfied with this, manufacturers of block and special stone, still drunk with the plastic 
 possibilities of so tractable a material, impressed it with the designs of rubber matting and 
 pressed-steel ceilings. These errors the industry is outgrowing. The most persistent crime 
 
' Sec. 2-876] 
 
 GENERAL METHODS OF CONSTRUCTION 
 
 1G3 
 
 against good taste is a rock-face block which is bad chiefly because it does not imitate success- 
 fully one of the least desirable types of natural stone. This much should be of record in this 
 connection. If block makers will now devote, as many of them are doing, as much thought to 
 making concrete look like itself as their predecessors have to making it look like the least desir- 
 able natural stone, a great future seems probable for the manufacturers of the material. 
 
 In making special units, the manufacturer will do well to follow the designs which archi- 
 tects work out for him. In making standard units, he will more frequently have the architects 
 on his side if he eliminates face designs and makes a plain unit whose title to beauty is in the 
 honesty of its appearance, in the tones and colors of its exposed aggregates, or in the light and 
 shade of a rugged texture. 
 
 87b. Facing Materials. — Most concrete block and special-dimension stone is 
 not of the same mix throughout. When such stone is made of a dry-tamp mixture, it is a simple 
 matter to face it, on the surfaces which are to be exposed, using a special aggregate which will 
 give the desired qualities in color and texture in the finished product. The facing mixture can 
 be backed up with plain concrete. When stone is cast in sand molds — in fact, in the manu- 
 facture of most wet-cast stone — the conditions in the work necessitate that the concrete shall 
 be the same all the way through. In such work, therefore, facing mixtures are not used. There 
 are some exceptions to this rule in the use of commercial equipment for the manufacture of con- 
 crete block in gang molds. One exception is in the coating of face plates with a thin film of 
 glue on which, after the glue has become sufficiently thick and sticky, the facing aggregate, 
 with no cement, is sprinkled so as to form a complete layer over the face plate. This plate, 
 placed in the bottom of a mold is filled with a wet mixture of concrete. The water loosens the 
 glue and the facing aggregate bonds with the wet backing. Another method of facing a wet- 
 cast material is in gang molds in which the product is made face up. The molds are filled with 
 a slushy mixture, not quite to the surface, then the facing mixture is spread on after the backing 
 concrete has partially hardened. This is troweled into place on the surface In general, 
 with regard to facing mixtures, it is common to make them of a 1 : 2 proportion of cement and 
 a special aggregate, or 1:23'^ or 1:3, depending not only upon the grading of the mixture but 
 upon the effect desired in smoothness or roughness in the product. Materials commonly used 
 include special sand, in white, buff, and so on, crushed marbles in the more choice work, crushed 
 granite, trap rock, even crushed cobble stones (which are within the reach of almost anybody 
 in a glacial country), and crushed limestone. Micaspar crystals, so-called, are used by many 
 manufacturers in producing a gray granite effect in either dark or light shades. Other facing 
 materials are on the market containing mica which gives a sparkle and life to the finished product. 
 It has been very common for most manufacturers to use fine sand for facing material, particu- 
 larly in tamp work, and to make a special effort to get a very smooth, fine face on their products. 
 Some architects are encouraging work in another direction by the interest which they have 
 shown in products having a rough texture. This is secured by using coarser aggregates not so 
 well graded — that is, not so much fine material to fill up all the spaces — and with just as little 
 cement as can be depended upon to bind the aggregates thoroughly in the face. It is common 
 in such work to use 1 : 3 mixtures and to use facing aggregates as large as }4 in. The results 
 which can be produced in this way are limitless and their effectiveness depends largely upon 
 the taste and judgment of the manufacturer. The use of yellow marble or black marble 
 (crushed trap rock is frequently used in place of black marble), red granites, and so forth, lead 
 to possibilities in colors in concrete stone which only need experiment to prove. 
 
 If a smooth block or polished finish is desired, the product should be made on as smooth 
 a surface as possible so that there will be few projections to work down. Where a well-graded 
 aggregate of a polishable material is used— as, for instance, granite or marble— and this material 
 is well distributed over the surface area with few spaces between, it is possible to polish a con- 
 crete product just as the granite or marble itself is polished. A recent development in the manu- 
 facture of concrete stone hes in the direction of obtaining a smooth surface by means of a sheet 
 of very heavy paraffined paper of glossy finish, which is placed in the bottom of the mold of 
 whatever kind is used; the facing material being placed in on top of this and the backing tamped 
 
 I 
 
164 
 
 CONCRETE ENGINEERS' HANDBOOK 
 
 [Sec. 2-87c 
 
 behind it. This gives a very smooth surface which requires only a minimum of rubbing to 
 give an excellent finish. 
 
 Aside from the special methods and devices in obtaining various products which have 
 already been described, facing mixtures are used chiefly with tamped products. In such work, 
 the facing mixture is generally a relatively dry mixture, placed on the bottom or face-side of 
 the mold box, and the backing tamped in behind it. When it is necessary to fill up the facing 
 material on the side of a mold of any kind, this can be done by piling it up 2 or 3 in. at a time 
 and filling in the backing behind it, or it may be done by the use of a thin piece of metal of a 
 size equal to the side of the mold which is to be faced. With the use of this dividing plate the 
 facing mixture is placed on one side and the backing on the other, the plate being gradually 
 raised as the two are tamped together to form a bond. This facing mixture, while relatively 
 dry, must neither be too dry nor too wet. It must be just sticky enough to hold its position when 
 pressed into shape. 
 
 Whenever a facing mixture is used, it is desirable to finish the work in such a way that the 
 special aggregates of whatever nature are used, are exposed to lend color and to give better 
 texture to the work by the removal of the cement which covers the surfaces, leaving the cement 
 in the matrix to bind the aggregates in their position. Various methods for finishing the stone, 
 not only that which is faced with a special mixture, but also that which is of like character 
 throughout, will be considered. 
 
 87c. Colors. — Great care should be exercised in the selection of colors. They 
 cannot be used in rich mixtures without destroying a great deal of the binding value of the 
 cement with which they are mixed, and the strongest colors cannot be obtained unless rich 
 mixtures are used. For both these reasons there has been some tendency to discourage the 
 use of mineral colors in cement mixtures. 
 
 J. H. Jackson, authority on colors, writes in Concrete as follows: 
 
 Mineral colors of the highest degree of purity are the only ones to use in coloring cement. The permanency 
 of shade or color obtained depends upon the elimination by the color manufacturer of anything in the color that 
 the cement itself will destroy. Few contractors realize that the more intense and brilliant are the colors, the more 
 quickly they fade, and, in more instances than one, help to disintegrate the concrete, for none of the mineral colors 
 useful in cement work, is found naturally brilliant and the addition of chemically prepared colors or the treatment 
 of native colors chemically to give them intensity is a positive detriment under all conditions. All true cement 
 colors will withstand the acid treatment (1 part acid, 5 parts to 6 parts water), scrubbing and troweling, but in 
 polishing after setting and trowel polishing before setting, special care should be taken when yellow, green and 
 similar colors are used for the metallic polishing is likely to darken the color. Never give a smooth finish to outside 
 concrete when color is used. 
 
 The standard proportions for colors generally used are 6 to QVz lb. of color to every 100 lb. of cement. The 
 amount of color can be increased slightly if a deeper shade is desired, but you should not use more than 10 lb. of 
 color to every 100 lb. of cement, for an excess of color reduces the binding power of the cement. 
 
 A table of color quantities and the results by L. C. Sabin from his ''Cement and Concrete" 
 is as follows: 
 
 Colored Mortars 
 
 Colors given to Portland-cement Mortars Containing 2 Parts River Sand to 1 Cement 
 
 Weight of dry coloring matter to 100 lb. cement 
 
 Dry material 
 used 
 
 Lamp black 
 
 Prussian blue 
 
 Yellow ochre 
 
 Ultramarine blue 
 
 Burnt umber ....... 
 
 Venetian red 
 
 Chatt. iron ore 
 
 Red iron ore 
 
 M lb. 
 
 Light slate 
 
 Light green slate . . . . 
 Light green 
 
 Licrht pinkish slate . . . 
 
 Slate, pink tinge 
 
 Light pinkish slate. . . 
 Pinkish slate 
 
 1 lb. 
 
 Light gray 
 
 Light blue slate 
 
 Light blue slate 
 
 Pinkish siate 
 
 Bright pinkish slate. . 
 
 Dull pink 
 
 Dull pink 
 
 2 lb. 
 
 Blue gray. 
 Blue slate 
 
 Blue slate 
 
 Dull lavender-pink. . . 
 
 Light dull pink 
 
 Light terra-cotta .... 
 Terra-cotta 
 
 4 lb. 
 
 Dark blue slate 
 Bright blue slate 
 Light buff 
 Bright blue slate 
 Chocolate 
 Dull pink 
 Light brick red 
 Light brick red 
 J 
 
Sec. 2-87(^1 
 
 (mNEUAL METHOD.'^ OF CON^HTRIJCTION 
 
 165 
 
 Coloring by absorption is described in the Concrete by Adolph Schilling: 
 
 To color in reds and in browns use sulphate of iron in a solution of 1 lb. of sulphate to 1 gal. of water; for 
 greens 1 lb. sulphate of copper to 3 gal. water. The older the concrete the longer the bath must continue. It 
 must be borne in mind that the coloring stops to a very great extent the hardening of the concrete. Therefore, 
 it is necessary to permit the concrete to attain such strength as is necessary in the ornamental work before im- 
 mersing it. Immersion may continue for a few minutes or for several days, depending upon the age of the con- 
 crete and upon the depth of color desired. The effects may be varied by using an aggregate which is not highly 
 absorptive so that this stands out while the matrix surrounding it is colored. Effects can be heightened in elaborate 
 designs by "picking out" certain parts, coloring them with a brush with cement stains. 
 
 Rough-textured work is sometimes beautified by having stain stippled on or brushed over the high points. 
 The manufacturer, of course, will always have to use his judgment in the application of these colorings. The great 
 danger is in overdoing. 
 
 Aside from the use of white cement to obtain light colors in concrete, it is possible to 
 make an ordinary gray cement surface lighter by rubbing with a concrete brick made of fine 
 materials. The work should be kept wet while being rubbed. 
 
 A method of obtaining white surfaces is described by John Oursler, in Concrete, as follows : 
 
 A wash made of 1 lb. of concentrated lye, 4 lb. of alum and 5 gal. of water, with enough cement added to make 
 the wash of a good consistency for spreading with a brush, has been used to give a white surface. 
 
 87c?. Spraying. — One method for removing the surface film of concrete block 
 and special stone where there is a special facing mixture, is the spraying method, described in 
 Concrete by the early user of the method, E. J. Thompson: 
 
 Immediately upon removing the block from the machine, place it where there will be a good light on the face 
 and spray it, using a fine vapor spray, such as is used in spraying fruit trees. The outlet holes in such sprays are 
 about the size of an ordinary pin, and, for the best results, should be used in connection with a water pressure of 
 40 lb. or more. This spray nozzle attached to a length of j2-in. hose is all the equipment needed. The spraying is 
 a simple operation. It can be done, after a little practice, by any intelligent laborer. The effect of the spray, 
 which lasts only momentarily, is to wash off the surface film of cement and expose the aggregate (see Fig. 117). 
 The spraying must not be continued until the surface begins to run or furrow, but just a little practice teaches the 
 operator when to stop. 
 
 Some manufacturers prefer spraying their products when they are lying face up; others 
 prefer to have them with the face perpendicular. The perpendicular method is the commonest, 
 but the face-up method has a tendency to leave all the cement on the surface, to wash it into 
 the pores. One manufacturer who advocates this method urges that it not only exposes the 
 aggregate but makes tke face of the product more dense. When this work is done on standard 
 concrete block, it is done very rapidly and adds very little to the cost of the block. 
 
 87e. Brushing. — Brushing the surfaces of concrete stone in standard and special 
 units is particularly desirable where a -rough-textured effect is desired (see Figs. 117 and 118). 
 It is commonly used where a graded aggregate, with larger particles than is ordinarily used 
 in facing mixtures, is employed. The brushing is done, ordinarily, while the product is com- 
 paratively green, using a fiber brush with stiff bristles and using considerable water while 
 the brushing is in progress. Care must be taken that brushing is not started too soon so that 
 the face of the products will be damaged, and on the other hand, the work must be done before 
 the product is too hard, or the work becomes expensive. The manufacturer will find that the 
 time for brushing depends a great deal upon the conditions of curing. For surfaces which have 
 been allov\^ed to become partially hardened, a brush about 4 in. wide made by clamping together 
 a number of sheets of wire cloth has been found more effective than the wire brushes which are 
 ordinarily sold for this purpose. Care must be taken in brushing not to injure the edges of 
 the products. Sometimes it is desirable for the operator to use a small frame or at least a 
 straight-edge to protect the edges of the product while brushing. 
 
 Fig. 119 shows three views of turning stand for handling green concrete block to be brushed. 
 This stand is used for handling products which are delivered on pallets face down. A block 
 on pallet is placed on stand as in the first position. The shelf is turned over, so that the block 
 
166 
 
 CONCRETE ENGINEERS' HANDBOOK 
 
 [Sec. 2-87/ 
 
 rests on its side as in the second position. The shelf which first supported the block is then 
 turned down leaving the face exposed for brushing or other treatment. 
 
 Where steam-curing is employed, it is common to run the block into the steam rooms for 
 a few hours and out again, when the brushing is done, and the products are returned to the cur- 
 ing rooms. 
 
 87/. Rubbing. — Rubbing, as a finish in concrete stone, is commonly used where 
 a limestone finish is desired. In wet-cast work when air bubbles and slight imperfections occur 
 on the surfaces of concrete stone, it is the usual practice to pour on the surface a creamy grout 
 
 of cement and water. This is rubbed in first with a brush or swabbed on with a cloth, and a 
 few minutes later rubbed in with a small wood block. When hard, the stone is rubbed down 
 with abrasives. The rubbing removes the effect of the painted surface. It is common to use 
 cement brick made with very fine material in rubbing concrete work. Ordinary commercial 
 abrasives are also employed. 
 
 For finishing marble concrete which may be cast in large blocks and cut up by gang saws 
 into slabs, or cast in thin sections in pressure machines as for floor tiles, the methods are like 
 those in factories where natural marble is handled. The pieces are first smoothed down on a 
 
 revolving rubbing bed, sand and water being 
 
 First position Second position Third position 
 
 Fig. 119. — Three views of turning stand for handling 
 green concrete block to be brushed. 
 
 fed to the grinding surface. Then power- 
 operated carborundum discs are employed, 
 or a second rubbing bed with a finer grit, 
 for a dull gloss surface, or when a polish 
 is desired, the work going under hand or 
 power-operated mops, employing oxalic acid 
 and putty powder in the process. 
 
 87gr. Tooling. — In considering 
 the tooling of concrete stone in various sur- 
 face treatments now put on it, there might 
 be considered all of the methods which are common to the finishing of natural stone. The 
 ))est quality of manufactured stone admits of the same treatments as are given to natural 
 stone, with similar results. The means at the disposal of the manufacturer of high-class con- 
 crete stone include the ordinary hand work with rasps and chisels, bush hammers, crandalls, 
 planers, polishers, and so on. To make possible the use of tools of this kind in obtaining 
 satisfactory finish, it is necessary that the concrete have a uniform texture, with no aggregd,tes 
 of unusual hardness; that is, the materials shall be of a like character throughout. The aggre- 
 gates used in such work, ordinarily pass a 3^-in. screen. They include crushed limestone, 
 
Sec. 2-87// J 
 
 GENE HAL METHODS OF CONSTRICTION 
 
 167 
 
 marble, granite, and trap rock. While faced products are fre(iuently tooled, it is more common 
 to find such finishing methods applied to products which are of like character throughout. 
 
 The use of pneumatic tools in bush-hammering and crandalling concrete surfaces is becom- 
 ing more common and resulting in great economy over the use of hand tools. A finish in parallel 
 grooves, very common on natural stone, is put on concrete stone using power equipment which 
 revolves a gang of thin carborundum wheels mounted together, so that considerable surface 
 is covered at one time. Methods of cutting which are common in the natural stone field and 
 which are used, though are not so common in the concrete stone industry, include the cutting 
 up of large slabs of stone by means of saws which make deep grooves, so that a slab is easily 
 split up on the job just before the stone is laid, thus preserving the edges and keeping theni 
 clean and sharp. 
 
 For fine ornamental details and for finishing undercuts which are not easily included in 
 patterns and molds, expert stone carvers are employed in some of the best plants making a 
 high quality of concrete stone. 
 
 87/i. Mosaics. — The subject of the surface treatment of concrete would not be 
 complete without the mention of the possibilities which lie in the use of mosaics in large and 
 small ornamental surfaces. These may be glued into a mold on strips of paper where the 
 concrete surface is to be flush with the mosaic itself; or the inlay, consisting of tile or bits of 
 marble or other stone, may be set in places which have been provided in the products when cast. 
 After the product is complete and before the curing period is entirely ended, the grooves or 
 spaces left where inlays are to be inserted are thoroughly wet and grouted in to hold the inlays 
 in place. 
 
 87i. Efflorescence. — Efflorescence is ordinarily considered in connection with 
 surfaces because it is a surface disfiguration. The cause, however, goes far back of the surface 
 treatment and lies in the fact that the concrete mixture is not so dense as should be obtained. 
 Efflorescence is, in reality, a disfiguration which is common to brick, to concrete and to 
 natural stone. It occurs, as a direct result of porosity in the material on which it appears. 
 An impervious substance is not subject to efflorescence. When a substance is porous, water 
 which is absorbed dissolves certain salts found in the material; as for instance in concrete 
 block and brick, the salt which is dissolved is principally lime carbonate. A concrete product 
 which soaks water like a sponge after a rain, subsequently dries out generally from exposure 
 to the sun. In drying out, any salt solution is brought to the surface and left there when the 
 water evaporates. The best remedy is in prevention by maintaining a dense mixture. Efflo- 
 rescence can usually be removed with a solution of muriatic acid and water although this is 
 not always successful. 
 
 87j. Air Bubbles. — Air bubbles, like efflorescence, are a part of the subject of 
 density. They are formed when there is insufficient care in placing the concrete. A wet 
 mixture should be spaded against the forms so that these little air pockets will not form. They 
 are also prevented by tapping the molds, or by vibrating platforms, actuated sometimes by 
 machinery. The object of such equipment, however, is not so much to avoid the pinholes, 
 which become surface blemishes, as to produce a dense concrete by consolidating the mass. 
 
 The removal of air bubbles is frequently accomplished by tooling the surface when the 
 concrete is removed, and sometimes it is done by filling the surface a cement paste being rublx^d 
 in, as is described in connection with rubbed surfaces. 
 
 87/c. Crazing. — Crazing, or the formation of hair-cracks in the surface of con- 
 crete stone, is more frequently encountered in wet mixtures than in dry mixtures; is more 
 common in rich mixtures than in lean mixtures; and it is said by some manufacturers that the 
 tendency to surface crazing diminishes greatly when a well-aged cement is used. The writer 
 has never known but one manufacturer using a wet mixture of concrete who claimed entire 
 freedom from crazing in his products. This manufacturer uses a graded mixture of trap rock. 
 
 The reason for the hair-checking is in the greater tendency of the surface of concrete stone 
 to undergo temperature changes, leaving the interior of the stone comparatively unaffected. 
 
168 
 
 CONCRETE ENGINEERS' HANDBOOK 
 
 [Sec. 2-88 
 
 These surface cracks do not cause any structural harm, as they extend scarcely more than 
 2 in. into the stone. They are entirely removed in ordinary tooling treatments to which 
 much of the wet-cast stone (in which crazing is most common) is subjected. While these 
 cracks are believed by some manufacturers to be characteristic of the very rich surface skin of 
 cement which forms on spaded mixtures of concrete or on very rich mixtures of fine material, 
 other manufacturers find that even in tooling this thin mortar skin on the surface, the crazing 
 is not entirely done away with as it may craze again later on. Its only disadvantage is in the 
 fact that it provides minute lodging spaces for dirt. It does not seem possible to say just how 
 this crazing can be avoided in concrete stone, because it is maintained by many manufacturers 
 that two pieces of stone made in the same way, from the same batch and cured in the same way, 
 will not show the same results, one piece being covered in a short time with surface cracks and 
 the other piece entirely free from them. A manufacturer of concrete monuments says that he 
 obviates surface crazing by keeping his products buried in damp sand for 30 days after manu- 
 facture. In manipulating the drier mixtures of concrete, manufacturers frequently insist that 
 workmen shall not trowel the surfaces of products after they are molded, this troweling having 
 a tendency to bring the fine cement particles to the surface and result in crazing, 
 
 88. Specifications of the American Concrete Institute. — Newly adopted (1917) specifi- 
 cations and building regulations of the American Concrete Institute for manufacture and 
 use of concrete architectural stone, building block and brick provide as follows: 
 
 1. Concrete architectural stone and building blocks for solid or hollow walls and concrete brick made in 
 accordance with the following specifications and meeting the requirements thereof may be used in building con- 
 struction. 
 
 2. Tests. — Concrete architectural stone, building blocks for hollow and solid walls and concrete brick must 
 be subjected to (a) compression and (b) absorption tests. All tests must be made in a testing laboratory of recog- 
 nized standing. 
 
 3. Ultimate Compressive Strength. — (a) In the case of solid stone, blocks, and brick, the ultimate compressive 
 strength at 28 days must average fifteen hundred (1500) lb. per sq. in. of gross cross-sectional area of the stone as 
 used in the wall and must not fall below one thousand (1000) lb. per sq. in. in any case. 
 
 (b) The ultimate compressive strength of hollow and two-piece building blocks at 28 days must average one 
 thousand (lOOO) lb. per sq. in. of gross cross-sectional area of the block as used in the wall, and must not fall below 
 seven hundred (700) lb. per sq. in. in any test. 
 
 4. Gnoss Cross-sectional Areas. — (a) Solid concrete stone, blocks and brick. The cross-sectional area shall 
 be considered as the minimum area in compression. 
 
 (b) Hollow building blocks. In the case of hollow building blocks, the gross cross-sectional area shall be 
 considered as the product of the length by the width of the block. No allowance shall be made for the air space 
 of the block. 
 
 (c) Two-piece building blocks. In the case of two-piece building blocks, if only one block is tested at a 
 time, the gross cross-sectional area shall be regarded as the product of the length of the block by one-half of the 
 width of the wall for which the block is intended. If two blocks are tested together, then the gross cross-sectional 
 area shall be regarded as the product of the length of the block by the full width of the wall for which the block is 
 intended. 
 
 5. Absorption. — The absorption at 28 days (being the weight of the water absorbed divided by the weight 
 of the dry sample) must not exceed ten (10) % when tested as hereinafter specified. 
 
 6. Samples. — At least six samples must be provided for the purpose of testing. Such samples must represent 
 the ordinary commercial product. In cases where the material is made and used in special shapes and forms too 
 large for testing in the ordinary machine, smaller specimens shall be used as may be directed. Whenever possible, 
 the tests shall be made on full-sized samples. 
 
 7 Compression Tests. — Compression tests shall be made as follows: The sample to be tested must be carefully 
 measured and then bedded in plaster of Paris or other cementitious material in order to secure uniform bearing 
 in the testing machine. It shall then be loaded to failure. The compressive strength in pounds per square inch 
 of gross cross-sectional area shall be regarded as the quotient obtained by dividing the total applied load in pounds 
 by the gross cross-sectional area, which area shall be expressed in square inches computed according to Art. 4. 
 
 When such tests must be made on cut sections of blocks, the pieces of the block must first be carefully meas- 
 ured. The samples shall then be bedded to secure uniform ])caring, and loaded to failure. In this case, however, 
 the compressive strength in pounds per square inch of net area must be obtained and the net area shall be re- 
 garded as the minimum bearing area in compression. The average of the compressive strength of the two portions 
 of blocks shall be regarded as the compressive strength of the samples submitted. This net compressive strength 
 shall then be reduced to compressive strength in pounds per square inch of gross cross-sectional area as follows: 
 
 The net area of a full-sized block shall be carefully calculated and the total compressive strength of the block 
 
Sec. 2-88] 
 
 GENERAL METHODS OF CONSTRUCTION 
 
 169 
 
 will be obtained by multiplying this area by the net compressive strength obtained above. This total gross com- 
 pressive strength shall be divided by the gross cross-sectional area as figured by Art. 4 to obtain the compressive 
 strength in pounds per square inch of gross cross-sectional area. 
 
 When testing other than rectangular blocks, great care must be taken to apply the load at the center of 
 (,1 gravity of the specimen. 
 
 I 8. Absorption Tests. — The samples shall be first thoroughly dried to a constant weight at a temperature not 
 
 to exceed two hundred and twelve (212) degrees Fahrenheit, and the weight recorded. After drying, the sample 
 shall be immersed in clean water for a period of 48 hr. The sample shall then be removed; the surface water wiped 
 off, and the sample reweighed. The percentage of absorption shall be regarded as the weight of the water absorbed 
 
 |! divided by the weight of the dry sample multiplied by one hundred (100). 
 
 i 9. Limit of Loading. — (a) Hollow walls of concrete building blocks. The load on any hollow walls of concrete 
 
 blocks, including the superimposed weight of the wall, shall not exceed one hundred and sixty-seven (107) lb. per 
 sq. in. of gross area. If the floor loads are carried on girders or joists resting on cement pilasters filled in place with 
 slush concrete mixed in proportion of one (1) part cement, not to exceed two (2) parts of sand and four (4) parts 
 of gravel or crushed stone, said pilasters may be loaded not to exceed three hundred (300) lb. per sq. in. of gross 
 
 i cross-sectional area. 
 
 I (6) Solid walls of concrete blocks. Solid walls built of architectural stone, blocks or brick and laid in Port- 
 
 i[ land-cement mortar or hollow block walls filled with concrete shall not be loaded to exceed three hundred (300) 
 j lb. per sq. in. of gross cross-sectional area. 
 
 10. Girders and Joists. — Wherever girders or joists rest upon walls in such a manner as to cause concentrated 
 loads of over four thousand (4000) lb. the blocks supporting the girders or joists must be made solid for at least 
 eight (8) in. from the inside face of the wall, except where a suitable bearing plate is provided to distribute the load 
 over a sufficient area to reduce the stress so it will conform to the requirements of Art. 9. 
 
 When the combined live and dead floor loads exceed sixty (60) lb. per sq. ft., the floor joists shall rest on a 
 steel plate not less than three-eighths {%) in. thick and of a width Vi to 1 in. less than the wall thickness. In 
 lieu of said steel plate the joists may rest on a solid block which may be three (3) or four (4) in. less in wail thick- 
 ness than the building wall, except in instances where the wall is eight (8) in. thick, in which cases the solid blocks 
 shall be the same thickness as the building wall. 
 
 11. Thickness of Walls. — (a) Thickness of bearing walls shall be such as will conform to the limit of loading 
 given in Art. 9. In no instance shall bearing walls be less than eight (8) in. thick. Hollow walls eight (8) in. 
 thick shall not be over sixteen (16) ft. high for one 
 
 story, or more than a total of twenty-four (24) ft. for TabLE I 
 
 two stories. 
 
 (6) Walls of residences and buildings com- 
 monly known as apartment buildings not exceeding 
 four stories in height, in which the dead floor load 
 does not exceed sixty (60) lb. or the live load over 
 sixty (60) lb. per sq. ft., shall have a minimum thick- 
 ness in inches as shown in Table I. 
 
 12. Variation in Thickness of Walls. — (a) 
 Wherever walls are decreased in thickness the top 
 course of the thicker wall shall afford a solid bear- 
 ing for the webs or walls of the course of the con- 
 crete block above. 
 
 13. Bonding and Bearing Walls. — Where the face wall is constructed of both hollow concrete blocks and 
 brick, the facing shall be bonded into the backing, either with headers projecting four (4) in. into the brickwork, 
 every fourth course being a header course, or with approved ties, no brick backing to be less than eight (8) in. thick. 
 Where the walls are made entirely of concrete blocks, but where said blocks have not the same width as the wall, 
 every fifth course shall overlap the course below by not less than four (4) in. unless the wall system alternates the 
 cross bond through the wall in each course. 
 
 14. Curtain Walls. — For curtain walls the limit of loading shall be the same as given in Art. 9. In no in- 
 stance shall curtain walls be less than eight (8) in. in thickness. 
 
 15. Party Walls. — Walls of hollow concrete blocks used in the construction of party walls shall be filled in 
 place with concrete in the proportion and manner described in .Art. 9. 
 
 16. Partition Walls.— UoUow partition walls of concrete blocks may be of the same thickness as required 
 in hollow tile, terra-cotta or plaster blocks for like purposes. 
 
 No. 
 of 
 stories 
 
 Base- 
 ment, 
 inches 
 
 First 
 story, 
 inches 
 
 Second 
 
 story, 
 
 inches 
 
 Third 
 story, 
 inches 
 
 Fourth 
 
 story, 
 
 inches 
 
 1 
 
 8 
 
 8 
 
 
 
 
 2 
 
 10 
 
 8 
 
 8 
 
 
 
 3 
 
 12 
 
 12 
 
 10 
 
 8 
 
 
 4 
 
 16 
 
 12 
 
 10 
 
 10 
 
 8 
 
SECTION 3 
 
 CONSTRUCTION PLANT 
 PREPARATION OF CONCRETE AGGREGATES 
 
 1. Preparation of Crushed -stone Aggregate. — The preparation of crushed-stone aggregate 
 has grown to be an industry of such size that marked refinements in methods have been in- 
 troduced in recent years in the better plants. The general scheme, however, is (1) the breaking 
 down of the ledge, by one means or another, into pieces which can be readily handled and fed 
 into crushing machinery; (2) the breaking of these larger pieces in crushers of one type or another ; 
 and (3) the separation of the crushed material into various sizes. 
 
 la. Preparation of Site for Quarrying. — When a ledge is located for quarrying, 
 it is necessary to strip off the overburden in order to expose clean rock and to prevent the work- 
 ings from being filled with dirt and debris. Quarrying is carried on in morc-or-less distinct 
 levels one above another, the overburden being stripped back a distance from the top and the 
 ledge quarried down a convenient depth, gradually working backward into the face. A lower 
 terrace will then be started, working backward for a distance, until the vertical face of the first 
 portion is encountered. Successive levels may be quarried in this way, the top ledge being 
 successively stripped back to greater distances and the lower ledges being worked again in the 
 same sequence. It is usually sought to quarry inward from the side face of a ledge whose top 
 is a considerable distance above the side of the crushing plant, in order that the rock may be 
 brought within reach by gravity. 
 
 16. Quarrying. — The general method of quarrying is to bring rock down by 
 charges of explosive set off in holes drilled in the rock ledge. These holes are put down to the 
 depth to which the rock is to be split. The requisite amount of powder is charged into the 
 hole, covered by sand, and fired by means of a fuse or by electricity. In larger operations 
 charges in a line of drilled holes are fired simultaneously by electricity. Gunpowder is the 
 explosive mostly used, although nitroglycerine and dynamite are often preferred both because 
 of the larger quantity of rock which can be brought down per batch and also because of the 
 shattering effect of these quick-acting explosives. Various refinements in minor details are of 
 great importance and have a distinct bearing on the effect of the explosive charge. Even such 
 an apparently small matter as the form of the bottom of the drill hole has a very marked effect. 
 When bored with a hand drill, the hole is triangular at the bottom and the blast in such a hole 
 will break rock in three directions. Explosives in a squared bottom hole have a more distinctly 
 lateral effect. An expert rock man will shoot approximately that portion of the rock which he 
 desires to bring down. 
 
 Ic. Drills. — The jumper is a drill similar to that used for drilling holes for plug 
 and feather work in dimension-stone quarries, except that it is larger and longer. It is usually 
 held by one man, who rotates it between the alternate blows from hammers in the hands of 
 two other men. Churn drills are long heavy drills measuring from 6 to 8 ft. in length. They 
 are raised by a workman, let fall, caught on the rebound, raised and rotated a little and then 
 dropped again, thus cutting a hole without being driven by hammer. They are more econom- 
 ical than jumpers as they cut faster and make larger holes. Machine rock drills bore much 
 more rapidly than hand drills and are more economical in most operations for preparing rock 
 for concrete aggregate, where the w^ork is of sufficient magnitude to justify the preliminary out- 
 lay. They drill in any direction and can often be used in boring holes so located that they 
 could not be bored by hand. They are worked either by steam or by compressed air, and may 
 
 in 
 
172 
 
 CONCRETE ENGINEERS' HANDBOOK 
 
 [Sec. Z-ld 
 
 be either percussion or rotary. The action of a percussion drill is the same as that of a churn 
 drill already described, a piston moved by steam or compressed air being attached to the drill 
 in such manner as to make a stroke at every complete movement of the piston, an automatic 
 device rotating the drill slightly at each stroke. Rotary drills may be either shot drills or dia- 
 mond drills and they are more often used for prospecting than for drilling holes for explosives, 
 inasmuch as in their use a core is obtained which is of value mainly as mdicating the strata 
 penetrated. 
 
 Id. Stone Crushers. — Crushers are of two general kinds: jaw crushers and gyra- 
 tory crushers. The former type is better adapted to small or portable plants, while the latter 
 is used in larger operations. A convenient size of jaw crusher for a portable plant is about 
 10 by 16 in. This will crush from 50 to 100 cu. yd. per day, depending upon the size of the stone 
 to be crushed. 
 
 Both types of crushers have means for regulating openings so that by using a proper open- 
 ing together with a proper crushing plate, almost any size of crusher product can be obtained, 
 the size being limited by a small opening at the crusher-plate end of the machine. The output 
 of any crusher will depend to a large extent upon the plant arrangement. Necessarily also, 
 the more finely a stone is crushed the more work must be done upon it and the less the output. 
 In deciding upon the type of crusher to be installed at any plant, it is best to get comparative 
 estimates, costs, and tables of weight and output from manufacturers of various types of ap- 
 paratus, balancing the advantages of one against those of another and finding the machine best 
 adapted to the purpose in mind. Machinery of this kind is constantly being improved and 
 changed in type, so that accurate data representative of the latest practice is difficult to give. 
 
 le. Screening and Grading of C|-ushed Stone. — As stone comes from the 
 crusher, it is carried by some elevating means, usually a bucket elevator, to revolving screens 
 fixed over bins. Elevating and screening plants can be furnished in either portable units 
 (in which case they are so arranged that they can be readily dismantled for transportation) 
 or in fixed units with the machinery more massive. The usual type of screen is a rotary screen 
 inclined on its longitudinal axis, screens of various-size holes disposed successively throughout 
 its length forming the screen barrel. The stone as received from the elevating buckets is fed 
 into the fine screen end. Through the openings in this screen the very fine materials, dust, 
 etc., are taken off; and as the stone progresses down the screen barrel, the several sizes fall 
 into bins arranged below them, from which they are drawn off into conveyances as required. 
 The storage bins vary in size from those having a capacity of 13 tons to those having a capacity 
 of 50 tons. In some of the modern types of bins, provisions are made so that a bin may be 
 raised to a height sufficient to permit wagons being driven under gate spouts. 
 
 1/. Washing Crushed Stone. — Crushed stone is often covered with a tenacious 
 film of dust of which it is very hard to get rid. Although seldom if ever done, it would be 
 advantageous to wash stone after crushing and screening, inasmuch as this dust is of such size 
 that it is impossible to coat it with cement, and so tenacious that it prevents the cement from 
 being in contact with the aggregate. 
 
 Ig. Crushed Limestone. — ^Limestone crushes with a flaky fracture and a con- 
 siderable amount of dust. If the finer screenings are to be used, it is well to roll them between 
 rollers, inasmuch as this flaky fracture renders them extremely friable and unsuited to the 
 production of concretes impervious to water or of high strength. 
 
 2. Screening of Sand and Gravel. — Screening of sand and gravel may be done by hand or 
 by machinery. Hand-screening is adapted to small jobs and light work. Power-screening is 
 adapted to handling larger quantities of material. Screening of gravel or sand containing large 
 amounts of coarse material can be done more cheaply by mechanical than by hand means, 
 using either revolving screens or fixed screens placed upon an incline. The type of screening 
 equipment is largely determined by location and natural topography, and the availability of 
 power. Revolving screens are most effective and economical for large quantities, the material 
 being conveyed by bucket elevators to the screens and then falling into bins provided with 
 gates convenient for unloading. 
 
Sec. 3-3] 
 
 CONSTRUCTION PLANT 
 
 173 
 
 Large and elaborate plants for screening of sand and gravel are being installed in increasing 
 numbers in situations where dredging of these materials from river beds is practicable. Many 
 of these plants are now turning out aggregate of exceedingly high quality, the screening and 
 grading operation being incidental to the elevation of this material. Necessarily also these 
 materials are washed while being screened and graded. 
 
 3. Washing of Sand and Gravel. — Gravel is not infrequently coated with a tenacious film 
 of material which, if not removed, may greatly reduce the strength of the concrete. Sand also 
 is not infrequently contaminated with clay, loam, or slit coatings of organic matter. Such 
 coatings are responsible for a great deal of difficulty and many defects in concrete; and their 
 removal, while not easy even by washing, is decidedly essential. 
 
 Various means have been proposed for washing sand and gravel. Attempts have been 
 made to wash them in piles with a hose but this is always difficult and usually impossible to 
 carry out properly, inasmuch as the materials washed from the upper layers are carried down 
 by the stream of water to the lower portions, where they are rarely dislodged. Another scheme 
 is shoveling sand into one end of a V-trough, washing it down with a hose and endeavoring 
 to carry off the finer materials in the runoff water. For large quantities of material a combina- 
 tion of a trough of this kind with an ejector has been successfully used. A concrete mixer has 
 also proven adaptable to this kind of work, water being turned in and the mixer allowed to 
 overflow while the drum is in rotation. Necessarily all of these processes are somewhat ex- 
 pensive and add to the cost of the aggregate, but where tests indicate that the quality of concrete 
 will be seriously affected by uncleanness of sand or stone, it should be undertaken without 
 hesitation even at the increased cost. 
 
 HANDLING AND STORAGE OF MATERIALS 
 
 4. General Considerations. — The handling of materials, and their storage and disposal is 
 of great economic importance in concrete work. Hundreds of tons are removed, loaded, trans- 
 ported, unloaded, piled and elevated oftentimes to considerable heights before the making of 
 concrete has begun; and all this material as concrete must be rehandled one or more times 
 before it is delivered to forms. The engineering problems prior to placement are, therefore, 
 highly important ; and their adequacy not only determines rate of progress, but their efficiency 
 may decide whether there is to be a quick turnover or a slow one, with corresponding profit or 
 possibly loss. 
 
 It is axiomatic that gravity should be employed in such work whenever possible by utiliz- 
 ing natural advantages of site to the fullest extent. It should also be borne in mind that there 
 is approximately twice the quantity of stone to be handled as sand; and twice the quantity of 
 sand as cement. In planning a job, careful routing and proportionate disposal of these materials 
 should therefore receive early and adequate attention. 
 
 It is also trite and almost unnecessary to say that suflficient storage room should be pro- 
 vided for supplies of materials, ample to insure continuous prosecution of work when desired. 
 Shipments may be held up through any number of unexpected and unforseen happenings, so 
 that unless there is reserve supply, a shutdown must result. Specifications often stipulate 
 that a certain reserve quantity of material shall be maintained on the job, but even where this 
 salutary provision is omitted, contractor and owner's engineer alike, for their individual and 
 mutual protection, should have regard for this very important feature. 
 
 It would seem that remarks as to providing proper care for materials on delivery to the 
 work were equally unnecessary, yet disregard of these important matters is seen every day. 
 Cement always reacts with water, whether such water comes from the mixer measuring tank, 
 or whether it comes from rain, or from condensed steam, or from dew, or from water absorbed 
 from the ground. Cement, therefore, should always be stored in weather-tight houses, having 
 floors raised at least 6 in. above the ground; and any cement directly at the work should be 
 
174 
 
 CONCRETE ENGINEERS' HANDBOOK 
 
 [Sec. 3-5 
 
 C/eats short or cont/nuous 
 nailed io cross t/es> which 
 
 To secure greater sfabiiity 
 the base of lower courses 
 may be made three to four 
 feet wide Crib shouid be 
 set up as material is 
 deposited 
 
 kept off the ground and carefully covered with an impervious covering, whether or not the 
 atmosphere seems damp, or rain actually falling, or the weather threatening. 
 
 5i Storage and Care of Stone. — One of the essential qualities of large aggregate for con- 
 crete is cleanness. Stone, therefore, should not be dumped indiscriminately, so that when it is 
 rehandled, dirt and rubbish are carried into the concrete. A thick layer of sand, or preferably 
 a platform of planks should be placed on the ground before stone is deposited ; and not only will 
 this precaution be found to keep the material clean, but it furthermore will often pay for its 
 own cost, both in the quantity of stone saved by this means in rehandling and also in assurance 
 as to freedom of the concrete from deleterious foreign matter. 
 
 6. Shoveling Materials Directly from Cars to Ground. — Where a siding extends to the 
 construction site, sand and stone may be shoveled directly from the cars to the ground, which 
 should have been previously smoothed. That portion of the ground designed for the stone 
 should first be spread with a layer of sand at least 1 in. thick, this layer serving to keep the stone 
 
 clean and also working economy in subse- 
 quent shoveling. In order that materials 
 may be piled high and the track be kept 
 may be spaced about 3'c toe clear, a bulkhead may be built of a double 
 row of 2 by 12-in. plank with 1 by 3-in. 
 cross-ties, having stops as indicated in Fig. 
 1. This method requires no fitting or cut- 
 ting of the plank. 
 
 7. Storage and Care of Sand. — Sand 
 should not be dumped directly on the 
 ground, but for like reasons, should receive 
 care similar to that described for stone. 
 Although sand is a common material — 
 "common as dirt" — all dirt is not sand and 
 dirt rarely, if ever, makes good concrete. It 
 is only by attention to such seemingly small matters that a job can be well organized, made 
 profitable, and the best results secured. 
 
 8. Conveyance Economics. — The type of vehicle in which sand and stone are conveyed 
 exercises a large influence on the economy of a job. As an excellent emploj'ment of gravity to 
 cut labor costs, bottom-dumping conveyances, whether horse-drawn or motor-driven, or 
 railroad cars, are conspicuous for economy. In certain situations, of course, gravity cannot be 
 employed. In these the use of a locomotive crane with grab bucket is increasingly prevalent; 
 and the type of car or barge used to convey materials within reach of the bucket will have a 
 pronounced effect on the amount of material the bucket can handle in a given time. 
 
 As an example of comparative conveyance economies, consider the following with respect 
 to two types of railroad cars: 
 
 Two quarries are located on different railroads. Railroad "A" is prepared to supply 
 only hopper-bottom cars; railroad ''B" can furnish only flat-bottom cars. Assume gravity 
 dump to be impossible, and that the plan of operation involves unloading the stone by shovehng. 
 Assume also that one quarry, on railroad No. ''A" quotes $1.30 per cu. yd.; and the other, on 
 railroad No. B" quotes SI. 32 per cu. yd. It will be found cheaper in the end to order material 
 at $1.32 per cu. yd. in flat-bottom cars for the reason that more than the difference in first 
 cost can be saved by lessened labor in unloading. 
 
 The reason for this is self-evident. A good man, under efficient superintendence, can 
 unload 2 cu. yd. per hr. from a flat-bottom car, as against 1J>^ cu. yd. per hr. from a hopper- 
 bottom car — a saving of at least 3 cts. per cu. yd. — and this proportional saving increases as 
 less efficient labor is employed. These figures represent average results. A good man working 
 under a yardage or carload system will perhaps average 50% more than the above, but the 
 
 idinj 
 
 Fig. 1. 
 
Sec. 3-9] 
 
 CONSTRUCTION PLANT 
 
 175 
 
 ratio will not change between the two types of cars. The often -neglected matter of shoveling, 
 therefore, becomes a matter of moment. 
 
 9. Unloading Economies. — ^Low-side cars are more economical for unloading than high- 
 side, except where a wagon loader (Figs. 2 and 3) is used. Where such a loader is emi)loyed, 
 
 Fig. 2. 
 
 the car side to which it is attached should either be of sufhcient height to give wagon clearance, 
 or stakes must be provided to raise the hopper to the proper level. The use of a hopper ex- 
 pedites unloading the car, relieving possible demuriage charges and cutting down team and 
 wagon hours. One, two, or three hoppers may be attached to a singh; car and one or more men 
 put in the car for each hopper, depending 
 upon the number of teams available. 
 Hoppers should be filled, ready to dump 
 when the teams range alongside the car; 
 and the more quickly materials are dis- 
 charged into wagons, possibly even with- 
 out stopping, the more economical will 
 be the process. 
 
 10. Proper Size and Type of Shovel. 
 — Although refinements in efficiency can 
 be carried to extremes, a potent factor 
 in securing a proper output of work in so 
 simple an operation as shoveling is a 
 proper size and type of shovel. A good 
 worker will always look for a good shovel, 
 which of itself is eloquent testimony as 
 to the importance of this tool. Frederick 
 W. Taylor found that a shovel adapted 
 to a load of 21 lb. gave the best results. 
 
 Such a shovel corresponds to the standard No. 4 size shovel. The No. 3 shovel is somewhat 
 smaller. The size of shovel should always be chosen with reference to the weight of the ma- 
 terials to be handled. * u i • 
 
 The better the quality of material in a shovel, the longer will it last. A shovel is a con- 
 veying tool and, at the same time, it is a cutting tool no less than a chisel or a drill. ISo con- 
 tractor would consider using an inferior steel for such cutting instruments, yet m buying shovels 
 
 Fig. 3. 
 
176 
 
 CONCRETE ENGINEERS' HANDBOOK 
 
 [Sec. 3-11 
 
 many consider first cost only, having little regard to performance or endurance. It is ele- 
 mentary economics to balance the value of a shovel costing $9 a dozen which lasts about 2 
 months, against the value of a shovel costing 15 a dozen, which lasts 1 month or even less, par- 
 ticularly when its performance, in addition to its absolute existence, is taken into account. 
 
 1 
 
 Fig. 4. 
 
 11. Clam-shell Buckets. — When gravity dumping is precluded, a most efficient unloading 
 device is the clam-shell bucket (Fig. 4), hung from derrick or locomotive crane. With this 
 combination a skilled engineman can unload a large quantity of material in a day, but care 
 should be taken that the bucket chosen is one suited to the work. Some buckets tend to ride 
 
 Fig. 
 
 the material when it is at all resistant, while others are so designed that they fill to the capacity 
 at each bite, with proportionate efficiency and economy. The cost of equipment of this 
 type is necessarily large, so that proportionate care should be taken in choosing the type of 
 
Sec. 3-12] 
 
 CONSTRUCTION PLANT 
 
 177 
 
 bucket. No matter how excellent the derrick, or crane, or engine, or operative, a poor bucket 
 will negative them all and run up handling costs at an alarming rate. An orange-peel bucket is 
 a digging rather than an unloading tool and is not well adapted to the usual rehandling of 
 materials. 
 
 12. Bucket Unloaders and Conveyors. — There are an almost endless variety of elevating 
 = bucket unloaders each suited to some particular need. When the quantity of materials to be 
 l] handled and stored is considerable, a bucket elevator installation (Fig. 5) may prove very 
 economical, the more particularly as it makes possible the use of gravity discharge from con- 
 veyance to elevating buckets as well as gravity discharge from storage bins through measuring 
 I hoppers to the mixer. The installation and power equipment required for such elevating 
 |; mechanisms is not necessarily expensive or extensive. Some lighter types are shown in Figs. 
 6 and 7, and from these, the plant may range up to the heavier types, capable of handling very 
 
 Fig. 6. 
 
 . large quantities. In each individual layout, the needs should be studied and the economy or 
 lack of it determined. No hard and fast recommendation can be made that will apply to all 
 situations. 
 
 13. Belt Conveyors.— In certain situations an endless belt, grooved to V-form by pidleys 
 over which it runs, furnishes a rapid and excellent means of handhng raw concrete materials. 
 It cannot, however, raise the materials so nearly vertically as can the bucket conveyor, its 
 elevating slope being limited to about 1 vertical foot for every 2,^ horizontal feet. The ap- 
 
 ! pUcations of conveying belts are without number, but, as in the case of the bucket conveyor, no 
 blanket recommendation can be made. The manufacturers of conveying machinery are in 
 position to advise with respect to types, appHcability, and relative economies of various types 
 
 I of conveyors for any given set of conditions, and should be consulted for individual needs. 
 
 I 14. Storage and Handling of Sack Cement.— Cement, because of case in handling, is usu- 
 
 i allv ordered in cloth or paper sacks, each holding 94 lb. Cloth sacks are less liable to rupture 
 
178 
 
 CONCRETE ENGINEERS' HANDBOOK 
 
 [Sec. 3-15 
 
 than are paper sacks, but their cost is greater. When sack cement is received, a chain of laborers 
 is formed between cars and storage house, each man shouldering one sack. Economies in such 
 procedure may be introduced by insuring ready entrance and exit both to cars and to storage 
 
 Fig. 7. 
 
 houses, with adequately wide gang planks to cars, so that confusion and interference may be 
 prevented. The speed at which men will work is then a question of the personality and driving 
 force of the superintendent or foreman. 
 
 Fig. 8. — Proper method of bundling cement sacks. 
 
 Upper left: A bundle of 50 cement sacks laid out flat with 2 ropes 40 in. long, under the pile, and with a longer 
 rope of about 8 ft., resting on top. 
 
 Upper right: The first operation in bundling is to bring two of the ropes over the pile, as shown, tying tightly. 
 
 Lower left: After the short ropes have been tied, the bundle is turned over, and the long rope brought around 
 and crosses in the middle of the bundle, engaging first the shorter ropes. 
 
 Lower right: Bundle of 50 cement sacks tied and tagged ready for shipment. 
 
 15. Bundling and Storage of Empty Cement Sacks. — Inasmuch as each cement sack has a 
 return value of 10 cts., it is important that none should be lost, damaged, or destroyed. Fur- 
 
Sec. 3-16] 
 
 CONSTRUCTION PLANT 
 
 170 
 
 thermore, the cleaning, bundling, and shipping of these sacks becomes an operation of impor- 
 tance on large jobs. Much cement is lost when bags are insufficiently shaken, and this retained 
 cement further adds to the weight and bulk of sacks bundled for return. The proper method 
 of bundling sacks for return shipment is shown in P'ig. 8. In showing a bundle of 50 sacks, it 
 has been with the purpose of emphasizing the greater convenience in handling with equal ad- 
 vantages as a counting unit of the 50-sack bundle, rather than the 100-sack bundle. 
 
 16. Storage and Handling of Water. — In most instances water in pipes is within reach of 
 a concrete job. Where this is not so and water must be pumped, reservoirs of adequate capacity 
 properly protected against contamination should be supplied. From such reservoirs, water 
 can be distributed in pipes to be used as needed. The storage of water is so simple and so easily 
 carried out that the needs of each individual situation are of greater determining importance 
 than any particular type of pumping or distributing apparatus. 
 
 17. A Typical Installation. — Fig. 9 illustrates storage arrangements and handhng facilities 
 as installed in connection with a large building for the Singer Manufacturing Co. at Elizabeths- 
 port, N. J. The available storage space in this instance was a narrow strip 20 ft. wide between 
 
 Elevation 
 
 Fig. 9. 
 
 the railroad track and the building, of which the work in question formed an extension. A 
 trench was made, extending the length of this available space, and this was sheathed and 
 converted into a tunnel by a covering of 2-in. plank. In the bottom of the trench was laid a 
 suitable track for the operation of a skip car. Gn the ledger pieces were mounted two saw- 
 toothed measuring hoppers, fitted with wheels. 
 
 Materials were unloaded from the cars and piled over the trench, sand and stone in alter- 
 nate piles. The saw-toothed measuring hoppers were kept continually against the toe of the 
 sand and stone piles to faciUtate charging" by breaking down of the face of the piles. Four 
 men in this way handled the materials for about 30 cu. yd. of concrete per hr. Materials 
 were automatically dropped from the measuring hoppers into the skip car as it passed below 
 the hoppers on its way to the mixer. The A-frame shown in Fig. 10 has been used to good 
 advantage as a substitute for the trench. With this arrangement one man only was needed to 
 charge the measuring hopper. 
 
Sec. 3-18] 
 
 CONSTRUCTION PLANT 
 
 181 
 
 CONCRETING PLANT 
 
 18. Plant Economics. — Necessarily much of the plant for the handling and storage of 
 materials must be considered as a part of the concreting plant proper. The economies of the 
 whole plant therefore depend upon the individual and collective economies of its elements. 
 The main factors affecting these are first cost, cost of installation, cost of operation, cost of main- 
 tenance, cost of removal, salvage, and interest on the investment. 
 
 18a. First Cost. — First cost of plant includes many items in addition to prices 
 for machinery. Bonus for quick delivery, where equipment is required in a hurry; express 
 charges; tracing charges; and a hundred other items all swell the quoted figures. And it is not 
 always best to consider first cost too closely. A plant that is cheapest in first cost may not be 
 the most economical; and labor losses due to delay speedily offset differences in price betw^een 
 good and poor equipment. Furthermore, low costs of operation and maintenance with higher 
 salvage returns still further reconcile any disparit}'-. 
 
 186, Cost of Installation. — Cost of installation varies with the character of the 
 plant, cost of labor, location of the work and a variety of factors which must be separately 
 considered for each situation. 
 
 18c. Cost of Operation. — Cost of operation depends both upon plant arrangement 
 and upon organization. The concreting plant should be of a type, size, capacity, and arrangement 
 to permit continuous operation during working hours, assuming an organization so coordinated 
 as to make this possible and desirable ; and the character and arrangement of plant will depend 
 to a large extent upon local conditions, such as contour of the ground, class of construction, 
 manner in which materials are delivered to the site, total yardage to be placed, time limit, bonus, 
 penalty and other financial considerations which permit the use of equipment more or less 
 expensive and elaborate. In addition, attention must be given to the time of the year during 
 which the work is to be done, the normal temperature at that season for the particular locality, 
 and the amount of land available for plant and material, since storage is always a factor in 
 operation. 
 
 18d. Cost of Maintenance. — Cost of maintenance includes upkeep of machines, 
 repairs, oil, etc., this being greater or less according to the mechanical excellence of the plant 
 and to its disposition and treatment. 
 
 18e. Cost of Removal. — Cost of removal includes clearing the site of the plant 
 and its appurtenances after completion of the work. This cost will vary, according as more or 
 less of the plant is sold or junked, with proportionate lessening of care and labor required in 
 loading on cars, and of transportation. 
 
 18/. Salvage. — The salvage value of machinery is always problematical. It 
 usually is worth what can be obtained for it. Certainly, depreciation on contractor's machinery 
 is very large and most estimates of salvage value should be liberally discounted. 
 
 19. Balancing the Plant.— The general layout of the work will probably be the determining 
 factor in the choice of means adopted for carrying out each portion of the work. The total 
 yardage of concrete will also have a pronounced effect, possibly suggesting two or more separate 
 installations of medium size, or a single installation of greater size, or a number of smaller 
 mixers placed on different parts of the work. Various factors must be balanced one against 
 the other and various layouts planned, with a following through from delivery of raw materials 
 to delivery of concrete in the forms, with juggling of one scheme with another until the most 
 advantageous result, consistent with allowable cost, is secured. Careful plannmg of plant 
 before starting the job is well repaid in results, and a well-balanced plant is far more profitable 
 than one poorly balanced. Installation of a mixer of double the capacity of the charging 
 facilities, or of a fraction of the capacity of the handling facilities for the mixed materials is 
 sheer waste. 
 
 20. Typical Plants.— Some typical examples of plants which have proven successful in 
 service, are given in the following paragraphs: 
 
182 
 
 CONCRETE ENGINEERS' HANDBOOK 
 
 [Sec. 3-20 
 
 Fig. 11 shows an arrangement on the work of Cramp & Co., Philadelphia. It will be noted 
 that materials are delivered in bottom-dump wagons upon the incline, and pass by way of 
 bucket elevator to the bins above the mixer. Once in the bins, it is a gravity process through 
 measuring hopper to mixer. Messrs. Cramp & Co. report 289 cu. yd. with this plant in 8)^ hr., 
 
 with a crew of eight men, and covering all handling from wagons and cement storage to delivery 
 bin on hoist tower. The mixer is }i-yd. capacity. 
 
 Fig. 12 illustrates an arrangement adopted by the Turner Construction Co. on the Bush 
 Stores, South Brooklyn. Materials were delivered to the work in standard cars, and unloaded 
 by shovel into special hoppers as indicated. These hoppers were readily portable, and each 
 
Sec. 3-20] CONSTRUCTION PLANT 183 
 
 I 
 
 I hopper had a capacity for approximately 1 cu. yd. Materials were drawn off as required into 
 I cars with a capacity of 6 cu. ft., and wheeled to the mixer which was set at a lower level so that 
 !! the fixed hopper was on a level with the ground. The bumping post A facilitated discharge 
 I of carts into the hopper. Carts were wheeled up against the bumper when a slight lift on the 
 
 handles did the trick. ^ 
 
 Fig. 13 illustrates the plant used in the erection of the buildings for Foster Armstrong Co. 
 
 at Despatch, N. Y. The work on these buildings was carried on through the winter months 
 . and the bins indicated provided the readiest means for heating the materials to the desired 
 fj temperature of 90 to 100°F. An auxiliary measuring tank took care of the salt solution used 
 ij in the concrete mixture. Steam coils also served to warm the water used in mixing the concrete. 
 
 When mixed, the concrete was placed immediately; in no case more than 10 min. elapsed, 
 i When the concrete had been placed, it was protected against the action of frost by a solid wood 
 
 covering, blocked up at least 6 in. above the surface of the floor in a manner to permit free 
 
 Fig. 12. 
 
 circulation of air beneath the covering. Heat was introduced beneath the floor (or in the 
 case of ground floors, beneath the board covering) by means of steam coils and salamanders, 
 provision being made for the escape of sufficient steam beneath the covering to prevent prema- 
 ture drying out of the concrete. Salamanders were sprinkled freely with water, thus producing 
 the necessary amount of moisture, and small openings were left in the floor slab to permit 
 the warm air to circulate over the upper surface of the floor. The sides of the floor were pro- 
 tected by canvas curtains which extended downward; to the floor next below. 
 
 There were placed beneath the floor and beneath the panels on top of the floor, at intervals 
 of 10 ft., self -registering thermometers, which in no case showed lower than 32°. This tempera- 
 ture was maintained until the test cubes which had been allowed to set on the floor and beneath 
 the top covering showed the strength used as a basis for the design. 
 
 The extra plant involved in carrying on winter work involves considerable outlay, and work 
 in freezing weather should not be undertaken without a thorough understanding of all that is 
 involved. 
 
 Fig. 14 indicates a more or less elaborate plant, designed fof hirgc work, or for crarnjx'd 
 
184 
 
 CONCRETE ENGINEERS' HANDBOOK 
 
 [Sec. 3-20 
 
 quarters. Tbe plant consists of a suitable bucket elevator, designed to handle the entire 
 aggregate. This elevator discharges the materials into trough screens as indicated. While 
 passing through these screens, the coarse materials are washed by a flow of water applied at the 
 head, and the sand is still more thoroughly washed by passing through the water boot and in- 
 clined worm. The washed materials are discharged from the first flight of trough screens upon 
 inclined troughs leading to a fixed measuring hopper at the mixer. The lower end of the 
 
 Pivot on which , 
 bucket tuma for 
 discharging 
 
 Trapvdoors for 
 tfnloadlngY^grs 
 
 Steel Bucket for ^ 
 hoisting concrete 
 
 Pivot for bucket- 
 
 Gates for i \ 
 
 'discharging sand 
 and gravel not showi 
 
 2 
 
 ^veliL^ 
 
 3rd .tloor leVel 
 
 2nd Irtoor level 
 
 Front guldo 
 
 - Crab for 
 hoisting hue 
 
 hi ii 
 
 /^jlansome^j j | 
 "-jpatenteilii . | 
 ^ '^^'|ooncretei !L- 
 ^ 1i miier, |jLr<j 
 
 lull 
 
 1 rJ, '? 
 
 |iTr?it;Tr;H7rr^i 
 I !i !; \ • 
 
 Fig. 13. 
 
 troughs is fitted with a gate to control the flow. Such excess material as cannot be cared for 
 by the measuring troughs falls to the ground in piles as indicated. Two belt conveyors operat- 
 ing in tunnels beneath the piles permit ready draft against these reserve piles, delivering materi- 
 als to the original elevator, and thence to the measuring troughs. With such a plant it is a 
 simple matter to handle upward of 50 cu. yd. per hr. with four men. 
 
Sec. 3-20] . CONSTRUCTION PLANT lg5 
 
 Figs. 15 and 16 illustrate two set-ups of practically the same plant, and illustrate forciblv 
 the importance of proper arrangement. With the arrangement shown in Fig. 16, 17 mei. 
 handled 72 cu. yd. in 43^ hr. With the arrangement shown in Fig. 15, 15 men handled 87 cu! 
 
 Fig. 14. 
 
 yd. in 43^ hr., a saving of approximately 13 cts. per cu. yd. In Fig. 15 the runway and plat- 
 form are too small, with the result that the men interfere with each other, and cannot work to 
 
 Fig. 15. 
 
 advantage. Furthermore, the use of the feed chute involves assembly of the batch in the 
 mixer drum. Water was fed to the machine a bucketful at a time, requiring an extra man for 
 this purpose. In Fig. 16 a charging hopper is substituted for the feed chute, and both platform 
 
186 
 
 CONCRETE ENGINEERS' HANDBOOK 
 
 [Sec. 3-21 
 
 and runway are increased in size. Water is fed to the mixer through a pipe, and all operating 
 levers are controlled by one man. Provision is also made to take care of any material working 
 down beneath the mixer, with the result that wear and tear on journals, etc., is reduced. 
 
 Side Ele-Yat'ioa 
 
 Fia. 16. 
 
 The difference in cost of the two arrangements amounted to $59. The illustrations are 
 taken from actual experience, and indicate results secured under different superintendents. 
 
 21. Machine vs. Hand-m.ixing. — Except in relatively small quantities, hand-mixing of 
 concrete is not to be economically considered. Furthermore, hand-mixing is inferior to ma- 
 
 chine-mixing, with no comparison in quantity output. The province of a mixing machine is 
 essentially the thorough incorporation of materials — one of the fundamentals in the production 
 of sound, enduring concrete. Mixing, therefore, should be accorded the respect due its impor- 
 tance, and the best possible means chosen for its accomplishment. 
 
Sec. 3-22] 
 
 CONST R UCTION PL A N T 
 
 187 
 
 I 22. Types of Mixers. — The general types of mixers which have endured and are on the 
 
 I market at the present time may be classified as drum mixers, trough mixers, gravity mixers, 
 
 i and -pneumatic mixers. 
 
 i 22a. Drum Mixers. — Drum mixers (Fig. 17) are essentially of a typo, differing 
 
 I mainly in excellence of mechanical construction and arrangement. The action of all of them 
 
 Fig. is. — Low charging drum mixer. 
 
 is about the same so far as mixing is concerned, the operation being accomplished by agitation, 
 lifting, and pouring of the several materials by blades and scoops attached to the inside of the 
 mixer drum. With the exception of tilting mixers, discharge of the materials from the drum 
 is accomplished by inserting a trough into one side of the drum, in such position as to catch 
 the concrete as it is poured from the mixing buckets. Minor differences in charging mechan- 
 isms and arrangements are to be noted in different makes, but the action of all is essentially 
 
 Fig. 19. — Small pot mixer. 
 
 the action of a churn, in which capacity they would function if filled with cream., instead of 
 with stone, sand, cement, and water. 
 
 Of the low-charging mixers, the mixer shown in Fig. 18 is typical. Small pot mixers 
 such as shown in Fig. 19 are excellent for small work. 
 
 226. Trough Mixers.— Trough mixers are paddle mixers of one type or another. 
 
CONCRETE ENGINEERS' HANDBOOK 
 
 [Sec. 3-22c 
 
 They may be batch mixers of theshovcUng type (Fig. 20), or continuous mixers (Fig. 21), in which 
 a sectional screw rotates in an open trough. Continuous mixers have not met with general 
 favor as have batch mixers since many engineers object to these mixers on the grounds of 
 uncertainty of mixing operation. 
 
 22c. Gravity Mixers. — Gravity mixers are essentially a series of large funnels 
 or pans suspended one above another with bottom gates which can be opened successively, 
 permitting materials to flow from one into the other with incidental mixing to a greater or 
 
 Fig. 20. — Batch mixer of the shoveling type. 
 
 less extent. Gravity mixers are often urged in preference to power-driven mixers on grounds 
 of cheapness in operation and low first cost, permitting their being scrapped when worn; but 
 many engineers do not advocate their use because of the inherent uncertainty of their mixing 
 operation and oftentimes the requirement of deterimental quantities of water to prevent the 
 mass sticking in the pans. 
 
 22d. Pneumatic Mixers. — Pneumatic mixers have been developed by various 
 inventors. At the present time there are two main types on the market. In some of these 
 
 Fig. 21. — Continuous mixer. 
 
 machines premixing is had before delivery, either mechanically or by the agitation of air pres- 
 sure, while in others the charge is introduced into a chamber, dependence for mixing being placed 
 on what may happen in transit through pipes under the delivering air pressure. Pneumatic 
 mixers have their own particular field — that of placing concrete in forms where access is par- 
 ticularly difficult — but because of the large compressor plant which must be installed for 
 each mixer, and for other reasons which are valid and of importance in many classes of work, 
 their use is relatively restricted. 
 
Sec. 3-23] 
 
 CONSTRUCTION PLANT 
 
 189 
 
 23. Machine Mixing. 
 
 23a. Time of Mixer Operations. — Considering the concreting plant proper as 
 an installation for mixing together raw materials to form concrete, the plant cycle can be 
 considered as complete in three operations, viz., charging, mixing, and discharging. 
 
 In charging and discharging the mixer, a time limit is imposed both by the physical laws 
 governing the flow of materials from one container to another, and also (in the case of power- 
 loading, or side-loading hoppers in particular) by the physical limitations of operatives and of 
 the mechanism itself. As plant refinements are given consideration (particularly with regard 
 to the gravity loading of measuring or charging hoppers from overhead bins) this loading 
 time is diminished; but when a side-loading hopper, or a measuring hopper is charged by wheel- 
 barrows, the time is lengthened more-or-lcss according to the perfection of the runway arrange- 
 ments and the speed at which the men work. 
 
 In the following table is given the result of timing of different types of mixers on different 
 ' classes of work. These studies were made both with a seconds clock, motion picture camera, 
 and with a stop-watch, and are the summary of a large number of observations. From this it 
 will be seen that the loading periods vary greatly; that the unloading periods have an equally- 
 great variation; that the mixing periods are usually dependent upon the time taken in loading 
 and unloading; and that successive operations overlap, the endeavor of the mixer man being 
 to get out his material on as near a batch-a-minute schedule as is possible. In a number of 
 instances it will be noted from this table, such a procedure gives a negative mixing time. 
 
 Summary of Timing Data on Concrete Mixers 
 Time given in minutes and seconds. 
 
 Run 
 
 Kind of 
 mixer 
 
 Loading 
 means 
 
 Load- 
 ing 
 
 Un- 
 loading 
 
 Actual 
 mixing 
 
 Actual 
 total 
 
 Loading 
 and 
 unloading, 
 total 
 
 Time of 
 mixing 
 batch, 
 minimum 
 schedule 
 
 1 
 
 Lakewood, 1 yd. . . 
 
 Batch hopper . 
 
 0:51 
 
 0:59 
 
 0:11 
 
 2:02 
 
 1:50 
 
 —0:50 
 
 2 
 
 Koehring, 1 yd ... . 
 
 Batch hopper . 
 
 0:36 
 
 0:34 
 
 0:42 
 
 1:51 
 
 1:10 
 
 —0:10 
 
 3 
 
 Smith, 3^ yd 
 
 Batch hopper . 
 
 0:15 
 
 0:19 
 
 0:25 
 
 1:01 
 
 0:34 
 
 +0:26 
 
 4 
 
 Foote, 3^ yd 
 
 Side loader . . . 
 
 0:16 
 
 0:17 
 
 0:20 
 
 0:54 
 
 0:33 
 
 +0:27 
 
 5 
 
 Foote, K yd.. . . . . 
 
 Side loader. . . 
 
 0:23 
 
 0:27 
 
 0:25 
 
 1:15 
 
 0:50 
 
 +0:10 
 
 6 
 
 Chain Belt, K yd. . 
 
 Side loader . . . 
 
 0:07 
 
 0:35 
 
 0:28 
 
 1:10 
 
 0:42 
 
 +0:18 
 
 7 
 
 Koehring, K yd. . . 
 
 Side loader . . . 
 
 0:12 
 
 0:32 
 
 0:21 
 
 1:05 
 
 0:44 
 
 +0:16 
 
 8 
 
 Lakewood, 1 yd . . . 
 
 Side loader . . . 
 
 0:11 
 
 1:02 
 
 1:11 
 
 2:25 
 
 1:12 
 
 —0:12 
 
 9 
 
 Ransome, 1 yd. . . . 
 
 Batch hopper. 
 
 0:35 
 
 0:40 
 
 1:40 
 
 2:57 
 
 1:15 
 
 —0:15 
 
 10 
 
 Ransome, yi yd. . . 
 
 Side loader . . . 
 
 0:08 
 
 0:12 
 
 0:54 
 
 1:10 
 
 0:20 
 
 +0:40 
 
 11 
 
 Chain Belt, 3'^ yd. 
 
 Side loader. . . 
 
 0:13 
 
 0:27 
 
 0:11 
 
 0:51 
 
 0:40 
 
 +0:20 
 
 12 
 
 Ransome, 3^ yd. . . 
 
 Side loader. . . 
 
 0:18 
 
 0:38 
 
 0:46 
 
 1:32 
 
 0:56 
 
 +0:40 
 
 13 
 
 
 Side loader . . . 
 
 0:17 
 
 0:29 
 
 0:20 
 
 1:06 
 
 0:46 
 
 + 0:14 
 
 
 
 0:21 
 
 0:33 
 
 0:28 
 
 1:29 
 
 0:53 
 
 +0:70 
 
 Average, omitting 8 and 9 
 
 0:19 
 
 0:29 
 
 0:17 
 
 1:17 
 
 0:49 
 
 +0:11 
 
 Note. — Mixer 8 had very poor blading. Mixer 9 was fed by derrick bucket. Long mix- 
 ing due to inability to get raw material and to dispose of mixed concrete. 
 
 236. Time of Mixing. — Insufficient time is given to the mixing operation itself 
 in most commerical work. Too long a period may possibly be indulged, but it usually is not; 
 and no fear need be entertained of injuring the concrete by a mixing interval up to and in- 
 cluding 30 min. The mixing operation proper comprehends not only admixture of materials, 
 but also reaction between cement and water with distribution of the products of this reaction 
 
190 
 
 CONCRETE ENGINEERS' HANDBOOK 
 
 [Sec. 3-23c 
 
 over the surfaces of sand and stone. The time required for such thorough incorporation, and, 
 to a certain extent, for the hastening of the reaction between cement and water, depends upon 
 the adequacy of the blading and cleanness of the mixer. Oftentimes mixers are put on work 
 with the drum (Fig. 22) half-choked with concrete or full of holes, or the blading so worn that 
 they cannot handle the materials. Necessarily such mixers will not produce the same result 
 as a clean mixer, properly bladed and having a tight drum. Also, mixers are not all equally 
 efficient. 
 
 So many factors enter into the making of good concrete, that a hard and fast rule applicable 
 to all cases cannot be made, but in general it may be said 30 sec. or even 1 min. of mixing is in- 
 adequate. It is far better, when it is desired to do a thoroughly first-class job, to employ 
 more mixers even at a higher first cost for equipment and work them on a longer schedule, than 
 it is to attempt with one mixer to get out concrete on a rapid-fire schedule. The latter method 
 often brings a chain of unfortunate consequences, for not only is the concrete inadequately 
 mixed and the cement insufficiently used, but also excess water is nearly always added in order 
 to make the mass free-working and to diminish the labor of mixing. 
 
 23c. Drum Speeds. — Extended experimentation has established standard 
 drum speeds for various sizes of mixers. Engines and motors as supplied with them are so 
 regulated as to maintain these speeds practically constant. Necessarily, as the art of concrete 
 making advances, changes will result, but the present rotational speeds of standard mixers 
 seem suited to the requirements of average practice. Obviously, a slower drum speed would 
 
 Fig. 22. — Mixer drum. Fig. 23. — Charging hopper mounted on mixer frame. 
 
 result in less thorough incorporation of materials and a greater speed might cause the materials 
 to stick to the drum through centrifugal action.^ It is best, therefore, to adhere to speed 
 ratings prescribed by manufacturers unless such, speeds are patently inefficient. Inasmuch 
 as any concrete mixer that will perform its operations better and more quickly than its com- , 
 petitors is sure to have correspondingly greater sales, it is safe to assume that mixer manu- | 
 facturers have adopted for their product the maximum speed consistent with proper operation. I 
 It is not well, therefore, for the user to attempt economies by changing speed of the mixer drum, j 
 2Zd. Loading the Mixer. — There are many time economies that may be effected ^ 
 in loading the charge of materials into the mixer. Various types of loading mechanism have 
 been designed to meet different conditions of service and the time cycle of each is different. ■ 
 A study of each type will show its adaptability to particular needs. i 
 
 Charging Hoppers. — Where a charging hopper mounted on the mixer frame can be used, 
 as in Fig. 23, the limitation to charging time is dependent upon the design of this hopper, upon 
 the slope of its sides and upon the size of opening from hopper to drum. Inasmuch as this 
 type of charging device is usually loaded by gravity from superposed measuring hoppers, like 
 
 1 For studies of nnxer actions see N. C. Johnson; Eng. Rcc, Dec. 4, 191o. 
 
Sec. Z-23d] 
 
 CONSTRUCTION PLANT 
 
 191 
 
 considerations must be taken into account in their design; and always there must be prompti- 
 tude in releasing of gates, etc. In some very large operations such as the Elephant Butte 
 Dam, pneumatic opening devices have been installed with an interlocking system, so that a 
 sequence of operations is carried out with almost perfect regularity and great efficiency. 
 
 Power Loaders. — Side loaders or power loaders are often attached to mixers in order to give 
 the advantages of low loading, as well as those of relatively high discharge of mixed materials. 
 The general type of mechanism employed is shown in Fig. 24. The type of loading hopper 
 or skip varies with different manufacturers, some hoppers having a raised back, requiring a 
 slight incline for wheelbarrows that must be dumped into the hopper, while others permit 
 running wheelbarrows directly on to the hopper back itself. Through a friction clutch, the 
 power loader is elevated by the same motive power which drives the mixer drum. Inasmuch 
 as it is required to hoist such loading skips to a considerable height before materials will run 
 
 Fig. 24. 
 
 from them into the mixer drum, it is essential that sufficient power be provided to hoist this 
 skip rapidly, as otherwise an undue amount of time will be consumed in this elevating operation. 
 The mixers of different manufacture vary widely as to speed of hoisting; and it will generally 
 be found that the more expensive mixers have a better and more rapid hoisting mechanism, 
 in addition to their other economies, than have the cheaper types of mixing machines. 
 
 Low-charging Mtxers.— Low-charging mixers (see Fig. 18), particularly in smaller units, 
 have of recent years been meeting with favor. In such mixers the opening at the charging 
 end is relatively larger than in other types of drum mixers and blading about this opening on 
 the interior of the drum is so disposed as to draw the materials within the drum from a relatively 
 small hopper of low height into which they are charged by wheelbarrows. With such mixers 
 an inclined runway platform of 2}^ to 3 ft. in height is required. Their advantages, therefore, 
 consist in a simplification of charging and the absence of hoisting mechanisms rather than in 
 
192 
 
 CONCRETE ENGINEERS' HANDBOOK 
 
 [Sec. 3-23e 
 
 any particular efficiency of mixing operation. Furthermore, these machines are relatively 
 low in price and a number of small units, gasoline or electric motor-driven, are often very 
 advantageous when distributed about the work. From a standpoint of thorough mixing and 
 flexibility of operation, there is much to recommend this practice, inasmuch as the needs of 
 one part of the work can be supplied without reference to other parts or causing an overdraft 
 on any one machine with consequent speeding up of operations as is the case when all parts 
 of the work are demanding concrete at the same time from a single, centralized plant. Without 
 disparaging in any way the importance of the time-cost element in concrete mixing operations, 
 it is yet to be regretted that considerations of quality and ultimate satisfaction of the customer, 
 rather than first cost, do not more often govern both the selection of the plant and its operation. 
 
 23e. Measuring Materials. — It is often taken for granted that measurement 
 of materials for a concrete batch is of little or no importance and that it can be accomplished 
 in almost any way. It is probable that the average mix varies at least 50% in its proportions 
 from those desired, and for this reason alone it is not to be wondered that much concrete found 
 on every hand is so variable in quality. 
 
 Materials should be measured either in bottomless boxes placed on wheelbarrows, or like 
 devices, or else in a barrow pan permitting of struck measurement. A measuring barrow of 
 known capacity permitting struck measurement is shown in Fig. 25. At the same time these 
 
 Fig. 25. 
 
 barrows are adapted by reason of their balance, to the conveyance of considerable quantities 
 of material at one time. Measuring hoppers of known capacity, if carefully filled, can be made 
 to function quite accurately; but where they are not struck, or where there is pronounced 
 variation in the moisture content of the sand, the quantities of materials obtained per batch 
 will be found surprisingly variable. 
 
 It is difficult to convince the average contractor that economies can result from careful 
 measurement. It may seem a useless task to confine field men to struck measure of sand and 
 stone, but if the comparative quantities of cement required for accurately proportioned and 
 inaccurately proportioned mixes were taken into account, the cost balance would usually be 
 found in favor of the careful proportioning and measurement. And in' addition, there should 
 be considered and there will be considered with increasing force with passage of time, the 
 ultimate performance and endurance of the concrete produced. The time is not long distant 
 when owners will demand of contractors guarantees as to the quality of the product which they 
 are to receive and only by careful proportioning and measurement and placing of the concrete 
 can a reasonable basis for such guarantees be established. 
 
 23/. Discharge of the Mixer. — A further economy of time can often be had by 
 giving attention to the proper and rapid discharge of materials from the mixer. Many mixers 
 
Sec. 3-24] 
 
 CONSTRUCTION PLANT 
 
 have inadequate and insufficient blading due to having become worn with passage of time. 
 Manj' also are partially choked by concrete hardened inside the drum. Both insufficient 
 blading and choked mixer drums mean a relatively slow discharge of materials from the dmm 
 which cuts into the essential mixing operation. 
 
 24. Transporting and Placing of Concrete. — Providing means for transporting mixed 
 concrete and for placing it properly in forms is an art in itself. These operations both in first 
 cost and in ultimate effect rank equal in importance with the operations of conveying, pro- 
 portioning, and of mixing raw materials. In mixed concrete, not only are the raw materials 
 to be handled and oftentimes conveyed to considerable distances, but in addition this must be 
 done at low unit cost and in such a manner and so expeditiously as to protect the mixed mass 
 from injury. 
 
 The means usually adopted for the conveyance and placing of concrete are some sort of 
 bucket or cableway, or else open spouts or chutes through which the concrete flows by gravity, 
 or else in barrows, carts, or cars. The particular means adopted in any case, will depend upon 
 the size of the operation, upon the physical conditions attendant and upon the financial limi- 
 tations to plant imposed by commercial considerations. 
 
 24a. Barrows. — As affecting perhaps the great bulk of concrete used today, 
 it will be proper to first consider the use of barrows or carts. This method involves less original 
 
 Fig. 26. 
 
 plant outlay than the others before enumerated. In many instances, the cost of installation 
 of an elaborate plant would cover not only the cost of the barrows themselves, but a great 
 part of the entire cost of distribution of the concrete by other means. 
 
 The ordinary wheelbarrow (Fig. 26) having a flat pan is not well adapted to the distri- 
 bution of concrete. In such a barrow a man can handle about 1 3^ to 2 cu. ft. of mixed concrete. 
 This load he can wheel about 25 ft. every 3 min., the objection to the pan wheelbarrow being 
 that the man's working rate is necessarily cut down by the care which is required to keep the 
 materials from slopping over the sides. Furthermore by the design of the barrow a large 
 proportion of the weight of the load is on the man's arms, rather than on the wheel. Deep 
 pan barrows have been designed to overcome this difficulty, but have not wholly accomplished 
 the desired end. 
 
 246. Concrete Carts.— Two-wheel concrete carts (Fig. 27) are better adapted 
 to this work than wheelbarrows, both because they can carry a larger load and also because this 
 load is balanced on the wheels themselves with Httle or no strain on the man. The usual 
 two-wheel concrete car is of 6-cu. ft. capacity in which about 41^ cu. ft. of mixed concrete can 
 be carried by one man. 
 
 In this comparison there are, however, certain cost offsets to be made. Wheelbarrows 
 13 
 
194 
 
 CONCRETE ENGINEERS' HANDBOOK 
 
 [Sec. 3-24r 
 
 require less scaffolding than do the heavier and wider carts, so that the cost of this runway must 
 be carefully estimated. When runways must be elevated, the showing becomes more favorable 
 for carts, as bents or supports for wheelbarrows must be practically of the same size and strength 
 as those for carts. Turnouts and gangways must in both cases be of ample width so that there 
 
 Fig. 27. 
 
 may not be congestion in the passing of full and empty carts going to and returning from the 
 forms. 
 
 24c. Buckets. — There is a great variety in types of buckets adapted to the 
 distribution of concrete. Some of these buckets are straight-side skips, as in Fig. 28, adapted 
 
 Fig. 28.— Tilting bucket. 
 
 Fig. 29. — Round self-tilting bucket. 
 
 to dump by overturning. Others are bottom-dumping buckets operated by a man at the form; 
 and these bottom-dumping buckets may be of various patterns, adapted to some particular 
 use. An example of this sort of bucket is shown in Fig. 31, in which the bottom is so constructed 
 
Sec. 3-24rf] 
 
 CONSTRUCTION PLANT 
 
 195 
 
 as to form a long narrow opening, actuated through a powerful lever mechanism. A groat 
 variety of these devices is on the market and the needs of each particular situation must b(; 
 studied and met by as specialized a product for that use, as financial considerations will permit. 
 
 24kd. Cableways and Buckets. — Cableways usually require large initial outlay 
 but on large operations they may be found very economical. Usually they consist of a strong 
 
 Fig. 30. — Bottom-dump bucket for Fig. 31. — Bottom-dump bucket for 
 
 large forms. narrow forms. 
 
 |i messenger cable or cables (carried between either fixed or movable towers) with actuating 
 ij cables to hoist and earry the buckets to any desired spot on the work. Cableway buckets may 
 t' be of various types, but that shown in Fig. 33 has proven its worth in many constructions. 
 ' The bucket shown in this illustration is a 2-yd. bucket. Its deep upper body and steep sloping 
 
 bottom provide capacity and free flow, while a gate controlled by levers regulates the discharge 
 
 of concrete. 
 
 Fig. 32.— Tower-hoist tilting bucket. Fig. 33.— Bottom-dump cableway bucket. 
 
 24e. Spouts or Chutes.— The handling of concrete through spouts or chutes 
 is a development of the last 8 years. This system at the present time is in more extensive 
 use than any of the foregoing methods of distribution, with the possible exception of distribution 
 
 it 
 
196 
 
 CONCRETE ENGINEERS' HANDBOOK 
 
 [Sec. 3-24/ 
 
 in carts. The economic features of spouting are undeniably attractive. To raise concrete 
 vertically in a tower by means of a skip bucket and engine located at the central mixer plant, 
 then distributing by gravity through channels which can be arranged in convenient sections 
 to cover any area with a radius from 10 to 300 ft. from the base of the tower, appeals strongly 
 both to engineering and to business sense. Further, the ease of handling by gravity is usually 
 greater and the time cost per cubic yard for placing is usually less than in transferring the same 
 quantity of material in hand-barrows, in cableway buckets, or in cars. Yet in spite of 
 its many good points, the convenience of spouting has brought about many abuses. 
 
 For instance, it is obvious that in order to flow readily through chutes, concrete must be 
 smooth and plastic, whereas the materials of which concrete is composed, with the exception of 
 water, are all exceedingly sharp and gritty. It is not to be wondered then that lubrication 
 and ease of flow secured by increased wetness, has encouraged the use of excess water, especially 
 where for reasons of cost, it is desired to erect only a relatively low tower causing the angle of 
 the spout to be comparatively flat. Furthermore, many spouting equipments have been in- 
 stalled with ease of distribution alone in view, the first cost of plant and rapid deterioration not 
 being taken into account, so that saving has later been sought by cutting comers to make 
 up for the initial mistake. 
 
 In all spouting installations, care must be taken to have the chutes at a workable inclina- 
 tion. Furthermore, it is important to maintain a uniform pitch throughout the entire line, 
 in order that the flow may be thorough and uninterrupted and not subject to slackening at one 
 part and accelerated flow in another. The pitch also must be greater when the material is to 
 be carried to a considerable distance than when it is to be carried only a short distance, for as 
 the distance increases, the friction of the concrete in a chute tends to overcome its initial mo- 
 mentum. Whereas, therefore, a wet concrete will flow 50 ft. with the pitch of 1 in 6 it becomes 
 necessary to increase this pitch to 1 in 4 for a distributing distance of 100 ft., while a distance 
 of 300 or 400 ft. will require a pitch of 1 in 3. The slopes as above described are based upon 
 chute rigidly supported having uniform pitch throughout; and it would be even better to in- 
 crease this pitch in order that concretes of a drier consistency may be used. 
 
 Various methods have been proposed for increasing the ease of flow of concrete in chutes. 
 Hydrated lime in one proportion or another has probably proven the most effective, but there 
 is no standard procedure in this regard, nor is the exact quantity of hydrated lime required for 
 any given concrete prescribable without experimental knowledge of the aggregates separately 
 and in combination. Hydrated lime added to concrete has some undesirable features, but 
 even aside from these it is an expensive diluent of inferior strength, and inasmuch as practically 
 the same effect through the same agency may be realized by a longer mixing of materials, the 
 wisdom of its use is not yet beyond question. 
 
 The unfortunate tendency, as before pointed out, is to add more water to spouting con- 
 cretes to make them flow freely. This, however, defeats its own end, inasmuch as segregation 
 takes place very readily from wet mixtures, so that there is initially a rapid rush of semifluid 
 materials down the chute, with afterward a slow dribbling of the heavier and harsher materials, 
 oftentimes requiring men in the rigging with hoes to keep the unwatered sand and stone from 
 stopping. With sloppy mixtures, therefore, not only is the quality of concrete impaired but 
 also the cost of delivery and placing is very largely increased. On the other hand, thoroughly 
 mixed concrete without excessive water may be successfully delivered through spouts disposed 
 at proper pitch without segregation or the loss in value attendant upon the use of excessively 
 wet mixtures. 
 
 24/. Sections Used in Spouting. — It is desirable that concrete spouting be arranged 
 in a series of units which may be assembled in various combinations. Continuous-line spouting 
 should be changeable to swivel-head, or swivel-head to continuous-line, as the conditions of 
 the work require, it being necessary, of course, to have in stock a supply of the necessary units. 
 This interchangeability is of great value in service, for spouts wear at the head and foot of each 
 unit of length. By reversing a trough section, end for end. when showing heavy wear at one 
 point, a new, unworn surface may be put at point of greatest wear. 
 
Sec. 3-24/] 
 
 CONSmVCTIOX PLANT 
 
 197 
 
 A standard trough section, Fig. 34, is made of No. 14 gage steel, forming a trough 8^2 in. 
 deep by 10 in. wide on top. The bottom is curved to practically a semicircle of 4-in. radius, 
 the upper part of the sides being straight and tangent to the curve. Each section is punched 
 with standard spacing, arranged for connecting all of the various attachments. 
 
 Fig. .34. 
 
 The hopper head, Fig. 35, attached at one end for receiving the concrete from the bin, or 
 from an upper trough section, forms one point of support of the next trough section. At 
 the other end is the splash hood, Fig. 36. By fastening the hopper head to the trough section 
 at one end, and the splash hood at the other, we have the complete trough section, Fig. 37. 
 
 Fig. 35. 
 
 These 24 by 24-in. hopper heads, as well as the splash hoods, can be bolted to either end of any 
 
 standard trough section. 
 
 Standard trough sections are joined for continuous-line spouting by bolting together their 
 , angle-iron yokes or flanges and bolting on the compression plate part. Thus, several sections 
 j are joined together, with a hopper head at one end of the entire group, and a splash hood at the 
 
 other end. 
 
198 
 
 CONCRETE ENGINEERS' HANDBOOK 
 
 [Sec. a-24/ 
 
 Fig. 38 shows the swivel-hook used in making the flexible joint between successive trough 
 sections for swivel-head spouting and shows one of these joints, in which the upper line of 
 spouting is supported by a fall and tackle attached to the bail on the splash hood; while the 
 lower line is supported by the swivel-hook, connecting the lower hopper head with the splash 
 hood of the upper line. The swivel-hook is kept clear of the path of the concrete. 
 
 In some cases it is desirable to have a flexible joint in continuous-line spouting. In this 
 case the two sections are put together in a different manner, Fig. 39, where both the hopper 
 
 Fig. 36. 
 
 head and the splash hood are dispensed with. The hanger plate is here used in conjunction 
 with a special yoke, after one of the angle-iron yokes has been removed. This allows a slight 
 movement sideways, without requiring the attachments for the swivel-head operation. 
 
 Various types of spouting have been tried, ranging from round pipe to rectangular troughs. 
 Best results have been secured from the use of 5-in. pipes, or 10-in. open troughs, the latter 
 
 Fig. 37. 
 
 having the preference for flat slopes, and the former where there is necessity for varying pitch, 
 with a Jikelihood of steeper pitch than named above. 
 
 With open spouting the use of remixing hoppers (Fig. 40), in connection with flexible 
 spouting (Fig. 41), accomplishes satisfactorily the necessary changes in pitch. 
 
 The greatest items of expense in spouting plants are first cost, installati.on, and mainte- 
 nance. Maintenance charges are particularly heavy. The ordinary stock spouting which is 
 made of No. 14 gage metal will seldom handle more than 2000 cu. yd. without renewal. This is 
 
Sec. 3-24/J 
 
 CONSTRUCTION PLANT 
 
 199 
 
 due to the abrasive action of the material, especially as affecting the rivets which join the various 
 sections. 
 
 A recent development is a spout made up of two or more longitudinal sections of the shape 
 indicated in Fig. 42. The various sections are interchangeable, and there are no bolts or rivets 
 
 Fig. 38. 
 
 extending through the spout, all joint bolts or rivets passing through the flanges, and the 
 various longitudinal joints made secure by fish plates. This type of spouting has the further 
 
 Fig. 39. 
 
 advantage of making possible renewal of worn sections severally, instead of renewing the 
 length of spout as a whole. This type furthermore ensures a spout which is stiff in all 
 directions, a point of considerable importance. 
 
200 
 
 CONCRETE ENGINEERS' HANDBOOK 
 
 [Sec. 3-24^ 
 
 24gr. Hoists.— Whether the distribution is by spouts, by carts, or by barrows, 
 it has become general practice on all work extending above ground to hoist the concrete. For 
 this purpose a tower is practically indispensable. 
 
 Pig. 40, 
 
 It will ordinarily be found advisable' to install the hoist at the beginning of operations, 
 since by so doing the mixer may readily be set so that the operation of charging may be facili- 
 tated, principally by cutting out inclines, with resultant saving in labor. 
 
 Fio. 41. 
 
 Towers are constructed of steel or wood. The hoist bucket should be constructed on 
 the simplest lines without catches or trips. A substantial bail made of two 3-in. Z-bars 
 back to back, is arranged to operate between two 5^^-in. wooden guides, and is fitted at the 
 
Sec. 3-24^] 
 
 CONSTRUCTION PLANT 
 
 201 
 
 lower end with journals in which rests the bucket trunnion. In setting up the tower and 
 bucket, it is advisable in all cases to set the bucket so that it is balanced, and to this end the 
 front guide should be so set as to be almost in contact with the nose of the bucket when the 
 latter is pushed back to a point where the load will tend slightly to press the stops on the sides 
 of the bucket backward against the bail. Friction of the nose against the guides is, by this 
 
 Fig. 42. 
 
 means, cut down. By removing the front guide at any point in the height of the tower, and 
 placing a block on the back of the latter, the bucket is canted forward so that it will drop its 
 contents out through the opening made by the removal of the front guide. The bucket auto- 
 matically rights itself, and is pulled back into position by the weight of the bail when the 
 operator releases the brake. 
 
 Fig. 43. 
 
 A typical hoist is shown in Fig. 43, operating in connection with a mixer, the power being 
 taken from an extension of the mixer shaft. The power equipment of the latter should be of 
 sufficient capacity to operate both mixer and hoist at the same time. A variation of this plant 
 showing a direct-connected hoist is shown in Fig. 44, but for ordinary conditions the first- 
 described arrangement is preferable. 
 
 At any desired height a bin or hopper is set, into which the material is discharged by the 
 
202 
 
 CONCRETE ENGINEERS' HANDBOOK 
 
 [Sec. 3-24:g 
 
 Fig. 45. 
 
Sec. 3-25] CONSTRUCTION PLANT 203 
 
 hoist bucket. From this point distribution may be effected by wheelbarrow, cart, car, or 
 spout, either separately or in combination. A concrete bin such as shown in Fig. 45 forms the 
 upper end of a spouting system, the gate of the hopper under manual control regulating the 
 y flow. 
 
 25. Spouting Plants. — Spouting plants may be classed as hoom jdanln, guy-line plants, and 
 tower plants. 
 
 f 25a. Boom Plants. — In boom plants, the first and second sections of spouting 
 
 are mounted on a bracket attached to the hoisting tower, the free end being moved by tag lines 
 to the position desired. This rig offers advantages of flexibility and freedom of movement 
 
 II not often obtained in placing concrete. Oftentimes open-throated booms (through which the 
 
 [i first section of spouting is carried) are used, these having the advantage of lending lateral 
 
 I stability to the spout itself as well as of economizing space. 
 
 25b. Guy-line Plants. — In guy4ine plants, the spout is suspended by blocks and 
 
 I falls from guy lines, or special cables suspended between towers, or other supports especially 
 set up for the purpose. The advantage of this type of spouting plant lies in its ready adapta- 
 bility. It is limited, however, in lateral movement unless its deficiencies are supplemented 
 by take-offs at various points with small boom plants or supplementary guy-line plants. 
 
 25c. Tower Plants. — Tower plants are of like general feature, but the spouting 
 line is supported at ends of successive sections by movable towers or tripods. A plant of this 
 kind is less flexible than a boom plant, but is more flexible than a guy-line plant, inasmuch as 
 the various supports in the line may be moved successively, rendering possible the covering of 
 a very wide area from a single hoisting tower. A guy-line plant, on the contrary, requires 
 under like circumstances that the whole line be dismantled and set up again in the new location. 
 
 I25d. Combinations of Spouting Systems. — Combinations of the above systems 
 i are used advantageously in one way and another in order to surmount special obstacles. Among 
 I such combinations may be mentioned a rehoisting tower which permits covering a wider area. 
 ' In such a plant the concrete is distributed from mixer and first tower through chutes to a hopper 
 at the base of the second tower, when it is again elevated and distributed throughout the work. 
 A careful study is required in order to make spouting plants thoroughly effective; and this study 
 should always be made before the job is started to make sure that the proper radius of delivery 
 and best arrangement is secured. 
 
 25e. Regulating Flow of Concrete in Spouting Plants. — It is quite essential for 
 the proper operation of the spouting plant that concrete should be uniformly and continuously 
 carried down the chutes. To this end a receiving hopper is placed at the head of the elevating 
 tower, with a man in control of its gate. Upon this man then depends to a large extent the 
 success of the operation. If he permits a proper amount of material to flow into the chutes, 
 they can usually be relied upon to carry it freely providing they are disposed at proper inclina- 
 tion. If he sees the line becoming choked, upon his slackening or shutting off the delivery 
 depends either a speedy clearing of the line with relatively continuous operation, or shutting 
 down for an indefinite period. 
 
 No matter what type of mixing equipment, or what system of distribution is adopted, 
 there should be kept in constatnt view the object to be attained, namely, the economical pro- 
 duction and placement, not merely of materials which will fill form spaces with possible accept- 
 ance, but rather of materials which in forms will solidify and endure under stress, whatever the 
 nature of such stress may be. Each yard of poor concrete carelessly placed gives concrete a 
 black eye. Each yard of good concrete properly placed is testimony as to the abilities of this 
 inherently wonderful material. 
 
SECTION 4 
 
 CONCRETE FLOORS AND FLOOR SURFACES, SIDEWALKS, 
 
 AND ROADWAYS 
 
 CONCRETE FLOORS AND FLOOR SURFACES 
 
 1. The Concrete -floor Problem. — Concrete floors in modern commercial buildings are of 
 peculiar importance. Not only is the floor slab an integral part of the structure, making pos- 
 sible its usefulness by supporting applied loads^ — such as machinery, or stored goods — but such 
 floors must further perform unusual service in withstanding severe concentrated stresses — as, 
 for instance, those due to passing trucks heavily loaded — and particularly must they withstand 
 at their top surfaces not only the crushing above referred to, but, in addition, a constant 
 and severe abrasion through the impact of shoes, or the movement of loads, with oftentimes 
 attack from chemicals used in manufacturing processes. 
 
 No part, therefore, either of aggregates or of the cement matrix in which they are embedded, 
 may give way without ''dusting," or progressive destruction, of the floor to greater or less 
 degree, causing not only annoyance and inconvenience, but possibly more serious consequences 
 by reason of the released particles being carried into machinery and manufactured products. 
 These particles are abrasive, gritty, and may be chemically injurious. The problem of satis- 
 factory concrete-floor surfaces becomes, therefore, essentially the problem of producing a 
 concrete (1) of a strength requisite to resist compression and shear due to floor loads; and (2) of 
 sufficient top-surface resistance to withstand the mechanical attack of normal service. Chem- 
 ical attack must be provided for by special supplementary treat- 
 ment with a resisting paint or varnish. 
 
 The requisite first named is met with relative ease. Den- 
 sity and strength through use of proper aggregates and good 
 cement, thoroughly mixed, without excess water, and carefully 
 placed are the procedures to be followed (see chapters on "Aggre- 
 gates" and "Water" in Sect. 1 and on "Mixing, Transporting, 
 and Placing Concrete" in Sect. 2). Porous floor slabs, such as 
 that shown in Fig. 1 do not make either for strength or for any of 
 the qualities desired in good concrete. 
 
 The requisite second named — that of producing a resisting 
 top surface is more difficult to meet with full satisfaction. If the Fig. i— Rough ^ Jracture 
 surface of concrete floors (or of concrete roads or sidewalks) could n^fied^2 dilmsO^°°'^ ^ ^ 
 be natural stone of proper quality molded with the same ease as 
 
 is concrete and made integral both as a monolith and by tying with steel to the rest of the 
 structure, there would be little cause for complaint on the score of dusting, wear, or struc- 
 tual functioning. Yet the aggregate employed in concrete is natural stone in fragments. 
 Since reinforcing steel does not affect wearing qualities at the surface, the difficulty must, 
 therefore, lie either in the choice of the natural materials, in the proportion of these materials 
 exposed as resistants to abrasion, or else in the quality or quantity of the cementing material 
 holding them vise-like against the abrading forces. 
 
 The ideal in artificial stone floors is the terazzo floor in which an even surface (95 7o or 
 more of which is natural stone with 5% or less of cementing material) is presented to wear. 
 Grinding concrete floors is necessarily expensive, but removing a surface layer by such means 
 
 205 
 
206 
 
 CONCRETE ENGINEERS' HANDBOOK 
 
 [Sec. 4-2 
 
 produces a superior result that is found to justify the cost. The significance of these facts, 
 together with what is known of the general top-surface character of concretes, leads to the 
 conclusion, which is fully borne out, that in the extreme top layers of a concrete floor is to be 
 found much of whatever difficulty is experienced. 
 
 2. "Dusting" of Concrete Floors. — Research has shown that ''dusting" floors are concretes 
 which are at least locally poor, such local weakness (most evident in the top coat) being shown 
 typically in Fig. 2, wherein the sand grains are seen to be uncertainly held in a loose and easily 
 abraded matrix of what appears to be of the nature of efflorescence. All dusting-floor surfaces, 
 however, are not identical with the one shown, nor are the causes of dusting necessarily the 
 same. Nevertheless, the process of progress of progressive destruction or ''dusting" is much 
 the same in most cases and is largely due to a loose condition of the cement binder which permits 
 the resistant sand grains to fall out, exposing fresh surfaces of soft, hydrated cement to attrition, 
 with repetition of the process until remedies are applied, or until resistant strata at irregular 
 
 depths below the surface are reached. These actions are aug- 
 mented at times by chemical action, either from moisture or from 
 other atmospheric or fluid agencies. 
 
 3. Making Good Concrete Floors and Floor Surfaces. — A 
 good concrete floor is essentially a good concrete. The principles 
 of making good concrete— the right materials and the right propor- 
 tions of same, including water; thorough mixing; careful placing; 
 careful curing — epitomize the making of good floors. 
 
 To these axiomatic and self-evident general principles should 
 be added the following: 
 •.3^ 1. Whenever possible, run top coat and base together. If this 
 
 Fig. 2. — Dusting surface of is not possible or advisable, remove top surface of base to a depth 
 di^Sf (Magnified 5 j^^^^ -^^^ r^^^^^ ^^^^^^ placing top coat, roughen the base 
 
 surface well and wash clean, using hose or brushes. This pro- 
 cedure is necessary to procure a proper bond between top coat and base. 
 2. Use coarse, rather than too fine material in the top coat.^ 
 4. Special Surface Finishes. • 
 
 4a. Surface Grinding. — "Granolithic" is a term applied alike to concrete 
 floors having cement and sand finish, and to those having a surface layer of crushed granite, 
 or other hard, enduring rock bonded with cement. As above noted, a desirable finish to 
 such floors, a finish that gives a pleasing appearance and removes much of any tendency there 
 may be to dusting or surface disintegration of any kind, may be produced by surface grinding 
 when the concrete is from 4 to 7 days old by means of a machine similar to that employed in 
 grinding terazzo floors. Such grinding removes any laitance or loose material from the surface, 
 produces a smooth though not polished surface and, by selection of aggregates before laying 
 with special reference to color, gives an unusually pleasing effect. 
 
 46. Integral Pigments. — Pigments of one coloration or another, chemically 
 inert toward concrete, can be had of a number of dealers. Inasmuch as surface color only is 
 desired, it is advantageous to apply them only in a relatively-thin mortar layer at the top 
 surface. This layer should be truly integral with the layers below, else it will scale off. It 
 should further be borne in mind that the coloring value in concrete of any integral pigment 
 will be affected strongly by the color of Portland cement, which is itself a pigment, white when 
 hydrated, gray-green when unhydrated. According, therefore, to the color added, the effect 
 of pigment in concrete will be more or less intense according to its percentage presence as 
 related to the percentage of cement in the mixture and to the degree of hydration of the latter. 
 The color of aggregates may also affect the result. Care should, therefore, be exercised in at- 
 tempting to secure a given intensity of final color, not to use pigments in such quantity as to be 
 detrimental to the concrete. Small trial batches will aid in securing the effect desired. 
 
 J See L. C. Wason: Tmns. A.S.M.E., 1914, p. 400. 
 
Sec. 4-4c] 
 
 CONCRETE FLOORS AND FLOOR SURFACES 
 
 207 
 
 4c. Finish Produced by Removal of Water From Surface. — A process of finishing 
 floors said to give excellent results is the abstraction of excess surface water through absorption 
 by dry cement laid on webbing over the soft floor. ^ This method should be advantageous 
 in many instances, particularly where excess water is used. The process is proprietary. 
 
 4:d. Integral Hardeners and Surface Compounds. — A number of compounds 
 are on the market designed to be incorporated with the surface to make it resistant. One of 
 these is carbide of silicon (carborundum) under one name or another. This material unques- 
 tionably has great abrasive resistance and if properly held in place by cement should produce 
 a surface capable of withstanding severe traffic. There is, however, no reason to expect better 
 conditions of manufacture attending its use than would obtain where it is omitted; and as 
 good quartz sand or crushed durable rock properly bedded in cement is capable of supplying 
 most needs; and inasmuch as the sparkling, glistening effect incident to the use of carborundum 
 is often objectionable, the advantages to be expected from it should be carefully looked into 
 before it is employed. 
 
 Common iron, powdered, is extensively marketed as a surface hardener for concrete floors. 
 A variety of claims, many of them conflicting, are advanced by its advocates. Some assert 
 that the iron oxidizes (rusts) with expansive filling of pores and prevention of further moisture 
 penetration. Soil-ammoniac may even be added to promote this rusting. Others claim no 
 rusting with the virtue residing in the superior hardness of such iron as remains at the surface. 
 
 It is unquestionably a fact that the average iron in contact with moist air will rust. This 
 produces a characteristic red stain of rust in the concrete, but it is doubtful if as an incident to 
 mixing or placing, this iron rust can be so directed, distributed, and placed as to constitute 
 a reliable pore filler; and it is further more likely to be an attractor of moisture than a preventive 
 of moisture penetration, since rust (Fe203-X H2O) is deliquescent. Further, iron is so inferior 
 in hardness to common quartz sand as to make the ratio of comparison about 230 (for quartz) 
 to 18 (for iron),'-^ so that with equally satisfactory embedment in cement, iron should prove 
 inferior to ordinary sand. In addition, factory grease is often not removed from the iron, so 
 that attachment of cement is hindered, if not inhibited. 
 
 There is no top coat superior in all-around qualities to good quartz sand of proper size and 
 grading, nor is there any additive at present known qualified per se to overcome initial deficien- 
 cies resulting from faulty manufacture or inferior materials. 
 
 5. Causes of Common Defects in Concrete Floors. — A statement of defects commonly 
 found in concrete floors and the causes which give rise to them is conversely an aid to the pro- 
 duction of floors of proper endurance. Avoidance of wrong practices is the surest guaranty 
 of success. Such a listing of the causes of defects, therefore, follows : 
 
 (a) Poor Cement. — This cause is infrequent. It is true that defective Portland cements 
 are occasionally manufactured and that they are marketed, but misuse of cement is more 
 frequent than deficiencies in the cement itself. Other causes should be sought and eliminated 
 before blame is attached to the cement. 
 
 (6) Poor Quality of Sand. — This cause is relatively frequent. Sands, as before noted, 
 are derived from the breakdown of natural rocks; and in most sand deposits the grains have 
 existed for millions of years, so that their inherent quality and endurance is vouched for, but 
 decomposing cementing materials, such as clay (uniting very small mineral particles to form the 
 larger grains), or organic matter, or dirt, or other impurities, may render the best sand unfit 
 for use in concrete. Be sure of the quality of sand before laying the floor (see Art. 30, Sect. 1). 
 
 (c) Poor Grading of Sand.— This cause is relatively frequent. It needs no demonstration 
 to prove that the greatest quantity of sand in a given volume will be obtained when the 
 |)articles are so graded in size that smaller grains will he between the spaces of larger grains, 
 progressively down the scale of sizes. Likewise, the least quantity of sand in a given volume will 
 be had when all the grains are of one uniform size. This truth is sometimes stated by saying 
 
 1 See P. M. Bruner: Proc. Am. Con. Inst., 1915. 
 
 2 Scleroscope scale. 
 
208 
 
 CONCRETE ENGINEERS' HANDBOOK 
 
 [Sec. 4-5 
 
 that "minimum voids" of " maximum density" is obtained with graded sizes of grains; and that 
 maximum voids" or ''minimum density" is obtained when the grains are all of one size. 
 Sand for concrete floors and particularly for the top coat should, therefore, be graded in size 
 in such manner as to give ''maximum density" with maximum quantity of enduring silicious 
 material of proper grain size at the surface. Be sure of the grading of sand before laijing the 
 floor. 
 
 (d) Dirty Water. — The occurrence of defects due to the use of dirty water is relatively 
 infrequent. Dirty water is not merely water that carries fine silty matter in suspension, but 
 more particularly contaminated water, carrying organic matter, such as stable or barnyard 
 drainage. Organic matter of this kind is very injurious to concrete and may even cause failure 
 to set, or total disintegration. Use only water of unquestionable cleanness. 
 
 (e) Too Much Water. — The use of too much water is of general occurrence. With excess 
 water in a mix, the fine particles of stone, and the fine particles of sand, and the finest particles 
 of floor cement separate and rise, forming a thick scum at the top of the slab. This is where 
 the slab should be most enduring, but if too much water is used, a material having about the 
 resistance and character of chalk is substituted for the enduring materials desired. Avoid 
 excess water. Use thorough mixing to obtain the plasticity desired. 
 
 if) Wrong Proportions of Materials. — Arbitrary proportions in concrete making have little 
 except careless convenience to recommend them. Through lack of understanding and be- 
 cause of the supposed difficulty of proper proportioning, they have remained in practice. It 
 should be borne in mind that each sand and each gravel has properties peculiar to itself; and 
 that the proportions in which they should be used in combination with cement and water 
 apply to them only, so that such proportions cannot be taken as a criterion for the use of other 
 sands or gravels. Proportion the materials so as to obtain maximum density (see chapter on "Ag- 
 gregates" in Sect. 1 and on "Proportioning Concrete" in Sect. 2). 
 
 {g) Insufficient Mixing. — One of the reasons that excess water is so commonly used in con- 
 crete is that it renders mixing easy. The desire for ease and rapidity of working tends to carry 
 the speeding-up process beyond allowable limits so that insufficient mixing is more often in- 
 dulged in than is realized. Whether the mixture is sloppy, plastic, or dry, mix it not less than 1 
 min. in a batch machine, or an equivalent amount if other form of mixing is employed. 
 
 (h) Too Much Tamping. — Concrete may be tamped too much. With a medium wet con- 
 crete or concrete of plastic consistency, tamping to a certain degree is desirable to compact the 
 mass. But tamping to more than the required degree brings unfortunate results. It should 
 be recognized that sand and stone and cement in a fluid or semifluid concrete are non-coherent 
 and that by the agitation of tamping, heavier materials sink and lighter materials rise. This 
 causes separation or "segregation," of necessity putting fine, chalky materials, not adapted 
 to resist abrasion, at the wearing surface. DonH flood the surface by tamping. 
 
 (0 Too Little Tamping. — It is frequently the case that a concrete floor is deposited in such 
 haste and with so little care that it does not compact; and particularly, it is not sufficiently 
 joggled to remove air from the mass. In all mixing operations, air is stirred into the plastic 
 mass, much as it might be into a stiff batter; and if such air is not removed in placing, a honey- 
 combed structure will result. Further, the tendency of entrapped air is to concentrate at or 
 near the upper, or wearing surface, so that when the cement has set, it will be molded as bubbles 
 in the mass. Needless to say, such bubbles are holes in the concrete; and holes offer very poor 
 resistance both to stress and to abrasion. Tamp enough to compact the concrete, but not enough 
 to flood it. 
 
 (j) Too Much Troweling. — After a concrete floor form is filled and screeded, the surface 
 is rough and irregular. When the slab has taken its initial set, the finisher rubs or floats the 
 surface to an even finish with a steel or wooden trowel. This process brings considerable water 
 to the surface, acting in a manner analogous to the tamping operation before referred to, so 
 that the very fine particles of cement and sand will rise to the surface. Don't flood the surface 
 by loo much troweling. 
 
Sec. 4-6] 
 
 CONCRETE FLOORS AND FLOOR SURFACES 
 
 209 
 
 {k) Use of Cement as a Surface Dryer. — It is not infrequently the case, particularly when 
 very wet concrete is used and it is desired to hasten finishing operations, that dry cement is 
 sprinkled over the surface of the partially set mass, and worked smooth with a trowel or float. 
 In this case, the liquid it is sought to absorb is a strong solution brought up from the body of the 
 slab; and to this solution the new cement further contributes like substances. This results 
 in a deposit of fine silicious material from the sand and hydrolized cement a little below the finish 
 with a skin of nearly pure cement directly at the surface. Necessarily, therefore, the body of 
 the concrete slab and this thin veneer of neat cement at the surface are separated by a loose, 
 non-adhering laitance film, so that scaling in patches of greater or less size soon results and will 
 continue indefinitely until loose portions are entirely removed. Use less water in mixing and 
 avoid adding cement as a dryer either neat or mixed with sand. 
 
 (l) Use of Retempered Concrete. — The setting of Portland cement takes place in two stages: 
 (1) a gradual stiffening, known as initial set; and (2) an attainment of rigidity, with later slow 
 hardening with passage of time. If the concrete for a floor has been mixed for some time, it 
 may have taken its initial set. Adding more water and reworking renders the mass again 
 plastic, so that it can be deposited much as might freshly mixed concrete. A certain valuable 
 property, however, has become lost by this retempering, or rewetting, inasmuch as among 
 other actions interlacing crystallization had already begun, and by rewetting and remixing, 
 these crystals have been damaged and their reticulation destroyed. Such a floor, therefore, 
 .will be of inferior strength and density and possibly crumbly. Avoid using concrete which has 
 taken its initial set. 
 
 (m) Loosening of Top Coat from Base. — Where floors are deposited in two layers and par- 
 ticularly where the under floor as deposited is overwet, the upper layer (top coat) will be sepa- 
 rated from the base by a scum of laitance deposited at the top surface of the under portion 
 previous to setting. If the top coat in such floors is of good quality and sufficiently thick, it 
 may stand ordinary shocks and wear, but if it is thin, or is subjected to sufficiently intense 
 stress, it will become loosened and at best be unpleasant in use, giving a hollow sound when 
 struck, or walked over, or in extreme cases, will become shattered and crumble into pieces of 
 greater or less size. Cast top coat and base at one operation wherever possible; and in other 
 cases, remove surface of base to a depth of 3^ in. or more and wash thoroughly, bonding to 
 top coat with layer of rich cement grout. Do not permit this latter to dry out or set before 
 top coat is applied. 
 
 (n) Placing Concrete in Freezing Weather without Protection. — Contact between Portland 
 cement and water results in a chemical reaction. As is well known, the speed of a chemical 
 reaction is a function of the temperature. Below 50°F. this reaction is very slow, and so long 
 as this temperature persists, there is comparatively slight formation of the binding or cement- 
 ing substance, though it may be formed later, after the temperature has risen. Furthermore, 
 when water freezes it expands with a force of approximately 300 tons per sq. in. with a volume- 
 tric increase of 8%, so that in frozen concrete there is a general disruption and dispersion 
 of components, possibly during the setting process, and to such an extent that in concrete 
 floors, due to lack of hydrostatic head, they rarely, if ever, become again fully consolidated, 
 regardless of how fully the cement may later react with water. Concrete floors which have been 
 frozen, therefore, are weak and scaly. Use heated aggregates, heated water, and proper protection 
 when concreting in freezing weather. Avoid the use of salt. Its advantages are not commensu- 
 rate with its disadvantages. 
 
 6. Remedial Measures. — Obviously, the best remedial measures, so far as the generality 
 of concrete floors is concerned, are the avoidance of improper procedures and the unceasingly 
 careful institution of proper ones. The foregoing list of the causes of defects is, therefore, a list 
 of remedial measures, so far as floors yet to be laid are concerned. Where, however, recognized 
 defects exist in floors in place, it is a matter of serious moment to effect their repair. 
 
 6a. Retopping.— Where head room and goods or machinery in place will permit, 
 removal of existing top down to sound concrete, with thorough chipping, roughening and 
 
 14 
 
210 
 
 CONCRETE ENGINEERS' HANDBOOK 
 
 [Sec. 4-66 
 
 cleansing of the exposed surface and the careful laying of a new top of proper quality is some- 
 times the most advisable and in the end, the cheapest procedure. Details of bonding top coat 
 to base have been previously given. 
 
 66. Chemical Hardeners. — Sodium or magnesium fluosilicate is marketed under 
 various trade names as a liquid hardener. A pronounced change is to be noted in the appear- 
 ance of concrete so treated with a greater or less hardening of the surface and corresponding 
 resistance to abrasion, according to the initial condition of the concrete. The use of fluosili- 
 cates, originally marketed as "fluates" has been known for many years, with a comparatively 
 recent revival through aggressive selling campaigns since the extension of uses for concrete 
 buildings, with corresponding increase in the number of defective floors. 
 
 6c. Use of Oils. — ^Linseed oil, both boiled and raw, with and without additions 
 and adulterants has been tried as a binder for dusting concrete floors, but has not proven ade- 
 quate for all uses. Its value may be gaged by the resistive and retentive values that an oxidized 
 film of a like oil might be expected to possess when subjected to the same traffic or other condi- 
 tions which gave rise to the original complaint. 
 
 China-wood oil, sometimes called Chinese wood-oil, or Tung oil, either alone or in com- 
 bination with linseed or other oils or resins has proven somewhat more effective than plain 
 
 linseed. It is today in extensive use as 
 a basis for various concrete floor, wall, 
 and water-resisting paints and in proper 
 combinations is very effective. But ex- 
 cellent as are the properties of this oil 
 it is not necessarily an effective remedy 
 for all dusting floors. The conditions 
 attendant on each individual case 
 should be minutely studied and the 
 chances of success estimated, rather 
 than attempting the haphazard appli- 
 cation of any palliative, proprietary or 
 public, on the chance that the result 
 desired will be secured. 
 
 Fig. 3.— Paint film on concrete floor. (Magnified 20 diams.) Qd. Floor CoatingS and 
 
 Paints. — It is oftentimes desirable for 
 reasons of sanitation or of appearance, on good as well as on dusting floors, to use a surface 
 coat of paint. A number of excellent floor paints are on the market which can be had in a 
 variety of colors, the surface obtained being hard and sufficiently resistant to abrasion for 
 most purposes and capable of withstanding the action of water almost perfectly. The man- 
 ner in which such a ffoor paint penetrates into the body of the concrete, leaving a hard re- 
 sistant film at the top, is shown in the micrograph of Fig. 3. 
 
 CONCRETE SIDEWALKS 
 
 7. Structural Functions. — Concrete floors and concrete sidewalks have similarity of func- 
 tioning as abrasion and impact-resisting surfaces and dissimilarity of functioning in that side- 
 walks have solid bedding and low concentrated load and are rarely called upon to act as beams. 
 There is further dissimilarity in that the perfection of surface of a concrete floor is not required 
 of the concrete sidewalk, since the latter is in the open where slight surface disintegrations do 
 no harm and are not noticeable. 
 
 8. Essential Qualities. — The same general conditions governing the manufacture of concrete 
 floors apply to the manufacture of concrete sidewalks since their requisites are held in common. 
 A concrete sidewalk is also subject to the same character of disintegrations as is a concrete 
 
Sec. 4-9] 
 
 CONCRETE SIDEWALKS 
 
 211 
 
 floor, aggravated by year-round exposure to weathering and like influences. Like precautions, 
 therefore, should be observed in its manufacture. 
 
 9. The Making of Concrete Sidewalks.— The disintegrations of sidewalks seen on every 
 hand point strongly to the need of either more careful procedures than those usually indulged 
 in or else to improved procedures, but in view of the many successful concrete sidewalks made 
 by careful following of accepted procedures, indications are that more care is a particularly in- 
 sistent demand. 
 
 9a. Porous Subbase. — One thing that must be always guarded against in a con- 
 crete pavement is the action of frost. For this reason, adequate drainage of subsoil must 
 be provided. The porous, or draining foundation for outdoor construction should, there- 
 fore, be from 6 to 12 in. thick, dependent upon the climate in which it is situated and upon 
 the character of soil. On an • extremely porous or sandy soil without frost, the foundation 
 may be omitted entirely, biit it is erring on the side of safety to use a porous subbase in all 
 cases. 
 
 Such a subbase may be cinders, coarse sand, or preferably broken stone or gravel. This 
 material should be thoroughly rammed to present a firm and unjdelding stratum and, at inter- 
 vals, drains of coarser sand or of gravel, or even open-tile drains may be advantageously in- 
 serted to carry off water which may collect. Cinders or sand should be thoroughly wetted and 
 compacted by ramming. 
 
 96. Concrete Base. — On top of this draining foundation is placed a slab of coarse 
 concrete rammed into place, of a quality at least equivalent to that obtained with arbitrary 
 proportions of 1 : 3 : 6. This base is usually 3 to 3)^ in. in thickness and it is separated into 
 blocks by form boards so that unequal settlement through freezing, rising through the penetra- 
 tion of tree roots or other vegetation, or buckling through temperature changes may not injure 
 the walk. Such divisions are quite essential and should not be omitted. 
 
 9c. Top or Wearing Surface. — Cast upon the concrete base and preferably made 
 integral with it is the top coat, the upper surface of which is the wearing surface. As in concrete 
 floors, the more silica or like rock material which can be exposed to wear in this surface, consist- 
 ent with proper gripping by cement, the more enduring will be this surface. The principles 
 stated for producing a surface of like character on concrete floors apply with equal or with even 
 greater force to concrete sidewalks. 
 
 M. Surface Finishing. — After screeding it is customary to finish the top surface 
 of pavements either by floating with a flat trowel, either of steel or of wood (with or without 
 later brooming to roughen the surface) or to groove or checker 
 the floated surface, both with the object of securing rough- 
 ness and of improving its appearance. All working of the 
 surface has an effect similar to that of floating, and as pointed 
 out in Art. 5 under ^'Floors" it may cause an excess of 
 water at the wearing surface. 
 
 As in making concrete floors, the use of dry cement, or 
 dry cement and fine sand sprinkled over an excessively wet 
 surface, is a common reliance for speeding-up the finishing 
 process. Destruction of top coat is often to be traced directly 
 to this practice. 
 
 9e. Surface Protection and Curing. — ^Laying 
 cement sidewalks in freezing weather is always attended with 
 risk. Even with heated aggregates and heated water (cement 
 is rarely, if ever heated) it is diflScult if not impossible to main- 
 tain the temperature of the mass at a point sufficiently high 
 
 and for a time sufl5ciently long to insure reaction between the water and cement so as to 
 prevent disruption either before final set, or so near that time that reconsolidation at elevated 
 temperatures will later take place. It should further be borne in mind, that lowered tempera- 
 
 Fig. 4. — Typical sidewalk disin- 
 tegration. (Public Square, Cleveland, 
 Ohio.) 
 
212 
 
 CONCRETE ENGINEERS' HANDBOOK 
 
 [Sec. 4-9/ 
 
 ture greatly prolongs the period of plasticity, leaving water uncombined and free to form 
 disrupting ice. 
 
 Even with heated aggregates, therefore, and later surface protection, great care must be 
 exercised to be sure that minute local disruptions (Fig. 4) due to frost have not made the con- 
 crete, particularly at the top, ready for further disintegrations, either through attrition or by 
 other actions. The best and safest plan is not to lay concrete sidewalks in freezing weather. 
 
 9/. Protecting Sidewalks in Hot Weather. — In view of what has been said in 
 Art. 5 under "Floors" with regard to the effects of floating and troweling in bringing water to 
 the surface and remembering further the requirements of cement in regard to quantities of 
 water necessary for slow hydration and chemical interactions, it is readily to be understood 
 that rapid evaporation in hot, dry weather will cause hair cracks, through a too rapid removal 
 of water, leaving behind and directly at the surface such of its products as may be in solution, 
 with further weakening of the concrete through deprivation of the remaining cement of its 
 necessary reacting water. 
 
 To guard against such happenings, as soon as possible after finishing, the pavement should 
 be covered with a moist protecting canvas supported above the surface by a frame, or by a 
 layer of moist sand, either canvas or sand being kept moist for several days. Only by observing 
 precautions such as these can success be obtained. 
 
 9g. Special Surface Finishes. — The same special finishes such as carborundum, 
 or iron, suggested for concrete floors (Art. 4) are advocated for concrete pavements by their 
 sponsors. The same remarks apply equally to both uses, as does a restatement of the fact 
 that no additive so far brought out is capable per se of overcoming defects in manufacture and 
 that none is superior in all-around qualities to good clean quartz of proper grading. 
 
 10. Vault-light Pavements. — Basement areas under sidewalks in cities are a valuable asset 
 but it is necessary that they be cheaply lighted. This desirable end is accomphshed through 
 the use of a combination of round or square lenses of thick glass set in cement mortar, formed 
 into pavement blocks of requisite size, reinforced with steel rods and supported on steel or 
 concrete girders. A template is used in spacing the lenses, the mortar being troweled in place 
 as in usual cement mortar work. From the standpoint of considerations previously presented, 
 this type of sidewalk should and does prove very enduring, by reason of the large area of resist- 
 ant glass exposed to abrasion and the lessened area of cement mortar, but care should be taken 
 to have the cement portion as carefully proportioned, mixed, and laid, as it would be where the 
 entire wearing surface is of cement mortar. Joints between adjacent blocks and adjoining 
 constructions are sealed with a waterproofing compound. Shattered vault hght sidewalks 
 are frequently seen, this shattering being due to the wheels of heavily laden hand trucks, or to 
 improper working or use of the cement mortar, with later disintegration through frost or 
 other actions, with loosening of lenses and permitting of continued spalling of their edges 
 until the light-transmitting properties of the glass are impaired, or the lens broken com- 
 pletely. These difficulties, however, do not reflect upon the value of this construction 
 properly executed. 
 
 11. Concrete Curbing. — Concrete in curb and gutter construction has met with much 
 favor. It proves generally excellent provided it is properly underdrained. Integral curb and 
 gutter blocks are made in lengths having definite cross-joints but no longitudinal joints, as such, 
 by the action of frost or vegetation, would invite separation by frost or by the action of pene- 
 trant vegetation. 
 
 12. Summary. — Concrete sidewalks, no less than other constructions, are structures of 
 concrete; and to be enduring, they must be good concretes. Furthermore, they must be 
 particularly protected against natural disintegrating forces, chief among which is frost. This 
 requires good subdrainage as a prime requisite; and of almost equal importance, a dense struc- 
 ture resistant to the penetration of water. To secure this, rigid observance of the principles 
 of mixing good concrete must be insisted upon. The man with "20 years' experience" may 
 only be one who persists in the least advisable procedures. 
 
Sec. 4-13] 
 
 CONCRETE ROADWAYS 
 
 213 
 
 CONCRETE ROADWAYS 
 
 13. Structural Functions. — Concrete sidewalks and concrete roadway pavements are 
 similar in that they are impact and abrasion-resisting surfaces bedded on a continuous sub- 
 base; and they are dissimilar in the perfection of the surface required and in the degree of 
 impact and abrasion which they must sustain. 
 
 14. Essential Qualities. — Concrete roads, no less than concrete floors, are, first of all, 
 concretes, so that their character is governed by the laws basically controlling the making of 
 good concrete. Particularly must provision be made against disintegration through weather- 
 ing, through the heavy impact of shod hoofs and tires of loaded vehicles, and through the 
 raveling action of fast motor traffic. The best means for this purpose at present known are: 
 (1) proper subdrainage; and (2) the securing of high-density concrete having adequate 
 cementation of adequate quantities of rock products in the wearing surface, through care in 
 selection and proper proportioning of materials; through adequate mixing, careful placing, and 
 proper curing, 
 
 15. One-course and Two-course Pavements. — The tendency at the present time is toward 
 the use of one-course, rather than two-course roadway pavements. A certain roughness of 
 surface is very desirable to prevent' slipping ; and in a pavement with the concrete properly 
 proportioned and mixed there is a requisite amount of rock material in the surface to withstand 
 abrasion, with the added advantage that the slab is monolithic, without separation planes, as 
 is so often the case where one course is laid upon another which has already set. The general 
 method of finish is, however, similar screeding, floating, and troweling being done in the usual 
 manner. 
 
 16. The Making of Concrete Roadways. 
 
 16a. Porous Subbase. — Inasmuch as a concrete roadway pavement should not 
 be called upon to sustain beam stresses, a continuous unyielding bed must be provided, else 
 cracking and unequal settlements will occur. Such a solid bedment of the concrete slab requires 
 special provision against upward heaving by frost action, which necessitates either a porous 
 subbase of a depth sufficient (together with side drains) to clear the ground of water below 
 frost line, or else one having a sufficiently yielding nature to permit local ground eruptions 
 without extension of the disturbance to the concrete slab above it, though this latter is almost 
 impossible to secure. 
 
 It is regretable that so little attention is paid to this important feature of subdrainage. 
 The majority of concrete roadways are placed directly on rolled ground, often without pretense 
 of drainage and in some cases, with even a pronounced dish toward the center. To counteract 
 the effect of settlement or frost action encouraged by this initial fault, reinforcement is embed- 
 ded in the slab, ^-lo of 1% of steel being the Joint Conference recommendation, but twice to 
 three times that amount being actually necessary to secure a modicum of assurance. Concrete 
 roadways should be capable of enduring and rendering splendid service without reinforcement. 
 Actually, even the reinforcement too often proves inadequate to the unnecessary task imposed 
 on it. 
 
 166. Proportioning and Selecting of Materials. — Arbitrary proportions are 
 generally used in pavement work, regardless of possible benefits obtainable by better grading. 
 For one-course work, average specifications call for 1:2:3 concrete; and for two-course work, 
 1 : 2yi : 5 concrete in the base with a topping of 1 : 2 mortar, although a concrete, using fine stone 
 instead of sand requiring even less cement, would be preferable. 
 
 The principles of proportioning and selection of materials for concrete roadways as laid 
 down by the Aggregate Committee of the 1914 National Conference are as follows: 
 
 1. For fine aggregate, use only sand or other fine aggregate free from very fine particles, and which has been 
 actually tested by mechanical analysis, and for the tensile strength of standard mortar. 
 
 2. Use coarse-grained sands or hard stone screenings with dust removed. 
 
214 
 
 CONCRETE ENGINEERS' HANDBOOK 
 
 [Sec. 4-16C 
 
 3. Use sand or other fine aggregate that is absolutely clean. 
 
 4. For coarse aggregate, use hard stone, such as granite, trap, gravel, or hard limestone. 
 
 5. If bank gravel or crushed stone is used, always remove the sand or screenings and remix in the proper 
 proportions. 
 
 If local conditions prevent following any of these rules, adopt some other material than concrete for your 
 pavement. 
 
 More detailed requirements for fine aggregate are: 
 
 The size of the fine aggregate shall be such that the grains will pass when dry a screen having H-in. openings. 
 In the field a ys-in. mesh or in some cases a }^-in. mesh screen may be used for this separation. 
 
 Not more than 10% of the grains below the H-in. size shall pass a sieve having 50 meshes to the linear inch, 
 and not more than 2 % shall pass a screen having 100 meshes to the linear inch. This is an exceptionally coarse 
 sand, but coarse sand is a necessity for a durable pavement. 
 
 16c. Joints. — Expansion joints must be provided to prevent cracking due to 
 temperature changes. These should be provided at linear intervals of from 30 to 50 ft. depend- 
 ing upon the climate of the region in which the pavement is situated. Joints should also be 
 placed at changes in grade, and longitudinally between curbs. The usual joint is from ^ to 
 }'2 in. wide and the National Conference recommends a preformed plastic filler. A variety 
 of special compounds for this purpose are on the market and also metal-and-plastic inserts 
 intended to reduce the surface exposure of joints to a minimum. 
 
 
 
 
 
 r ■■■ r ■ ! 
 
 Proper consistency for 
 concnete road work 
 
 
 
 
 
 
 
 h 
 
 
 
 
 
 
 
 
 
 
 
 
 
 be obi 
 
 
 \ 
 
 
 
 
 
 
 
 
 
 
 
 I 
 
 
 
 
 
 
 
 
 
 
 
 
 1 
 
 1 
 
 1 
 
 
 
 
 
 'With this consistency about 
 one-hatf the strength Js lost 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 -1 
 
 
 
 
 
 
 
 
 
 
 
 
 
 -1 
 
 
 Wifh the sloppy concnefe sometime 
 used in road worlc and m building cor 
 strudion, t^o- thirds to three three- 
 fourths of Hie possible strength erf 
 the concrete is lost. 
 
 ■ y 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 70 80 90 \00 110 120 130 140 150 160 170 180 130 
 Percent of water giving maximum strength 
 
 Fig. 5. — Proper consistency of concrete for road work.^ 
 
 IQd. Curing. — Difficulties that have arisen in the use of concrete pavements 
 have forced recognition of some usually neglected though well-known principles of concrete 
 making. Among these is protection of the pavement for a period of time against drying in 
 warm weather and frost in cold weather. For the purpose first named, earth dams permitting 
 flooding the pavement with water have been used with success; also protecting canvases, or 
 layers of sand, or earth, or sawdust have been used each being kept moist. In the winter time 
 protection against frost has been obtained by the use of hay or straw and sometimes manure. 
 The use of the latter, however, is dangerous, inasmuch as manure not only discolors the concrete, 
 but may bring about disintegration through penetration and decomposition of organic acids. 
 
 16e. Consistency. — Observation indicates a general tendency to mix pavement 
 concretes too wet (see Fig. 5). The proper consistency is plastic, permitting compacting and 
 molding, with surface finishing, but without runoff, or separation of ingredients. Overwet 
 concretes in roadway pavements cannot be expected to have better qualities than the same 
 character of mix possesses in other types of structures. 
 
 1 D. A. Abrams: Bulletin Portland Cement Assoc. » 
 
SECTION 5 
 
 PROPERTIES OF CEMENT MORTAR AND PLAIN CONCRETE 
 
 By Adelbert P. Mills^ 
 
 STRENGTH OF CEMENT MORTAR AND PLAIN CONCRETE 
 
 1. Strength in General. — The strength of mortar and concrete made with a given cement 
 is dependent primarily upon: (1) the inherent strength of its aggregates, particularly the large 
 aggregates; (2) the proportion of cement per unit of volume of mortar or concrete; (3) the 
 degree of compactness or densit}^ of the mortar or concrete; and (4) the time afforded subsequent 
 to final deposition in molds or forms. 
 
 (1) . With a given aggregate (sand, or sand and broken stone or gravel) the mortar or con- 
 crete strength will, other things being equal, increase with increased proportion of cement in 
 the mixture. 
 
 (2) . With a given proportion of cement in the mixture the mortar or concrete strength will, 
 other things being equal, increase with increased density of the mixture. 
 
 (3) . Strength normally increases with age for an indefinite period. 
 
 The first of these rules does not apply in comparing mortars or concretes which have been 
 made with different aggregates, with different cements, with different proportions of water 
 used in gaging, or with different methods of mixing or placing the mixture, nor in comparing 
 mortars or concretes which have cured under different conditions. 
 
 The density of the mixture secured will depend primarily upon the gradation in sizes of 
 the aggregate, but may also be affected by the manner and extent of manipulation of the mate- 
 rial in mixing and placing, and by varying the proportion of water used in gaging. 
 
 The second rule fails as a basis of comparison when different cements are used, when the 
 aggregates possess a different mineral character or are to a differing degree contaminated with 
 mineral or organic impurities, and may not apply when methods or conditions of making, 
 placing, and curing of the mixture vary. 
 
 The third rule is independent of most other factors except that certain circumstances 
 such as the proportion of water used in gaging and the conditions of curing may affect the rate 
 of increase of strength with age. An apparent falling off or retrogression in tensile strength 
 after from 3 to 6 months is commonlj^ noted in tests of neat cement, and less commonly in 
 tests of mortars. A less-marked retrogression in compressive strength is also frequently ob- 
 served in tests of neat cement, and a slight retrogression in compressive strength of mortars 
 and concretes is not infrequently observed after long periods. 
 
 A number of empirical formulas have been derived by various experimentors which attempt 
 to express a definite mathematical relation between strength of mortar and concrete and the 
 absolute volume of cement and sand, or cement, sand, and coarse aggregate, in the mixture. 
 Owing to the fact that the effect of many factors, some of which are mentioned above, cannot 
 be taken into account by such formulas, their application is restricted to certain classes of 
 laboratory investigations which do not involve many of the variables encountered in the use of 
 similar materials on construction work. 
 
 2. Laboratory Tests, Their Use and Significance.— The value of cement mortars and 
 concretes as structural materials depends primarily upon their ni('chani<-nl strength and dura- 
 
 1 Assistant Professor of Materials, Cornell University. Author, "Materials of Construction." 
 
 215 
 
216 
 
 CONCRETE ENGINEERS' HANDBOOK 
 
 [Sec. 5-2 
 
 bility when hardened. The conditions encountered in practical use, however, are necessarily 
 so variable as to exclude the possibility of the establishment of standards based directly upon 
 practical experience. 
 
 The only established American standards for mortars are those for tensile strength (see 
 Appendix A). These standards merely fix values of tensile strength, determined under labora- 
 tory conditions, which experience has shown may be expected of cements and sands found 
 satisfactory in practical construction work, in order that inferiority in any particular mortar 
 materials may be detected by deviation from such standards. In other words, knowing that 
 good concrete or mortar sands will in laboratory tests show a tensile strength not inferior to 
 that of standard sand mortars, it is assumed that any sand which exhibits like tensile properties 
 in the laboratory will not fail to satisfactorily meet the conditions of structural use. 
 
 The conditions encountered by a mortar in a structure are not duplicated in the laboratory, 
 but the laboratory method is so standardized that the external conditions of the test may be 
 duplicated elsewhere or at a different time. 
 
 Two factors operate to lessen the importance of laboratory tests of mortars as an indication 
 of suitability of the material for construction uses: (1) the closeness of the relation between the 
 results of tensile tests and the qualities which a mortar in a structure will be called upon to 
 show may properly be considered open to question, and (2) laboratory tests of this class of 
 material cannot be made with any great degree of precision, and the results may be very 
 much in error if the work is not performed under proper conditions by a skilled operator. 
 
 Mortars are never used structurally in such a way that they will be depended upon to 
 carry tensile stress, while they are commonly used to carry compressive stress. From this 
 circumstance it may be argued that a compressive test in the laboratory will afford a more 
 direct indication of the structural qualities of the material than does the tensile test. The 
 compressive test is less easily made than the tensile test, however, and calls for more elaborate, 
 more expensive, and less portable equipment. It is not easy to establish the relationship 
 between laboratory test results and structural qualities of mortars, but the tensile test has been 
 made so much more generally than the compressive test, that the average man has no experience 
 by which he may judge the value of the latter, while he has observed that materials which pass 
 the tensile test infrequently prove unsatisfactory in a structure. Whether the compressive 
 test will fail any less frequently as an indication of unsuitability remains to be conclusively 
 shown, but the somewhat inadequate data available seem to point toward this conclusion. 
 It is undoubtedly a fact that certain natural impurities in sands, as well as certain classes of 
 material sometimes intentionally added to mortars for special purposes such as decreasing 
 permeability or altering the appearance, reveal their injurious character to a much more 
 marked extent in compressive tests than they do in tensile tests. 
 
 The results of laboratory tests of mortars are affected by a number of factors, not all 
 of which are readily subject to control. It is never possible to determine precisely the relative 
 qualities of mortar materials tested in different laboratories or by different operators. The 
 atmospheric condition of the laboratory, with respect to both temperature and humidity, is 
 one important factor which ought not to be subject to variation, but is so nevertheless. The 
 most important consideration, however, is the fact that very slight variations in the detail 
 methods of manipulation of the materials in making test specimens affect the test results to 
 so great an extent that a very close check between the results of two different operators is 
 impossible. 
 
 An experienced operator may be able to check his own results within say 5%, but a second 
 equally-experienced operator who can also check his own results thus closely, may not be able 
 to check the first man within less than 15 or 20%. Each man has developed invariable methods 
 of manipulation, but the methods of the two men will never be identical. This fact need not 
 invalidate comparative tests, however, and the results of tests of standard mortar and mortar 
 made by the same experienced operator with a commercial sand substituted for the standard I 
 sand should be truly comparable. 
 
Sec. 5-3] 
 
 CEMENT MORTAR AND PLAIN CONCRETE 
 
 217 
 
 Bearing in mind the considerations above discussed, it may be concluded: (1) that labora- 
 tory acceptance tests of mortar must not be considered to show the absolute strength which 
 may be developed in a structure, but merely to indicate the approximate relative value of th(? 
 proposed materials and materials whose suitability has been proved; and (2) that laboratory 
 tests must never be entrusted to anyone who has not had a large experience in making this 
 particular kind of test with the advantage of a fully equipped laboratory. The testing of 
 concrete materials is not the job for a novice, and the average field laboratory is not a fit place 
 to do the work. 
 
 3. Neat, Mortar, and Concrete Strength Compared. — A comparison of the strength of 
 cement, cement mortar, and concrete can only be made when the many variable factors which 
 influence the results of tests have been eliminated so far as is practically possible. This means 
 that only tests made with identical materials under the same auspices are truly comparable. 
 
 9CX ■ 
 
 
 
 
 
 
 
 
 
 
 
 
 
 miiiii 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 rNerrt cement H 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 600- 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 I 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 1 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 700- 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 ¥ 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 600- 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 500- 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 3 
 
 5+ 
 
 3ir 
 
 d 
 
 j.r 
 
 d 
 
 
 nc 
 
 ir 
 
 
 a 
 
 or 
 
 
 
 r 
 
 ^ ronriTTiercial sand mortar 
 
 
 
 
 
 400 
 
 > 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 \ ^ 
 
 - 
 
 or 
 
 A 
 
 nr 
 
 ne 
 
 rc 
 
 io 
 
 1 
 
 so 
 
 n 
 
 i 
 
 m 
 
 3r 
 
 
 r 
 
 
 
 
 
 
 
 i 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 \ 
 
 -= 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 300 
 
 i 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 7 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 eoo 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 »oo 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 0 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 7da eedlo 3 Mo 6 Mo 12 Mo 
 
 Age of specimens 
 
 Fig. 1— Relation between tensile strength of neat cement and cement mortars (9 brands of cement). 
 
 The strengths of the cement, mortar, and concrete mixtures shown graphically by the diagrams 
 of Figs. 1 and 2 are based upon one series of tests made by the Technologic Branch of the U. S. 
 Geological Survey at the Structural Materials Testing Laboratories formerly located at St. 
 Louis, Mo. The complete report of these tests is contained in Bulletin 344 of the U. S. Geolog- 
 ical Survey and Technologic Paper 2 of the U. S. Bureau of Standards. 
 
 The diagrams for neat cement and 1 : 3 standard sand mortar are averages of three tests 
 of each mixture for each of nine separate brands of cement. For the commercial sand mortars 
 and all of the concrete mixtures, a blend of these nine cements was used. The diagrams aver- 
 age the results of three tests of each mortar and 18 to 21 tests of each concrete. (The irregu- 
 larities shown by the diagrams for the commercial sand mortars would doubtless not appear 
 if they represented the average of a larger number of tests.) The same commercial sand was 
 used in the mortars and all of the concretes. It is described as Merrimac River sand and is 
 composed of flint grains having comparatively smooth surfaces. It has a fairly well-graded 
 composition, its void content is not particularly high, but it is finer than is desirable. The four 
 coarse aggregates are typical well-graded aggregates of the classes indicated. It should not 
 be concluded from these tests that granite or gravel aggregate can be depended upon to excel 
 limestone in all cases. Very slight differences in two apparently similar materials of the same 
 class will often make a great difference in their value as concrete aggregate. 
 
218 
 
 CONCRETE ENGINEERS' HANDBOOK 
 
 [Sec. 5-4 
 
 It is not intended that the diagrams of Figs. 1 and 2 shall form a basis for definite con- 
 clusions concerning the relative strengths of various cement mixtures, but only to show in a 
 general way the results of tests of typical materials in various mixtures. The value of any 
 given material can be determined only by tests of its qualities regardless of the qualities other 
 materials of the same type may show. The more important of the many factors which affect 
 mortar and concrete strength are considered below. 
 
 7dci. E8dx 
 
 3 Mo. 
 
 6 Mo 
 
 Age of specimens 
 
 Fig. 2. — Relation between compressive strength of neat cement, cement mortars, and concretes 
 
 (9 brands of cement). 
 
 4. Aggregates of Mortar and Concrete. — Testing of the cement used in all important 
 
 concrete structures has been common practice for many years, but the importance of the 
 quality of the aggregate, in its relation to the quality of mortar and concrete of which it forms 
 a constituent, has not yet come to be adequately appreciated by the majority of engineers, 
 architects, and contractors. While cement of good quality is essential to the making of good 
 concrete, its manufacture has today been standardized to such an extent by exacting speci- 
 fications that in the average case it is actually safer to assume, without tests, that the cement 
 is satisfactory than to assume that the aggregate materials most readily available may properly 
 be used without careful experimental determination of their quality. 
 
 The principal requisites for concrete aggregates are structural strength and durability, 
 a proper gradation of sizes of particles, and cleanliness or freedom from deleterious matter. 
 
 The unsuitability of a weak, soft, or porous material is quite obvious, but a well-graded 
 limestone aggregate may make better concrete than a harder granite aggregate whose void 
 content is high, and a coating of matter partly or wholly of organic origin upon the particles 
 
Sec. 5-5] 
 
 CEMENT MORTAR AND PLAIN CONCRETE 
 
 219 
 
 of the best-graded granite aggregate obtainable may cause the concrete in which it is used to 
 have exceptionally poor qualities. 
 
 The physical testing of aggregates is, unfortunately, not controlled by any generally- 
 accepted specifications, but methods are at the present time undergoing standardization by the 
 technical committees of the most interested National engineering societies. Some of the 
 largest engineering organizations, as well as State, Federal, and municipal public service 
 commissions, employing large quantities of concrete, make a regular practice of subjecting 
 all aggregate materials used to a systematic physical examination. The variability of aggregate 
 materials available in different localities, and even of an aggregate from a single source of 
 limited extent, magnifies the importance of tests not only in choosing the most suitable aggre- 
 gates from those available for construction work in any given locality, but also in certifying 
 the quality of all shipments of that aggregate to the job. This entails a considerable expense 
 for the testing of materials which possess a very low intrinsic value. It is an expense which 
 may be justified, however, by the direct benefit gained by a thorough knowledge of how avail- 
 able materials may best be utilized. An unsatisfactory material may be greatly improved by 
 washing, perhaps, or by screening and readmixture of the different grades in different propor- 
 tions, or the judicious mixture of two available materials may be found to yield a material 
 greatly superior to either one alone. 
 
 It is very desirable, also, that the proportions of the mixture of cement and fine and 
 coarse aggregate be not rigidly specified, but only that the physical properties of the result- 
 ing concrete shall be up to a fixed standard of strength or, in some cases, density or im- 
 permeability. This will often mean that when the local materials are inferior, a concrete of 
 the required quality may be attained either by using the local materials in a rather rich 
 mixture, or by importing better aggregate from a distance, using a leaner mixture. Thus 
 the relative costs of the additional cement used with the local materials, or the freight charges 
 on imported aggregate will be the factor which determines the choice. 
 
 For tests, specifications, and properties of aggregates, see chapter on "Aggregates" in 
 Sect. 1. 
 
 5. Effect of Mineral Character of Aggregates. — The structural strength and durabihty 
 of concrete aggregates is dependent upon the mineral character of the rock from which it is 
 derived. In the case of coarse aggregate of artificially crushed stone, the original qualities of 
 the rock have obviously not been altered. When the rock has been broken down into gravel 
 or sand through the operation of natural agencies, the structural qualities of the individual 
 particles of the material will still be identical with those of the parent rock except for possible 
 changes effected by chemical agencies. 
 
 The principal classes of rocks from which concrete aggregates are derived are granites, 
 trap-rocks, limestones, and sandstones. Granite is an igneous rock of variable structure 
 and texture, whose principal mineral constituents are quartz and feldspar with varying 
 amounts of mica, hornblende, etc. The structural qualities of granites vary greatly but 
 granites as a class rank among the hardest, strongest and most durable stones. The term 
 trap-rock is commonly used to include basalt, diabase, and a number of other igneous rocks 
 possessing similar chemical and physical properties. The principal mineral constituents 
 of most of these rocks are pyroxene and feldspar. They are commonly rather fine-grained, 
 hard, tough, and durable. Limestones contain carbonate of lime, calcite, or carbonate of 
 lime together with the double carbonate of lime and magnesia, dolomite, as the essential 
 constituent. Sand and clay are common impurities and some varieties contain large 
 amounts of shells and other fossils. Limestones vary greatly in structure, strength, hard- 
 ness, and durability. Some of the limestones are superior in structural qualities to some 
 of the granites, but the average limestone is inferior to the average granite or the average 
 trap-rock as concrete aggregate. Sandstones consist of grains of varying sizes, chiefly 
 quartz, bound together by silica or iron oxide or, less frequently, by lime carbonate or 
 clay. The structural qualities, strength, hardness, and durability of sandstones vary 
 
220 
 
 CONCRETE ENGINEERS' HANDBOOK 
 
 [Sec. 5-5 
 
 greatly according to the texture, and the class of the binder. A silicious binder excels 
 any other in all respects; an iron oxide binder is usually, though not always, superior to 
 one of lime carbonate ; and sandstones having a clay binder are in all respects least valuable 
 as concrete aggregate. 
 
 Fig. 3, which is based upon tests made at the U. S. Bureau of Standards {Tech. Paper 58), 
 shows the great variability in strength exhibited by concretes made with different classes of 
 coarse aggregate and with different aggregates of the same general class. The proportions 
 were 1 : 2 : 4 in all cases, and the same cement, a blend of nine standard brands of Portland 
 cement, was used throughout. Two river sands of similar character and somewhat similar 
 granular analysis were used in making the specimens of each coarse aggregate. It will be noted 
 that, except in the case of the granite concretes which included only four different aggregates, 
 the range in strength of concretes with different aggregates of the same class is often more than 
 100% of the average strength. 
 
 All sands are derived from rocks which have been broken down or disintegrated 
 through the operation of purely-physical agencies, without change of mineral identity, 
 or which, in addition to disintegration, have been more or less decomposed by chemical 
 
 1*50001 
 
 5 
 
 I eooo 
 
 0 Ave fbr aggrtjott of max. strtngrti 
 a Ave fcraotrnMot* of min. stnfnafh _ 
 
 
 jtie compressive strength mJb.persq.in 
 
 
 
 
 1! 
 
 t; 
 
 
 O 
 
 Aytfij 
 
 r all aggn 
 
 ^ote. 
 
 
 
 
 Jt 
 
 g 
 
 
 
 
 1 
 
 
 
 
 ? 
 
 i 
 
 
 
 
 c 
 
 
 0< 
 
 
 « 
 
 
 ! 
 
 
 cr 
 
 1 
 
 
 
 
 S 
 
 
 
 T 
 
 % 
 > 
 
 •s 
 
 
 
 ones 
 
 -1 
 
 V) 
 01 
 
 
 
 M 
 
 
 
 1 
 
 E 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 Ultimc 
 
 1 1 
 
 
 
 
 
 
 
 1:2:4 granite concrete 
 
 Ave. of i or more tests of 
 each qranite 
 
 
 
 1-2:4 limestone concrete 
 Ave. of 3 or more tests of 
 each limestone 
 
 
 I. 7000 
 
 £ 6000 
 
 P 5000 
 
 E 3000 
 g 2000 
 
 1000 
 
 
 
 
 
 
 
 
 
 1 
 
 
 
 —o 
 a> 
 
 a 
 
 
 ■o 
 
 
 
 <u 
 a 
 
 J£ 
 
 1 1 
 
 Is teste 
 
 > 
 
 P 
 
 
 cr 
 
 
 5 
 o 
 
 0) 
 
 & 
 
 rO 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 \ Z-A gravel concrete 
 
 Ave. of 3 or more tests of 
 each qravel 
 
 
 13 26 52 4 13 86 52 4 '3 
 
 Age in weeks Age in weeks Age in weeks 
 
 Fig. 3. — Comparative strengths of concrete with various types of aggregate. 
 
 agencies involving the formation of new compounds. The principal disintegrating agen- 
 cies are temperature changes, which are operative because of the unequal expansion and 
 contraction of the component minerals and because of frost action, and abrasion caused 
 by the flow of water, glacial action, or by winds. Chemical decomposition is accom- 
 plished through the solvent power of water, facilitated often by the presence of various 
 chemically-active substances, acids, etc., carried by the water. 
 
 Quartz is the mineral which makes up the bulk of the particles of most sands. This 
 is due to the fact that only the harder constituents of rocks survive* as sand, and quartz is 
 not only a very common constituent of rocks, but is also very hard and resists chemical 
 decomposition. The fact that quartz is the principal constituent of a sand does not 
 insure its suitability for concrete, however. Comparatively small amounts of certain 
 minerals like mica or even feldspar or hornblende, or very small amounts of organic impuri- 
 ties will render a quartz sand altogether unfit for use. Sandstone is a common source of 
 quartz sand, and the quality of the sand will depend upon the character of the binder of 
 the original rock, since the individual particles of sand are made up of still smaller parti- 
 cles of quartz bound together by silica, iron oxide, lime carbonate, or clay according to 
 the class of the rock. Sands are seldom derived from trap-rock or granite directly, 
 
Sec. 5-6] 
 
 CEMENT MORTAR AND PLAIN CONCRETE 
 
 221 
 
 though sand beds may often be largely made up of the constituent minerals of these 
 rocks. Pyroxene and hornblende are complex silicates possessing a degree of hardness, 
 strength, and durability slightly inferior to that of quartz. Hornblende particularly has 
 inferior weathering qualities. Feldspars are essentially silicates of alumina with potash, 
 soda, or lime. They are considerably less strong and durable than quartz. Mica is a 
 very objectionable constituent of sands for concrete. It is soft, has low strength, particu- 
 larly in shear, and its laminated structure promotes the percolation of water. Its surface 
 is also of such a character that the bond secured by cement is very poor. Limestones do 
 not serve as a source of concrete sands although calcite and dolomite may occur in sands 
 derived from sandstones having a calcium carbonate binder. Limestone screenings or 
 crusher dust are also sometimes used as fine aggregate though the concrete made there- 
 with is usually inferior in strength to that made with an average sand, and it is also apt 
 to be, or will in time become, more permeable, owing to the solubility of calcite and dolomite. 
 
 Sand deposits being rarely of a residual character, but usually deposited by stream or 
 glacial action, and being also of such a character that the percolation of surface waters 
 through the beds is very easy, 
 the material is often contami- ^ 
 nated by matter, much of it 
 of organic origin, carried in 
 suspension by water. Thus 
 the coating of the grains by ^ 650 
 such substances as tannic acid c 
 is frequently encountered. j_ ^'^^ 
 The effect of such impurities "0^550 
 is extremely detrimental and £ 
 the difficulty with which their 500 
 presence may be detected in- 
 creases the importance of care- 
 ful tests of concrete sands. 
 
 6. Effect of Shape and Size 
 of Aggregates. — Specifications 
 frequently call for a sharp sand, 
 i.e., one composed of rough 
 angular particles, in spite of 
 the fact that in many localities 
 
 -® 450 
 
 S 400 
 
 350 
 
 90 
 
 \80 
 
 Age in days 
 
 Fig. 4a. — Effect of size of sand upon tensile strength 1 : 1 mortar. 
 (Tests of R. P. Davis.) 
 
 river or beach sands having somewhat rounded particles are the only ones obtainable, and 
 have been used with perfect satisfaction. The shape of the particles is chiefly important in 
 so far as it affects the void percentage of the material, and rough, irregular particles do not 
 compact better than rounded particles. On the contrary, rounded particles which afford 
 no opportunity for bridging will compact into the densest mass. In some instances the 
 surface facets of angular particles may afford a better adherence for cement than the 
 rounded surfaces of water worn particles, but this factor is less influential than is 
 the density of the mass, and mortars and concretes made with aggregates composed of 
 worn, rounded particles are not inferior in strength to those made with "sharp" aggregates, 
 and very often excel the latter in strength. The same considerations apply in the case of crushed 
 stone vs. gravel as coarse aggregate. Concrete of excellent quality or very inferior quality 
 may be made with either class of material, the mineral character of the particles and the 
 gradation of sizes being much more influential factors than the shape of the particles of 
 aggregate. The requirement of sharpness is based upon an erroneous idea of the positive 
 advantage gained; it often works hardship, or injury, or is unenforcible, and should be 
 omitted from specifications. 
 
 The prevailing size of the particles of aggregates is more important than the shape, but 
 
222 
 
 CONCRETE ENGINEERS' HANDBOOK 
 
 [Sec. 5-7 
 
 far less important than the gradation of sizes because of the direct relation of the latter to 
 density and strength. A comparatively coarse sand is always preferable to fine sand. It 
 has a smaller surface to be coated with cement per unit of volume and therefore requi-res 
 less cement to produce a mortar of a given strength. It is less difficult to fill its interstices 
 with cement than in the case of fine sand, and a denser mass is usually secured with the 
 
 same proportion of cement. 
 1 I I n I n A composition of coarse sand 
 and finer material which will 
 serve to fill the voids in the 
 coarse material will usually 
 excel either a very coarse or 
 a very fine sand alone and 
 will lead to economy of ce- 
 ment. This is particularly 
 true when permeability is im- 
 portant, a mortar made with 
 a combination of coarse and 
 
 7 28 
 
 90 
 
 Age in days 
 
 Fig. Ah. — Effect of size of sand upon tensile strength 1: 2 mortar, 
 (Tests of R. P. Davis.) 
 
 360 
 
 sizes of particles are well 
 graded from coarse to fine, 
 being less permeable than one 
 made with either exclusively 
 
 coarse or exclusively fine material. 
 
 The relation between size of sand and tensile strength of mortar is shown by Figs. 4a, 
 46, and 4c for 1 : 1, 1 : 2, and 1 : 3 mortars, respectively. These diagrams are based upon 
 tests made in the laboratories of the College of Civil Engineering, Cornell University, by R. P. 
 Davis {" Materials of Construc- 
 tion," by A. P. Mills, pp. 152- 
 153). The various sands used 
 were prepared artificially by 
 separating a natural beach sand 
 of nearly pure quartz into eight 
 sizes or combinations of sizes. 
 It is shown that the sand which 
 passes the 20-mesh sieve and is 
 retained on the 30-mesh sieve 
 produces the strongest mortar 
 for all mixtures and at all ex- 
 cept the early ages. The sand 
 of all sizes finer than that pass- 
 ing the 10-mesh sieye ranks 
 second for all mixtures, the 
 blend of equal amounts of 10- 
 20 and 30-40 sand ranks third, 
 10-20 sand ranks fourth, and 
 
 180 
 
 Age in days 
 
 Fig. 4c. 
 
 •Effect of size of sand upon tensile strength 1: 3 mortar. 
 (Tests of R. P. Davis.) 
 
 all finer sizes of sand rank 
 lower in the order of their fine- 
 ness. 
 
 7. Relation Between Density and Strength. — The term density is employed referring to 
 mortars and concretes, meaning the ratio of the sum of the absolute volumes or absolutely 
 solid substance of the individual constituents contained in a measured unit volume of mortar 
 or concrete to the measured unit volume of the materials combined in the form of mortar or 
 
Sec. 6-7] 
 
 CEMENT MORTAR AND PLAIN CONCRETE 
 
 223 
 
 concrete, water being neglected as an individual constituent. In other words the density is the 
 solidity ratio, the ratio of the volume of solids to the volume of the mass of mortar or concrete. 
 
 Many experimental studies have shown that the strength of mortars and concretes is 
 directly proportional to the density of the mixture. The density of the mixture is dependent 
 partly upon the thoroughness of mixing, the amount of water used in gaging, etc., but is pri- 
 marily dependent upon the gradation of sizes of the aggregates. Aggregates in which the 
 relative amount of particles of different sizes is such that the particles of one size just suffice 
 
 sr 
 
 C300 
 
 s. 
 
 
 
 
 
 
 
 
 
 
 92 
 
 
 
 
 
 '1 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 19 
 
 
 
 
 t| 
 
 li » 
 
 es s 
 
 
 
 
 
 
 
 
 
 
 it 
 
 
 
 101 
 
 
 
 
 
 
 16 .rf 
 
 
 1i_ 8. 
 
 
 10 
 
 
 
 
 
 
 
 
 " '!« 
 
 
 "' ^'o 
 
 °59 
 1?3 890 
 
 " o' M 
 o 
 
 
 
 
 69 46, 66 
 15° 
 
 ,12 
 
 
 09 
 ofS 
 
 0 24 
 
 ° 
 
 0°' 
 
 
 
 
 108 
 
 
 
 
 0119 
 
 ft 
 
 89 
 102 ">5 
 
 o 
 
 ni ° 
 
 
 
 
 56 
 
 
 40 
 
 ;a 
 
 
 
 
 
 \U 130 
 
 
 15J 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 .600 £10 £20 .630 .640 .650 £60 .670 .680 .690 .700 .710 .720 730 .740 .750 760 .770 .780 .790 
 
 Density of mortar 
 
 Fig. 5a. — Relation of "density" or solidity ratio of 1:3 mortar to tensile strength. Age, 13 weeks. 
 
 to fill the voids of the next larger size will have a minimum void space, and will therefore re- 
 quire a minimum proportion of cement to secure a product of given strength. The ideal con- 
 ^ Crete of maximum density would be made with aggregates of this character, and, in addition, 
 would be so proportioned that the mortar just suffices to coat the particles and fill the voids of 
 the coarse aggregate, and the cement just suffices to coat the particles of sand and fill its voids. 
 This ideal can of course only be approximated in practice because the larger particles of each 
 
 - 6000 
 +- 
 
 ^ sooo 
 
 n, 4000 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 19° 
 
 
 
 
 ^ 37 
 
 3! 
 
 
 E 
 
 
 
 
 
 
 
 
 
 
 
 
 ^^ 
 
 
 
 36 0 
 
 3 25 
 
 3 
 
 f4. 
 
 
 IT 
 
 
 
 
 
 
 
 
 
 
 44 ? 
 
 
 5» 
 74 O19 
 
 ,69 64 
 
 14 3, 
 
 ° 23 
 
 
 
 
 
 
 
 
 
 oll4 
 
 
 102 
 
 99 
 
 
 
 ■ 55 < 
 
 sr « 
 
 
 30 
 
 9? 
 
 >Z9 
 
 
 10 
 
 
 
 'J? 
 
 
 139 
 Cbl52 
 
 
 . 109; 
 113 "' 
 
 '""'^ 
 
 ^o'°o 
 
 120 
 
 O|07 
 
 n 2« 
 0 84 
 78 
 
 
 
 'J 
 
 
 "70 
 
 
 
 
 
 
 151 
 
 144 
 
 0 
 
 149 
 
 
 
 
 "110 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 .590 .600 .610 .620 .630 .640 .650 .660 .670 .680 j690 .700 .710 .7£0 .730 740. ,750 .760 J70 
 
 Densi+y 
 
 Fig. 56. — Relation of "density" or solidity ratio to compressive strength of 1:3 mortars. Age, 13 weeks 
 
 aggregate will not closely approach each other. Owing to the wedging action of the smaller 
 particles the larger stones and grains of sand are forced apart so that the density of the mixture 
 is certain to be less than that theoretically possible. A slight excess of mortar over that theo- 
 retically required is usually beneficial to density and strength. 
 
 The relation of density of 1 : 3 mortars to tensile strength is shown by Fig. 5a which com- 
 I prises tests of 157 different natural sands made by the U. S. Bureau of Standards {Tech. Paper 
 
224 
 
 CONCRETE ENGINEERS' HANDBOOK 
 
 [Sec. 5-8 
 
 58). The numbers on the diagram indicate the order of fineness of the sands, No. 1 being the 
 coarsest, and No. 157 the finest. The corresponding relation of density and compressive 
 strength for the same mortars is shown by Fig. 55. Note that Figs. 5a and 56 show that 
 the densitv of mortars is in general inversely proportional to the fineness of the sand used, the 
 finer the sand, the lower the density. 
 
 4200f 
 4000 
 3800 
 
 3000 
 .E 2800 
 
 I 
 
 E,400 
 
 ^ leoo 
 
 1000 
 600 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 5 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 -5 — ' 
 
 
 
 y 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 5 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 y 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 Density 
 
 Fig. G. — Relation between density and compressive strength of concrete. (Mix 1:2:4. Age, 4 weeks.) 
 
 The relation of density to compressive strength of 1:2:4 concretes made with various 
 aggregates is shown by Fig. 6. The diagram is based upon tests made by the U. S. Bureau of 
 Standards {Tech. Paper 58, Tables 23-26). The data are not extensive enough to definitely 
 fix the relation sought because of the inevitable wide variation in test results due to variable 
 
 7da IMo. 3 Mo. 
 
 9 Ma >£Ma 
 Fig. 7. — Effect of mica upon tensile strength of 1 : 3 standard sand mortar. 
 
 6 Mo. 
 
 Age 
 
 factors other than density. That the relation is a direct one, approximately that shown by 
 the straight line upon the diagram, is, however, a justifiable conclusion. i 
 8. Effect of Mica, Clay, and Loam in Aggregates.— The occurrence of mica, clay, and^ 
 loam in aggregates has been explained in connection with the consideration of mineral 
 
,Sec. 6-9] 
 
 CEMENT MORTAR AND PLAIN CONCRETE 
 
 225 
 
 _ '5 ZO 
 
 Percen+age of clay in sand 
 
 Fig. 8. — Effect of clay upon strength of 1 : 3 
 commercial sand mortar. (All mixtures artifici- 
 ally made in the laboratory.) 
 
 composition in Art. 5. The very detrimental effect of mica upon the strength of 1 : 3 
 standard sand mortar is shown by Fig. 7. The diagrams are based upon tests made by W. 
 N. WilHs {Eng. News, vol. 54, p. 145). The loss in strength amounts to 15 to 25% vvith 
 2H% of mica in the sand, 25 to 45% with 5%, of mica, and becomes still more marked as 
 the proportion of mica is increased. Mr. Willis also observed that increasing the proportion 
 of mica increased the voids in the sand from 37% with no mica, to 67% with 20% of mica. 
 The weight was at the same time lowered 20%o, and the amount of water required in gaging 
 the mortar was 3 times that required in gaging ^ 
 mortars free from mica. 
 
 The effect of clay in sands is dependent upon 
 its state of subdivision and the uniformity with 
 which it is distributed through the sand. In most 
 laboratory tests the addition of clay in moderate 
 amounts has been found to be beneficial. Typical 
 results of laboratory tests are exhibited by Fig. 8 
 which is derived from tests made by F. L. Koman 
 (Eng. & Cont., vol. 43, p. 403). With the materials 
 here used the maximum increase in strength due 
 to clay additions was observed to be about 20%, and was secured with additions of 10 to 15% 
 of clay. Similar tests made by L, T. B. Southwick and G. A. Wellman {Eng. Rec, vol. 63, 
 p. 332) show that maximum strengths of 1 : 13-2, 1:3, 1 : 4)^, and 1 : 6 mortar mixtures are 
 secured with 3%, 10%, 15%,, and 20% of clay, respectively. These laboratory results do not 
 prove that similar percentages of clay will be beneficial or harmless in natiual clayey sands. 
 The manner of distribution and degree of fineness of the clay in concrete sands will be the de- 
 termining factors, and the amount permissible will not usually approach the above limits. 
 Lumps of clay do not become broken up in concrete mixing, and should be carefully excluded 
 from aggregates. 
 
 Loam, in the usual acceptance of the term, is earth which is made up of vegetable mold 
 together with clay or sand or both. It is extremely injurious to mortars and concretes because 
 
 of its content of organic matter. Fig. 9, derived 
 from the series of tests of F. L. Roman above re- 
 ferred to, shows the effect of loam and organic 
 matter in sand upon strength. From those tests 
 it appears that 5% of this loam (about 1.5% 
 organic matter in the sand) reduces the strength 
 of the mortar about 20%, and other amounts are 
 nearly proportionally detrimental. Organic 
 matter naturally occurring in sands is frequently 
 Percentage of loam in sand found detrimental to an even greater extent than 
 
 Percentage of organic matter in sand Caoprox.) is indicated by these tests, wherein the organic 
 Fk;. O.-Effect of organic^loam upon strength of 1:3 j.^^^^ -^^ ^j^^ ^j^^^p^ powdor was mixed with 
 
 the sand. Organic matter not infre(iuently covers 
 sand particles with a film which is not easily perceptible, but which tremendously retards the 
 normal rate of hardening and gaining strength. An investigation made by Sanford E. Thomp- 
 son {Trans. Am. Soc.C.E., vol. 65) led to the conclusion that organic matter constituting over 
 10% of the silt and at the same time over 0.1% of the sand is distinctly. injurious. 
 
 9. Effect of Consistency. — The important relation of consistency to the strength of mortar 
 and concrete is shown by Figs. 10 and 11. The amount of water is expressed as a percentage 
 of the total dry weight of aggregates and includes any moisture carried by the sand or coarse 
 aggregate in its natural condition. These diagrams are based upon tests made in the labora- 
 tories of the Sheffield Scientific School under the direction of Prof. Barney {Eng. and Cont., 
 vol. 42, p. 244). 
 15 
 
226 
 
 CONCRETE ENGINEERS' HANDBOOK 
 
 [Sec. 5-9 
 
 These tests show that a quite definite percentage of water is required to produce a mortar 
 or concrete of maximum strength with given materials. For the particular materials used in 
 this case the maximum 1 : 2 mortar strength was attained with about 15.6% of water, and the 
 maximum strength of 1:2:4 concrete with about 8.4% of water. For other materials or 
 other mixes these consistencies for maximum strength will not remain the same, but for any 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 \ 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 N 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 V 
 
 
 
 
 
 < 
 
 y 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 14 15 lb 17 18 19 20 21 
 
 Percentage of wafer based on fotal dry weight of 
 cement and sand 
 Fig. 10. — Effect of consistency upon strength of 1 : 2 mortar. 
 
 mixture of given materials there is a critical consistency which will be productive of higher 
 strength than any drier or wetter consistency. This fact is particularly important in view of 
 the common practice of using extremely wet concrete mixes in order to be able to deliver the 
 concrete on the work cheaply and expeditiously by the use of chuting devices between mixer 
 and forms. The fact should be understood that such wet mixtures may be used only with a 
 
 considerable sacrifice in strength of the concrete placed. On the other hand, these tests indicate 
 that nothing is gained by making a concrete so dry that it must be rammed or tamped in place 
 instead of being puddled. Concretes containing 8 to 9% of water are of a sufficiently mushy 
 consistency to be readily puddled, but from 12 to 15% of water, or even more, is commonly 
 used to produce the fluid consistency desirable for chute or spout delivery. 
 
Sec. 5-10] 
 
 CEMENT MORTAR AND PLAIN CONCRETE 
 
 227 
 
 For harmful effects from the use of excess water and for suggested procedures, see chapter 
 on ''Water" in Sect. 1. 
 
 10. Compressive and Tensile Strengths Compared. — The compressive and tensile strengths 
 of the same mortar mixture may be contrasted by a comparison of the diagrams of Figs. 1 
 and 2, pages 217 and 218. A direct comparison for 1:3 standard sand mortar is afforded 
 by Fig. 12 which is based upon tests of seven brands of cement made in the Structural Materials 
 Laboratory formerly maintainedat St. Louis, Mo., by the U. S. Geological Survey (U. S. Geol. Surv. 
 
 360 
 
 Age in days 
 
 Fig. 12. 
 
 ■Ratio of compressive to tensile strength of 1 
 of 30 tests of one brand. 
 
 3 standard Portland cement mortar. 
 Curve is average of 7 brands.) 
 
 (Each result is average 
 
 Bull. 331). The specimens used were standard tension briquettes and 2-in. compression cubes. 
 It is shown by the diagram that the average value of the ratio of compressive strength to tensile 
 strength is far from being constant as the age increases because of the relatively more rapid 
 rate of gain in tensile strength during the first few weeks and the very slight gain or actual 
 retrogression which characterizes the tensile strength after the first 6 months. The average 
 
 160 200 240 
 
 Age in days 
 
 Fig. 13. — Ratio of compressive to tensile strength of 1 : 3 mortar using blend of 7 Portland cements and 22 com- 
 mercial sands. 
 
 value of the ratio for all cements is about 6 between the 1-month and the 6-month periods, 
 but the individual brands of cement show variations of from 15 to 40% from this average. 
 
 A similar series of tests, made under the same auspices, with a blend of the above 7 brands 
 of cement and 22 commercial sands in 1 : 3 mortars has been made the basis of the diagrams of 
 Fig. 13. Because of the very wide variations shown by the 22 different mortars, only the 
 average for all the mortars, the average for the 5 mortars showing the highest value of the 
 
228 
 
 CONCRETE ENGINEERS' HANDBOOK 
 
 [Sec, 5-10 
 
 ratio and the average for the 5 mortars showing the lowest value of the ratio are plotted. The 
 same variation in the ratio of compressive to tensile strength with age shown by standard sand 
 mortars is exhibited by the commercial sand mortars. 
 
 An inspection of Figs. 12 and 13 will show that the tensile strength of mortars is not more 
 than a very approximate indication of the probable compressive strength of similar mortars 
 with the same cement, and even then the age, the mixture, and the sand used are sources of 
 variation which must be taken account of. 
 
 The tensile strength of concrete is a property of little importance because, being low in 
 comparison with the compressive strength, concrete is practically never designed to carry 
 tensile stress. When concrete is used in situations involving tensile stress it is more economical 
 
 Tensile and Compressive Strengths of Concrete 
 
 Character of coarse 
 aggregate and mix 
 
 Age, months (approx.) 
 
 Tensile 
 tests 
 
 Compression 
 tests 
 
 Compressive 
 strength 
 (lb. per sq. in.) 
 
 Tensile 
 strength 
 (lb. per sq. in.) 
 
 Ratio 
 tensile strength 
 
 compressive strength 
 
 Limestone 1 :2:4.. 
 
 Sandstone 1 : 2 : 4 
 
 Sandstone 1 :2}4 -5. 
 
 Average . 
 
 Average . 
 
 Average . 
 
 2,206 
 2,708 
 2,500 
 
 2,505 
 
 1,069 
 1,375 
 1,417 
 1,722 
 2,000 
 2,139 
 
 1,620 
 
 1,028 
 1,639 
 972 
 889 
 1,042 
 2,083 
 1,472 
 1,889 
 1,639 
 
 1,406 
 
 278 
 308 
 253 
 306 
 264 
 257 
 
 278 
 
 149 
 142 
 133 
 178 
 158 
 128 
 153 
 150 
 161 
 
 150 
 
 121 
 114 
 106 
 158 
 114 
 97 
 179 
 129 
 139 
 
 129 
 
 11.1% 
 
 9.3% 
 
 9.1% 
 
Sec. 6-11] 
 
 CEMENT MORTAR AND PLAIN CONCRETE 
 
 229 
 
 to use steel reinforcement than to use the very large sections which would be required if the 
 concrete were depended upon to carry tension. 
 
 An indication of the comparative strength of concrete in tension and compression is afforded 
 by the table shown on page 228. These data were derived in tests made by the writer in the 
 laboratories of the College of Civil Engineering, Cornell University. The concrete was 
 mixed and the specimens molded in the field. 
 
 Note that the values of the ratio of tensile to compressive strength in this table would 
 have been somewhat lower had the specimens been tested in compression at the same age they 
 were in tension. 
 
 11. Strength of Plain Concrete Columns. — The strength of plain concrete columns, as 
 determined by tests of laboratory specimens whose dimensions are comparable with those of 
 columns used in structures, is usually not less than 75 nor more than 90% of the strength of 
 cubes of the same concrete, the column length not exceeding 10 to 12 diameters. 
 
 A number of series of tests of plain concrete columns are tabulated below and on page 
 230. The strength of cubes of similar concrete is indicated where data from comparable tests 
 are available. 
 
 Strength of Plain Concrete Columns 
 
 Watertown Arsenal Tests ^ 
 
 Mixture and 
 character of 
 coarse aggregate 
 
 Age, 
 months 
 
 Com- 
 pressive 
 strength 
 (lb per 
 sq. in.) 
 
 Cross- 
 section, 
 
 inches 
 (approx.) 
 
 Length, 
 feet 
 
 1 
 
 1 
 
 Mortar 
 
 6 
 
 5,011 + 
 
 12 
 
 5 X 12 
 
 5 
 
 8 
 
 1 
 
 2 
 
 Mortar 
 
 6 
 
 3,652 
 
 12 
 
 5 X 12 
 
 5 
 
 8 
 
 1 
 
 2 
 
 Mortar 
 
 6 
 
 2,488 
 
 12 
 
 5 X 12 
 
 5 
 
 8 
 
 1 
 
 3 
 
 • Mortar 
 
 6 
 
 2,062 
 
 12 
 
 5 X 12 
 
 5 
 
 8 
 
 1 
 
 3 
 
 Mortar 
 
 6 
 
 2,692 
 
 12 
 
 5 X 12 
 
 5 
 
 8 
 
 1 
 
 4 
 
 Mortar 
 
 6 
 
 1,564 
 
 12 
 
 5 X12 
 
 5 
 
 8 
 
 1 
 
 4 
 
 Mortar 
 
 6 
 
 1,471 
 
 12 
 
 5 X 12 
 
 5 
 
 8 
 
 1 
 
 5 
 
 Mortar 
 
 6 
 
 1,038 
 
 12 
 
 5 X 12 
 
 5 
 
 8 
 
 1 
 
 5 
 
 Mortar 
 
 6 
 
 1,082 
 
 12 
 
 5 X12 
 
 5 
 
 8 
 
 1 
 
 1 
 
 : 2 (Pebbles) 
 
 5 
 
 1,525 
 
 12 
 
 5 X 12 
 
 5 
 
 8 
 
 1 
 
 1 
 
 :2 (Pebbles) 
 
 8 
 
 1,720 
 
 12 
 
 5 X 12 
 
 5 
 
 8 
 
 1 
 
 1 
 
 :2 (Trap rock) 
 
 5 
 
 3,900 
 
 12 
 
 5 X12 
 
 5 
 
 8 
 
 1 
 
 2 
 
 :3 (Pebbles) 
 
 8 
 
 1,769 
 
 12 
 
 5 X 12 
 
 5 
 
 8 
 
 1 
 
 2 
 
 :4 (Pebbles) 
 
 3)^ 
 
 1,710 
 
 12 
 
 5 X12 
 
 5 
 
 8 
 
 1 
 
 2 
 
 :4 (Pebbles) 
 
 5 
 
 1,506 
 
 12 
 
 5 X 12 
 
 5 
 
 8 
 
 1 
 
 2 
 
 : 4 (Trap rock) 
 
 5 
 
 1,750 
 
 12 
 
 5 X12 
 
 5 
 
 8 
 
 1 
 
 2 
 
 :4 (Trap rock) 
 
 6 
 
 1,990 
 
 12 
 
 5 X 12 
 
 5 
 
 8 
 
 1 
 
 2 
 
 :5 (Pebbles) 
 
 3 
 
 1,100 
 
 12 
 
 5 X 12 
 
 5 
 
 8 
 
 1 
 
 3 
 
 :6 (Pebbles) 
 
 5 
 
 700 
 
 12 
 
 5 X 12 
 
 5 
 
 8 
 
 1 
 
 3 
 
 :6 (Pebbles) 
 
 8 
 
 462 
 
 12 
 
 5 X 12 
 
 5 
 
 8 
 
 1 
 
 3 
 
 : 6 (Trap rock) 
 
 4 
 
 1,350 
 
 12 
 
 5 X 12 
 
 5 
 
 8 
 
 1 
 
 2 
 
 :4 (Cinders) 
 
 
 871 
 
 12 
 
 5 X 12 
 
 5 
 
 8 
 
 1 
 
 3 
 
 :6 (Cinders) 
 
 5 
 
 1,060 
 
 12 
 
 5 X12 
 
 5 
 
 8 
 
 » "Tests of Metals," 1904, 1905. 
 
230 ^ CONCRETE ENGINEERS' HANDBOOK [Sec. 5-12 
 
 University of Illinois Tests^ 
 
 Ratio 
 col. strength 
 
 Mixture 
 (coarse 
 
 aggregate, 
 crushed 
 
 limestone) 
 
 Compressive 
 strength of 
 columns 
 (lb. per sq. in.) 
 
 Age 
 columns, 
 months 
 (approx.) 
 
 Compressive 
 strength of 
 cubes 
 (lb. per sq. in.) 
 
 Age 
 cubes, 
 months 
 (approx.) 
 
 Cross- 
 section, 
 
 inches 
 (approx.) 
 
 Length, 
 feet 
 
 cube strength 
 
 (approx.) 
 
 
 1:1>^:3 
 1:1M:3 
 
 2,120 
 2,480 
 
 2 
 
 2 
 
 
 12 in. cyl. 
 12 in. cyl. 
 
 10 
 
 95.3 
 
 2 
 
 2,600 
 
 2 
 
 10 
 
 69.9 
 
 1:2:3% 
 
 1,710 
 
 2 
 
 2,443 
 
 2 
 
 12 X 12 
 
 12 
 
 
 1:2:3% 
 1:2:3% 
 
 2,004 
 1,610 
 1,709 
 1,189 
 
 2 
 
 
 
 9X9 
 
 12 
 
 
 2 
 
 
 
 12 X 12 
 12 X 12 
 
 12 
 
 
 1:2:3% 
 
 2 
 
 
 
 12 
 
 6 
 
 60.5 
 
 1:2:3% 
 
 2 
 
 1,962 
 
 .2 
 
 12 X 12 
 
 
 1:2:3% 
 1:2:3% 
 1:2:3% 
 1:2:4 
 
 1,079 
 2,650 
 2,770 
 1,165 
 2,000 
 2,210 
 1,590 
 
 2 
 
 
 
 9X9 
 12 X 12 
 
 6 
 
 
 12 
 
 
 
 12 
 
 
 16 
 
 
 
 12 X 12 
 
 12 
 
 
 2 
 
 
 
 12 in. cyl. 
 12 in. cyl. 
 12 in. cyl. 
 12 in. cyl. 
 
 10 
 
 
 1:2:4 
 
 2 
 
 
 
 10 
 
 
 1:2:4 
 
 2 
 
 
 
 10 
 
 78.1 
 
 1:2:4 
 
 2 
 
 2,035 
 
 2 
 
 10 
 
 
 1:2:4 
 
 1,945 
 1,460 
 
 2 
 
 
 
 12 in. cyl. 
 12 in. cyl. 
 
 10 
 
 78.3 
 
 1:2:4 
 
 2 
 
 1,865 
 
 2 
 
 10 
 
 
 1:2:4 
 
 1,810 
 1,925 
 
 2 
 
 
 
 12 in. cyl. 
 12 in. cyl. 
 
 10 
 
 80.4 
 
 1:2:4 
 
 6 
 
 2,390 
 
 6 
 
 10 
 
 99.7 
 
 1:2:4 
 
 1,845 
 
 6 
 
 1,850 
 
 6 
 
 12 in. cyl. 
 
 10 
 
 99.7 
 
 1:2:4 
 
 1,770 
 
 6 
 
 1,775 
 
 6 
 
 12 in. cyl. 
 
 10 
 
 99.8 
 
 1:2:4 
 
 2,680 
 
 6 
 
 2,685 
 
 6 
 
 12 in. cyl. 
 
 10 
 
 85.3 
 
 1:2:4 
 
 2,160 
 
 6 
 
 2,530 
 
 6 
 
 12 in. cyl. 
 
 10 
 
 74.6 
 
 1:2:4 
 
 1,770 
 
 6 
 
 2,370 
 
 6 
 
 12 in. cyl. 
 
 10 
 
 
 1:3:6 
 
 955 
 
 2 
 
 
 
 12 in. cyl. 
 12 in. cyl. 
 12 in. cyl. 
 12 in. cyl. 
 
 10 
 
 
 1:3:6 
 
 1,110 
 575 
 
 2 
 
 
 
 10 
 
 
 1:4:8 
 
 2 
 
 
 
 10 
 
 
 1:4:8 
 
 575 
 
 2 
 
 
 
 10 
 
 
 
 
 
 
 
 
 
 TJniversi ty 
 
 OF Wisconsin Tests^ 
 
 
 
 84.0 
 
 1:2:4 
 
 2,040 
 
 2 
 
 2,427 
 
 2 
 
 12 X 12 
 
 10 
 
 88.0 
 
 1:2:4 
 
 2,110 
 
 2 
 
 2,395 
 
 2 
 
 12 X 12 
 
 10 
 
 91.7 
 
 1:2:4 
 
 2,055 
 
 2 
 
 2,240 
 
 2 
 
 12 X 12 
 
 10 
 
 88.1 
 
 1:2:4 
 
 2,080 
 
 2 
 
 2,360 
 
 2 
 
 12 X 12 
 
 10 
 
 1 Bulls. 10 and 20 of the Univ. of 111. Eng. Exper. Station. 
 
 2 Data from tests of cubes made at ages which do not correspond even approximately to the age of the column 
 made from the same concrete have been omitted. 
 
 3 Bull. 300. 
 
 12. Effect of Method of Mixing. — The method of mixing mortars and concretes may- 
 vary with respect to: (1) amount of water used; (2) duration of mixing operation; and (3) 
 detail method of manipulation. The effect of variation in the amount of water used is con- 
 sidered in Art. 9. The effect of the duration of the mixing operation is shown by Fig. 14 
 which is based upon tests by Prof. Scofield of Purdue University {Eng. and Cont., Jan. 17, 1915). 
 All of the concrete was mixed in a Chicago Cube Mixer of 23'^-cu. ft. capacity, run at the rate 
 of 26 revolutions per min. These concretes are all much stronger than the average com- 
 mercial concrete but this fact does not affect the significance of the test results. It appears 
 that with the particular materials used a very decided advantage is gained by operating this 
 mixer much longer than the usual period of mixing. The actual time of mixing most advan- 
 tageous to the quality of concrete produced under given conditions will probably vary greatly 
 
Sec. 6-13] 
 
 CEMENT MORTAR AND PLAIN CONCRETE 
 
 231 
 
 with different materials and with different mixers. It is very probable, however, that the 
 average concrete used in every day practice would be considerably improved in quality, if it 
 were mixed for a longer period. This is certainly true of concretes which are turned out at a 
 rate of a batch per minute as is sometimes the case. A distinct advantage is gained by mixing 
 beyond the point which produces a batch of even color. The mass becomes more viscous; 
 there is less danger of separation of fine and coarse material; for a given water content it appears 
 to possess a wetter consistency and flows better in transporting by chute and in depositing in 
 the forms; and it forms a concrete of greater density, less permeability, and greater strength. 
 
 It cannot be said that machine-mixing will invariably produce better concrete than hand- 
 mixing, but for all except the smallest work it is less expensive and is, therefore, generally pre- 
 ferred. Hand-mixing is more apt to be severely slighted than machine-mixing because of the 
 heavy labor involved and the comparatively long time required. 
 
 'cubes^ r 
 
 Fig. 14. 
 
 20 22 24 £6 28 30 
 
 Total time of mixing in minutes 
 
 -Effect of time of mixing upon strength of concrete. 
 
 In comparative tests of concretes made with the same materials, weighed and molded by 
 employees of the laboratory of the U. S. Bureau of Standards, but mixed in the field by three 
 different contractors, each being permitted to use his own methods of mixing, both hand and 
 machine, all conditions being the same except the actual mixing of the materials, variations 
 of as much as 70% in compressive strength were obtained {Tech. Paper 58, U. S. Bureau of 
 Standards). 
 
 13. Effect of Method of Placing.— The importance of the effect of the methods of manipula- 
 ting and molding laboratory specimens of mortar upon the qualities of the specimens shown 
 by tests has been discussed in Art. 2. The same factors are operative in the case of molding 
 laboratory specimens of concrete, and their disturbing effect becomes even more pronounced 
 when the work is done in the field. Experimental data are lacking which might show the extent 
 of the effect of variations in molding methods, but an indication of the importance of this fac- 
 tor is afforded by any series of tests of mortar or concrete specimens made from the same batch 
 of material and stored and tested in an identical manner. A number of such apparently iden- 
 tical specimens may perhaps vary in strength less than 5% if made by an experienced operator, 
 but a second equally-experienced operator using the same materials and the same general 
 
232 
 
 CONCRETE ENGINEERS' HANDBOOK 
 
 [Sec. 5^14 
 
 methods will often obtain results which are not within 20% of those of the first man. This 
 deviation must be attributed primarily to slight differences in detail methods of molding the 
 specimens. 
 
 When special methods of handling the materials are considered, only very scanty compara- 
 tive data are available. The following tests reported by R. E. Goodwin of the Materials 
 Testing Division of the New York Public Service Commission indicate that concrete placed in 
 mass may possess greater strength than when it is cast in molds of the size ordinarily use<l • 
 {Eng. News, Feb. 18, 1915). Several pieces of concrete were cut from existing subway struc- 
 tures at places designated before the concrete was placed in the forms. While the concrete 
 was being poured in the forms, samples from the same batches were cast in molds. A portion 
 of the specimens thus molded were stored in moist sand on the work, while others were stored 
 in the laboratory moist room. In addition similar specimens were made from the same mate- 
 rials brought from the work and mixed and molded in the laboratory. The pieces of concrete 
 taken from the work were cut from various portions of 12-in. walls at the age of 2 months 
 and after being rough-dressed were polished smooth to exact dimensions. All tests were made 
 at the age of 90 days ; the concrete was a 1 : 2 : 4 mixture ; and all specimens were 6 by 6 by 12 in. 
 
 Compressive Strength of Field and Test-sample Concrete 
 
 Conorote made on the work 
 (All in one line are from the same batch) 
 
 Concrete made in the laboratory 
 (Samples of same materials from the work used) 
 
 Specimens 
 cut from 
 12-in. wall 
 
 Specimens 
 poured in 
 molds and 
 stored on 
 the work 
 
 Specimens 
 poured in 
 
 molds aiul 
 stored in 
 
 moist room 
 
 Consistency 
 of batch 
 
 Specimens 
 
 made in 
 laboratory 
 and stored 
 on the work 
 
 Specimens 
 
 made in 
 laboratory 
 and stored 
 in moist room 
 
 Consistency 
 of batch 
 
 (4) 3,095 
 
 (4) 2,410 
 (4) 2,415 
 (4) 2,760 
 
 (4) 2,880 
 
 
 
 (2) 2,585 
 (4) 2,020 
 
 (4) 2,870 
 (4) 1,720 
 
 (3) 2,060 
 
 (4) 1,775 
 
 wet 
 very wet 
 
 wet 
 very wet 
 
 (4) 2,135 
 (4) 2,225 
 (4) 1,980 
 (4) 1,900 
 
 (4) 2,080 
 (4) 1,930 
 (4) 1,705 
 (4) 1,910 
 
 wet 
 wet 
 wet 
 wet 
 
 Average 
 (16) 2,670 
 
 Average 
 (10) 2,480 
 
 Average 
 (15) 2,110 
 
 
 Average 
 (16) 2,060 
 
 Average 
 (16) 1,910 
 
 
 (Figures in parentheses indicate number of specimens averaged for each result.) 
 
 The quality of concrete deposited under water is usually considered to be decidedly in- 
 ferior to that of concrete placed under normal conditions where water is not encountered. 
 This is doubtless true if the material is permitted to fall freely through the water, or if the cir- 
 cumstances are such that the formation of laitance is facilitated. That first quality concrete 
 can be made in subaqueous construction was shown, however, during the construction of the 
 Detroit River Tunnel. In this case concrete of 1:3:6 mix was deposited at a depth of 60 
 to 80 ft. below the water surface through 12-in. tremies. Test cores cut from the tremie- 
 deposited concrete by a 6-in. shot drill showed a compressive strength of from 2740 to 4000 lb. 
 per sq. in. 1 year after deposition. Other specimens in the shape of roughly-cut 6-in. cubes of 
 1:2:4 tremie-deposited concrete developed a strength of from 1800 to 3040 lb. per sq. in., 
 and it was believed that their strength was impaired by the operation of cutting {Trans. Am. 
 Soc. C. E., vol. 74, p. 338). The matter of the pressure under which this concrete was deposited 
 probably has some bearing upon the quality, for in this case the hydrostatic pressure at the 
 bottom of a tremie tube was about 30 lb. per sq. in. 
 
 14. Effect of Regaging. — The Joint Committee on Concrete and Reinforced Concrete 
 recommends that "the remixing of mortar or concrete that has partly set should not be per- 
 mitted" (Proc. Am. Soc. C. E.. Dec, 1916, p. 1673), and most engineers specify that mortar or 
 
Sec. 5-14] 
 
 CEMENT MORTAR AND PLAIN CONCRETE 
 
 233 
 
 concrete shall be used within 1 hr. or even ^ hr. after it is gaged. It is undoubtedly gonerallv 
 advisable on construction work to adhere to the practice of not permitting regaging and it is 
 particularly important that concrete which has stood undisturbed for some time be not per- 
 mitted to get into the form in ^ _ 
 
 GAGING UPON STRENGTH OF MoRTARS Ag 
 
 4 Months 
 Office of Public Roads Tests ^ 
 
 Effect of Regaging 
 
 its non-plastic condition, but 
 the harmful effect of regaging 
 is often less pronounced than 
 is commonly believed, and 
 exceptions to the general rule 
 may under certain circum- 
 stances properly be made. 
 
 The data on this page 
 show that the strengthof Port- 
 land-cement mortars is not in- 
 juriously affected by allowing 
 them to stand for periods of 
 from 1 to 3 hr., and then re- 
 gaging and molding. In fact 
 the delayed treatment ap- 
 pears slightly beneficial owing 
 probably to the increased 
 amount of working given the 
 material. This effect is one 
 that has been shown with singular unanimity by a considerable number of experimenters using 
 all classes of cement. One fact brought out particularly by the tests of Mr. Sabin is that the 
 material regaged should not merely be remixed, but should have sufficient water added so 
 that the original consistency will be restored after regaging. 
 
 Mix 
 
 Mortar 
 made up 
 into briquettes 
 immediately 
 after mixing 
 
 Mortar broken up 
 after initial set and 
 made into bricjuettes. 
 W^ater added to 
 restore normal 
 consistency 
 
 ^lortar l)roken up 
 after final set and 
 made into briciuettes. 
 
 restore normal 
 consistency 
 
 
 
 Tensile strength 
 
 in lb. per sq. in. 
 
 Neat 
 
 657 
 
 653 
 
 540 
 
 1:1 
 
 628 
 
 678 
 
 563 
 
 1:2 
 
 504 
 
 554 
 
 499 
 
 1:3 
 
 407 
 
 326 
 
 353 
 
 Initial set 
 
 1 hr. 42 min. 
 
 
 
 Final set 
 
 7 hr. 15 min. 
 
 
 
 Effect of Regaging upon Tensile Strength of 1 : 2 Mortar — Age 1 Year 
 
 Tests of L. C. Sabin2 
 
 Molded 
 imme- 
 diately 
 
 Stood 
 1 hr., 
 
 regaged 
 and 
 
 molded 
 
 Stood 3 hr., 
 regaged 
 
 each hour 
 and then 
 molded 
 
 stood 3 hr., 
 regaged each hour 
 with water added 
 to restore original 
 consistency and 
 then molded 
 
 Stood 5 hr., 
 regaged 
 each hour 
 and then 
 molded 
 
 Stood 5 hr., 
 reiaged each hour 
 with water added 
 to restore original 
 consistency and 
 then molded 
 
 Stood 
 5 hr., 
 regaged 
 and then 
 molded 
 
 Stood 
 
 5 hr. 
 regaged 
 and then 
 molded 
 
 No water 
 
 579 
 Water a 
 554 
 
 added 
 
 565 
 dded to 
 579 
 
 in regaging 
 
 569 
 restore ori 
 
 ginal consistency 
 627 
 
 570 
 
 624 
 
 568 
 
 560 
 
 Natural cements and quick-setting Portland cements appear to be less capable of showing 
 an undiminished strength after regaging than do normal or slow-setting Portlands. 
 
 The most pronounced effect of regaging of mortars and concretes is in the direction of 
 retarding the set and delaying the hardening, thus reducing the strength at early periods. 
 Candlot (''Ciments et Chaux Hydrauliques, " 1898, p. 358) found that mortars regaged after 
 attaining their final set all required 8 to 10 hr. to set regardless of the rapidity or slowness with 
 which the mortar originally set. This effect of regaging alone will often be a sufficient cause 
 
 1 U. S. Department of Agriculture, Bull. 235. 
 - "Cement and Concrete," p. 253. 
 
234 
 
 CONCRETE ENGINEERS' HANDBOOK 
 
 [Sec. 5-15 
 
 for prohibiting regaged mortar or concrete on construction work requiring an early assumption 
 of load. 
 
 Candlot also found that regaging had a very detrimental effect upon adhesive strength of 
 mortars to stone, the loss being often 50%. Sabin ("Cement and Concrete," p. 290) also 
 found that regaging was detrimental to adhesive strength of mortar to stone, the effect being 
 more pronounced with rich mortars. Earnest McCullough {Eng. News, Jan. 11, 1906) found 
 that regaged mortars showed a loss in power to adhere to old mortar or concrete, but found that 
 the addition of 10 to 12% of lime to the regaged mortar produced a material whose adhesive 
 strength considerably excelled that of mortar placed when freshly mixed. 
 
 15. Effect of Curing Conditions. — The principal variations in curing conditions which 
 affect the process of hardening and gaining strength of mortars and concretes are: (1) varia- 
 tions of moisture conditions, and (2) variations of temperature. 
 
 The effect of different conditions of exposure to moisture is shown by the following data 
 based upon tests made at the U. S. Bureau of Standards {Tech. Paper 58). The test specimens 
 were 8 by 16-in. cylinders.. 
 
 Effect of Variation of Moisture Condition in Curing Period 
 Tests of Bureau of Standards^ 
 
 Alix, class of concrete, and curing conclitions 
 
 Compressive strength (lb. per sq. in.) 
 
 1 week 
 
 4 weeks 
 
 13 weeks 
 
 26 weeks 
 
 52 weeks 
 
 1 : 6 gravel (quaking) : In damp closet entire period. 
 
 
 1,898 
 
 1,968 
 
 2,172 
 
 2,400 
 
 1:6 gravel (quaking): 4 weeks in damp closet, 
 then removed. 
 
 
 1,648 
 
 1,825 
 
 2,063 
 
 2,220 
 
 1:2:4 trap rock (quaking) : Immersed immedi- 
 ately after molding. 
 
 
 2,851 
 
 .3,570 + 
 
 4,094 + 
 
 3,956 
 4,247 + 
 
 
 
 3,978 + 
 
 3,978 + 
 
 4,100 
 
 8 weeks in damp closet, then immersed 
 
 
 
 3,190 
 
 3,457 
 
 3,389 
 
 1:2:4 gravel (mushy) : Sprinkled daily for 1 week, 
 then stored indoors in dry room. 
 
 481 
 
 1,104 
 
 1,469 
 
 
 4 weeks in damp room, then placed in open, exposed 
 to weather. 
 
 
 1,834 
 
 2,500 
 
 
 1:2:4 gravel (quaking) : In damp closet entire period 
 
 2,612 
 2,085 
 
 
 
 
 These tests indicate: (1) that concrete specimens cured in the moist air of the damp closet 
 until tested become somewhat stronger than ones immersed in water after a longer or shorter 
 period in the damp closet, or ones immersed immediately after molding. (This may be due in 
 part at least to the fact that the immersed specimens were tested while still holding much more 
 water than the damp closet specimens.) (2) Specimens cured in the damp closet are con- 
 siderably stronger than ones cured in the comparatively dry air of the laboratory, even though 
 
 1 Tech. Paper 58. 
 
Sec. 5^15] 
 
 CEMENT MORTAR AND PLAIN CONCRETE 
 
 235 
 
 the latter were sprinkled daily for the first week. (3) Specimens gained strength in the open 
 air exposed to the weather to a considerably greater extent than did others cured indoors in a 
 comparatively dry room, but not to as great an extent as ones stored in the damp closet. 
 
 The relation between the mean temperature encountered during the curing period and the 
 strength of 1:2:4 concrete is shown by the diagram of Fig. 15. This diagram was con- 
 structed by A. B. Mc Daniel as a summary of 
 the results of tests made at the Engineering Ex- 
 periment Station of the University of Illinois {Bull. 
 81). The concrete used was a 1:2:4 mixture by 
 weight (1:2.2:3.6 by volume), the coarse aggre- 
 gate being crushed limestone. A portion of the 
 specimens were 6 by 6-in. cylinders, some were 
 6-in. cubes and the balance were 8 by 16-in. 
 cylinders. All values were reduced to an equiva- 
 lent value for 8 by 16-in. cylinders, however. 
 Ten sets of specimens were made, and each set was 
 subjected to a different mean temperature through- 
 out the period of curing. The mean temperatures 
 employed were 26.5°, 27.1°, 30°, 34.7°, 35.5°, 
 48.5°, 68°, 71.8°, 72.8°, and 95.6° respectively. 
 The marked effect of low temperatures in at least 
 delaying, if not permanently preventing, the 
 hardening process is excellently shown by the 
 
 diagram of Fig. 15. The diagram represents the results obtained with only one mixture of 
 one class of materials, but the relative effect of various temperatures on other concretes 
 may be expected to at least approximate the relation found for this concrete, and the dia- 
 gram should furnish sugges- 
 
 Effect of Curing 1 : 4 Mortar under Steam Pressure tive information useful in es- 
 (8 X 16-in. cylinders) 
 Tests of Bureau of Standards^ 
 
 20 ZZ 24 26 28 
 
 Fia. 15. — Effect of temperature of curing upon 
 compressive strength of 1:2:4 concrete. (The 
 temperatures given are the mean temperatures 
 encountered during the period of curing.) 
 
 timating the strength of con- 
 crete cured at abnormal tem- 
 peratures. 
 
 Certain classes of con- 
 crete products such as tiles 
 and blocks are subjected to 
 an accelerated hardening 
 treatment by the use of steam. 
 The effect of such treatment 
 upon strength is shown by 
 data given on this page from 
 tests of the U. S. Bureau of 
 Standards {Tech. Paper 58). 
 
 The results show that up 
 to 80 lb. per sq. in. gage pres- 
 sure, steam has an accelerat- 
 ing action upon the harden- 
 ing of cement mortar, and 
 
 that the compressive strength increases with the pressure as well as with the time of exposure 
 to steam. A compressive strength considerably (in some cases over 100%) in excess of that 
 obtained normally after aging for 6 weeks may be obtained in 2 days by using steam under 
 pressure for curing. Furthermore, the steam permanently accelerates the hardening of the 
 
 Gage 
 pressure 
 (lb. per 
 sq. in.) 
 
 Tempera- 
 ture 
 (Fahr.) 
 
 Duration 
 of exposure 
 to steam 
 (hours) 
 
 Compressive strength 
 
 2 days 
 
 7 days 
 
 14 days 
 
 28 days 
 
 Not 
 
 
 
 613 
 
 1,296 
 
 1,528 
 
 1,727 
 
 steamed. 
 
 
 
 
 
 
 
 0 
 
 212 
 
 48 
 
 1,267 
 
 
 
 
 2 
 
 218 
 
 24 
 
 1,808 
 
 1,792 
 
 1,805 
 
 
 10 
 
 239 
 
 24 
 
 1,786 
 
 1,555 
 
 1,701 
 
 1,902 
 
 20 
 
 258 
 
 24 
 
 2,139 
 
 2,284 
 
 2,740 
 
 
 40 
 
 286 
 
 24 
 
 3,292 
 
 3,381 
 
 3,984 
 
 
 80 
 
 323 
 
 24 
 
 3,964 
 
 3,966 
 
 4,433 
 
 
 80 
 
 323 
 
 24 
 
 4,487 
 
 4,187 
 
 4,840 
 
 
 1 Tech. Paper 58. 
 
236 
 
 CONCRETE ENGINEERS' HANDBOOK 
 
 [Sec. 6-16 
 
 concrete which subsequently increases in compressive strength with age upon exposure to the 
 atmosphere. 
 
 It was noted that the steam-cured concrete was more uniform in appearance and lighter 
 in color than normally-aged mortar from the same materials. These tests were made upon 
 Portland-cement mortars, but the same conclusions were found to apply to concretes. The 
 mortar or concrete should obtain an initial set before exposure to the steam treatment. 
 
 16. Effect of Freezing. — The effect of low temperatures in delaying or permanently pre- 
 venting the hardening of mortar and concrete has been shown by Fig. 15. In the event of the 
 temperature being close to the freezing point of water from 4 to 8 times as long a period 
 is required to obtain a final set as is required at normal room temperatures. If water in mortar 
 or concrete freezes before the cement has set, it is not available for the chemical action of setting 
 and hardening and hence the concrete or mortar will not set at all until the ice melts. These 
 facts must be borne in mind when removing forms from concrete placed during cold weather. 
 If the temperature hovers above the freezing point for some time after concrete is deposited, 
 there is a possibility of the water drying out before the greatly delayed setting has taken place. 
 If, however, the concrete has begun to set before the temperature drops considerably below the 
 freezing point, the expansion of the water in solidifying produces an expansive force in excess 
 of the cohesive strength of the green concrete. This action results in a destruction of the bond 
 and crumbling of the concrete when the ice melts. If the temperature does not fall more than 
 a very few degrees below freezing, the result may simply be the further delaying of the set 
 without appreciable injury. 
 
 Two factors operate to lessen the injurious effect of freezing upon mortar and concrete: 
 (1) concrete is a rather poor heat conductor, the outer portion therefore serving as an insulation 
 for the bulk of the material and preventing an injurious lowering of the temperature in the 
 interior of th^ mass; and (2) the chemical action of setting and hardening of cement being an 
 endothermic reaction, the heat evolved serves to raise the temperature of the material and 
 so offsets to a degree the loss of heat by radiation and conduction. Experimental data secured 
 during the construction of the Arrow Rock Dam and the Kensico Dam (see Art. 42) indicate 
 that mass concrete shows a rise in internal temperature of from 20 to 40°F. above the initial 
 temperature within a period of from 15 to 30 days. 
 
 Serious injury is oftenr suffered by concrete which encounters temperatures considerably 
 below the freezing point within the first few hours after placing, but this injury is usually con- 
 fined to the outermost portion of the work and seldom penetrates more than an inch or two of 
 depth. A very frequent form of injury is a scaling off of a very thin crust of rich material 
 which has been flushed to the surface in finishing the work. 
 
 Numerous experimental studies of the effect of frost action on mortars have been made and 
 have led to somewhat conflicting conclusions, but practically all of these have involved the uSe 
 of such small specimens, briquettes, 2-in. cubes, etc., that the condition of exposure is compar- 
 able only to that of the outermost surface layer of concrete. 
 
 Authorities are quite in accord in prescribing that "concrete should not be mixed or de- 
 posited at a freezing temperature, unless special precautions are taken to avoid the use of 
 materials containing frost or covered with ice crystals, and to provide means to prevent the 
 concrete from freezing after being placed in position and until it has thoroughly hardened." 
 
 17. Effect of Salts. — Common salt (NaCl) is frequently used as an ingredient of the mixes 
 of concrete or mortar which must be placed in cold weather. Its primary effect is the lowering 
 of the temperature at which water will freeze. Approximately 1 % of salt in the mixing water 
 lowers the freezing point 1°F. Calcium chloride (CaCla) is also used to a lesser extent to serve 
 the same purpose. 
 
 The effect of common salt and calcium chloride upon the strength of a 1 : 2 : 4 limestone 
 concrete is shown by the diagrams of Fig. 16 which are derived from tests made by H. E. 
 Pulver and S. E. Johnson of the University of Wisconsin {The Wisconsin Engineer, October, 
 1913). The specimens were 4-in. cubes, and the tests were made in duplicate, one series of 
 
Sec. 6-17] 
 
 CEMENT MORTAR AND PLAIN CONCRETE 
 
 237 
 
 specimens being cured indoors at normal temperatures (60°-75°F.), the other cured out of 
 doors or in a refrigerator at temperatures below 32°F. The amounts of the salts used are 
 expressed as percentages by weight of the mixing water. 
 
 The tests show that common salt used alone is quite injurious to the strength of concrete 
 cured at normal temperatures, the loss being roughly proportional to the amount of salt used. 
 With concrete cured at temperatures below freezing, however, it facilitates the hardening 
 process. The tests show an increase of strength for the addition of NaCl up to 12%, after 
 which there is a decrease. Common salt retards the setting of concrete to a considerable degree. 
 
 Calcium chloride used alone is beneficial to the strength of concrete, whether cured at 
 normal temperatures or below freezing, up to about 4 % CaClj, at which point the maximum 
 
 u 
 c 
 
 t 
 5,1 
 
 m 
 in 
 
 I 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 K 
 
 
 - 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 CD 
 
 1000 
 
 3 0 y \c 
 Perceni-age NaCl 
 
 o 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 li: 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 Pr 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 12 Wk 
 
 
 
 
 6<7oCaClE 1 
 
 cr 
 in 
 v. 
 
 3000 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 r 
 
 
 s 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 '1 
 
 
 
 
 
 
 
 
 
 — < 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 h 
 
 
 '2 1 
 
 
 
 
 1 6«7oCaC 
 
 0 3 
 
 ^ ^ 9 12 
 Percentage NaCI 
 
 0 3 
 
 Fig. 16. — Effect of salts upon strength of 1 : 2 : 4 concrete. 
 
 specimens cured at normal temperatures (60 to 75° F. 
 
 specimens cured at low temperatures (below 32° F.). 
 
 . 6 9 12 
 
 Percentage NaCf 
 
 15 
 
 strength is obtained. This maximum strength, however, in the case of the cold-cured specimens 
 is only about one-half the maximum strength of the cold-cured specimens having 12% of com- 
 mon salt. Calcium chloride accelerates the setting of concrete. 
 
 With concretes cured at low temperatures the best effect was obtained with a mixture of 
 2% of CaClz and 9% of NaCl. This mixture gave about the same strength as the cold-cured 
 concrete having 12%, of NaCl alone, and was not as detrimental to the strength of normally 
 cured concrete as the NaCl alone seemed to be. 
 
238 
 
 CONCRETE ENGINEERS' HANDBOOK 
 
 [Sec. 5-18 
 
 18. Effect of Hydrated Lime and Waterproofing Compounds. — The addition of hydrated 
 lime in small percentages has not a very marked effect upon the strength of laboratory specimens 
 of mortar and concrete. Fig. 17, which is based upon tests made by Prof. Harry Gardner 
 of the University of Kansas (Eng. Rec, vol. 64, p. 309), shows the effect of varying percentages 
 by weight Of hydrated lime upon the tensile strength of 1 :3 standard sand mortars. The 
 replacement of the cement by hydrated lime appears to be slightly beneficial to strength up to 
 about 15%, except in the case of tests made at the ages of 3 and 7 days wherein the effect was 
 generally detrimental in proportion to the amount of lime used. Other tests, notably those of 
 H. S. Spackman {Concrete-Cement Age, vol. 4, p. 112 and Eng. Rec, vol. 69, p. 25) have shown 
 that small amounts of hydrated lime sometimes appear to affect the strength favorably, some- 
 times unfavorably. The most pronounced effect of hydrated lime added to mortars and con- 
 cretes is its producing a more plastic, better-working material. The fat, viscous mortar 
 produced spreads better under the trowel, and in the case of concrete the presence of a small 
 amount of lime hydrate tends toward the production of a mixture of greater uniformity by 
 
 prevention of the separation of fine and coarse 
 materials. This fact may constitute a distinct ad- 
 vantage in the case of the average construction 
 job which would not be noted under the ideal 
 conditions of mixing and molding in the laboratory. 
 
 The effect of a large number of commercial 
 waterproofing compounds upon the tensile and 
 compressive strengths of mortars in 1 : 4, 1 : 6, and 
 1 : 8 mixtures has been investigated b^'- the U. S. 
 Bureau of Standards {Tech. Paper 3). Figs. 18o 
 and 186 show the results of tests of the effect of 15. 
 such compounds upon the compressive strength of 
 1 : 4 mortar. These results are typical of the re- 
 sults obtained with all mortar mixtures, and the 
 results obtained in tensile tests do not differ 
 greatly from those obtained in compression. The 
 proprietary compounds are, of course, not identi- 
 fied by name, but they are classified as follows: 
 No. 27 is dolomitic hydrated lime; No. 28 is a 
 solid chemically active filler designed to form an 
 insoluble lime resinate void filler; Nos. 29 to 36 
 inclusive are water-repelling solid substances con- 
 sisting essentially of stearic acid with soda and potash or lime, designed to form an insoluble 
 lime soap; No. 37 is cement containing a water-repelling substance; Nos. 38 to 40 inclusive 
 are chemically active liquid fillers designed to fill the voids with either tar, insoluble lime sili- 
 cate, etc. 
 
 19. Effect of Sea Water Used in Gaging. — The use of sea water to gage cement mortars 
 and concretes is almost invariably forbidden by specifications for work done in localities where 
 the use of sea water might be convenient. It has not been conclusively shown, however, that 
 the use of sea water instead of fresh water has a particularly harmful effect. Messrs. Taylor 
 and Thompson (''Concrete, Plain and Reinforced," 1916, p. 166) found by a very Hmited number 
 of tests of 1 :2 : 4 concrete cubes that there was no appreciable difference in strength of specimens 
 gaged with sea water and other specimens gaged with fresh water. 
 
 Results obtained in comparative tensile tests of mortar briquettes made by Cloyd M. 
 Chapman are shown by Figs. 19,a, 19,& and 19,c {Eng. News, vol. 63, p. 291). These tests 
 were made upon three sets of specimens : series (A) specimens were gaged with fresh water and 
 cured in fresh water; series (B) specimens were gaged with fresh water and cured in sea water; 
 and series (C) specimens were gaged with sea water and cured in sea water. The specimens 
 
 Fig. 
 
 Percen+age hydra+ed lime 
 00 95 90 85 60 75 10 
 
 Percentage cement 
 
 17. — Effect of hydrated lime upon tensile 
 strength of 1 : 3 standard mortar. 
 
Sec. 5-19] 
 
 CEMENT MORTAR AND PLAIN CONCRETE 
 
 239 
 
 were tensile briquettes and the sea-water curing was done in tanks in the laboratory using a 
 frequently changed bath of sea water. These tests indicate that the use of sea water for gaging 
 
 g-3500 
 S.3000 
 S 2500 
 
 S 1000 
 
 I 
 
 € 500 
 
 s 
 
 
 
 
 
 
 
 /,' 
 
 
 
 
 
 iy^y.^ ■ 
 
 --^rr-^ 
 
 
 
 
 
 
 
 
 
 
 
 
 
 )3 26 ^ 52. 
 
 Age of test pieces in weeks 
 
 350Q 
 3000 
 2500 
 2000 
 1500 
 1000 
 500 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 1 Z 4 
 
 52 
 
 13 26 
 
 Age of test pieces in weeks 
 & 
 
 Fig. 18. — Compressive strength of waterproofed mortars. (One part Portland cement to 4 parts No. 1 sand.) 
 
 Age in mon-t-hs 
 
 Age in months 
 
 Age in months 
 
 o 6 c . 
 
 Fig. 19.— Effect upon tensile strength of gaging cement mortars with sea water, a, Specimens gaged with 
 fresh water — cured in fresh water: 6, specimens gaged with fresh water— cured in sea water; c, specimens gageu 
 with sea water — cured in sea water. 
 
 is not particularly detrimental to the tensile strength of mortars except in very lean mixtures. 
 The condition of curing of these specimens was probably not as severe a test as actual immer- 
 
240 
 
 CONCRETE ENGINEERS' HANDBOOK 
 
 [Sec. 6-20 
 
 sion in moving sea water would have been. On the other hand, the section of the specimens 
 was so small that any superficial or surface effect of the sea water would appear to have an 
 injurious effect much greater than that suffered by concrete of large bulk exposed in sea water. 
 
 20. Effect of Oils Used in Gaging.— The use of certain classes of mineral residual oils in 
 gaging mortars and concretes with the object of dampproofing them or reducing their permea- 
 
 0 5 10 15 20 E5 0 5 10 15 20 Z5 
 
 Perc,en+age of oil based on weigh+ Percentage of oil based on weight 
 
 of cement of cement 
 
 Fio. 20. — Effect of mineral residual oils upon compressive strength of 1 : 3 mortar. 
 • (Tests of Logan Waller Page.) 
 
 bility lends importance to the consideration of the effect of such oils upon the strength of the 
 mortars or concretes in which they are used. Figs. 20 and 21 present the results of tests 
 made by Logan Waller Page, Director of the Office of Public Roads (U. S. Dept. of Agriculture, 
 Bull. 230). It appears from these tests that the use of mineral oils up to from 5 to 10% of 
 
 3000 
 ^2500 
 i. EOOO, 
 
 C 2000 
 SI 1500 
 
 -t= 500 
 
 r 
 
 
 
 
 1 ■ 2 4 concrete 
 
 
 
 
 
 
 
 
 
 
 
 / 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 1.3.5 concrete 
 ^ — 1 
 
 
 
 
 
 
 
 
 
 
 
 
 r lY 
 
 
 
 
 lYr- 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 1 ; 3 .'6 concrete 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 1 
 
 
 8 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 3 5 10 15 20 25 
 
 f^rcentage of oil based on weight 
 of cement 
 
 -?000 
 1500 
 1000 
 500 
 
 2000 
 1500 
 1000 
 ■ 500 
 
 
 
 
 M'.3;5 concreto 
 
 
 
 
 L 
 
 
 
 
 
 
 
 
 
 lYr"!] 
 
 
 
 
 
 
 
 ia. 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 L 3 :6 con 
 water c 
 
 Crete 
 jred _ 
 
 
 
 
 Yr 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 0 5 10 \5 20 25 
 
 Percentage of oil based on weight 
 of cemenf 
 
 Fig. 21. — Effect of mineral residual oils upon 
 
 compressive strength of concrete. (Tests of Logan Waller Page.) 
 
 the cement lowers the compressive strength to a moderate degree only but that larger amounts 
 may be very injurious. 
 
 Other tests made by Arthur Taylor and Thomas Sanborn {Trans. Am. Soc. C. E., vol. 
 76, p. 1094) using Western asphaltic oils showed a more marked falling off in strength of mortars 
 
-Sec. &-21] 
 
 CEMENT MORTAR AND PLAIN CONCRETE 
 
 241 
 
 than was observed in Page's tests. At 28 and 50 days the compressive strengths of 3-in. cubes 
 of 1 : 3 mortar made by the incorporation of Western oils were as follows : 
 Many oils used for various 
 
 Effect of Asphaltic Oils upon Strength of 1 : 3 
 Mortar 
 Compressive Strength 
 
 Character and percentage 
 of oil used 
 
 28 days 
 
 50 days 
 
 (lb. per 
 sq. in.) 
 
 Relative 
 value 
 
 (lb. per 
 sq. in.) 
 
 Relative 
 value 
 
 No oil 
 
 
 3,950 
 
 100.0 
 
 4,400 
 
 100.0 
 
 Boiler fuel 
 
 5 
 
 2,435 
 
 61.6 
 
 3,620 
 
 82.4 
 
 Boiler fuel 
 
 10 
 
 1,780 
 
 45.1 
 
 2,460 
 
 56.0 
 
 Boiler fuel 
 
 15 
 
 1,460 
 
 37.0 
 
 2,000 
 
 45.5 
 
 Boiler fuel 
 
 25 
 
 712 
 
 18.2 
 
 1,000 
 
 22.8 
 
 Richmond fuel 
 
 10 
 
 1,640 
 
 41.5 
 
 
 
 Road oil No. 6 
 
 10 
 
 1,235 
 
 31.2 
 
 
 
 Liquid asphalt 
 
 10 
 
 1,080 
 
 27.4 
 
 
 
 commercial purposes contain animal 
 oils, vegetable oils or admixtures 
 of these. Such oils have been found 
 to be capable of not only weaken- 
 ing cement mortars and concretes, 
 but actually to disintegrate con- 
 crete in some cases, the effect being 
 most pronounced in the early stages 
 of setting and hardening (see tests 
 of James C. Hain, Eng. News, 
 March 16, 1905). 
 
 21. Effect of Laitance. — Lai- 
 tance is a whitish substance which 
 is washed out of concrete and sub- 
 sequently deposited as a scum when 
 there is an excess of water used in 
 mixing (see chapter on "Water" in 
 Sect. 1), or when concrete is depos- 
 ited in water, or when water collects 
 
 in pools on the surface of freshly laid concrete. The laitance consists of the finest flocculent 
 matter in the cement together with some silt and clay from the aggregates. Its occurrence 
 is explained by the formation of amorphous hydrates in the early stages of the setting of cement. 
 The composition of laitance is practically identical with that of cement, but it hardens only 
 very slowly and never acquires much strength. As a consequence, if not removed by water 
 
 and brushes or by a steam jet, it forms a distinct 
 plane of weakness between successive layers of 
 concrete. The washing out of a portion of the 
 finest part of the cement means the loss by the 
 concrete of just so much of its most valuable 
 constituent, because it is the impalpably fine 
 portion of the cement which is most active in 
 binding together the inert particles of the aggre- 
 gate. This same material alone does not develop 
 great cohesiveness, however. A familiar example 
 of this fact is afforded by the relative behavior 
 of very finely ground cements and cements of 
 only average fineness in neat and mortar mixtures. 
 The cement of average fineness will greatly excel 
 7-28 7-90 7-180 7^60 in neat strength, but the cement which is ground 
 
 Age interval in days so finely that the proportion of impalpably fine 
 
 Fig. 22. — Rate of increase in tensile strength of material is very large will form mortars of greatly 
 
 standard 1 : 3 mortar. (Each result the average of 9 . , , i 
 
 to 70 tests.) superior strength. 
 
 22. Rate of Increase in Mortar Strength 
 —Retrogression.— The relation between the early test strength and the subsequent gain in 
 strength is shown by Figs. 22 and 23 which are based upon the tests of the former Structural 
 Materials Laboratory of the U. S. Geological Survey in St. Louis (U. S. Geol. Surv. Bull. 
 331). The rate of gain in both tensile and compressive strength for these 1 : 3 mortars (made 
 with seven typical Portland cements) is shown to be approximately inversely proportional at 
 
 16 
 
242 
 
 CONCRETE ENGINEERS' HANDBOOK 
 
 [Sec. 6-23 
 
 all ages to the strength at 7 days, those mortars which show the lowest tensile or compressive 
 strength at 7 days maintaining the best rate of gain in strength at all ages. 
 
 500 
 
 0 i 1 I I i I I I I I > I ' I I I I I I I 
 7-Za 7-90 7-180 7-3&0 
 
 Age jn+erval in days 
 
 Fig. 23. — Rate of increase in compressive strength of standard 1 : 3 mortar. 
 (Each result the average of 20 to 70 tests.) 
 
 The following table indicates the relation between the early strength of mortar and sub- 
 sequent retrogression in strength as determined by the Structural Materials Laboratory Tests 
 above quoted : 
 
 Retrogression in Strength of 1:3 Standard Mortars 
 
 Tests of Structural Materials Laboratory 
 
 Strength at 7 days 
 (lb. per sq. in.) 
 
 % showing retrogression between 
 ages of 
 
 7 and 28 
 
 days 
 
 28 and 
 
 90 days 
 
 90 and 
 180 days 
 
 180 and 
 
 360 days 
 
 
 
 Tension 
 
 
 Below 200 
 
 0 
 
 0 
 
 86 
 
 86 
 
 200- 250 
 
 0 
 
 0 * 
 
 62 
 
 71 
 
 250- 300 
 
 0 
 
 0 
 
 48 
 
 100 
 
 300- 350 
 
 0 
 
 0 
 
 57 
 
 100 
 
 
 
 Compression 
 
 
 Below 800 
 
 0 
 
 0 
 
 0 
 
 20 
 
 800- 900 
 
 0 
 
 0 
 
 0 
 
 14 
 
 900-1,000 
 
 0 
 
 0 
 
 8 
 
 25 
 
 1,000-1,100 
 
 0 
 
 0 
 
 0 
 
 12 
 
 1,100-1,200 
 
 0 
 
 0 
 
 0 
 
 20 
 
 1,200-1,500 
 
 0 
 
 0 
 
 22 
 
 0 
 
 Above 1,500 
 
 0 
 
 40 
 
 40 
 
 20 
 
 23. Transverse Strength. — The transverse strength of granular brittle materials Hke 
 mortars and concretes is best expressed by the Modulus of Rupture. The modulus of rupture 
 
Sec. 5-24] 
 
 CEMENT MORTAR AND PLAIN CONCRETE 
 
 243 
 
 is the apparent stress in the extreme fiber of a transverse test specimen under the load which 
 produces rupture. For specimens of rectangular section of breadth b and height h, loaded 
 centrally on a span I, the breaking load being W, the modulus of rupture is computed by the 
 formula 
 
 Modulus of rupture = r 
 
 The extreme fiber stress thus computed is not the actual fiber stress because the formula 
 involves the inaccurate assumption that the material deforms elastically for all stresses up to 
 rupture. The comparative relations between results are not affected by this inaccuracy of 
 •the formula, however, when the tests compared are made upon specimens of similar material, 
 because the computed values of the modulus of rupture are very nearly proportional to the 
 actual stresses. 
 
 Since the extreme fiber stresses on the tension side and on the compression side of a beam 
 of homogeneous material are equal, and the tensile strength of mortar or concrete is only a small 
 fraction of the compressive strength, the transverse strength of mortar or concrete is almost 
 wholly dependent upon the tensile strength. The modulus of rupture found in transverse 
 tests will invariably be considerably in excess of the tensile strength, however, because the 
 computed stress in the extreme fiber considerably exceeds the actual stress. 
 
 The data on this page constitute a summary of a portion of an extensive series of tests of 
 transverse strength of mortars and concretes made by Wm. B. Fuller. (" Concrete, Plain and 
 Reinforced" by Taylor and Thompson, p. 334.) The tests were made upon specimens 6 by 6 in. 
 
 in section, supported on spans of 30 and 60 in. 
 crushed trap-rock aggregate were used 
 throughout the series of tests. The speci- 
 mens were broken at the age of 1 month. 
 
 24. Shearing Strength.— The shear- 
 ing strength of mortars and concretes 
 possesses great significance because com- 
 pressive failure of short compressive speci- 
 mens or structural members is usually 
 failure by shearing on a diagonal plane, 
 and because shearing stresses are impor- 
 tant considerations in all cases of concrete 
 beams. It is very difficult, however, to 
 make experimental determinations of 
 pure shearing strength because most 
 methods and devices which may be used 
 to make shearing tests involve either a 
 cutting action, bearing pressures, or beam 
 stresses. The data on the shearing strength 
 of mortars given on page 244 are derived 
 from tests made by Feret. The speci- 
 mens used were prisms, 2 by 2 cm. in sec- 
 tion, subjected to single shear, the condi- 
 tions being such that beam stresses prob- 
 ably affected the results considerably. 
 Specimens were tested after 5 months 
 curing. 
 
 One brand of cement and the same sand and 
 
 Transverse Strength of Mortars and 
 Concretes 
 Tests of William B. Fuller 
 
 Proportions 
 by weight, 
 
 
 Proportions 
 bv volume. 
 
 Modulus of rupture 
 (lb. per sq. in.) 
 
 cement: sand: 
 stone 
 
 cement: sand: 
 stone 
 
 Maxi- 
 mum 
 
 Mini- 
 mum 
 
 Aver- 
 age of G 
 
 1 
 
 0:0 
 
 1 
 
 0:0 
 
 968 
 
 850 
 
 906 
 
 1 
 
 1:0 
 
 1 
 
 1.17:0 
 
 866 
 
 628 
 
 734 
 
 1 
 
 2:0 
 
 1 
 
 2.34:0 
 
 640 
 
 592 
 
 616 
 
 1 
 
 3:0 
 
 1 
 
 3.51:0 
 
 432 
 
 392 
 
 418 
 
 1 
 
 4:0 
 
 1 
 
 4.68:0 
 
 294 
 
 262 
 
 279 
 
 1 
 
 5:0 
 
 1 
 
 5.85:0 
 
 180 
 
 170 
 
 173 
 
 1 
 
 6:0 
 
 1 
 
 7.02:0 
 
 94 
 
 92 
 
 93 
 
 1 
 
 1:2 
 
 1 
 
 1.17: 2.06 
 
 798 
 
 646 
 
 710 
 
 1 
 
 1:3 
 
 1 
 
 1.17: 3.09 
 
 732 
 
 573 
 
 655 
 
 1 
 
 2:4 
 
 1 
 
 2.34: 4.12 
 
 480 
 
 399 
 
 439 
 
 1 
 
 2:5 
 
 1 
 
 2.34: 5.17 
 
 413 
 
 349 
 
 380 
 
 1 
 
 3:5 
 
 1 
 
 3.51: 5.17 
 
 308 
 
 262 
 
 285 
 
 1 
 
 3:6 
 
 1 
 
 3.51: 6.21 
 
 246 
 
 213 
 
 228 
 
 1 
 
 4:8 
 
 1 
 
 4.68: 8.25 
 
 158 
 
 156 
 
 157 
 
 1 
 
 6:10 
 
 1 
 
 7.02:10.34 
 
 91 
 
 87 
 
 89 
 
244 
 
 CONCRETE ENGINEERS' HANDBOOK 
 
 [Sec. 5-24 
 
 Shearing Strength of Cement Mortars 
 Tests of R. Fereti 
 
 C haracter of sand 
 
 Approximate pro- 
 portions by weight 
 
 Ultimate strength (lb. per sq. in.) 
 
 Ratio of 
 shear to 
 compres- 
 
 Cement 
 
 Sand 
 
 Shear 
 
 Tension 
 
 Compres- 
 
 Very coarse granite sand 
 
 1 
 
 18.6 
 
 170 
 
 69 
 
 240 
 
 0.71 
 
 
 1 
 
 9.9 
 
 570 
 
 146 
 
 870 
 
 0.66 
 
 
 
 6.9 
 
 1,070 
 
 212 
 
 1,540 
 
 0.70 
 
 
 
 5.2 
 
 1,440 
 
 258 
 
 2,350 
 
 0.61 
 
 
 
 4.1 
 
 2,000 
 
 314 
 
 3,320 
 
 0.60 
 
 
 
 3.2 
 
 2,560 
 
 367 
 
 4,170 
 
 0.61 
 
 
 1 
 
 2.5 
 
 2,790 
 
 421 
 
 5,210 
 
 0.54 
 
 
 
 1.8 
 
 3,580 
 
 480 
 
 5,970 
 
 0.60 
 
 
 1 
 
 1.2 
 
 3,930 
 
 537 
 
 6,670 
 
 0.59 
 
 
 
 0.7 
 
 3,640 
 
 563 
 
 6,810 
 
 0.65 
 
 Medium-sized very shelly sand. . 
 
 1 
 
 12.9 
 
 256 
 
 81 
 
 310 
 
 0.83 
 
 
 1 
 
 7.0 
 
 . 669 
 
 182 
 
 950 
 
 0.70 
 
 
 1 
 
 5.0 
 
 1,040 
 
 240 
 
 1,510 
 
 0.69 
 
 
 
 4.1 
 
 1,350 
 
 278 
 
 1,990 
 
 0.68 
 
 
 
 3.1 
 
 1,810 
 
 320 
 
 2,720 
 
 0.67 
 
 
 
 2.5 
 
 2,250 
 
 368 
 
 3,430 
 
 0.66 
 
 
 
 2.0 
 
 2,650 
 
 415 
 
 4,380 
 
 0.61 
 
 
 
 1.4 
 
 2,750 
 
 521 
 
 5,440 
 
 0.50 
 
 
 
 0.9 
 
 3,580 
 
 541 
 
 6,100 
 
 0.59 
 
 
 1 
 
 0.5 
 
 3,540 
 
 602 
 
 6,720 
 
 0.53 
 
 Very fine silicious sand 
 
 1 
 
 12.3 
 
 156 
 
 67 
 
 160 
 
 0.97 
 
 
 1 
 
 5.8 
 
 370 
 
 126 
 
 540 
 
 0.69 
 
 
 
 3.5 
 
 768 
 
 214 
 
 1,230 
 
 0.62 
 
 
 
 2.4 
 
 1,410 
 
 302 
 
 1,940 
 
 0.73 
 
 
 
 1 8 
 
 9 1 
 
 oD'± 
 
 
 0.75 
 
 
 I 
 
 1.3 
 
 2,570 
 
 436 
 
 3,710 
 
 0.69 
 
 
 
 1.0 
 
 2,750 
 
 510 
 
 5,000 
 
 0.55 
 
 
 \ 
 
 0.7 
 
 3,070 
 
 574 
 
 5,760 
 
 0.53 
 
 
 
 0.5 
 
 3,570 
 
 647 
 
 6,500 
 
 0.55 
 
 
 
 0.3 
 
 4,120 
 
 691 
 
 7,110 
 
 0.58 
 
 Equal parts of coarse medium and 
 
 
 5.0 
 
 1,720 
 
 328 
 
 2,350 
 
 0.73 
 
 fine ground quartzite 
 
 
 3.0 
 
 3,100 
 
 450 
 
 4,010 
 
 0.77 
 
 
 
 2.0 
 
 3,070 
 
 518 
 
 4,810 
 
 0.64 
 
 20-31-mesh ground quartzite. . . 
 
 
 3.0 
 
 
 456 
 
 3,640 
 
 
 Neat Portland cement 
 
 
 0.0 
 
 3,680 
 
 698 
 
 8,040 
 
 0.46 
 
 The data at the top of page 245, showing the results of shearing tests, are derived from tests 
 made at the Massachusetts Institute of Technology under the direction of Prof. C. M. Spofford. 
 The specimens were cylinders 5 in. in diameter and 153^^ in. long. The ends were securely 
 
 1 Data taken from "Concrete, Plain and Reinforced" by Taylor and Thompson, p. 136, 1909 Edition. 
 
I Sec. 6-24] CEMENT MORTAR AND PLAIN CONCRETE 245 
 
 clamped in cylindrical bearings and the load was applied along the middle third of the length 
 by a semi-cylindrical block. The final failure appeared to be by true shear. 
 
 The following data are 
 taken from tests made at Shearing Strength of Concrete 
 
 the University of Illinois Summary of Massachusetts Institute of Technology Tests i 
 
 Engineering Experiment 
 Station under the direction 
 of Prof. A. N. Talbot {Bull 
 8). Two methods of testing 
 were used. In the first a 
 6-in. hole was punched in a 
 concrete plate or block; in 
 the second, a short beam 4 
 by 4 in. in cross-section was 
 securely clamped at the ends 
 and the middle third of the 
 d length loaded. Three forms 
 jj of specimens were used in 
 the punching tests: (1) plain 
 concrete plate; (2) recessed 
 
 Proportions 
 
 Method 
 of 
 
 Shearing strength 
 (lb. per sq. in.) 
 
 Compressive 
 strength 
 (lb. per sq. in.) 
 
 Ratio of 
 shear to 
 
 storing 
 
 Max. 
 
 Min. 
 
 Ave. 
 
 5 by 15^^2-in. 
 cylinders 
 
 compres- 
 sion 
 
 1:2:4 
 
 air 
 
 1,630 
 
 960 
 
 1,310 
 
 2,070 
 
 0.63 
 
 1:2:4 
 
 water 
 
 2,090 
 
 1,180 
 
 1,650 
 
 2,620 
 
 0.63 
 
 1:3:5 
 
 air 
 
 1,590 
 
 890 
 
 1,240 
 
 1,310 
 
 0.94 
 
 1:3:5 
 
 water 
 
 1,380 
 
 840 
 
 1,120 
 
 1,360 
 
 0.32 
 
 1:3:6 
 
 air 
 
 1,450 
 
 950 
 
 1,180 
 
 950 
 
 1.25 
 
 1:3:6 
 
 water 
 
 1,200 
 
 1,030 
 
 1,120 
 
 1,270 
 
 0.88 
 
 Shearing Strength of Concrete 
 
 Summary of University of Illinois Tests 
 
 Propor- 
 tions 
 
 Form of specimens 
 
 Method of 
 storing 
 
 Shearing 
 strength 
 (lb. per 
 sq. in.) 
 
 Compressive strength 
 (lb. per sq. in.) 
 
 Ratio of shear 
 to compression 
 
 Cube 
 
 Cylinder 
 
 Cube 
 
 Cylin- 
 der 
 
 1:3:6 
 
 
 air 
 
 679 
 
 1,230 
 
 
 0.55 
 
 
 1:3:6 
 
 
 water 
 
 729 
 
 1,230 
 
 
 0.59 
 
 
 1:3:6 
 
 
 damp sand 
 
 905 
 
 2,428 
 
 1,322 
 
 0.37 
 
 0.68 
 
 1:3:6 
 
 
 damp sand 
 
 968 
 
 1,721 
 
 1,160 
 
 0.56 
 
 0.83 
 
 1:2:4 
 
 
 damp sand 
 
 1,193 
 
 3,210 
 
 2,430 
 
 0.37 
 
 0.49 
 
 1:3:6 
 
 
 air 
 
 796 
 
 1,230 
 
 
 0.65 
 
 
 1:3:6 
 
 
 water 
 
 692 
 
 1,230 
 
 
 0.56 
 
 
 1:3:6 
 
 
 water 
 
 879 
 
 1,230 
 
 
 0.71 
 
 
 1:3:6 
 
 
 damp sand 
 
 1,141 
 
 2,428 
 
 1,322 
 
 0.47 
 
 0.86 
 
 1:3:6 
 
 
 damp sand 
 
 910 
 
 1,721 
 
 1,160 
 
 0.53 
 
 0.78 
 
 1:2:4 
 
 Recessed block 
 
 damp sand 
 
 1,257 
 
 3,210 
 
 2,430 
 
 0.39 
 
 0.52 
 
 1:3:6 
 
 Reinforced recessed block 
 
 air 
 
 1,051 
 
 1,230 
 
 
 0.86 
 
 
 1:3:6 
 
 
 damp sand 
 
 1,821 
 
 2,428 
 
 1,322 
 
 0.75 
 
 1.38 
 
 1:3:6 
 
 
 damp sand 
 
 1,555 
 
 1,721 
 
 1,160 
 
 0.90 
 
 1.39 
 
 1:2:4 
 
 Reinforced recessed block 
 
 damp sand 
 
 2,145 
 
 3,210 
 
 2,430 
 
 0.67 
 
 0.88 
 
 1:3:6 
 
 Restrained beam 
 
 damp sand 
 
 1,313 
 
 2,428 
 
 1,322 
 
 0.54 
 
 1.00 
 
 1:3:6 
 
 
 damp sand 
 
 1,020 
 
 1,721 
 
 1,160 
 
 0.59 
 
 0.88 
 
 1:2:4 
 
 Restrained beam 
 
 damp sand 
 
 1,418 
 
 3,210 
 
 2,430 
 
 0.44 
 
 0.58 
 
 1 Taken from Bull. 8, Uni. of 111. Eng. Exp. Sta. 
 
246 
 
 CONCRETE ENGINEERS' HANDBOOK 
 
 [Sec. &-25 
 
 Adhesion of New to Old Concrete 
 Transverse Strength of Joints — Tests of Hector St. 
 George Robinson 
 
 Method employed to secure 
 a bond 
 
 Computed ten- 
 sile stress 
 in extreme 
 fiber 
 (lb. per 
 sq. in.) 
 
 Efficiency of 
 bond, % 
 
 Sohd specimens with no 
 joint 
 
 302 
 362 
 289 
 340 
 352 
 
 
 Average. 
 
 
 329 
 
 100.0 
 
 Surface (molded against 
 rough board) merely 
 wetted 
 
 140 
 78 
 130 
 110 
 172 
 
 
 Average. 
 
 
 126 
 
 38.3 
 
 Surface roughened with a 
 chisel, cleaned and 
 wetted 
 
 194 
 170 
 205 
 142 
 165 
 234 
 
 
 Average. 
 
 
 185 
 
 56.2 
 
 Surface roughened, 
 cleaned, and thoroughly 
 coated with neat cement 
 grout 
 
 325 
 272 
 280 
 248 
 
 
 Average. 
 
 
 281 
 
 85.5 
 
 Surface treated with hy- 
 drochloric acid, washed, 
 brushed, and wetted 
 
 oUU 
 248 
 260 
 201 
 340 
 271 
 
 
 Average 
 
 
 270 
 
 82.0 
 
 
 filled with additional fresh concrete. All specimens 
 
 concrete block; (3) recessed concrete 
 block reinforced outside of the area 
 subjected to the direct action of the 
 punch. 
 
 25. Adhesive Strength. — The ad- 
 hesion of mortars to various building 
 materials is a matter of much impor- 
 tance which has, however, been insuffi- 
 ciently investigated. Fig. 24 presents 
 the results of tests made by General 
 E. S. Wheeler (Report Chief of Engi- 
 neers, U. S. A., 1895, p. 3019, and 1896 
 pp. 2799, 2834). Discs of the material 
 concerned, 1 by 1 in. square and in. 
 thick, were prepared and inserted in 
 the center of briquette molds. The 
 molds were subsequently filled with 
 mortar and the specimens were tested 
 in the usual manner in tension. Mr. 
 Wheeler found that a consistency wet- 
 ter than that which gives a maximum 
 tensile strength is required to give a 
 maximum adhesive strength of mortar 
 to stone, even though the surface of 
 the stone be saturated with moisture. 
 Irregularities of the surface of stone or 
 brick appeared not to affect adhesive 
 strength, but a dirty surface, or in- 
 sufficient moistening of the surface 
 greatly reduced adhesion. 
 
 25a. Adhesion to Con- 
 crete Previously Placed. — The adhe- 
 sion of concrete to old work of the 
 same character constitutes an impor- 
 tant problem in many classes of con- 
 struction work, but few experimental 
 determinations of the bond between 
 new and old work have been made. 
 The data on this page have been derived 
 from a series of tests made by Hector 
 St. George Robinson in 1912 {Proc. In- 
 stitute of Civil Engineers, vol. 189, p. 
 310). The specimens used were prisms 
 of 1:2:4 concrete 30 in. long and 4 
 by 4 in. in section. One set were solid 
 prisms. The remainder were made by 
 placing a stop-board in the mold 8 in. 
 from one end and allowing the con- 
 crete molded in this end to harden for 
 7 days before the stop-board was re- 
 moved and the balance of the mold 
 were tested after further hardening for 
 
Sec. 5-256] 
 
 CEMENT MORTAR AND PLAIN CONCRETE 
 
 247 
 
 28 days. Four different treatments accorded the face of the old concrete to improve the 
 bond are enumerated. The tests were made by rigidly clamping the 8-in. portion of 
 I each beam in a fixed support and loading the cantilever beam at a point 20 in. from the sup- 
 port. The relative strengths of the various joints thus tested was determined by computing 
 the extreme fiber stress on the tension side of the joint (modulus of rupture). The ap- 
 parent tensile strength thus computed is much higher than the actual tensile stress, but the 
 relative efficiencies of the various methods of securing a bond are nevertheless shown. 
 
 Fig 24 — Adhesive strength of Portland cement mortar. 1 part cement; 1 part crushed quartz (Nos. 20-30). 
 
 (Wheeler, Report of Chief of Eng'r's, 1895.) 
 
 256. Adhesion or Bond to Steel. — See Art. 2, Sect. 6. 
 26. Strength of Natural Cement Mortar and Concrete. — The production and use of natural 
 cement in the United States has declined so rapidly since 1899, when the amount produced 
 reached its maximum of nearly 10,000,000 bbl. per year, and greatly exceeded the output of 
 Portland cement, that the present importance of natural cement as a material of engineering 
 construction is almost negligible in comparison with that of Portland cement. The reasons 
 for the great decline in importance of natural cement are briefly: (1) the great improvement in 
 quality and lowering of cost of Portland cement; (2) the inferiority of the average natural ce- 
 
 FlG. 
 
 „ _ 10 T2 14 16 la 20 22 
 Age . in months 
 
 25a. — Tensile strength of natural cement— average of 10 brands. (Sabin.) 
 
 ment to the average Portland cement in structural qualities; (3) the great variability in quality 
 shown by natural cements owing to the lack of close control of the manufacturing process; 
 and (4) a general distrust of natural cement among engineers and others which is often alone 
 responsible for its use being forbidden by specifications. 
 
 Natural cement, mortars, and concretes vary greatly in strength owing to a great varia- 
 bility in both composition and constitution of the cement. " This variation is found not only 
 in comparing cements from different locaUties, but even in comparing samples taken at ditterent 
 times from the output of any one locality. The only general statements that may be made 
 
248 
 
 CONCRETE ENGINEERS' HANDBOOK 
 
 [Sec. 5-27 
 
 concerning their strength is that natural cements rarely show more than half the tensile strength 
 of Portland cements of the same age, and their compressive strength rarely exceeds one- 
 third that of Portland cement in similar mixtures" (" Materials of Construction," by A. P. 
 Mills). The diagrams of Figs. 25a and 256 average the results obtained from tensile 
 tests of mortars of ten representative brands of natural cement made by L. C. Sabin ("Report 
 Chief of Engineers," 1895, p. 2937), and compressive tests of eight to nine brands of natural 
 cement made by Clifford Richardson {Brickhuilder, vol. 6, p. 253). 
 
 Very few data are available showing the strength of natural cement concretes. Tests 
 made at the Watertown Arsenal in 1899 show the following strengths of 12-in. cubes of 1 : 2 : 4 
 
 3000 
 
 Is 
 
 > X 
 
 w 0-1500 
 E 500 
 
 28 
 
 Age in days 
 
 90 
 
 Fig. 256. — Compressive strength of natural cement — average of 8 to 9 brands (1 brand only at 90 days). 
 
 {Richardson.) 
 
 trap-rock concrete made with one brand of typical natural cement ("Tests of Metals," 1899 
 and 1901): 
 
 
 Max. 
 (lb. per sq. in.) 
 
 Min. 
 (lb. per sq. in.) 
 
 Average of 5 
 (lb. per sq. in.) 
 
 Compressive strength at 3 months 
 
 Compressive strength at 14 months 
 
 460 
 914 
 
 263 
 585 
 
 332 
 715 
 
 Sabin ("Cement and Concrete," p. 314) quotes the following tests made by A. W. Dow for 
 the Engineer Commissioner of the District of Columbia in 1897. The specimens were 12-in. 
 cubes of 1:2:6 concrete, made with six different coarse aggregates and tested at the age of 
 12 months. Comparative tests of a Portland-cement concrete made with the same aggregates 
 with the same proportions are also reported. 
 
 
 Max. 
 
 Min. 
 
 Average 
 
 
 (lb. per sq. in.) 
 
 (lb. per sq. in.) 
 
 (lb. per sq. in.) 
 
 Compressive strength of 1 : 2 : 6 
 
 
 
 
 Natural-cement concrete 
 
 915 
 
 763 
 
 844 
 
 Compressive strength of 1 : 2 : 6 
 
 
 
 
 Portland-cement concrete 
 
 3,060 
 
 1,850 
 
 2,670 
 
 The standard specifications of the American Society for Testing Materials (A. S. T. M. 
 Standards, 1916) require that the minimum tensile strength of 1:3 natural cement mortar 
 made with standard Ottawa sand shall be: 
 
 7 days (1 day in moist air, 6 days in water) — 50 lb. per sq. in. 
 28 days (1 day in moist air, 27 days in water) — 125 lb. per sq. in. 
 
 27. Strength of Cinder Concrete. — The compressive strength of a number of mixes of 
 concrete made with anthracite coal cinders and six different Portland cements is shown by 
 
I Sec. 5-27] CEMENT MORTAR AND PLAIN CONCRETE 240 
 
 the following summaries of two series of tests made at the Watertown Arsenal ("Tests of 
 Metals," 1898, 1903, and 1904). The specimens of each test series were 12-in. cubes, and the 
 I average values given are means of from two to four tests. 
 
 Strength of Cinder Concrete 
 
 Watertown Arsenal Tests — Tests Made in 1898 
 
 Brand of 
 cement 
 
 Proportions 
 of mixture 
 
 Average compressive 
 strength 
 (lb. per sq. in.) 
 
 
 
 
 1 month 
 
 3 months 
 
 A 
 
 1 
 
 1:3 
 
 1,466 
 
 2,001 
 
 B 
 
 1 
 
 1:3 
 
 1,032 
 
 1 393 
 
 C 
 
 1 
 
 1:3 
 
 2,329 
 
 2'834 
 
 D 
 
 1 
 
 1:3 
 
 1,602 
 
 2,414 
 
 E 
 
 1 
 
 1:3 
 
 1,438 
 
 1,890 
 
 F 
 
 1 
 
 1:3 
 
 1,379 
 
 1,788 
 
 A 
 
 1 
 
 2:3 
 
 1,098 
 
 1,634 
 
 A 
 
 1 
 
 2:4 
 
 904 
 
 1,325 
 
 A 
 
 1 
 
 2:5 
 
 769 
 
 1,084 
 
 B 
 
 1 
 
 2:5 
 
 471 
 
 685 
 
 C 
 
 1 
 
 2:5 
 
 940 
 
 1,600 
 
 D 
 
 1 
 
 2:5 
 
 696 
 
 1,223 
 
 E 
 
 1 
 
 2:5 
 
 744 
 
 880 
 
 A 
 
 1 
 
 :3:6 
 
 529 
 
 788 
 
 Tests Made in 1903 and 1904 (One Brand of Cement) 
 
 Proportions of 
 mixture 
 
 Compressive strength (lb. per sq. in.) 
 
 5 weeks 
 
 32 weeks 
 
 1 year, 15 weeks 
 
 Max. 
 
 Min. 
 
 Ave. of 3 
 
 Max. 
 
 Min. 
 
 Ave. of 2 
 
 Max. 
 
 Min. 
 
 Ave. of 4 
 
 1:2 :4 
 1:2M:5 
 1:3 :6 
 
 2,430 
 1,400 
 1,200 
 
 1,950 
 1,570 
 1,350 
 
 2,143 
 1,457 
 1,293 
 
 2,600 
 2,020 
 1,730 
 
 2,500 
 1,980 
 1,560 
 
 2,550 
 2,000 
 1,645 
 
 2,610 
 1,950 
 1,400 
 
 2,410 
 1,480 
 1,290 
 
 2,488 
 1,700 
 1,363 
 
 f More recent tests of cinder concrete, the results of which should be indicative of the 
 range of quality of the cinder concrete used in building construction, are summarized in the 
 table on page 250. These tests constitute a portion of a study of " Cinder Concrete Floor Con- 
 struction" by Harold Perrine and by George E. Strehan {Trans. Am. Soc. C. E., vol. 79, 
 p. 523). The specimens were 8 by 16-in. cylinders made by competent men with laboratory 
 training, but the material was taken from that going into the floors of various structures 
 then in process of construction in New York City. The samples were taken without advance 
 
250 
 
 CONCRETE ENGINEERS' HANDBOOK 
 
 [Sec. 5-28 
 
 Strength of Cinder Concrete 
 Perrine and Strehan Tests 
 
 Class of concrete 
 
 Compressive strength (lb. per sq. in.) 
 
 1 month 
 
 2 months 
 
 6 months 
 
 1 year 
 
 1 
 
 2:5 continuous-mixer concrete, low-grade cinders. . 
 
 407 
 
 701 
 
 933 
 
 913 
 
 1 
 
 2 : 5 batch-mixer concrete, good cinders 
 
 818 
 
 1,254 
 
 1,744 
 
 1,465 
 
 1 
 
 
 980 
 
 1,035 
 
 1,478 
 
 1,475 
 
 1 
 
 2 : 5 hand-mixed concrete, good cinders 
 
 507 
 
 662 
 
 754 
 
 813 
 
 Strength of Cinder Concrete 
 Structural Materials Laboratories' Tests 
 
 notice been given the contractor, and the specimens, after being molded on the job, were 
 tested in the Columbia University Laboratory. 
 
 Tests of 1:2:4 cinder concrete 
 made at the Structural Materials 
 Testing Laboratories at St. Louis in 
 1909 are summarized on this page 
 {Tech. Paper 2, U. S. Bureau of 
 Standards). Tests of 21 cylinders 8 
 by 16 in. are averaged. 
 
 28. Working Stresses. — For work- 
 ing stresses recommended by the Joint 
 Committee, see Appendix B. 
 
 
 Compressive strength 
 (lb. per sq. in.) 
 
 At age of 
 52 weeks 
 
 4 weeks 
 
 13. weeks 
 
 26 weeks 
 
 Max 
 
 1,964 
 
 2,445 
 
 2,792 
 
 2,958 
 
 Min 
 
 1,499 
 
 1,981 
 
 2,187 
 
 2,493 
 
 Ave 
 
 1,647 
 
 2,217 
 
 2,525 
 
 2,761 
 
 ELASTIC PROPERTIES OF CEMENT MORTAR AND CONCRETE 
 
 29. Stress-strain Curves for Mortars and Concretes. — Typical stress-strain curves for 
 a number of classes of 1:2:4 concrete at the age of 1 year are presented by Fig. 26. These 
 curves average the results of tests of 21 specimens (8 by 16-in. cylinders) for each class of con- 
 
 .E 3200 
 
 ■<n eeoo 
 
 15 2400 
 o) 2000 
 5n 1600 
 
 0) 
 
 ^(200 
 O 600 
 
 u 
 
 400 
 0 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 C 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 J 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 7 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 / 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 f 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 / 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 - 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 / 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 i 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 v 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 -Yield Doint 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 t 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 I 
 
 
 
 
 
 
 
 
 
 
 
 
 
 --Yield point 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 f 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 t 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 i 
 
 
 
 
 
 
 
 
 
 
 
 
 i 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 ■Y 
 
 eld r)oin+ 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 t 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 r 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 \ i .A Granl+e 
 concrete 
 
 
 
 
 
 V.Z A Limestone 
 concrete 
 
 
 
 V 
 
 2 '4 Grovel 
 
 
 / 
 
 
 
 
 
 
 
 
 
 \:e:4 Cinder 
 concrete 
 
 
 
 
 
 
 
 
 
 
 concrete 
 
 
 
 
 
 
 
 
 
 
 
 0 .0005 O0\0 .0005 /DO 10 0005 .0010 .0005 0010 .0015 
 
 Strain in inches per inch 
 Fig. 26. — Stress-strain curves for concretes. (Curves are averages for 21 tests. Age 12 mo.) 
 
 Crete, made at the Structural Materials Laboratories at St. Louis (U. S. Bureau of Standards 
 Tech. Paper 2). At earlier ages tests of these same concretes resulted in stress-strain curves 
 closely resembling these 1-year-test curves except that the slope of the curves is less, and the 
 curvature greater, at the earlier test periods. 
 
i| Sec. 6-30] 
 
 CEMENT MORTAR AND PLAIN CONCRETE 
 
 251 
 
 Stress-strain curves for mortars exhibit the same characteristics as do these curves fv)r 
 concretes. 
 
 30. Yield Point. — Mortars and concretes are not perfectly elastic materials for any range 
 of loading, there being a slight decrease in the proportion of stress to strain as the stress in- 
 creases, and a slight permanent set for very low stresses. Even the first portion of the stress- 
 strain curve is, therefore, not a perfectly straight line, but for purposes of curve plotting it is 
 sensibly so up to a certain point. This point where deviation from a practically straight-line 
 relation between stress and strain is first perceived is designated the yield point. Beyond the 
 yield point the slope of the curve decreases at an increasingly rapid rate, and the permanent 
 set increases correspondingly. 
 
 The following table gives values of the yield point in tests of mortars of three proportions 
 made by the Bureau of Standards {Tech. Paper 58). 
 
 Yield Point of Mortars 
 
 
 Age 4 weeks 
 
 Age 13 weeks 
 
 Proportions 
 by volume 
 
 Yield point 
 (lb. per sq. in.) 
 
 Modulus of 
 elasticity 
 (lb. per sq. in.) 
 
 Ultimate 
 strength 
 (lb. per sq. in.) 
 
 Yield point 
 (lb. per sq. in.) 
 
 Modulus of 
 elasticity 
 (lb. per sq. in.) 
 
 Ultimate 
 strength 
 (lb. per sq. in.) 
 
 1:1 
 1:2 
 
 1,834 
 
 4,243,000 
 
 5,613 
 3,070 
 1,432 
 
 2,600 
 1,833 
 700 
 
 4,153,000 
 4,673,000 
 2,200,000 
 
 6,739 
 4,560 
 1,663 
 
 1:4 
 
 400 
 
 2,120,000 
 
 The yield points of various classes of 1:2:4 and 1:3:6 concretes at ages up to 1 year 
 are shown by Fig, 27 which is based upon tests of the Bureau of Standards {Tech. Paper 58). 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 — 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 Ultimate sfrenoth' 
 
 
 
 
 
 
 
 
 
 
 ( 
 
 — 1 
 
 
 3-6 concretes 
 
 Fig 
 
 13 26 
 Age in weeks 
 
 27.— Yield point of 1 
 
 2 : 4 and 1 
 
 13 26 
 
 Age in weeks 
 
 3 : 6 concretes. 
 
 52 
 
 Values of ultimate compressive strength for the same concretes are shown for purposes of com- 
 parison. These tests indicate that the yield point of the average concrete is in the neighborhood 
 of three-tenths of the compressive strength at 1 month and about four-tenths of the compressive 
 strength at 1 year. 
 
 31. Modulus of Elasticity.— The modulus of elasticity of an elastic material is the quotient 
 obtained by dividing unit stress by the corresponding unit deformation or strain, the limit 
 of elastic behavior not being exceeded. In American practice the unit of measurement is 
 pounds per square inch. Since mortars and concretes are not perfectly elastic materials, the 
 deformation not bearing a constant relation to the stress for any range of loading, the quotient 
 
252 
 
 CONCRETE ENGINEERS' HANDBOOK 
 
 [Sec. 5-32 
 
 of stress divided by strain will vary, decreasing as the stress increases. Properly speaking an 
 inelastic material has no modulus of elasticity, but practice has sanctioned the use of the term 
 in connection with mortars and concretes, meaning the quotient of any small stress increment by 
 the corresponding strain increment. The value of E thus computed is, therefore, the slope of a 
 short chord of the stress-strain curve, or if the stress increment be very small, E at any point on 
 the stress-strain curve is represented by the tangent to that curve. Within the limits of work- 
 ing stresses for mortars or con- 
 
 8 
 
 o 
 
 c 
 
 4000 
 
 3000 
 
 .E6 
 
 (J) ' 
 
 ^ 5000 
 j3 
 
 2000 
 
 1000 
 
 4000 
 
 3000 
 
 20OO 
 
 cretes the value of E changes 
 only very slightly, and its initial 
 value may, therefore, properly be 
 used for purposes of design. This 
 initial value of E may most con- 
 veniently be determined by not- 
 ing the slope of the tangent to 
 the first portion of the stress-strain 
 curve. 
 
 The values of the initial mod- 
 ulus of elasticity for the three 
 mortar mixtures mentioned in the 
 last preceding chapter are given 
 in the above table. Values of E 
 found for the various concretes 
 also mentioned above are given in 
 Fig. 28. Little may be said by way of generalization concerning E for concretes. It increases 
 with age and with the richness of the mix, but varies greatly with different classes of aggre- 
 gate materials and with different aggregates of the same general class. E for cinder concrete 
 appears to be something less than one-half the average value found with rock concretes. 
 
 The design values of E recommended by the Joint Committee are given in Appendix B. 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 Y 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 — 1: 
 
 2:4c 
 
 sncre+es — 
 
 1000 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 -t:3 
 
 :6 con 
 
 cre+«s 
 
 26 
 
 52 
 
 Fig. 
 
 Age in weeks 
 
 28. — Modulus of elasticity of 1 
 
 4 \^ 26 S( 
 
 Age in weeks 
 
 4 and 1:3:6 concretes. 
 
 CONTRACTION AND EXPANSION OF CEMENT MORTAR AND 
 
 CONCRETE 
 
 32. Coefficient of Expansion. — Mortars and concretes expand as the temperature is raised 
 and contract as the temperature is lowered. The coefficients of linear expansion per degree 
 Fahrenheit for a series of mortars 
 and crushed stone concretes tested 
 under the writer's direction in the 
 laboratories of the College of Civil 
 Engineering, Cornell University, in 
 1916 are listed in the table on this 
 page. All of the specimens were 
 molded in the shape of bricks 8 in. 
 long, 4 in. wide, and 2 in. thick. 
 Heating was done in a specially-con- 
 structed resistance type of electric 
 furnace, and distortion was meas- 
 ured by a special extensometer 
 actuated by fused quartz contact 
 bars extending through the furnace walls. Measurements were made to the nearest 
 0.000,000,9 in. per in. of length of the specimen. The test results listed are averages of from 
 two to ten tests of each class of material, the ages varying from 1 to 8 months. The range of 
 temperatures employed was from about 70° to 212°F. 
 
 Mortars 
 
 Concretes 
 
 Mixture 
 
 Coefficient of 
 expansion per 
 degree Fahrenheit 
 
 Mixture 
 
 Coefficient of 
 expansion per 
 degree Fahrenheit 
 
 Neat 
 
 0.000,007,83 
 
 1:13^:3 
 
 0.000,006,77 
 
 1:1 
 
 0.000,007,43 
 
 1:2 :4 
 
 0.000,006,60 
 
 1:2 
 
 0.000,006,00 
 
 1:23^:5 
 
 0.000,005,58 
 
 1:3 
 
 0.000,006,05 
 
 1:3 :6 
 
 0.000,005,37 
 
 1:4 
 
 0.000,005,94 
 
 
 
 1:5 
 
 0.000,005,77 
 
 
 
Sec. 5-33] 
 
 CEMENT MORTAR AND PLAIN CONCRETE 
 
 253 
 
 •- c: 
 «o o 
 
 It ° 
 
 O 
 
 05 
 
 0.00 
 
 -0.05 
 -0.10 
 
 Specimen No.I46-Cl 
 
 These tests show that the coefficient of expansion of mortars and concretes increases with in- 
 crease of richness of the mix, but that the range of values between a very lean concrete and neat 
 cement is comparatively short. An average concrete will have a temperature coefficient almost 
 exactly equal to that of the average steel. This fact is one of great importance in all cases of 
 I reinforced concrete. 
 
 33. Moisture Changes. — Mortars and concretes expand in volume if kept wet or immersed 
 in water, and contract if exposed in air. Experiments made by Prof. A. H. White at the 
 University of Michigan {Proc. Am. 
 Soc. Test. Mat., vols. 11 and 14) +0.10 
 indicate that this property is not 
 confined to the early hardening 
 period but is characteristic of mor- 
 tars and concretes even after 20 
 years in service. Fig. 29a shows 
 the variations in length observed 
 in two 1-in. by 1-in. by 4-in. bars 
 of 1:3 mortar tested by Prof. 
 White. These bars appear to 
 have suffered no impairment of the 
 ability to expand immersed and 
 contract when exposed to air even 
 after a period of nearly 5 years. 
 These particular specimens, when 
 about 4 years old, show increases in length of about 0.05% in a 3 to 4-month period when 
 placed in water after thorough drying out, and their contraction in air is scarcely less rapid. 
 Specimens of the same mixes made with other cements showed in a number of cases more ex- 
 tensive volume changes than do the ones shown in Fig. 29a. One mortar stored in air con- 
 tracted about 1.10% in the first 3 months. A specimen cut from the rich mortar top coat 
 of a sidewalk which had been in service for 20 years expanded about 0.16% when stored in 
 
 o Immersed in water 
 @ In air of room 
 ^ in olessicator 
 
 In water- sat ura+ed air 
 
 Specirnen No. I46-Q2 
 
 Time in years 
 
 Fig. 29a. 
 
 -Expansion and contraction of 1 : 3 mortars when alter- 
 nately wet and dried. 
 
 50 60 70 80 90 
 
 Time in days 
 contraction of concretes. 
 
 * Fig. 296. — Expansion and contraction of concretes. Air and water storage. 
 
 W 
 
 water for 2 years, and a portion from the gravel concrete base of the same walk expanded 
 about 0.12% in the same period. 
 
 The results of a number of tests of concrete made under various auspices are shown by Fig. 
 296. Curves I, II, and III show the changes in length of concretes tested by A. T. Goldbeck 
 in the laboratory of the Office of Public Roads, U. S. Department of Agriculture {Proc. Am. Soc. 
 Test. Mat., vol. 11). The specimens were 8 in. square and 5 ft. long. Specimens I and II 
 
 L 
 
254 
 
 CONCRETE ENGINEERS' HANDBOOK 
 
 [Sec. 5-34 
 
 were stored in air; specimen III was wrapped in burlap which was kept moist continuously. 
 Curves IV, V, VI, and VII are derived from tests niade by Prof. H. C. Berry in the laboratories 
 of the University of Pennsylvania {Proc. Am. Soc. Test. Mat., vol. 11). The specimens were 
 9 in. square and the gaged length was 20 in. Specimens IV and V were stored in air while speci- 
 mens VI and VII were stored in water. Curve VIII is the record of a test of a slab of concrete 
 12 in. wide, 6 in. deep, and 10 ft. long between gage points, made by Prof. B. P. Fleming of the 
 State University of Iowa. The slab was exposed in the air of the laboratory throughout the 
 period of observation. Its behavior is notable in one respect in that a pronounced expansion 
 was observed for the first 10 days, amounting to about ,0.012%. After about 12 days it con- 
 tracted continuously for about 75 days longer, at which time it shows a net contraction of about 
 0.03%. Mr. Goldbeck states that some of his mixtures showed a tendency to expand in the 
 early hardening period of a few days but the amount of expansion was very slight. 
 
 The tests of Fig. 296 indicate that the changes in volume of concretes when exposed in 
 either a wet or a dry situation are rather variable with different cements and aggregates. It 
 appears, however, that a concrete which dries out in the air may be expected to contract from 
 0.02 to 0.05%, and when immersed in water may expand at least half this amount. If the 
 concrete in a structure is so restrained that it is not free to expand or contract, it is possible 
 therefore that stresses amounting to from 400 to 1000 lb. per sq. in. in tension may occur, E 
 being considered to be 2,000,000 lb. per sq. in. This means that the tensile strength of concrete 
 is exceeded and the concrete will, and commonly does, crack. Difficulty caused by the expan- 
 sion of concrete in a damp or wet situation is not so commonly encountered, and the stresses 
 introduced will never cause compressive failure. They may, however, cause a buckling action 
 in the case of continuous surfaces of large extent. 
 
 DURABILITY OF CEMENT MORTAR AND CONCRETE 
 
 34. Fire -resistance Properties. — Concrete ranks highly as a fire-resistant and fireproofing 
 material principally because it possesses a low rate of heat conductivity and has a low coeffi- 
 cient of expansion practically equal to that of steel, in addition to being incombustible. Other 
 masonry materials like some of the natural stones and terra-cotta are no less incombustible 
 than concrete, but are inferior to the latter as a fireproofing material because they possess either 
 greater conductivity or a higher coefficient of expansion. 
 
 Tests of the conductivity of concretes made by Prof. Ira H. Woolson {Proc. Am. Soc. Test. 
 Mat., vols. 5, 6, and 7) led to the conclusions: " That all concretes " — stone gravel, and cinder — 
 "have a very low thermal-conductivity, and herein lies their ability to resist fire. That when 
 the surface of a mass of concrete is exposed for hours to a high heat, the temperature of the 
 concrete 1 in. or less beneath the surface will be several hundred degrees below the outside. 
 That a point 2 in. beneath the surface would stand an outside temperature of 1500°F. for 2 hr. 
 with a rise of only 500° to 700°, and points with 3 in. or more of protection would scarcely be 
 heated above the boiling point of water." 
 
 The low thermal-conductivity of concrete is partly due to its porosity, air spaces affording 
 efficient protection against conduction, and partly due to the absorption of heat of vaporiza- 
 tion by the water of combination in the set cement when the temperature of dehydration of the 
 latter is reached. This dehydration probably begins at about 500°F. and is completed at about 
 900°F. (S. B. Newberry, Cement, May, 1902, p. 95). The absorption of heat by the surface 
 material in becoming itself dehydrated retards the dehydration of the underlying material. 
 The surface concrete which is injured by heat, but which remains in place, affords protection 
 for the material beneath, for it becomes a poorer conductor than the original concrete. The 
 Joint Committee on Concrete and Reinforced Concrete recommends that "metal be protected 
 by a minimum of 2 in. of concrete on girders and columns, 1}^ in. on beams, and 1 in. on floor 
 slabs." 
 
 The experience gained in great conflagrations like the Baltimore fire, the San Francisco 
 
Sec. 6-35] 
 
 CEMENT MORTAR AND PLAIN CONCRETE 
 
 255 
 
 fire, the Edison plant fire, etc., has been that concrete exposed to intense heat for considerable^ 
 periods becomes calcined to a depth of from 3^ to in. but shows no tendency to spall ofT 
 except at exposed corners and edges (see reports of Captain J. S. Sewell to the Chief of Engineers, 
 U. S. A., and of Prof. Norton to the Insurance Engineering Experiment Station, on the Balti- 
 
 j; more fire, Eng. News, March 24, and June 2, 1904; the report of S. A. Reed to the National Board 
 of Fire Underwriters on the San Francisco fire, Eng. News, vol. 56, p. 137; the comments of 
 various engineers upon the Edison plant fire, Eng. News, vol. 73, p. 38; and the report on the 
 Edison fire of the National Fire Protective Association, obtainable in booklet form from the 
 New York Board of Fire Underwriters). 
 
 Prof. Norton and others have concluded that there is little difference in the action of fire 
 
 I on stone concrete and cinder concrete. 
 
 35. Weathering Qualities. — The principal agencies affecting the durability of concretes 
 and mortars which are classed as weathering agencies are changes of atmospheric temperature, 
 wind and rain, and changes of atmospheric moisture. The expansion and contraction of mor- 
 tars and concretes subjected to variations of temperature and moisture conditions are respon- 
 sible for practically all failures of these materials under conditions of exposure to the weather. 
 Either temperature effects or moisture effects may be alone operative, or both effects may be 
 combined. Temperature stresses caused by the shrinkage of continuous large surface areas 
 are particularly apt to cause cracking. This cannot be wholly prevented, but cracks can be 
 made less harmful by the use of steel reinforcement so placed that a multitude of small cracks, 
 which do not open up much, replace a few large and deep cracks. In the average situation the 
 introduction of dangerous stresses caused by a tendency to expand or contract is more apt to 
 be due to moisture changes than to temperature changes, because the volumetric changes in 
 the latter case are less marked. The expansion and contraction of rich mortars and concretes 
 is considerably more extensive than that of leaner mixes when the moisture condition varies, 
 and the same thing appears to be true to a lesser extent when the temperature varies. This 
 circumstance is responsible for the difficulty often encountered in causing a surface coating of 
 comparatively rich material plastered upon a leaner base material to adhere permanently if 
 the bond between the two is at all defective. The surface material tends to expand and con- 
 tract more than the underlying material not only because it is richer in cement, but also be- 
 cause it protects the underlying material from as extensive temperature and moisture changes 
 as it itself experiences. The result is the introduction of excessive stresses in the surface 
 material, the opening up of tension cracks, or buckling due to compressive stress, and the 
 ultimate spalling off of the surface layer. The principal preventive measures which may bo 
 adopted are the use of as lean a surface coat as is practicable, the use of as thin a plaster coating 
 as possible thus favoring the formation of many small cracks rather than a few large ones, 
 and the adoption of all measures tending to make a strong bond between the two classes of 
 material. An excessive amount of troweling of surfaces is to be avoided because of the flush- 
 ing to the surface of a film of nearly neat cement which will tend to peel off in some cases. 
 
 36. Abrasive Resistance. — The abrasive resistance of mortars "is primarily of importance 
 in the determination of the best mortar for use in the top coat of concrete floors, walks, and pave- 
 ments. Resistance to abrasion will always be dependent not only upon the cement, as regards 
 the tenacity with which it clings to the sand grains, which will be largely dependent upon its 
 fineness and its lime content, but also upon the hardness of the sand used. Abrasion either 
 wears away the cement and the sand grains, or it pulls the sand grains out of the cement matrix. 
 
 "With soft sand particles the resistance to abrasion with a given cement decreases con- 
 stantly as the percentage of sand is increased. With hard sand grains the abrasive resistance 
 increases as the proportion of sand increases, until the volume of cement becomes relatively 
 too small to bind the sand grains together thoroughly. This limit is found to be reached when 
 the mortar contains not more than two parts of sand to one of cement." (" Materials of Con- 
 struction," by A. P. Mills.) The abrasive resistance of concretes is dependent almost wholly 
 upon that of the mortar made by its cement and fine aggregate. The coarse aggregate is prac- 
 
256 
 
 CONCRETE ENGINEERS' HANDBOOK 
 
 [Sec. 6-37 
 
 tically always forced back from the surface of concrete exposed to abrasion. If it is exposed, 
 the same considerations of relative hardness of the stone particles and proportion of the mix 
 applicable in the case of mortars apply to the concrete. 
 
 37. Action of Sea Water. — The behavior of concrete in sea water is a problem which 
 has occupied much of the attention of engineers for many years. The question has often 
 been discussed, and many attempts have been made to determine experimentally the exact 
 action of sea water upon concrete, and the causes of that action. The amount of accurate in- 
 formation available is rather meager, however, and the results of experimental investigations 
 are inconclusive and often contradictory. Many concrete structures in sea water out of the 
 range of frost action have remained intact and uninjured for many years. Others have been 
 seriously disintegrated, particularly between high and low tide levels. The disintegration is 
 evidently often due in part to frost action, but chemical action is frequently indicated by the 
 softening of the mortar, and the complete disintegration of mortar and concrete specimens 
 by subjection to the action of sea water at normal temperatures in the laboratory has been 
 accomplished. The exact nature of the chemical action involved cannot be definitely stated. 
 It is commonly believed, however, that the magnesium sulphate in the sea water is the most 
 injurious constituent, and that the magnesium chloride and calcium chloride are somewhat 
 less active. The magnesium sulphate attacks the lime in the cement, also the alumina, form- 
 ing large and rapidly growing crystals of hydrated magnesia and calcium sulpho-aluminate. 
 Both magnesium chloride and sodium chloride attack the silicates of the cement. 
 
 The chemical action is accompanied by various physical phenomena. Sometimes the mass 
 swells, cracks, and gradually falls apart; sometimes the mortar softens and becomes disinte- 
 grated leaving the coarse aggregate exposed and finally permitting it to fall away; and occa- 
 sionally a crust forms on the surface which later cracks off. 
 
 An important symposium of European investigators' studies of the problem of concrete 
 in sea water is afforded by the several papers of Chapter XVII of the Proceedings of the Sixth 
 Congress of the International Association for Testing Materials held in New York in 1912. 
 
 The conclusions arrived at from a study of important German and Scandinavian tests, 
 have been expressed, in part, as follows {Concrete and Constructional Engineering, January, 1910) : 
 
 1. Good Portland cements such as are now on the European market, are very resistant to the action of sea 
 water. A marked difference in the behavior of cements of slightly different composition has not been found, ex- 
 cept that a high proportion of aluminates tends to cause disintegration. 
 
 2. In a dense mortar, the chemical action is confined to an outer layer of small depth, further action being 
 checked by the slowness of diffusion. A porous mortar, by admitting salt water to the interior, is apt to crack by 
 expansion owing to chemical change. 
 
 3. The main agency in the destruction of mortar and concrete in marine embankments, harbor works, groynes, 
 etc., is not chemical action, but the alternations of saturation, drying in the sun, freezing, etc., due to the alternate 
 exposure and covering by the rise and fall of the tide. 
 
 4. The denser the mortar the better (1 cement :3 sand is too poor). An admixture of fine sand with the 
 ordinary sand increases the closeness of the mixture. A well-graded aggregate would be advantageous for the 
 same reason. 
 
 5. The addition of finely ground silica or trass to the cement before mixing is possibly advantageous in the 
 case of weaker mortars. It is very doubtful whether anything is gained by adding trass to the richer mortars. 
 
 6. The destructive action of the sea being mainly physical and mechanical, and not chemical, tests by mere 
 immersion in still sea water are of very little value in determining the behavior of concrete in marine engineering 
 works. A mixture which disintegrates under this test is certainly useless, but a mixture which passes the test 
 may disintegrate under the more stringent conditions of practical use. 
 
 7. As long a period as is practicable should be allowed for the hardening of concrete blocks before placing 
 in the sea. 
 
 8. The behavior of test specimens for the first 12 months is very irregular, and definite conclusions can only 
 be drawn from the results of long-period tests. 
 
 The most notable American investigations of the subject are one made by the Bureau 
 of Standards {Tech. Paper 12) and one begun in 1908 by the Aberthaw Construction Co. in 
 cooperation with the United States Navy Department (reported in a pamphlet issued by the 
 Aberthaw Construction Co. in 1914; also in Eng. Rec, March 21, 1914). Few general conclu- 
 
Sec. &-38] 
 
 CEMENT MORTAR AND PLAIN CONCRETE 
 
 257 
 
 sions may be drawn from the results of the first 5 years' observations of the Aberthaw tests. 
 This is particularly true since European experience has shown that the first indications of 
 injury to many concretes appear only after from 5 to 10 years' exposure, and in some cases only 
 after more than 20 years. The most notable indications afforded by the Aberthaw tests after 
 5 years' exposure in a latitude involving wide variations of temperature and frequent freezing 
 and thawing, are: 
 
 1. Concrete which is alternately immersed and exposed as the tide rises and falls is most 
 subject to injury. 
 
 2. Lean mixtures (1:3:6) are very much more subject to attack than rich mixtures (1:1 
 :2), and medium mixtures (1:2:4) are more vulnerable than rich ones. 
 
 3. Concrete mixed with a plastic or even a very wet consistency appears to withstand sea 
 water attack better than concrete of a dry consistency. 
 
 4. The relative immunity from attack by sea water of concretes made with cements of 
 various classes, including a low-iron cement, high- low- and average-alumina cements, an iron- 
 ore cement, and a slag Portland cement, has not been conclusively established. 
 
 The Joint Committee makes the following recommendations for concrete placed in sea 
 water: 
 
 To effect the best resistance to sea water, the concrete must be proportioned, mixed, and placed so as to 
 prevent the penetration of sea water into the mass or through the joints. The aggregates should be carefully 
 selected, graded, and proportioned with the cement so as to secure the maximum possible density; the concrete 
 should be thoroughly mixed; the joints between old and new work should be made water-tight; and the concrete 
 should be kept from exposure to sea water until it is thoroughly hard and impervious. 
 
 38. Action of Alkali. — The effect of alkali on concrete is a problem resembling in many 
 respects that of the action of sea water on concrete. The problem is of especial interest in 
 connection with concrete construction in the arid regions of the West, where soluble salts are 
 present in the soil to an extent not usually found elsewhere. 
 
 The principal salts encountered in alkali waters usually include: magnesium sulphate, 
 calcium sulphate and sodium sulphate, magnesium chloride, sodium chloride, and potassium 
 chloride, together with carbonates of magnesium, sodium, and potassium. Of these the 
 sulphates appear to be most active in causing disintegration of concrete; the chlorides also are 
 active, while the carbonates appear to be without effect. 
 
 The attempts at an explanation of the manner of attack of these salts upon concrete 
 have hitherto encountered the same difficulty found in the case of sea water — an unsatisfactory 
 knowledge of the constitution of cement. From the physical point of view the action exactly 
 resembles the action of frost except that it is more rapid. There exists, apparently, a disruptive 
 force which quickly destroys the bond and causes disintegration. This action appears to 
 proceed most rapidly in the parts of a structure subjected to alternate wetting with alkali 
 water and drying in the air. In porous concrete the action proceeds much more rapidly than in 
 dense concrete, where, indeed, it may make no progress at all. 
 
 As in the case of the injurious action of sea water on concrete, instances of failure caused 
 by alkali waters are merely isolated ones, presenting an interesting field for study, but not 
 constituting a very serious menace to the future of concrete construction in the arid regions 
 of the West. The remedy in the present state of our knowledge is, as in the case of marine 
 structures, a matter of the possible physical precautions only — the securing of the densest 
 possible concrete, thus preventing injury by the exclusion of the salt-bearing waters. 
 
 39. Action of Acids, Oils, and Sewage.— The Joint Committee Report makes the following 
 statements concerning the effect of acids and oils upon concrete : 
 
 Dense concrete thoroughly hardened is affected appreciably only by acids which seriously injure other 
 materials. Substances like manure, that contain acids, may injuriously affect green concrete, but do not affect 
 concrete that is thoroughly hardened. 
 
 Concrete is unaffected by such mineral oils as petroleum and ordinary engine oils. Oils which contain fatty 
 acids produce injurious effects, forming compounds with the lime which may result in a disintegration of the con- 
 crete in contact with them. 
 17 
 
258 
 
 CONCRETE ENGINEERS' HANDBOOK 
 
 [Sec. 5-40 
 
 The use of concrete sewer pipes has led to considerable study of the effect of sewage and 
 sewage gases upon concrete. Sidney H. Chambers concluded from an investigation reported 
 before the Concrete Institute (Great Britain) in 1910: 
 
 That the gases in solution in sewage and those expelled from it, arising from its decomposition, do act in- 
 juriously upon Portland-cement concrete, notwithstanding the fact that the concrete is constituted of sound and 
 good materials, when the following conditions prevail: (1) A high degree of putrescence of the sewage; (2) a mois- 
 tened surface, which held or absorbed the putrid gases; (3) the presence of a free air supply. Further, that in the 
 absence of one or the other of the above-enumerated factors little danger from erosion need be feared. 
 
 Rudolph Hering is responsible for the following statements concerning the effect of the 
 acids in sewage upon concrete (quoted from a report to the President of the Borough of Brook- 
 lyn in 1908 by Gustave Kaufman in Proceedings of the National Association of Cement Users, 
 vol. 8, 1912, p. 725). 
 
 Portland cement used for the manufacture of concrete pipes is attacked by certain strong acids, such as 
 sulphuric acid, which converts the carbonate into sulphate of lime, which is comparatively soft and easily eroded. 
 Therefore cement pipe cannot be used where strong acids are known to enter the sewers. 
 
 The acid question should be viewed in a reasonable light. When the dilution of sewage is sufficient the 
 discharge of a small amount of even strong acid will not cause objectionable effects, as evidenced by European 
 cities where the use of concrete sewers is almost exclusive in some cities, as Paris and Vienna. In England concrete 
 sewers are also very common. 
 
 The greasy substance which is usually found to coat the perimeter of a sewer under the water line tends 
 to protect the cement from the action of acids to some extent. 
 
 Over 400 miles of concrete sewer pipe laid in the City of Brooklyn during a period of over 
 50 years are giving eminent satisfaction. 
 
 40. Electrolysis in Concrete. — Experience and laboratory tests have shown that under 
 certain conditions concrete may be seriously damaged by electrolytic action caused by the 
 flow of electric current between the concrete and iron or steel embedded therein. The phe- 
 nomena assumes importance under certain conditions of use of reinforced concrete, also the use 
 of concrete foundations and footings in which the bases of columns of buildings, bridges, and 
 elevated railway structures are embedded. A very important laboratory and field study of 
 the entire problem has been made by the U. S. Bureau of Standards, and the conclusions arrived 
 at in this investigation constitute the authority for the statements made here {Tech. Paper 
 18, Bureau of Standards). 
 
 The electrolytic effect differs according to the direction of flow of the current. If elec- 
 trically positive iron or steel is in contact with concrete the iron will become corroded provided 
 the concrete is moist or wet and the potential gradient is high enough to heat the junction to a 
 temperature not below about 45°C. (113°F.). The minimum potential gradient found effective 
 in causing corrosion in moist concrete was about 60 volts per ft. Iron when corroded expands 
 to about 2.2 times its original volume and causes mechanical pressure found in some cases to 
 reach values as high as 4700 lb. per sq. in. This causes cracking of the concrete. The passivity 
 of iron below 45°C. is due chiefly to the inhibiting effect of Ca(0H)2 in the concrete, and on 
 this account old concrete in which the Ca(0H)2 has been largely carbonated is probably more 
 susceptible to electrolysis than new concrete in the same moisture condition. For air-dried 
 concrete a much higher potential gradient is required to produce the temperature at which 
 corrosion becomes dangerous than for moist concrete. Under actual conditions, therefore, 
 corrosion from stray currents may be expected only under special or extreme conditions. 
 Normal wet concrete carrying current also increases its resistance a hundredfold in the course 
 of a few weeks, owing partly to the precipitation of CaCOs which fills up the pores. This 
 further lessens danger of trouble. 
 
 Electrolysis of the concrete in contact with negative iron is manifested in a different way. 
 The concrete near the cathode becomes softened, beginning at the cathode surface, and extend- 
 ing to a depth of in. or more. This softening practically completely destroys the bond 
 between iron or steel and-concrete. While the anode effect becomes serious in normal concrete 
 only on comparatively high voltages, decreases much more rapidly than the voltage, and almost 
 
Sec. 5-41] 
 
 CEMENT MORTAR AND PLAIN CONCRETE 
 
 259 
 
 I disappears at voltages likely to be encountered in practice, the cathode effect develops at all 
 I voltages, the rate being roughly proportional to the voltage. The softening effect is due to 
 the gradual concentration of alkalies, Na and K, near the cathode, these finally becoming strong 
 enough to attack the cement. The cathode effect is wholly limited to the vicinity of the cath- 
 ode, the strength of the mass of the concrete not being affected. 
 
 Salt or calcium chloride, even in very small amounts (a fraction of 1%), multiplies the 
 rate of corrosion of iron at the anode many hundredfold because it increases the conductivity 
 of wet concrete, destroys the passivity of iron at ordinary temperatures, and prevents the 
 increase in resistance with flow of current by preventing the precipitation of CaCOs. Salt 
 should therefore not be used in structures subject to electrolytic action, and special considera- 
 tion should be given to the possibility of electrolytic action in the cases of all concretes exposed 
 to sea water or salt brine. 
 
 The danger of electrolysis of reinforced-concrete structures through the operation of 
 stray currents has been overestimated. Certain precautions are necessary under special 
 conditions, but there is no cause for serious alarm. Non-reinforced-concrete structures are 
 1 practically immune from injury by electrolysis. 
 
 I The precautions to be adopted in the special cases where electrolysis is to be feared include 
 avoidance of grounds in direct-current circuits in buildings; providing insulating joints in 
 
 'pipe lines which enter buildings outside the walls; completely isolating the buildings by in- 
 sulating joints in pipe lines which enter the building and also continue on beyond; providing 
 a copper cable shunt around the building if the potential drop is large; isolating from the con- 
 crete lead-covered cables entering the building; and interconnecting all metal work in the 
 building if practicable, but without connecting this metal work to ground plates or to pipe 
 lines outside of the insulating joints. 
 
 41. Effect of Manure. — Manure, is occasionally used to cover up fresh concrete in freezing 
 weather, not only because it is a poor conductor of heat when rather dry, but also because its 
 decomposition is a source of heat. Experience has shown (see Eng. Neivs, vol. 49, pp. 11, 104, 
 126, 127, and 175; also Journal of the New England Waterworks Association, vol. 22, p. 242) 
 that manure not onlj^ discolors the work, but that it also has a marked disintegrating effect if 
 placed in contact with freshly placed concrete. The injury is especially pronounced if rain 
 wets the manure during the early hardening period and carries the uric acid into the concrete. 
 
 ij The use of manure as a preventive of freezing of fresh concrete may be considered permissible 
 j only if the work is covered first by a material which will be sufficiently impermeable to prevent 
 Ij the seepage of acid into the concrete. 
 
 j Concrete which has once thoroughly hardened appears not to be susceptible to injury 
 by contact with manure except that it is in some cases somewhat discolored. 
 
 MISCELLANEOUS PROPERTIES OF CEMENT 
 j MORTAR AND CONCRETE 
 
 42. Rise of Temperature in Setting. — The chemical combination of water with Portland 
 
 cement is an endothermic reaction, the heat evolved being sufficient to materially raise the 
 temperature of mortars and concretes during the period of setting and hardening. The total 
 rise in temperature, the rate of increase, and the time interval before the maximum temperature 
 
 I is reached are all variable, depending upon the character of the cement used, the proportions of 
 the mixture, the size of specimen or bulk of material involved (in so far as this determines the 
 distance of the point at which temperatures are measured from any exposed surface of the 
 
 i material), external temperature conditions, the amount of water used in mixing, etc. 
 
 j [ The results of observations of temperatures acquired at the center of 12-in . cubes of cements 
 and mortars tested at the Watertown Arsenal (''Tests of Metals," 1901, p. 493), are shown by 
 Fig. 30a. With these specimens the maximum temperature was attained in from 12 to 18 
 
260 
 
 CONCRETE ENGINEERS' HANDBOOK 
 
 [Sec. 5-42 
 
 hr. except in the case of natural cements which reached their maximum temperature much ear- 
 lier. The heating effect is less than one-half as great with 1 : 1 mortar as with neat cement, and 
 is quite small with leaner mixtures. 
 
 £12 
 194 
 
 1 158 
 .0 140 
 
 -|l04 
 S.86 
 ,2? 68 
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 100 
 90 
 80 
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 p 60 
 
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 30 
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 ind 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 Portia 
 
 nd 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 p 
 
 jrti 
 
 ind 
 
 i:3 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 le 14 16 16 EO 22- 24 26 E8 30 32 34 36 38 40 
 
 Time in hours 
 
 Fig. 30a. — Rise in temperature of cement and mortar in setting and hardening. (12-in. cubes.) 
 
 Radiation has a great deal to do with the temperatures observed at the centers of these 
 comparatively small specimens, as is shown by comparison of the curves for mortars with Fig. 
 306 which shows the rise in temperature of a 1 : 23^ : 6 concrete, the temperature having been 
 measured in the midst of a very large mass of concrete with the thermometer covered by a 
 
 Si 
 
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 l5 
 
 <n 
 
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 90 
 86 
 82 
 78 
 74. 
 70 
 66 
 62 
 58 
 54 
 
 50 
 46 
 
 r 25' 
 
 Approximate depth 
 
 4' 6.5' 
 
 of concrete over tViermometer 
 
 6.5' 9' 
 
 
 
 
 
 
 -. 
 
 
 J 
 
 
 
 t..: 
 
 L 
 «i 
 
 
 
 
 h 
 
 -\ 
 
 - 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
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 -f- 
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 u 
 
 
 
 r— 
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 icrete 
 
 
 
 
 hermomel 
 
 
 
 
 
 
 
 
 4. 
 
 r 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 Q) 
 
 1 
 
 r 
 
 1 
 
 r 
 
 
 
 lover themnome 
 
 -k 
 
 k 
 
 u 
 
 % 
 
 
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 t- 
 
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 ^8 
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 A- 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
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 4 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
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 1- 
 
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 8 
 
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 V 
 
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 1 with fresh concrete | 
 
 
 t 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 1 
 
 
 
 
 \ 
 
 
 
 
 
 
 
 
 r 
 
 
 
 
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 -f 
 
 
 
 
 
 
 
 
 + 
 
 
 
 
 
 
 Xj 
 
 
 
 
 
 
 
 
 
 
 
 
 
 H 
 
 I 
 
 
 
 
 
 
 i 
 
 
 
 
 
 
 
 
 
 
 
 
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 X- 
 
 
 
 
 
 
 
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 1 
 
 
 
 \ 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 1 
 
 
 
 T 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 s 
 
 ^ 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 EG 
 
 40 
 
 60 80 100 
 
 Time in hours 
 
 120 
 
 140 
 
 160 
 
 180 
 
 Fig. 306. — Rise in temperature of mass concrete in setting and hardening. 
 
 II part sand-cement, 
 parts sand. 
 D parts gravel, 
 2 parts large cobbles. 
 Dotted lines indicate daily variation in atmospheric temperature. 
 
 depth of from 1 to 9 ft. of concrete as indicated. Fig. 306 constitutes a portion of a report 
 upon an investigation of the temperature changes in mass concrete made during the construc- 
 tion of the Arrowrock Dam by the United States Reclamation Service near Boise, Idaho ("Tem- 
 perature Changes in Mass Concrete" by Charles H. Paul and A. B. Mayhew, Tran^, Am. Soc. 
 
Sec. 5-43] CEMENT MORTAR AND PLAIN CONCRETE 261 
 
 C. E., vol. 79, p. 1225, 1915). A summary of the results obtained in these tests is given in the 
 following table. 
 
 Temperature Changes in Mass Concrete 
 Arrowrock Dam Tests 
 
 Proportions of mix (parts by volume) 
 
 Distance 
 to nearest 
 face (ft.) 
 
 Depth of 
 concrete over 
 
 Rise in 
 temperature 
 
 Highest 
 temperature 
 
 Time 
 reaching 
 highest tem- 
 perature, hr. 
 
 Sand cement 
 
 Sand 
 
 Gravel 
 
 Cobbles 
 
 thermometer 
 (ft.) 
 
 in degrees 
 Fahr. 
 
 in degrees 
 Fahr. 
 
 *1 
 
 2 
 
 4 
 
 23^ 
 
 1.0 
 
 2.0 
 
 2.5 
 
 78.0 
 
 1 
 
 *1 
 
 2 
 
 4 
 
 
 2.0 
 
 1.5 
 
 3.6 
 
 77.6 
 
 1 
 
 
 
 5 
 
 2 
 
 10.0 
 
 35.0 
 
 16.6 
 
 91.6 
 
 32 
 
 
 2 
 
 
 1^ 
 
 3.0 
 
 3.0 
 
 19.5 
 
 74.5 
 
 5 
 
 
 2^ 
 
 6 
 
 2 
 
 19.5 
 
 3.5 
 
 20.6 
 
 64.6 
 
 15 
 
 
 2M 
 
 4 
 
 
 3.5 
 
 6.0 
 
 26.9 
 
 69.8 
 
 5 
 
 
 2^ 
 
 6 
 
 2 
 
 76.0 
 
 15.5 
 
 27.5 
 
 94.0 
 
 31 
 
 
 23^ 
 
 
 2^ 
 
 31.0 
 
 80.0 
 
 35.7 
 
 96.2 
 
 101 
 
 
 2^ 
 
 6 
 
 2 
 
 20.0 
 
 28.5 
 
 36.7 
 
 86.2 
 
 45 
 
 * Thermometer bedded in fresh concrete in shallow trench dug in concrete 11 days old. 
 
 Similar tests made during the construction of the Kensico Dam at Valhalla, N. Y., by- 
 George T. Seabury, are reported to have established that {Proc. Am. Soc. C. E., vol. 29, p. 1247- 
 1253): 
 
 With the 1:3:6 concrete used, thermometers being inserted as soon as the concrete was placed and immedi- 
 ately covered to as great a depth as possible, the rise in temperature was uniformly about 40°F. 
 
 The maximum concrete temperature reached was often well above 100°F. in summer, in one instance, 118.5". 
 This maximum was usually reached in about 15 days. 
 
 The rate of increase in temperature was about 1** per hr. for 4 or 5 hr. gradually increased to 8°-10° per hr. 
 during the period of final set of the cement, and then dropped suddenly to about H** per hr. for a considerable 
 time. A total rise in temperature of from 25" to SO^F. often occurred subsequent to the period of final setting. 
 
 43. Porosity. — The porosity of a mortar or concrete is expressed by the percentage of void 
 space (space filled by air or uncombined water) in terms of the total volume. It is determined 
 experimentally by subtracting from the total apparent volume the volume of solid matter and 
 dividing by the total apparent volume. The total apparent volume may be determined by 
 direct measurement or by determination of the volume of water it displaces, absorption being 
 prevented by a waterproof coating of grease or varnish. The volume of solid matter is deter- 
 mined by weighing the specimen dry in air, and subsequently in water after the pores of the 
 material have been thoroughly impregnated with water. The difference in these weights 
 divided by the weight of a unit volume of water is the volume of solid matter. Extreme accu- 
 racy in determining porosity is not possible because of the difficulty encountered in completely 
 fiilling all the voids in the material. 
 
 The porosity of mortars and concretes is principally dependent upon the consistency of the 
 mixture, and the granularmetric composition of the aggregates. Plastic or wet consistencies 
 will in general produce mortars having less void space, and therefore lower porosity, than dry 
 consistencies, and a well-graded aggregate will form less porous mortar than one whose particles 
 are not well graded in size. A concrete will usually be considerably less porous than a mortar 
 because of its proportion of comparatively non-porous coarse aggregate, and a fine-sand aggre- 
 gate will in general produce more porous mortar than a coarse sand because of the larger amount 
 of water required in gaging the latter. 
 
 The porosity of mortars will usually be found within the limits of 15% and 30%, an average 
 1 : 2 mortar showing 20 to 25 % porosity. Concretes show from about 12 to about 20 % porosity, 
 
262 
 
 CONCRETE ENGINEERS' HANDBOOK 
 
 [Sec. 5-44 
 
 the lower figure applying to concretes having a relatively large proportion of coarse aggregate 
 and the higher figure to concretes having a relatively low proportion of coarse aggregate. 
 
 44. Permeability and Absorptive Properties. — The permeability of mortar or concrete is a 
 measure of the rate at which water under a given pressure will pass through a given thickness 
 of the material. The absorptive properties of a mortar or concrete constitute a measure of the 
 rate at which moisture will be absorbed when the material is exposed in damp situations or 
 covered with water under negligibly small heads. 
 
 Permeability is an important consideration where water-tightness of walls, etc., is re- 
 quired and percolation of water is not admissible. 
 
 Absorptive properties of a mortar determine its value as a dampproofing coat, particularly 
 in the event of its use as a plaster over metal lath, which must be protected to prevent corrosion. 
 In view of the disintegrating effect of expansion and contraction of mortars used as a plaster, 
 etc., the moisture content (which largely affects this expansion and contraction) should not be 
 greatly variable. Thus the least absorptive mortar will be most durable, up to the limit 
 reached when the cement content is relatively so high that the expansion and contraction is 
 disproportionately increased. 
 
 The determining of precise information concerning each of these properties is dependent 
 upon a standardization of methods of conducting tests. Such standard methods have not 
 yet been adopted, and it is therefore impossible to quote data as to the absolute permeability 
 or absorptive power of mortars. 
 
 Tests to determine the relative permeability and absorptive power of mortars were made 
 at the Structural Materials Laboratory as St. Louis in 1909, and are reported in Technologic 
 Paper 3 of the Bureau of Standards. Owing to the small number of tests made and certain 
 unsatisfactory features of the testing method employed, only a few general conclusions will 
 be drawn from the report of these tests. (1) Permeability decreases rapidly for all mixtures 
 with increase in age of the specimens when tested; (2) permeability decreases considerably 
 with the continuation of the flow; (3) permeability increases with the leanness of the mixture, 
 the dryness of the mixture, and increased coarseness of the sand. 
 
 Absorption was found to be dependent upon the same factors: it decreased with the age 
 of the mortar as a rule, but not as rapidly as did the permeability (especially with the leaner 
 mixtures); it decreased but slightly with increased richness of the mixtures; and the wetter 
 mixtures were slightly less absorptive than the dryer mixtures. 
 
 The permeability of mortars and concrete is closely related to the porosity, but the relation- 
 ship is not always direct, and is by no means constant, since the continuity and size of the pores 
 determines permeability more than does the actual percentage of voids. 
 
 The Joint Committee Report says concerning the permeability and the waterproofing of 
 concrete : 
 
 Many expedients have been resorted to for rendering concrete impervious to water. Experience shows, 
 however, that when concrete or mortar is proportioned to obtain the greatest practicable density and is mixed to 
 the proper consistency, the resulting mortar or concrete is impervious under moderate pressure. 
 
 On the other hand, concrete of dry consistency is more or less pervious to water, and though compounds of 
 various kinds have been mixed with the concrete or applied as a wash to the surface, in an effort to offset this 
 defect, these expedients have generally been disappointing, for the reason that many of these compounds have at 
 best but temporary value, and in time lose their power of imparting impermeability to the concrete. 
 
 In the case of subways, long retaining walls and reservoirs, provided the concrete itself is impervious, cracks 
 may be so reduced, by horizontal and vertical reinforcement properly proportioned and located, that they will 
 be too minute to permit leakage, or will be closed by infiltration of silt. 
 
 Asphaltic or coal-tar preparations applied either as a mastic or as a coating on felt or cloth fabric, are used 
 for waterproofing, and should be proof against injury by liquids or gases. 
 
 For retaining and similar walls in direct contact with the earth, the application of one or two coatings of 
 hot coal-tar pitch, following a painting with a thin wash of coal-tar dissolved in benzol, to the thoroughly dried 
 surface of concrete is an efficient method of preventing the penetration of moisture from the earth. 
 
 45. Protection of Embedded Steel From Corrosion.^ — The Joint Committee Report says 
 concerning the corrosion of metal reinforcement in concrete : 
 
Sec. &-46] 
 
 CEMENT MORTAR AND PLAIN CONCRETE 
 
 263 
 
 Tests and experience indicate that steel sufficiently embedded in good concrete is well protected against 
 corrosion, no matter whether located above or below water level. It is recommended that such protection be not 
 less than 1 in. in thickness. If the concrete is porous so as to be readily permeable by water, as when concrete is 
 laid with a very dry consistency, the metal may corrode on account of the presence of moisture and air. 
 
 The historic tests made by Prof. C, L. Norton for the Insurance Engineering Station in 
 Boston in 1902 led to the following conclusions the validity of which has never been disproved by 
 either tests or experience {Eng. News, vol. 48, p. 333) : 
 
 1. Neat Portland cement, even in thin layers, is an effective preventive of rusting. 
 
 2. Concretes, to be effective in preventing rust, must be dense and without voids or cracks. They should 
 be mixed quite wet where applied to the metal. 
 
 3. The corrosion found in cinder concrete is mainly due to the iron oxide, or rust, in the cinder, and not 
 to the sulphur. 
 
 4. Cinder concrete, if free from voids and well rammed when wet, is about as effective as stone concrete in 
 protecting steel. 
 
 Further tests made by Prof. Norton in 1903 showed conclusively that steel reinforcement, 
 corroded before being embedded in concrete, does not corrode further, provided only that it 
 has a continuous unbroken coating of concrete. This fact is important since it is almost impos- 
 sible to prevent exposure of reinforcing steel on construction work to the elements, and a large 
 proportion of such steel is therefore corroded when placed in the work. 
 
 Experience has abundantly shown that if concrete be mixed sufficiently wet so that it will 
 flow about the reinforcement with only a moderate amount of puddling, the thin film of rich 
 mortar which coats the steel affords a perfect preventive of corrosion. 
 
 46. Weight of Mortar and Concrete. — The weight of mortars and concretes varies with the 
 proportions of the mixture, the consistency used in mixing, and the character and granular- 
 metric composition of the aggregates. William B. Fuller found the following range of weights 
 of mortars of various proportions made with the same sand and cement : 
 
 
 1:1 
 
 1:2 
 
 1:3 
 
 1:4 
 
 1:5 
 
 1:6 1:7 
 
 
 145.1 
 
 143.3 
 
 140.0 
 
 137.7 
 
 138.6 
 
 135.5 137.6 
 
 The Bureau of Standards found the following relation between weight of gravel concrete 
 and the proportions of mixture (Tech. Paper 58): 
 
 Proportions of mixture 
 
 1:1:2 
 
 1:13^:3 
 
 1:2:4 
 
 1:23^:5 
 
 1:3:6 
 
 1:4:8 
 
 Ave. weight (lb. per cu. ft.) 
 
 147 
 
 145 
 
 144 
 
 143 
 
 142 
 
 140 
 
 The same series of tests of the Bureau of Standards indicate the following relations to hold 
 between weight and consistency: 
 
 The cinder concrete appears to 
 be slightly heavier the wetter the 
 mixture, but all of the rock con- 
 cretes are slightly heavier the drier 
 the mixture. No universally appli- 
 cable relation between weight of 
 concrete and the class of aggregate 
 used is shown by tests. Different 
 gravel concretes, for instance, will 
 show greater variations in weight 
 than the difference in average weight of gravel concretes and granite concretes. For pur- 
 poses of design the weight of any class of stone concrete may be assumed to be 150 lb. per 
 cu. ft., while cinder concrete may be assumed to weigh 115 lb. per cu. ft. 
 
 Proportions by 
 volume 
 1:2:4 
 Coarse aggregate 
 
 Weight per cubic foot 
 
 Watery or 
 
 fluid 
 consistency 
 
 Soft, mushy 
 consistency 
 
 Stiff quaking 
 consistency 
 
 Cinder 
 
 115.2 
 
 114.9 
 
 113.1 
 
 Granite 
 
 147.6 
 
 147.7 
 
 148.9 
 
 
 139.6 
 
 142.7 
 
 144.5 
 
 Limestone 
 
 144.7 
 
 145.9 
 
 147.8 
 
SECTION 6 
 
 GENERAL PROPERTIES OF REINFORCED CONCRETE 
 
 1. Advantages of Combining Concrete and Steel.— Steel can be put into a form to resist 
 a given tensile stress much more cheaply than to resist an equal amount of compressive stress. 
 This comes from the fact that the solid bar is well adapted to take tensile stresses, while for 
 compressive stresses the steel must be made into forms of more extended cross-section in order 
 to provide sufficient lateral rigidity. Other facts to be noted are the lack of durability of 
 steel in many locations and its failure to stand up under a high heat. 
 
 Concrete, on the other hand, cannot be used in tension except to a very limited extent, 
 but its compressive strength is sufficiently high to be of structural importance. It is also a good 
 fireproof material and has great durability. In addition, concrete is a cheaper material than 
 I steel, can readily be obtained in almost any locality, and tests and the results of observations 
 show that it thoroughly protects embedded steel from corrosion. 
 
 From the above considerations it follows that the advantages to be gained by using con- 
 crete reinforced with steel instead of either material separately will vary with different types 
 of structures. In structural members subjected to both tension and compression, as in all 
 forms of beams, the proper combination of the two materials meets with the best success. 
 Steel rods embedded in the lower side of the beam carry the tensile stresses while the compressive 
 stresses are carried by the concrete. Here the steel is used in its cheapest form and the con- 
 struction may be made strong, economical, and very durable. In compressive members of 
 appreciable length, such as columns, a combination of the two materials is also quite advanta- 
 geous, although to a varying degree depending upon whether the reinforcing steel is used in 
 the form of small rods or as structural-steel shapes. 
 
 2. Bond Between Concrete and Steel. — Most of the tests on bond have been made by 
 embedding a short reinforcing bar in a block of concrete and pulling it out in a testing machine. 
 In the kind of tests referred to, the concrete is in compression and conditions do not correspond 
 to those ordinarily encountered in beams and slabs. Pull- 
 out tests of this character, however, have been found to be 
 of valuable aid in determining actual values for bond. This 
 conclusion was reached through an extended series of ex- 
 periments at the University of Illinois during the period 
 1909-1912; a series which include both pull-out tests and Fig. l. 
 
 beam tests, as described further on in this article. 
 
 In a series of experiments made at the University of Wisconsin, test beams were arranged 
 as shown in Fig. 1, the reinforcing bars being embedded only a short distance from each end, 
 leaving the middle portion exposed. The stress in the rods were computed from the observed 
 deformations. The beam was prevented from failing in the early stages of the tests, by an 
 upper set of auxiliary rods. Failure finally occurred by the pulling out of the lower rods, as 
 intended. Pull-out tests were also made with concrete cylinders and rods arranged as shown 
 in Fig. 2. Cylinders designated as (a) were tested in the ordinary way with their upper surfaces 
 bedded against the lower face of the puUing head. In cylinders (6) tension was applied to 
 both upper and lower rods, bringing tension also in the concrete. The principal conclusions 
 arrived at were as follows (the word static is used in connection with beams progressively 
 tested to failure under loads gradually applied, and the word repeated occurs in connection 
 with those beams subjected to repeated loadings) :^ 
 
 1 Bull. 5, vol. 5, University of Wisconsin. 
 
 265 
 
266 
 
 CONCRETE ENGINEERS' HANDBOOK 
 
 [Sec. 6-2 
 
 The static bond between 1:2:4 concrete and plain round steel rods increases with age at least up to 6 months. 
 About 80% of the 6 months' bond strength is developed in 1 month. 
 
 Owing to the variation in the results of individual tests and the difference between laboratory and practical 
 working conditions, it does not seem as though the maximum static bond between concrete of the class used and 
 plain rods less than % in. diameter should be assumed greater than 250 lb. per sq. in. or for the rods or larger size 
 200 lb. per sq. in. 
 
 The method of making bond tests by pulling a rod from a cylinder of concrete in such a manner that the 
 concrete around the rod is compressed gives results which are neither of quantitative nor qualitative value. The 
 results obtained are dependent largely upon the compressive stress acting on the head of the cylinder. Cylinder 
 tests in which the rod and concrete are both subjected to a tensile stress give results more in accord with the bond 
 values obtained from beam tests. 
 
 The static bond between the class of concrete employed and corrugated bars is about twice as great as that 
 which can be developed with plain round rods of about the same size. The static bond between concrete and rusted 
 rods is very much greater than that obtained where plain round rods are used. 
 
 From the tests under repeated loadings it seems evident that 50 to 60% of the static bond between concrete 
 and plain round rods may be repeated a large number of times without failure in bond; that 60 to 70% are the 
 corresponding figures for corrugated bars. Under repeated loadings the bond between concrete and rusted round 
 rods is considerably greater than that between concrete and plain round rods. 
 
 Considering the severity of the tests made there seems to be no valid reason for believing that the bond be- 
 tween concrete and plain round rods will be destroyed under repeated loadings, providing a proper working value is 
 used. Such a value for concrete of the class used in these tests should not be over 50 lb. per sq. in. 
 
 1Q In tests at the University of Illinois, attention was given to 
 _ . j-y obtaining accurate measurement of slip of bar through the concrete 
 i! i as the loading progressed, in both the ordinary pull-out tests and in 
 ■—-^ '■ ' '• tests on beams. In the beams of this series the concrete was not cut 
 away from the rods as in the tests at the University of Wisconsin. 
 
 In the pull-out tests the amount of movement of the free end 
 of the embedded bar was measured by means of an Ames gage. 
 In the beam tests, the Ames gage was used to measure center deflec- 
 tions and the movement of the ends of the reinforcing bars. In 
 many of the tests, observations were also made on the amount of 
 slip of the reinforcing bar with respect to the adjacent concrete at 
 several points along the length of the beam. 
 
 The concrete blocks used in the pull-out tests were usually 8 in. 
 in diameter and 8 in. long, with the bar embedded axially. The 
 beams tested were 8 by 12 in. in section with an effective depth of 10 
 in. The span length was generally 6 ft. All beams were tested 
 with two symmetrical loads, generally at the one-third points of the 
 span. With the exception of six tests, the longitudinal reinforce- 
 ment consisted of a jingle bar of large diameter placed horizontally throughout the length of 
 the beam. The principal results of these tests and the conclusions reached were as follows:^ 
 
 (a) 
 
 Fig. 2. 
 
 Pull-out Tests 
 
 Bond between concrete and steel may be divided into two principal elements, adhesive resistance and sliding 
 resistance. The source of adhesive resistance is not known, but its presence is a matter of universal experience 
 with materials of the nature of mortar and concrete. Sliding resistance arises from inequalities of the surface of 
 the bar and irregularities of its section and alignment together with the corresponding conformations in the con- 
 crete. The adhesive resistance must be overcome before sliding resistance comes into action. In other words, 
 the two elements of bond resistance are not effective at the same time at a given point. Many evidences of the 
 tests indicate that adhesive resistance is much the more important element of bond resistance. 
 
 Relation of Bond Stress to Slip of Bar as Load Increases. — Pull-out tests with plain bars show that a con- 
 siderable bond stress is developed before a measurable slip is produced. Slip of bar begins as soon as the adhesive 
 resistance is overcome. After the adhesive resistance is overcome, a further slip without an opportunity of rest 
 is accompanied by a rapidly-increasing bond stress until a maximum bond resistance is reached at a definite amount 
 of slip (see Fig. 3). 
 
 1 Bull. 71, Engineering Experiment Station, University of Illinois. 
 
Sec. 6-2] 
 
 GENERAL PROPERTIES OF REINFORCED CONCRETE 
 
 267 
 
 The true relation of slip of bar to bond stress can best be studied by considering the action of a bar over a 
 very short section of the embedded length. The difficulties arising from secondary stresses made it impracticable 
 to conduct tests on bars embedded very short lengths. The desired results (Fig. 3) were obtained by varying 
 the forms of the specimens in such a way that the effect of different combinations of dimensions could be studied. 
 
 Pull-out tests with plain bars of the same size embedded different lengths furnish data which suggest the 
 values of bond resistance over a very short length of embedment, or indicate values of bond resistance which are 
 independent of the length of embedment. Tests with bars of different size which were embedded a distance 
 proportional to their diameters give the true relation when the effect of size of bar is eliminated. Two series of 
 tests of this kind on plain round bars of ordinary mill surface gave almost identical values for bond resistance after 
 eliminating the effect of length of embedment and size of bar, and we may consider that these values represent the 
 stresses which were developed in turn over each unit of area of the embedded bar as it was withdrawn by a load 
 applied by the method used in these tests. These tests showed that for concrete of the kind used (a 1 : 2 : 4 mix, 
 stored in damp sand and tested at the age of about 60 days) the first measurable slip of bar came at a bond stress 
 of about 260 lb. per sq. in., and that the maximum bond resistance reached an average value of 440 lb. per sq. in. 
 If we conclude that adhesive resistance was overcome 
 at the first measurable slip, it will be seen that the ad- 
 hesive resistance was about 60% of the maximum bond 
 resistance. This ratio did not vary much for a wide 
 range of mixes, ages, size of bar, condition of storage, 
 etc. 
 
 Sliding resistance reached its maximum value 
 for plain bars of ordinary mill surface at a slip of 
 about 0.01 in. The constancy in the amount of slip 
 corresponding to the maximum bond resistance for a 
 wide range of mixes, ages, sizes of bar, conditions of 
 storages, etc., is a noteworthy feature of the tests. 
 With further slip the sliding resistance decreased 
 slowly at first, then more rapidly, until with a slip of 
 0.1 in. the bond resistance was about one-half its maxi- 
 mum value. 
 
 Bond Resistance in Terms of Compressive 
 Strength of Concrete. — Pull-out tests with plain round 
 bars show end slip to begin at an average bond stress 
 equal to about one-sixth the compressive strength of 
 6-in. cubes from the same concrete; the maximum bond 
 resistance is equal to about one-fourth the compres- 
 sive strength of 6-in. cubes. These values were about 
 the same for a wide range of mixes, ages, and conditions 
 of storage. In terms of the compressive strength of 8 
 by 16-in. concrete cylinders these values would be about 
 13%for first end slip and 19% for the maximum bond 
 resistance. 
 
 Distribution of Bond Stress Along a Bar. — The 
 
 tests indicate that bond stress is not uniformly dis- 
 tributed along a bar embedded any considerable length 
 and having the load applied at one end. Slip of bar 
 begins first at the point where the bar enters the con- 
 crete, and the bond stress must be greater here than elsewhere until a sufficient slip has occurred to develop the 
 maximum bond resistance at this point. SUp of bar begins last at the free end of the bar. After slip becomes 
 general, there is an approximate equality of bond stress throughout the embedded length. 
 
 Variation of Bond Resistance with Size, Shape, and Condition of Surface of Bar. — The maximum bond 
 resistance was not materially different for bars of different diameters. 
 
 Rusted bars gave bond resistances about 15% higher than similar bars with ordinary mill surface. 
 
 The tests with flat bars showed wide variations of bond resistance and were not conclusive. Square bars 
 gave values of unit stress about 75 % of those obtained with plain round bars. 
 
 T-bars gave lower unit bond resistance than plain round bars, but gave about double the bond resistance 
 per unit of length that was found for the plain round bars of the same sectional area. 
 
 With polished bars the bond resistance is due almost entirely to adhesion between the concrete and steel. 
 Numerous tests with polished bars embedded in 1:2:4 concrete and tested at 60 days indicated a maximum bond 
 resistance of about 160 lb. per sq. in., or about 60% of the bond resistance of bars of ordinary surface at small 
 amounts of slip. 
 
 Adhesive resistance must be destroyed, sliding resistance largely overcome, and the concrete ahead of the 
 projections must undergo an appreciable compressive deformation before the projections on a deformed bar 
 become effective in taking bond stress. The tests indicate that the projections do not materially assist in re- 
 sisting a force tending to withdraw the bar until a sUp has occurred approximating that corresponding to the 
 
 1 
 to 200 
 
 100 
 
 Fig. 3. 
 
 • /g-/n corrugated rounds, 
 
 Embedmeni- varable. 
 
 ♦ l^-in plain rounds, 
 
 Embedment- mriable. 
 o Plain rounds, D/amefer 
 and embednient yar/ab/e 
 
 IS -4 concref-e. 
 Age about 2 monttis. 
 
 I I I I L 
 
 .01 
 
 .OZ .03 .04- .05 
 
 Slip of Bar -Inches 
 
 -Relation of bond to slip of bar as load increases. 
 
268 
 
 CONCRETE ENGINEERS' HANDBOOK 
 
 [Sec. 6-2 
 
 maximum sliding resistance of plain bars. As slip continues a larger and larger portion of the bond stress is taken 
 by direct bearing of the projections on the concrete ahead. 
 
 In determining the comparative merits of deformed bars, the bar which longest resists beginning of slip 
 should be rated highest, other considerations being equal. 
 
 The concrete cylinders of the pull-out specimens with deformed bars were reinforced against bursting or 
 splitting, because it was desired to study the load-slip relation through a wide range of values. In only a few tests 
 was the maximum bond resistance reached at an end slip less than 0.1 in. It should be recognized that, in general, 
 the bond stresses reported for deformed bars at end slip of 0.05 and 0.1 in. could not have been developed with 
 bars embedded in unreinforced blocks. These high values of bond resistance must not be considered as available 
 under the usual conditions of bond action in reinforced-concrete members. In the tests in which the blocks were 
 not reinforced, evidence of splitting of the blocks was found at end slips of 0.02 to 0.05 in. 
 
 The normal components of the bearing stresses developed by the projections on a deformed bar may pro- 
 duce very destructive bursting stresses in the surrounding concrete. The bearing stress between the projections 
 and the concrete in the tests with certain types of commercial deformed bars was computed to be from 5800 to 
 14,000 lb. per sq. in. at the highest bond stresses considered in these tests. The large slip and the high bearing 
 stresses developed in the later stages of the tests show the absurdity of seriously considering the extremely high 
 values that are usually reported to be the true bond resistance of many types of deformed bars. 
 
 Round bars with standard V-shaped threads gave much higher bond resistance at low slips than the com- 
 mercial deformed bars. The average bond resistance at an end slip of 0.001 in. was 612 lb. per sq. in. The maxi- 
 mum bond resistance was 745 lb. per sq. in. These were the only deformed bar tests in which failure came by 
 shearing the surrounding concrete. 
 
 The 1-in. twisted square bars gave a bond resistance per unit of surface at an end slip of 0.001 in., only 88% 
 of that for the plain rounds. Following an end slip of about 0.01 in., these bars showed a decided decrease in bond 
 resistance, and a slip of 5 to 10 times this amount was required to cause the bond resistance to regain its first 
 maximum value. After this, the bond resistance gradually rose as the bar was withdrawn. Some of the bars were 
 withdrawn 2 or 3 in. before the highest resistance was reached. The apparent bond stresses at these slips were 
 very high; but, of course, such stresses and slips could not be developed in a structure and could not have been 
 developed in the tests had the blocks not been reinforced against bursting. Such values are entirely meaningless 
 under any rational interpretation of the tests. 
 
 Anchoring of Reinforcing Bars. — The tests with plain round bars anchored by means of nuts and with washers, 
 only showed that the entire bar must slip an appreciable amount before these forms of anchorage come into action. 
 Anchorages of the dimensions used in these tests did not become effective until the bar had slipped an amount 
 corresponding to the maximum bond resistance of plain bars. With further movement the apparent bond re- 
 sistance was high, but was accompanied by excessive bearing stresses on the concrete. 
 
 The load-slip relation for bars anchored by means of hooks and bends was not determined.^ 
 
 Influence of Method of Curing Concrete. — Tests on specimens stored under different conditions indicate that 
 concrete stored in damp sand may be expected to give about the same bond resistance and compressive resistance 
 as that stored in water. Water-stored specimens gave values of maximum bond resistance higher in each instance 
 than the air-stored specimens; the increase for water storage ranged from 10 to 45%. The difference seemed to 
 increase with age. The presence of water not only did not injure the bond for ages up to 3 years, but it was an 
 important factor in producing conditions which resulted in high bond resistances. However, it was found that 
 specimens tested with the concrete in a saturated condition gave lower values for bond than those which had been 
 allowed to dry out before testing. The bars in specimens which had been immersed in water as long as 3H years 
 showed no signs of rust or other deterioration. 
 
 Influence of Freezing of Concrete. — Specimens made outdoors in freezing weather, where they probably 
 froze and thawed several times during the period of setting and hardening, were almost devoid of bond strength. 
 
 Influence of Age and Mix of Concrete. — Pull-out tests made at early ages gave surprisingly high values of 
 bond resistance. Plain bars embedded in 1 : 2 : 4 concrete and tested at 2 days did not show end slip of bar until a 
 bond stress of 75 lb, per sq. in. was developed. Bond resistance increases most rapidly with age during the first 
 month. The richer mixes show a more rapid increase than the leaner ones. The tests on concrete at ages of 
 over 1 year showed that the bond resistance of specimens stored in a damp place may be expected ultimately to 
 reach a value as much as twice that developed at 60 days. 
 
 The load-sUp relation of leaner and richer mixes was similar to that for 1:2:4 concrete. For a wide range of 
 mixes the bond resistance was nearly proportional to the amount of cement used. This relation did not obtain 
 in a mix from which the coarse aggregate had been omitted. 
 
 Effect of Continued and- Repeated Load. — When the application of load was continued over a considerable 
 period of time or when the load was released and reapplied, the usual relation of slip of bar to bond resistance was 
 considerably modified. The few tests which were made indicate that the bond stress corresponding to beginning 
 of slip is the highest stress which can be maintained permanently or be reapplied indefinitely without failure of 
 bond. 
 
 1 Other tests have shown that a semicircular hook of 4 times the diameter of the bar and well embedded 
 in concrete may be assumed to develop the elastic limit of the steel without exceeding the bearing strength of 
 the concrete. The curved ends should consist of bends through 180 deg. with a short length of straight rod 
 beyond the bend, A short cross rod aids greatly in distributing the bearing stress in the concrete. Short 
 square hooks upon the ends of bars are not of great value. 
 
Sec. 6-2] 
 
 GENERAL PROPERTIES OF REINFORCED CONCRETE 
 
 Effect of Concrete Setting Under Pressure. — Bond resistance of plain bars is greatly increased if the con- 
 crete is caused to set under pressure. With a pressure of 100 lb. per sq. in. on the fresh concrete for 5 days after 
 molding, the maximum bond resistance was increased 92 % over that of similar bars in concrete which had set, 
 without pressure. The greater density of the concrete and its more intimate contact with the bar seems to be 
 responsible for the increased bond resistance. Light pressures gave an appreciable increase in bond resistance. 
 With polished bars the effect of pressure was slight. 
 
 As might have been expected, the compressive resistance of concrete setting under pressure was increased in 
 much the same ratio as the bond resistance. At the age of 80 days the initial modulus of elasticity in compression 
 for concrete which set under a pressure of 100 lb. per sq. in. was about 37 % higher and the compressive strength 
 was increased by about 73 % over that of concrete which had set without pressure. The density of the concrete, 
 as determined by the unit weights, was increased about 4 % by a pressure of 100 lb. per sq. in. on the fresh con- 
 crete. The increase in strength and density was relatively greater for the low than for the high pressures. A 
 pressure continued for 1 day, or until the concrete had taken its final set and hardening had begun, seems to have 
 produced the same effect in increasing the strength and elastic properties of the concrete as when the pressure 
 was continued for a much longer period. 
 
 Beam Tests 
 
 
 Number 
 of tests 
 
 First end 
 slip of bar 
 
 End slip 
 of 0.001 
 in. 
 
 Maximum 
 bond 
 stress 
 
 1 and l^-in. plain round. . . 
 
 28 
 
 245 
 
 340 
 
 375 
 
 ^4 -in. plain round 
 
 3 
 
 186 
 
 242 
 
 274 
 
 ^i-in. plain round 
 
 3 
 
 172 
 
 235 
 
 255 
 
 1-in. plain square 
 
 6 
 
 190 
 
 248 
 
 278 
 
 1-in. twisted square 
 
 3 
 
 222 
 
 289 
 
 337 
 
 l^i-in. corrugated round. . . . 
 
 9 
 
 251 
 
 360 
 
 488 
 
 The mean computed values for bond stresses in the 6-ft. beams in the series of 1911 and 1912 were as given 
 below. All beams were of 1 : 2 : 4 concrete, tested at 2 to 8 months by loads applied at the one-third points of the 
 span. Stresses are given in pounds per square inch. 
 
 In the beams reinforced with 
 plain bars end slip begins at 67 % of 
 tlie maximum bond resistance; for 
 the corrugated rounds this ratio is 
 51%, and for the twisted squares, 
 66%. 
 
 The bond unit resistance in 
 beams reinforced with plain square 
 bars, computed on the superficial 
 area of the bar, was about 75% of 
 that for similar beams reinforced 
 with plain round bars of similar size. 
 
 Beams reinforced with twisted 
 square bars gave values at small slips about 85% of those found for plain rounds. At the maximum load, the 
 bond-unit stress with the twisted bars was 90 % of that with plain round bars of similar size. 
 
 In the beams reinforced with IJ'i-in. corrugated rounds, slip of the end of the bar was observed at about the 
 same bond stress as in the plain bars of comparable size. At an end slip of 0.001 in., the corrugated bars gave a 
 bond resistance about 6 % higher and at the maximum load, about 30 % higher than the plain rounds. 
 
 The beams in which the longitudinal reinforcement consisted of three or four bars smaller than those used in 
 most of the tests gave bond stresses which, according to the usual method of computation, were about 70% of 
 the stresses obtained in the beams reinforced with a single bar of large size. It seems probable that the lower 
 computed bond stresses in these tests are due to errors in the assumptions made as to the distribution of bond 
 stress and not to actual differences of bond resistance in the bars of different size. 
 
 The tests on beams with the loads placed in different positions with respect to the span gave little variation 
 in bond resistance during the early stages of the tests. The maximum bond resistances increased rapidly as the 
 load approached the supports. These tests indicate that the variation in the maximum bond stresses must be due 
 to the presence of other than normal beam action. 
 
 The bond stresses developed in the beam tests indicate that with beams of the same cross-section the bond 
 stresses are distributed in the same way during the early stages of the test in beams varying widely in span length 
 and loading. During the later stages of the test, the distribution of bond stress seems to depend largely upon 
 the conditions of stress in the concrete through the region of the span where beam bond stresses are high. The 
 distribution of bond stresses in beams of different cross-section apparently varies with the relative dimensions of 
 the beam and the reinforcing bars. 
 
 In the reinforced-concrete beams it was found that very small amounts of slip at the ends of the bar repre- 
 sented critical conditions of bond stress. For beams failing in bond the load at an end slip of 0.001 in. was 89 to 
 94 % of the maximum load found in beams reinforced with plain bars, and 79 % of the maximum load for similar 
 beams reinforced with corrugated bars. As soon as slip of bar became general, other conditions were introduced 
 which soon caused the failure of the beam. 
 
 The bond stresses developed in a reinforced-concrete beam by a load applied as in these tests varies widely 
 over the region in v/hich beam bond stresses are present. High bond stresses are developed just outside the load 
 points at comparatively low loads. The load which first developed a bond stress nearly equal to the maximum 
 bond resistance in the region of beam bond stresses produced a stress near the support which was not more than 
 about 15 to 40 % of the maximum bond resistance. As the load is increased, the region of high bond stress is thrown 
 nearer and nearer the support, and at the same time the bond stress over the region just outside the load point 
 
270 
 
 CONCRETE ENGINEERS' HANDBOOK 
 
 [Sec. 6-3 
 
 becomes steadily smaller. This indicates a piecemeal development of the maximum bond stress as the load is 
 increased. The actual bond stresses in certain tests varied from less than one-half to more than twice the average 
 bond resistance computed in the usual manner. 
 
 Slip of bar in a reinforced-concrete beam has a marked influence in increasing the center deflection during 
 the later stages of loading. 
 
 The comparison of the bond stresses developed in beams and in puU-out specimens from the same materials 
 is of interest. Such a comparison should be made for similar amounts of slip. In the pull-out tests the maximum 
 bond resistance came at a slip of about 0.01 in. for plain bars. The mean bond resistance for the deformed bars 
 tested was not materiallv different from that of the plain bars until a sHp of about 0.01 in. was developed; with a 
 continuation of slip the projections came into action and with much larger slip high bond stresses were developed. 
 The beam tests showed that about 79 to 94 % of the maximum bond resistance was being developed when the bar 
 had slipped 0.001 in. at the free end; hence the bond stress developed at an end slip of 0.001 in. was used as a basis 
 of the principal comparisons in the pull-out tests. However, it is recognized that, under certain conditions, the 
 stresses developed at larger amounts of slip may have an important bearing on the effective bond resistance of 
 the bar. 
 
 The pull-out tests and beam tests gave nearly identical bond stresses for similar amounts of slip in many 
 groups of tests, but it seems that this was the result of a certain accidental combination of dimensions in the two 
 forms of specimens and did not indicate that the computed stresses in the beams were the correct stresses. How- 
 ever, it is believed that a properly designed pull-out test does give the correct value of bond resistance, and gives 
 values which probably closely represent the bond stresses which actually exist in a beam or other member as 
 slipping is produced from point to point along the bar. The relative position of the bar during molding may be 
 expected to influence the values of bond resistance found in the tests. 
 
 A working bond stress equal to 4 % of the compressive strength of the concrete tested in the form of 8 by 16- 
 in. cylinders at the age of 28 days (equivalent to 80 lb. per sq. in. in concrete having a compressive strength of 
 2000 lb. per sq. in.) is as high a stress as should be used. This stress is equivalent to about one-third that causing 
 first slip of bar and one-fifth of the maximum bond resistance of plain round bars as determined from pull-out tests. 
 The use of deformed bars of proper design may be expected to guard against local deficiencies in bond resistance 
 due to poor workmanship and their presence may properly be considered as an additional safeguard against ulti- 
 mate failure by bond. However, it does not seem wise to place the working bond stress for deformed bars higher than 
 that used for plain bars. 
 
 3. Length of Embedment of Reinforcing Bars to Provide for Bond. — Let fs be the working 
 tensile strength of the steel, As the area of bar, o the circumference of bar, d the diameter or 
 thickness of bar, u the working unit bond strength, and x the required length of embedment 
 (or grip) for the above values of f, and u. Then, to develop the strength of the steel, using 
 either round or square bars, 
 
 XOU = A,fs J» 
 
 or 
 
 4. Ratio of the Moduli of Elasticity. — ^Let fs = unit stress in steel, fc = unit stress in con- 
 crete, Es = modulus of elasticity of steel, and Ec = modulus of elasticity of concrete. Since 
 the modulus of elasticity of a material is the ratio of stress to deformation, it follows that, for 
 equal deformations, the stresses in the steel and concrete will be as their moduli of elasticity. 
 Thus, 
 
 fs ^Es 
 fo Eo 
 
 This ratio of the moduli is generally denoted by the letter n, or 
 
 fs = nfc 
 
 The equation just given shows that if the stress in either the steel or concrete of a concrete 
 column is known, the stress in the other material can be found, and this relation is made use of 
 in the derivation of column formulas. Fig. 26, Sect. 5, page 250, shows that the modulus of 
 elasticity of concrete in compression is less for the greater loads, and hence the value of n is 
 greater. Thus, it is plain that with increasing loads in concrete columns the steel receives a 
 greater proportionate stress, the variation in the amount carried by the steel depending on the 
 variation in the value of n. In order to take account of the fact that under increasing loads the 
 
Sec. 6-5] GENERAL PROPERTIES OF REINFORCED CONCRETE 271 
 
 steel receives an increasing proportion, it is desirable to use a value of n in the computations 
 for design somewhat larger than that which is obtained by taking a value of Ec corresponding 
 to working loads on small prisms (about 10). A value of 15 for n may well be used for the ordi- 
 nary 1:2:4 mix. 
 
 In concrete beams, experiments show that the tension which remains in the concrete just 
 below the neutral axis, and properly not allowed for in the derivation of the beam formulas, 
 has its effect in the position of the neutral axis and the strength of the beam. It is found that 
 a value of 15 for n is not too large for calculations of strength of beams, assuming the ordinary 
 1 :2 :4mix, although great accuracy in this respect is not necessary. This value of 15 for 
 n is the one most generally used, but a value of 12 is also frequently employed. The value of 
 15 corresponds to a value of Ec of 2,000,000 which is somewhat low as determined by compressive 
 tests. 
 
 For the proper values of n to use for other mixtures see recommendations of the Joint 
 Committee, Appendix B. 
 
 Comparatively few tests have been made on the elasticity of concrete in tension, but these 
 seem to indicate that for small stresses, it is practically the same as in compression, although 
 probably slightly less. 
 
 5. Behavior of Reinforced Concrete Under Tension. — Early tests indicated that the ulti- 
 mate stretch of reinforced concrete in tension is as much as 10 times that of plain concrete, 
 but such results were due to the fact that it was found extremely difficult to determine just when 
 the concrete begins to crack. Cracks do not become noticeable, even on very close examination, 
 until a stretching occurs corresponding to a tensile stress much beyond the ultimate tensile 
 strength of the concrete. The steel causes a uniform elongating of the concrete so that the 
 cracks which open up are very small and remain invisible for some time. 
 
 A method of detecting minute cracks in the tensile side of beams was accidentally discovered 
 in 1901-02 in some experiments made at the University of Wisconsin. It was found that when 
 beams were hardened in water and only partially dried before testing, very fine hair-cracks be- 
 came noticeable at a moderate load. Before these cracks occurred, however, dark wet lines 
 appeared across the beam, and it was observed that each of these lines was later followed by a 
 very fine crack. These water-marks were proven to be incipient cracks by the sawing out of 
 a strip of concrete along the outer part of the beam. Careful measurements of extension showed 
 that these streaks, or water-marks, occurred at practically the same deformation at which the 
 concrete ruptured when not reinforced. This same phenomenon has since been observed by 
 many careful experimenters, and the fact is now generally established that concrete, reinforced 
 with steel, does not elongate under tensile stress to any greater extent before cracking than plain 
 concrete. 
 
 A reinforced-concrete beam for working loads is usually more heavily stressed on the ten- 
 sion side than the ultimate tensile strength of plain concrete — enough steel being usually em- 
 bedded near the lower face to permit the full allowable compressive strength of the concrete 
 to be utilized. The presence, then, of the cracks above referred to, accurring long before a 
 reinforced-concrete beam has obtained its working load, must seriously affect the tensile strength 
 of the concrete. The moment formulas now in most general use for the design of reinforced- 
 concrete beams neglect entirely the tensile strength of the concrete. 
 
 Experiments have shown that concrete when well placed and mixed somewhat wet, com- 
 pletely protects the steel in the tensile side of a beam from corrosion, even when the unit stress 
 in the steel somewhat exceeds the elastic hmit. 
 
 6. Shrinkage and Temperature Stresses. — In reinforced-concrete structures which are 
 free to contract and expand, the stresses occurring from temperature changes and from shrink- 
 age in hardening are due wholly to the mutual action of the steel and concrete. Of the stresses 
 produced from these two causes, those which result from hardening are the greater, but experi- 
 ments show that even these are not sufficient to be of practical importance. In regard to the 
 
272 
 
 CONCRETE ENGINEERS' HANDBOOK 
 
 [Sec. 6-7 
 
 temperature stresses, they are negligible by reason of the nearly equal rates of expansion of the 
 two materials. 
 
 On the other hand, if reinforced-concrete structures are restrained by outside forces, or if 
 they are of such dimensions that they cannot be considered as sufficiently well bonded to act 
 as a unit — such as long retaining walls — then the stresses resulting are much greater, and the 
 tensile strength of the concrete will be reached (this will occur with a drop in temperature some- 
 where between 10 and 20°F.), thus producing cracks, called contraction cracks. To prevent 
 plainly noticeable cracks due to shrinkage and lowering of the temperature, all exposed surfaces 
 should be reinforced with about 0.3 of 1 % of steel, based on the cross-section of the concrete. 
 This is less than the amount required theoretically, but experience shows this amount to give 
 very satisfactory results where the foundations are stable. If the structure is fixed in two 
 directions, the reinforcement must be placed accordingly. The above percentage of steel 
 should be figured for an area of cross-section of maximum thickness of about 12 in. 
 
 No amount of reinforcement can entirely prevent contraction cracks. The steel can, 
 however, if of small diameter and placed close to the surface, force the cracks to take place at 
 such frequent intervals that the required deformation occurs without any one crack becoming 
 large. No cracks will open up to be plainly noticeable until the steel is stressed beyond its 
 elastic limit. The amount of steel should be such, then, that without being stressed beyond its 
 elastic limit, it will withstand the tensile stress resulting from the maximum fall of temperature 
 (usually considered to be 50°) in the steel itself plus the tensile stress necessary to crack the con- 
 crete. A high elastic-limit steel is thus advantageous. 
 
 The size and spacing of the cracks will also depend upon the bond strength of the rein- 
 forcing rods. The distance between cracks in any given case will be the length required to 
 develop a bond strength equal to the tensile strength of the concrete. Thus, bars with ir- 
 regular surfaces which provide a mechanical bond with the concrete are in general more effective 
 than -smooth bars. 
 
 7. "Weight of Reinforced Concrete. — Reinforcing steel in the usual proportions adds from 
 3 to 5 lb. to the weight of plain concrete per cubic foot. The weight of plain concrete for the 
 various kinds of aggregate may be found on page 263. Reinforced concrete is usually assumed 
 as 150 lb. per cu. ft. in making computations for design. 
 
SECTION 7 
 
 BEAMS AND SLABS 
 
 RECTANGULAR BEAMS AND SLABS 
 
 1. Forces to be Resisted. — As expressed by the Joint Committee the forces to be resisted 
 are those due to : 
 
 1. The dead load, which includes the weight of the structure and fixed loads and forces. 
 
 2. The live load, or the loads and forces which are variable. The dynamic effect of the 
 live load will often require consideration. Allowance for the latter is preferably made by a 
 proportionate increase in either the live load or the live-load stresses. The working stresses 
 recommended (see Appendix B) are intended to apply to the equivalent static stresses thus 
 determined. 
 
 2. Distribution of Stress in Homogeneous Beams. — The following statements and for- 
 mulas are in accordance with the theory of homogeneous beams : 
 
 1. At any cross-section the internal forces, or stresses, may be resolved into normal and 
 tangential components. The components normal to the section are stresses of tension and 
 compression, while the tangential components add together and form a stress known as the 
 resisting shear. 
 
 2. The shear at any cross-section is borne by the tangential stresses in that section. The 
 moment at any section is borne by the component stresses normal to that 
 section. 
 
 3. The neutral axis passes through the center of gravity of the cross- 
 section. 
 
 4. The intensity of stress normal to the section increases directly with 
 the distance from the neutral axis and is a maximum at the extreme fiber 
 (Fig. 1). The intensity of this stress at any given point in the cross-sec- 
 tion is given by the formula 
 
 Compressi on 
 
 / = 
 
 My 
 
 in which / = unit fiber stress at distance y from neutral axis. 
 
 M = external bending moment at section in inch-pounds. 
 y = distance in inches from neutral axis to any fiber. 
 / = moment of inertia of the cross-section about the neutral axis. 
 5. The general formula which gives the longitudinal shear per square inch (y) at any 
 desired point in the cross-section is 
 
 VQ 
 
 in which V 
 ' Q 
 
 = total shear at the section in pounds. 
 
 = statical moment about the neutral axis of that portion of the cross-section lying 
 either above or below (depending upon whether the point in question is above 
 or below the neutral axis) an axis drawn through the point in question parallel 
 to the neutral axis. 
 
 = moment of inertia of the cross-section about the neutral axis. 
 
 = width of beam at the given point. 
 
 273 
 
 18 
 
274 
 
 CONCRETE ENGINEERS' HANDBOOK 
 
 [Sec. 7-2 
 
 a^ y/////////. 
 
 Area of 
 cross-hcrtched 
 portion- A' 
 
 Fig. 2. 
 
 In the above formula, by the term statical moment is meant the product of the area mentioned 
 by the distance between its center of gravity and the neutral axis. For example, the longi- 
 tudinal shearing intensity at a point c in a rectangular beam. Fig. 2, may be expressed as 
 follows : 
 
 VA'r 
 
 For rectangular beams and all beams of uniform width, the largest value of v for any given sec- 
 tion will occur at the neutral axis since the statical moment 
 Q has its maximum value for a point on this axis, and h is 
 constant. 
 
 6. If a beam is of constant cross-section throughout, the 
 maximum values of / and v will occur at the section where 
 M and V respectively have maximum values. 
 
 7. In addition to the longitudinal or horizontal shear at 
 any point there coexists a vertical shear and the intensity of 
 this vertical shear is equal to the intensity of the horizontal 
 shear. 
 
 8. The intensity of the shear at the top and bottom of a beam is zero and the intensity 
 of shear (horizontal and vertical) along a vertical cross-section for a rectangular beam varies 
 as the ordinates to a parabola, as shown graphically in Fig. 3. The maximum value occurs 
 
 the average intensity, or 2 • 
 
 9. At the neutral plane there exists a tension and compression at angles of 45 deg. to the 
 horizontal, and the intensity of these forces is equal to that of the shear. 
 
 10. At the end of a simply supported beam where the shear is a 
 maximum and the bending moment a minimum, the stresses lie prac- 
 tically at 45 deg. to the horizontal throughout the entire depth of 
 beam. 
 
 11. At the section of maximum moment, the shear is zero and the 
 stresses are horizontal. 
 
 12. If / represents the intensity of horizontal fiber stress and v the intensity of vertical 
 or horizontal shearing stress at any point in a beam, the intensity of the inclined stress will be 
 given by the formula 
 
 at the neutral axis and is 
 
 Fig. 3. 
 
 and the direction of this stress by the formula 
 
 tan 2K = y 
 
 where K is the angle of the stress with the horizontal. 
 
 13. At any given point maximum compressive 
 stress and maximum tensile stress make an angle of 
 90 deg. with each other. 
 
 14. The directions of the maximum stresses for 
 a simply supported beam uniformly loaded are as 
 given in Fig. 4. The general direction of the stresses 
 in a beam with any given loading may be determined 
 by means of the formulas for t and K given above. 
 
 15. The common theory of flexure gives the unit stress correctly at the important section 
 of maximum moment and also for the extreme fibers in other sections, since at these points 
 the shear is zero. Where the shear is not zero an inclined stress is the result and the flexure 
 formula gives only the horizontal component of this stress — namely, the fiber stress. 
 
 Lines of maximum tensidn 
 Lines of maximum compression 
 
 Fig. 4. 
 
Sec. 7-3] 
 
 BEAMS AND SLABS 
 
 27.5 
 
 Fig. 5. 
 
 Fig, 6. 
 
 3. Assumptions in Theory of Flexure for Homogeneous Beams. — The two main assump- 
 tions in the common theory of flexure are : 
 
 1. If, when a beam is not loaded, a plane cross-section be made, this cross-section will still 
 be a plane after the load is put on and bending takes place (Navier's hypothesis). 
 
 2. The stress is proportional to the deformation — namely, to the 
 amount of elongation or compression per unit of length (Hooke's Law). 
 
 From the first assumption it follows that the unit deformations of the 
 fibers at any section of a beam are proportional to their distance from 
 the neutral axis. By means of the second assumption the important 
 principle is established that the unit stresses in the fibers are also pro- 
 portional to the distances of the fibers from the neutral axis. 
 
 4. Plain Concrete Beams. — The first assumption in the common 
 theory of flexure, as given in the preceding article, may be applied directly 
 to plain concrete and also to reinforced-concrete beams. Careful measurements seem to show 
 some deviation from a plane, but in general this assumption seems to be warranted. From 
 this fact it follows (as stated above) that deformations of the fibers are proportional to the 
 distances of the fibers from the neutral axis. OS in Fig. 5 is the stress-deformation diagram 
 
 for concrete in compression with the deformations repre- 
 sented vertically. The curve OT is the stress-deforma- 
 tion diagram for concrete in tension. For working loads 
 the curves OS and 07" do not vary materially from straight 
 lines and the unit stresses in the fibers at any section of a 
 plain concrete beam may thus be assumed to vary directly 
 as the deformations and consequently as the distances of the fibers from the neutral axis. 
 Hence, the common flexure formula for homogeneous beams applies when the loads are work- 
 ing loads. For ultimate loads, however, ths formula does not strictly apply. 
 
 A plain concrete beam will fail by cracks opening up along the uneven lines which are 
 shown in Fig. 4 on account of the low strength of concrete in tension. If concrete were only 
 stronger in tension, then the plain concrete beam might be of 
 some structural value. In order to offset this disadvantage of 
 plain concrete, steel is used. 
 
 6. Purpose and Location of Steel Reinforcement. — Steel 
 reinforcement should have the general directions shown in Fig. Fig. 7. 
 
 6 in order to take the tension in the beam and prevent the 
 
 cracks starting along the lines indicated. Fig. 7 is the simplest method of reinforcement 
 and quite often used for light loads. In beams highly stressed, curved or inclined reinforce- 
 ment is needed, in addition to the horizontal rods. The most common method is to use several 
 bars for the horizontal reinforcement and then to bend up some of these at an angle of from 30 
 to 45 deg. as they approach the end of the beam and where they are not needed to resist bending 
 
 stresses. The concrete is depended upon to take care of 
 the compressive and pure shearing stresses, its resistance 
 to such stresses being large. 
 
 6. Tensile Stress Lines in Reinforced-concrete 
 Beams. — ^Lines of maximum tension in the concrete of re- 
 inforced-concrete beams are considerably inchned imme- 
 diately above the line of the steel. The inclination of 
 these lines is greater, the greater the shear, and the less 
 the horizontal tension. The inclination, therefore, increases toward the end of the beam. At 
 points nearer the neutral plane, the horizontal tensile stresses become less and the inclined ten- 
 sion approaches the value of the shearing stress, while its inclination approaches 45 deg. Fig. 
 8 is an attempt to represent roughly the general direction of the inclined tensile stresses in a 
 simply supported beam uniformly loaded and with horizontal reinforcement. 
 
 Diagonal-tension -'' 
 cracks UKely to occur 
 
 -Very little tension in 
 the concrete here (if any) 
 on account of concrete 
 cracking across the 
 tension face 
 
 Pig. 8. 
 
27.0 
 
 CONCRETE ENGINEERS' HANDBOOK 
 
 [Sec. 7-7 
 
 7. Flexure Formulas for Reinforced-concrete Beams. — A great many varieties of flexure 
 formulas have been proposed from time to time to be used in the design of reinforced-concrete 
 beams. As might be expected, many of the earlier formulas considered the concrete to carry 
 its share of the tension which we know now cannot be done with safety. Only two classes of 
 flexure formulas are at the present time in practical use. In each of these classes, tension in 
 the concrete is neglected and a plane section before bending, is assumed to be a plane after 
 bending takes place. 
 
 The formulas almost universally used and made standard by the Joint Committee 
 relate to working stresses and safe loads, and are based on the straight-line theory of stress 
 distribution. The other formulas referred to above relate to ultimate strength and ultimate 
 loads and the stress-deformation curve for concrete in compression is assumed to be a full 
 parabola. Ultimate-load formulas are used to such a limited extent that only a few pages of 
 this handbook are devoted to their consideration — ^namely. Arts. 10 and 11. 
 
 8. Assumptions in Flexure Calculations. — The following assumptions are made in deriving 
 the flexure formulas: (1) the adhesion of concrete to steel is perfect within the elastic limit of 
 
 the steel; (2) no initial stresses are con- 
 ^ sidered in either the concrete or the steel 
 
 Tl ^M^W^ to contraction or expansion; (3) the 
 
 applied forces are parallel to each other 
 and perpendicular to the neutral surface 
 of the beam before bending; (4) sectional 
 planes before bending remain plane sur- 
 faces after bending within the elastic limit 
 of the steel ; (5) no tension exists in the con- 
 crete; (6) modulus of elasticity of concrete 
 is constant. 
 
 9. Flexure Formulas for Working Loads — Straight-line Theory. — The unit stress in the 
 steel is within the elastic limit, and the unit stresses in the concrete at the given section of the 
 beam are considered to vary as the ordinates to a straight line (see Fig. 9). Tension in the 
 concrete is neglected. The formulas follow^ (see Notation, Appendix D): 
 
 1 The formulas may be derived as follows: 
 
 Total compressive resistance = total tensile resistance, or 
 
 Vzfckbd = Asf^ (o) 
 From the assumption that deformations vary as the distances of the fibers from the natural axis and assuming 
 stress proportional to deformation 
 
 fc fs 
 
 Diagram 
 
 Fig. 9. 
 
 Eckd E,da 
 
 k) 
 
 which reduces to 
 
 fs = fen 
 
 ■* or fc 
 
 fsk 
 
 n(l - k) 
 
 , or k = 
 
 The total resisting moment of the beam is the sum of the moments of the total compressive stresses and of 
 the total tensile stresses about the neutral axis, or 
 
 M = %kd{y2fckhd) + d(l - k)Asfs 
 = \ifckW + Asfsdil - k) 
 
 Eliminating k between equations (o) and (6), the following formula for steel ratio results 
 
 fc^nfc^ ) 
 
 Introducing the value of fs from equation (6) into equation (a), we have 
 
 Yik'^hd - A^nCl - A;) = 0 
 y2k% - p6n(l - A;) = 0 
 
Sec. 7-9] 
 
 BEAMS AND SLABS 
 
 277 
 
 k = ■\/2pn -f (pn)2 — pn = 
 
 V = 
 
 J = 1 - 3^ A; (2) 
 hd fs (fs ^ , \ 2/, (3) 
 
 ff4 + l) 
 fc \nfa / 
 
 M.^VMd^ or or /. = (4) 
 
 Ms = ?>/J(6rf2), or bd^ or /. = (5) 
 
 /-^^ - ;^.) («) 
 
 The above formulas show that for a given ratio of p and k remain. the same for all sizes 
 
 fc 
 
 of beams. The formula for Mc gives the resisting moment when the maximum allowable value 
 of fc is introduced as the limiting factor and the formula for Ms gives the resisting moment 
 when the maximum allowable value of fs is the limiting factor. The lesser of these two re- 
 sisting moments, when proper working values are assigned to fc and fs, is the safe resisting 
 moment of the beam in question. 
 
 Unlike steel beams, reinforced-concrete beams require a preliminary formula to be solved 
 before the formula for resisting moment may be employed. Solving this preliminary formula 
 locates the position of the neutral axis which is in the same position only for beams of a given 
 percentage of steel reinforcement. 
 
 The method of procedure in flexure formulas is to determine the vertical section of the beam 
 where the moment is a maximum and apply the formulas at that section. Either formula for 
 p, containing the values of fc and fs, determines the amount of steel reinforcement which is 
 needed to cause the beam to be of equal strength in tension and compression. The formulas 
 for resisting moment determine the bending moment which a beam will safely withstand (for an 
 existing structure) or the size of the beam needed to resist a given bending moment (for a pro- 
 posed structure). 
 
 If a beam is over-reinforced, its resisting moment depends on Mc, and if under-reinforced 
 on Ms. 
 
 If it is desired to find the fiber stresses in concrete and steel of a given beam, the formulas 
 
 /, = — ~and/c = -r~- (or /c = ^{'^ ^ should be used, where M is the external bending 
 Asjd kjbd^ \ k / 
 
 2M M 
 
 moment in each case. For a given external M, either bd^ = or bd"^ = may be used to de- 
 
 fckj pfsj 
 
 from which 
 
 k = \/2pn + (pn.)^ — pn 
 Substituting the value of Asfs from (o) into (c), we get 
 
 Mc = V2fck(l - H k)bd^ 
 
 Mc = Hfckjbd^ 
 
 Substituting the value of fc from (a) into (c), and remembering that As = pbd, 
 
 Ms = pfsjbd^ 
 
 Equation (a) may be solved to give 
 
 2f,p fck 
 /c = — , or p = 
 
278 
 
 CONCRETE ENGINEERS' HANDBOOK 
 
 [Sec. 7-9 
 
 termine cross-section, when the p used is obtained from the formula p = 
 
 fck . . 
 V = — ' in which k = 
 2/. 
 
 fc \nfc I 
 
 or from 
 
 1 + 
 
 Illustrative Problem. — What will be the resisting moment (Af) for a beam whose breadth (b) is 8 in. with 
 a distance from the center of the reinforcement to the compression surface {d) of 12 in., the area of steel section being 
 0.96 sq. in.? Assume n = 15; /<, = 650 lb. per sq. in.; and /s = 16,000 lb. per sq. in. 
 
 O.f 
 
 (8) (12) 
 
 0.01 
 
 From (1) 
 
 From (4) 
 From (5) 
 
 k 
 J 
 
 Mc 
 Ms 
 
 \/(2)(0.01)(15) + (0.01)2(15)2 
 0.861 
 
 (0.01)(15) = 0.418 
 
 1/^(650) (0.418) (0.861) (8) (12)2 ^ 134,700 in.-lb. 
 (0.01) (16,000) (0.861) (8) (12)2 ^ 158,700 in.-lb. 
 
 Mc is the lesser of the two resisting moments and hence controls in the design. 
 
 Illustrative Problem. — Assume the beam of the preceding problem to be 14 in. deep and subjectetl to a 
 bending moment of 130,000 in.-lb. Compute the maximum unit stresses in the steel and concrete. 
 
 0.96 
 (8)(14) 
 
 = 0.0086 
 
 From (1) 
 
 From (4) 
 
 From (5) 
 
 k 
 j 
 
 130,000 
 
 fc 
 
 130,000 
 
 a/(2) (0.0086) (15) + (0.0086)2(15) = 
 0.868 
 
 (0.395) (0.868) (8) (14)2 
 480 lb. per sq. in. 
 
 (0.0086) Us) (0.868) (8) (14)2 
 11,100 lb. per sq. in. 
 
 (0.0086) (15) = 0.395 
 
 Illustrative Problem. — A beam is to be designed to withstand a bending moment of 300,000 in.-lb. and to 
 have equal strength in tension and compression. A 1 : 2 : 4 concrete will be used with Ec = 2,000,000 and/c = 600 
 lb. per sq. in. The pull in the steel is to be limited to 14,000 lb. per sq. in. Its modulus of elasticity Es is 30,000,000. 
 
 ^» _ 1 ^ ^ _ Z2 
 Ec~ fc ~ 3 
 
 From (1) and (2) 
 
 From (3) 
 
 ^ + (15) (600) 
 (600) (0.391) 
 
 0.391 and j = 0.870 
 
 0.0084 
 
 (21)(14,000) 
 
 Either (4) or (5) may now be used in determining b and d since the amount of steel to be employed will cause 
 simultaneous maximum working stresses. 
 
 From (5) 
 
 6^2 = 
 
 300,000 
 
 2930 
 
 293, or d = 17H in. 
 
 (0.0084) (14,000) (0.870) 
 Many different values of b and d will satisfy the last equation. If b is taken as 10 in., then 
 „ 2930 
 
 =-Io- 
 
 Finally 
 
 As = (0.0084) (10) (17.25) = 1.45 sq. in. 
 If IH in. is allowed between the tension surface of the concrete and the center of the steel, the entire depth of the 
 beam should be 19 in. 
 
Sec. 7-10] 
 
 BEAMS AND SLABS 
 
 279 
 
 10. Flexure Formulas for Ultimate Loads. — The stress-deformation curve for concrete in 
 compression is assumed to be a full parabola. Experiments show this to be very nearly the 
 case for ultimate loads. The amount of reinforcement is considered as sufficient to develop 
 the full compressive strength of the concrete without stressing the steel beyond its yield point. 
 Failure under such conditions (Fig. 10) will occur by crushing the concrete. 
 
 Average compressive 
 stress - I fc 
 
 Total, compressive 
 stress- 1 fcbkd. 
 
 Fig 10. 
 
 The formulas follow (see Notation, Appendix D) : 
 
 ^ = \ 3pn + (^pn\ ^ - ^pn = 
 
 1 -h; 
 
 2nfa 
 
 V = 
 
 Mo = Hfckjibd^), or bd' 
 Ms = pfsjihd^), or bd' 
 
 2 fck 
 M 
 
 Hfckj 
 
 fc 
 
 2k 
 
 M 
 
 Pfsj 
 
 (1) 
 
 (2) 
 (3) 
 
 (4) 
 (5) 
 (6) 
 
 In the above formulas, /, = elastic limit of the steel, and fc = ultimate compressive strength 
 of concrete. 
 
 When using the above formulas, it should be remembered that the amount of steel in the beam 
 is assumed as sufficient to cause the ultimate resisting moment to be due to the concrete. Thus, 
 the resisting moment of the beam may be figured by using the formula for Mc. Xf an amount 
 of steel is used such that the ultimate strength of the concrete and the elastic limit of the. steel 
 would be reached simultaneously, either Mc or Ms may be used to determine the ultimate 
 resisting moment. If a less amount of steel is used than the amount just mentioned, the condi- 
 tions of the assumption do not hold, and the formulas given above cannot be used. When this 
 happens the ultimate moment may be figured by means of formulas based on a parabolic varia- 
 tion of compression in the concrete and applicable for any load up to the ultimate. The para- 
 bola for such a case is not a full one and the formulas are cumbersome to use and not at all 
 fitted for practical use. 
 
 The formulas for ultimate loads, however, can readily be employed to design a beam for 
 
 equal strength in tension and compression. The method is to first find the required amount 
 
 MM 
 
 or bd^ = — . may be used to determine the size of beam 
 
 of steel. Then either bd^ = 
 
 fckj 
 
 necessary. 
 
 Illustrative Problem. — A beam is to be designed to have equal strength in tension and compression and 
 to safely withstand a bending moment of 150,000 in.-lb., the ultimate compressive strength of the concrete being 
 taken at 2000 lb. per sq. in. and the elastic limit of the steel at 40,000 lb. per sq. in. Assume n = 15. 
 
280 
 
 CONCRETE ENGINEERS' HANDBOOK 
 
 [Sec. 7-11 
 
 k ^ — 
 
 1 + 
 
 20 
 
 0.698 
 
 30 
 
 1 - Hk = 0.775 
 2 0.598 
 
 J 
 
 ^ " 3 ' 20 
 
 With a factor of safety of 4, the ultimate bending moment is 600,000 in.-lb. and 
 
 0.02 
 
 With & = 8 in., then 
 
 Also, 
 
 d2 
 
 600,000 
 
 (^3X2000) (0.598) (0.775) 
 
 = 972 
 
 972 
 8 
 
 121.5, or d = 11 in. 
 
 As = (0.02) (8) (11) = 1.76 sq. in. 
 
 11. Flexure Formulas for "Working Loads and for Ultimate Loads Compared. — Formulas 
 for ultimate loads are open to the objection that when a factor of safety is applied which will 
 bring the stress in the concrete to about a good working stress, the stress in the steel becomes 
 unduly low from a standpoint of economy. A factor of safety of 3 or 4, as is usually taken, 
 leaves a high stress in the concrete with the stress in the steel far below what is usually consid- 
 ered a safe stress. Beams designed by the ultimate load formulas will generally be of smaller 
 cross-sectional dimensions than when the straight-line formulas are employed; but, on the other 
 hand, a larger amount of steel is required. Practically identical results will be obtained by the 
 two classes of formulas if about 15% lower compressive stress is permitted in the concrete by 
 the ultimate load formulas than by the formulas based on the straight-line theory. There 
 seems to be no good reason why the simple formulas based on the theory of straight-line stress 
 variation should not be used for purposes of design, safe working stresses being employed. 
 
 12. Lengths of Simply-supported Beams. — The span length for beams and slabs simply 
 supported should be taken as the distance from center to center of supports, but need not be 
 taken to exceed the clear span plus the depth of beam or slab. 
 
 13. Shearing Stresses. — In Fig. 11 is shown a small portion of a concrete beam, so short 
 that no appreciable portion of the load on the beam acts directly upon it. The opposing total 
 
 compressive forces are denoted by C and C; and the tension 
 in the steel on each face by T' and T. The tension in the 
 concrete may be neglected. Let V be the total shear on 
 this small portion of the beam. From conditions of equili- 
 brium, C = T' and C = T. The total horizontal shearing 
 stress upon a horizontal section immediately above the steel 
 is T' — T, and if h denotes the breadth of the beam and v 
 the unit shear (horizontal or vertical) at any point between 
 the neutral axis and the steel, then 
 
 T 
 
 Neaf rai <t 
 
 Fig. 1] 
 
 hx 
 
 (1) 
 
 The various couples acting upon the element produce equilibrium ; hence 
 
 Vx = {T - T)jd 
 
 or 
 
 Vx 
 
 Substituting this value in equation (1) there results 
 
 V (2) 
 
 which is the value of shear intensity at any point between the neutral axis and the steel. 
 
Sec. 7-14] 
 
 BEAMS AND SLABS 
 
 281 
 
 The value of j for working loads varies within narrow limits and v will change but sHghtly 
 if the different values of j are inserted in equation (2). The average value of j for beams in 
 ordinary construc{ion is Using this value, equation (2) reduces to 
 
 bd 
 
 (3) 
 
 Fig. 12. 
 
 I 
 
 IP Shearing stress is the same at all points between the neutral axis and the steel, and above 
 the neutral axis it follows the parabolic law. Fig. 12 represents the distribution of shearing 
 stress on a vertical cross-section assuming no tension in the concrete. 
 
 The longitudinal tension in the concrete near the end of beam modifies the distribution 
 of the shear, increasing the shearing stress somewhat at the neutral axis and decreasing it at 
 the level of the reinforcement. Equation (3), however, gives results ^ 
 which are sufficiently accurate and are derived for beams having the 
 horizontal bars straight throughout. When any web reinforcement is 
 used, the distribution and the amount of the shearing stresses at the end 
 of a simply supported beam are materially different from the foregoing. 
 The analysis of the stresses becomes more complex and a determination 
 of their value impracticable. Even here, however, the above formula 
 serves a useful purpose. It is found that shear is the chief factor in the 
 failure of a beam by diagonal tension and either formula (2) or formula (3) may be used in de- 
 sign if properly controlled by the results of experiments. 
 
 Failure by the actual shearing of the concrete in a beam is not a likely occurrence under 
 any conditions as the shearing strength of concrete is at least one-half the crushing strength. 
 
 14. Methods of Strengthening Beams Against Failure in Diagonal Tension. — The in- 
 tensity of the diagonal tensile stress at any point in a beam depends upon the shear and hori- 
 zontal tension in the concrete, with shear as the chief factor. The percentage of horizontal 
 reinforcement must also be considered, since the amount of steel employed affects the hori- 
 zontal deformation and consequently the tension in the 
 concrete. Thus beams may be strengthened against 
 failure in diagonal tension by keeping the horizontal 
 tension small through the use of considerable horizontal 
 steel at points of heavy shear, by avoiding heavy shear- 
 ing stresses, and by providing some type of web rein- 
 forcement. A low unit working stress in whatever 
 type of web reinforcement is employed is also much to 
 be preferred. 
 
 The most unfavorable part of a beam as regards 
 diagonal tension is at points of excessive shear com- 
 bined with considerable bending moment. A sufficient 
 number of reinforcing rods should be extended horizon- 
 tally to the ends of the beam to provide for bending 
 with low unit stresses in the steel. In small beams, ver- 
 tical stirrups looped about the horizontal rods may be employed throughout for web reinforce- 
 ment but in large beams under heavy shearing stresses, both stirrups and bent rods should be 
 used. The stirrups in large beams should be securely fastened to the longitudinal rods in 
 such a way as to prevent slipping of bar past the stirrup. Inclined web members may also 
 be used in place of vertical stirrups if securely attached to the horizontal rods. Vertical 
 stirrups may be made in various forms, as indicated in Fig. 13. 
 
 15. Moment and Diagonal-tension Tests — General.^ — When a beam begins to fail by 
 yielding of the steel at, or near, the section of maximum bending moment, any further load 
 
 1 For detailed treatment of this subject, see "Concrete, Plain and Reinforced" by Taylor and Thompson 
 (1916 Edition). 
 
282 
 
 CONCRETE ENGINEERS' HANDBOOK 
 
 [Sec. 7-15 
 
 rapidly increases the deformation, large cracks open up in the concrete on the tension side, the 
 neutral axis rises on this account, and the ultimate failure soon occurs by the crushing of the 
 concrete. A steel tension failure is found to occur when the amount of steel used is less than 
 the amount determined by theoretical formulas which makes the beam of equal strength in 
 tension and compression. This result agrees, then, with what is expected. Likewise it is 
 found that with a larger amount of steel than is theoretically required, the yield point of steel 
 is not reached and the beam fails directly by crushing of the concrete. Beams with no web 
 reinforcement and with the existence of large shearing and moment stresses, 
 fail by inclined cracks opening up in the concrete, thus substantiating to a 
 considerable degree the theoretical deductions regarding the internal stresses in 
 beams. 
 
 The results of breaking tests on reinforced beams with different percent- 
 ages of steel reinforcement compare well with the results derived from the- 
 oretical formulas. Considering the nature of the material, the calculations by 
 the two assumptions of stress variation are found to agree sufficiently with 
 the experimental results to justify their use in problems of design. 
 
 Another method of testing reinforced-concrete beams is by the use of ex- 
 tensometers to measure distortions, so that the deformation of the steel and of 
 the extreme fiber of the concrete may be calculated and the neutral axis determined. In 
 making beam tests it is customary to place equal loads at points dividing the length into 
 three equal parts. The advantage of this arrangement lies in the fact that the bending 
 moment is practically uniform between the loads and, if measuring devices are attached, the 
 deformations of the fibers at the top and at the bottom may be easily determined. If in Fig. 
 14 the deformations be aa' and hb', the neutral axis 0 is located by connecting 
 a' and b' with a straight line, intersecting ab at 0. 
 
 In moment calculations, the position of the neutral axis is of prime im- 
 portance and once this is known, the actual strength may be determined with 
 little uncertainty. The formula for k shows that the position of the neutral 
 axis depends only upon the percentage of steel employed and upon the ratio 
 E 
 
 or n. The value of Ec is the only value in the formula which is uncertain. 
 
 It might be well to take the value as determined by the ordinary compression 
 test for use in theoretical formulas, but closer results can be obtained from 
 these formulas if the value of n is taken so that for average conditions the neu- 
 tral axis is in as nearly as possible the same position theoretically and experimentally. The 
 reason that Ec, as thus determined, will give better results at working loads, is due to the effect 
 of the remaining tension in the concrete below the neutral axis — a stress which is properly not 
 allowed for in the resisting moment. 
 
 In making the experiments above described, it was observed that the neutral axis raised 
 as the loading increased, k being approximately H at working loads. It was also noted that 
 
 for the lower loads the neutral axis as deter- 
 mined from the theoretical formulas is more 
 uncertain and generally lower in the beam 
 than for the higher loads. This is undoubtedly 
 due to the relatively large influence of the ten- 
 sile strength of the concrete in such cases. This 
 rise of the neutral axis as the load increases is 
 shown in Fig. 15. Consider aibi to be the plane 
 ab after a bending takes place just sufficient to bring the maximum tensile stress in the concrete 
 to its ultimate value. When loads are applied which cause a greater bending moment, the 
 concrete in tension becomes broken by fine cracks, and the steel takes a greater part of the 
 tensile stress. The elongation at b now increases faster than at a, and the neutral axis rises 
 
 Fig. 16. 
 
Sec. 7-15] 
 
 BEAMS AND SLABS 
 
 283 
 
 rapidly. When working loads are reached, the position of the neutral axis moves but little, 
 and the steel takes all the tension. 
 
 Fig. 16 illustrates a typical diagonal tension failure in beams reinforced with only horizontal 
 bars. The initial crack forms at a and branches toward b. A little later the concrete begins to 
 fail in a horizontal tension crack just above the rods, running from a toward the end of the beam. 
 This horizontal crack is brought about by the new conditions which exist after the concrete 
 has become cracked along the diagonal line and the normal diagonal tension has thus ceased 
 to act. Sometimes this horizontal crack does not extend to the end of the beam — the final 
 failure occurring either by the diagonal crack extending to the top of the beam or the horizontal 
 rods pulling out. Thus final failure often occurs from stresses which are developed after initial 
 failure has occurred. However, the initial failure and its cause is what is of importance in 
 design. 
 
 Tests show that it is possible to provide sufficient web reinforcement by means of stirrups 
 and bent bars to develop the full strength of the beam whether governed by the crushing strength 
 of the concrete or the elastic limit of the steel. It is found that part of the diagonal tension is 
 taken by the concrete so that web reinforcement need not be designed to take all the diagonal 
 tensile stresses. 
 
 Vertical stirrups spaced a distance apart equal to, or greater than, the depth of beam help 
 but little in preventing diagonal cracks between successive stirrups. They may prevent final 
 failure, however, by preventing the extension of a crack horizontally along the reinforcing rods. 
 Stirrups are found by tests to be most effective when spaced a distance apart equal to one-third 
 the depth of beam. To give the best results they 
 should be securely fastened to the longitudinal bars 
 in the tension side of the beam. 
 
 Tests in which curved and inclined rods were 
 used, but in which no rods continued straight for 
 the entire length of the beam, showed results very 
 little better than for straight rods. 
 
 Vertical stirrups and bent rods combined are 
 found by tests to give the very best results. Tests also seem to indicate that too much re- 
 liance should not be placed upon one or two bent rods. For this reason, even if one or two 
 rods are bent up properly to take the diagonal tension, it would be good design to consider this 
 rod, or rods, as not taking any diagonal tensile stress and to provide a thorough web reinforce- 
 ment by means of stirrups. 
 
 Tests show that bent bars may be inclined at any angle between 30 and 45 deg. without 
 the beam showing any marked difference in strength. Beams having sharp bends in the rein- 
 forcing bars are found to have less strength than beams with bars having circular bends of a 
 radius about 12 diameters. 
 
 Fig. 17 represents the conditions which developed in the test of a beam. The cracks are 
 numbered in the order of their appearance, final failure occurring at crack No. 4 and being due 
 to inadequate web reinforcement. The stirrups were stressed beyond their yield point. 
 
 It appears from tests of beams in which bent rods were employed with a good anchorage at 
 their ends, that the anchorage is quite advantageous in increasing web resistance. This form 
 of construction is also found to be an insurance against failure at low loads through defective 
 concrete or insufficient bond. 
 
 Hooks at the ends of the horizontal tensile bars prevent slipping of the bars in the concrete 
 and are found to increase the strength of the beam materially. 
 
 The results of experiments show that the ultimate compressive strength of concrete in a 
 beam is at least equal to its crushing strength as determined by tests on cubes hardened under 
 similar conditions; also, that the yield point of the steel should be regarded as ultimate strength 
 as far as reinforced beams are concerned. When the steel reaches its yield point, the beam de- 
 flects, and failure soon occurs by the crushing of the concrete. 
 
284 
 
 CONCRETE ENGINEERS' HANDBOOK 
 
 [Sec. 7-16 
 
 16. Bond Stress. — The tension in the horizontal steel near the lower surface of a reinf orced- 
 concrete beam is a maximum near the center of beam and decreases each way toward the end. 
 The difference in the tension between any two points is transmitted to the concrete by the bond 
 between the steel and the concrete. 
 
 A formula for bond may be derived for beams in which the reinforcement is horizontal 
 or straight throughout. The total shearing stress per linear inch between the steel and the 
 concrete, considering a length of beam equal to x, is 
 
 T' - T 
 
 X 
 
 From Fig. 11 
 
 Vx = (T' - T)jd 
 
 or 
 
 T' — T V 
 
 = -rr (bond stress per linear inch) 
 
 V 
 
 and the bond stress per square inch of the surface of the steel bars is divided by the sum 
 
 in inches of the circumference of the bars at the given vertical cross-section. If u = unit 
 bond stress, and 2o the total circumference of all bars in a beam at the given section, then 
 
 V 
 
 The above formula shows that theoretically the bond stress is a simple function of the 
 shear and varies with the shear. Thus, shear diagrams may be used to represent the variation 
 of bond stress along a beam. When using the above formula, the average value of j = J-^ 
 may be taken. 
 
 If we consider simply supported beams, tests on rectangular and T-beams loaded at the 
 quarter points show that when stirrups are used the beam is stiffened and the bond stress along 
 the horizontal rods near the end of beam is somewhat reduced. A reason for this may be 
 shown in the fact that, after the concrete begins to crack from diagonal tension, the stirrups 
 aid in carrying part of the tensile stress which results from the bending moment then existing 
 at the line of the diagonal crack; the stress in the horizontal rods at the end of beam is thus 
 reduced and likewise the liability of failure through bond. A greater reduction of the bond 
 stress has been found to exist when the web reinforcement is provided by means of bent rods 
 and stirrups. The reduction becomes considerable when about one-half of all the rods are bent 
 up, provided, however, that a sufficient number of rods be thus employed. Results seem to 
 indicate that no reduction should be considered in design unless the number of rods bent 
 be greater than two or three and that the bends be made at least at two points at each end of 
 beam. Tests show that for conditions especially favorable, an average of 50% more bond stress 
 may safely be allowed on the horizontal rods at the end of beam than would be considered 
 safe by the above formula. No allowance should be made when only stirrups are employed 
 for the web reinforcement. 
 
 It has been found in the testing of simply supported T-beams with steel straight throughout 
 and loaded at the one-third points, that the bond stress along the horizontal steel is affected by 
 the presence of tensile stresses in the concrete and that this bond stress is usually a maximum 
 just outside the load points. The observed bond stress at these points was in some cases as 
 much as 50% greater than the computed stress. 
 
 In beams reinforced for diagonal tension the bond stresses along the horizontal bars are 
 not distributed as uniformly as in beams having the reinforcement horizontal or straight through- 
 out. The bond stresses are found to be concentrated at and near stirrups and at and near 
 points of bending of the longitudinal rods. Such concentration of bond stress causes local slip 
 of the longitudinal bars unless the web reinforcement is well distributed along the beam. If a 
 stirrup is not rigidly attached to the horizontal rods and local slip of bars occurs, the effective- 
 ness of the stirrup is somewhat impaired. 
 
Sec. 7-17] 
 
 BEAMS AND SLABS 
 
 285 
 
 The bond stress in continuous beams is treated in Art. 39. 
 
 The bond strength of vertical (or inclined) stirrups may be insufficient to develop the re- 
 quired strength of the stirrups with respect to tension. This possibiHty must also be investi- 
 gated in the design of beams having web reinforcement in the form of bent rods. Tests show 
 that it is safe to assume that the stress in a stirrup or bent-up bar may be transferred to the 
 concrete above a point 0.6 the depth of beam from the upper surface. In most cases it is found 
 that stirrups must have hooked ends. 
 
 For illustrative problem, see page 298. 
 
 17. Web Reinforcement in General. — Inclined web reinforcement may be separate mem- 
 bers firmly connected with the horizontal reinforcement to prevent slipping, or some of the 
 horizontal bars may be bent up near the ends of the beam where they are not needed to resist 
 bending. The vertical reinforcement may be used separately or in combination with inclined 
 reinforcement, depending upon the preference of the designer and upon the amount of diagonal 
 tension to be provided for. Vertical stirrups should be looped around the horizontal bars and 
 in important beams should also be firmly secured to these bars by wiring or otherwise. Stir- 
 rups should usually be looped or hooked at the top in order to prevent slipping due to insuffi- 
 cient bond (see Art. 19). 
 
 The proportioning of web reinforcement cannot be done with any degree of exactness since 
 very little experimental work has been performed along this line. Howler, rough determina- 
 tions of what is required may be obtained on rational grounds. The only information from 
 tests is the value of the maximum shearing stress which measures diagonal tension failure — (1) 
 for beams with horizontal bars only, and (2) for beams having an effective system of web rein- 
 forcement. Also, tests on beams, with and without web reinforcement, show that when rein- 
 forcement is provided for diagonal tension, the concrete may be assumed 
 to carry its full value of the shear and the steel the remainder. It is I t I 
 
 generally conceded as safe practice in the designing of beams to use 
 only two-thirds of the external vertical shear in making calculations of 
 the stresses to be taken by the web reinforcement. 
 
 Consider now Fig. 18, in which V represents the average total shear 
 over the portion s of the beam. Let v' represent average unit horizon- 
 tal shear on any plane below the neutral axis. Then (see Art. 13) 
 
 bjd 
 
 The total shear over any such horizontal plane is v'bs', whence 
 
 V OS = -TT 
 
 Jd 
 
 The function of stirrups, either vertical or inclined, is to resist by their tensile strength 
 that portion of the above shearing stress which is not carried by the concrete. 
 
 Assume a vertical stirrup to be placed at the section A- A, and to oppose the shear over the 
 portion of the beam. The total stress in the stirrup is Asfs (in a U-shaped stirrup, As is the sum 
 of the areas of the two legs), and it is produced by that part of the total shear over the horizontal 
 plane bs not taken by the concrete. Assuming the steel to take two-thirds of the total shear, 
 then 
 
 1 I 
 
 U.-J 
 
 Fig. 18. 
 
 AJs = 
 
 'bs = ^--V, 
 
 Vs 
 jd 
 
 Solving 
 
 As = 
 
 s 
 
 fsjd 
 
 (vertical stirrups) 
 
 (1) 
 
 (2) 
 
286 
 
 CONCRETE ENGINEERS' HANDBOOK 
 
 [Sec. 7-18 
 
 For inclined members and bent-up bars, the lines on a beam representing the direction in 
 which the diagonal tensile cracks are likely to occur, are crossed more times per unit of length 
 for a given horizontal spacing than would be the case if vertical stirrups were employed; that is, 
 a given amount of inclined steel is much more effective in taking diagonal tension than the same 
 amount of vertical steel. It may be assumed that the allowable stress in the inclined bars is 
 
 approximately — and the required area of steel, assuming the steel to take two-thirds of 
 sin 4o 
 
 the shear, is 
 
 2 0.7(Vs) 
 
 or 
 
 s 2 0.77 
 
 Since tests show that bent bars may be inclined at any angle between 30 and 45 deg. without 
 a beam showing any marked difference in strength (see Art. 15), the Joint Committee recom- 
 mends that the longitudinal spacing of vertical stirrups should not exceed one-half (3^) the depth 
 of beam and that of inclined members and bent-up bars should not exceed three-fourths {^i) 
 of the depth of beam. 
 
 18. Region Where No Web Reinforcement is Required. — There is a region near the center 
 of most beams in which the shear does not exceed that permissible for plain concrete. In this 
 region no shear reinforcement is required. The distance from one support to a point beyond 
 which no stirrups are required may be found as follows for a uniformly loaded beam. 
 
 Let 
 
 I = span of beam in feet. 
 w = uniform load in pounds per foot. 
 
 Xi = distance in feet from left support beyond which no stirrups are required. 
 
 Vi = unit working shear for plain concrete, 
 
 Vi = total working shear, producing unit shear of Vi. 
 From equation (2) on page 280, it is obvious that, where Vi = v, no stirrups are recjuired. 
 At this point 
 
 ^ Wd 
 
 But 
 Whence 
 
 or, in terms of v at the end of beam, 
 
 Xi 
 
 wl 
 
 _ I Vibjd 
 ~ 2 wT 
 
 Xi 
 
 When V at the end of beam equals 3^1, then 
 
 / 
 
 = 3 
 
 This derivation applies only to a static uniformly distributed load over the whole span. 
 
 Suppose a 10-ft. beam {h = 10 in. and d = 20 in.) is uniformly loaded with a static load of 2900 lb. per ft. 
 and assume vi = 40 lb. per sq. in. according to recommendation of Joint Committee for 2000-lb. concrete. Also 
 assume jd = lid. Then 
 
 10 (40)(10 )(17.5) 
 XI = y 290"0 = 2.6 ft. 
 
 When designing for floor systems it is often more proper to consider the uniform load to 
 
Sec. 7-18] 
 
 BEAMS AND SLABS 
 
 287 
 
 Diagram I 
 
 10 15 5 
 
 Spacing of U-5tirrups in inches 
 
 80 30 10 
 
 Spacing of W- Stirrups »n inches 
 
 0 5 10 15 
 
 Spacing of U-Stirrupjs m inches 
 30 0 lO 20 30 
 
 Spacing of W-S+irrups in Inches 
 
 Diagram II 
 
 10 \5 5 
 
 Spacing of U-5tirrups in inches 
 
 80 30 10 20 
 
 Spacing of W-5tirrups in inches 
 
 Diagram III 
 
 0 5 10 15 
 
 Spacing of U-5firrups in inches 
 0 10 20 30 
 
 Spacing of W-SfirrupS in inches 
 
 VO \5 5 
 
 Spacing of U-Stirrups in inches 
 
 20 30 \0 
 
 Spacing of W-Sfirrops in inches 
 
 0 5 10 15 
 
 Spacing of U-5tirrups in inches 
 O 10 20 30 
 
 Spacing of W-5tirrupS in inchea 
 
288 CONCRETE ENGINEERS' HANDBOOK [Sec. 7-18 
 
 Diagram IV 
 
 P 5 10 15 5 10 15 0 5 10 15 
 
 Spacing of U-5tirrups In inches Spacing of U-Stirrups in inches 
 
 O 10 20 30 10 20 30 0 (0 20 30 
 
 Spacing of W- Stirrups in inches Spacing of W-Stirrups in inches 
 
 Diagram V 
 
 0 5 10 15 5 10 15 0 5 10 15 
 
 Spacing of U-Stirrups in inches Spacing of U-Sttrrups in inches 
 
 0 10 20 30 10 ?0 30 0 10 20 30 
 
 Spacing of W-Stirfups in inches Spacing of W-Stirrups in inches 
 
 Diagram VI 
 
 spacing of U-Stirrups in inches Spacing of U-S+irrups m inches 
 
 O 10 20 30 10 20 30 O 10 ZO 3Q 
 
 Spacing of W-5tirrvp6 in inches Spacing of W-Stirrwp* in jnche^ 
 
Sec. 7-19] 
 
 BEAMS AND SLABS 
 
 289 
 
 Diagram VII 
 
 10 15 5 10 
 
 Spacing of U-5+irrups in inches 
 
 20 30 10 ZO 
 
 5p<Jcing of W-S+trrups in inches 
 
 Diagram VIII 
 
 15 O 5 10 15 
 
 Spacmg of U-S+irrups in fnches 
 
 30 0 10 20 3Q 
 
 Spacing of W- Stirrups in inches 
 
 10 IS 5 
 
 Spacing of U-Sfirrups <n Inches 
 
 20 30 10 
 
 Spacing of W-Stirrups In inches 
 
 10 15 0 5 \0 15 
 
 Spacing of U- Stirrups in inches 
 
 ZO 30 O 10 20 30 
 
 Spacing of W- Stirrups in inches 
 
 be a moving load. In this case the shear at the center of the span is not zero, but is equal at 
 
 maximum to • where w' is the live load per foot. When plotting the shear diagram this value 
 
 should be plotted at the center of the beam and the total end shear at the support. The varia- 
 tion in shear between these two points may be safely assumed to be a straight line. The shear 
 to be carried by the concrete being known, the point in the beam beyond which no shear rein- 
 forcement is required may be quickly located. 
 
 19. Vertical Stirrups. — The required total area of cross-section of a vertical stirrup may 
 be determined by the formula (see page 285) 
 
 A -^-^ 
 ' 3 hjd 
 
 assuming the web reinforcement to carry two-thirds of the total shear. (For U-shaped stirrup, 
 As is the sum of the areas of the two legs.) With a given size of stirrup, this formula may be 
 solved to give the spacing required, or 
 
 _ 3 Asfsjd 
 ^ " 2' V 
 
 19 
 
290 
 
 CONCRETE ENGINEERS' HANDBOOK 
 
 [Sec. 7-19 
 
 The value of V should be taken at the section where the spacing is desired. This spacing 
 formula may be solved directly by means of Diagrams 1 to VIII inclusive for three sizes of 
 stirrups.^ 
 
 If the shear diagram is drawn for any given beam, it is convenient to use the above formula 
 in the form 
 
 3 A,f..jd 
 2' s 
 
 V = 
 
 and, for various even-inch spacings (s) of the stirrups, to solve for the corresponding total 
 external shears. At the point where the ordinate to the shear diagram scales a computed total 
 shear, there the spacing may be made the even inch used for s in the formula. 
 
 Tests have shown that little or no value is derived from stirrups spaced a distance apart 
 equal to, or greater than, d (see page 283). A practical limit suggested by the Joint Com- 
 mittee is one-half the depth. 
 
 In restrained beams the first stirrup should be placed no farther than one-half the mini- 
 mum spacing from the edge of support and, in beams simply supported, the first stirrup should 
 be placed not farther than one-half the minimum spacing from the center of support. 
 
 The variation of shear intensity along a uniformly loaded beam is shown in Fig. 19, The 
 
 area ABCD represents the total stress to be taken 
 by the stirrups at each end of beam. The ordi- 
 nate AB represents two-thirds of the shear at the 
 support per 1-in. length of beam. 
 
 Some attention must be paid to the diameter 
 of stirrup which it will be possible to employ in any 
 given case. The diameter should not be so small 
 that the stirrups will be placed too close together 
 for convenience in construction, yet not so far 
 apart that the limiting value }4d is exceeded. 
 Fig. 19. But, in addition to such consideration, the bond 
 
 strength of the stirrup must be investigated if the 
 stirrups are not made with hooked ends, since the danger of slipping determines the maxi- 
 mum diameter which may be employed. 
 
 The distribution of bond stresses developed on the surface of the stirrups is indeterminate. 
 Evidently it must not be expected that tension will be transferred through bond to the con- 
 crete until the compression area of the beam is reached, or until a point but little below is 
 reached. Experiments show that it is safe to assume the grip of a stirrup to be 0.6 the depth 
 of beam. Using notation on page 270. 
 
 fsAs = 0.6 dou 
 
 A A 
 
 i -§ 
 
 • -l<o 
 
 1 +B 
 
 ■ i 
 
 y Y . 
 
 ^^^^ 
 
 mmMmMWMM concrete>^ 
 
 A' 
 
 
 
 < 
 
 
 
 =0.6^^ 
 
 0 fs 
 
 But, for round or square stirrups, letting i = maximum diameter of stirrup, 
 
 Then 
 
 1 . 
 
 (-3 
 
 If each end of the stirrup is bent into a prong or hook, then stirrups of larger diameter may be 
 used than is indicated by the above formula. Tests show that if hooks with a semicircular 
 
 1 Diagrams similar to those by Frank S. Bailey in E7ia. News. Oct. 12. 1916. 
 
Sec. 7-20] 
 
 BEAMS AND SLABS 
 
 291 
 
 bend of 4 diameters are well embedded in concrete, the stress in the bar will reach the elastic 
 limit before slipping takes place (see page 268). 
 
 It is considered good practice to use ^ie-'m. stirrups with hooked ends for beams from 10 
 to 25 in. deep, ^^-in. stirrups for beams from 25 to 40 in. deep, and 3^-in. stirrups for beams from 
 40 to 60 in. deep. 
 
 Illustrative Problem. — A simply supported beam is 9 by 16 in. in cross-section and the tension rein- 
 forcement is 2 in. above the lower face of the beam. Span of the beam is 8.5 ft. Uniform load of 1800 lb. per ft. 
 If necessary, the web is to be reinforced against diagonal tension using vertical stirrups. Allowable = 10,000; 
 J, = 40; w = 80. 
 
 V 7650 
 
 =6^ = T9)moA) = p^'- 
 
 The allowable shear is 40 lb. per sq. in., hence stirrups are necessary. 
 
 The diameter of a stirrup without any prong or hook should not exceed 
 
 80 
 
 0.27 in. 
 
 If the stirrups are to be bent at the upper end, Me-in. round bars may be considered secure against slipping. 
 Stirrups are unnecessary at a distance from support equal to 
 
 0- 'i) 
 
 1.80 ft. 
 
 The minimum spacing of stirrups (U-shape) will occur at the supports, or 
 
 3 2(0.077) (10,000) (14) 
 
 3.70 in. 
 
 2 7650 
 
 The shear diagram for one-half the beam is shown in Fig. 20. For a 4-in. stirrup spacing 
 
 V 
 
 2(0.077) (10,000) (7;/8) (14) 
 
 7090 lb. 
 
 The point where V = 7090 lb. is easily found by scaling to the shear diagram. For a5-in. stirrup spacing V = Yo 
 (7090) = 5670 lb. For a 6-in. spacing V = (7090) = 4730 lb., etc. Time can be saved by using Diagram I 
 in finding values of V directly. 
 
 ■R)in+ where s+irnjp spacing (s)= 4ii 
 
 •S'Stn. 
 s = 6in. 
 
 Shear Diagrami 
 
 •■■-stirrups not required be+ween 
 this point and center of bearr 
 
 Fig. 20. 
 
 Fig. 21. 
 
 20. Method of Placing Stirrups from the Moment Diagram. — It is a well-known principle 
 of mechanics that the difference in moment between any two points along a beam is equal to 
 the average total shear over the distance between the points multiplied by that distance. Thus, 
 from Fig. 21, 
 
 Vs = {Ma - Mc) 
 
 or, 
 
 V = (^-^ - Mc) ^) 
 s 
 
 For loads concentrated at points along a beam this law is not strictly true, unless in each 
 case the concentration occurs at a point midway between the transverse sections chosen; but 
 in the case of concentrated loadings, by beams cast against girders in concrete construction, 
 and even by loadings on slabs transmitted to the floor beams, the concentration may not be 
 sharply defined, and there is no determinate law of shear variation over such a region. More- 
 over, as this discussion will show later, the distance s is relatively small where shear is large. 
 
292 
 
 CONCRETE ENGINEERS' HANDBOOK 
 
 [Sec. 7-20 
 
 Within the limits of actual conditions in reinforced-concrete construction, therefore, the 
 above statement may be considered very approximate to the truth, for the beam loaded with 
 concentrated loads. 
 
 Substituting the above value for V in equation (1) on page 285, we have 
 
 (Ma - Mc) ^ 3 Asfsjd 
 s 2 ' s 
 
 whence 
 
 Mi = (Ma - Mc) = 1.5AJ,jd (2) 
 
 in which Mi is the increment of moment between the ends of the portion of the beam to be re- 
 inforced with the stirrup. 
 
 If the stirrup is inclined at an angle d with the horizontal, then 
 
 sm d ^ ^ 
 
 When e = 45 deg. (or, accurately enough, any angle between 30 and 45 deg.) 
 
 Mi = 2.1 Asfsjd (4) 
 
 Consider the portion of the beam shown in Fig. 22a, loaded in such a manner as to pro- 
 duce the moment curve OP. It is desired to reinforce the portion shown with vertical stirrups, 
 keeping in mind the principles just laid down. A certain stirrup has been adopted which, for 
 this particular beam, gives the value of Mi from equation (2) equal to the vertical distance 
 
 0.4 
 
 ii 
 
 Cji_i 
 
 1 
 
 (6) 
 
 Fig. 22. 
 
 shown in Fig. 22a. The first increment intercepts the portion Om of the curve OP; the second, 
 mn, and so on Let one of these intercepts, as mn, be projected on the beam, thereby defining 
 the area ABDC on the diagram of the beam. This area is the portion of the beam over which 
 the adopted stirrup will exactly carry the shear. The length of the portion is seen to vary as 
 the shear varies along the beam. Since the stirrup is required to carry the shear for this 
 portion of the beam, it should be placed through the center of gravity of the shear area for this 
 portion. Likewise, each other portion of the beam defined by the projection of Mi would 
 have a stirrup through the center of gravity of its shear area. 
 
 To eliminate the feature of having to locate this center of gravity, the following method is 
 proposed: Lay off as the first value }i Mi (Fig. 22&). Let all other spaces be equal to Mi as 
 before. These increments have m', n', etc., for points of intersection on the moment curve. 
 Let these points be projected on the beam. Each projection will thus determine the position 
 of a stirrup. This scheme gives very closely the same results as before, the stirrup being 
 placed slightly nearer the support than when placed through the center of gravity of the shear 
 area for the portion of the beam; this error, however, is well inside the accuracy of placing this 
 part of the reinforcing. 
 
 Diagrams IX to XII inclusive are plotted for a ready solution of equations (2) and (4) 
 given above. Their use is explained in an illustrative problem on page 295. 
 
Sec. 7-201 
 
 BEAMS AND SLABS 
 
294 CONCRETE ENGINEERS' HANDBOOK [Sec. 7-20 
 
 spunod-i^sui J.0 eptipsnom ui dnaaiis joj. lusLugjoui 4.u9luolu Buipuag 
 
 t CO fM (\1 
 
 (p) ssqoui ui Luoaq jo qidap aAi+Da^^g 
 
 spunod-qDU! spuDsnoqi ui dnjj.^s aoj. 4.uauiaj3U! 4U8Luoai Buipusg 
 
Sec. 7-201 
 
 BEAMS AND SLABS 
 
 295 
 
 A simple method of constructing the parabola or moment diagram for uniform loading is 
 as follows: Let AG, Fig. 23, represent the base of the parabola, with a middle ordinate of 4.5 
 at D. Divide the base into any desired number of equal parts, as for instance, six. Number 
 these points from each end beginning with zero. Divide the middle ordinate by the product 
 
 4.5 
 
 of the numbers at that point, as g = 0.5. This constant, if multiplied by the product of 
 
 the pair numbers at any point gives the ordinate at that point. For example, the ordinate at 
 (7 is 2 X 4 X 0.5 = 4.0. If an ordinate is de- 
 sired at a point between the equal divisions, as 
 A^, the fractional part of the division may be 
 expressed for the point from each way. At X 
 the distance from A is 2.6 units, and from G, 
 3.4 units. The ordinate at X is 2.6 X 3.4 X 
 0.5 = 4.42. If the middle ordinate does not 
 fall at an even division, as would be the case if 
 
 an odd number of units were used, the fractional values for the mid-point would be used the 
 same as the full values in the above case. 
 
 Illustrative Problem. — Determine the size and spacing of vertical stirrups for shear reinforcement in a 
 beam loaded as shown in Fig. 24a. 
 
 For the static uniform load of 1000 lb. per ft., the moment at the center is 
 
 For the concentrated loads 
 
 ivl^ (1000) (21)2 (12) 
 ^ = -r = 8 
 
 M = (1000) (7) (12) 
 
 = 662,000 in.-lb. 
 840,000 in.-lb. 
 
 which moment obtains through the central third of the beam. The moment curves for each condition are plotted 
 
 separately and the ordinates of the two combined to form 
 the curve of total moments, shown in Fig. 24b. 
 
 The total end shear is (1000) (21) H + 10,000 = 
 20,500 lb. Just outside the third point the shear is 20,500 
 - (1000) (7) = 13,500 lb. Just inside the third point it is 
 13,500 - 10,000 = 3500 lb. At the center the shear is zero. 
 The shear diagram is shown in Fig. 24c. 
 
 In flexure the following values will be adopted: fs = 
 16,000, fc = 650; n = 15. It is found that for equal 
 strength in tension and compression a depth {d) of 26 in. is 
 required when 6 = 20 in. Also j = 0.875. 
 
 The allowable shear to be carried by the concrete 
 alone will be taken as 30 lb. per sq. in. The concrete will 
 then carry (30) (20) (26) = 15,600 1b. This value is plotted 
 as point a on the shear diagram. Fig. 24c; and the distance 
 from the support to point a is the distance along the beam 
 where shear reinforcement is required. In this problem the 
 stirrups will be hooked at the ends and an allowable stress of 
 10,000 lb. per sq. in. will be assumed. It will be necessary 
 to select a size of stirrup that will not be so small that 
 the spacing will be too small for convenient construction, 
 nor yet large enough to cause the spacing to be greater 
 than \<id, a limit recommended by the Joint Committee. 
 To accomplish this, let a vertical line be projected from a, 
 Fig. 24c, to the moment curve, as at h. Assume some dis- 
 tance along the moment curve, as he, whose horizontal 
 projection is approximately Md, and note the value of its 
 vertical projection (in this case 200,000 in.-lb.). With this 
 value for vertical stirrups, enter Diagram IX at the right and 
 move to the left until the value of ; = 0.875 is reached; then 
 horizontal line from d = 26 in. This point is found to give a 
 
 (c) Shear Diagrcim 
 
 Fig. 24. 
 
 vertically until an intersection is made with 
 stirrup area equal to 0.58 sq. in. 
 
 Four Me-in. round rods give a combined area of 0.601 sq. in., and by entering Diagram IX at d = 26 in., 
 thenoe horizontally to A, = 0.60 sq. in., thence vertically to; = 0.875 and finally to the right, M = 205,000 in.- 
 lb. for vertical stirrups. 
 
206 
 
 CONCRETE ENGINEERS' HANDBOOK 
 
 [See. 7-21 
 
 Beginning as in Fig. 226, (205,000) is laid off at the base of the moment curve, as de, Fig. 246, after 
 which equal increments of 205,000 are laid off until the point 6 is passed. Through the points thus fixed, hori- 
 zontal lines are drawn to intersect the moment curve. From each intersection a vertical line projected to the 
 beam locates a stirrup. 
 
 21. Bent Bars and Vertical Stirrups for Web Reinforcement. — The ends of the horizontal 
 bars in a reinforced-concrete beam may usually be bent up to assist vertical stirrups in pre- 
 venting diagonal tension failure. Although in some cases these bars may be found theoretically 
 to take all the diagonal tensile stresses not taken by the concrete, vertical stirrups are always 
 desirable, as shown by tests. 
 
 Plain rods bent up to provide web reinforcement often lack sufficient bond strength to 
 render them fully effective. Where bent up at a considerable angle they should be turned again 
 horizontally and extend some distance along the upper part of the beam, as shown in Fig. 25. 
 In heavy construction the ends of all bars should be bent into a hook. The most convenient 
 method of using reinforcement is to bend up two rods at a time and make all the bars inclined 
 at the same angle with the horizontal. The bars bent should theoretically be such as to keep 
 the center of gravity of the beam cross-section in the line drawn vertically through the center 
 
 of the section. An exception occurs to the bending of 
 two rods at a time, in the case of an odd number of 
 horizontal rods. Here, one of the bends may consist 
 of either one or three rods. 
 
 If bent rods are not required to provide for di- 
 agonal tension, then the horizontal rods may be dis- 
 pensed with at the points, beyond which they are not 
 needed to provide for tension due to bending. This 
 method of stopping off the horizontal rods is not de- 
 sirable, however, as the bond in the concrete near the 
 middle of the beam is not as good as would be the 
 case near the end where the moments are smaller. 
 Also, when a bar is discontinued, the stress in those 
 which remain is immediately increased tending still 
 further to impair the bond between the steel and the 
 a hook is employed on the discontinued rods. With 
 The horizontal components of the upturned bars act 
 with the bars unbent in taking the tension due to bending, and so in general the tension in the 
 horizontal rods decreases somewhat more gradually toward the end of beam as it should. 
 
 At the bend in a horizontal rod, the unit stress in the concrete may become excessive if the 
 bend is too abrupt. Tests indicate that the strength of beams with bars having sharp bends is 
 less than for beams with bars having a radius of bend equal to about 12 diameters. 
 
 The bent rods, if of the same diameter, should be so arranged that each rod will take an 
 equal part of the diagonal tension — that is, if they can be bent in this way and still provide 
 satisfactorily for the horizontal tension. The points where the bent rods should cross the 
 neutral axis of the beam may be found by any of the methods given for determining the spacing 
 of vertical stirrups, remembering that a rod bent at 45 deg. (or, as tests show, at any angle be- 
 tween 30 and 45 deg.)^ is q-^ = 1.43 times as effective per unit area as a vertical stirrup. If 
 
 the rods cannot be bent as desired, then vertical stirrups must be used to provide for some of the 
 diagonal tension, and it would be preferable in all cases to use stirrups even in that part of the 
 beam where the bars are bent up. 
 
 Consider uniform loading and let ABCD in Fig. 25 be the diagonal tension area. Assume 
 that four rods may be bent near the end of beam and assume also that the first two bars cannot 
 be bent up nearer the center of beam than the point K, Fig. 25. The area cdef should be made 
 
 > CD 
 
 
 \ 
 
 i 
 i 
 
 i 
 i 
 
 ■f 
 
 
 a 1 d ;t e D 
 
 ..- ■2 Bars--. ; 
 
 "^"^N ' k \^ ' 
 
 _ \ <^^_ "^-i^AV |-N?ytr^lane3< J 
 
 ^. — — _ J 
 
 
 Fia. 25. 
 
 concrete. This is true whether or not 
 bent up rods a better condition exists. 
 
 1 See p. 286. 
 
Sec. 7-22] 
 
 BEAMS AND SLABS 
 
 297 
 
 equal to 1.43 times the allowable tensile stress in the two No. 2 bars; likewise area ahcd should 
 represent 1.43 times the allowable tensile stress in the No. 1 bars. (The vertical rt should theo- 
 retically pass through the center of gravity of the area and, if dt is made ^2 and te J-f 2 of de, this 
 will be practically accomplished. The error is not serious if dt is made equal to te.) If the No. 
 1 bars cannot be bent up so that the areas abed and edef have the line cd in common, then a ver- 
 tical stirrup, or stirrups, will be needed to take care of the space between these areas. If the areas 
 overlap, the No. 1 bars may be bent up nearer the end of beam if no other condition governs. 
 In any case the distance s should not be greater than ^id — that is, the maximum spacing at 
 which inclined web reinforcement can be considered effective. Stirrups will at least be needed 
 to the left of ab and to the right of ef. The area taken care of by a vertical stirrup is equal to 
 its tensile value. 
 
 The Joint Committee recommends that bent bars be considered as adding to diagonal 
 tension resistance for a horizontal distance from the point of bending equal to ^d. The Joint 
 Committee also recommends for the case where stirrups are needed in combination with bent- 
 up bars, that the stresses in the stirrups be determined by finding the amount of the total shear 
 which may be allowed by reason of the bent-up bars, and substracting this shear from the total 
 external vertical shear. Two-thirds of the remainder is recommended as the shear to be con- 
 sidered as carried by the stirrups. 
 
 The bond strength of inclined bars must be investigated. This strength should be pro- 
 vided in the upper portion of the beam. As with vertical stirrups, it is arbitrarily assumed that 
 no stress is transmitted from the steel to the concrete below a point which is 0.6d below the upper 
 surface of the beam. 
 
 Assume that the stress in an inclined bar is its working stress. This gives the maximum 
 condition. Using the notation of page 270 and I' for length 
 
 Vou = Asfs 
 
 and for round or square bars, 
 
 or 
 
 /, 
 
 V = ~- diameters 
 
 If the allowable /« = 10,000 lb. per sq. in. and the allowable u = 80 lb. per sq. in., then V 
 (as shown at No. 2 bars in Fig. 25) should equal 31 diameters by the above formula. If 1' = 
 16,000 lb. per sq. in., V = 50 diameters. 
 
 The value of hooks, on the ends of bars is discussed on page 268. 
 
 See illustrative problem on page 298. 
 
 22. Points Where Horizontal Reinforcement May be Bent. — For uniformly loaded beams 
 the bending-moment curve is a parabola as shown in Fig. 26. 
 Let a = area of bars required at the center of beam. 
 02 = area of bars to be bent. 
 
 P2 = 100 ^ = % of total steel that may be bent up. 
 
 M = maximum moment —— • 
 <p 
 
 X2 = distance from support to point where bars may be bent. 
 Mx^ = moment at distance X2 from support. 
 
 Then 
 
 M ^ a 
 Mx^ a — an 
 
 Substituting values of M and Mx^, and solving for Xi, 
 
298 
 
 CONCRETE ENGINEERS' HANDBOOK 
 
 [Sec. 7-22 
 
 Fig. 26,' based on the above formula, indicates the points at which rods may be bent up for 
 three types of beams with the maximum bending moments specified. To illustrate the use of 
 
 the diagram, assume that a beam designed for M 
 
 10 
 
 requires 3.5 sq. in. of steel at the 
 
 center'. To find the point where 40 % of the steel may be bent up and still leave sufficient steel 
 to carry the tension, trace horizontally from the 40% mark at the right to the curve M = 
 
 and then vertically to the lower margin where 0.22Z is read. 
 
 10 
 
 100 
 
 90 
 80 
 70 
 
 e 
 
 5 60 
 i 50 
 
 I 
 
 t 
 
 o 
 u 
 
 I- 
 
 40 
 30 
 20 
 10 
 
 10 \ 
 o 
 
 SZ 
 
 40+3 
 60^ 
 
 70 
 
 80 
 
 90 
 
 100 
 
 0.1 02 0.3 0.4 
 
 Location of section in terms of span length 
 
 0.5 
 
 Fig. 26. 
 
 If it is desired to bend up a number of rods two or more at a time, then Xt should be deter- 
 mined for each bend. After this is done, the remaining horizontal bars should be secure 
 against slipping. 
 
 For concentrated and unsymmetrical loading, the maximum moments at various sections 
 will need to be determined, in order to ascertain the points where the horizontal bars may 
 
 be bent up. From these maximum moments obtain 
 the required area of horizontal rods at the different 
 points (1, 2, 3, and 4, Fig. 27). Plot a curve to scale, 
 as shown. Thus, ah represents the area required at the 
 point a. On the center ordinate lay off the required 
 areas of the rods, and draw horizontals as shown. The 
 rods may be bent up where these horizontals cut the 
 curve but it would be better, however, to carry them a short distance beyond the theoretical 
 points. 
 
 Illustrative Problem. — Design the left end of a simply supported beam to span 15 ft. and to support the 
 loads shown in Fig. 28 with equal strength in tension and compression. Allowable Sc = 650; w = 80; » = 40 
 without web reinforcement and 120 with an effective web reinforcement. 
 
 1 Diagram taken from an article in Eng. News, Aug. 19, 1916, by Karl D. Schwendener. 
 
 Fig. 27. 
 
Sec. 7-22] 
 
 BEAMS AND SLABS 
 
 299 
 
 The reactions are readily found and are given in the sketch. The maximum moment occurs at the center 
 load since the shear passes through the value zero at this point. TVe shall assume the weight of beam included in 
 the uniform load of 1000 lb. per ft. 
 
 M = (25,500) (8) (12) - (23,000) (4) (12) 
 It is found that a depth (d) of 28 in. is required when 6 = 16 in. 
 25,500 
 
 1,344,000 in.-lb. 
 
 _ V_ 
 
 ~ bjd ~ (16)(J^)(28) ~ 
 
 Thus web reinforcement is needed. 
 
 As = (16) (28) (0.0077) 
 
 65 lb. per sq. in. 
 
 3.45 sq. in. 
 
 We shall select eight %-in. round rods = 3.53 sq. in. Bond for one rod 
 at the left end of beam is 
 
 25,500 
 
 443 lb. per sq. in. 
 
 (2.356) (Ji) (28) 
 
 For plain rods, the number which must extend straight to the left end 
 of beam is, for bond conditions especially favorable (see page 284), 
 443 
 
 (1.5) (80) 
 
 = 4 
 
 8'"0* - - > 
 
 1 
 
 1 
 
 i 
 
 
 Un'rfonn \oad-\QO(? 
 1 
 
 <- ^— 15- 
 
 1 
 1 
 
 including ^ 
 weigh+ of beam 
 
 >' - > 
 
 1 
 1 
 1 
 
 tit in 
 
 
 Fig. 28. 
 
 Thus, at this end of beam four rods may be bent up. 
 
 The concrete will be found to take care of any diagonal tension between the concentrated loads since concrete 
 will take (40) (16) = 640 lb. per lin. in. Horizontal shear (which measures diagonal tension) at the support is 
 
 V 25,500 
 
 jd = Gi)(28) = ^0^0 P^"- 
 and to the left of the adjacent concentrated load, it is 
 
 2J^00 
 (^/^)(28) 
 
 880 lb. per lin. 
 
 The total diagonal tension to be taken by the web re- 
 inforcement is represented by a trapezoid (Fig. 28), 
 the parallel sides of which are (1040) = 690 lb. and 
 % (880) = 590 lb. and the length 4 ft. Hence, total 
 stress in this part of beam to be taken by the con- 
 crete and by the web reinforcement is 
 
 690 + 590 
 
 X 4 X 12 
 
 30,720 lb. 
 
 Fig. 29. 
 
 is required, but it would seem advisable in a design 
 by the dotted lines. The length of embedment of the 
 
 Since four rods are to be bent, their compara- 
 tive tensile value is 
 
 (4) (0.4418) (16,000) (1.43) = 40,400 1b. 
 
 Thus, the tensile value of the rods is in excess of the 
 stress to be provided for. 
 
 An investigation must now be made to see 
 whether the tensile stresses in the bottom of the beam 
 will permit the bending of the bars. Fig. 29 shows 
 the bending-moment curve plotted to scale, and the 
 points where the bars may be bent up are determined 
 by the method described on page 298. It is clear 
 that the bars cannot be bent up as desired to provide 
 thoroughly for diagonal tension. The points where 
 the bars are actually bent up are about 2 in. beyond 
 the theoretical points as determined by moment. 
 
 If each bent-up bar is assumed to take diago- 
 nal tension to the left only of its point of bending, 
 then the area bANc remains unprovided for. Stirrups 
 will be provided to take diagonal tension between 
 the point t, where the line be produced meets the 
 neutral line, and the adjacent load. Only one stirrup 
 of this kind to also place stirrups at the positions indicated 
 inclined bars should be 50 diameters, or 37)^ in. 
 
300 
 
 CONCRETE ENGINEERS' HANDBOOK 
 
 [Sec. 7-23 
 
 23. Transverse Spacing of Reinforcement. — The shearing stress along ab, Fig. 30, should 
 equal the amount of stress transmitted by bond along bed. If bond and shearing strengths 
 
 were equal, ab should equal bed, and the clear space between bars should be ^'^2^^ diameters 
 
 = 1.57 diameters. But shearing strength here employed is controlled by the diagonal tension. 
 For y = 90 lb. per sq. in. and u = 80, ab should be at least 1,40 diameters; and for v = 120 lb. 
 per sq. in. and u = 80, ab should be at least 1.05 diameters. There is likely to be more or less 
 tension in the concrete surrounding the bars, and, besides, since the concrete is not easily placed 
 between the rods, it may have a lower strength in that vicinity. A clear spacing of 1^ to 
 2 diameters is advisable unless it is determined by computation that the bond stress is very 
 much lower than the allowable. In the above discussion plain bars only have been considered. 
 Deformed bars, if stressed to their full bond value, should be spaced farther apart than plain 
 bars. 
 
 The Joint Committee recommends that the lateral spacing of parallel bars should not be less 
 than 3 diameters, center to center, and that the distance from the side of the beam to the center 
 of the nearest bar should not be less than 2 diameters. In order that concrete 
 may be readily placed between bars and also give sufficient concrete on the sides of 
 the beam for fire protection, it is also advisable to require that the spacing of rods 
 be not less than 1 in. in the clear (if the maximum size of aggregate does not ex- 
 ceed 1 in.) and that in. in the clear be also considered the minimum distance 
 Fig. 30. of the rods from the sides of the beam. Thus, the least width of beam should be 
 the greater of the two values determined from the following formulas: 
 
 b = [3(n - 1) + 4]d, 
 
 b = ag{n — 1) + ndi + 3 
 
 in which b = least width of beam in inches. 
 
 di = thickness of the rods in inches. 
 
 n = maximum number of rods which occurs in a horizontal layer. 
 ag = maximum size of aggregate in inches. 
 For an aggregate with a maximum size of 1 in., the width of beam for all rods greater than 
 in. in diameter will ordinarily be governed by the first formula and for ^^-in. rods and less, 
 by the second formula. 
 
 Where two or more layers of rods are used, the rods should be so placed as to permit the 
 mortar to run between them. The Joint Committee specifies a limiting clear space of 1 in. and 
 does not recommend the use of more than two layers ''unless the layers are tied together by 
 adequate metal connections, particularly at and near points where bars are bent up or bent 
 down." The Joint Committee also advises that "where more than one layer is used, at least 
 all bars above the lower layer should be bent up and anchored beyond the edge of the support." 
 
 24. Depth of Concrete Below Rods. — Tests show that a 2-in. thickness of concrete is 
 necessary to thoroughly protect embedded steel from the direct action of flames. Flat slabs 
 are found to be affected to a less depth than projecting members such as beams and 
 columns. The Joint Committee suggests that ''the metal in girders and columns be protected 
 by a minimum of 2 in. of concrete; that the metal in beams be protected by a minimum of 1^ 
 in. of concrete; and that the metal in floor slabs be protected by a minimum of 1 in. of 
 concrete." 
 
 The following depths of concrete below the center of steel may ordinarily be employed 
 except where conditions are unusually severe. 
 
 Slabs 
 
 T-i i i. wj\ Depth below center 
 
 Depth to steel (d) ^ steel 
 
 3^^ in. and under in. 
 
 Between 3^ in. and 4^^ in 1 in. 
 
 4^4 over. 1>^ in. 
 
Sec. 7-25] BEAMS AND SLABS 301 
 
 Beams and Girders 
 
 Depth to steel (d) Depth in the clear 
 
 below steol 
 
 10 in. and under 1 in. 
 
 Between 10 in. and 20 in 1}^ in. 
 
 20 in. and over 2 in. 
 
 26. Ratio of Length to Depth of Beam for Equal Strength in Moment and Shear. — With 
 given working stresses in concrete and steel, there is a definite ratio of length to depth of beam 
 which will give equal strength in moment and shear. First, consider beams simply supported. 
 For a single concentrated load at the center of span 
 
 I ^ 2f,p 
 d vi 
 
 in which Vi = allowable shearing stress and fs = working stress in steel. For a uniformly- 
 
 distributed load 
 
 d Vi 
 
 For beams loaded with equal loads at the third points, 
 
 I ^ SfsP 
 d vi 
 
 Taking for example, vi = 40 lb. per sq. in., /, = 16,000, fc = 650, n = 15, and, using an 
 average value of ]4 for we have the following ratios for ^• 
 
 For concentrated load at center of span J = 6.16 
 
 For uniformly distributed load J = 12.32 
 
 For equal loads at the third points 4 = 9.24 
 
 It should be clear that the strength of beams of greater relative length than obtained by the 
 formulas will be determined by their moment of resistance, while that of shorter beams by their 
 shearing resistance. 
 
 In the case of continuous beams the above formulas will apply if I is taken as the length 
 between points of inflection. It is often convenient to know the extreme limit in design. The 
 Joint Committee recommends 120 lb. per sq. in. for the shearing strength of concrete when 
 adequately reinforced against diagonal tension. This is a low figure but is adopted in order to 
 prevent any likelihood of cracks opening up in the concrete. Suppose then, it is required to 
 
 know the minimum value of ^ for a given beam, uniformly loaded. From the formula for 
 
 uniform load, using the working stresses given above, 
 
 l_ ^ (4) (16,000) (0.0077) ^ 
 d 120 
 
 At the same time that the ratio of length to depth is being investigated for moment and 
 shear, there are other conditions which must be considered. For instance, the ratio of length to 
 breadth of beam should not exceed a value of about 25 if the beam is not supported laterally. 
 The reason for this is found in the fact that the upper part of the beam is a column, and to 
 prevent additional stress due to side bending the length should not exceed about 25 times the 
 width. On the other hand, the best-shaped beam is one in which h lies between }4d and ^id. 
 In any given case, to satisfy all requirements and arrive at a satisfactory design, two or three 
 trials may be required. 
 
302 
 
 CONCfRETE ENGINEERS' HANDBOOK 
 
 [Sec. 7-26 
 
 26. Economical Proportions of Rectangular Beams. — Without taking the cost of web 
 reinforcement into consideration, it can be shown mathematically that the cost of a rectangular 
 reinforced-concrete beam to resist a given bending moment and be of equal strength in tension 
 and compression, varies inversely with the depth, directly with the square root of the breadth, 
 and directly with the cube root of the ratio of breadth to depth. 
 
 The breadth and depth of a rectangular beam to be of equal strength in tension and com- 
 pression may be found by means of the formulas (see page 277) 
 
 1+4 
 
 nfc 
 
 ^ Jcl^J PJsJ 
 
 If b or d, or the ratio ^ is decided upon, the proportions and steel area of a beam to resist a 
 
 given bending moment are definitely determined. Thus for a fixed breadth, fixed depth, or 
 fixed ratio of breadth to depth, the cost of a beam will vary with the working stresses employed 
 since values of k, j, and p depend wholly on values of fc, fs, and n. 
 Where the depth is fixed, it is found that, if the ratio 
 
 _ cost of s teel per unit volume 
 cost of concrete per unit volume 
 
 does not exceed a value of 60 to 80 (60 a common value), no economy results from using/s greater 
 than 16,000 lb. per sq. in. when fc = 600 to 700 lb. per sq. in., or from using fs greater than 
 12,000 lb. per sq. in. when fc = 400 to 500 lb. per sq. in. Somewhat higher values than these 
 may be economically used for /« when the breadth of beam is fixed. In both cases cost de- 
 creases as fc increases. When the ratio ^ is fixed, no economy results from using /« greater 
 
 than 16,000 to 18,000 when fc = 400 to 600, or from using fc greater than 14,000 when 
 fe = 700. In this cag'e cost increases as fc increases. 
 
 In the case where the cross-section of beam is determined by shear, the maximum depth 
 theoretically permissible is that for which bd is just large enough to carry the shear. With a 
 beam designed for moment alone, the cost decreases as the depth increases, but the area of the 
 cross-section becomes less. A point must be reached when the beam will be of just the required 
 strength in moment and shear (see Art. 25). The question which now arises is whether or not 
 a still greater depth will result in greater economy. The quantity bd must now remain constant 
 for the greater depths. But bd^, on the other hand, is increased and the concrete stress (fc) 
 decreased. A smaller value for fc permits the use of a smaller percentage of steel, and the cost 
 is still further reduced. Thus it should be clear that the proportions of a beam will not be 
 determined by shear excepting as to minimum cross-section — an increase in depth always result- 
 ing in a gain in economy. It should be noted in this connection, however, that although deep 
 beams are economical of concrete, the wooden forms cost more than. they do for shallow beams. 
 
 27. Rectangular Beams with Steel in Top and Bottom. — Compressive stresses are usually 
 carried by concrete more economically than by steel. It is sometimes desirable, however, to 
 place steel in the compression as well as in the tension side of the beam. When a rectangular 
 beam is limited as to size, double reinforcement is sometimes the result, and in such cases the 
 value of the steel reinforcement on the compressive side needs to be known. The effectiveness 
 of steel in compression has sometimes been questioned, but the results of tests indicate that 
 the steel does its share of the work. 
 
 The Joint Committee recommends that "the reinforcing bars for compression in beams 
 should be straight and should be 2 diameters in the clear from the surface of the concrete. For 
 the positive bending moment, such reinforcement should not exceed 1% of the area of the 
 concrete." 
 
Sec. 7-27] 
 
 BEAMS AND SLABS 
 
 303 
 
 The formulas which are used in the design of double-reinforced rectangular beams an^ 
 derived by means of the same fundamental principles as for beams with single reinforcement. 
 In deriving the following formulas the compression in the concrete is assumed to follow the 
 linear law and the tension in it is neglected ; the formulas then apply to working conditions only. 
 
 Area of -..., 
 compressive 
 steel = A' 
 
 Area of - 
 tensile 
 Steel = A, 
 
 Cross- section Stress Diagram 
 Fig. 31. 
 
 Let A' = cross-sectional area of compressive reinforcement (Fig. 31). 
 
 d' = distance from the compressive face of the beam to the center of the compressive 
 reinforcement. 
 
 p' = ratio of cross-section of steel in compression to cross-section of beam above the 
 
 1 A' 
 
 tensile steel = t~i • 
 bd 
 
 /'s = compressive unit stress in steel. 
 
 1 +^ 
 
 nfc 
 
 (1) 
 (1-4) 
 
 (2) 
 
 •'' A,jd pjbd^ ^ ' 
 
 */ _ fcujl - k) ' . (6) 
 
 k 
 
 Ms = bd%pj CO 
 
 '-m ' 
 
 The cases which may be met with in practice, with the method of solution in each instance 
 indicated, are as follows: 
 
 1. To determine fiber stresses. 
 
 Compute p, p', and ^• 
 
 Solve for k from formula (1) and j from formula (2). 
 
 Substitute value of j in formula (3) and value of A' in fornmlas (4) and (5)., 
 Solve directly for fiber stresses. 
 2. To determine moment of resistance. 
 
 Compute p, p', and ^- 
 
304 
 
 CONCRETE ENGINEERS' HANDBOOK 
 
 [Sec. 7-27a 
 
 Solve for k and j. 
 
 Substitute value of k in formula (6) and find value of /« when the maximum allowable 
 
 value of fc is substituted. 
 Solve formula (7) for Ms using either the value of fs determined from formula (6) or 
 
 the allowable value of fs, whichever is the lesser. 
 
 3. To determine d for a given b and given values of As, — „ fc, and /«. 
 
 (Trial method. Best shown by use of diagrams. See illustrative problem, page 347.) 
 
 4. To determine p and p', or only p' (see Art. 27a). 
 
 Formulas for shear, bond, and web reinforcement are the same for double-reinforced beams 
 as for beams with tensile steel only. When using formulas for shear and bond stress along 
 horizontal tension rods of beams double-reinforced, an average value of j = 0.85 may be taken. 
 
 27a. Formulas for Determining Percentages of Steel in Double -reinforced 
 Rectangular Beams. ^ — The formulas given below are based on the fundamental fact that for 
 any given values of fc and fs, k has exactly the same value regardless of the shape or type of 
 beam. This single value for all beams is expressed by the formula 
 
 1 
 
 k = - 
 
 1 + 4 
 
 nfc 
 
 It follows from this that if steel is added to the section without changing the extreme fiber 
 stresses, this added tensional and compressive steel must form a balanced couple whose stresses 
 conform to the stresses already in the section. 
 
 Let pi = steel ratio for the beam without compressive steel. 
 P2 = steel ratio for the added tensional steel. 
 
 p = Pi + P2. 
 
 p' = steel ratio for compressive steel. 
 Ml = moment of the beam without compressive steel. 
 M2 = moment of the added steel couple. 
 
 M = Ml + M2. 
 
 Then 
 
 k=-^-j- (1) 
 
 I 7A 
 
 (3) 
 
 = fsPi ( 1 - I) 
 
 M2 = M - Ml (4) 
 
 p = Pl + P2 ^ (6) 
 
 P' = P.-i^, . (7) 
 (See page 348 for illustrative problem.) 
 
 28. Deflection of Rectangular Beams. — Fig. 32 gives the general form of a deflection dia- 
 gram for a reinforced-concrete beam. The portion AB shows the deflection before the con- 
 crete has begun to fail in tension near the center of the beam, BC shows the deflection during 
 
 1 From thesis by Robert S. Beard submitted to graduate school of University of Kansas in partial fulfill- 
 ment of the requirements for the Master's Degree. 
 
Sec. 7-28a] 
 
 BEAMS AND SLABS 
 
 305 
 
 a second or readjusting stage, and CD the deflection with the steel near the center of beam 
 carrying practically all the tension. 
 
 28a. Maney's Method. ^ — The deflection of a reinforced-concrete beam of whatever 
 shape may be determined by the formula 
 
 D = c ^ (ec + es) 
 
 where 
 
 D = maximum deflection (if desired in inches, the units specified be- 
 low should be used). 
 I = span (inches). 
 
 d = depth of the beam to the center of the steel (inches). 
 
 fc 
 
 Be = unit deformation in extreme fiber for the concrete = ^r' a 
 
 . Deflection 
 Cs = unit deformation in extreme fiber for the steel = 4- Fig. 32. 
 
 c = — in which 
 
 Ci = the numerical coefficient in the formula for deflection of homogeneous beams, 
 
 D = Ci ~^Y^ depending on the loading and on how the ends are supported. 
 
 C2 = the numerical coefficient in the formula for bending moment, M = C2wP, 
 
 for a simple beam loaded at center, c = 3l2 or 0.0833 
 uniformly loaded, c = or 0.1041 
 loaded at the third points, c = or 0.1065 
 
 for a beam with fixed ends, loaded at center, c = or 0.0416 
 uniformly loaded, c = H2 or 0.0313 
 loaded at the third points, c = ^{44 or 0.0347 
 
 28b. Turneaureand Maurer's Method.- — Turneaureand Maurer recommend that 
 8 to 10 be used for n in the formulas which they have derived, and which are given below. They 
 also state that the formulas presented are the result of modifying the deflection formulas for 
 homogeneous beams in accordance with the following assumptions : 
 
 1. The representative or mean section has a depth equal to the distance from the top of 
 the beam to the center of the steel. 
 
 2. It sustains tension as well as compression, both following the linear law. 
 
 3. The proper mean modulus of elasticity of the concrete equals the average or secant 
 modulus up to the working compressive stress. 
 
 4. The allowance for steel in computing the moment of inertia of the mean section should 
 be based on the amount of steel in the mid-sections, since stirrups and bent-up rods do not 
 affect stiffness materially for working loads. 
 
 The following are the deflection formulas for rectangular reinforced concrete beams: 
 
 W73 ^ (1) 
 
 a = }i[k^ + (1 - ky + Snpil - fc)2] (2) 
 
 k = 141^ (3) 
 2 + 2np 
 
 From equations (2) and (3), the value of a for any values of p and n may be computed, and then 
 the deflection from equation (1). The notation employed in the above formulas is as follows: 
 
 1 See paper by G. A. Maney, presented before the seventeenth annual meeting of the American Society for 
 Testing Materials. 
 
 2 "Principles of Reinforced Concrete Construction" 2d Edition, p. 116. 
 20 
 
306 
 
 CONCRETE ENGINEERS' HANDBOOK 
 
 [Sec. 7-29 
 
 D = maximum deflection (if desired in inches, the units specified below should be used). 
 b = breadth of the beam (inches). 
 
 d = depth of the beam to the center of the steel (inches). 
 W = total load (pounds). 
 I = span (inches). 
 p = steel ratio. 
 
 Es = modulus of elasticity of the reinforcing steel (pounds per square inch). 
 n = ratio of the moduli of elasticity of steel and concrete, 
 a = a numerical coefficient depending on p and n. 
 k = proportionate depth of the neutral axis. 
 
 Ci = the numerical coefficient in the formula for deflection of homogeneous beams, 
 
 Ci -^jj depending on the loading and support. For example, 
 
 for a cantilever loaded at the end, Ci = }i 
 
 for a cantilever uniformly loaded, Ci = 3^^ 
 
 for a simple beam loaded at center, Ci = }"^g 
 
 for a simple beam uniformly loaded, Ci = ^is4 
 
 for a beam with fixed ends, load at the center, Ci = '^{^2 
 
 for a beam with fixed ends, uniformly loaded, Ci = ^84 
 
 The following are the deflection formulas for reinforced concrete T-beams (referred to later): 
 c 1 WP n 
 Es ' biP ' p 
 
 ft = Vslk^ - ( 1 - (fc - ' + (1 - ky + Spna -k)^ 
 
 «[r-r(r+©i' 
 
 np + 
 
 ,b' b'/t\ , t 
 
 in which /3 is a coefficient depending upon the steel ratio and n, and other symbols as before. 
 
 29. Slabs. — A reinforced-concrete slab should be figured in the same manner as a rectangu- 
 lar beam, the bending moment being usually computed for a width of slab equal to 1 ft. The 
 ratio of steel in a slab is most readily found by dividing the cross-section of one bar by the 
 area between the centers of two adjacent bars, this area being the spacing of the bars multi- 
 plied by the depth of steel below the top of slab. 
 
 Slab bars should not be placed too far apart to properly take stress directly nor yet should 
 they be spaced so close that the concrete cannot be properly placed between them. The main 
 tensile reinforcement should not be spaced farther apart than 2}^ times the thickness of the slab. 
 The minimum limit should be about the same as in beams. 
 
 Shearing failures are not usually important in slabs, but in special cases of heavy loading 
 the same care should be used as in the design of large beams. ^ 
 
 29o. Moments in Continuous Slabs. — For uniformly loaded spans, fully 
 continuous over two or more intermediate supports, a moment of }i2U>l^ may be used both in the 
 centers of all spans and over all supports, for both dead and live loads. For slabs continuous 
 for two spans only, with ends restrained, the bending moment both at the center support and 
 
 near the middle of span should be taken ^sj^' For very unequal spans or spans of unusual 
 
 length, the moments should be computed more accurately. 
 
 295. Provision for Negative Moment in Continuous Slabs. — Slabs having 
 spans of any appreciable length should be reinforced against negative moment. This may be 
 done by bending up a part of the rods in the spans on each side of a support and extending each 
 
 1 See p. 344 for illustrative problems using diagrams. For flat slab floors, see chapter in Sect. 11. 
 
Sec. 7-29c]* 
 
 BEAMS AND SLABS 
 
 307 
 
 set of bent rods along the top of beam into the adjoining span. The bend in the bars should 
 be near the }i points in the span, and usually at an angle of 30 deg. with the horizontal. Too 
 sharp an angle may tend to crack the slab. 
 
 When placing slab reinforcement in long spans, a top reinforcement at least to the third 
 point will be desirable. In ordinary sp.ans the steel should at least be lapped a sufficient dis- 
 tance over supports to provide adequate bond strength, and the steel should be bent up far 
 enough from the support to provide properly for negative moment. 
 
 29c. Floor Slabs Supported Along Fom Sides. — When a floor panel is square, 
 or nearly so, the slab may advantageously be reinforced in both directions. Exact analysis 
 of stresses in such a case is impossible, but some important facts have been brought out by ap- 
 proximate solutions for uniform loading. The theory applied in such an analysis depends upon 
 the fact that the load at any point on the slab is distributed to the two systems of reinforcing 
 bars at that point, in proportion to the stiffness of the beam elements lying in those directions. 
 
 The following recommendations are from the report of the Joint Committee: 
 
 Floor slabs having the supports extending along the four sides should be designed and reinforced as con- 
 tinuous over the supports. If the length of the slab exceeds times its width, the entire load should be carried by 
 transverse reinforcement. 
 
 For uniformly distributed loads on square slabs, one-half the live and dead load may be used in the cal- 
 culations of moment to be resisted in each direction. For oblong slabs, the length of which is not greater than I'/i 
 times their width, the moment to be resisted by the transverse reinforcement may be found by using a proportion of 
 the live and dead load equal to that given by the formula 
 
 I 
 
 r = - 0.5 
 
 b 
 
 where I = length and b = breadth of slab. The longitudinal reinforcement should then be proportioned to carry 
 the remainder of the load. 
 
 In placing reinforcement in such slabs account may well be taken of the fact that the bending moment is 
 greater near the center of the slab than near the edges. For this purpose two-thirds of the previously calculated 
 moments may be assumed as carried by the center half of the slab and one-third by the outside quarters. 
 
 29(i. Cross-reinforcement in Slabs. — When the length of a floor panel is large 
 compared to its breadth, the longitudinal reinforcement (that is, reinforcement parallel with 
 the length) is of little value in carrying loads, but a small amount is nevertheless generally 
 desirable in preventing shrinkage and temperature cracks and in binding the entire structure 
 together. It is more important for wide beam spacing than when the beams are closely spaced. 
 The amount of steel to use is usually selected somewhat arbitrarily, and Y^-\^. or Y^-va. rods 
 spaced 18 to 24 in. apart is common practice. The top of the slab over a girder should be re- 
 inforced transversely not only for stiffening the girder, but also to provide for the negative 
 bending moment produced with the bending of the slab at right angles to the direction of the 
 principal slab steel. 
 
 T-BEAMS 
 
 30. T-Beams in Floor Construction. — When a slab and beam (or girder) are built at the same 
 time and thoroughly tied together by means of stirrups, bent-up rods, and cross-slab reinforce- 
 ment, a part of the slab may be considered to act with the upper part of the beam in compression. 
 This form of beam is called a T-beam, and the extra amount of concrete in the compressive 
 part of such a beam makes possible a considerable saving over the rectangular form. The 
 thickness of the flange is fixed by the thickness of slab required to support its load, but the 
 width of slab which can be taken as effective flange width must be selected somewhat arbitrarily. 
 
 31. Tests of T-beams.— T-beams are found to fail under essentially the same condi-. 
 tions as rectangular beams, and the same general principles apply. Tests show that the maxi- 
 mum load carried can be materially increased by placing cross bars in the top of slab and 
 by adding fillets between the flange and the beam. Cross bars are found to be actually needed 
 to insure T-beam action but fillets are not required in ordinary cases. 
 
308 
 
 CONCRETE ENGINEERS' HANDBOOK 
 
 [Sec. 7-32 
 
 T-beams with projections of flange on each side of web of 10.5 times the thickness of the 
 slab have been found to carry a load only 5% larger than beams with projections of 6.8 times 
 the thickness of the slab. No appreciable difference was found in the latter beams between the 
 deformations at the edge of flange and the deformations in the flange at the web. 
 
 32. Flange Width. — The Joint Committee has recommended the following rules for deter- 
 mining flange width : 
 
 (a) It shall not exceed one-fourth of the span length of the beam. 
 
 (6) Its overhanging width on either side of the web shall not exceed 6 times the thickness of the slab. 
 
 Beams in which the T-form is used only for the purpose of providing additional compression area of concrete 
 should preferably have a width of flange not more than 3 times the width of the stem and a thickness of flange 
 not less than one-third of the depth of the beam. 
 
 33. Bonding of Web and Flange. — The web and flange of a T-beam can be considered well 
 tied together when slab reinforcement crosses the beam and when the web reinforcement extends 
 well up into the slab. The bonding should be especially well looked after near the end of beam, 
 and this is generally accomplished by means of the bent rods and stirrups brought up as high 
 as possible, in addition to the slab reinforcement (as mentioned) acting at right angles to the 
 
 length of the beam. Along the center of the 
 beam the differential stresses between the 
 beam and slab are not large, but it is better 
 to insert vertical stirrups extending up into 
 the slab at occasional intervals since shrinkage 
 of the concrete is apt to part the slab from the 
 beam if there is not some means to hold the 
 two together mechanically. The thinner the 
 sections, the more thorough should be the 
 bonding. 
 
 34. Flexure Formulas. — With a T-beam 
 it is necessary to distinguish two cases; namely, (1) the neutral axis in the flange, and (2) 
 the neutral axis in the web. 
 
 Case I. The Neutral Axis in the Flange. — All formulas for "moment calculations" of 
 rectangular beams apply to this case. It should be remembered, however, that h of the for- 
 
 A A 
 
 mulas denotes flange width, not web width, and p (the steel ratio) is not ^ (Fig. 33). 
 
 Case IL The Neutral Axis in the Weh. — The amount of compression in the web {aaaa, 
 Fig. 33) is commonly small compared with that in the flange, and is generally neglected. The 
 formulas to use, assuming a straight-line variation of stress and neglecting the compression in 
 the web, are: 
 
 1 
 
 
 <• -^c >j 
 
 \ 
 
 ■t rol plane ^ 
 
 
 
 ■i i 
 
 
 
 Stress DiQ9ram 
 
 Fig. 33. 
 
 M = 
 
 k = 
 
 jd = 
 
 1 +-4- 
 nfc 
 
 2ndK + ht^ 
 2nA, + 2bt 
 
 + 
 
 (a 
 
 (1) 
 (2) 
 
 (3) 
 
 (4) 
 (5) 
 
 (6) 
 
Sec. 7-34a] 
 
 BEAMS AND SLABS 
 
 309 
 
 " A;) 
 
 /.=^^,-^' (9) 
 M, = S,A,id (10) 
 
 Approximate formulas can also be obtained. From the stress diagram, Fig. 33, it is clear 
 that the arm of the resisting couple is never as small as — 3^f, and that the average unit 
 compressive stress is never as small as ^/c, except when the neutral axis is at the top of the 
 web. Using these limiting values as approximations for the true ones, 
 
 Mo = y2fcht{d - y2t) (a) 
 
 M 
 
 M. = A.f.{d - Ht), or A, = (j^-^^^ _ y^^^ (6) 
 
 The errors involved in these approximations are on the side of safety. 
 
 Formulas which take into account the compression in the stem are recommended where the 
 flange is small compared to the stem. Such formulas may be found in the report of the Joint 
 Committee, and are as follows: 
 
 kd 
 
 z 
 
 ^2ndAs + (b - b')t'' ^ ^As + (b - b')t\j ^ 
 
 (kdt' - HP)h + [{kd - ty {t + H^kd - i))W 
 t{2kd - t)b + {kd - tyb' 
 jd = d — z 
 
 Asjd 
 
 2Mkd 
 
 nAs + (6 - b')t 
 
 [{2kd - t)bt + {kd - tyb']jd 
 
 34a. Case II Formulas for Determining Dimensions and Steel Ratio for 
 Given Working Stresses.^ — The following formulas are sometimes useful in the design of T- 
 beams when the neutral axis is in the web : 
 
 t - VR2' - 12fcRi 
 
 d = 2r; ^^^^ 
 
 in which 
 
 Ri=Ic+^- 
 n 
 
 and 
 
 ©©-©■(!:) 
 
 (Formula (12) should be solved by exact methods as the slide rule does not give satisfactory 
 results.) 
 
 If the depth of beam is fixed by the headroom available, formula (13) gives directly the 
 proper percentage of steel for given working stresses. 
 
 1 From thesis by Robert S. Beard submitted to graduate school of University of Kansas in partial fulfillment 
 of the requirements for the Master's Degree. 
 
310 
 
 CONCRETE ENGINEERS' HANDBOOK 
 
 [Sec. 7-36 
 
 35. Designing for Shear. — Since a T-beam will usually have ample strength in compression 
 for any ordinary depth of beam likely to be selected, the design of the stem of the T, or the beam 
 below the slab, is largely a question of providing sufficient concrete to take care of the shearing 
 stresses and to give a good layout of the tension rods. The manner of providing reinforce- 
 ment for shearing stresses in T-beams is similar to that described for rectangular beams. In 
 T-beams, however, the reinforcement for shear should run well up into the slab in order to tie 
 the beam and slab together. The shearing strength of a T-beam is about the same as that of a 
 rectangular beam of the same depth and a width equal to the width of the stem of the T. 
 
 36. General Proportions of T-beams. — T-beams should not be made too deep in propor- 
 tion to the width of stem as such forms are relatively weak at the junction of stem and flange. 
 The width should preferably be from one-third to one-half the depth in ordinary cases. For 
 large beams the width may be made from one-third to one-fourth the depth. 
 
 All re-entrant angles in concrete are points of weakness and such angles should, therefore, 
 be avoided. 
 
 37. Economical Considerations. — When a floor slab forms the flange of a T-beam, it is 
 possible to determine economical proportions for the stem. 
 
 Consider a portion of a rectangular beam one unit in length. 
 
 Let c = cost of concrete per unit volume. 
 
 r = ratio of cost of steel to cost of concrete per unit volume. 
 • C = cost of beam per unit length. 
 d = depth of beam below slab. 
 
 Then 
 
 using the approximate formula (5) on page 309. 
 
 When d' is fixed by the headroom available, the cost will be a minimum when b' is made as 
 small as possible, and its value will then be determined by the shearing stress or by the space 
 required for the rods. The expression also shows that the cost will decrease with increased 
 values of fs, and that with a fixed value of b'd' the cost decreases with increase in depth. If the 
 value of b' is assumed as fixed, then there is a definite value of d which will give minimum cost. 
 The following expression has been deduced from the preceding equation and will give the value 
 of d for minimum cost when the value of 6' is fixed : 
 
 From this expression the best depths for various assumed widths may readily be determined and 
 the desirable proportions finally selected. 
 
 The following table is convenient in determining values of r: 
 
 38. Conditions Met with in De- 
 sign of T-beams. — In practice the de- 
 sign of T-beams will take one of the 
 following forms with method to be 
 followed in each case indicated: 
 
 1. To find moment of resistance 
 or fiber stresses. 
 The values of k and j may be 
 found from equations (3) and 
 (6), or from equations (2), (4) 
 and (5), on page 308, and then 
 the values of the fiber stresses 
 from equations (7) and (8), or 
 the moment of resistance from 
 
 Cost of 
 
 Cost of concrete, dollars per cubic yard 
 
 
 
 
 
 
 
 
 
 
 steel, 
 
 
 
 
 
 
 
 
 
 cts. per lb. 
 
 5 
 
 G 
 
 7 
 
 8 
 
 ■ 9 
 
 10 
 
 11 
 
 12 
 
 1 
 
 26 
 
 22 
 
 19 
 
 16 
 
 
 
 
 
 IK 
 
 40 
 
 33 
 
 28 
 
 25 
 
 22 
 
 
 
 
 2 
 
 63 
 
 44 
 
 38 
 
 33 
 
 29 
 
 26 
 
 
 
 
 66 
 
 66 
 
 47 
 
 41 
 
 37 
 
 33 
 
 30 
 
 
 3 
 
 80 
 
 66 
 
 67 
 
 60 
 
 44 
 
 40 
 
 36 
 
 33 
 
 
 92 
 
 78 
 
 66 
 
 68 
 
 61 
 
 46 
 
 42 
 
 38 
 
 4 
 
 
 88 
 
 76 
 
 66 
 
 69 
 
 63 
 
 48 
 
 44 
 
Sec. 7-39] 
 
 BEAMS AND SLABS 
 
 311 
 
 equations (9) and (10), When the moment of resistance depends upon the concrete, 
 equation (9) is useful in determining the value of to use in equation (10). To obtain 
 this value, the maximum allowable value of fc should be inserted in equation (9). (If 
 
 the value of k is found to be less than ^, then the problem falls under Case I and the 
 
 formulas for rectangular beams apply.) 
 
 2. To design a T-beam in which the flange forms a portion of a floor slab. 
 
 Depth and width of stem of beam should be selected with reference to shearing strength, 
 space for necessary rods, and other considerations. The depth hashing been selected, 
 the amount of steel may be approximately determined by equation (6). The amount 
 of steel being known, the value of j may be determined by equation (6). The value 
 of k should also be found from equation (2) or (3) in order to ascertain if the beam 
 falls under Case I or Case II. The stress in the concrete, corresponding to the allow- 
 able working stress in the steel, is then found from equation (8). 
 
 3. To find the minimum depth for a single-reinforced T-beam. 
 
 The value of ^, or d, may be found from equation (12) for given working stresses, and 
 
 the amount of steel from equation (13). The width of stem should then be selected 
 with reference to shearing strength, etc. 
 
 4. To design a T-beam not connected with a floor system. 
 First method : 
 
 First, select suitable proportions for the web. A flange thickness is then assumed 
 
 such as to give satisfactory proportions between t and d. The value of ^ is then 
 
 known and k and j can be determined from equations (1), (4), and (5). The 
 area of steel and the breadth of flange are then found from equations (10) and 
 (11) respectively. 
 Second method: 
 
 For any assumed thickness and width of flange, the depth of beam may be de- 
 termined by equation (12) and the percentage of steel from equation (13). The 
 width of stem should then be selected with reference to shearing strength, etc. 
 (When making approximate computations for shear or bond stress along the horizontal 
 tension rods, an average value of j = may be assumed, as for rectangular beams.) 
 
 39. Design of a Continuous T-beam at the Supports. — Negative bending moment exists 
 at the supports of continuous beams and tensile steel must be placed in the top of beams over 
 supports to prevent cracks opening up at these points. For the usual case of equal spans and 
 indefinite live load, the common method of providing for this negative moment is by bending 
 up one-half of the rods on each side and extending each set over the supports into the adjoin- 
 ing span. The remaining lower horizontal rods in each span are carried horizontally through 
 the supporting columns. 
 
 In a design of continuous T-beams at the supports it should be noted that the flange is 
 under tension, that the stress in the concrete is negligible above the neutral axis and that 
 a rectangular section may be considered at such points. The method of design is thus similar 
 to the design of a double-reinforced rectangular beam at the center of span with the exception 
 that the compressive and tensile stresses about the neutral axis are inverted (see page 302). 
 
 Since a T-beam in the center of span becomes a rectangular beam over supports, the stress 
 in the tensile steel at the support will generally be greater in ordinary designing than the corre- 
 sponding stress at the center of beam; that is, this stress- will be greater if half the rods are bent 
 up on each side and lap over the support. For this reason, then, when selecting the steel at 
 the center of span, a little more than the required amount at that point should be chosen. It 
 should be noticed, however, that the column has some strengthening action at the support 
 and it will not be necessary to keep t'^o closely to the allowable stress. 
 
312 
 
 CONCRETE ENGINEERS' HANDBOOK 
 
 [Sec. 7-39 
 
 A higher compressive stress may be allowed in the concrete at the supports than at the 
 middle of span, because of the fact that the negative moment decreases very rapidly and only 
 a short section is under maximum stress. Also, a slight excess of stress at this point does not 
 in any way endanger the structure but merely increases somewhat the positive moment on the 
 beam. 
 
 There are three methods of reducing the compressive stress in the concrete at the bottom 
 of the beam over supports: (1) by increasing the amount of compressive steel in the bottom 
 of the beam; (2) by increasing the area of compressive concrete, which may or may not require a 
 flat haunch depending upon the width of the support; and (3) by increasing the areas of both 
 steel and concrete. 
 
 The bond stress along the horizontal rods at the top of a continuous beam over supports 
 may be found by the same formula as is employed for the tension rods at the end of a simply 
 supported beam. However, if bent up rods are employed for web reinforcement and if these 
 same rods are employed to take the tension over supports, the beam is greatly stiffened and the 
 bond stress along the top rods is undoubtedly reduced appreciably below that given by the 
 theoretical formula. This bond stress is affected by the amount of web reinforcement used in a 
 somewhat similar manner to the way the bond stress is affected along the rods at the end of a 
 simply supported beam (see page 284). 
 
 In beams considered uniformly loaded, the rods which are bent should extend beyond the 
 center of support to about the fourth point, or in beams of very definite live load to the third 
 point (point of zero moment varies for different loadings), to provide thoroughly for negative 
 moment, and this length should be increased if it is not sufficient to transfer to the concrete 
 through bond, the greatest allowable tensile stress in the rods. 
 
 If half of the rods from each span are used over the support, then half of the total number 
 will extend to about the fourth point where the tension due to negative moment becomes zero. 
 At this point the shear is only one-half of what it is at the end of span, if the beam is considered 
 uniformly loaded. Since bond stress due to increment (or decrement) tension varies as shear, 
 a sufficient number of rods are thus run out to the fourth point, and with the bent rods being 
 added gradually to this number until all the rods are acting in this manner at the support, it 
 is clear that this method of design is satisfactory even when the bond stress at the support 
 is the maximum allowable. 
 
 Rods should be bent in a position to take as much diagonal tension as possible, usually 
 at an angle between 30 and .45 deg., and the points where the rods are no longer needed at the 
 bottom of beam to resist tension may be found as explained on page 297. It is also necessary 
 to determine where the rods over supports may be bent down. It will be on the safe side, 
 and sufficiently accurate, to consider the curve for negative moment as a straight line between 
 the support and the point of zero moment at the fourth point. (For a very definite live load, 
 zero moment should be assumed at the third point.) With this variation of the moment in 
 mind, it is an easy matter to find where the rods may be bent down at the top of the beam. 
 The designer must use his judgment in the matter, but this much may be said: if a bend 
 cannot be made in a rod, as proposed, due to the controlling points for bending at the top 
 and bottom, a greater number of rods may be employed at the center of span in order to 
 make this bending possible, and the design governed accordingly. It is evident from the 
 above that it will not always be possible to place the rods so as to take all the diagonal ten- 
 sion, in which case both stirrups and bent rods must be used. 
 
 Another point to be noted in the design of a continuous beam at the supports is the bond 
 stress of the compressive reinforcement. It can be shown that the bond stress per square inch 
 for the tension and compression rods will be proportional to the product of the diameters by 
 their distances from the neutral axis. Since the compressive steel will generally be nearer the 
 neutral axis than the tensile steel, it follows that, if the compression bars are no larger in di- 
 ameter than the tension bars, the bond stress per square inch will always be less than that of 
 the tension bars. It is sufficient to consider simply the compressive stress in the steel and 
 
Sec. 7-40] BEAMS AND SLABS 313 
 
 provide a sufficient length from this point to the end of the bar to transmit this stress. The 
 working strength of the steel in compression cannot be reached without exceeding the com- 
 pressive strength of the concrete in which it is embedded. Consequently, in common design 
 it will be satisfactory to provide a lap beyond the support sufficient to take care of compress- 
 ive stress in the steel equal to the maximum as determined by the concrete. 
 
 40. T-beams with Steel in Top and Bottom. — The following formulas correspond to those 
 
 for rectangular beams given on page 303. ^ = ^' 
 
 A(2k - A) - (2k - 2A) + 2p'n (k - (l - 
 A(2k - A) +2p'n(^k-~^ 
 
 (1) 
 
 (2) 
 
 Asjd pjbd^ 
 
 = r^k 
 
 _ fcfijl - k) , . 
 
 Ms = bd'fspj (7) 
 
 t 
 
 T-beams.^ A = 
 
 40a. Formulas for Determining Percentages of Steel in Double -reinforced 
 
 i • 
 
 d 
 
 1 
 
 1 + ^- 
 
 6-6A+2A^ + A3(^) 
 
 J = 
 
 (2) 
 
 (3) 
 
 6 - 3A 
 
 Ml = fsp^jbd^ (4) 
 M2 = M - Ml ' (5) 
 
 M2 
 
 P2 = 
 
 bd^ 
 
 (6) 
 
 P = Pi + P2 ■ (7) 
 
 1 - k 
 
 P (8) 
 
 d 
 
 41. Deflection of T-beams. — Formulas are given in Art. 28 
 
 1 From thesis by Robert S. Beard submitted to graduate school of University of Kansas in partial fulfillment 
 of the requirements for the Master's Degree. See p. 304 for notation, etc. 
 
314 
 
 CONCRETE ENGINEERS' HANDBOOK 
 
 [Sec. 7-42 
 
 SPECIAL BEAMS 
 
 42- Wedge-shaped Beams. ^ — The analysis of wedge-shaped beams is useful chiefly in 
 the design of counterforts and buttresses, and in the design of cantilever beams for overhanging 
 sidewalks or roadways on deck bridges. Formulas follow for the general case shown in Fig. 34: 
 
 k = 
 
 V2pn cos , p^n^ cos^ /3t 
 cos2 /3« cos* /3c 
 
 pn cos |3< 
 
 cos 2 /3; 
 
 1 + 
 
 + 1 
 
 Mc = 
 
 fc U/< 
 
 Hfckj (bd^) cos2 /3c, or bd^ = 
 
 ^ fck / cos^ fic \ 
 2f, \cos ^t) 
 
 ^COS Pt. 
 
 2M 
 
 Mc = pfsj (bd^) COS ^t, or bd'^ 
 
 M 
 
 p/J(cos /SO 
 COS /3i 
 
 _ 2/.P/ COS 
 A; \cos=^^c/ 
 
 0' 
 , or/s 
 
 or/c = 
 
 nfc 
 
 2M 
 
 M 
 
 Asjd (cos /St) 
 
 ^cos2 /3c/ n(l - /c) 
 
 When the compression side of the beam is horizontal, cos /3c becomes equal to unity. 
 
 Likewise, when the tension side of the beam is 
 horizontal, cos /3t = 1. With both top and bot- 
 tom faces of the beam horizontal, the above 
 formulas reduce to those for a rectangular beam 
 given in Art. 9. The above formulas should not 
 be applied to beams having /3c greater than 45 
 deg. on account of approximations in the theory 
 which depart more and mare from accuracy as 
 /3c increases. 
 
 Let Vi represent the total shear to be taken 
 by concrete and web reinforcement, and let V 
 
 Fig. 34. 
 
 represent total shear on section computed as for a rectangular beam. Then 
 
 M 
 
 (tan /3c + tan /Sj 
 
 jSc and /3 1 are to be taken as positive when they bear the same relation to the direction of V as 
 shown in Fig. 34, and are to be taken as negative when they bear the reverse relation. The 
 formulas for shear, bond, stirrup spacing, etc. in rectangular beams apply to wedge-shaped 
 beams if V in formulas is replaced by Vi. 
 
 43. Beams of Any Complex or Irregular Section. 
 
 43a. Analytical Method. ^ — Long and cumbersome formulas for the analysis 
 of complex sections can be avoided by a simple application of general methods of solution to 
 the homogeneous transformed section. Such a general solution is here presented, based upon 
 the standard notation and using the homogeneous section easily obtained by multiplying the 
 steel areas by n, the ratio of the moduli of elasticity of steel and concrete. Thus an equivalent 
 concrete section is obtained, to which the ordinary principles of analysis for homogeneous 
 beams are applied. An equivalent steel section could as well be used if desired. 
 
 Adopting the usual fundamental assumptions for the analysis of reinforjped-concrete 
 
 1 See Appendix I of "Earth Pressure, Retaining Walls and Bins' 
 "Bridge Engineering" by J. A. L. Waddell. 
 
 2 Method as given by J. H. Cissel in Eng. Rec, March 24, 1917. 
 
 by William Cain. See also vol. 
 
 of 
 
Sec. 7-43a] 
 
 BEAMS AND SLABS 
 
 315 
 
 beams, based upon the straight-line theory, it can easily be shown that an equivalent homo- 
 geneous concrete section will be obtained by multiplying the steel area by n. The neutral 
 axis can then be located at the centroid, or center of gravity, of the transformed section. This 
 is done most conveniently by equating the statical moment of the equivalent area on one side 
 to the statical moment of that on the other side. 
 
 Knowing the position of the neutral axis, the moment of inertia of the transformed section 
 
 fl 
 
 with respect to this axis may be calculated, and the well-known fundamental formula M 
 
 y 
 
 used to find the resisting moments for a given section, or the fiber stresses produced by a given 
 bending moment. 
 
 Fig. 35 gives the dimensions and notation used for the exact analysis of a reinforced-concrete T-beam, in- 
 cluding the effect of the compression in the concrete of the stem above the neutral axis. To compute the safe 
 resisting moment for this beam when As = 4 sq. in., n = 15, /, = 16,000 and fc = 650, the compressive area is 
 divided, for convenience, into the parts shown in the figure. Equating the statical moments with respect to the 
 neutral axis of the transformed steel and the concrete, 
 
 (32 X 4)(?/ - 2)+ ^= 60(24 - y) 
 
 Solving, 
 
 y = 7.31 and d - y = 16.69 
 The moment of inertia about this axis can then be computed as follows: 
 n/s = 15 X 4 X (16.69)2 = 
 
 1, 
 
 Ic = (44 X 7.313 X H) - (32 X 3.31^ 
 I = Ic = nis = 
 
 16,713 
 5,342 
 
 22,055 
 
 Fig. 35. 
 
 Fig. 36. 
 
 The resisting moments are therefore: 
 
 fj 
 
 y 
 
 n(d — y) 
 
 = M„ = 
 
 Ms 
 
 650 X 22,055 
 7.31 
 
 16,000 X 22,055 
 
 = 1,960,000 in.-lb. 
 = 1,410,000 in.-lb. 
 
 15 X 16.69 
 
 The safe resisting moment is thus 1,410,000 — given by the steel. 
 
 Given the rectangular beam shown in Fig. 36 subjected to a positive bending moment of 760,000 in.-lb 
 with n = 15, As = 2.25 sq. in., and A' = 1.69 sq. in., to compute the fiber stresses in the steel and concrete, proceed 
 as follows: 
 
 The transformed area of compressive steel = 15 X 1.69 = 25.4 
 The transformed area of tensile steel ' = 15 X 2.25 = 33.8 
 Equating statical moments of tensile and compressive areas about the neutral axis (using transformed section) : 
 ^ + 25.4(2/ - 2) = 33.8(22 - y) 
 Solving, y = 7.58 in. and d — y = 14.42 in. 
 
 Then the moment of inertia will be: 
 
 nl's = 15 X 1.69 X (5.58)2 = 789 
 nIs = 15 X 2.25 X (14.42)2 = 70I8 
 Zc = 12 X (7.58)3 X }i =1742 
 'l = Ic + nIs + nfs = 9549 
 
 Therefore the fiber stresses are:- 
 
 fc = -r 
 
 My _ 
 
 T ~ 
 
 nM(y 
 
 760,000 X 7.58 
 
 = 605 
 
 2) 
 
 nM(22 - y) 
 
 9549 
 
 760.000 X 5.58 X 15 
 9549 
 
 760,000 X 14.42 X 1 
 9549 
 
 6680 
 
 = 17,300 
 
316 
 
 CONCRETE ENGINEERS' HANDBOOK 
 
 [Sec. 7-436 
 
 436. Graphical Method.^ — Application of graphical methods to the location 
 of the neutral axis and the determination of effective depth and resisting moment of reinforced- 
 concrete beams is here proposed for cases of complex sections where the steel is distributed 
 as in beams reinforced for compression or where several layers of rods are used. It is assumed 
 that graphical methods are famiHar to the reader, and these methods will be illustrated both 
 for investigating a given section and for designing a section to resist a given bending moment. 
 
 67,^OOX 17.3= 1.161600 -M. \ 
 
 / 
 
 / 
 
 mi / 
 
 / 
 
 (h) 
 
 Fig. 37. 
 
 ^ j 30^ 
 
 To Find the Neutral Axis. — Consider the problem of finding the neutral axis and resisting moment 
 of a given reinforced-concrete beam, reinforced on the compression side, as in Fig. 37a. Applying the 
 usual method of finding the centroid of an area, the compression side of the beam is divided into thin 
 slices, taken small in order to avoid large errors, and each slice is represented by a force Ai, A2, etc. In 
 order to obtain a homogeneous section, the area of the steelis multiplied hy n = 15. The force diagram for 
 
 these "area forces" is then drawn. Fig. 376, beginning with nAs at one 
 end, Ai next, etc., as shown. 
 
 The funicular polygon for locating the resultant of these forces is 
 then drawn, Fig. 37c, and the intersection O of the line NO, with the lines 
 parallel to the rays for the compressive areas, gives the position of this 
 resultant, or the centroid of the section, and hence locates the neutral axis. 
 Note that none of the area below O is used, according to the usual assump- 
 tion that tension in concrete is neglected. 
 
 To Find the Resisting Moment. — Now in Fig. 37e the point P is on 
 the neutral axis just found, RS is drawn to convenient scale to represent the 
 allowable stress in the concrete, 700 lb. per sq. in., and the straight line PS 
 is drawn to represent the variation of stress across the section. By scaling the ordinates to this line at centers 
 of the slices considered, and multiplying by their areas, the values of the resisting forces were obtained, after noting 
 that the steel stress in tension is UV X 15, or 13,400 lb., which is less than the allowable value, showing that the 
 concrete governs. If the steel stress were found to exceed the allowable value, the line UV should be made 
 16,000/n to locate the stress line PS, and the steel would govern. The compressive forces are plotted in the 
 force polygon, Fig. 37d, in order to locate their resultant by the polygon, Fig. 37/ at X. 
 
 Knowing the magnitude of the compressive forces, by summation, to be 67,200 lb. the effective depth is 
 scaled off as 17.3 in. and the resisting moment is then 67,200 X 17.3 = 1,162,600 in.-lb. This has been checked 
 closely by the usual formulas. 
 
 Fig. 38. 
 
 » Method as given by W. S. Wolfe in Eng. Rec. March 24, 1917, and June 27, 1917. 
 
Sec. 7-436] 
 
 BEAMS AND SLABS 
 
 317 
 
 Application to complex sections, as Fig. 38a, can be made as shown by Figs. 386 and 38c, locating the 
 neutral axis. 
 
 Designing Double-reinforced Beams. — To design a double-reinforced beam for a given bending moment, first 
 assume the position of the steel and the allowable total depth, and assume a convenient width of, say, 10 in. and 
 
 Given : ^^6S0 i%=/6,000 
 <■■ Assume Jb=lO 5 
 
 M=I,OCO,000 
 
 Assume d'= 01 
 or A' =1.75°" 
 
 A, - : 
 
 Allowable depfh over all 20" 
 
 — T580— 870 0 
 
 \455_ / 15.70 0 
 
 ' 3,550 
 
 Required b-- 10 %^;ooo ^ ^^-^ ^ay eo" 
 A'=/a9 X 17.5 X.OI =3.43°" 
 A=19.9 X /75 A. 01/8 =4.10°" 
 
 Fig. 39. 
 
 a proper value for the compression steel ratio p'. The stress line US is then drawn in Fig. 39</ by making RS 
 equal to the allowable compressive stress in the concrete (650) and UV equal to the allowable stress in the steel in 
 tension (16,000) divided by n (15), or 1066. The intersection P then locates the neutral axis. 
 
 y=az5-/>\ 
 i (d)/ 
 
 I / I=H^H>'Y 
 I / ^I00(l0)(9f5) 
 I / 1 = 9^50 in* 
 
 1/ 
 
 / 5: I \ \\ \ 
 / ; n \\\. - . 
 / i I \ \\\\\\\ 
 
 Fig. 40. 
 
 1 
 
 The compression side is divided into a number of slices as shown, and the areas, A\, A2, etc., computed, 
 knowing the value of A2, because the compressive steel area is known from the given ratio p'. To find the area of 
 tension steel required, the value of nAs is obtained from the polygon. Fig. 39b, and the force diagram Fig. 39c, 
 
318 
 
 CONCRETE ENGINEERS' HANDBOOK 
 
 [Sec. 7-44 
 
 using the closing line NO. nAg is then scaled off (31) and divided by n (15), giving 2.06 sq. in. for the area desired. 
 The steel ratio p is then easily computed. The resisting moment is obtained as before, using the diagrams, Figs. 
 39d, 39e, and 39/, by which a resisting moment of 502,000 in. -lb. is obtained, as shown. 
 
 Now, if the width is increased and the steel ratios are kept constant, the resisting moment will increase 
 directly as the increase in width. Therefore, if the assumed width be multiplied by the ratio of the required moment 
 to the moment obtained for this width, the required width is found. This computation, given on the diagram, 
 shows that a 20-in. width is necessary. The required areas of steel in compression and tension can then be com- 
 puted as given, knowing the values of the steel ratios p and p'. 
 
 Moment of Inertia of Complex Beam Sections. — Fig. 40a shows the cross-section of a double-reinforced 
 concrete beam, the moment of inertia of which is desired. The compression side of the beam is divided into small 
 slices, the area of the steel being multiplied by 15, and the areas of these slices are laid off in the force polygon. 
 Fig. 40b, together with nAs, or 15 times the area of the tension steel. For convenience the pole p is taken 
 with the pole distance H equal to 100 sq. in. to the scale at which the areas were laid out in the force polygon. 
 From Fig. 406 the funicular polygon Fig. 40c is drawn locating the neutral axis by the intersection O. It will 
 be noted that all the strings in this funicular polygon are extended until they intersect the neutral axis, the axis 
 about which / is desired, at points 1, 2, 3, etc. Now for convenience the pole p' is taken so that the pole distance 
 //' equals 10 in. to the scale at which the section of the beam was drawn, and p' is connected with points 1, 2, 3, 
 etc. Now, parallel to these rays the corresponding strings in the funicular polygon. Fig. 40d, are drawn, thus 
 following Culmann's approximate method for finding the moment of inertia graphically. From this construction 
 we get I = H X H' X V = 100 X 10 X 9.25 = 9250 in.*, from which fc and /« can easily be obtained for any 
 given bending moment by using the formulas 
 
 SHEAR AND MOMENT IN RESTRAINED AND CONTINUOUS BEAMS 
 
 44. Span Length for Beams and Slabs. — The Joint Committee recommends the following: 
 
 The span length for beams and slabs simply supported should be taken as the distance from center to center 
 of supports, but need not be taken to exceed the clear span plus the depth of beam or slab. For continuous or 
 restrained beams built monolithically into supports, the span length may be taken as the clear distance between 
 the faces of supports. Brackets should not be considered as reducing the clear span in the sense here intended, 
 except that when brackets which make an angle of 45 deg. or more with the axis of a restrained beam are built 
 monolithically with the beam, the span may be measured from the section where the combined depth of beam 
 and bracket is at least one-third more than the depth of the beam. Maximum negative moments are to be con- 
 sidered as existing at the end of the span as here defined. 
 
 When the depth of a restrained beam is greater at its ends than at mid-span and the slope of the bottom 
 of the beam at its ends makes an angle of not more than 15 deg. with the direction of the axis of the beam at 
 mid-span, the span length may be measured from face to face of supports. 
 
 45. Recommendations of Joint Committee as to Positive and Negative Moments. — In 
 
 computing the positive and negative moments in beams and slabs continuous over several 
 supports, due to uniformly distributed loads, the Joint Committee recommends the following 
 rules : 
 
 wl- 
 
 (a) For floor slabs, the bending moments at center and at support should be taken at for both dead and 
 live loads, where w represents the load per linear unit and I the span length. 
 
 (6) For beams, the bending moment at center and at support for interior spans should be taken at ^-^ and 
 
 for end spans it should be taken at ~^ for center and interior support, for both dead and live loads. 
 
 (c) In the case of beams and slabs continuous for two spans only, with their ends restrained, the bending 
 moment both at the central support and near the middle of the span should be taken as . 
 
 (d) At the ends of continuous beams, the amount of negative moment which will be developed in the beam 
 will depend on the condition of restraint or fixedness, and this will depend on the form of construction used. In 
 the ordinary cases a moment of jg- may be taken; for small beams running into heavy columns this should be 
 
 wl- 
 
 increased, but not to exceed — • 
 
 For spans of unusual length, or for spans of materially unequal length, more exact calculations should be 
 made. Special consideration is also required in the case of concentrated loads. 
 
 Even if the center of the span is designed for a greater bending moment than is called for by (o) or (6), the 
 negative m 'ment at the support should not be taken as less than the values there given. 
 
 46. Theorem of Three Moments. — By means of the theorem of three moments, the moments 
 
 I 
 
 (15) 
 
Sec. 7-46] 
 
 BEAMS AND SLABS 
 
 319 
 
 at the supports of a continuous beam may be deduced. The theorem, assuming level supports, 
 is in its general form as follows (Fig. 40 A)^: 
 
 M^kh + 2M3 (^2/3 + ^3/2) + M4 ^3/2 = - P2 h'h (ko-k^') - P3 ^3-12 m,-'Sk,^W) (1) 
 
 1 Derivation of the theorer,: of three moments is as follows: 
 
 Let the origin of coordinates be B (Fig. A), with x measured positively toward the left. Considering only 
 the deformation due to the bending moment, and neglecting the deformation due to shearing forces, the equation 
 of the elastic curve is given by 
 
 d^y M 
 
 dx2 ~ EI 
 
 in which M is the bending moment at any point x, y. If this expression is integrated once, there results an ex- 
 pression of the slope of the tangent to the elastic curve at any point x, y. Thus the slope of the tangent at 
 
 P is 
 
 P Mdx 
 EI 
 
 and for the whole member the change in slope of the tangent becomes 
 
 Mdx 
 A EI 
 
 (a) 
 
 Fig. a. 
 
 In Fig. B is shown a beam resting on several supports, all on 
 the same level. Let us consider the portion of the beam between .4. 
 and B, by cutting it out close to the support by the planes m and n. 
 
 The part of the beam cut out is shown in Fig. C. Each end has the same shear and moment acting upon it as it 
 did in its original position, thus causing stresses throughout the portion AB identical with those acting before the 
 beam was cut. The moment at any point L, to the left of the load P, is 
 
 
 Ml = A/2 + Vox 
 
 \ I 1 
 
 
 
 H — 
 
 A 
 
 [ t ti 
 
 9 C D 
 
 Fig. B. 
 
 r 
 
 •jr 
 
 Fig. C. 
 
 > 
 
 The moment at any point R to the right of the load is 
 
 Mr = M2 + V-ix" - Pix" - kl) 
 By taking moments first about .4, and then about B, the values of and V3 are found to be 
 
 Ml - Ms + Pkl 
 
 Mz - M2 + P(l - kl) 
 
 1 t^^^.__. 
 
 :) 
 
 (c) 
 id) 
 
 It is apparent that the moment at any point in the beam may be found if the moment at the reaction can 
 be found. We will, therefore, proceed with the solution of a general case of a continuous girder with concentrated 
 loads. 
 
 Consider now a beam resting on several supports on the same level, and loaded as shown in Fig. D. The 
 slope of the tangent to the deflection curve at B, considering the portion of the beam to the left of B, is ^2; and 
 that for the tangent to the deflection curve at B, considering the portion to the right of B, is 08. Since the de- 
 flection curve of a beam must necessarily be continuous, MN is a straight line, and 
 
 <l>2 = — <l>3 
 
 (/) 
 
320 
 
 CONCRETE ENGINEERS' HANDBOOK 
 
 [Sec, 7-46 
 
 This is the general equation of three moments for one concentrated load in each span. 
 When there is more than one concentrated load in each span, the loading may be broken up into 
 a series of cases similar to the above. For each set of loads the right-hand portion of equation 
 (1) is solved, and the results finally combined, and placed equal to the left-hand portion of 
 equation (1). 
 
 The theorem may be applied to a beam with uniform loads. For the loading shown in 
 Fig. 41 the theorem has the form 
 
 M2l2h + 2Mz(l2h 
 
 ■Mr 
 
 3/2) + M,IJ 
 
 4 
 
 ^3, 
 
 Ms, 
 
 (2) 
 
 Fig. 40A. 
 
 For a constant moment of inertia and all spans equal, equation (1) reduces to 
 
 M2 + 4Af3 + iif4 = - P2Z(fc2 - W) - PsK^ks - 3/C32 + /C33) 
 
 When k2 = kz = 0.5 this equation reduces to 
 
 M2 + 4ikr3 + M4 = - O.375P2Z - 0.375P3Z 
 
 When in equation (2) the moment of inertia is a constant and all spans carry the same uniform 
 load 
 
 M2I2 + 2M3(72 + h) + M,h 
 
 and when h = h, 
 
 2 
 
 M2 + 4ilf3 -h M4 = 
 
 1000 lb. per ff: 
 
 Fig. 41. 
 
 800 lb. per ft 
 
 ■ZO'- 
 
 1000 lb. per H 
 
 ■I8'- 
 
 Fig. 42. 
 
 When using any of the foregoing formulas for continuous girders, it should be borne in 
 mind that the supports are assumed to be on the same level throughout the process of loading. 
 
 Illustrative Problem. — Compute the moments at the supports and the magnitude of the reactions for 
 the beam shown in Fig. 42. Assume a constant section throughout, whence Ii = I2 = I3. Consider first the 
 two spans between Ri and Rs. From equation (2) 
 
 I6M. + 2M.(16 + 20) + 20M. = - - 
 
 If the values of M in equations (6) and (c) are substituted into equation (a), and the latter integrated from A to 
 P2, and from P2 to B, and finally the values of V2 and F3 as given in equations (d) and (e) are incerted, there re- 
 sults for 4>2 the value 
 
 M2Z22 + 23/i?22 + P2h3(k2 - k2^) 
 
 In a similar manner the values of ^3 may be found by considering C as the origin of x, and integrating toward the 
 left, whence 
 
 ^4^32 + 2MiU^ + PzhK2k3 - 3^32 + ks^) 
 ^3 _____ 
 
 Inserting these values into equation (/) there results 
 
 M2hl3 4- 2^3(^2/3 + Z3/2) + ikf4Z3/2 = - P2l2^l3{k2 - A;2») - Pzh''l2{2kz - Zkz^ + kz^) ig) 
 A development precisely similar to the foregoing may be applied to a beam with uniform loads. 
 
Sec. 7-46 J 
 
 BEAMS AND SLABS 
 
 321 
 
 20M> + 2M3(20 + 18) + I8A/4 = - 
 
 whence 
 
 IMi + iHMi + oMi = - UoO.OOO 
 Applying equation {2) to the two spans between Ri and Ra, and noting the change in subscripts accordingly, we 
 obtain 
 
 (80 U)(2U)-i _ (100())(18)3 
 4 4 
 whence 
 
 IOM2 + 38^3 + 9M4 = - 1,529,000 {h) 
 Since all supports are free, Mi = Mi = 0. Noting this, and solving (a) and {b) simultaneously, 
 
 90ilf2 + 25M3 = - 3,280,000 
 90 M2 + 34 Ms = - 3,761,000 
 317M3 = - 10,481,000 
 Afs = - 33,100 ft.-lb. 
 
 ^2 
 
 M4 
 
 M2 
 
 M3 
 
 M4 
 
 (a) 
 
 1 1 • 
 
 yyjperfA 
 
 
 
 (b) 
 
 
 W2 perft\ 
 
 
 A 
 
 2 
 
 M4 
 
 \W2perft 
 
 per ft 
 
 M3 
 
 M4 
 
 > 
 
 per ft 
 
 rn 
 
 — 1 perff. 
 
 
 <•■ •■> 
 
 
 V-./p-i- ..>| 
 
 iVj per ft 
 ~ (f) 
 
 Mb 
 
 
 > 
 
 
 
 
 ^>2> per ft 
 
 
 
 < 4 ■ 
 
 
 M3 
 
 M4 
 
 ■r/y| 1 iVjperft 
 
 A/^ /l/j /^^ 
 
 Mz 
 
 M4 
 
 > 
 
 Wz per ft. 
 
 k^Z 
 
 k/h\ 
 
 ["!■ 
 
 
 perft\ 
 
 
 <.. ^z - > 
 
 
 M3 M4 
 Fig. 43. 
 
 (j) 
 
 Substituting this into (6) we find 
 
 We may now find the reactions. 
 
 16Z?i 
 
 M2 
 
 - - 45,600 
 
 45,000 ft.-lb. 
 
 16/2i = 128,000 - 45,600 = 82,400 
 Ri = 5150 lb. 
 
 (5150) (36) - (1000) (16) (28) + 2^Ri 
 
 (800) (20)-' 
 2 
 
 = - 33,100 
 
 20i?2 = - 33,100 - 185,400 + 448,000 + 160,000 = 389,500 
 222 = 19,470 lb. 
 (1000) (18)2 
 
 18«4 
 
 33,100 
 
 21 
 
 Rk = 7,160 lb. 
 
322 
 
 CONCRETE ENGINEERS' HANDBOOK 
 
 [Sec. 7-46 
 
 (7160) (38) - (1000) (18) (29) + 20R, 
 
 (800) (20) 2 
 
 45,600 
 
 20/23 
 
 45,600 - 272,000 + 552,000 + 160,000 
 Rs = 18,220 lb. 
 
 As a check, 
 
 Ri + Ri + R3 + Ri 
 
 total load = 50,000 lb. 
 
 Very often a continuous girder may have a uniform load over one or more spans. Fig. 43 
 will be found to give the possible cases, and the corresponding formulas follow. Should no 
 load be on one of the spans, w for that span becomes zero, and the term containing it will drop 
 out. Only the right-hand side of the general equation is affected by the loading. 
 
 Let M 2/2 + 2M3 {h + Z3) + Mih be denoted by r. 
 
 Then (a) r 
 
 (b) r 
 
 (c) r 
 
 (d) r 
 
 (e) r 
 
 W3I3 
 
 [2 4.) 4: 
 
 U . 2 ^ 4 / 4 • 
 
 ki' + 
 
 ki'^ 
 
 I 
 
 w= 2000 lb. per n 
 J<--/<5'---5|^ 18'--^ /<?'-->i 
 
 Fig. 44. 
 
 (/) r = 
 (.9) r = 
 (h) r = 
 {i) r = 
 U) r = 
 
 
 w= 2000 lb. per ft 
 
 
 
 » 16' >|< 18'- ■>}< 18'- > 
 
 
 Fig. 45. 
 
 2Po?/(A: - - w;3Z3-^(^ - fci^ + k^^ - • 
 
 Spans similarly located and similarly loaded have the same representative term. Cases 
 {%) and (jf) are given to show how concentrated loads may be considered with uniform loads. 
 
 When a beam has fixed ends — ^that is, when the slope of the tangent to the elastic curve at 
 the end is constant for all loading — ^the theorem of three moments may readily be applied. This 
 is accomplished by supplying a span of zero length at each fixed end, and then by proceeding as 
 before. This satisfies the requirement made in the above definition for fixed ends. 
 
 Illustrative Problem. — Determine the moments at the supports and the magnitude of the reactions for 
 the beam shown in Fig. 44. In Fig. 45 the fixed ends have been removed and the spans /„ and V put in their places. 
 The equations now become, remembering that Ro = R' = 0, and Mi = M2 = 0, 
 
 2 Ml + M2 = - \wl2 
 
 All + 4M2 + M3 
 
Sec. 7-47] 
 
 BEAMS AND SLABS 
 
 323 
 
 M2 + 4A/3 + Mi = - ~wli 
 
 If these equations are solved simultaneously, there results 
 
 Ml = M2 = M 
 Ri = = 18,000 lb 
 Rz = Rs = 36,000 lb 
 
 54,000 ft.-lb. 
 
 -./06 ^ 
 
 _ - 077, 
 
 
 --<-C77 
 
 
 
 NT 
 
 
 2 
 
 3 
 
 4 
 
 5 
 
 Fig. 46. — Moments in continuous beams; supported ends; uniform load on all spans; spans all equal. 
 
 Coefficients of {wl-). 
 
 of7 
 
 6 
 
 10 
 
 6\S 
 JO 
 
 S\6 
 
 /a 
 
 4\o 
 
 10 
 
 
 1 2 
 
 3 4 
 
 
 0 t// 
 
 I7\ /S /3\l3 
 
 
 //to 
 
 28 
 
 28 28 
 
 2d 
 
 28 
 
 1 
 
 2 3 
 
 A 
 
 5 
 
 23\20 
 38 
 
 /8 I /9 
 38 
 
 A S3 
 
 63 \' 
 
 49] S/ 
 
 ^0 
 
 ofse 
 
 d4Z 
 
 86^75 67'\70 72~T7/ 
 
 7S\86 
 
 MZ J42 J42 i42 i42 142 i42 
 
 kFiG. 47. — Shears in continuous beams; supported ends; uniform load on all spans; spans all equal. 
 Coefficients of {wV). 
 
 47. Uniform Load Over All Spans. — In Fig. 46 are given the moments in continuous beams 
 for a uniform load over all spans. Supported ends and equal spans are assumed. Positive 
 moments are plotted above the beam. The shears on each side of the supports are given in 
 Fig. 47. The reaction at any given support is the sum of the two shears at that support. 
 
 The maximum moments at (or near) the center of many of the interior spans are not given, 
 as they are small and do not vary greatly from those given for the continuous beam of four 
 spans. Maximum positive moment occurs for zero shear as in simple beams. 
 
 In continuous beams with fixed ends, and with uniform load and equal spans (the same as 
 
324 
 
 CONCRETE ENGINEERS' HANDBOOK 
 
 [Sec. 7-48 
 
 above), the iriaximum positive moment occurs at the center of each span and its value in every 
 case is The negative moment over each support equals ^2 Shears close to the supports 
 
 are all equal with a value of 3-2^'^- 
 
 The maximum positive moment on a beam of one span, with one end fixed and the other 
 end free, and uniformly loaded is ^{23'^^^ and the negative moment at the fixed end is }^iwl^. 
 
 48. Fixed and Moving Concentrated Loads. — Continuous beams of one, two, and three 
 equal spans and any span length may be figured for fixed and moving concentrated loads by 
 means of the influence lines of Figs. 48 to 52 inclusive. 
 
 48a. Influence Lines. — As a load moves over a beam, the shear and moment at 
 a given section will vary. If the value of moment at any point A is plotted as an ordinate at 
 the point where the load is applied, and this process repeated for each position of the load, the 
 result is called an influence diagra7n for the moment at point A ; and the curve generated by the 
 extremities of all ordinates is called an influence line for the moment at point A. Similar lines 
 may be drawn for shear and for deflections. In structures, influence lines may also be drawn 
 for stress intensities at a given point. The curve gets its name because of the fact that for any 
 
 chosen point, it gives the influence on a certain function at that point, for varied positions of the 
 load. 
 
 It should be noted that the influence line for moment — for a simple beam, for instance — • 
 differs from the moment diagram for that beam. The moment diagram gives the moment at 
 any point for one position of the load ; while the influence hne for moment gives the moment at 
 one point for any position of the load. For each point in the beam there may be drawn an in- 
 fluence line, but each influence line is descriptive of but one point. In Fig. 53 there is drawn an 
 
 influence line for moment at A. The moment at ^ is - ^ ' and that is the value of the ordinate 
 
 Pah X 
 
 at A. The ordinate at B is —f- • - and is the moment at A when the load P is at B. 
 
 I a 
 
 Suppose the beam to have a load of 1 lb, moving across it. The ordinate at A is then-^ ' 
 
 Usually influence lines are drawn for unit loads. The ordinate at B is then the moment at .1 
 when a unit load is placed at B. If the load at B is not unity, then the moment at A will be 
 equal to the load times the ordinate at B for the 1-lb. load. 
 
 If the beam is loaded with a uniform load, the moment at A is equal to the load per foot 
 
Sec. 7-4Sol 
 
 BEAMS AND SLABS 
 
 325 
 
 Moment over 
 Center Support 
 
 End Reaction 
 and End Shear | 
 
 Center 
 Reaction 
 
 Shear at 
 Center Support 
 
 oooo'ocio od^oociddoooo 
 
 Fig. 49. — Influence lines for two equal spans, supported ends. 
 
 Moment over 
 End Support 
 
 o 
 
 000 
 
 Moment at"M' 
 
 Moment over 
 Center Support 
 
 End Reaction 
 and 
 End Shear 
 
 Center 
 Reaction 
 
 Shear at 
 Center Support 
 
 000000000 
 
 ciooooddoo 
 
 to §1 
 o o 
 
 o o o o o 
 
 OOOOOuOOO 
 
 Fig. 50.— Influence lines for two equal spans, fixed ends. (Spans, 10 ft. Load, 
 
 unity.) 
 
326 
 
 CONCRETE ENGINEERS' HANDBOOK 
 
 [Sec. 7-48a 
 
 Fig. 51. — Influence lines for three equal spans, supported ends. (Spans, 10 ft. Load, unity.) 
 
 Fig. 52. — Influence lines for three equal spans, fixed ends. (Spans, 10 ft. Load, unity.) 
 
Sec. 7-48a] 
 
 BEAMS AND SLABS 
 
 327 
 
 times the area of the influence diagram for the moment at A. In Fig. 53 this is \ w --y 'l'2j 
 
 w 
 
 or ^ • ab, which is readily recognized as the moment at A for a uniform load. For a partial 
 
 uniform loading, the load per foot multiplied by the area of the influence diagram for the loaded 
 portion will give the moment at A. 
 
 
 
 
 
 < X 
 
 
 
 
 < a •■■ 
 
 
 
 Fig. 53. 
 
 Influence lines have been constructed in Figs. 48 to 52 inclusive for continuous beams of 
 equal spans, each of 10 units in length, with the outer ends either fixed or simply supported. 
 If the influence lines are desired for equal spans other than 10 ft, in length, they may be con- 
 
 Hz 
 
 w ib.per Jin.ft 
 
 ■e 
 
 Mz 
 
 Ms 
 
 3 
 
 Moment 
 
 .096 wl 
 
 063 y/t^ 
 
 Shear 
 
 le 
 
 wi 
 
 Supported Ends 
 
 Rj = -i 
 
 Moment 
 
 Shear 
 
 .0S4 
 
 ..021 
 
 Fixed Ends 
 
 Fig. 54. — Moving uniform load, two equal spans. 
 
 structed by regarding one unit of length as one-tenth the span. The ordinates will be the same 
 as those plotted here. The ordinates for positive moment are plotted above the line. 
 
328 
 
 CONCRETE ENGINEERS' HANDBOOK 
 
 [Sec. 7-48a 
 
 For two equal spans with ends either supported or fixed, it is readily seen that the greatest 
 positive moment at any point M will be obtained when the load covers only the span in which 
 
 uj lb. perl in ft. 
 
 
 My 
 
 UJ lb. per lin. ft 
 
 
 
 — > 
 
 <■ 
 
 
 < / > 
 
 
 
 
 
 ^4 
 
 ,-.101w-t^ 
 
 Moment 
 
 Shear 
 
 Supported Ends 
 
 MoTTient 
 
 Shear 
 
 -.059 u/^^ 
 
 Fixed Ends 
 
 Fi(i. ^>^^. — Moving uniform load, three equal spans. 
 
 ^ w Ib.perlin.ft 
 
 Moment 
 
 Shear 
 
 20, 
 
 -.028 W 
 
 Supported Ends 
 iZ ....069 rri^ 
 
 Moment 
 
 Shear 
 
 . -.OS 8 w^^ 
 
 P2=pj- 2 we 
 
 Fixed Ends 
 
 Fig, 56. — Moving uniform load, three equal spans. 
 
 M occurs, since the area for that span is positive. The greatest end reaction and end shear will 
 also be obtained when the load is over one span. The greatest center reaction, negative mo- 
 
Sec. 7-48a] 
 
 BEAMS AND SLABS 
 
 329 
 
 ment (over center support), and shear at center support, will be obtained by fully loading both 
 spans. 
 
 For three equal spans, the uniform live load should cover alternate spans to give the 
 greatest positive moment in any span, and to give the greatest end reaction and end shear. P'or 
 
 w lb per I in. ft 
 
 ^2 
 
 >K- -- -e 
 
 .073 wi' 
 
 .053 we' 
 
 Moment 
 
 Shear 
 
 60 
 
 ^4 = -jW 
 
 Supported Ends 
 
 .039 wi' 
 
 056 yvi"- 
 
 .022 wi' 
 
 Moment 
 
 Shear 
 
 . ■32 we 
 
 Mo 
 
 .045 w^^ 
 
 ISO 
 
 ■81 we 
 
 180 
 
 Fixed Ends 
 
 Fig. 57. — Moving uniform load, three equal spans. j 
 
 intermediate reactions and negative moment over intermediate reactions, the spans adjacent; 
 to the reaction in question should be fully loaded, 
 
 49. Moving Uniform Loads. — If a uniform load is considered, influence lines indicate the 
 spans which should be loaded in order to obtain the maximum values of the given functions. 
 
 
 
 
 Mb 
 
 
 
 
 
 Afc 
 
 \ 
 
 
 
 
 
 
 Fig. 58. 
 
 Fig. 54 represents the variation in moment and shear for a uniform load on one span of a beam 
 of two equal spans, both fixed and supported ends. Figs. 55, 56 and 57 give values for various 
 loadings with three equal spans. 
 
330 
 
 CONCRETE ENGINEERS' HANDBOOK 
 
 [Sec. 7-50 
 
 To illustTate the effect of loads on various spans upon the bending moments, influence hnes 
 have been drawn for six equal spans (Fig. 58), for moments at the centers of span 1-2 and 3-4, 
 and at supports 2 and 4. A maximum moment at the center of a span requires each alter- 
 nate span to be loaded, and a maximum moment at the support requires the two adjacent 
 spans to be loaded and then each alternate span. The effect of loads on remote spans is 
 seen to be small. Influence lines are not drawn for shears since, in a large number of spans, 
 the shears do not differ greatly from those in simple beams. 
 
 50. Maximum Moments from Uniform Loads. — The following table gives the values of 
 maximum negative and positive moments for girders of equal spans. Each column headed 
 Fixed load" gives the maximum moment at the point under consideration for a uniform load 
 covering the entire length. The columns headed ''Moving load" give the maximum moment 
 at the point under consideration when the uniform load is placed on certain spans to cause the 
 maximum moment. 
 
 Maximum Moments in Continuous Beams; Supported Ends; Uniform Fixed and Moving 
 
 Loads 
 Coefficients of (wP) 
 
 No. of spans 
 
 Intermediate spans 
 
 End spans 
 
 Fixed load 
 
 Moving load 
 
 Fixed 
 
 load 
 
 Moving load 
 
 At 
 center 
 + 
 
 At 
 support 
 (-) 
 
 At 
 center 
 + 
 
 At 
 support 
 (-) 
 
 At 
 center 
 + 
 
 At 2d 
 support 
 (-) 
 
 At 
 center 
 + 
 
 At 2d 
 support 
 (-) 
 
 Two 
 
 
 
 
 
 0.070 
 0.080 
 
 0. 125 
 0.100 
 
 0.096 
 0.101 
 
 0.125 
 0.117 
 
 Three 
 
 0.025 
 
 
 0.075 
 
 
 
 0.036 
 
 0.071 
 
 0.081 
 
 0.107 
 
 0.077 
 
 0.107 
 
 0.098 
 
 0.120 
 
 
 
 
 
 
 
 
 
 (0.115)1 
 
 Five 
 
 0.046 
 
 0.079 
 
 0.086 
 
 0.111 
 
 0.078 
 
 0.105 
 
 0.099 
 
 0.120 
 
 
 
 
 
 (0.106)1 
 
 
 
 
 (0.116)1 
 
 Six.... 
 
 0.043 
 
 0.086 
 
 0.084 
 
 0.116 
 
 0.078 
 
 0.106 
 
 0.099 
 
 0.120 
 
 
 
 
 
 (0.106)1 
 
 
 
 
 (0.116)1 
 
 
 0.044 
 
 0.085 
 
 0.084 
 
 0.114 
 
 0.078 
 
 0.106 
 
 0.099 
 
 0.120 
 
 
 
 
 
 (0.106)1 
 
 
 
 
 (0.116)1 
 
 1 Where two adjacent spans only are loaded. 
 
 The fixed-load coefficients will apply to the dead load when finding the maximum coefficients 
 due to a moving uniform load — the case ordinarily encountered in building construction. In- 
 asmuch as the theoretical 
 maximum moments in con- 
 tinuous beams of five or more 
 spans would involve unreason- 
 able assumptions as to posi- 
 tion of the live loads, the 
 values of moment coefficients 
 in small table may be taken. 
 
 Combining the dead and 
 live loads into a single unit 
 for the purpose of determin- 
 ing general moment coefficients which will apply to all ordinary cases, we obtain the values 
 given in table on page 331. 
 
 Nature of load 
 
 Intermediate spans 
 
 End spans 
 
 At 
 center 
 
 At 
 support 
 
 At 
 center 
 
 At 2d 
 support 
 
 
 0.046 
 
 0.086 
 
 0.080 
 (0.070) 
 
 0.101 
 (0.096) 
 
 0.107 
 
 (0.125) 
 0.117 
 
 (0.125) 
 
 
 0.086 
 
 0.107 
 
Sec. 7-51] 
 
 BEAMS AND SLABS 
 
 331 
 
 In continuous-beam com- 
 putations, the beam is as- 
 sumed as freely supported at 
 the interior supports and the 
 assumption is made that the 
 supports are of no appreci- 
 able width. For beams in 
 concrete construction, there- 
 fore, the coefficients in the 
 third column of the table 
 should be reduced, and could 
 very well be taken equal to 
 those in the second column. 
 The live load will generally 
 range from two to five times 
 the dead load, but the ratio of 10: 1 
 is given to show the slight varia- 
 tion in moment coefficients for ra- 
 tios above 5:1. 
 
 From a study of the table, re- 
 ducing the bcnding-moment coeffi- 
 cients at the interior supports as 
 above proposed, it is seen that the 
 bending moment at the center and 
 at the support for interior spans 
 
 Ratio of live to dead 
 
 Intermediate spans 
 
 End spans 
 
 At 
 center 
 
 At 
 support 
 
 At 
 center 
 
 At 2d 
 support 
 
 Three or more spans 
 2:1 
 5:1 
 10:1 
 
 0. 073 
 u . u/ y 
 0.082 
 
 0. 100 
 n 1 r\A 
 
 U . iU't 
 
 0.105 
 
 0.094 
 
 U . yJvo 
 
 0.099 
 
 0.114 
 
 U . 1 i O 
 
 0.116 
 
 Two spans 
 2:1 
 5:1 
 10:1 
 
 
 
 0.087 
 0.092 
 0.093 
 
 0.125 
 0.125 
 0.125 
 
 
 
 
 
 
 
 may be taken as (0,083 
 
 and "for end spans it may be taken 
 as Jq- for center and adjoining sup- 
 port, where w includes both dead 
 and live loads. In the case of two 
 spans only, the bending moment at 
 the center support may be taken 
 
 as -TT, and near the middle of the 
 
 span as 
 
 10 
 
 Where the ends of 
 
 a two-span beam are restrained, 
 the bending moment may well be 
 
 taken as jj^, both at the center 
 
 support and near the middle of the 
 span. 
 
 The shear at each support of 
 continuous beams with fixed ends 
 may be taken as one-half the span 
 load. If the ends are simply sup- 
 ported, the shear in the end spans 
 near the second support will be 
 approximately O.Qwl. 
 
 51. Beam Concentrations. — 
 Floor systems of beam-and-girder 
 
 Loads at Middle Fbmts 
 
 ■.67 
 
 p / 
 
 
 
 
 
 \i / 
 
 /// 
 
 
 Loads at Th.rd Points 
 
 .1.97 .267 
 
 1 40 .'SO /SQ 
 
 Loads at Middle and Quarter Points 
 Fig. 59. — Moments and shears in continuous beams; supported 
 ends; spans all equal; concentrated loads as shown. Coefficients of 
 (PI) for moment. Coefficients of (P) for shear. 
 
332 
 
 CONCRETE ENGINEERS' HANDBOOK 
 
 [Sec. 7-51 
 
 construction impose concentrated loads on the girders at the ends of the floor beams. 
 There may be one or more floor beams built into each girder, depending upon the shape of 
 
 the panels. 
 
 Figs. 59 and 62 inclusive give the shears and moments 
 caused by beam concentrations on continuous girders. The 
 girder spans are assumed equal and the ends of the girders 
 as simply supported. In girders with fixed ends and with 
 full loading as shown in Fig. 59, the maximum positive mo- 
 ment for loads at the middle points is the same as the maxi- 
 mum negative moment and equals 0.125P/ in every case. 
 For loads at the third points, the maximum positive moment 
 is 0.110 PI and the maximum negative moment is exactly 
 twice this value. For loads at the middle and quarter 
 points, the maximum positive moment is 0.187PZ and the 
 maximum negative is 0.313PZ. The maximum shears in all 
 cases are the same as in simple beams. 
 
 In the following table are given the maximum positive 
 and negative moments due to beam concentrations. Ends 
 of beams are assumed as simply supported. 
 
 It is quite evident that the variation of moment co- 
 efficients is very nearly the same as shown in the table pre- 
 viously given for uniform loads. 
 
 Maximum Moments in Continuous Girders Due to Beam Concentrations; 
 
 Supported Ends 
 Coefficients of (PI) 
 
 No. of spans 
 
 Intermediate spans 
 
 End spans 
 
 Fixed load 
 
 Moving load 
 
 Fixed load 
 
 Moving load 
 
 At 
 center 
 + 
 
 At 
 support 
 (-) 
 
 At 
 center 
 + 
 
 At 
 support 
 (-) 
 
 At 
 
 center 
 + 
 
 At 2d 
 support 
 (-) 
 
 At 
 center 
 + 
 
 At 2d 
 support 
 
 (-) 
 
 Loads at 
 middle points 
 Two 
 
 
 
 
 
 0.156 
 0.175 
 0.171 
 
 0.187 
 0. 150 
 0.158 
 
 0.203 
 0.213 
 0.211 
 
 0.187 
 0.175 
 0.1741 
 
 
 0.100 
 0.130 
 
 
 0.175 
 0.191 
 
 
 Five 
 
 0.119 
 
 0.156^ 
 
 Loads at 
 third points 
 Two 
 
 
 
 
 
 0.222 
 0.244 
 0.240 
 
 0.333 
 0.267 
 0.281 
 
 0.278 
 0.289 
 0.286 
 
 0.333 
 0.311 
 0.3091 
 
 
 0 .066 . 
 0.122 
 
 
 0.200 
 0.228 
 
 
 Five 
 
 0.211 
 
 0.2761 
 
 Loads at 
 middle and 
 quarter points 
 Two 
 
 
 
 
 
 0.267 
 0.314 
 0.303 
 
 0.465 
 0.372 
 0.394 
 
 0.383 
 0.406 
 0.401 
 
 0.465 
 0.438 
 0.4351 
 
 
 0.128 
 0.204 
 
 
 0.312 
 0.352 
 
 
 Five 
 
 0 . 296 
 
 0.3891 
 
 1 Two adjacent spans only are loaded. 
 
 Fig. 60. — Concentrated loads as 
 shown; loads at middle points; two 
 and three equal spans; supported ends. 
 Coefficients of (PI) for moment. Co- 
 efficients of (P) for shear. 
 
Sec. 7-51 
 
 BEAMS AND SLABS 
 
 333 
 
 By similar reasoning to that employed in deriving mom(nit coefficients for uniform loads, 
 we obtain the following values : p p 
 
 Intermediate spans 
 
 Nature of load 
 
 At center 
 
 Loads at middle points 
 Dead load 
 
 (two spans) 
 
 Live load 
 
 (two spans) 
 
 Loads at third 
 Dead load 
 
 (two spans).. . 
 Live load 
 
 (two spans).. . 
 
 pomis 
 
 Loads at middle and 
 
 quarter points 
 Dead load 
 
 (two spans) 
 
 Live load 
 
 (two spans) 
 
 0.130 
 0.191 
 
 0.122 
 0.228 
 
 0.204 
 0.352 
 
 At 
 support 
 
 0.119 
 0.156 
 
 0.211 
 0 . 276 
 
 0.296 
 0.389 
 
 End spans 
 
 At center 
 
 0.175 
 (0.156) 
 
 0.213 
 (0.203) 
 
 0.244 
 (0.222) 
 
 0.289 
 (0.278) 
 
 0.314 
 (0.267) 
 0.406 
 
 (0.383) 
 
 At 2d 
 support 
 
 0.158 
 (0.187) 
 
 0.175 
 (0.187) 
 
 0.281 
 (0.333) 
 
 0.311 
 (0.333) 
 
 0.394 
 (0.465) 
 
 0.438 
 (0.465) 
 
 ■ 87 
 
 Fig. 61. — Concentrated loads as 
 shown; loads at third points; two and 
 three equal spans; supported ends. 
 Coefficients of (P) for shear. 
 
 Combining the dead and 
 live loads into a single unit for 
 the purpose of determining 
 general moment coefficients, 
 we obtain the following: 
 
 .38J 
 
 .438 — 
 /.94 
 
 Fig. 62. — Concentrated loads as 
 shown; loads at third points; two and 
 three equal spans; supported ends. 
 CoeflScients of {PI) for moment. Co- 
 efficients of {P) for shear. 
 
 Ratio of live to dead 
 
 Loads at middle points 
 2:1 
 5:1 
 (Two spans) 
 2:1 
 5:1 
 
 Loads at third points 
 2:1 
 5:1 
 (Two spans) 
 2:1 
 5:1 
 
 Loads at middle and 
 quarter points 
 2:1 
 5:1 
 (Two spans) 
 2:1 
 5:1 
 
 Intermediate spans 
 
 At center 
 
 0.171 
 0.181 
 
 At 
 support 
 
 0.143 
 0.150 
 
 0.193 
 0.210 
 
 0.303 
 0.327 
 
 End spans 
 
 At center 
 
 0.200 
 0.207 
 
 0.187 
 0.195 
 
 At 2d 
 support 
 
 0.169 
 0.172 
 
 0.187 
 0.187 
 
 0.254 
 0.265 
 
 0.274 
 0.281 
 
 0.259 
 0.269 
 
 0.301 
 0.306 
 
 0.333 
 0.333 
 
 0.358 
 0.374 
 
 0.375 
 0.391 
 
 0.344 
 0.364 
 
 0.423 
 0.431 
 
 0.465 
 0.465 
 
334 
 
 CONCRETE ENGINEERS' HANDBOOK 
 
 [Sec. 7-52 
 
 The following table shows that a floor girder carrying one or more beams and subjected 
 to an indefinite live load may be computed with sufficient accuracy by considering it simply 
 supported and then reducing the maximum moment so found (and an equal negative moment) 
 by the same ratio of reduction used with uniform loading. For example, suppose the maximum 
 moment due to given concentrated loads is K (considering the beam supported), then if 
 }^i2^l^ is used for uniform loading instead of /-sivP, ^{2 of K, or %K, may be used for the con- 
 centrated loads. The table gives moment coefficients according to this rule. These coeffi- 
 cients should be compared with those in the preceding table. 
 
 No. of spans 
 
 Intermediate 
 spans 
 
 End span and 
 adjoining 
 support 
 
 Loads at middle points 
 
 Three or more spans 
 
 Two spans 
 
 0.167 
 
 0.208 
 
 0.208 
 0.250 
 
 Loads at third points 
 Two spans 
 
 0.222 
 0.278 
 
 0.278 
 0.333 
 
 Loads at middle and 
 quarter points 
 
 0.333 
 0.417 
 
 0.417 
 0 . 500 
 
 In a beam loaded at the middle points we find that the moment at the center of intermediate 
 spans may have a value about 8% greater than the recommended value for use in design. 
 (This, however, is considering the beam as freely supported at the interior supports, and the 
 supports of no appreciable width.) All other values for this loading are somewhat less than 
 those recommended. In fact, the specified moment coefficient for the center support of a two- 
 span beam may be reduced and made the same as for the center of span. In beams of three or 
 more spans and for the same loading as just mentioned, the moment coefficient for the inner 
 supports of end spans may be reduced so as to have the same value as specified for the interior 
 spans. 
 
 The moment at interior supports in beams loaded at the third points may have a value 
 about 19% greater than that specified, but the width and monolithic character of the supports 
 will offset this to a considerable extent. It is preferable, however, to make some allowance 
 for this in design although for simplicity this has not been done in this handbook. The same 
 is true for the inner supports of the end spans although the increase in moment over that speci- 
 fied is about 10%. 
 
 The reader may draw his own conclusions in regard to moment coefficients in beams loaded 
 at the middle and quarter points. 
 
 The recommendations for shear given for uniform loads will apply in the case of beam con- 
 centrations. 
 
 52. Negative Moment at the Ends of Continuous Beams. — The amount of negative mo- 
 ment at the ends of continuous beams depends upon the manner in which the ends are restrained. 
 A beam cannot be entirely fixed unless the restraint is sufficient to cause the neutral surface at 
 the ends to be horizontal. The moment coefficient must be left to the judgment of the designer, 
 but the shear and moment diagrams shown in Figs. 54 to 57 inclusive (for beams with fixed 
 ends) and in Figs. 70 and 72 will prove useful in this connection (see also recommendations 
 Qf Joint Committee on page 318). 
 
Sec. 7-53] 
 
 BEAMS AND SLABS 
 
 335 
 
 63. Bending Up of Bars and Provision for Negative Moment.— In Figs. 63 to 65 inclusive 
 
 are given bending-moment curves which apply to continuous beams, supported ends, for uni- 
 form loads (both live and dead) on two, three, and four equal spans. The dead-load curves 
 are the same as shown in Fig. 46 for uniform load over all spans. In plotting the live-load 
 curves, the loadings were considered which give maximum and minimum values at each of the 
 
 Fig. 63. — Moment curves for uniform load; two spans; supported ends. 
 
 one-tenth division sections. Thus, these curves do not represent any one condition of loading, 
 but may be used to determine the extreme values of the live-load moment at any given point 
 in the span. 
 
 It should be noted that some portions of the live-load curves are quite different from those 
 shown in Figs. 46 to 57 inclusive. For example, the part of the maximum live-load curve close 
 to the center support in the beam of two spans (Fig. 63) is quite different from a similar portion 
 
 0.12 
 
 0.14 I I I [ \ L_J I I \ \ I \ \ \ 
 
 0 0.2 a4. 0,6 0.8 l.O 1.2 1.4 
 Fig. 64. — Moment curves for uniform load; three spans; left-hand half; supported ends. 
 
 in any of the curves shown in Figs. 46 and 54. This is due to the fact that for these sections 
 maximum and minimum moments are caused by only partial loading, and not by having either 
 one or both spans fully loaded. If influence lines were plotted for these sections, this point 
 would be clearly brought out. 
 
 In the two-span beam shown in Fig. 66, maximum and minimum bending-moment curves 
 
336 
 
 CONCRETE ENGINEERS' HANDBOOK 
 
 [Sec. 7-53 
 
 are given for a 3 : 1 ratio of live to dead load. To obtain these curves from Fig. 63, points should 
 be determined for each one-tenth of the span. The following notation will be employed: 
 
 tVd = dead load per unit of length. 
 
 wi = live load per unit of length. 
 
 Wt = Wd + wi = total load per unit of length. 
 
 I^Ki- 6r). — Moment curves for uniform load; four spans; left-liand half; supported ends. 
 
 Consider a point in either span of a two-span })eam (Fig. 63) at a distance 0.21 from the center 
 support. The moment will vary between 
 
 - Omwdl"^ + omwii^ 
 
 and 
 
 0.12 
 
 0.14 I 1 i 1 1 \ \ 1 1 1 1 1 1 1 I 1 1 \ \ 1 1 
 
 O 02 0.4 06 0.8 t.O 1.2 U (.6 1.8 ZO 
 
 Fig. 66. — Moment curves for continuous beams of two spans; supported ends. Ratio of live load to dead load 3 : 1. 
 
 Assuming wi = Swa, the moment varies from + 0.07wdP to - 0.17iVdP; that is (since Wd = 
 Huh) from + 0.0175ty«Fto + 0m25wtl'. 
 
 Curves such as shown in Fig. 66 show what positive and negative moments should be pro- 
 vided for at any point in the span. They also indicate in what manner the steel may safely 
 be bent up from the lower side of the beam. The curves in Fig. 66 show that a negative moment 
 
Sec. 7-53] 
 
 BEAMS AND SLABS 
 
 337 
 
 is likely to occur over one-half of each span of a two-span beam. Any fixing of the ends, how- 
 ever, will reduce this length. 
 
 Referring to Figs. 66, 67 and 68, it is clear that negative moment may, under extreme 
 conditions, occur entirely across the beam. For ordinary cases, then, it would seem that top 
 
 0.12 
 
 0.10 
 
 % 0.08 
 
 *5 0.06 
 in 
 
 ^ 0.04 
 ^ O02 
 
 .E 0 
 
 4- 
 
 S 002 
 O 004. 
 <^ 0.06 
 ift 008 
 
 0) 
 
 3 0.10 
 ^ 0.1 E 
 0.14 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 V — 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 + 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 0.2 
 
 0.4 
 
 0.6 
 
 0.8 
 
 I.O 
 
 Pig f,7 — Moment ourves for continuous beams of three spans; left-hand half; supported ends. Ratio of live load 
 * to dead load 3:1. 
 
 reinforcement should extend to at least the fourth point. In special cases, however, it may 
 be desirable to provide for negative-tension reinforcement over the entire span. If reinforcing 
 frames are used, the top rods employed for handling and for fastening the stirrups into a unit 
 will aid materially in taking care of any tensile stresses which may occur in the top of the beam. 
 
 0 0.2 04 0,6 0.8 1.0 l.E 1-4- 16 "0 20 
 Fig. 68. — Moment curves for continuous beams of four spans; left-hand half; supported ends. 
 
 to dead load 3:1. 
 
 Ratio of live load 
 
 It is also true that in monolithic floor construction, the adjoining slab will help considerably in 
 preventing top tensile stresses at the center of the span. 
 
 In view of the above considerations, it would seem that rods may be bent up with sufficient 
 accuracy (for ordinary cases where uniform live load is somewhat indefinite), by applying the 
 method of Art. 22. For special cases, curves should be drawn similar to those shown in Figs. 
 66 to 68 inclusive. 
 22 
 
338 
 
 CONCRETE ENGINEERS' HANDBOOK 
 
 [Sec. 7-53 
 
 Maximum and minimum moment curves for concentrated loads are shown in Figs. 69 to 
 73 inclusive. Three-span beams only are considered and curves are given for both supported 
 and fixed ends. From a study of these curves and in view of the considerations presented 
 previously in this article, it would seem (assuming the usual indefinite live load) that rods in 
 girders loaded at the third points may safely be bent up as explained in the floor-bay design of 
 Art. 11, Sect. 11. The curves are given for a 4 :1 ratio of live to dead load. The moment 
 
 Co. 
 
 Eh- 
 
 •si 
 
 CD 1- 
 
 0.16 
 O.08 
 0. 
 O.08 
 0.16 
 
 
 
 
 /- 
 
 
 
 
 
 
 
 
 / 
 
 
 
 
 
 H 
 
 ■ 
 
 
 
 
 
 
 
 
 h 
 
 
 
 
 
 
 
 
 
 
 
 \:: 
 
 
 
 
 
 
 
 
 
 \ — 
 
 
 
 
 
 
 
 
 
 
 
 7 - 
 
 
 0' 
 
 0 
 
 2 
 
 OA 
 
 \ 0. 
 
 
 Q» t.O 
 
 15 
 
 
 Fig. 69. — IVIoment curves for continuous 
 beams of three spans; supported ends; loads at 
 middle points. Ratio of live load to dead load 
 4 : 1. 
 
 Fig. 70. — Moment curves for continuous 
 beams of three spans; fixed ends; loads at 
 middle points. Ratio of live load to dead 
 load 4 : 1. 
 
 due to dead weight of girder stem has little effect in considerations regarding the bending up of 
 rods. 
 
 The formulas given below may be of use when considering the variation of moment in 
 beams of many equal spans for different kinds of loading. In deriving the formulas for maxi- 
 mum positive moment, alternate spans were considered as covered with live load. This gives 
 the worst condition of loading for positive moment. Likewise in deriving the formulas for 
 maximum negative moment, the worst condition of loading for negative moment was assumed 
 
 0.32 
 
 0.24 
 0. 16 
 
 .e 0.08 
 
 o O.06 
 
 % 016 
 
 § 0.2^ 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 i 
 
 
 
 
 
 
 
 
 . 
 
 
 
 + 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 M 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 P 
 
 
 
 
 
 i 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 \ 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 l- 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 \ 
 
 / 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 <u a- 
 
 0.2 0.4 0.6 0.8 I.O I.2_U 
 
 024 
 
 0.16 
 0.08 
 0 
 0.08 
 0.16 
 0.24 
 0.32 
 
 55 
 
 ^2 
 
 02 0.4 0.6 08 1.0 1.2 1.4 
 
 Fig. 71. — Moment curves for continuous 
 beams of three spans; supported ends; loads at 
 third points. Ratio of live load to dead load 4 : 1 
 
 Fig. 72. — Moment curves for continuous beams of 
 three spans; fixed ends; loads at third points. Ratio 
 of live load to dead load 4:1. 
 
 — that is, two adjacent spans were assumed as fully loaded alternating with one span unloaded. 
 For the same load on each span, the formulas reduce to simple terms, and give the moments on 
 continuous beams with fixed ends and with any number of spans. 
 The following notation will be employed: 
 
 maximum positive moment at center of span, 
 minimum positive moment (or maximum negative moment) at 
 center of span. 
 
 maximum negative moment at the support. 
 
 M max. 
 M,nin. 
 
 M max. 
 
Sec. 7-54] 
 
 BEAMS AND SLABS 
 
 339 
 
 Uniform loads: 
 
 Loads at middle points: 
 
 Mr, 
 
 M, 
 
 Mr, 
 
 Mr 
 
 Mr 
 
 Mr 
 
 24 
 
 {2wt - Wd) 
 
 12 
 
 -3g(4^. 
 
 (3P, 
 
 J_ 
 16 
 _| 
 24 
 
 (4P, 
 
 Loads at third points: 
 
 Mmax. 
 Mmin. 
 M max- 
 
 (2P, - Pd) 
 
 (Pt - 2Pd) 
 
 Pd) 
 
 Triangular distribution of load : 
 51 
 
 Mr 
 
 Mr 
 
 = (2.2TF. - Wd) 
 
 2.2Tr,) 
 
 - Wd) 
 
 Pa) 
 
 0.40 
 
 0.E4 
 £0.16 
 ^ 0.08 
 0 
 
 4- 
 
 £0.08 
 E 
 
 J 0.16 
 
 in 
 a> 
 
 ^^0.40 
 
 
 
 
 
 
 
 
 
 -r 
 
 
 
 vl 
 
 
 
 
 r- 
 
 
 
 
 
 
 -/ 
 
 
 
 
 
 
 B 
 
 /: 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 f 
 
 
 
 
 
 
 - d-i 
 
 
 
 
 
 
 
 
 r 
 
 
 
 0 0.2 0.4 0.6 08 1.0 1.2 1.4 
 
 Fig. 73. — Moment curves for con- 
 tinuous beams of three spans; supported 
 ends; loads at middle and quarter points. 
 Ratio of live load to dead load 4 : 1. 
 
 64. Continuous Beams with Varying Moment of 
 Inertia. — It is the practice of some designers to place 
 more steel between the supports of continuous beams 
 than the amount just sufficient to resist the bending 
 moment specified by the Joint Committee. They consider that by doing this the stresses 
 over the supports are reduced and the design is more economical. They maintain, also, that 
 this method of procedure is advisable in order to provide for imperfect continuity of the 
 beam and to take care of unknown stresses caused by unequal settlements of the supports. 
 It is important, therefore, to determine the actual moments which occur at the center and 
 supports in such cases. 
 
 A study of this kind, assuming uniform loads, has been made by R. E. Spaulding^ for beams 
 with fixed ends and for beams with one end fixed and one end free. The curves shown in Fig. 
 74 are the result of his investigations and apply to either rectangular or T-beams. Mr. Spauld- 
 ing assumed a uniform moment of inertia for that portion of the beam over the support in which 
 negative moments exist, and another moment of inertia for the central portion of the span — 
 that is, between points of inflection. 
 
 The condition of fixed ends is obtained for full loading on the lower floors of a building hav- 
 ing very heavy columns. The case of one end fixed and one end free is found in either span of a 
 two-span beam with the same uniform load on both spans. 
 
 Referring to the curves of Fig. 74 it is clear that with both ends fixed and with the same 
 
 moment of inertia throughout, the moment at the center is 0.042ii'Z2 = — and at the end is 
 
 1 Eng. News, Jan. 13, 1910. 
 
340 
 
 CONCRETE ENGINEERS' HANDBOOK 
 
 [Sec. 7-54 
 
 12' 
 
 as is well known. If, as is sometimes done, a small amount of steel is placed 
 
 It 
 
 over the supports, such that j = 0.20, and the full bending moment is provided for at 
 
 55 % of -g- j and at 
 
 the support will be O.OSGtt'Z^. Such a condition stresses the steel at the support to a value 2.2 
 times the working stress, since provision is made at the support for a moment of only (0.20) 
 mwP) = Howl^ = 0.025wP — that is, of course, assuming the amount of steel used to be pro- 
 portional to the moments of inertia. Again, suppose the amount of steel at the support of a 
 beam with fixed ends to be made one-half that at the center and that the center be designed 
 for XowP- The actual moment at the center will be 0.053 wP and the end bending moment will 
 be 0.07SwP. Thus, with this distribution of the reinforcement, the steel at the support will 
 
 0073 
 
 be overstressed to an amount ^ ^^^ = 1.5 times the assumed working stress. It may be seen 
 
 from the curves of Fig. 74 (for beams with fixed ends) that if a beam is figured as simply sup- 
 ported, then about 0.6 of the amount of steel 
 that is employed in the middle of the beam 
 should be placed over supports in order to 
 make as economical a design as possible under 
 the conditions. For example, with this dis- 
 tribution of steel, provision is made at the sup- 
 port for a moment of 
 
 O.ISO 
 
 0.075u;Z2 
 
 0.5 1.0 
 
 Values of 1^ 
 I 
 
 "Mf denotes Max. Moment at Support 
 
 M " » " iDef^veen Supports 
 
 If moment of inertia at support 
 
 J = " " " y center of spari 
 
 Fig. 74. — Curves of maximum bending moments in 
 beams with different moments of inertia at end and 
 centers. 
 
 and this is exactly the actual moment caused 
 by such an arrangement. The assumption 
 which is made that the moments of inertia are 
 proportional to the amount of steel used makes 
 practically no difference in these comparative 
 results. 
 
 In the above discussion no compressive 
 steel has been considered at the supports. In 
 rectangular beams, of course, there is none 
 needed but in T-beams the section becomes 
 rectangular at the supports and the stress in 
 the concrete needs attention. Referring to Table 2, page 355, for n = 15, it may be seen that for 
 an allowable compressive stress in the concrete at the support of 750 lb. per sq. in. and 16,000 
 lb. in the steel, the required percentage of tensile steel is practically 1%. It is thus possible 
 to get along without any additional strengthening of the compressive part of the beam at 
 the support only when the beam is reinforced at the center of span with less than 1% -r- 
 0.6 = 1.7% of the area of the stem (including the portion of the slab directly above it), 
 
 voP 
 
 whether or not 1.7% is less than the amount determined for the full bending moment of 
 
 The question naturally arises whether it is more economical to design for the full bending 
 
 voP 
 
 moment of -g- and provide steel over supports equal in amount to 0.6 of the steel at the middle 
 
 wP 
 
 of beam, or, as recommended by the Joint Committee, to design for - ^ both at the center of 
 
 span and over supports. Consider a given case where a continuous beam designed as simply 
 supported requires 3 sq. in. of steel at the center of span. Of this amount (0.6) (3) = 1.8 sq. 
 in. should be bent up and carried over the top of the support (disregarding amount of steel 
 required for bond), the rest of the rods being horizontal in the bottom of the beam. The 
 
Sec. 7-55] 
 
 BEAMS AND SLABS 
 
 341 
 
 approximate volume of this steel may be taken as the area of the steel in the middle times the 
 span in inches, or 3 sq. in, X I = SI cu. in. The same beam designed according to the Joint 
 Committee's recommendations would require 2 sq. in. of steel at the center of span and 2 sq. in. 
 over the top at the support, and the volume of steel may be taken as the volume of the long 
 rods, the area of which amounts to 2 sq. in., plus the volume of steel carried over from the 
 adjoining spans and extending to say the one-fourth point of the span, or approximately (2 sq. 
 in.)(Z) + (2)(3^/) = cu. in. The reader should notice in the above comparison that the 
 first design is favored to the extent that no top steel is assumed to extend beyond the center 
 of the support. Assume this steel. to extend to only the one-fifth point of the span, then the 
 volume of steel in the first case becomes equal to 31 + (1.8) (3-^0 = 3.4Zcu. in. Thus it is clearly 
 
 seen that a beam designed for a moment -g- in the center of span requires approximately 25% 
 
 more steel than the beam designed with for both positive and negative moments. 
 
 Mr. Spaulding in his study of continuous beams (referred to above) assumed that the 
 moment of inertia is constant between the points of inflection, and there changes abruptly to 
 another value which is constant for the ends of the beam. This assumption neglects in all 
 cases the value of concrete below the neutral axis because it is in tension and is liable to crack. 
 This is true, however, only for sections subjected to the maximum bending moments. In fact, 
 the variation of the moment of inertia throughout the beam may be represented by a curve 
 with maximums at the points of inflection and minimums at the middle of the beam and at 
 the supports. Sanford E. Thompson in an article published in the issue of Engineering News 
 of Jan. 13, 1910, under the title "Continuity in Reinforced-concrete Beams," states that exten- 
 sive studies made in his office, considering the moment of inertia to vary in this way, gave results 
 which substantially agree with those obtained by Mr. Spaulding and prove the latter's assertion 
 that the assumption of constant moment of inertia between points of inflection is sufficiently 
 accurate for all practical purposes. 
 
 On page 340 it is shown that a beam reinforced for the full bending moment of -g- at 
 
 the center of span will induce a bending moment (if properly designed) of approximately 
 
 O.OTSifZ^ over supports. This is only about 10% less than the moment of recommended 
 
 by the Joint Committee. 
 
 DESIGNING TABLES AND DIAGRAMS FOR BEAMS AND SLABS 
 
 55. Illustrative Problems. — The use of designing tables and diagrams can best be explained 
 by giving the solutions of typical designing problems. The following working stresses will 
 be assumed throughout: 
 
 fc = 650. fs = 16,000. n = 15. v = 40 without web reinforcement 
 and 120 when thorough web reinforcement is provided. 
 
 Bond stress will not be considered here. 
 
 . Design a rectangular beam to span 4:0 ft. and to support a load of 600 lb. per ft. {including 
 weight of beam) . Beam is assumed to be simply supported. 
 Solution using Tables. 
 
 From Table 2, for n = 15, /. = 16,000 and = 650 
 K = 107.4 
 
 ^ ^ (600X40^2) ^ 4 ,0 
 8 8 ' ' 
 
342 
 
 CONCRETE ENGINEERS' HANDBOOK 
 
 [Sec. 7-55 
 
 Assume d = 1.56. Then Table 4 shows that 6 = 18 in. and d = 27^ in. will be satisfactory. 
 Area of cross-section, bd = (18) (27.5) = 495 sq. in. 
 
 As = (495) (0.0077) = 3.81 sq. in. 
 
 We shall select four l^^-in. round rods = 3.98 sq. in. (see Table 1 or Table 5). 
 
 V 12,000 
 
 '^bfd^ (18) (^) (2-7:5) = 2^ P^^.^^- 
 
 Web reinforcement is not theoretically needed. 
 The beam may be reviewed as follows: 
 
 P = T->^^ = 0.0080 
 bd 495 
 
 From Table 3, for this value of p, 
 
 k = 0.384 j = 0.872 
 
 Then, 
 
 1,440 ,000 irmniK 
 
 = (3:98)(a^)T2^ ^ '^''^^ 
 . (2) (15,100) (0.0080) 
 
 fc = Qgg^ = 630 lb. per sq. in. 
 
 Solution Using Diagrams. — In Diagram 2,^ the intersection of the curves fc = 650 and 
 fs = 16,000 is first found. Tracing down, p is found to be 0.0077, and tracing horizontally 
 
 is found to be 107.3. The solution then follows as in the use of tables given above. 
 
 Diagrams 1 and 2 may also be employed to determine the safe resisting moment of a given 
 beam and the greatest unit stresses in the steel and concrete due to a given bending moment. 
 
 To determine the safe resisting moment of a given beam, the value of p should be computed. 
 After finding this value on the lower margin, trace vertically, stopping at the first of the two 
 curves /c = 650 and/s = 16,000 (assuming these the allowable stresses). Now trace horizon- 
 tally to either side margin and the value of K is found. Then, M = Kbd"^. Consider a beam 
 of the above dimensions to have 1 % of steel. Tracing vertically from this value on the lower 
 margin of Diagram 2, the 650 curve is the first curve to be reached and at a value of K = 117.0. 
 ThenM = (117)(18)(27.5)2 = 1,593,000 in.-lb. 
 
 To determine the greatest unit stresses in the steel and concrete of a given beam due to a 
 given bending moment, the value of p should be computed as before. Also, K should be com- 
 
 M 
 
 puted from the formula K = With these values of p and X, find the intersection of the 
 
 vertical and horizontal lines through these values respectively, and from the adjacent steel and 
 concrete curves the values of /c and/ may be estimated. Consider a beam of the above dimen- 
 sions and with 0.7% of steel, to be subjected to a bending moment of 1,200,000 in.-lb. or 
 
 K = ^Qy2 ~ 88.2. The intersection of the vertical and horizontal lines through these 
 
 values respectively in Diagram 2 gives fc = 550 and fs = 14,400. This procedure is followed 
 in reviewing beam design. 
 
 Diagrams 1 and 2 may also be employed to find minimum allowable depth of beam for 
 a given percentage of steel and various assumed widths, also to find the amount of steel for 
 a beam with given loading. 
 
 To find the depth of beam for a given percentage of steel and given allowable stresses, 
 select the lower value of K determined by the intersection of the allowable stress curves with 
 
 1 Diagram first given by Arthur W. French, Trans. Am. Soc. C. E., vol. 56, 1906, p. 362. 
 
Sec. 7-55] 
 
 BEAMS AND SLABS 
 
 343 
 
 the vertical line representing the given steel percentage. This value of K substituted in formula 
 
 M = Kbd^, ord = 
 
 \ oK. 
 
 gives the smallest permissible depth for various assumed widths. 
 
 To find the percentage of steel for a given beam, compute the value of K from formula 
 M 
 
 K = Locate this value at the left of the diagram. Trace horizontally to the right until 
 
 the proper allowable stress curve is reached. Thus, ii K = 80, the curve of fc = 650 intersects 
 the horizontal through the given value of K at a value for p of 0.003, but fs for this percentage 
 is seen to be over 22,000. The desired percentage is 0.0056, determined by the intersection of 
 the curve fs = 16,000 with the horizontal in question. 
 
 2. A beam of 16-m. width, having its compression face inclined at an angle of 30 deg. and 
 its tension face at an angle of 20 deg., is to be designed for a moment of 3,000,000 in.-lb. What 
 depth and percentage of reinforcement are necessary? 
 
 Diagram 2 gives K = 107.3 and p = 0.0077 (see preceding problem). 
 
 COS^ jSc 
 
 These values should be multiplied by cos^jSc and respectively. The products may 
 
 cos P( 
 
 be obtained directly from Diagram 3. 
 
 Entering the diagram with a value of 10.73 (resetting the decimal in X = 107.3) on the 
 lower margin, trace vertically to inclined line for = 30 deg. and then horizontally to the left- 
 hand margin where a value of 10.73 cos^ (30 deg.) = 8.05 is found. Pointing off properly, the 
 proper value of K to use is 80.5. Then 
 
 3,000,000 , . 
 
 To find 0.0077 enter the diagrams with a value of 7.7 on the lower margin, trace 
 
 vertically to inclined line for I3c = 30 deg., then horizontally to inclined line for /3f = 20 deg., 
 and then vertically upward to the upper margin where a value of 6.1 is found. The proper 
 value of p to use is 0.0061. 
 
 Diagram 3 may be employed when either /3c = 0 or )3( = 0. The procedure would be 
 the same as above. 
 
 3. A beam with ?> = 16 in., d = 45 in., and p = 0.007 is subjected to a moment of 2,500,000 
 in.-lb. Assuming the compression face inclined at an angle of 20 deg. and the tension face at 25 deg., 
 what are the unit stresses fc and fsf 
 
 ^ 2,500,000 ^ 
 . (16) (45)2 ''-^ 
 
 Before using Diagram 2, the values of K and p should be multiplied by — and ^^^^ 
 
 cos Pc cos Pc 
 
 respectively. These products may be obtained directly from Diagram 3. 
 
 Entering the diagram with a value of 7.72 on the left-hand margin, trace horizontally to 
 inclined line for jSc = 20 deg. and then vertically downward to the lower margin where a value 
 7 72 
 
 of (.os2(20 deg ) ^ ^'"^^ found. Pointing off properly, the proper value of K to use in Diagram 
 2 is 87.5. 
 
 To find 7.00 , enter the diagram with a value of 7.00 on the upper margin, trace 
 
 cos Pc 
 
 vertically downward to inclined line for j8j = 25 deg., then horizontally to inclined line for 
 /3c = 20 deg., and then vertically downward to the lower margin where a value of 7.25 is found. 
 . The proper value of p to use in Diagram 2 is 0.0072. 
 
 Using K = 87.5 and p = 0.0072 in Diagram 2 gives /c = 540 and/.. = 13,700. 
 
 Diagram 3 may be employed when either /3c = 0 or /3j = 0. The procedure would be the 
 same as above. 
 
344 
 
 CONCRETE ENGINEERS' HANDBOOK 
 
 [Sec. 7-55 
 
 4. Determine the approximate weight of a rectangular beam 24 in. wide, with a clear span of 
 25 ft. and carrying a load of 5000 lb. per ft. 
 
 The load per foot length per inch width will be : 
 
 wl^ 
 
 From Diagram 4^ we obtain c = 0.885. If we take M = — , then the weight of the beam per 
 
 o 
 
 foot length will be: 
 
 iv' = [0.885 + (25) (0.003)]^^^^^-^-? = 1200 1b: per ft. 
 
 5. What safe load per square foot {including dead weight) can be supported by a slab 6 in. 
 deep {d = 4^^ in.) and lO-ft. span reinforced with ^i-'m. round rods placed 8 in. apart? The 
 slab is simply supported and reinforced in only one direction. 
 
 0.1963 
 = (8X4.75) ^^-^^^^ 
 
 Referring to Diagram 6, Part 2, and tracing vertically from this value of p on the lower 
 margin to an intersection with the curve of d = 4:-^-i in., and then tracing horizontally to the 
 left-hand margin, a l)ending moment of 20,100 in. -lb. is found. 
 
 Select this value of the bending moment on the left-hand margin of Diagram 5 and trace 
 horizontally to the right to an intersection with a vertical line through 10, denoting span 
 
 wl^ 
 
 length. The safe load, based on M = can now be estimated directly by means of the 
 curved lines and is found to be 168 lb. per sq. ft. 
 
 ♦ 
 
 (168) (0.80) = 1343^ lb. per sq. ft., safe load for slab simply supported. 
 
 6. Design a slab to span 6 ft. and to carry a live load of 250 lb. per sq. ft. Slab is to be fully 
 continuous and reinforced in only one direction. 
 
 Assume the weight of slab at 50 lb. per sq. ft. Total load for slab is thus 300 lb. per sq. ft. 
 From Diagram 5 for this span length and load per square foot, a bending moment of 11,000 
 wl^ 
 
 in. -lb. is found, based on y^' Diagram 6, Part 1, shows that a depth (d) of 3 in. will be ample 
 
 — total depth 3^^ in. Also, As = 0.275 sq. in. 
 
 From Table 6, we may use ^-^-in. round rods spaced 4% in. on centers. 
 
 The assumed and actual dead weights are close enough, and the slab need not be redesigned. 
 The slab should be reinforced against negative moment at the supports. The slab should also 
 be reinforced transversely in order to prevent shrinkage and temperature cracks. Shear at 
 ends of slab in direction of reinforcement is (300) (3) = 900 lb. per ft. of breadth. Allowable 
 shear = (12)(3)(40) = 1440. Thus no web reinforcement is needed, as is usually the case 
 except for excessive loading. 
 
 7. Design a slab for a 10 by lO-ft. panel to carry a live load of 250 lb. per sq. ft. Slab is to 
 be fully continuous and reinforced in both directions. 
 
 The dead load will be assumed at 60 lb. per sq. ft. Total moment to be resisted in each 
 direction according to recommendations of the Joint Committee (Art. 29c, page 307) is 
 
 wl"^ ... ^ 
 
 (Diagram 5 may be used, assuming M = j^"? dividing result by 2.) 
 
 Using Diagram 6, Part 1, the slab is seen to be of very nearly equal strength in tension 
 and compression when d = S}'2 in. and As = 0.32 sq. in. The required spacing for center half 
 
 1 DiaRram and formulas taken from article by M. J. Lorente in Eng. News, March 20, 1913. 
 
Sec. 7-55] 
 
 BEAMS AND SLABS 
 
 345 
 
 of slab, then, is 4 in. on centers for ^^-in. round bars. The bars should be spaced the same 
 throughout the center half of slab and then the spacing gradually increased to the edge of the 
 slab, using one-half as many bars in the outside quarters. The slab should be reinforced against 
 negative moment at the supports. 
 
 The depth of the slab should be made 5 in. in order to have the upper reinforcing system 
 at the minimum distance 33-^ in. from the surface of the slab. The lower system will then be 
 slightly stronger than necessary. The dead weight is approximately that assumed. For 
 safety in construction, it is preferable to require the two systems of reinforcement to be fastened 
 together at frequent intervals. Web reinforcement is not necessary. 
 
 8. Design the center cross-section of a T-beam in a floor system; the beam is to have a snan 
 of 12 ft. and be fully continuous. Maximum shear {live plus dead) is closely equal to 12,200 lb. 
 Maximum moment {live plus dead) = 356,300 in.-lb. Supported slab is Q-in. thick. 
 
 The only purpose of the concrete below the neutral axis is to bind together the tension and 
 compression flanges, and consequently its section is determined by the shearing stresses in- 
 volved and space for the necessary bars. The shearing stress v should not be greater than 120. 
 The area b'd (unless the value of j should turn out to be less than '^-^^ 
 
 12,200 . 
 
 = 116 sq. m. 
 
 (%)(120) 
 
 The following formula of Art. 37, gives the most economical depths for various assumed 
 web widths: 
 
 fsb' 2 
 Assuming r as 60, then 
 
 for b' = 9 in. d = 15.2 
 
 for // = 10 in. d = 14.6, etc. 
 
 Some rough calculations show that if four bars are to be used and all in one row, the breadth 
 of stem necessary for the bars controls. A breadth b' of 10 in. and a depth d of 15 in. (total 
 depth 17 in.) will be tried. 
 
 Diagrams 7 and 8 cannot be employed to solve for the resisting moment of a given beam 
 but are useful in designing. Formula (11), Art. 34, may be put in the following form 
 
 M 
 
 bd^ 
 
 ( ^ 2kd} d 
 
 k and j in this equation are functions of fc and fs, and hence the variables are /c, and the ratio 
 ^- The curves at the left in Diagrams 7 and 8 are plotted from this equation with a fixed value 
 
 M t 
 
 of /s = 16,000 lb. per sq. in. Values of fc may be determined for various values of ^ and -^j or 
 
 M t 
 values of may be determined for various values of fc and It must not be overlooked, 
 
 however, that these diagrams will apply only when the amount of steel is such that/s = 16,000 
 lb. per sq. in. This amount of steel may be easily determined when the corresponding j is 
 
 M t 
 
 found from the curves at the right of the diagram. Suppose ^^^2 j ^ 
 
 intersection of horizontal and vertical lines through these values respectively in Diagram 8 
 shows /c to equal 600, and then tracing from this intersection horizontally to the right until the 
 vertical line is reached indicating fc = 600 (at the right-hand side of the diagram), we find j 
 
 equal to 0.91. Finally, As = j^, in which j = 0.91, /. = 16,000, and M and d are known. 
 
 Diagram 8 will now be employed in working out the problem stated at the beginning of this 
 discussion. 
 
346 
 
 CONCRETE ENGINEERS' HANDBOOK 
 
 [Sec. 7-55 
 
 The breadth of the flange is controlled by one-fourth the span,^ or 36 in. Assuming a 
 depth (d) of 15 in. . 
 
 M 356,300 
 
 bd^ (36) (15)2 
 
 = 44 
 
 For this value of ^ and for ^ = = 0.40, we find from the diagram that this beam falls 
 
 under Case I; that is, the neutral axis is in the flange. 
 
 Diagrams 1 and 2 may be used for T-beams under Case I. In the problem at hand, a 
 horizontal line through the value 44 for K, in Diagram 2, intersects the oblique line fs = 16,000 
 at a value of fc = 370. The value of p corresponding is 0.003. Then As = pbd = (0.003) (36) 
 (15) = 1.62 sq. in. Table 5 shows that four M-in. round bars will give the required steel area. 
 
 9. The flange of a T-beam is 24 in. wide and 4 in. thick. The beam is to sustain a bending 
 moment of 480,000 in.-lb. What depth of beam, and amount of steel are necessary? 
 
 We will try d ^ IS in. 
 
 M 480,000 
 
 bd^ (24)(18)2 ^^-^ 
 
 I 
 
 M t 4 
 
 For this value of ^ and for ^ = = 0.222, we find from Diagram 8, fc = 485 lb. per sq. in. 
 and j = 0.910. Then 
 
 A - 480,000 _ 
 
 ~ (16,000) (0.910) (18) ~ ^-^^ sq- 
 
 The stress in the concrete of 485 is permissible and the beam as designed will be considered 
 satisfactory. (Formula (12) in Art. 34a may be used to find minimum depth for a given flange 
 width without trial.) 
 
 Suppose 2.0 sq. in. of steel were inserted in a beam of the above dimensions, and suppose 
 that the safe resisting moment is desired. Diagram 10 must be used for this case. 
 
 ^ = 0.222 as before. Tracing vertically from this value on the lower margin of the left diagram 
 
 to a value of p = 0.0046 and then tracing horizontally to the left margin, we find a value of k 
 = 0.32. In a similar manner we find j equal to 0.91 
 
 Ms =f.Asjd = (16,000) (2.0) (0.91) (18) = 525,000 in.-lb. 
 
 _ fsk _ (16,000) (0.32) _ 
 ~ nil - k) - (15)(1 -0.32) - 
 
 or, from Table 8, ■ 
 
 fc = (0.0314) (16,000) = 502 lb. 
 
 Since fc is less than 650, the resisting moment depends upon the steel, or Ms = 525,000 in.-lb. 
 
 10. A continuous T-beam, uniformly loaded, has a bending moment at the center of each span 
 of 356,300 in.-lb. Negative bendi7ig moment at the supports and the positive bending moment at 
 
 wl^ 
 
 the ceriter of span are figured by the formula, M = -^^^ tensile steel at the center of span 
 
 consists of four %^-in. round bars, h' = 10 in. d = 15 in. Design the supports. 
 
 At the supports the flange of the T-beam, being in tension, is negligible and the T-beam 
 changes into a rectangular beam with steel in top and bottom. Two of the tension bars on 
 each side of the supports will be bent up and made to lap over the top of the supports, while the 
 other two bars on each side will be continued straight and lapped over supports at the bottom 
 of beam. 
 
 The ratios of steel in tension and compression are the same, and are respectively: 
 
 1 See recommendations of the Joint Committee, Art. 32. 
 
Sec. 7-551 BEAMS AND SLABS 347 
 
 (1.77 in above equation taken from Table 5.) 
 
 = A 
 d 15 
 
 From Diagram 12, knowing = p, we obtain 
 
 0.133 
 
 Thus, 
 
 ^ d' _ \k ^ 0.361 
 
 For ^ =0.10.... j. ^^ggg 
 
 ^ d' _ ^ ^ I = 0.377 
 
 For = 0.15 \ . „ 
 
 I J = 0.866 
 
 ^ d' „ f /c = 0.372 
 
 For = 0.133...] .^^_g^3 
 
 (It is usually well within the precision of the actual work, and on the safe side, to use the curves 
 for the value of ^ next larger than the actual value. Thus in this problem the values of k and 
 
 j for ^ =0.15 could be used with sufficient accuracy.) 
 Then 
 
 , M 356,300 i^.nmK 
 
 ^' = Ajd=' (1.77) (0.873) (15) = ^^'^^^ 
 
 and, using Table 9, 
 
 Jc = ^ = (0.0394) (15,400) = 607 lb. per sq. in. 
 
 The stresses in the concrete and steel are within the allowable and no haunch or additional 
 steel are necessary. 
 
 The moment of resistance at the supports may be found as follows: 
 
 . fMl - k) 650 
 
 = ^ = 0;039^ = 16,500 lb. per sq. m. 
 
 Thus the moment of resistance depends on the steel and 
 
 Ms = bd^fspj = (10) (15)2 (16,000) (0.0118) (0.873) = 371,000 in. -lb. 
 
 11. At the support of a continuous T-beam the following values are known: b = 12 in., 
 p = p', As = 3.0 s^. in., M = 750,000 in. -lb., fc = 750 and fs = 16,000. Find the required depth 
 of beam. 
 
 Assume i- = 0.10 
 
 d 
 
 From formula on Diagram 12, 
 
 /. =- 
 
 Assume j = 0.87. Then d = 17.95 
 
 Adopting d = 18 in. p = ^^^HIS) ^ ^'^^^^ 
 
 Diagram 12 for ^ =0.10 and p = = 0.0139 shows j = 0.886 
 Assume j = 0.886. Then d = 17.6 in. 
 Adopting d= 17M in. p = jYfyifJK) = ^"^^^^ 
 
 . _ M M 75,000 _ 
 
 - Asjd Asfs ~ (3) (16,000) - ^^-^ 
 
348 
 
 CONCRETE ENGINEERS' HANDBOOK 
 
 [Sec. 7-56 
 
 Diagram 12 shows j ~ 0.886 
 
 d' 
 
 Thus d = 17§^ in is satisfactory provided ^ is approximately 0.10, 
 
 12. In a double-reinforced rectangular beam b = 12 m., d! = 18 in., = 0.10, M = 750,000, 
 
 fc — 650 and fs = 16,000. Determine the required percentages of tensile and compressive steel. 
 Following the method outlined in Art. 27a, we have, using Table 2, 
 
 k = 0.378, Pi = 0.0077, and K = 107.4 
 Ml = 107.4 (12) (18)2 = 418,000 in.-lb. 
 M2 = 750,000 - 418,000 = 332,000 in.-lb. 
 
 332,000 nnnr^n 
 
 = 16,000 (0.9)(12)(18)2 = ^-^^'^ 
 P = Pi + P's = 0.0136 
 
 7>' 
 
 = (0-0059) (oW'^) = 
 
 56. Leffler*s Comprehensive Beam Chart. ^ Simple Rectangular Beams. — The procedure 
 to design a beam simply reinforced to carry a certain bending moment M with certain allowable 
 stresses is as follows: Divide the allowable fs by the allowable fc thus obtaining the value of 
 
 fs . . . fs- 
 
 T' Find this value on the chart (Diagram 13) on the curve . • From it move vertically to the 
 
 Jc Jc 
 
 p' = 0 curve. This point on the p' = 0 curve lies where some p curve intersects the p' = 0 
 curve. The value of this p curve gives the p to use. The abscissa of this point on the p' 
 
 M 
 
 = 0 curve gives the value of L., to be used in Ls = Equation (I) given on the diagram 
 
 can then be solved for bd"^ which completes the solution for moment. To b and d such values 
 can then be assigned as shear requirements may demand. If shear requirements are so large 
 as to govern the dimensions of the beam, the design for moment is accomplished by solving (I) 
 for Ls, fs being used at its allowable value. The p to use is that of the p curve which intersects 
 the = 0 curve at the point whose abscissa is Ls. 
 
 Doubly Reinforced Rectangular Beams. — Suppose the dimensions b and of a beam to 
 carry a certain M, with certain allowable stresses, are arbitrarily fixed by such limitations as 
 headroom, clearance, architectural effects, shear requirements, etc., the procedure would then 
 be as follows: Solve equations (I) and (II) for Ls and Lc. Using Ls as an abscissa and Lc as 
 an ordinate, plot a point on the chart. If the point falls below or on the p' = 0 curve, the beam 
 can be designed as a simple beam; the value of p to use being that of the p curve which intersects 
 the p' = 0 curve at the point whose abscissa is Ls. If the point falls well above the p^ = 0 curve, 
 the beam must be a doubly reinforced one. The values of p and p' to use are those of the p 
 and p' curves on which the point falls. If the point falls above the p' = 0 curve and beyond the 
 scope of the chart, the beam is probably impossible as a reinforced concrete beam and recourse 
 must be had to structural steel. 
 
 If the point falls but a short distance above the p' = 0 curve, it is possible to design the 
 beam as a simply reinforced one. The procedure is as follows: Proceed horizontally from the 
 point until the p' = 0 curve is intersected. The value of the p carve that meets the p' = 0 
 curve at this point of intersection gives the value of p to use for designing the beam with ten- 
 sile reinforcement only. The actual fc will be the same as the allowable fc, but the actual/., 
 will be less than the allowable fs. Carefully made cost figures are the only means of determining 
 whether this simply reinforced beam will be cheaper than a doubly reinforced one. 
 
 To find M when p, p', b, d, the allowable fs, and the allowable fc are given, proceed as 
 follows: Read off the abscissa Ls of the intersection of the p and p' curves, then evaluate equa- 
 
 1 By Ralph R. Lepft>er, Chicaeo, 111. 
 
Sec. 7-56] 
 
 BEAMS AND SLABS 
 
 349 
 
 tion (I) for M; read off the ordinate Lc of the intersection of the ]) and p' curves and evakiate 
 equation (II) for M. Use the smaller value. 
 
 T-beams. — A T-beam can be regarded as a large rectangular beam minus two rectangular 
 beams, or, if we combine the two small beams, as a large rectangular beam minus a smaller 
 rectangular beam. Referring to the beam-section on the chart, ABCD is the large rectangular 
 beam and the empty rectangular spaces under the flanges of the T-beam added together form 
 the smaller rectangular beam. 
 
 Let M, the allowable and the allowable fc be given. First, design a simple rectangular 
 beam that will carry the given M with the given allowable /, and fc. Having found hcP, assign 
 to 6 a value in accordance with some rules, such as those of the Joint Committee, and then solve 
 for d. Ascertain the amount of tensile steel reinforcement required and make the width of the 
 stem such that it can be properly arranged. We now have the approximate dimensions of the 
 T-beam. At this point it is well to investigate the shear in different sections of the beam. As- 
 suming that we have found that the shear is sufficiently well taken care of, we can proceed to 
 finish the design for moment. 
 
 The expression for the value for k can ])e written as a function of "j (assuming n = 15). 
 This means that for every value of y there will be found on the same vertical line a point on the 
 
 k curve (or straight line) whose ordinate gives the value of k corresponding to the value of 
 
 Having found the value of A;, locate the neutral axis of the T-beam. If kd is less than, 
 equal to, or ver}^ nearly equal to / (the thickness of the flange usually determined by the floor 
 slab design) the neutral axis lies in the flange, at the bottom of the flange or but a short distance 
 below the flange, in which case the design of our T-beam for moment is already complete. If 
 kd is much greater than t, the neutral axis lies well down in the web. To secure a correct theo- 
 retical solution it then is necessary to ascertain how much M is carried by the two imaginary 
 beams under the flanges. At this point a little thought discloses that since these imaginary 
 beams cause a loss of resisting ilf , it is necessary to deepen d. Having depeened d according 
 to our best judgment, the next thing is to find by the formula M = LJ)d~fs how much resisting 
 M the deepened beam is capable of sustaining at the allowable unit stresses, the beam being 
 considered as a simple rectangular beam, of width b, and depth d, equal to the deepened d. 
 The p to use is that of the p curve which intersects the p' = 0 curve at the point whose abscissa 
 is Ls. Again the neutral axis is located by "the same process as before, and as before, if the 
 neutral axis is in the flange, in the lower edge of the flange or very close up to the bottom of the 
 flange the design for moment is complete. If the neutral axis comes well down in the web it is 
 necessary to ascertain how much resisting moment is carried by the imaginary beams under the 
 
 kd — t 
 
 flanges. From the stress diagram, A'l = — ' .the svmbols for the imaginary beam being 
 
 distinguished by the subscript 1. The Lsi corresponding to this ki is the abscissa of the ki 
 on the chart. Solve the equation Mi = Lsibidrfs in which bi is the combined width of the two 
 imaginary beams and di their depth. This Mi is the resisting moment carried by the two im- 
 aginary beams. The pi to use is that of the p curve which intersects the p' = 0 curve at the 
 point whose abscissa is L,i. If to Mi we add the M caused by the loading and thus obtain a 
 sum equal to the resisting M of the enlarged beam, our solution for moment is complete. If 
 their sum is considerably different from it, d should be increased or decreased until they are 
 closely equal, the same procedure being gone through as before. The amount of tensile steel 
 needed for the T-beam is equal to pbd — pibidi. If ki should be so small as not to come within 
 the scope of the chart. Mi can be obtained from the equation 
 
 fcikibidi/^ kidA fcikibjidi^ fckidi ■ , , Mi 
 
 Ml =^—2- (^^ - -J-) 2^mwhichA.. Also A.i = pMi 
 
 in which is the same a« in the large beam ABCD for the of the large beam is the same as 
 the fs of the smaller imaginarj^ beam. 
 
350 
 
 CONCRETE ENGINEERS' HANDBOOK 
 
 [Sec. 7-57 
 
 Note that this solution of a T-beam does not neglect the stress in the stem. 
 
 Doubly Reinforced T-beams. — After the reader has thoroughly grasped the foregoing solu- 
 tions of the "Doubly Reinforced Rectangular Beams" and the "Simply Reinforced T-beam" 
 he will find it easy to devise a method, parallel to that given for simple T-beams, for designing 
 doubly reinforced T-beams; it being only needful to remember that the resisting M of the larger 
 
 fs 
 
 beam is that of a doubly reinforced beam and that is a function of j only, and as such is pri- 
 
 Jc 
 
 marily independent of the amount or location of the compressive steel reinforcement. It is 
 j that is primarily dependent on the amount and location of the compressive steel reinforcement. 
 
 57. Beard and Schuler's Comprehensive Charts. ^ Rectangular Beams and Floor Slabs. — 
 The moment caused by a uniformly distributed load at any point on any beam which is fixed, 
 partially restrained, or simply supported at the ends may be expressed in inch-pounds by the 
 w . . 
 
 formula, M = 12 — in which w is the uniform load in pounds per foot, I is the span in feet, and 
 <p 
 
 (f> is the moment denominator. ^ is 8 for the moment at the center of a simply supported 
 beam. 
 
 The formula for the resisting moment of a simple rectangular reinforced-concrete beam 
 is M = Kbd"^ in which K —l^^ fckj for the compression couple and fspj for the tension couple. 
 When the beam is supporting a uniformly distributed load the general formula may be expanded 
 into the two forms 
 
 12'^ f ^ fsivbd^ 
 <p 
 
 and 
 
 i2'{f = yijc-kjbd-' 
 
 In the upper left-hand quadrant of Diagram 14, which is called the moment chart, the 
 logarithmic abscissas and ordinates represent the spans in feet and moments in inch-pounds 
 
 w w 
 
 respectively. Each sloping line on the diagram represents a particular value of - • With- a 
 
 • <p <p 
 
 constant, the formula takes the form M = Cp and, when expressed in logarithms, the form log 
 M = log C + 2 log j. The moment chart is a graphical representation of this family of curves, 
 which are parallel straight lines. 
 
 The upper right-hand quadrant, or the depth chart, is the logarithmic plat of the equation 
 M = Kbd^. In this case also the logarithmic ordinates represent the moments but the loga- 
 rithmic abscissas represent values of K. Each sloping line represents a particular value of bd"^. 
 b is taken as 12 in. in all cases and the line is designated by the corresponding value of d. 
 
 This plat then represents the general equation 
 
 log M = log C + log A' 
 
 It is a series of parallel 45-deg. lines. 
 
 The lower right-hand quadrant or the stress chart is a logarithmic plat of the two families 
 of curves: 
 
 K = f4>j and K = Mfckj 
 
 In the stress chart the logarithmic abscissas and ordinates represent values of K and p 
 respectively, and the sloping lines represent particular values of fc and fs. 
 
 As = pbd. In the lower left-hand quadrant or the steel chart, the lines sloping upward 
 to the right represent constant values of d and the abscissas and the ordinates, as numbered 
 at the right-hand side of the diagram, represent values of As and p respectively. 
 
 \2<x 
 
 When the cross-sectional area of a round rod is a and the rod spacing is s, = — ^ • Each 
 1 By Robert S. Beard and Don B. Schuler. 
 
Sec. 7-57] 
 
 BEAMS AND SLABS 
 
 351 
 
 line sloping upward to the left in the steel chart is marked with the diameter of the rod whose 
 area it represents. The abs3issas represent the steel area and the ordinates, as marked at 
 the left side of the diagram, represent the round-rod spacing. The spacing for square rods is 
 4 
 
 - times the spacing of round rods of the same thickness. The k and j curves are also platted 
 in this steel chart. 
 
 Illustrative Problem. — Suppose that it is required to design a slab to carry a total live 
 and dead load of 400 lb. per sq. ft. over a simple span of 20 ft. with limiting stresses of fc = 050 
 and/s = 16,000 1b. 
 
 Before entering Diagram 14 the load per square foot, 400 lb., must be divided by the 
 moment denominator which is 8 in this case. 
 
 400 
 8 
 
 50 
 
 In the stress chart, the intersection of the values fc = 650 and fs = 16,000 gives a value 
 oi K = 107. Any designer who uses a particular set of stresses constantly remembers the 
 corresponding value of K, and omits this operation. 
 
 w 
 
 Now in the moment chart find the intersection of the sloping line — = 50 with the line 
 
 representing a span of 20 ft., as indicated in Fig. 75. The ordinate of this intersection corre- 
 sponds to a moment of 240,000 in. -lb. Follow this ordinate into 
 the depth chart to its intersection with K = 107. At this intersec- 
 tion d = 13.6 in. It is decided to use a depth of 14 in. which cor- 
 responds to K = 102 for this moment. Follow the line K = 102 
 into the stress chart. At the point where fc = 650, fs = 17,600, 
 at the point where fs = 16,000, fo = 630. This second set are there- 
 fore the limiting working stresses. At this point p = 0.0073. Follow 
 this abscissa into the steel chart to = 14 in. At this intersection 
 As = 1.23 sq. in. Follow this As line to the 1-in. line. The required 
 spacing for 1-in. round rods is 7.7 in. Use a spacing of 7}^ in. If 
 
 . . 4 
 
 it is required to use 1-in. square rods, the necessary spacing is - X 
 
 7.7 = 9.8 in. Use 9^:4-in. spacing. 
 
 If it is desired to find the resisting moment when every other 
 rod is turned up, this process is reversed. In the steel chart follow 
 
 the rod spacing of 15 in. to the 1-in. line. As = 0.626 sq. in. Follow this As line lo d = 
 14 in., p = 0.00372. Follow this p line to fs = 16,000, fc = 430, K = 54. Follow the K = 54 
 line to d =14 in', in the depth chart, M = 128,000 in. -lb. 
 
 T-heams. — The moment caused by a uniformly distributed load at any point on any beam 
 
 may be expressed in inch -pounds by the formula, M = ^' 1-. 
 
 Fig. 75. 
 
 The resisting moment of the steel reinforcement in 
 
 T-beam is Rbt^ where R = and 
 
 A = 
 
 ^- Then when the T-beam is supporting a uniformly distributed load 
 
 M = 12- f= Rbt^ 
 
 0 
 
 In order to make Diagram 15 of more general application the moment is divided by b, the 
 breadth of the beam in inches. The moment formula then takes the form 
 
 M 
 
 'b 
 
 I2w 
 
352 
 
 CONCRETE ENGINEERS' HANDBOOK 
 
 [Sec. 7-57 
 
 B is used to express the width of the beam in feet, then expresses the uniformly distributed 
 
 load in terms of live load per square foot of flange. 
 
 In the upper left-hand quadrant of Diagram 15 the logarithmic abscissas and ordinatei 
 represent the spans in feet and moments in inch-pounds divided by the breadth of beam i 
 
 w . w 
 
 inches respectively. Each sloping line represents a particular value of With — ^ a 
 
 M 
 
 stant, the formula takes the form =Cp. The moment chart is a logarithmic graphica. 
 
 representation of this family of curves. 
 
 The upper right-hand quadrant or the slab chart is the logarithmic plat of the equation 
 M , . _ M 
 
 Rt^. The abscissas represent values of R, and the ordinates, values of 
 
 The sloping 
 
 lines correspond to particular values of t. 
 
 The lower right-hand quadrant or steel chart is a logarithmic plat of the equation R = fs-^^^' 
 
 VJ 
 
 In this steel chart the abscissas and ordinates represent values of R and ^ or y respectively. 
 
 The sloping lines represent particular values of /s. 
 
 In the lower left-hand quadrant or proportional chart, the lines sloping upward to the 
 
 right are the logarithmic plat of the family of curves y = 
 
 V3 
 
 The abscissas and ordinates of 
 
 the proportional chart represent values of p and y respectively, and 
 the sloping straight lines represent particular values of A. 
 
 The curved lines on the proportional chart have been platted 
 from this formula by solving for the values of p corresponding to 
 
 fc 
 
 particular values of 
 
 fs 
 
 e and A, and then drawing the d curves 
 
 through the intersections of these values of p with the correspond- 
 ing A lines in the proportional chart. The curve A = /c is the line 
 of division between the T-beam and the simple beam. 
 
 Illustrative Problem. — Design a simply supported T-beani 
 with the following factors predetermined: I = 40ft., w = 40001b. 
 per lin. ft., fc = 600, /, = 15,000, t = 8 in., and b = QA in. 
 
 Before entering Diagram 15, the total load per linear foot of 
 4000 lb. must be divided by both the moment denominator, 8, and 
 the breadth of the beam in feet, 5}i 
 
 IV 4000 ^„ „^ „ 600 
 
 Fig. 76. 
 
 <f>B 8 X SVs 
 
 157000 
 
 0.04 
 
 Now perform the operations on the T-beam chart indicated in Fig. 76. At the inter- 
 im M 
 section of^ = 93.75 with the line I = 40 ft., the ordinate is = 150,000 at the intersection 
 
 of this ordinate with / = 8 in., R = 2350. At the intersection of the abscissas, R = 2350 
 with fs = 15,000, y = 0.157. Now follow the ordinate y = 0.157 to its intersection with the 
 line d = 0.04. At this point A = 0.179 and p = 0.00548. 
 
 '^ = 1 =0.179= 
 A, = pbd = 0.00548 X 64 X 44.7 = 15.67 sq. in. 
 
 The T-beam chart is worked in the reverse direction when it is desired to know what 
 resisting moment a given section can exert. Buppose that the bending up of four rods has 
 reduced the steel ratio to 0.00358. 
 
 i 
 
Sec. 7-57] 
 
 BEAMS AND SLABS 
 
 353 
 
 In the proportional chart follow the abscissa, p = 0.00358 to its intersection with the 
 sloping line A = 0.1791. At this point 6 = 0.00285 and y = 0.103. 
 
 If /c = 600, 0=5^? = 0.00285, /. - 21,100 lb. 
 
 J s 
 
 lifs = 15,000, d = = 0.00285, fc = 428 lb. 
 
 This second group are the working stresses. Now follow the ordinate y = 0.103 to its inter- 
 section with the value fc = 15,000. At this point R = 1540. Trace this abscissa, R = 1540, 
 
 M 
 
 to its intersection with the line i = 8 in. At this point = 98,000. Then M = 98,000 X 64 = 
 6,270,000 in.-lb. 
 
 The handling of problems on both Diagrams 14 and 15 is facilitated by the use of two 
 pointers. The last value found is held by one pointer while the other is used to pick out 
 the value determined by the next step in the problem. 
 
 23 
 
354 
 
 CONCRETE ENGINEERS' HANDBOOK 
 
 [Sec. 7-57 
 
 Table 1. — Areas, Perimeters, and Weights of Rods 
 
 Round rods 
 
 , Square rods 
 
 
 
 Area 
 
 Perimeter 
 
 Weight per 
 
 
 Area 
 
 Perimeter 
 
 Weight per 
 
 Siz6 (incliGs) 
 
 (square inches) 
 
 (inches) 
 
 foot (pounds) 
 
 (square inches) 
 
 (inches) 
 
 foot (pounds) 
 
 /4 
 
 n 
 u 
 
 r»/i 01 
 
 u 
 
 / OO 
 
 U 
 
 1 7 
 
 n 
 u 
 
 UDZO 
 
 i 
 
 uu 
 
 0.21 
 
 He 
 
 U 
 
 
 U 
 
 
 U 
 
 OR 
 ZD 
 
 u 
 
 nQ77 
 
 1 
 
 ZO 
 
 0.33 
 
 % 
 
 0 
 
 1104 
 
 1 
 
 178 
 
 0 
 
 38 
 
 0 
 
 1406 
 
 1 
 
 50 
 
 0.48 
 
 
 0 
 
 1503 
 
 1 
 
 374 
 
 0 
 
 51 
 
 0 
 
 1914 
 
 1 
 
 75 
 
 0.65 
 
 /2 
 
 U 
 
 1 OAQ 
 
 lyoo 
 
 i 
 
 0/1 
 
 U 
 
 fl7 
 
 U 
 
 zouu 
 
 Z 
 
 uu 
 
 0. 85 
 
 
 U 
 
 Z4oO 
 
 i 
 
 / D/ 
 
 u 
 
 QK 
 oO 
 
 U 
 
 o lD-± 
 
 Z 
 
 ZO 
 
 1.08 
 
 H 
 
 0 
 
 3068 
 
 1 
 
 964 
 
 1 
 
 04 
 
 0 
 
 3906 
 
 2 
 
 50 
 
 1.33 
 
 
 0 
 
 3712 
 
 2 
 
 160 
 
 1 
 
 26 
 
 0 
 
 4727 
 
 2 
 
 75 
 
 1.61 
 
 /4 
 
 U 
 
 i i 1 Q 
 441o 
 
 o 
 Z 
 
 Q KA 
 
 1 
 i 
 
 KH 
 
 ou 
 
 U 
 
 OOZO 
 
 Q 
 
 o 
 
 UU 
 
 1 .91 
 
 
 U 
 
 KICK 
 
 o 
 
 z 
 
 OOO 
 
 1 
 
 7A 
 
 0 
 
 DOUZ 
 
 6 
 
 Zo 
 
 2.25 
 
 Vs 
 
 0 
 
 6013 
 
 2 
 
 749 
 
 2 
 
 04 
 
 0 
 
 7656 
 
 3 
 
 50 
 
 2.60 
 
 
 0 
 
 6903 
 
 2 
 
 945 
 
 2 
 
 35 
 
 0 
 
 8789 
 
 3 
 
 75 
 
 2.99 
 
 
 0 
 
 7854 
 
 3 
 
 142 
 
 2 
 
 67 
 
 1 
 
 0000 
 
 4 
 
 00 
 
 ^ 40 
 
 
 0 
 
 9940 
 
 3 
 
 534 
 
 3 
 
 38 
 
 1 
 
 2656 
 
 4 
 
 50 
 
 4.30 
 
 iM 
 
 1 
 
 2272 
 
 3 
 
 927 
 
 4 
 
 17 
 
 1 
 
 5625 
 
 5 
 
 00 
 
 5.31 
 
 
 1 
 
 4849 
 
 4 
 
 320 
 
 5 
 
 05 
 
 1 
 
 8906 
 
 5 
 
 50 
 
 6.43 
 
 
 1 
 
 7671 
 
 4 
 
 712 
 
 6 
 
 01 
 
 2 
 
 2500 
 
 6 
 
 00 
 
 7.65 
 
 \% 
 
 2 
 
 0739 
 
 5 
 
 105 
 
 7 
 
 05 
 
 2 
 
 6406 
 
 6 
 
 50 
 
 9.98 
 
 1% 
 
 2 
 
 4053 
 
 5 
 
 498 
 
 8 
 
 18 
 
 3 
 
 0625 
 
 .7 
 
 00 
 
 10.41 
 
 m 
 
 2 
 
 7612 
 
 5 
 
 891 
 
 9 
 
 39 
 
 3 
 
 5156 
 
 7 
 
 50 
 
 11.95 
 
 2 
 
 3 
 
 1416 
 
 6 
 
 283 
 
 10 
 
 68 
 
 4 
 
 0000 
 
 8 
 
 00 
 
 13.60 
 
 
 3 
 
 9761 
 
 7 
 
 069 
 
 13 
 
 52 
 
 5 
 
 0625 
 
 9 
 
 00 
 
 17.22 
 
 
 4 
 
 9087 
 
 7 
 
 854 
 
 16 
 
 69 
 
 6 
 
 2500 
 
 10 
 
 00 
 
 21.25 
 
 2M 
 
 5 
 
 9396 
 
 8 
 
 639 
 
 20 
 
 20 
 
 7 
 
 5625 
 
 11 
 
 00 
 
 25.72 
 
 3 
 
 , 7 
 
 0686 
 
 9 
 
 425 
 
 24 
 
 03 
 
 9 
 
 0000 
 
 12 
 
 00 
 
 30.09 
 
BEAMS AND SLABS 355 
 
 Table 2. — Use for Rectangular Beams and Slabs 
 
 k \'2 fkj 
 i = 1 — Q' V J^TT \ ' ^ = Vf^jj or-|r- (from Formula M = Khd'^) 
 
 Ratio of Moduli n = 12 
 
 
 fc 
 
 A; 
 
 j 
 
 P 
 
 K 
 
 fs 
 
 fc 
 
 k 
 
 j 
 
 P 
 
 K 
 
 
 500 
 
 0 
 
 332 
 
 0 
 
 889 
 
 u . uuoy 
 
 73 
 
 6 
 
 
 500 
 
 0 
 
 273 
 
 0 
 
 909 
 
 0 
 
 0043 
 
 62 
 
 0 
 
 
 550 
 
 0 
 
 354 
 
 0 
 
 882 
 
 n nnci 
 
 U . UUOi 
 
 85 
 
 7 
 
 
 oou 
 
 0 
 
 292 
 
 0 
 
 903 
 
 0 
 
 0050 
 
 72 
 
 2 
 
 
 600 
 
 0.375 
 
 0 
 
 875 
 
 u . uuyt 
 
 98 
 
 4 
 
 
 Ann 
 uuu 
 
 0 
 
 310 
 
 0 
 
 897 
 
 0 
 
 0058 
 
 83 
 
 2 
 
 
 650 
 
 0 
 
 394 
 
 0 
 
 869 
 
 U . UiU/ 
 
 111 
 
 3 
 
 
 650 
 
 0 
 
 328 
 
 0 
 
 891 
 
 0 
 
 0067 
 
 95 
 
 0 
 
 12,000 
 
 
 
 
 
 
 
 
 
 16,000 
 
 
 
 
 
 
 
 
 
 
 
 Tnri 
 /UU 
 
 0 
 
 412 
 
 0 
 
 863 
 
 u . yjizu 
 
 124 
 
 4 
 
 
 •ynn 
 / uu 
 
 0 
 
 344 
 
 0 
 
 885 
 
 0 
 
 0075 
 
 106 
 
 2 
 
 
 800 
 
 0 
 
 444 
 
 0 
 
 852 
 
 0.0148 
 
 151 
 
 3 
 
 
 800 
 
 0 
 
 375 
 
 0 
 
 875 
 
 0 
 
 0094 
 
 131 
 
 3 
 
 
 900 
 
 0 
 
 475 
 
 0 
 
 842 
 
 0.0177 
 
 178 
 
 8 
 
 
 900 
 
 0 
 
 403 
 
 0 
 
 866 
 
 0 
 
 0113 
 
 156 
 
 5 
 
 
 500 
 
 0.300 
 
 0 
 
 900 
 
 0.0054 
 
 67 
 
 5 
 
 
 500 
 
 0 
 
 261 
 
 0 
 
 917 
 
 0 
 
 0038 
 
 59 
 
 2 
 
 
 550 
 
 0 
 
 320 
 
 0 
 
 893 
 
 0.0063 
 
 78 
 
 6 
 
 
 550 
 
 0 
 
 280 
 
 0 
 
 907 
 
 0 
 
 0045 
 
 69 
 
 4 
 
 
 600 
 
 0 
 
 340 
 
 0 
 
 888 
 
 0 . 0073 
 
 90 
 
 6 
 
 
 600 
 
 0 
 
 298 
 
 0 
 
 901 
 
 0 
 
 0052 
 
 79 
 
 6 
 
 
 650 
 
 0 
 
 358 
 
 0 
 
 881 
 
 0.0083 
 
 102 
 
 5 
 
 
 650 
 
 0 
 
 314 
 
 0 
 
 895 
 
 0 
 
 0060 
 
 91 
 
 3 
 
 14,000 
 
 
 
 
 
 
 
 
 
 17,000 
 
 
 
 
 
 
 
 
 
 
 
 700 
 
 0 
 
 375 
 
 0 
 
 875 
 
 0.0094 
 
 114 
 
 8 
 
 
 700 
 
 0 
 
 331 
 
 0 
 
 890 
 
 0 
 
 0068 
 
 102 
 
 9 
 
 
 800 
 
 0 
 
 407 
 
 0 
 
 864 
 
 0.0116 
 
 140 
 
 4 
 
 
 800 
 
 0 
 
 361 
 
 0 
 
 880 
 
 0 
 
 0085 
 
 127 
 
 1 
 
 
 900 
 
 0 
 
 435 
 
 0 
 
 855 
 
 0.0140 
 
 167 
 
 5 
 
 
 900 
 
 0 
 
 390 
 
 0 
 
 870 
 
 0 
 
 0103 
 
 152 
 
 2 
 
 
 500 
 
 0 
 
 286 
 
 0 
 
 905 
 
 0.0048 
 
 64 
 
 7 
 
 
 500 
 
 0 
 
 230 
 
 0 
 
 923 
 
 0 
 
 0029 
 
 53 
 
 1 
 
 
 550 
 
 0 
 
 306 
 
 0 
 
 898 
 
 0.0056 
 
 75 
 
 4 
 
 
 550 
 
 0 
 
 248 
 
 0 
 
 917 
 
 0 
 
 0034 
 
 62 
 
 4 
 
 
 600 
 
 0 
 
 325 
 
 0 
 
 892 
 
 0.0065 
 
 86 
 
 7 
 
 
 600 
 
 0 
 
 264 
 
 0 
 
 912 
 
 0 
 
 0040 
 
 72 
 
 2 
 
 
 650 
 
 0 
 
 343 
 
 0 
 
 886 
 
 0.0074 
 
 98 
 
 4 
 
 
 650 
 
 0 
 
 280 
 
 0 
 
 907 
 
 0 
 
 0046 
 
 82 
 
 4 
 
 1.5,000 
 
 
 
 
 
 
 
 
 
 20,000 
 
 
 
 
 
 
 
 
 
 
 700 
 
 0 
 
 360 
 
 0 
 
 880 
 
 0.0084 
 
 110 
 
 3 
 
 
 700 
 
 0 
 
 295 
 
 0 
 
 902 
 
 0 
 
 0052 
 
 93 
 
 3 
 
 
 800 
 
 0 
 
 391 
 
 0 
 
 870 
 
 0.0105 
 
 135 
 
 7 
 
 
 800 
 
 0 
 
 324 
 
 0 
 
 892 
 
 0 
 
 0065 
 
 115 
 
 6 
 
 
 900 
 
 0 
 
 418 
 
 0 
 
 861 
 
 0.0125 
 
 161 
 
 5 
 
 
 900 
 
 0 
 
 351 
 
 0 
 
 883 
 
 0.0079 
 
 139 
 
 5 
 
 
 
 
 
 
 Ratio of Moduli n = 
 
 = 15 
 
 
 
 
 
 
 
 
 
 /. 
 
 fc 
 
 fc 
 
 3 
 
 p 
 
 1 
 
 /. 
 
 fc 
 
 it 
 
 J 
 
 V 
 
 K 
 
 
 500 
 
 0 
 
 384 
 
 0 
 
 872 
 
 0.0080 
 
 83 
 
 7 
 
 
 500 
 
 0 
 
 319 
 
 0 
 
 894 
 
 0 
 
 0050 
 
 71 
 
 3 
 
 
 550 
 
 0 
 
 407 
 
 0 
 
 864 
 
 0 . 0093 
 
 96 
 
 4 
 
 
 550 
 
 0 
 
 339 
 
 0 
 
 887 
 
 0 
 
 0058 
 
 82 
 
 3 
 
 
 GOO 
 
 0 
 
 428 
 
 0 
 
 857 
 
 0.0107 
 
 110 
 
 0 
 
 
 600 
 
 0.358 
 
 0 
 
 881 
 
 0 
 
 0067 
 
 94 
 
 4 
 
 12,000 
 
 
 
 
 
 
 
 
 
 16,000 
 
 
 
 
 
 
 
 
 
 
 
 650 
 
 0 
 
 448 
 
 0 
 
 851 
 
 0.0121 
 
 123 
 
 6 
 
 
 650 
 
 0 
 
 378 
 
 0 
 
 874 
 
 0 
 
 0077 
 
 107 
 
 4 
 
 
 700 
 
 0 
 
 467 
 
 0 
 
 844 
 
 0.0136 
 
 138 
 
 0 
 
 
 700 
 
 0 
 
 397 
 
 0 
 
 868 
 
 0 
 
 0087 
 
 120 
 
 6 
 
 
 750 
 
 0 
 
 484 
 
 0 
 
 839 
 
 0.0151 
 
 152 
 
 0 
 
 
 750 
 
 0 
 
 414 
 
 0 
 
 862 
 
 0 
 
 0097 
 
 133 
 
 8 
 
 
 800 
 
 0 
 
 501 
 
 0 
 
 833 
 
 0.0167 
 
 166 
 
 9 
 
 
 800 
 
 0 
 
 429 
 
 0 
 
 857 
 
 0 
 
 0107 
 
 146 
 
 7 
 
 
 500 
 
 0 
 
 348 
 
 0 
 
 884 
 
 0 . 0062 
 
 76 
 
 7 
 
 
 500 
 
 0 
 
 306 
 
 0 
 
 898 
 
 0 
 
 0045 
 
 68 
 
 5 
 
 
 550 
 
 0 
 
 372 
 
 0 
 
 876 
 
 0.0073 
 
 89 
 
 5 
 
 
 550 
 
 0 
 
 326 
 
 0 
 
 892 
 
 0 
 
 0053 
 
 80 
 
 4 
 
 
 600 
 
 0 
 
 391 
 
 0 
 
 870 
 
 0 . 0084 
 
 102 
 
 0 
 
 
 600 
 
 0 
 
 346 
 
 0 
 
 885 
 
 0 
 
 0061 
 
 91 
 
 8 
 
 14 000 
 
 
 
 
 
 
 
 
 
 17,000 
 
 
 
 
 
 
 
 
 
 
 
 650 
 
 0 
 
 410 
 
 0 
 
 863 
 
 0 . 0095 
 
 114 
 
 8 
 
 
 650 
 
 0 
 
 365 
 
 0 
 
 878 
 
 0 
 
 0070 
 
 103 
 
 5 
 
 
 700 
 
 0 
 
 428 
 
 0 
 
 857 
 
 0.0107 
 
 128 
 
 3 
 
 
 700 
 
 0 
 
 382 
 
 0 
 
 873 
 
 0 
 
 0079 
 
 116 
 
 1 
 
 
 750 
 
 0 
 
 446 
 
 0 
 
 851 
 
 0.0120 
 
 142 
 
 3 
 
 
 750 
 
 0 
 
 398 
 
 0 
 
 866 
 
 0 
 
 0088 
 
 129 
 
 5 
 
 
 800 
 
 0 
 
 462 
 
 0 
 
 846 
 
 0.0132 
 
 156 
 
 3 
 
 
 800 
 
 0 
 
 415 
 
 0 
 
 862 
 
 0 
 
 0097 
 
 141 
 
 9 
 
 
 500 
 
 0 
 
 334 
 
 0 
 
 889 
 
 0 . 0056 
 
 74 
 
 1 
 
 
 500 
 
 0 
 
 272 
 
 0 
 
 909 
 
 0 
 
 0034 
 
 61 
 
 8 
 
 
 550 
 
 0 
 
 355 
 
 0 
 
 882 
 
 0.0065 
 
 86 
 
 1 
 
 
 550 
 
 0 
 
 292 
 
 0 
 
 903 
 
 0 
 
 0040 
 
 72 
 
 2 
 
 
 600 
 
 0 
 
 375 
 
 0 
 
 875 
 
 0.0075 
 
 98 
 
 3 
 
 
 600 
 
 0 
 
 311 
 
 0 
 
 897 
 
 0 
 
 0047 
 
 83 
 
 7 
 
 15,000 
 
 
 
 
 
 
 
 
 
 20,000 
 
 
 
 
 
 
 
 
 
 
 
 650 
 
 0 
 
 393 
 
 0 
 
 869 
 
 0.0085 
 
 111 
 
 3 
 
 
 650 
 
 0 
 
 328 
 
 0 
 
 891 
 
 0 
 
 0053 
 
 94 
 
 4 
 
 
 700 
 
 0 
 
 411 
 
 0 
 
 863 
 
 0.0096 
 
 124 
 
 2 
 
 
 700 
 
 0 
 
 344 
 
 0 
 
 885 
 
 0 
 
 0060 
 
 106 
 
 2 
 
 
 750 
 
 0 
 
 429 
 
 0 
 
 857 
 
 0.0107 
 
 137 
 
 9 
 
 
 750 
 
 0 
 
 359 
 
 0 
 
 880 
 
 0.0067 
 
 117 
 
 9 
 
 
 800 
 
 0 
 
 445 
 
 0 
 
 852 
 
 0.0118 
 
 151 
 
 2 
 
 
 800 
 
 0 
 
 374 
 
 0 
 
 875 
 
 0.0075 
 
 130 
 
 9 
 
 Sec. 7-57 
 
 k = 
 
 1 + 
 
 nfc 
 
356 CONCRETE ENGINEERS' HANDBOOK [Sec. 7-57 
 
 Table 3. — Values of k and j for Rectangular Beams and Slabs 
 k = \/ 2pn + ipn)^ — pn j = I — H k 
 
 V 
 
 
 n = 
 
 12 
 
 
 
 n = 
 
 15 
 
 
 
 
 n = 
 
 = 12 
 
 71 = 
 
 = 15 
 
 
 A; 
 
 j 
 
 
 
 
 3 
 
 k 
 
 i 
 
 k 
 
 j 
 
 n 
 u 
 
 UUlw 
 
 0 
 
 
 0 
 u 
 
 
 0 
 
 158 
 
 0 
 
 947 
 
 0 
 u 
 
 OOQO 
 
 0 
 
 ■^70 
 
 0.877 
 
 0.402 
 
 0 
 u 
 
 OOO 
 
 n 
 u 
 
 
 u 
 
 lOO 
 
 0 
 U 
 
 
 0 
 
 169 
 
 0 
 
 944 
 
 u 
 
 00Q9 
 uuyz 
 
 0 
 u 
 
 Q7Q 
 o < o 
 
 0.876 
 
 0.405 
 
 0 
 u 
 
 oOO 
 
 u 
 
 (\(\^A 
 
 UU11 
 
 0 
 u 
 
 iOD 
 
 0 
 U 
 
 
 0 
 
 181 
 
 0 
 
 940 
 
 0 
 u 
 
 OOQ/t 
 uuyi 
 
 0 
 u 
 
 O / o 
 
 0.875 
 
 0.407 
 
 0 
 u 
 
 004: 
 
 u 
 
 
 0 
 u 
 
 1 77 
 
 0 
 u 
 
 
 0 
 
 192 
 
 0 
 
 936 
 
 0 
 u 
 
 
 0 
 
 O 1 i7 
 
 0.874 
 
 0.411 
 
 0 
 u 
 
 OOO 
 
 0 
 
 0018 
 
 0 
 
 186 
 
 0 
 
 938 
 
 0 
 
 202 
 
 0 
 
 933 
 
 0 
 
 0098 
 
 0 
 
 381 
 
 0.873 
 
 0.414 
 
 0 
 
 862 
 
 u 
 
 uuzu 
 
 u 
 
 1 QA 
 
 lyo 
 
 u 
 
 yoo 
 
 0 
 
 217 
 
 0 
 
 928 
 
 u 
 
 01 00 
 UlUU 
 
 0 
 u 
 
 OoO 
 
 0.872 
 
 0.418 
 
 0 
 u 
 
 OOl 
 
 u 
 
 
 u 
 
 90zL 
 
 0 
 u 
 
 0*^9 
 yoz 
 
 0 
 
 222 
 
 0 
 
 926 
 
 0 
 u 
 
 01 09 
 UlUZ 
 
 0 
 u 
 
 Q87 
 Oo / 
 
 0.871 
 
 0.420 
 
 0 
 u 
 
 oou 
 
 n 
 u 
 
 
 u 
 
 91 9 
 Zl^ 
 
 u 
 
 Q9Q 
 yzy 
 
 0 
 
 231 
 
 0 
 
 923 
 
 0 
 u 
 
 01 04. 
 
 0 
 
 u 
 
 QQ1 
 oy 1 
 
 0.870 
 
 0.423 
 
 0 
 u 
 
 ooy 
 
 n 
 u 
 
 009fi 
 
 0 
 
 990 
 ^ZU 
 
 0 
 
 Q97 
 yz I 
 
 0 
 
 240 
 
 0 
 
 920 
 
 u 
 
 01 Ofi 
 
 yj i uo 
 
 0 
 u 
 
 oyi 
 
 0.869 
 
 0.426 
 
 0 
 
 OOO 
 
 0 
 
 0028 
 
 0 
 
 227 
 
 0 
 
 924 
 
 0 
 
 248 
 
 0 
 
 917 
 
 0 
 
 0108 
 
 0 
 
 396 
 
 0.868 
 
 0.429 
 
 0 
 
 857 
 
 n 
 U 
 
 \j\}o\) 
 
 u 
 
 
 u 
 
 Q99 
 yzz 
 
 0 
 
 258 
 
 0 
 
 914 
 
 u 
 
 Olio 
 
 Ul lU 
 
 0 
 U 
 
 oyo 
 
 0.867 
 
 0.432 
 
 0 
 u 
 
 OOO 
 
 n 
 u 
 
 OOQ9 
 
 u 
 
 9zl1 
 
 Z'il 
 
 u 
 
 Q90 
 yzu 
 
 0 
 
 263 
 
 0 
 
 912 
 
 0 
 u 
 
 0119 
 Ui IZ 
 
 0 
 u 
 
 /109 
 lUZ 
 
 0.866 
 
 0.434 
 
 0 
 u 
 
 OOO 
 
 u 
 
 00^4 
 
 u 
 
 9/18 
 
 0 
 u 
 
 Q1 7 
 y 1 < 
 
 0 
 
 271 
 
 0 
 
 910 
 
 0 
 u 
 
 0114 
 
 Ul 1^ 
 
 0 
 u 
 
 4-04. 
 
 0.865 
 
 0.437 
 
 0 
 
 u 
 
 ^^A 
 
 OOt: 
 
 u 
 
 UUoD 
 
 u 
 
 
 u 
 
 Q1 ^ 
 
 y lo 
 
 0 
 
 277 
 
 0 
 
 908 
 
 u 
 
 01 1 A 
 Ul lO 
 
 u 
 
 ACYJ 
 
 0.864 
 
 0.440 
 
 0 
 u 
 
 OOO 
 
 0 
 
 0038 
 
 0 
 
 260 
 
 0 
 
 913 
 
 0 
 
 284 
 
 0 
 
 905 
 
 0 
 
 0118 
 
 0 
 
 410 
 
 0.863 
 
 0.443 
 
 0 
 
 852 
 
 u 
 
 
 u 
 
 9AA 
 ZOO 
 
 u 
 
 Q1 1 
 
 y 1 1 
 
 0 
 
 292 
 
 0 
 
 903 
 
 u 
 
 01 90 
 UIZU 
 
 u 
 
 /t1 9 
 ^iz 
 
 0.863 
 
 0.446 
 
 0 
 u 
 
 OOl 
 
 n 
 u 
 
 OOzL9 
 
 u 
 
 970 
 Z / U 
 
 0 
 u 
 
 Q1 0 
 y lu 
 
 0 
 
 297 
 
 0 
 
 901 
 
 0 
 u 
 
 01 99 
 \J 1 zz 
 
 0 
 u 
 
 4.1 ^ 
 
 4:10 
 
 0.862 
 
 0.448 
 
 yj 
 
 OO 1 
 
 u 
 
 (\r\AA 
 
 u 
 
 97A 
 Z / O 
 
 u 
 
 yuo 
 
 0 
 
 303 
 
 0 
 
 899 
 
 n 
 u 
 
 Ol 9/1 
 
 u 
 
 4.1 7 
 "±1 < 
 
 0.861 
 
 0.451 
 
 n 
 u 
 
 oou 
 
 u 
 
 nO/LA 
 UU'iO 
 
 u 
 
 9C1 
 Zol 
 
 u 
 
 QOA 
 
 yuo 
 
 0 
 
 309 
 
 0 
 
 897 
 
 u 
 
 01 9fi 
 UIZO 
 
 u 
 
 4.1 Q 
 
 4:iy 
 
 0.860 
 
 0.454 
 
 n 
 
 u 
 
 Ot:c7 
 
 0 
 
 0048 
 
 0 
 
 286 
 
 0 
 
 904 
 
 0 
 
 315 
 
 0 
 
 895 
 
 0 
 
 0128 
 
 0 
 
 422 
 
 0.859 
 
 0.457 
 
 0 
 
 848 
 
 u 
 
 no f^o 
 
 U 
 
 901 
 
 zyi 
 
 u 
 
 QOQ 
 
 yuo 
 
 0 
 
 320 
 
 0 
 
 893 
 
 u 
 
 Ol QO 
 UloU 
 
 u 
 
 AOA 
 
 0.859 
 
 0.459 
 
 0 
 u 
 
 84-7 
 
 04: / 
 
 u 
 
 OOPi9 
 
 U 
 
 zyo 
 
 u 
 
 Q01 
 
 yul 
 
 0 
 
 324 
 
 0 
 
 892 
 
 u 
 
 Ol Q9 
 UloZ 
 
 u 
 
 4.97 
 IZ i 
 
 0.858 
 
 0.461 
 
 0 
 u 
 
 R4.fi 
 
 OtcU 
 
 u 
 
 OOCiA 
 UUO'l 
 
 u 
 
 QOO 
 oUU 
 
 u 
 
 QOO 
 
 yuu 
 
 0 
 
 329 
 
 0 
 
 891 
 
 u 
 
 01 Q/1 
 
 U104: 
 
 u 
 
 A9Q 
 izy 
 
 0.857 
 
 0.464 
 
 n 
 
 04:0 
 
 U 
 
 OOtiA 
 UUOO 
 
 U 
 
 QO/t 
 oU'± 
 
 u 
 
 oyy 
 
 0 
 
 333 
 
 0 
 
 889 
 
 u 
 
 Ol QA 
 UloO 
 
 n 
 u 
 
 /LQ9 
 lOZ 
 
 0.856 
 
 0.466 
 
 0 
 u 
 
 RA^ 
 
 04:0 
 
 0 
 
 0058 
 
 0 
 
 309 
 
 0 
 
 897 
 
 0 
 
 337 
 
 0 
 
 888 
 
 0 
 
 0138 
 
 0 
 
 434 
 
 0.855 
 
 0.468 
 
 0 
 
 844 
 
 U 
 
 OOAO 
 
 u 
 
 Q1 A 
 Oil 
 
 u 
 
 oyo 
 
 0 
 
 344 
 
 0 
 
 885 
 
 u 
 
 Ol /to 
 
 Ui'iU 
 
 u 
 
 AQA 
 '±oO 
 
 0.855 
 
 0.471 
 
 0 
 u 
 
 04:0 
 
 U 
 
 OOA9 
 
 u 
 
 
 u 
 
 CQ/l 
 
 oy'l 
 
 0 
 
 348 
 
 0 
 
 884 
 
 u 
 
 Ol A9 
 Ul'iZ 
 
 u 
 
 AQ7 
 
 0.854 
 
 0.473 
 
 0 
 u 
 
 04:0 
 
 U 
 
 OOAA 
 
 u 
 
 Q99 
 OZZ 
 
 n 
 u 
 
 oyo 
 
 0 
 
 352 
 
 0 
 
 883 
 
 n 
 u 
 
 Ol AA 
 U144 
 
 u 
 
 AACi 
 
 0.853 
 
 0.475 
 
 0 
 u 
 
 R4-9 
 
 U 
 
 OOAA 
 UUOO 
 
 u 
 
 oZO 
 
 u 
 
 QQ9 
 
 oyz 
 
 0 
 
 356 
 
 0 
 
 881 
 
 u 
 
 Ol zLA 
 UIttO 
 
 u 
 
 AA9 
 
 0.853 
 
 0.477 
 
 0 
 u 
 
 R4.1 
 
 O^l 
 
 0 
 
 0068 
 
 0 
 
 330 
 
 0 
 
 890 
 
 0 
 
 360 
 
 0 
 
 880 
 
 0 
 
 0148 
 
 0 
 
 444 
 
 0.852 
 
 0.479 
 
 0 
 
 840 
 
 0 
 
 0070 
 
 0 
 
 334 
 
 0 
 
 889 
 
 0 
 
 365 
 
 0 
 
 878 
 
 0 
 
 0150 
 
 0 
 
 446 
 
 0.861 
 
 0.481 
 
 0 
 
 840 
 
 0 
 
 0072 
 
 0 
 
 338 
 
 0 
 
 887 
 
 0 
 
 369 
 
 0 
 
 877 
 
 0 
 
 0152 
 
 0 
 
 449 
 
 0.850 
 
 0.483 
 
 0 
 
 839 
 
 0 
 
 0074 
 
 0 
 
 342 
 
 0 
 
 886 
 
 0 
 
 372 
 
 0 
 
 876 
 
 0 
 
 0154 
 
 0 
 
 451 
 
 0.850 
 
 0.485 
 
 0 
 
 838 
 
 0 
 
 0076 
 
 0 
 
 345 
 
 0 
 
 885 
 
 u 
 
 QTA 
 O/O 
 
 u 
 
 575 
 
 0 
 
 0156 
 
 0 
 
 453 
 
 U . o4y 
 
 
 0 
 
 838 
 
 0 
 
 0078 
 
 0 
 
 349 
 
 0 
 
 884 
 
 0 
 
 380 
 
 0 
 
 873 
 
 0 
 
 0158 
 
 0 
 
 455 
 
 0.848 
 
 0.489 
 
 0 
 
 837 
 
 0 
 
 0080 
 
 0 
 
 353 
 
 0 
 
 882 
 
 0 
 
 384 
 
 0 
 
 872 
 
 0 
 
 0160 
 
 0 
 
 457 
 
 0.848 
 
 0.493 
 
 0 
 
 836 
 
 0 
 
 0082 
 
 0 
 
 356 
 
 0 
 
 881 
 
 0 
 
 387 
 
 0 
 
 871 
 
 0 
 
 0170 
 
 0 
 
 467 
 
 0.845 
 
 0.502 
 
 0 
 
 833 
 
 0 
 
 0084 
 
 0 
 
 360 
 
 0 
 
 880 
 
 0 
 
 390 
 
 0 
 
 870 
 
 0 
 
 0180 
 
 0 
 
 476 
 
 0.841 
 
 0.513 
 
 0 
 
 829 
 
 0 
 
 0086 
 
 0 
 
 363 
 
 0 
 
 879 
 
 0 
 
 394 
 
 0 
 
 869 
 
 0 
 
 0190 
 
 0 
 
 485 
 
 0.838 
 
 0.522 
 
 0 
 
 826 
 
 0 
 
 0088 
 
 0 
 
 366 
 
 0 
 
 878 
 
 0 
 
 398 
 
 0 
 
 867 
 
 0 
 
 0200 
 
 0 
 
 493 
 
 0.836 
 
 0.531 
 
 0 
 
 823 
 
Sec. 7-S7 
 
 BEAMS AND SLABS 
 
 357 
 
 Table 4. — Values of hd^ for Different Values of b and d 
 
 Values 
 of 
 
 b = 
 
 = 12 
 
 b = 
 
 = 10 
 
 
 d 
 
 bd 
 
 d 
 
 bd 
 
 
 3 
 
 36 
 
 3 
 
 3 
 
 32 
 
 8 
 
 192 
 
 4 
 
 48 
 
 4 
 
 4 
 
 43 
 
 7 
 
 300 
 
 5 
 
 60 
 
 5 
 
 5 
 
 54 
 
 8 
 
 432 
 
 6 
 
 72 
 
 6 
 
 6 
 
 65 
 
 7 
 
 588 
 
 7 
 
 84 
 
 7 
 
 7 
 
 76 
 
 5 
 
 / Do 
 
 8 
 
 96 
 
 8 
 
 8 
 
 87 
 
 5 
 
 972 
 
 9 
 
 108 
 
 9 
 
 9 
 
 98 
 
 5 
 
 1,200 
 
 10 
 
 120 
 
 10 
 
 9 
 
 109 
 
 4 
 
 1,452 
 
 11 
 
 132 
 
 12 
 
 1 
 
 120 
 
 2 
 
 1,728 
 
 12 
 
 144 
 
 13 
 
 1 
 
 131 
 
 2 
 
 
 13 
 
 156 
 
 14 
 
 3 
 
 142 
 
 3 
 
 2,352 
 
 14 
 
 168 
 
 15 
 
 4 
 
 153 
 
 6 
 
 2,700 
 
 15 
 
 180 
 
 16 
 
 4 
 
 164 
 
 0 
 
 3,072 
 
 16 
 
 192 
 
 17 
 
 5 
 
 175 
 
 0 
 
 3,468 
 
 17 
 
 204 
 
 18 
 
 6 
 
 186 
 
 0 
 
 Q QCC 
 
 o,ooo 
 
 18 
 
 216 
 
 19 
 
 7 
 
 197 
 
 0 
 
 4,332 
 
 19 
 
 228 
 
 20 
 
 8 
 
 208 
 
 0 
 
 4,800 
 
 20 
 
 240 
 
 21 
 
 9 
 
 218 
 
 8 
 
 5,292 
 
 21 
 
 252 
 
 23 
 
 0 
 
 229 
 
 8 
 
 5,808 
 
 22 
 
 264 
 
 24 
 
 1 
 
 240 
 
 6 
 
 0,<54fS 
 
 23 
 
 276 
 
 25 
 
 2 
 
 252.0 
 
 6,912 
 
 24 
 
 288 
 
 26 
 
 3 
 
 263 
 
 0 
 
 7,500 
 
 25 
 
 300 
 
 27 
 
 4 
 
 274 
 
 0 
 
 8,112 
 
 26 
 
 312 
 
 28 
 
 5 
 
 285 
 
 0 
 
 8,748 
 
 27 
 
 324 
 
 29 
 
 6 
 
 295 
 
 8 
 
 y,4Uo 
 
 
 ooD 
 
 30 
 
 7 
 
 307 
 
 0 
 
 10,092 
 
 29 
 
 348 
 
 31 
 
 9 
 
 318 
 
 2 
 
 10,800 
 
 30 
 
 360 
 
 32 
 
 9 
 
 328 
 
 3 
 
 11,532 
 
 31 
 
 372 
 
 33 
 
 9 
 
 339 
 
 4 
 
 12,288 
 
 32 
 
 384 
 
 35.0 
 
 350 
 
 0 
 
 13,068 
 
 33 
 
 396 
 
 36 
 
 2 
 
 361 
 
 8 
 
 13,872 
 
 34 
 
 408 
 
 37 
 
 2 
 
 372 
 
 1 
 
 14,700 
 
 35 
 
 420 
 
 38 
 
 3 
 
 383 
 
 3 
 
 15,552 
 
 36 
 
 432 
 
 39 
 
 4 
 
 394 
 
 0 
 
 16,428 
 
 37 
 
 444 
 
 40 
 
 6 
 
 406 
 
 0 
 
 17,328 
 
 38 
 
 456 
 
 41 
 
 6 
 
 416 
 
 0 
 
 18,252 
 
 39 
 
 468 
 
 42 
 
 7 
 
 427 
 
 1 
 
 19,200 
 
 40 
 
 480 
 
 43 
 
 8 
 
 438 
 
 0 
 
 d bd 
 
 d = 0.56 
 
 bd 
 
 bd 
 
 1.5b 
 
 bd 
 
 d = 1.76 
 
 6 I d bd 
 
 d = 2b 
 
 b d bd 
 
 2.7 
 3.6 
 4.5 
 5.4 
 6.3 
 
 7.2 
 8.1 
 8.9 
 9.8 
 10.7 
 
 11.6 
 12.6 
 13.5 
 14.3 
 15.2 
 
 16.1 
 17.0 
 17.9 
 18.8 
 19.7 
 
 20.6 
 21.5 
 22.4 
 23.2 
 24.1 
 
 25.0 
 25.9 
 26.8 
 27.7 
 28.6 
 
 29 
 30 
 31 
 32 
 33 
 
 33 
 34 
 35 
 
 40 
 54 
 67 
 81 
 
 94, 
 
 108 
 121 
 133 
 147, 
 160 
 
 174 
 
 189 
 202 
 214 
 228 
 
 241 
 
 255 
 268 
 282 
 295 
 
 309 
 323 
 336 
 348 
 361, 
 
 375, 
 388, 
 402 
 416 
 429, 
 
 442 
 456 
 469, 
 482, 
 495, 
 
 508, 
 522, 
 536. 
 
 3.8 
 4.6 
 5.3 
 6.0 
 6.7 
 
 7 
 
 9.2 
 10.6 
 12.0 
 13.4 
 
 14.6 
 15 
 
 17.0 
 18.0 
 19.0 
 
 20.010. 
 21.010, 
 22.011 
 23.011 
 24.012 
 
 25.012 
 25.812 
 26.813 
 27.813 
 28.614 
 
 29.414 
 30.215 
 31.015 
 32.016 
 32.816 
 
 33.616 
 
 34.4 
 35.217 
 35.817 
 36.618 
 
 17.2 
 
 6 
 9 
 3 
 
 37.418.7 
 38.219.1 
 38.8 19.4 
 39.619.8 
 40.420.2 
 
 41.2 20.6 
 41.820.9 
 42.4121.2 
 
 28.8 
 41.4 
 56.2 
 72.0 
 89.8 
 106.5 
 124.9 
 144.5 
 162.0 
 180.5 
 
 200.0 
 220.2 
 242.0 
 264.2 
 288.0 
 
 312.2 
 332.7 
 359.5 
 387.0 
 409.0 
 
 7 432.0 
 
 456.0 
 480.0 
 522.0 
 537.5 
 
 565.0 
 592.0 
 620.0 
 642.0 
 670.0 
 
 700.0 
 730.0 
 753.0 
 784.0 
 816.0 
 
 847.0 
 875.0 
 900.0 
 
 4.8 
 5.8 
 6.7 
 7.6 
 8.4 
 
 9.2 
 
 10 
 11 
 
 12, 
 
 12 
 13 
 13 
 14 
 
 15, 
 
 15 
 16 
 16 
 17 
 18 
 
 18 
 19 
 19 
 20 
 20 
 21 
 21 
 22.1 
 22.6 
 23.1 
 
 23.6 
 24.1 
 24.5 
 25.0 
 25.4 
 
 25.9 
 26.4 
 26.8 
 
 23 
 33 
 44, 
 57, 
 70. 
 
 84, 
 98. 
 114, 
 129, 
 144, 
 
 161 
 176 
 193 
 213 
 231, 
 
 246, 
 265, 
 285, 
 306 
 324 
 
 345 
 364 
 384 
 .2 408 
 424 
 
 449 
 . 7 470 
 
 488 
 510 
 533 
 
 557 
 580 
 600 
 625 
 645 
 
 670 
 697 
 718 
 
 3.6 
 4.4 
 5.1 
 5.8 
 6.4 
 
 7.010 
 7.611 
 
 8.1 
 
 8.713 
 9.213 
 
 9.714 
 10.215 
 10.716 
 11.216 
 11.617 
 
 12.1 
 
 12.518 
 12.919 
 13.319 
 13.8 20 
 
 14.221 
 14.621 
 15.022 
 15.423 
 15.823 
 
 16.124 
 16.524 
 16.925 
 17.326 
 17.626 
 
 18. 
 
 18 
 18 
 19 
 19 
 
 19 
 20.1 
 20 
 
 5.4 
 6.6 
 7.7 
 8.7 
 9.6 
 
 12 
 
 .4 
 
 .0 
 .5 
 
 7 28.0 
 128.6 
 4 29.1 
 
 .027 
 
 .3 27 
 
 8 29 
 
 7 
 
 30.1 
 6 
 
 4 30 
 
 19.8 
 29.0 
 39.2 
 50.1 
 61.5 
 
 73.5 
 86.7 
 98.0 
 114.0 
 127.0 
 
 141.6 
 156.0 
 172.0 
 188.2 
 202.0 
 
 219.0 
 235.0 
 249.0 
 265.0 
 286.0 
 
 302.0 
 320.0 
 338.0 
 356.0 
 375.0 
 
 388.0 
 410.0 
 430.0 
 0 450.0 
 
 465.0 
 
 186.0 
 503.0 
 524.0 
 546.0 
 565.0 
 
 588.0 
 605.0 
 624.0 
 
 5.7 
 6.9 
 8.0 
 9.1 
 10.0 
 
 10.9 
 11.8 
 12.7 
 13. 
 14. 
 
 19.4 
 28.3 
 37.6 
 49.2 
 59.0 
 
 69.8 
 81.4 
 95.2 
 .0 
 .0 
 
 6109 
 3120 
 
 15.2136. 
 15.9149. 
 16.7163. 
 17.4177. 
 
 8 
 3 
 5 
 5 
 
 192.0 
 
 18.1 
 
 18.8207, 
 19.5224 
 20 . 2 240 
 20.8254 
 
 3.0 
 3.6 
 4.2 
 4.8 
 5.3 
 
 5.8 
 6.3 
 6.7 
 7.2 
 7.6 
 
 8.0 
 8.4 
 8.8 
 9.2 
 9.6 
 
 9.9 
 10.3 
 10.7 
 
 6.0 
 7.3 
 8.4 
 9.5 
 10.6 
 
 11.5 
 12.5 
 13.4 
 14.3 
 15.2 
 
 15.9 
 16.8 
 17.6 
 18.3 
 19.1 
 
 18.0 
 26.3 
 35.2 
 45.6 
 56.2 
 
 66.8 
 78.8 
 89.8 
 103.0 
 115.6 
 
 127.0 
 141.0 
 155.0 
 168.0 
 184.0 
 
 19, 
 20, 
 21 
 11.0 22 
 
 21.5273.0 
 
 11.4 
 
 22 
 
 196.0 
 6212.0 
 3228.0 
 0 242.0 
 7 258.5 
 
 22.1288, 
 22.8306, 
 8 23.4323 
 24.0339 
 
 5 24.6356 
 
 25.2373 
 25.8 392 
 26.4410 
 27.0429 
 27.6447 
 
 28 . 1 464 
 28.7 485 
 29 . 3 504 
 
 6 29.8525 
 9 
 
 30 . 4 544 , 
 30.9562 
 31.4581 
 32.0602 
 
 11.7 23 
 
 12.1 
 12.4 
 12.7 
 
 13.3 
 13.7 
 
 274.0 
 292.0 
 7 306.0 
 323.0 
 
 24. 
 24. 
 25.4 
 13.0 26.0338.0 
 
 6 354.0 
 3 374.0 
 390.0 
 5408.0 
 2 424.0 
 
 14.0 27.9 
 28 
 
 14.3 
 14.6 29 
 
 29 
 30 
 30 
 31 
 32 
 
 16.4 32 
 16.6 
 16.9 
 
 14.9 
 15.2 
 15.5 
 15.8 
 16.1 
 
 7 443.0 
 4462.0 
 9478.0 
 5498.0 
 
 517.0 
 
 537.0 
 2 552 .0 
 
 8 572.0 
 
 Table o. — Number of Rods and Sectional Area in Square Inches for Beam and Column 
 
 Reinforcement 
 
 Size of rod 
 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 10 
 
 11 
 
 12 
 
 13 
 
 14 
 
 15 
 
 16 
 
 
 in. 
 
 / round. . . 
 
 0 
 
 39 
 
 0 
 
 58 
 
 0 
 
 78 
 
 0 
 
 98 
 
 1 
 
 18 
 
 1 
 
 37 
 
 1 
 
 57 
 
 1 
 
 77 
 
 1 
 
 96 
 
 2 
 
 16 
 
 2 
 
 36 
 
 2 
 
 55 
 
 2 
 
 75 
 
 2 
 
 94 
 
 3.10 
 
 'A 
 
 \ square . . 
 
 0 
 
 50 
 
 0 
 
 75 
 
 1 
 
 00 
 
 1 
 
 25 
 
 1 
 
 50 
 
 1 
 
 75 
 
 2 
 
 00 
 
 2 
 
 25 
 
 2 
 
 50 
 
 2 
 
 75 
 
 3 
 
 00 
 
 3 
 
 25 
 
 3 
 
 50 
 
 3 
 
 75 
 
 4.04 
 
 
 in. 
 
 f round. . . 
 
 0 
 
 49 
 
 0 
 
 74 
 
 0 
 
 99 
 
 1 
 
 24 
 
 1 
 
 49 
 
 1 
 
 74 
 
 1 
 
 99 
 
 2 
 
 24 
 
 2 
 
 48 
 
 2 
 
 73 
 
 2 
 
 98 
 
 3 
 
 23 
 
 3 
 
 48 
 
 3 
 
 73 
 
 3.98 
 
 
 1 square. . 
 
 0 
 
 63 
 
 0 
 
 94 
 
 1 
 
 27 
 
 1 
 
 58 
 
 1 
 
 90 
 
 2 
 
 21 
 
 2 
 
 53 
 
 2 
 
 85 
 
 3 
 
 16 
 
 3* 
 
 48 
 
 3 
 
 80 
 
 4 
 
 11 
 
 4 
 
 43 
 
 4 
 
 75 
 
 5.06 
 
 
 in. 
 
 ! round. . . 
 
 0 
 
 61 
 
 0 
 
 91 
 
 1 
 
 23 
 
 1 
 
 53 
 
 1 
 
 84 
 
 2 
 
 15 
 
 2 
 
 45 
 
 2 
 
 76 
 
 3 
 
 07 
 
 3 
 
 37 
 
 3 
 
 68 
 
 3 
 
 99 
 
 4 
 
 30 
 
 4 
 
 60 
 
 4.91 
 
 
 \ square . . 
 
 0 
 
 78 
 
 1 
 
 07 
 
 1 
 
 56 
 
 1 
 
 95 
 
 2 
 
 34 
 
 2 
 
 73 
 
 3 
 
 12 
 
 3 
 
 52 
 
 3 
 
 91 
 
 4 
 
 30 
 
 4 
 
 69 
 
 5 
 
 08 
 
 5 
 
 47 
 
 5 
 
 86 
 
 6.25 
 
 
 in. 
 
 / round. . . 
 
 0 
 
 74 
 
 1 
 
 11 
 
 1 
 
 48 
 
 1 
 
 86 
 
 2 
 
 23 
 
 2 
 
 60 
 
 2 
 
 97 
 
 3 
 
 34 
 
 3 
 
 71 
 
 4 
 
 08 
 
 4 
 
 45 
 
 4 
 
 83 
 
 5 
 
 20 
 
 5 
 
 57 
 
 5.94 
 
 
 \ square . . 
 
 0 
 
 94 
 
 1 
 
 41 
 
 1 
 
 89 
 
 2 
 
 36 
 
 2 
 
 84 
 
 3.31 
 
 3 
 
 78 
 
 4 
 
 25 
 
 4 
 
 73 
 
 5 
 
 20 
 
 5 
 
 67 
 
 6 
 
 15 
 
 6 
 
 62 
 
 7 
 
 09 
 
 7.56 
 
 
 in. 
 
 / round. . . 
 
 0 
 
 88 
 
 1 
 
 32 
 
 1 
 
 77 
 
 2 
 
 21 
 
 2 
 
 65 
 
 3.09 
 
 3 
 
 53 
 
 3 
 
 98 
 
 4 
 
 42 
 
 4 
 
 86 
 
 5 
 
 30 
 
 5 
 
 74 
 
 6 
 
 19 
 
 6 
 
 63 
 
 7.07 
 
 H 
 
 \ square . . 
 
 1 
 
 12 
 
 1 
 
 68 
 
 2 
 
 25 
 
 2 
 
 81 
 
 3 
 
 38 
 
 3 
 
 94 
 
 4 
 
 50 
 
 5 
 
 06 
 
 5 
 
 62 
 
 6 
 
 19 
 
 6 
 
 75 
 
 7 
 
 31 
 
 7 
 
 88 
 
 8 
 
 44 
 
 9.00 
 
 
 in. 
 
 / round. . . 
 
 1 
 
 03 
 
 1 
 
 55 
 
 2 
 
 07 
 
 2 
 
 59 
 
 3 
 
 11 
 
 3 
 
 63 
 
 4 
 
 15 
 
 4 
 
 67 
 
 5 
 
 18 
 
 5 
 
 70 
 
 6 
 
 22 
 
 6 
 
 74 
 
 7 
 
 26 
 
 7 
 
 78 
 
 8.30 
 
 me 
 
 \ square. . 
 
 1 
 
 32 
 
 1 
 
 98 
 
 2 
 
 64 
 
 3 
 
 30 
 
 3 
 
 96 
 
 4 
 
 62 
 
 5 
 
 28 
 
 5 
 
 94 
 
 6 
 
 60 
 
 7 
 
 26 
 
 7 
 
 92 
 
 8 
 
 58 
 
 9 
 
 24 
 
 9 
 
 90 
 
 10.56 
 
 
 in. 
 
 f round. . . 
 
 1 
 
 20 
 
 1 
 
 80 
 
 2 
 
 41 
 
 3 
 
 01 
 
 3 
 
 61 
 
 4 
 
 21 
 
 4 
 
 81 
 
 5 
 
 41 
 
 6 
 
 01 
 
 6 
 
 61 
 
 7 
 
 22 
 
 7 
 
 82 
 
 8 
 
 42 
 
 9 
 
 02 
 
 9.62 
 
 H 
 
 1 square . . 
 
 1 
 
 53 
 
 2 
 
 29 
 
 3 
 
 06 
 
 3 
 
 83 
 
 4 
 
 59 
 
 5 
 
 36 
 
 6 
 
 12 
 
 6 
 
 89 
 
 7 
 
 66 
 
 8 
 
 42 
 
 9 
 
 19 
 
 9 
 
 95 
 
 10 
 
 72 
 
 11 
 
 48 
 
 12.25 
 
 
 [\n. 
 
 / round. . . 
 
 1 
 
 38 
 
 2 
 
 07 
 
 2 
 
 76 
 
 3 
 
 45 
 
 4 
 
 14 
 
 4 
 
 83 
 
 5 
 
 52 
 
 6 
 
 21 
 
 6 
 
 90 
 
 7 
 
 59 
 
 3 
 
 28 
 
 8 
 
 97 
 
 9 
 
 66 
 
 10 
 
 35 
 
 11.04 
 
 
 \ square . . 
 
 1 
 
 75 
 
 2 
 
 63 
 
 3 
 
 52 
 
 4 
 
 39 
 
 5 
 
 27 
 
 6 
 
 15 
 
 7.03 
 
 7 
 
 91 
 
 8 
 
 79 
 
 9 
 
 67 
 
 10 
 
 55 
 
 11 
 
 43 
 
 12 
 
 30 
 
 13 
 
 18 
 
 14.06 
 
 
 in. 
 
 f round. . . 
 
 1 
 
 57 
 
 2 
 
 35 
 
 3 
 
 14 
 
 3 
 
 93 
 
 4 
 
 71 
 
 5 
 
 50 
 
 6 
 
 28 
 
 7 
 
 07 
 
 7 
 
 85 
 
 8 
 
 64 
 
 9 
 
 43 
 
 10 
 
 21 
 
 11 
 
 00 
 
 11 
 
 78 
 
 12.57 
 
 1 
 
 1 square . . 
 
 2.00 
 
 3 
 
 00 
 
 4 
 
 00 
 
 5 
 
 00 
 
 6 
 
 00 
 
 7 
 
 00 
 
 S.OO 
 
 9 
 
 00 
 
 10 
 
 00 
 
 11 
 
 00 
 
 12 
 
 00 
 
 13 
 
 00 
 
 14 
 
 00 
 
 15 
 
 00 
 
 16.00 
 
 
 in. 
 
 / round. . . 
 
 1 
 
 98 
 
 2 
 
 98 
 
 3 
 
 98 
 
 4 
 
 97 
 
 5 
 
 96 
 
 6 
 
 96 
 
 7 
 
 95 
 
 8 
 
 95 
 
 9 
 
 94 
 
 10 
 
 94 
 
 11 
 
 93 
 
 12 
 
 92 
 
 13 
 
 92 
 
 14 
 
 91 
 
 15.90 
 
 m 
 
 \ square. . 
 
 2 
 
 53 
 
 3 
 
 79 
 
 5.06 
 
 6 
 
 33 
 
 7 
 
 59 
 
 8 
 
 86 
 
 10 
 
 12 
 
 11 
 
 39 
 
 12 
 
 66 
 
 13 
 
 92 
 
 15 
 
 19 
 
 16 
 
 45 
 
 16 
 
 72 
 
 18 
 
 98 
 
 20.25 
 
 
 in. 
 
 f round. . . 
 
 2 
 
 45 
 
 3 
 
 68 
 
 4 
 
 91 
 
 6 
 
 14 
 
 7 
 
 36 
 
 8 
 
 59 
 
 9 
 
 82 
 
 11.04 
 
 12 
 
 2.7 
 
 13 
 
 50 
 
 14 
 
 73 
 
 15 
 
 95 
 
 17.18 
 
 18 
 
 41 
 
 19.64 
 
 VA 
 
 1 square. . 
 
 3 
 
 12 
 
 4 
 
 68 
 
 6 
 
 25 
 
 7 
 
 81 
 
 9 
 
 37 
 
 10 
 
 94 
 
 12 
 
 50 
 
 14 
 
 06 
 
 15 
 
 62 
 
 17 
 
 19 
 
 18 
 
 75 
 
 20 
 
 31 
 
 21.87 
 
 23 
 
 44 
 
 25.00 
 
CONCRETE ENGINEERS' HANDBOOK 
 Table 6. — Spacing of Round Rods in Slabs 
 
 [Sec. 
 
 Diam- 
 
 Sectional area of steel per foot of slab when spaced as follows: 
 
 eier 
 (inches) 
 
 2 in. 
 
 21/^ 
 
 in. 
 
 3 
 
 in. 
 
 
 in. 
 
 4 
 
 in. 
 
 41/2 
 
 in. 
 
 5 in. 
 
 5}^ in. 
 
 6 
 
 n. 
 
 7 in. 
 
 8 
 
 n. 
 
 9 in. 
 
 10 in. 
 
 12 
 
 in. 
 
 
 0 
 
 29 
 
 0 
 
 23 
 
 0.20 
 
 0 
 
 17 
 
 0 
 
 15 
 
 0 
 
 13 
 
 0.12 
 
 
 
 
 
 
 
 
 
 
 
 
 He 
 
 0 
 
 46 
 
 0 
 
 36 
 
 0 
 
 31 
 
 0 
 
 26 
 
 0 
 
 23 
 
 0 
 
 20 
 
 0.18 
 
 0.17 
 
 0 
 
 15 
 
 0.13 
 
 
 
 
 
 
 
 H 
 
 0 
 
 66 
 
 0.53 
 
 0.44 
 
 0 
 
 38 
 
 0.33 
 
 0 
 
 29 
 
 0.26 
 
 0.24 
 
 0 
 
 22 
 
 0.19 
 
 0 
 
 17 
 
 0.15 
 
 0.13 
 
 
 
 Vie 
 
 0 
 
 90 
 
 0.72 
 
 0.60 
 
 0.51 
 
 0.45 
 
 0.40 
 
 0.36 
 
 0.33 
 
 0 
 
 30 
 
 0.26 
 
 0 
 
 23 
 
 0.20 
 
 0.18 
 
 0 
 
 15 
 
 y2 
 
 1 
 
 18 
 
 0 
 
 94 
 
 0.78 
 
 0 
 
 6/ 
 
 0 
 
 59 
 
 0 
 
 52 
 
 0.47 
 
 0.43 
 
 0 
 
 39 
 
 0.34 
 
 0 
 
 29 
 
 0.26 
 
 0.24 
 
 0 
 
 20 
 
 He 
 
 1 
 
 49 
 
 1 
 
 19 
 
 0 
 
 99 
 
 0 
 
 85 
 
 0 
 
 75 
 
 0.66 
 
 0.60 
 
 0.54 
 
 0 
 
 50 
 
 0.43 
 
 0 
 
 37 
 
 0.33 
 
 0.30 
 
 0 
 
 25 
 
 'A 
 
 1 
 
 84 
 
 1 
 
 47 
 
 1 
 
 23 
 
 1 
 
 05 
 
 0 
 
 92 
 
 0.82 
 
 0.74 
 
 0.67 
 
 0 
 
 61 
 
 0.53 
 
 0 
 
 46 
 
 0.41 
 
 0.37 
 
 0 
 
 31 
 
 ^He 
 
 2 
 
 23 
 
 1 
 
 78 
 
 1 
 
 48 
 
 1 
 
 27 
 
 1 
 
 11 
 
 0 
 
 99 
 
 0.89 
 
 0.81 
 
 0 
 
 74 
 
 0.64 
 
 0 
 
 56 
 
 0.49 
 
 0.45 
 
 0 
 
 37 
 
 H 
 
 2 
 
 65 
 
 2 
 
 12 
 
 1 
 
 77 
 
 1 
 
 51 
 
 1 
 
 32 
 
 1 
 
 18 
 
 1.06 
 
 0.96 
 
 0 
 
 88 
 
 0.76 
 
 0 
 
 66 
 
 0.59 
 
 0.53 
 
 0 
 
 44 
 
 ^He 
 
 3 
 
 11 
 
 2 
 
 48 
 
 2 
 
 07 
 
 1 
 
 78 
 
 1 
 
 56 
 
 1 
 
 38 
 
 1.24 
 
 1.13 
 
 1 
 
 04 
 
 0.89 
 
 0 
 
 78 
 
 0.69 
 
 0.62 
 
 0 
 
 52 
 
 14 
 
 /8 
 
 3 
 
 61 
 
 2 
 
 88 
 
 2 
 
 40 
 
 2 
 
 06 
 
 1 
 
 80 
 
 1 
 
 60 
 
 1.44 
 
 1.31 
 
 1 
 
 20 
 
 1.03 
 
 0 
 
 90 
 
 0.80 
 
 0.72 
 
 0 
 
 60 
 
 
 4 
 
 14 
 
 3 
 
 31 
 
 2 
 
 76 
 
 2 
 
 37 
 
 2 
 
 07 
 
 1 
 
 84 
 
 1.66 
 
 1.51 
 
 1 
 
 38 
 
 1.18 
 
 1 
 
 03 
 
 0.92 
 
 0.83 
 
 0 
 
 69 
 
 1 
 
 4 
 
 71 
 
 3 
 
 77 
 
 3 
 
 14 
 
 2 
 
 69 
 
 2 
 
 36 
 
 2 
 
 09 
 
 1.88 
 
 1.71 
 
 1 
 
 57 
 
 1.35 
 
 1 
 
 18 
 
 1.05 
 
 0.94 
 
 0 
 
 78 
 
 
 
 
 4 
 
 77 
 
 3 
 
 98 
 
 3 
 
 41 
 
 2 
 
 98 
 
 2 
 
 65 
 
 2.39 
 
 2.17 
 
 1 
 
 99 
 
 1.70 
 
 1 
 
 49 
 
 1:33 
 
 1.19 
 
 0 
 
 99 
 
 m 
 m 
 
 
 
 
 
 4 
 
 91 
 
 4 
 
 21 
 
 3 
 
 68 
 
 3 
 
 27 
 
 2.95 
 
 2.68 
 
 2 
 
 45 
 
 2.10 
 
 1 
 
 84 
 
 1.64 
 
 1.47 
 
 1 
 
 23 
 
 
 
 
 
 5 
 
 09 
 
 4 
 
 45 
 
 3 
 
 96 
 
 3.56 
 
 3.24 
 
 2 
 
 97 
 
 2.55 
 
 2 
 
 23 
 
 1.98 
 
 1.78 
 
 1 
 
 48 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 5 
 
 30 ! 4 
 
 71 
 
 4.24 
 
 3.86 
 
 3 
 
 53 
 
 3.03 
 
 2 
 
 65 
 
 2.36 
 
 2. 12 
 
 1 
 
 77 
 
 
 
 
 
 
 
 
 
 
 
 Table 7. — Spacing of Square Rods in Slabs 
 
 Di- 
 
 Sectional area of steel per foot of slab when spaced as follows: 
 
 sion 
 (inches) 
 
 2 in. 
 
 2H 
 
 in. 
 
 3 
 
 in. 
 
 3H 
 
 in. 
 
 4 
 
 in. 
 
 4 1/2 
 
 in. 
 
 5 
 
 in. 
 
 in. 
 
 6 in. 
 
 j 7 in. 
 
 8 in. 
 
 9 in. 
 
 10 in. 
 
 12 in. 
 
 
 0.37 
 
 0.30 
 
 0.25 
 
 0 
 
 .21 
 
 0 
 
 .19 
 
 0 
 
 .17 
 
 0 
 
 .15 
 
 0.13 
 
 0.12 
 
 
 
 
 
 
 Vie 
 
 0.59 
 
 0.47 
 
 0.39 
 
 0 
 
 .33 
 
 0 
 
 .29 
 
 0 
 
 .26 
 
 0 
 
 .23 
 
 0.21 
 
 0.19 
 
 0.17 
 
 0.15 
 
 0.13 
 
 
 
 % 
 
 0.84 
 
 0.67 
 
 0 
 
 .56 
 
 0.48 
 
 0 
 
 .42 
 
 0 
 
 .37 
 
 0 
 
 .34 
 
 0.31 
 
 0.28 
 
 0.24 
 
 0.21 
 
 0.19 
 
 0.17 
 
 0.14 
 
 He 
 
 1.15 
 
 0 
 
 92 
 
 0 
 
 77 
 
 0.66 
 
 0 
 
 .57 
 
 0 
 
 51 
 
 0 
 
 .46 
 
 0.42 
 
 0.38 
 
 0.33 
 
 0.29 
 
 0.25 
 
 0.23 
 
 0.19 
 
 
 1.50 
 
 1 
 
 20 
 
 1 
 
 00 
 
 0 
 
 86 
 
 0 
 
 75 
 
 0 
 
 67 
 
 0 
 
 60 
 
 0.55 
 
 0.50 
 
 0.43 
 
 0.37 
 
 0.33 
 
 0.30 
 
 0.25 
 
 He 
 
 1.90 
 
 1 
 
 52 
 
 1 
 
 27 
 
 1 
 
 08 
 
 0 
 
 95 
 
 0 
 
 84 
 
 0 
 
 76 
 
 0.69 
 
 0.63 
 
 0.54 
 
 0.47 
 
 0.42 
 
 0.38 
 
 0.32 
 
 % 
 
 2.34 
 
 1 
 
 87 
 
 1 
 
 56 
 
 1 
 
 34 
 
 1 
 
 17 
 
 1 
 
 04 
 
 0 
 
 94 
 
 0.85 
 
 0.78 
 
 0.67 
 
 0.59 
 
 0.52 
 
 0.47 
 
 0.39 
 
 'He 
 
 2.84 
 
 2 
 
 27 
 
 1 
 
 99 
 
 1 
 
 62 
 
 1 
 
 42 
 
 1 
 
 33 
 
 1 
 
 13 
 
 1.03 
 
 0.94 
 
 0.81 
 
 0.71 
 
 0.66 
 
 0.57 
 
 0.47 
 
 H 
 
 3.37 
 
 2 
 
 70 
 
 2 
 
 25 
 
 1 
 
 93 
 
 1 
 
 69 
 
 1 
 
 50 
 
 1 
 
 35 
 
 1.23 
 
 1.12 
 
 0.96 
 
 0.84 
 
 0.75 
 
 0.67 
 
 0.56 
 
 'He 
 
 3.96 
 
 3 
 
 17 
 
 2 
 
 64 
 
 2 
 
 26 
 
 1 
 
 98 
 
 1 
 
 76 
 
 1 
 
 58 
 
 1.44 
 
 1.32 
 
 1.13 
 
 0.99 
 
 0.88 
 
 0.79 
 
 0.66 
 
 Vs 
 
 4.59 
 
 3 
 
 67 
 
 3 
 
 06 
 
 2 
 
 62 
 
 2 
 
 30 
 
 2 
 
 04 
 
 1 
 
 84 
 
 1.67 
 
 1.53 
 
 1.31 
 
 1.15 
 
 1.02 
 
 0.92 
 
 0.77 
 
 'He 
 
 5.27 
 
 4 
 
 22 
 
 3 
 
 52 
 
 3 
 
 01 
 
 2 
 
 64 
 
 2 
 
 34 
 
 2 
 
 11 
 
 1.92 
 
 1.76 
 
 1.51 
 
 1.32 
 
 1.17 
 
 1.05 
 
 0:88 
 
 1 
 
 6.00 
 
 4 
 
 80 
 
 4 
 
 00 
 
 3 
 
 43 
 
 3 
 
 00 
 
 2 
 
 67 
 
 2 
 
 40 
 
 2.18i 2.00 
 
 1.71 
 
 1.50 
 
 1.33 
 
 1.20 
 
 1.00 
 
 iH 
 
 
 6.08 
 
 5 
 
 06 
 
 4 
 
 34 
 
 3 
 
 80 
 
 3 
 
 37 
 
 3 
 
 04 
 
 2.76 
 
 2.53 
 
 2.17 
 
 1.89 
 
 1.69 
 
 1.52 
 
 1.27 
 
 I'A 
 
 m 
 
 
 
 
 6 
 
 25 
 
 5 
 
 36 
 
 4 
 
 69 
 
 4 
 
 17 
 
 3 
 
 75 
 
 3.41 
 
 3.12 
 
 2.68 
 
 2.34 
 
 2.0^ 
 
 1.87 
 
 1.56 
 
 
 
 
 
 6 
 
 48 
 
 5 
 
 67 
 
 5 
 
 04 
 
 4 
 
 54 
 
 4.12 
 
 3.78 
 
 3 24 
 
 2.84 
 
 2.52 
 
 2.27 
 
 1.89 
 
 IK 
 
 
 
 
 
 
 6 
 
 75 
 
 6 
 
 00 
 
 5 
 
 40 
 
 4.91 
 
 4.50 
 
 3.86 
 
 3.37 
 
 3.00 
 
 2.70 
 
 2.25 
 
 1 
 
 
 
 
Sec. 7-57] 
 
 BEAMS AND SLABS 
 
 359 
 
 Table 8. — Values of 
 
 k k 
 
 - — j-r IN Formula = , /, 
 
 - k) n(l - A;) 
 
 Applies to Rectangular Beams, T-beams, and Beams With Steel Top and Bottom 
 
 Based on n = 12 
 
 0.200 
 0.202 
 0.204 
 0.206 
 0.208 
 
 0.210 
 0.212 
 0.214 
 0.216 
 0.218 
 
 0.220 
 0.222 
 0.224 
 0.226 
 0.228 
 
 0.230 
 0.232 
 0.234 
 0.236 
 0.238 
 
 0.240 
 0.242 
 0.244 
 0.246 
 0.248 
 
 0.0214 
 0.0216 
 0.0219 
 
 0.0222 0.260 
 0.0224 0.262 
 0.0227 0.264 
 0.0230 0.266 
 0 . 0232 0 . 268 
 
 0.0208 0.250 
 0.0211 0.252 
 
 0.254 
 0.256 
 0.258 
 
 0 . 0235 
 0.0238 
 0.0240 
 0 . 0243 
 0 . 0246 
 
 0 . 0249 
 0.0252 
 0 . 02.54 
 0.0257 
 0 . 0260 
 
 0.0263 
 0.0266 
 0.0269 
 0.0272 
 0.0275 
 
 0.270 
 0.272 
 0.274 
 0.276 
 0.278 
 
 0.280 
 0.282 
 0.284 
 0.286 
 0.288 
 
 0.290 
 0.292 
 0.294 
 0.296 
 0.298 
 
 0278 
 0281 
 0284 
 0287 
 0290 
 
 0293 
 0296 
 0299 
 0302 
 0305 
 
 0308 
 0311 
 0315 
 0318 
 0321 
 
 0324 
 0327 
 0331 
 0334 
 0337 
 
 0341 
 0344 
 0347 
 0350 
 0354 
 
 300 
 302 
 304 
 306 
 308 
 
 310 
 312 
 314 
 316 
 318 
 
 320 
 322 
 324 
 326 
 328 
 
 330 
 332 
 334 
 336 
 338 
 
 340 
 342 
 344 
 346 
 348 
 
 0.0357 
 0.0360 
 0.0364 
 0.0367 
 0.0371 
 
 0.0375 
 0 . 0378 
 0.0381 
 0 . 0385 
 0.0389 
 
 0 . 0392 
 0 . 0396 
 0 . 0400 
 0 . 0403 
 0 . 0406 
 
 0.0410 
 0.0414 
 0.0417 
 0.0421 
 0.0426 
 
 0 . 0429 
 0 . 0433 
 0.0437 
 0 . 0440 
 0.0444 
 
 0 . 350 
 0 . 352 
 0 . 354 
 0 . 356 
 0.358 
 
 0.0448 0. 
 0.0452 0. 
 0 . 0457 
 0.0461 
 0.0465 
 
 0 . 360 0 . 0469 
 0 . 362 0 . 0473 
 0.364 0.0477 
 0.366 0.0481 
 0.368 0.0485 
 
 0.370 
 0.372 
 0 . 374 
 0.376 
 0.378 
 
 0.380 
 0.382 
 0.384 
 
 0 . 0489 
 0 . 0493 
 0.0497 
 0 . 0502 
 0 0507 
 
 0.0511 
 0.0515 
 0.0520 
 
 0.386 0.0524 
 0.388 0.0529 
 
 400 1 0. 0556 0, 
 40210.0560 0, 
 404:0.056510, 
 406l0.0570|0, 
 408 0.0575 0, 
 
 410 
 412 
 414 
 416 
 418 
 
 420 
 422 
 424 
 426 
 428 
 
 430 
 432 
 434 
 
 0.057910. 
 0.058310. 
 0.0588:0. 
 0.0593|0. 
 0.0598 0, 
 
 0 . 0603 
 0 . 0608 
 0.0613 
 0.0618 
 0 . 0623 
 
 0.0629 
 0.0634 
 0 . 0639 
 436 0.0644 
 438 0.0650 
 
 0.390 0.0533 0 
 0.39210.0537 0 
 0.394 0.0542 0 
 0.396 0.0546 0, 
 0 . 398 0 . 05.50 0 
 
 440 0.0655 
 442 0.0660 
 444 0 . 0665 
 446 0.0671 
 448 0.0677 
 
 450 
 452 
 454 
 456 
 458 
 
 460 
 462 
 46i 
 466 
 468 
 
 4-70 
 472 
 474 
 476 
 478 
 
 480 
 482 
 484 
 486 
 488 
 
 0 . 0682 0 . 500 0 . 0834 
 0.0687 0.502 iO. 0840 
 0.0693 0.504,0.0846 
 0.0697 0.506,0.0853 
 0.0704 0.508 0.0860 
 
 0.0710 0.510 
 0.0716 0.512 
 
 0.0721 
 0.0727 
 0.0733 
 
 0.0739 
 0 . 0744 
 0 . 0750 
 0 . 0756 
 0 . 0763 
 
 0.0770 
 0 . 0776 
 0.0781 
 0 . 0788 
 0.0795 
 
 490 0.0801 
 492 0.0807 
 494 0.0813 
 496 0.0820 
 498 0.0828 
 
 0.514 
 0.516 
 0.518 
 
 0.520 
 0.522 
 0 . 524 
 0.526 
 0.528 
 
 0 . 530 
 0.532 
 0.534 
 
 0.0867 
 0 . 0873 
 0 . 0880 
 0 . 0888 
 0.0895 
 
 0.0903 
 0.0910 
 0.0917 
 0.0925 
 0.0933 
 
 0.0940 
 0.0947 
 0.0954 
 
 0.536 0.0962 
 0.538 0.0970 
 
 0.540 0.0979 
 0.542 1 0.0985 
 0.-544 0.0913 
 0.546 0. 1002 
 0.548 0.1011 
 
 5.W,0. 
 552 iO. 
 
 554 
 556 
 558 
 
 500 
 562 
 564 
 566 
 568 
 
 570 
 572 
 574 
 576 
 578 
 
 580 
 582 
 584 
 ■586 
 588 
 
 590 
 592 
 .594 
 596 
 598 
 
 1019 
 1027 
 1035 
 1043 
 1052 
 
 1060 
 1070 
 1079 
 1089 
 1097 
 
 1105 
 1112 
 1 122 
 1132 
 1143 
 
 1152 
 1160 
 1170 
 1180 
 1190 
 
 1200 
 1210 
 1220 
 1230 
 1240 
 
 Table 9. — Values of 
 
 k 
 
 IN Formula fc = 
 
 n(l - k) ^ — — jc _ 
 
 Applies to Rectangular Beams, T-beams, and Beams With Steel Top and Bottom 
 
 Based on n = 15 
 
 0.200 
 0.202 
 0.204 
 0.206 
 0.208 
 
 0.210 
 0.212 
 0.214 
 0.216 
 0.218 
 
 0.220 
 0.222 
 0.224 
 0.226 
 0.228 
 
 0.230 
 0.232 
 0.234 
 0.236 
 0.238 
 
 0.240 
 0.242 
 0.244 
 0.246 
 0.248 
 
 0.0166 
 0.0169 
 0.0171 
 0.0173 
 0.0175 
 
 0.0177 
 0.0179 
 0.0181 
 
 0.0183 
 0.0186 
 
 0.0188 
 0.0190 
 0.0192 
 0.0194 
 0.0196 
 
 0.0199 
 0.0201 
 0 . 0203 
 0.0205 
 0 . 0207 
 
 0.0210 
 0.0212 
 0.0214 
 0.0217 
 0.0219 
 
 0.250 
 0 . 252 
 0 . 2.54 
 0.256 
 0.258 
 
 260 
 262 
 264 
 266 
 268 
 
 0 . 0222 
 0.0224 
 0.0226 
 0.0229 
 0.0231 
 
 0.0234 
 0 . 0236 
 0 . 0238 
 0.0241 
 0.0243 
 
 0.300 
 0.302 
 0 . 304 
 0.306 
 0.308 
 
 0.310 
 0.312 
 0.314 
 0.316 
 0.318 
 
 270 0.0246 
 272 0.0248 
 27410.0251 
 27610.0254 0.326 
 278 0.0257 0.328 
 
 0 . 320 
 0.322 
 0.324 
 
 280 
 282 
 284 
 286 
 288 
 
 290 
 292 
 294 
 296 
 298 
 
 0.0259 
 0.0261 
 0 . 0264 
 0.0267 
 0 . 0269 
 
 0.0272 
 0.0275 
 0.0278 
 0.0280 
 0.0283 
 
 0.330 
 0.332 
 0.334 
 0.336 
 0.338 
 
 0.340 
 0.342 
 0.344 
 0.346 
 0.348 
 
 0.0286 
 0.0288 
 0.0291 
 0.0293 
 0.0296 
 
 0 . 0299 
 0.0301 
 0 . 0305 
 0.0.308 
 0.0311 
 
 0.0314 
 0.0317 
 0.0319 
 
 0.0322 0.376 0 
 0.0325 0.378,0 
 
 0.0328 0.380 0 
 0.0331 0.382 0 
 0.0334 0.384 0 
 0.0337 0.386 0, 
 0.0340:0.388 0 
 
 0.034410. 390,0 
 0.0347 0.392 0, 
 0.0350 0.394 0, 
 0.0353 0.396 0, 
 0.0356 0.398 0, 
 
 0 
 
 350 0 
 
 0 
 
 352 0 
 
 0 
 
 354 0 
 
 0 
 
 356 
 
 0 
 
 0 
 
 358 
 
 0 
 
 0 
 
 360 
 
 0 
 
 0 
 
 302 
 
 0 
 
 0 
 
 364 
 
 0 
 
 0 
 
 366 
 
 0 
 
 0 
 
 368 
 
 0 
 
 0 
 
 370 
 
 0 
 
 0 
 
 372 
 
 0 
 
 0 
 
 374 0 
 
 0.400 
 0.402 
 0.404 
 0.406 
 0.408 
 
 . 0375 
 .0378 
 .0381 
 .0384 
 .0387 0 
 
 0391 
 0394 
 ,0398 
 0401 
 0404 
 
 0408 
 0411 
 0415 
 0419 
 0422 
 
 0426 
 0429 
 0433 
 0437 
 0440 
 
 410 
 412 
 414 
 410 
 418 
 
 0.0444 
 0.0447 
 0.0451 
 0.0455 
 0.0459 
 
 0.0463 
 0 . 0406 
 0.0470 
 0 . 0474 
 0.0478 
 
 420 0.0482 
 422 0 . 0486 
 424 0.0490 
 426 i 0.0494 
 428|0.0498 
 
 430 ' 0.0.502 
 432^0. 0506 
 434 0.0511 
 436 0.0515 
 4.38|0.0520 
 
 440 0.0524 
 442 1 0.0528 
 444 0.0532 
 446 0.0536 
 448 0.0540 
 
 450 
 452 
 454 
 456 
 458 
 
 0.0666 
 0.0671 
 0.0677 
 0 . 0682 
 0.0562 0.508 0.0688 
 
 0.0545 0.500 
 0.0549 0.502 
 0 . 05.54 0 . 504 
 0 . 0558 0 . 506 
 
 460 0.0567 
 402 0.0571 
 0.0576 
 0.0581 
 0.0586 
 
 464 
 466 
 468 
 
 470 0 
 472 0 
 
 474,0 
 476|0 
 478 0 
 
 480 0 
 
 48210 
 484 0 
 486i0 
 488 iO 
 
 .0591 
 .0,595 
 .0600 
 . 0005 
 .0610 
 
 .0615 
 .0620 
 .0625 
 .0630 
 . 0635 
 
 490 0.0640 
 492 0.0645 
 494 0.0650 
 496 0.0656 
 498 0.0660 
 
 510 
 512 
 514 
 516 
 518 
 
 0.0694 
 0 . 0699 
 0 . 0705 
 0.0711 
 0.0717 
 
 550 
 5.52 
 554 
 556 
 558 
 
 560 
 562 
 564 
 566 
 568 
 
 0.0815 
 0.0821 
 0 . 0827 
 0 . 0834 
 0.0841 
 
 0 . 0848 
 0 . 0855 
 0 . 0862 
 0 . 0869 
 0.0870 
 
 520 0.0723 0. 
 522 0.0728 0. 
 524 0.0733 0. 
 
 52610.0739 
 528 0.0745 
 
 530 
 532 
 5.34 
 536 
 538 
 
 540 
 542 
 544 
 546 
 548 
 
 0.0751 
 0.0757 
 0.0763 
 0.0770 
 0.0776 
 
 0.0783 
 0.0789 
 0 . 0795 
 0.0802 
 0 . 0808 
 
 570 0.0884 
 572 0.0891 
 574 0 . 0898 
 576 0.0905 
 578 0.0912 
 
 580 
 582 
 584 
 586 
 588 
 
 590 
 592 
 594 
 596 
 598 
 
 0.0920 
 0.0927 
 0.0935 
 0.0943 
 0.0951 
 
 0 . 0959 
 0.0967 
 0.0975 
 0.0984 
 0.0993 
 
Sec. 7-57 
 
 BEAMS AND SLABS 
 
 361 
 
 Diagram 3. — Use for Wedge-shaped Beams. 
 
 In designing, use Diagram 1 or 2 and then multiply the values of K and p obtained by cos-Pc and -- ^^ f " respec- 
 
 cospi 
 
 lively. (These products may be obtained directly from diagram given below.) 
 
 In reviewing, determine K and p in the usual manner and then multiply these values by and S^^^L 
 
 COS-pc CO.S'^c 
 
 respectively before using Diagram 1 or 2. (These products may be obtained directly from diagram given below.) 
 
 Values "C" =(Va lues 'A' x -§||^ ) 
 17 16 15 U 13 12 n -10 9 6 7 6 5 4 3 2 1 0 
 
 O I 2 3 -4- 5 6 7 8 9 10 U 12 13 14 15 16 17 
 
 Values X = (Values "CVf§|^)=CVaIue5'B"x c5i%c) 
 
 Diagram 4. —Use for Finding Approximate Weight of Rectangular Beams. 
 
 0.70 0.75 0.80 085 090 095 1.00 1.05 1.10 K15 
 
 Values j>f V 
 
CONCRETE ENGINEERS' HANDBOOK [Sec. 7-g' 
 
 DiAGKAM 5. 
 
 Bending Moments in Slabs for Uniformly Distributed Loads. 
 
 Span \n Feei- 
 
 14 13 12 f1 10 9 8 .7 6 5 4 3 
 
 0 H M I II M M 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 n I 1 1 1 1 1 n M M II 1 1 inrrmi 
 
 3 4 5 6 7 8 9 10 11 IE 13 14 15 
 
 Span in Feet 
 
 362 
 
 15 
 
Sec. 7-57] 
 
 BEAMS AND SLABS 
 
 363 
 
 o in o in 
 
 epuno^ qoui j.o spuosnoty^ u! 4.usuuov\ Suipusg 
 
CONCRETE ENGINEERS' HANDBOOK 
 
 [Sec. 7 
 
 Diagram 7. 
 Use for T-beams. 
 
 Based on fs = 16,000; n = 12. 
 .3 ■ .4, 300 400 SOO 600 
 
 700 
 
 800 
 
 Values of ^ 
 
 500 600 VOO 
 
 Values of fc 
 
 Diagram 8. 
 Use for T-beams. 
 
 Based on fs = 16,000; n = 15. 
 .4 30d 400 
 
 500 
 
 600, 
 
 700 
 
Sec. 7-57] 
 
 BEAMS AND SLABS 
 
 365 
 
 Diagram 9. 
 Use for T-beams. 
 
 Values of k and j for various percentages of steel. Based on n = 12. 
 
 Values of J 
 
 Values of 3 
 
 Diagram 10. 
 Use for T-beams. 
 
 Values of k and j for various percentages of steel. Based on n = 15. 
 
 .65 
 
 ^^^^ 
 
 3 AO 
 35 
 
 20 
 
 :2t 
 
 Vciiwes of -I 
 
 Values of J 
 
366 CONCRETE ENGINEERS' HANDBOOK [Sec. 7 
 
 Diagram 11. 
 
 Use for Rectangular Beams With Steel in Top and Bottom. 
 
 Based onn = 12. 
 
 Vofluea of p' 
 
Sec. 7-57] BEAMS AND SLABS 367 
 
 Diagram 11. — (Continued) 
 
 Values of d' 
 
368 
 
 CONCRETE ENGINEERS' HANDBOOK 
 
 [Sec. 7 
 
 Diagram 12. 
 
 Use for Rectangular Beams With Steel in Top and Bottom. 
 
 Based on re = 15. 
 
370 
 
 CONCRETE ENGINEERS' HANDBOOK 
 
 [Sec. 7-57 
 
 Diagram 13. 
 Leffler's Comprehensive Beam Chart. 
 
 Curves for -r 
 a 
 
 0.10 and n = 15. 
 
 Values of Ls 
 
 .005 006 .007 .008 .009 OlO 
 
 D25 fi30 .030 
 
 .lol I M M M M I 
 
 V X Zf'Zf Along p'O asl'TSOandg 
 
 8 2 a I Li .ncneases U ^8 
 
 s g Along p=0 osl-eSOortl- 
 
m I 
 
 » i 
 
 lis 
 
 lili 
 
 II 
 o a 
 
I 
 
SECTION 8 
 
 COLUMNS 
 
 1. Column Types. — Concrete columns are of four principal types: 
 
 1. Plain concrete columns or piers. 
 
 2. Columns reinforced with longitudinal rods only. 
 
 3. Columns reinforced with both hoops and longitudinal rods. 
 
 4. Columns reinforced with structural-steel shapes. 
 
 2. Plain Concrete Columns or Piers. — The Joint Committee does not consider any com- 
 pression member a column unless it is reinforced and has a ratio of unsupported length to least 
 width greater than four (see Art. 7 and Appendix B). Compression mem- 
 bers in which the unsupported length to least width is four or less are referred 
 to as piers. Piers may or may not be reinforced depending upon the stresses 
 to which they are subjected. 
 
 Bending stresses in columns due to eccentric loads must be provided for 
 by increasing the section until the maximum stress does not exceed the allow- 
 able. A formula for homogeneous columns or piers follows. Formulas applic- 
 able to reinforced-concrete columns or reinforced piers are given under "Bend- 
 ing and Direct Stress," Sect. 9. 
 
 The ordinary formula for the compressive fiber stress due to eccentric 
 loading upon solid rectangular columns or piers of homogeneous materials 
 (Fig. 1) is as follows: 
 
 N = total load. 
 A = area of column. 
 Xq = eccentricity. 
 t = breadth of column. 
 
 I 
 
 i 
 
 Fig. 
 
 fc = total unit pressure on outer fiber nearest to line of vertical pressure. 
 
 Then 
 
 0 + "?) 
 
 and the additional intensity of compressive stress due to eccentric loading is seen to be equal 
 N 6xo 
 
 ^""A'-T' 
 
 3. Columns with Longitudinal Reinforcement. — Since the modulus of elasticity of a mate- 
 rial is the ratio of stress to deformation, it follows that for equal deformations the stresses in the 
 steel and concrete of a concrete column will be as their moduli of elasticity. Thus 
 
 or fs - ScU 
 
 fs ^ E. 
 fc Ec 
 Let A = total net area. 
 Ac = area of concrete. 
 As = area of longitudinal steel. 
 
 As 
 
 p = steel ratio = ^ • 
 
 fs = tensile unit stress in steel. 
 
 fc = compressive unit stress in concrete. 
 
 371 
 
372 
 
 CONCRETE ENGINEERS' HANDBOOK 
 
 [Sec. 8-4 
 
 Es 
 
 P = total strength of a reinforced column for the stress fc. 
 
 Then 
 
 P =fcAc +fsAs =fc{A - vA) +fcnpA 
 
 or 
 
 P = fcA [1 + (n - 
 
 Tests^ on columns with vertical steel bar reinforcement indicate that the steel may be 
 counted upon in design to take its portion of the loading as computed from the above equation. 
 
 The economy of steel reinforcement is dependent upon the working stresses permissible 
 in the concrete and the value of n, since the stress in the steel = fcTi. The stresses in the steel 
 will be relatively low except in the unusual combination of high working stresses in the com- 
 crete with a large value of n. 
 
 4. Columns with Hooped and Longitudinal Reinforcement. — Whenever a material is 
 subjected to compression along one axis, then, as a consequence, there will be an expansion of 
 the material along axes which are perpendicular to the one first considered. Thus, if the mate- 
 rial of a column is held laterally, then lateral compressive stresses are developed which tend to 
 neutralize the effect of the longitudinal compressive stresses and thus to increase the resistance 
 against failure. This is the principle involved in the use of spiral or banded reinforcement. 
 The addition of bands or spirals to columns having longitudinal reinforcement does not have 
 much effect upon the deformation of such -columns up to the point of failure without hooping. 
 In fact the elastic limit and rigidity of the column appears to be decreased if anything. The 
 effect of such hooping, however, raises slightly the ultimate strength and increases the capacity 
 of the column to deform at loads beyond the elastic limit, so that a somewhat higher working 
 stress may be employed on the concrete than for plain concrete columns. Tests show that about 
 1 % of a closely spaced spiral of high-carbon steel is sufficient to prevent the longitudinal rods 
 from bulging outward and will provide a satisfactory amount of toughness with a corresponding 
 raising of the ultimate strength beyond the elastic limit. 
 
 The following notation will be used for hooped reinforcement. 
 
 Let di = diameter of enclosed concrete to center line of hooping. 
 Ah = sectional area of one strand of hooping for given pitch, 
 s = pitch allowed. 
 
 Ih = length of hooping in 1 ft. in height of column, 
 p = percentage of hooping. 
 
 Then 
 
 For p = 1% 
 Also 
 
 Ah= ^ 
 
 spdi 
 
 (0.0025) (s)((ii) or 
 
 AhIh ='^'(12) (0.01) 
 
 0.0025di 
 
 37.7rfi 
 
 ih = — 
 
 s 
 
 Results for banded and spiral reinforcement will not differ appreciably and the above 
 formulas may be used for both bands and spirals. 
 
 5. Columns Reinforced with Structural-steel Shapes. — If a structural-steel column is 
 designed to take all the load and then is simply fireproofed with a covering of concrete, it 
 
 » For tests on columns, see "Concrete, Plain and Reinforced," by Taylor and Thompson, 3d Edition, 1916. 
 
Sec. 8-6] 
 
 COLUMNS 
 
 373 
 
 cannot properly be called a reinforced-concrete column. To be classed under this heading the 
 steel must be designed so that it takes a load in combination with the concrete; that is, the 
 steel must be figured in the same way as vertical rods and the stresses determined by the formu- 
 las previously given. 
 
 Structural-steel reinforcement is sometimes in the form of a cross in the center of the column 
 or more often angles are employed connected by riveted latticing. Tests of columns of this 
 character generally show lower ultimate strength than similar columns reinforced with the same 
 quantity of steel in the form of vertical rods. This is most likely due to the difficulty of prop- 
 erly placing the concrete around the steel, and, furthermore, to the fact that the adhesion of 
 concrete to steel where the latter presents broad flat surfaces is not good. 
 
 To be able to count upon the concrete in columns reinforced with structural forms, the 
 steel should be well enclosed either by the form itself or by means of bands or hooping. How- 
 ever, when the amount of steel becomes very large, the relative value of the concrete becomes 
 more uncertain, and it would be good design to neglect its element of strength. 
 
 6. Working Stresses. — Working stresses recommended by the Joint Committee are given 
 in Art. 7. 
 
 7. Recommendations of the Joint Committee. — The form given below is essentially that 
 of the Special Committee on Concrete and Reinforced Concrete of the American Society of 
 Civil Engineers, Proc. Am. Soc. C. E., Dec, 1916, p. 1688. 
 
 By columns are meant compression members of which the ratio of unsupported length to least width 
 exceeds about four, and which are provided with reinforcement of one of the forms hereafter described. 
 
 It is recommended that the ratio of unsupported length of column to its least width be limited to 15. 
 
 The effective area of hooped columns or columns reinforced with structural shapes shall be taken as the area 
 within the circle enclosing the spiral or the polygon enclosing the structural shapes. 
 
 Columns may be reinforced by longitudinal bars; by bands, hoops, or spirals, together with longitudinal bars; 
 or by structural forms which are sufficiently rigid to have value in themselves as columns. The general effect of 
 closely spaced hooping is to greatly increase the" toughness of the column and to add to its ultimate strength, 
 but hooping has little effect on its behavior within the limit of elasticity. It thus renders the concrete a safer and 
 more reliable material, and should permit the use of a somewhat higher working stress. The beneficial effects of 
 toughening are adequately provided by a moderate amount of hooping, a larger amount serving mainly to increase 
 the ultimate strength and the deformation possible before ultimate failure. 
 
 Composite columns of structural steel and concrete, in which the steel forms a column by itself, should be 
 designed with caution. To classify this type as a concrete column reinforced with structural steel is hardly per- 
 missible, as the steel generally will take the greater part of the load. When this type of column is used, the con- 
 crete should not be relied upon to tie the steel units together nor to transmit stresses from one unit to another. The 
 units should be adequately tied together by tie-plates or lattice bars, which, together with other details, such as 
 splices, etc., should be designed in conformity with standard practice for structural steel. The concrete may exert 
 a beneficial effect in restraining the steel from lateral deflection and also in increasing the carrying capacity of the 
 column. The proportion of load to be carried by the concrete will depend on the form of the column and the 
 method of construction. Generally for high percentages of steel, the concrete will develop relatively low unit 
 stresses, and caution should be used in placing dependence on the concrete. 
 
 The following recommendations are made for the relative working stresses in the concrete for the several 
 types of columns: 
 
 (a) For concentric compression on a plain-concrete pier, the length of which does not exceed four diameters, 
 22.5% of the compressive strength may be allowed. 
 
 (6) Columns with longitudinal reinforcement to the extent of not less than 1 % and not more than 4 %, and 
 with lateral ties of not less than J-i in. in diameter, 12 in. apart, nor more than 16 diameters of the longitudinal 
 bar: the unit stress recommended for (a). 
 
 (c) Columns reinforced with not less than 1 % and not more than 4 % of longitudinal bars and with circular 
 hoops or spirals not less than 1 % of the volume of the concrete and as hereinafter specified: a unit stress 55% 
 higher than given for (a), provided the ratio of unsupported length of column to diameter of the hooped core is not 
 more than 10. 
 
 The foregoing recommendations are based on the following conditions: 
 
 It is recommended that the minimum size of columns to which the working stresses may be applied be 12 in., 
 out to out. 
 
 The hoops or bands are not to be counted on directly as adding to the strength of the column. 
 
 Longitudinal reinforcement bars should be maintained straight, and shall have sufficient lateral support 
 to be securely held in place until the concrete has set. 
 
 Where hooping is used, the total amount of such reinforcement shall not be less than 1 % of the volume of the 
 column, enclosed. The clear spacing of such hooping shall be not greater than one-sixth the diameter of the en- 
 
374 
 
 CONCRETE ENGINEERS' HANDBOOK 
 
 [Sec. 8-8 
 
 closed column, and preferably not greater than one-tenth, and in no case more than 2H in. Hooping is to be 
 circular and the ends of bands must be united in such a way as to develop their full strength. Adequate means 
 must be provided to hold bands or hoops in place so as to form a column, the core of which shall be straight and 
 well-centered. The strength of hooped columns depends very much upon the ratio of length to diameter of hooped 
 core, and the strength due to hooping decreases rapidly as this ratio increases beyond five. The working stresses 
 recommended are for hooped columns with a length of not more than 10 diameters of the hooped core. The Com- 
 mittee has no recommendations to make for a formula for working stresses for columns longer than 10 diameters. 
 
 Bending stresses due to eccentric loads, such as unequal spans of beams, and to lateral forces, must be pro- 
 vided for by increasing the section until the maximum stress does not exceed the values above specified. Where 
 tension is possible in the longitudinal bars of the column, adequate connection between the ends of the bars must 
 be provided to take this tension. 
 
 8. Tables and Diagrams. — Table 1 gives the allowable load P for different sizes and shapes 
 of columns reinforced with longitudinal bars and reinforced with both longitudinal bars and spiral 
 reinforcement. The area of longitudinal steel is given for each size of column and percentage 
 of reinforcement. It is assumed that the longitudinal bars in the round and octagonal columns 
 are arranged to form a circle in the cross-section of the column with the hooping immediately 
 outside this circle. 
 
 Table 2 gives the sectional area of hooping and length of hooping per foot of column for 
 a maximum pitch of one-sixth (3-^) the diameter of the enclosed concrete but not to exceed 2}-2 
 in. according to the recommendations of the Joint Committee (see Art. 7). For some diam- 
 eters, values are given for a pitch of about one-tenth {}io) the diameter of enclosed concrete. 
 This table can be used for spiral reinforcement without material error. 
 
 Table 3 gives the volume of column in cubic feet per foot of length for diameter D. 
 
 The column diagram given can be used for either designing or reviewing designs of columns. 
 
 Ili^ustrative Problem. — What size of square column reinforced with 2% of longitudinal steel and with the 
 required number of lateral ties will be required to support a centrally applied load of 900,000 lb.? A 3000-lb. 
 concrete is to be used and the unsupported length of column is less than 15 diameters. 
 
 From Art. 3, 
 
 P 
 
 P = fcA[l + (n - Dp] 
 
 Ml + {n- Dp] 
 
 From Art. 7(6), fc = 22.5% of -3000 lb. = 675 lb. per sq. in. Also n = 10 (see Appendix B). Then the 
 effective area of column 
 
 900.000 _ 900,000 _ 
 
 ~ 675[1 + (10 - 1)(0.02^] ~ 797 ~ ^^^^ 
 
 The value of /c[l -f (n — 1)] = 797 may be obtained directly from the column diagram. 
 The side of effective square area 
 
 d = ■v/1130 = 33.6 in., say 34 in. 
 
 A column 37 in. square will suffice considering D/i in. of concrete all around as fireproofing. 
 
 The problem may be readily solved by using Table 1. For a 3000-lb. concrete and p = 0.02, we find that a 
 column 36 in. square will support 868,000 lb. and a 37-in. square column will support 922,000 lb. Thus a column 
 37 in. square is ample. 
 
 Illustrative Problem. — What size of round column and area of longitudinal steel will be required to sup- 
 port a load of 1,100,000 lb.? A 3000-lb. concrete is to be used with 1 % of spiral reinforcement. Unsupported 
 length is less than 10 diameters. Take p = 0.025 and n = 10. 
 
 From Art. 7(c) we find that the value of /c may be taken 55% greater than for (a), or (1.55) (3000) (0.225^) = 
 1050 lb. per sq. in. 
 
 Then from Art. 3 
 
 P = fcA[l + (n - Dp] 
 The column diagram shows /c[l + (w — l)p] = 1287. Therefore 
 
 P = 1287(A) or A = ^'^^287^^ ^ 
 
 !^^ = 855 d = \[^^ = SSiu. D = 36in. 
 
 4 ^ TT 
 
 As = (0.025) (855) = 21.4 sq. in. 
 
 From Table 1 we find directly that a 36-in. column will be of sufficient size and that As = 21.4 sq. in. 
 
 Table 2 shows that for a column with d = 33 in., the pitch of spiral should not exceed 2V2 in., with a sectional 
 area o£ spiral of 0.206 sq. in. and with a length of spiral per foot of column height of 498 ia. 
 
Sec, 8-8] 
 
 COLUMNS 
 
 375 
 
 Table 1.— Use for Round, Octagonal and Square Columns 
 P - fcA[l + (n - l)p] A> = pA 
 D = outside diameter. For octagonal columns D = short diameter. For square colomns D = side 
 
 A » for round and octagonal columns. 
 A = for square columns. 
 
 of square. 
 
 
 
 1:2:4 (2,000-lb. concrete) 
 
 1 : 11/2 : 3 
 
 (2,500-lh. concrete) 
 
 1:1:2 
 
 ( 3, 000-1 b. concrete) 
 
 
 
 
 
 
 
 n = 15 
 
 
 
 n = 12 
 
 
 
 n = 10 
 
 
 A, 
 (sa. in.) 
 
 og 
 
 eg 
 
 
 
 1 % spiral 
 /c = 7001b. 
 
 
 
 1 % spiral 
 /e = 8701b. 
 
 
 
 1 % spiral 
 /c= 1,050 
 
 
 
 
 ^§ 
 
 fc = 4^0 lb. 
 
 fc = 565 lb. 
 
 fc = 675 lb. 
 
 
 
 
 amet( 
 umns 
 
 ctive 
 olum 
 
 per sq 
 
 . in. 
 
 per sq. in. 
 
 per sq. in. 
 
 per sq. in. 
 
 per sq. in. 
 
 lb. per 
 sq. in. 
 
 0) 
 
 i-i 
 
 o 
 
 onal 
 
 
 
 
 
 
 
 
 
 
 C3 
 
 -d 
 
 M 
 
 Qg 
 
 
 Square 
 
 Round 
 
 Round 
 
 Square 
 
 Round 
 
 Round 
 
 Square 
 
 Round 
 
 Round 
 
 3 
 D* 
 
 c 
 
 3 
 
 03 
 
 
 W o 
 
 or oct. 
 
 or oct. 
 
 or oct. 
 
 or oct. 
 
 or Oct. 
 
 or oct. 
 
 
 O 
 « 
 
 O 
 
 
 
 P (lb.) 
 
 P (lb.) 
 
 P (lb.) 
 
 P (lb.) 
 
 P (lb.) 
 
 P (lb.) 
 
 P (lb.) 
 
 P (lb.) 
 
 P (lb.) 
 
 
 
 
 P = 0.01 
 
 10 
 
 7 
 
 25,100 
 
 19,700 
 
 30,700 
 
 30,700 
 
 24,100 
 
 37,200 
 
 36,100 
 
 28,300 
 
 44,100 
 
 0.5 
 
 0 
 
 4 
 
 11 
 
 8 
 
 32,800 
 
 25,800 
 
 40,100 
 
 40,100 
 
 31,500 
 
 48,600 
 
 47,100 
 
 37,000 
 
 57,600 
 
 0.6 
 
 0 
 
 5 
 
 12 
 
 9 
 
 41,500 
 
 32,600 
 
 50,800 
 
 50.800 
 
 39,900 
 
 61,500 
 
 59,600 
 
 46,800 
 
 72,800 
 
 0.8 
 
 0 
 
 6 
 
 13 
 
 10 
 
 51,300 
 
 40,300 
 
 62,700 
 
 62,700 
 
 49,300 
 
 75,900 
 
 73,600 
 
 57,800 
 
 89,900 
 
 1.0 
 
 0 
 
 8 
 
 14 
 
 11 
 
 62,100 
 
 48,800 
 
 75,800 
 
 75,800 
 
 59,600 
 
 91,800 
 
 89,100 
 
 70,000 
 
 103,800 
 
 1.2 
 
 1 
 
 0 
 
 1 5 
 
 12 
 
 73 900 
 
 58 100 
 
 90,200 
 
 90,200 
 
 71,900 
 
 109,300 
 
 106,000 
 
 83 300 
 
 129 700 
 
 1.4 
 
 1 
 
 I 
 
 16 
 
 13 
 
 86! 700 
 
 681200 
 
 105^900 
 
 105^900 
 
 83^500 
 
 128,' 200 
 
 124^400 
 
 97;600 
 
 152^000 
 
 1.7 
 
 1 
 
 3 
 
 17 
 
 14 
 
 100,700 
 
 79,000 
 
 122,300 
 
 122,800 
 
 96,600 
 
 148,700 
 
 144,200 
 
 113,200 
 
 176,000 
 
 2.0 
 
 1 
 
 5 
 
 18 
 
 15 
 
 115,500 
 
 90,800 
 
 141,000 
 
 141,000 
 
 110,800 
 
 171.000 
 
 166,000 
 
 130,000 
 
 ;j03,000 
 
 2.3 
 
 1 
 
 3 
 
 19 
 
 16 
 
 131,500 
 
 103,200 
 
 161,000 
 
 161,000 
 
 126,100 
 
 194.000 
 
 183,000 
 
 148,000 
 
 230,000 
 
 2.6 
 
 2 
 
 0 
 
 20 
 
 17 
 
 148,300 
 
 116,400 
 
 181.000 
 
 181,000 
 
 142,300 
 
 219,000 
 
 213.000 
 
 167,000 
 
 260,000 
 
 2.9 
 
 2 
 
 3 
 
 21 
 
 18 
 
 166,000 
 
 130,500 
 
 203,000 
 
 203,000 
 
 159,000 
 
 246.000 
 
 239,000 
 
 188,000 
 
 291,000 
 
 3.2 
 
 2 
 
 6 
 
 22 
 
 19 
 
 185,000 
 
 145,500 
 
 220,000 
 
 226,000 
 
 178,000 
 
 274,000 
 
 266,000 
 
 209.000 
 
 325,000 
 
 3.6 
 
 2 
 
 8 
 
 23 
 
 20 
 
 205.000 
 
 161,000 
 
 251,000 
 
 251,000 
 
 197,000 
 
 303,000 
 
 295,000 
 
 231.000 
 
 360,000 
 
 4.0 
 
 3 
 
 1 
 
 24 
 
 21 
 
 226,000 
 
 178,000 
 
 276,000 
 
 276,000 
 
 217,000 
 
 335,000 
 
 325,000 
 
 255,000 
 
 397,000 
 
 4.4 
 
 3 
 
 5 
 
 25 
 
 22 
 
 249,000 
 
 195,000 
 
 303,000 
 
 303,000 
 
 235,000 
 
 367,000 
 
 356,000 
 
 280,000 
 
 435,000 
 
 4.8 
 
 3 
 
 8 
 
 26 
 
 23 
 
 272,000 
 
 213,000 
 
 332,000 
 
 332,000 
 
 260,000 
 
 401,000 
 
 389,000 
 
 306,000 
 
 476,000 
 
 5.3 
 
 4 
 
 2 
 
 27 
 
 24 
 
 295,000 
 
 232,000 
 
 361,000 
 
 361,000 
 
 283,000 
 
 437,000 
 
 424,000 
 
 333,000 
 
 518,000 
 
 5.8 
 
 4 
 
 5 
 
 28 
 
 25 
 
 321,000 
 
 252,000 
 
 392,000 
 
 392,000 
 
 308,000 
 
 474,000 
 
 460,000 
 
 361,000 
 
 562,000 
 
 6.3 
 
 4 
 
 9 
 
 29 
 
 26 
 
 347,000 
 
 272,000 
 
 424,000 
 
 424,000 
 
 333,000 
 
 513,000 
 
 498,000 
 
 381,000 
 
 608,000 
 
 6.8 
 
 5 
 
 3 
 
 30 
 
 27 
 
 374,000 
 
 294,000 
 
 457,000 
 
 457,000 
 
 359,000 
 
 553,000 
 
 537,000 
 
 422,000 
 
 656,000 
 
 7.3 
 
 5 
 
 7 
 
 31 
 
 28 
 
 403,000 
 
 316,000 
 
 491,000 
 
 491,000 
 
 386,000 
 
 595,000 
 
 577,000 
 
 453,000 
 
 705,000 
 
 7.8 
 
 6 
 
 2 
 
 32 
 
 29 
 
 432,000 
 
 339,000 
 
 527,000 
 
 527,000 
 
 414,000 
 
 638,000 
 
 619,000 
 
 486,000 
 
 757.000 
 
 8.4 
 
 6 
 
 6 
 
 33 
 
 30 
 
 462,000 
 
 362,000 
 
 564,000 
 
 564,000 
 
 443,000 
 
 683,000 
 
 663,000 
 
 520,000 
 
 808,000 
 
 9.0 
 
 7 
 
 1 
 
 34 
 
 31 
 
 493,000 
 
 387,000 
 
 602,000 
 
 602,000 
 
 473,000 
 
 729,000 
 
 708,000 
 
 555,000 
 
 864,000 
 
 9.6 
 
 
 6 
 
 35 
 
 32 
 
 526,000 
 
 413,000 
 
 642,000 
 
 642,000 
 
 504.000 
 
 777,000 
 
 755,000 
 
 592,000 
 
 922,000 
 
 10.2 
 
 I 
 
 0 
 
 36 
 
 33 
 
 559,000 
 
 438,000 
 
 683,000 
 
 683,000 
 
 536,000 
 
 826,000 
 
 802,000 
 
 630,000 
 
 980,000 
 
 10.9 
 
 8 
 
 6 
 
 37 
 
 34 
 
 593,000 
 
 406,000 
 
 725,000 
 
 725,000 
 
 569,000 
 
 877,000 
 
 851,000 
 
 668,000 
 
 1,039,000 
 
 11.6 
 
 
 1 
 
 38 
 
 35 
 
 629,000 
 
 494,000 
 
 768,000 
 
 768,000 
 
 603,000 
 
 929,000 
 
 901,000 
 
 708,000 
 
 1,100,000 
 
 12.3 
 
 I 
 
 6 
 
 39 
 
 36 
 
 665,000 
 
 523,000 
 
 812,000 
 
 812,000 
 
 638,000 
 
 983,000 
 
 953,000 
 
 750,000 
 
 1,170,000 
 
 13.0 10 
 
 2 
 
 40 
 
 37 
 
 703,000 
 
 542,000 
 
 858,000 
 
 858,000 
 
 675,000 
 
 1,040,000 
 
 1.007,000 
 
 792,000 
 
 1,230,000 
 
 13.7 10 
 
 8 
 
 41 
 
 38 
 
 742,000 
 
 582,000 
 
 905,000 
 
 905,000 
 
 711,000 
 
 1,100,000 
 
 1,060,000 
 
 835,000 
 
 1,300,000 
 
 14.4 11 
 
 3 
 
 42 
 
 39 
 
 780,000 
 
 613,000 
 
 954,000 
 
 951,000 
 
 749,000 
 
 1,150,000 
 
 1,120,000 
 
 879,000 
 
 1,370,000 
 
 15.2 12 
 
 0 
 
 p = 0.015 
 
 10 
 
 7 
 
 26,700 
 
 21,000 
 
 32.600 
 
 32,200 
 
 25,300 
 
 39,000 
 
 37,500 
 
 29,500 
 
 45,900 
 
 0.7 
 
 0 
 
 6 
 
 11 
 
 8 
 
 34,900 
 
 27,400 
 
 42,600 
 
 42,100 
 
 33,100 
 
 51,000 
 
 49,100 
 
 38,500 
 
 59,900 
 
 1.0 
 
 0 
 
 8 
 
 12 
 
 9 
 
 44,100 
 
 36,700 
 
 53,900 
 
 53,300 
 
 41,800 
 
 64,500 
 
 62,100 
 
 48,800 
 
 75,800 
 
 1.2 
 
 1 
 
 0 
 
 13 
 
 10 
 
 54,500 
 
 42,800 
 
 66,500 
 
 65,800 
 
 51,700 
 
 79,600 
 
 76,600 
 
 60,200 
 
 93,600 
 
 1.5 
 
 1 
 
 2 
 
 14 
 
 11 
 
 65,900 
 
 51,800 
 
 80,500 
 
 79,700 
 
 62,500 
 
 96,400 
 
 92,700 
 
 72,800 
 
 113,200 
 
 1.8 
 
 1 
 
 4 
 
 15 
 
 12 
 
 78,500 
 
 61,700 
 
 95,800 
 
 94,800 
 
 74,400 
 
 114,700 
 
 110,300 
 
 86,600 
 
 134,800 
 
 2.2 
 
 1 
 
 7 
 
 16 
 
 13 
 
 92,100 
 
 72,400 
 
 112,400 
 
 111,100 
 
 87,400 
 
 134,600 
 
 129,400 
 
 101,700 
 
 158,000 
 
 2.5 
 
 2 
 
 0 
 
 17 
 
 14 
 
 106,800 
 
 83,900 
 
 130,400 
 
 129,000 
 
 101,300 
 
 156,000 
 
 150,000 
 
 118,000 
 
 183,000 
 
 2.9 
 
 2 
 
 3 
 
 18 
 
 15 
 
 122,700 
 
 96,300 
 
 150,000 
 
 148,000 
 
 116,400 
 
 179,000 
 
 172,000 
 
 135,300 
 
 211,000 
 
 3.4 
 
 2 
 
 7 
 
 19 
 
 16 
 
 139,500 
 
 109,600 
 
 170,000 
 
 169,000 
 
 132,300 
 
 204,000 
 
 196,000 
 
 154,000 
 
 240,000 
 
 3.8 
 
 3 
 
 0 
 
 20 
 
 17 
 
 157,500 
 
 123,700 
 
 192,000 
 
 190,000 
 
 149,400 
 
 230,000 
 
 221,000 
 
 174,000 
 
 271,000 
 
 4.3 
 
 3 
 
 4 
 
 21 
 
 18 
 
 176,500 
 
 138,600 
 
 216,000 
 
 213,000 
 
 168,000 
 
 258,000 
 
 249,000 
 
 195,000 
 
 304,000 
 
 4.9 
 
 3 
 
 8 
 
 22 
 
 19 
 
 196,800 
 
 155,000 
 
 240,000 
 
 237,000 
 
 187,000 
 
 288,000 
 
 277,000 
 
 217,000 
 
 338,000 
 
 5.4 
 
 4 
 
 3 
 
 23 
 
 20 
 
 218,000 
 
 171,000 
 
 266,000 
 
 263,000 
 
 207,000 
 
 319,000 
 
 307,000 
 
 241,000 
 
 374,000 
 
 6.0 
 
 4 
 
 7 
 
 24 
 
 21 
 
 240,000 
 
 189,000 
 
 293,000 
 
 290,000 
 
 228,000 
 
 351,000 
 
 338,000 
 
 265,000 
 
 413,000 
 
 6.6 
 
 5 
 
 2 
 
 25 
 
 22 
 
 264,000 
 
 207,000 
 
 322,000 
 
 319,000 
 
 250,000 
 
 385,000 
 
 371,000 
 
 291,000 
 
 453,000 
 
 7.3 
 
 5 
 
 7 
 
 26 
 
 23 
 
 288,000 
 
 227,000 
 
 352.000 
 
 348,000 
 
 273,000 
 
 422,000 
 
 415,000 
 
 318,000 
 
 495,000 
 
 7.9 
 
 6 
 
 2 
 
 27 
 
 24 
 
 314,000 
 
 247,000 
 
 383,000 
 
 379,000 
 
 297,000 
 
 459,000 
 
 442,000 
 
 347,000 
 
 538,000 
 
 8.6 
 
 6 
 
 8 
 
 23 
 
 25 
 
 341,000 
 
 267,000 
 
 416,000 
 
 411,000 
 
 323,000 
 
 498,000 
 
 479,000 
 
 376,000 
 
 585,000 
 
 9.4 
 
 7 
 
 4 
 
 29 
 
 26 
 
 368,000 
 
 289,000 
 
 450,000 
 
 444,000 
 
 349,000 
 
 538,000 
 
 518,000 
 
 407.000 
 
 633,000 
 
 10. 1 
 
 8 
 
 0 
 
 30 
 
 27 
 
 397,000 
 
 312,000 
 
 485,000 
 
 480,000 
 
 377,000 
 
 581,000 
 
 558,000 
 
 439,000 
 
 682,000 
 
 10.9 
 
 8 
 
 6 
 
 31 
 
 28 
 
 428,000 
 
 335,200 
 
 522,000 
 
 516,000 
 
 405,000 
 
 624,000 
 
 601,000 
 
 465,000 
 
 733,000 
 
 11.8 
 
 9 
 
 2 
 
 32 
 
 29 
 
 458,000 
 
 360,000 
 
 559,000 
 
 554,000 
 
 434,000 
 
 670,000 
 
 644,000 
 
 506,000 
 
 787,000 
 
 12.6 
 
 9 
 
 9 
 
 33 
 
 30 
 
 481,000 
 
 385,000 
 
 599,000 
 
 593,000 
 
 465,000 
 
 716,000 
 
 690,000 
 
 541,000 
 
 841,000 
 
 13.5 
 
 10 
 
 6 
 
 34 
 
 31 
 
 524,000 
 
 412,000 
 
 639,000 
 
 632,000 
 
 497,000 
 
 765,000 
 
 736,000 
 
 577,000 
 
 899,000 
 
 14.4 
 
 11 
 
 3 
 
 35 
 
 32 
 
 559,000 
 
 438,000 
 
 681,000 
 
 674,000 
 
 529,000 
 
 816,000 
 
 775,000 
 
 616,000 
 
 953,000 
 
 15.4 
 
 12.1 
 
 36 
 
 33 
 
 594,000 
 
 466,000 
 
 724,000 
 
 717,000 
 
 563,000 
 
 867,000 
 
 835,000 
 
 655,000 
 
 1,019,000 
 
 16.3 
 
 12 
 
 8 
 
 •37 
 
 34 
 
 630,000 
 
 495,000 
 
 769,000 
 
 761,000 
 
 598,000 
 
 921,000 
 
 887,000 
 
 696,000 
 
 1,081,000 
 
 17.3 
 
 13 
 
 6 
 
 38 
 
 35 
 
 668,000 
 
 525,000 
 
 815,000 
 
 806,000 
 
 634,000 
 
 976,000 
 
 938,000 
 
 737,000 
 
 1,146,000 
 
 18.4 
 
 14 
 
 4 
 
 39 
 
 36 
 
 706,000 
 
 555,000 
 
 862,000 
 
 853,000 
 
 670,000 
 
 1,032,000 
 
 993,000 
 
 770,000 
 
 1,212,000 
 
 19.4 
 
 15 
 
 3 
 
 40 
 
 37 
 
 746,000 
 
 586,000 
 
 911,000 
 
 901,000 
 
 708,000 
 
 1,090,000 
 
 1,050,000 
 
 824,000 
 
 1,280,000 
 
 20.5 
 
 16 
 
 1 
 
 1 4^ 
 
 38 
 
 787,000 
 
 618,000 
 
 961,000 
 
 950,000 
 
 746,000 
 
 1,150,000 
 
 1,110,000 
 
 868,000 
 
 1,350,000 
 
 21.7 
 
 17 
 
 0 
 
 (42 
 
 39 
 
 829,000 
 
 651,000 
 
 1,012,000 
 
 ] ,000,000 
 
 786,000 
 
 1,211,000 
 
 1,170,000 
 
 915,000 
 
 1,420,000 
 
 22.8 
 
 17 
 
 9 
 
376 
 
 CONCRETE ENGINEERS' HANDBOOK 
 
 [Sec. 8-8 
 
 
 
 1:2:4 
 
 (2,000-lb. concrete) 
 
 1: VA:3 
 
 (2,500-lb. concrete) 
 
 1:1:2 
 
 (3,000-lb. 
 
 concrete) 
 
 
 
 
 
 
 n = 15 
 
 
 n = 12 
 
 
 
 n = 10 
 
 
 (sq. 
 
 in.) 
 
 
 
 
 
 1 % spiral 
 A = 700 lb. 
 
 
 
 1 % spiral 
 /c = 8701b. 
 
 
 
 1 % spiral 
 
 
 fc = 450 lb. 
 
 fc = 565 lb. 
 
 fc = 675 lb. 
 
 /c= 1,050 
 lb. per 
 sq. in. 
 
 
 
 Diamet< 
 columns 
 
 tive 
 lum: 
 
 per sq 
 
 . in. 
 
 per sq. in. 
 
 per sq. in. 
 
 per sq. in. 
 
 per s 
 
 q. m. 
 
 (U 
 
 
 
 
 
 
 
 
 
 
 
 
 ta 
 
 73 O 
 
 of c- 
 
 Square 
 
 Round 
 or oct. 
 
 Round 
 or oct. 
 
 Square 
 
 Round 
 or oct. 
 
 Round 
 or oct. 
 
 Square 
 
 Round 
 or oct. 
 
 Round 
 or oct. 
 
 Squi 
 
 Roun 
 octag 
 
 
 
 P (lb.) 
 
 P (lb.) 
 
 P (lb.) 
 
 P (lb.) 
 
 P (lb.) 
 
 P (lb.) 
 
 P (lb.) 
 
 P (lb.) 
 
 P (lb.) 
 
 
 V = 0.02 
 
 10 
 
 7 
 
 28,200 
 
 22,200 
 
 34,500 
 
 33,800 
 
 26.500 
 
 40,800 
 
 .39,100 
 
 30,700 
 
 47,700 
 
 1.0 
 
 0.8 
 
 11 
 
 8 
 
 36,900 
 
 29,000 
 
 45,000 
 
 44,100 
 
 34,600 
 
 53,300 
 
 51,000 
 
 40,100 
 
 62,300 
 
 1.3 
 
 1.0 
 
 12 
 
 9 
 
 46,700 
 
 36,700 
 
 57.000 
 
 55,700 
 
 43,800 
 
 67,500 
 
 64,600 
 
 50,700 
 
 78,800 
 
 1.6 
 
 1.3 
 
 13 
 
 10 
 
 57,600 
 
 45,300 
 
 70,400 
 
 68,900 
 
 54,100 
 
 83,300 
 
 79,700 
 
 62,600 
 
 97,400 
 
 2.0 
 
 1.6 
 
 14 
 
 11 
 
 69,600 
 
 54,800 
 
 85,200 
 
 83,400 
 
 65,500 
 
 100,800 
 
 96,400 
 
 75,800 
 
 117,800 
 
 2.-^ 
 
 1.9 
 
 15 
 
 12 
 
 83,000 
 
 65,200 
 
 101,300 
 
 99,300 
 
 78,000 
 
 120,000 
 
 114,600 
 
 90,100 
 
 140,000 
 
 2.9 
 
 2.3 
 
 16 
 
 13 
 
 97,300 
 
 76,500 
 
 118,900 
 
 116,500 
 
 91,500 
 
 140,800 
 
 134,800 
 
 105,800 
 
 165,000 
 
 3.4 
 
 2.7 
 
 17 
 
 14 
 
 113,000 
 
 88,700 
 
 137,900 
 
 135,000 
 
 106,100 
 
 163,000 
 
 156,000 
 
 123,600 
 
 191,000 
 
 3.9 
 
 3.1 
 
 18 
 
 15 
 
 129,600 
 
 101,800 
 
 153,000 
 
 155,000 
 
 121,800 
 
 187,000 
 
 179,000 
 
 140,900 
 
 219,000 
 
 4.5 
 
 3.5 
 
 19 
 
 16 
 
 147,400 
 
 115,900 
 
 180,000 
 
 176,000 
 
 138,600 
 
 213,000 
 
 204,000 
 
 160.000 
 
 249,000 
 
 5,1 
 
 4.0 
 
 20 
 
 17 
 
 166,000 
 
 130,700 
 
 203,000 
 
 199,000 
 
 156,000 
 
 241,000 
 
 230,000 
 
 181,000 
 
 281,000 
 
 5.8 
 
 4.5 
 
 21 
 
 18 
 
 187,000 
 
 146,500 
 
 228,000 
 
 223,000 
 
 175,000 
 
 270,000 
 
 258,000 
 
 203.000 
 
 316,000 
 
 6 5 
 
 5 1 
 
 22 
 
 19 
 
 208,000 
 
 163,000 
 
 254,000 
 
 249,000 
 
 195,000 
 
 301,000 
 
 288,000 
 
 226,000 
 
 351,000 
 
 7.2 
 
 5^7 
 
 23 
 
 20 
 
 230,000 
 
 181,000 
 
 281,000 
 
 276,000 
 
 217,000 
 
 ,333,000 
 
 319,000 
 
 250,000 
 
 389,000 
 
 8.0 
 
 6 3 
 
 24 
 
 21 
 
 254,000 
 
 200,000 
 
 310,000 
 
 304,000 
 
 239,000 
 
 367,000 
 
 351,000 
 
 276,000 
 
 430,000 
 
 8.8 
 
 6.9 
 
 25 
 
 22 
 
 279,000 
 
 219,000 
 
 341,000 
 
 333,000 
 
 262,000 
 
 403,000 
 
 386,000 
 
 303,000 
 
 471,000 
 
 9,7 
 
 7.6 
 
 26 
 
 23 
 
 305,000 
 
 239,000 
 
 372,000 
 
 364,000 
 
 286,000 
 
 441,000 
 
 422,000 
 
 331,000 
 
 515,000 
 
 10.6 
 
 8.3 
 
 27 
 
 24 
 
 332,000 
 
 261,000 
 
 400,000 
 
 397,000 
 
 311,000 
 
 480,000 
 
 459,000 
 
 361,000 
 
 561,000 
 
 11.5 
 
 9.0 
 
 28 
 
 25 
 
 360,000 
 
 283,000 
 
 440,000 
 
 431,000 
 
 338,000 
 
 521,000 
 
 498,000 
 
 391,000 
 
 608,000 
 
 12.5 
 
 9.9 
 
 29 
 
 26 
 
 389,000 
 
 306,000 
 
 476,000 
 
 466,000 
 
 366,000 
 
 563,000 
 
 538,000 
 
 423,000 
 
 658,000 
 
 13.5 
 
 10.6 
 
 30 
 
 27 
 
 420,000 
 
 330,000 
 
 513,000 
 
 503,000 
 
 395,000 
 
 607,000 
 
 581,000 
 
 456,000 
 
 710,000 
 
 14.6 
 
 11.5 
 
 31 
 
 28 
 
 452,000 
 
 355,000 
 
 552,000 
 
 540,000 
 
 424,000 
 
 653,000 
 
 625,000 
 
 491,000 
 
 763,000 
 
 15.7 
 
 12.3 
 
 32 
 
 29 
 
 485,000 
 
 380,000 
 
 592,000 
 
 579,000 
 
 455,000 
 
 701,000 
 
 670,000 
 
 526,000 
 
 818,000 
 
 16.8 
 
 13.2 
 
 33 
 
 30 
 
 519,000 
 
 407,000 
 
 633,000 
 
 620,000 
 
 487,000 
 
 750,000 
 
 717,000 
 
 563,000 
 
 876,000 
 
 18.0 
 
 14. 1 
 
 34 
 
 31 
 
 554,000 
 
 435,000 
 
 676,000 
 
 662,000 
 
 520,000 
 
 801,000 
 
 766,000 
 
 603,000 
 
 935,000 
 
 19.2 
 
 15.1 
 
 35 
 
 32 
 
 590,000 
 
 464,000 
 
 721,000 
 
 706,000 
 
 5.54,000 
 
 853,000 
 
 817,000 
 
 641,000 
 
 997,000 
 
 20.5 
 
 16.1 
 
 36 
 
 33 
 
 628,000 
 
 493,000 
 
 766,000 
 
 750,000 
 
 589,000 
 
 907,000 
 
 868,000 
 
 682,000 
 
 1,060,000 
 
 21.8 
 
 17.1 
 
 37 
 
 34 
 
 666,000 
 
 523,000 
 
 814,000 
 
 797,000 
 
 625,000 
 
 963,000 
 
 922.000 
 
 723,000 
 
 1,125,000 
 
 23.1 
 
 18.2 
 
 38 
 
 35 
 
 706,000 
 
 555,000 
 
 862,000 
 
 844,000 
 
 663,000 
 
 1,021,000 
 
 976,000 
 
 767,000 
 
 1,190,000 
 
 24.5 
 
 19.2 
 
 39 
 
 36 
 
 746,000 
 
 587,000 
 
 912,000 
 
 893,000 
 
 702,000 
 
 1,080,000 
 
 1,032,000 
 
 811,000 
 
 1,260,000 
 
 25.9 
 
 20.4 
 
 40 
 
 37 
 
 789,000 
 
 620,000 
 
 963,000 
 
 943,000 
 
 742,000 
 
 1,141,000 
 
 1,090,000 
 
 857,000 
 
 1,330,000 
 
 27.4 
 
 21.5 
 
 41 
 
 38 
 
 832,000 
 
 653,000 
 
 1,016,000 
 
 985,000 
 
 782,000 
 
 1,200,000 
 
 1,150,000 
 
 904,000 
 
 1,410,000 
 
 28.9 
 
 22.7 
 
 42 
 
 39 
 
 876,000 
 
 688,000 
 
 1,070,000 
 
 1,048,000 
 
 824,000 
 
 1,270,000 
 
 1,210,000 
 
 952,000 
 
 1,480,000 
 
 30.4 
 
 23.9 
 
 p = 0.025 
 
 10 
 
 7 
 
 29,800 
 
 23,400 
 
 36,370 
 
 35,300 
 
 27,700 
 
 42,700 
 
 40,500 
 
 31,800 
 
 49,500 
 
 1.2 
 
 1.0 
 
 11 
 
 8 
 
 38,900 
 
 30,600 
 
 47,500 
 
 46,100 
 
 362,00 
 
 55,800 
 
 52,900 
 
 41,600 
 
 64,700 
 
 1.6 
 
 1.3 
 
 12 
 
 9 
 
 49,300 
 
 38,700 
 
 60,100 
 
 58,200 
 
 45,800 
 
 70,600 
 
 67,000 
 
 52,600 
 
 81,800 
 
 2.0 
 
 1.6 
 
 13 
 
 10 
 
 60,800 
 
 47,800 
 
 74,200 
 
 72,000 
 
 56,500 
 
 87,100 
 
 82,700 
 
 64,500 
 
 101,000 
 
 2.5 
 
 2.0 
 
 14 
 
 11 
 
 73,500 
 
 57,800 
 
 89,800 
 
 87,100 
 
 68,500 
 
 105,400 
 
 100,000 
 
 78,600 
 
 122,000 
 
 3.0 
 
 2.4 
 
 15 
 
 12 
 
 87,600 
 
 68, sop 
 
 106,900 
 
 103,700 
 
 81,500 
 
 125,400 
 
 119,000 
 
 93,500 
 
 145,600 
 
 3.6 
 
 2.8 
 
 16 
 
 13 
 
 102,900 
 
 80,700 
 
 125,000 
 
 121,800 
 
 95,600 
 
 147,200 
 
 140,000 
 
 109,800 
 
 171,000 
 
 4.2 
 
 3.3 
 
 17 
 
 14 
 
 119,200 
 
 93.600 
 
 145,000 
 
 141,200 
 
 111,000 
 
 171,000 
 
 162,000 
 
 127,200 
 
 198,000 
 
 4.9 
 
 3.9 
 
 18 
 
 15 
 
 136,900 
 
 107,500 
 
 167,000 
 
 162,000 
 
 127,000 
 
 196,000 
 
 186,000 
 
 146,100 
 
 227,000 
 
 5.6 
 
 4.4 
 
 19 
 
 16 
 
 156,000 
 
 122,300 
 
 190,000 
 
 184,000 
 
 145,000 
 
 223,000 
 
 212,000 
 
 166,000 
 
 259,000 
 
 6.4 
 
 5.0 
 
 20 
 
 17 
 
 176,000 
 
 138,000 
 
 215.000 
 
 208,000 
 
 163,000 
 
 252,000 
 
 239,000 
 
 188,000 
 
 292,000 
 
 7.2 
 
 5.7 
 
 21 
 
 18 
 
 197,000 
 
 155,000 
 
 240,000 
 
 233,000 
 
 183,000 
 
 282,000 
 
 268,000 
 
 211,000 
 
 328,000 
 
 8.1 
 
 6.4 
 
 22 
 
 19 
 
 219,000 
 
 173,000 
 
 268,000 
 
 260,000 
 
 204,000 
 
 314,000 
 
 298,000 
 
 235,000 
 
 365,000 
 
 9.0 
 
 7.1 
 
 23 
 
 20 
 
 243.000 
 
 191,000 
 
 297,000 
 
 288,000 
 
 226,000 
 
 348,000 
 
 331,000 
 
 260,000 
 
 404,000 
 
 10.0 
 
 7.9 
 
 24 
 
 21 
 
 268,000 
 
 211,000 
 
 327.000 
 
 317,000 
 
 249,000 
 
 384,000 
 
 365,000 
 
 286,000 
 
 445,000 
 
 11.0 
 
 8.7 
 
 25 
 
 22 
 
 295,000 
 
 231,000 
 
 359,000 
 
 348,000 
 
 274,000 
 
 422,000 
 
 400,000 
 
 314,000 
 
 489,000 
 
 12.1 
 
 9.5 
 
 26 
 
 23 
 
 322,000 
 
 253,000 
 
 393,000 
 
 381,000 
 
 299,000 
 
 461,000 
 
 438,000 
 
 344,000 
 
 534,000 
 
 13.2 
 
 10.4 
 
 27 
 
 24 
 
 351,000 
 
 275,000 
 
 428,000 
 
 415,000 
 
 325,000 
 
 .502,000 
 
 476,000 
 
 374,000 
 
 582,000 
 
 14.4 
 
 11.3 
 
 28 
 
 25 
 
 380,000 
 
 299,000 
 
 464,000 
 
 450,000 
 
 353,000 
 
 544,000 
 
 517,000 
 
 406,000 
 
 631,000 
 
 15.6 
 
 12.3 
 
 29 
 
 26 
 
 418,000 
 
 323,000 
 
 502,000 
 
 486,000 
 
 382,000 
 
 589,000 
 
 559,000 
 
 439,000 
 
 683,000 
 
 16.9 
 
 13.3 
 
 30 
 
 27 
 
 443,000 
 
 348,000 
 
 541,000 
 
 525,000 
 
 413,000 
 
 635,000 
 
 603,000 
 
 474,000 
 
 736,000 
 
 18.2 
 
 14.3 
 
 31 
 
 28 
 
 477,000 
 
 374,000 
 
 582,000 
 
 565,000 
 
 443,000 
 
 683,000 
 
 649,000 
 
 509,000 
 
 792,000 
 
 19.6 
 
 15.4 
 
 32 
 
 29 
 
 512,000 
 
 459,000 
 
 594,000 
 
 606,000 
 
 476,000 
 
 733,000 
 
 696,000 
 
 546,000 
 
 850,000 
 
 21.0 
 
 16.5 
 
 33 
 
 30 
 
 548,000 
 
 430,000 
 
 668,000 
 
 648,000 
 
 508,000 
 
 784,000 
 
 744,000 
 
 535,000 
 
 908,000 
 
 22.5 
 
 17.7 
 
 34 
 
 31 
 
 585,000 
 
 459,000 
 
 713,000 
 
 692,000 
 
 544,000 
 
 837,000 
 
 795,000 
 
 624,000 
 
 970,000 
 
 24.0 
 
 18.9 
 
 35 
 
 32 
 
 628,000 
 
 489,000 
 
 760,000 
 
 738,000 
 
 579,000 
 
 892,000 
 
 848,000 
 
 665,000 
 
 1,034,000 
 
 25.6 
 
 20.1 
 
 36 
 
 33 
 
 663,000 
 
 520,000 
 
 808,000 
 
 784,000 
 
 616,000 
 
 949,000 
 
 901,000 
 
 708,000 
 
 1,100,000 
 
 27.2 
 
 21 .4 
 
 37 
 
 34 
 
 703,000 
 
 553,000 
 
 858,000 
 
 833,000 
 
 654,000 
 
 1,007,000 
 
 958,000 
 
 751,000 
 
 1,170,000 
 
 28.9 
 
 22.7 
 
 38 
 
 35 
 
 745,000 
 
 585,000 
 
 909,000 
 
 882,000 
 
 693,000 
 
 1,070,000 
 
 1,012,000 
 
 796,000 
 
 1,240,000 
 
 30.6 
 
 24.1 
 
 39 
 
 36 
 
 789,000 
 
 619,000 
 
 962,000 
 
 933,000 
 
 733,000 
 
 1,130,000 
 
 1,110,000 
 
 842,000 
 
 1,310,000 
 
 32.4 
 
 25.5 
 
 40 
 
 37 
 
 833,000 
 
 654,000 
 
 1,016,000 
 
 985,000 
 
 775,000 
 
 1,190,000 
 
 1,130,000 
 
 889,000 
 
 1,380,000 
 
 34.2^26.9 
 
 41 
 
 38 
 
 878,000 
 
 690,000 
 
 1,070,000 
 
 1,040,000 
 
 817,000 
 
 1,260,000 
 
 1,190,000 
 
 938,000 
 
 1,460,000 36.0 28.4 
 
 42 
 
 39 
 
 925,000 
 
 727,000 
 
 1,130,000 
 
 1,095,000 
 
 860.000 
 
 1,320,000 
 
 1,260,000 
 
 988,000 
 
 1,540,000 38.0 29.9 
 
Sec. 8-8] 
 
 COLUMNS 
 
 377 
 
 
 
 1:2:4 
 
 (2,000-lb concrete) 
 
 1 : : 3 (2,500-lb. concrete) 
 
 1:1:2 
 
 (3,000-lb. concrete) 
 
 
 
 
 
 
 
 
 n = 15 
 
 
 
 71 = 12 
 
 
 
 n = 10 
 
 
 
 A 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 0) OQ 
 
 is 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 1 % spiral 
 /c = 7001b. 
 
 
 
 1 % spiral 
 /c = 8701b. 
 
 
 
 1 ^0 Spiral 
 
 
 
 
 
 'tl 
 
 fc = 450 1b. 
 
 fc= 505 lb. 
 
 fc = 075 lb. 
 
 /c= 1,050 
 
 
 
 
 
 s § 
 
 
 per sq. in. 
 
 per sq. in. 
 
 per sq. in. 
 
 per sq. in. 
 
 per sq. in. 
 
 sq. in. 
 
 
 
 
 Dia 
 coll 
 
 Effec 
 
 of CO 
 
 Square 
 
 Round 
 or Oct. 
 
 Round 
 or oct. 
 
 oquare 
 
 Round 
 or oct. 
 
 Round 
 or Oct. 
 
 oquare 
 
 Round 
 or Oct. 
 
 Round 
 or oct. 
 
 Squai 
 
 
 Round 
 octago: 
 
 
 
 P (lb.) 
 
 P (lb.) 
 
 P (lb). 
 
 P (lb.) 
 
 P (lb.) 
 
 P (lb.) 
 
 P (lb.) 
 
 P(lb.) 
 
 P (lb.) 
 
 
 
 
 
 
 
 
 
 P = 0.03 
 
 
 
 
 
 
 
 
 10 
 
 7 
 
 31,300 
 
 24.600 
 
 38,200 
 
 30,800 
 
 02 Qnr> 
 
 44,500 
 
 42 000 
 
 00 000 
 
 t3o,UUU 
 
 51,400 
 
 1 
 
 5 
 
 1 
 
 2 
 
 
 g 
 
 41,900 
 
 32,100 
 
 50,000 
 
 48^100 
 
 0 / ,oUU 
 
 58,200 
 
 01,oUU 
 
 A'i ^ no 
 
 'iO, lUU 
 
 67,100 
 
 1 
 
 9 
 
 1 
 
 5 
 
 12 
 
 9 
 
 51,800 
 
 40^600 
 
 03,200 
 
 00,700 
 
 1 / ,oUU 
 
 73,000 
 
 no 400 
 
 uy,'-rUU 
 
 54 500 
 
 84,900 
 
 2 
 
 4 
 
 1 
 
 9 
 
 13 
 
 10 
 
 63,900 
 
 50,200 
 
 78,100 
 
 75,100 
 
 oy ,uuu 
 
 yu,yu() 
 
 85 700 
 
 07 300 
 
 104,700 
 
 3 
 
 n 
 
 2 
 
 4 
 
 14 
 
 1 1 
 
 77,300 
 
 00,700 
 
 94,500 
 
 90,800 
 
 / I , 'JUU 
 
 1 1 n nnn 
 1 10,000 
 
 1 0*^ 700 
 luo, / uu 
 
 81 45C 
 
 126,700 
 
 3 
 
 6 
 
 2 
 
 9 
 
 15 
 
 12 
 
 92,100 
 
 72^300 
 
 112,400 
 
 losiloo 
 
 oo,uuu 
 
 1 Q 7 nnn 
 1 0 1 , UUO 
 
 1 9"^ "^00 
 
 t70, tjyjyj 
 
 151,000 
 
 4 
 
 Q 
 
 0 
 
 3 
 
 4 
 
 16 
 
 13 
 
 108,000 
 
 84,800 
 
 132,000 
 
 127,000 
 
 QQ 7nn 
 
 t ^ 1 nnn 
 
 144 800 
 
 1 1 Q COO 
 1 io,ouu 
 
 177,000 
 
 c 
 0 
 
 1 
 
 4 
 
 0 
 6 
 
 17 
 
 14 
 
 125,200 
 
 98,200 
 
 153,000 
 
 147^200 
 
 lie: nnn 
 
 1 7c nnn 
 1 /o,000 
 
 1 fis 000 
 
 1 Uo,UUU 
 
 1 "^9 000 
 
 205,000 
 
 5 
 
 Q 
 
 y 
 
 4 
 
 18 
 
 15 
 
 143,900 
 
 113^000 
 
 170,000 
 
 109^000 
 
 1 oz, / uu 
 
 ociA nnn 
 
 1 n Q 000 
 1 y«3,uuu 
 
 1 K-l 000 
 10 X ,UUU 
 
 236,000 
 
 0 
 
 0 
 0 
 
 5 
 
 3 
 
 19 
 
 16 
 
 164,000 
 
 128^400 
 
 200,000 
 
 192,000 
 
 10 1 ,uuu 
 
 00 Q nnn 
 
 91 Q 000 
 
 172 000 
 
 268,000 
 
 7 
 i 
 
 1 
 
 6 
 
 0 
 
 20 
 
 17 
 
 185,000 
 
 145,000 
 
 220,000 
 
 2 17] 000 
 
 171,000 
 
 203,000 
 
 04.0 oon 
 
 1 04 000 
 
 1 £7^,UUW 
 
 303,000 
 
 8 
 
 7 
 i 
 
 0 
 
 8 
 0 
 
 21 
 
 18 
 
 207^000 
 
 1 03,000 
 
 253,000 
 
 24 si 000 
 
 191,000 
 
 294,000 
 
 070 000 
 
 / OjUUU 
 
 910 000 
 z 1 y ,uuu 
 
 339,000 
 
 0 
 y 
 
 7 
 
 7 
 
 22 
 
 19 
 
 231^000 
 
 181^000 
 
 282,000 
 
 271^000 
 
 213,000 
 
 328,000 
 
 30Q 000 
 
 94*^ 000 
 
 378,000 
 
 1 0 
 
 lU 
 
 0 
 
 8 
 
 5 
 
 23 
 
 20 
 
 256,000 
 
 201,000 
 
 312,000 
 
 300^000 
 
 230,000 
 
 303,000 
 
 '^4'^ 000 
 
 9fiQ 000 
 
 419,000 
 
 1 0 
 
 n 
 u 
 
 9 
 
 4 
 
 24 
 
 21 
 
 282,000 
 
 22l[000 
 
 344,000 
 
 331^000 
 
 200,000 
 
 401,000 
 
 000 
 
 0 / 0, WO\7 
 
 9Q7 000 
 
 462,000 
 
 13 
 
 2 
 
 10 
 
 .4 
 
 25 
 
 22 
 
 310!000 
 
 243^000 
 
 378,000 
 
 3 03! 000 
 
 286,000 
 
 430,000 
 
 415 000 
 
 320,000 
 
 507,000 
 
 14 
 
 0 
 
 11 
 
 .4 
 
 26 
 
 23 
 
 338,000 
 
 205^000 
 
 413,000 
 
 397^000 
 
 312,000 
 
 481,000 
 
 453 000 
 
 350,000 
 
 555.000 
 
 15 
 
 Q 
 
 y 
 
 12 
 
 .5 
 
 27 
 
 24 
 
 368,000 
 
 289!000 
 
 450,000 
 
 433!000 
 
 349,000 
 
 523,000 
 
 494!000 
 
 387^000 
 
 004,000 
 
 17 
 
 3 
 
 13 
 
 .6 
 
 28 
 
 25 
 
 A no oon 
 40U,UUu 
 
 314,UUU 
 
 488,000 
 
 Add r\f\n 
 409.000 
 
 309,000 
 
 508,000 
 
 530,000 
 
 421,000 
 
 055,000 
 
 18 
 
 
 14 
 
 .7 
 
 29 
 
 26 
 
 432,000 
 
 339,000 
 
 528.000 
 
 508,000 
 
 399,000 
 
 014,000 
 
 579,000 
 
 455,000 
 
 708,000 
 
 20 
 
 
 15 
 
 .9 
 
 30 
 
 27 
 
 466,000 
 
 306,000 
 
 509,000 
 
 548,000 
 
 430,000 
 
 002,000 
 
 025,000 
 
 491,000 
 
 705,000 
 
 21 
 
 g 
 
 17 
 
 .2 
 
 31 
 
 28 
 
 501,000 
 
 393,000 
 
 012,000 
 
 588,000 
 
 402,000 
 
 712,000 
 
 072,000 
 
 528,000 
 
 822,000 
 
 23 
 
 
 18 
 
 .5 
 
 32 
 
 29 
 
 537,000 
 
 422,000 
 
 057,000 
 
 032,000 
 
 496,000 
 
 704,000 
 
 721,000 
 
 500,000 
 
 8^1,000 
 
 25 
 
 2 
 
 19 
 
 .8 
 
 33 
 
 30 
 
 575,000 
 
 452,000 
 
 703,000 
 
 070,000 
 
 531,000 
 
 818,000 
 
 772,000 
 
 005,000 
 
 943,000 
 
 27 
 
 .0 
 
 21 
 
 .2 
 
 34 
 
 31 
 
 614,000 
 
 482,000 
 
 750.000 
 
 722,000 
 
 507,000 
 
 873,000 
 
 824,000 
 
 040,000 
 
 1,006,000 
 
 28 
 
 .8 
 
 22 
 
 .0 
 
 35 
 
 32 
 
 655,000 
 
 514,000 
 
 799.000 
 
 709,000 
 
 004,000 
 
 931,000 
 
 878,000 
 
 089,000 
 
 1,072,000 
 
 30 
 
 .7 
 
 24 
 
 .1 
 
 36 
 
 33 
 
 695,000 
 
 540,000 
 
 850.000 
 
 818,000 
 
 043,000 
 
 990,000 
 
 933,000 
 
 733,000 
 
 1,140,000 
 
 32 
 
 .7 
 
 25 
 
 .7 
 
 37 
 
 34 
 
 739,000 
 
 580,000 
 
 902,000 
 
 807,000 
 
 082,000 
 
 1,050,000 
 
 991,000 
 
 778,000 
 
 1,210,000 
 
 34 
 
 .7 
 
 27 
 
 .2 
 
 38 
 
 35 
 
 783,000 
 
 015,000 
 
 950,000 
 
 920,000 
 
 723,000 
 
 1,113,000 
 
 1,050,000 
 
 825,000 
 
 1,280,000 
 
 30 
 
 .8 
 
 28 
 
 .9 
 
 39 
 
 36 
 
 829,000 
 
 050,000 
 
 1,012,000 
 
 973,000 
 
 765,000 
 
 1,178,000 
 
 1,120,000 
 
 873,000 
 
 1,360,000 
 
 38 
 
 .9 
 
 30 
 
 .5 
 
 40 
 
 37 
 
 875,000 
 
 087,000 
 
 1,009,000 
 
 1,028,000 
 
 808,000 
 
 1,240,000 
 
 1,170,000 
 
 921,000 
 
 1,440,000 
 
 41 
 
 .1 
 
 32 
 
 .3 
 
 41 
 
 38 
 
 923,000 
 
 725.000 
 
 1,127,000 
 
 1,084,000 
 
 852,000 
 
 1,310,000 
 
 1,240,000 
 
 972,000 
 
 1,510,000 
 
 43 
 
 .3 
 
 34 
 
 .0 
 
 42 
 
 39 
 
 972,000 
 
 704,000 
 
 1,187,000 
 
 1.140.000 
 
 898,000 
 
 1,380,000 
 
 1,300,000 
 
 1,023,000 
 
 1,590,000 
 
 45 
 
 .6 
 
 35 
 
 .8 
 
 p = 0.035 
 
 10 
 
 7 
 
 32,900 
 
 25,800 
 
 40,200 
 
 38,400 
 
 30,100 
 
 40,400 
 
 43,500 
 
 34,200 
 
 53.200 
 
 1 
 
 .7 
 
 1 
 
 .4 
 
 M 
 
 8 
 
 42,900 
 
 33,800 
 
 52,400 
 
 50,200 
 
 39,400 
 
 00,000 
 
 50,800 
 
 44,000 
 
 69,400 
 
 2 
 
 2 
 
 1 
 
 .8 
 
 12 
 
 9 
 
 54,300 
 
 42.700 
 
 00,400 
 
 03,300 
 
 49,800 
 
 70,700 
 
 72,000 
 
 50,500 
 
 87,900 
 
 2 
 
 8 
 
 2 
 
 .2 
 
 13 
 
 10 
 
 67,100 
 
 52,700 
 
 81,900 
 
 78,300 
 
 61,500 
 
 94,000 
 
 88,800 
 
 09,800 
 
 108,500 
 
 3 
 
 5 
 
 2 
 
 .8 
 
 14 
 
 11 
 
 81,100 
 
 03.800 
 
 99,100 
 
 94,800 
 
 75,500 
 
 114,500 
 
 107,500 
 
 84,500 
 
 131,300 
 
 4 
 
 2 
 
 3 
 
 .3 
 
 15 
 
 12 
 
 96,500 
 
 75.900 
 
 118,000 
 
 112,800 
 
 88,600 
 
 136,300 
 
 127,800 
 
 100,500 
 
 156,000 
 
 5 
 
 0 
 
 4 
 
 0 
 
 16 
 
 13 
 
 113,400 
 
 89.100 
 
 138.400 
 
 132,100 
 
 104,000 
 
 160,000 
 
 150,000 
 
 117,900 
 
 183,000 
 
 5 
 
 9 
 
 4 
 
 0 
 
 17 
 
 14 
 
 131,500 
 
 103.300 
 
 101.000 
 
 153,000 
 
 120,000 
 
 186,000 
 
 174,000 
 
 130,800 
 
 213,000 
 
 6 
 
 9 
 
 5 
 
 4 
 
 18 
 
 15 
 
 151,000 
 
 118.700 
 
 184,000 
 
 170,000 
 
 138,400 
 
 213,000 
 
 200,000 
 
 155,000 
 
 244,000 
 
 7 
 
 9 
 
 0 
 
 2 
 
 19 
 
 16 
 
 172,000 
 
 135,000 
 
 210,000 
 
 201.000 
 
 158,000 
 
 242,000 
 
 227,000 
 
 179,000 
 
 279,000 
 
 9 
 
 0 
 
 7 
 
 0 
 
 20 
 
 17 
 
 194,000 
 
 152,000 
 
 237,000 
 
 220.000 
 
 178,000 
 
 274,000 
 
 257,000 
 
 202,000 
 
 313,000 
 
 10 
 
 1 
 
 7 
 
 9 
 
 21 
 
 IS 
 
 217,000 
 
 171,000 
 
 205,000 
 
 254,000 
 
 199,000 
 
 307.000 
 
 289,000 
 
 220,000 
 
 352,000 
 
 11 
 
 3 
 
 8 
 
 9 
 
 22 
 
 19 
 
 242.000 
 
 190,000 
 
 296,000 
 
 283,000 
 
 222,000 
 
 342,000 
 
 320,000 
 
 252,000 
 
 391,000 
 
 12 
 
 7 
 
 9 
 
 9 
 
 23 
 
 20 
 
 268,000 
 
 211.000 
 
 328,000 
 
 313,000 
 
 246,000 
 
 379.000 
 
 355,000 
 
 279,000 
 
 434,000 
 
 14 
 
 0 
 
 11 
 
 0 
 
 24 
 
 21 
 
 296.000 
 
 232,000 
 
 301,000 
 
 345,000 
 
 271,000 
 
 417,000 
 
 392,000 
 
 308,000 
 
 478,000 
 
 15 
 
 4 
 
 12 
 
 1 
 
 25 
 
 22 
 
 325,000 
 
 255,000 
 
 390,000 
 
 379,000 
 
 298,000 
 
 458,000 
 
 430,000 
 
 338,000 
 
 525,000 
 
 17 
 
 0 
 
 13 
 
 3 
 
 26 
 
 23 
 
 355,000 
 
 279,000 
 
 433,000 
 
 414,000 
 
 325,000 
 
 501,000 
 
 470,000 
 
 309,000 
 
 500,000 
 
 18 
 
 5 
 
 14 
 
 5 
 
 27 
 
 24 
 
 386.000 
 
 303,000 
 
 472,000 
 
 451,000 
 
 354,000 
 
 545,000 
 
 512,000 
 
 402,000 
 
 025,000 
 
 20. 
 
 2 
 
 15 
 
 8 
 
 28 
 
 25 
 
 419,000 
 
 329,000 
 
 512,000 
 
 490,000 
 
 384,000 
 
 592,000 
 
 555,000 
 
 435,000 
 
 078,000 
 
 21 . 
 
 9 
 
 17 
 
 2 
 
 29 
 
 26 
 
 453,000 
 
 350,000 
 
 554,000 
 
 529,000 
 
 416,000 
 
 640,000 
 
 000,000 
 
 471.000 
 
 733,000 
 
 23. 
 
 7 
 
 18. 
 
 0 
 
 30 
 
 27 
 
 489,000 
 
 384,000 
 
 597,000 
 
 571,000 
 
 448,000 
 
 090,000 
 
 048,000 
 
 509.000 
 
 791,000 
 
 25. 
 
 5 
 
 20. 
 
 0 
 
 31 
 
 28 
 
 520,000 
 
 413,000 
 
 042,000 
 
 014,000 
 
 482,000 
 
 742,000 
 
 097,000 
 
 547,000 
 
 850,000 
 
 27. 
 
 4 
 
 21. 
 
 5 
 
 32 
 
 29 
 
 504,000 
 
 443.000 
 
 089,000 
 
 059,000 
 
 517,000 
 
 796,000 
 
 740,000 
 
 580,000 
 
 912,000 
 
 29. 
 
 5 
 
 23. 
 
 1 
 
 33 
 
 30 
 
 004,000 
 
 404,000 
 
 737,000 
 
 705,000 
 
 553,000 
 
 852,000 
 
 800,000 
 
 028,000 
 
 970,000 
 
 31. 
 
 5 
 
 24. 
 
 7 
 
 34 
 
 31 
 
 045.000 
 
 500,000 
 
 787,000 
 
 752,000 
 
 591,000 
 
 910,000 
 
 854.000 
 
 070,000 
 
 1,042,000 
 
 33. 
 
 7 
 
 20. 
 
 4 
 
 35 
 
 32 
 
 088,000 
 
 540,000 
 
 839,000 
 
 802,000 
 
 020,000 
 
 969,000 
 
 910.000 
 
 714.000 
 
 1,110,000 
 
 35.8 
 
 28. 
 
 1 
 
 36 
 
 33 
 
 731,000 
 
 574,000 
 
 892,000 
 
 853,000 
 
 070,000 
 
 1,031,000 
 
 908.000 
 
 700,000 
 
 1,181,000 
 
 38. 
 
 1 
 
 29. 
 
 9 
 
 37 
 
 34 
 
 770,000 
 
 010,000 
 
 947,000 
 
 905,000 
 
 711,000 
 
 1,094,000 
 
 1,027,000 
 
 816,000 
 
 1,254,000 
 
 40. 
 
 4 
 
 31. 
 
 8 
 
 38 
 
 35 
 
 822,000 
 
 040,000 
 
 1,003,000 
 
 959,000 
 
 754,000 
 
 1,159,000 
 
 1,088,000 
 
 855,000 
 
 1,330,000 
 
 42. 
 
 8 
 
 33. 
 
 7 
 
 39 
 
 36 
 
 870,000 
 
 083,000 
 
 1,002,000 
 
 1,015,000 
 
 798,000: 
 
 l,230,00()i 
 
 1,150,000 
 
 904,000 
 
 1,410,000 
 
 45. 
 
 3 
 
 35. 
 
 6 
 
 40 
 
 37 
 
 919,000 
 
 722.000 
 
 1,120,000 
 
 1,072,000 
 
 842,000 
 
 1,300,0001 
 
 1,220,000 
 
 955.000 
 
 1,490,000 
 
 47. 
 
 8 
 
 37. 
 
 0 
 
 41 
 
 38 
 
 969,000 
 
 701,000 
 
 1,180,000 
 
 1,130,000 
 
 888,0001 
 
 1,370,000 
 
 1,280,000 
 
 1,007,000 
 
 1,570,000 
 
 50. 
 
 6 
 
 39. 
 
 7 
 
 42 
 
 39 
 
 1,021,000 
 
 802,000 
 
 1,250,000 
 
 1,190,000 
 
 930,000 
 
 1,440,000 
 
 1.350.000 
 
 1,061,000 
 
 1,050,000|53.2 
 
 41. 
 
 8 
 
378 
 
 CONCRETE ENGINEERS' HANDBOOK 
 
 [Sec. 8-8 
 
 
 
 1:2:4 
 
 (2,000-lb concrete) 
 
 1 : IH : 3 
 
 (2,500-lb. concrete) 
 
 1:1:2 
 
 (3,000-lb. concrete) 
 
 
 
 
 
 
 
 
 n = 15 
 
 
 
 n = 12 
 
 
 
 n = 10 
 
 
 
 A 
 
 8 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 1 % spiral 
 fc = 700 lb. 
 
 
 
 1 % spiral 
 /c = 8701b. 
 
 
 
 1 % spiral 
 
 
 
 
 
 O m 
 
 ^ c 
 
 fc = 450 lb. 
 
 fc = 565 lb. 
 
 fc = 675 lb. 
 
 /c== 1,050 
 
 
 
 
 
 "S 2 
 11 
 
 tive 
 •lum 
 
 per sq 
 
 . in. 
 
 per sq. in. 
 
 per s 
 
 q. in. 
 
 per sq. in. 
 
 per sq. in. 
 
 lb. per 
 sq. in. 
 
 u 
 
 Round or 
 octagonal 
 
 q2 
 
 Effec 
 
 of CO 
 
 Square 
 
 Round 
 or Oct. 
 
 Round 
 or oct. 
 
 Square 
 
 Round 
 or oct. 
 
 Round 
 or oct. 
 
 Square 
 
 Round 
 or Oct. 
 
 Round 
 or oct. 
 
 Squa 
 
 
 
 P (lb.) 
 
 P (lb.) 
 
 P (lb). 
 
 P (lb.) 
 
 P (lb.) 
 
 P (lb.) 
 
 P (lb.) 
 
 P (lb.) 
 
 P (lb.) 
 
 
 
 
 
 V = 0.04 
 
 10 
 
 7 
 
 34,400 
 
 27.000 
 
 42,000 
 
 39,900 
 
 31.300 
 
 48,200 
 
 45,000 
 
 35,300 
 
 55,000 
 
 2 
 
 0 
 
 1 
 
 5 
 
 11 
 
 8 
 9 
 
 44.900 
 
 35.300 
 
 54,900 
 
 52,100 
 
 41,000 
 
 63,000 
 
 58,800 
 
 46,200 
 
 71,800 
 
 2 
 
 6 
 
 2 
 
 0 
 
 12 
 
 56,900 
 
 44.600 
 
 69,500 
 
 65,800 
 
 51,800 
 
 79, 700 
 
 74,400 
 
 58,400 
 
 90,900 
 
 3 
 
 2 
 
 2 
 
 5 
 
 13 
 
 10 
 
 70,200 
 
 55.100 
 
 85,800 
 
 81,400 
 
 64,000 
 
 98,400 
 
 91,800 
 
 72,100 
 
 112,200 
 
 4 
 
 0 
 
 3 
 
 1 
 
 14 
 
 11 
 
 84,900 
 
 66.700 
 
 103.800 
 
 98,500 
 
 77,400 
 
 119,100 
 
 111,000 
 
 87,300 
 
 135,800 
 
 4 
 
 8 
 
 3 
 
 8 
 
 15 
 
 12 
 
 101,000 
 
 79,400 
 
 123,500 
 
 117,200 
 
 92,000 
 
 141,700 
 
 132,200 
 
 103,800 
 
 162,000 
 
 5 
 
 8 
 
 4 
 
 5 
 
 16 
 
 13 
 
 118,600 
 
 93,200 
 
 144,900 
 
 137,600 
 
 108,100 
 
 166,000 
 
 155,000 
 
 121,900 
 
 190,000 
 
 6 
 
 8 
 
 5 
 
 3 
 
 17 
 
 14 
 
 137,500 
 
 108,000 
 
 168,000 
 
 160,000 
 
 125,300 
 
 193,000 
 
 180,000 
 
 141,400 
 
 220,000 
 
 7 
 
 8 
 
 6 
 
 2 
 
 18 
 
 15 
 
 158,000 
 
 124,000 
 
 193.000 
 
 183,000 
 
 143.900 
 
 221,000 
 
 207,000 
 
 162,000 
 
 252,000 
 
 9 
 
 0 
 
 7 
 
 .1 
 
 19 
 
 16 
 
 180,000 
 
 141,200 
 
 220,000 
 
 208,000 
 
 164,000 
 
 252,000 
 
 235,000 
 
 185,000 
 
 287,000 
 
 10 
 
 2 
 
 8 
 
 0 
 
 20 
 
 17 
 
 203,000 
 
 159,000 
 
 248,000 
 
 235,000 
 
 185,000 
 
 234,000 
 
 265,000 
 
 208,000 
 
 324,000 
 
 11 
 
 6 
 
 9 
 
 .1 
 
 21 
 
 18 
 
 227,000 
 
 179,000 
 
 278,000 
 
 264.000 
 
 207,000 
 
 319,000 
 
 297.000 
 
 234,000 
 
 364,000 
 
 13 
 
 0 
 
 10 
 
 .2 
 
 22 
 
 19 
 
 253,000 
 
 199,000 
 
 310,000 
 
 294,000 
 
 231,000 
 
 355,000 
 
 331.000 
 
 261,000 
 
 405,000 
 
 14 
 
 4 11 
 
 .3 
 
 23 
 
 20 
 
 281,000 
 
 221,000 
 
 343,000 
 
 326,000 
 
 256,000 
 
 394,000 
 
 367.000 
 
 289,000 
 
 448,000 
 
 16 
 
 0 12 
 
 6 
 
 24 
 
 21 
 
 310,000 
 
 243,000 
 
 378,000 
 
 349,000 
 
 282,000 
 
 434,000 
 
 405.000 
 
 318,000 
 
 495,000 
 
 17 
 
 6 
 
 13 
 
 .9 
 
 25 
 
 22 
 
 830,000 
 
 267,000 
 
 415,000 
 
 394,000 
 
 309,000 
 
 476,000 
 
 445,000 
 
 349,000 
 
 543,000 
 
 19 
 
 4 15 
 
 .2 
 
 26 
 
 23 
 
 371,000 
 
 292.000 
 
 454,000 
 
 430,000 
 
 338,000 
 
 521,000 
 
 486,000 
 
 382,000 
 
 593,000 
 
 21 
 
 2 16.6 
 
 27 
 
 24 
 
 404,000 
 
 317.000 
 
 494,000 
 
 468,000 
 
 368,000 
 
 567,000 
 
 529,000 
 
 415,000 
 
 646,000 
 
 23 
 
 0 18 
 
 1 
 
 28 
 
 25 
 
 439,000 
 
 345,000 
 
 536,000 
 
 509,000 
 
 399,000 
 
 615,000 
 
 574,000 
 
 451,000 
 
 701,000 
 
 25 
 
 0 19 
 
 .6 
 
 29 
 
 26 
 
 474,000 
 
 373,000 
 
 580,000 
 
 550,000 
 
 432,000 
 
 665.000 
 
 620,000 
 
 487,000 
 
 758,000 
 
 27 
 
 0 21 
 
 .2 
 
 30 
 
 27 
 
 512,000 
 
 402,000 
 
 625,000 
 
 593,000 
 
 466,000 
 
 717,000 
 
 670,000 
 
 526,000 
 
 818,000 
 
 29 
 
 2 
 
 22 
 
 .9 
 
 31 
 
 28 
 
 551,000 
 
 432,000 
 
 672,000 
 
 638,000 
 
 501,000 
 
 772,000 
 
 720,000 
 
 565,000 
 
 880,000 
 
 31 
 
 4 
 
 24 
 
 .6 
 
 32 
 
 29 
 
 591,000 
 
 464,000 
 
 721,000 
 
 685,000 
 
 538,000 
 
 828,000 
 
 772,000 
 
 606,000 
 
 943,000 
 
 33 
 
 6 
 
 26 
 
 4 
 
 33 
 
 30 
 
 632,000 
 
 496,000 
 
 772,000 
 
 733,000 
 
 575,000 
 
 886,000 
 
 826,000 
 
 648,000 
 
 1,009,000 
 
 36 
 
 0 
 
 28 
 
 .3 
 
 34 
 
 31 
 
 675,000 
 
 530,000 
 
 824,000 
 
 782,000 
 
 614,000 
 
 946,000 
 
 882,000 
 
 693,000 
 
 1,080,000 
 
 38 
 
 4 
 
 30 
 
 2 
 
 35 
 
 32 
 
 719,000 
 
 665,000 
 
 878.000 
 
 834,000 
 
 655,000 
 
 1,008,000 
 
 940.000 
 
 738,000 
 
 1,150,000 
 
 41 
 
 0 
 
 32 
 
 2 
 
 36 
 
 33 
 
 764,000 
 
 601.000 
 
 934.000 
 
 886,000 
 
 698,000 
 
 1,072,000 
 
 1.000.000 
 
 785,000 
 
 1,220,000 
 
 43 
 
 6 
 
 34 
 
 2 
 
 37 
 
 34 
 
 811,000 
 
 637,000 
 
 991.000 
 
 941,000 
 
 739,000 
 
 ■ 1,140,000 
 
 1.060.000 
 
 834,000 
 
 1,300,000 
 
 46 
 
 2 
 
 36 
 
 .3 
 
 38 
 
 35 
 
 860,000 
 
 675,000 
 
 1,051,000 
 
 996,000 
 
 784,000 
 
 1.210,000 
 
 1,130,000 
 
 883,000 
 
 1,370,000 
 
 49 
 
 0 
 
 38 
 
 .5 
 
 39 
 
 36 
 
 910,000 
 
 715,000 
 
 1,112,000 
 
 1,054,000 
 
 828,000 
 
 1,280,000 
 
 1,190,000 
 
 935,000 
 
 1,450,000 
 
 51 
 
 8 
 
 40 
 
 .7 
 
 40 
 
 37 
 
 961,000 
 
 755,000 
 
 1,170,000 
 
 1,110,000 
 
 876,000 
 
 1,350,000 
 
 1,260.000 
 
 987,000 
 
 1,540,000 
 
 54 
 
 0 
 
 43 
 
 .0 
 
 41 
 
 38 
 
 1,014,000 
 
 796,000 
 
 1,240,000 
 
 1,180,000 
 
 923,000 
 
 1.420.000 
 
 1,330,000 
 
 1,0.0,000 
 
 1,620,000 
 
 57 
 
 8 
 
 45.4 
 
 42 
 
 39 
 
 1,068,000 
 
 838,000 
 
 1,300,000 
 
 1,240,000 
 
 973,000 
 
 1,500,000 
 
 1,400,000 
 
 1,097,000 
 
 1,710,000 
 
 60 
 
 .8 
 
 47 
 
 .8 
 
 V = 0.045 
 
 10 
 
 7 
 
 36,000 
 
 28,100 
 
 43,900 
 
 41,400 
 
 32,500 
 
 50,100 
 
 46,500 
 
 36,500 
 
 56,800 
 
 2 
 
 2 
 
 1 
 
 .7 
 
 11 
 
 8 
 
 47,000 
 
 36,900 
 
 57,400 
 
 54,100 
 
 42,500 
 
 65,400 
 
 60,700 
 
 47,700 
 
 74,200 
 
 2 
 
 9 
 
 2 
 
 .3 
 
 12 
 
 9 
 
 59,500 
 
 46.700 
 
 72,600 
 
 68,400 
 
 53,800 
 
 82,800 
 
 71,800 
 
 60,300 
 
 93,800 
 
 3 
 
 7 
 
 •2 
 
 .9 
 
 13 
 
 10 
 
 73,400 
 
 57.500 
 
 89,600 
 
 84,500 
 
 66,400 
 
 101,200 
 
 94,800 
 
 74,500 
 
 115,900 
 
 4 
 
 5 
 
 3 
 
 .5 
 
 14 
 
 11 
 
 88,800 
 
 69,700 
 
 108,400 
 
 102,200 
 
 80,300 
 
 123,600 
 
 114,800 
 
 90,100 
 
 140,200 
 
 5 
 
 4 
 
 4 
 
 3 
 
 15 
 
 12 
 
 105,700 
 
 83,000 
 
 129,000 
 
 121,700 
 
 95,500 
 
 147,000 
 
 136,500 
 
 '107,200 
 
 167,000 
 
 6 
 
 5 
 
 5 
 
 1 
 
 16 
 
 13 
 
 124,000 
 
 97.500 
 
 151,000 
 
 142,800 
 
 112,200 
 
 173,000 
 
 160,000 
 
 126,000 
 
 196,000 
 
 7 
 
 6 
 
 6 
 
 0 
 
 17 
 
 14 
 
 143,800 
 
 113,000 
 
 176,000 
 
 165,600 
 
 130,000 
 
 200,000 
 
 186,000 
 
 146,000 
 
 227,000 
 
 8 
 
 8 
 
 6 
 
 9 
 
 18 
 
 15 
 
 165,000 
 
 130.000 
 
 202,000 
 
 190,000 
 
 149,000 
 
 230,000 
 
 213,000 
 
 168,000 
 
 261,000 
 
 10 
 
 2 
 
 8 
 
 0 
 
 19 
 
 16 
 
 188,000 
 
 148,000 
 
 229,000 
 
 216,000 
 
 170,000 
 
 262,000 
 
 243,000 
 
 191,000 
 
 297.000 
 
 11 
 
 5 
 
 9 
 
 0 
 
 20 
 
 17 
 
 212,000 
 
 167,000 
 
 259,000 
 
 244,000 
 
 192,000 
 
 295,000 
 
 274,000 
 
 215,000 
 
 335.000 
 
 13 
 
 0 
 
 10.4 
 
 21 
 
 18 
 
 238,000 
 
 187,000 
 
 290,000 
 
 274,000 
 
 215,000 
 
 331,000 
 
 307,000 
 
 242,000 
 
 376.000 
 
 14 
 
 6 
 
 11 
 
 5 
 
 22 
 
 19 
 
 265,000 
 
 208,000 
 
 324,000 
 
 305,000 
 
 240,000 
 
 369,000 
 
 342,000 
 
 269,000 
 
 418,000 
 
 16 
 
 3 
 
 12 
 
 8 
 
 23 
 
 20 
 
 294,000 
 
 231,000 
 
 358,000 
 
 338,000 
 
 265,000 
 
 409,000 
 
 389,000 
 
 298,000 
 
 463.000 
 
 18 
 
 0 
 
 14 
 
 1 
 
 24 
 
 21 
 
 323,000 
 
 254,000 
 
 395,000 
 
 372,000 
 
 293,000 
 
 451,000 
 
 418,000 
 
 329,000 
 
 511.000 
 
 19 
 
 9 
 
 15 
 
 6 
 
 25 
 
 22 
 
 356,000 
 
 279,000 
 
 434,000 
 
 409,000 
 
 321,000 
 
 495,000 
 
 459,000 
 
 361,000 
 
 561,000 
 
 21 
 
 8 
 
 17 
 
 1 
 
 26 
 
 23 
 
 388,000 
 
 305,000 
 
 474,000 
 
 447,000 
 
 351,000 
 
 541,000 
 
 502,000 
 
 394,000 
 
 613.000 
 
 23 
 
 8 
 
 18.7 
 
 27 
 
 24 
 
 423,000 
 
 332,000 
 
 516,000 
 
 486,000 
 
 384,000 
 
 589,000 
 
 546,000 
 
 418,000 
 
 668.000 
 
 25 
 
 9 
 
 20 
 
 3 
 
 28 
 
 25 
 
 459,000 
 
 365,000 
 
 560,000 
 
 528,000 
 
 415,000 
 
 639,000 
 
 593,000 
 
 465,000 
 
 724.000 
 
 28 
 
 1 
 
 22 
 
 1 
 
 29 
 
 26 
 
 496,000 
 
 390.000 
 
 606,000 
 
 571,000 
 
 448.000 
 
 691,000 
 
 641,000 
 
 503,000 
 
 783.000 
 
 30 
 
 4 
 
 23 
 
 9 
 
 30 
 
 27 
 
 535,000 
 
 420.000 
 
 653,000 
 
 616,000 
 
 484,000 
 
 745,000 
 
 691,000 
 
 543,000 
 
 845.000 
 
 32 
 
 8 
 
 25 
 
 8 
 
 31 
 
 28 
 
 576,000 
 
 452,000 
 
 703,000 
 
 663,000 
 
 520,000 
 
 801,000 
 
 744,000 
 
 584,000 
 
 908.000 
 
 35 
 
 3 
 
 27 
 
 7 
 
 32 
 
 29 
 
 617,000 
 
 485,000 
 
 754,000 
 
 711,000 
 
 558,000 
 
 859,000 
 
 805,000 
 
 626,000 
 
 974.000 
 
 37 
 
 9 
 
 29.7 
 
 33 
 
 30 
 
 661,000 
 
 518,000 
 
 807,000 
 
 760,000 
 
 597,000 
 
 920,000 
 
 854,000 
 
 670,000 
 
 1,042.000 
 
 40 
 
 5 
 
 31 
 
 8 
 
 34 
 
 31 
 
 705,000 
 
 554,000 
 
 861,000 
 
 812,000 
 
 638,000 
 
 982,000 
 
 911,000 
 
 715,000 
 
 1,110,000 
 
 43. 
 
 3 
 
 34 
 
 0 
 
 35 
 
 32 
 
 752,000 
 
 590,000 
 
 918,000 
 
 865,000 
 
 679,000 
 
 1,046,000 
 
 971,000 
 
 762,000 
 
 1,190,000 
 
 46. 
 
 1 
 
 36 
 
 2 
 
 36 
 
 33 
 
 800,000 
 
 628,000 
 
 976,000 
 
 920,000 
 
 723,000 
 
 1,110,000 
 
 1,032,000 
 
 812,000 
 
 1.260.000 
 
 49. 
 
 0 
 
 38 
 
 5 
 
 37 
 
 34 
 
 849,000 
 
 666.000 
 
 1,036,000 
 
 977,000 
 
 767,000 
 
 1,180,000 
 
 1,100,000 
 
 861,000 
 
 1,340.000 
 
 51. 
 
 9 
 
 40 
 
 8 
 
 38 
 
 35 
 
 899,000 
 
 706.000 
 
 1,098,000 
 
 1,035,000 
 
 813,000 
 
 1,250,000 
 
 1,160,000 
 
 913,000 
 
 1,4'?0,000 
 
 55. 
 
 2 
 
 43 
 
 3 
 
 39 
 
 36 
 
 951,000 
 
 747.000 
 
 1,160,000 
 
 1,094,000 
 
 860,000 
 
 1,320,000 
 
 1,230,000 
 
 965,000 
 
 l,5u0,000 
 
 58. 
 
 3 
 
 45 
 
 7 
 
 40 
 
 37 
 
 1,005,000 
 
 790.000 
 
 1,230,000 
 
 1,160,000 
 
 910,000 
 
 1,400,000 
 
 1,300,000 
 
 1,020,000 
 
 1,590,000 
 
 61. 
 
 5 
 
 48.4 
 
 41 
 
 38 
 
 1,060,000 
 
 833,000 
 
 1,290,000 
 
 1,220,000 
 
 958,000 
 
 1,480,000 
 
 1,370,000 
 
 1,075,000 
 
 1,670,000 
 
 65. 
 
 0 
 
 51. 
 
 0 
 
 42 
 
 39 
 
 1,109.000 
 
 877,000 
 
 1,360,000 
 
 1.290,000 
 
 1,010,000 
 
 1,550,000 
 
 1,440,000 
 
 1,130,000 
 
 1,760,000 
 
 68. 
 
 4 
 
 53. 
 
 8 
 
I Sec. 8-8J COLUMNS 379 
 
 
 
 1:2:4 
 
 (2,000-lb. concrete) 
 
 1: 1}^ : 3 (2,500-lb. concrete) 
 
 1:1:2 
 
 (3,000-lb. concrete) 
 
 
 
 
 
 
 TO = 15 
 
 
 n = 12 
 
 
 
 re = 10 
 
 
 At 
 
 
 
 
 
 
 
 
 
 
 
 
 fan. in."> 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 meter of 
 mns (D) 
 
 
 
 
 1 % spiral 
 /c = 7001b. 
 
 
 
 1 % spira! 
 /c = 8701b 
 
 
 
 1 % spiral 
 
 
 
 t3 on 
 
 fc = 450 lb. 
 
 fc = 565 lb. 
 
 fc = 675 lb. 
 
 /c= 1,050 
 
 
 
 > 6 
 
 per sq. in. 
 
 per sq. in. 
 
 per sq. in. 
 
 per sq. in 
 
 per sq. in. 
 
 lb. per 
 sq. in. 
 
 V 
 
 (.< 
 
 0 ee 
 
 51 
 
 
 Square 
 
 Round 
 
 Round 
 
 Square 
 
 Round 
 
 Round 
 
 Square 
 
 Round 
 
 Round 
 
 cfl 
 3 
 O* 
 
 c be 
 
 
 
 or oct. 
 
 or oct. 
 
 or oct. 
 
 or oct. 
 
 or oct. 
 
 or oct. 
 
 tc 
 
 
 
 
 P (lb.) 
 
 P (lb.) 
 
 P (lb.) 
 
 P (lb.) 
 
 P (lb.) 
 
 P (lb.) 
 
 P (lb.) 
 
 P (lb.) 
 
 P (lb.) 
 
 
 p = 0.05 
 
 10 
 
 7 
 
 37,500 
 
 29,400 
 
 45,800 
 
 42,900 
 
 oo TAA 
 
 51,900 
 
 48,000 
 
 0 T TAA 
 
 o7,700 
 
 58,600 
 
 2 . 5 
 
 1 .9 
 
 11 
 
 8 
 
 49,000 
 
 38,500 
 
 59,800 
 
 56, 100 
 
 44,000 
 
 67,800 
 
 63,300 
 
 49,200 
 
 76,600 
 
 3 . 2 
 
 2.5 
 
 12 
 
 9 
 
 62,000 
 
 48,700 
 
 75,700 
 
 70,800 
 
 55,700 
 
 85,800 
 
 80,200 
 
 62,300 
 
 96,900 
 
 4 . 1 
 
 3.2 
 
 13 
 
 10 
 
 76,500 
 
 60,100 
 
 93,500 
 
 87,600 
 
 63,800 
 
 1 A^ AAA 
 
 1U().0U0 
 
 97.900 
 
 Tr* AAA 
 
 76,900 
 
 119,600 
 
 5 . 0 
 
 3.9 
 
 14 
 
 11 
 
 92,600 
 
 72,700 
 
 11^,000 
 
 i AC AAA 
 
 lOb.OUU 
 
 O O O AA 
 
 128.000 
 
 1 18.400 
 
 93,000 
 
 144,900 
 
 6 . 1 
 
 4 .8 
 
 15 
 
 12 
 
 110,201) 
 
 86,500 
 
 1c54,dOU 
 
 1 Ofi AAA 
 
 An AAA 
 
 153.000 
 
 142.400 
 
 1 1 A TAA 
 
 1 10, /OO 
 
 1 TO AAA 
 
 172,000 
 
 7 . 2 
 
 5 . 7 
 
 16 
 
 13 
 
 129,400 
 
 101,600 
 
 ICO f\r\f\ 
 158,000 
 
 t A O AAA 
 
 1 1 ^5 OAA 
 
 1 lb, oUU 
 
 1 TA AAA 
 
 1 Ci'7 OAA 
 
 Io7,.ii00 
 
 1 OA AAA 
 
 loO,OOU 
 
 OAO AAA 
 
 202,000 
 
 8 . 5 
 
 6.6 
 
 17 
 
 14 
 
 150,000 
 
 1 17,oUU 
 
 TOO AAr\ 
 
 18o,UUU 
 
 172,000 
 
 1 O 'I QAn 
 
 OAO AAA 
 
 1 AO AAA 
 
 151,000 
 
 OOyI AAA 
 
 zc>4,000 
 
 9 . 8 
 
 7 . 7 
 
 18 
 
 15 
 
 172,000 
 
 IOC onn 
 
 zlU.UuU 
 
 1 A 7 AAA 
 
 ly / ,uuu 
 
 155.000 
 
 OOO AAA 
 
 OOA AAA 
 
 zz(J,(jUU 
 
 1 "70 AAA 
 
 OCA AAA 
 
 zby,ooo 
 
 11.3 
 
 8.8 
 
 19 
 
 16 
 
 196,000 
 
 154,000 
 
 j,5y,ouo 
 
 OOi AAA 
 
 Z24,UUU 
 
 1 Tfi AAA 
 
 OTI AAA 
 
 0 C 1 AAA 
 
 zol .OOU 
 
 1 Cl'y AAA 
 
 iy7,uoo 
 
 OA^^ AAA 
 
 oOb,000 
 
 12.8 
 
 10. 1 
 
 20 
 
 17 
 
 221,000 
 
 174,000 
 
 270,000 
 
 253,000 
 
 199.000 
 
 306.000 
 
 283.000 
 
 222.000 
 
 346,000 
 
 14.5 
 
 11.4 
 
 21 
 
 18 
 
 248,000 
 
 195,000 
 
 303,000 
 
 284,000 
 
 223.000 
 
 343.000 
 
 317.000 
 
 250.000 
 
 388,000 
 
 16.2 
 
 12.7 
 
 22 
 
 19 
 
 278,000 
 
 217,000 
 
 337,000 
 
 316.000 
 
 248.000 
 
 383,000 
 
 353.000 
 
 278.000 
 
 432,000 
 
 18.1 
 
 14.2 
 
 23 
 
 20 
 
 306,000 
 
 241,000 
 
 374,000 
 
 350.000 
 
 275,000 
 
 424.000 
 
 392.000 
 
 308.000 
 
 478,000 
 
 20.0 
 
 15.7 
 
 24 
 
 21 
 
 337,000 
 
 265,000 
 
 412,000 
 
 386.000 
 
 303.000 
 
 467.000 
 
 432.000 
 
 339.000 
 
 528,000 
 
 22.1 
 
 17.3 
 
 25 
 
 22 
 
 371,000 
 
 291,000 
 
 452,000 
 
 424.000 
 
 333.000 
 
 513.000 
 
 474.000 
 
 372.000 
 
 579,000 
 
 24.2 
 
 19.0 
 
 26 
 
 23 
 
 405,000 
 
 318,000 
 
 494,000 
 
 464.000 
 
 364.000 
 
 560,000 
 
 518.000 
 
 406.000 
 
 633,000 
 
 26.5 
 
 20.8 
 
 27 
 
 24 
 
 441,000 
 
 346,000 
 
 548,000 
 
 505.000 
 
 396.000 
 
 610,000 
 
 564.000 
 
 443.000 
 
 689,000 
 
 28.8 
 
 22.6 
 
 28 
 
 25 
 
 478,000 
 
 376,000 
 
 584,000 
 
 548.000 
 
 430,000 
 
 662.000 
 
 612,000 
 
 481.000 
 
 748,000 
 
 31.3 
 
 24.5 
 
 29 
 
 26 
 
 517,000 
 
 406,000 
 
 632,000 
 
 .592.000 
 
 465.000 
 
 716.000 
 
 662,000 
 
 520.000 
 
 809,000 
 
 33.8 
 
 26.6 
 
 30 
 
 27 
 
 558,000 
 
 438,000 
 
 680,000 
 
 639,000 
 
 512.000 
 
 772.000 
 
 714.000 
 
 561,000 
 
 872,000 
 
 36.5 
 
 28.6 
 
 31 
 
 28 
 
 600,000 
 
 471,000 
 
 733,000 
 
 686.000 
 
 539,000 
 
 831.000 
 
 768.000 
 
 603,000 
 
 938,000 
 
 39.2 
 
 30.8 
 
 32 
 
 29 
 
 644,000 
 
 501,000 
 
 786.000 
 
 737,000 
 
 578.000 
 
 891.000 
 
 823,000 
 
 647,000 
 
 1,006,000 
 
 42.1 
 
 33.0 
 
 33 
 
 30 
 
 688,000 
 
 541,000 
 
 841,000 
 
 788,0f)0 
 
 619.000 
 
 954,000 
 
 881,000 
 
 692,000 
 
 1,077,000 
 
 45.0 
 
 35.3 
 
 34 
 
 31 
 
 735,000 
 
 578,000 
 
 898.000 
 
 841,000 
 
 666.000 
 
 1,018,000 
 
 941,000 
 
 739,000 
 
 1,150,000 
 
 48.1 
 
 37.7 
 
 35 
 
 32 
 
 784,000 
 
 615,000 
 
 957,000 
 
 897,000 
 
 704,000 
 
 1,085,000 
 
 1,002.000 
 
 787,000 
 
 1,230,000 
 
 51.2 
 
 40.2 
 
 36 
 
 33 
 
 833,000 
 
 655,000 
 
 1,018,000 
 
 954.000 
 
 750,000 
 
 1,150,000 
 
 1,066,000 
 
 837,000 
 
 1,300,000 
 
 54.5 
 
 42.8 
 
 37 
 
 34 
 
 885,000 
 
 695,000 
 
 1,080,000 
 
 1,012,000 
 
 795.000 
 
 1,220,000 
 
 1,130,000 
 
 889,000 
 
 1,380,000 
 
 57.8 
 
 45.4 
 
 38 
 
 35 
 
 937,000 
 
 736,000 
 
 1,140,000 
 
 1,073,000 
 
 843,000 
 
 1,300,000 
 
 1,200,000 
 
 942,000 
 
 1,470,000 
 
 61.3 
 
 48.1 
 
 39 
 
 36 
 
 992,000 
 
 779,000 
 
 1,210,000 
 
 1,130,000 
 
 892,000 
 
 1,370,000 
 
 1,270,000 
 
 996,000 
 
 1,550,000 
 
 64.8 
 
 50.9 
 
 40 
 
 37 
 
 1,047,000 
 
 823,000 
 
 1,280,000 
 
 1,200,000 
 
 942,000 
 
 1,4.50,000 
 
 1,340,000 
 
 1,052,000 
 
 1,640,000 
 
 68.5 
 
 53.8 
 
 41 
 
 38 
 
 1,105,000 
 
 879.000 
 
 1,350,000 
 
 1,270,000 
 
 993.000 
 
 1,530,000 
 
 1,410,000 
 
 1,110,000 
 
 1,730,000 
 
 72.2 
 
 56.7 
 
 42 
 
 39 
 
 1,160,000 
 
 914,000 
 
 1,420,000 
 
 1,330,000 
 
 1,046,000 
 
 1,610,000 
 
 1,490,000 
 
 1,170,000 
 
 1,820,000 
 
 76.1 
 
 59.7 
 
380 CONCRETE ENGINEERS' HANDBOOK [See. 8-8 
 
 Table 2. — Hooped Column Reinforcement 
 
 Diameter of 
 enclosed con- 
 crete to 
 center line 
 of hooping 
 (inches) 
 
 Pitch 
 (inches) 
 
 Sectional 
 area of 
 hooping 
 (square 
 inches) 
 
 Length of 
 hooping in 
 I ft. in height 
 (inches) 
 
 Diameter of 
 encosed con- 
 crete to 
 center line 
 of hooping 
 (inches) 
 
 Pitch 
 (inches) 
 
 Sectional 
 area of 
 hooping 
 (square 
 inches) 
 
 Length of 
 hooping in 
 1 ft. in height 
 (inches) 
 
 8 
 
 1 
 
 max. 
 
 0.020 
 0.025 
 
 1 
 
 302 
 242 
 
 24 
 
 2H 
 
 2)4 max. 
 
 0.142 
 0.150 
 
 381 
 362 
 
 9 
 
 1 
 
 max. 
 
 0.022 
 0.034 
 
 339 
 226 
 
 25 
 
 2}^ max. 
 
 0. 156 
 
 377 
 
 10 
 
 1 
 
 l^i max. 
 
 0.025 
 0.041 
 
 377 
 232 
 
 26 
 
 2}y-2 max. 
 
 0. 162 
 
 392 
 
 11 
 
 m 
 
 l^i max. 
 
 0.031 
 0.048 
 
 369 
 237 
 
 27 
 
 2>^ max. 
 
 0. 169 
 
 407 
 
 12 
 
 m 
 
 2 max. 
 
 0.037 
 0.060 
 
 362 
 226 
 
 28 
 
 23-^ max. 
 
 0. 175 
 
 422 
 
 13 
 
 2}^i max. 
 
 0.045 
 0.069 
 
 356 
 230 
 
 29 
 
 2}-^ max. 
 
 0.181 
 
 437 
 
 14 
 
 2}i max. 
 
 0.048 
 0.079 
 
 384 
 234 
 
 30 
 
 23^ max. 
 
 0. 187 
 
 452 
 
 15 
 
 2}i max. 
 
 0.056 
 0.094 
 
 377 
 226 
 
 31 
 
 23-^ max. 
 
 0.194 
 
 467 
 
 16 
 
 2}^ max. 
 
 0.065 
 0.100 
 
 371 
 241 
 
 32 
 
 23^ max. 
 
 0.200 
 
 483 
 
 17 
 
 2)-^ max. 
 
 0.074 
 0.106 
 
 366 
 256 
 
 33 
 
 23^ max. 
 
 0.206 
 
 498 
 
 18 
 
 2}^ max. 
 
 0.084 
 0.112 
 
 362 
 271 
 
 34 
 
 23^^ max. 
 
 0.212 
 
 513 
 
 19 
 
 2^2 max. 
 
 0.089 
 0.119 
 
 382 
 287 
 
 35 
 
 2}i max. 
 
 0.219 
 
 528 
 
 20 
 
 2 
 
 23^ max. 
 
 0. 100 
 0.125 
 
 377 
 302 
 
 36 
 
 23^ max. 
 
 0.225 
 
 543 
 
 21 
 
 2H 
 
 2}4 max. 
 
 0.112 
 0.131 
 
 373 
 317 
 
 37 
 
 23^ max. 
 
 0.231 
 
 558 
 
 22 
 
 2)^^ max. 
 
 0.124 
 0.137 
 
 369 
 332 
 
 38 
 
 2}'2 max. 
 
 0.238 
 
 574 
 
 23 
 
 2H ■ 
 2)^ max. 
 
 0.130 
 0.144 
 
 386 
 347 
 
 39 
 
 23^ max. 
 
 0.244 
 
 588 
 
I Sec. 8-9] COLUMNS 381 
 
 Table 3. — Volume of Column in Cubic Feet per Foot of Length for Diameter D 
 
 n 
 U 
 
 Square 
 
 xvouncl 
 
 vJct agonal 
 
 
 Square 
 
 Round 
 
 Octagonal 
 
 10 
 
 0.69 
 
 0.55 
 
 0.58 
 
 27 
 
 5.06 
 
 3.98 
 
 4.19 
 
 11 
 
 0.84 
 
 0.66 
 
 0.70 
 
 28 
 
 5.44 
 
 4.28 
 
 4.51 
 
 12 
 
 1.00 
 
 0.79 
 
 0.83 
 
 29 
 
 5.84 
 
 4.62 
 
 4.84 
 
 13 
 
 1.17 
 
 0.92 
 
 0.97 
 
 
 
 
 
 14 
 
 1.36 
 
 1.07 
 
 1.13 
 
 30 
 
 6.25 
 
 4.91 
 
 5.18 
 
 
 
 
 
 31 
 
 6.68 
 
 5.24 
 
 5.53 
 
 15 
 
 1.56 
 
 1.23 
 
 1.29 
 
 32 
 
 7.11 
 
 5.58 
 
 5.88 
 
 16 
 
 1.78 
 
 1.40 
 
 1.47 
 
 33 
 
 7.55 
 
 5.94 
 
 6.26 
 
 17 
 
 2.01 
 
 1.58 
 
 1.66 
 
 34 
 
 8.02 
 
 6.31 
 
 6.64 
 
 18 
 
 2.25 
 
 1.77 
 
 1.87 
 
 
 
 
 
 19 
 
 2.51 
 
 1.97 
 
 2.08 
 
 35 
 
 8.50 
 
 6.67 
 
 7.05 
 
 
 
 
 
 36 
 
 8.98 
 
 7.05 
 
 7.46 
 
 20 
 
 2.78 
 
 2.18 
 
 2.30 
 
 37 
 
 9.49 
 
 7.46 
 
 7.88 
 
 21 
 
 3.06 
 
 2.41 
 
 2.54 
 
 38 
 
 10.00 
 
 7.87 
 
 8.30 
 
 22 
 
 3.36 
 
 2.64 
 
 2.78 
 
 39 
 
 10.60 
 
 8.29 
 
 8.75 
 
 23 
 
 3.67 
 
 2.89 
 
 3.05 
 
 
 
 
 
 24 
 
 4.00 
 
 3.14 
 
 3.32 
 
 40 
 
 11.10 
 
 8.71 
 
 9.20 
 
 
 
 
 
 41 
 
 11.70 
 
 9.15 
 
 9.67 
 
 25 
 
 4.34 
 
 3.41 
 
 3.59 
 
 42 
 
 12.30 
 
 9.61 
 
 10.10 
 
 26 
 
 4.70 
 
 3.68 
 
 3.89 
 
 
 
 
 
 9. Reduction Formula for Long Columns. — Where long columns must be used, the reduc- 
 tion formula which follows, taken from the Los Angeles Building Ordinance, may be safely 
 employed in the design of columns whose unsupported length {I) is between 15 and 30 times 
 the least dimension of effective section (d). Let r represent the quantity by which the working 
 
 stress for columns with ^ less than 15 should be multiplied to give a working stress which may 
 
 be used for long columns. Then 
 
 r = 1.6-K;5(J) 
 
 10. Columns Supporting Bracket Loads. — A column supporting a roof is frequently made 
 to carry a traveling crane which runs on a track supported by side brackets or at one side of the 
 column. To compute the maximum stress in such a column, it is necessary to find the maximum 
 bending moment at whatever section it occurs, and then combine the stresses due to bending 
 and thrust by the general method explained in Sect. 9. 
 
 The maximum bending moment occurs at the load, and depends upon the height at which 
 the load is placed and the end conditions of the column. To simplify the calculations, the 
 depth of the bracket will be considered small in comparison with the length of the column. 
 This is not strictly correct but the error involved will be on the safe side since any increase in 
 the depth of a bracket reduces the maximum amount. 
 
 Columns supporting bracket loads will be considered for the three conditions: (1) both 
 ends free to turn, (2) both ends fixed, and (3) one end fixed and the other end free. Columns 
 in practice will have conditions intermediate to these and good judgment as to flexibility of the 
 end connections is necessary to arrive at correct results in any particular case. 
 
 Column Free to Turn at Both Ends.— The bending-moment diagram for this case is shown 
 in Fig. 2. The bending moment Px of the eccentric load is resisted by horizontal forces at 
 the ends of the column which form a couple, the value of which is also Px. The maximum pos- 
 
382 
 
 CONCRETE ENGINEERS' HANDBOOK 
 
Sec. 8-10] 
 
 COLUMNS 
 
 383 
 
 sible value of the bending moment is evidently Px, which would occur with tiie load at either 
 end of the column. The minimum value of the maximum bending moment occurs when 
 
 a = b and equals VzPx. 
 
 Column Fixed at Both Ends. — Analysis shows that the least value of the maxinuin bending 
 moment is and occurs when h has the following values: 0.21 IL, 0.500L, and 0.789L. The 
 
 Fig. 2. 
 
 moment diagrams for these cases are shown in Fig. 3, The greatest value of the maximum 
 moment is Px and occurs when the bracket is either at the top or bottom of the column. 
 
 The following general formulas may also be obtained : 
 
 At A (Fig. 3) 
 
 Just above C, 
 Just below C, 
 or 
 
 At O, 
 
 M 
 M 
 M 
 
 Ma - Ra 
 
 Ma - Ra -\-Px 
 
 Mo + Rh 
 
 Column Fixed at One End and Free to Turn at the Other.— The minimum value of the maxi- 
 mum bending moment equals }iPx and occurs when b is equal to 0.258L or 0.605L. The bend- 
 
384 CONCRETE ENGINEERS' HANDBOOK [Sec. 8-10 
 
 ing-moment diagrams for these cases are shown in Fig. 4 which also shows the condition to 
 make the bending moment at the base zero, and the particular case of b = L. 
 The following general formulas apply : 
 
 Fig. 4. 
 
 Just above C (Fig. 4), 
 
 M = - Ra 
 
 Just below C, 
 
 M = Px - Ra 
 
 At O 
 
 Mo = Px - %Px ■ -j) 
 
SECTION 9 
 
 BENDING AND DIRECT STRESS 
 
 1. Theory in General. — If a beam is acted upon by forces which are all normal to its length, 
 then the stresses resulting are due to simple bending, and the formulas deduced in Sect. 7 may 
 be employed. If, however, any of the forces acting throughout the length of a beam be in- 
 clined, or if additional forces be applied at the ends, then our beam formulas for simple bending 
 will not apply. Likewise, in columns, if the load be eccentrically applied or if lateral pressure 
 be exerted, both bending and direct stresses will result and the ordinary column formulas given 
 in Sect. 8 cannot be used except to give approximate results when the amount of bending is 
 small. 
 
 The same combination of stresses occurs also in arch rings and may occur in special cases. 
 The formulas to be derived can be employed in any type of reinforced-concrete structure pro- 
 vided the normal component of the resultant thrust on the given section acts with a lever arm 
 about the center of gravity of the section. In long beams and columns, the deflection resulting 
 from flexure should be given consideration when determining the 
 eccentricity of the axial and inclined forces. 
 
 Let us first consider structures of plain concrete. The distri- 
 bution of pressure on any section due to a resultant pressure act- 
 ing at different points will be explained. Consider a section 
 represented in projection by EF, Fig. 1. When the resultant R 
 acts at the center of gravity 0, the intensity of stress is uni- 
 form over the section and is equal to the vertical component of 
 
 N 
 
 R divided by the area of section, or If ^ acts at any other 
 point, as Q, and if the projection of the section is taken such 
 
 that the distance xo represents the true lever arm of N about the center of gravity, then 
 the force N is equivalent to an equal N at O and a couple whose moment is A^.Co. The in- 
 tensity of the uniformly varying stress due to this bending moment at a distance x from 0 
 
 is (by the common flexure formula for homogeneous beams) in which / is the moment 
 
 I 
 
 At 
 
 of inertia of the section about an axis through 0 at right angles to the plane of the paper, 
 the edges E and F this intensity = Regarding compressive and tensile stresses as posi 
 
 tive and negative respectively, the intensity of stress at edge E is 
 
 r _ N NXqXi 
 
 •^-^ " A + / 
 
 At edge F it is 
 
 NxoXi 
 
 If the stress fj comes out minus, the value obtained is the maximum tension as shown in Fig. 
 2. In plain concrete construction a greater tension than about 50 lb. per sq. in. should not be 
 allowed . 
 
 When we come to reinforced concrete, which is composed of two materials (c'oncrete and 
 steel) with different values of E, then the steel area at any given cross-section may be replaced 
 25 385 
 
386 
 
 CONCRETE ENGINEERS' HANDBOOK 
 
 [Sec. 9-1 
 
 by an area of concrete equal to n times the area of the steel, placed in the plane of the steel 
 reinforcement. This section may be called the transformed section, or section of concrete 
 theoretically equivalent in resistance to the actual section. Under this heading rectangular 
 sections only will be considered and Fig. 3 represents a transformed section as referred to above. 
 
 Thus, if Ac is the area of the concrete, and Ao is the area of the steel = As -\- A'; then the 
 equivalent area 
 
 A = Ac nAo ^ bt + n{As + A') 
 
 If Ic is the moment of inertia of the concrete about the gravity axis, and Is is the moment of 
 inertia of the steel about the same axis, then 
 
 and 
 
 (/o) 
 
 UV) 
 
 I = Ic +-nh 
 
 N ( + ) Nxoxi 
 
 Ac-\- nAo ( - ) Ic + nis 
 
 A A' 
 
 If we denote p and p' by ^ and respectively, then the distance from the face most highly 
 stressed to the center of gravity of the transformed section is (by moments) 
 
 u =■ 
 
 bt^ + nAsd + nA'd' 
 
 bl^ 
 
 + nAsd + nA'd' 
 
 bt + n{As + A') 
 
 t/2 + npd + np'd' 
 
 1 -\- np 
 
 np 
 
 
 
 N 
 
 
 
 
 
 Cornpres; 
 
 
 V 
 
 > 
 
 f 
 
 L 
 
 Q 
 
 <. — 
 
 < 
 
 u J 
 
 toi 
 
 1 
 
 'x| 
 
 ?i,.--Center o 
 gravity 
 
 1 Pi 
 
 f 
 
 n 
 y 
 
 
 d ? 
 
 1 
 
 
 
 Fig. 3. 
 
 Ic = Hbu' + }ib(t - uy = ^^u' + (t - uy^ 
 
 Is = As{d - uy + A'(u - d'y 
 
 I ^ Ic + nis =^^u^ + {t - + 7iAs(d - uy + nA'iu - d'y 
 
 t 
 
 If the rpinforcement is symmetrical, then u 
 
 and 
 
 / = y^^bt^ + 2nAsiy2t - d'y = }i2bt'' + 2npbt {Vzt - d'Y 
 
Sec. 9-2] 
 
 BENDING AND DIRECT STRESS 
 
 387 
 
 Since, A = bt -\- n{As + A') = bt -\- nbt(p + p') 
 
 (fc) 
 
 N 
 
 ( + ) 
 
 Nxo 
 
 ifc') bt + nbt(p + p') (-) yi2bt' + 2npbt(y2t - d'y 
 
 2. Analytical Determination of Stresses in Rectangular Sections. 
 
 2a. Compression Over the Whole Section — Steel Top and Bottom (Case I). — 
 The formulas developed in preceding article apply when the stress is either compression over 
 the entire section, or when there is compression over a portion of the section with a tension 
 over the remainder not exceeding the allowable tensile stress in the concrete. The formulas 
 we shall use will apply to rectangular sections with symmetrical reinforcement and are given 
 in the following form for convenience, letting po denote the 
 quantity p -\- p'\ 
 
 r = ; - d' 
 
 (fc) 
 (fc) 
 
 r 1 (+) M 1 
 
 Ll + npo(-)r^ + 12npor^} 
 
 (1) 
 (2) 
 
 By referring to Fig. 4 it will be clear that the stress in the steel 
 is always less than n X fc] thus, if /c is kept within its allow- 
 able value, the steel is sure to be safely stressed. 
 
 Equation 2 gives a means of determining the eccentricity 
 of the resultant force, or Xo, for which there can be neither 
 tension nor compression at the surface opposite to that near 
 which the thrust acts. To obtain the value of Xo which gives a 
 zero value to //i equate the two terms within the brackets, 
 and solve. 
 
 1 6a;o^ 
 
 1 -\- n(p + p') ■ + 12npor2 
 
 Xo = 
 
 + 24:npr^ 
 1 + n(p + p') 
 
 (3) 
 
 If n is assumed to be 15, and, if the steel is embedded in the concrete one-tenth of the total 
 depth from each surface so that 2r = '^it, formula (3) becomes 
 
 1 + 28. 
 
 If the values n 
 
 t 6 + 90po 
 
 15 and 2r = ^^^rare substituted in formula (1), this formula becomes 
 
 •"^ bt Ll + 15po t 
 or if the expression in the brackets is denoted by K 
 
 1 + 28.8poJ 
 
 NK 
 bt 
 
 (4) 
 
 (5) 
 
 (6) 
 
 Diagrams 1 to 6 inclusive give values of K for various values of po, and y > and for both 
 n = 12 and n = 15. The termination of the curves are determined in Diagram 5 by equation 
 (4) and in the other diagrams by similar equations. For greater values of Case I does not 
 apply; that is, there is tension in the concrete and Case II must be employed. 
 
388 
 
 CONCRETE ENGINEERS' HANDBOOK 
 
 [Sec. 9-2a 
 
 «j ^0 s an I DA 
 
Sec. d-2al 
 
 BENDING AND DIRECT STRESS 
 
 389 
 
 ~ 9 S 
 
 Q Q Q 
 
390 
 
 CONCRETE ENGINEERS' HANDBOOK 
 
 [Sec. ^2a 
 
Sec. 9-2a] 
 
 BENDING AND DIRECT STRESS 
 
 391 
 
 y JO senpA 
 
392 
 
 CONCRETE ENGINEERS' HANDBOOK 
 
 [Sec. 9-^ a 
 
Sec. 9-2a] BENDING AND DIRECT STRESS 393 
 
 55 
 O 
 
 M 
 
 o 
 
 )^ 
 
 o 
 
 I 
 
 > 
 O 
 
 o 
 
 Q i 
 
394 
 
 CONCRETE ENGINEERS' HANDBOOK 
 
 [Sec. 9-26 
 
 2b. Tension Over Part of Section — Steel Top and Bottom (Case II).— It will 
 be on the safe side and convenient as regards the construction of working diagrams to consider 
 that, when any tension exists in the concrete, the steel carries all the tensile stresses. In this 
 case there are three unit stresses to be determined: namely, maximum unit compression in 
 concrete fc, maximum unit compression in steel //, and maximum unit tension in steel fs. The 
 general formulas developed in Art. 1 are not applicable to this case and the following method 
 may be used : 
 
 Referring to Fig. 5, it follows that 
 
 ktj 
 
 and 
 
 (7) 
 
 (8) 
 
 Sin-ce the resultant fiber stress equals 
 
 ^ ^ fs'pobt 
 
 + 
 
 fchkt 
 
 fsPobt 
 2 
 
 Eliminating fs' and fs by means of equations (7) and (8) 
 
 N 
 
 _ fcht k^ + 2npok — npo 
 
 ~~2' k ■ 
 
 fc.bt _|_ 2nkpo — npo 
 
 ~2" k 
 
 (9) 
 
 The moment of the stresses about the gravity axis, eliminating // 
 and fs as before, is 
 
 /c6i{^f'+^(3 -2A:)] (10) 
 
 M 
 
 or, if the quantity within the brackets is designated by L, then 
 
 M =fM'L, or/. =-^, (11) 
 
 Fig. 5. 
 
 The position of the neutral axis must be determined before 
 equation (11) can be used. Since Nxa = M we may multiply 
 equation (9) by Xq and equate it to equation (10). Proceeding in this manner the following 
 equation results 
 
 A;3 _ 3 1^/^ _ + Qynp.k'^ = Snpo (~ + 2^') (12) 
 
 Diagrams 7, 8, 9, 11, 12 and 13, based on equation (12), give values of k for various values 
 Xo d' 
 
 of po, "y' y' both n = 12 and n = 15. Diagrams 10 and 14 give values of L. 
 
 The method of procedure in solving problems under Case II is as follows: (1) Determine k 
 from the proper diagram; (2) find L from Diagram 10 or 14; (3) solve equation (11) for /.; 
 (4) find unit stresses in the steel from formulas (7) and (8). 
 
 Illustrative Problem. — A beam is 9 in. wide and 20 in. deep. The reinforcement both above and below 
 consists of one steel rod 1 in. in diameter embedded at a depth of 2 in. At a certain section, the normal component 
 of the resultant force is 60,000 lb., acting at a distance of 3.4 in. from the gravity axis. Assume n = 15. Compute 
 the maximum unit compressive stress in the concrete. 
 
 As + A' 
 
 (2) (0.7854) 
 
 (9) (20) 
 
 = 0.0087 
 
 3^ 
 20 
 
 0.17 
 
Sec 9-26] 
 
 BENDING AND DIRECT STRESS 
 
 395 
 
 J.O S8n|lD/\ 
 
396 
 
 CONCRETE ENGINEERS' HANDBOOK 
 
 [Sec. 9-26 
 
Sec. 9-26) 
 
 BENDING AND DIRECT STHESS 
 
 397 
 
 s s9 S3 2 o 
 
 C> CJ O Q O Q P 
 
 tu 
 
 o o o 
 o o o 
 
 5 § 8 § o. 
 
398 
 
 CONCRETE ENGINEERS' HANDBOOK 
 
 [Sec. 9-25 
 
 ^bgS? S S. S iO R lS S IJ^ S !5 5 » ^ ^. ^ 
 •»» ■ • ' ■ • . , • • 
 
Sec. 9-26] 
 
 BENDING AND DIRECT STRESS 
 
 399 
 
 •d io can I OA 
 
400 
 
 CONCRETE ENGINEERS' HANDBOOK 
 
 [Sec. 9-2?) 
 
Sec. 9-26] 
 
 BENDING AND DIRECT STRESS 
 
 401 
 
 :5: 
 
 m 
 
 S 5S S 2 
 o o o o o 
 
 1 1 I 
 
 26 
 
402 
 
 CONCRETE ENGINEERS' HANDBOOK 
 
 [Sec. 9-25 
 
Sec. 9-2c] 
 
 BENDING AND DIRECT STRESS 
 
 403 
 
 For these values of po and Diagram 5 gives K = 1.70 and shows that the problem falls under Case I 
 
 Then by formula (6) 
 
 NK 
 bt 
 
 (60,000) (1.70) 
 (9) (20) 
 
 567 lb. per sq. in. 
 
 Illustrative Problem. — Change the eccentricity of the preceding problem to 6 in. and solve. 
 
 Xo 
 
 For Po = 0.0087 and 
 
 0.30, 
 
 Diagram 5 shows that is too great for the problem to come under Case 
 
 I. 
 
 The method of procedure for Case II must then be followed. 
 
 Xo 
 
 Diagram 12 gives k = 0.73 for the values of po and 
 Diagram 14 shows L to be 0.123. Solving equation (11) 
 
 _ J/_ _ (60,000) (6) 
 ~ Lbt^ ~ (0.123) (9) (20)2 
 
 given above. With k = 0.73 and po = 0.0087, 
 
 815 lb. per sq. in. 
 
 Using formula (8) gives 
 
 /. = nfc (^^ - l) = (15)^815) 
 
 18 
 
 73 X 20 
 
 l) = 2830 lb. per sq. 
 
 The stress /«' may be found by formula (7) but is always less than n X fc- 
 
 Illustrative Problem. — An arch is 20 in. deep and is reinforced with three rods H in. in diameter to 
 each foot of width, both above and below. If the rods are embedded to a depth of 2 in. and the normal com- 
 ponent of the resultant thrust on a section is 100,000 lb. for 1-ft. width of arch, with an eccentricity of 3.4 in., 
 determine the maximum intensity of compressive stress on the concrete. Assume n = 15 
 
 (6) (0.4418) 
 
 PO = 
 
 = 0.0110 
 
 Diagram 5 gives K 
 
 (12) (20) 
 Xo 3.4 
 
 T = -20 = ^-^^^ 
 
 1.63 and the problem comes under Case I. Then by formula (6) 
 
 . _ NK^ _ (100,000) (1.63) 
 bt ~ 
 
 679 lb. per sq. in. 
 
 (12)(20) 
 
 2c. Tension Over Part of Section — Steel in Tension Face Only (Case III). 
 Referring to Fig. 6 and taking moments about the center of the steel we 
 have 
 
 Ne' = y2fckjbd^ 
 
 Since the algebraic sum of the compressive and tensile forces must equal 
 N, we may write 
 
 N = Mfckbd - fsVhd 
 We also know (see page 276) that 
 
 Fig. 6. 
 
 Sc 
 
 n(l - k) 
 
 From these three equations may be obtained the formulas: 
 
 k^ - 
 3 = 
 V = 
 
 K = 
 
 fc = 
 
 2pn (1 
 
 1 -Hk 
 k^ 
 
 k) = k^j 
 
 .d 
 
 2n{l 
 ^Ne 
 
 bd^ 
 2Ne 
 
 =y2fckj 
 
 bd 
 
 kjbd^ 
 
 Diagrams 15 and 16^ may be used in designing and reviewing structures subjected to bend- 
 ing and direct stress with steel in tension face only. 
 
 1 Scheme of diagrams proposed by Robert S. Beard, Asst. City Engineer, Kansas City, Mo. 
 
Sec. 9-26] 
 
 BENDING AND DIRECT STRESS 
 
 405 
 
406 
 
 CONCRETE ENGINEERS' HANDBOOK 
 
 [Sec. 9-3 
 
 Illustrative Problem. — The vertical wall of a cantilever retaining wall is subjected to an earth pressure 
 of 2400 lb. applied at a distance of 4.54 ft. above the top of footing. The weight of vertical wall is 2200 lb. 
 which can be considered as applied 5 in. in front of the steel. Determine the unit stresses fc and /«, assum- 
 ing n = 15, p = 0.0077 and d = 10.5 in. 
 The moment at the top of footing 
 
 M = (2400) (4.54J (12) + (2200) (5) = 141,700 in.-lb. 
 _ _141^^700 _ 
 
 ^ - (12) (10757^ - 
 
 e' 141.700 
 
 d (2200)(10.5) 
 
 Entering Diagram 16 with a value of p = 0.0077 on the lower right-hand margin and tracing vertically to 
 a value of ^ = 6.1, then horizontally to the left to a point vertically above K = 107, we find /« = 14,000 and 
 fc = 610. 
 
 Illustrative Problem. — Design the vertical wall of the retaining wall described in the preceding problem 
 so that fc = 750 and fs = 16,000. Assume the weight of wall at 2000 lb. 
 
 For these unit stresses the left-hand part of Diagram 16 shows K = 133.8. Then 
 
 \(133.i 
 
 (133.8) (12) 
 141,700 
 
 9.4 in., say 9^/^ in. 
 
 d (2000) (95) 
 
 = 7.45 
 
 Following across the diagram horizontally to the right to a value of ^ = 7.45 and then vertically downward to the 
 lower right-hand margin we find p — 0.0085. 
 
 3. Graphical Determination of Stresses. ^ 
 
 3a. Rectangular Sections. — Fig. 7a shows the cross-section of a column, re- 
 inforced with twelve 1-in. square bars, and of dimensions as shown. This column is loaded with 
 an eccentric load of 75,000 lb., acting 1 in.- outside the edge of the column. It is desired to find 
 the maximum unit stress in the concrete and steel, assuming that the load is symmetrically 
 placed about the axis XX, Fig. 7a. In this solution the effect of the bending moment pro- 
 duced by the eccentricity of the load will first be considered, after which the effect of the direct 
 load will be added. 
 
 The neutral axis is located by means of Figs. 76 and 7c, and the base line ef is drawn in 
 Fig. 7d. The length eh is measured off to represent some convenient number of pounds per 
 square inch, in this case 400, and the line her is drawn. From the intercepts between the lines 
 cf and cr, the stress in the tension steel may be found. Figs. 7e and 7g locate the resultant 
 of the compressive stresses. Figs. 7/ and 7h locate the resultant of the tensile stresses, and 
 the effective depth is found to be 14.6 in., from which 656,000 in.-lb. is found to be the resisting 
 moment when the maximum stress in the concrete is 400 lb. per sq. in. The eccentricity of N 
 measured from the neutral axis is approximately 8.25 in., therefore the load produces a moment 
 of 619,000 in.-lb. Now, if the resisting moment is 656,000 in.-lb. when the maximum concrete 
 stress is 400 lb. per sq. in,, by proportion the maximum stress in the concrete produced by a 
 bending moment of 619,000 in.-lb. will be 377 lb. per sq. in., shown to scale as ea in Fig. 7d. 
 
 The effect of the direct load will now be considered. A direct load applied at the neutral 
 axis will have the effect of moving the base line ef to the left, thus increasing the compressive 
 stresses and decreasing the tensile stresses. Assume the base line ef to be moved to eji, a 
 distance of 100 lb. per sq. in. The increase of compressive stresses is added to the decrease in 
 the tensile stresses and found to be 33,600 lb., which is laid off to scale along the line Cig, thus 
 locating the point g. In other words, a load of 33,600 lb. applied approximately at the neutral 
 axis would move the base line ef to ei/i, increasing the compressive stresses in the concrete 100 
 lb. per sq. in. and decreasing the tensile stress in the tension steel lOO(n) = 1500 lb. per sq. in. 
 The base line is now moved another 100 lb. per sq. in. to the left, and the additional increase in 
 compressive stresses plus the decrease in the tensile stresses is added to the 33,600 lb. already 
 
 ^ Method as given by W. S. Wolfe, Instructor in Architectural Engineering, University of Illinois, in Eng. d: 
 Cont., April 25, 1917, and May 23, 1917. 
 
Sec. 9-36] 
 
 BENDING AND DIRECT STRESS 
 
 407 
 
 obtained, making 71,400 lb. which is laid off along Cih, thus locating h. Again the base line is 
 moved 100 lb. per sq. in. to the left, reaching the position 63/3 and i is found to be out a distance 
 of 112,400 lb. Now draw the curve eghi and locate k, the point where the 75,000-lb. hne cuts; 
 then draw the base line kxij, which shows ax = 587 lb. per sq. in. = the maximum stress in the 
 concrete, and 7jd(n) = the maximum tensile stress in tensile steel. That is, a load of 75,000 lb. 
 applied approximately along the neutral axis will move the base line ef to the position xy, thus 
 increasing the compressive stress in the concrete 210 lb. per sq. in. and decreasing the tensile 
 stress in the steel ijf(n) = 210(15) = 3150 lb. per sq. in. Now a slight approximation has been 
 made in assuming that a load applied at the neutral axis moves the base line directly to the left. 
 This assumption is exactly correct to start with, but as the base line moves to the loft, more 
 area comes under compression and the position of the load must move slightly toward the ten- 
 
 FiG. 7. 
 
 sion side of the beam in order to keep the base lines parallel. This deviation from the neutral 
 axis, however, is only slight as long as there is very much tension in the column. For example, 
 when the base line has moved to e 1/1 the load should act through Ci in place of through c; when 
 the base line has moved to 62/2 the load should act through C2; etc. 
 
 As a graphical check on the work the compressive and tensile stresses in the column when 
 the base line is xy are found and laid out in the force polygon, Fig. 7j, from which the funicular 
 polygon. Fig. 7i, is drawn, locating their resultant, which, of course, should have the same 
 action line as the applied load A^. The error is found to be z, which is just about as large as 
 the error made in assuming that the eccentricity was the distance from the neutral axis to the 
 action line of N. If it is desired to make a correction for this error, add z to the eccentricity 
 used. This will increase M slightly, which will increase the length of ea, in this case from 377 
 to 383 and the maximum stress in the concrete is 583 lb. per sq. in. 
 
 36. Hollow Circular Sections. — When the section of a reinforced-concrete chim- 
 ney, or any hollow circular section such as shown in Fig. 8a is considered as having an eccen- 
 tric load, we have a very difficult problem as far as an analytical solution is concerned. ^ How- 
 
 1 See analytical method used in the design of chimneys, Art. 15, Sect. 18. 
 
 I 
 
408 
 
 CONCRETE ENGINEERS' HANDBOOK 
 
 [Sec. 9-36 
 
 ever, when the graphical analysis is applied, we find that the solution is comparatively simple, 
 being but very little more complicated than the solution of a solid square column reinforced on 
 all sides. 
 
 In Fig. 8a the concrete is divided into a number of small slices. From the approximate 
 centroids of a number of these slices, on the compression side, lines are drawn to the right on 
 which are shown the areas of the various slices, which areas may be called their compression 
 values. Assuming that the concrete does not take tension, the tension value of each of these 
 slices would be zero. From the steel rods, lines are also drawn on which are marked n times 
 the steel area (assuming n = 15), which is the tension value of the rods and also their compres- 
 sion value. These tension and compression values are laid off in the regular way in the force 
 polygon Fig. 86, from which the funicular polygon Fig. 8c is drawn, locating the neutral 
 axis by the intersection 0. The base line ef in Fig. Sd is now drawn, eh is made equal to 
 
 V 
 
 Fig. 8. 
 
 400 lb. per sq. in., for convenience, and the line bcr is drawn. From the intercepts between 
 the lines her and ef and, from the areas of the slices and rods, the compressive forces and also 
 the tensile forces are computed. 
 
 At this point it is desirable to make a check on the work which has been accomplished. 
 We know that the summation of the compressive forces produced by a bending moment should 
 equal the sum of the tensile forces, therefore by making these two summations and comparing 
 the results a check will be obtained. This has been done at the right of Fig. Sd and the error 
 found to be less than }i of 1 %, which is satisfactory. The resultant of the compressive forces 
 is now located by the force polygon Fig. Se and the funicular polygon Fig. 8/; also the result- 
 ant of the tensile forces is located by the force polygon Fig. Sg and the funicular polygon Fig. 
 Sh. From these results the effective depth is found to be 50.2 in. and the resisting moment, 
 when the maximum stress in the concrete is 400 lb. per sq. in., is computed and found to be 
 
Sec. 9-3c] 
 
 BENDING AND DIRECT STRESS 
 
 409 
 
 approximately 9,675,000 in.-lb. The moment produced by the eccentric load AT, which has 
 been assumed to act 12 in. outside of the section and to have a magnitude of 230,000 lb., is 
 7,770,000 in.-lb., which by proportion would produce a maximum stress in the concrete of 321 
 lb. per sq. in. 
 
 The effect of a direct load of 230,000 lb. at the neutral axis will now be considered; the base 
 line e/ is moved 1001b. to the left, to the position ej/i, and eig is computed and found to be 
 142,000 lb., which is equal to 15 times the area of all the rods times 100 lb., plus the summation 
 of the area of each and every slice times the average increased compression over it. The base 
 line is now moved 100 lb. farther to the left, to the position 62/2, and is found to be 300,200 
 lb. In like manner esi is found to be 473,300 lb. and the curve eghi is drawn, k is now lo- 
 cated so that kX = N = 230,000 lb. and the true base line is obtained, giving Xa 
 (the total maximum stress in the concrete) = 158 + 321 = 479 lb. per sq. in. The intersection 
 of XF with ad gives n which locates the axis of zero stress. The maximum stress in the steel 
 is rY = 710 X 15 = 10,650 lb. per sq. in. 
 
 As a graphical check on the work, the intercepts between the lines ad and XY times the 
 various areas will give the resisting forces produced in these areas by N. These forces were 
 found and laid off in the force polygon Fig. Si, from which was drawn the funicular polygon 
 Fig. Sj, locating their resultant which should have the same action line as N. The distance 
 z shows the error found. 
 
 3c. Solid Circular and Other Sections. — A solid circular column may be handled 
 in the same way as the hollow section just considered, the only difference being that the areas 
 of some of the slices will be relatively larger because of the absence of the hole. Also the con- 
 struction for sections of other shapes will be just the same except that the relative size and posi- 
 tion of the different slices and rods will vary. 
 
 i 
 
SECTION 10 
 
 MOMENTS IN RIGID BUILDING FRAMES 
 
 1. Importance of the Subject. — The reinforced-concrete building frame differs from other 
 frames particularly in its rigidity at the junction of members, when proper provision has been 
 made for continuity. This applies not only to columns extending from the basement to the roof, 
 and to beams continuous for more than one span; but more particularly to the rigidity existing 
 between the column system and the continuous beams. Many recent tests^ upon completed 
 structures show that the concrete building frame is a rigid one, and, if properly reinforced, may 
 well be designed as such. Many tests also show that, whether so designed or not, the reinforced- 
 concrete frame, due to its rigidity, causes column stresses hitherto not considered, and of a mag- 
 nitude to cause more careful investigation during future design. 
 
 2. Method of Analysis. — The method of analysis which follows is that of slope-deflections 
 developed with the aid of the principle of area moments. 
 
 M 
 
 From the property of the curve for beams we have the following important laws 
 
 1. The change in angle between the tangents at any two points on the elastic curve of a 
 member, which change is caused by the action of moments on the members at those points, is 
 
 M 
 
 equal to the area on the diagram of these moments, included between the two points. 
 
 2. The lateral displacement of one end of a member, when that member is acted upon by 
 
 M . 
 
 moments, is equal to the statical moment of the ^ diagram about the displaced end. This 
 is the principle of area-moments (see Mich. Technic, 1910). 
 
 1 Bulls. 64 and 84, Eng. Exp. Sta., Univ. of 111.; Jour. Am. Cone. Inst., vol. II, No. 6, 1914. 
 
 » For any small portion ds of a member (see Fig. A), there is a certain value of x which may be designated by 
 dx. Let tangents be drawn to the curve at the extremities of the portion ds; then the angle between these tangents 
 may be called d<j>. For every value of c?<^ there is a corresponding elementary portion dy intercepted on B'B. 
 This distance dy is equal to x • d4> (since the curvature is in reality very slight) ; and, similarly, yi is equal to 
 
 r r r Mxdx 
 
 (See Art. 46, Sect. 7.) 
 
 Suppose that a curve be plotted such that the ordinate at any point represents the bending moment at the 
 corresponding point in the member, divided by EI. Thus, for any point P, 
 
 Mp M 
 the ordinate to the curve is The area under this curve for a distance 
 
 M 
 
 dx along its axis (shaded in the figure) is equal to-^j • dx; likewise, under the 
 whole curve it is 
 
 Mdx 
 EI 
 
 C — 
 J A EI 
 
 But this is the expression for the change of slope 4> of the tangent at A 
 with respect to the tangent at B; hence the first law is evident. 
 
 The shaded area in Fig. A has a moment about the end B equal to 
 
 ^jdx 'X. This is seen to be the value of dy. The moment about B of the 
 M 
 
 total area under the curve is 
 
 Fig. a. 
 
 i: 
 
 Mxdx 
 
 U EI 
 
 which is the value of y, the linear deflection of one end normal to the original position of the member. The second 
 law then follows. 
 
 411 
 
412 
 
 CONCRETE ENGINEERS' HANDBOOK 
 
 [Sec. 10-2 
 
 Before proceeding with the appUcation of these laws or principles, it is necessary to estab- 
 lish rules concerning the signs of moments and deflections. 
 
 The moment at the end of any member will be considered positive if the external moment 
 at that end acts in a clockwise direction. The subscript of the moment will determine the mem- 
 ber and end in question; thus Mab is the moment at the end A of the member AB. 
 
 The change in slope of the tangent to the elastic curve of a member will be considered posi- 
 tive when the tangent has turned in a clockwise direction. 
 
 The deflection of any point in the member will be measured normal to, and away from, the 
 base line or line of original position. The sign of the deflection will be considered positive when 
 measured in the same direction from the base line as are positive slopes. 
 
 M 
 
 diagrams will be plotted on the tension side of the member. 
 
 In Fig. la there is shown a member AB acted upon by moments and shears at the ends. 
 Let AB' be the original position when considering the member AB, and A'B the original posi- 
 tion when considering the member BA. The angles 6 a 
 I and Ob are negative, as is also d. According to the 
 
 ^ scheme of signs, the moments at both ends are positive. 
 
 However, when the diagram is plotted with respect to 
 
 the member, as indicated above, there is seen to occur 
 
 I ^.rtTTTTTIlTll '^aA ^ point of inflection. Fig. Ih is the diagram for the 
 
 •^'iili^^ member under the loading shown. 
 
 £Z Since the end A is not fixed, the total deflection d of 
 
 B is due partly to the change in slope at A and partly 
 Fig. 1. to the flexure of the member. Thus, the distance d — 
 
 BjJ, is the deviation due to flexure, and is equal to the 
 
 statical moment of the diagram about B. The area of the ^ diagram is seen to be the 
 
 algebraic sum of the areas a' ah and a'hh' ; whence 
 
 The change in slope of the member from 5 to A is 0^ — Qb^ and is equal to the area of 
 the ^ diagram, or 
 
 Qa - Qb = 2^ {Mab + Mba) 
 Eliminating Mba from the two equations above, there results 
 
 Substituting K for ^ and R for 
 
 Mab = 2EK{2dA + Ob - SR) (1) 
 Mba = 2EK{2dB + Oa - SR) (la) 
 This is the general equation for moment at one end of a member carrying no transverse load, 
 in terms of the relative changes in position of the ends.^ 
 
 When there is a transverse load upon the member an expression similar to equation (1) 
 may be derived. Consider the member shown in Fig. 2a. The deflection of B from the tan- 
 
 M 
 
 gent at A is d — 6aI and equals the moment of the diagram about B (see Fig. 2b). 
 
 1 The above method for the general treatment of rigid frames first appeared in Bulletin 1, University of 
 Minnesota, "Secondary Stresses and other Problems in Rigid Frames," by G. A. Maney, March, 1915. The 
 method was applied to the special case of wind stresses in office buildings, Bulletin 80, University of Illinois, by 
 W. M. Wilson and G. A. Maney (June, 1915). The notation here used is from the latter bulletin. 
 
Sec. 10-2] 
 
 MOMENTS IN RIGID BUILDING FRAMES 
 
 413 
 
 This diagram is similar to that shown in Fig. 16, except that the area caused by the load is 
 added to the diagram of the other moments at A and B. 
 
 \ MabI MbaI ^ Pab 
 El\_ Z ^ & 
 
 M 
 
 The difference in slope between the two ends is (Oa — Ob), and is equal to the area of the 
 
 diagram. 
 
 0 = 
 
 I 
 
 2EI 
 
 (^Mab + Mba + ^) 
 
 EI 
 
 From these two equations Mba may be eliminated, whence 
 
 Mab = 2EK{2dA +dB - SR) 
 
 (2) si^:::------ 
 
 Similarly 
 
 Mba = 2EK{2dB -\- Oa - ZR) + ^ (2a) ^Jl^A 
 
 If the transverse load on the member is symmetrical, then ' 
 equation (2) reduces to 
 
 Mab = 2EK{2dA + Ob - 3 R) - 
 Mba = 2EK(2dB + Oa - SR) + 
 
 (3) ^ 
 Ell 
 
 (3a) 
 
 in which F is the area of the moment diagram for the 
 given svmmetrical loading on a simple span.^ For 
 
 F 
 
 different types of symmetrical load the values of y are given below: 
 
 Values of 7", Moment Factor for Symmetrical Loads* 
 
 Loadinc 
 
 1 
 
 P P 
 
 K/-4-#4-?-H 
 f \ 
 
 "p ^ 'p 
 
 V - ^ ---4 
 
 .p «p >p ■ 
 
 Ib.perff. 
 
 I* £7 - ■>)<■• i ■ ■>|<-(7 
 
 I w Ib perffy I 
 
 f lb per -ft-. ' f 
 
 Moment diagram 
 
 .Pa 
 
 Tp^ 
 
 wa '- 
 '2 
 
 Moment factor 
 
 8 
 
 — P£ 
 
 wa' 
 17 
 
 * Prepared from a fable by rE.Richarfj Masfers Thesis, I9IS, UnuoflU 
 
 It should be noted that in equations (2) and (3) the sign of the last term is negative. 
 In equations (2a) and (3a) the sign of the last term is positive. A general rule results : 
 
 1 The last term of equations (2) and (3) is the moment at the end of a fixed beam loaded similarly to 
 member AB. 
 
414 
 
 CONCRETE ENGINEERS' HANDBOOK 
 
 [Sec. 10-3 
 
 If the load, independent of the member, tends to rotate around the joint under considera- 
 tion in a clockwise direction, the sign of the last term of equations (2) and (3) is negative. If 
 the load, independent of the member, tends to rotate around the joint under consideration in 
 a counter-clockmse direction, the sign of the last term of equations (2) and (3) Is positive, as 
 in equations (2a) and (3a). 
 
 By comparison of equations (1) to the succeeding ones, the last term in equations (2) 
 and (3) are noted to be ''load terms"; that is, they are the modifying factors of the equations 
 due to the presence of the loads. 
 
 A number of special equations arise from those in (1), (2) and (3) when various end re- 
 straints are imposed upon the member. These, for convenience, have been tabulated below. 
 The above statement concerning signs applies here also. 
 
 Member and loading 
 
 e 
 
 
 Mba 
 
 Eq-No. 
 
 ^ d) 
 
 
 Zero 
 
 
 lb 
 
 ^) 
 
 
 
 
 Ic 
 
 ¥ 1) 
 
 
 
 
 Id 
 
 ^) 
 
 ^A '^B 
 
 
 6EH9b 
 
 Je 
 
 
 
 Zero 
 
 3EKe^i-^(Uci) 
 
 2b 
 
 
 
 
 4EKe^'i-^ 
 
 2c 
 
 1^ Symmehical load 
 
 ^\ ^ 
 
 
 -Zero 
 
 3EKe^^\^ 
 
 3b 
 
 
 ^A^'^B 
 
 
 BEKe^i-j- 
 
 3c 
 
 ^^Symmefrical lead ^ 
 
 
 
 4EKeg+-j 
 
 3d 
 
 3. Application of Method of Analysis to Simple Cases. — 1. Let it be desired to determine the 
 moment at the central support of the beam shown in Fig. 3. 
 
 Referring to equation 26 in the above table of special moment equations. 
 
 Mba = 3EKidB + ^ (h + a) 
 
 2h^ 
 K. 
 
 t^. JJ.....M...Lj 
 
 r ' T T and from equation 16, 
 
 Fia. 3. Mbc = ZEKt^b 
 
 Since the summation of moments around the joint B must equal 
 
 zero for equilibrium, 
 
 Mba + Mbc = 0 
 {3EKi + SEK,)eB + ^(^1 + a) = 0 
 
 Ob = 
 
 SEKi + 3EK, 
 
 [ga: + a)] 
 
Sec. 10-3] 
 
 MOMENTS IN RIGID BUILDING FRAMES 
 
 415 
 
 whence 
 
 Pah 
 
 (^1 + 
 
 either of which equations gives tension at the top of the beam over the support B. 
 If Ki = and li = h, 
 
 Mbc = - Mba = - 
 
 Pab 
 
 4/i2 
 
 (^1 + a) 
 
 2. Let it he desired to find the moment at the top of the lower column in the frame shown in 
 Fig. 4. Let the values of K for the column he the same, and let K for the girders, also, be alike. 
 From equation Sd in the above table, 
 
 F 
 
 Mba = 4EKidB + y (o) 
 
 From equation Id, 
 
 Mbd = Mbe = 4EK2dB 
 
 and 
 
 Mbc = 4:EKidB 
 
 Since the sum of the moments around the joint B 
 must equal zero, 
 
 Mba + Mbc + Mbd + Mbe 0 
 {SEKi + SEK2)dB +7 = 0 
 1 F 1 
 
 ^ ^ wl^ 
 
 SEKi + SEKo ' I SEKi + SEK2 ' 12 
 
 Substituting into the above equations there results 
 
 D 
 
 
 
 
 n lb per ft 
 
 
 wmm. 
 
 
 
 B K, 
 
 
 
 E 
 
 7/ 
 
 (") 
 
 
 Y 
 
 
 
 
 
 Momenf Diacpvm 
 
 Fig. 4. 
 
 Mbd = Mbe 
 Mbc 
 
 wP 
 12 
 
 /K^ + 2K2 \ 
 = \2Kr + 2K2I 
 
 \2Ki+ 2K.J 12 
 
 \2Ki + 2K2) 12 
 
 If K2 = 0, that is, a simple support replaces the columns at B, 
 Mba - - Mbc 
 
 wl^ 
 24 
 
 which is the moment at the central support of a continuous girder with fixed ends loaded on one 
 span with a uniform load. 
 
 4. Conception of Rigidity of Building Frames. — Suppose a point in 
 space to be held to some infinitely rigid body by several elastic members, 
 each of which is fixed at the end farthest from the point in question. 
 Thus, in Fig. 5, the point A is the rigid junction of the members. The 
 members 1 to 5 are fixed at points a, h, c, d and e, and are rigidly con- 
 nected to each other at A. Let AB be a loaded member, tending to 
 turn point A in a clockwise direction. If the members 1 to 5 are infinitely 
 rigid or any one of them is infinitely rigid, the slope of the tangent to 
 AB at A will remain in a fixed position throughout loading, and AB 
 would be said to be fixed at A. Again, if the members 1 to 5 were not 
 at all rigid, A would turn and there would be no bending restraint upon AB, through the 
 members meeting at point A. If there is no load acting upon members 1 to 5, there are two 
 limiting conditions of rigidity at the joint A — namely: (1) zero restraint and- (2) fixity, due, 
 
416 
 
 CONCRETE ENGINEERS' HANDBOOK 
 
 [Sec. 10-5 
 
 respectively, to whether the members 1 to 5 have zero stiffness, or have infinite stiffness. It 
 is evident, since the sum of the effects of each of the members is the total effect of all of the 
 members in restraining the rotation of point A, that the effect of all together may be thought 
 of as the effect of but one member capable of giving the same restraint. 
 
 In a building frame of reinforced concrete, any beam is restrained a certain amount at 
 each end, due to the rigid connection existing between it and the columns above and below, 
 and the girder beyond. The restraint causes negative moments in the ends of the beam, and 
 in turn the load on the beam, in causing a tendency for a rotation of A, causes flexural stresses 
 in the restraining members. 
 
 Fig. 6 shows a uniform load over one span in a building frame, and the deformations caused. 
 The members radiating from A and B are restrained at their outer ends to an extent varying 
 between a hinged and a fixed condition. If loads were put on aA and Bd, the deflections in 
 the columns at A and B would be practically removed, and AB would be practically fixed. 
 Suppose, however, that instead of this the spans beyond a and d were loaded. This would add 
 
 to the deflections shown. Further deformation could be 
 obtained by loading alternate panels in all directions. 
 
 A different effect could be obtained by loading the 
 panels above and below AB. This would develop a point 
 of contraflexure in the column, but would give a constant 
 moment in the unloaded girders. By loading alternate 
 bays the maximum effect of this kind would be obtained. 
 
 It will be possible to remove the portion ahcdef of the 
 frame, and impose upon the extremities of the members 
 such moments as will imitate any condition of full or partial 
 loading. It will be assumed in these cases that the three 
 girders have equal K's; that the lower columns have equal K's; and that the upper columns 
 have equal K's. This will be recognized as a common condition; and though cases may arise 
 which are different, this case will aid the judgment of the designer. 
 
 5. Moments at Interior Columns in Beam-and-girder Construction. 
 
 6a. All Terminals Hinged (Case I). — From the fact that the frame is symmet- 
 
 
 
 
 
 
 
 
 b .... c 
 
 
 
 
 a 
 
 
 d 
 
 
 
 
 A—-'B 
 
 
 
 
 
 
 e 
 
 
 Fig. 6. 
 
 9c 
 
 K 
 Ho ^ 
 
 tiomenf diagram 
 
 rically loaded and is symmetrically rigid 6 a = — Qb] hence, from equation 3c, 
 
 F 
 
 From equation 16, 
 
 Mab = 2EKidA - 
 MAf = SEKoOa 
 
 MAa = SEKidA 
 
 MAb = SEKzdA 
 
 But for equilibrium, 
 
 Ma/ + MAa + MAb + Mab = O 
 Substituting and solving, 
 
 _ F 1 
 
Sec. 10-56] 
 
 MOMENTS IN RIGID BUILDING FRAMES 
 
 417 
 
 I 
 
 Substituting this value into the above moment equations, 
 
 I ISKo + 5Ki + SK,i 
 
 _ FT SKo 1 
 
 I IsKo + 5Ki + ^K^] 
 
 5b. All Terminals Fixed (Case II). — As in the preceding development, 
 
 do) 
 
 (lb) 
 
 Symhad! 
 
 
 
 f 
 
 
 i i 
 
 
 From equation Id, 
 
 Since 
 
 it is found that 
 
 By substitution, 
 
 Mab = 2EKidA 
 
 Ma/ = ^EKodA 
 
 MAa = ^EK.dA 
 
 MAb = ^EKsdA 
 
 Ma/ + MAa + MAb + Mab = O 
 
 dA = 
 
 I 4Ko + 6Ki + 4K, 
 
 I l2Ko + SKi + 27^:3] 
 
 (Ila) 
 (116) 
 
 Equation (Ila) could have been derived from (la) by replacing the SK of each outstanding 
 member by 4:K, as would be clear from a study of equations 16 and Id 
 
 From these two cases, Diagram 1 on page 421 was prepared. Instead of plotting the values 
 
 K K 
 
 of K directly, the ratio of each K to Ki was plotted. Thus Ko' = and K/ = The 
 
 diagram shows clearly the small difference in the negative moment at the end of the loaded girder 
 between hinged and fixed terminals. When Ks^ = 0 there are no upper columns, as is the case 
 in a viaduct bent, or a one-story deck structure. When K3' = Ko' = 0, the girder becomes one 
 on simple intermediate supports. 
 
 If a structure of two stories in height were extended indefinitely in either direction and 
 alternate spans were loaded, Cases I and II would be modified by giving the two outside girders 
 a constant moment over their length, and having their terminals neither fixed nor hinged. 
 Expressions for Mab and Ma/ for each case follow. 
 27 
 
418 
 
 CONCRETE ENGINEERS' HANDBOOK 
 
 [Sec. 10-5c 
 
 6c. Columns Hinged. Outer Girders with Constant Moment (Case III).- 
 
 5d. Columns 
 
 Mab 
 
 Mas 
 Fixed. 
 
 f<3 
 
 3Xo + + 3K3 I 
 3A^ 1 
 
 Outer Girders with Constant Moment 
 
 (Ilia) 
 
 (III6) 
 (Case IV).— 
 
 If 
 
 BK, 
 
 2Ko + 2A'i + 2/^. 
 
 (IVa) 
 
 (1V6) 
 
 Equations (Ilia) and (IVa) are plotted on Diagram 2. It indicates, as before, the small dif- 
 ference in the value of Mab between fixed and hinged columns. 
 
 A third condition which will develop high stresses in the columns, and which will give a 
 high degree of rigidity to joints A and B, is a case of loading in which alternate bays are loaded, 
 and all the spans of one bay for the full height of the structure are loaded. When all panel 
 loads are the same, or when the loads are such as to give the same rotation at each joint, the 
 point of inflection will be at the center of the column. The girders may have their extremities 
 either hinged or fixed, or the outer girders may have constant moments over their length. 
 Each of these conditions is treated in a following case. 
 
 be. Point of Inflection at Center of Columns. Outer Girders Hinged (Case V). — 
 
 a ff, A 
 
 1. 
 
 Mab 
 
 Mas = 
 
 1 
 
 
 f 
 
 .TTTTTTiTl^ 
 
 ^ i 
 
 
 II6K0 + 5Ki + 6i^3. 
 
 (Va) 
 
 r.] 
 
 6/. Point of Inflection at Center of Columns. 
 
 Outer Girders Fixed (Case VI) 
 
 (Vb) 
 
 K, A. 
 
 1 
 
 
 f 
 
 
 i i 
 
 
Sec. 10-5^] 
 
 MOMENTS IN RIGID BUILDING FRAMES 
 
 419 
 
 Mab = 
 
 Ma/ = 
 
 F rSKo + 2K, + SK, 
 
 ii 
 
 3Ko + SKi + SK, 
 
 5g. Point of Inflection at Center of Columns. 
 Moment (Case VII).— 
 
 I (Via) 
 (VI6) 
 
 Outer Girders with Constant 
 
 ^3 
 
 ■p. I 
 
 Mab 
 Mas 
 
 e 
 
 
 
 f 
 
 llllllllllll^ 
 
 
 
 3Ko + Ki + SK 
 
 l ISK 
 
 SKo + 2Ki + 3A' 
 
 + 2Ki + 3^3. 
 
 (Vila) 
 (VII6) 
 
 Equations (Va), (Via) and (Vila) have been plotted in Diagram 3. It should be noted, ac- 
 cording to Diagrams 2 an^ 3, that the effect of the constant moment along the outer girders is 
 to cause a rotation in addition to that caused normally by the load; and that this additional 
 rotation causes a decrease in negative moment in the girder AB from what it would have been 
 had the girders been either hinged or fixed at their outer ends. This is consistent with the 
 fundamental conception of rigidity. 
 
 Diagram 4 has been plotted to show comparatively the results of the foregoing cases. 
 Since both Ko' and Ks' have the same coefficients in the foregoing expressions for Mab, Diagram 
 4 is plotted between the moment and the sum of Ko' and Ks'. The resulting curves are identical 
 with those for Kz' = 0 in Diagrams 1 to 3. 
 
 F 
 
 The variation of moment is for nearly all cases not over 10% of the load factor y For 
 
 values of (Ko' + K/) > 5, the moment is proportional to the value of (Ko' + K/). The small 
 difference in moments between hinged and fixed terminals is again emphasized. All cases 
 
 F 
 
 approach a moment Mab = y as a maximum limit. 
 
 On the first floor there may arise a condition in which the first tier of columns may be 
 hinged or fixed, while the second tier of columns may have a central point of inflection. Since 
 the greatest stress will arise from fixity of the lower tier, rather than from a hinged condition, 
 it will be investigated. The outer girders will be located as in Cases V to VII above. 
 
 6h. Point of Inflection at Center of Upper Columns. Lower Columns Fixed. 
 Outer Girders with Constant Moment (Case VIII). — 
 
 — ^Kjllllllllli 
 
 _ _F \2 Ko -\- K, + ZK{ \ 
 '^^^ I l2Ko + 2Ki + 3A3J 
 
 M, 
 
 F V SK, 1 
 
 I l2Ko + 2Ki + 3X3J 
 
 (Villa) 
 (VIII6) 
 
 5i. Point of Inflection at Center of Upper Columns. Lower Columns Fixed. 
 Outer Girders Hinged (Case IX).— 
 
420 
 
 CONCRETE ENGINEERS' HANDBOOK 
 
 [Sec. lo-sy 
 
 i7* 
 
 d K, A 
 
 *7V 
 
 Mab 
 
 Mai, 
 
 F V 4:K^ + 3Xi + eiiCs l 
 Z L4i^o + SXx + 6K3J 
 
 rL4Ko + 5^1:1 + 6X31 
 
 (IXa) 
 
 (1X6) 
 
 6j. Point of Inflection at Center of Upper Columns. Lower Columns Fixed. 
 Outer Girders Fixed (Case X).— 
 
 ^/r: Ai 
 
 Mab = 
 = 
 
 Ho Ka 
 
 Z L2/^o + 3/^1 + 3X3] 
 3X3 1 
 
 f[ 
 
 2Ko + 3Ki + 3/!:= 
 
 (Xa) 
 (X6) 
 
 The moment Mab as given by each of the above three cases is plotted in Diagram 5. There 
 is little difference between the moments on this diagram and those on Diagram 3, for corre- 
 sponding Kz. 
 
 A review of the foregoing ten cases shows that the negative moments in the loaded girders 
 cannot in any way exceed those for a continuous girder whose ends are similarly restrained. 
 It is justifiable, therefore, to design the girders as continuous, so far as negative moments are 
 concerned; and for either positive or negative moments the recommendations of the Joint 
 Committee (see Art. 45, Sect. 7) are entirely adequate as a basis of computation. 
 
 The effect on the columns of unbalanced floor load is shown for each of the ten cases on 
 Diagrams 6 to 9 inclusive. The greatest moment in the interior columns is found at either 
 end of the second-tier columns when the first tier has fixed bases (Case VIII, Diagram 9). 
 
 F 
 
 The value of this moment is shown on Diagram 10. Very rarely will it exceed 60% of y> 
 
 which for uniform loads gives 0.05 wl"^. 
 
 6. Moments at Exterior Columns in Beam-and-girder Construction. — Since exterior 
 columns must resist all the moment in the end of the joining beam, the moments will be liable 
 to be somewhat higher than in interior columns, where the girder beyond takes its share of the 
 moment. 
 
 Syjnmefrical 
 
 It has already been shown (page 418) that when moments are applied at a and h such 
 that points of inflection are caused at the mid-height of the columns, the columns more firmly 
 retard the rotation of the joint A. The moment, therefore, is greater in the columns when 
 the moments at a and h are thus applied. The condition of restraint of B may vary from 
 
Sec. 10-6] 
 
 MOMENTS IN RIGID BUILDING FRAMES 
 Diagram 1 
 
 100 
 
 % 
 t 
 
 Cu.|r- 
 
 o 
 
 .1 60 
 ^ 50 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 'Mill 1 ' t' ,,. '\ 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 c 
 
 ur 
 
 ve 
 
 s 
 
 
 * V 
 
 ah 
 
 
 > or 
 
 
 
 
 
 
 
 
 
 
 r1 
 
 
 
 — 
 
 
 
 
 - 
 
 
 
 
 
 
 
 
 
 - 
 
 — 
 
 
 
 
 
 
 
 
 
 — 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 - 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 -< 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 / 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 K3 
 
 
 negative moment at 
 ends of airder AB 
 
 
 
 / 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 ' Columns and qinders 
 
 7 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 • hinged at ends- - Case I 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 - — columns ana giraers 
 
 ^=k; fixed at ends- Case U 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 e 3 
 Values of (Lower column) 
 
 Diagram 2 
 
 100 
 
 90 
 
 c 
 
 K 
 
 c 
 
 (y^ 70 
 
 il 
 
 I 
 
 Z 
 
 60 
 
 -50 
 
 - K, 
 
 Sym. 
 
 K, 
 
 A 
 Ko 
 
 
 
 Negative moment 
 at ends of girder AS 
 
 at ends Case III 
 
 Columns fixed 
 
 ^=K' at ends Case lY 
 
 z 
 
 60 
 
 50 
 
 Values of Ko (Lower columin) 
 Diagram 3 
 
 TTT 
 
 jS^^ Negative moment M*© 
 •<fKq \ at ends of girder 
 
 'ex- 
 
 ^ ^. treme ends CaseV 
 
 K. K. Girders fixed at ck- 
 
 V K . treme ends.- CaseYI 
 
 —•-k; ^»k; Girders with con - 
 
 stent moments. Case TH 
 
 Values of Ki (Lower column) 
 
422 CONCRETE ENGINEERS' HANDBOOK [Sec. 10-6 
 
 Diagram 4 
 
 0 2 4 6 8 10 , 12 14 16 18 20 
 
 Values of (k: + KJ 
 
 Diagram 5 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 1 1 1 1 1 
 
 1 
 
 
 
 
 
 1 
 
 
 L 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 Vc 
 
 
 jes 
 
 pf 
 
 H 
 
 
 
 i 
 
 1 
 
 ST 
 
 ■ 
 
 £01 
 
 
 n 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 g. 90 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 11 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 S i- 
 
 80 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 A 
 
 J" 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 y 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 / 
 
 
 
 / 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 /■ 
 
 w 
 
 K3 
 
 
 Necrative momerrT m^b 
 
 Negative 
 of 
 
 
 
 
 
 / 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 I 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 Ki 
 
 
 K, 
 
 
 
 
 
 - rtirders with cons-forrf 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 A 
 
 K. 
 
 B 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 Ke 
 
 
 
 Ko 
 
 
 moments - 
 
 
 
 -Case YIU 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 ? 
 
 
 
 
 
 - Girders hinged-Case IX 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 K, 
 K 
 
 
 
 
 
 
 
 - Girders f ixec 
 
 - 
 
 CaaeX 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 = 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 0 I 2 3 4 5 
 
 Values of Kq (Lower column) 
 
 Diagram 6 
 
 Values of K3 (Upper coluTnn) 
 
Sec. 10-6] 
 
 MOMENTS IN RIGID BUILDING FRAMES 
 Diagram 7 
 
 423 
 
 100 
 
 c 
 
 O O 80^*- 
 
 •^=k; -^ = k; fixed Case VI 
 
 5ym 
 
 K3 Moment MAf 
 
 Columns 
 
 hinged •• Case HI 
 - — Columns 
 
 Values of K3 (Upper column ) 
 
 Diagram 8 
 
 100 
 
 80 
 
 P 
 
 20 
 
 "5 ^ 
 I 0 
 
 Moment M^^f 
 
 Girders hinged-Case^ 
 
 Girders with con 
 
 stanf momen+ CaseYI 
 
 ^ = k; ^-Ka Girders fixed-Case-ra 
 
 Values of K3 (upper column) 
 Diagram 9 
 
 . < 
 
 100 
 
 Moments in second 
 tier columns 
 
 Outer girders with 
 
 constant moments-Case YlII 
 
 Outer girders 
 
 hinged Case IX 
 
 K. K. ^,< Outer girders 
 
 Values of KJ, (Lower column) 
 
424 
 
 CONCRETE ENGINEERS' HANDBOOK 
 
 [Sec. 10-6 
 
 Diagram 10 
 
 l.O , 1.5 
 Values of 
 
 Diagram 11 
 
 Moments in 
 exterior columns 
 
 K3 Jm}oad^ 
 
 Values of '^X^j 
 
Sec. l(H6al 
 
 MOMENTS IN RIGID BUILDING FRAMES 
 
 425 
 
 fixity to a hinged support. It will be assumed that A and B do not receive translation during 
 the loading of the frame. The following moments at the head of the lower column result: 
 6a. Inner End of Girder Hinged (Case XI). — 
 
 66. Inner End of Girder Fixed (Case XII).— 
 
 Diagram 11 was plotted from these two cases. Case XI gives an increase in moment 
 of approximately 40% over Case XII, and this shows the greatly undesirable result of putting 
 an expansion joint in the girder system only one span from the exterior columns. If the girders 
 are continuous past the second column from the end, few cases will arise in which the moment 
 
 at the head of the lower exterior column (MAb) will exceed 60% of the load factor ^0 , which 
 
 for a uniform load over AB gives 0.05 wP. 
 
 7. Moments in Columns in Flat-slab Construction. — In flat-slab construction, it is neces- 
 sary to estimate the stiffness of the floor before it is possible to compute the moments at the 
 heads of the column. This may be done by replacing the floor slab by an "equivalent girder." 
 Just what portion of the floor slab may be considered as a girder resisting deformation in the 
 columns depends upon the type of reinforcing employed. 
 
 Consider a flat-slab structure in which four-way reinforcing has been employed. If one 
 column is submitted to a bending moment, the rigidity of the joint at its end depends upon the 
 rigidity of the column beyond and upon the stiffness 
 of the floor slab. The moment taken by the flat 
 slab may be taken partially by the diagonal system ^ 
 of reinforcement to columns lying diagonally across ^ a,>^Nil/'x,c 
 
 the panel from the column in question, and partly i | a 
 
 by the rectangular system to columns in the same (jT—. 2>^^|^4^^^f!m) 
 bent as the column in question. If this same struc- ^ 
 ture is loaded on its floor in alternate panels, the Fig. 7. 
 
 moment caused in the slab will be carried to the 
 columns by the two systems of reinforcement. 
 
 Consider now an exterior column as shown in Fig. 7. Let two adjacent spans be loaded 
 .with a uniform load. Moment is brought to the column by the three bands of reinforcement 
 a, b, and c. As the structure is loaded symmetrically to the right and to the left of band 6, 
 the moments brought by the bands a and c will be equal. The proportion of moment carried 
 by bands a and c to that carried by band b depends upon the rigidity of the two systems of 
 paths. 
 
 Suppose there exist no diagonal bands a and c. The reinforcement would now follow a 
 two-way system and reinforcing for the central portions of the panel would cross over the bands 
 of reinforcing between columns. If in this system the load is applied symmetrically with re- 
 spect to the band &, the moment of the load carried to A would reach A largely through the beam 
 action of the bands of reinforcing extending bet\veen columns. It is here assumed that the 
 band 6 has a width equal approximately to the width of the depressed head. 
 
 Whereas, in the case of the four-way reinforcement a considerable amount of moment 
 is brought by bands a, b and c to column A, which in the two-way reinforcement would have 
 reached the column A through the band b and through the band along the outer wall, it will 
 be assumed for purposes of estimation that the amount of moment brought to A may be deter- 
 mined by assuming a beam action of the rectangular reinforcement 
 
426 
 
 CONCRETE ENGINEERS' HANDBOOK 
 
 [Sec. 10-8 
 
 It will be considered first that the stiffness between columns A and B is furnished by a 
 strip of floor slab the width of band b. Later a discussion concerning the error involved in this 
 assumption will be undertaken. 
 
 The moment brought to the head of the column will be estimated by considering a load 
 over the area between A and B having the width of one panel. The frame in sectional view 
 appears on page 420. The moment Mab is given under Case XII. It will be sufficient in com- 
 puting the value of Ii for Ki to consider only the concrete area, neglecting the cross-section 
 of the reinforcement. In the columns, however, the reinforcement should be considered in 
 computing the moment of inertia. Having solved for the moment by formula (XII), the neces- 
 sary flexural reinforcement at the ends of the column may be computed. The moments in the 
 various bands in the slab are found in the computations on flat slab floors (see Art. 20, Sect. 
 11). 
 
 Concerning the portion of the slab between columns A and B, it has been assumed above 
 as having a width equal to the width of the band b. It should be noted that the moment of 
 inertia Ii of the cross-section of this strip varies directly with its width. Hence Ki also varies 
 directly with its width. An examination of formula (XII) indicates that considering the values 
 Ko and Ks as constants, the moment is decreased by an increase in Ki, that is to say, by an in- 
 crease of the width of the strip. Just how extensive this variation may be can readily be deter- 
 mined from Diagram 11. 
 
 8. Criteria for Maximum Combined Stresses in Columns. — Maximum stresses in columns 
 may occur from two sources: first, from a maximum bending moment and direct stress; and, 
 secondly, from a maximum deflection of the column together with direct stress. Maximum 
 deflection of the column, and maximum moment in the column occurring from moment intro- 
 duced into the column from the girder, do not occur simultaneously. Although it is not 
 always true, it is usually the case that the maximum moment, together with the combined 
 stress, will give the greatest fiber stress in the column. 
 
 It may be necessarj^ to design certain columns for a given static load, or it may be necessary 
 to investigate columns for special loading in a given panel or panels. For that reason the 
 following criteria are given. 
 
 8a. Interior Columns. — Maximum moments are caused by loading the bay 
 adjacent to the column in question for the full height of the structure, and, where possible, by 
 loading alternate bays in both directions from that bay. The maximum moment will be found 
 in the second-tier columns. The maximum stress in these columns will be found by combining 
 the stress caused by this maximum moment with the axial stress produced by the load on the 
 floors above the first floor. 
 
 Maximum deflections may be caused in the interior columns by loading spans alternate in 
 all directions. In the case of very slender columns, the deflection may add a moment caused 
 by the direct load to that moment causing the deflection. The stress thus produced, when com- 
 bined with the dead load and such live load as is transmitted to the column in question will 
 produce the maximum fiber stress for that case of loading. 
 
 8b. Exterior Columns. — ^Loading to produce the maximum stress due to moment 
 in the exterior columns is the same as that for interior columns. Likewise the loading causing 
 the maximum deflection in the exterior column is applied in the same manner as the loading 
 causing the maximum deflection in the interior columns. Exterior columns, however, will 
 receive more moment than interior columns because of the fact that the columns are not assisted 
 by a girder beyond in restraining the moment generated. Whatever negative moment there is 
 in the end of the girder in the exterior bay, it must be balanced by a moment in the exterior 
 column above and below that girder. The stresses due to this moment combined with the 
 stresses due to axial loading will cause the maximum fiber stress in the column. 
 
 A very important consideration in the exterior columns is the moment caused in the corner 
 columns. When the floor is constructed of slabs and beams, the moment may be considered 
 as being introduced into the column by the two girders meeting at right angles to each other, 
 
Sec. 10-9] 
 
 MOMENTS IN RIGID BUILDING FRAMES 
 
 427 
 
 and these moments may then be combined to give a diagonal resulting moment. The column 
 in such construction is usually square, and there will, therefore, be a change of axes for this 
 resultant bending. When the floor is a flat slab, moments are brought into the corner column 
 from three sources (in four-way reinforcing systems), from the lintel beams and from the 
 diagonal band of reinforcing that crosses the corner panel. It has been found from tests on 
 buildings that this diagonal band carries extremely high stresses in the reinforcing and that these 
 high stresses introduce into the corner columns very large moments. An estimate of the mo- 
 ment in the corner column can be made by considering an equivalent beam action (see page 425) 
 to replace this diagonal band. The negative moment at the extremity of this equivalent beam 
 may be computed and this moment may then be combined with the resultant moment of the 
 moments at the ends of the lintel beams. It should be here noted that the length of the 
 equivalent girder is the length of the diagonal across the panel. If the corner column is not 
 round, the transfer of axes is necessary before computing maximum fiber stress. 
 
 9. Wind Stresses in Building Frames. — It is often necessary to determine the stresses set 
 up in a large building frame by wind pressure on some exposed face of the building. Exact 
 methods^ of the analysis of these stresses are very long and tedious, and some of them are so 
 laborious that it is impracticable if not impossible to apply them to a building of any consider- 
 able height. 
 
 An exhaustive study of stresses due to wind pressures in buildings has been made in Bulletin 
 80 of the Engineering Experiment Station of the University of Illinois by Prof. W. M. Wilson 
 and G. A. Maney. In this bulletin a number of approximate methods have been studied with 
 a view to the determination of their accuracy, and a comparison of exact methods has been made 
 to determine their applicability, after which the writers of the bulletin present an original 
 procedure for exact analysis by slope-deflections. 
 
 Tall building frames which are not symmetrical about a vertical center line or which have 
 a considerable variation in the sizes of adjacent members should be analyzed by some one 
 of the exact methods, of which that of slope-deflections is probably the most usable. 
 
 For bents of a given building frame having practically equal spans and equal column 
 sections, an approximate method based on the following assumptions has been found to give 
 quite accurate results.^ 
 
 1. Points of inflection in columns are located at their mid-height. 
 
 2. The point of inflection of each girder is at its mid-length. 
 
 3. The shear on each interior column is equal, and the shear on each exterior column is 
 equal to one-half the shear on any interior column. 
 
 The wind is considered as being applied at each floor level. If W represents the total wind 
 force applied to the structure above a given floor, then according to assumption (3), the shearing 
 force W is distributed among the several columns just above this floor in such a manner that 
 each interior column takes an equal amount and each exterior column takes half of the amount 
 
 W 
 
 taken by an interior column. Thus if n is the number of panels, — is the amount taken by 
 W 
 
 each interior column, and is the amount taken by each exterior column. 
 
 The direct stresses in the girders are relatively small and are usually neglected. The 
 direct stresses in the columns are important in very tall narrow buildings and may be com- 
 puted by considering' the frame of the building to act as a cantilever beam with its fixed 
 end at the ground. The direct stress in any given column will be proportional to the 
 
 1 The reader is referred to the following articles, any one of which gives an exact solution of the problem: 
 "Wind Stresses in the Frames of Office Buildings," by Albert Smith, Journal Western Society of Engineers, 
 
 vol. 20, No. 4, p. 341. 
 
 "Stresses in Tall Buildings," by Cyrus A. Melick, Bull. 8, College of Engineering, University of Ohio. 
 
 "The Theory of Frameworks with Rectangular Panels and its Application to Buildings Which Have to 
 Resist Wind," by Ernst F. Johnson, Trans. Am. Soc. C. E., vol. 55, p. 413. 
 
 "Wind Stresses in the Steel Frames of Office Buildings," Bull. 80, Engineering Experiment Station, Uni- 
 versity of Illinois. 
 
 2 "Wind Stresses in Frames of Office Buildings," by Albert Smith of Purdue University, Journal of Western 
 Society of Engineers, vol. 20, No. 4, p. 341, 
 
428 
 
 CONCRETE ENGINEERS' HANDBOOK 
 
 [Sec. 10-10 
 
 77 
 
 Fig. 8. 
 
 distance of the column from the center line of the building. To be able to find these direct 
 
 stresses, the moment of the total wind pressure about a given 
 floor level must be placed equal to the moment of the direct 
 stresses in the columns about the center line of building. The 
 direct stress in a column due to wind should be combined with 
 the stress due to vertical loading. 
 
 Since, according to assumptions 1 and 2 above, points of 
 inflection occur at the center of each column and at the center 
 of each girder, a single joint may be removed as a free body- 
 as in Fig. 8. The moments in ^the members about the joint 
 may readily be found from such a figure. 
 The couple caused by the shears on the columns must be resisted by the shears on the 
 girders. The girders are assumed as having the same length. Then 
 
 2ln 
 
 (1) 
 
 assuming a and b to refer to two adjacent stories. It is sufficiently exact in making com- 
 putations by this method and in keeping with the assumption of equal column sections to 
 consider Wa and h2 as equal to Wb and hi respectively. 
 
 Wbhi 
 
 The moment at the base of the upper column is equal to ? that at the top of the 
 
 lower column, The moment at the end of either girder due to the wind is 
 
 For an exterior panel (Fig. 9) the value of V is given by yy. 
 equation (1). The moment at the base ot the upper column is 2/? 
 
 equal to 5 at the top of the lower column, ] and at the 
 
 J ^ ^ .J 
 end of the girder, -y* 
 
 10. Roof Frames. — It would be impractical to give here a 
 complete set of formulas for the large range of possible roof 
 frames. It will be of use, however, to give some of the more 
 common types of frames. 
 
 Type I 
 
 7-^ 
 
 ■Case I. 
 
 
 
 
 
 
 
 > 
 
 
 Fig. 9. 
 
 Va = 
 
 H 
 
 r 
 
 ho 
 
 wl^ R(3ho + 5h) 
 
 64 ■ ho^S + R(3hoh + hi^) 
 
 S = ^ 
 
 %wl 
 
 Case 11. 
 
 A 
 
 ■4" 
 
 Type n— Case /. 
 
 ♦unTTiunTrn 
 
 Hb 
 Ha 
 
 h 
 
 h, 
 Va 
 
 H 
 
 V_VWh,{S +3R) + bhi^R{3ho+h) + 5ho^hS-{- WR{ho+ h)' 
 16L ho^S + R{3hoh + hi^) 
 
 Vh - Hb 
 
 wl 
 2 
 
 R(l + A;) 
 
 = S 
 
 wl 
 
 ShilSd + k^) + R(l -j-k 
 
 -hk^)] 
 
Sec. 10-11] 
 
 MOMENTS IN RIGID BUILDING FRAMES 
 
 429 
 
 Case II. 
 
 Hb 
 
 phiV 5S + R{4: -\- 2k) 1 
 8 U(l +k') +R{1 + k + k^)j 
 
 7^ Ha = phi - Hb 
 
 Case III. 
 
 Hi 
 
 I 
 
 _Pkr 2k^S + R{1 + 2k) 1 
 2 L'S^Cl +/c3) +72(1 +k-\-k^)\ 
 
 P - H. 
 
 Type III.— Case /. 
 
 r 
 
 21 
 
 wlx^ 4l,R + S(l + Uj) 
 8lh' R -\- S 
 
 S = 
 
 Ii 
 
 Case II. 
 
 Va 
 
 Hb 
 
 ^ 21 
 
 ph 4I2R + Sjh + 5I2) 
 81' R + S 
 
 11. L-frames. — The L-frame may occur either with a fixed or hinged column base; or with 
 a fixed or hinged girder-end. In the four cases which follow, the girder-end is fixed and the 
 column base hinged. The frame may, with its loading, be revolved through 90 deg. to suit the 
 reverse of the cases shown. Frames of this nature which are fixed at both extremities seldom 
 occur in practice as a separate structure, and hence are not given here. 
 
 Case I. 
 
 R = 
 
 Ha = 
 
 SPam^ 
 
 Va 
 
 Rh^ {SR + ^S) 
 
 2S (a + 26) +SR(b + 2a) 
 
 {3R + 4S) 
 Mb = - HaK Mc = - HAh + VaB 
 
 Pri 
 
 - HAh + VAa 
 
430 
 
 CONCRETE ENGINEERS' HANDBOOK 
 
 [Sec. 10-11 
 
 Case II. 
 
 wb 
 
 A' 
 
 mmmmc 
 
 <.t.....^..::-^...>i 
 
 Mx,, (max.) = 
 
 2w 
 
 Case III. 
 B 
 
 1 
 
 i 
 
 Case IV. 
 
 A 
 
 Ha 
 Va 
 
 SPa [2h^S - aS (26 + m)] 
 ' Rh^{SR + 4:S) 
 
 2P [ISRh^ + a^Si2b + m)] 
 
 Havi 
 
 {SR + 4^') 
 
 Mb = HAh 
 
 Pa 
 
 Mc = Mb + VAb 
 
 Ha = 
 
 Sph (R + S) 
 
 V - 2p 
 
 A very common form of the L-frame is the one which is hinged at both extremities. It is 
 not uncommon for the beam to have some slope other than horizontal. Both cases follow 
 briefly: 
 
 Mac = ZEKxQa - N 
 Mae = 3EK2dA 
 Mas = — Mac 
 Mab 
 
 A 
 
 5« 
 
 Load 
 
 Oa 
 
 SEK, 
 
 Mab 
 Mac = -Fr— ' Ki 
 
 N = - MAi 
 
 K2 
 
 N 
 
 Mac 
 
 NK2 
 
 Pah 
 21^ 
 
 Suppose the "load" to be a concentrated load P, placed a distance a from C. Then = 
 (I + a). If the "load" is a symmetrically placed load, with respect to the member AC, 
 
 N 
 
 = y, the moment at the end of a fixed beam which carries that particular loading (see page 
 
 41.3 for values of j for various loads). 
 
 When the beam is not horizontal, but slopes upward to the end C, the solution of the 
 fram^ may be obtained by letting h in Type II of "Roof Frames" become zero. 
 
SECTION 11 
 
 BUILDINGS 
 FLOORS— GENERAL DATA 
 
 1. General Types of Concrete Floors. — There are four general types of concrete floors: 
 (1) monolithic beam and girder construction, (2) flat-slab construction, (3) unit construction 
 and (4) steel-frame construction with concrete slabs. In the fourth type mentioned, the beams 
 and girders are usually covered with concrete for fire protection. 
 
 2. Floor Loads. — The following extract from the Seattle Building Code illustrates good 
 practice : 
 
 All floors shall be constructed to bear a safe live load per superficial square foot of not less than the following 
 amounts. 
 
 Pounds 
 
 Public buildings 100 
 
 Detention buildings, in cells or wards 60 
 
 Churches, chapels, theatres, assembly halls or court rooms with permanent seats 80 
 
 Lobbies, passageways, corridors and stairways of the same 100 
 
 Assembly halls with movable seats 100 
 
 Halls used for dancing, or roller skating 150 
 
 Lobbies, passageways, corridors and stairways of the same 100 
 
 Stables 80 
 
 Dwellings, apartment houses, flat buildings and lodging houses 50 
 
 Class rooms in schools 60 
 
 Assembly rooms in schools 80 
 
 OflBce buildings and hotels, ground floor 125 
 
 For floors above the ground floor 75 
 
 Store buildings for light merchandise, ground floor 125 
 
 For floors above the ground floor 100 
 
 Store buildings for heavy merchandise, such as grocery stores or hardware stores 150 
 
 Warehouses 200 
 
 Factories and workshops, when the nature of the work is general 125 
 
 Machine shops, armories, drill rooms and riding schools 250 
 
 Floors in a building to be used for the sale, storage or manufacture of heavy machinery, shall be propor- 
 tioned to the load they may have to carry. 
 
 In addition to specifying minimum floor loadings for which the various types of buildings 
 must be designed, the Rochester Building Code requires also the following : 
 
 The weight placed on the floors of any building, now or hereafter constructed, shall be safely distributed 
 thereon. The bureau may require the owner or occupant of any building or portion thereof to redistribute the 
 load on any floor, or to lighten such load when deemed necessar.y, even if not greater than the minimum in this 
 section prescribed. 
 
 To prevent overloading in all warehouses, storehouses, factories, workshops and stores, now or hereafter 
 constructed, where heavy materials are kept or stored, or machinery introduced, the weight that each floor will 
 safely sustain upon each square foot thereof, or upon each varying part of such floor, shall be estimated by a 
 competent person employed by the owner or occupant, or the bureau may make such estimate, and said estimate 
 shall be placed permanently on a stone or metal tablet in a conspicuous place in the hallway of each story or vary- 
 ing parts of each story of the building to which it relates. 
 
 No person shall place or permit to be placed on the floor of any building, now or hereafter constructed, any 
 greater load than the safe load thereof as correctly estimated and ascertained as herein provided. 
 
 The working stresses usually employed, and those recommended by the Joint Committee 
 are intended to apply to static loads only. Proper allowance for the dynamic effect of the live 
 
 431 
 
432 
 
 CONCRETE ENGINEERS' HANDBOOK 
 
 [Sec. 11-3 
 
 load should be taken into account by adding the desired amount to the live load to produce an 
 equivalent static load before applying the unit stresses in proportioning parts. An allowance 
 for impact will be necessary only in special cases, as in the case of floors supporting heavy ma- 
 chinery. The amount to add to the live load because of impact will vary all the way from 25 
 to 100% depending upon the proportion of the specified live load which may be subject to 
 motion. 
 
 The dead load of any floor may be estimated from the following approximate data — weights 
 are per square foot of floor surface: 
 
 Wooden wearing surfaces, 4 to 6 lb. Plaster, 5 lb. 
 
 Screeds or nailing strips, 2 lb. Suspended ceiling, 10 lb. 
 
 Cinder-concrete filling (2 in. thick), 15 lb. Cinders, 7 lb. 
 
 Hollow tile, see Art. 12. 
 
 3. Economic Considerations. — The size of floor bays depends upon the loading, the uses 
 to which the building is to be put, and the size and shape of the ground area. In a large build- 
 ing it is frequently worth while to make several comparative estimates with different layouts 
 of the beams, girders and columns, so as to obtain the most economical arrangement under 
 the given conditions. 
 
 The cost of changing forms to meet changes in size of beams and columns in different stories 
 must be kept constantly in mind. It is more economical to vary the depths of beams from floor 
 to floor than the widths, on account of the slab panels ; and to vary square columns in one dimen- 
 sion rather than in two, both on account of the columns themselves and the beams which frame 
 into them. It is not advisable to change the size of members from floor to floor for small differ- 
 ences in computed dimensions; nor is it advisable on any floor, if simplicity in field work is 
 desired, to make slight changes in the sizes of beams and in the thicknesses and reinforcement 
 of the slabs, unless such variations occur over large areas. When slabs of considerable variation 
 in span alteFn,^irej it is better to vary the thickness and keep the reinforcement the same, than 
 to vary the reinforcement and use the same thickness of slab. Although slight differences in 
 dimensions are not desirable, the designs made should be considered carefully in every detail. 
 
 4. Floor Surfaces. — A concrete floor usually has a mortar or granolithic finish as wearing 
 surface. Such a surface if not allowed to set rapidly is hard and practically impervious to 
 water. 
 
 ^he usual proportions for granolithic finish are 1 part Portland cement, 1 part sand, and 
 ^j^t crushed stone which passes through a K-in. mesh screen. This mortar surfacing is laid 
 ^^ff the concrete slab and troweled to a hard finish. If placed before the concrete below has 
 ^t, it may be from to 1 in. thick but where the concrete of the slab becomes old before the 
 granolithic finish is laid, the thickness should be at least 2 in. 
 
 Granolithic finish is screeded to grade with a straight-edge, smoothed with a wooden 
 float, and finished with a steel trowel. It is often marked off into blocks, or sections, of suitable 
 size by shallow grooves. The object in dividing the surface into panels is purely ornamental, 
 since reinforcement against shrinkage cracks is provided in the lower portion of the concrete 
 slab. 
 
 For a complete treatment of concrete-floor surfaces, see Sect. 4. 
 
 Some engineers prefer to lay a cinder-concrete base, not less than 2 in. thick, before placing 
 the cement finish. The advantage of this method is that should there be any reason for remov- 
 ing a section of the finished floor, it can be accomplished without injury to the reinforced-con- 
 crete slab. On account of the porosity of the cinder concrete it is difficult, however, to obtain 
 a satisfactory cement finish if the same is placed before the cinder concrete base has set. 
 
 If a wood floor is desired, 2 by 3-in. or 2 by 4-in. sleepers are usually laid on top of the 
 rough concrete slab, and cinder concrete or stone concrete poor in cement is run between these 
 nailing strips. For ordinary cases, % to 13^ -in. maple flooring nailed to 2 by 3-in. sleepers, 16 in. 
 apart, is found satisfactory. The proportions of cinder concrete specified vary considerably — 
 average proportions are 1 : 3 : 6. The sides of sleepers are usually beveled, but this does not 
 
Sec. 11-5] 
 
 BUILDINGS 
 
 433 
 
 prevent them from becoming loose if the wood shrinks. The best method of holding the sleepers 
 in place is to drive 40d. nails in the sides of the sleepers at intervals of 3 ft. on alternate sides. 
 These nails key with the concrete and prevent movement of the sleeper. 
 
 There has been much dissatisfaction with the above method of constructing wooden floor 
 surfaces on account of the liability of the sleepers to decay by dry rot. To lessen decay the 
 sleepers should be well seasoned and might advantageously be impregnated with preservative 
 of some sort, and care should be taken to permit no moisture to reach them even after the top 
 flooring is in place. Sufficient time should be allowed the concrete to dry out before the finished 
 flooring is laid. Dry rot is a fungus disease of wood that thrives in the presence of moisture 
 and absence of circulating air. It makes little difference in result whether the moisture is the 
 sap of the green wood, or water that soaks into the wood after it has been dried. 
 
 Sleepers are sometimes laid in dry cinder fill. In such construction the chance of dry rot 
 is probably small, since there is some opportunity for air circulation through the porous cinders. 
 Cinders also tend to absorb moisture. 
 
 A sand-and-tar base for plank flooring on reinforced-concrete floor slabs has been employed 
 to some extent to avoid any danger from dry rot. A layer of sand mixed with coal tar is spread 
 about in. thick on the floor slab and leveled while still warm and soft with a straight-edge. 
 On this layer is then placed first a layer of 2-in. plank, then %-in. roiigh pine board, and finally 
 a wearing surface of % to 13^:^-in. maple. The different layers of planking should be placed in 
 different directions. 
 
 The following construction has been used in schoolhouse floors with satisfactory results: 
 At the completion of the floor slab, a 1-in. footing of sand is placed on top of the same and brought 
 to a true level by screeding. On top of the sand are placed 13'^-in. boards laid diagonally and 
 nailed together at the edges. The form lumber used for centering is generally employed for 
 this work. On top of the l^-^-in. boards is then placed building paper, which in turn is covered 
 by the finished flooring. 
 
 So-called "nailable concretes" have been attempted but without substantial success. 
 They are, in effect, poor concretes, made from cinders and sand with a small amount of asbestos 
 or like substance, into which flooring nails may be driven. Their prolonged retention of water, 
 generation of acid from wet cinders and faulty hold on nails has prevented their extensive use. 
 
 In hotels and similar establishments, linoleum or carpeting over a fairly smooth cement 
 base seems to give satisfaction. If the floor is very wet and alkaline, saponification may take 
 place, with destruction of the linoleum. A thoroughly good cement is insurance against such 
 occurrences. Sodium silicate should not be used, as it liberates free sodium. 
 
 Flat tiles are sometimes employed as a floor surface in reinforced-concrete construction. 
 The most durable and sanitary tile is the vitreous-clay tile. The ceramic tile is perhaps the most 
 often used and is a vitreous-clay tile manufactured in square, hexagonal, and round shapes. The 
 squares range in size from }^ to in. and the hexagonal shapes from to 1 in. Tiles are 
 laid some little time after the floor slab is poured and, if set in Portland-cement mortar, should 
 be embedded in a layer not less than 2 to in. thick in order to prevent curling of the tile due 
 to shrinkage of the base. There are now on the market, however, some patented bases which 
 seem satisfactory and which will allow a bedding thickness of only 1 in. over and above the 
 thickness of the floor slab proper. 
 
 5. Small Floor Openings. — The methods of arranging slab reinforcement around small 
 openings, such as for chimneys and ventilators, needs some consideration. Both a wrong and 
 a right method are shown in Fig. 1. The openings are shown against a wall, and the floor 
 slab is reinforced in only one direction. 
 
 The method shown of placing the reinforcing rods parallel to the opening makes a neat 
 looking job, but it should be evident that no proper provision is necessarily made to carry the 
 load which would ordinarily come on the wall at the openings. The right method shown is 
 less artistic, but, as a general rule, is vastly more efficient. The cross rods serve only to rein- 
 force the slab locally. All such designs should be carefully studied to make sure that the 
 28 
 
434 
 
 CONCRETE ENGINEERS' HANDBOOK 
 
 [Sec. 11-6 
 
 m// 
 
 trr 
 
 I I 
 
 nil 
 1 1 1 1 
 
 LLLL'i 
 
 TT 
 
 mm 
 ii 1*1 
 
 II Mil 
 
 Oil 
 
 I I I 1 1 
 Mill 
 
 1 1 1 1 
 
 I'l l 
 iliiil 
 
 II!'" 
 
 I . I 
 
 mm 
 
 f 
 
 /II 
 
 I 
 
 1^1 
 
 required strength is secured at all points. Slab rods should never be curved horizontally to 
 dodge floor openings as a curved rod will tend to rotate. 
 
 Wrought-iron and galvanized-iron sleeves are built into the construction work for all 
 steam, return, sprinkler, sewer, gas, and similar pipes. All floor sleeves should be flush with 
 the ceiling line and should extend about 2 in. above the floor line. Pipe-risers should not be 
 allowed to come up through columns as repairs and alterations are difficult, if not impossible, 
 under such an arrangement; electric conduits, however, form an exception to this rule. Special 
 
 shafts with fireproof walls are sometimes 
 used for plumbing and vent pipes, and this 
 practice has much to commend it since a 
 floor to be a perfect fire cutoff should be 
 solid from wall to wall, with stairways, 
 elevators, and all openings enclosed in ver- 
 tical fireproof walls. 
 
 6. Provision for the Attachment of 
 Shaft -hangers and Sprinkler Pipes. — There 
 are many different methods of attaching 
 shafting, sprinkler pipes, and machinery to 
 the under side of a concrete floor. The 
 method adopted should be as flexible as 
 possible in order to accommodate future 
 changes in the line of shafting, additions, or 
 improved machinery. All bolts and sockets 
 
 1 
 
 JJl 
 
 3ecr/7? ••' 
 
 Wrong Method 
 
 Fia. 1. 
 
 Sea/7?- 
 
 Right Method 
 
 Method of placing reinforcement around small 
 
 floor openings. 
 
 should be placed before the concrete is run, as drilling is expensive and reinforcing bars are 
 liable to be encountered, which would cause more or less difficulty, and might lead to serious 
 trouble if cut off. 
 
 A convenient method is to place suitable bolts at intervals, usually 4 ft. on centers, with 
 their threaded ends projecting from the concrete. By means of these projecting bolts, timbers 
 may be fastened wherever desired and the shaft-hangers lag-screwed in place. 
 
 —• Stirrups 
 
 Han^r ^ItSj--^.^ 
 
 Shaft hanger ... 
 hcf scretyecf "" 
 
 Section, 
 
 Elevation 
 
 Fig. 2. 
 
 The heads of bolts projecting into concrete should be enlarged or bent so that the bolt 
 will not tear out. If desired, a washer or plate may be employed for this purpose underneath 
 the head of the bolt. Figs 2 and 3 illustrate the use of large washer nuts. These are tempo- 
 rarily held in position by a thin iron tube resting on the centering and by a bolt projecting 
 through the hole in the bottom of the beam box. The washer nut is tightened down to the top 
 of the tube which secures it firmly in an upright position (see detail shown in Fig. 4). When 
 the beam boxes or centering are removed, the bolts are easily unscrewed from the nut, leaving 
 a clear passage through the concrete to the nut above. The stirrups shown are to prevent 
 excessive deformation in the beam. By the method shown in Fig. 3, the shaft-hangers may 
 be securely fastened by bolts. 
 
Sec. 11-6] 
 
 BUILDINGiS 
 
 435 
 
 When projecting bolts are used, it is seldom that all are made use of, and those not used 
 present an unsightly appearance. This is the main objection to the scheme shown in Fig. 5. 
 Saddles and check nuts serve to hold the long bolts in position. 
 
 The Unit socket is shown in Fig. 6. This device serves also as a support and spacer for 
 the beam rods, keeping them all properly spaced from each other and from the beam box or 
 centering. 
 
 A form of adjustable socket for use in beams, principally, is shown in Fig. 7. These castings 
 are made in convenient lengths and the slot in the bottom makes it possible to place hangers 
 
 
 
 
 
 
 \^-5tirrups 
 
 
 
 Holes punched 
 J)oJts cfh^pecf 
 through before Cis 
 bolted to beam, 
 
 d and \ \ 
 
 Section 
 
 Elevation 
 
 Fig. 3. 
 
 or bolts at any desired location along the length of the insert. The casting can be anchored 
 as securely in the concrete as may be necessary by the use of stirrups passed through the open 
 spaces. 
 
 Figs. 8 and 9 illustrate two other methods used for attaching shaft-hangers to beams. 
 Fig. 8 is what is known as a pipe-slot hanger. Fig. 9 can be made a strong and serviceable 
 support for motors and machinery. 
 
 The schemes shown in Figs. 10 and 11 have been used quite extensively by the Turner 
 Construction Co. of New York City. 
 
 Fig. 12 shows a hanger socket mainly for use in slabs. The casting varies in length with 
 the depth of the slab and is made smaller at the tapped end than at the top, so there will be no 
 possibility of the binding of the tap-screw 
 when screwed in. The cross-pin shown 
 passes transversely through a cored hole in 
 the upper end of the casting. Fig. 13 
 shows a somewhat similar type of socket. 
 A bolt through a hole in the form secures 
 it in position during concreting. Figs. 14 
 and 15 show two other methods used for 
 slabs between beams. 
 
 A flexible method of attachment is 
 shown in Fig. 16 — a method used in the 
 shops of the United Shoe Machinery Co., 
 Beverly, Mass. The anchor bolts were 
 spaced 3 ft. on centers in all transverse 
 
 floor and roof girders. A transverse line of bolts alternately spaced 
 was also built in the middle of each floor and roof panel. 
 
 2". 3\ fP/. ffappecf) 
 
 Fig. 4. — Detail of socket and bolt. 
 
 1 ft. and 6 ft. centers 
 In the girder arrangement, the 
 nuts on the lower ends of the anchor bolts engage cast-iron saddles which clamp against 
 pairs of angles with wood fillers. Fig. 16 shows how the "T "-headed attaching bolts may 
 move freely along the slot formed by the angles. In the floor panels, the bolts have simple 
 heads or nuts upon the upper ends which bear upon flat washers. In thin slabs the heads bear 
 upon washer plates in the upper surface of the slab (Fig., 16). 
 
436 
 
 CONCRETE ENGINEERS' HANDBOOK 
 
 [Sec. 11-6 
 
 \ Harcf P/ne 
 
 ^^"niy Bent bar 
 
 "^S? — 
 Fig. 5. 
 
 Fig. 6. 
 
 
 1 
 
 
 liol 
 
 -<• 
 
 
 
 1 
 
 
 \-Slof 
 
 Fig. 7. 
 
 bolt 
 
 Fig. 8. 
 
 Fig. 9. Fig. 10. Fig. 11. 
 
Sec. 11-7] 
 
 BUILDINGS 
 
 437 
 
 It is not good practice to depend upon reinforcing rods for direct support of shaft-hangers, 
 as thereby too great a strain is placed locally on the rods and the concrete under them will 
 become loosened from the continuous pull and vibration of the shafting. 
 
 A'Pin.fZ'hn^ 
 
 Washer 
 
 Fig. 12. 
 
 Elevgiiior> Plan 
 
 Fig. 13. 
 
 ,4W^ f yyasher 
 
 Fia. 16. 
 
 7. Bedding Machinery. — Machines can readily be connected to the concrete by drilling 
 small holes in the floor at the proper points and setting lag-screws or through-bolts. If through- 
 bolts are required, the holes should be located before the floor is concreted. A permanent 
 level base may be obtained by leveling the machine an inch or two above the foundation proper 
 and grouting it in place. A dam of sand is first built around the machine. Then the grout, 
 
438 CONCRETE ENGINEERS' HANDBOOK [Sec. 11-8 
 
 in proportions 1 part cement to 1 or 2 parts of sand mixed to the consistency of thick cream, 
 is poured so as to run under the casting. As the mortar sets it is rammed with a rod to prevent 
 shrinkage. 
 
 In most cases machinery runs better and lasts longer if placed on solid foundations, but 
 there are exceptional cases where machines seem to require more or less give on their bases, as 
 in certain textile mills. In such cases a wood-finished floor can be laid or wooden timbers 
 fastened to the concrete floor. Sheet cork and linoleum carpet have also been used for this 
 purpose. 
 
 8. "Waterproof Floors. — In case of fire, a water-tight floor in factory buildings prevents 
 damage from water to the machinery or materials in the stories below. According to Insurance 
 Engineering, the damage from water after a fire in fireproof factory buildings is much greater 
 than that from the fire itself. A concrete floor with granolithic surface is practically impervious 
 to water but unless a floor is made self-draining, water will get down through the floor open- 
 ings. Raised sills should be provided around all openings, whether or not a cement finish is 
 used, and scuppers of ample size should be placed in the outside walls to 
 carry water [from the floors to the outside of the building. Fig. 17 is a 
 /7 ^'^^^l detail of wall scuppers used in the Liberty Silk warehouses. New York 
 City. The waterproofing of floors may be accomplished by using an 
 impervious finished flooring or by using an undercoating of asphalt felt 
 laid in hot asphalt. 
 
 9. Tests. — Within the last few years full-sized floor panels of both 
 the monolithic beam and girder and the flat-slab types, have been 
 tested for the elastic deformation of the concrete and the steel under 
 increasing loads. The tests consisted in loading, with an increasing 
 load, a number of consecutive panels of a reinforced-concrete build- 
 ing, and of observing the exact contraction or expansion of certain 
 portions of the steel and of the concrete by means of delicate instru- 
 ments. In order to make such observations, the concrete was re- 
 moved from the steel in given places, and drill holes were made in the 
 steel thus uncovered. Into these holes the extensometer points were placed. Small holes 
 were also drilled in the concrete for the same purpose. From the elastic deformations ob- 
 tained for moduli of elasticity of the steel and concrete, the stresses in the members were 
 computed and the results compared with those for which the building was designed. Much 
 valuable data pertaining to proper methods for design have been obtained from these 
 tests. 
 
 10. Basement Floors. — A basement floor in dry ground is usually made of 1 : 3:5 concrete, 
 3 or 4 in. thick. Usually no wearing surface is needed other than the ordinary concrete troweled 
 to a hard finish, but, where considerable wear is expected, the usual mortar coat may be laid 
 as on the upper floors. To prevent shrinkage cracks, the floor should be divided into blocks 
 about 8 or 10 ft. square. This may be accomplished by laying alternate blocks, and then 
 filling in the intermediate ones after the adjoining concrete has set. Basement floors are 
 constructed in the same manner as concrete sidewalks (see Sect. 4). 
 
 It is not safe to depend upon the concrete itself being water-tight. If the basement is 
 below tide-water or ground-water level, a layer of waterproofing consisting of three to six layers 
 of waterproof felt (cemented together and to the concrete by coal-tar pitch or asphalt) should 
 be spread on the concrete and carried up in continuous sheets on the walls to above water level. 
 The whole surface should then be covered with another layer of concrete at least 3 or 4 in. 
 thick. 
 
 The earth under the basement floor should be well drained, and drains of tile pipe, or of 
 screened gravel and stone, may be placed in trenches just below the floor. Sometimes it is 
 necessary to cover the entire area with cinders or stone; and sometimes the concrete must be 
 made extra thick, or reinforcement added, to resist the upward pressure of the water. 
 
Sec. 11-11] 
 
 BUILDINGS 
 
 439 
 
 MONOLITHIC BEAM-AND-GIRDER CONSTRUCTION 
 
 11. Ordinary Type of Beam-and-girder Construction. — The theory and methods of design 
 of slabs, beams, and girders are given in Sect. 7. If a floor slab is reinforced in one direction 
 only, the load will practically all be transmitted to the beams at right angles to the direction 
 of the reinforcing rods. A small part, however, will be transferred directly to the girders at 
 the sides of the panels, but this may well be neglected in the calculations for cross-beams. In 
 fact, even with reinforcement in two directions, the load should be assumed as all transferred 
 to the cross-beams unless the panel is nearly square. 
 
 If panels, nearly square, are reinforced in both directions, the loads carried to the cross- 
 beams and girders will not be uniformly distributed over the length of such beams and girders, 
 but may be assumed to vary in accordance with the ordinates of a triangle. This assumption 
 is surely on the safe side in regard to moment, if the area of the triangle is made equal to that 
 part of the total load on the panel which is transmitted to the beam in question — as determined 
 by the formula of Art. 29c, Sect. 7. Assumptions of this load being either uniformly distributed, 
 or varying as the ordinates of a parabola, give a lower resulting moment than the triangle 
 method. 
 
 Let w be the uniform load per unit of area on the slab, and W2 and Wi the parts of this unit 
 load that go to the shorter and longer beams respectively. Applying the loads in the form of a 
 triangle having its apex at the middle of the beam, the maximum moment will be for the longer 
 beam, and, if this beam is considered simply supported and as carrying the load from one panel 
 only, 
 
 M = }i2Wihl^ 
 
 and for the shorter beam 
 
 M = li2W2bH 
 
 w 
 
 If the slab is square, Wi is and M = wl^. For beams made continuous, the bending 
 
 moment may be multiplied by the coefficient % or as the case may be. With beams or 
 girders common to two panels, the bending moment should be multiplied by 2. 
 
 Unless the panel is nearly square, floor slabs should not be reinforced in two directions, 
 as it is evident that no economy results from double reinforcement when the ratio of length to 
 breadth of panel is greater than about 1.2. If the length of the slab exceeds 1.5 times its 
 breadth, the entire load should surely be carried by the transverse reinforcement. 
 
 When the floor surface is given a granolithic finish, this finish under certain conditions 
 may be considered to act with the slab proper in taking the stresses under loading. Usually, 
 however, the designer has no way of finding out whether the finish will be placed at the same 
 time as the concrete, or run a number of hours later, and this alone should call for caution on 
 the part of the designer in figuring the finish as a part of the slab. If the superintendent on 
 the job is known as a careful man and if there is to be very careful supervision, the floor may 
 be sometimes figured in this way, but never without putting an underscored note on the drawing 
 to the effect that the finish shall be run immediately after the pouring of the concrete slab. 
 Also the surface of the floor should be blocked off only along the center line of columns and no 
 joint should be made between column lines, as such joints would affect the needed strength of 
 the slab. 
 
 Where care in construction is not assured, or where any appreciable wear on the floor is 
 expected, the finish should properly not be included in the effective slab thickness. It is also 
 advisable not to figure this way for a winter job under any conditions. 
 
 It is possible, by taking great precautions, to bond a wearing surface to a concrete slab after 
 the concrete in the slab has set. This requires special treatment including thorough cleaning 
 and soaking of the old concrete, providing a bond layer of neat cement mortar, and placing 
 the surface before this neat cement has begun to harden. Poor joints occur quite frequently, 
 
440 
 
 CONCRETE ENGINEERS' HANDBOOK 
 
 [Sec. 11-11 
 
 however, and it is advisable not to figure the finish as a part of the slab in a case of this kind, 
 even though considerable care is assured at the time of construction. 
 
 If it is deemed advisable in any given design to consider the finish as an integral part of 
 the slab, the maximum fiber stress in compression may be determined by the method outlined 
 in Fig. 18. The distance b is made equal to the maximum allowable stress on the ordinary 
 concrete, and the distance a which represents the maximum stress on the cement finish is then 
 computed on this ratio. The later stress, however, should not be in excess of the maximum 
 allowable stress for cement finish. 
 
 Fig. 19 shows an arrangement for slab steel such that there is the same steel area at the top 
 of slab over the cross-beams as at the bottom of the slab midway between these beams. All 
 rods are bent and are identical in shape — namely, straight at one end, bent up at the center 
 over a beam, and bent over a beam at the other end. The required arrangement is effected by 
 
 &^ i 
 
 Neutral Plane 
 
 fMax. allow- 
 able fiber ( 
 stress In 
 concrete 
 
 Fig. 18. 
 
 Fig. 19. 
 
 shifting alternate rods 7 ft. ahead. Some designers bend up all the rods over supports, but a 
 better method is to continue some steel at the bottom of slab and thus make sure that no point 
 in tension is unprovided with steel. The rods arranged as in Fig. 19 may be carried over three 
 spans and still obtain the same amount of steel over supports as in the center of span, but the 
 amount of steel at the bottom of slab near the supporting beams becomes very small. 
 
 Fig. 20 shows another arrangement for the slab steel. Both straight and bent-up rods 
 are employed, each rod extending over three slab spans. The joints in the bent rods occur 
 over supports and the steel is lapped a sufficient distance to provide adequate bond strength. 
 
 Fig. 20. 
 
 Fig. 21. 
 
 This lapping is so arranged that two-thirds as much steel occurs over the supports as in the 
 center of span. 
 
 Fig. 21 shows an arrangement of steel which gives three-fourths the center-of-span area 
 over supports. The arrangement is similar to that shown in Fig. 20 except that the rods extend 
 over only two spans. 
 
 The arrangement shown in Fig. 20 requires the least steel and that shown in Fig. 19 re- 
 quires the most. Fig. 19 shows undoubtedly the best design for spans over 6 or 7 ft. 
 
 For moments in beams and girders due to building acting as a rigid frame, see Sect. 10. 
 
 IiiiiTTSTRATrvE PROBLEM. — Design an interior floor bay to support a live load of 250 lb. per sq. ft. with the 
 columns spaced 21 ft. by 21 ft. on centers and with two intermediate beams. Working stresses recommended by 
 the Joint Committee for a 2000-lb. concrete (see Appendix B) are to be employed throughout. (Tables and 
 diagrams referred to are those of Sect. 7.) 
 
 Since an interior floor bay has been assumed, the bending moment — may be employed for the three parts 
 
 of the floor bay, namely: the slab, the beam, and the girder. Fig. 22 shows the proposed arrangement of beams 
 and girders. 
 
Sec. 11-11] 
 
 BUILDINGS 
 
 441 
 
 The floor surface will be given a granolithic finish, consisting of a layer of 1 : 2 mortar, 1 in. thick, spread upon 
 the surface of the concrete slab before it has begun to set, and troweled to a hard finish. For simplicity, the weight 
 of finish will be assumed as included in the specified live load of 250 lb. per sq. ft. Of course it should be readily 
 understood that in practice, where a definite live load is required, the finish should be considered separately as a 
 superimposed dead load. 
 
 Slab. — The main reinforcement will be placed in the direction A A', Fig. 19, and the span of slab will be 7 ft. 
 Slab is to be fully continuous and its total depth will be taken to the nearest Yz in. 
 
 Diagrams 5 and 6, Sect. 7, show that a 4H-in. (d = ZYi-in.) slab with span of 7 ft. will sustain a load (live plus 
 dead) of approximately 325 lb. per sq. ft. Corresponding weight of slab is 56 lb. per sq. ft., and the total load per 
 square foot for the slab to carry is therefore 250 + 56 = 
 306 lb. 
 
 For a slab with d = ZYi in.. Diagram 6, Part 1, 
 gives As = 0.325 sq. in. Referring to Table 6, it is 
 evident that %-in. round rods spaced 4 in. on centers 
 will give sufl5cient steel area. If desired, the span of 
 slab may be taken as the clear distance between faces 
 of supports (see Art. 44, Sect. 7). Four round rods % 
 in. in diameter will be placed transversely in each 7-ft. 
 panel to prevent shrinkage and temperature cracks, and 
 to bind the entire structure together. 
 
 Cross-beams (Eight-rod Design). — The cross-beams 
 have a span of 21 ft. The beams and slab will be poured 
 at the same time and thoroughly tied together so that 
 a T-beam section may be considered. The distance be 
 tween beams is 7 ft., and the dead and live loads of the 
 slab per foot length of beam is equal to 7 X 306 = 2140 
 lb. Assume dead load of the stem of beam at 240 lb. per 
 lin. ft. Then the total loading per foot of length equals 
 2380 lb. The maximum shear 
 
 
 T! TT r* — — ■ r-i 1 
 
 ' ' « 1 ! 1 '1 
 i-i i_l r'-'-i-_ 
 
 
 
 s-r^ Tr^^'i — V^" 
 
 
 A 
 
 j 1 I 1 j 1 1 1 
 
 A- i 
 
 
 i' ii i' ji 
 
 j.l li l| |! 
 
 |- ^'-^'-ffi 7'.o' iji 
 
 c> 
 
 Si 
 
 
 ik--- I{ 2/'- <?'•[■■•' Ij! 
 
 i'l !' 1' !' 
 
 1 
 
 
 1 1 1 1 1 
 ! " i ' ! 
 
 ' II Girder- | 1 
 
 — J-'-h 1-1 a^- -1- J r^'-u-. 
 
 • 1 h-i-i 
 
 1 
 
 *. 
 
 
 ..^ -u^^- 
 
 ! 1 1 1 ! ! ! ! 
 
 
 V = 
 
 (2380) (21) 
 
 and the maximum moment 
 (2380) (21)2 (12) 
 
 M = 
 
 12 
 
 25,000 lb. 
 
 1,050,000 in.-lb. 
 
 N — u — m^- 
 
 Cross section A-A*, 
 Fig. 22.^ 
 
 (If desired, the span of beam may be taken as the clear distance between faces of supports. 
 The required cross-section of web as determined by shear 
 
 See Art. 44., Sect. 7.) 
 
 25,000 
 105 
 
 238 sq. in. 
 
 The following formula of Art. 37, Sect. 7, gives the most economical depths for various assumed web widths: 
 
 with r in this design as 60, then 
 
 jj ^ 
 
 It«p 
 
 
 
 
 
 
 Cross- section of CrossDeam 
 
 Fig. 23. 
 
 d = 
 
 rM t 
 Ub' 2 
 
 for b' = 9 in. 
 for b' = 10 in. 
 for 6' = 11 in. 
 
 d = 23.1 in. 
 d = 22.0 in. 
 d = 21.2 in., etc. 
 
 In order to provide for most of the diagonal tension by means of 
 bent-up rods, it is proposed to use eight rods placed in two rows, 
 four rods to a row. ' A value of b' = 10 in. may be satisfactory as re- 
 gards rod spacing, but b' X d as given above is not great enough to 
 provide for shear. It will be more economical to deepen the beam 
 of 10-in. width to a depth of 23li in. than to adopt a width of 11 
 in. and a depth of about 21.2 in. Fig. 23 shows the arrangement 
 of the steel which will be tried in the bottom of the beam at the 
 center of the span. This exact arrangement will also be tried at 
 the top of beam over supports. It is quite likely that there will be 
 eight rods in the girder of about 1-in. or IJi-in. diameter and, since 
 the cross-beam rods should fit nicely with the girder rods over sup- 
 porting columns, the two layers of rods will be placed 2 in. center 
 to center as shown. 
 
 The weight of the stem is 
 
 (10) (22.5) (150) 
 144 
 
 = 235 lb. per ft. and the weight assumed thus is satisfactory. 
 
442 
 
 CONCRETE ENGINEERS' HANDBOOK 
 
 [Sec. 11-11 
 
 1,050,000 
 
 The width of the flange of the T-beam is controlled in this design (see Art. 32, Sect. 7) by 12 times the 
 thickness of slab plus the width of stem, or 64 in. Then 
 
 4.5 
 
 = 0.19, Diagram 8 shows fc = 300 lb. per sq. in. and j = 0.93. Then 
 
 As = 
 
 = 29.7 
 
 M t 
 For this value of -rj: and for j 
 
 (64) (23.5)2 
 
 0.19, Diagram 8 shows fc = 300 lb. per sq. in. and j 
 1,050,000 
 
 (16,000) (0.93) (23.5) 
 
 = 3.00 sq. in. 
 
 Four H-in. round rods and four ^^-in. round rods will be selected (Fig. 23), having a total area of 3.00 sq. in. (see 
 
 Table 5). Eight ^yie-in. round rods would do, 
 
 Elevation of Crossbeam 
 
 
 
 ...ft a c 
 
 
 
 
 L zs" ■ 
 
 ■4- 
 
 zs' A 
 
 but the Yi~ and ^^-in. rods are more likely to be found in stock. 
 
 The four ?4-in. rods will be placed in the lower row 
 and it will be sufficiently accurate to consider the 
 center of gravity of the steel area as midway between 
 the two rows. It is evident now that the width of 
 beam for rod spacing was correctly assumed. (Note 
 the rod spacing recommended by the Joint Com- 
 mittee in Art. 23, Sect. 7.) 
 
 Four rods will be bent up and lap over the top 
 of the support. The other four will be continued 
 straight and lap over support at the bottom of beam 
 (see Fig. 24). The bond stress along the eight rods 
 at the top of beam near support 
 
 Plan of Crossbeam 
 
 (Bent Oars nar^^oaaj' 
 Fig. 24. 
 
 25,000 
 
 [(4) (2.356) + (4) (1.964)1(0.85) (23.5) 
 
 = 73 lb. 
 per sq. in. 
 
 which is satisfactory. 
 
 (The average value of j is taken in the above equation, see Art. 27, Sect. 7.) The proposed arrangement of 
 rods is shown in Fig. 24. 
 
 The rods at the top of beam over supports will have the same effective depth (d) as the rods at the bottom 
 of beam at the center of span (Figs. 23 and 24). Then 
 
 d' 3.5 
 
 23.5 
 
 = 0.149 
 3.00 
 
 P = 
 
 (10) (23.5) 
 
 The following values may be obtained from Diagram 12: 
 k = 0.384 
 j = 0.864 
 
 Then, using also Table 9, 
 1,050,000 
 
 0.0128 = p' 
 
 fs 
 
 ~ (3.00) (0.864) (23.5) 
 fc = (17,200) (0.0415) 
 
 17,200 lb. per sq. in. 
 = 715 lb. per sq. in. 
 
 , point- -H 
 
 -ax. 
 
 support 
 
 (One fhird po/nf- 
 
 Allowable compression in concrete at the support 
 may be 750 lb. per sq. in., and hence no haunch or 
 extra steel is necessary (see Appendix B). The ten- 
 sile stress in the steel is greater than the allowable 
 but will be considered as satisfactory here simply 
 for the purpose of presenting several comparative 
 designs using different numbers of rods. If a little 
 more than the required amount of steel had been 
 selected at the center of span — say eight 94-in. rounds 
 — the stress in the steel over supports would not 
 have figured out greater than the allowable (see Art. 39, 
 
 Sect. 7). The diagram given on page 298 may be employed to find the points in the beam where the lower horizontal 
 
 S3"- 
 
 80"- >j 
 
 .1 
 
 • 30"- 
 ■■)(.,= 82" 
 
 Fig. 25. 
 
 rods may be bent up. Bending up the first two rods is equivalent to bending up 
 
 (2) (0.30 7) 
 3.00 
 
 = 0.205 or 20}^% of 
 
 the steel. After bending up the next two rods 50 % of all the steel is bent. The diagram above referred to shows 
 that those bends may be made at 0.318i (80 in.) and 0.210i (53 in.) from the center of support. 
 
 The rods at the top near support should be bent down as explained in Art. 39, Sect. 7 ; the first two rods at a 
 
 distance not less than 
 
 (2) (0.442) , I 
 
 3.00 
 
 of 
 
 25 in. from the center of support (assuming zero moment at the third point): 
 
 and the next two at a distance }^ of ^ = 42 
 
 from the same point (see Fig. 25). 
 
Sec. 11-11] 
 
 BUILDINGS 
 
 443 
 
 The distance from the support to the point where web reinforcement is not needed (see Art. 18, Sect. 7) 
 _ 21 _ (40) (10) (0.93) (23.5) 
 ^' ~ 2 2380 
 
 = 6.86 ft. = 82 in. 
 
 Also, 
 
 and 
 
 AB = 
 
 25,000 
 (0.85) (23.5) 
 
 = 830 lb. 
 
 CZ) = 3 • (40) (10) = 270 lb. 
 Fig. 25 shows the diagonal-tension trapezoid. 
 
 The points to bend rods at the top of beam control the design, as shown in Fig. 25. The horizontal distance 
 between bent rods is about the allowable — namely, about Hd. Since bending of longitudinal reinforcing bars at 
 an angle across the web of beam may be considered as adding to diagonal tension resistance for a horizontal dis- 
 tance from the point of bending equal to %d (see Art. 21, Sect. 7), stirrups will be needed only for the areas ABbc 
 and abCD. Diagonal tension represented by the area abCD will be cared for by the additional stirrups which will 
 be inserted to secure good T-beam action. The distance from the center of support to the point where stirrups are 
 not necessary scales 27 in. The stirrups near the end of beam will be looped about the upper rods, and hence will 
 be in an inverted position to those in a simply supported beam. 
 
 We shall use ^i-in. round U-shaped stirrups bent at the ends (see Art. 19, Sect. 7). The minimum spacing 
 of stirrups will occur at the support, and this spacing is given in Diagram IV, Sect. 7, page 288, as 4.2 in. Spac- 
 ing at other points along the beam may be found readily by means of this diagram. The first stirrup will be placed 
 2 in. from the edge of girder, assuming the girder to have a less width than the column. In order to secure good 
 T-beam action, the web and flange will be tied together with vertical stirrups placed about 18 in. on centers and 
 looped about the lower rods for the center half of beam. 
 
 The bars bent over the support should run to the third point of the adjoining span to provide thoroughly for 
 negative moment, assuming a very definite live load (see Art. 39, Sect. 7). The allowable stress in the compres- 
 sion rods at the support is 715 X 15 = 10,725 lb. per sq. in., and the necessary length for bond of H-in. round rods 
 . (10,725) (0.7 5) 
 (4) (80) 
 
 Cross-beams (Six-rod Design). — Four li-in. round rods and two ^^-in. round rods will give the required area 
 of 3.00 sq. in. The arrangement shown in Fig. 26 will be adopted. 
 
 = 25 in. (see Art. 21, Sect. 7). This length is shown in Fig. 24. 
 
 b \ a b 
 
 Fig. 26. 
 
 
 
 
 
 5 VC 4 C 4 C 1 
 
 ^1 
 
 Fig. 27. 
 
 The three rods in the upper row will be bent up and lap over supports as shown in Fig. 27. The other three 
 will lap over support at the bottom of beam. The bond stress along the six rods at the top of beam near support 
 
 25,000 
 
 per sq. 
 
 ~ [(4) (2.749) + (2) (1.964)1(0.85) (23.5) 
 which will be considered satisfactory for the arrangement of rods shown 
 
 The first 5^-in. rod may be bent at 93 in. from 
 center of support and the next two Ji-in. rods may be 
 bent at 53 in. from center of support. 
 
 The two Ji-in. rods may be bent down at a dis- 
 (2)(0.6013) . I 
 
 84 lb. 
 
 n Fig. 27 (see Art. 
 
 16, Sect. 7). 
 
 3.02 
 
 of 
 
 33 in. from the 
 
 <-— - ss ■ --J 
 
 and the %-\x\. rod at \^ of ^ =42 
 
 n:1 
 
 \ 
 
 jpoint 
 
 66' 
 
 \ 
 
 tance not less than 
 
 center of support, 
 
 from the same point. 
 
 The points to bend rods at the top of beam con- 
 trol the location of the bends of the two ^^-in. rods. 
 The 56-in. rod will be bent so as to be about Y^d from 
 the other two rods (see Fig. 28). The spacing of stir- 
 rups, and the distance from the center of support to the 
 
 point where stirrups are not necessary may be found in the same manner as for the eight-rod design. 
 
 Cross-beams {Four-rod Design). — Four J^-in. square rods will give an area of 3.06 sq. in. which is only slightly 
 more than is required. These rods will all be placed in one row as shown in Fig. 29. i Fig. 30 shows the complete 
 
 Fig. 28. 
 
 1 The beam is slightly narrow according to the rod spacing recommended by the Joint Committee. 
 
444 
 
 CONCRETE ENGINEERS' HANDBOOK 
 
 [Sec. 11-11 
 
 design. Stirrups are provided to take all the diagonal tension — the strengthening action of the bent rods not being 
 considered. It should be noted that the design is not conservative with respect to the negative-tension reinforce- 
 ment, since the upper rods run only to the fourth point and the curve for negative moment has not been con- 
 sidered in bending down the rods (see Art. 39, Sect. 7). Nevertheless a design of this kind is generally considered 
 good. In fact, it is all that is desired under ordinary conditions in roof design because of the character and amount 
 of the live load. 
 
 Of course, it should be realized that some latitude may generally be allowed in that part of beam design 
 referred to above, on account of the improbability of obtaining maximum conditions, but it is a good idea to have 
 
 Fig. 29. Fig. 30. 
 
 some conservative plan in mind, and to live up to that plan as nearly as circumstances will permit. At any rate, 
 designs should never become so radical as to include rods bent up close to the support in the computations for 
 negative reinforcement. 
 
 Fig. 31 shows a common continuous-beam design using separate straight rods over the supports. All the 
 diagonal tensile stresses are cared for by vertical stirrups. 
 
 Girder. — The girder has a span of 21 ft. with concentrated loads at the third points. The weight of the stem 
 >4rill be assumed at 500 lb. per lin. ft. Reaction of concentrated loads = 2 X 25,000 = 50,000 lb. 
 
 Fig. 31. 
 
 Maximum moment of concentrated loads with ends of beam simply supported would be 
 
 (50,000) (7) (12) = 4,200,000 in.-lb. 
 this moment reduces to 
 
 Taking M = 
 
 ^^(4,200,000) = 2,800,000 in.-lb. 
 Moment of dead load = 220,500 in.-lb . 
 Total moment 
 
 3,020,500 in.-lb. 
 
 If desired, the span of girder may be taken as the clear distance between faces of supports (see Art. 44, 
 Sect. 7). 
 
 The total maximum shear 
 
 V = 50,000 + 10.5(500) = 55,300 lb. 
 The cross-section of web as determined by shear = ^ 105^ ~ ^■^^ Using the formula for economical depths, 
 
 we have: 
 
 It is quite probable that a breadth of 12 in. will give proper rod spacing, but the 
 527 
 
 corresponding depth of -^2" = i^- likely to be too great when the cost of 
 columns and walls is taken into consideration. i Besides, the depth would be 
 
 1 In order to include the effect of columns and walls in the formula d = ^J~p 
 
 + first determine the total horizontal area covered by the stems of the T-beams 
 
 under consideration and the total horizontal sectional area of the columns and 
 walls at the level of the beams. If the cost per cubic foot of the columns and walls 
 is greater or less than the cost per cubic foot of the T-beam stems, increase or de- 
 crease their area in proportion to the difference in cost. Then increase the cost 
 per cubic foot of the T-beam stems in the ratio which the total corrected area bears to the area of the T-beam 
 stems in order to obtain the unit cost c which is used to determine the ratio r. 
 
 
 d 
 
 b' X d 
 
 (inches) 
 
 (inches) 
 
 (square 
 inches) 
 
 12 
 
 33. 1 
 
 397 
 
 14 
 
 30.8 
 
 431 
 
 15 
 
 29.8 
 
 447 
 
 16 
 
 29.0 
 
 464 
 
Sec. 11-11] 
 
 BUILDINGS 
 
 445 
 
 great in proportion to the breadth of stem and the beam would be relatively weak at the junction of stem and flange. 
 Illumination from windows must also be considered. The following cross-section will be taken as satisfactory: 
 
 15 in. 
 
 32^2 in. 
 
 The breadth of the flange of the T-beam is controlled in this case by 12 times the thickness of slab plus 
 the width of stem, or 69 in. Then 
 
 M. _ 3,020,000 _ 
 
 hd^ ~ (69)(32.5)2 - "^^-^ 
 
 M t 4 5 
 
 For this value of ^ and for ^ = = 0.14, Diagram 8 shows /c = 415 lb. per sq. in., and j = 0.935. Then 
 
 3,020,000 
 
 (16,000) (0.935) (32.5) 
 
 6.21 sq. in. 
 
 Eight 1-in. rouyd rods (total area 6.28 sq. in.) will be chosen. The bond stress along the eight rods at the top of 
 beam close to the support, 
 
 55,300 
 
 ^ = (8) (3.14) (0.85) (32.5) = P^'' 
 
 which is satisfactory. 
 
 Fig. 32 shows sketch of adopted cross-section. The weight of the stem is 
 
 (15)(31.5)(150) 
 
 144 
 
 and the value assumed is on the safe side. 
 
 492 lb. per lin. ft. 
 
 S.4 
 
 abba 
 
 d d c 
 
 IS" >J 
 
 Cross- section of Girder 
 Fig. 32. 
 
 Fig. 33. 
 
 j. r 
 
 The rods at the top of girder over supporting columns will be placed as shown in Fig. 33 in order to fit in 
 nicely with the cross-beam rods. It should be noticed that the value of d at the support is 33.5 in., or 1 in. more 
 than at the center of span. Then 
 
 6.28 
 
 V = l^ -N/QQ r;^ = 0.0125 
 
 (lo)(33.5) 
 V' = 0.5p (see Fig. 34) 
 
 The following values are obtained from Diagram 12 and Table 9, 
 
 h 
 
 k = 0.403 
 
 Then 
 
 M 
 
 Asjd 
 
 j = 0.882 
 
 3,020,000 
 
 = 0.0449 
 
 (6.28) (0.882) (33.5) 
 
 n(l - k) 
 = 16,200 lb. per sq. in. 
 
 fc = (16,200) (0.0449) = 730 lb. per sq. in. 
 
 These values will be considered satisfactory. 
 
 It is proposed to have bent rods take as much of the diagonal tension as possible. The total maximum 
 shear = 55,300 lb. The shear on the support side of the third point = 55,300 - 500(7) = 51,800 lb. On the side of 
 the third point toward the center of span, the shear = 51,800 - 50,000 = 1800 lb., or « = 4 lb. per sq. in. Thus, 
 web reinforcement is needed only from the support to the point where the beam intersects the girder. 
 
 Horizontal shear (measures diagonal tension) at the support 
 
 and at the third point, it is 
 
 V _ 55,300 
 jd (0.85) (33.5) 
 
 51,800 
 (0.93) (32.5) 
 
 1950 lb. per lin. in. 
 1720 lb. per Jin. in. 
 
446 
 
 CONCRETE ENGINEERS' HANDBOOK 
 
 [Sec. 11-11 
 
 The total diagonal tension is represented by a trapezoid, the parallel sides of which are 1950 lb. and 1720 lb., and 
 the length 7 ft. Hence total diagonal tension is 
 
 '°°°+'^^° (7.0)02) - 154.100 lb. 
 
 Two-thirds of this amount, or 102,700 lb., will be taken by the web reinforcement. If six rods are to be bent, their 
 tensile value is 
 
 (6) (0.785) (16,000) (1.43) = 108,000 1b. 
 which is in excess of the stress to be provided for. 
 
 Now shear is nearly uniform between the supports and the third point, and, as far as diagonal tension is 
 concerned, it would be sufficiently accurate to give equal spacing to the inclined rods. Since the size of columns 
 is not given, an 18-in. diameter of column will be considered. The spacing suggested above, then, would be taken 
 between a point 9 in. from the center of support (that is, at the edge of column) and a point where the center of the 
 beam intersects the girder. The plan proposed is to bend six rods', two at a time, and the points to bend for diagonal 
 tension should be laid off on a line approximately midway between the neutral axes for positive and negative 
 moment — as MM, Fig. 34. These points (1, 2, and 3) may be determined by dividing the distance mentioned 
 above into three equal parts and locating a point at the center of each part. 
 
 An investigation must now be made to determine whether or not the tensile stresses in the beam will permit 
 the bending of the rods as above suggested. From a study of moment curves of continuous beams loaded at the 
 third points, for different conditions, it is found sufficiently on the safe side, as regards bending up rods, to con- 
 sider the point of zero moment to occur at a distance of ^•^ of I from the third point measured toward the support 
 
 I 
 
 or of g (56 in. in 
 
 (see Art. 53, Sect. 7). Also when considering the bending down of rods, the same distance, 
 
 this case), may be safely taken as the distance out from the support to the point of zero moment. The curve of 
 bending moments in each case is to all practical purposes a straight line. Thus, the point where the first two 
 
 rods may be bent up, using the above data, is about - — = 14 in. from the center of the intersection of the 
 
 cross-beams. Allowing say 4 in. beyond the theoretical point for bending, this distance becomes 18 in. 
 
 As regards diagonal tension, the rod to intersect the center line at point 1 should be bent at r, as shown by the 
 dotted line. Since rods cannot be bent at r, stirrups will be employed to take the diagonal tension between the bent 
 
 rods and the cross-beam. (Stirrups will also 
 
 J'^sr/rru/ts [* -■ S6' - —^S6' - >| be placed at occasional intervals throughout 
 
 the girder to bind together the web and 
 flange.) Round rods of Vz-in. diameter may 
 be used for stirrups if bent at the upper end. 
 The tensile value of each stirrup (U-shape) is 
 (2) (0.196) (16,000) = 6270 lb. The shear to 
 be provided for in 1-in. length of beam is 
 1720 lb. and it will be necessary to space the 
 
 stirrups ^^20 ~ ^'^ ^P^^^. as 
 
 shown in Fig. 34. 
 
 It should be noticed that in bending up 
 the lower rods attention should be paid to the 
 points where the upper rods may be bent 
 down. In the design at hand, using 45-deg. 
 angle bends, the rods may be bent approxi- 
 mately as planned and the design will be ac- 
 cepted. The horizontal spacing of the bent 
 rods should not be greater than the distance 
 between points 1 and 2, or between points 2 and 3, unless stirrups are provided where such spacing occurs. 
 
 The rods at the top of girder should extend each side of the center of support far enough to obtain their full 
 strength in bond, which is 50 in. 
 
 The maximum stress in the compression rods at the support is 730 X 15 = 11,000 lb. per sq. in. The 
 
 necessary length for bond of 1-in. rods is ^^^^^^'^g^^ = 35 in., say 36 in. This length is shown in Fig. 34. 
 The top of the slab over the girder will be reinforced transversely with %-in. rods spaced 12 in. 
 
 Plan 
 
 (Senr bars not shown) 
 Fig. 34. 
 
 to c, in 
 
 order to provide for the negative bending moment produced with the bending of the slab next to the girder. 
 
 Designing Plates. — Plates I to IV inclusive give different complete designs for the 21 by 21-ft. floor bay 
 in question. If desired, all the cross-beam and girder reinforcement may be made into frames, except the cross- 
 beam reinforcement running into columns in the designs of Plates I and III. Even for these beams, however, the 
 stirrups may be rigidly spaced if wired at the upper turns to longitudinal rods of small diameter. After the stirrup 
 steel is suspended in the forms, the bent rods can then be easily slipped into place and wired. 
 
 Fig. 35 and Plate II show a girder design such that both girder and cross-beam reinforcement may be built 
 into frames. In the arrangement shown, the girder frames (with the exception of the rods over supports) would be 
 put in place first, then the cross-beam frames running into columns, next the negative tension rods of the girders, 
 
Sec. 11-12] 
 
 BUILDINGS 
 
 447 
 
 and finally the two intermediate cross-beam frames. The arrangement as regards strength is not as ideal as in 
 the girder design of Plates I, III, and IV since the bent-up rods do not extend over supports, but in the design shown 
 there is a surplus of diagonal-tension reinforcement which offsets this v.eakness. To be sure the bent rods are 
 anchored by transverse rods placed between layers of steel, but the danger lies in the liability of these bars to slip 
 horizontally when a force inclined to the vertical is exerted upon them. The additional amount of diagonal- 
 tension reinforcement, however, over and above that figured will make the design a safe one. The liability to 
 slip applies more especially to the lower transverse rods, as the upper rods are long and well anchored in the slab. 
 
 If the croBs-beam steel in Plate III could have been placed over the top of the girder steel at columns, then 
 all reinforcement shown on this plate could have been made up into frames before being placed in the forms. The 
 reason why the cross-beam rods were not placed above the girder rods in the design in question was because the 
 cross-beam rods would then interfere with the main reinforcement of the slab. To prevent this, the girder rods 
 over supports would need to be lowered and this could not be accomplished in the design in question because of 
 the compressive stress in the concrete being already a maximum at this point. It could be accomplished, how- 
 ever, by either forming a flat haunch, by inserting extra compressive steel, or by deepening the girder. 
 
 Placing of the reinforcement in the forms in frames insures accurate location of all steel. If a loose-rod method 
 is used, great care is required in construction to make sure that the steel is not disturbed during the pouring of the 
 concrete. Usually small diameter rods are needed to make the frames, in addition to the main reinforcement. 
 
 The placing of inverted stirrups near the supports has not been considered in the above discussion. These 
 are placed after all other reinforcement is in its proper pjosition and are slipped down over the negative-tension 
 steel and wired to it. The continuous stirrup shown in some of the designs is convenient for this purpose. In 
 schemes 3 and 4 the continuous stirrup is also used in place of some of the upright U-stirrups. 
 
 Spacing rods should be placed in all beams and girders between the upper and lower rows of steel. These 
 are plain rods of the desired size and should be placed transversely not more than 5 ft. on centers. Special frame 
 
 
 ^ . S6' --- 
 
 K SS'- 
 
 |< 40' ■ 
 
 
 56' - -- v 
 
 f^rods Z'-0"k 
 
 /7y 
 
 Stirrups 
 
 "cfoc 
 
 
 r 1 1 loX , 
 
 / / // 
 
 
 
 
 t 
 
 
 
 " ei±a_ 
 
 T 
 
 
 4 
 
 
 >k- 
 
 <- • 40' .-. >!<..- 
 
 *.— so' > 
 
 <- - - 7'-0 
 
 ■ 58'- 
 
 ... 44'.- 
 <-..- U' 
 
 <— IS' Column y 
 
 ' Z-y^rods s'/ont 
 
 - 7'-0' >■ 
 
 r 
 
 Fig. 35. 
 
 supports may be provided, if desired, or U-shaped stirrups may be employed if the length of the hook is made 
 sufficient to permit the stirrups to rest on the slab form. If special supports are used, they should be spaced about 
 5 ft. on centers. In the steel schedules given, special frame supports and spacing rods are omitted for simplicity. 
 
 The width of stirrups is given in the bending schedules as the clear width inside of the outer strands, and the 
 vertical height is given inside the turns. A little thought will make clear that these are the dimensions needed 
 in bending. 
 
 12. Hollow-tile Construction. — Hollow-tile construction is used to a considerable extent 
 for light buildings such as modern store buildings and office structures. 
 
 Fig. 36 shows a typical one-way hollow-tile slab and Fig. 37 a two-way tile construction. 
 No cross-beams are employed in the one-way type except the small ribs of the floor slab formed 
 between the rows of hollow tile. In the two-way type, cross-beams are placed at the columns. 
 The tiles are placed directly upon the forms with the reinforcing rods in the spaces between 
 them, and the concrete is filled in between the tiles and poured over the top to form the floor. 
 The ribs form a series of comparatively light T-beams side by side with flanges usually 2 or more 
 inches in thickness. The main beams or girders are also of T-shape. The flanges of these beams 
 or girders are usually of the same thickness as the floor slab, but lighter tiles are sometimes used 
 near the stem, in which case the flange becomes thinner than when the tiles are entirely omitted 
 at this part of the floor. The function of the tiles is simply to create a void in the concrete 
 and thus to decrease the dead weight of slab, and they do not enter into the calculations for 
 strength of floor. 
 
 Either hard-burned or semi-porous tile may be used in reinforced-concrete floor construe- 
 
448 
 
 CONCRETE ENGINEERS' HANDBOOK 
 
 [Sec. 11-12 
 
 Plate I 
 
 J-l| I I ILL 
 
 z/'-o" 
 
 I 111 
 
 'si 5 
 
 ^1 
 
 I I IK 
 I I I 
 
 si Slab 
 I 'gran 
 
 / *rods ii\ \c roc 
 S'-O' long I 
 of Slab over 
 entire g/r'aer 
 
 — 1-| 
 
 5/^ is', 36' 3 /" 
 f'Grano/ithic ■■ , 
 
 Cross secTion X-x 
 (Enlarged) 
 
 Gt is'.j$' a- 1"* 
 
 ■ Upper row of rods 
 Lower roiv 
 
 Note • Inverred sfirrLips are markec/ o 
 Concrete 12 4 mix 
 
 OETAIUS OP 
 AN INTERIOR FLOOR BAY 
 
 /Irrangemem Steel in Crossbeam 
 
 Arrangement of Steel in Gircfer 
 
 Plate 1(A) 
 
 BEAM AND GIRDER RODS 
 
 Bends 
 
 6-6" 
 
 Lengffh 
 
 30 Ol 
 
 For 
 beam 
 
 girder 
 
 BZ 
 
 82 
 
 vxantBd 
 ■for 
 each 
 beam 
 
 tin 
 
 Length 
 
 Number 
 wonted 
 
 ■for 
 each 
 beam 
 
 
 
 S\' 3'. 6' \S 
 
 
 
 ■ ■ /7'-6"' ' ' 
 
 j'0Slat>3tee/ 7bTa//en^ n'/O' 
 
 STEEL BENDING SCHEDULE 
 roR 
 
 AN INTERIOR FLOOR BAY 
 SCHEME I 
 
Sec. 11-12] 
 
 BUILDINGS 
 
 449 
 
 Plate II 
 
 / Proa's /a"\c n>c 
 S'-o'lon^ \aTTvp 
 of slab oyer 
 entire girder 
 
 II 
 
 Reinforcemenr 
 
 , Zro(. 
 
 s 
 
 
 T-0- 
 
 
 finish . 
 
 zie ^ 
 
 1'* stirrup i ,' /'" 
 
 
 UlL 
 
 
 
 
 I'll 
 
 i 
 
 
 
 4; 
 
 
 J'-O" 
 
 69 about- /-7 
 
 -J'-O' 
 
 
 
 
 Cross section X-X 
 J/, {Enlarged) 
 
 j ^ Upper row of rods ~ 
 
 Note ■ Inverted stirrups are marked a 
 Concrete 12 4 mix 
 
 DETAILS OF 
 
 AN INTERIOR FLOOR BAY 
 
 Arrangement of Steel 'a Crossbeam 
 
 /irrangement of Steel in Girder 
 
 Plate 11(A) 
 
 BEAM AND GIRDER RODS 
 
 Straight 
 
 36- 4' 
 
 52 
 
 3/ 
 
 /$'-3" 
 
 Total 
 
 numcer 
 wanted 
 
 U- STIRRUPS 
 
 Leng'th 
 
 S'-9'' 
 
 7'- 6" 
 
 Beam 
 
 or 
 Girder 
 
 82 
 
 Number 
 ■for 
 eaci-i 
 
 TotO» 
 
 No- 
 vo n-fed 
 
 CONTINUOUS STIRRUPS 
 
 a 
 
 b 
 
 Spacers 
 
 Size 
 
 Length 
 
 Beam 
 
 No for 
 each 
 
 Total 
 No 
 
 
 
 
 N? 
 
 Deem 
 
 wantec 
 
 
 7 
 
 4-4@5 
 
 
 29'- 6" 
 
 Bl 
 
 2 
 
 2 
 
 ■ 4 
 
 24 
 
 
 a4'- 7" 
 
 B2 
 
 1 
 
 2 
 
 2 
 
 ^4 
 
 // 
 
 3@/Z 
 
 
 28'- 4" 
 
 6/ 
 
 1 
 
 2 
 
 2 
 
 V — yn~ 
 
 7^-o" \s\s'a" 
 
 
 > 
 
 / 
 
 7-^" 
 
 
 STEEL BENDING SCHEDULE 
 
 AN INTERIOR FLOOR BAY 
 SCHEME 2 
 
 29 
 
450 
 
 CONCRETE ENGINEERS' HANDBOOK 
 
 [Sec. 11-12 
 
 Plate III 
 
 _U±L 
 
 IP! 
 
 V 
 
 ■f rods /e\"\c roc 
 S'-O" long \ \af K 
 of s!ab over ei 
 girder | 
 
 ReinfcrcemenT 
 spaced 4"c to c 
 
 ^ /"granolithic fmisn 
 
 ~T\ \-\ 
 
 Gh IS", 36' 8-i'" 
 "Oranolifhic - ^k^, 
 
 Lj-jJi-i. LU J. 
 
 I " Grano/ifhic 
 
 finish > |- 
 ^ ii" Stirrups a 
 
 Kl 
 
 6 g> abcuT I'-e" 
 
 \ B2 
 
 10". ay" 4- 
 
 Cross section X X 
 ( £nlorgec/J 
 
 / granolithic 
 finish-. 1 1' If. ' 
 
 ■■. I '^stirrups 
 
 Gl- 15", ^S' S-i" 
 
 z 
 
 Upper row of rods 
 Lower ran 
 
 Arrangement of Steel m Crossbeam 
 
 Arrangement of Steel in Girder' 
 
 Note' Inverted stirrups are marked a 
 Concrete 15 4 mix 
 
 DETAILS OF 
 
 AN INTERIOR FLOOR BAY 
 
 Plate III (A) 
 
 BEAM AND GIRDER RODS 
 
 MI 
 
 Lengi-h 
 
 Bi 
 
 29'- 
 
 30' 
 
 28'- 2-^" 
 
 61 
 
 61 
 
 Number 
 wanTed 
 for 
 each 
 Deam 
 
 numcer 
 wonted 
 
 Spacers 
 
 U- STIRRUPS 
 
 Bends 
 
 
 Length 
 
 For 
 
 beam 
 girder 
 
 Number 
 wonted 
 for each 
 beam 
 
 Total 
 
 number 
 wan-ted 
 
 
 No 
 
 
 7" 
 
 
 
 
 s'e" 
 
 Bl 
 
 2 
 
 15 
 
 ■45 
 
 
 
 B2 
 
 1 
 
 IS 
 
 
 i/'' 
 
 
 
 i"0 
 
 7'- 6" 
 
 6/ 
 
 / 
 
 5 
 
 S 
 
 CONTINUOUS STIRRUPS 
 
 a 
 
 b 
 
 Spacers 
 
 S>ze 
 
 Length 
 
 For 
 beomor 
 qirder 
 
 Number 
 for each 
 beam 
 
 Totol 
 numDer 
 wanted 
 
 24 
 
 7 
 
 4-4@S 
 
 8 
 
 29-6" 
 
 Bl 
 
 2 
 
 Z 
 
 6 
 
 4@S 
 
 e4'- 7" 
 
 B2 
 
 1 
 
 2 
 
 
 // 
 
 
 
 53'-2" 
 
 Gl 
 
 1 
 
 2 
 
 2 
 
 
 
 27'-7" 
 
 ? 
 
 ^ 
 
 
 
 
 
 
 
 7'J 
 
 
 
 
 
 
 
 17'- 
 
 6" 
 
 
 
 ^"^5iai> 5 fee/ Total len^fii /7'/0' 
 
 Continuous Stirrup 
 
 STEEL BENDING SCHEDULE 
 FOR 
 
 AN INTERIOR FLOOR BAY 
 SCHEME 3 
 
Sec. 11-12] 
 
 BUILDINGS 
 
 451 
 
 Plate IV 
 
 II I I ll-L 
 
 I I ' 
 
 V 
 
 I I I I 
 I I I I 
 
 S/a£> 
 
 / Proofs /^'{c ^pc 
 S'-0"long\ \aT ; 
 of slab oyer 
 entire ^irtier 
 
 Retnforcemen t 
 
 ■ncluding \ 
 gronolithic 
 
 "1"! 
 
 Gi is', 36' e 
 I'Orano/ifhic :. \ 
 finisn 
 
 Note Both upright 
 and inverted stirrups 
 at points marked "b' 
 
 Cross-secTion X-X 
 ( Enlarged] 
 
 Upper row of rods 
 Lomr rotv.. 
 
 /Arrangement of Steel in Crossbeam 
 
 /Irrangewent of Steel in Girder 
 
 Note ■ Inverted stirrups are markect a 
 Concrete I' 2 -4 mix 
 
 DETAILS OF 
 
 AN INTERIOR FLOOR BAY 
 
 Plate IV(A) 
 
 BEAM AND GIRDER RODS 
 
 Straight 
 
 Lengti- 
 
 30'- 0-2 
 
 28-Z-2 
 
 BZ 
 
 Number 
 wan+ed 
 ■for eoch 
 
 Toral 
 No 
 
 wonted 
 
 Spacers 
 
 U-STIRRUPS 
 
 Bends 
 
 Size 
 
 Length 
 
 For 
 beorn 
 
 or 
 gi rder 
 
 Number 
 for 
 each 
 beam 
 
 Total 
 No 
 
 wonted 
 
 
 No 
 
 
 
 
 
 
 
 S'-9" 
 
 3i 
 
 2 
 
 15 
 
 45 
 
 B2 
 
 1 
 
 IS 
 
 
 
 
 
 
 ■ 
 
 
 
 
 
 2 
 
 7'-e" 
 
 c-/ 
 
 1 
 
 S 
 
 S 
 
 
 
 CONTINUOUS STIRRUPS 
 
 a 
 
 b 
 
 Spacers 
 
 Size 
 
 Length 
 
 Beam or 
 Girder 
 
 No for 
 each 
 beam 
 
 TotQl 
 NO 
 
 wanted 
 
 
 
 24 
 
 7 
 
 4-S@5- 
 
 /> 
 
 so'-o 
 
 4S'->" 
 
 Bl 
 
 2 
 
 Z 
 
 6 
 
 B2 
 
 1 
 
 2 
 
 ^4 
 
 
 
 i"0 
 
 SS'2" 
 
 6/ 
 
 1 
 
 2 
 
 2 
 
 3@9 
 
 27'- 7" 
 
 
 2 
 
 *^ \ 
 
 7'- 0 " 
 
 y 
 
 5 
 
 _ 2'- 8" 
 
 s\ s'-e" U 
 
 " 2'-8''S 
 
 /7'-6" 
 
 3"'^ 5/at> Stee/ Tata/ length /vW 
 
 Continuous Stirrup 
 
 STEEL BENDING SCHEDULE 
 
 FOR 
 
 AN INTERIOR FLOOR BAY 
 SCHEME 4 
 
452 
 
 CONCRETE ENGINEERS' HANDBOOK 
 
 [Sec. 11-12 
 
 tion. Hard-burned tile, due to its density, has a higher crushing strength and will, therefore, 
 undergo a greater stress without any sign of failure, but it does not seem to be as good a fire- 
 resisting material as the semi-porous. 
 
 The following table gives average weights of the common sizes of hollow tile : 
 
 Weights of Hollow Tile 
 
 4 by 12 by 12 16 lb. 8 by 12 by 12 30 lb. 
 
 5 by 12 by 12 20 lb. 9 by 12 by 12 33 lb. 
 
 6 by 12 by 12 22 1b. 10 by 12 by 12 35 1b. 
 
 7 by 12 by 12 27 lb. 12 by 12 by 12 40 lb. 
 
 H- J I I L JL JL-jML JULJUL ' j ^ 
 ]C] 1 I LJLJrjuJiTj 
 
 -inrTr-ir-TrTr- 
 L JLjl jljLji-JL 
 nr-irT^ 
 
 JLJLjLJL_J j I LJU 
 
 JLJL. 
 
 •ir-ir 
 
 •Tru-TrTr-in" — I 
 
 L"]L:i[:in!:]:]rj!:][:]r]nc]L]L]q i □ 
 
 r-"nt-->nrTnnnr-,rnrTr-^rnr-|. n 1 r-i 
 
 r-ir-tr-iT-.f^r- 
 
 JL_ll_ 
 
 ... , . -\r-ir 
 
 _iLjL jLJlU 
 
 JLJ1-J1.JUJLJLJ1.J I Ljd 
 
 Tr-,rnnr-|rn(— in 1 | nrl 
 JL Ji_1i_jlJLjljL J , LJLJ 
 
 H-ir-1i->r-;rnr--.rT 
 
 iJl 
 
 -inr->rT"i-inr 
 
 (-JI-JLJuJl-ULJI- JLjl_Jl.Jl_Jl_JLJLJL J L 
 
 LTJr:irj:D[]rx][::[:]::irjrx]:: [ 
 
 Plan, 
 
 77/e--' 
 Half Section A-A' 
 
 Fig. 36. 
 
 Hollow 'Tile 
 
 Section A- a' 
 
 Fig. 37. 
 
 The commercial sizes of tiles are usually 12 by 12 in. in plan and vary in depth from 4 to 
 16 in. The depth of a tile concrete floor should be designed so as to allow for these commercial 
 sizes. The standard sizes manufactured by the National Fire Proofing Co. are all 12 by 12 in. 
 in plan with the following depths : 4 in., 5 in., 6 in., 7 in., 8 in., 9 in., 10 in., 12 in., 15 in. Special 
 sizes of tile may be obtained if the order is of sufficient size to warrant the manufacturing of the 
 same. 
 
 Tiles are likely to vary in. from the dimensions specified so that the plans should show 
 the full thickness of the floor and the minimum amount of concrete topping. If the tiles are 
 small, due to shrinking in burning, the thickness of floor should be made up in concrete. 
 
 Unless the tile is thoroughly sprinkled before the floor is poured, slight depressions will 
 occur over the ribs. This is because the hollow tile absorbs the moisture in the concrete of the 
 top coat, causing it to set more quickly than the rib with its greater body of concrete and greater 
 shrinkage. Sprinkling of the tile should be insisted upon, especially in hot weather. 
 
 Hollow-tile floors are generally plastered on the underside as it is only in the roughest kind 
 of work that this is not done. The surface of the tiles should be deeply scored so that the 
 plaster will bind firmly. 
 
 Ordinary hollow tile is open at both ends and cannot be used when the floor is reinforced 
 in both directions. For such floors two-way tile should be procured. 
 
Sec. 11-12] 
 
 BUILDINGS 
 
 453 
 
 Hollow-tile jfloors are used mostly in long-span construction, and where the loads are light 
 and distributed. The reason for this is the small dead load, the flat ceiling, and the simplicity 
 of the form work. The cost of the sohd type of beam and girder construction for the conditions 
 of long span and light loads is generaly much greater than for hollow tile. 
 
 Illustrative Problem. — Design an interior panel of a one-way hollow-tile floor to carry a live load of 100 
 lb. per sq. ft. with the rows of girders spaced 21 ft. on centers. The recommendations of the Joint Committe 
 will be followed for a 2000-lb. concrete (see Appendix B). The ratio of the unit cost of steel in place to unit cost 
 of concrete in place (r) will be taken at 70, with 20 cts. as the cost of concrete per cubic foot. 
 
 The finished flooring will consist of maple boards nailed to 2 by 3-in. sleepers. The sleepers will be 
 
 placed on the concrete slab and cinder concrete in proportions 1:3:6 filled in between them. The following weights 
 will be taken in pounds per square foot of floor area: wooden floor, 5; sleepers or nailing strips, 2; concrete filling, 
 15; plaster, 5. 
 
 Ribs 4 in. wide will be assumed making a 16-in. width of flange for the small T-beams. The concrete topping 
 will be made 2 in. If a 9-in. tile is assumed, the total load per linear foot of beam will be 274 lb., made up of the 
 following items: 
 
 Live load = 100 X ^y\2 = 133 lb. 
 Wood floor = 5 X H = 7 lb. 
 Sleepers = 2X%= Z\h. 
 Concrete filling = 15 X ^ = 20 lb. 
 Concrete topping = 25 X = 33 lb. 
 
 Tile = 33 lb. 
 
 Stem = X (150) = 38 lb. 
 
 Plaster = 5 X = 7 lb. 
 
 Total = 274 lb. 
 
 The bending moment. 
 
 wl'^ (274) (21 ) (21) (12) 
 M = Y2 = 12 ^ 121,000 m.-lb. 
 
 The economical depth of floor now needs to be determined. Using the same notation as in Art. 37 of Sect. 7 
 and, in addition, using the term c< to represent the variation in cost of tile in place per 1-in. change in depth, we 
 have 
 
 crM 
 
 C = cb'd' + ■ ■ — + d'ct 
 
 + 2) 
 
 as the total cost of the small T-beams per unit length. The following expression has been deduced from the pre- 
 ceding equation by the aid of the calculus, and will give the value of d for minimum cost when the value of b' 
 is fixed: 
 
 Mfsib'Cc + 
 
 M t 
 + 0 
 
 144c0 ' 2 
 
 The term Cc in this formula means the cost of concrete per cubic foot. A value of IH cts. will be given to cr 
 Then 
 
 d = a/ (20)(70)(121,000) + 2_ = 7.4 in. 
 \ (16.000U260) ^ 2 
 
 (16,000) (260) 
 
 The effective depth d must be taken so as to provide for a commercial depth of tile. A 7\^-\n. effective depth 
 will be tried with a IJ^-in. fireproof covering below the center of steel. This assumption would permit of a 7-in. 
 tile. 
 
 Proceeding with the design, however, we find that 
 
 M 121,000 
 
 and 
 
 6d2 (16) (7.5) 
 t 2 
 
 I = 135 
 
 - 7.5 - 
 
 Diagram 8, Sect. 7, shows the stress in the concrete to be above 650 lb. per sq. in., which cannot be allowed. By 
 trial it is found that a Q'^-in. effective depth is needed to bring the concrete stress to an allowable value. For this 
 depth, 
 
 M _ 121,000 _ 
 6d2 ~ (16) (9.5)2 - 
 
454 
 
 CONCRETE ENGINEERS' HANDBOOK 
 
 [Sec. 11-12 
 
 From Diagram 8, j = 0.91. Then 
 
 121,000 
 
 (16,000)(0.91)(9.5) 
 
 Two round rods will be employed in each rib — one will lie straight and the other will be bent up at both ends 
 
 and extend along the top of beam to the quarter point of the adjoining span. This arrangement will give the same 
 steel over supports as in the center of span. 
 The total shear close to support 
 
 y.^i^ = 2880 1b. 
 
 The bond along the two rods at the top over supports 
 
 2880 
 
 76 lb. per sq. in. 
 
 (2) (2.36) (0.85) (9.5^ 
 The distance from the support to where stirrups are unnecessary 
 21 (40) (4) (0.91) (9.5) 
 
 274 
 
 = 5.4 ft. = 65 in. 
 
 All the diagonal tension not taken by the concrete will be provided for by vertical stirrups; in other words, the 
 strengthening action of the bent rod will not be considered. Round stirrups H-in. diameter will be employed, 
 bent at the ends. Stirrup spacing near the support: 
 
 3 (0.049) (2) (16,000) (0.85) (9.5) 
 " = 2 2880 = 
 
 The spacing adopted is shown in Plate V. 
 
 The moment, shear, and bond considerations given above do not take into account the strengthening action 
 of the flange of the T-shaped girders, and allowance may be made for this when thought necessary. It is quite 
 evident that the concrete is not overstressed in compression over supports, but the stress at the edge of girder 
 flange should be investigated. For the width of girder flange shown in Plate V, the bending moment may be 
 taken at ^/j (121,000) = 104,000 in. -lb. This value is obtained by considering the point of zero moment at the 
 third point. 
 
 Diagram 12 and Table 9 give the values: 
 
 k 
 
 k = 0.440 j = 0.848 _ = 0.0524 
 
 Then 
 
 M 104,000 ,.^nniu 
 
 Ajd= (0.88) (0.848) (9.5) = ^^'^^^ 
 
 fc = (14,700) (0.0524) = 770 lb. per sq. in. 
 
 The load on the floor is (274) (i^ie) = 205 lb. per sq. ft., and the girder therefore caTries a load of (205) (21) = 
 4300 lb. per lin. ft., to which should be added the weight of the girder itself. This weight will be assumed at 360 
 lb. making a total load, which may be considered uniform, of 4660 lb. per lin. ft. The bending moment 
 
 (4660) (21) (21) (12) 
 M = j2 = 2,0o5,000 in.-lb. 
 
 Assume the total depth of girder to be limited to 36 in., effective depth to 32}^ in. Then ^ = = 0.34. 
 
 t M 
 Diagram 8 shows that, for j = 0.34 and fc = 650, ^ = 107, or 
 
 2,055,000 
 
 = 18.2 m. 
 
 (32.5)2(107) 
 
 M 
 
 An arbitrary value of 24 in. will be adopted by b, which makes ^ = 81. Diagram 8 shows the neutral axis to 
 
 lie in the flange. Diagram 2 shows p = 0.0057, or 
 
 As = (0.0057) (24) (32.5) = 4.4 sq. in. 
 
 Eight %-in. round rods will be selected, with a total area of 4.81 sq. in. The width of stem will be made 14 in. to 
 proviile properly for shear. The remaining computations for stresses in the steel and concrete at the support, and 
 
Sec. 11-12] 
 
 BUILDINGS 
 
 456 
 
 Plate V 
 
 
 1 
 
 7'-0" 
 
 */m ips 
 
 7'o ^ 
 2rocrs a 
 
 
 / 
 
 
 
 
 
 H 
 
 ■ /s'Co/- 
 
 r" 
 
 
 Jti9' 1 
 
 Stirrup spacing 
 ?' Se> about /'-I/" 
 
 M 
 
 
 Z'Conavte 5/ab 
 
 F/n/shed F/oor 
 
 ^1 J 
 
 Crossbeam ^'t//' Z-^'^ 
 
 S'j-J'S/eepers 
 
 -/ j^'.'je g//7gigy concrete f///- 1 S 6 mix 
 
 Note: Inserted stirrups are marAea !r' 
 Concrete IZ4 mix 
 
 TT 
 
 Section A/t 
 T/7roug/7 Girder 
 
 Section BB 
 
 DESIGN OF 
 AN INTERIOR FLOOR BAY 
 
 FOR A 
 
 ONE-WAY HOLLOW TILE FLOOR 
 
 Plate V(A) 
 
 SEAM AND GIRDER RODS 
 
 Bends 
 
 8'-s" 
 
 > 
 
 > - 
 
 \/ 
 
 S'-J" ^ 
 
 
 
 
 S/'-6" 
 
 
 Straight 
 
 Size 
 
 Length 
 
 30'-//'' 
 
 Beam 
 
 or 
 Girder 
 
 No 
 wanted 
 
 for 
 each 
 beam 
 
 Total 
 
 No. 
 wanTed 
 
 \l 
 
 ■ Spacers 
 
 Continuous Stirrup 
 
 U- STIRRUPS 
 
 Bends 
 
 Length 
 
 7-8' 
 
 Beam 
 
 or 
 Girder 
 
 No 
 wanted 
 
 for 
 each 
 beam 
 
 CONTINUOUS STIRRUPS 
 
 a 
 
 b 
 
 
 Size 
 
 Length 
 
 Beam 
 
 or 
 Girder 
 
 No for 
 each 
 beam 
 
 5/' 
 
 iO 
 
 e&s"- 
 
 -3<S>4- -& 
 
 8 
 
 69- O" 
 
 O 
 
 Z 
 
 9 
 
 4 
 
 6-e-6-7 
 
 
 iO'-3f 
 
 B 
 
 
 STEEL BENDING SCHEDULE 
 
 FOR 
 
 AN INTERIOR FLOOR BAY 
 ONE WAY HOLLOW TILE FLOOR 
 
456 
 
 CONCRETE ENGINEERS' HANDBOOK 
 
 [Sec. 11-12 
 
 for shear and bond, are similar in every way to those given under cross-beam design in two-intermediate beam 
 construction. In Plate V an 18-in. column is assumed for convenience. 
 
 As in solid concrete floors, the pwoper depth for girders in hollow-tile construction depends upon the use to 
 which the building is to be put, and the cost of all columns and walls in the building per unit increase in height. 
 The cost of formwork should also receive attention. Of course, as regards the materials in the girder itself, cost 
 decreases with depth. The problem resolves itself into finding the limiting depth of the girder in view of the many 
 conditions which must be considered. Very often a perfectly flat ceiling is desired, and very wide girders must 
 then result. 
 
 The following table has been prepared to simplify the computations in the design of the small T-beams of a 
 hollow-tile floor. The table shows at a glance, for any given thickness of concrete topping, the minimum depth of 
 tile needed for any given bending moment, and the corresponding steel area required. Greater depths of tile may 
 sometimes prove more economical. 
 
 It is quite evident from a study of the table that different thicknesses of concrete topping should be con- 
 sidered in order to arrive at the most economical design. When the rib width does not change in the economical 
 considerations, the economy of the various designs may be based on the cost per foot length of floor having a 
 width, the distance center to center of ribs ; but when the steel area is such that the width of rib varies, the economy 
 of the designs should be based on the cost per square foot of floor. 
 
 Table for Hollow-tile Floors 
 
 Depth of tile 
 (inches) 
 
 — 
 
 Based on fc = 650; fs = 16,000; n = 15 
 in. allowed from center of steel to bottom of slab) 
 
 Bending moments and steel areas for various depths of tile and concrete 
 (Moments in thousands of inch-pounds; steel areas in square inches) 
 
 4-in. rib 
 
 For any other width of rib multiply all values in table by distance center to center of ribs 
 
 and divide by 16 
 
 Thickness of concrete topping (inches) 
 
 1.5 
 
 2.0 
 
 2.5 
 
 3.0 
 
 3.5 
 
 4.0 
 
 4.5 
 
 5.0 
 
 4 
 
 27/ 
 
 / 
 
 /0.49 
 
 • 
 
 35/ 
 / 
 
 /0.55 
 
 43/ 
 
 / 
 
 /0.62 
 
 52/ 
 / 
 
 /0.68 
 
 
 
 
 
 
 41/ 
 
 / 
 
 /0.58 
 
 52/ 
 /0.68 
 
 69/ 
 / 
 
 /0.74 
 
 72/ 
 /0.80 
 
 * 
 
 84/ 
 
 / 
 
 /0.86 
 
 
 
 
 6 
 
 56/ 
 / 
 
 /0.65 
 
 71/ 
 
 / 
 
 /0.78 
 
 84/ 
 / 
 
 /0.86 
 
 * 
 
 96/ 
 
 / 
 
 /0.92 
 
 • 
 
 110/ 
 
 / 
 
 /0.98 
 
 * 
 
 124/ 
 
 / 
 
 /1. 04 
 
 
 
 7 
 
 71/ 
 
 / 
 
 /0.70 
 
 90/ 
 
 / 
 
 /0.84 
 
 107/ 
 
 / 
 
 /0.96 
 
 124/ 
 / 
 
 /1. 04 
 
 139/ 
 
 / 
 
 /1. 10 
 
 155/ 
 /1. 17 
 
 • 
 
 172/ 
 / 
 
 /1. 23 
 
 
 8 
 
 86/ 
 /0 . 73 
 
 110/ 
 
 / 
 
 /0.90 
 
 131/ 
 
 / 
 
 /1. 03 
 
 152/ 
 
 / 
 
 /1. 14 
 
 171/ 
 / 
 
 /1. 23 
 
 • 
 
 189/ 
 / 
 
 /1. 29 
 
 • 
 
 208/ 
 / 
 
 /1. 35 
 
 227/ 
 / 
 
 /1. 41 
 
 9 
 
 104/ 
 /0.76 
 
 130/ 
 / 
 
 /0.94 
 
 156/ 
 / 
 
 /1. 09 
 
 181/ 
 
 / 
 
 /1. 21 
 
 204/ 
 / 
 
 /1. 32 
 
 226/ 
 
 / 
 
 /1. 41 
 
 248/ 
 / 
 
 /1. 48 
 
 * 
 
 268/ 
 /1. 53 
 
 10 
 
 117/ 
 
 / 
 
 /0.78 
 
 150/ 
 
 / 
 
 /0.98 
 
 181/ 
 
 / 
 
 /1. 14 
 
 210/ 
 /1. 28 
 
 233/ 
 /1. 40 
 
 243/ 
 /1. 51 
 
 290/ 
 / 
 
 /1. 59 
 
 314/ 
 
 / 
 
 /1. 67 
 
 12 
 
 147/ 
 /0.81 
 
 190/ 
 / 
 
 /1. 03 
 
 231/ 
 /1. 22 
 
 269/ 
 / 
 
 /1. 38 
 
 309/ 
 / 
 
 /1. 52 
 
 340/ 
 /1. 65 
 
 374/ 
 /1. 77 
 
 407/ 
 / 
 
 /1. 87 
 
 • Neutral axis in flange. 
 
Sec. 11-13] 
 
 BUILDINGS 
 
 457 
 
 FLAT-SLAB CONSTRUCTION 
 
 By Walter S. Edge^ 
 
 13. General Description. — The term flat-slab construction as here employed may be con- 
 sidered to include that type of building construction employing a reinforced-concrete slab in 
 which the load upon the floor is carried directly to the columns without the agency of other 
 elements, such as beams or girders. 
 
 As commonly constructed, a reinforced-concrete floor slab of uniform thickness (for all 
 or the greater part of its area) is supported symmetrically upon columns provided with wide 
 conical-shaped capitals at their junction with the under side of slab (see Fig. 38.) The slab 
 may be uniform in thickness from the edge of one capital to another; or a portion of it, sym- 
 metrical with respect to the column, may be increased in thickness, forming a drop panel (see 
 Fig. 39). Another form occasionally used, carries the thickened slab from column to column, 
 thus forming in reality shallow beams between columns, and giving the effect of a paneled 
 ceiling. 
 
 The methods of reinforcing the slab and columns differ radically in different systems, and 
 are described in Art. 17. In general it may be said that the slab reinforcement consists of a 
 
 Fig. 38. — Velie Motor Vehicle Co.'s factory building, Moline, 111. 
 
 large number of comparatively small-diameter rods, a considerable percentage of which radiate 
 from the center of the various columns and are commonly located in the bottom of the slab at 
 the center of the span, and in the top of the slab over the column head. While in some respects 
 flat-slab construction is structurally the simplest form of concrete-steel design, it is of compara- 
 tively recent development, and has a present-day use out of all proportion to the time that has 
 elapsed since its first introduction. Since the erection in 1903 of the first flat-slab building, 
 their relative number has steadily increased until at present, probably 80% of new reinforced- 
 concrete construction (in which the live load is 100 lb. per sq. ft. or more) is of this type. 
 
 14. Advantages Over the Beam-and-girder Type.— The reasons for its popularity as 
 might be supposed are economic ones, and may be summarized as follows : 
 
 1. The ceiling, being flat or practically so, offers no obstruction to the passage of light; and, 
 as the windows may extend to the underside of floor, good daylight illumination may be obtained 
 when desired for the maximum width of building (see Figs. 38 and 39). 
 
 2. The failure, or partial failure, of concrete structures under fire has been first to develop 
 at the corners of the columns, beams, and girders where spalling of the concrete is apt to take 
 place. A flat-slab floor supported upon round concrete columns offers practically no sharp 
 
 I Consulting Engineer, New York City. 
 
458 
 
 CONCRETE ENGINEERS' HANDBOOK 
 
 [Sec. 11-15 
 
 angles for spalling to begin, and experience shows that such construction will suffer little damage 
 where a beam-and-girder building would be seriously injured. 
 
 3. The automatic sprinkler system is after all the final safeguard where inflamable materials 
 are stored in the building, and a much more efficient installation can be made with the fiat 
 ceiling, than with an unfinished ceiling of the beam-and-girder type (see Fig. 39). 
 
 4. A considerable saving in story height and in total height of a multi-storied building 
 (or an increase in clear story height with the same height of building) may be secured by the 
 use of flat-slab construction as compared with the common form of beam-and-girder design 
 (see Fig. 40). 
 
 5. The danger of sudden collapse from excessive overload — particularly in the case of the 
 so-called four-way type of floor, due to the interlacing of a great number of small steel rods run- 
 ning in four directions — is much less than in beam-and-girder type of construction. 
 
 Fig. 39. — Stewart Warner Speedometer Corporation building, Chicago, 111. 
 
 6. The slab formwork is much simplified and no added complication is introduced by the 
 round columns or ornamental column head, since it is now common practice to use metal 
 forms which are fairly well standardized and may be rented from a number of firms making a 
 specialty of this work. 
 
 7. For average conditions, the flat-slab type of construction is more economical than the 
 beam-and-girder type for live loads of 100 lb. per sq. ft. and over, and the economy increases 
 with the load. 
 
 15. Classes of Buildings to Which Adapted. — The many advantages which the flat-slab 
 system of floor construction possesses has developed an excess of enthusiasm for its use and has 
 caused its adoption in some cases where an adherence to the beam-and-girder type would un- 
 doubtedly have been the part of wisdom. The stern realities, however, of the economic side 
 of the case, due to the present high cost of structural-steel shapes, has brought about the 
 adoption of flat-slab floors in many types of buildings which were heretofore almost universally 
 
Sec, 11-16] 
 
 BUILDINGS 
 
 459 
 
 constructed with structural-steel skeleton frames. Among these may be mentioned apartment 
 houses, stores, hotels, and loft buildings for light manufacturing. 
 
 Structures to which this system is best adapted may be summarized as follows : 
 
 1. Warehouses. 5. Wharves. 
 
 2. Factories. 6. Coal-storage bins, etc. 
 
 3. Cold-storage plants. 7. Railroad terminals. 
 
 4. Garages and auto service stations. 
 
 The conditions which make for -economy in the use of flat-slab floors are as follows : 
 
 1. An approximately uniform and equal spacing of supporting columns. 
 
 2. The absence of frequent large openings in the floor. 
 
 3. A superimposed live load of 100 lb. per sq. ft. or more. 
 
 FLAT SLAB CONSTRUCTION BEAM AND QIRDER CONSTRUCTION 
 
 Fig. 40. 
 
 While other considerations may make the use of flat-slab construction advisable, it will 
 frequently be found that other types — such as steel-tile construction or combination terra-cotta 
 block and concrete — will prove to be more economical for lighter loads or irregular spans. 
 
 In the great majority of buildings in which this system of floor construction is employed, 
 utilitarian considerations alone prevail; nevertheless, it can, on occasion, be combined with a 
 simple scheme of decoration to give a very pleasing effect. 
 
 16. Remarks Regarding Design. — The methods used in the design of flat-slab floors are 
 almost entirely based on the results of extensometer measurements of the deformation of con- 
 crete and steel in actual buildings under test loads (see Art. 19). The theoretical analyses of 
 stress which have been made have proved to be ultr.aconservative as compared to test results, 
 due undoubtedly to the fact that it is extremely difficult to take all factors into consideration 
 in any formula. No theoretical method of analysis has been developed which really meets 
 
460 
 
 CONCRETE ENGINEERS' HANDBOOK 
 
 [Sec. 11-16 
 
 the conditions of actual design, and it seems highly improbable that any method will be devised 
 that will give more than a close approximation, which the present methods of empirical design 
 undoubtedly give. The reason for this lies in the materials which go to make up the con- 
 struction. Steel is a thoroughly tested material and may be secured with well-determined 
 physical properties. The quality of concrete, however, is far more uncertain and its physical 
 properties can only be controlled within certain limits and cannot be accurately forecast. 
 Even if samples are taken and tested, it cannot be safely predicted that the quality of the mate- 
 rial in the structure will agree with that in the sample to the degree of accuracy that a theoretical 
 analysis would require. Further than this, there are shrinkage and temperature stresses set 
 up in any structure having considerable area, and these stresses are not subject to accurate de- 
 termination beforehand since the methods of construction cannot be always known in advance. 
 
 Flat-slab floors, as ordinarily designed, are figured to carry uniform loads or, at most, 
 only moderate concentrations, such as light partitions, etc. When concentrated loads must 
 be carried in addition to the uniform live load, it is common to introduce beams for this purpose. 
 These may be either of customary sections or they may be of the wide and shallow type involv- 
 ing only a small loss in clear story height of the building. The same practice is followed with 
 reference to openings of stairs, etc. 
 
 The calculation of beams of this class involve many considerations. Under some condi- 
 tions, safe results will be secured if the beam is computed for the concentrated load alone, but 
 usually a portion of the live and dead load of the slab should be included. Where the con- 
 tinuity of the slab is destroyed by an opening, the marginal beams should be figured to carry 
 their share of the floor in addition to the concentrated loads and, of course, their own weight. 
 
 When necessary to secure drainage, flat-slab floors or roofs may be pitched and this is 
 common practice in warehouse construction. When it is necessary to introduce steps or sudden 
 changes of slope, however, special treatment will be required. 
 
 Probably the great majority of floors of this type are laid with a cement flnish. If this is 
 laid at the same time or very shortly after the pouring of the structural slab and before the 
 concrete has taken its flnal set, it may be safely considered as a part of same and the reinforcing 
 steel so calculated. In the absence of positive information to this effect, it is better to include 
 the weight of the finish but not its thickness in the computations for reinforcement, for in the 
 majority of cases the finish is laid at some later time and does not bond with the floor slab. 
 
 Exterior columns, particularly in the upper stories of buildings, require special reinforce- 
 ment to resist bending. Interior columns are also subject to bending stresses but usually they 
 are more heavily reinforced than the exterior columns and, further, the slab on the unloaded 
 side of the column acts to a certain extent to relieve the bending in the column. While this 
 point should be given due consideration it is seldom that the interior columns need extra steel 
 if the recommendations of the Chicago Code or the American Concrete Institute (A. C. I.), as 
 regards size, are followed. The interior columns under the roof are really very unlikely to 
 be subjected to unbalanced live load and in the lower stories where it may occur, the direct 
 load and column diameter are both greater, tending to reduce the bending effect. 
 
 A majority of the designs of flat-slab structures have been made by specialists who have 
 made a special study of this particular form of construction. It is well that such is the case, 
 for the subject is a specialty and, while simple enough in many cases, the production of a safe 
 and economical design is a task that demands a skill not quickly acquired. It is manifestly 
 impossible in a book of this character to embody and arrange sufficient information so that 
 any one could proceed with safety with its aid to lay out any and every problem, and such is 
 not its purpose. 
 
 Failures of flat-slab structures have occurred and other structures which have so far es- 
 eaped will certainly give trouble if they ever receive their full designed load. Causes for such 
 shameful occurrence are usually easily to be found. The first and probably most common cause 
 is a fierce commercial competition which, in the absence of strict building code requirements, 
 has led unscrupulous designers to place a sublime confidence in thin sections which no tests can 
 
Sec. 11-17] 
 
 BUILDINGS 
 
 461 
 
 justify. Another cause has been poor construction which, of course, is fatal to any design but 
 probably in no class of construction is first-class concrete work more essential to success. A 
 third contributing cause has been gross ignorance. 
 
 It is the part of wisdom and true economy, therefore, to entrust work of this class only 
 to those who have had experience under a competent designer or, better still, to have the design 
 made by or reviewed by a consulting engineer who has successful constructions of this class to 
 his credit. 
 
 17. Systems. — A number of systems of designing and reinforcing flat-slab floors have been 
 developed which, so far as the external appearance of the finished building is concerned, are 
 very similar but differ radically in structural features and in method of design. They may be 
 divided into four general classes with regard to the method of reinforcement which, stated in 
 the order of the total number of buildings constructed of each class, are: (1) The four-way 
 system; (2) the two-way system; (3) the circumferential system; and (4) the three-way system. 
 
 In the four-way system the slab may be designed either of a uniform thickness, or drop 
 panels may be used at the column capitals. The reinforcement is placed in four direct bands 
 running in two directions and two diagonal bands which pass diagonally across the panel from 
 column to column. In this system the center line of each band passes over the center of the 
 supporting columns. A portion of the rods are usually bent up over the column capital. 
 Several systems of this type do not follow the rule in this respect, however. Additional trans- 
 verse steel is placed in the top of the slab over the direct bands in many cases. 
 
 In the two-way system the reinforcement is all placed in two directions. The direct bands 
 of reinforcement are carried from column to column and the rectangular area remaining is rein- 
 forced in both directions parallel to the direct bands by similar bars which pass across them. 
 It is common practice in this system to carry the bars of the direct bands in the top of the slab 
 at the column head and in the bottom of the slab at the center of span. Also the slab bars are 
 commonly bent up where they pass over the direct bands. 
 
 The circumferential system makes use of both radial and circumferential reinforcement 
 around the column head. The region between column heads which is commonly occupied 
 by the direct bands is also reinforced with concentric rings which overlap with those around 
 the column head. Finally, the slab in the center of the panel is reinforced in a similar manner. 
 
 The three-way system requires a special arrangement of columns and, while it possesses 
 certain theoretical advantages over the four-way and two-way systems, its use so far has been 
 comparatively limited. In this system the interior columns are placed in such a manner that 
 the lines connecting their center lines form equilateral triangles. By this arrangement all the 
 bands of reinforcement have equal spans and all pass over the column heads. 
 
 The majority of the systems herein described are operating under license from the Flat 
 Slab Patents Co. of Chicago and are protected by special detail patents. 
 
 The more important systems will now be described in detail. 
 
 17a. Barton Spider Web System. — The Barton Spider Web system is similar 
 to other flat-slab systems as to the arrangement of columns, column heads, and drop panels, 
 ' but differs radically in the type of reinforcement used. As regards the slab, it is a four- way 
 system and over the column head it is a two-way system. It will be seen by referring to Fig. 41 
 that the slab reinforcement is made up of straight rods of small diameter which run from column 
 to column and are not continuous. The negative reinforcement over the column head con- 
 sists of two systems of bent rods at right angles to each other which are placed in the top of 
 slab and have their looped ends bent down. 
 
 Two methods of fabricating this steel are in use. In the first, that shown in Fig. 41, the 
 column-head steel is in the form of a fabricated mattress of bars in one direction which comes 
 to the job bent and ready to place. In this case the mat serves to space and support the floor- 
 slab reinforcement. In the other type the column head steel consists of loose bars supported from 
 the forms by blocks of concrete cast in the field. A zig-zag stirrup hangs from these bars and 
 
462 
 
 CONCRETE ENGINEERS' HANDBOOK 
 
 [Sec. ll-17a 
 
 supports the ends of the direct belts of steel in the bottom of the slab. The negative moment 
 in the slab across the line between columns is taken care of by a belt of steel in the top of the 
 slab at right angles to and over the direct belt. 
 
 The designers of this system recognize the necessity of additional vertical reinforcement 
 in the exterior columns to resist bending, particularly in the upper stories. 
 
 In order to govern the design, it is assumed that the loads will seek to enter the columns by 
 the shortest possible route and will, therefore, cause a radiation of stress out from the column in 
 all directions. Also, it is assumed that these stresses will pass through a point of inflection 
 on account of the slab being fixed. This point of contrafiexure cannot be definitely located, 
 inasmuch as variations in the loading, such as naturally occurs on all floors, will cause the point 
 to shift either toward or away from the columns. 
 
 To meet the above conditions the Barton Spider Web system is a four-way system in 
 which all the major slab reinforcement radiates from the column head and also is composed 
 of separate units of negative (column head) and positive (slab center) steel, which lap past 
 
 Fig. 41. 
 
 each other so as to provide steel at top and bottom of the slab in the region of the shiftin 
 point of inflection. 
 
 So far as the methods of arriving at bending moments in the different belts are concerned 
 the engineers of the Barton Spider Web system believe the American Society method to b 
 the most mathematically correct, but that their coefficients are needlessly conservative. The 
 believe the Chicago Code to be the most satisfactory for practical use and, as every test made o 
 these floors in that city gave a deflection of less than H200 of the span for a test load o 
 twice the live load plus a superimposed dead load and showed perfect elasticity, it is felt tha 
 the Chicago coefficients are amply safe. The Cleveland Code with respect to columns i 
 favored above others by the designers of this system. 
 
 Fig. 39 is a typical interior of a building constructed according to this system. 
 
 Fig. 42 is copied from the working plans of a building designed under this system and 
 illustrates its practical application. Much labor is saved on drawings by the method of lettering 
 bands here adopted. The method of framing openings is by the use of wide, shallow beams, 
 which method is quite commonly used by other designers. 
 
 For cuts and descriptive matter, the writer is indebted to the Barton Spider Web system. 
 
Sec. 11-176] 
 
 BUILDINGS 
 
 463 
 
 176. Cantilever Flat-slab Construction. — Flat-slab floors are designed by the 
 Concrete Steel Products Co., Consulting Engineers, Chicago, under the trade name of ''Canti- 
 
 1-> 
 
 III r , 
 
 shm 
 
 ^A^^// y^ef i 
 
 MM 
 
 Second floor slab s-hsel 
 
 
 'of 
 
 m 
 
 Detail of be! 
 
 t 
 
 
 Rentarks 
 
 Nod 
 
 Zize 
 
 Le^ 
 
 Hi 
 
 a 
 
 19 
 
 
 
 e'o' 
 
 
 T-6' 
 
 
 b 
 
 IE 
 
 5 
 
 
 i2'-if 
 
 
 T-6' 
 
 
 6 
 
 V 
 
 i^ff 
 
 F, 
 
 c 
 
 15 
 
 3 
 
 
 It-O' 
 
 
 I'-e' 
 
 
 d 
 
 II 
 
 5 
 
 
 ib'-e 
 
 
 1-6" 
 
 
 e 
 
 18 
 
 8 
 
 f 
 
 
 
 f-ff 
 
 
 f 
 
 
 9 
 
 
 ^o'-o- 
 
 
 
 
 0 
 
 ^ 
 
 
 t 
 
 iz'o' 
 
 
 3'- 6* 
 
 
 
 35 
 
 9 
 
 a.- 
 
 9-0- 
 
 
 d'-o' 
 
 top bars 
 
 I I n n n l^o^'^- The last leHer 
 Vat A W\T\\ of the mark ofa/ayer 
 I I I U U K o/^ column headshel 
 This indir.qf^5 map" gives ihe locafion oT 
 of Sonne B indicates boffom layer -'T 
 indicarl'es fop layer All layers or column 
 .head slwel or mats are called ibr like 
 Cffher belts of bars, i.e. al riqhf anafes 
 with bars. ^ 
 
 JpB^'rs"^ spBars Typicql slab steel layout 
 
 Section "X-X* 
 
 Fig. 42. 
 
 Fig. 43. 
 
 lever Flat-slab Construction." In their designing, use is made of either a true flat slab or a 
 slab stiffened by the use of drop panels around the column capitals as is done with other systems. 
 
464 
 
 CONCRETE ENGINEERS' HANDBOOK 
 
 [Sec. ll-17fc 
 
 Earlier designs prepared by this company made use of radial rods, rings around the column 
 heads, and column rods bent down into the slab ; but as extensometer tests proved these to be 
 ineflScient, their use has been discontinued in later work. 
 
 The system of reinforcement commonly employed is the four- way system which was 
 developed and perfected by the engineers of this company. On account of the greater stiffness 
 and economy of materials secured, the drop-panel type of floor is always favored, although 
 many floors have been designed and built of the true flat type. 
 
 Section A-B 
 
 Fig. 44. 
 
 Great stress is laid on the accurate placing and anchoring of the reinforcement in place. 
 The negative reinforcement passing over the column head is supported by head rods which in 
 turn are carried on concrete blocks and in the center of span the accurate spacing of bars is 
 maintained by bar spacers, tying devices, or wiring. A view of a typical floor is shown iii Fig. 43. 
 
 Hundreds of buildings have been constructed according to this system and a number of 
 these located in Chicago have been tested with very satisfactory results (see Art. 19). 
 
 For information and cut the writer is indebted to the Concrete Steel Products Co. of 
 Chicago. 
 
Sec. ll-17cl 
 
 BUILDINGS 
 
 465 
 
 17c. Simplex System. — The Simplex system of flat-slab construction developed 
 by the Concrete Steel Co. of New York is a four-way system with certain added refinements 
 in the way of devices for anchoring the rods in place (see Figs. 44 and 45). It is designed on 
 conservative Hues — particularly with respect to the stresses on the concrete which are not 
 allowed to exceed the New York City Building Code requirements. In the great majority 
 of cases a drop head is used around the column capital. The steel reinforcement is commonly 
 calculated on the basis of the Pittsburgh or Chicago Ruling, although it may be designed to meet 
 the requirements of any building code. In the design of the interior and exterior columns 
 the Chicago Ruling is used. 
 
 0^ (D, ^ Headra/.. 0 
 
 METHOD OF PLACING 
 FLAT SLAB STEEL 
 
 OPERATION 
 
 No. I. P/ace head rod supports using eigh-h Hy- Chairs 
 
 per inferior column. 
 No. 2. Place head rods on supporfs. 
 No.a Half way befween columns.place Slab Spacer 
 
 for short cross band bars. 
 No A Place shorf cross band bars being carefiil 
 fo'hickey'fhem so fhai- fhey lie fled across 
 head rods. 
 
 Na5. Half way beftveen columns place Slab Spacer 
 
 for long crossband bars 
 No.6. Place long cross band bars; being careful fo 
 'hickey'fhem fo lie flaf across shorf cross 
 band bars over columns. 
 No. 7. Tie long and shorf cross band bars wifh No. I 
 
 Bar-Tys af iheir infer secf ion over columns. 
 No. 8. Place diagonal band bars, being carefi/l fo 
 "hickey'fhem fo lie flaf across long crossband 
 bars over columns.. 
 No.9. Place second diagonal band bars bein^ car^ 
 ful fa 'hickey"fhem so fhai fop offhese ban 
 is I' below fop ofcancrefe slab. 
 No. 10. Tie and support diagonal band bars rrifh 
 Slab Spacer af fheir infersecfion af cenfsrof 
 panel 
 
 Note - Ty-Chairs, ten in number may be used in 
 operation No. 10 instead or Slab Spacer 
 
 KEY. 
 
 ■I denotes Hy- Chair, 
 ^-v-- denotes Slab Spacer, 
 
 « denotes No.t. Bar-Ty, 
 
 o demfea Ty- Chair, 
 
 FLAT SLAB CONSTRUCTION 
 
 COncrelte: steel co. 
 
 Fig 45. 
 
 The concrete sizes to use with the system for the Pittsburgh and Chicago Rulings have 
 been computed and are given in the tables of Art. 21. The dimensions there given apply only 
 to square interior panels. Exterior panels may require special treatment. 
 
 The method of placing the steel reinforcement is explained in detail in Fig. 45. In practice 
 it is frequently found that the weight of the rods themselves will give them sufficient sag so 
 that Uttle or no bending is required. 
 
 This system has been used very extensively and with great success and, since the policy 
 of its sponsors has always been a conservative one, it has never failed to give satisfaction. 
 
 VJd. The Mushroom System. — The Mushroom system was developed by C. A. 
 P. Turner and was in very extensive use until quite recently. Further use of the system, 
 unless licensed by the Flat Slab Patents Co., was prevented by order of Court on account of 
 alleged infringement of the Norcross patent (see Art. 18). 
 30 
 
466 
 
 CONCRETE ENGINEERS' HANDBOOK 
 
 [Sec. ll-17e 
 
 The system, as originally developed, was a four-way system with certain added features. 
 In some cases the column bars were bent down into the slab around the column head to form 
 a radial reinforcement. In other cases special elbow rods were inserted in the column head 
 for the same purpose. A series of ring rods, spaced by radials forming a spider, were also used 
 as a reinforcement at the column head and as a support to the slab steel. In some cases a flat 
 spiral was used in the same position. The use of this system has been pushed with much 
 business enterprise, but the severe service to which this class of buildings is almost sure to be 
 subjected has shown beyond question, that the concrete sections used in many of the earher 
 
 N i6 xSXoncrefe block 
 to suppo rt arv ie bar j'/^/rj/ii 
 
 Formulae ■•-('^or bending moTients in foot pounds) 
 Negafive bending moment at A (to be resisted by one cross belt and i 
 fhsitive bending moment at B (to be resisted by one cross belt. 
 
 Negative bending moment at B (to tie resisted By one center belt)- ■■ -^/^ 
 
 Positive bending moment at C[to be resisted Joy one diagonal Mt) w 
 
 KT ' total had per sq ft (bath dead and live) 
 
 L- Side ofsq. panel{in feet). L-mean of sides in rectangular panels (in feet) 
 When the long side exceeds the short side by more than 10%. the steel in the 
 belts shall be proportioned according to the cutie of the rario of the side of 
 thp panel to the side of the assumed square panel 
 
 Fig. 46. 
 
 Li 
 
 Note-.' h- 
 
 { Belt bars notsupport- 
 \ ed directly ontbrmi-^ 
 \ bars are to be win}^ 
 to the first bar, aboH 
 
 i'xC'CorKmte 
 
 All belt bars to be 
 depressed atthecenler 
 
 aepi 
 
 of the panel so as to 
 
 . , . - provide 4' concrete 
 
 -i or below which is so Section thru in the clear, below 
 
 concrete 
 
 . _ ^ iki ■ ■ 
 
 I supported. column head ^^f- 
 
 i Provide a sufficient nunber of concrete blocks to insure 
 ! against excessive deflection of the forming ban 
 
 J '^fprminij 
 
 6W Concrete block 
 f ^rCenter belt JsXkS!^ 
 
 Width of all bona 
 
 Section taken on center line 
 of bay showing center 
 belta 
 
 Order of placing steel 
 Belts are designated as shown on plan 
 
 1. Lay circular form bars 
 
 2. Lay all cross belts running in one direction 
 
 3. Lay all cross belts running in opposite direction 
 A. Lay all diagonal belts running in one direction 
 5. Lay all diagonal belts running in 'opposite diPBcffon 
 ^- '^^^ form bars for center belts 
 7. Lay all center belts running in one direction 
 
 A Q Lay all center belts running in opposite direction 
 ® The cross belts and diagonal belts are in the top or 
 the slab over the colunnn head, and in the bottom of the 
 slab between the columns. The center belts are in 
 the top of the slab over the cross belts an(J In the bof^ 
 torn of the slab over the diagonal belts 
 5+andard four-way flat slab construction 
 
 designs were entirely too light, since sagging and cracking of floors have resulted. The right 
 requirements of the various building codes are intended to and do remedy this condition, so 
 that now all systems must compete on a substantially uniform basis. 
 
 Vie. Watson System. — A modified type of the four-way system is that developed 
 by Wilbur J. Watson & Co. of Cleveland, Ohio. This system differs from the standard four- 
 way system in the introduction of center belts or bands of reinforcement which do not pass 
 over the column heads, in addition to the usual direct and diagonal bands. 
 
 The details of this system together with the methods of computation and methods of 
 placing are clearly shown in Fig. 46. 
 
Sec. 11-17/] 
 
 BUILDINGS 
 
 467 
 
 17/. Akme System. — The Akme system of girderless-floor construction was 
 developed by the Condron Co., Structural Engineers of Chicago. It is a two-way system of 
 very simple construction and has had a wide and successful use. 
 
 When building codes are in use which have special rulings regarding flat-slab floors, the 
 system is designed to meet their requirements but, in order to govern design where such rulings 
 do not apply, the Condron Co. have prepared a set of instructions which are given below. 
 These are to be used in connection with design standards (see Figs. 47 and 48). 
 
 Typical square panet 
 
 Main Dand i 
 
 P-or>35i 
 
 'Mid band 
 
 "Spain of mid 
 Slab 
 
 ^yercpTunihj, 
 
 I tomp. wiTfh for\\ 
 \mginbcin iafcenk[\ 
 
 Typical recfangular g:inel 
 
 C=or>./ 
 
 'A 
 
 Moments sauare panels Moments rectangular panels 
 
 Moment mam bandar cof. =^ =• M Compute moments forsquare panel with L ''La 
 Moment mainbandat center=^=Mc T^tl^Scj^et^c = Moments long span. 
 Moment mid band at center -^j^--m i^M &-^>Mc = f^oments^ sport span. 
 For determining moments at olher --/i ((^^)u^Gs = iz ■(^^') = Moment of long span 
 sections for mam bands assume rn,=jl(^\ufG,Gf=^^(j^)^Moment of short span 
 line of contra-flexure 0.2S from J ^^'^^^'^n '^^^^t^'^ , ^ 
 capital and a stmightlitle variation ^^^"^^^ allowable punching shear =100 Ib.persq.in. 
 betmen edge cfcap and this line. ^T^^^'^ '^'ff^? ^1% n.,a,v- , < . 
 
 •C^rs.ncLed.ithmain bands. 'jTml^^^/^t^^^^^^^^^ 
 
 -4- 
 
 1^ 
 
 K 
 
 I 
 
 roof 
 
 :^,al5omln = 6'' 
 
 Approximate length of bar L+0.075^-3" 
 3"= approximately amount taken up by bend 
 'd"= nominal size of bar Stagger =0.075 
 'C'= tvidth of plate. Dimensions gi/en are apprvximode and 
 may be changed slightly to sum conditions 
 
 U— H 
 
 uf=(LL+D0per5q.ft 
 W=(LL+DL)(L'-C')i^ 
 if head is not square use 
 C=5ide of equivalent square. 
 5= L-C= Clear span. 
 G = Clear' span behveen main bands 
 L/r^''^= Average span. 
 Sa<L,-C) 
 Recommended unit stresses 
 750 lb per sq in for. concrete with 
 ultimate strength ofdftOO 
 18, 000 lb. per sq. in tor steel with elas- 
 tic limit of ar least 50,000 lb.pers(j. in 
 
 ■"JiolQ^ \0p5 
 
 — > 
 
 Fig. 47. — Akme system. Drop panel type. Design standards, 
 
 ■0015 0.145-M..^. 
 
 <- Line of col. capital- 
 
 Condron Co. 
 
 RULES FOR THE DESIGN OF GIRDERLESS FLOORS 
 
 (To Accompany Akme Design Standards) 
 
 The term girderless floors as herein used refers to flat slabs of uniform or varying thickness supported without 
 beams or girders on columns having flaring heads. 
 
 Flat-slab Type. — In this type the slab thickness is uniform between column heads. 
 
 Drop-panel Type. — In this type the lower face of the slab is dropped so as to increase the thickness of the slab 
 above the column head. The lateral dimensions of this portion of the slab, which is usually made square, should 
 be not less than 0.35L. 
 
 Paneled-ceiling Type. — This type may conform in general to either of the above types with the exception that 
 the slab is reduced in thickness in the central portion of the panel. 
 
468 
 
 CONCRETE ENGINEERS' HANDBOOK 
 
 [Sec. 11-17/ 
 
 Columns. — The diameter or side of any interior concrete column shall be not less than one-thirteenth of the 
 panel length or one-twelfth of the clear story height, except that for columns supporting roofs only this dimension 
 shall be not less than one-fifteenth of the panel length. In any case the diameter or side of the column shall be 
 not less than 12 in. 
 
 Bending in Columns. — Exterior or wall columns supporting floors or roofs shall be designed to resist, in addi- 
 tion to direct load, 40 % of the negative bending moment for exterior floor panels or 80 % for exterior roof panels. 
 
 Column Head. — The diameter of the column head, measured where it intersects the underside of the slab, 
 should be approximately 0.235L, but may vary to suit conditions. It shall have a vertical face below the slab of 
 IH in., below which the surface of the head shall have a slope of 45 deg. to the vertical face of the column shaft. 
 If other shapes of column head are used, the surface of the same shall nowhere fall inside of the surface of the 
 above-defined conical head. Heads may be round, octagonal, or square. 
 
 Typ/cal square panel 
 
 1_ 
 
 lypical necf angular parie\ 
 
 '7X: 
 
 -f- 
 
 Main band T WW i \ y^dMn^^^^ 
 
 Comp- widfl^lxincif-4t 
 maxoffL] 
 
 , G 
 
 Span of mid 
 
 slab 
 
 Si 
 
 C3: 
 
 Comp widfh over col. 
 
 t)and+4fbutnof 
 
 r 
 
 TL 
 
 .omp. wia 
 "'n ml/7 1 
 
 it; 
 
 \mmo 
 
 -K 
 
 Hi:— 
 
 Main^band \ 
 Wiolfh ofmainbar^0.4Ls 
 
 Midband 
 
 O.EL> 
 
 [< -- Ls^Shorf dimension 5^ 
 
 Moments rectangular panels 
 
 Momenfs square panels 
 
 Momenl main band a f col.=j^=M 
 Moment main band afcenfer=j^,-- 
 Moment mid band af- center = -^^^ 
 fbr determining moments af other 
 
 sections for main bands assume ^ wq^/^f =Momentof5ho-tspan 
 
 ine of contra- fexure 0.25 from a/ ^^Z*^^' v J '^^^^t^L^ , — >r— 
 capita/ anda straight line. anatlon ^frn.TnTrnf^^^^^^^ 
 
 ""rn'^^^'V'^ 7nTmtAr sL^^^^^^^^^^ Recommend.dunitstr.5ses 
 C bars included with mam bands. ""'"T"" T =%,a/somin ^6^ 750ib.per5a.ln for concr^e mth 
 
 Thickness of plate =.4f with a minimum of 4" ultimate strength of^fiOOIbpersqin. 
 
 18,000 lb. per so. in. for steel with elas- 
 
 Approximate length of bar L^0.075-f3" tic limit of at least 50,000 Ib.persq.in. 
 
 ur=(LL-/-DL)per3a.ft 
 
 W=(LLWL)(L^-C9^2 
 
 if head is not square use 
 
 C= Side of equivalent square. 
 
 5 = L-C = C/earspan. 
 
 G=Clear spanbetween main bands 
 
 L/'f''^^ Average span. 
 
 3"= approximately amount taken up by bend 
 Y= nominal size of bar Stagger =0.075 
 "0"=^ widtb'of plate. Dimensions giyen are approx/maf& and 
 may be changed slightly to surf conditions. 
 
 \.o.eL 
 
 f^^- '^ Une of col capital • ■ ->l 
 
 -S 
 
 Fig. 48. — Akme system. Flat slab design standards, Condron Co. 
 
 If round or octagonal heads are used, the diameter of head to be used in the slab calculations shall be the side 
 of an equivalent square. Where a square plate is used as part of the column head and its lateral dimension is 
 within the 45 deg. slope of the conical head, the size of said square plate shall be used as the diameter of column 
 head in making slab calculations, provided the thickness of said plate is equal to or greater than one-half the thick- 
 ness of the slab and not less than 4 in. 
 
 Slab Thickness. — The minimum thickness of the slab (except in paneled-ceiling type) shall be not less than 
 
 ^ for floors and ^ for roofs, nor less than given by the following formula: 
 
 t = O.OIQLVu' + in. 
 where t = total slab thickness in inches; L = panel length in feet; and w 
 per square foot. 
 
 total live and dead load in pounds 
 
Sec. 11-17/] 
 
 BUILDINGS 
 
 469 
 
 In the paneled-ceiling type the thickness of the enclosed panel shall be not less than one thirty-seoond of its 
 clear span. 
 
 Drop Panel. — The depth of drop panel where used shall be determined by using its width at the section 
 considered as the full width to resist compression resulting from negative moment. 
 
 Panel Strips — For purposes of computation each panel of the slab is to be divided into two sets of strips called 
 A (main slab strips) and B (mid-slab strips). Strips A extend from column to column and have a width equal to 
 
 — , and strips B occupy the space between strips A, and likewise have a width of 
 
 Reinforcement in strips A shall be placed symmetrically about column centers for a width of approximately 
 0.4L at mid-span and approximately 0.5L over columns. The width for compression shall be taken as the width of 
 the belts of reinforcement, plus 4 times the thickness of the slab, but shall not exceed YzL. The width of main 
 belts of reinforcement over the columns shall not exceed twice the width of the column head. 
 
 B ending-moment Coefficients, Interior Panels. — For the flat-slab type the negative bending moment taken at 
 
 WS 
 
 a cross-section of each strip A at the edge of a column head shall be The positive bending moment taken 
 
 at a cross-section of each strip A midway between column supports shall be "jg" The positive and negative 
 
 WS 
 18 
 
 bending moments taken at a cross-section of each strip B at the middle of the panel on the center line of columns, 
 
 w 
 
 respectively, shall be • 
 
 WS WS 
 
 For the drop-panel type the corresponding moments at the above-mentioned section shall be ~2o' 
 and j^- 
 
 For paneled-ceiling type the moment coefficients shall be the same as for the flat-slab type. 
 
 For determining moments at other sections of main strips A, the line of contraflexure shall be assumed to be 
 
 at a distance equal to from the center of column, with a straight-line variation moment between the edge of the 
 
 head and the said line of contraflexure. 
 
 In the above W = one-half total live and dead load on the panel, exclusive of the area over the column head ; 
 S = the clear span in feet between column heads; w = total live and dead load per square foot; and G = the 
 clear distance in feet between main belts of bars at the section midway between columns. 
 
 B ending-moment Coeffi,cients, Exterior Panels. — For exterior panels without cantilever overhang, where wall 
 columns with flaring heads or brackets are used, and for other spans not continuous over both supports, the positive 
 bending moment coefficients shall be increased 20%. 
 
 When bearing walls or piers and girders are substituted for the above wall columns with flaring heads or 
 brackets, compute the moments for the exterior panels of such construction by assuming the distance from the face 
 of column head to inside face of wall or girder as S; and the distance between the first interior main belt and the 
 inside face of wall or girder as G. 
 
 Oblong Panels. — For oblong panels the moments shall first be determined for an assumed square panel with 
 sides equal to the mean of the length and breadth of the oblong panel. The moments thus found for strips A 
 shall be multiplied by the ratio of the square of the span in question and the square of the span of the assumed 
 square panel, and the moments thus found used in determining the steel required in strips A. 
 
 The moments for strips B shall be computed as follows: The load carried by the long and short span strips B 
 shall be in the proportion of the ratio of the square of the short span to sum of squares of long and short spans and 
 the ratio of the square of the long span to sum of squares of long and short spans respectively. The moments shall 
 
 then be found as for square panel using the proportion of w carried by the span in question instead of 
 
 When the length of panel does not exceed the breadth by more than 5 % , all computations may be made on 
 the basis of a square with sides equal to the mean of the length and breadth. The rules given herein shall not be 
 used for rectangular panels in which the length exceeds four-thirds of the breadth, but special consideration shall 
 be given to such cases. 
 
 Stresses in Steel and Concrete. — The stresses shall be calculated on the basis of the straight-line formula, 
 neglecting the tension value of the concrete. The depth of the slab for calculation of stresses shall be taken as the 
 distance from the compressive face to the center of gravity of the belt of reinforcement in a given strip. The 
 tensile stress in steel reinforcement should not exceed 16,000 lb. per sq. in. for structural-steel grade nor 18,000 
 lb. per sq. in. for cold-twisted or high-carbon deformed bars. The maximum allowable compression in the concrete 
 shall not exceed 750 lb. per sq. in. The allowable punching shear on the perimeter of the column head shall not 
 exceed 100 lb. per sq. in. Where governing ordinances or laws require lower allowable unit stresses, such unit 
 stresses shall be substituted for the above. 
 
 Walls and Openings. — Where necessary, slabs shall be thickened or girders or beams shall be used under walls 
 and around openings to carry concentrated loads. 
 
 Placing of Reinforcement. — Reinforcement shall be rigidly held in its designed position while pouring con- 
 crete. The bars in the upper portion of the slab should be rigidly supported by frames or transverse bars resting 
 on concrete blocks of proper height. Bars in the lower portion of the slab should be raised from the forms and 
 
470 
 
 CONCRETE ENGINEERS' HANDBOOK 
 
 [Sec. ll-17j^ 
 
 held ih proper position, preferably by a continuous combined spacing and raising device. The lateral spacing of 
 bars shall not exceed lyi times the thickness of the slab, nor more than 12 in. 
 
 Bars shall be bent to conform to the bending diagrams shown in Figs. 47 and 48 and shall be so placed in the 
 slab that they will not be nearer than % in. from the face of the concrete. 
 
 A great number of buildings have been constructed under this system and many of them 
 have been tested with very satisfactory results. Fig. 49 shows one of these floors during the 
 steel-placing process in which the simplicity of the arrangement is apparent. For data and 
 cuts the writer is indebted to the Condron Co. of Chicago. 
 
 Fig. 49, 
 
 llg. Corr -plate Floors. — A type of flat-slab floor known commercially as the 
 Corr-plate floor, developed by the Corrugated Bar Co. of Buffalo, N. Y., is having a very wide 
 and successful use. 
 
 The general features of the slab and columns as used with this system are similar to those 
 used with other types, and it may be designed either with or without drop heads. When 
 drop heads are used, they are frequently made somewhat shallower than is the practice of 
 other designers. In other respects there is little to distinguish the system from others so far 
 as the exterior appearance goes. 
 
 It has been the practice of the engineers of this system to give due consideration to the 
 reinforcement of the exterior columns to resist their share of the bending, the necessity of which 
 has been demonstrated by numerous load tests of actual buildings. 
 
 The method of reinforcement used is that known as the two-way system and was developed 
 primarily from a series of very interesting tests performed upon a small model. For information 
 regarding this the reader is referred to Art. 19. 
 
 The practice of the designers of this system has been somewhat modified as the result 
 of tests which have been made on actual Corr-plate floors but only as regards minor details, 
 the original findings having been demonstrated to be substantially correct. 
 
 With regard to the design of exterior columns and lintel beams, the engineers of the Cor- 
 rugated Bar Co. feel that a universal practice should be laid down which should be followed by 
 all, such as the recommendations of the American Concrete Institute or the Special Committee 
 of the A. S. C. E. This system may be designed to meet the requirements of any code. 
 
 Among its advantages are the use of bars of moderate length and extending over one span 
 only (Fig. 49 A). It is also common practice to use bars of a larger diameter than is customary 
 with other systems, thus giving a saving in unit price for slab reinforcement. The slab reinforce- 
 
Sec. 11-17A] 
 
 BUILDINGS 
 
 471 
 
 ment extends in two directions at right angles to each other. A portion of the rods in the center 
 of the slab are bent up over the center line of columns so that extra rods are not needed at this 
 point in the top of slab. 
 
 llh. S-M-I System. — This system, more commonly known as the Smulski 
 system, was invented and patented by Edward Smulski and is the best-known if not the only 
 type of circumferential system. Its introduction is of very recent date but it has been quite 
 extensively used, particularly in certain portions of the Eastern States. 
 
 The feature which differentiates this system from others that have been described is the 
 arrangement of the reinforcement for, as regards the slab, the proportions are much the same 
 in all systems. Use is made of circumferential and radial reinforcement in both top and bottom 
 of the slab with only a small amount of steel passing from column to column. For the following 
 description and theoretical discussion the writer is indebted to the inventor of the system : 
 
 Fig. 49A. 
 
 Description of the System 
 
 The reinforcement of a typical interior panel, fully illustrated in Figs. 50 and 51, consists of three types of 
 
 units. 
 
 1. Unit C at the column head composed of cings and radial bars in the shape of hair pins, the upper prong of 
 which resists tension while the lower prong resists compression (see Fig. 52). 
 
 2. Unit A between columns consisting of two trussed bars and rings. 
 
 3. Unit B in central portion consisting of four diagonal trussed bars and rings. 
 Units T are sometimes used as shown in "Top Reinforcement," Fig. 50. 
 
 The radial bars are provided with a semicircular hook of sufficient dimensions to transfer the stresses into 
 the concrete by bond and bearing. The center ring which they sometimes engage keeps them in place and forms 
 an additional factor of safety. 
 
 The trussed bars of Units A and B are bent up near the points of inflection and carried near the top and 
 parallel to the surface of the slab to the column head, where they engage the center ring. The bent portion resists 
 shear and binds the column-head section to the rest of the slab. The straight portion of the trussed bars in the 
 center of the slab and at the column head resists tension due to the positive and negative bending moments 
 reBpectively. 
 
472 
 
 CONCRETE ENGINEERS' HANDBOOK 
 
 [Sec ll'-VJk 
 
 I / 5ec+ion thru column hed 
 Top reinforcement 
 
 Fig. 50. 
 
 >1 Plan , 
 If required unit T can placed over unmA 
 
 Section thru column hea 
 
 Typical panel 
 
 showing top and bottom re^Trforcem^nf 
 . Fig. 51, 
 
Sec. 11-17/?] 
 
 BUILDINGS 
 
 473 
 
 The trussed bar extends into the column head a sufficient distance beyond the point of maximum stress 
 {i.e., the edge of the column head) to develop, in combination with the hook, their full tensile strength. The ring 
 which they engage serves to distribute the bearing stresses laterally on to a large area of concrete. 
 
 Position of Units. — Units A and B are placed near the bottom while unit C is near the top of the slab. 
 
 Compression Reinforcement. — By introduction of compression reinforcement in the shape of lower prongs of 
 the radials, the slab is stiffened at the support, and the compression stresses in concrete reduced. If desired, 
 therefore, it is possible to omit the drop panel at the column head and use an altogether flat ceiling. This is often 
 desirable either for the sake of appearance or to simplify shafting or piping. 
 
 Secondary Reinforcement. — Sometimes to prevent cracks on the top of the slab between columns, additional 
 secondary reinforcement consisting of short gtraight bars, and called Units T, is used. These bars are usually 
 placed after the concrete of the slab is poured. 
 
 Fig. 52. 
 
 Theoretical Discussion 
 
 The scientific basis of the S-M-I system is evident from the following discussion of the action of a flat slab 
 under load. 
 
 Shape of the Slab after Deflection. — After deflection a flat slab assumes a composite shape, namely, the shape 
 of an umbrella at the column head, and the shape of a saucer in the central portion. 
 
 Lines of Equal Deflection. — The shape of a deflected slab can be seen better from Fig. 53, which shows in 
 sections the deflection curve along the side and the diagonal of the panel, and in plan the lines of equal deflection. 
 The lines of equal deflection, which are based on tests, were obtained by connecting the points which deflected 
 an equal distance below their original position. 
 
 Direction of Stresses and Reinforcement. — By referring to the plan and the sections, it is evident that deforma- 
 tion of fibers are equal and therefore the fiber stresses act perpendicularly to the lines of equal deflection as indicated 
 by arrows in Fig. 53. The best method of resisting these stresses, or preventing the deformation, is either by 
 placing the bars perpendicularly to the lines of equal deflection, or by enclosing them by means of a ring, the hooping 
 action of which is explained later. Fig. 54 shows the deflection lines in light dash lines and the reinforcement 
 according to the S-M-I system in heavy lines. The radials and trussed bars are perpendicular to the lines of equal 
 deflection. The rings either intersect the deflection lines at angles close to 90 deg., or they enclose the same and pre- 
 vent the enclosed concrete from spreading. The combination therefore fulfills all the requirements of efficient 
 and economical reinforcement. 
 
474 
 
 CONCRETE ENGINEERS' HANDBOOK 
 
 [Sec. 11-1 7/i 
 
 Continuous Construction Separated into Simple Parts. — In continuous beams and slabs the bending moment 
 at the points of inflection is zero. Therefore, it is possible to separate the structure at these points without chang- 
 
 4 s/.- 'Cb^un^n Mad 
 
 Line of I 
 irrfivcfion h— — ^ 
 
 .V-Li-bL.ti 
 
 I I 1 I J i^r-\ ■Infermedlaf^ lines 
 
 Arrows in sections and p/an 
 indicafff direction of fensile stress 
 
 Fig. 53. 
 
 ing the stresses in the remainder of the structure. This may be accomplished by insertion of a hinge or other 
 connection capable of transferring shear. The same is possible in flat-slab construction. 
 
 Separating Flat Slab into Simple Parts. — As ex- 
 plained above, a flat slab may be separated into 
 simple parts, namely: Circular cantilevers at the 
 column head; slabs between columns; and slabs sup- 
 ported at four points subjected to stresses in all direc- 
 tions. 
 
 In designing the reinforcement it is permissible 
 to treat the separate parts independently and to 
 provide in each of them a sufficient amount of steel 
 to resist the particular bending moments to which 
 they may be subjected. 
 
 The unit shear at the points of inflection" is 
 always low, not exceeding 40 lb. per sq. in., so that 
 concrete is capable of taking care of the shearing 
 stresses. Since it is not advisable to rely on con- 
 crete alone, the parts of the slab subjected to posi- 
 tive bending moment and reinforced by Units A and 
 B are tied securely to the circular cantilever at the 
 column head by the bent portions of the trussed bars 
 and by overlapping of Units A and C. 
 
 The position of the points of inflection is vari- 
 able for different positions of the live load. To 
 provide for this and also to prevent secondary cracks, 
 due to temperature and shrinkage, the reinforcement 
 in the various units overlaps, thereby tying the slab 
 together and enabling it to act as a whole if such ac- 
 tion is required by any contingencies. 
 
 Column-head Section. — At the column head the 
 portion of the slab within the points of inflection acts 
 like a circular cantilever loaded uniformly over its area, and also along its circumference by the loads transferred 
 to it from the rest of the slab. This portion is subject to negative bending moments: i.e., the particles in the 
 upper part of the slab elongate while the particles in the lower part compress. 
 
 For explanai-ion of action of rings see text 
 Heavy lines shov/ reinforcement 
 Light dashed lines show curves of equal deflection, 
 fr^inlbrcement at column head is near loo of slab. 
 
 Fig. 54. 
 
Sec. 11-1 7/i] 
 
 BUILDINGS 
 
 475 
 
 Section "A-A" 
 Fig. 55. 
 
 The negative bending moment at the column head is larger than the positive bending moment in the center 
 of the slab. The amount of steel required there is consequently larger than in any other part of the slab. 
 
 The most unfavorable condition of loading for the column-head section is when all the spans surrounding the 
 column are loaded. In such case the shape of the cantilever will be as shown in Fig. 55. Since after deflection any 
 circle increases its radius as well as its circumference, the par- 
 ticles must elongate in radial as well as in circumferential di- 
 rection and are therefore subjected to radial and circumferen- 
 tial stresses. The most effective tensile reinforcement is by 
 means of rings and radial bars. 
 
 Compressive stresses act also in radial and circumferen- 
 tial direction. The compression acting radially composes 
 the bulk of compressive stresses and may be resisted by the 
 compressive prongs of radial bars in combination with con- 
 crete. The efficiency of steel in resisting compression is well 
 established by tests of Prof. Withey in America and Prof. 
 Bach in Germany. These tests are described in Taylor and 
 Thompson's "Concrete, Plain and Reinforced, 3d Edition. 
 
 Slabs Between Columns. — The principal stresses in this 
 part act mainly in one direction which at first is parallel to the 
 edge of the panel and then gradually becomes inclined, as is 
 evident from Figs. 53 and 54. In addition secondary stresses 
 due to cross-bending and also due to shrinkage and tem- 
 perature changes act across the principal stresses. 
 
 The advantages of using rings in this part to resist the 
 various stresses are as follows: (1) They intersect the lines of 
 equal deflection more nearly at right angles than straight bars 
 (see Fig. 54) ; (2) they bind the Units A and B, thereby pre- 
 venting secondary cracks; (3) the rings in the two units sup- 
 plement each other; and (4) the arrangement is economical 
 as the rings cover the whole surface without waste of material. 
 
 Central Part of Slab. — The central portion acts like a 
 slab supported at four corners and loaded with uniform load. 
 
 The bending moment is positive so that the top is in compression and the bottom in tension. As evident from 
 Fig. 53, the stresses act in all directions; the reinforcement consisting of rings, therefore, is fully effective. 
 
 Action of Rings. — The following discussion demonstrates that rings resist effectively stresses acting within the 
 ring in any direction. 
 
 In considering the action of rings it must be remembered that they are filled with solid concrete which governs 
 
 their shape. The deformation and 
 
 ,fbinf3 £'^-^^^2__^^j^!!;^e^'nf/ecf/on ^ ^ stresses in the rings are caused by the 
 
 ' " " *■ ^~ ' pressure of the concrete on their cir- 
 
 cumference so that the rings assume 
 the shape of the solid concrete which 
 they enclose. 
 
 Fig. 56 shows parts of the slab 
 at the column and also in the center 
 of the slab, subjected to stresses in all 
 directions. AA is a section in any 
 direction. 
 
 Under load the slab compresses 
 near one surface and expands near 
 the other surface so that 0-3 shortens 
 by 3-3', while 0-4 expands by 4-4'. 
 The same is true of a section in any 
 direction; therefore the circle 4-4 tends 
 to assume the shape of 4'-4'. Before 
 assuming the new position the con- 
 crete must stretch the ring by which 
 it is enclosed; i.e., increase its radius 
 and therefore its circumference. The 
 concrete exerts a pressure along the 
 circumference of the ring similar to 
 Since the modulus of elasticity of the steel is different from that of the 
 
 It stretches, how- 
 
 Section "A-A" 
 R/ngs in unit C 
 
 ^ , ^ ^ Section '>\-A" 
 
 4- 4 = elongation of distance 0-4 
 Z- 2'- elongation cf distance 0-Z 
 
 lis 4-4^pli^5 elongation of 
 
 distance 2-4) 
 
 Rings in unit B 
 
 Fig. 56. 
 
 the pressure of water in a reservoir. 
 
 enclosed concrete, the steel ring by its tensile resistance prevents partially the movement, 
 ever, to some extent, causing tensile stresses in the steel. 
 
 Considering the second ring, it is evident that the movement of point 2 consists not only of the elongation 
 
476 
 
 CONCRETE ENGINEERS' HANDBOOK 
 
 [Sec. 11-172 
 
 of the distance 0-4 but also of the elongation of the distance 2-4. The outside ring therefore shares the stresses 
 with the ring inside. Any deformation of the concrete irrespective of its direction is talcen up at once by all the 
 rings placed outside of the place of deformation. All rings are therefore effective in resisting stresses (see also Art. 
 19). 
 
 Forces Acting in All Directions. — Where the forces act in all directions as in the center of the slab and at the 
 column head, the ring stretches uniformly along its circumference. After deformation the shape of the ring remains 
 substantially circular and the stresses are uniform along its circumference. 
 
 Forces Acting Principally in One Direction. — If the forces act principally in one direction, as in unit A, the 
 condition is similar to that of a solid disc of concrete with a tight-fitting steel ring around it, subject to a force in 
 one direction. Under the pressure of the enclosed concrete the shape of the ring changes gradually into an oblong 
 curve with the concrete following and still pressing tightly on the ring. In this case the stresses in the ring are a 
 maximum at the sections cut by a diameter perpendicular to the direction of the stress, and decrease to zero at 
 points 90 deg. from the point of maximum stress. 
 
 From the above it is evident that the stresses due to the principal bending moment are small in the parts of 
 rings of Unit A which are near the column head, so that they can resist stresses in the diagonal direction in places 
 where they run almost parallel to the diagonal trussed bars. 
 
 Fig. 57. 
 
 The advantages claimed for this system by its sponsors are: (1) An economy of steel 
 over other systems; (2) freedom from obstruction over the column head, an advantage when 
 structural-steel column cores are used; and (3) ease of pouring column head and slab due to 
 the large diameter and comparatively small number of rods used. 
 
 The bending of the circular slab rods may be done in a shop and the large rings shipped in 
 two parts with a liberal lap provided, or it may be bent on the job with a form of tire bender. 
 
 This system has been used in the construction of about 70 buildings to date, all of which 
 have given satisfactory service. 
 
 m. Three-way System. — The three-way system was invented and patented 
 (Patent No. 1,064,850) by David W. Morrow, Civil and Architectural Engineer of Cleveland, 
 Ohio, and has been successfully used in buildings for the Cleveland Railway Co. of Cleveland. 
 
 The principal novelty of this design lies in the arrangement of interior columns which are 
 located at the apices of equilateral triangles. Under this arrangement the bands of steel 
 reinforcement are all of equal span. Fig. 57 is a plan of a building laid out on this system, 
 and Fig. 58 shows the detail at the column head for the same design. 
 
Sec ll-17i] 
 
 BUILDINGS 
 
 477 
 
 avoided 
 60 deg., 
 carriers 
 
 possible 
 
 The advantages claimed for this system are as follows: Right-angle turns are 
 and in passing from one aisle to another it is only necessary to turn through an angle of 
 thus making it easy for the running of vehicles or the free movement of overhead 
 handling long material. 
 
 The reinforcing steel over the column head is placed in three layers, thus giving it a 
 slight advantage over the four-way system in effective 
 depth at this point. 
 
 The interior column spacing being equal, the spans 
 and consequently the amount of steel and length of rods 
 in the bands will be equal. 
 
 It is claimed that the three-way system is par- 
 ticularly adapted to garages on account of the ease 
 with which a car may be turned into one of the aisles 
 between columns, the plan being, of course, to park the 
 cars in the diagonal aisles. 
 
 The following data with regard to one of the build- 
 ings designed, according to this system, has been fur- 
 nished by Mr. Morrow and may be useful to those who 
 may wish to compare this type with others. 
 
 The floor, in what is known as the storeroom, is 184 
 ft. long and 120 ft. wide, the columns being spaced 23 
 ft. c. to c, giving 20 ft. wide longitudinal and diagonal 
 aisles. It was designed for a live load of 350 lb. 
 
 The floor slab is WH in. thick reinforced with twenty-one 
 H-in. round rods per band, spaced 6 in. c. to c, all rods lapping 
 over the column head and extending 5 ft. 6 in. beyond the center of 
 columns, the rods being approximately 34 ft. long. 
 
 The floor slab is supported on 20-in. spirally reinforced 
 columns with a flaring cap 5 ft. in diameter, and a hexagonal drop 
 panel 4^^ in. thick and 8 ft. across. The outside edge of the slab 
 
 rests on a 17-in. brick wall with pilasters at column points and is bounded with a depressed slab in. thick and 
 4 ft. wide. 
 
 Buildings designed by Mr. Morrow have been designed in accordance with the Cleveland 
 Building Code and, of course, none of the building codes really cover this system. 
 
 The Cleveland law requires that flat-slab construction shall be figured with a bending 
 
 moment in any quadrant over the column 
 
 head of not less than 
 
 27 
 
 in foot-pounds, in 
 
 Fig. 59. 
 
 which W equals total weight per square foot, 
 dead and live load, and L equals length in feet 
 of a side of an equivalent square in rectangular 
 panels, and the side of a square in square 
 panels. The length L shall be taken center 
 to center of columns. For the three-way 
 system, L is taken as the side of a square which 
 
 has the same area as a parallelogram panel (see Figs. 59a and 596). 
 
 The bending moment of the Cleveland Code in the quadrant over the head is -^^y . There 
 
 fore the bending moment in the sextant = X substituting the value of L in terms 
 
 WP 
 
 of I, we have for the bending moment in the sextant, M = at the column cap. The bending 
 
 moment at the center of span is taken to be one-half of the above, or ikf = • 
 
478 
 
 CONCRETE ENGINEERS' HANDBOOK 
 
 [Sec. 11-1 7i 
 
 42 
 
 entire 
 
 Mr. Morrow has also figured this system on what might be called a cantilever and sus- 
 pended-span method. In this he assumes the line of inflection to be out three-tenths of the span 
 from the center of column and that the minimum size of the column cap is two-tenths of the 
 span. 
 
 The bending moment on a sextant at the edge of the head by this method is M = 
 which is more conservative than the Cleveland Code. 
 
 By the Pittsburgh Ruling we would have M = for the center of span for 
 
 periphery which would be equal to -g^- for a single band at the center of span. The method 
 
 of computing the steel required over the column head by the Pittsburgh Ruling depends upon 
 the size of column cap, but since the steel in the center of span is usually the determining 
 element, it is evident that this ruling checks the Cleveland Code quite closely in this case. 
 
 In all the buildings so far constructed under this system, a depressed head has been used 
 to assist in taking care of the bending stress and shear in the concrete at the edge of the column 
 
 cap. It has been the practice to lap all rods 
 
 ^ ,^3./ over the columns, extending them three-tenths 
 
 of the span beyond the center. 
 
 A computation made by Mr. Morrow 
 would tend to show that the bending moment 
 in the three-way system is less than that in a 
 four-way system and that the diagonal bands 
 are less efficient than the direct bands in the 
 four-way, while the equal spans -and equal loads 
 of the three-way system are probably more effi- 
 cient. The demonstration of this is as follows. 
 
 
 / • 
 
 ....> 
 
 
 
 
 
 
 
 
 / 
 
 
 I!— 
 
 / 
 
 
 (o) 
 
 Comparing two panels of equal area [see Fig. 60(a) and (6)] the one (a) a square panel supported by the 
 ordinary arrangement of columns, and the other (6) a parallelogram panel such as is found in the three-way system. 
 Let (a) represent a panel having four-way reinforcing. Then the loads in the panel are carried to the columns 
 by four half bands on the sides and two full bands on the diagonals, making four full bands per panel. 
 
 Let (6) represent a parallelogram panel having three-way reinforcing. In this case the loads are carried by 
 four half bands around the edge, and one full diagonal band, making three full bands per panel. The average span 
 of the bands in the square panel is 
 
 The span of the bands in the parallelogr am are all of the same length being 1.075Z/. 
 
 Assuming that the bending moment varies as the span, we can make a comparison of the bending moments 
 in the two panels by comparing their average span: 
 1.207L = average span in square panel. 
 1.075L = average span in parallelogram panel. 
 
 0.132L 
 1.075L 
 
 = 0.12.3 = 12.3% 
 
 Therefore, if the above assumptions are correct, the bending moment in a square panel is 12.3% more than 
 the bending moment in a hexagonal panel carrying the same load. 
 
 The question naturally arises as to whether or not one-half of the load of the panel goes 
 to the diagonals. 
 
 If we lay out a square panel so that the sides of the diagonal bands intersect on the sides 
 of the square bands, all bands being equal to 0.414L in width, and figure the area in the panel 
 falling in common over side and diagonal bands, we will find them to be as follows : 
 
 Area in panel falling in common over side and diagonal bands, 48.5%. 
 
 Area in panel falling over side bands, 17.2%. 
 
 Area in panel falling only over diagonal bands, 34.3%. 
 
 This would apparently tend to show that a large portion of the load falls on the diagonals 
 
Sec. 11-18] 
 
 BUILDINGS 
 
 479 
 
 and that they, due to their long span, are not able to carry it and that it is transferred to the 
 side bands producing an inverse moment over their center. As is well known, cracks usually 
 appear at these points in loading tests of flat-slab floors which are not reinforced to resist this 
 moment. 
 
 18. Patents. — O. W. Norcross, on Nov. 22, 1901, applied for a patent on a "new and useful 
 flooring for buildings." On April 29, 1902, he was granted Patent No. 698,542 covering the 
 following claims: 
 
 1. The combination of separate posts or supports, and a flooring consisting of a metallic network formed 
 by strips of wire netting enclosed therein, so as to radiate from the posts or supports on which the floor rests. 
 
 2. A flooring resting on separate supports and consisting of concrete with metallic network so arranged therein 
 that the amount of metal will be greatest at the points where the greatest tensile and shearing strains are to be 
 supported. 
 
 3. A flooring resting on separate posts, and consisting of metallic network formed by strips of wire netting 
 laid from post to post, to cross each other in cob-house fashion, and concrete enclosing the metallic network. 
 
 4. A flooring resting on separate posts, and consisting of metallic network formed by strips of wire netting 
 laid from post to post, and in the diagonals of the figures outlined by the posts, and concrete enclosing the metallic 
 network. 
 
 5. A flooring resting on separate supports, consisting of concrete having a metallic network enclosed in the 
 bottom layer thereof, with the body portion of said concrete of lighter material than the bottom layer thereof. 
 
 On the same date, April 29, 1902, Patent No. 698,543 was also granted to Mr. Norcross, 
 covering flooring for buildings, the claims of which are similar to No. 698,542 except that 
 instead of columns, walls or other longitudinal supports are specified. 
 
 Norcross Patent No. 698,542 was sustained by the United States Circuit Court of Appeals 
 for the Eighth Judicial District on Dec. 10, 1914, in the case of John L. Drum vs. C. A. P. Turner. 
 Upon petition of defendant, that Court decided not to reopen the case or to permit the intro- 
 
 I duction of additiohal evidence offered by the defendant. On June 1, 1915, the Supreme Court 
 
 I of the United States refused to set aside the decision of the lower court or grant the defendant 
 any relief whatever. On June 9, 1915, the United States District Court for the District of 
 Minnesota, entered a decree against the defendant, C. A. P. Turner, for infringement of the 
 Norcross patent and an injunction against further infringement. 
 
 On Aug. 3, 1915, the United States District Court for the District of Minnesota, held 
 that Mr. Turner's ''Spiral Mushroom System" was an infringement of the said Norcross 
 patent, and imposed a fine for the violation of the injunction. On Jan. 21, 1916, the Circuit- 
 Court of Appeals for the Eighth Circuit unanimously refused to reconsider the decision of the 
 
 it District Court on the question of infringement, and denied Mr. Turner's petition for a writ 
 
 I of certiorari. 
 
 On Oct. 4, 1916, the United States District Court for the District of Minnesota in the case 
 of C. A. P. Turner, plaintiff, and Deere Webber Building Co. and Deere Webber Co., defendants, 
 decided "that the defendants' structure does not infringe any of claims 1, 2, 4, 6 or 8" of C. A. P. 
 Turner Patent No. 985,119, and "that said claims and each of them are void for lack of inven- 
 I tion in view of the prior art, as held in the case of Turner vs. Moore supra." 
 
 In the United States District Court in the District of New Jersey in the case of C. A. P. 
 Turner, plaintiff, vs. Lauter Piano Co. and American Concrete Steel Co., defendants, the Court 
 held that the claims of Patents, Nos. 985,119 and 1,003,384, granted to the plaintiff and which 
 were relied upon by him in this suit are invalid. 
 
 As the matter now stands, therefore, Mr. Turner is under injunction both as to his original 
 construction and his so-called "Spiral Mushroom System." 
 
 The leading flat-slab promoters in the United States are now licensed under the Norcross 
 patent. This is true of two-way, three-way and four-way flat-slab advocates alike. 
 
 Besides the Norcross patent which has been declared basic, there are several other patents 
 which have been granted covering special methods of construction and reinforcement. Some 
 I of these have been tested in the courts. It is evident, therefore, that the field for the designer 
 of flat slabs is not a free one and, unless he can invent a method of reinforcement that is entirely 
 
 I 
 
480 
 
 CONCRETE ENGINEERS' HANDBOOK 
 
 [Sec. 11-19 
 
 new and an advance in the art, he must be licensed under one of the ''Systems" so-called. The 
 possibilities are so well covered that it is difficult to see how a really new system can be devised. 
 Rather should we seek to advance by improving our materials or methods of construction, 
 or both. 
 
 19. Loading Tests. — Since the first introduction of the flat-slab type of construction 
 there have been erected in the United States many hundreds of buildings of this kind. The 
 majority have been subjected to test loads over one or more floor panels of at least 1^ times j 
 the designed live load, and have passed the test without cracking or serious deflection., 
 There have been a few cases of flat-slab floors which have given trouble just as there have been' 
 cases of poor design with other systems, but it is undoubtedly true that the flat-slab type will ! 
 stand more abuse in this respect than any other form of reinforced-concrete construction. 
 The continuous bands of small steel rods running in two or more directions and all passing over 
 the column heads form a network of remarkable strength. When a structure of this type is 
 loaded to destruction it shows a very slow yielding, and it is safe to say that a properly designed 
 flat slab with continuous bands of steel passing over the column heads cannot show a sudden 
 collapse either under test or in actual service. 
 
 Besides the loading tests before referred to, which have been made to satisfy architects 
 and building superintendents, in which at most the deflection of the floor slab only was measured, 
 there have been numerous extensometer tests (so-called) made of complete buildings under a 
 variety of conditions of loading. 
 
 In these tests a portion of the floor slab in a completed building has been subjected to 
 load very much as a beam is tested in a testing laboratory, and the actual deformations of 
 exposed portions of the reinforcing steel at various points in the slab have been measured. 
 Also, measurements have been made of the deformations of the concrete of the slab. Measure- 
 ments of the deformations of the steel and concrete in both the exterior and interior columns 
 have also been made in many of the recent tests. 
 
 Much of this information has been held as confidential by various designers, but the results 
 of many such tests have been published and are thus available for the use of all. Tests of 
 this kind giving as they do actual distortion in the individual steel rods and in the concrete 
 itself, give a very accurate indication of the stresses in the structures at the time of making the 
 test due to the applied load. Of course, there are initial stresses due to dead load and tempera- 
 ture which can be computed with more or less accuracy and stresses due to the shrinkage of 
 concrete, which are more uncertain, which must be allowed for in addition to the live-load stress. 
 
 The performing of tests of this kind entailed a great amount of physical labor in their 
 actual execution. A high order of ability was shown in planning and organizing the operation, 
 and an equally high degree of skill was required to secure satisfactory results in the use of 
 laboratory instruments under such trying conditions. The cost of these tests was great, but 
 the value of the results obtained and the advance given to the art of design has justified the 
 cost in money and in personal sacrifice. 
 
 A list of the more important tests relating to flat-slab floors is given below : 
 
 Experiments on Models. — An interesting series of experiments upon laboratory models 
 was performed by the research department of the Corrugated Bar Co. to obtain a scientific 
 basis for the design of flat-slab floors. A rubber plate 0.5 in. thick was stretched over a box 
 which had cylindrical plugs projecting from the bottom to just touch the rubber sheet, so spaced 
 that they would, acting as miniature columns, divide the slab up into nine equal panels. Half | 
 rounds were used at the sides to act as wall columns. Load was applied by exhausting air I 
 from the box so that the rubber sheet deflected just as a flat slab is supposed to do and this load I 
 was maintained very exactly by means of a water jet aspirator and U-tube manometer. Deflec- ' 
 tion readings were taken at points 0.1 of the span apart. 
 
 Test of Flat-slab Floor of the Deere & Webber Co.'s Building, Minneapolis, Published December, 
 1910. — The type of floor is four-way and is typical of the standard practice of the Concrete 
 Steel Products Co. at that time. Deformation measurements were made on concrete and steel 
 
Sec. 11-19] 
 
 BUILDINGS 
 
 481 
 
 in the slab only. Tests are described by Arthur R. Lord in papers entitled "A Test of a Flat- 
 slab Floor in a Reinforced Concrete Building" and "A Discussion of the Basis of Design of 
 Reinforced-Concrete Flat Slabs," presented before the Annual Convention of the National 
 Association of Cement Users (Am. Cone. Inst.). These articles appeared in Engineering 
 News, Dec. 22 and 29, 1910 and Jan. 12, 1911. 
 
 Test of Franks Building, Chicago. — Floors were constructed according to the Cantilever 
 Flat-slab system by the Leonard Co. of Chicago. The panel size was 19 ft. 4 in. by 20 ft. 3 in. 
 Four interior panels located on the tenth (top) floor were tested. The slab thickness was 9)^ 
 in., and 133^ in. at the drop panels. 
 
 Test of Powers Building, Minneapolis. — Described in a paper by F. J. Trelease read before 
 the American Concrete Institute in March, 1912 (vol. viii, Proceedings). 
 
 Test of Larkin Warehouse. — Test of a four-way slab described in a paper presented by 
 Arthur R. Lord before the American Concr&te Institute in December, 1912. Test was also 
 published in the Engineering Record, January, 1913 and in the Cement Era, January, 1913. 
 
 Test of Soo Line Terminal, Chicago, '^Te&i of single-story structure which carries railroad 
 tracks on the upper deck. For complete description of this building see Engineering Record, 
 Aug. 16, 1913, Engineering News, Aug. 21, 1913, and Railway Age Gazette, Aug. 22, 1913. For 
 description of test see article in Bulletin 84 of the Engineering Experiment Station of the 
 University of Illinois. 
 
 Test of Schulze Baking Co.'s Building, Chicago. — Floor slab is of the four-way type with 
 short transverse bars in top of slab on the center line of columns. Test is described in Bulletin 
 84 of the Engineering Experiment Station of the University of Illinois. 
 
 Tests of S-M-I Flat Slabs. — Test made under the supervision of Sanford E. Thompson 
 of a slab 20 ft. square supported upon columns spaced 12 ft. on centers so that the slab projected 
 4 ft. on all sides in order that a condition of continuity over the columns would exist. 
 
 Test of Shredded Wheat Factory Niagara Falls, N. Y. — Type of floor is two-way, designed 
 by the Corrugated Bar Co. and known commercially as Corr-plate Floor. Test is described 
 in Bulletin 84 of the Engineering Experiment Station of the University of Illinois. It is also 
 reported in the Journal of the American Concrete Institute, vol. ii, No. 6, 1914, in an article by 
 W. A. Slater. 
 
 Tests of Circumferential Cantilevers. — Tests were made upon octagonal slabs each carried 
 upon an 8-in. square column having an octagonal flare head 2 ft. in diameter. For description 
 of tests see Engineering Record, vol. 73, page 249. 
 
 Worcester Slab Test. — The structure upon which this test was performed was constructed 
 especially for that purpose at Worcester, Mass. The objects were to determine the effect 
 of different steel arrangements and the effect of variation in size of the column capital. A com- 
 plete description of this test will be found in Bulletin 84 of the Engineering Experiment Station 
 of the University of Illinois. 
 
 Test of Schwinn Building, Chicago. — The floors are of the four-way type and were designed 
 before the present Chicago ruling was adopted. Description of this test is given in an article 
 by Arthur R. Lord in vol. xiii of the Proceedings of the American Concrete Institute, 1917. 
 
 Test of Curtis-Ledger Factory, Chicago. — Floor tested was designed according to the Barton 
 Spider Web system. Summary of this test is given in Bulletin 84 of the Engineering Experiment 
 Station of the University of Illinois. 
 
 Test of Sears-Roebuck Building, Seattle. — Floor was designed according to the Akme Two- 
 way system. Test was performed under the supervision of the Building Department of the 
 City of Seattle and is reported in the Proceedings of the Pacific North west Society of Civil 
 Engineers, vol. xv for January and February, 1916, in a paper presented by D. E. Hooker. 
 
 Test of Bell Street Warehouse, Seattle. — The type of floor used in this structure is a four-way 
 system designed according to the Mushroom system by C. A. P. Turner. The first test made 
 on this building is described in an article by D. E. Hooker in Engineering Record, vol. 73, page 
 647. 
 
 31 
 
482 
 
 CONCRETE ENGINEERS' HANDBOOK 
 
 [Sec. 11-19 
 
 Since the slab tested was cast in freezing weather and the temperature near the freezing 
 point for several days, it was claimed by the designers that the concrete was not thoroughly 
 cured at the time of making the test and that, therefore, conclusions drawn from the test were 
 unwarranted. In order to satisfy all concerned a second test was made upon four panels 
 of the same floor slab previously tested. Description of the second test is given in Engineering 
 News— Record, April 19, 1917. 
 
 Test of Building of Pierce Arroiv Motor Car Co., Buffalo, N. Y. — Floor is a four-way system 
 designed according to the Chicago Code. 
 
 ■ Test of S-M-I Flat Slab at Purdue University, April 19, 1917. — Test was performed upon 
 a structure erected for the purpose and did not form a part of a building built for commercial 
 use. Test was reported by Prof. F. K. Hatt of Purdue University to Edward Smulski. 
 
 Discussion of Tests. — To those who are familiar with the accurate results which are to be 
 obtained in laboratory testing, the results secured in field tests of buildings are apt to be a 
 disappointment. 
 
 Many factors enter into these tests which cannot be entirely controlled and inconsistencies 
 develop in the measured results which cannot always be explained. In the first place the 
 structure is composed of two materials of radically differing properties. The steel possesses 
 definite qualities, properties, and areas which can be accurately measured. Its accurate 
 location in the structure also can be secured by the exercise of proper care. The concrete, on 
 the other hand, as at present manufactured, does not possess qualities which can be accurately 
 forecast either in similar structures or in different parts of the same structure and, as is well 
 known, those which it does possess are subject to change with time, but even this change cannot 
 be known in advance. Accurate test results are, therefore, not to be expected. 
 
 The tested slab in the majority of cases form but a small portion of the floor and the 
 distribution of stress to the surrounding portions has a modifying effect upon the results. Also, 
 the relative size and stiffness of the columns above and below the floor will cause differing 
 results in the tests of floors otherwise similar. Shrinkage and temperature changes produce 
 effects which are difficult to measure and eliminate, and additional complications are introduced 
 by settlement of the columns, either due to yielding of the foundation or to shortening in the 
 columns themselves under direct load. 
 
 A brief summary of the more important results of tests is given below: 
 
 1. Tests of wide beams and later of flat-slab buildings proved that a beam may be made 
 much wider than the support, with only a moderate loss of efficiency. 
 
 2. Tests of models and of actual buildings have demonstrated the advisability of transverse 
 reinforcement placed on the edge of the panel in the top of slab, particularly where no drop 
 panel is used. 
 
 3. For the true flat slab, i.e., where no drop heads are used, the critical section so far as 
 the steel is concerned, is at the edge of the column capital. 
 
 4. With sections and reinforcement as specified under the Chicago ruling, surprisingly 
 low stresses have been measured in the steel. This is particularly true of slabs having drop 
 panels. 
 
 5. Eccentric action of the load produces a marked bending action in interior and particu- 
 larly in exterior columns which should be reinforced accordingly. Columns for single-story 
 structures, where unbalanced live load is carried on the roof, should receive attention in this 
 regard. 
 
 6. Footings for single-story structures of this class should be designed with a liberal factor 
 of safety as the danger of settlement is apparently greater in this class of structure. 
 
 7. Structures with relatively thick slabs show low steel stress as compared to thinner 
 slabs designed in the same manner. 
 
 8. Stresses due to dead load can be measured if proper precautions are taken during 
 construction. 
 
Sec. 11-19] 
 
 BUILDINGS 
 
 483 
 
 9. Different arrangements of load have different effects upon the same panel and single- 
 panel loading does not necessarily develop the greatest stresses. 
 
 10. In flat-slab design, deflection rather than stress in the steel controls because large 
 deflection results in serious cracking of the slab. Floors designed under the Chicago ruling 
 do not exceed one-eight-hundredth of the span and most of them show but slightly more than 
 one-half of this amount. 
 
 11. Relative deformations in the concrete are of value as a comparison, but as an accurate 
 indication of stress they have little weight due to lack of uniformity in the elastic properties 
 of concrete. 
 
 12. Buildings designed according to the flat-slab rulings of any of our better cities, 
 under any of the systems, and constructed with a reasonable amount of intelligent supervision, 
 give entire satisfaction in service. Of these rulings, that of Chicago is the best and most 
 conservative. 
 
 . 13. Many more tests on four-way reinforced slabs are available than of other systems and 
 for this reason comparisons, which are largely a matter of judgment, cannot be made 
 with certainty. Analyses of published data made by a number of engineers would indicate that 
 given the same span, load, and concrete sections, the circumferential system will show the 
 lowest stress per pound of steel, the four-way system next, and the two-way system last. This 
 is not necessarily their order with regard to total economy, however. Bending and placing 
 costs and the unit price of steel are contributing factors which modify the final result. 
 
 14. The ruling adopted by the American Concrete Institute meets certain objections to 
 the Chicago ruling and harmonizes better in some respects with the results of later tests. It 
 is recommended for use as the best that is now available. 
 
 15. The ruling adopted by the Special Committee of the American Society of Civil Engi- 
 neers, while theoretically sound in some respects, is incorrect in others, and adopts coefficients 
 which are unnecessarily severe and not justified by the facts. This ruling, if generally 
 adopted, would, in the author's opinion, impose a needless burden on the industry. 
 
 The following very able analysis of this subject based on four-way tests is taken from a 
 paper by Arthur R. Lord, Structural and Testing Engineer of Chicago, read before the South- 
 western Cement Association at Kansas City, Feb. 22, 1917. 
 
 The total positive and negative moment is taken in the A. C. I. report as 0.09 whih — qc)'^ and the division 
 of this moment between the various sections is given in Table I. The Joint Committee report increases the total 
 moment to O.lOTwhih — %c)^- and allows a much less flexible design than the A. C. I. report. Reduced to the 
 same basis the Chicago Ruling total moment is 0.092i(;Zi(l2 — ^6c)2. 
 
 Table P 
 
 Code 
 
 Type 
 of flat 
 slab 
 
 Total 
 — and 
 
 + 
 mom., 
 coef.2 
 
 Total 
 — mom., 
 
 % 
 
 Total 
 + mom., 
 
 % 
 
 Distribution limits 
 
 Col. head 
 section, % 
 
 Middle 
 section, % 
 
 Outer 
 section, % 
 
 Inner 
 section, % 
 
 Sum, 
 
 % 
 
 1 
 
 2 
 
 3 
 
 4 1 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 10 
 
 Chicago 
 Ruling. . . 
 
 4-way 
 2-way 
 
 0.087 
 0.093 
 
 66.7 
 62.5 
 
 33.3 
 37.5 
 
 53.2 
 50.0 
 
 13.4 
 12.5 
 
 20.0 
 25.0 
 
 13.4 
 12.5 
 
 100 
 100 
 
 A. C. I. 
 
 report 
 
 Drop 
 No drop 
 
 0.090 
 0.090 
 
 70 . 0 to 60 . 0 
 70 . 0 to 50 . 0 
 
 30.0 to 40.0 
 30.0 to 50.0 
 
 60 . 0 to 50 . 0 
 60.0 to 40.0 
 
 10.0 to 20.0 
 10.0 to 30.0 
 
 18.0 to 28.0 
 18.0 to 38.0 
 
 12.0 to 22.0 
 12.0 to 32.0 
 
 100 
 100 
 
 J. C. 
 
 report 
 
 Drop 
 No drop 
 
 0.107 
 0.107 
 
 62.5 
 62.5 
 
 37.5 
 37.5 
 
 50.0 
 50.0 to 40.6 
 
 12.5 
 12.5 to 21.9 
 
 28 . 1 to 22 . 5 
 28 . 1 to 20 . 6 
 
 9.4 to 15.0 
 9.4 to 16.9 
 
 100 
 
 100 
 
 Total Moment Requirement. — It will be seen from Table I that the A. C. I. report and the Chicago Ruling 
 adopt the same total moment coefficient. In passing it may be mentioned that this coefficient is the highest called 
 
 1 Table taken from discussion by T. L. Condbon at the A. C. I. convention at Chicago, Feb. 8, 1917, 
 
 2 Coefficient in formula. Total mom. = Kwl\{h — qc)^. 
 
484 
 
 CONCRETE ENGINEERS' HANDBOOK 
 
 [Sec. 11-19 
 
 for in any building code in the United States so far as I know and the highest heretofore used in practical designing. 
 It represents the most conservative past practice. The J. C. report increases this moment 18.5%. Let us investigate 
 what extensometer tests of buildings erected under Chicago regulations show as to the actual stresses developed in ij 
 flat slabs. Table II gives a summary of four Chicago tests, all of four-way flat slabs of the drop type — that is, with 
 
 Table II 
 
 Test 
 
 S 
 
 presses over 
 
 column headi 
 
 Steel stresses 
 
 In steel 
 
 In concrete 
 
 At panel center^ 
 
 Top rods 
 
 . Tests 
 
 Chicago* 
 
 Test 5 
 
 Chicago* 
 
 Tests 
 
 Chicago* 
 
 Tests 
 
 Chicago* 
 
 1 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 
 9,200 
 
 23,700 
 
 1,110 
 
 1,400 
 
 10,100 
 
 31,000 
 
 
 
 Larkin 
 
 8,500 
 
 23,000 
 
 800 
 
 1,120 
 
 16,000 
 
 37,000 
 
 
 
 Schulze 
 
 7,100 
 
 22,500 
 
 530 
 
 1,230 
 
 6,200 
 
 34,000 
 
 4,900 
 
 71,500 
 
 Schwinn^ 
 
 9,300 
 
 38,000 
 
 300 
 
 1,160 
 
 18,900 
 
 39,000 
 
 9,900 
 
 70,000 
 
 a greater thickness adjacent to the support than in the slab proper. All these floors were loaded to or in excess of 
 twice the total dead and live design load as indicated in Table III. It certainly seems to me that these tests 
 indicate eminently safe and conservative design. The average steel stress at the support measured about 9000 lb. 
 per sq. in. against a computed stress of about 27,000 lb. per sq. in., and the concrete stress about 700 lb. per sq. in. 
 against a computed value of about 1250 lb. per sq. in. At the panel center the average steel stress was about 13,000 
 lb. per sq. in. against a computed value of 37,000 lb. per sq. in. The concrete stress at the center is always very 
 low — less than half that at the supports. The average stress in the top rods was less than 8000 lb. per sq. in. 
 against a computed value of over 70,000 lb. per sq. in. Taking into account the possible and reasonable effect 
 of various loadings and of long-continued loading, it would appear that a factor of safety of four was largely 
 exceeded. 
 
 Table III 
 
 Test 
 
 Panel size 
 
 Length Breadth 
 
 Slab thickness 
 
 Slab Drop 
 
 Test load: 
 live 
 
 dead total 
 
 Area loaded 
 
 Twice 
 design 
 load in 
 place 
 
 Floor 
 tested 
 
 Franks . 
 
 20'-3" 
 
 Larkin 
 
 24'-2' 
 
 20' -0" 
 
 9.0" 
 
 13.00' 
 
 624 
 115 
 
 739 
 
 9.0" 
 
 15.75' 
 
 618 
 
 120 738 
 
 Schulze 
 
 20'-0" 
 
 17' 
 
 Schwinn . 
 
 26'-0" 
 
 25'-0" 
 
 9.0' 
 
 15.00" 
 
 10.0' 
 
 .00" 
 
 722 
 
 122 844 
 
 450 
 125 
 
 575 
 
 24 hr. 
 
 10th (top) 
 
 24 hr. 
 
 2d 
 
 m 
 
 24 hr. 
 
 379 days 
 
 6th (top) 
 
 1 Average of readings in fully loaded area. 
 
 2 Average of readings on direct and diagonal bands, 
 s Observed steel stresses as reported in tests. 
 
 * Stresses calculated by Chicago RuHng for a design load equal to applied test load. 
 
 ^ Concrete stresses reduced to 80% of those reported for the teste and computed on the basis of the initial 
 modulus of elasticity. 
 
 8 Stresses given are for readings taken with full test load in place over 1 year. 
 
Sec. 11-19] 
 
 BUILDINGS 
 
 485 
 
 Table III gives information about the Chicago tests referred to in this paper. Table IV gives further data 
 including a calculation of the various resisting moments from the measured steel stresses. In considering these 
 tests the stress in slab steel of direct bands at the edge of the loaded area has been increased so that the stress in 
 rods outside the loaded area has been added to the stress in the rods under the load and the total stress figured on 
 this basis. Also, in column 12 of Table IV, all the head rods are considered as stressed as highly as the slab rods, 
 though they lie much nearer the neutral axis of the slab. Even on this extremely conservative basis the average total 
 moment is only 0.025WL, equal to 0.035wh(h — qc)-, as against the value of Q.09xvh{h — qc)^ used in the Chicago 
 and A. C. I. Codes. It is possible that other loadings may increase even the sum of the moments somewhat and 
 that very long continued loading may have its effect. One test involved a full year under load and the increase 
 in the total moment was about 7%, with no increase during the last 3 or 4 months. But taking all influences into 
 account I do not see how any one can fairly ask for a more conservative moment coefficient than that adopted by the 
 A. C. I. and Chicago Codes. In one test at Seattle of a flat-slab building built on a very deficient basis as compared 
 with either of these codes, and in which as a result exceedingly high stresses were developed, reaching as high as 
 50,0001b. per sq. in. in the steel, and in which modifying actions were reduced to a minimum, the total resisting 
 moment on the basis of the A. C. I. Code was 0.066 whih — qc)- against the code specification of O.OQwhih — qc)"^. 
 
 Table IV 
 
 Test 
 
 Section 
 considered 
 
 Right area of all 
 slab rods and ef- 
 fective laps 
 
 Effective section, 
 all slab rods and 
 effective laps 
 
 d, for steel areas 
 in columns 3 
 1 and 4 
 
 Right area of 
 slab rods, laps 
 and head rods 
 
 Effective section, 
 slab rods, laps 
 and head rods 
 
 d, for steel areas 
 given in 6 and 7 
 
 Resisting moment of observed steel stresses 
 
 All slab rods and 
 effective laps 
 
 Slab rods, laps and 
 head rods 
 
 Right 
 area 
 
 Effective 
 section 
 
 Right 
 area 
 
 Effective 
 section 
 
 1 
 
 2 
 
 3 in. 
 
 4 
 
 sq. in. 
 
 5 
 
 sq. in. 
 
 6 
 
 sq. in. 
 
 7 
 
 sq. in. 
 
 8 in. 
 
 9 
 
 10 
 
 11 
 
 12 
 
 Franks. . . 
 
 Over column 
 
 Center of panel 
 Top rods 
 
 10.40 
 
 6.88 
 
 13.31 
 8.33 
 
 10.88 
 7.62 
 
 11.98 
 
 14.88 
 
 10.68 
 
 0.0129PFL 
 0.00681FL 
 
 0.0197irL 
 
 0.0165TFI/ 
 0.0081IFL 
 
 0 . 0246 WL 
 
 0. 0149 PFL 
 0.0068PFL 
 
 0. 0217 WL 
 
 0. 0185 PFL 
 0.0081 TFL 
 
 0. 0266 PFL 
 
 
 
 
 
 
 
 
 Larkin. . . 
 
 Over column 
 
 head 
 
 Center of panel 
 Top rods 
 
 11.33 
 7.75 
 
 13.70 
 9.37 
 
 13.87 
 7.84 
 
 12.53 
 
 14.89 
 
 13.75 
 
 0. 0123 PFL 
 0.0090PFL 
 
 0.0213TFL 
 
 0.0149PFL 
 0.0109]FL 
 
 0.0258PFL 
 
 0.0135PFL 
 0.0090PFL 
 
 0.0225TFL 
 
 0.0161PFL 
 0.0109T^^L 
 
 0.0270PFL 
 
 
 
 
 
 
 
 
 Schulze. . 
 
 Over column 
 Center of panel 
 
 9.20 
 5.95 
 1.10 
 
 10.84 
 7.18 
 1.10 
 
 12.10 
 7.75 
 7.88 
 
 9.20 
 
 10.84 
 
 12.10 
 
 0.0105PFL 
 0.0038PFL 
 0.0006TFL 
 0.0149TFL 
 
 0.0123TFL 
 0.0046PFL 
 0.00061FL 
 0.0175TFL 
 
 0.0105TFL 
 0.0038TFL 
 0.0006PFL 
 0.0149PrL 
 
 0.0123TFL 
 0.0046TFL 
 0.0006TFL 
 0.0175PFL 
 
 
 
 
 
 
 
 
 
 
 
 
 Schwinn . 
 
 Over column 
 
 head 
 
 Center of panel 
 Top rods 
 
 13.74 
 9.57 
 1.77 
 
 16.42 
 11.58 
 1.77 
 
 10.40 
 8.85 
 8.75 
 
 13.74 
 
 16.42 
 
 10.40 
 
 0.0102PFL 
 0. 0122 PFL 
 0.00121FL 
 0.0236PyL 
 
 0.0122PFL 
 0.0148TFL 
 Q.0Q\2WL 
 0.0282TFL 
 
 0. 0102 IFZ, 
 0.0122WL 
 0. 0012 PFL 
 0.0236TFL 
 
 0.0122PFL 
 0.0148IFL 
 0.0012PFL 
 0.0282PFZ/ 
 
 
 
 
 
 
 
 
 
 
 
 
 In view of the extensometer test record and the fact that hundreds of other buildings designed on the Chicago 
 basis have indicated by deflections under test load that they have equally low stresses and moments, I believe the 
 position of the Joint Committee is ultraconservative in this requirement. 
 
 Distribution of the Total Moment. — The question as to the amount of total moment to be specified in design 
 is, as I view it, one of conservative vs. idtraconservative opinion. The distribution of the total moment, however, 
 does not admit of so many opinions. The J. C. report makes a rigid division of the moment into 62.5 % negative 
 and 37.5% positive for any or all types of flat slab. The A. C. I. report leaves a portion of the moment to be as- 
 signed in accordance with the physical details of the slab and its reinforcement. I cannot see that there is any room 
 for debate on this difference. If the J. C. division is right for a flat slab with a drop and supported by stiff columns, 
 it is wrong for a flat slab without a drop and supported by relatively slender columns. The evidence of the tests 
 also is positive and conclusive on this. For four panels loaded uniformly, the tests show a distribution of approxi- 
 
486 
 
 CONCRETE ENGINEERS' HANDBOOK 
 
 [Sec. 11-19 
 
 mately — and H +, with individual cases in which as high as 70% of the total moment has been negative 
 (see Table V). These figures are for buildings with a much deeper drop than the J. C. permits. "With other con- 
 ditions of loading, certainly the center or positive moment would be greater and the ratio of negative moment to 
 total moment reduced. And with no drop present, the change would be even greater as shown in the test of the 
 Bell Street Warehouse at Seattle where only 40 % of the total moment was negative and 60 % positive — practi- 
 cally a reversal of the conditions where a large drop is used. This data gives ample basis for the A. C. I. recom- 
 mendation that permits an extreme division of 50% — and 50% + where the drop is not used, and also permits 
 from 60 to 70% negative where the drop is used. This division accords much more closely with the facts than the 
 rigid J. C. distribution, which forces the designer to either use an unbalanced design or employ additional material 
 where the total material already involved is more than ample but improperly distributed. 
 
 Table V 
 
 Test 
 
 Steel area in square inches 
 
 Observed steel stress 
 
 Total tension carried 
 
 % of whole 
 
 Two* 
 diagonal 
 bands 
 
 Long 
 direct 
 band 
 
 Short 
 direct 
 band 
 
 Diagonal 
 band 
 
 Long 
 direct 
 band 
 
 Short 
 direct 
 band 
 
 Diagonal 
 bands 
 
 Direct 
 bands 
 
 Diagonal 
 bands 
 
 Direct 
 bands 
 
 Franks 
 
 Schwinn 
 
 10.00 
 11.13 
 8.48 
 13.89 
 
 3.54 
 5.10 
 3.45 
 5.10 
 
 3.14 
 2.55 
 2.55 
 4. 12 
 
 6,950 
 12,900 
 
 4,220 
 21,900 
 
 7,350 
 10,200 
 
 8,400 
 16,000 
 
 10,100 
 24,200 
 7,200 
 18,600 
 
 69,000 
 143,800 
 
 36,000 
 304,000 
 
 57,000 
 113,800 
 
 47,400 
 159,000 
 
 54.8 
 55.9 
 43.2 
 65.6 
 
 45.2 
 44.1 
 56.8 
 34.4 
 
 
 
 
 
 
 
 
 
 
 54.9 
 
 45.1 
 
 
 
 
 
 
 
 
 
 
 * Effective section. 
 
 The J. C. also assigns a fixed proportion of the moment — 12.5% of the total moment — to the mid-section in 
 flat slabs with a drop. The whole report shows no conception of the fact that the distribution of the moment is 
 different in two-way and four-way flat slabs — a fact very clearly brought out in tests. In four-way flat slabs 
 tested in Chicago as shown in Table II, columns 8 and 9, the calculated stress across the mid-section was over 
 70,000 lb. per sq. in. and the actual stress under 10,000 lb. per sq. in. even after a year. In the test of the Sears- 
 Roebuck Building at Seattle, a two-way flat slab gives a computed stress at this section of 27,200 lb. per sq. in. and 
 an actual stress of 23,500 lb. per sq. in. This would indicate that the moment across the mid-section is several 
 times as great with the two-way arrangement of the steel as it is with the four-way arrangement. The J. C. assign- 
 ment of moment to this section is too large for four-way slabs and too small for two-way slabs, and again the result 
 is unbalanced design. The A. C. I. report permits a distribution that will fit either type. 
 
 Customary practice in four-way design makes a distribution of the steel between direct and diagonal bands 
 varying from a 1 : 1 ratio to a 1 .5 : 1 ratio and the results of tests as shown in Table V show that these ratios are about 
 right. The average distribution of the moment was about even between direct and diagonal bands and this 
 would result in a 1.4 : 1 ratio between direct and diagonal bands. The J. C. report specifies as a minimum that 
 60% of the positive moment must be taken by the outer section and this gives a minimum relation of 2.1 times as 
 much cross-sectional steel area in the direct as in the diagonal band. Just why all previous practice is passed 
 over and a novel arrangement required, I am at a loss to know after studying the available data. 
 
 Arbitrary Limitations. — Arbitrary limits, established in addition to the natural limitations of the moment 
 and shear requirements, should be reduced to a minimum. With flat-slab construction the moment at the center 
 is too small to establish a slab thickness such as we deem advisable from the standpoint of deflection and some 
 arbitrary limit is desirable here, as found in all three codes cited above. At the support, however, the shear 
 determines an adequate thickness and no other limit is needed. 
 
 An investigation of the Chicago buildings tested in the past does not appear to call for the limitation of 0.4L 
 as the minimum drop dimension nor for one-half the slab thickness as the maximum drop thickness. These and 
 other J. C. limitations handicap the designer in the economic use of his materials and do not seem to be founded 
 on any experience in the past. The A. C. I. report contains all the limitations that seem necessary to safe design. 
 
 Column Moment. — The amount of moment in the column will depend on the type of flat slab and on the stiff- 
 ness of the columns above and below the slab. For the Franks test, made on an upper floor and with 20-in. columns 
 on a 20-ft. span, the observed moment was 0.0138wZi(^2 — gc)^. For small columns in tipper stories of buildings the 
 A. C. I. specification of O.Q22wli{h — Qc)^ should be ample even where the drop type of flat slab is used as was the 
 case in the Franks building. For lower stories the column moment undoubtedly largely increases but it would not 
 be right to specify these larger moments as a minimum for all columns. This is a matter where the judgment 
 and calculations of the engineer must decide on the proper increase but the specification of a definite minimum 
 moment by the A. C. I. Committee seems to me a step in the right direction. The J. C. report and the Chicago 
 Ruling leaves this in very indefinite shape, with the result that many designers acting in good faith will entirely 
 overlook the design of the columns for bending moment. 
 
Sec. 11-20] 
 
 BUILDINGS 
 
 487 
 
 20. Methods of Design and Problems. — Many attempts have been made to develop a 
 scientific method of analysis of the stresses which will be developed in flat-slab floors as at 
 present constructed. On account of the complex nature of the problem and the various modi- 
 fying factors which enter into it no satisfactory method of design, so far as the writer is aware, 
 has been presented. 
 
 A very comprehensive testing program of flat slabs has been carried out in which actual 
 stresses have been measured (see Art. 19). With these results as a foundation, rules governing 
 the design of such structures have been adopted by various city building departments and 
 technical societies, and the adequacy of these rulings has been proven in turn by tests of build- 
 ings designed in accordance with their provisions (see Appendix C). A study of the provisions 
 of the various rulings reveals at once the fact that all methods of design now in common use 
 are to a large extent empirical. 
 
 The following general remarks regarding the design of flat-slab buildings may be of aid 
 to those who are inexperienced in this field. 
 
 1. The live load for which the structure is to be designed must be established either by 
 the architect or the engineer, or, if the structure comes under the jurisdiction of a building 
 department of a city, the Building Code will be found to contain rules which will apply. 
 Frequently, it is necessary to obtain a special ruling from the building department in a particu- 
 lar case. 
 
 2. The story heights, etc., having been decided upon with due consideration of the use to 
 which the building is to be put, the next question to be settled is the column spacing. Fre- 
 quently this is decided by the owner or the architect but in industrial buildings, the machinery 
 to be housed may be the controlling factor. In garages and service stations of moderate width, 
 a wide central span and narrow side panels may be necessary. However, the following general 
 rules may be stated. 
 
 (a) In general the short span construction is the cheapest, but it is very seldom that 
 spans less than 16 ft. are used, and probably a 20-ft. panel represents the average. 
 (6) The ideal design has square panels. 
 
 (c) The next best arrangement consists of identical rectangular panels. 
 {d) Oblique panels have been used, but their use is not recommended, 
 (e) The outer spans are frequently made shorter than the interior spans in order to make 
 the bending moment in exterior panels equal to that in interior panels. 
 
 3. The method of design to be used for the slab depends on whether the structure comes 
 under the jurisdiction of a city building department. If it does, and the department has 
 adopted a ruling on flat-slab design, that ruling must be followed in so far as it applies. If no 
 ruling has been adopted, then much time and trouble will be saved by visiting the building 
 examiners and obtaining a statement as to just what they will accept. In cases where no city 
 authorities have jurisdiction, the designer is at liberty to use any of the rulings which have been 
 adopted by our larger cities or technical societies, but the ruling of the American Concrete 
 Institute is recommended. 
 
 The following recommendations regarding designs of flat-slab floors are made by Arthur 
 R. Lord after a thorough study of all the available test data and an analysis of the various 
 rulings which have been adopted (see paper by A. R. Lord entitled, " Extensometer Test Data and 
 Recommended Practice in the Design of Flat Slabs," read before the Southwestern Cement 
 Association, Feb. 23, 1917). The intimate connection which Mr. Lord has had with the develop- 
 ment of this branch of engineering and his thorough knowledge of the practical construction side 
 entitles him to speak with authority and with his recommendations we are thoroughly in 
 accord. 
 
 For engineers who desire to be conservative and who are designing for normal variations in the conditions 
 of manufacture of the concrete in various localities, I would personally advise the use of the A. C. I. Committee 
 report as the best and most thoroughly considered design basis now available. For engineers designing structures 
 to be erected by the most improved means under positive controls, ensuring a superior grade of concrete, I would 
 
488 
 
 CONCRETE ENGINEERS' HANDBOOK 
 
 [Sec. ll-20a 
 
 approve the use of the A. C. I. teport with a moment coefEdent of 0.08u;^i(/2 5c) 2 instead of the value given in 
 their report 0.09w/i(Z2 — qcy-. 
 
 For four-way flat slabs with drops which is the economical type to use, I should advise the following dis- 
 tribution of the total moment: 
 
 Column-head section 50% 
 
 Mid-section 10 % 
 
 Outer section (direct bands) 20 % 
 
 Inner section (diagonal bands) 20 % 
 
 Where no drop is used, I should advise taking from 6 to 10 % (depending on the column stiffness) of the total amount 
 away from the column-head section and distributing it equally over the inner and outer sections. 
 
 For two-way flat slabs with drops, I would advise the following distribution of the total moment: 
 
 Column-head section 50 % 
 
 Mid-section 15% 
 
 Outer section 23% 
 
 Inner section 12%. 
 
 The writer agrees with these recommendations and knows them to be very conservative. 
 With first-class construction and supervision, satisfactory results have been obtained with the 
 four-way system using drop panels and concrete sections as given in the tables of Art. 21 under 
 the Pittsburgh Ruling, and using low steel stresses (never over 16,000 lb. per sq.in.). This 
 is somewhat less than the minimum recommended by Mr. Lord, both as regards concrete and 
 steel, and its universal adoption is not recommended. 
 
 The Pittsburgh Ruling may be regarded as giving the minimum design which has been 
 used with satisfactory results. With a uniformly high grade of concrete assured and properly 
 seasoned before loading, it is the writer's opinion that the Pittsburgh Ruling would give 
 good results. With conditions as they are, Mr. Lord's recommendations above quoted 
 represent the best practice. 
 
 A series of designs of typical panels will now be given in order to illustrate the method of 
 application of the various rulings. 
 
 It is not the intention to compare the relative economy of the various systems and the 
 reader should be slow to make such comparisons for, unless all factors are considered, they 
 are sure to be misleading. As a matter of fact there is very little difference between different 
 designs which are made upon exactly the same basis. 
 
 20a. Computations for Akme System. — The following computations are 
 published through the courtesy of the Condron Co. of Chicago and are based on the Chicago 
 Ruling. For method of analysis and moment distribution see Art. 17/; also Appendix C. 
 
 Span 22 ft. 0 in. by 25 ft. 0 in. 200 lb. L.L. 91/2-in. slab Slab = 119 lb. per sq. ft. 
 
 „ . , , WL^ L.L. = 200 lb. per sq. ft. 
 
 Col. head section = -:j-r~ ft.-lb. w = load per sq. ft. . — ^ 
 
 ^^/^ Total = 319 lb. per sq. ft. 
 
 Outer section = -g^- ft.-lb. W = 
 
 Inner & mid. section = ft.-lb. 
 
 Head = (0.225) (23.5) = 5.29, say 5 ft. 6 in. round head. 
 Plate = (22) (0.35) = 7.70, say 8 ft. 0 in. short direction. 
 
 Plate = (25) (0.35) = 8.75, say 9 ft. 0 in. long direction. Use 9 ft. 0 in. square 
 22 -f 25 ^ 
 2 = 23.5 avg. span. 
 
 , u J . (319) (23.5)3 (12) f 18,000 lb. per sq. in. 
 
 Col. head moment = = 1,655,980 in.-lb. 7 ' ^ 
 
 (2) (15) fc = 700 lb. per sq. in. 
 
 Cubes 
 2' = 1 
 23.5' = 12,978 
 
 —3 (1,655,980)(15,625) , . 1 
 
 22 = 10,648 ^2 978 ^ 1,995,000 in.-lb., long span 
 
 25" - 15.626 a^|0Kia6«) . ,,360,000 in.-lb., short span 
 
See. 11-206] 
 
 BUILDINGS 
 
 489 
 
 Long span Short span 
 
 Col. head section 1,995,000 1,360,000 in. 
 
 Outer section 997,500 680,000 m in. 
 
 Inner & mid. section 340,000 498,750 
 
 997,500 
 Long span (85^)2(141) 
 
 Short span 
 
 Long span 
 
 Short span 
 
 Slab 
 
 Long bands 
 
 680,000 
 
 (89i)H14l) 
 
 1.995.0C0 
 
 (157;i)-^(108) 
 
 101 = X p = 0.63% 
 
 (0.0063) (141) (85^) = 7.45 sq. in. 
 
 17 — Va(I> = 7.51 sq. in. 
 
 = m = K 0.42 % 
 
 (0.0042) (85i) (141) = 4.96 sq. in. 
 
 12 — H(f> = 5.30 sq. in. 
 = 74 = X 0.46 % 
 
 SH in. 
 
 Assume plate. . . . 7H in. 
 7li + m = 17 in. 
 
 IH in. 
 d = 15^^ in., high bars 
 
 ^4 in. 
 15\i in. 
 
 low bars 
 
 1.360,000 
 
 (0.0046) (15^i) (108) = 7.88 sq. in. 
 17 - 9i0 + 2 - y2<f> = 7.90 sq. in. 
 
 55 
 
 0.34 % 
 
 (15^^)2(108) 
 
 (0.0034) (15H) (108) = 5.56 sq. in. 
 12 - ^l<^ + 2 - ^y^(t> = 5.69 sq. in 
 
 — 1 in. = 8H-in., short bands 
 
 — VA in. = 8-in., long bands 
 340,000 
 
 (8) (0.9) (18,000) 
 498,750 
 (8}-^) (0.9) (18,000) 
 
 = 2.62 sq. 
 = 3.63 sq. 
 
 14 - 1.^0 = 2.74 sq. in. 
 19 - i/2<^ = 3.73 sq. in. 
 
 Make long main band 10 ft. 0 in. 
 
 Make short main band 9 ft. 6 in. 
 
 Mid. 
 Mid. 
 
 band 10 ft. 0 in. 
 
 band S ft. 6 in. 
 
 206. Computations for Corr -plate Floors. 
 
 be taken as typical of the method used in the design 
 of Corr-plate floors based on the Standard Building 
 Regulations of the American Concrete Institute, 1917. 
 As will be noted, the designs are for typical interior 
 panels only, and special conditions and exterior panels 
 will require a different treatment. The computa- 
 tions were furnished the writer through the courtesy 
 of the Corrugated Bar Co. of Buffalo, N. Y. 
 
 The moment distribution usually adopted in 
 Corr-plate floors is given in Fig. 61. In the prob- 
 lems here presented the moment distribution recom- 
 mended by the American Concrete Institute is used. 
 
 Typical Interior Panel (Fig. 62). — 
 
 Data: Panel = 20 ft. 4 in. by 20 ft. 4 in. 
 Live load = 175 lb. per sq. ft. 
 
 fs = 18,000 fc = 750 
 ^1 = ^2 = 20.33 
 
 c = 4.5 qc = %(4.5) = 3 ft. 0 in. 
 
 Assuming that a 754 -in. slab will be required, we find from the 
 thickness formula that the total slab thickness 
 
 -The following computations may 
 
 Va/ues oFZ m fhe equation M= 
 
 t = 0.02 X 20.33 X \/l75 + 93 -f 1 = 7.65 
 
 Therefore the assumption made is satisfactory. 
 
 The sum of the positive and negative moments in 
 pounds or the total moment 
 
 = KO.Og) (268) (20.33) (17.33)2(12) 
 - 1.767,500 in.-lb. , 
 
 inch- 
 
 Disthbution across ends of pane/ 
 Fig. 61. 
 
490 
 
 CONCRETE ENGINEERS' HANDBOOK 
 
 [Sec. 11-206 
 
 Corr-plate moment distribution to agree with A. C. I. Standards of 1917 
 
 A. C. I. section 
 
 Negative moment in 
 % of total moment 
 
 A. C. I. section 
 
 Positive moment in 
 % of total moment 
 
 Col. head section | 
 
 Middle section 
 
 2 bands A = 31 % 
 2 bands B = 21 % 
 1 band C = 13% 
 
 
 2 bands A = 13% 
 2 bands B = 9 % 
 1 band C = 13 % 
 
 Negative Moment Steel in Each Direction. — 
 
 At drop panel, jdfs = (0.875) (9.5) (18,000) = 149,625 
 At outer section, jWs = (0.895) (6.5) (18,000) = 104,100 
 (1.767,500) (0.31) 
 
 5fr.f5 ctoc. 
 
 —/o'-z"- ^.^^-^--^il 
 
 c — - 
 
 I 
 I 
 
 1 
 
 1 
 I 
 I 
 
 4 
 
 Fig. 62. 
 
 Positive Moment Steel in Each Direction. — 
 
 (1.767,.500)(0.13) 
 
 2 bands A = 
 2 bands B = 
 1 band C = 
 
 104,100 
 (1.767,500')(0-09) 
 
 104,100 
 (1,767,500)(0.13) 
 
 104,100 
 
 2.21 sq. in. Use 12 — J-i-in. rounds, bend 10. 
 1.53 sq. in. Use 8 — J^^-in. rounds, bend 6. 
 2.21 sq. in. Use 12 — Vi-in. rounds, bend 6. 
 
 The area of steel required for negative moment in the several bands is secured by bending up the necessary 
 number of bars from the bottom of the slab or by bending up alternate bars and supplying the deficiency in nega- 
 tive reinforcement by introducing straight bars in the top of the slab. 
 
 (0.3) (20.33)^268) 
 
 (122) (0.875) (9.5) = ^"•^ 
 
 The unit shearing stress u = 
 
Sec. H-206] 
 
 BUILDINGS 
 
 491 
 
 Typical Interior and Exterior Panel (Fig. 63). — 
 
 Data: Panel = 20 ft. 4 in. by 22 ft. 6 in. 
 Live load = 175 lb. per sq. ft. 
 
 fs = 18,000 fc = 750 
 
 h = 20.33 ft. h = 22.5 ft. 
 
 c = 5.0 qc = %{5.0) = 3.33 
 
 L = 21.42 
 
 Assuming an 8-in. slab 
 
 < = 0.2 X 21.42 X Vl75 -f- 96+ 1 = 8.08. 
 The assumption made is therefore satisfactory. 
 Design for the Long Span, h = 22.5 ft. — 
 
 Total moment = (0.09)(271)(20.33)(19.17)2(12) 
 = 2,182,000 in.-lb. 
 
 Fia. 63. 
 
 Note. — The same distribution of moment applies for the rectangular panel as that given for the square panel. 
 Negative Moment Steel. — 
 
 At drop panel jdf, = (0.875) (10.5) (18,000) = 165,400 
 At outer section yd/s = (0.895) ( 7.0) (18,000) = 112,800 
 
 2 bands A = 
 2 bands B = 
 
 1 band C = 
 Positive Moment Steel. — 
 
 2 bands A = 
 
 (2, 182,000) (0.31) 
 
 165,400 
 (2,182,000) (0.21) 
 
 165,400 
 (2,182,000) (0.13) 
 112,800 
 
 (2,182,000) (0.13) 
 112,800 
 
 = 4.08 sq. in. 
 = 2.77 sq. in. 
 = 2.52 sq. in. 
 
 = 2.52 sq. in. Use 9 — H-in. square, bend up 8 
 
 2 bands B = (^'182,^000)^(0.09) ^ ^ ^ _ ^^^ .^ square, bend up 6 
 
 1 band C = (^•182g)0K0.13) ^ ^ 52 gq. in. Use 10 - H-in. square, bend up 5 
 
492 
 
 CONCRETE ENGINEERS' HANDBOOK 
 
 [Sec. 11-20C 
 
 Design for the Short Span, h = 20.33 fl. — 
 
 Total moment = (0.09)(271)(22.5)(17)2(12) 
 = 1,903,000 in.-lb. 
 
 Negative Moment Steel. — 
 
 At drop panel jd/^ = (0.875) (11) (18,000) = 173,200 
 At outer section jd/s = (0.895) (6.5) (18,000) = 104,700 
 (1,903,000) (0.31) 
 
 173,200 
 (1,903.000) (0.21) 
 
 2 bands D = 
 2 bands E = 
 
 1 band F = 
 Positive Moment Steel. — 
 
 2 bands D = 
 
 2 bands E = 
 1 band F = 
 
 173,200 
 (1,903,000) (0.13) 
 104,700 
 
 (1,903,000) (0.13) 
 
 104,700 
 (1,903.000) (0.09) 
 
 104,700 
 (1,903.000) (0.13) 
 
 3.41 sq. in. 
 2.31 sq. in. 
 2.36 sq. in. 
 
 2.36 sq. in. Use 10 — yi-xn. square, bend up 7 
 1.63 sq. in. Use 6 — J^-in. square, bend up 5 
 2.36 sq. in. Use 10 — M-in. square, bend up 5 
 
 104,700 
 
 Note. — For exterior panels — provided the span lengths remain unchanged — the positive moment steel, and 
 the negative moment steel at the first row of interior columns, in the bantis normal to the wall is to be increased by 
 20%. 
 
 20c. Computations Based on Pittsburgh Ruling. — The following design of a 
 typical panel according to the Pittsburgh Ruling is here inserted to illustrate this method of 
 
 1- 
 
 . L -.- -r 
 
 Fig. 64. 
 
 computation and it is not recommended for universal adoption. In order to give a design 
 the equal of that recommended by the American Concrete Institute, the value of M and M' 
 will have to be increased or a lower steel stress used. It will be observed that the concrete 
 sections required are thinner and the amounts of steel computed are less than that required 
 by either the Chicago Code or the American Concrete Institute Rulings. 
 
 Panel = 20 ft. 10 in. by 22 ft. 2}4 in. 
 Assume SJ-i-in. slab and a 4y2-in. drop. 
 Column cap = 4 ft. 9 in. diameter. • 
 
 Live load = 250 lb. per sq. ft. 
 Dead load = 106 lb. per sq. ft. 
 Total load = 356 lb. per sq. ft. 
 
 Using the Pittsburgh Ruling which is based on the radial distribution of stress theory, the total bending moment 
 
 WL 
 
 in the center of the suspended slab ia M = "yg"' where W is the total live and dead load on the panel and L is the 
 
 pa^iel side (Fig. 64). fs = 16,000 lb. per sq. in. 
 
 For the short direct bands, /c = 750 lb. per sq. in. 
 
 ^ _ (3561.20.83)H20.83)a2) ^ 
 lb 
 
 d = 8Vi in. — 1 in. = 7}'^ in. Area of steel = 22.54 sq. in. 
 
 113 — J-i-in. round rods 
 
Sec. ll-20c] 
 
 BUILDINGS 
 
 493 
 
 113 
 
 This steel is distributed among eight bands and the proportion for one short direct band is — g- = 14 — 
 
 in. round rods per band. 
 
 For the long direct bands we have 
 
 ^ _ (356) (22.18)^(22.18)(12) _ 
 lb 
 
 d = 7y2 in. Area = 26.90 sq.in. = 135 - Vz-in. rounds 
 135 
 
 — g- = 17 — J'^-in. rounds per band 
 
 The diagonal bands in an oblong panel are calculated on the basis of the average panel length. 
 
 (20.83) (22.18) 
 Average panel = 2 — ^1.5 ft. 
 
 (356) (21 .5)^) (21. 5) (12) „ 
 M = = 2,650,000 in.-lb. 
 
 d = 7^2 in. Area of steel = 24.52 sq. in. = 123 — J^-in. round rods 
 1 23 
 
 — g- = 16 — y2-in. round rods per band. 
 
 Check of steel over column head. 
 
 Using the same ruling as before, the bending moment at the column head to be resisted by eight bands is 
 W'L' 
 
 M = where W is the total panel load exclusive of that over the column capital and L' is the span measured 
 
 from edge of cap to edge of cap. In this case, as the difference of spans is small, we can use the average span for L' 
 
 Area of slab = (20.83) (22. 18) = 462.0 
 Area of col. cap. = 17.7 
 
 444.3 
 
 W = (444.3) (356) = 158,000. L' = 21.5 - 4.75 = 16.75 ft. 
 
 (158,000) (16.75) (12) „^ ^ . „ 
 
 M = -^^ = 2,892,000 in.-lb. 
 
 d = 13 in. — 2 in. = 11 in. Area of steel = 18.77 sq. in. = 94 — i-^-in. round rods 
 
 The total band width is 0.4 of the average span, which is 8.6 ft. The 94 — Vz-xn. round rods are not to be dis- 
 tributed over the whole band width but over the width of the column cap plus twice the effective depth. This is 4 
 ft. 9 in. + 2 X 11 in. = 6 ft. 7 in. 
 
 The code specifies that over the column head the computed spacing must be maintained for the full band width 
 
 The steel required then in the eight bands radiating from the column capital will be (94) = 123 — }^-in. round 
 rods 
 
 We actually have 
 
 2 short direct bands 14 = 28 
 2 long direct bands @ 17 = 34 
 4 diagonal bands @ 16 = 64 
 
 126 — Yz-in. round rods 
 which gives a margin of safety over the required amounts. 
 
 Shear at the edge of the column cap = (179)^]^) = 80 lb. per sq. in. with an allowable punching shear of 
 120 lb. per sq. in. 
 
 Stirrups are not necessary in this form of construction as all the slab rods are in the top of the slab over the 
 drop head and are in the bottom at the center of the span. 
 
 18.77 
 
 V = X 11 ^ 0.0069 (allowable = 0.0097 for 16,000 and 750) O. K. for fc 
 
 The percentage of steel in the center of the span is always low and compression of that point is never a con- 
 trolling factor. * 
 
 If standard concrete sections are used, it will seldom be necessary to check other portions 
 of the slab design, except under special conditions. When such is not the case, the compres- 
 sion in the concrete should be checked at the edge of the drop head. Where a smaller depth of 
 drop head is used or is omitted altogether, the compression around the edge of the column cap 
 will be the controlling feature and, in order to keep to the allowable stress, compression steel 
 may have to be inserted. 
 
 Where continuity cannot be secured, as for example at the exterior columns, the direct 
 and diagonal bands ending at the exterior columns should be increased 20%.. 
 
494 
 
 CONCRETE ENGINEERS' HANDBOOK 
 
 [Sec. ll-20d 
 
 20d. Computations Based on Chicago Ruling. — The computations of a simple 
 panel which follow are inserted to illustrate the method of applying the Chicago Ruling to a 
 four- way system. Results by this method are very similar to those obtained by the A. C. I. 
 Ruling. The across direct bands which are called for under this ruling are not required under 
 the Pittsburgh Ruling and many designers who use the Chicago method do not put these rods 
 in the top of the slab across the column center lines. While a small amount of reinforcement is 
 required at this point, it is doubtful whether its omission would entail serious cracking along 
 the edge of the panel in the drop-head type of construction where conservative concrete sections 
 are used. 
 
 Panel 24 ft. 8 in. by 24 ft. 8 in. 
 
 Assume a lOJ'i-in. slab 
 
 and a 63^-in. drop. 
 Drop head 8 ft. 9 in. by 8 ft. 9 in. 
 Col. cap 5 ft. 9 in. diameter. 
 
 Live load = 200 lb. 
 Finish = 12 lb. 
 Dead load = 138 lb. per sq. ft. 
 350 
 
 fs = 16,000 lb. per sq. in. 
 fc = 750 lb. per sq. in. 
 
 The formula for the minimum thickness of slab as given by the Chicago Code is t = 0.023 L-\J w where t is 
 the total thickness of slab in inches, L is the panel length in feet and mo is the total 
 1^ , I live and dead load in pounds per square foot. 
 
 t = 0.023 X 24.67 \/350 = 10.6 in. so our assumption is satisfactory, = 
 
 9.25 in., the other code requirement, is exceeded. 
 
 The column cap must be at least 0.225 X L = 5.56 ft. which the assumed dimen- 
 sion of 5 ft. 9 in. satisfies. 
 
 The width of strips A and B (Fig. 65) as used in the moment computations will 
 be 12 ft. 4 in. W, used in the formulas, will be (12.33) (350) = 4315 lb. 
 Negative moment for strip A. 
 
 WL'^ _ (4315) (24.67)'' (12) 
 15 ~ 
 
 
 
 
 
 1 — 
 
 <• 
 
 
 
 
 \— 
 
 .1, 
 
 i 
 
 
 h 
 
 
 M 
 
 15 
 
 = 2,103,000 in.-lb. 
 
 Fig. 65. 
 Positive moment for strip B 
 
 Positive moment for strip A 
 
 ELI _ 
 40 
 
 M = 
 
 (4315) (24.67)2 (12) 
 
 Negative moment for strip B 
 
 M 
 
 M 
 
 60 
 
 TFL2 
 60 
 
 (4315) (24.67)2 (12) 
 60 
 
 525,700 in.-lb. 
 
 40 
 
 = 525,700 in.-lb 
 
 = 787.500 in.-lb. 
 
 The negative moment in strip A is resisted by one diagonal and one direct band. 
 
 2,103,000 
 
 Area of steel = (o.87)(l7 - 2)(16,o"00) = ^^"^^ 
 
 40 — V2-\r\. sq. rods. 
 
 which are not necessarily divided equally between one direct and one diagonal band. For distribution see the 
 following computation: 
 
 The positive moment in each strip A is resisted by the steel in one cross or direct band. 
 
 d = lOJ-i in. - 1 in. = 9}i in. 
 787,500 
 
 Area of steel = 
 
 (0.91) (9.5) (16,000) 
 
 5.70 sq. in. 
 
 23 — H-in. sq. rods. 
 
 The positive moment of each strip B shall be resisted by the steel in one diagonal band. 
 Area of steel 
 
 525,700 
 
 (0.91)(9.5)(16,000) 
 
 3.785 sq. in. 
 16 — I'^-in. sq. rods. 
 
 The negative moment of strip B is resisted by the across direct bands which are short rods placed in the top 
 of the slab over and at right angles to the direct bands. 
 
 Area of steel = 3.785 sq. in. 
 
 = 16 — Vi-'m. sq. rods. 
 
Sec. ll-20e] 
 
 BUILDINGS 
 
 495 
 
 The proper steel distribution will then be: 
 24 — H-in. sq. rods in direct bands. 
 16 — Vi-in. sq. rods in diagonal bands. 
 16 — H-\n. sq. rods across direct bands. 
 
 For exterior panels the amount of reinforcement is to be increased by the use of the moment coefficients 
 specified for this case. 
 
 If sections given in the tables are used, it will not be necessary to check the shears or concrete stresses except 
 in special cases. 
 
 The methods of computation used for rectangular panels are clearly explained in the text so that the detailed 
 method will not be given here. In fact the entire ruling is very clearly written and covers the vital points of design 
 very well. 
 
 20e. Computations Based on Ruling of American Concrete Institute. — The 
 
 following design of a simple panel of a four-way flat slab under the A. C. I. Ruling will serve to 
 make clear the wording of the ruling. As in all other problems in this chapter, the ordinary 
 straight line stress variation is assumed and, as in other cases, the critical sections so far as 
 compression in the concrete goes are either around the column capital or at the edge of the drop 
 panel. This method gives a very similar steel distribution and general results much like those 
 obtained by the Chicago Ruling. 
 
 Given an interior panel 25 ft. 6 in. by 25 ft. 6 in. to be designed to carry a live load of 175 lb. per sq. ft. 
 plus a 1-in. cement finish. Assume a slab thickness t = 10?4 in. and a drop G\i in. thick and 9 ft.O in. by 9 ft. 
 0 in. The column capital will be 5 ft. 9 in. diameter. (Refer to Fig. 3, Appendix C, page 858.) 
 
 Testing the assumed dimensions by the ruling we have as follows: 
 
 t = 0.02L v^"- + 1 in. Live load = 175 lb. per sq. ft. 
 
 t = 0.02 X 25.5\/3T6 + 1 in. Dead load = 129 lb. per sq. ft. 
 
 t = 10.07 in. . Finish load = 12 lb. per sq. ft. 
 
 Total load w = 316 lb. per sq. ft. 
 ~= = ^-"^^"^ = ^•^'^ = 10,000 lb. per sq. in. 
 
 fc = 750 lb. per sq. in. 
 The assumption of i = lOH in. is thus seen to be satisfactory. 
 
 The drop where used shall not be less than 0.3L = (0.3) (25.5) = 7.65 ft. and the assumed size of drop 9 
 ft. by 9 ft. need not be changed. 
 
 The sum of the positive and negative moments shall not be less than O.OQwhih — qc)-, where w is the unit 
 live and dead load in pounds per square foot, li is the span in feet center to center of columns parallel to sections 
 at which moments are considered, ^2 is the span in feet center to center of columns perpendicular to sections 
 at which moments are considered, c the average diameter of column capital in feet at the point where its thickness 
 is l}'Hn.,and q is the distance from center line of capital to the center of gravity of the periphery of the half 
 
 capital divided by | (for our case q = 
 
 Total moment = (0.09) (316) (25.5) (25.5 - %X 5.75)2(12) = 4,086,600 in.-lb. 
 
 Under the ruling 50 % of this must be resisted by the steel over the column head. (Refer to Fig. 3, page 858.) 
 Not less than 10 % shall be resisted in the mid-section, not less than 18 % shall be resisted in the outer sections, 
 and not less than 12% of the total moment shall be resisted on the inner section. 
 The following distribution is adopted: ■ 
 
 Column-head section 50% 
 
 Mid-section 10% 
 
 Outer section (direct bands) 20 % 
 
 Inner section (diagonal bands) 20% 
 
 Total 100% 
 
 The moments to be taken by the different sections will be as follows: 
 
 Column-head section 2,043,300 in.-lb. 
 
 Mid-section 408,660 
 
 Outer section 817,320 
 
 Inner section 817,320 
 
 Total 
 
 4,086,600 in.-lb. 
 
496 
 
 CONCRETE ENGINEERS' HANDBOOK 
 
 [Sec. 11-20/ 
 
 The column-head section includes two diagonal bands and one direct band. 
 
 The effective depth is IGYz in. — 2 in. = 14H in. (assuming 2 in. to the center of gravity of the steel). With 
 fs = 16,000 and fc = 650, we have the area for three bands in top of slab over the column head 
 
 2,043,300 
 
 ~ (14.5) (0.874) (16,000) - 10-08 sq. in. 
 which is equal to 40 — H-in. sq. bars. 
 
 For the mid-section d = lOH— IH = 9 in. 
 
 408,660 
 ' ~ (9) (0.91) (16,000) ~ ^-^^ 
 
 which is equal to 13 — J-^-in.sq. bars in the across direct bands located in the top of the slab across the sides of 
 the panel. 
 
 In the outer section d = 9 in. 
 
 _ 817,320 _ o., . 
 
 ~ (9) (0.89) (16,000) - ^-^^ ^"^^ 
 
 which is equal to 26 — H-in. sq. rods for each direct band. 
 
 Under this system of moment distribution, the effective area of steel for the inner section — i.e., the two diag- 
 onal bands — will be the same as that for the direct band or 6.23 sq. in. The actual area then of each diagonal 
 
 band must be ( ' 2)(o 707) ^ ^-'^'^ which is 18 — J-^-in. sq. rods. 
 
 Assuming that all the diagonal rods are bent up over the head to form negative reinforcement, the amount 
 of steel in a direct band which will have to be bent up will be 10.08 — 6.33 (effective area of two diagonal 
 bands) = 3.75 sq. in., which is equal to 15 — }^-in. sq. rods. As there ai;e 26 — H-in. sq. rods in each direct band, 
 1 1 may be stopped at the edge of the column capital or carried through as compression reinforcement. Another 
 allowable arrangement would be to bend up all the bars in the direct band over the column head, which would re- 
 quire an effective area of 10.08 — 6.33 = 3.75 sq. in. in the two diagonal bands to be bent up, which equals 
 3.75 
 
 (2) (0 707) ~ 2.65 sq. in., or 11 — J'^-in. sq. rods per diagonal band. These rods should be carried (0.35) (25.5) = 
 8.92 ft., or 8 ft. 11 in. past the center line of the column. 
 
 The balance of the rods in the diagonal band should extend (0.35) (25.5) = 8 ft. 11 in. on either side of the 
 center of panel — that is, they must be at least 17 ft. 10 in. long in this case. 
 
 It is common practice to use rods in the direct bands which are long enough to cover two spans and when 
 this is done the laps should be staggered. Since they are spliced over the column head, each bar should extend 
 40 diameters or 20 in. past the center line of splice in this case. 
 
 Still another possible arrangement of steel would be to arrange the diagonal bands as just outlined and then 
 to only bend up half the bars in the direct bands, lapping them 8 ft. 11 in. past the center line from each side so as 
 to make up the necessary area over the head. This, of course, only applies where the steel in the direct bands 
 extends for one span only. Where rods covering two spans are used in the direct bands, this arrangement will 
 have to be modified or extra short rods will have to be added over the column head. 
 
 20/. Roof Design. — In the design of flat-slab roofs it should be remembered 
 that the deflection rather than the strength may be and usually is the controlling factor. For 
 this reason very thin slabs should not be used. The Chicago and A. C. I. Rulings limit the 
 thickness of the roof slab to one-fortieth of the span as a minimum. For a 20-ft. span this 
 would give a thickness of 6 in. which has been found by experience to be about right for a roof 
 slab of this span. 
 
 There are two methods of constructing roofs in common use. The first method is to pitch 
 the slab, laying the roofing surface directly on the concrete. The second method is to cast 
 the roof slab level and obtain the required pitch by means of a cinder-concrete fill (of varying 
 depth) which is surfaced with a cement finish and upon which the roofing membrane is laid. 
 The first method involves a comparatively small dead load and, as the roof live load is small, 
 — generally 40 lb. per sq. ft. — the total load for which the slab will be designed is small. In the 
 second case the dead load may be very considerable in certain parts of the slab where a deep 
 cinder fill is necessary. This matter should obviously receive consideration in the design. 
 
 The writer has made it a practice in the first case mentioned — i.e., a pitched slab roof — 
 to never design for a lighter live load than 75 lb. per sq. ft. This means a total load of 75 lb. 
 plus the dead load of the slab and roof finish. Some other engineers use even more than this. 
 The reason for such a rule is that the roof slab, being subjected to the maximum temperature 
 variation, both daily and seasonal, is very apt to crack if the percentage of steel is too small and 
 designing with light loads will not give sufficient reinforcement to resist these stresses. 
 
Sec. Il-20g7] 
 
 BUILDINGS 
 
 497 
 
 Frequently in roof design the question of the best design to accommodate sawtooth sky- 
 Hghts will arise. An economy in the use of concrete and a more rational design will often 
 result if a beam-and-girder design is used instead of the flat-slab type for such cases. Some 
 designers prefer the older type for the roofs of all flat-slab buildings. 
 
 20g. Beams in Flat-slab Floors. — Two general cases of the design of beams 
 forming a part of fiat slabs arise. The first is that of beams located between the exterior 
 columns and supportmg the edge of the slab and carrying either a low wall under the windows, 
 or, where no windows occur, a wall extending from the floor to the underside of the beam above. 
 The stresses to be withstood are torsional, due to the deflection of the panel, and tension, due 
 to bending under applied vertical load. For this reason a wide shallow beam as nearly square 
 in section as possible is to be preferred. Where a drop panel is used, it is advantageous to 
 make the beams the same depth as the drop. Lighting consideration make it advisable to. 
 keep the depth as small as possible. Parallel and adjacent to the beam there is a half band of 
 steel in the bottom of the slab which has a smaller effective depth than the reinforcement of 
 the wall beam. It is evident, therefore, that a part at least of this band cannot be stressed 
 to its full value until the steel in the bottom of the beam is overstressed. 
 
 It is the practice of some designers to calculate the steel in the wall beams for the wall 
 load and beam dead load only, and there are several cases where such a practice has given 
 satisfactory results. With this method, however, there is every chance of having cracked 
 beams and exterior walls. The writer believes that in addition to the load above specified, 
 the live load on a portion of the floor should be included. It is recommended that in addition 
 to the weight of the beams and wall, the exterior beams be calculated to carry, as a uniform 
 load, the live load of a strip of floor adjacent to the beam, of length equal to the span, and width 
 equal to one-sixth the span at right angles to the exterior wall of the building. 
 
 For the second-class beams located in interior panels, the same rule may be used, except 
 that where the beam carries concentrated or stair loads these must be provided for in addition. 
 Some designers prefer the wide, shallow type of beam with a depth equal to the drop head which 
 has, besides the advantage of increased clearance, the added advantage of less liability of 
 unsightly cracking under excessive deflection. As will be evident from a study of any building 
 design, cases of this kind are bound to arise which are indeterminate. Under these conditions 
 the application of such theoretical analysis as will apply must be modified by experience based 
 on former designs, successful or otherwise. 
 
 Continuity should be secured in all beams whenever possible, particularly in the wall 
 beams. Adequate temperature reinforcement in exposed concrete surfaces should always 
 be provided. Where the wall beams do not extend below the floor slab, it is usually possible 
 to utilize the entire wall below the window as a beam and a very stiff construction can be secured. 
 The danger of cracking, due to torsional stresses, is greater, however, in this type of design. 
 
 20h. Columns. — The calculation of the columns supporting a flat slab is one 
 of the most important parts of the entire design and is more frequently overlooked than any 
 other portion of the work. 
 
 Tests have shown conclusively that both interior and exterior columns below and above 
 the loaded floor are stressed by an unbalanced live load in the panel which they support. This 
 point is well covered by the A. C. I. Ruling and was also covered in a way by the Chicago and 
 Pittsburgh Rulings, but in many cases little consideration has been given to it. 
 
 There is a provision in the Chicago Ruling that no column shall be less than one-twelfth 
 the panel length or one-twelfth the clear height in thickness. For the interior columns in all 
 stories below the roof story, this rule is good and should be used even though it will often give 
 a larger section than the direct load will require. An exception may be made where structural- 
 steel cores are used. In the story below the roof somewhat smaller columns may be used as 
 there is little possibility of unbalanced live load on the roof, but it is the writer's opinion that 
 12 in. diameter should be the minimum size for an interior column. 
 
 For the exterior columns the Chicago Ruling will frequently give rather thicker sections 
 32 
 
498 CONCRETE ENGINEERS' HANDBOOK [Sec. ll-20i 
 
 than are necessary, particularly when wide columns are desired on account of the exterior 
 appearance of the building. The following minimum thicknesses are recommended for exterior 
 reinforced-concrete columns: 
 
 16-ft. span 14 in. 
 
 20-ft. span 16 in. 
 
 24-ft. span 18 in. 
 
 28-ft. span 20 in. 
 
 The method of computing the moment in and designing columns supporting flat-slab 
 buildings as recommended by the American Concrete Institute will give conservative results 
 and should meet with universal adoption. At present commercial competition has led to a 
 slighting of the reinforcement to resist bending in the exterior columns in many cases, with 
 the result that hair cracks have developed on the outside of the column at the base of the 
 bracket. No serious consequences accompany such cracks, but if the adjacent panels were 
 overloaded, a dangerous condition would probably develop. 
 
 As a practical minimum for vertical reinforcement in columns, the equivalent of four ^^-in. 
 round rods has been adopted by some designers and is a safe minimum for general use. 
 
 The bending stresses in columns will be found to be more important in the upper 
 stories than in the lower floors on account of the increase in the direct load as the footings are 
 approached. 
 
 For moments in columns in fiat-slab construction, see also Art. 7, Sect. 10. The methods 
 of computing stresses in columns due to bending and direct stress are given in Sect. 9. 
 
 20i. Brick Exterior Wall Supports. — Several of the building codes allow the 
 use of brick bearing walls for the exterior support of flat-slab floors. The fact that such a 
 support cannot develop a large negative moment is offset by providing an excess of 40% of 
 steel in the adjacent slabs as against 20% where reinforced-concrete columns are used. Some 
 structures of this class have not given entire satisfaction and one failure is attributed to the 
 weakness of the brick supporting walls {Engineering News, vol. 76, page 262). Two difficulties 
 arise in this connection ; one is to determine what thickness of wall is necessary and the other 
 is to secure first-class brickwork. Another element which is almost certain to cause trouble 
 is that the interior columns and brick exterior walls settle at different rates which, in a multi- 
 story building, will introduce serious strains in the slabs. Where such a construction must 
 be used, it is recommended that pilasters of a total thickness at least equal to one-twelfth of 
 the span be used and that a heavy continuous beam be carried around the edge of the floor. 
 
 It is inadvisable to use corbels or brackets on the brick pilasters as they will increase the 
 bending stress in the pilaster and may actually endanger the structure. If they are required 
 for shear, the slab drop or beam should be increased in size to take care of it and no attempt 
 should be made to restrain the slab at a brick wall or pilaster bearing. Usually there is little 
 if any saving in total cost by the use of brick, and the reinforced-concrete skeleton building 
 veneered with brick is superior structurally. 
 
 21. Tables. Pittsburgh Ruling. — The following tables have been computed for a standard 
 four-way flat slab having square panels, drop heads, and reinforced according to the Pittsburgh 
 Ruling. 
 
 The stresses used are as follows : 
 /. = 16,000 lb. per sq. in. 
 
 fc = not more than 750 lb. per sq. in. at the edge of column capital. 
 The shear at the edge of the drop head 
 V 
 
 V = ^ is not greater than 60 lb. per sq. in. 
 For 100 lb. per sq, ft. superimposed load, the drop head is 0.35L square. 
 
Sec. 11-21] 
 
 BUILDINGS 
 
 499 
 
 For 150 to 400 lb. per sq. ft. superimposed load, the drop head is 0.4L square, where L is 
 the center to center distance between columns. 
 
 The superimposed load means the live or live and dead load above the structural slab and 
 may or may not include the weight of the cement finish, depending upon whether it is laid at 
 the same time as the slab or at a later date if it forms a part of the structure at all. 
 
 The computations are based on a typical interior panel and the amounts of steel given 
 in the tables must be increased in exterior panels of the same span in accordance with the Pitts- 
 burgh Code requirements. Where concrete exterior columns are used and are so constructed 
 and reinforced that they will withstand the bending stresses developed by the unbalanced load, 
 the amount of steel given in the tables should be increased 20% for all the bands which end at 
 the outside of the building or are parallel to and adjacent to the wall beam. 
 
 Bands which extend from wall to wall, such as diagonal bands at corner panels, should 
 be increased 40% over the values given in the tables. It is the writer's opinion that the same 
 method should be followed in treating exterior panels where brackets are not provided at the 
 wall columns and insufficient vertical steel is used in them to properly resist the bending stresses 
 due to the unbalanced loading, i.e., the slab reinforcement should be increased 40%. 
 
 Where reinforced-concrete interior columns are used and the exterior panels are carried 
 on brick piers or walls, the Pittsburgh Ruling recommends that the steel in the exterior panels 
 be increased 40%. 
 
 Where, as is frequently the case, the span of the exterior panels is greater or less than that 
 of the interior panels, due consideration must be given to these conditions, and the preceding 
 remarks regarding exterior panels will be modified thereby. 
 
 These tables are given here principally as a check on design and as an aid to estimating, 
 and it is not the intention that they should take the place of a carefully considered design in 
 any special case. Particularly is this true where conditions, due to irregular panels, openings, 
 or concentrated loading make an accurate analysis of actual stresses imperative. 
 
 Where panels are not square and where the angles between column center lines are not 
 right angles, the tables do not apply and cannot be used without modification. 
 
 For intermediate spans and loads the proper values (for square panels) may be obtained 
 from the tables approximately by interpolation. 
 
 Flat-slab Panels — Pittsburgh Regulations 
 Interior Panels — Superimposed Load 100 lb. per sq. ft. 
 
 Panel 
 
 Capi- 
 tal 
 
 Head 
 
 T 
 
 Total 
 drop 
 
 t 
 
 Concrete 
 in cu. ft. 
 
 Steel in each band 
 
 Steel 
 in. lb. 
 
 diame- 
 ter 
 
 Slab 
 
 per 
 sq. ft. 
 
 Direct 
 
 Diagonal 
 
 per 
 sq. ft. 
 
 16' X 16' 
 
 3'-6" 
 
 5'-7" X 5'-7" 
 
 8" 
 
 5" 
 
 0.465 
 
 
 11-% '> 
 
 1.460 
 
 17' X 17' 
 
 3'-9" 
 
 6'-0" X 6'-0" 
 
 
 5>^" 
 
 0.492 
 
 
 12-% "0 
 
 1.491 
 
 18' X 18' 
 
 4'-0" 
 
 6'-4" X 6'-4" 
 
 9" 
 
 5M" 
 
 0.513 
 
 14-^r'<^ 
 
 14-%"</> 
 
 1.510 
 
 19' X 19' 
 
 4'-3" 
 
 6'-8" X 6'-8" 
 
 
 6" 
 
 0.535 
 
 15-M"<^ 
 
 15-%"<^ 
 
 1.610 
 
 20' X 20' 
 
 4'-6" 
 
 7'-0" X 7'-0" 
 
 
 6M" 
 
 0.559 
 
 
 17-% "(^ 
 
 1.700 
 
 21' X 21' 
 
 4'-9" 
 
 7'-4" X 7'-4" 
 
 lOM" 
 
 6M" 
 
 0.580 
 
 15-% 
 
 15-% "□ 
 
 1.780 
 
 22' X 22' 
 
 5'-0" 
 
 7'-8" X 7'-8" 
 
 lOM" 
 
 7" 
 
 0.622 
 
 17-% "□ 
 
 17-% 
 
 1.980 
 
 23' X 23' 
 
 5'-3" 
 
 8'-l" X 8'-l" 
 
 
 
 0.644 
 
 19-% "□ 
 
 19-% "□ 
 
 2.140 
 
 24' X 24' 
 
 5'-6" 
 
 8'-5" X 8'-5" 
 
 
 
 0.669 
 
 15->^> 
 
 15->^"0 
 
 2.190 
 
 25' X 25' 
 
 5'-9" 
 
 8'-9" X 8'-9" 
 
 12M" 
 
 
 0.691 
 
 16-K"</> 
 
 16->^"</> 
 
 2.260 
 
 26' X 26' 
 
 5'-9" 
 
 9'-2" X 9'-2" 
 
 
 
 0.740 
 
 18-H"</> 
 
 18->^"</> 
 
 2.430 
 
 27' X 27' 
 
 6'-0" 
 
 9'-6" X 9'-6" 
 
 13M" 
 
 
 0.763 
 
 19-K"<A 
 
 19->^"</> 
 
 2.480 
 
500 
 
 CONCRETE ENGINEERS' HANDBOOK 
 
 [Sec. 11-21 
 
 Interior Panels — Superimposed Load 150 lb. per sq. ft. 
 
 
 
 
 Capi- 
 
 
 
 
 T 
 
 
 Concrete 
 
 Steel in each band 
 
 Steel 
 
 Panel 
 
 tal 
 
 
 Head 
 
 Total 
 drop 
 
 t 
 
 in cu. ft. 
 
 
 
 in lb. 
 
 diame- 
 
 
 Slab 
 
 per 
 
 
 
 per 
 
 
 
 
 ter 
 
 
 
 
 
 sq. ft. 
 
 Direct 
 
 Diagonal 
 
 sq. ft. 
 
 16' 
 
 X 
 
 16' 
 
 or ntt 
 6 — O 
 
 o — / 
 
 X 
 
 5'-7" 
 
 y 
 
 
 0.4/0 
 
 11-^^ 
 
 11-^^ 
 
 1 '70 
 
 1 . 7o 
 
 17' 
 
 X 
 
 17' 
 
 3'-9" 
 
 6'-0" 
 
 X 
 
 6'-0" 
 
 9>^" 
 
 
 0.518 
 
 12-^^ "□ 
 
 
 1.84 
 
 18' 
 
 X 
 
 18' 
 
 4'-0" 
 
 6'-4" 
 
 X 
 
 6'-4" 
 
 10" 
 
 6" 
 
 0.543 
 
 13-^r'n 
 
 13-^r'a 
 
 1.91 
 
 19' 
 
 X 
 
 19' 
 
 4'-3" 
 
 7'-0" 
 
 X 
 
 7'-0" 
 
 lOH" 
 
 6M" 
 
 0.571 
 
 15-^^ "□ 
 
 i5-M"a 
 
 2.06 
 
 20' 
 
 X 
 
 20' 
 
 4'-6" 
 
 7'-4" 
 
 X 
 
 7'-4" 
 
 11" 
 
 
 0.594 
 
 17-^r'n 
 
 
 2.15 
 
 21' 
 
 X 
 
 21' 
 
 4'-9" 
 
 7'-8" 
 
 X 
 
 7'-8" 
 
 ny2" 
 
 7" 
 
 0.635 
 
 
 18-^^ "□ 
 
 2.21 
 
 22' 
 
 X 
 
 22' 
 
 5'-0" 
 
 8'-l" 
 
 X 
 
 8'-l" 
 
 
 
 0.662 
 
 
 15->^"<^ 
 
 2.44 
 
 23' 
 
 X 
 
 23' 
 
 5'-3" 
 
 8'-5" 
 
 X 
 
 8'-5" 
 
 
 
 0.705 
 
 16^3^ "0 
 
 16-H''<^ 
 
 2.50 
 
 24' 
 
 X 
 
 24' 
 
 5'-6" 
 
 8'-9" 
 
 X 
 
 8'-9" 
 
 13>^" 
 
 8K" 
 
 0.744 
 
 17-^^ "0 
 
 
 2.52 
 
 25' 
 
 X 
 
 25' 
 
 5'-9" 
 
 9'-2" 
 
 X 
 
 9'-2" 
 
 13^^" 
 
 83^" 
 
 0.770 
 
 19-3^ "<A 
 
 19-K"0 
 
 2.70 
 
 26' 
 
 X 
 
 26' 
 
 5'-9" 
 
 9'-8" 
 
 X 
 
 9'-8" 
 
 14M" 
 
 8M" 
 
 0.800 
 
 17-3^ "□ 
 
 i7->^"n 
 
 2.91 
 
 27' 
 
 X 
 
 27' 
 
 6'-0" 
 
 10'-6" 
 
 X 
 
 10'-6" 
 
 15>^" 
 
 9" 
 
 0.830 
 
 18-3^ "□ 
 
 i8->^"n 
 
 2.98 
 
 Flat-slab Panels — Pittsburgh Regulations 
 
 Interior Panels 
 
 Superimposed load 
 
 200 lb. per sq. ft. 
 
 
 
 250 lb. per sq. 
 
 ft. 
 
 
 Side 
 of 
 panel 
 
 Capi- 
 tal 
 diam- 
 eter 
 
 Side 
 of 
 head 
 
 T 
 Total 
 drop 
 
 t 
 
 Slab 
 
 Con- 
 crete 
 in 
 
 cu. ft. 
 per 
 
 sq. ft. 
 
 Steel in each band 
 
 Steel 
 in lb. 
 
 per 
 sq. ft. 
 
 T 
 Total 
 drop 
 
 t 
 
 Slab 
 
 Con- 
 crete 
 in 
 
 cu. ft. 
 per 
 
 sq. ft. 
 
 Steel in each band 
 
 Steel 
 in lb. 
 
 per 
 sq. ft. 
 
 Direct 
 
 Diag- 
 onal 
 
 Direct 
 
 Diag- 
 onal 
 
 16' 
 
 3'-6" 
 
 6'-5" 
 
 10" 
 
 6" 
 
 0.555 
 
 12-% "□ 
 
 12-% "□ 
 
 2.00 
 
 10%" 
 
 
 0.600 
 
 i3-%"n 
 
 13-%" □ 
 
 2.16 
 
 17' 
 
 3'-9" 
 
 6'-10" 
 
 mi" 
 
 6H" 
 
 0.578 
 
 13-% 
 
 13-% 
 
 2.02 
 
 mi" 
 
 7" 
 
 0.645 
 
 14-%" □ 
 
 14- %"□ 
 
 2.18 
 
 18' 
 
 4'-0" 
 
 7'-3" 
 
 mi" 
 
 w 
 
 0.620 
 
 i5-%"n 
 
 15-%"D 
 
 2.19 
 
 12" 
 
 IM" 
 
 0.684 
 
 16-%" □ 
 
 16-%"n 
 
 2.34 
 
 19' 
 
 4'~3" 
 
 7'-8" 
 
 iw 
 
 7" 
 
 0.645 
 
 ]6-%"n 
 
 16-% "□ 
 
 2.27 
 
 12M" 
 
 7%" 
 
 0.710 
 
 18-%" □ 
 
 18-%" □ 
 
 2.92 
 
 20' 
 
 y-6" 
 
 8'-0" 
 
 12" 
 
 7>i" 
 
 0.665 
 
 18-% "□ 
 
 18-% "□ 
 
 2.34 
 
 mi" 
 
 8\i" 
 
 0.755 
 
 14-H"<A 
 
 14-M"<^ 
 
 2.54 
 
 21' 
 
 4'-9" 
 
 8'-5" 
 
 mi" 
 
 iVi" 
 
 0.696 
 
 mH"4> 
 
 15-M"</> 
 
 2.56 
 
 mi" 
 
 8M" 
 
 0.778 
 
 16-M",/, 
 
 16->^"0 
 
 2.74 
 
 22' 
 
 5'-0" 
 
 8'- 10" 
 
 mi" 
 
 7H" 
 
 0.720 
 
 17-K"<A 
 
 17-M"<A 
 
 2.78 
 
 mi" 
 
 9" 
 
 0.825 
 
 17-H"<^ 
 
 17-M"</. 
 
 2.78 
 
 23' 
 
 5'- 3" 
 
 9'-3" 
 
 14" 
 
 8}i" 
 
 0 . 765 
 
 18-K'V 
 
 18-H"<A 
 
 2.88 
 
 15" 
 
 9M" 
 
 0.867 
 
 19-M'V 
 
 19-M"</' 
 
 2.99 
 
 24' 
 
 5'-6" 
 
 9'-8" 
 
 iw 
 
 8H" 
 
 0.81] 
 
 i6-M"n 
 
 i6-H"n 
 
 302 
 
 15%" 
 
 10" 
 
 0.908 
 
 i6-K"n 
 
 16-1^^" □ 
 
 3.02 
 
 25' 
 
 5'-9" 
 
 lO'-O" 
 
 mi" 
 
 9K" 
 
 0.849 
 
 i7-M"n 
 
 17-H"a 
 
 3.06 
 
 16>^" 
 
 10>^" 
 
 0.955 
 
 i8-M"n 
 
 18-3-^" □ 
 
 3.24 
 
 26' 
 
 5'-9" 
 
 10'-5" 
 
 mi" 
 
 OH" 
 
 0.901 
 
 i8-K"n 
 
 i8-H"n 
 
 3.08 
 
 171^^2" 
 
 11" 
 
 1.010 
 
 i9-H"n 
 
 i9-M"n 
 
 3.28 
 
 27' 
 
 6'-0" 
 
 lO'-lO" 
 
 mi" 
 
 mi" 
 
 0.946 
 
 20-K"n 
 
 20-H"n 
 
 3.30 
 
 18H" 
 
 iiM" 
 
 1.050 
 
 2i-M"n 
 
 2i-H"n 
 
 3.48 
 
Sec. 11-21] BUILDINGS 501 
 
 Flat-slab Panels — Pittsburgh Regulations 
 
 Interior Panels 
 
 Superimposed load 
 
 300 lb. per sq. ft. 
 
 
 
 350 lb. per sq. 
 
 ft. 
 
 
 Side 
 of 
 panel 
 
 Capi- 
 tal 
 diam- 
 eter 
 
 Side 
 
 ot 
 head 
 
 T 
 Total 
 drop 
 
 t 
 
 Slab 
 
 Con- 
 in 
 
 cu. ft. 
 per 
 
 sq. ft. 
 
 Steel in each band 
 
 Steel 
 in. lb. 
 
 per 
 sq. ft. 
 
 ToUl 
 drop 
 
 t 
 
 Slab 
 
 Con- 
 in 
 
 cu. ft. 
 per 
 
 sq. ft. 
 
 Steel in each band 
 
 steel 
 in lb. 
 
 per 
 sq ft 
 
 Direct 
 
 Diag- 
 
 Direct 
 
 Diag- 
 onal 
 
 16' 
 
 3'-6" 
 
 6'-5" 
 
 113-^" 
 
 
 0.665 
 
 13-%" □ 
 
 i3-M"n 
 
 2.16 
 
 
 8" 
 
 0.725 
 
 13-% "□ 
 
 i3-%"n 
 
 O Id 
 ^ . ID 
 
 17' 
 
 3'-9" 
 
 6'-10" 
 
 12" 
 
 
 0.702 
 
 15-%" □ 
 
 i5-%"n 
 
 2.33 
 
 
 o>2 
 
 0.765 
 
 15-% "□ 
 
 15-%" □ 
 
 z . OO 
 
 18' 
 
 4'-0" 
 
 7'-3" 
 
 12H" 
 
 8K" 
 
 0.750 
 
 16-M"n 
 
 16-%" □ 
 
 2.34 
 
 13>^" 
 
 9" 
 
 0.810 
 
 17-%" □ 
 
 17-% "□ 
 
 2.49 
 
 19' 
 
 4'- 3" 
 
 7'-8" 
 
 ISH" 
 
 SH" 
 
 0.790 
 
 18-?^ " □ 
 
 18-%" □ 
 
 2.48 
 
 1*>4 
 
 y>2 
 
 0.860 
 
 
 14-M"<^ 
 
 2 . 69 
 
 20' 
 
 4'-6" 
 
 S'-O" 
 
 U}4" 
 
 
 0.840 
 
 15-M"</> 
 
 15-H"<f> 
 
 2.72 
 
 15" 
 
 10" 
 
 0.900 
 
 
 15^>^"</, 
 
 2.72 
 
 21' 
 
 4'-9" 
 
 8'-5" 
 
 15" 
 
 9^i" 
 
 0.882 
 
 
 16-M"<p 
 
 2.75 
 
 15%" 
 
 lOM" 
 
 0.945 
 
 i7-y2"<i> 
 
 17->^"</> 
 
 2.92 
 
 22' 
 
 5'-0" 
 
 S'-IO" 
 
 15>^" 
 
 lOK" 
 
 0.928 
 
 
 i8-y2"<i> 
 
 3.03 
 
 16M" 
 
 11" 
 
 0.995 
 
 iB-yu 
 
 18-M"«^> 
 
 2.94 
 
 23' 
 
 5'-3" 
 
 9'-3" 
 
 16" 
 
 loy^" 
 
 0.970 
 
 15-H"D 
 
 i5-M"n 
 
 2.98 
 
 17>i" 
 
 11%" 
 
 1.058 
 
 i6-K"n 
 
 ]6->^"n 
 
 3.16 
 
 24' 
 
 5'-6" 
 
 9'-8" 
 
 17" 
 
 mi" 
 
 1.020 
 
 
 i7-M"n 
 
 3.20 
 
 18" 
 
 12M" 
 
 1.115 
 
 i7-K"n 
 
 i7-i,^"n 
 
 3.20 
 
 25' 
 
 5'-9" 
 
 lO'-O" 
 
 17M" 
 
 iw 
 
 1.055 
 
 
 i8-M"n 
 
 3.26 
 
 18%" 
 
 13>i" 
 
 1.180 
 
 18-1^^" □ 
 
 i8-M"n 
 
 3.24 
 
 26' 
 
 5'-9" 
 
 10'-5" 
 
 18^" 
 
 12H" 
 
 1.105 
 
 2o-M"n 
 
 2o-M"n 
 
 3.46 
 
 20" 
 
 14" 
 
 1.250 
 
 2o-i.^"n 
 
 2o-H"n 
 
 3.46 
 
 Flat-slab Panels — Pittsburgh Regulations 
 
 Interior Panels 
 
 Superimposed load 
 
 400 lb. per sq. ft. 
 
 Side 
 of 
 
 Capi- 
 tal 
 
 diam- 
 eter 
 
 Side 
 of 
 
 T 
 Total 
 
 t 
 
 Slab 
 
 Con- 
 crete 
 
 in 
 cu. ft. 
 
 per 
 sq. ft. 
 
 Steel in each band 
 
 steel 
 in lb. 
 
 panel 
 
 head 
 
 drop 
 
 Direct 
 
 Diag- 
 onal 
 
 per 
 sq. ft. 
 
 16' 
 
 3'-6" 
 
 6'-5" 
 
 12%" 
 
 8^4" 
 
 0.785 
 
 14-% "□ 
 
 14-% "□ 
 
 2.31 
 
 17' 
 
 3 '-9" 
 
 6'- 10" 
 
 13M" 
 
 9M" 
 
 0.828 
 
 i5-%"n 
 
 15-% "□ 
 
 2.33 
 
 18' 
 
 4'-0" 
 
 7'-3" 
 
 UH" 
 
 9%" 
 
 0.878 
 
 17-%" □ 
 
 17-%" □ 
 
 2.49 
 
 19' 
 
 4 '-3" 
 
 7'-8" 
 
 
 mi" 
 
 0.924 
 
 14->^",^ 
 
 14-K'V 
 
 2.69 
 
 20' 
 
 4'-6" 
 
 8'-0" 
 
 16" 
 
 mi" 
 
 0.968 
 
 16-M"</> 
 
 16-H''0 
 
 2.90 
 
 21' 
 
 4'-9" 
 
 8'-5" 
 
 16%" 
 
 iiM" 
 
 1.025 
 
 17-M"</> 
 
 17-H"«A 
 
 2.92 
 
 32' 
 
 5'-0" 
 
 8'-10" 
 
 171.^2" 
 
 mi" 
 
 1.090 
 
 i5-K"n 
 
 15->^"D 
 
 3.10 
 
 23' 
 
 5 '-3" 
 
 9 '-3" 
 
 183^" 
 
 13" 
 
 1.155 
 
 16-1^" □ 
 
 16-M"a 
 
 3.16 
 
 24' 
 
 5'-6" 
 
 9'-8" 
 
 19" 
 
 13%" 
 
 1.220 
 
 i7-M"n 
 
 i7-H"n 
 
 3.20 
 
 25' 
 
 5'-9" 
 
 10 '-0" 
 
 19%" 
 
 14M" 
 
 1.280 
 
 i9-M"n 
 
 i9-M"n 
 
 3.44 
 
 26' 
 
 5'- 9" 
 
 10'-5" 
 
 21" 
 
 15" 
 
 1.333 
 
 2i-M"n 
 
 2i->^"n 
 
 3.63 
 
 Chicago Ruling. — The following tables have been computed for a four-way flat slab having 
 square panels, drop heads, and reinforced according to the Chicago Ruling. They include 
 designs for from 100 to 400 lb. per sq. ft. live load and spans of from 16 to 27 ft. in some cases. 
 Being based on square interior panels they will not apply without modification to other con- 
 ditions and are useful mainly for estimating and as a check on actual designs. The general 
 remarks given under the head of Pittsburgh Ruling apply in this case. 
 
 For methods of computation see Art. 20(i and Appendix C. 
 
 The stresses used are: /« = 18,000 and fc = not over 750 lb. per sq. in. 
 
502 
 
 CONCRETE ENGINEERS' HANDBOOK 
 
 [Sec, 11-21 
 
 O OJ " 
 
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 K^ 
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 c^\ 
 
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 1-H 1-H (M CO Ttl 
 
 K-<i< KiM \* \^<» K(M 
 
 * COK rHK rHK C^ r-K rHK 
 
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Sec. 11-211 BUILDINGS 503 
 
 Flat-slab Panels — Chicago Regulations 
 
 Interior Panels — Superimposed Load 300 lb. per sq. ft. 
 
 Panel 
 
 Capital 
 diam- 
 
 Head 
 
 T 
 total 
 drop 
 
 t 
 
 Concrete 
 in cu. ft. 
 per sq. ft. 
 
 Steel in each band 
 
 Steel 
 in lb. 
 
 slab 
 
 Direct 
 
 Across 
 direct 
 
 Diagonal 
 
 per 
 sq. ft. 
 
 16' X 16' 
 
 3'-6" 
 
 5'_7" X 5'-7" 
 
 1 1 14 " 
 
 IVa" 
 
 0 648 
 
 19-%" □ 
 
 12-%" □ 
 
 i3-%"n 
 
 
 17' V 17' 
 Li a i ' 
 
 3'— 9" 
 
 fi'-o" V 6'— n" 
 
 123,';" 
 
 8" 
 
 0 715 
 
 20-%"n 
 
 13-% "□ 
 
 14-% "□ 
 
 0 on 
 
 18' X 18' 
 
 4'-0" 
 
 6'-4" X 6'-4" 
 
 13^" 
 
 SH" 
 
 0.766 
 
 
 
 
 3. 13 
 
 19' X 19' 
 
 4 '-3" 
 
 o -8 X o -8 
 
 14>^" 
 
 9" 
 
 0 . 810 
 
 18-H"<A 
 
 12-1.-2 
 
 13-H"<^ 
 
 3.23 
 
 20' X 20' 
 
 4 '-6" 
 
 7'-0" X 7'-C" 
 
 15M" 
 
 
 0.854 
 
 2O->^"0 
 
 13-M"<^ 
 
 14->^"0 
 
 3.27 
 
 21' X 21' 
 
 4'-9" 
 
 7'-4" X 7'-4« 
 
 163^^" 
 
 10" 
 
 0.899 
 
 22->^"<^ 
 
 15-K"<A 
 
 16-M"0 
 
 3.57 
 
 22' X 22' 
 
 5'-0" 
 
 7'- 8" X 7'-8" 
 
 16H" 
 
 lOM" 
 
 0.945 
 
 20-M"n 
 
 i3->^"n 
 
 i3-M"n 
 
 3.71 
 
 23' X 23' 
 
 5'-3" 
 
 8'-l" X 8'-l" 
 
 18" 
 
 11" 
 
 0.990 
 
 2i-K"n 
 
 i4-M"n 
 
 i5-M"n 
 
 3.88 
 
 24' X 24' 
 
 5'-6" 
 
 8'-5" X 8'-5" 
 
 19" 
 
 11%" 
 
 1.055 
 
 23-M"n 
 
 i5-M"n 
 
 i6-M"n 
 
 4.00 
 
 25' X 25' 
 
 5'-9" 
 
 8'-9" X 8'-9" 
 
 20" 
 
 12K" 
 
 1.100 
 
 25-K"n 
 
 i6->^"n 
 
 i7-M"n 
 
 4.11 
 
 26' X 26' 
 
 5'-9" 
 
 9'-2" X 9'-2" 
 
 20^" 
 
 12H" 
 
 1.153 
 
 27->^"n 
 
 i8-M"n 
 
 i9-M"n 
 
 4 . 34 
 
 
 Interior Panels- 
 
 —Superimposed Load 350 lb. per sq. ft. 
 
 
 
 Panel 
 
 Capital 
 diam- 
 eter 
 
 Head 
 
 T 
 total 
 drop 
 
 t 
 
 Concrete 
 in cu. ft. 
 per sq. ft. 
 
 Steel in each band 
 
 bteel 
 in lb. 
 
 slab 
 
 Direct 
 
 Across 
 direct 
 
 Diagonal 
 
 per 
 sq. ft. 
 
 16' X 16' 
 
 3'-6" 
 
 5'-9" X 5'- 9" 
 
 12>^" 
 
 8" 
 
 0.718 
 
 i9-%"n 
 
 13-%" □ 
 
 13-%" □ 
 
 2 91 
 
 17' V 17' 
 
 3'-9" 
 
 6'-l" X 6'-l" 
 
 13^^" 
 
 8M" 
 
 0.768 
 
 2i-%"n 
 
 14-%" □ 
 
 15-%" □ 
 
 2 97 
 
 18' X 18' 
 
 4'-0" 
 
 6'-6" X 6'-6" 
 
 
 9" 
 
 0.810 
 
 my2"<}> 
 
 
 12->^"<^ 
 
 3.10 
 
 19' X 19' 
 
 4' -3" 
 
 6'-ll" X 6'-ll" 
 
 15M" 
 
 9>^" 
 
 0.855 
 
 19-M"<^ 
 
 13-K"0 
 
 14-M"<A 
 
 3.48 
 
 20' X 20' 
 
 4'-6" 
 
 7'-4" X 7'- 4" 
 
 
 10" 
 
 0.905 
 
 21-K"0 
 
 14-1.^ "0 
 
 15-M"0 
 
 3.53 
 
 21' X 21' 
 
 4'-9" 
 
 7'-5" X 7'-5" 
 
 mi" 
 
 mi" 
 
 0.965 
 
 i9-M"n 
 
 \2-y2"n 
 
 13-K"n 
 
 3.77 
 
 22' X 22' 
 
 5'-0" 
 
 8'-0" X 8'-0" 
 
 18" 
 
 mi" 
 
 1.010 
 
 21-1.^" □ 
 
 i3-M"n 
 
 
 3.92 
 
 23' X 23' 
 
 5'-3" 
 
 8'-4" X 8'-4" 
 
 19" 
 
 12" 
 
 1.075 
 
 22-M"n 
 
 my2"n 
 
 15-H"n 
 
 4.00 
 
 24' X 24' 
 
 5'-6" 
 
 8' -9" X 8'-9" 
 
 20>i" 
 
 mv 
 
 1.130 
 
 24-H"n 
 
 mH"o 
 
 17-M"n 
 
 4 . 12 
 
 25' X 25' 
 
 5'-9" 
 
 9'-3" X 9'-3" 
 
 21" 
 
 13" 
 
 1.175 
 
 27-H"n 
 
 i8->^"n 
 
 i8-M"n 
 
 4.40 
 
 26' X 26' 
 
 5'-9" 
 
 9'-5" X 9'-5" 
 
 22>r' 
 
 13%" 
 
 1.240 
 
 28-M"n 
 
 .i9-M"n 
 
 20-M"n 
 
 4.54 
 
 
 Interior Panels- 
 
 —Superimposed Load 400 lb. per sq. ft. 
 
 
 
 Panel 
 
 Capital 
 diam- 
 eter 
 
 Head 
 
 T 
 
 total 
 drop 
 
 t 
 
 Concrete 
 in cu. ft. 
 per sq. ft. 
 
 Steel in each band 
 
 Steel 
 in lb. 
 
 slab 
 
 Direct 
 
 Across 
 direct 
 
 Diagonal 
 
 per 
 sq. ft. 
 
 1 A' N/ 1 A' 
 ID A lO 
 
 3'-6" 
 
 6'-l" X 6'-l" 
 
 13" 
 
 8M" 
 
 0.746 
 
 20-% "□ 
 
 14-%" □ 
 
 14-%" □ 
 
 3 .09 
 
 17' S/ 17' 
 1 < A 1 # 
 
 3'-9" 
 
 6'-4" X 6'-4" 
 
 143^" 
 
 9" 
 
 0.814 
 
 22-% "□ 
 
 i5-%"n 
 
 16-% "□ 
 
 3 24 
 
 18' X 18' 
 
 4'-0" 
 
 6'-9" X 6'-9" 
 
 15" 
 
 
 0.855 
 
 25-% "□ 
 
 17-% "□ 
 
 18-% "□ 
 
 3.47 
 
 19' X 19' 
 
 4'-3" 
 
 7'-2" X 7'-2" 
 
 mi" 
 
 10" 
 
 0.905 
 
 21-M"<> 
 
 13-K"</, 
 
 14-M"0 
 
 0 . Do 
 
 20' X 20' 
 
 4'-6" 
 
 7'-6" X 7' -6" 
 
 17M" 
 
 10%" 
 
 0.978 
 
 22-H"0 
 
 15-H"<^ 
 
 16-H"0 
 
 3.77 
 
 21' X 21' 
 
 4'-9" 
 
 8'-0" X 8'-0" 
 
 mi" 
 
 mi" 
 
 1.022 
 
 25-H"0 
 
 16-K'> 
 
 17-K"0 
 
 3.91 
 
 22' X 22' 
 
 5'-0" 
 
 8'-4" X 8'-4" 
 
 mi" 
 
 12" 
 
 1.090 
 
 2i->^"n 
 
 i4->^"n 
 
 14-M"n 
 
 3.98 
 
 23' X 23' 
 
 5'-3" 
 
 8'-8" X 8'-8" 
 
 20" 
 
 12K2" 
 
 1.130 
 
 24-M"n 
 
 
 i6-H"n 
 
 4.29 
 
 24' X 24' 
 
 5'- 6" 
 
 9'-2" X 9'-2" 
 
 21" 
 
 mi" 
 
 1.200 
 
 26-<^"n 
 
 i7-K"a 
 
 i8-H"n 
 
 4.52 
 
 25' X 25' 
 
 5'-9" 
 
 9'-5" X 9'-5" 
 
 22 1.^" 
 
 14" 
 
 1.270 
 
 28-H"n 
 
 i8-H"a 
 
 i9-H"n 
 
 4.60 
 
 26' X 26' 
 
 5'-9" 
 
 9'-ll" X 9'-ll" 
 
 23>i" 
 
 14M" 
 
 1.315 
 
 3o-H"n 
 
 20-H"O 
 
 20-wn 
 
 4.71 
 
 Corr-plate Floors. — The following table gives data regarding designs which are based on 
 the Corr-plate method of computation for square interior panels and apply only to such cases. 
 
CONCRETE ENGINEERS' HANDBOOK 
 
 [Sec. 11-21 
 
 ft' 
 
 H 
 
 51 
 
 35315 
 
 6 6 6 6 <6 
 
 mil 
 
 > > 
 
 Hill 
 
 iifii 
 
 18565 
 
 6 6 6 6 6 
 
 m 
 
 _ 
 
 o o o o o 
 
 b b b b b 
 
 ;;;;; 
 
 o o o o o 
 
 5;8SS§2 
 
 ;;;;; 
 
 o o o o o 
 
 Why 
 
 S 2 2 S 2 
 iiiii 
 
 mil 
 
 I 
 
 llili 
 
 inn 
 
 fffff 
 
 T — in; — 
 
 fffff 
 
 \ « o w ^ 
 
 illl 
 
 Hill 
 
 fffff 
 
 fffff 
 
 Hill 
 
 nisi 
 
 illll 
 
 Illll 
 Illll 
 
 f- f. E. f. 
 
 fffff 
 
 Hill 
 
 Illll 
 
 iim 
 
 >>>>> 
 
 mil 
 Illll 
 
 1. 1, 
 
 XX XXX 
 
 « Bo Bo 00 00 
 X X X X X 
 
 Bo QO 00 X « 
 
 X X X X X 
 
 00 00 00 Bo Bo 
 
 X X X X X 
 
 Bo Bo Bo 00 Bo 
 
 I. ^ S 
 X X X X X 
 
 b b b b b 
 
 X X X X X 
 
 nm 
 
 HI 
 
 ml 
 
 ml 
 
 I 
 
 b 
 
 X 
 
 b 
 
 ? 
 
 X 
 
 ? 
 
 Bo 
 
 X 
 
 X 
 
 X 
 
Sec. 11-21] 
 
 BUILDINGS 
 
 505 
 
 The stresses used are 
 
 Ss = 18,000 lb. per sq. in. 
 fc = 700 lb. per sq. in. 
 
 They are based on the Corr-plate method of moment distribution and, of course, will not 
 apply where other rulings are in force. The main value of tables of this kind is for estimating 
 and as a check on design. The general remarks under the head of Pittsburgh Ruling apply 
 in this case. 
 
 For methods of computation see Arts. 17 g and 206. 
 
 Akme System. — The designs given in the following tables were not computed by the 
 Condron Co., but were worked out by one of the construction companies who have used the 
 Akme system extensively. As given here the steel stresses govern, the compression in the 
 concrete being always less than the allowable. 
 
 The data here given is useful for estimating and as a check on design, and applies only to 
 square interior panels using the stresses specified. For any other conditions special treat- 
 ment will be required. 
 
 The stresses used are 
 
 fs = 18,000 lb. per sq. in. 
 fc = 750 lb. per sq. in. 
 
 These tables should be used in connection with Figs. 47 and 48, pp. 467 and 468. For 
 method of computation see Arts. 17/ and 20a. 
 
 Akme System — Flat Slab 
 
 
 100 lb. per sq. ft. 
 
 150 lb. per sq. ft. 
 
 Size of 
 panel 
 
 Capi- 
 tal 
 
 diam- 
 eter 
 
 Side 
 
 
 
 Reinforcing steel 
 
 Con- 
 crete, 
 
 
 
 Reinforcing steel 
 
 Con- 
 crete, 
 
 drop 
 head 
 
 Drop 
 
 Slab 
 
 Band A 
 
 Band B 
 
 cu. ft. 
 
 per 
 sq. ft. 
 
 Drop 
 
 Slab 
 
 Band A 
 
 Band B 
 
 cu. ft. 
 
 per 
 sq. ft. 
 
 15' X 15' 
 16' X 16' 
 17' X 17' 
 18' X 18' 
 19' X 19' 
 20' X 20' 
 21' X 21' 
 22' X 22' 
 23' X 23' 
 24' X 24' 
 25' X 25' 
 
 3'-6" 
 3'-9" 
 4'-0" 
 4'-3" 
 4'-6" 
 4'-6" 
 4 '-9" 
 5'-0" 
 5'-3" 
 5'-6" 
 5'-9" 
 
 5'-3" 
 5'-9" 
 6'-0" 
 6'-6" 
 6'-9" 
 7'-0" 
 7'-6" 
 7'-9" 
 8'-0" 
 8'-6" 
 8'-9" 
 
 4" 
 4" 
 4" 
 4" 
 4" 
 4" 
 4" 
 
 w 
 
 5" 
 5" 
 5" 
 
 6" 
 6" 
 
 6M" 
 7" 
 
 7y2" 
 
 7>^" 
 
 8" 
 
 8M" 
 
 9" 
 
 9" 
 
 9>^" 
 
 12->^<^ 
 
 15-^g<^ 
 12-H<f> 
 
 5- y2<f> 
 
 6- M</. 
 
 10- >^</> 
 
 11- M<^ 
 
 12- >'2> 
 
 9- ys<i> 
 
 10- ys<t> 
 
 0.541 
 0.543 
 0.583 
 0.626 
 0.667 
 0.666 
 0.709 
 0.755 
 0.800 
 0.802 
 0.843 
 
 4" 
 
 .4" 
 *4" 
 
 4" 
 4" 
 4" 
 4" 
 
 4>^" 
 6" 
 5" 
 5" 
 
 6" 
 6" 
 
 6M" 
 7" 
 
 73-^" 
 7M" 
 8" 
 
 8H" 
 
 9" 
 
 9" 
 
 9M" 
 
 10-^^0 
 
 I2^ys<^ 
 10-?4 4> 
 
 12- ^4 <i> 
 
 13- ^4 </. 
 
 14- ^4 <A 
 
 15- K<^ 
 12-Ii<t> 
 
 6- H<t> 
 
 7- y2<t> 
 s-y2 4> 
 
 9- y2<f> 
 
 10- M<^ 
 
 i2-y2<i> 
 
 l2-%<t> 
 
 0.541 
 0.543 
 0.583 
 0.626 
 0.667 
 0.666 
 0.709 
 0.755 
 0.800 
 0.802 
 0.843 
 
 
 200 lb. per sq. ft. 
 
 250 lb. per sq. ft. 
 
 15' X 15' 
 16' X 16' 
 17' X 17' 
 18' X 18' 
 19' X 19' 
 20' X 20' 
 21' X 21' 
 22' X 22' 
 23' X 23' 
 24' X 24' 
 25' X 25' 
 
 3'- 6" 
 3'-9" 
 4'-0" 
 4'-3" 
 4'-6" 
 4'- 6" 
 4'-9" 
 5'-0" 
 5'-3" 
 S'-e" 
 5'-9" 
 
 5'-3" 
 5'-9" 
 6'-0" 
 6'-6" 
 6'-9" 
 7'-0" 
 7'-6" 
 7'-9" 
 8'-0" 
 8'-6" 
 8'-9" 
 
 4" 
 4" 
 
 4H" 
 4H" 
 4>^" 
 4>^" 
 5" 
 
 6" 
 
 6H" 
 W 
 7" 
 
 7H" 
 
 8" 
 
 8" 
 
 8>^" 
 9" 
 
 93-^" 
 9>^" 
 10" 
 
 \i-H<i> 
 
 \2~y<t> 
 \o-y4^4> 
 n-yi<j> 
 is-yi<t> 
 
 14- ^8 (A 
 
 15- ^^0 
 
 7-y2<t> 
 
 9->^0 
 10-H<A 
 
 n-H<t> 
 
 9- ys<i> 
 
 10- H<t> 
 
 10-^0 
 
 12- ^0 
 
 13- ^^0 
 
 14- ^ 
 
 0.533 
 0.584 
 0.630 
 0.674 
 0.714 
 0.713 
 0.756 
 0.802 
 0.847 
 0.849 
 0.895 
 
 4" 
 4" 
 
 4H" 
 
 5" 
 
 5" 
 
 5" 
 
 5" 
 
 5>^" 
 
 6" 
 
 6" 
 
 W 
 
 W 
 7" 
 
 7H" 
 8" 
 
 83-^" 
 
 9" 
 
 9" 
 
 9>^" 
 10" 
 10>^" 
 11" 
 
 11- %4> 
 
 12- %4> 
 
 n-y4.<i> 
 
 12- H<t> 
 
 13- M<^ 
 
 16-^4 
 
 is-Vs<t> 
 1^%<I> 
 
 9- y2<t> 
 n-y2<i> 
 
 12- H</. 
 
 10- ^^0 
 
 i2-ys<}> 
 
 0.583 
 0.626 
 0.672 
 0.721 
 0.761 
 0.801 
 0.803 
 0.846 
 0.892 
 0.938 
 0.983 
 
506 
 
 CONCRETE ENGINEERS' HANDBOOK 
 
 [Sec. 11-22 
 
 
 300 lb. per sq. ft. 
 
 350 lb. per sq. ft. 
 
 Size of 
 panel 
 
 Capi- 
 tal 
 diam- 
 eter 
 
 Side 
 
 
 
 Reinforcing steel 
 
 Con- 
 crete, 
 
 
 
 Reinforcing steel 
 
 Con- 
 crete, 
 
 drop 
 head 
 
 Drop 
 
 Slab 
 
 Band A 
 
 Band B 
 
 cu. ft. 
 
 per 
 sq. ft. 
 
 Drop 
 
 Slab 
 
 Band A 
 
 Band B 
 
 cu. ft. 
 
 per 
 sq. ft. 
 
 15' X 15' 
 16' X 16' 
 17' X 17' 
 18' X 18' 
 19' X 19' 
 20' X 20' 
 21' X 21' 
 22' X 22' 
 23' X 23' 
 24' X 24' 
 25' X 25' 
 
 3'-6" 
 3'-9" 
 4'-0" 
 4'-3" 
 4'-6" 
 4' -6" 
 4'-9" 
 5'-0" 
 5'-3" 
 5'-6" 
 5'-9" 
 
 5'-3" 
 5'-9" 
 6'-0" 
 6'-6" 
 6' -9" 
 7'-0" 
 7'-6" 
 7'-9" 
 8'-0" 
 8'-6" 
 8'-9" 
 
 4" 
 
 43-^" 
 4M" 
 5" 
 
 5M" 
 5H" 
 5M" 
 6" 
 
 6>^" 
 
 7" 
 
 7" 
 
 7" 
 8" 
 
 8K" 
 9" 
 
 9H" 
 
 10" 
 
 10>^" 
 
 11" 
 
 iiM" 
 
 i2-ys<t> 
 
 12- Ii4, 
 11-^^0 
 
 13- 1 
 
 9->^0 
 9-M<A 
 10-M<A 
 12-3-^ 0 
 9-^^<A 
 
 10- ^^ <A 
 
 11- M</> 
 
 12- ^^ <A 
 10-M<A 
 1 
 
 0.624 
 0.673 
 0.713 
 0.763 
 0.808 
 0.848 
 0.850 
 0.895 
 0.941 
 0.990 
 1.030 
 
 4>^" 
 5" 
 5" 
 5" 
 
 5>^" 
 
 5>^" 
 
 6" 
 
 6" 
 
 7" 
 
 73-^" 
 73-^" 
 
 73-^" 
 8" 
 
 W2" 
 9" 
 
 93-^" 
 10" 
 10" 
 10>^" 
 
 11" 
 
 113-^" 
 12" 
 
 I3-H0 
 
 10- ^^0 
 
 11- ^^0 
 
 10-^80 
 
 12- 7:^0 
 
 13- ^^0 
 
 14- ^:^0 
 
 15- 7:i0 
 
 12- 10 
 
 13- 10 
 
 9->^0 
 
 10- 3-^0 
 
 11- 3-^0 
 
 12- 3-^0 
 
 9- M0 
 
 12-^0 
 
 10- ^^0 
 
 10- % 0 
 
 11- M0 
 
 12- ^0 
 
 0.671 
 0.721 
 0.760 
 0.840 
 0 .849 
 0.889 
 0.897 
 0.937 
 0.982 
 1.037 
 1.076 
 
 400 lb. per sq. ft. 
 
 15' X 15' 
 16' X 16' 
 17' X 17' 
 18' X 18' 
 19' X 19' 
 20' X 20' 
 21' X 21' 
 22' X 22' 
 23' X 23' 
 24' X 24' 
 25' X 25' 
 
 3'-6" 
 3'-9" 
 4'-0" 
 4'-3" 
 4'-6" 
 4'-6" 
 4'-9" 
 5'-0" 
 5'-3" 
 5'- 6" 
 5'-9" 
 
 5'-3" 
 5'-9" 
 6'-0" 
 6'-6'' 
 6'-9" 
 7'-0" 
 7'- 6" 
 7'-9" 
 8'-0" 
 8'-6" 
 8'-9" 
 
 5" 
 5" 
 
 5H" 
 
 6" 
 
 6" 
 
 6M" 
 7" 
 
 8" 
 
 8" 
 
 8>^" 
 9" 
 
 W 
 10" 
 
 mi" 
 11" 
 
 iiM" 
 
 12" 
 
 12H" 
 
 10- M0 
 
 12- M0 
 
 13- M«A 
 
 11- Ji<> 
 
 12- 7:^0 
 
 13- ^i<A 
 
 l5-7:^<^ 
 
 12- 1./. 
 
 13- l.A 
 
 14- l</> 
 
 9-K2 <A 
 10->^ 0 
 12-M<A 
 
 9-^^<A 
 
 9-K<A 
 
 10- H<P 
 
 11- %0 
 
 12- M0 
 I2-M0 
 
 0.718 
 0.762 
 0.807 
 0.851 
 0.896 
 0.936 
 0.944 
 0.984 
 1.029 
 1.078 
 1.123 
 
 
 
 
 
 
 22. Construction Methods and Safeguards. — The care with which the actual construction 
 of a reinforced-concrete building is carried out has more to do with its success than any other 
 part of the operation and is certainly far more important than the design. This is just as 
 true of the flat-slab type as of any other. The present practice, therefore, of several of the 
 larger bar companies in furnishing design without inspection is to be regretted. 
 
 The more important features regarding construction as they appear to the writer may be 
 summarized as follows: 
 
 1. Column Forms. — The present practice is to use metal forms for the interior columns, 
 including the column capital, which are commonly leased from one of the steel form com- 
 panies. Forms for the exterior columns are commonly built of wood, although in some cases 
 metal forms are also used. Great care should be taken to see that these forms are tight, 
 particularly around the capital, as leakage at this point always results in a poor casting which 
 never can be properly repaired. The same is true where beams join the interior columns. 
 
 2. Pouring Columns. — Columns should be cast to the bottom of the column capital and 
 allowed to set 24 hr. before casting the capital and slab. Otherwise, there is likely to be a 
 separation at this point. 
 
 3. Floor Forms. — Both matched and edged lumber is used for floor forms with satisfactory 
 results. In the latter case, however, more care must be exercised to prevent leakage which is 
 always attended with serious consequences. 
 
 4. Methods of Securing Reinforcing Steel in Position. — In all reinforced-concrete construction 
 the accurate location and securing of the reinforcing steel in its proper position until the cast- 
 ing of the concrete is completed is of the utmost importance. Until recently, this fact was not 
 generally realized and some of the older constructions, which have been lately demolished, 
 
Sec. 11-22] 
 
 BUILDINGS 
 
 507 
 
 exhibit a remarkable displacement of both major and secondary reinforcement. Beam bars 
 were crowded together in one side of the beam or forced up several inches above their correct 
 position and stirrups were in almost all conceivable positions. 
 
 The necessity of the positive anchoring of the reinforcement in position during the cast- 
 ing period is even greater in the case of flat-slab floors than with other forms of construction, 
 because a small amount of vertical displacement will have a much greater relative effect. 
 
 This fact has been well understood by engineers engaged in this class of construction 
 and the result has been the development of numerous special devices for spacing rods uniformly 
 and maintaining them at the proper distance above the forms. Cuts of some of these devices 
 are shown in Art. 73, Sect. 2. These are carried in stock by, or can be quickly obtained from, 
 their various makers and are all of proven merit. It is strongly recommended that devices of 
 this type or of equal efficiency be specified and their use insisted on as experience has shown 
 that slipshod methods have no place in this class of construction. 
 
 5. Casting Floor Slab. — The ideal floor is one that is cast at a single operation. It is not 
 always possible to do this but the fewer construction joints in the floor, the better. It is usual 
 to make these in the center of panels and it is the practice of some engineers to use extra short 
 steel rods across these joints. A thorough cleaning of the finished surface together with the 
 liberal use of neat cement on this surface before casting the adjoining section is advantageous. 
 Care should be taken, by the frequent use of grade points, to see that the slab is cast to the 
 thickness specified. Variations of an inch or more have frequently been observed in practice. 
 
 6. Floor Finish. — Two methods of applying the cement finish to the floor slab are in 
 common use. The first, known as the monolithic method, consists in smoothing down the 
 structural slab and applying a thin finish before the concrete has thoroughly set. In this 
 case the finish is properly considered as a part of the slab in the computations for strength. 
 
 In the second method, more commonly used in cold-weather work, the finish to a thick- 
 ness of % in. or more is applied after the structural slab has set and cannot be considered as 
 an integral part of it for in many cases it does not bond uniformly with it but, of course, adds 
 to the dead load. 
 
 7. Wall Beams. — Where the wall beams do not extend above the slab they are cast with 
 it, but in some cases where they extend above they are cast part with the floor and the balance 
 afterward. When this is the case, it is advisable to provide additional stirrups to bond the 
 two sections together. 
 
 8. Concrete. — Very wet concrete should not be permitted and the remarks on this subject 
 in Sects. 1 and 2 apply here with full force. A mixture no leaner than 1:2:4 should be used in 
 the slabs with an aggregate not coarser than 1 in. For columns the writer prefers a richer 
 mixture and has recommended 1 : IJ^ :3 wherever possible. 
 
 9. Steel. — Deformed or square twisted bars are to be preferred for all flat-slab work on 
 account of their superior bonding qualities. Where market conditions permit, hard-grade 
 steel is recommended for slabs on account of its higher elastic limit. It has been the usual 
 practice in four-way flat slabs to specify the exact points of lapping of the various bars. Just 
 as good results may be secured by allowing the splices to come where they will but so arrang- 
 ing the bars that the splices are well staggered. The ability to use steel of random lengths 
 frequently results in a considerable saving. 
 
 10. Removal of Forms. — It is not advisable to remove the support from a flat-slab floor as 
 soon after casting as with the beam-and-girder type. Especial care must be exercised in this 
 respect in cold-weather work. Yielding of the green concrete in the slab may be sufficient to 
 crack the exterior columns, a thing which has happened in the past. 
 
 It is usually possible to so arrange the floor forms that they may be removed without 
 disturbing some of the props which support the slab, and this practice should be followed. It 
 is unwise to leave this matter to the judgment of an inexperienced contractor, who will likely 
 as not pull out the forms and put in the props afterward. No rule can be given but it may be 
 said that the time that the slab should be supported is a function of the mean temperature, 
 
508 
 
 CONCRETE ENGINEERS' HANDBOOK 
 
 fSec. 11-23 
 
 provided it has been prevented from actually freezing, and cases are on record where 40 days 
 was insufficient. 
 
 In winter work heating must frequentlj'" be resorted to, both of the materials before mixing 
 and of the slab after pouring, but unless the heated air can pass over the top of the slab, but 
 little benefit will be derived from the latter method. 
 
 UNIT CONSTRUCTION 
 
 23. Method of Construction in General. — The two main systems of ''unit" construction 
 for buildings are the Unit-hilt system and the Ransome Unit system. In the first system 
 mentioned, all members are cast in forms on the ground and set in place, when hard, by a 
 derrick. The Ransome system differs principally in that the slab is poured in place after the 
 unit beams, girders, and columns have been erected. In the Unit-hilt system all the units are 
 tied together by virtue of bars projecting into pockets or open spaces in which concrete is 
 poured. 
 
 24. Advantages of the Unit Method. — The greatest advantage obtained from the use 
 of separately molded members occurs when a large number of the same size of beams, columns, 
 
 and girders are to be employed. The same forms can 
 then be used over and over again. From this it 
 follows that the unit type of construction is not likely 
 to have universal application due to the multiphcity 
 of shapes in complex structures. Study of designs, 
 however, should in most cases reduce the number of 
 shapes to a workable minimum. Restriction in sizes 
 of discontinuous members is also likely to be a con- 
 trolling factor and, under some conditions, narrow the 
 application of this system to certain layouts governed 
 by the capacities of the handling apparatus. 
 
 Where a large number of members of the same 
 ^lui sys^;m^o7reTntTced-oo""f^^^^^ ^re required, «m( construction is likely to 
 
 be cheaper than monolithic construction for three 
 main reasons: (1) the greatly reduced amount of falsework required as compared with the 
 monolithic type, (2) the reduction in the number of men required for the work of construc- 
 tion, and (3) the chance to carry on the work under cover in all kinds of weather. 
 
 Shrinkage cracks do not usually occur in buildings of unit construction since all the shrink- 
 age has taken place in the individual units before their incorporation into the structure. Every 
 element may be inspected and approved before it is placed, or, if desired, a given percentage 
 of the units may be tested before erection. 
 
 Side Eleva-1-ion 
 Fig. 67. — Typical girders for Unit-bilt system. 
 
 25. "Unit-bilt" System. — Figs. 66 to 69 inclusive, and Plate VI show the principle of 
 construction of the Unit-hilt system used by the Unit Construction Co. of St. Louis. The 
 columns are provided with brackets for the support of the girders, and the girders are set 
 on a mortar bed at a distance back from the center of column sufficient to allow the column 
 
Sec. 11-26] 
 
 BUILDINGS 
 
 509 
 
 rods to overlap. The girder rods to take negative moment over supports project into the space 
 over the columns — the girders being cut back so as to give the necessary length for embedment 
 (Fig. 67 and Plate VI). The ribbed slabs rest on shelves or ledges on the sides of the girder, 
 with the top of slab above the top of girder so as to permit the slab reinforcing rods to project 
 into the space thus formed and provide for negative tensile stresses. The outer face of each 
 side rib of slab has a groove, so that the two grooves of adjacent slabs will form a key for the 
 grout filling. No attempt is made in the field to obtain a complete bond between the filled 
 concrete and the concrete of the separately molded members, since the filled concrete is used 
 either in direct shear or in compression, or as a means for bonding the projecting rods in the 
 space between the units. 
 
 It would be impossible to enumerate all constructions that have been carried out by Unit- 
 hilt methods,. The more important applications are as follows: sea walls, caissons, docks, 
 retaining walls, tunnel linings, culverts, pipe, sewers, fences, fence 
 posts, telegraph posts, piles, lamp posts, warehouses, factories, 
 elevators, cotton-handling plants, cotton mills, bridges, and via- 
 ducts, fireproof residences, school buildings, railroad stations, 
 roundhouses and train sheds. 
 
 Railroad work offers a wonderful field for the development of 
 the ''unit" method. If it were possible to standardize such struc- 
 tures as engine houses, freight sheds, snowsheds, train-sheds and 
 small stations, the construction of these by the Unit method on 
 a factory basis at central locations would effect great economies. 
 
 i^""rTi 
 
 1~ 
 
 
 
 Longi+udinaf Sec-tlon ' 
 Fig. 68. — Typical floor slab for Unit-hilt system. 
 
 Bottcm Plan SecTion 
 
 Fig. 69. — Details of column con- 
 struction, Unit-hilt system. 
 
 26. Ransome Unit System. — The main details of the Ransome Unit system are shown in 
 Plate VII. The usual reinforcement is placed near the sides of the columns and, in addition, 
 a longitudinal rod is inserted in a central cored hole extending lengthwise through the column. 
 The grouting of the column is done from the top after the setting of the girders and beams. 
 The cored hole is enlarged and flared out at the bottom in order to insure an even bed for the 
 column. The main column reinforcement is not continuous from story to story and the caps 
 and bases of the columns are enlarged so that, at these points, the concrete alone will be able 
 to transfer the weights. 
 
 The main girders are placed on top of the columns— the ends of each girder being enlarged 
 so as to almost cover the cap of the column. The beams are made with dove-tailed ends and 
 fit into pockets in the girders (Plate VII). In the design of this system, the vertical stirrups 
 
510 
 
 CONCRETE ENGINEERS' HANDBOOK [Sec. 11-26 
 
 Plate VI 
 
 Plate VII 
 
 ^ U-bar exTencf/n^ 
 from co/umn 
 
 Recess to receJv^e 
 ■f/oor s/ab 
 
 i' II! !i| ''' 
 
 Hr--i!r"i— CT^Ir 
 
 Sec-rional P\an. 
 
 in F/oor concre^ 
 
 Sectional Ellevai-ion 
 
 -Top afShb 
 
 ■Pane/ . 
 
 Derails of 
 Construction of the 
 'Ransome Unit System* 
 
 •~ Bo/rs ■■ ■^ 
 
 Section through SlaO find Beam showing arrangement of centering 
 
Sec. 11-27] 
 
 BUILDINGS 
 
 511 
 
 must be so arranged as to thoroughly bind the slab and beam, and make these members act 
 as a unit. The beams and girders are usually cast of a depth equal to the distance from the 
 bottom to the neutral axis only (Plate VII). The slab forms are erected between the beams 
 (which are usually spaced about 4 ft. on centers) and rest upon ledgers bolted to the sides of 
 the beam. The beam and girders are joined to the slab with a beveled joint, giving a slight 
 draft to the forms, thus affording considerable concrete around the connecting rods. 
 
 There is no shoring placed under the floor during construction so that the beam-and-girder 
 units should be designed strong enough to carry their own weight plus that of forms and wet 
 concrete slabs, in addition to an allowance for impact from buckets and so forth. The slab 
 panels may be removed in a much shorter time than would be permitted on monolithic con- 
 struction, since the previously molded beams and 
 girders take care of the entire load up to the time 
 when the full live load is brought on the floors. 
 
 The beams in some buildings have been con- 
 nected by means of tie bars as shown in Fig. 70. At 
 about the quarter points and in the top of each beam 
 and girder a hole was cast, extending down into the L 
 beam about 3 in. From this to the end of beam a 
 slot was formed. With two beams end to end, a rod 
 with a right-angle bend at each end could then be 
 slipped in place before the slab was poured. In the more recent developments of the system 
 (Plate VII), the ends of the reinforcing rods project above the tops of the beam and girder 
 units and a loose rod is inserted in the slab to provide for negative moment. The beams 
 and girders are figured as simply supported although, with proper design, the continuity 
 might probably be taken advantage of to some extent. 
 
 STEEL-FRAME CONSTRUCTION WITH CONCRETE SLABS 
 
 27. Types of Construction. — The steel skeleton consists of columns, girders and cross- 
 beams — the same arrangement as in the monolithic beam-andrgirder construction — the beams 
 usually being spaced about 6 ft. on centers. The many types of this form of construction 
 differ from each other in the form of slab steel used, the position of the concrete relative to the 
 beam, and in the use of curved or flat slabs. 
 
 28. Wrapping of I-beams. — I-beams completely enclosed in concrete should be wrapped 
 with wire, wire mesh, or metal lath, to prevent the concrete below the bottom flange from crack- 
 ing and dropping off. The material used should be of sufficient size to render efficient service 
 even though it should, by some accident, become corroded. It should be wrapped securely to 
 the I-beam flange, and, at the same time, sufficient space should be left to effect a concrete 
 clinch between the wrapping material and the beam. 
 
 29. Types Illustrated. — Fig 71 shows the floor slab placed directly on the tops of the steel 
 beams. The reinforcement of the slab may be either small rods, a wire fabric, or sheet re- 
 
 ^ inforcement. The slab as shown must be calculated as a simple 
 
 jk^^^jt.;:^^^^;^>^^^ beam, since reinforcement is not provided for negative moment 
 ± 1 over the supports. 
 
 Fia. 71. Fig. 72 is a very common form of concrete floor supported by 
 
 steel girders. The form for constructing same is also shown. 
 The haunches of the slab are carried down to the lower flange of the I-beam, the under 
 surface of which may be covered with metal lathing and plaster for fire protection (see Figs. 
 73 and 74). The I-beam is sometimes entirely encased in the concrete but it is difficult to 
 place the material under the lower flange. 
 
 Concrete floors are sometimes laid as continuous slabs with only the upper part of the I- 
 beams in the concrete, and sometimes the slab is so located with reference to the I-beams that 
 
512 
 
 CONCRETE ENGINEERS' HANDBOOK 
 
 [Sec. 11-30 
 
 the metal is placed between the beams instead of running over them, as in Figs. 75 and 76. 
 Still another type of floor consists of arches sprung between the lower flanges of the I-beams 
 and filled to the floor level with cinders. 
 
 ^ ■■ B'e"c foe 
 
 nana nu-r 
 
 6* iVooci washer 
 
 Fig. 72. 
 
 • f^laster 
 
 Fig. 73. 
 
 Fig. 74. 
 
 The Roebling slab floor is of many types, a common form being shown in Fig. 77. The 
 reinforcement is flat bars, which are bent at the beams so as to connect with the flange as 
 
 shown. Spacers, which supply the place of dis- 
 tributing rods, are fitted into slots in the bars. A 
 m 16-ft. span may be constructed with this type of 
 floor. A flat ceiling is obtained by suspending 
 metal lathing from beam to beam and plaster- 
 
 FiG. 75. 
 
 FiG. 76. 
 
 ing. A Roebling segmental concrete arch floor is shown in Fig. 78. 
 
 Sfee/ rorf. spacer 
 
 ,F/at bar 
 
 4 
 
 ,-Sleeper 
 
 Part Plan 
 
 Plastering 
 
 Part Longitudinal Sectiort 
 
 Fig. 77. — Roebling slab floor. 
 
 ROOFS 
 
 30. Structural Design. — Concrete roofs of the usual type are designed in the same manner 
 as floors. Any likelihood of vapor condensing on the underside of roofs in buildings where 
 steam-laden or moist air is to be found may be avoided by proper insulation and good ventilation. 
 
 When roof girders or frames are built monolithic with the columns in order to reduce 
 sizes of members, the method and formulas given in Sect. 10 may be employed to determine the 
 resulting moments. 
 
 31. Loading. — The roof of a reinforced-concrete building should be designed to carry the 
 weight of roof covering and snow which may come upon it. If the roof has considerable pitch, 
 wind pressure should also be considered. A roof load commonly assumed in temperate 
 climates for flat roofs is 40 lb. per sq. ft., in addition to the weight of the concrete itself. 
 
 . The snow load varies with the latitude and the humidity. As a maximum it is approxi- 
 mately 30 lb. per horizontal sq. ft. in Canada and northern Wisconsin, 20 lb. in the City of 
 Chicago, 10 lb. in Cincinnati, and rapidly diminishes southward. 
 
Sec. 11-32] 
 
 BUILDINGS 
 
 513 
 
 The wind load, which acts horizontally, varies with the velocity of the wind. A pressure 
 of 30 lb. per sq. ft. of vertical surface is usually assumed. Several formulas are in existence for 
 determining wind pressure on inclined surfaces. Duchemin's formula which follows, is pre- 
 ferred by many engineers as it is based upon carefully conducted experiments : 
 
 2 sin A 
 'l + sin2A 
 
 where P = normal pressure of wind in pounds per square foot of inclined surface. 
 Pi = pressure of wind in pounds per square foot on a vertical surface. 
 A = angle of inclination of the roof. 
 
 ' The dead load of any roof may be estimated quite closely from the following data — weights 
 are per square foot of roof surface : 
 
 Five-ply felt and gravel roof, 6 lb. 
 Four-ply felt and gravel roof, 5H lb. 
 Three-ply ready roofing, 0.6 to 1 lb. 
 
 Slates, ViQ in. thick, 7/4 lb,; ]'i in. thick, 9.6 lb. (the common thickness is Vie in. for sizes up to 10 by 20 in.). 
 
 Shingle clay tiles, 11 to 14 lb. 
 Spanish tile, 8 lb. 
 
 Vitrified roofing tile, 1 in. thick, 9 lb. (including asphalt and five thicknesses of felt, 11^^ lb.). 
 Slate tile, to 1 in. thick, 13 lb. (including asphalt and felt, Id^i lb.). 
 Tin roofing, sheets or shingles, 1 lb. 
 Copper roofing, sheets, V/z lb.; tiles IVi lb. 
 Corrugated iron, 1 to 3 lb. 
 
 Skylights with galvanized-iron frames, H-in. glass, lb.; Me-in., 5 lb.; %-m., 6 lb. 
 Plaster, 5 lb. 
 
 Suspended ceilings, 10 lb. 
 Cinders, 45 lb. per cu. ft. 
 Cinder concrete, 112 lb. per cu. ft. 
 
 32. Prevention of Condensation on Concrete Roof Slabs. ^ — In storage warehouses and 
 buildings of a similar nature, where no artificial heating is required, condensation can be 
 almost eliminated by proper ventilation. Buildings of this type may be classed among those 
 requiring little or no insulation for concrete roofs. Power houses, paper mills, roundhouses 
 and similar structures with concrete roofs, however, are a class of buildings which require the 
 best of insulation and ventilation to prevent condensation. Between these two extremes lie 
 many manufacturing and industrial buildings which, if built of concrete, will require a more 
 or less positive type of insulation for the roof slab. 
 
 With these facts in mind it can be seen at once that it would be folly to use the same kind 
 of insulation for all classes of buildings and expect to obtain good results and at the same time 
 exercise economy. In one case, a certain method of insulation may meet all the requirements, 
 while in another, on account of different conditions, the results may be entirely unsatisfactory. 
 
 1 By Albert M. Wolf, C. E., in Concrete-cement Age, May, 1914, 
 33 
 
514 
 
 CONCRETE ENGINEERS' HANDBOOK 
 
 [Sec. 11-32 
 
 For this reason the writer will refer so far as possible to the particular kind of condensation 
 insulation which is applicable to each type of building. 
 The types of insulation to be discussed are as follows : 
 
 1. Roofing felts and quilts. 
 
 2. Cinder fill (with cement finish upon which the roofing is laid). 
 
 3. Cinder-concrete fill (covered with roofing). 
 
 4. Hollow tile (with mortar top coat upon which roofing is laid). 
 
 5. Combination hollow tile and cinder fill. 
 
 6. Double concrete roof (light concrete slab above the main roof slab). 
 
 7. Suspended ceilings. 
 
 Roofing Felts and Quilts. — If the concrete roof is pitched or sloped to provide for drainage 
 and the building is to be used for ordinary light manufacturing, warehouse or storage purposes 
 and very little steam or moisture is present, a heavy roofing quilt or insulator placed under the 
 tar-and-gravel or prepared waterproof roofing, will furnish sufficient insulation. Such insu- 
 lation is easily applied, is of light weight and low first cost. The cost in general amounts to 
 about 1}4: cts. per sq. ft. in place. It has a disadvantage which somewhat offsets the low cost, 
 that of being soft and cellular and therefore easily pierced or broken down, thus destroying 
 its insulating value and necessitating a renewal. 
 
 Cinder Fill. — One of the most common methods of insulating roofs is to place a fill of 
 steam-boiler cinders on the roof pitched to provide for drainage and covered with a coating of 
 cement mortar about 1 in. thick upon which the roofing is placed. This method of insulation 
 permits the use of a level roof slab, which in itself is quite a saving. The extra cost of form- 
 work with a roof sloped in one general direction amounts to about 3^ to 1 ct. per sq. ft., while 
 if the surface is warped this extra cost will amount to 3 or 4 cts. per sq. ft. 
 
 The cinders, which should be a porous grade of steam-boiler cinders, free from refuse or 
 slag, should be wet down thoroughly, then placed on the roof, arranged to the proper slopes and 
 tamped' to an even surface. The minimum thickness of cinders at any place should be not 
 less than 3 in. Before the cinders have had a chance to dry out a 1 : 3 cement mortar coat, 
 mixed quite wet, should be placed on the cinders to a depth of about 1 in. and given a smooth 
 float finish. After the mortar has thoroughly set the roofing may be placed. 
 
 It is essential that the cinders be wet down before hoisting to the roof, for if this is done 
 after placing on the roof slab the excess water will stand on the slab and cause trouble by seeping 
 through the ceiling. If the cinders are not wet down before placing, they do not tamp well and 
 when the surface finish of cement mortar is applied, the dry cinders will take up the water in 
 the mortar and decrease its strength and value. 
 
 With insulation of this sort it is necessary to provide for expansion of the top surface. 
 The top portion of the fill and the mortar finish should therefore be kept 1 in. or so from all 
 parapet walls to allow for expansion joints to be filled with some bituminous or asphalt paving 
 pitch. 
 
 A cinder fill weighs on an average from 65 to 75 lb. per cu. ft. and it is therefore important 
 that the downspouts be so arranged as to keep the average depth to a minimum, which will 
 usually be about 12 in. The cost of this type of insulation for an average depth of 12 in. is 
 ibout 9 or 10 cts. per sq. ft. 
 
 This insulation is solid enough to bear all the usual weights coming upon it, gives no trouble 
 'rom expansion and can be readily used on concrete slabs which are designed as future floors 
 m case of the desire for the addition of future stories because of the ease with which it can be 
 torn up and re-used. It has been used extensively on concrete warehouses and manufacturing 
 buildings and is a very satisfactory insulation for any type of building except power houses, 
 paper mills and other similar buildings where much steam is present. 
 
 Cinder-concrete Insulation. — A porous cinder concrete mixed in proportions of 1 part by 
 volume of cement to 8 parts or 10 parts of porous screened steam-boiler cinders has been used 
 to a considerable extent as an insulator. It should be placed carefully so as not to lose the 
 
Sec. 11-32] 
 
 BUILDINGS 
 
 515 
 
 porosity, in much the same manner as a cinder fill, and finished off with a mortar coat on which 
 to lay the roofing. Expansion joints should be provided at all walls the entire depth of the 
 fill, since on hot days such a fill expands and exerts considerable pressure. Many parapet 
 walls have been pushed out of place because of failure to observe this rule. A cinder-concrete 
 fill insulation should primarily be very porous, with a rich mortar finish to seal the dead air 
 space in the fill. 
 
 A cinder-concrete fill is not so efficient an insulator as a cinder fill, the cost is higher and 
 the danger of expansion is greater. The excessive weight, about 100 lb. per cu. ft., is the main 
 disadvantage, which means that the roof construction must be considerably heavier in order 
 to carry the load. The cost of the cinder-concrete fill with an average depth of 12 in. will vary 
 from 13 to 15 cts. per sq. ft., depending on the height of building and the cost of cinders. 
 
 Hollow-tile Insulation. — Hollow clay tile laid on a concrete roof slab and covered with a 
 cement-mortar finish upon which the roofing is laid forms a good insulation against heat and 
 cold and resulting condensation. The tile used are 3 or 4 in. thick, of the ordinary soft clay 
 partition-tile variety, with scratched surfaces to furnish a key for the cement-mortar surfacing 
 about ^-i or 1 in. thick. The tile should be laid end to end to form continuous air spaces, 
 with tar or asphalt expansion joints at all walls. 
 
 This insulation can be used on sloping roofs only and in fact is the ideal one for such roofs, 
 since it combines the advantages of light weight (32 to 35 lb. per sq. ft.) comparatively low 
 cost and positive insulation. It can be constructed very rapidly and easily and can be used 
 for almost any type of structure. The average cost of this construction will be about 10 to 
 12 cts. per sq. ft. 
 
 Combination Hollow Tile and Cinder Fill. — Without doubt the best insulator is the com- 
 j bination hollow tile and cinder fill, for it combines and augments the advantages of each method 
 ' considered separately. It is constructed in the same manner as the cinder fill except that 
 ' the hollow tile are first placed end to end on the roof slab, and the cinders and mortar finish 
 placed thereon. The weight of this construction for an average depth of 12 in. amounts to 
 from 70 to 75 lb. per sq. ft. and the cost is about 12 to 13 cts. per sq. ft. 
 
 This insulation is as nearly perfect as can be made without the use of expensive cork insu- 
 lation combined with tile, etc. For power houses, paper mills, roundhouses and structures of a 
 similar nature it is as nearly positive as can be constructed, and with proper ventilation no 
 trouble should be experienced from condensation. The dead air space directly over the roof 
 slab afforded by the tile, and the protecting covering of cinders (which at the same time forms 
 the roof slopes) to keep the temperature of the air in the dead air space at the normal tempera- 
 ture, allow little chance of condensation except when the ventilation is insufficient. The 
 principal objection is the weight of this type of insulation, but where a positive insulator is 
 I required the advantages cited overcome the disadvantages of weight. 
 
 ; Double-roof of Concrete Slabs. — A somewhat new and little-used type of construction is 
 j that of secondary concrete roof slab pitched for drainage, supported on a wooden framework 
 I resting on the concrete roof slab. The framework can be built up of 2 by 6-in. stringers set 
 ' on edge on the slab and in turn supporting by means of short struts and braces, 2 by 6-in. 
 
 rafters directly above at the required height and slope. The several frames thus formed are 
 
 tied together with longitudinal and diagonal braces at such distances apart as can be spanned 
 ' by the thin concrete slab to be placed on the stiffened metal lath fastened to the top of frames. 
 
 The ribbed metal lath should be lapped at the sides and the ends, and securely fastened together. 
 . It will be found most economical to use the maximum span allowable for the heaviest metal 
 I lath obtainable. This is a No. 24-ga. stiffened metal lath which will support without other 
 
 centering a 2-in. concrete slab before the same has set on a span of 6 ft. After hardening, 
 
 such a slab will readily carry a live load of 25 lb. per sq. ft. 
 
 The concrete of a 1 : 2 : 4, or better still, a 1 : 13^2 : 3 mixture should be mixed rather dry, 
 ' for if there is an excess of water some of the cement will be carried away when the water drips 
 
 through the mesh. The coarse aggregate used should be a crushed stone or gravel passing a 
 
516 
 
 CONCRETE ENGINEERS' HANDBOOK 
 
 [Sec. 11-33 
 
 ^-in. mesh. The slab should be given a smooth float finish ready to receive roofing or water- 
 proofing, which should be of the best, for if any moisture reaches the metal mesh it will soon 
 rust and the wood supports rot out and the roof be destroyed. This construction weighs about 
 30 to 35 lb. per sq. ft. and will cost on an average about 16 cts. per sq. ft. 
 
 This construction gives a continuous dead air space over the roof slab and if the work is 
 well done and the ends effectively closed it provides a very effective insulation for any type of 
 building. It has the disadvantages of high first cost and the use of a woodeii framework, which 
 does not bring the construction within the fireproof classification. 
 
 Suspended Ceilings. — Suspended ceilings are used quite frequently to prevent heat radiation 
 and condensation on concrete roofs. In beam-and-girder construction good insulation can be 
 obtained by fastening a metal lath to the bottoms of the beams with wires or expansion bolts 
 and applying a cement plaster. Where the spans are short an ordinary metal lath will suffice 
 but for long spans (over 2 ft.) a stiffened metal lath should be used. The lath should be lapped 
 and wired together at sides and ends so as to form a stiff surface. The type of ceiling just 
 described will cost about 6 to 7 cts. per sq. ft. 
 
 For flat-slab roofs without beams or girders a different type of suspended ceiling is used. 
 This construction consists mainly of a No. 26-ga. stiffened metal lath ceiling wired at every rib 
 to 3^ by 13^-in. flats or l>^-in. channels, 5 ft. c. to c, running at right angles to the ribs, the 
 flats or channels being supported by No. 7 wire hangers or by ^ by 1-in. flats spaced about 5 
 ft. c. to c. hung from the concrete roof slab and placed at the time of pouring the latter. If a 
 lighter or No. 24-ga. stiffened metal lath is used, the supports should be not more than 4 ft. 
 c. to c. Such a ceiling will cost on an average 103'^ cts. per sq. ft. 
 
 These ceilings should be plastered with a 1:5: 12 plaster consisting of hydrated lime, 
 Portland cement and sand, respectively, thoroughly mixed while dry before adding water. 
 Long cow-hair should be used in the plaster. 
 
 On account of the dead air space between roof and ceiling this type of construction gives 
 a very positive insulation. The metal lath, however, has a tendency to rust and experience 
 has shown that suspended ceilings will break down in hot fires. Then again the roof slab must 
 be sloped or other provision made for drainage, which adds to the cost. Suspended ceilings 
 have been used as insulators in nearly every class of buildings including power houses, mills 
 and roundhouses and have in general given good service. 
 
 33, Concrete Roof Surfaces. — Although concrete roofs have been designed to be impervious 
 without a covering of any other material, it seems to be the general opinion that it is difficult to 
 secure absolute imperviousness in this type of roof because of the likelihood of shrinkage cracks. 
 Some engineers, however, believe that the cracks may be kept so minute, by properly reinforcing 
 against all the tensile stresses due to expansion and contraction, that the roof will stay water- 
 proof. In fact, some large buildings have been erected on this theory, and have shown no 
 signs of leakage. There is no doubt, however, that a concrete roof without a special covering 
 of any sort is at least well adapted to all structures where absolute imperviousness is not essen- 
 tial, as in train sheds, reservoir roofs, and in out-of-door structures in general. 
 
 Shrinkage joints are sometimes introduced in concrete roof slabs. These joints are made 
 water-tight by the introduction of a trough-shaped strip of copper or lead which will take con- 
 traction and expansion. If a sufficient number of joints of this nature are made, shrinkage 
 cracks will undoubtedly be reduced to a minimum. To make shrinkage joints the slab must 
 be poured separately from the beams and girders, and no bond should be allowed between the 
 slab and the roof framing. If steel beams are used, there is no danger, but in all-concrete con- 
 struction, galvanized strips, well oiled, or some equivalent should be introduced. 
 
 A maximum amount of reinforcement against shrinkage cracks will be of little avail in a 
 vioncrete roof unless the concrete itself is impervious. Methods of waterproofing concrete 
 may be roughly classified as follows: (1) Use of a rich-concrete mixture of such proportions of 
 sand and stone as to produce maximum density; (2) use of waterproof coatings or washes; 
 
Sec. 11-34] 
 
 BUILDINGS 
 
 517 
 
 (3) admixture of substances designed to produce impermeability. It is often advisable to 
 combine two or more of these methods. 
 
 Methods of proportioning concrete for maximum density are given in Sect. 2. Plastering 
 a concrete roof surface with a layer of rich mortar has been found effective when care has been 
 taken to place the same before the concrete has taken its final set. If such care is not exercised, 
 variation in temperature and moisture between the concrete and plaster will be almost certain 
 to cause a separation. Troweling concrete to a dense, hard surface, in the same manner that 
 granolithic work is troweled, makes concrete impervious and nearly equal to a surfacing of rich 
 mortar. 
 
 The methods employed to waterproof concrete by using washes or by introducing foreign 
 ingredients into the mixture are given in chapter on "Waterproofing Concrete" in Sect. 2. 
 The general remarks made there apply to roofs as well as to other types of structures. 
 
 A hard wearing surface is sometimes required in roof construction — for example, when it 
 forms the floor of a roof garder. The roof surface in such a case should have a granolithic 
 finish the same as in floors, and a form of bitumen emulsion mixed with cement mortar may be 
 used in making the surface impervious. 
 
 It is not good practice to use a concrete roof without covering on steep slopes, as the 
 slabs must then either be laid quite dry (which does not favor imperviousness) or else top 
 forms must be used to retain the concrete. The latter method is slow and expensive. 
 
 34. Separate Roof Coverings. — The felt and gravel roof is the most common and efficient 
 form of roof covering. The covering is composed of layers of waterproof felt, cemented to- 
 gether and to the concrete by coal-tar pitch or asphalt. The method followed is to lay from 
 four to eight thicknesses of felt over the roof surface, each ply being cemented to the preceding 
 layer, and the entire surface then floated with a heavy, flowing coat of pitch or asphalt, in 
 which (while hot) clean, dry, uniformly screened gravel or slag is embedded, sufficient in 
 quantity to cover the surface thoroughly. 
 
 Asphalt has in the past been preferred to coal-tar pitch as a binding medium, but at the 
 present time coal-tar products appear to be satisfactory when made to contain a large percent- 
 age of carbon, and are being used by many in preference to asphalt. Asphalt, however, seems 
 to be much superior to coal-tar pitch in ready roofings. 
 
 The following specification for a five-ply tar and gravel or slag roof has been taken from 
 the Proceedings of the American Railway Engineering Association, 1910 and is essentially 
 the Barrett Co.'s specification. 
 
 There shall be used five (5) thicknesses of saturated felt weighing not less than fourteen (14) lb. per one 
 hundred (100) sq. ft., single thickness; not less than two-hundred (200) lb. of pitch; and not less than four hun- 
 dred (400) lb. of gravel or three hundred (300) lb. of slag from H to size, free from dirt, per one hundred 
 (100) sq. ft. of completed roof. 
 
 The material shall be applied as follows: First, coat the concrete with hot pitch mopped on uniformly. 
 Second, lay two (2) full thicknesses of tarred felt, lapping each sheet seventeen (17) in. over the preceding one, and 
 mop with hot pitch the full width of the seventeen (17) in. lap, so that in no case shall felt touch felt. Third, coat 
 the entire surface with hot pitch, mopped on uniformly. Fourth, lay three (3) full thicknesses of felt, lap- 
 ping each sheet twenty-two (22) in. over the preceding one, mopping with hot pitch the full width of the 
 twenty-two (22) in. lap between the piles, so that in no case shall felt touch felt. Fifth, spread over the entire 
 surface of the roof a uniform coat of pitch, into which, while hot, imbed the gravel or slag. The gravel or slag 
 in all cases must be dry. 
 
 The coal-tar pitch or asphalt is the life of the roof, particularly in the topcoating. The 
 gravel or slag should be applied liberally, in order to completely bury the coat of waterproof 
 material and protect it from injury due to walking on the roof, and from the action of the sun. 
 In the following statements asphalt only will be mentioned, but it must be understood that 
 the same statements will apply to coal-tar pitch. 
 
 In waterproofing, the object of using saturated felt is merely to provide a medium to 
 hold the asphalt together, and thus allow for expansions, contractions and settlings. The 
 greatest care and judgment must be exercised in laying felt to see that it is properly stretched. 
 
518 
 
 CONCRETE ENGINEERS' HANDBOOK 
 
 [Sec. 11-35 
 
 laid smoothly, and that no wrinkles appear. All of the roofing felts on the market are made of 
 wool or flax, saturated with a preserving material that will harmonize with the asphalt. Satu- 
 rated wool roofing felts are compact, and although they absorb little of the asphalt, they hold 
 it in repeated layers to constitute the body material of the roof. The flax felt is porous and 
 of strong texture, and absorbs the asphalt, holding it within its fibers. Unsaturated burlap 
 or canvas can be made to hold asphalt if first run through a bath of liquid asphalt or tar. 
 
 The greater the amount of felt and asphalt used, the greater the life of the roof. How- 
 ever, if too much asphalt is used on top, it will run; hence the flatter the roof, the greater the 
 life. No roof should have less than 100 lb. of asphalt per 100 sq. ft. 
 
 What is known as a felt and gravel roof should not be used in cold climates, on a surface 
 of greater pitch than about 3 in. to the foot. In hot climates 1 in. to the foot is about the 
 proper maximum. Where the roof is of greater pitch, the gutters may be put in with galvanized 
 iron, copper, tin or piles of felt and asphalt, and the steeply pitched surface covered with 
 clay tile or slate shingles with lap joints. Vitrified clay tile are made in a great variety of 
 forms, flat; ribbed, and corrugated; but those of some interlocking pattern are best. Nailing 
 strips (1 by 2 in.) should be inserted in the concrete under each row of tile or slate, but it is 
 important that they should be so placed as not to affect the strength of the roof slab. Slate 
 or tile should not be used on a surface with a pitch of less than 6 in. to the foot. 
 
 A most durable roof covering for flat surfaces (slope not exceeding 3^ in. to the foot) is 
 vitrified roofing tile embedded in asphalt. The tile consists of flat rectangular terra-cotta 
 tile about 1 to 13-^ in. thick, bedded in hot asphalt, on top of four to six thicknesses of felt of 
 the kind mentioned above. The joints between the tiles should be filled with asphalt. Slate 
 tiles also make a good wearing surface. They are usually % to 1 in. thick, by 12 by 12 in. in 
 area. 
 
 Tin, corrugated iron, and copper roofings are sometimes placed on reinforced-concrete 
 buildings. These roofings do not have a long life if laid directly on the concrete, but seem to 
 give satisfaction when a wooden sheathing is used as a separator. Copper is expensive as 
 compared with other types of roofing and is usually employed only on very costly buildings. 
 
 There are many roofings on the market which come ready to lay, made on the principle 
 of the built-up roof. These roofings are especially valuable for use in small and isolated build- 
 ings, as they do not require the expert help which is necessary for a built-up roof. In a prepara- 
 tion which comes rolled, however, it is impossible to obtain a great deal of waterproof material, 
 and there is an additional weakness in some coverings of this type due to the fact that the 
 waterproofing sheets are nailed together with a short exposed lap. Expansion and contraction 
 of the body material in the course of time opens holes alongside the nails and permits water to 
 enter. In spite of adverse criticism, however, there are many ready roofings which seem to 
 give satisfaction and are used frequently on pitched surfaces, and to some extent on flat slopes. 
 
 A number of types of metal and composition shingles are on the market. Reinforced- 
 concrete slabs from 5 to 6 ft. long have been used for this purpose. 
 
 35. Drainage. — Felt and gravel roofs should have a pitch of at least in. to 1 ft. in order 
 to provide proper drainage. Flat tile, however, may be employed on surfaces with a very 
 slight pitch — preferably not over in. per ft. 
 
 Gussets in flat roofs are generally formed by placing a well-tamped cinder filling over the 
 concrete slab and then laying a 1 to 2-in. surface of cinder or stone concrete. This type of 
 roof generally permits the use of the floor forms without much change and there is the ad- 
 vantage of having all the column heights of the top story the same. Stone-concrete surfacing 
 seems to be the most satisfactory but a cinder mixture is sometimes specified. 
 
 If a flat ceiling is not required, or if it is the intention to employ a suspended ceiling (Fig. 
 79), the roof beams and roof girders may be so inclined as to give the required roof pitch. This 
 method is advantageous when cold weather is likely to overtake the work before it can be 
 closed in, for, with the use of cinders as a filling, there is delay in placing the final roof surface. 
 
 Where concrete walls project above the roof proper, grooves or reglets about 1 by 1)^ in. 
 
Sec. 11-35] 
 
 BUILDINGS 
 
 519 
 
 mast be left in the concrete wall in which to insert the edge of the flashing (see Fig. 80). To 
 make a reglet, a strip of wood is nailed to the forms and the strip is taken out when the forms 
 are removed. The flashing is keyed with the reglet, as shown, and cemented up with a rich 
 cement — sometimes rubber cement. 
 
 When metal standing flashing is specified on felt roofs, it is necessary to nail through the 
 metal and felt into wooden strips embedded in the concrete slab in order to hold the flashing 
 in place (see Fig. 80). Since expansion and contraction will soon loosen the nails, the nails 
 and the flange of the flashing should be covered with a felt strip prior to coating the roof with 
 the top coat of asphalt and gravel. A reinforce- 
 ment of flax felt, mopped on solidly, appears to be ^-^--m^^m 
 
 as satisfactory as metal flashing in every way (see 
 Fig. 81). All such felt flashings should be turned 
 
 — ^i^y. — . 
 Fig. 79. 
 
 Air// rf7rouff/f A7o/e 
 /n hanc^er 
 
 •z''.;^' Hanger 
 
 Mefa/'Lafh and fVaster ' 
 
 ....^.4'-o'. 
 
 -Suspended ceiling construction. 
 
 up at the parapet walls and curbs at least 4 in. at' 
 the highest points of the roof, and not less than 12 
 in. high as the outlets are approached, in order to 
 avoid overflows should the outlets become clogged. 
 
 Fig. 80 shows the method employed in apply- 
 ing a galvanized-iron flashing for a felt and gravel roof on a large reinforced-concrete ware- 
 house. The reglet was made to taper slightly — that is, wider at the outside — in order to per- 
 mit the easy withdrawal of the wood strip that was nailed to the forms. The groove was 
 made 10 in. above the roof line. When the concrete was being poured for the roof floor, a 
 heavy strip of wood was laid with its upper surface flush with the top surface of the concrete, 
 and 6 in. from, and parallel with, the parapet wall. This strip was employed to nail the edge 
 of the flashing to the roof after the roof had been covered with felt and asphalt. The nails and 
 the flange of the flashing were covered with a felt strip prior to coating the roof with the top 
 coat of asphalt and gravel. 
 
 Croove to be -f/h'ed 
 yyifh cement mortar 
 
 •' _^Gafvan/zecl irorj 
 
 Nailing h 1 
 Strip 
 
 J/a///nj strip 
 Fig. 80. 
 
 7b be mecf with 
 f cement mortar 
 
 •^Counterf /ashing 
 Felt and Gravel 
 
 The flashing was secured in the reglets by band-iron clips, as shown. These bands (2 in. 
 long) were bent midway to an angle of about 30 deg. After insertion they were struck with 
 the hammer on the bend, which straightened them out and jammed them in place, thereby 
 holding the flashing. The reglet was pointed up with a rich cement to insure a water-tight 
 connection. Figs. 81 to 84 inclusive, show different methods of flashing which, may be employed 
 (see also Figs. 89 and 90, and Art. 36). 
 
 In flat-roof construction with parapet walls, all valleys in the roof surface should lead to 
 drain boxes. Fig. 85 shows the drainage scheme for part of the roof of a shoe factory at Cam- 
 bridge, Mass. 
 
 Drain boxes, or conductor boxes as they are usually called, have been made in many ways — 
 lead or copper being generally used. Fig. 86 shows a double copper box for conductors used 
 on a factory building at Roxbury, Mass. Fig. 87 is a sketch of a conductor box installed in a 
 
520 
 
 CONCRETE ENGINEERS' HANDBOOK 
 
 [Sec. 11-35 
 
 warehouse at Portland, Maine. Wire screens or strainers should always be specified since they 
 prevent the downspouts from becoming clogged by leaves, etc. The downspouts may be 
 
 F/crt nie 
 
 '"-■i^. Nailing Strips 
 
 Fig. 82. 
 
 carried down inside or outside the building — depending upon preference and sewer conditions. 
 The conductors should be cast iron when carried inside the buildings, while corrugated iron, 
 
 Stab 
 
 / \ 71 • \'i *• 
 
 / Na///nq ' ' • ' ••'••I 
 
 •strips 
 Fig. 83. 
 
 6/atecf Tile Copin^.,^ 
 Brick W/-.... 
 
 Composition 
 roofing-. 
 
 Fig. 84. 
 
 due to its ability to expand without breaking, is better on the outside of the building where 
 freezing of water in the conductors may be expected. All vapor and soil pipes extending 
 
 ■Sky fight over E/evator 
 
 Fig. 85. 
 
 through the roof should have an expansion sleeve soldered on and counterflashed with an in- 
 verted copper cone attached to pipe. For discussion of conductor heads for roof drainage, see 
 article by A. M. Wolf in Engineering News, May 11, 1916, page 901. 
 
Sec. 11-36] 
 
 BUILDINGS 
 
 521 
 
 Fig. 88 shows the method of draining a roof surface of considerable pitch — a condition 
 under which parapet walls could not be used. This is called hanging-gutter construction. 
 The gutters should have a slope of about 1 in. in 16 ft. 
 
 Copper wire basket 
 Copper box .. 
 
 Copper wire basket--'' 
 
 Outside funne/' 
 inside fwnnel 
 
 Cast Jron ,^..> 
 Conductor pipe 
 
 Plastic slate 
 
 Cinder conctefe 
 
 ■•• H'oof slab 
 
 Brass sleeye soldered 
 to outside copper casing 
 
 Fig. 86. 
 
 The following table will serve as a guide by which to proportion the size of gutters and 
 downspouts : 
 
 Span of roof 
 
 Gutter 
 
 Conductor 
 
 Up to 50 ft. 
 50 to 70 ft. 
 70 to 100 ft. 
 
 6 in. 
 
 7 in. 
 
 8 in. 
 
 4 in. every 40 ft. 
 
 5 in. every 40 ft. 
 
 6 in. every 40 ft. 
 
 Copper Strainer , 
 
 There is wide variation in practice in the 
 number of square feet of roof surface to 1 sq. 
 in. of leader opening, but the above table 
 should give some idea of the general practice. 
 
 pis 
 
 "^"Copper 
 
 Caiicin^ Joint 
 
 Downspout 
 Fig. 87. 
 
 Head coyensd 
 with voider 
 
 I'xS" Nailing strip Spikes 
 ■•. used for anchors 
 
 S'-l&oz. Copper 
 OuTter 
 
 ^ i",/' m>od Screed 
 
 Fig. 88. 
 
 36. Parapet Walls. ^ — Parapet walls are of two general types, brick or reinforced concrete, 
 or a combination of both, the type used depending upon the architectural treatment desired. 
 Where a roof is enclosed by parapet walls, overflow openings in the latter should be provided 
 at suitable points to prevent flooding of the roof in case the downspouts become stopped up. 
 Such openings can be provided by simply leaving a hole in the parapet, or better still, the 
 
 By Albert M. Wolf, C. E., in Concrete, Dec, 1916, 
 
522 
 
 CONCRETE ENGINEERS' HANDBOOK 
 
 [Sec. 11-36 
 
 openings (4 to 8 in. square) can be lined with sheet metal or a pipe projecting at least 1 in. 
 beyond the face of the wall. 
 
 Brick Para-pet Walls. — Brick parapet walls have been used extensively, but unless properly 
 built, they are very likely to give more or less trouble, and if built to avoid this, the usual 
 economy over concrete (in localities where brick are comparatively cheap) is lost. This has 
 given rise to the use of concrete parapets either exposed or covered with a veneer of brick or 
 terra-cotta. 
 
 The vital part of the brick parapet wall is the inner side, and usually this is made up of 
 common brick laid up in ordinary lime mortar. As a result, many brick parapets in a few 
 years become a crumbling mass, owing to the freezing of the brick just above the roof flashing, 
 after being saturated with water splashing up on them during rains, or from snow piled on the 
 roof. Such a condition increases the cost of maintaining the roof in proper condition, for 
 once the brick start crumbling, the flashing becomes loosened, water finds its way back of the 
 roofing and down through the slab. This means that the parapet must be rebuilt, the flashing 
 and probably the roofing also, must be replaced. 
 
 Since the life of a good roofing is materially shortened by the conditions cited above, first- 
 class roofers now recommend that the inner side of the brick parapets be built of hard-burned 
 vitrified or paving brick laid up in cement mortar, and covered with a waterproof coping. In 
 addition to this, the coating of the parapet (on roof side) with roofing tar or pitch up to the 
 underside of the coping will do much toward making it more permanent. 
 
 Where cinder or cinder-concrete fills are used to form the drainage slope, special attention 
 should be paid to the anchoring of parapet walls to the concrete slab and the provision of expan- 
 sion joints between walls and roof filling. When cinder fills are used, a coating of cement 
 mortar from 1 to IH in. thick is placed on the top after grading the cinders to the proper slopes, 
 to produce a firm foundation for the roofing. The mortar coat expands considerably, as does 
 cinder-concrete filling, which is sometimes used, and expansion joints from 1 to 2 in. wide should 
 therefore be placed at all walls extending down through the mortar topping or cinder concrete. 
 These joints should be filled with a tar or asphalt paving pitch which will perform its function 
 of completely filling the joint under all conditions of weather. 
 
 Lack of proper provision for expansion of roof fills has been the cause of the pushing out 
 of line, and rendering dangerous many brick parapets. For this reason, in addition to the 
 expansion joints, it is well to anchor the brick walls to the spandrel slabs or girders by means of 
 stub bars bent up from the roof slab or spandrel. No rules for exact size and spacing of such 
 anchor bars can be given, but in general J-^ in. diameter bars projecting up into the wall for 
 about 2 ft. at a spacing of from 18 in. to 2 ft. will do the work.- Of course it is highly essential 
 that the wall be built in solid around the rods with good cement mortar. 
 
 Another detail of importance is the flashing slot and strip left in the parapet to provide 
 a means of fastening the flashing and counterflashing and making waterproof the roofing at 
 the parapets. A very satisfactory detail for flashing of ordinary tar and gravel roofing at 
 parapets is shown in Fig. 89. This detail, recommended as good practice by the American 
 Railway Engineering Association, makes use of a 2 by 4-in. timber with one edge beveled, laid 
 continuous in the parapet at the proper height in place of a stretcher course of brick. This 
 serves as a nailing strip for a light wooden strip holding the flashing and counterflashing in 
 place. After placing the flashing the slot is completely sealed up with cement grout or roofing 
 cement. 
 
 Concrete Parapets. — For the proper flashing of concrete parapet walls the detail shown in 
 Fig. 90 can be recommended. A 2 by 4-in. piece of lumber is ripped on the diagonal as shown 
 and then placed in the forms at the desired height, the upper strip being securely nailed thereto, 
 so as to insure its removal when forms are taken down, while the lower piece is just tacked to 
 forms (from outside) with wires or nails driven into it as shown to anchor it to the concrete. 
 The flashing and counterflashing are then placed in the same manner as for brick walls. 
 
 As generally designed, concrete parapets, in addition to retaining or masking the drainage 
 
Sec. 11-36] 
 
 BUILDINGS 
 
 523 
 
 slopes, carry a portion of the roof load as beams, but owing to the fact that they are generally 
 much deeper than required to simply carry the load, other considerations besides the load must 
 be taken into account. That is, enough reinforcing must be provided and distributed in a 
 manner which will prevent the formation of expansion and contraction cracks resulting from 
 the excessive changes of temperature to which side walls are subjected. Ordinarily parapets 
 are at least 3 ft. deep overall, and usually this depth is much more than is required to resist 
 the bending moments induced therein, especially in flat-slab construction where the portion 
 of the flat slab adjacent to the parapet carries most of the roof panel load. This means that a 
 very low percentage of steel will be required to resist the tensile stresses produced by bending, 
 in fact, it may be such small amount as to be incapable of resisting the stresses set up by tem- 
 perature changes. For this reason, it is always well to so detail the reinforcement of the portion 
 of the parapet above the roof as to have not less than 25 to 30% of longitudinal reinforcement 
 arranged somewhat as shown in Fig. 91 with plenty of vertical reinforcement in the form of 
 
 
 
 
 Z'xA'Confimous 
 
 ^■-Sfrlp 
 
 \-Counfer 
 f (ashing 
 
 V' Flashing 
 ^Sjoofing 
 
 [■Hasfic cemenf 
 or pitch 
 
 
 Cinder fill 
 
 
 Consf.joinf- 
 
 ^'orlaiwrsfn.^ t^- 
 Lap ar mid span : 
 
 Z-i"5tr or larger 
 at Cols, only 
 
 l''Clear- 
 
 T ifori'Sfn^ighf- 
 
 \ i'orm" Stirrups 
 ■ ltfol8"cfo c. 
 
 - -. , i T ^'oni'Str. Lap 
 '^r^:^-^ --''af mid span- 
 
 \Consf Joinf 
 \ Roof line 
 
 Fig. 89. 
 
 Fig. 90. 
 
 Fig. 91. 
 
 stirrups to tie together the portions of parapet poured at different times (portion below and 
 above roof line). 
 
 Concrete parapets should always be designed and reinforced as fully or partially continu- 
 ous beams, depending upon the location thereof and the condition of end supports, for unless 
 this is done unsightly cracks are sure to appear near the supports. If the parapet walls are of 
 such form as to require pouring in two operations (as in Fig. 91) they will, of course, not be so 
 strong as if poured in one operation, and, therefore, if the total depth is to be considered effect- 
 ive a bending moment coefficient somewhat lower than used in the formula for fully continuous 
 
 beams, viz., M =^2' ^^^^1^ be used. In the writer's opinion, this should be, for beams of 
 
 the type in question, M = for interior spans and M = for end spans at support and 
 mid-span. 
 
 Parapets seldom require very much diagonal tension reinforcement owing to the depth 
 of same and the relatively light loads to be carried, and the use of bent bars is therefore seldom 
 warranted, since the stirrups used to tie the portions of wall together can be made of sufficient 
 number to care for all diagonal tension stresses in excess of that which the concrete alone will 
 resist. At corners extra horizontal bars should be provided, bent around the corner so as to 
 lap with the main bars, for unless this is done, cracks are sure to develop, owing to expansive 
 and contractive forces acting at right angles to each other. In large buildings it will be found 
 
524 
 
 CONCRETE ENGINEERS' HANDBOOK 
 
 [Sec. 11-36 
 
 advisable to provide expansion joints in parapets about every 200 ft. over columns, the spans 
 adjacent to such joints being designed and reinforced same as end spans. 
 
 Where concrete parapets are covered with brick or terra-cotta which forms a part of the 
 architectural treatment, they must be reinforced to resist the twisting action produced by the 
 
 Fig. 92. Fig. 93. 
 
 weight of the material hung from the concrete (see Figs. 92 and 93). When the concrete wall 
 supporting the ornamental part of the parapet or cornice is not very high, as in Fig. 92, the 
 stresses thus produced can be taken care of by placing vertical bars or stirrups so as to reinforce 
 
 8 "» /4 " 3 earns.. . glp " 6-4 "e-O^ " 
 
 Section through Machine Shop 
 
 l"'/?oc/s9"c^^p7\ ^^.^ fRods ll"c.to 
 
 No. 7z A Fabric 
 
 Section B-B 
 Fig. 
 
 the wall as a cantilever with sufficient longitudinal reinforcement to resist the tensile stresses 
 due to beam action in a vertical plane. If the concrete portion of the parapet is relatively 
 high, and the ornamental stonework or terra-cotta projects a considerable distance beyond 
 
Sec. 11-36] 
 
 BUILDINGS 
 
 525 
 
 the face of the wall, or, in other words, if the center of gravity of the entire parapet lies outside 
 the face of the concrete wall, buttresses should be used at the columns and the wall reinforced 
 as a counterfort wall between the buttresses. The main reinforcement in the buttresses (at the 
 back) should be well anchored into the floor construction or into the column below and after 
 extending up the back of the buttress be bent down in front of the longitudinal wall bars, thus 
 giving a positive support for the continuous wall slab. Buttressed parapet walls should also 
 be reinforced to take care of the ordinary beam action which may develop due to the weight 
 of the parapet. The same bending-moment coefficients are recommended as for ordinary 
 roof parapets previously described. 
 
 Fig. 94B. 
 
 If the covering of brick or terra-cotta is only a few inches thick, it can be supported by 
 spandrel angles anchored to the concrete, and placed wherever offsets occur in the veneer. 
 In addition to these angles, corrugated metal brick ties set in the concrete should be built 
 into the brick joints to aid in holding the brick in place. Terra-cotta or stonework which 
 projects 9 in. or more beyond the face of concrete should be anchored to the parapet by bolts 
 passing through the webs of the blocks of terra-cotta or by bent plate anchors securely fastened 
 into the stone courses. These anchors should be used in addition to the spandrel angle supports 
 previously mentioned (see Figs. 92 and 93). When a well-designed concrete parapet is used, 
 there is no danger of failure of the main support of the cornice, and if the spandrel angles and 
 anchors are properly placed and well covered with mortar when placing the veneer, practically 
 all danger of part of the cornice falling is removed. 
 
526 
 
 CONCRETE ENGINEERS' HANDBOOK 
 
 [Sec. 11-37 
 
 37. Sawtooth Construction. — Sawtooth roofs, in general, have been found especially 
 well adapted for machine shops and factories, where it is desirable to have a uniform daylight 
 
 Fig. 95. — Cross-section detail of sawtooth roof. 
 
 illumination over the entire floor area. In reinforced concrete, unfortunately, this type of 
 construction is expensive because of the irregularities of the forms, and has been employed only 
 
 Fig. 96. 
 
 to a limited extent on that account. Typical designs of sawtooth roofs are shown in Figs. 94A, 
 94B, 95 and 96. The skylights are usually arranged to face the north, as the sun's rays would 
 be undesirable and would cause excessive heat in the building in the summer time. 
 
 
 poO>^^>^ \\ 
 
 Frame ^^^^^ \\ 
 
 fGrouT 
 
 
 'A 
 
 7/e beam ■ 
 
 -f -1 
 
 Girder-- '' 
 
 
 
 
 Fig. 97. — Cross-section showing typical arrangement of units in sawtooth construction, Unit-hill system. 
 
 Fig. 96 is a sketch of a sawtooth roof used in a. cotton mill at East Boston, Mass. The 
 girders supporting the sawtooth were made of sufficient stiffness so that no horizontal tie rods 
 
Sec. 11-38] 
 
 BUILDINGS 
 
 527 
 
 were necessary. The lines of columns for the sawtooth span are 20 ft. c. to c. and two girders 
 are located in each bay. 
 
 A sawtooth roof construction using separately molded members has been developed by the 
 Unit Construction Co. of St. Louis (see Art. 25). Fig. 97 is a cross-section showing the typical 
 arrangement of units. The roof portion of the sawtooth rests at its lower end on a ledge in 
 the girder and at the upper end on a ledge in the skylight frame. The lower end of the frame 
 also rests on a ledge in the girder and the horizontal, beams are provided in order to tie in the 
 building, and for the possible support of shafting or other installations. The skylight frame 
 extends from column to column and consists essentially of a large flat plate into which the 
 window frames are cast. The connections are made in practically the same manner as the 
 floor connections for this system of unit construction as described in Art. 25. 
 
 Transverse sectior> Typical elevation 
 
 Fig. 98. 
 
 Sawtooth skylights are usually glazed with wire glass, held in metallic frames, so that the 
 entire construction is fire-resisting. 
 
 38. Trainshed of "Unit-bilt" Construction. — A trainshed built by the Unit-hilt system 
 is shown in Fig. 98 (see Art. 25). 
 
 COLUMNS 
 
 39. Details of Design. — Columns exceeding 15 diameters in length should be avoided as 
 
 but few tests have been made on columns of such slender proportions. Fortunately, however, 
 columns longer than 15 diameters are quite rare except in roof stories where a light roof load 
 often requires very light sections. Where long columns must be used, the reduction formula 
 given in Art. 9, Sect. 8 may be used in their design. 
 
 A high percentage of longitudinal reinforcement — say above 4 to 6% — is undesirable on 
 account of the uncertainty of the strength of such columns. In fact, increase in the proportion 
 of cement is usually much to be preferred to a high steel percentage, not only because the cement 
 gives a more definitely known increase in strength, but also because it is a cheaper column rein- 
 forcement than steel. In no case should a leaner concrete mixture than 1 : 2 : 4 be used. 
 
 All columns should be reinforced with at least four %-in. round rods or four ^^-in. square 
 bars whether or not such reinforcement is theoretically required. This should be done to 
 guard against the possibility of eccentric loading and for safety in construction. 
 
 Provision should be made in all cases for the bending moment which will be developed 
 
528 
 
 CONCRETE ENGINEERS' HANDBOOK 
 
 [Sec. 11-39 
 
 by unequally loaded panels, eccentric loading, or uneven spacing of columns. Especially is 
 this true for wall columns and corner columns. Moment in columns due to unsymmetrical 
 floor loading may be found as explained in Sect. 10. Moments in columns due to bracket 
 loads may be found by the formulas of Art. 10, Sect. 8. The stresses and amount of reinforce- 
 ment required for a column subjected to bending as well as direct stress may be found as 
 explained in Sect. 9. 
 
 The maximum allowable stress in a column due to bending and direct stress may be taken 
 considerably greater than for direct stress only, since the maximum stress does not occur 
 over the whole cross-section of the column but simply at the outside edge. The bending 
 moment also rapidly diminishes along the column from the maximum value used in the stress 
 computations. A unit stress 15% greater than for direct compression may be permitted pro- 
 vided the unit stress due to the maximum loading centrally applied is not greater than the 
 allowable for direct compression. 
 
 In structures with inflammable contents where a special fireproofing is not provided around 
 concrete columns, l}i in. on all sides should be considered as protective covering and then this 
 thickness should not be included in the calculations for strength. A less thickness than 1^ in. 
 should be sufficient where the contents of a building are not especially inflammable. In no 
 case, however, should the steel be nearer the surface than 1^ to 2 in. since it is desirable to 
 prevent any tendency of the vertical rods to buckle under working loads. For square or 
 rectangular columns the corners should be beveled or rounded, as sharp corners are more seriously 
 affected by fire than round ones. 
 
 It is advisable in all cases to place occasional horizontal hoops around the vertical steel. 
 Although tests do not show that they are absolutely necessary if a 1 ^ to 2-in. concrete covering 
 is provided, yet it is good practice to employ them as an additional precaution against any 
 buckling of the rods under small loads. Such hoops, also, serve to keep the rods in place during 
 the pouring of the concrete, and should not be farther apart than 12 in. nor exceeding about 
 16 times the diameter of the vertical rods. For columns of moderate size, J^-in. wire hoops 
 are generally used 12 in. on centers. 
 
 Bands, hoops or spirals to be considered effective should have a clear spacing not greater 
 than one-sixth (preferably one-tenth) the diameter of the enclosed column — in no case, however, 
 more than 23^ in. — and should be equal in amount to at least 1 % of the volume of the column 
 inside the hoops (see report of the Joint Committee in Art. 7, Sect. 8). It should be noted 
 that the Joint Committee recommends an increase in the allowable working stress on the con- 
 crete of hooped columns only when the ratio of the unsupported length of column to the 
 diameter of the hooped core is not greater than 10. 
 
 It should always be kept in mind that hoops and spirals are tension reinforcement and 
 subject to all the rules governing the design of such steel. If hoops are used, the ends should 
 lap a sufficient distance to secure the requisite grip, or else the ends should be bent toward the 
 center of the column to accomplish the same purpose. When spiral reinforcement is employed, 
 it is important to have one continuous piece from top to bottom, or else joints should be made 
 as just explained. 
 
 Hooping should extend from the bottom of the column to the under side of the slab above 
 and adequate means should be provided to hold it rigidly in place so as to form a column, the 
 core of which will be straight and well-centered. For ordinary percentages of hooping, tests 
 show that wire of high-carbon steel gives a much larger increase in the ultimate strength of 
 column than wire of mild steel. 
 
 The core of a hooped column should preferably be circular in cross-section, but square and 
 rectangular hoops are sometimes employed in practice. (The Joint Committee recommends 
 that only circular hoops be employed.) In such cases, the hoops are probably less effective, 
 but it is not known how much. The ideal condition, theoretically, is hooped columns of cir- 
 cular cross-section with circular cores. 
 
 When light vertical rods are used they may be spliced by lapping a sufficient distance above 
 
Sec. 11-39] 
 
 BUILDINGS 
 
 529 
 
 the floor level (about 30 diameters) to develop the requisite bond strength. They should 
 be bent slightly inward at the top in order to come just within the rods of the column above, 
 thus preventing any tendency of the load coming down from the upper rods to bulge the 
 column. Rods much over an inch in diameter should be milled square and butted together 
 in a secure manner. 
 
 The splicing of large rods may be effected by joining them in rather closely fitting pipe 
 sleeves. All such splices should be made just above the floor level and not more than 12 in. 
 above the same. Since each rod should find a bearing on a rod below, the number of rods 
 increases downward in the building and, unless the loading is eccentric, great care should be 
 taken to have the rods symmetrically arranged in each story. A splice of this kind is good 
 for compression in the rods but is of no use when the stress is tension. Rods in tension may 
 be spliced by using short extra rods at the splice. These short rods should extend above and 
 below the joint far enough to develop the requisite bond strength. Rods may extend through 
 several stories if so desired, but it is difficult to hold them in place while concreting the lower 
 lengths. 
 
 At the bottom of the columns, the loads from the rods should be distributed over the 
 footing by means of a steel bearing plate. In order not to overstress the concrete in the column, 
 the concrete immediately below the plate should be enlarged so as to bring the average pressure 
 within the allowable. Tests show that if considerable lengths of the longitudinal reinforcing 
 rods are bent outward into a reinforced-concrete footing, the strength of column will be about 
 as great as when these rods are bedded on a metal plate. The results, however, also show that 
 the use of metal plates leads to greater uniformity and strength. Base plates should be set 
 in a thin grout of proportions about 1 : 2 in order to prevent hollow places under the 
 plates. 
 
 In our modern city buildings where the floor space on the first floors is valuable, the use 
 of a steel column core presents a very economical and efficient construction. To be able 
 to count upon the concrete in such columns, the steel itself should be sufficiently rigid to act 
 as a column and the concrete should be well enclosed either by the steel form itself or by means 
 of bands or hooping. However, if the amount of steel is very large, the relative value of the 
 concrete is more uncertain, and its element of strength should be neglected. The best conc^- 
 nection between column and beam is a structural-steel seat riveted to the column, and of suffi- 
 cient 'size to carry the entire load in bearing. These seats do not appear in the finished surface 
 of rooms and their stiffness is considerably assisted by the concrete shell. Where structural- 
 steel columns are used, the floor beams and girders should be figured as simply supported unless 
 the usual continuity rods are run through the column. 
 
 Wherever concrete walls are to be built against concrete columns at some later time, 
 weather breaks or key ways should be formed in the columns. This is accomplished by nailing 
 small strips on the inside of the forms. Where windows are to be set into column reveals, 
 window ties should be embedded. In order to obtain a low cost for formwork it is better 
 to vary square columns in one dimension rather than in two, both on account of the columns 
 themselves and the beams framing into them. 
 
 For the same reason it is advisable to keep the story heights uniform so that the forms will 
 not have to be cut off or spliced from story to story. If the story heights vary it is better to 
 have the high stories at the bottom as it is cheaper to cut off the forms than to splice them. 
 
 Cinder-concrete cylinders have been used to some extent as a protective covering for rein- 
 forced-concrete columns and also to take the place of column forms. In Factory No. 1 of the 
 plant of the Bush Terminal Co., South Brooklyn, N. Y., the interior columns are cylindrical 
 and composed of an outside shell of cinder concrete 23^ in. thick. These cinder-concrete 
 cylinders were prepared in advance in 2-ft. lengths, in a zinc mold, with spiral hooping and 
 expanded metal forming the inner surface. After hardening, they were set one upon another 
 in the building and filled with concrete. 
 
 Hooped concrete columns with cast-iron cores known as the Emperger columns have been 
 34 
 
530 
 
 CONCRETE ENGINEERS' HANDBOOK 
 
 [Sec. 11-40 
 
 \ 
 
 W 
 
 7^ 
 
 / 
 
 X 
 
 receiving some attention. It is claimed that by using the hooped reinforcement high working 
 stresses may be used, thus reducing the size of column (see Engineering Record, March 3, 1917). 
 
 Hollow reinforced-concrete columns have been used in heating and ventilation systems, 
 the air heating and fan systems being in a pent house on the roof. The heated air is forced 
 down the hollow columns and is distributed to the different floors through openings in the 
 columns. 
 
 40. Loading. — In addition to its own dead weight, a column should be designed to carry 
 the live and dead loads of the roof and floors above. The full live load on floors must be used 
 in designing columns for warehouses or buildings for heavy mercantile and manufacturing 
 purposes. In other buildings, however, of five stories or more in height, where it seems reason- 
 able to suppose that a full live load will never occur on all floors simultaneously, reductions 
 may be made as follows: 
 
 The live load on the floor next below the top floor may be assumed at 95 % of the allowable live load, on the 
 next lower floor at 90%, and on each succeeding lower floor at correspondingly decreasing percentages, provided 
 that in no case shall less than 50% of the allowable live load be assumed. i 
 
 41. Column Brackets. — Column brackets, such as are shown in Fig. 99, are serviceable 
 in stiffening the building frame and in decreasing the stress in the girders. They should not, 
 
 however, be considered as decreasing the girder span, 
 but simply as so much additional protection against 
 failure. 
 
 Brackets are usually required in high buildings 
 in order to brace the frame properly against wind 
 stresses. If, however, buildings are both high and 
 narrow, the stresses in the members due to wind 
 should be figured and proper provision made for 
 the same. For method of computing such stresses, 
 see Art. 9, Sect. 10. 
 
 A column supporting a roof may also serve as a crane post or to carry a heavy load on a 
 bracket. For method of computing the bending moment due to such eccentric loading see 
 Art. 10, Sect. 8 
 
 Illustrative Problem. — In order to show in detail the method of column design, the computation^ will be 
 given for the design of a typical interior column of a three-story building assuming that no bending moment is 
 likely to be caused by unequally loaded panels or eccentric loading. The columns will be spaced 21 ft. on centers 
 both ways and the two intermediate beam design of Plate I, page 448, will be adopted as the typical floor-bay 
 design for all three floors. 
 
 The height from finished floor to finished floor will be made 13 ft., except the basement which will be made 
 11 ft. A concrete mixture of 1 : 1}-^ : 3 will be used, which will allow a working stress of 565 lb. per sq. in. when longi- 
 tudinal steel only is employed and 870 lb. per sq. in. when eff'ective hooping is provided. As recommended in the 
 final report of the Joint Committee (see Appendix B) the concrete will be assumed to have a modulus of elasticity 
 one-twelfth that of the steel. The size of octagonal columns will be given in terms of the short diameter. 
 
 The load coming from the roof may be found by multiplying the total shear on both girder and cross-beam by 
 2, and adding. Assume end reaction from roof beam at 16,100 lb. and from roof girder at 36,200 lb. Then the 
 roof load is 
 
 (2) (36,200) + (2) (16,100) = 104,600 lb. 
 
 Similarly, each floor load is 
 
 (2) (55,300) + (2) (25,000) = 160,600 lb. 
 
 The building in question will be considered as for manufacturing purposes and no reduction in live load will 
 be m^de. The complete design of column is given on page 531. Tables of Sect. 8 will be used in the design. 
 
 The recommendations of the Joint Committee will be followed and the amount of vertical steel is in every 
 case less than 4% and more than 1 %. The pitch of the spiral hooping is given to the nearest \i in., as the mills 
 that fabricate spirals can manufacture them to such a pitch. The weight of column has been figured for a length 
 2 ft. less than the story height in order to compensate somewhat for taking the weight of beams and girders as 
 extending from center to center of supports. It is the intention to splice all rods, not lapped, by means of a tightly 
 fitting pipe sleeve about 12 in. long. 
 
 Fig. 99. 
 
 From 1916 New York City Building Code, Bureau of Manhattan. 
 
 I 
 
Sec. 11-41] 
 
 BUILDINGS 
 
 531 
 
 Third-floor Columns. — Assume weight of column at 2700 lb. Then the total load is 104,600 + 2700 = 
 107,300 lb. 
 
 Taking the percentage of longitudinal steel at 0.025, Table 1 shows that a column whose effective 
 diameter is 14 in. (total diameter 17 in.) is good for 111,000 lb. Table 3 shows weight of column per foot to 
 be (1.66) (150) and the total weight (1.66) (150) (11) = 2740 lb. Table 1 shows the steel area to be 3.9 sq. in. 
 and from Table 4 we get seven Ji-in. round rods. 
 
 Second-floor Columns. — Assume weight at 3800 lb., then total weight is 107,300 + 160,600 + 3800 = 
 271,700 lb. 
 
 Table 1 shows that with 1% spiral and 3.5% longitudinal rods, a column whose effective diameter is 17 in. 
 (total 20 in.) is satisfactory. Table 3 shows the weight to be (2.30) (150) (11) = 3800 lb. Table 1 gives the steel 
 area of 7.9 sq. in. and Table 4 shows that fourteen Ys-in. round rods will be sufficient. From Table 2 for pitch of 
 2]^^ in., the area of spiral is 0.106 sq. in. (say H-in. round) and the length of spiral per foot is 256 in. For mini- 
 mum pitch of l^i in. the area of spiral is 0.088 sq. in. (say Vs-in. round) and the length per foot is 366 in. The 
 11 X 12 
 
 value of 
 
 d'^ 18 
 
 First-floor Columns.— Total load is 271,700 + 160,600 + 6500 = 438,800 lb. 
 
 Table 1 shows that a column of 26 in. outside diameter and with 2 % longitudinal steel will answer. Weight is 
 (3.89) (150) (11) = 6420 lb. Steel area is 8.3 sq. in., or fourteen li-in. round. Area of spiral is 0.144 sq. in. (say 
 ^^6-in. or ^6-in. round) and length per foot is 347 in. For minimum pitch of 2^4 in., area is 0.13 sq. in. (say Ji^-in. 
 round) and the length per foot is 386 in. 
 
 Basement Columns. — Total load is 438,800 + 160,600 + 7500 = 606,900 lb. 
 
 Size of column from Table 1 is 31 in. outside diameter or 28 in. effective for p = 1.5%. Steel area is 9.2 sq. 
 in. Use fourteen 1-in. round rods. Weight of column is (5.53) (150) (9) = 7500 lb. Area of spiral is 0.175 sq. in. 
 (say H-in. round) and the length of spiral per foot is 422 in. 
 
 Column Schedule 
 
 = 7H- 
 
 Story 
 
 Kind of 
 load 
 
 Amount 
 of load 
 
 Shape and 
 size of 
 column 
 (short 
 
 diameter) 
 
 Vertical 
 
 steel 
 (rounds) 
 
 Spiral (core 
 3 in. less 
 than diameter 
 of column) 
 
 Third. . . . 
 
 Roof 
 Column. . 
 
 104,600 
 2,700 
 
 17 in. 
 
 octagonal 
 
 7-li in. 
 
 
 
 Total. . . . 
 
 107,300 
 
 
 
 
 Second. . . 
 
 Floor 
 
 Column. . 
 
 160,600 
 3,800 
 
 20 in. 
 octagonal 
 
 14-Ji in. (lap 
 above 27 in.) 
 
 dia. = in. 
 pitch = 2>^ in. 
 
 
 Total.. . . 
 
 271,700 
 
 
 
 
 First 
 
 Floor. . . . 
 Column. . 
 
 160,600 
 6,500 
 
 26 in. 
 octagonal 
 
 14-% in. (lap 
 above 27 in.) 
 
 dia. = J'ie in. 
 pitch = 23^^ in. 
 
 
 Total.. . 
 
 438,800 
 
 
 
 
 Basement 
 
 Floor. . . . 
 Column. . 
 
 160,600 
 7,500 
 
 31 in. 
 
 octagonal 
 
 14-1 in. 
 
 dia. = >2 in. 
 pitch — 2]y^ in. 
 
 
 Total.. . 
 
 606,900 
 
 
 
 
 Illustrative Problem. — It will be of interest to investigate the columns in the foregoing problem concerning 
 stresses due to unbalanced load. Since the columns just designed represent standard practice, the following investi- 
 gation will serve to point out the effect of unbalanced loading upon such a design. AH diagrams referred to in this 
 problem will be found in Sect. 10. 
 
 From a comparison of Diagrams 6 to 10, it may be seen that the largest moments in interior columns due to 
 unbalanced loads will occur in the second tier columns. 
 
 Assuming the neutral axis of the girder in Plate I to be governed by the fact that the beam is doubly rein- 
 forced, kd is found to be 13.1 in. 
 I = Ic + nis 
 
 (15)(13.1)3 (15)(22.9)3 
 
 Ic = 
 
 71.100 in." 
 
 nia 
 
 47,300 in.* 
 
532 
 
 CONCRETE ENGINEERS' HANDBOOK 
 
 ISec. 11-42 
 
 Whence 
 
 I = 118,400 in.4 and Ki = ^^^^ = 471 in.3 
 
 For the basement column, similarly, 
 
 I = 41,760 in.4 and = 316 in.3 
 For the first tier column, 
 
 I = 20,300 in.4 and K3 = 131 in.^ 
 ^0'= 0.675 Kz' = 0.278 Ko'/K/ = 2.42. 
 
 From an examination of Diagram 10 it may be seen that, for these values of Ka' and Ko' Kz', the moment 
 
 F 
 
 at the end of the first story columns will not exceed 20% of -j* Therefore, for the live-load moment, 
 ^ My 2/ (11 0,250)(21)(11.5)(0.20) 
 
 f^— = Wl-j = = 262 lb. per sq. in. 
 
 This is the extreme compressive unit stress in the concrete due to unbalanced loading. It may be noted that the 
 panels either way from the loaded one have no live load. Thus, referring to the design of this column in the pre- 
 vious problem, the total load carried by the column under present conditions is 
 
 438,800 - (^^^"^^^^Q) = 383.800 lb. 
 
 Now for a given column, in which no load is assigned to the spiral steel, the concrete unit stress due to direct load is 
 
 I 
 
 A[l +{n- 1) p] 
 
 Then in the above case 
 
 383 800 
 
 •^^ = (415.5)[1 + (0.02)(11)] = "^^^ ^'5- 
 This stress, when combined with that caused by the unbalanced load, gives 
 
 /c(max) = 262 + 757 = 1019 lb. per sq. in. 
 
 A re-design is advised. 
 
 Illustrative Problem. — The moment in an outer column of the same size as the interior column of the 
 preceding problems may be computed in the following manner. The dead-load moment may be computed with 
 the girder-end fixed, and the live-load moment with the girder-end hinged. The two results may then be added 
 together as the most probable moment in the column. For the live load (Case XI), 
 
 F r 3(131) T ^ F 
 
 l2(131)-f 471 + 2(48.3). 
 = (0.474) fl 10,250) (21)= 1,097,000 in.-lb. 
 
 r r n^idL) -1 t 
 
 " 7 L2(131)-f 471 + 2(48.3)J " ^-^^ 1 
 = {OA 
 
 For the dead load (Case XII), 
 
 1- 0.265? 
 
 I L3(131) + 2(471) + 3(48.3) J ' I 
 = (0.265) (50,350) (21) = 280,000 in.-lb. 
 Total moment = 1,377,000 in.-lb. 
 ^ „ , (1,377,000) (8. 5) 
 
 For flexure / = 300 = ^'^^ P^^" sq- in. 
 
 which is the stress due to flexure only, at the top or bottom of the first-story column. The direct load on this 
 outside column will be 438,800 2, plus, say, 50,000 lb. for the spandrel load above the first floor. The direct 
 stress is 
 
 . 269,4 00 
 
 = (415.5) [1 -h (0.2) (12)] = P^'" 
 
 Thia, added to the flexure stress above, gives for a total stress 
 
 /c(max) = 576 + 523 = 1099 lb. per sq. in. 
 
 WALLS AND PARTITIONS 
 
 42. Bearing Walls. — The two principal types of reinforced-concrete office and factory 
 buildings differ only in the character of the outside walls. A skeleton type is the common form 
 of construction but bearing walls are built occasionally. 
 
 Although the bearing-wall type of construction is naturally the only type employed in 
 concrete residences, it has not been found in much favor for manufacturing and commercial 
 buildings in general. Of course, a building with monolithic wall construction possesses great 
 
Sec. 11-43] 
 
 BUILDINGS 
 
 533 
 
 rigidity when properly designed and reinforced, but it cannot be erected as rapidly as buildings 
 with other wall types, and, unless great care is employed in construction, there is a likeli- 
 hood of defects occurring in casting the work. There is also difficulty in obtaining a uniform 
 color for the wall surfaces. 
 
 Concrete walls are either of single thickness, or of double thickness with an air space 
 between. One method of obtaining an air space is by building in a central course of hollow- 
 tile blocks. Double walls render the interior of the building less subject to changes in tem- 
 perature and more completely moisture-proof, but, in the ma- 
 jority of office and manufacturing buildings, single walls are not 
 objectionable and make a great saving in floor space. Occa- 
 sional projections or pilasters improve the appearance and add to 
 the strength of a single wall. Building codes usually specify that 
 concrete bearing walls are to be made the same thickness as 
 brick bearing walls. 
 
 Except in residences, bearing-wall footings must usually 
 be reinforced. A cantilever projection is formed on each side of 
 the wall and the amount of reinforcement may be determined as 
 for a simple cantilever beam (see Art. 5, Sect. 12). 
 
 43. Curtain Walls. — In the usual type of construction, wall 
 girders are placed at each floor and the reinforced-concrete walls 
 (called curtain walls) are designed merely to fill in the panels be- 
 tween the columns and girders which form the skeleton frame 
 of the building. They are not intended to carry any weight 
 but need to be strong enough to withstand wind pressure of 30 
 lb. per sq. ft. Such walls may be designed as slabs carrying a 
 uniformly distributed load and supported on all four sides. 
 Figuring in this way, using the above value for the wind pres- 
 sure, a wall in ordinary building construction will never exceed 
 33^ to 4 in. in thickness, which is the practical limit for ease in 
 construction. In fact, 6-in. walls will generally be required in 
 order to obtain sufficient imperviousness to moisture. Also, a 
 6-in. wall is usually cheaper than a 4-in. wall owing to the greater 
 ease of pouring the concrete and placing the reinforcing steel in 
 position. A 4-in. wall, however, is all that is needed for fire 
 protection. Building codes usually specify a minimum thickness 
 of 8 in. 
 
 In addition to the consideration of wind pressure, concrete 
 walls are reinforced generally for the purpose of preventing cracks 
 and guarding against accidents during or immediately after con- 
 struction, except, of course, where lateral pressures occur, when 
 
 Ca/. 7/ 
 
 the 
 
 First Floor Plan 
 
 Fig. 100. — Lang building, 
 Haverhill, Mass. 
 
 beam action must be considered. The reinforcement gener- 
 ally consists of yi to 3'^-in. rods placed both horizontally and ver- 
 tically from 12 to 18 in. apart. 
 
 Slots in the columns are made by nailing a strip on the inside of the column forms. In 
 this way the curtain walls are mortised into the columns and a contraction joint is formed, so 
 that the contraction to be provided for is due only to the curtain wall itself. 
 
 When curtain walls are employed with a small percentage of window area, it is customary 
 to use both horizontal and vertical reinforcement, the same as when the wall covers the entire 
 panel, and to place one or more rods near the edge of the slab about all openings. When wire 
 fabric is used for reinforcement, the edges of the openings are stiffened by bending back the 
 fabric into a U-shape. 
 
 Where a large percentage of the outside wall of a building is used for windows, as is gener- 
 
534 
 
 CONCRETE ENGINEERS' HANDBOOK 
 
 [Sec. 11-43 
 
 4'/?ffof S/ab ■ ^"*^ocfs 9*c foe. 
 
 /z'.Zo'Cof. 
 ^'(* Hoops 
 
 /e'.zo'Co/ 
 
 4-S''<f Pods /Z'-3' 
 
 /0-f^//00pS //'',/7* 
 
 4-/^'P Rods /3'-0' 
 /0-^'''^//oops /7'/7' 
 
 zs'-o' 
 
 20'xSO'Co/ 
 8-M"<f/?ads /S^-3" 
 J2'i"^/yoops/3^/7^ 
 
 {20'x20'Co/ 
 \d-/j"^J?ods/si6 
 
 /e"x26"Co/. 
 
 HOOPS /7^.y7' 
 
 6- ^^Rod^ 
 
 ^' * Hoops /2'cibC 
 
 4- /"^Pods /S'-O^ 
 /O'^"^ Hoops /3{y7^ 
 
 /S"x24'Co/. 
 
 4-$°^Rods/Z'-3'' 
 
 /0-:^'^HoopsJj'My 
 
 /S''x24''Co/ 
 
 4-^''^A;ds/g*r^ 
 
 /0-;^''fiHoopsAfiJ7'^ 
 
 /0-;f'^Hoops /3!£/^ 
 
 ^*,24''Co/. 
 4-^'*/?ods/2'-s'^ 
 
 Hoops /3"m/7" 
 
 /6',24'Co/ 
 4-f^/?ods/2'-6" 
 /O-^'* Hoops /3''*2/' 
 
 ^"x24''Co/ 
 /2-;l:'^ Hoops /3'lc/S^' 
 
 F/n/s/7ed f/rst f/oor 
 
 Dnvewqy 
 
 Earth /7// 
 
 TYPICAL CROSS SECTION THROUGH BUILDING 
 Fig. 101. — Lang building, Haverhill, Mass. 
 
Sec. 11-43] 
 
 BUILDINGS 
 
 535 
 
 ally the case in factory buildings, the wall proper consists of little more than columns with 
 deep wall beams or wall girders forming belt courses between them. The spandrels, or walls 
 directly beneath the window sills, may be reinforced so as to act as the upper part of the wall 
 beams, but the usual method is to consider this portion as separate from the beam and merely 
 reinforce with small rods (in some cases with wire fabric) so as to prevent cracks and to reinforce 
 the construction during the setting of the concrete. If this is done, the spandrels may be put 
 
 *^ 1 1 Finished BHJ^ F/oor 
 
 F/n. ^'^r/oar 
 
 l£-I"<f Hoops 4 ^-^^ 
 
 ffi 
 
 r., Section 
 \^^E!rL3Lfloor co(s. 58-49 Ind.SI 
 " 53-66 
 
 i ^Hi! 
 
 Fin /KiF/oor 
 
 1=^ 
 
 Section opposite Windov?3 
 
 Elevation of 
 Typical Interior Column 
 
 (27 Cols; 
 
 4' 4" 
 
 ' l"x4 "m-ather Brea^ 
 
 4 Rods /g"'^ IS- J" ^ 
 /Z-y^floops /z"c.foc is"/S 
 
 Elevation of ^C^^XBldg \Une 
 
 Typical Exterior Column ^ L-^e-'J 
 
 Section opposite Panel Walls 
 For Cols.2-)3 Incl. 21-32 Incl 
 72-99 Incl. and 15 (53 Cols.) 
 
 — *-i 
 
 "^'^7 Foadyyay Beam 
 ^\fvt>e put in /afer 
 
 •■^J BracJfef 
 
 i- - ForCois 2-6 inc/ 
 
 :" 
 ^ tVl 
 
 7-J2 
 
 Bracket 
 
 Fin. Basement F/oor 
 
 Details of East Cols 
 
 Cols. I & 7/ 0pp. Windows 
 
 ■I6-: 
 
 3^ 
 
 - ~~—f,i 4 Keyyyay 
 
 4"^ifoc>ps 
 l-ie'ct^c. 
 Cols I a 71 Opposite wall Panels 
 Col. 71 as shown 
 Col. f Opposite Hand 
 
 ^^""S/ab distributing rods 
 y'' Stair landing s/ab rods ^ 
 turn up into pane/ yyaZ/s S-6 
 2--^" conrinuous rods at stair 
 /andin^s. 2- to tar sides 
 ■ .-Fin Bl!^ F/. of columns 
 
 i."mod 
 
 Detail of "Typical Wall Panel 
 
 Fig. 102. — Details of Lang building, Haverhill, Mass. 
 
 in after the main structural parts have been cast, in the same manner as complete curtain walls. 
 This saves time in the erection of the building and allows the use of more care in obtaining a 
 neat finish on the spandrel walls. 
 
 In Fig. 102 two spandrel sections are shown, one where wall beams are employed and 
 the other where only the floor slab and spandrels form the exterior belt course. Figs. 100 to 
 106 inclusive show many of the details of the Lang Building at Haverhill, Mass. Many of 
 the details are referred to elsewhere. 
 
536 
 
 CONCRETE ENGINEERS' HANDBOOK 
 
 [Sec. 11-43 
 
 £/crsf/c cement 
 
 4 "Concrete 75p screecfecf fo Xme/ 
 
 yyi7terprc>of/r7£r and -SiJ/ic/ ^ "'^^^'^'^ 
 
 £/asf/c Ce/77e/7f: 
 
 £d^e of Concrete Si// 
 PIpn of Elevator Door Opening 
 
 Bo/red to co/omn 
 Fig. 103. — Details of Lang building, Haverhill, Mass. 
 
Sec. 11-43] 
 
 BUILDINGS 
 
 ELEVATlOn 
 
 SECTION 
 
 1 I 8 NAILING 
 
 
 ^ ■ % EDGE or 5ILL 
 
 
 'cast 
 
 
 
 SLOCK 
 
 5-4'oPETlING 
 
 
 
 8'-0i' 
 
 
 
 • POIlTlQH'Or'nWT'n,GDR.'PLAH ' 
 
 ' jnownie;LOCATioN:orDa5UiJHown'AwvB 
 
 Fig. 104. — Details of Lang building, Haverhill, Mass. 
 
538 
 
 CONCRETE ENGINEERS' HANDBOOK 
 
 [Sec. 11-43 
 
Sec. 11-44] 
 
 BUILDINGS 
 
 539 
 
 Reinforced concrete is well adapted to the construction of walls where considerable strength 
 is required, but in very thin walls, such as curtain walls, it becomes a relatively expensive 
 building material on account of the cost of forms. Brick, concrete blocks, or terra-cotta are 
 often used on this account, not only for spandrel walls, but to cover entire wall panels. 
 
 44. Brick and Other Veneer. — From an architectural standpoint it sometimes becomes 
 necessary to cover the exterior columns and walls with brick, terra-cotta, marble, limestone, or 
 other material. The facing or veneer is usually laid after the structural frame is completed. 
 
 •Stairs at Rear of Building 
 
 PIPE RAIL DETAILS 
 Fig. 106. — Details of Lang building, Haverhill, Mass. 
 
 A brick veneer generally consists of one thickness of brick, and this is tied into the concrete 
 work by means of copper or galvanized-iron ties. These ties are usually about % in. wide 
 and are tacked to the form boards in such a manner that about 4 in. of the tie will be embedded 
 in the concrete work. When the forms are removed, the portion of the tie lying flat against the 
 forms is bent out as shown in Fig. 107. A tie should be provided for about every 4 sq. ft. of 
 
540 
 
 CONCRETE ENGINEERS' HANDBOOK 
 
 [Sec. 11-45 
 
 wall surface. Brick facings should be supported, at least at every floor, by a ledge formed in 
 the concrete. Angle irons are usually employed to carry stone and brickwork over door and 
 window openings. 
 
 Fig. 108 shows one method of supporting a stone lintel. The anchor at the top should be 
 noted. 
 
 Fig. 107. 
 
 The arrangement of the ties and supports for terra-cotta varies greatly with the design. 
 The facing is usually supported by projections in the concrete fitting into openings in the terra- 
 cotta, forming a sort of dovetailing. Iron anchors then tie the two together. It is very im- 
 portant that suitable play be provided for, as terra-cotta cannot be made to exact dimensions. 
 Stone facings are also supported by projections in the concrete, the same as terra-cotta, but 
 the stone can be made to more nearly fit any simple shape given to the concrete ledge. 
 
 Fig. 108. 
 
 45. Window Openings. — The arrangement of floor slab and wall beam in Fig. 108 makes 
 it possible to obtain the maximum amount of light within the building; that is, the window 
 frames are run as close as possible to the underside of the floor slab. It should be noted that 
 to accomplish this, the beams should either be run parallel with the walls of the building, so 
 that none of the beams will take bearing on the lintel, or else the floor arrangement should 
 be similar to that shown in Fig. 101 where only parallel girders are employed spaced relatively 
 close together. If beams run parallel with the walls, the spacing of the beams may be returned 
 at the corners by a diagonal girder as illustrated in Fig. 109. By so doing the windows may 
 be made the same height throughout. 
 
Sec. 11-45] 
 
 BUILDINGS 
 
 541 
 
 Reinforced-concrete buildings to be considered fire-resisting should have windows of wire 
 glass, held in metallic frames. Wire glass is either ribbed, rough, maze, or polished plate 
 having wire embedded in its center during the process of manufacture. The wire netting 
 used for this purpose is similar to ordinary chicken netting with about a 1-in. mesh. It is 
 embedded in the glass at a very high temperature 
 which insures adhesion between the netting and glass, 
 and the two materials become one and inseparable. 
 
 If the glass is broken by shock or by intense heat, it J-^r^ .v. '^^.T^ 
 
 remains intact. It is this property which gives wire 
 glass its fire-retarding qualities. 
 
 Metal frames are made in a great variety of 
 forms to meet all purposes. Sashes may be station- 
 ary, sliding, pivoted either horizontally or vertically, 
 
 GJa^s-. 
 
 \ 
 
 \ T 
 
 \^ L,_ 
 
 Norma/ 6/ refer, 
 
 [_. 
 
 Normal Beams 
 
 Fig. 109. 
 
 Fig. 110. 
 
 hinged, or double hung with weights like an ordinary window. Sash may also be obtained 
 which will close automatically and lock under fire by the fusing of a link or other means to ac- 
 complish the same result. A number of sections of steel-sash windows are shown in Figs. 110, 
 
 , 111, and 112. These sections are included to show 
 ' ' , ' ' a few of the different methods of fastening metal 
 frames to the walls or wall columns. 
 
 When properly constructed, a tin-covered wood 
 shutter is a most effective window protection. 
 There is one objection, however, to the use of shutters 
 on window openings and that is, they must neces- 
 sarily be open while the building is in use, and, when 
 
 /inchor 
 
 Secrion 
 through 
 Jamb 
 
 i>race T- frame 
 
 Finished sill 
 ■formed after 
 sash are set 
 Line of rou^h yyalhr. 
 
 Grout 
 
 ! Glazing C//p 
 
 \7a 
 
 Vertical 
 Section 
 
 Jamb 
 
 Fig. 111. 
 
 Fig. 112. 
 
 the need comes for them, they are apt to be overlooked and not closed. They, moreover, 
 hide a fire and are unsightly for many locations. Frames and sash of wood covered with metal 
 are sometimes used — the wood furnishing the strength and the metal the protection. Note the 
 combination of wood, tin and steel in the window design shown in Fig. 113, 
 
542 
 
 CONCRETE ENGINEERS' HANDBOOK 
 
 [Sec. 11-45 
 
 Automatic steel rolling fire shutters, placed either on the outside of window openings or 
 in the window reveals immediately in front of the window frames and sash, form the least 
 objectionable appearing type of shutter protection for windows, and seem to be entirely ade- 
 
 J 
 
 Countersunk 
 rap Screyrs in 
 J "' /" Stops 
 
 Fig. 113. 
 
 Fiu. 114. 
 
 Fig. 115. 
 
 Section through 
 Jamb above 
 Transom Bar 
 
 Section through 
 Box and Jamb 
 
 Weight Box 
 
 rrr 
 
 Fio. 116. 
 
 Section 1-hrough 
 Basement Window 
 Fig. 117. 
 
 quate for all ordinary conditions. Open sprinklers, or water curtains as they are frequently 
 called, are sometimes used in connection with shutters or wire glass windows, but do not seem 
 to be very efficient in themselves except in very moderate exposures. 
 
Sec. 11-46] 
 
 BUILDINGS 
 
 543 
 
 Fig. 114 shows a window construction with ordinary wooden sash employed in a large 
 hospital in the Middle West. Fig. 115 shows the window construction in a shoe factory at 
 Cambridge, Mass. Double-hung window details for a building at Boston, Mass., are shown 
 in Fig. 116. A detail of a basement window opening is given in Fig. 117. Fig. 118 shows a 
 common window and wall construction. 
 
 Tar ar?cf ffraye/ Poof" 
 
 /6 oz. Copper-' 
 
 i"°Pods 4'-0^/or7^ /S^c.foc. 
 
 Wood Sash and 
 
 tY/rp 6/ass 
 
 -Cast Stone St// 
 
 Bhc/r Wa/F 
 befyreen 
 
 /^" co/umns 
 
 Floor Plan 
 
 Fig. 118. — Truck garage. Pacific Mills, Lawrence, Mass. 
 
 46. Door Openings. — Doors in a reinforced-concrete building should be fire-resisting, the 
 same as windows, and may be of either the hinged, sliding, or rolling type. If tin-covered wood 
 doors are provided on openings in interior partition walls, they are generally hung on inclined 
 tracks so that they will close automatically. Where it is desirable to keep them open, an 
 automatic release operated by a fusible link is provided. 
 
544 
 
 CONCRETE ENGINEERS' HANDBOOK 
 
 [Sec. 11-46 
 
 Sliding and rolling doors are generally used only for large openings such as driveway and 
 freight-elevator entrances and. are usually arranged to close automatically in case of fire. Slid- 
 ing doors should have the rails on which they operate protected by metal shields to prevent 
 obstruction.. Steel rolling doors are made in horizontal, jointed, sectional strips, which wind 
 
 Section 
 through 
 Head 
 
 Hin^e 
 
 Section 
 through 
 Transom Bar 
 
 Section 
 through 
 Head 
 
 Sash 
 
 Section throu3h 
 Trcjngom Bar 
 
 Section 
 through Jamb 
 
 "Sectfon through 
 Jamb 
 
 Strap Hinge 
 
 ^OQuHle shecifhing 
 battsns ip form panels 
 Fig. 120. 
 
 up on a roller placed in a pocket above the opening, the ends moving in metal grooves to hold 
 them in place (see Fig. 118). There are a number of types of vertically operated doors. A 
 common type for freight elevators is shown in Fig. 103. 
 
 Fig. 121, 
 
 Fig. 119 shows the details of an ordinary outside wooden door. Fig. 120 gives the details 
 of large swinging driveway doors used in a building at Boston, Mass. The arrangement around 
 the main-street entrance doors to the same building is shown in Fig. 121. Similar door details 
 in the Lang Building (Fig. 104) should also be noted. 
 
Sec. 11-47] 
 
 BUILDINGS 
 
 545 
 
 47. Basement Walls.— Basement walls to sustain earth may be designed as simply sup- 
 ported slabs, the earth-pressure reactions being usually taken by the basement and first floors. 
 The common construction is to employ curtain walls at least 8 in. thick between wall columns, 
 and, in addition to reinforcing them vertically to take the earth pressure, to place rods near 
 the bottom of the wall so as to make the wall carry itself as a beam from footing to footing. 
 This type of wall thus requires no foundation of its own and may be built in the same way as 
 the curtain walls in the upper stories. Sometimes it becomes necessary to reinforce hori- 
 zontally for the earth pressure. This brings a lateral force on the columns, but the resulting 
 eccentricity of column loading may be disregarded for ordinary cases unless a very light wall 
 column is used. It is customary in the design of basement walls to assume the earth pressure 
 as due to a fluid weighing 30 lb. per cu. ft. 
 
 Sidewalk lights are formed of circular pieces of plate glass set in reinforced-concrete slabs, 
 and supported by steel beams. A typical vault-light construction, using concrete beams is 
 shown in Fig. 122. The spacing of the beams should be governed by the thickness of the slab — 
 the usual spacing being 3 to 4 ft. 
 
 Concrete beam 
 
 Prisms " 
 
 Expansion Joint 
 fil/ed yvit/7 
 traterproofin^ 
 
 Fig. 122. 
 
 48. Partitions. — Solid concrete partition walls may be made 3 or 4 in. thick if reinforced 
 in a similar manner to exterior curtain walls. Extra rods should be placed near the edges of all 
 openings, and rods should project into the floor and ceiling for anchorage. It is usually con- 
 venient to pour the concrete after the floors are laid, and, where partitions are not located under 
 beams, this may be done by leaving a slot in the floor at the proper place. 
 
 Concrete blocks have been used to some extent, but their thickness is often objectionable. 
 In a warehouse at Nashville, Tenn., concrete blocks were employed 8 by 8 by 24 in. in size 
 with two hollow spaces. The blocks around the elevators were 4 by 4 by 6 in. solid. Rabbets 
 were formed in each end and in top ^nd bottom surfaces, and these were filled with cement 
 mortar as the blocks were laid, in order to secure as perfect a bond as possible. 
 
 A solid concrete wall 4 in. thick makes a very efficient fire-resisting partition, but is heavy, 
 and difficult to install. For this reason metal lath and plaster, terra-cotta tile, and plaster- 
 block partitions are generally used in preference to concrete. 
 
 Metal lath and plaster partitions are extensively employed in concrete buildings and 
 make a fairly good (although not first-class) fire-resisting construction. Some tests of plaster 
 partitions in actual fires have shown such constructions to be reliable under fairly severe con- 
 ditions, while other tests have shown failure, due to the difficulty of obtaining sufficient bond 
 between the plaster and the metal lath to resist successfully the combined action of fire and 
 water. 
 
 The ordinary form of metal lath and plaster partition consists of channel-iron or other 
 steel studding set crosswise of the partition, and connected at the top and bottom with the 
 floor. The metal lath is then fastened to both sides of the studding and each side plastered 
 35 
 
546 
 
 CONCRETE ENGINEERS' HANDBOOK 
 
 [Sec. 11-48 
 
 with a mixture of lime and cement mortar. Three coats of plaster are generally applied — the 
 first or scratch coat containing the usual quantity of hair or fiber. Openings are cased on all 
 sides with a steel framework to which the lath is firmly fastened. This type of a partition is 
 usually made from 4 to 6 in, thick, and made either hollow or with a central body of cinder 
 concrete. A 2-in. solid partition, however, is sometimes used and consists of only one thickness 
 of metal lath wired to the steel uprights. 
 
 Metal lath and plaster partitions made by the Roebling Construction Co. are shown in 
 Figs. 123, 124 and 125. The studs are fastened top and bottom by means of knees. Where 
 flats are used for uprights, the knees are formed by simply bending the ends of the studs. 
 Where cement-finished floors are used, wood plugs or expansion bolts are inserted for fastening. 
 Wood furrings, where required for nailing purposes, are attached to the studs by means of 
 
 iC 
 
 Elevation 
 
 /Va 18 Galy. Wire Lacing 'Rough Door 
 
 Frame 
 
 Enlarged Section A-B 
 Fig. 123. 
 
 staples. Furrings are usually required to receive the baseboards, chair rail, and picture mold- 
 ing. They are not needed, however, where cinder concrete is filled in between the lath since 
 cinder concrete will receive nails readily. The vertical members at door openings are made 
 to extend the full height from floor to ceiling. Such members around openings are punched 
 at intervals with holes to permit the fastening of wood frames or other trim. Of course, from 
 a standpoint of fire-resistance, metal or plaster trim is preferable to wood. 
 
 Two types of plaster partitions erected by the Expanded Metal Cos. are shown in Figs. 
 126 and 127. A metal and plaster partition known as the hy-rih construction is shown in Fig. 
 128. The ribs of the hy-rib take the place of studs and eliminate the necessity of attaching 
 sheets of lath to studs. The manner of attaching a hy-rib partition to floor and ceiling is shown. 
 
Sec. 11-48] 
 
 BUILDINGS 
 
 547 
 
 y/ x8 C/ranrre/ 
 
 No. 18 6a/y, Wire 
 
 Enlarged Section with Door Casing 
 
 Fig 124. 
 
 Expanded Meftr/ or 
 
 Fig. 125. 
 
 Fig. 126, 
 
548 
 
 CONCRETE ENGINEERS' HANDBOOK 
 
 Sec. 11-48] 
 
 Where hollow partitions are required, two thicknesses of hy-rib are used, separated by means 
 of rib studs. 
 
 Terra-cotta tile partitions are very satisfactory light-weight partitions if made of semi- 
 porous or porous material. Partition blocks are made in thicknesses varying from 2 to 12 in., 
 the 4-in. block being the most common. Plaster on both sides of a terra-cotta partition in- 
 creases the thickness by 13^ in., that is, in. to a side. Two-in. and 3-in. tile partitions are 
 not recommended for dependable efficiency in case of fire. The safe height of a terra-cotta 
 partition may be approximated by multiplying the thickness of the block in inches by 40. 
 This will give the safe height in inches. 
 
 Full-porous terra-cotta blocks are slightly more expensive than semi-porous, as they 
 weigh more per square foot, and have heavier faces and webs, but they make a partition which 
 
 is decidedly more dependable under fire test. At 
 least a portion of a partition should be built of 
 full-porous blocks in order to provide for the nail- 
 ing on of wood trim. 
 
 Terra-cotta partitions should be securely 
 braced with slate at the ceihng. The blocks 
 should be wet before being laid and also before 
 being plastered, and should be laid in a mortar 
 composed of 1 part lime-putty, 2 parts cement, and 2 or 3 parts sand. 
 
 Plaster blocks as used in partition construction are made principally of gypsum or plaster- 
 of-Paris, with an admixture of wood fiber, reeds, or other suitable material. They are extremely 
 light, easy to handle, and can readily be cut and sawed, but possesses several disadvantages 
 in actual use. They also offer poor resistance to hose streams in case of fire. 
 
 The following table gives weights of various forms of partitions. The weights for the 
 block partitions do not include the plastered surfaces. If a partition is to be plastered on both 
 sides, add 10 lb. per sq. ft. 
 
 Fig. 127. 
 
 Exparrcfecf 
 Me fa/.. 
 
 Weights of Partitions 
 
 Ffoor 
 
 Hy-rib 
 
 My-Rib 
 
 Steel Channe/ or 
 //■ood strip may be 
 substitLrted -tbrang/e 
 
 Expans/'or? Bo/t or 
 .' kVoocf P/u^ and Screw 
 
 Kind of partition 
 
 Thickness 
 (inches) 
 
 Weight 
 (lb. per 
 sq. ft. of 
 partition) 
 
 Solid plaster .... 
 
 /2 
 
 20 
 
 
 32 
 
 Hollow plaster. . 
 
 
 4 
 
 22 
 
 
 
 ' 2 
 
 12 to 14 
 
 
 
 3 
 
 15 to 17 
 
 Terra-cotta 
 
 
 4 
 
 16 to 18 
 
 
 
 5 
 
 18 to 20 
 
 
 
 6 
 
 > 
 
 24 to 26 
 
 
 
 2 
 
 7 
 
 Plaster block. . . . 
 
 
 4 
 
 12 
 
 
 
 .8 
 
 22 
 
 Fig. 128. 
 
 Large buildings should be divided by fire walls of either concrete or brick, so that if fire 
 occurs in any part of the structure it will be kept from spreading. A fire wall to be thoroughly 
 effective should run from basement to roof and be provided with automatic fireproof doors. 
 Severe exposures require fire doors on each side of such walls. 
 
[Sec. 11-49 
 
 BUILDINGS 
 
 549 
 
 Stairs and elevator shafts should be enclosed in either concrete or brick partitions. Where 
 considerations of appearance or light prevent the use of opaque enclosures, the fire hazard 
 should be reduced by using wire glass in metal frames. Open grille-work around passenger 
 elevators should be of this construction. 
 
 STAIRS 
 
 49. General Design. — The usual type 
 of reinforced-concrete stairway consists of 
 an inclined slab with the steps formed upon 
 its upper surface. The design of such stairs 
 is a simple problem, the slab being figured 
 as freely supported and with a span equal 
 to the horizontal distance between sup- 
 ports. Transverse reinforcement is used 
 only for stiffening and to prevent shrinkage 
 cracks. A common load used in designing 
 stairs in commercial and manufacturing 
 buildings is 100 lb. per horizontal sq. ft. 
 
 A simple method of design is to support 
 the ends of a stairway slab directly on floor 
 beams or floor girders, or on some special 
 beam or beams inserted for the purpose. 
 This, however, cannot always be accomp- 
 lished conveniently and it is common de- 
 sign to make the span of a stairway slab 
 to include a platform slab as well. Stresses 
 
 in slabs of this type are somewhat indeterminate on account of the angle which occurs at 
 the edge of the platform; but many such slabs, however, have been computed as simple 
 slabs freely supported and have given satisfaction in every case. 
 
 /Vc^foT/t^r) stairway design, some arrangement is 
 
 ^""^."^^^^ \ usually made whereby short spans may be em- 
 ployed, as slab construction is not suitable for 
 long flights. When it becomes necessary to use 
 
 Fig. 129 a. 
 
 Section at A-A 
 
 Fig. 1295. 
 
 stair spans of great length, however, without any chance of intermediate support, a side 
 girder construction may be employed. 
 
550 
 
 CONCRETE ENGINEERS' HANDBOOK 
 
 Sec. 11-50] 
 
 Stairways should be enclosed in fire-resisting partitions in order to prevent the spread of 
 fire (see Art. 48). 
 
 
 
 IIP 
 
 rv 
 
 D- 
 
 iiiwi 
 
 \\-- - - 1 
 
 ■•Ion of Stairs under 4^F] 
 A 
 
 l.J 
 
 1 
 
 
 iitit 
 
 Ml 
 
 4d 
 
 
 
 Detail of Stair Beam 
 ond Hanger 
 
 Fig. 130. 
 
 50. Methods of Supporting Stairs. — The slab method of construction is usually the cheaper 
 and employed wherever possible. In long straight flights, a beam is often used to shorten 
 
[Sec. 11-51 
 
 BUILDINGS 
 
 551 
 
 the span. This beam may run between the regular columns which make up the structural 
 frame, or additional short posts may be provided. Fig. 105, page 538, shows a stairway sup- 
 ported in this way. The design is that employed near columns 1 and 2 of the Lang Building, 
 Fig. 100, page 533. 
 
 Sars 8 c toe S'-o' ^ 
 ' Stirrups /B'cYoc 
 
 Fig, 131. 
 
 Where a stairway must be broken into two or more runs to the story, it is customary to 
 employ either rod hangers or beams intermediate in the story height. Figs. 129 A and 1295 
 show a stairway supported by rod hangers only. In Fig. 130 both intermediate beams and 
 rod hangers are employed. 
 
 Stairways are generally constructed at some convenient time after the structural frame 
 is completed. Where stair-runs start from floor beams or floor girders, 
 a plank should be nailed to the side of the beam forms to cause rab- 
 bets in the concrete, and dowels should also be provided. 
 
 The ends of a stairway slab are usually fixed to more or less 
 extent and the dowels provided in the course of construction may 
 well be made of such length and in such numbers as to give a suffi- 
 cient reinforcement for negative moment. In short spans fixed in 
 this way, the moment at the center of slab may be computed with 
 
 safety using the formula M = 
 
 10 
 
 Fig. 132. 
 
 A stairway supported by brick walls is shown in Fig. 131. The 
 same arrangement would be employed with concrete bearing walls. 
 
 Steps are sometimes molded after the rough concrete stair slab is in place. The steps may 
 then be molded separately and set on the slab, or they may be poured in place. In the former 
 case, rods should be embedded in each step to permit of handling without injury, while in the 
 latter case the steps are all poured at once in a similar manner to the way the pouring is done 
 when they form an integral part of the slab. 
 
 61, Stair Details. — In factory construction stair treads are usually surfaced with a 1 or 
 
552 
 
 CONCRETE ENGINEERS' HANDBOOK 
 
 [Sec. 11-52 
 
 l)^-in. ceUient top finish. A method of constructing the steps is illustrated in Fig. 132. The 
 form board for the riser is arranged as shown at A and the top of this board is beveled so that 
 by placing an upper form board on the outside a nosing or projection in the concrete work is 
 formed. 
 
 Another type of step with cement finish is shown in Fig. 130. This step makes a good 
 appearance, but is a trifle more difficult to construct than the step previously mentioned. 
 
 The outer edge of concrete steps are protected in many cases by a light steel angle which 
 runs the entire width of the stairs. When linoleum is used, this angle is raised so as to have 
 the upper leg flush with the finished tread. Metal non-slipping treads embedded in the concrete 
 are sometimes employed. 
 
 Fig. 1295 shows the method of connecting concrete stairs with a wood-finished floor. Fig. 
 133 gives the details of wood-covered concrete stairs employed in a fireproof residence. In the 
 best buildings, treads and risers are often covered with marble slabs having plaster soffits. 
 
 The rise of a stair is the height from the top of one step to the top of the next. The run is 
 the horizontal distance from the face of one riser to the face of the next. The run is usually 
 less than the width of tread on account of the nosing. To secure a comfortable stair, the run 
 must bear a certain relation to the rise. One rule is that the sum of the rise and run should 
 be equal to from 17 to 17>^ in. For ordinary use, a rise of 7 to 7K in. is about right. Stairs 
 
 having a rise greater than 7^^ in. are steep. Properly designed stairs without nosings should 
 have at least 12-in. treads to be comfortable and the above rule does not apply to this type 
 of stair. 
 
 The common railing is of galvanized-iron pipe put together with malleable-iron fittings 
 and having the stanchions rigidly secured in sockets in the concrete work. Fig. 106 gives 
 details of a railing of this type used in the Lang Building. The hand-rail should be placed at 
 a height of about 2 ft. 6 in. above the tread on line with the face of riser. Rails monolithic 
 with the stairs are sometimes used. 
 
 52. Elevator-shaft Pits. — Elevator-shaft pits should not be less than 3 ft. below the base- 
 ment floor. When oil-cushion buffers are employed, however, the pit depth should vary accord- 
 ing to the speed of the elevator and the stroke of the buffer designed for that particular speed. 
 With a car speed of 600 ft. per min., the pit should be about 11 ft. deep at the center where the 
 buffer is placed. 
 
 If the first floor is made the bottom landing, a pit pan may be suspended from the first 
 floor, and buffers, if used, may be placed on the basement floor and arranged to project up into 
 the pit through special openings in the pan. This method is generally employed when it is 
 advantageous to gain headroom in the basement underneath the pit pans. All footings for 
 columns and foundations adjoining elevator shafts should be kept below the floor of the elevator- 
 shaft pit, and the pit itself should be finished to plumb-line dimensions. 
 
 In localities where the water level in the soil is very high, or where there are underground 
 
 Stcr/r Land/r?^ 
 
 TbpF/oor, 
 
 , Sub- F/oon 
 2"S/eeper and 
 , Cinder F/// 
 
 Fig. 133. 
 
 ELEVATOR SHAFTS 
 
Sec. 11-53] 
 
 BUILDINGS 
 
 553 
 
 springs, the pits should be thoroughly waterproofed. This may be done in the same manner 
 as for basement floors (see Art. 47, Sect. 11). If the water pressure is very great, waterproof 
 pans may be employed as shown in Fig. 134. 
 
 The sides of the pits are usually made of concrete 6 to 8 in. thick and the bottom of the pit 
 is made a slab of the same thickness as the basement floor. The sides of the pit are supported 
 
 Basemenf Leye/ 
 
 Top of 
 P/f Ptrn 
 
 /re/nforc/r?^ "-^ 
 
 yvaterproofinj 
 Fig. 134. 
 
 at the top and bottom, and may be figured as a simple slab acted upon by the earth pressure 
 and by a pressure due to the live load on the basement floor. The steel used in the side walls 
 should be run into the basement floor in order to take the reaction at the upper end of the slab. 
 
 53. Pent Houses. — The heights of pent houses over elevator shafts vary according to the 
 number of sets of sheave beams and the size of the sheaves. If the machine is located in the 
 
 Terr and Grave/ ^oof 
 
 Pe^/et for 
 ff/ashin^ 
 
 8 "Is 
 
 # " ^Stirrups 
 2' /""Bars 
 
 Longitudinaf 
 
 Sect-Ion 
 
 Fig. 135. — Pent-house details. 
 
 Transverse Section 
 
 basement or on any intermediate floor, the height of the pent house will vary according to the 
 location of the counterweights with respect to the machines. If the machine is placed over 
 the shaft, the size and height of the pent house may be affected by the dimensions of the 
 machine itself. Figs. 135 and 136 give details of actual pent-house constructions. 
 
 In all machine rooms, doors should be provided of sufficient size to permit any part of the 
 
554 
 
 CONCRETE ENGINEERS' HANDBOOK 
 
 [Sec. 11-54 
 
 machinery to be removed in case of repairs. The construction should also be such as to admit 
 of easy access for the purpose of oiling sheave bearings, etc. 
 
 Local regulations of some localities require that a grating be installed in all elevator shafts, 
 just below the overhead machinery. This grating should be constructed and properly sup- 
 ported to sustain the weight of a number of men. 
 
 S^S' 
 
 ^- tVooe/ Skj//ffhi- Curb 
 
 , 4 "S/ab, . i "°/?ods e'c.toc. 
 
 j Dowels 
 "/d'ctoc 
 
 ■3-0 
 Door 
 
 2 Pitch 
 
 4-6 
 
 'I- I // 
 
 °-6" Wa// 
 
 Section A-A 
 
 Section B-B 
 
 Fig. 136. — Pent-house details. 
 
 PROVISION FOR CONTRACTION AND EXPANSION 
 
 64. Methods Employed. — There is no well-established practice of providing for contraction 
 and expansion in reinforced-concrete buildings as a whole. Most engineers construct curtain 
 
 . Position of /ead 
 when p/acec^ 
 against the form 
 
 6 
 
 Fig. 137. 
 
 walls into keyways leaving only the wall beams and wall girders subject to contraction, and it 
 has been found that these can usually be reinforced sufficiently to prevent cracking. Tempera- 
 
Sec. 11-54] 
 
 BUILDINGS 
 
 555 
 
 ture reinforcement in wall beams should be placed horizontally near the outer surface and 
 should be well distributed throughout the beam depth. This longitudinal reinforcement 
 should be well lapped at the corners of the building. Buildings of a length of 300 ft. have been 
 constructed in this manner with no expansion joints and have not cracked. 
 
 When used, expansion joints to be a success should completely separate the building from 
 bottom to top. They should preferably be made by means of a double column supporting a 
 double girder, and the joints should be waterproofed. Some engineers specify that a complete 
 separation shall be made every 100 ft. in length of the building. 
 
 Rather than provide sufficient steel to take contraction in long walls, expansion joints 
 are often resorted to. Fig. 137 shows different methods of making such joints. In type A 
 
 Strip 
 
 . Sliding P/af^ 
 
 Center Line^ 
 of Bu//dincf 
 
 Hoof 
 
 Fig. 138. 
 
 grooves are simply cast in the end of one section and coated with cold-water paint or pitch. 
 In building the next section a tongue is formed fitting the groove. This type of joint is generally 
 reinforced with burlap when placed in front of an earth fill. In type Fig. 137, sheet lead is 
 used. The projecting portion of the sheet is bent up against the form while concrete is being 
 placed in the first section. After removing the forms it is bent down parallel with the wall 
 surface so as to extend into the abutting concrete. A U-shaped bend is formed in the lead 
 sheet at the concrete joint. In type C the tongue and groove is formed as in A, and in addition 
 a bent lead or copper strip is inserted. 
 
 Fig. 138 shows the manner of providing expansion joints in the United Shoe Machinery 
 Co.'s factory at Beverly, Mass. 
 
 Fig. 139 shows expansion joints in a sawmill at Waycross, Ga. An expansion joint occurs 
 directly on every column line. This was done by pouring columns and all transverse beams 
 on column line monolithic, and by making all intermediate beams and girders distinct members. 
 Recesses were made to receive these. The slab was also an after consideration and was poured 
 independently of beams and girders. The walls were poured after the columns had been strip- 
 ped. The monolithic feature of concrete construction was entirely eliminated. This was done 
 not only for contraction and expansion, but to permit the pouring of the concrete at convenient 
 times. The roof slabs were cut on all column and ridge lines for movement, and afterward 
 capped. No other covering than concrete was used for the roof and this was waterproofed by 
 introducing a foreign ingredient into the concrete mixture. 
 
SECTION 12 
 
 FOUNDATIONS 
 
 1. Bearing Capacity of Soils. — The bearing capacity of soils depends upon their composition, 
 the degree to which they are confined, and the amount of moisture which they contain. An 
 approximate idea of the loads which may be safely placed upon uniform strata of considerable 
 thickness may be obtained from the table on page 583. 
 
 There are many kinds and mixtures of soils and it is not always possible to judge the safe 
 bearing capacity of any given soil by reference to a table such as that above mentioned. When 
 there is any doubt, tests or borings should be made. After opening the trenches, the best 
 method is to load known areas and observe the settlement, but in interpreting the results, the 
 fact should not be overlooked that a small area will bear a larger load per unit of area for a 
 short time than a larger area will perpetually. Thus, the area tested should be as large as 
 practicable and the test should continue for some time. 
 
 Ordinary soils will usually bear more weight the greater the depth, owing to the fact 
 that they become more condensed from the superimposed load. With clay, depth is especially 
 important as there is less liability of its being displaced laterally due to other fexcavations in 
 the immediate vicinity, and also because at greater depths the amount of moisture in it is not 
 subject to so much variation. In any soil, the bed of the foundation should be below the reach 
 of frost. 
 
 It is safe to say that any rock in its native bed will bear the heaviest load that can be brought 
 upon it by any masonry construction. It scarcely ever happens that rock is loaded with the 
 full amount of weight which it is capable of sustaining. In preparing a rock bed, all loose and 
 decayed portions should be cut away and the bed dressed to a plane surface as nearly perpen- 
 dicular to the direction of the pressure as is practicable. Any fissures or seams in the rock 
 should be filled with concrete. A sloping surface should be stepped or the foundation designed 
 with sufficient toe to prevent sliding. 
 
 Sand when dry, or wet sand when prevented from spreading laterally, forms one of the 
 best beds for a foundation. However, porous, sandy soils are easily removed by running water 
 and require extreme care at the hands of the constructor. 
 
 Piles are used in such soils as are not able to bear the weight of structures without an exces- 
 sive spread of the footing base. If a pile is driven so that its lower end rests upon a hard stratum, 
 the loading is limited by the strength of the pile considered as a column. The load on an ordi- 
 nary bearing pile is carried by the friction of the earth on the sides of the pile. The only for- 
 mulas in anything like general use for friction piles are the following, known as the Engineering 
 News formulas : 
 
 For a pile driven with a drop hammer, P = 
 
 For a pile driven with a steam hammer, P = 
 
 ^ ' s + 0.1 
 
 in which P is the safe load in pounds, W the weight of the hammer in pounds, h the fall of the 
 hammer in feet, and s the penetration or sinking in inches under the last blow, assumed to be 
 sensible and at an approximately uniform rate. The above formulas were deduced for wood 
 piles, but they are the best there are for concrete piles. They are also claimed to be safe, for 
 ordinary weights of hammer and the usual height of fall, for a pile that acts as a column. 
 
 557 
 
558 
 
 CONCRETE ENGINEERS' HANDBOOK 
 
 [Sec. 12-2 
 
 In the driving of pviles, Joseph R. Worcester of Boston, advises for piles which meet a hard 
 resistance, a penetration of 1 in. under a 2000-lb. hammer falHng 10 ft.; and for piles held by- 
 friction a penetration of 3 in. under a 2000-lb. hammer falhng 15 ft. Ordinary piles of spruce 
 and Norway pine will usually sustain 10 tons by friction and 15 tons in bearing. These piles 
 should never be less than 6 in. in diameter at the small end and never more than 18 in. at the 
 large end. They may be driven 2 to 3 ft. apart depending upon their length, the hardness of 
 the soil and the size of the butts. It has been found that little or no additional bearing power 
 is secured if the spacing is much less than 2 ft. on centers. Concrete piles are manufactured 
 in different sizes and shapes, and their bearing capacity should be thoroughly considered for 
 each case. 
 
 2. Pressure on the Soil. — A footing must be spread until the safe bearing capacity of the 
 soil is not exceeded. An effort need not be made to eliminate all settlement, but rather to so 
 plan the structure that whatever settlement does take place will be uniform. In other words, 
 the center of gravity of the loads from the columns should coincide with the center of gravity 
 of the upward reactions, or with the center of gravity of the base if the base rests directly on 
 the soil. 
 
 In buildings subject to shock or constant live load, the area of the footing should be propor- 
 tioned for the full live and dead loads. In other buildings, that footing should be chosen in 
 which the live load bears, the highest percentage to dead load and its area determined for the 
 total load at the allowable bearing in the soil; the pressure of the dead load per unit area should 
 then be determined and the area of all other footings should be proportioned for dead load only 
 with this unit pressure. Of course, the preceding statement applies without change only when 
 the soil is uniform throughout. 
 
 The pressure on the foundation of a tall building should be considerably less than that which 
 may be allowed on the foundation of a low one-story structure. In the former case a slight 
 inequality of bearing power, and consequent unequal settling, might imperil the safety of the 
 entire building, while in the latter case no serious harm would result. 
 
 3. Plain Concrete Footings. — The depth of a plain concrete footing must be sufficient so 
 that the allowable tensile strength of the concrete is not exceeded. If the area of the base of 
 the footing is considerably greater than the required area for the top, the footing may be stepped 
 or sloped. The top area required depends upon the unit bearing pressure allowed on the con- 
 crete (see recommendation of Joint Committee in regard to bearing pressure in Appendix B). 
 The base area required is governed by the safe bearing capacity of the soil. 
 
 In finding the maximum tensile stress in the concrete, the projection should be treated as 
 a cantilever loaded uniformly by the soil reaction. In stepped footings, the depth of the steps 
 should be made such that the tension in the concrete nowhere exceeds the allowable. 
 
 In a series of tests on unreinforced-concrete column footings made at the University of 
 Illinois,^ the bending moment was computed by the method shown in Art. 7 for reinforced 
 footings and the moduli of rupture were calculated by using a resisting moment based upon the 
 full width of the footing; that is, by considering the fiber stress in the concrete at the bottom of 
 the footing to be uniform over the length of a section passing through the face of the wall, in- 
 stead of taking into account the variation of stress across the section. As is usually the case 
 when plain concrete is used in flexure, the unreinforced footings showed considerable variation 
 in results. The variations were such as not to permit a method of determining the effective 
 width of resisting section to be established or to obtain a formula for resisting moment. Based 
 upon the full section of the footing, the moduli of rupture obtained were considerably less (aver- 
 aging about one-third less) than the moduli of rupture of control beams made with the same 
 concrete. 
 
 4. Advantages in Using Reinforced Concrete for Foundations. — Reinforced concrete is 
 well adapted to the construction of foundations. As compared with plain concrete, its advan- 
 
 1 See Bull. 67, University of Illinois, Engineering Experiment Station. 
 
Sec. 12-5] 
 
 FOUNDATIONS 
 
 559 
 
 tages for spread footings are a reduction in the amount of excavation required, a saving in 
 material, and a reduction in the weight of the foundation itself. 
 
 5. Wall Footings. — Except in residences, bearing-wall footings must usually be reinforced. 
 A cantilever projection is formed on each side of the wall and the amount of reinforcement may 
 be determined as for a simple cantilever beam. 
 
 In figuring maximum bond stress, tests show that the total external shear at the wall-face 
 section should be used in the formulas of Art. 16, Sect. 7. For diagonal tension, the shear 
 should be computed at a distance from the wall face equal to the effective depth of the footing. 
 It is good practice to design small wall footings so that no diagonal tension reinforcement is 
 required. In stepped and sloping footings the depth to steel at a distance (d) from the face 
 of wall is less than in footings of uniform depth so that diagonal tension is quite likely to control 
 in such footings. In large important footings, where diagonal tension is a critical element, 
 web reinforcement should be employed preferably made up in some type of unit frame for con- 
 venience in construction and to ensure the accurate placing of the steel. 
 
 6. Types of Column Footings. — Foundations for columns are of four principal types: (1) 
 single footings; (2) combined footings, including two or more columns; (3) cantilever footings; 
 and (4) raft foundations covering the whole foundation area. 
 
 7. Single Column Footings. — A single symmetrical slab either square or rectangular is 
 the most common form of spread footing. The reinforcing bars are placed at the bottom of 
 the footing and run in either two or four directions. 
 
 Depth for Punching Shear. — The punching shear on a column footing, on an area equal to 
 the perimeter of the column times the depth to the steel should not exceed the allowable value, 
 which for a 1 : 2 : 4 concrete is 120 lb. per sq. in. (see Report of Joint Committee, Appendix B). 
 The load producing punching shear may be found by multiplying the column load by the ratio 
 
 footing area m inus column area 
 footing area 
 
 
 
 6 -> 
 
 £ 
 
 
 
 c 
 
 Fig. 1. 
 
 Maximum Bending Moment. — The maximum bending 
 moment occurs at the face of the column. Referring to Fig. 
 1, the bending moment at BC is due to the load on the area 
 A BCD. The distance out from BC to the center of gravity of 
 the area A BCD — distance x, Fig. 2 — may be formed for a 
 square footing by the following formula 
 
 A I i 
 
 ac 
 "2 
 
 a + c 
 
 Then, for a square footing with square or round column 
 
 (b - a)2(26 + a) 
 2462 
 
 (6 -a)2(26 + a) 
 
 where P is the total column load 
 then 
 
 If we let Ci = 
 
 2462 
 
 Fig. 2. 
 
 M = CiP 
 
 Diagram 1 gives values of Ci for various values of a and 6. 
 
CONCRETE ENGINEERS' HANDBOOK 
 Diagram 1 
 
 [Sec. 12-7 
 
 6 8 10 12 W 16 18 
 
 Length of footing side (b) in feet 
 
 Diagram 2 
 
 20 22 24 
 
 Rectangular footings for 
 square or round colunnns 
 
 M=C2dP 
 For rectangular columns 
 
 M=C3dP 
 
 .20 
 
 30 
 
 Values of f (or|-) 
 
 AQ 
 
 .5C> 
 
Sec. 12-7] 
 
 FOUNDATIONS 
 
 561 
 
 For a rectangular footing with a rectangular column U....Q.....j^ 
 (Fig. 4), the moment at BC is Fig. 3. 
 
 M = C^dP V d > 
 
 Diagram 2 gives values of Cz for various values of v and v 
 
 G. 4. 
 
 Width of Footing to Use in Flexure Computations for Two-way Reinforcement.'^ — With two- 
 way reinforcement evenly spaced over the footing, it seems that the tensile stress is approxi- 
 mately the same in bars lying within a space somewhat greater than the width of the pier 
 and that there is also considerable stress in the bars which lie near the edges of the footing. 
 For intermediate bars stresses intermediate in amount will be developed. For footings having 
 two-way reinforcement spaced uniformly over the footing, the method proposed for determining 
 the maximum tensile stress in the reinforcing bars, is to use in the calculation of resisting mo- 
 ment at a section at the face of the pier the area of all the bars which lie within a width of 
 footing equal to the width of pier plus twice the thickness of footing, plus half the remaining 
 distance on each side to the edge of the footing. This method gives results in keeping with 
 the results of tests. When the spacing through the middle of the width of the footing is closer, 
 or even when the bars are concentrated in the middle portion, the same method may be applied 
 without serious error. Enough reinforcement should be placed in the outer portion to prevent 
 the concentration of tension cracks in the concrete and to provide for other distribution of stress. 
 
 No failures of concrete have been observed in tests and none would be expected with the 
 low percentages of reinforcement used. 
 
 Bond Stresses.^ — The method proposed for calculating maximum bond stress in column 
 footings having two-way reinforcement evenly spaced, or spaced as noted in the preceding 
 paragraph, is to use the ordinary bond stress formula, and to consider the circumference of all 
 the bars which were used in the calculation of tensile stress, and to take for the external shear 
 that amount of upward pressure or load which was used in the calculation of the bending 
 moment at the given section. 
 
 Bond resistance is one of the most important features of strength of column footings, and 
 probably much more important than is appreciated by the average designer. The calculations 
 of bond stress in footings of ordinary dimensions where large reinforcing bars are used show 
 that the bond stress may be the governing element of strength. Tests show that in multiple- 
 way reinforcement a special phenomenon affects the problem and that lower bond resistance 
 may be found in footings than in beams. Longitudinal cracks form under and along the rein- 
 forcing bar due to the stretch in the reinforcing bars which extend in another direction, and 
 these cracks act to reduce the bond resistance. The development of these cracks along the 
 reinforcing bars must be expected in service under high tensile stresses, and low working bond 
 
 1 From Bull. 67, University of Illinois, Engineering Experiment Station. 
 
 36 
 
562 
 
 CONCRETE ENGINEERS' HANDBOOK 
 
 [Sec. 12-7 
 
 stresses should be selected. An advantage will be found in placing under the bars a thickness 
 of concrete of 2 in., or better 3 in., for footings of the size ordinarily used in buildings. 
 
 Difficulty may be found in providing the necessary bond resistance, and this points to 
 an advantage in the use of bars of small size, even if they must be closely spaced. Generally 
 speaking, bars of M-in. size or smaller will be found to serve the purpose of footings of usual 
 dimensions. The use of large bars, because of ease in placing, leads to the construction of 
 footings which are insecure in bond resistance. Column footings reinforced with deformed 
 bars develop high bond resistance. Curving the bar upward and backward at the end in- 
 creases the bond resistance, but this form is awkward in construction. Reinforcement formed 
 by bending long bars in a series of horizontal loops covering the whole footing gives a footing 
 with high bond resistance. 
 
 The use of short bars placed with their ends staggered increases the tendency to fail by 
 bond and cannot be considered as acceptable practice in footings of ordinary proportions. 
 In footings in which the projection is short in comparison with the depth, the objection is very 
 great. 
 
 Diagonal Tension.^ — As a means of measuring resistance to diagonal tension failure, the 
 vertical shearing stress should be calculated by using the vertical sections formed upon the 
 square (assuming square column) which lies at a distance from the face of the pier equal to the 
 depth of the footing. This calculation gives values of the shearing stress, for footings which 
 failed by diagonal tension, which agree fairly closely with the values which have been obtained 
 
 V . 
 
 in tests of simple beams. The formula used in this calculation is v =tt3' where V is the total 
 
 vertical shear at this section taken to be equal to the upward pressure on the area of the foot- 
 ing outside of the section considered, h is the total distance around the four sides of the section, 
 and jfi is the distance from the center of reinforcing bars to the center of the compressive stresses. 
 The working stress now frequently specified for this purpose in the design of beams, 40 lb. 
 per sq. in., for 1:2:4 concrete, may be applied to the design of footings. 
 
 Four-way Reinforcement.^ — Footings having reinforcement placed in the direction of the 
 diagonals as well as parallel to the sides (four-way reinforcement) give good tests. The sig- 
 nificance of the results is so obscured by the variety of manner of failure (bond, diagonal tension, 
 and perhaps tension) and by variations in the quality of the concrete, that a comparison with 
 two-way reinforcement on the basis of loads carried would not be of value. This type of 
 distribution of reinforcement should be included in further tests. Measurements of deforma- 
 tion in the bars are needed to determine the division of stress among the four sets of bars. 
 
 Stepped and Sloping Footings.^ — In stepped footings, the abrupt change in the value of the 
 arm of the resisting moment at the point where the depth of footing changes may be expected to 
 produce a correspondingly abrupt increase of stress in the reinforcing bars. Where the step is 
 large in comparison with the projection, the bond stress must become abnormally large. It is evi- 
 dent that the distribution of bond stress is quite different from that in a footing of uniform thick- 
 ness. The sloped footing also gives a distribution of stress which is different from that in a 
 footing of uniform thickness. However, for footings of uniform thickness the bond stress is a 
 maximum at the section at the face of the pier; in a sloped footing the bond stress at the sec- 
 tion at the face of the pier would be less accordingly than in a footing of uniform thickness, 
 and a moderate slope may be found to distribute the bond stress more uniformly throughout 
 the length of the bar. This is not of advantage if the full embedment of the bar is effective in 
 resisting any pull due to bond. 
 
 Illustrative Problem. — Design a single square footing for a round column of 24 in. diameter carrying 
 300,000 lb., when the safe bearing capacity of the soil is 2 tons per sq. ft. Use a 2000-lb. concrete with medium 
 steel reinforcement. Two sets of bars are to be used placed at right angles to each other and parallel with the 
 sides of the footing. 
 
 » From Bull. 67, University of Illinois, Engineering Experiment Station. 
 
Sec. 12-7] 
 
 FOUNDATIONS 
 
 563 
 
 The required area of footing is found by dividing the load on the column plus the assumed weight of footing 
 
 336,000 
 
 (36,000 lb.) by tlie safe bearing capacity of the soil, or 
 (area 84.0 sq. ft.). 
 
 The load producing punching shear 
 
 4000 
 
 84 sq. ft. A base 9 ft. 2 in, square will be selected 
 
 84.0 - 3.1 
 
 84.0 
 
 (300,000) = 289,000 lb. (The weight of footing need not 
 
 be considered except in determining the area of base of footing as the upward reaction of the soil due to weight of 
 footing is equal and opposite to this weight.) The minimum depth, then, for punching shear is 
 
 289,000 
 
 (2)(3.14)(12)(120) 
 
 = 32 in, 
 
 Shear, as measuring diagonal tension, is measured at a distance from 
 face of column equal to the depth of the footing to the steel, or 32 in. in 
 this problem. The reaction of the soil on the cross-hatched area. Fig. 5, is 
 what causes shear on the vertical planes through EFGH. 
 
 84.0 - (7.33)' 
 
 Total shear on EFGH 
 
 84.0 
 
 bjd 
 
 (4) (88) (0.875) (32) 
 
 (300,000) = 108,000 lb. 
 
 = 11 lb. per sq. in. 
 
 108,000 
 
 Thus the footing needs no stirrups for a depth (d) of 32 in. 
 
 Whenever the unit shear is found too great at the given distance out 
 from the face of column, the plane where the shear is just the allowable 
 "should be determined. Then the total tension to be taken by stirrups 
 divided by the tensile value of one stirrup gives the number of stirrups 
 required. 
 
 In nearly all cases a slight deepening of the footing is all that is necessary to avoid using stirrups. The placing 
 of stirrups is troublesome and footings should be designed so that web reinforcement will not be needed, if this can 
 be done without greatly increasing the depth. 
 
 The bending moment acting on each set of rods is, by means of Diagram 1, 
 
 M = 6.2(300,000) 
 
 . , _ Ji 
 
 fsjd (16,000) (0.875) (32) 
 
 = 1,860,000 in.-lb. 
 1,860,000 
 
 4.2 sq. in. 
 
 Fourteen 56-in. round bars will be employed in each band, of width equal to diameter of column plus twice the thick- 
 ness of footing, plus half the remaining distance on each side to the edge of the footing. The effective width in 
 this problem is 8 ft. 3 in. (see Fig. 5). We will place 16 bars throughout the 
 entire width of footing, as shown in the complete design, Fig. 6. The bond 
 stress 
 
 y 289,000(H) 
 
 (1.96) (14) (0.875) (32) 
 
 = 94 lb. per sq. in. 
 
 either de- 
 
 in 
 
 /!6 deformed bars ineachsef 
 \l spaced 7" c. to c 
 
 9-2" 
 
 H?-H+H-l-M+f-H4 
 
 Since the allowable bond stress for plain bars is 80 lb. per sq. in. 
 formed bars must be used or a larger number of smaller plain bars. 
 
 Illustrative Problem. — Design a single square sloping footing for a 
 round building column of 30-in. diameter, carrying 610,000 lb. Consider the 
 safe bearing capacity of the soil at 2\'i tons per sq. ft. and use four-way rein- 
 forcement, /s = 16,000. /c = 650. 
 
 It will be assumed that satisfactory soil may be found at the required depth 
 below the basement floor. A base plate will be provided under the column 
 rods, and this plate will be placed on top of the finished footing. The top of 
 the footing will be made with an area of about twice that of the column and a 
 bearing pressure of 700 lb. per sq. in. will be permitted (see recommendations 
 of the Joint Committee, Appendix B). Investigation shows that this stress is 
 not exceeded. ' If desired, the base plate may be placed some distance down in 
 the footing, but under such an arrangement the column rods for the first story 
 must be placed while the footing is being poured, which is troublesome. Some- 
 times, however, this is avoided, especially where the footing is at considerable 
 depth, by inserting only short column rods of such a length that they may be 
 
 properly spliced immediately above the basement floor. If this is done, the first floor can be laid entire and the 
 first-story columns started above it. 
 
 The top of the footing will be made square, 42 in. on a side. This provides a 6-in. ledge all around so that, 
 if it is desired to erect the first-story columns before pouring the basement floor, the column forms will have some 
 place on which to rest. The depth of the footing at the outer edge will be made 12 in. The dead weight of the 
 footing will be taken at 60,000 lb. 
 
 Fig. 6. 
 
564 
 
 CONCRETE ENGINEERS' HANDBOOK 
 
 [Sec. 12-^7 
 
 The required area of footing is found by dividing the total load by the safe bearing capacity of the soil. 
 
 670,000 
 (2.5) (2000) = 
 
 We shall select an area 11 ft. 6 in. square. 
 
 Four-way reinforcement for single column footings is shown in Fig. 8. After deducting the area of the 
 column base, the remainder of the footing slab is considered as eight cantilevers, four running parallel to the sides 
 and four on the diagonals. 
 
 Depth for punching shear, 
 
 / 132.2 - 4.9 n 610,000 . 
 
 ~ { 132.2 / (3.14)(2.o)(12)(120) ~ 
 
 Fig. 7. Fig. 8. 
 
 Referring to Fig. 7, the reaction of the soil on the cross-hatched area is what causes shear on the vertical 
 planes EFGH 
 
 ,' = 52-k = 52- = 9.8 in. 
 
 132 2 — (11 17)2 
 Total shear on EFGH = ^^^^ ' — - (610,000) = 34,200 lb. 
 
 34,200 
 
 ' = (4) (134) (0.875) (9.8) = ^'^ 
 The total moment acting around the entire column perimeter is found fromDiagram 1 to be 
 M = (4) (7.8) (610,000) = 19,030,000 in.-lb. 
 
 _ 19,032,000 
 
 ~ (16,000) (0.875) (52) ~ 26.2 sq. in. 
 
 This total amount of steel will be divided by eight in order to determine the amount of steel in each band. If 
 desired, more bars can be placed parallel to the sides of the footing than in a diagonal direction. In this design 
 six 1-in. round bars will compose each set. 
 
 In view of tests made on footings with two-way reinforcement it would seem that the bars need not all pass 
 directly under the column. In fact, if the width of bands needs to be increased in order to cover the entire area of 
 the footing, it would be conservative to say that this may be done provided that the increase in the width of each 
 band is not great. Until more tests have been made along this line, it would seem wise (especially in footings in 
 which the depth decreases toward the outer edge) not to be too radical in the design of such important structures 
 as footings. Where a small increase in the width of bands does not suffice to cover the entire area of footing, a 
 few short cross rods may be employed to span the open spaces. Increase in band width should preferably be made 
 in the bands parallel to the sides of the footing as, in these bands, the lengths of the rods do not change. 
 
 It should be noticed that the diagonal rods have a greater length subjected to cantilever action than the 
 rods parallel to the sides of the footing. Since it is difficult to calculate accurately the stresses in a square footing, 
 it does not seem proper to attempt any but equal division of moment between the rods in the different directions. 
 Some designers prefer an octagonal shape of footing so that all rods will have approximately the same length. The 
 rods should be cut from 2 to 4 in. shorter than the total width of footing, but if desired, the rods may be cut 
 considerably shorter and staggered so as to allow for the decrease in bending moment from the column toward 
 the edges of the footing. 
 
Sec. 12-8] 
 
 FOUNDATIONS 
 
 565 
 
 The bond stress along the horizontal twision bars must now be investigated. 
 /132.2 - 4.9\ 610,000 
 " = { 132.2 ) (48)(3.14)(0.875)(52) = 
 
 Deformed bars will be used. The complete design is shown in Fig. 8. 
 
 Where, on account of soil conditions, a greater depth is needed beiow the basement floor than that required 
 for the footing, the pier or column between the top of the footing and the basement floor should be flared in a 
 similar manner (but inverted) to the flare in columns just below flat-slab floors, as by so doing the bending moment 
 and shear on the footing ma'y be decreased. This should be clear from the discussion on flat-slab floors. 
 
 8. Combined Column Footings. — It is sometimes necessary to support a column on, or 
 very near, the edge of a property line in order not to encroach upon adjacent property. In 
 such single symmetrical footing cannot be used. The nearest interior column is 
 
 usually selected and a combined footing constructed under both columns. Sometimes a 
 combined. footing will include more than two columns. 
 
 The design of combined footings consists in constructing a base of such shape that the 
 center of gravity of the loads will coincide with the center of gravity of the upward reaction. 
 In addition, the base must have sufficient area so that the allowable pressure on the soil will 
 not be exceeded. 
 
 The combined footing may be either a slab of uniform thickness or an inverted T-beam. 
 If the slab type of combined footing is used with two columns, the slab must have a trape- 
 zoidal shape when the columns are placed at opposite ends of the footing. A rectangular 
 shape may be used where a longitudinal projection is possible beyond the heavier load of a 
 sufficient length to cause the center of gravity of the rectangle to coincide with the center of 
 gravity of the loads. Transverse reinforcement is needed when the width of footing is much 
 larger than the width of columns. 
 
 Fig. 9. 
 
 Illustrative Problem. — Design a combined footing for columns 1 and 2, Fig. 9. Column 1 is 20 in. square 
 and sustains a load of 280,000 lb. Column 2 is 24 in. square and sustains a load of 380,000 lb. Distance 
 between centers of columns is 14 ft. Allowable soil pressure is 6500 lb. per sq. ft. Assume a 2000-lb. concrete 
 with medium steel reinforcement. 
 
 Load on column 1 = 280,000 lb. 
 Load on column 2 = 380,000 lb. 
 
 660,000 lb. 
 
 Assumed weight of footing = 55,000 lb. 
 
 Total = 715,000 lb. 
 
 Pressure on the soil due to column loads only is 
 
 715,000 
 
 6500 
 
 660,000 
 
 lio 
 
 ■■ 110 sq. ft. 
 6000 lb. per sq. ft. 
 
566 
 
 CONCRETE ENGINEERS' HANDBOOK 
 
 [Sec. 12-8 
 
 The lengths of the parallel sides are unknown and two equations will be needed to solve. First equation may 
 be obtained from the formula that the area of a trapezoid equals the average of the sum of the parallel sides multi- 
 plied by its length. The second equation may be found from the principle stated above — that the center of gravity 
 of the trapezoid must coincide with the center of gravity of the combined column loading. This must be so in 
 order that the pressure on the soil, and the consequent settling (if any), may be uniform — also to prevent dangerous 
 transverse stresses in the columns. 
 
 Area of footing = ^L±_^ (16.83) = 110 
 fci + 62 = 13.07 
 
 '■-■- = IS 
 
 or 
 
 h = 7.44 ft. 
 
 Using the common equation for the center of gravity in a trapezoid 
 
 Solving equations (1) and (2) for 61 and 62 
 
 61 = 8.82 ft. and 62 = 4.25 ft. 
 
 The values of 61 and 62 may also be found by considering the footing 1 ft. wide and using the formulas on 
 page 582 to find intensity of pressure at each end. Thus, if we let pi denote the intensity at the column 2 end 
 and p2 the intensity at the column 1 end, we have 
 
 0(1^-7.44)-] 
 
 = 52,900 lb. 
 
 660,000 
 
 [. 
 
 .83 ' 16.83 
 
 '16.83 
 
 16.83 L 16.83 J 
 
 16.83 ' ^'''^^ 
 
 Then 
 
 8.82 - 4.25 1 
 - (2)(16.83) '''l"""" - ^^"■'^ 
 
 To find the necessary depth of footing and amount of steel required, we must find the section maximum 
 moment (which is the section of zero shear) and then determine the center of gravity of the area to one side of this 
 section as an aid in finding the value of this maximum moment. 
 
 8.82 - 4.25 
 (2) (16.83) 
 h - 0.0154^22 = 7.18 
 h -- 8.22 
 
 And by trial, 61 - 62 = 8.82 - 4.25 = 4.57 
 
 b3 - b2 ^ 16.83 -- 8.22 
 
 4.57 16.83 
 bs - b2 = 2.34 
 
 b3 = 2.34 + 4.25 = 6.59 
 
 ^ h 63 + 2bi _ 8^ 6.59 + 2(8.82) _ . 
 ' ~ 3 ' 63 + 61 ~ 3 ■ 6.59+8.82 ~ 
 
 We can now compute the value of the maximum moment. 
 
 M = 380,000 (8.22 - 1.5 - 4.31) = 915,800 ft.-lb. 
 
 or 
 
 M = (915,800) (12) = 10,989,600 in.-lb. 
 
 The moment for 1 in. of width along the line of maximum moment, M = ^r ^nf/fo!^ = 139,000 in.-lb. The 
 
 (6.59)(12) 
 
 footing must be considered as an inverted beam at this section, and the required depth and the area of the steel 
 may be computed by the usual methods. 
 
 If the moment had been taken about a line through the center of gravity of the entire trapezoid, the result 
 would be 133 000 in.-lb., or an error of only about 2.2 %. This approximate solution is often used in practice as the 
 error is small. 
 
 From Diagram 2, page 360, K = 107.4 and p = 0.0077 for the stresses as recommended by the Joint 
 Committee. 
 
 , ^ 139,000 
 d = V-ToT^ - 36.0 in. 
 
 \ 1( 
 
 As = (0.0077) (12) (6.59) (36) = 21.9 sq. in. for the whole width. 
 Eighteen IH-in. round rods will be needed. 
 
Sec. 12-8] 
 
 FOUNDATIONS 
 
 567 
 
 The depth required for punching shear at column 2 is 
 , 380,000 - (6000) (4) 
 
 and at column 1 it is 
 
 (24)(4)(120) 
 
 280,000 - (6000) (1.67) 2 
 (20) (4) (120) 
 
 31 in. 
 
 28 in. 
 
 In each case the depth is less than that required for moment. 
 
 The depth required for diagonal tension must now be determined. The shear at column 2 equals 380,000 
 lb. minus the part of this load distributed on the soil over the area mrsn, Fig. 11, or 
 
 and 
 
 The shear at column 1 is 
 
 and 
 
 V = 
 d = 
 
 V = 
 d = 
 
 380,000 - (8. 14) (2.5) (6000) = 258,000 1b. 
 
 258,000 . 
 
 (8.14)(12)(0.875)(120) ~ 
 
 280,000 - (4.80) (2.17) (6000) = 217,500 1b. 
 217,500 
 
 (4.80) (12) (0.875) (120) 
 
 36 in. 
 
 which is just that required for moment. 
 
 Adopting a depth (d) of 36 in. and assuming that deformed rods are used, then the number of rods required 
 for bond will be (assuming u = 100) 
 
 or number of rods 
 
 258,000 
 
 21 
 
 Soid' (3.93) (0.875) (36) (100) 
 
 Thus the bond controls the number of horizontal rods required and 21 rods will be employed (see Fig. 10). 
 
 Stirrups are af Cached b Uf^r hngi-hjclinal rods 
 
 -^J^^T " 
 
 6-ir^4'-07g\.,^^ 
 
 ie-f^-3fchc about ea'c. foe. £0-§°^2^'c.foc. 
 
 Fig. 10. 
 
 -31 i. 
 
 Transverse reinforcement should be provided to prevent bending of the projections of the footing. Consid- 
 ering column 2, the width of distributing beam will be taken at 3 ft. 6 in. Now the loading for which this cantilever 
 
 should be designed is equal to one-half of the column loading multiplied by — ^rFK~* The moment arm equals 
 
 one-half of the length of the projection, and 
 
 /380,000\ /8.82 - 2n 
 
 8.82 
 
 M = 
 d = 
 
 K».»z — ■z\ /3.4l\ , 
 -Sli2-)(—)»2) -3,010. 
 
 000 in.-lb. 
 
 3,010,000 
 (107.4)(3.5)(12) 
 
 = 25. 
 
 The depth is smaller than the depth of the whole slab and, since a greater depth is used, the amount of steel 
 found as follows: 
 
 _ JW^ 3,010,000 
 
 " ~ fsjd ~ (16,000) (0.875) (36) " ^ ^ 
 
 Referring to table on page 357, fourteen %-in. round rods will be needed. 
 
568 
 
 CONCRETE ENGINEERS' HANDBOOK 
 
 [Sec. 12^9 
 
 In the same manner, the distributing reinforcement for column 1 is determined 
 /280,000\ /4.25 
 
 ^= (— 2— )(— 
 
 d 
 
 ^ ■^^)(^)(12) = 658,000 in.-lb. 
 
 \ 658,000 ^ 
 
 \(107.4)(3.5)(12) "^-^^ 
 
 Since the depth of the whole slab must be used, the necessary steel is found as above. 
 
 658,000 
 
 As = 
 
 = 1.3 sq. in. 
 
 (16,000) (0.875) (36) 
 
 It sometimes happens that the required depth of the distributing beam may be larger than the depth of the 
 whole slab. In such cases the footing may have an increased thickness under the column or else steel may be 
 introduced at the top and bottom. Double reinforcement, however, should not be employed when excavation 
 can readily be made. 
 
 The bond stress along the rods of the distributing beams must now be investigated. Considering the beam 
 under column 2, the maximum shear is 
 
 380,000\ /3.82 - 2.0 
 
 ) = 147,000 lb. 
 
 2 / \ 8.82 
 
 Assuming that deformed rods are used, then the number of rods required for bond will be (u = 100) 
 
 or number of rods 
 
 147,000 
 
 (2.36) (0.875) (36) (100) 
 
 20 
 
 In a similar 
 beam under 
 
 Thus the number determined for bond controls, 
 manner, 12 rods are needed in the distributing 
 column 1. 
 
 Fig. 10 shows the spacing of the transverse reinforcement. 
 Steel should be introduced between the end distributing beams 
 about 24 in. c. to c. in order to make the footing more rigid and 
 more capable of directly transmitting the loads. 
 
 Web reinforcement is not needed in the distributing beams, 
 as each distributing beam and column has a similar load to that 
 of a single footing, and for such footings we know that the in- 
 tensity of shearing stress, as measuring diagonal tension, may be 
 computed on a section at a distance out from the face of column 
 equal to the depth of the footing to the steel. In a longitudinal 
 direction, however, shear should be computed on a section close 
 to the support, as mn, Fig. 11, since in this direction the tension steel is at the top of the footing and cracks may 
 open up in the web as in simple beams. The spacing of stirrups is given in Figs. 10 and 11, determined as in a 
 simple beam. 
 
 The loads beyond the centers of the columns in a longitudinal direction cause negative moment at the sup- 
 ports; that is, the load to the left of column 1 and to the right of column 2 cause tension in the bottom of the 
 footing at the sections AB and CD respectively (Fig. 9). This tension, however, does not exceed the tensile strength 
 of the concrete. At column 1, using the common flexure formula. 
 My _ (8.82) (1.5) (6000) (9) (18) (12) 
 ^ ~ I ~ 
 
 33 lb. per sq. in. 
 
 (8.41)(12)(36)3 
 
 A few rods 4 ft. long will be placed at the bottom of footing across the sections AB and CD in order to provide 
 for any possible defects in the bed of the foundation. 
 
 9. Cantilever Footings. — The cantilever type of construction may be employed in place 
 of a combined footing under the usual conditions; that is, when encroachment upon adjacent 
 property must be avoided and when, at the same time, it becomes necessary to make use of 
 the land close to the property line. In cantilever construction, the wall-column footing and 
 the footing of the nearest interior column are connected by a beam or strap, and this strap is 
 extended so as to support the wall column. To save excavation, the top of the strap is usually 
 placed at the same level as the tops of the footings. 
 
 Illustrative Problem. — Let us assume the size of columns and column loading as shown in Fig. 12. The 
 distance from the center of interior column to the property line will be made 12 ft. and the outer edge of the wall 
 column will be placed on the property line. In addition to the above, we shall consider that piles are necessary 
 and that it is not feasible to drive such piles closer than 2 ft. apart in any direction. The safe bearing capacity 
 of each pile will be taken at 10 short tons. The working stresses as recommended by the Joint Committee for a 
 2000-lb. concrete and medium steel will be adopted. 
 
 Approximate figuring shows that about 12 piles will be needed under the wall footing, and 18 piles under the 
 
Sec. 12-9] 
 
 FOUNDATIONS 
 
 569 
 
 interior footing; that is, if the center of the wall footing is placed about as shown in Fig. 12. Knowing the number 
 of piles required, the shape of the footings may be easily determined. The arrangement showO for the piles insures 
 a spacing of at least 2 ft. in every direction. 
 
 On account of cantilever action, the uplift on the interior column may be due to the combined live and dead 
 load of the wall column, but, since the live load may not always be present, the uplift from the dead load only 
 
 should be considered. Disregarding the strap weight to the 
 right of the center of wall footing, this uplift is 
 (130,000) (2.33) - (1200) (9) (4.5) 
 
 
 
 
 
 9' 
 '•■9' 
 
 J'r9. 
 
 
 
 
 /? 
 
 ■ ■'e' 
 
 '0''- 
 
 ■■> 
 
 
 
 
 
 
 
 
 
 / 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 ' 
 
 
 H 
 
 
 
 
 
 
 
 * 
 
 
 
 
 
 — 0 
 
 
 
 , i' 
 
 
 
 
 1 
 
 tS;' 
 
 
 
 
 
 K 
 
 
 
 
 
 
 
 
 
 
 
 
 \ 
 
 
 
 
 
 T. % _ V 
 
 Plan 
 
 < s'oo ' 
 
 < // '33 ■— 
 
 <-..., izloo— 
 
 4 
 
 Properry.,. 
 Line 
 
 9.00 
 
 = 28,000 lb. 
 
 The strap will now be designed for the total live and 
 dead load on the exterior column since this loading gives 
 maximum conditions. The total load on the wall footing is 
 (175,000) (11. 33) + (1200) (12) (6) 
 
 ».00 
 
 230,700 lb. 
 
 Stirrups amched 10 ippernxfs^^iyp 4S00O 
 \De ad 130000 
 
 ''3 
 
 ^ ■ t — tlv _ 
 
 The uplift on the interior column may be found in a similar 
 manner to that for dead load only, or more accurately (tak- 
 ing moments about the outer edge of the wall column) 
 (230,700) (3.00) - (175,000) (0.67) - (1200) (12) (6) 
 
 12.00 
 
 40,700 lb. 
 
 The loads on the strap are shown in Fig. 13. The maximum 
 moment occurs where the shear is zero, or accurately enough, 
 
 Elevation 
 
 triple- 
 looped 
 
 stirrups 6'c.ToOf 
 
 (175,000) (6^ 
 230,700 
 
 4.5 ft. 
 
 IS-i'"*4rc he 
 
 iPmiiiili^flK 
 iTTtiTTTtrrnw..'' 
 ittllttiHtitft^:-- 
 
 Total /TSOOOlb. 
 Uniforrp load /200 lb. per ft 
 
 Plan of Steel in Footings 
 
 Plan of Steel in Strap 
 Fig. 12. 
 
 Point of Max Momknt. 
 
 ■msr 
 
 A /ora; 230. 700/0 
 
 m\\\\\\\ 
 
 k- 4'S ■> 
 
 Fig. 13. 
 
 from the outside edge of wall column. 
 
 M = (175,000) (3.83) 
 
 = 3,372,000 in.-lb. 
 Then, assuming b = 39 in. 
 
 3,372,000 
 '(107.4) (39) 
 
 (^^^^) (4.25) (2.25) = 281,000 ft.-lb. 
 
 28K2 in. 
 
 The shear in the strap at the inside edge of wall column should not exceed 120 lb. per sq. in. with an effective sys- 
 tem of web reinforcement. This is due to the fact that tension exists in the upper portion of the strap; in other 
 words, the inward spread of the wall-column load will not aid to any appreciable extent in reducing the tendency to 
 fail through diagonal tension. For the dimensions determined for moment 
 
 (175,000) 
 
 '230,700 
 
 ) (1.33) 
 
 (39) (0.875) (28.5) 
 123,900 
 
 127 lb. per sq. in. 
 
 (39) (0.875) (28.5) 
 
 The strap will be deepened so that d = 30 in. and total depth = 32 in. Stirrups will be needed from the inner 
 edge of the exterior column to a point 3.5 ft. from the property line where the shear becomes 10 lb. per sq. in. 
 Triple-looped y2-in. round stirrups will be employed and these will be spaced 6 in. on centers, or 
 _ 3 Asfsjd _ 3 (6) (0.196) (16,000) (0.875) (30 ) 
 ^ ~ 2 V ~ 2 123,900 
 
 = 6 in. 
 
570 
 
 CONCRETE ENGINEERS' HANDBOOK 
 
 [Sec. 12-10 
 
 The unit shear at the inner edge of wall footing is 
 
 (40,700) + (1200) (6) 
 (39) (0.875) (30) = 
 
 Two triple-looped }^-in. round stirrups placed just to the left of this point will make the design satisfactory. 
 The number of horizontal rods will now be determined. 
 
 Jlf 3,372,000 
 
 ^ ^ fsjd ~ (16,000) (0.875) (30) ~ 
 
 Fourteen Ji-in. round deformed rods will be used. 
 
 123,900 
 
 (14) (2.75) (0.875) (30) 
 
 122 lb. per sq. in. 
 
 This value will be considered satisfactory on account of bending the ends of the rods as shown. 
 
 The horizontal tension rods should have a sufficient length for straight bond to the left of the section of 
 maximum moment. Alternate rods may stop off at 10 ft. from the property line, which is about 5 in. beyond the 
 point where they are no longer needed in tension. 
 
 The interior footing may be designed in the same manner as explained in Art. 8, except that in this problem 
 we have the moment from concentrated loads instead of from uniform loads. 
 
 M = (80,000) (2.37) (2) + (20,000) (0.62) (2) + (20,000) (1.87) (2) + (40,000) (0.87) (2X 
 = 548,400 ft.-lb., or 6,581,000 in.-lb. 
 
 The depth to steel will be made 30 in. in order to provide properly for the strap. Web reinforcement is not needed 
 
 M 6,581,000 
 
 f,jd (16,000) (0.875) (30) 
 
 15.6 sq. in. 
 
 The steel will be placed in two directions. Thus, for moment, 3.9 sq. in. or seven ^^-in. round rods will be needed 
 in each band. Let us now determine the number of Ji-in. round deformed rods in each set that will be required 
 for bond. 
 
 (16) (20,000) 
 
 ^ = S^' """^^^^ '•^^^ = (100)(2.75)V.875)(30) 
 
 This number controls. 
 
 The weight of footing, as designed, deducting the weight of strap already considered, is approximately 22,000 
 lb. This gives a total pressure on the piles of 360,000 + 22,000 - 28,000 = 354,000 lb. Thus the number of 
 piles was correctly chosen. In this design there is no danger of failure by direct shear above the tops of the piles 
 as for this kind of shear about one-half the compressive strength of the concrete may be allowed. There i? also 
 no danger from punching shear around the base of the column. 
 
 The footing under the wall column acts as a simple cantilever. The moment on each edge of strap 
 M = (60,000) (17) = 1,020,000 in.-lb. 
 As in the interior footing, the depth to steel will be made 30 in. 
 
 1,020,000 
 
 ' ~ (16,000) (0.875) (30) ~ ' 
 Twelve li-in. round rods are satisfactory for moment. For bond 
 
 , , , 60,000 _ 
 
 number of deformed rods = (loO) (1.57) (0.875) (30) = 
 
 A few cross rods will also be employed to better distribute the load. 
 
 The weight of wall footing deducting the weight of strap is approximately 10,000 lb., giving a total pressure 
 on the piles of 230,700 + 10,000 = 240,700 lb., or almost exactly 10 tons per pile, as planned. 
 
 It is generally considered good practice to lay the concrete directly upon the heads of the piles. The ground 
 is usually excavated around the piles, the depth depending upon soil conditions, and then a layer of broken stone 
 is spread and rammed before the concrete is laid. The supporting power of the soil between the piles is thus 
 utilized. 
 
 10. Raft Foundations. — The raft foundation in building construction may be considered 
 as an extension of the single or combined footing until the foundation covers at least a con- 
 siderable, if not the entire, area of the building. Its use is limited to sites where the allowable 
 pressure on the soil is very small or where the building is supported by piles sustained by 
 friction. 
 
 Raft foundations may be divided into the following three general classes: 
 
 1. Flat slabs of plain or reinforced concrete. 
 
 2. Beams or girders with a slab underneath. 
 
 3. Beams or girders with a slab on top. 
 
Sec. 12-11] 
 
 FOUNDATIONS 
 
 571 
 
 A flat-slab foundation may be designed in 
 20, Sect. 11). The foundation is considered as 
 
 |T]]jii ^us'. 
 
 '■•These t?ars in boffvm of Lj 
 s/at> -for a// Canf-i/et'er S/abs 
 
 Plan 
 
 the same manner as a flat-slab floor (see Art. 
 an inverted flat slab loaded by the reaction of 
 the ground and supported by the columns. 
 The column base should be made large 
 enough to prevent excessive moments and 
 shears in the concrete. 
 
 Basement Floor Level 
 1 
 
 
 
 f /S'-o'^ ^-T'-o'y 
 
 4' Concrete F/cx>r S/ab 
 
 
 
 J) \Cinder [i'^ 
 
 
 MM 
 
 Section A-B. 
 Fig. 14. 
 
 I Ml /// 
 
 < 
 
 • 7'-e'' 
 
 ■2'-'f-> 
 
 '•S-p-6'croc, 
 
 Fig. 15. 
 
 Fig. 14 shows a design of a raft foundation of the second class. The action of forces is 
 exactly as in an inverted floor and calls for similar treatment in designing. As compared with 
 Class 3, Class 2 permits a T-beam design, but on the other hand, requires an extra fill and 
 separate floor surface in the basement. 
 
 11. Examples of Column Footings. — Fig. 15 
 shows a design of a single footing for the typical 
 interior columns of a factory building for the 
 Bradley Knitting Co., Delavan, Wis. The design 
 of the top of footing should be noted. 
 
 Fig. 16 is an example of a combined footing 
 employed in an automobile depot in Boston. 
 
 Fig. 17 is a special foundation of novel design 
 used in the Garage Building of the Decanville 
 Automobile Co., New York City. 
 
 /nf^nor Co/ m/l Co/. 
 
 Baseme n f F/oor Line_ 
 
 Z4'-o'' 
 
 Y-s'-o- 
 
 rods 
 
 Y\4-6>\ >-A 
 
 / Z / 7\7^/ 
 
 \ 
 
 . ■ - . 4-1"^ rods \ \ \ ) 
 i ^' -39- i" 'Prods 8k"c 
 
 6" ■ /!9' e' 
 
 L-5-i?'J 
 
 ElevoTiop erf Foundation Beam 
 Fig. 16. 
 
572 
 
 CONCRETE ENGINEERS' HANDBOOK 
 
 [Sec. 12-12 
 
 12. Concrete Piles,-— Concrete piles are usually employed where wooden piles would be 
 subject to decay or- to destruction by the action of marine worms. By using one of the types 
 of concrete piles, they may be employed under almost any circumstance where wooden piles 
 would be suitable. Wooden piles, to insure permanency must be cut off below permanent 
 
 water line, as, when subjected to an atmosphere which is alter- 
 nately wet and dry, they will decay. Concrete piles are not 
 so restricted, as they are permanent above water line as well 
 as below it, and the cutoff line can therefore be as high as is 
 consistent with the depth actually required in the foundations 
 which they support. The cost of the concrete piles, them- 
 selves, is somewhat greater than that of wooden piles, but 
 this cost is, in a great many cases, more than offset by the 
 fact that, due to their size and shape, they are capable of sus- 
 taining greater loads than wooden piles, and by the saving in 
 excavation, pumping, sheeting and masonry made possible by 
 their use. 
 
 Reinforced-concrete sheet piling has been used quite ex- 
 tensively for permanent bulk-heads, piers and the like, and is 
 designed to meet the requirements of the particular situation 
 where it is employed. This sheet piling may be designed with 
 an interlocking feature, or without, as may best and most 
 economically meet the particular requirement. 
 
 Concrete piles are of two general types — ^those molded in 
 place and those molded before driving. 
 
 12a. Piles Molded in Place. — The Raymond, 
 Simplex and Pedestal piles are the forms of this type which are 
 most widely used. 
 
 Raymond Piles. — Raymond concrete piles are made by 
 driving a tapering reinforced steel shell to refusal, by means of 
 a collapsible steel core, withdrawing the core and thereupon filling the shell with concrete. 
 There is thus a shell or form for every pile and the concrete within the shell may be reinforced 
 or not, as may be desired. Fig. 18 shows a partly and an entirely completed Raymond pile. 
 Raymond piles are of 8-in. diameter at the point and taper 0.4 in. per ft. up to 37 ft. in 
 length. 
 
 The shell of a Raymond pile is made of sheet steel of a thickness depending upon the 
 elasticity and crushing tendency of the earth, and is reinforced by means of steel wire spirally 
 wound on the interior of the shell on a 3-in. pitch and held in place by 
 grooving the shell around the wire and fastening the ends by electric 
 welding. The weight of both shell and wire may be varied to suit con- 
 ditions, but 24-gage steel in the shell and No. 3 wire are most com- 
 monly used. 
 
 A boot or point of pressed steel is placed on the end of the core or 
 mandrel and completely encloses the bottom of the shell. 
 
 After the core is collapsed and withdrawn, the shell is inspected to 
 insure its perfection and it is then filled with concrete. 
 
 A mechanical problem that for several years taxed the ingenuity 
 of engineers connected with the Raymond system, was to provide a 
 shell of sufficient strength to withstand the back pressure of the soil 
 after the driving core had been withdrawn. This difficulty has been successfully overcome 
 by means of the spirally reinforced steel shell, which is made on machines specially designed 
 for the purpose. This shell is now universally used in connection with this method and is 
 illustrated in Fig. 19. 
 
 Fig. 18. — Raymond pile core 
 collapsed and partly withdrawn. 
 Completed Raymond pile without 
 reinforcement. 
 
Sec. 12-12a] 
 
 FOUNDATIONS 
 
 573 
 
 Simplex Piles. — In the Simplex pile a hollow cylindrical steel tube or form, 16 in. in diameter 
 and ^ in. thick, is driven to a suitable bearing. When the required depth is reached, the form 
 is filled with concrete to a sufficient height, and then withdrawn. The driving point, depend- 
 ing on soil conditions, may be either a conical cast-iron point that is left in place, or a hinged 
 cutting edge called an alligator point which opens as the tube is withdrawn. Fig. 20 shows the 
 standard Simplex pile which fills the great majority of requirements and is adaptable in all 
 ordinarily unreliable ground. The method shown is termed ''Standard" as about 75% of 
 Simplex piling stands on the detachable cast-iron point. The alligator point is limited in its 
 use to a certain class of soil — ^that is, one that 
 stands well after penetration. As generally used, 
 no reinforcement is required, but the system ad- 
 mits of placing any desired reinforcing members 
 within the form. In very wet soil a permanent 
 casing of slightly less diameter than the inside of 
 the cylindrical form is used to prevent washing or 
 displacement of the concrete. Sometimes a pile is 
 molded on the surface and lowered through the 
 form, which is then withdrawn, leaving the molded 
 pile in position. 
 
 Pedestal Piles. — Pedestal piles are made in the 
 following manner: (1) a core and steel casing are 
 driven to the required depth, (2) the core is re- 
 moved and a charge of concrete is dropped in, (3) 
 the core is used as a rammer to compresg the con- 
 crete into the surrounding soil, (4) this process is 
 
 •Pu//ing Cab/es 
 }h~—Po'/Ar?^ C/amp 
 
 i.---Stee/ Dn^in^ 
 Form 
 
 Fig. 20. — Method of constructing Simplex concrete piles. 
 
 34' Sheet Pile 
 
 Fig. 21. — Details of piles used in ore dock 
 construction at Cleveland. 
 
 repeated until a pedestal base about 3 ft. in diameter is formed, (5) the casing is withdrawn 
 leaving the completed pile with an enlarged base. 
 
 Composite Wood and Concrete Piles. — In order to eliminate the great cost of placing extremely 
 long concrete piles, a method has been devised by the Ryamond Concrete Pile Co. to install 
 what is know as a composite pile. This consists of a Raymond pile superimposed upon a 
 wood pile. The length of the concrete pile is determined by the depth to which it is necessary 
 to drive the wood pile in order to insure its cutoff being below water. A dowelled connection is 
 
574 
 
 CONCRETE ENGINEERS' HANDBOOK 
 
 [Sec. 12-12c 
 
 provided between the wood and concrete pile which is strong enough to withstand any lateral 
 displacement. By this method long wood piles can be used, where necessary, with short 
 concrete piles superimposed upon them, thus avoiding the necessit}^ of excavation, sheeting 
 and pumping in material which is apt to be extremely difficult and costly to handle. 
 
 126. Piles Molded Before Driving. — Cast piles must be reinforced to permit 
 handling, to withstand the shock due to driving, and to withstand the lateral strains, if con- 
 ditions are such as to require it. 
 
 Precast piles are usually designed to meet the particular conditions under which they are 
 to be used. In some cases they are provided with cast- or wrought-iron or steel driving points 
 and may also have an iron pipe cast in the center for jetting. Usually, however, the iron or 
 steel point is omitted, and jetting, if found necessary, is done by means of pipes, which are not 
 fastened to the pile. 
 
 To protect the pile from shattering under the severe blows oi tne hammer, laminated 
 wood blocks may be used, although some contractors provide a special driving head in which 
 a ciishion of sand, rope, or other material is placed between a driving block of wood and the 
 concrete to prevent crushing the head of the pile. These piles can be built in any required 
 size, and have been placed up to 90 ft. in length, though considerably shorter piles are more 
 frequently employed. On dock and pier work premolded concrete piles are used both as 
 supporting piles and as sheet piles to retain the ground behind them. Such piles must not 
 only be reinforced against driving and handling stresses, but must be designed to safely support 
 the soil load to which they will be subjected. These piles must ordinarily be cured for from 
 30 to 60 days after casting before they can be used, which, on rush work, is considerable detri- 
 ment to their use. It is also extremely difficult to predetermine the exact length which will be 
 required, and if it is necessary to qut off any unused portion of the pile this must be done by 
 breaking away the concrete from the reinforcement and either bending this latter down into 
 the superimposed caps or cutting it off with hack saws or acetylene torches. 
 
 Fig. 21 shows the details of the concrete piles employed in the construction of ore docks 
 for the Pennsylvania Railroad Co. at Cleveland, Ohio. The piles, octagonal in shape and 
 reinforced with eight 1-in. round rods, were cast vertically in steel forms, a cast-iron shoe 
 being fitted into the form and becoming a part of the finished pile. A 1:2:4 mixture of both 
 gravel and broken stone was used. The forms were removed in from 12 to 24 hr. after pouring, 
 depending upon the weather. No pile was driven before it was at least 30 days old. Sheet 
 piles were also used, as shown in Fig. 21. 
 
SECTION 13 
 
 RETAINING WALLS 
 
 A retaining wall is a wall of masonry — such as stone, plain concrete, or reinforced concrete — 
 built to sustain the lateral pressure of earth or of other material possessing more or less frictional 
 stability. In stone masonry and plain concrete walls the section must be made heavy enough 
 so that the weight of the structure will prevent overturning, whence the name gravity section. 
 In reinforced-concrete walls the weight of a considerable part of the sustained material is 
 utilized to maintain stability and, in addition, the sections may be designed to more nearly 
 develop the full strength of the concrete. 
 
 The treatment of retaining walls here given deals (1) with the earth thrust or pressure which 
 acts against the wall, (2) with the forces which maintain the stability of the wall under this 
 thrust, and (3) with the design of walls to withstand these attacking and stabilizing forces. 
 
 1. Earth Pressure. — In Fig. 1 let AB represent the back of a retaining wall, and AC 
 the surface of the ground. The earth has a tendency to break away and come down some 
 line, as CB, thus producing pressure on the wall. The weight of the 
 earth tends to cause this breaking away while the resisting forces are 
 the friction on the face AB and on the plane BC (the latter called 
 internal friction); the cohesion along the line BC; and the resistance of 
 the wall due to its stability against overturning and sliding. The 
 coefficients of these frictions, and of cohesion, vary with slope of the 
 surface AC, the fineness of the retained material, and its moisture 
 content. Cohesion is influenced greatly by moisture and the vibration 
 of moving loads, and seldom obtains in a newly made untamped fill. 
 Most earth pressure theories, therefore, treat the limiting case of an 
 ideal granular mass possessing no cohesion.^ This has the effect of reducmg the curve of 
 rupture BC to a straight line. 
 
 All earth pressure theories given here assume (1) that the surface of rupture BC is a 
 plane, (2) that the point of resultant pressure is at one-third the height of the wall from the 
 bottom when the surface of the material meets the top of the wall, and (3) that the resultant 
 pressure makes some definite angle with the horizontal. These theories result in a triangular 
 distribution of pressure against the face of the wall when the surface of the material meets 
 the top of the wall; and since the resultant pressure must pass through the centroid of this 
 pressure-distribution triangle, it must act at one-third the height of the wall from the bottom. 
 Its inclination varies with different conditions and theories. 
 
 Table 1. — Angles of Repose and Weights per Cubic Foot for Various Earths^ 
 
 Material 
 
 Slope 
 
 Angle of 
 repose, degrees 
 
 Weight in lb. 
 per cu. ft. 
 
 Sand, dry 
 
 2.8 
 
 1 to 1 
 
 4 
 
 1 
 
 20 to 35 
 
 90 to 110 
 
 
 1.75 
 
 1 to 
 
 1 
 
 1 
 
 30 to 45 
 
 100 t o 110 
 
 Sand, wet 
 
 2.8 
 
 1 to 1 
 
 2 
 
 1 
 
 20 to 40 
 
 110 to 120 
 
 
 2.8 
 
 1 to 
 
 1 
 
 1 
 
 20 to 45 
 
 80 to 100 
 
 
 2.1 
 
 1 to 
 
 1 
 
 1 
 
 25 to 45 
 
 80 to 100 
 
 Ordinary earth, wet 
 
 2.1 
 
 1 to 1 
 
 75 
 
 1 
 
 25 to 30 
 
 100 to 120 
 
 Gravel, round to angular 
 
 1.75 
 
 1 to 0 
 
 9 
 
 1 
 
 30 to 48 
 
 100 to 135 
 
 Gravel, sand and clay 
 
 2.8 
 
 1 to 1 
 
 3 
 
 1 
 
 20 to 37 
 
 100 to 115 
 
 1 For theory including cohesion, see Cain's "Earth Pressure, Walls and Bins." It is valuable for the investi- 
 gation of stability of existing walls backed by earth which has been compacted in some manner. 
 From Cain's "Earth Pressure, Walls and Bins," p. 9. 
 
 575 
 
576 CONCRETE ENGINEERS' HANDBOOK [Sec. 13-la 
 
 Table 2. — Coefficients of Internal Friction^ 
 
 Kind of material 
 
 Tangent 
 of angle 
 of internal 
 friction 
 
 Appi 
 corre 
 
 Angle, 
 degrees 
 
 oximate 
 sponding 
 
 Slope 
 
 Authority 
 
 Coal, shingle, ballast, 
 
 
 
 
 
 
 etc 
 
 1 
 
 423 
 
 54 
 
 0. 7 to 1 
 
 B. Baker 
 
 Bank sand 
 
 1 
 
 423 
 
 54 
 
 0.7 to 1 
 
 Goodrich 
 
 Riprap 
 
 1 
 
 097 
 
 48 
 
 0 . 9 to 1 
 
 Goodrich 
 
 Earth 
 
 1 
 
 097 
 
 48 
 
 0.9 to 1 
 
 B. Baker 
 
 Quicksand, 100 up . . . 
 
 0 
 
 895 
 
 42 
 
 1 . 1 to 1 
 
 Goodrich 
 
 Clay 
 
 0 
 
 895 
 
 42 
 
 1 . 1 to 1 
 
 B. Baker 
 
 Quicksand, 50-100 
 
 0 
 
 750 
 
 37 
 
 1 . 3 to 1 
 
 Goodrich 
 
 Earth 
 
 0 
 
 750 
 
 37 
 
 1 . 3 to 1 
 
 Steel 
 
 Bank sand 
 
 0 
 
 750 
 
 37 
 
 1 . 3 to 1 
 
 Wilson 
 
 Sand, 50-100 
 
 0 
 
 549 
 
 29 
 
 1 . 8 to 1 
 
 Goodrich 
 
 Bank sand 
 
 0 
 
 549 
 
 29 
 
 1 . 8 to 1 
 
 Goodrich 
 
 Clay 
 
 0 
 
 474 
 
 25 
 
 2.1 to 1 
 
 Goodrich 
 
 Cinders 
 
 0 
 
 474 
 
 25 
 
 2.1 to 1 
 
 Goodrich 
 
 Gravel, 3'^-in 
 
 0 
 
 474 
 
 25 
 
 2.1 to 1 
 
 Goodrich 
 
 Gravel, 3^-in 
 
 0 
 
 350 
 
 19 
 
 2.9 to 1 
 
 Goodrich 
 
 Bank sand 
 
 0 
 
 350 
 
 19 
 
 2.9 to 1 
 
 Goodrich 
 
 Sand, 30-50 
 
 0 
 
 258 
 
 14 
 
 3.9 to 1 
 
 Goodrich 
 
 Sand, 20-30 
 
 0 
 
 179 
 
 10 
 
 5.6 to 1 
 
 Goodrich 
 
 la. Rankine's Formula for Resultant Active Earth Pressure. — Rankine has 
 developed the following formula^ for the case in which (1) the total active thrust P acts uponi 
 a vertical plane, (2) acts parallel to the surface of the earth for all cases in which e>0, (3) actsi 
 in a material of indefinite extent, and (4) the earth carries no load except its own weight (Fig. 2): 
 
 ^ wh^ 
 
 in which 
 
 P = total active thrust of earth against the vertical plane as described 
 above. 
 
 w = weight per cubic foot of retained material. 
 h = height of vertical section considered, as AC. 
 cos d — \/ cos^ d — cos2 </) 
 
 Fig. 2. 
 
 cos ^ + V cos2 d — cos2 0 
 
 where 
 
 6 = angle of surcharge. 
 0 = angle of internal friction. 
 Diagram 1 gives the values of Ce for various values of 9 and <l>. It should be noted that P is 
 
 parallel to the surface AB, when 6 is either positive, or zero; and that it acts at a point D, ^ 
 
 (AC\ 
 
 » E. P. Goodrich: Trans. Am. Soc. of C. E., vol. 53, p. 301. 
 
 * For derelopment see Baker's "Masonry Construction," 10th Ed., p. 493, 
 
Sec. 13-16] 
 
 When d = <p, then 
 When 6=0 
 
 in which 
 
 RETAINING WALLS 
 
 P = y^wh^ ■ cos (f> 
 
 577 
 
 Ce' 
 
 Ce' = tan2 (450 _ 3^^) 
 Diagram 1 
 
 Values of Ce m P - Ce"^' 
 
 The following table gives values of C/ for varying values of <t>. 
 
 .60 .90 
 
 
 Ce' 
 
 4> 
 
 Ce' 
 
 4> 
 
 Ce' 
 
 1 
 
 Ce' 
 
 20° 
 
 0.490 
 
 30" 
 
 0.3333 
 
 40° 
 
 0.2174 
 
 50" 
 
 0.1325 
 
 25° 
 
 0.406 
 
 35° 
 
 0.2710 
 
 45*^ 
 
 0.1718 
 
 55° 
 
 0.0994 
 
 16. Coulomb's Wedge of Maximum 
 Pressure. — Coulomb advanced the theory that the 
 wedge ACF (Fig. 3), lying between the surface of the 
 earth AF and the plane of rupture CF would move 
 down against the vertical plane AC due to its own 
 weight, causing the resultant pressure P'. The prism 
 itself is in equilibrium through a force acting upward 
 against CF, and making the angle <f) with CF ; and a force 
 opposing P' due to the resistance of the wall BC and of 
 the prism BAC (if BC is not vertical). If angle BCA 
 is relatively large, there will be found a plane of rupture 
 SC, in the prism BAC, tending to force a portion of this 
 
 prism upward. This plane of rupture makes angle /S with AC, which is dependent upon 
 and (f). Its value is given in Table 3. 
 37 
 
578 CONCRETE ENGINEERS' HANDBOOK [Sec. 13-16 
 
 Table 3 
 
 0 
 
 
 /3 
 
 Batter 
 
 of /3 
 (inches 
 per foot) 
 
 0 
 
 4> 
 
 
 Batter 
 
 of j8 
 (inches 
 per foot) 
 
 15° 
 
 15° 
 
 0° 0' 
 
 0 
 
 20° 
 
 20° 
 
 0° 0' 
 
 0 
 
 
 20 
 
 20° 25' 
 
 
 
 25 
 
 15° 30' 
 
 3H 
 
 
 25 
 
 21° 05' 
 
 
 
 30 
 
 18° 25' 
 
 4 
 
 
 30 
 
 21° 55' 
 
 
 
 35 
 
 19° 10' 
 
 4:14 
 
 
 35 
 
 21° 30' 
 
 
 
 40 
 
 18° 55' 
 
 414 
 
 
 40 
 
 20° 35' 
 
 
 
 45 
 
 18° 05' 
 
 Z14 
 
 "/8 
 
 
 45 
 
 19° 15' 
 
 4M 
 
 
 50 
 
 16° 45' 
 
 3% 
 
 "/8 
 
 
 50 
 
 17° 35' 
 
 
 
 55 
 
 15° 10' 
 
 
 
 55 
 
 15° 45' 
 
 
 
 
 
 
 
 
 
 
 30° 
 
 30 
 
 0° 0' 
 
 0 
 
 25° 
 
 25 
 
 0° 0' 
 
 0 
 
 
 35 
 
 12° 05' 
 
 214 
 
 
 30 
 
 13° 35' 
 
 27^ 
 
 
 40 
 
 14° 25' 
 
 314 
 
 "78 
 
 
 35 
 
 16° 15' 
 
 ^ / s 
 
 
 45 
 
 15° 0' 
 
 3Vi 
 
 
 40 
 
 16° 55' 
 
 "/8 
 
 
 50 
 
 14° 40' 
 
 314 
 
 
 45 
 
 16° 40' 
 
 "/8 
 
 
 55 
 
 13° 40' 
 
 "7a 
 
 
 50 
 
 15° 45' 
 
 
 
 
 
 
 
 55 
 
 14° 25' 
 
 "/8 
 
 40° 
 
 40 
 
 0° 0' 
 
 0 
 
 
 
 
 
 
 45 
 
 9° 45' 
 
 2 
 
 35° 
 
 35 
 
 0° 0' 
 
 0 
 
 
 50 
 
 11° 30' 
 
 2% 
 
 
 40 
 
 10° 50' 
 
 2M 
 
 
 55 
 
 11° 40' 
 
 
 
 45 
 
 10° 55' 
 
 
 
 
 
 
 
 50 
 
 10° 15' 
 
 
 45° 
 
 45 
 
 0° 0' 
 
 0 
 
 
 55 
 
 12° 45' 
 
 
 
 50 
 
 8° 40' 
 
 m 
 
 
 
 
 
 
 55 
 
 10° 05' 
 
 2% 
 
 Let the point G on BC be so located that CG = }iBC. This will be the point of applica- 
 tion of the resultant earth pressure P. Its line of action as applied to the wall makes the 
 angle 7J with a normal to the wall at G. 
 
 Fig. 4. 
 
 When the face BC of the wall is battered such that a < ^, Construction I should be followed : 
 When a = follow Construction I if Z > 0; otherwise follow Construction II. 
 When a > /3, foUow Construction II. 
 
 Construction I. — ^Let the wall BC (Fig. 4) retain material whose surface slope is d, and whose 
 angle of internal friction is <j>. It will be assumed that the resultant thrust P will be applied 
 
Sec. 13-16] 
 
 RETAINING WALLS 
 
 579 
 
 at the third point D, so that it makes the angle </> with a normal at D. [It may be assumed 
 to make the angle of friction (Z of the material against the wall) with the normal when that 
 angle does not exceed 4>.] Beginning at B, lay off on BG arbitrary distances, equal, for con- 
 venience, as Ba, ab, etc., and connect these points a, h, c, etc. with C. Compute the weight 
 of these prisms thus formed, with length of 1 ft. normal to the drawing, and lay them off to 
 some convenient scale, on CG, beginning at C, as Wa, Wb, etc. If the distances on BG are 
 equal, those on CG will likewise be equal. Draw CS making an angle k with CG, where k is the 
 angle made by P with the vertical. Draw a'Wa, b'Wb, etc., all parallel to CS, and through these 
 points a', h', c', etc., thus obtained, draw a curve. A tangent to this curve parallel to CG is 
 tangent at m', through which Cm may be drawn. Thus the prism BCm is that which causes the 
 maximum pressure P for the conditions assumed; and m'n, drawn parallel to CS, and scaled 
 off to the scale of the weights on CG, gives the maximum value of this thrust P. 
 
 When 6=0 the surface of fill is level, and Cm then bisects angle BCG, and may thus be 
 drawn at once. The weight of prism BCm may be laid off on CG and m'n drawn to get the 
 corresponding value of P. 
 
 Construction II (Fig. 5). — When it has been found from Table 3 that the wall BC makes 
 a greater angle than /3 with the vertical, it becomes necessary to determine the pressure P' 
 
 Fig. 5. Fig. 6. 
 
 acting parallel to AB, against a vertical plane AC through C. Lay off for convenience equal 
 distances Aa, ab, etc., and connect a, b, c, etc. with C. Compute the weights of the prisms 
 thus formed, and plot these weights to some convenient scale on CG (which makes an angle 
 <l> with a horizontal through C), as CWa, Wa, Wb, etc. Draw through these points the lines 
 Waa', Wbb', etc., making the angle k' (equal to 90° — d) with CG. A smooth curve may then 
 be drawn through a', 6', c', etc. Point r' is located by drawing a tangent to this curve parallel 
 to CG. The plane rC is the plane of rupture for the wedge BCr, which might be forced up- 
 ward due to the pressure P' and the resistance of the wall. The resultant pressure P' acting at 
 D in either direction on the line FD is equal to sr' scaled to that of the plotted weights on CG. 
 
 Let the inclination of BC to the vertical be a. Extend CG beyond C and lay off a counter- 
 clockwise; then lay off 90 deg. The line thus obtained will make an angle q with the plane of 
 rupture Cr. 
 
 Now if q is less than Z, where Z is the angle of friction of the material on the wall, produce 
 P' to F, on BC. Through F the resultant P on the face BC will act, such that its angle of in- 
 clination with the vertical (d) equals ZrCG, or what is the same thing, such that it makes the 
 angle q with a normal to BC. The magnitude of P is r'S scaled to that of the weights on CG. 
 
 If q is greater than Z, perform Construction I making P slope the angle Z with the normal 
 to BC, Fig. 4. 
 
580 
 
 CONCRETE ENGINEERS' HANDBOOK 
 
 [Sec. 13-lc 
 
 When 6 is negative and not large, or when oc is negative but not to exceed 10 deg., the con- 
 structions I and II may still be used. 
 
 Rebhann's Construction. — The value of P by the Coulomb theory may be found in the 
 following manner: Let BC, Fig. 6, be the back face of a retaining wall with the earth surface 
 on the line BJ. AC is the vertical plane against which P acts, at point D, y^AC from C. 
 CJ makes the angle of internal friction </> with the horizontal. Draw AF so that ZCAF = 6 
 + </). With O as the center and FJ as the diameter, describe the arc FMJ. Draw CM tangent 
 to this arc. (The point of tangency M may be located accurately by describing the semicircle 
 CMO on the diameter CO.) Make CN = CM, and through N draw QN \\ AF. Making RN 
 = QN forms the triangle QRN. The magnitude of P is 
 
 P = (iy)(Area QRN) 
 
 Its direction corresponds to Rankine's thrust, parallel to AJ. 
 
 Point Q may be located by making a construction similar to that for finding N. Draw TC \\ AF. Describe 
 arc A UJ on the diameter AJ. Locate U by drawing arc T US on diameter TS, whence TU is tangent to A UJ at U. 
 Make TQ = TU. 
 
 When 6 = ^, A J and CJ do not meet; also, Z.CAF = 2^. 
 When 6=0, ZCAF = <f>, and P is horizontal. 
 
 Ic. Comparison of Coulomb and Rankine Results. — Rebhann's construction, 
 also that of Fig. 4, gives results which check those of the Rankine formula, upon a vertical plane, 
 when the earth extends indefinitely. The theories differ somewhat for other cases. 
 
 Id. Useful Interpretation of Results of Earth Pressure Theories. — Assume a 
 wall BC, Fig. 7, to retain a fluid. Since the pressure of fluid is dependent upon the depth, the 
 intensity of the pressure at any depth y = wy, in which w is the weight 
 per cubic foot of the fluid in pounds. Then the pressure at CD is wh. 
 Thus the variation in pressure intensity is as a straight line, BD. The re- 
 sultant of this triangle of pressure is F = ^if^A^. It acts horizontally, normal 
 to BC and through the centroid of BCD. 
 
 Suppose the top of the wall to be covered at a depth h'y Fig. 8, 
 with this liquid. The pressure at J5 is = wh'. Similarly, CD = w{h' + h). 
 The total pressure on BC is equal to the area of the trapezoid of pressure 
 BPDC ; and its resultant P acts through the centroid of this trapezoid dis- 
 tance y from CD. It acts normal to BC. The distance y may be found 
 from the formula 
 
 Fig. 7. 
 
 y = 
 
 h h + 3h' 
 
 3 h + 2h' 
 
 Now the precise difference between earth (granular mass) and fluid 
 pressures, is that in the case of earth there is not equal pressure in all di- yig 8 
 
 rections at a given point. If E„ represents vertical pressure at a point in 
 an earth mass, and Ei the lateral pressure at that point, then Ei = CeEv. Since in each of 
 the foregoing formulas it is obvious that the intensity of pressure varies with the depth, th 
 pressure areas caused by the earth are similar to those caused by a fluid. Thus, from 
 Diagram 1 on page 577, if w' = CeW, then P would equal the pressure caused by a fluid whos 
 weight is w', and whose direction of acting upon the plane considered would be parallel to 
 the direction of P. In like manner any of the foregoing developments may be converted 
 into fluid-like action. 
 
 2. Live Load on Top of Fill — Equivalent Surcharge. — When a live load is apphed in a 
 direction normal to the face of the wall, as for instance upon a track or roadway at right angles 
 to a bridge abutment. Fig. 9, the earth thrust P is larger than before. Let the weight of th 
 appUed live load (including impact) be replaced by an equal weight represented by the prisE 
 AMN, of the same material as that retained by the wall. Let the depth of this ''equivalen 
 surcharge" be h\ 
 
Sec. 13-3] 
 
 RETAINING WALLS 
 
 581 
 
 The total pressure on MC is given by the trapezoidal pressure ACSR, whose resultant is 
 P. It may be obtained by determining an equivalent fluid of weight w' as described in the 
 preceding article; or it may be determined graphically by finding the thrust on M C, then on MA, 
 
 and taking their difference. In usual cases 
 being less than 5% on the safe side. As before, 
 
 h h^+Sh 
 3 * h 
 
 P will be nearly equal to + ^)^> the error 
 
 y 
 
 + 2h' 
 
 The thrust P is finally prolonged to meet the face BC, where 
 it is combined with the weight of the prism BAG, to determine 
 the final thrust against the wall. 
 
 Construction I, page 578 could have been employed to find 
 directly the thrust against BG by placing the surcharge up 
 to B. 
 
 When roadways, tracks, or other live loads are placed 
 close to, and parallel to, the wall, the method described 
 
 above will apply. If these live loads are remote from the wall, the method of procedure 
 described in the following article is recommended. It will, of course, apply to both cases, 
 but is a saving for the conditions cited. 
 
 3. Live Load on Top of Fill — Pressure Distribution. — ^Let AD, Fig. 10, represent a track 
 
 parallel to the wall BG. Tests ^ show that the pressure on AD is 
 practically all distributed between AF and DG, which make 30 deg. 
 with the vertical. Assuming uniform distribution throughout this 
 region, the pressure per square foot on the horizontal plane FG is 
 computed, F being the point where AF strikes the wall. Let the in- 
 tensity of pressure on ah, without the load AD, be shown by the tri- 
 angle abc. Let de (= c/) equal the unit pressure on FG due to AD 
 multiplied by some factor N dependent upon the material in the fill. 
 
 The resultant of ahc, acting ^ above G, and the resultant of dejc, act- 
 ing y^GF above C, are then combined to get the final thrust on BG. 
 
 The factor N just referred to may be determined from the table on page 577. This fac- 
 tor is the ratio of lateral unit pressure to vertical unit pressure, and is thus dependent upon 
 the value of the angle of internal friction. Thus N = Ge in the table referred to. 
 
 4. Stability of a Retaining Wall. — Two motions of the wall tend to result due to the action 
 of the earth thrust P: (1) a tendency to slide forward; and (2) a tendency 
 to tip forward about some point on the base. 
 
 The tendency for the wall to slide forward may be stated as being 
 equal to the tangential component of the resultant force R acting upon 
 the base, or plane of bearing. In walls having a horizontal plane of 
 bearing it is equal to the horizontal component Rh, Fig. 11. The re- 
 sistance to sliding may be developed in three ways: (1) by the friction on 
 the plane AD, (2) by the depth of A below the surface of the ground in 
 front of the wall,^ and (3) by a key wall projecting downward from the 
 plane AD. The frictional resistance of the base plane AD may be taken 
 
 as the total component normal to AD multiplied by the coefficient of friction of the wall material 
 upon the supporting soil (see Table 4). The key wall will cause compression in the soil before 
 it, the intensity of which should not exceed seven-tenths of the maximum working unit pressure. 
 
 The resistance to overturning the wall is afforded by a distributed reaction of the bearing 
 soil upward against the base of the wall. Since the bearing capacity of the soil must not be 
 exceeded, it is necessary to study the distribution of this reaction, and to determine simple 
 rules which may govern to secure stability. 
 
 1 See tests by Prof. M. L. Enger, Eng. Rec, Jan. 22, 1916, p. 106-8. 
 
 2 See Art. 15, Sect. 17, page 763. 
 
582 
 
 CONCRETE ENGINEERS' HANDBOOK 
 
 [Sec. 13-4 
 
 Table 4. — Coefficients and Angles of Friction 
 Between Earth and Other Materials 
 
 Materials 
 
 / = 
 
 tan <l> 
 
 4> 
 
 Masonry upon masonry 
 
 0.65 
 
 33° 
 
 Masonry upon wood, with grain 
 
 0.60 
 
 31° 
 
 Masonry upon "wood, across grain. . . . 
 
 0.50 
 
 26° 40' 
 
 Masonry on dry clay 
 
 0.50 
 
 26° 40' 
 
 Masonry on wet clay 
 
 0.33 
 
 18° 20' 
 
 
 0.40 
 
 21° 50' 
 
 Masonry on gravel 
 
 0.60 
 
 31° 
 
 Fig. 12. 
 
 Consider the base of a 1-ft. section of wall to be represented in projection by AB, Fig. 12. 
 When R acts at the center of gravity O, the intensity of pressure is uniform over the base 
 
 R R 
 
 plane and is equal to the vertical component of R divided by the base area, ~ R 
 
 acts at any other point, as Q, the force Rv is equivalent to an equal Rv at O and a couple whose 
 moment is RvXq. At any point distant x from O the intensity of the pressure due to this moment 
 
 is ^"^"^ ' in which / is the moment of inertia of the base plane about an axis through 0 at right 
 angles to AB and lying in the base plane. At the edges A and B this intensity = = 
 
 QRvXo 
 
 ^2 The intensity of pressure at edge A is 
 
 R 
 
 Pi 
 
 QRvXo 
 
 and at edge B it is 
 
 P2 
 
 Rvf 6^o\ 
 
 Rvl _6^o\ 
 hV h) 
 
 (lb. per sq. ft.) 
 
 (lb. per sq. ft.) 
 
 Since the base plane cannot resist tension, the second term of ^2 must not exceed the 
 first. As a limiting condition, if the two terms are just equal, p2 = 0, or 
 
 Rv _ ^RvXq 
 
 in which Xo is the distance to Q when the eccentricity Rv just causes this equality. Solving 
 
 b 
 6 
 
 Xo 
 
 (ft.) 
 
 The rule thus determined follows: The resultant f orce acting upon the base plane must strike 
 it back {toward B) of the forward third point of the base plane, if no tension is to be taken by that 
 plane. 
 
 When ?>2 = 0, 
 
 Pi = (lb. per sq. ft.) 
 
 The relation between the point of application of the resultant R upon the base and the 
 unit stress in the soil under the toe is shown in Diagram 2. The rapid rise in unit pressure due 
 to a small forward movement should be noted. Since it has been advised in this discussion 
 to limit the resultant to the forward third point, the unit pressure under the toe for that condi- 
 tion is taken as 100%. Its maximum value depends upon the decision of the engineers re- 
 garding the quality and supporting power of the soil. If R falls inside of the middle third, the 
 
Sec. 13-4] 
 
 RETAINING WALLS 
 
 583 
 
 pressure at the toe will be less than 100 %. The difference between its percentage and 100 % will 
 
 give the unit pressure at the heel in % of 
 
 Table 5 gives allowable unit pressures upon various soils. If high walls are to be built, 
 tests should be made at the site, as described in Art. 1, Sect. 12. The tendency of the 
 pressure on the bearing soil to heave the earth in front of the wall should also be investigated. 
 
 Diagram 2 
 
 •£ 200 
 I 
 
 ^ 300 
 
 D 
 
 It 400 
 
 O 20 40 60 
 
 Distance of K from toe in %oib 
 
 Table No. 5. — Safe Bearing Capacity of 
 Short Tons per Square Foot 
 
 Soils 
 
 When the soil pressure under 
 the toe is greater than the allow- 
 able unit pressure, the base area 
 should be increased by extending 
 the toe forward. On a solid wall 
 this may be done by projecting a 
 toe from the front face, whose 
 top surface slopes from 30 to 60 
 deg. with the face of the wall, 
 and whose bottom surface is an 
 extension of the base plane. 
 Such a projecting toe must be 
 designed for shear and moment, 
 the same as the toe on a rein- 
 forced-concrete wall. 
 
 On a reinf orced-concrete wall 
 the unit pressure under the toe 
 may be reduced by extending the 
 toe farther from the face of the wall 
 weight of fill. 
 
 When a base slab is used under the body of the wall for capping piles, or for providing 
 suitable bed, the unit pressure at the front edge of the wall at the top surface of the base slab 
 must not exceed the allowable compressive unit stress in the concrete. Should it exceed this 
 value, provision may be made as above described. Care should always be exercised in properly 
 
 Kind of material 
 
 Mini- 
 mum 
 
 Maxi- 
 mum 
 
 Rock, the hardest, in thick layers in 
 
 
 
 
 200.0 
 
 
 Rock equal to best ashlar masonry .... 
 
 25.0 
 
 30 
 
 
 15.0 
 
 20 
 
 Rock equal to poor brick masonry 
 
 5.0 
 
 10 
 
 
 6.0 
 
 8 
 
 Clay in thick beds, moderately dry .... 
 
 4.0 
 
 6 
 
 
 1.0 
 
 2 
 
 Gravel and coarse sand, well cemented. 
 
 8.0 
 
 10 
 
 Sand, dry, compact and well cemented. 
 
 4.0 
 
 6 
 
 
 2.0 
 
 4 
 
 
 0.5 
 
 1 
 
 Some designers extend the back slab to obtain more 
 
584 
 
 CONCRETE ENGINEERS' HANDBOOK 
 
 [Sec. 13-4a 
 
 keying the body of the wall against sliding or overturning on the base slab. Keyed joints and 
 dowel rods may be used for this purpose. 
 
 4a. So-called " Factor of Safety." — In masonry walls it has been customary to 
 define the factor of safety against overturning by the relation 
 
 moment of resisting forces 
 
 factor of safety = 7 — 7^ 1 ? 
 
 moment of overturnmg forces 
 
 when these moments are taken about the toe of the base. This would assume a rigid bed for 
 the wall. When the bed is of yielding material, the wall will not rock on the toe because the 
 earth under the forward third of the base will crush, allowing the wall to settle as it tips. This 
 factor of ''safety" will not be used in this discussion. 
 
 46. Factor of Limitation. — Two considerations resulting from the foregoing 
 discussion, and from long experience, are (1) that the resultant of pressures stay within the 
 middle third, and (2) that uniform settlement should, as far as possible, be provided for by 
 bringing the resultant as near as practicable to the center of the base. These conditions make 
 it desirable that in the usual condition of loading the resultant should pass through the middle 
 third near its center, and that for the unexpected or unusual loadings it should never exceed 
 the forward limit of the middle third. The "factor of limitation" is that factor by which the 
 thrust of the earth may be multiplied to determine that thrust which will cause the resultant 
 to pass through the forward third point. In other words, 
 
 factor of limitation = 
 
 limiting thrust 
 actual thrust 
 
 It is of interest to note that for a wall of given height, supporting a level fill, the factor of 
 
 limitation may be increased a certain percentage by 
 Diagram 3 increasing the width of base a definite amount, which 
 
 amount is the same for plain walls and for reinforced- 
 concrete cantilever or counterfort walls. This fact is 
 illustrated in Diagram 3. For example, suppose a given 
 wall with a 30-ft. base to have been designed with a 
 factor of limitation of one. If 12.5 ft. were added to 
 the width of base, the factor of limitation for the wall 
 would be doubled. 
 
 5. Types of Retaining Walls. — Retaining walls of 
 concrete may be of plain concrete, or of reinforced con- 
 crete. Plain concrete walls have proportions similar to 
 those of masonry, and their section is usually the 
 gravity section. Reinforced-concrete walls, however, 
 are not as massive. The various types in common use 
 are: (1) the T- or cantilever wall; (2) the counterforted 
 cantilever or Wall; (3) the buttressed wall; and (4) the cellular wall. 
 
 Other special forms have been developed for various 
 purposes. 
 
 The following discussion is intended to give preliminary proportions before the final inves- 
 tigation is made for stabihty. For the cases here given no further designing need be done; 
 but for irregular embankments and other unusual conditions the cases cited will aid in choosing 
 the preliminary form of wall. 
 
 6. Design of Plain Concrete Walls. 
 
 6a. Formulas and Diagrams for the Two Principal Types.- 
 
 0 5 10 15 
 
 Additional width in feet 
 
 Level fill against plain concrete walls 
 and reinforced concrete 
 counterfort walls. 
 
 -For type (a) shown 
 
 m Fig, 13, let 
 
Sec. 13-6a] 
 
 RETAINING WALLS 
 
 5S5 
 
 A = area of cross-section. 
 }Y = weight of 1 ft. of wall 
 
 width at top = g 
 
 150A for concrete. 
 
 width at base, 
 height of wall. 
 
 angle between P and horizontal. 
 
 distance from outer third point Q to centroid of section, 
 factor of limitation, or the number of times Ph may be in- 
 creased before the resultant passes through the third point Q. 
 Assuming the resultant to be passing through Q, then the algebraic 
 sum of the moments about Q equals zero, or XMq = 0, 
 
 Wc + 
 
 2P,b 
 
 When 
 
 and W = 150A, then c = 0.3266, whence, since P = 
 
 28.56^ + 0.333 fw'hh sin e - 0.im7fw'h^ .cos e 
 Diagram 4 
 
 Ratio of width of base to total height » 
 
 Many designers prefer to disregard the frictional component Pv It cannot well be devel- 
 oped upon a vertical face. Walls should have a slight batter on the back, although presumably 
 vertical. Letting e = 0, 
 
 I = 0.0764 V/u/ 
 
 When e = 30 deg. 
 
 28.5 + 0.1667/w;' t - 0.1442/w)' = 0 
 
 Diagram 4 gives values of ^ for various values of and /, when e = 0 and 30 deg. 
 
586 
 
 CONCRETE ENGINEERS' HANDBOOK 
 
 [Sec. 13-66 
 
 Type (b), {Fig. 14). — For comparison with type (a), a level filling will be assumed. It 
 may be noted that a prism of earth BAC will add to the stability for this type. However, 
 the wall in general is less stable because its weight W lies forward from Q. 
 As before, assuming XMq = 0, there results, 
 
 I = 0.10\/fw' when w = 100 lb. per cu. ft. 
 
 and 
 
 I = 0.0915ViV when w = 120 lb. per cu. ft. 
 
 in which w is the weight per cubic foot of the material comprising the 
 material in prism ABC. Diagram 5 gives values of ^ for these two 
 cases when / and w' have varied values as well as with e = 30 deg. or zero. 
 Diagram 5 
 
 •-> 
 
 
 
 \ ^^I 
 
 
 i 
 
 
 , p 
 
 
 
 Y 
 
 
 < ^ > 
 
 C 
 
 Fig. 14. 
 
 .6 .7 .8, • .9 
 
 Ra+io' of widfh Of base toto+al height' -^i 
 
 Ratio of width of base to total height 
 
 Ratio of height 
 
 of earth to 
 height of wall 
 above ground 
 
 Ratio of thick- 
 ness of base to 
 total height of 
 
 wall = r- 
 
 h, 
 
 Ratio of height 
 of earth to 
 height of wall 
 above ground 
 
 r 
 
 Ratio of thick- 
 ness of base to 
 total height of 
 
 wall = — 
 
 n 
 
 1.0 
 
 0.35 
 
 2.0 
 
 0.58 
 
 1.1 
 
 0.42 
 
 2.5 
 
 0.60 
 
 1.2 
 
 0.46 
 
 3.0 
 
 0.62 
 
 1.3 
 
 0.49 
 
 4.0 
 
 0.63 
 
 1.4 
 
 0.51 
 
 6.0 
 
 0.64 
 
 1.5 
 
 0.52 
 
 9.0 
 
 0.65 
 
 1.6 
 
 0.54 
 
 14.0 
 
 0.66 
 
 1.7 
 
 0.55 
 
 25.0 
 
 0.68 
 
 1.8 
 
 0.56 
 
 or more 
 
 
 65. Trautwine's Table.— 
 
 Table on this page taken from Traut- 
 wine's ''Civil Engineer's Pocketbook" 
 
 gives values of ^ for various heights 
 
 of surcharge, as for instance, road- 
 beds. Values here given are empirical, 
 and may be used for average condi- 
 tions. They are the result of long 
 practice in retaining-wall design. 
 The earth is assumed to slope up 
 from the top of wall till it reaches a 
 level at the height indicated by the 
 ratio in the first column. 
 
Sec. 13-6c] 
 
 RETAINING WALLS 
 
 587 
 
 Trautwine recommends that when the backing is somewhat consoHdated in horizontal 
 layers, each of these thicknesses may be reduced, but that no rule can be given for this. He 
 also states that since sand and gravel have no cohesion, the full dimensions as above should 
 be used with these materials, even though the backing be deposited in layers. A mixture of 
 sand, or earth with a large proportion of pebbles, bowlders, etc., will exert a greater pressure 
 against the wall than the materials ordinarily used for backing ; and hence when such backing 
 has to be used, the above thicknesses should be increased, say, about }i to }i part. 
 
 6c. Selection of Preliminary Section. — The above methods of proportioning 
 the gravity section enables the selection of a preliminary wall against which the earth pressures 
 
 Diagram 6 
 
 1.0 
 
 (D 
 D 
 
 1.5 
 
 e.0 
 
 e.5 
 
 ^^slPlain concrete walls 
 _ 5 Gravity sections 
 /a)and(b) 
 
 Reinforced concrete walls 
 Canti fever, counterfort, 
 or buttress walls_ 
 
 30 
 
 40 50 30 40 50 60 
 
 Distance of resultant from toe in percent of b 
 
 may be determined, and the actual resultant obtained. The resulting soil pressures are then 
 determined and compared to the allowable pressures. Analysis should be made for the lateral 
 pressure / • Ph (the resultant for which is to pass through the forward third point) as well as for 
 Ph. Likewise resistance to sliding should be great enough to prevent sliding under the action 
 of /-P^. 
 
 Since the thrust of / • Ph will cause the resultant to pass through the forward third point, 
 the actual thrust Ph will cause a resultant much nearer the center. Diagram 6 has been drawn 
 to show the effect of various values of/ upon the actual position of the resultant. For instance, 
 for type (a), by designing with / = 1.75, the actual resultant will pass through a point 0.4756 
 from the toe. If the actual thrust, causing this actual resultant, should be multiplied by 1.75, 
 the resultant thus caused would pass through the forward third point. 
 
 Having the .position of the resultant (0.4756 from the toe), 
 the unit soil pressure under the toe may be found, from Diagram 
 
 2, to be 55 % of that is, 55 % of the allowable maximum unit 
 
 pressure assigned to the case when R passes through the third 
 point. 
 
 7. Design of Cantilever or T-walls of Reinforced Concrete. — 
 
 A cantilever or T-wall of reinforced concrete consists of a can- 
 tilever stem AD, Fig. 15, rigidly attached to a base slab, BC. 
 Since stability greatly depends upon the weight W2 of the earth 
 ADNM, it is likewise affected by the distance x of the face of the 
 stem from the toe B. 
 
 la. To Determine Approximate Base Width. — As an approximate determination 
 (within 10%), we will assume the weight of the wall and earth above DN to be equal to a block 
 of earth of section A (6 — x), of the same weight, w, of the earth in the fill.^ Assuming the 
 
 1 See article by H. M, Gibb, Eng. News, July 24, 1913. 
 
588 
 
 CONCRETE ENGINEERS' HANDBOOK 
 
 [Sec. 13-7a 
 
 resultant to pass through the forward third point, the total moment about that point must 
 equal zero. The weights of the length x of the base will be neglected. Thus, 
 
 ^ (6 - x){Sx + 6) = y2fw'h^ 
 
 Whence 
 
 in which 
 
 Vfi 
 
 Vw(l ~+ 2k - 3/b2) 
 
 = CVfi 
 
 C2 = 
 
 1 
 
 lOVl + 2k 
 1 
 
 Sk'- 
 
 for w = 100 lb. per cu. ft. 
 forw = 120 lb. per cu. ft. 
 
 ' 10-95 Vl +2k - 3/c2 
 Values of Ci and C2 (to be introduced in place of C above) may be taken from Diagram 7. 
 
 Diagram 7 
 
 0.14 
 
 20 AO j60 
 
 Values oT k 
 
 Diagram 8 
 
 Proportions of cantilever, counterfort, or buttress retaining wall. 
 
 It is important to note in Diagram 7 that for any given value of w' and /, the minimum 
 
 value of ^ is that for k = }4; and that when k = }i it is very nearly the same. Since it is 
 
 obvious that resistance to sliding is increased by increasing the weight on the base, it is common 
 practice where possible to place the stem so that fc = 3^. 
 
Sec. 13-76] 
 
 RETAINING WALLS 
 
 580 
 
 The width of base of a cantilever wall may be taken from Diagram 8, by entering with a 
 I value of w' and moving to the right to the selected value of/; thence down to the desired value 
 
 of k] thence to the right margin where the proper value of ^ may be read. 
 
 Illustrative Probsem. — Desired cantilever wall, /t = 20 ft,; A; = H, / = 2, For material to be retained, 
 w = 100, e = 0, <t> = 29 deg. Determine width of base. From the table on page 577 C/ = 0.348. From Art. Id, 
 w' = Ce'w or w' = 34,8. Entering Diagram 8 with thi.« value, thence horizontally to / = 2.00, thence downward 
 to A; = H (for w = 100), thence horizontally to the right margin, ^ = 0.70. Therefore 6 = 20 X 0.7 = 14 ft. 
 
 From the right-hand side of Diagram 6 it is found that for / = 2 and k = H, the resultant will strike the base at a 
 point 0.50 X 14 = 7.00 ft. from the toe. The corresponding pressure under the toe for that position is, from 
 Diagram 2, about 50% of the working value 
 
 76. Stem. — The tendency of the earth pressures 
 is to break the stem of the wall, similar to a cantilever beam. 
 This moment is greatest at the junction of the wall with the 
 base. Fig. 16 shows a cantilever wall, with the total earth pres- 
 sure area divided into two parts — the pressure on the stem, and 
 the pressure on the base slab. The thrust Pi is the same as that 
 for a depth of earth equal to the height of the stem above the 
 upper side of the base slab. The bending moment per foot of 
 wall at its junction with the base surface is equal to the moment 
 of Pi about that same line; or 
 
 M = f Fi cos e . ^ = — g— • cos e 
 
 But for balanced stresses in the reinforced concrete, M = Kbd'^ (see Diagram 2 of Sect. 7, 
 page 360). Whence for a 12-in. width of stem, 
 
 d = 0.408/ii''''^^'- cos e _ (in.) 
 
 Fig. 16. 
 
 (ft.-lb.) 
 
 in which d = effective depth of stem slab in inches, hi = total height of stem in feet, and 
 w' = fluid equivalent in pounds per cubic foot. 
 When e = 0, 
 
 d = OAOShi'' yj^-^ (in.) 
 
 When e = 30 deg., 
 
 d = 0.380Ai''^ (in.) 
 The vertical reinforcing will be on the face next to the backing. 
 
 Diagram 9 gives the effective depth d for various values of fw' and K. The latter term 
 includes the selection of working stresses and the percentage of steel. To this effective depth 
 is added sufficient concrete to properly imbed the steel. The thickness of wall at the top is 
 arbitrary, but never less than 6 in. 
 
 Illustrative Problem. — The stem of a wall is 16 ft. above the top of the base slab. Required its maximum 
 thickness for w' = 40, / = 1.50, n = 15, /s = 16,000, and fc = 650. Slope of surcharge (hence e) is 30 deg. From 
 Diagram 2, Seet. 7, K = 107, p = 0.0077. 
 
 Entering Diagram 9 with fw' = (40) (1.50) = 60; thence to iv = 107; thence vertically to hi = 16; thence 
 to right margin, where d = 19.5 in. Due to the fact that e = 30 deg., from the correction curve at the top it is 
 found that d for this case should be 93.2% of 19.5, or 18.2 in. Total thickness, therefore, should be 21 in. 
 
 It may be necessary in some cases to take into account the weight of the vertical slab in 
 finding the maximum thickness required. The resultant of the earth pressure and weight of 
 
590 
 
 CONCRETE ENGINEERS' HANDBOOK 
 
 [Sec. 13-7c 
 
 stem may readily be found graphically and then the eccentricity of this resultant on the section 
 at the junction with the base slab may be scaled. The required thickness and percentage of 
 reinforcement may then be found by means of Diagram 15 or 16 of Sect. 9, pages 404 and 405. 
 
 It is not necessary to carry all the steel to the top of the stem. The moment diagram 
 should be plotted for the stem and some of the rods cut off a sufficient distance above the 
 point where they are not needed to secure anchorage (see Arts. 16 and 22, Sect. 7). 
 
 Diagram 9 
 
 Angle e in degrees 
 0 5 to 15 20 g5 . 30 35 4 0 45 
 
 7c. Base Slab. — The greatest stress in the base slab will be developed when the 
 resultant is at the most forward position— that is, in the present discussion at the forward 
 third point. Under this condition the toe has a maximum upward pressure, while the heel 
 has a maximum downward pressure combined with a minimum upward pressure. 
 
 The toe slab MN (Fig. 17) must be designed for the moment and shear at caused by the 
 pressure area A . Investigation must also be made for bond stress. Reinforcing will be along the 
 bottom for this portion of the base. Diagram 14 (see page 596) may be used in the shear in- 
 vestigation.. Shear usually governs toes of length ^ or less. 
 
 The heel slab carries the upward pressure represented by the area B; the downward com- 
 ponent of Pi shown hy Pz'; and the dead weight of all earth above the slab RS. The thrust 
 
Sec. 13-7c] 
 
 RETAINING WALLS 
 
 591 
 
 Plate I 
 
 /L4'' 
 
 Grade £/ey /OO 0 
 
 VAyJ^Rods \ I 1 
 
 84 c toe 
 
 rrifiifrlnifrif 
 
 Section A-A 
 
 A- 
 
 lO'-O" 
 
 Rear dievarion 
 
 til' 
 
 \ ' 1 AO 
 
 k-- -» /-,/^. 
 •/fco' Mark Fi 
 
 !i"R.--. 
 Bend 90 \ 
 
 . A3 ^14-8"- 
 
 i"°Pod ;-io''lg. Mark A, 
 I"' •• i2'-io" ■■ A, 
 
 r 
 
 Steel Bending uist 
 
 Concrete i i;-4fnjx 8.1 cuyd per lOft lengTti of tvalf. 
 Steel- 770 Jb per /Oft. length of wall. Deformed oars. 
 Use /Vo. /6 yVire f Steel Wire Gauge J for ty/ng. 
 
 To splice /ong/tudinal reinforcement overlap 2 ft and wrap with wire 
 
 CANTILEVER TYPE 
 
 RETAINING WALL 
 
 CORRECT 
 «PPROVED 
 
 Plate II 
 
 r. 
 
 A3- 
 
 t: 
 
 'Rods 
 
 3d 
 
 Tdds 
 
 ■Iilihi 
 
 — lO-O 
 
 Rear Elevation 
 f Footing rods not shown/ 
 
 Concrete ■/•■ ■ f mix /6 4 cuyd. per 10 ft of yrall 
 Steel: 1290 la per 10 ft. of wall Deformed bars ' 
 Use No. 16 Wre f Steel Wire Gauge.] for tying 
 
 To splice longitudinal reinforcement overlap 2' and wrap with wire 
 
 12 R- 
 
 < 8-0 ' >l 
 
 "'/^od 11-2 "fg. Foofing 
 
 7^1" 
 
 I2'-I"...- 
 
 n^l"- 
 
 I " " Rod 8 - 8 "Ig. Mark Ai 
 3"' " IJ-8" " " 
 ■l'^ " l8'-e" " " A3 
 
 Steel Bending List 
 
 L- SHAPED 
 
 RETAINING WALL 
 
 SCALE DATE 
 DRAWN* BV TRACCD ev 
 COSRtCT 
 
592 
 
 CONCRETE ENGINEERS' HANDBOOK 
 
 [Sec. IZ-M 
 
 P2 has the distribution shown by area C, varying from /i at S to /2 at R. Vertical components 
 // and fz of /i and /2 respectively, designate the pressure area D. The pressure of the weight 
 of earth above RS is not shown. Reinforcing for moment requires rods near the upper surface. 
 Shear and bond should be provided for. 
 
 Sliding on the base may be figured from Rv and the coefficient of friction of the base ma- 
 terial on the soil (see Table 4, page 582). If /-P causes more tendency to slide than resisted by 
 friction, small projecting key walls should be cast on the under side of the base and anchored 
 to it by dowel rods (see page 581). 
 
 Fig. 17. Fig. 18. 
 
 Diagram 10 7d. Expansion Joints. — Expansion 
 
 joints should divide the structure into sections to 
 relieve continuity and to relieve temperature and 
 shrinkage stresses. These latter stresses should be 
 cared for by the addition of about 0.3 of 1% of 
 steel placed horizontally in the stem, to confine 
 cracking of any extent to the joints. The joints 
 should be lock-joints, and in general should be 
 waterproofed to prevent unsightly stains. 
 
 Typical designs are shown in Plates I and II. 
 8. Design of Counterforted Walls. — For walls 
 above about 20 ft. in height, the cantilever type of 
 retaining wall requires a wall of great thickness to 
 be self-supporting. A great saving is effected in the 
 amount of material used for high walls by placing 
 ribs, or counterforts, at intervals on the back side of 
 the wall, tying it to the back of the base slab (see 
 Fig. 18). The vertical wall is therefore a slab con- 
 tinuous over the counterforts, and loaded by hori- 
 zontal loads, thus saving much material in the wall 
 itself. 
 
 The width b of the base and height of the 
 wall may be found as though the wall were of 
 the cantilever type, by using Diagrams 1 and 8. 
 Minimum material is required when the vertical 
 
 wall is from 2 3 from the toe, as was shown for 
 
 cantilever walls in Diagram 7. 
 8a. Thickness and Spacing of Counterforts. — The counterforts should have a 
 thickness equal to one-twentieth of the height of the wall, and perferably never less than 12 
 in. Their spacing should be such as to give the minimum amount of material required for the 
 wall. This spacing s, measured in feet fropa center to center, should be 
 
 Spacing in feet (s) c.io c. 
 of counterforts 
 
 = 2.46 + 0.216A 
 
 (ft. 
 
Sec. 13-86] 
 
 kETAlNING WALLS^ 
 
 593 
 
 and should not be less than 7 ft. for walls under 20 ft. in height. The spacing may be ob- 
 tained directly from Diagram 10. 
 
 The reinforcing of the counterfort will be discussed in an illustrative problem. 
 
 86. Vertical or Face Wall. — The face wall, as noted above, is a slab supported 
 at its junction with the counterforts and the base slab. Its thickness is commonly computed 
 by considering it to be made up of horizontally loaded continuous beams, as AB, Fig. 19. 
 At any depth the load per square foot would be hx - w', and the 
 
 From this the thickness at that 
 
 moment at the center would be hxw' 
 
 10 
 
 depth could be computed. The maximum thickness would be required 
 in a strip at the bottom of the vertical wall, when hx would equal h', 
 if we neglect any restraint offered by the base slab, 
 
 I2fw'h's\ 
 
 Mr 
 
 10 
 
 Then 
 = Khd'' 
 
 Since 6 = 12 in. for convenience. 
 
 0.32s 
 
 I fw'h 
 \ K 
 
 (in.) 
 
 tvhich is the effective depth of the slab at the bottom. The total thickness is found by adding 
 protective covering for the steel. The thickness of the wall varies from this value at the 
 bottom to a thickness not less than 3-^o^ at the top. The equation given above is the basis for 
 Diagram 11. 
 
 If it is desired to design for ""j^^ , use 0.9 of the effective depth given in Diagram 11. 
 
 Diagram 11 
 
 Illustrativk Problem. — Given f ^ 2, w' = 50, h' = 36 ft. To find thickness of face wall. Counterforts 
 are spaced 7 ft. 6 in. c. to c. Use fs = 15,000, fc = 650, whence K = 110. Enter Diagram 11 with {fw'h') = 3600, 
 thence to right to K = 110; thence vertically to s = 7.5 ft. ; thence to right where d is given as 13.7 in. Total thick- 
 ness, say, 15 in. 
 
 Sc. Back Floor Slab. — This is the portion of "the base slab to the rear of the 
 vertical wall. It must be designed to resist the upward reaction of the soil against that por- 
 tion of the base; and the weight of the earth immediately above it, together with the vertical 
 38 
 
594 
 
 CONCRETE ENGINEERS' HANDBOOK 
 
 [Sec. 13-8c 
 
 component of the inclined earth thrust if the surcharge is sloped. Fig. 20 shows a counter- 
 forted wall with the earth sloped to the line AD. The total pressure on the plane DC is P, 
 whose effect above the base slab is represented by the triangle of pressure Djh. The portion 
 Ddg (with resultant Pi) is transferred to Ah, as Abe. The portion dghj (resultant P2) is trans- 
 ferred to hj as hcjh. Vertical components of he and jh, at h and j, would determine the down- 
 ward component pressure of Po. This is the same as was given for the cantilever wall, Fig. 17. 
 
 Fig. 20. Fig. 21. 
 
 The various pressures acting are shown in Fig. 21. Trapezoid abhg represents the vertical 
 component aedh of P2, plus the pressure eghd of the earth immediately above gh (ADjb, Fig. 
 20). The area jmn represents the upward pressure of the soil. 
 
 When the surcharge is level, ahdc = 0. 
 
 The whole floor slab may now be designed for either shear at the counterforts, or moments 
 in the slab treated as continuous over the counterforts. In either case the pressure on a strip 
 at the back edge hm would determine the depth of the back slab. If the slab is not reinforced 
 
 Diagram 12 
 
 adequately against shear and diagonal tension, by the addition of bent-up bars or stirrups, 
 then the allowable shearing unit stress will be low, and shear will govern the depth of the slab. 
 In this case, the effective depth of slab in inches is given by the relation . 
 
 sh" 
 
 d = {w + fw' sin d) 
 
 in which h" is the depth in feet oi the earth above the slab at the heel. Diagram 12 is based 
 on this formula. 
 
Sec. 13-8c] 
 
 RETAINING WALLS 
 
 595 
 
 Illustrative Problem. — Given: Stem of wall 20 ft. high, surcharge slope, 6 = 30 deg. ; angle of internal 
 friction, <j> = 40;/ = 1.75; length of base of counterfort = 7.5 ft.; spacing of counterforts, center to center = 7 
 ft.; weight of earth, w => 120 lb. per cu. ft. Find: depth of back floor slab when v = 3Q lb. per sq in. 
 
 = 20 + (7.5) (tan 30 deg.) = 24.4 ft. 
 From Diagram 1, Ce => 0.316; hence = CgW = 38 lb. per cu. ft. 
 
 iw + fw'sin e) = 120 + (1.75) (38) (0.5) = 153.3 lb. per cu. ft. 
 
 h"{w -\- fw' sin0) =» (24.4) (153.3) = 3745 lb. per sq. ft. This is the downward pressure per square foot 
 at the top of the back edge of the slab. Entering Diagram 12 from the left margin with this value, we move toward 
 the right to v = 30; then vertically to s = 7; then out to the right margin, where we find d = 41.9 in., say 44 in. 
 total depth, allowing proper covering for the stefel at the lower face. This is the total thickness required for shearing 
 forces. Investigation must of course be made of this for unit stresses in moment and bond. 
 
 Diagram 13 
 
 When stirrups or bent-up bars are placed in the slab to adequately resist the shearing 
 forces, then the bending moments will govern the thickness of the slab, for walls up to at least 
 40 ft. high, since the allowable shearing unit stress will be high. Under these conditions, since 
 the base slab, like the vertical wall, is continuous over supports, 
 
 Khd'' = 1.2{w +fw' sin d)s^ 
 
 Whence 
 
 d = 0.32s^J ^"'+-^^'^ 
 
 in whieh d is the effective depth in inches. The effective depth may be taken directly from 
 Diagram 13, which is based upon the above formula. The use of the diagram is essentially the 
 same as the use of Diagram 11. . 
 
 If it is desired to design for a moment of {w + fw' sin ^)y2' ^-^ effective depth 
 
 given in Diagram 13. 
 
 Illustrative Problem. — Given same wall as above, but with v = 120 lb. per sq. in. In this case the thick- 
 ness will be governed by the moment in the slab. Let K = 95. Entering Diagram 13 with (w + fw' sin 6) = 
 153.3, proceed to the right to K = 95; then vertically to s = 7_ft.; then to the right margin where d = 9 in. Use 
 11 in. total thickness. 
 
 Some designers favor placing a beam along the back edge of the back floor slab, to stiffen 
 it. The design of this beam is arbitrary when a panel of the back floor is either square, or 
 as is usually the case, has its shortest span between counterforts. 
 
596 
 
 CONCRETE ENGINEERS' HANDBOOK 
 
 [Sec. 13-8(/ 
 
 The methods of reinforcing the back slab are given in Art. 8e with a discussion of the rein- 
 forcement of the base slab as a whole, 
 
 M. Cantilever Toe Slab. — When the resultant pressure R strikes the base at 
 the forward third point, the upward pressure on the base is triangular, and the pressure per 
 square foot at the toe A, Fig. 22, is equal to twice the vertical com- 
 ponent of R divided by the area of the base over which R acts, ex- 
 pressed in square feet. Thus the total shear on the section CD, neg- 
 lecting the small downward weight of AD, is equal to Rv{2 — k)k, 
 for a length of 1 ft. of wall (pierpendicular to the paper). From Art. 
 
 13, Sect. 7, V = whence, assuming ji" = 
 
 hjd 
 
 Fig. 22. 
 
 0.095 
 
 R,\2 - k)k 
 
 in which d is the effective depth of the toe slab in inches, at the section CD. It may be tapered 
 toward A. Diagram 14 has been prepared to give this depth from the above formula. 
 
 iLLnsTRATivE Problem. — Given i2„ = 18,0001b.; A; = 0.3; v = 35 lb. persq. in. Find d. Enter Diagram 
 14 at the left with Ry = 18,000; then on a horizontal line to k = 0.3; then vertically to v = 35; and lastly, to the 
 right margin where d = 25 in. This must be compared to the value obtained from the moment requirement. 
 
 Diagram 14 
 
 8e. Methods of Reinforcing Counterforted Wall. — In the preceding discussion 
 only the critical sections of the main units of the wall have been dealt with. It should be 
 noted that those same sections should be investigated for diagonal tension and for bond, as in 
 the design of continuous slabs. Important considerations in the arrangement of the rein- 
 forcement will be discussed for each part of the wall. 
 
 Vertical Slab. — The reinforcing bars in the vertical wall should be at the outer face at the 
 centei of the span, and at the back face at the counterforts. All of the reinforcement needed 
 
Sec. 13-8e] 
 
 RETAINING WALLS 
 
 597 
 
 at the front in the center of the span is needed at the back face at the counterfort, since the 
 moment at the support should be taken as that at the center. It is not advisable, however, 
 to bend all of the rods to the back face at the supports, but to have about one-fourth of them 
 run through. This amount of steel should be supplied at the back face by short rods running 
 about }i s each way from the counterfort, or further if necessary for bond. 
 
 About 0.3 of 1% of steel should be placed vertically in the wall, and continued into the 
 base slab. This serves two purposes: it provides a support for the horizontal steel before pour- 
 ing; aild it resists the tendency to form large horizontal cracks, particularly near the bottom 
 of the face wall. 
 
 Base Slab. — The toe, after having been proportioned for shearing forces, must be reinforced 
 for bending moment at the section CD, Fig. 22. Rods should be placed near the bottom face 
 of the toe and anchored at its outer end. Part of these rods should extend the entire width 
 of the base, near its lower face; and part of them may well be bent up, with a long radius, to 
 the back face of the vertical wall, extending upward fromt he base a distance of about ^ s, 
 or more. About 0.3 of 1 % of steel should run parallel to the face wall, near the bottom of the toe. 
 
 The principal reinforcement at the back slab runs parallel to the face wall. The rods 
 at the center of the span s are at the bottom of the slab, and at the supports (counterforts) near 
 the top. The points of bending the rods should be between the quarter and third points of the 
 span, as determined from Fig. 26, Sect. 7, page 298. Not all of the rods should be bent up at 
 these points, but some should run straight through near the bottom face. The extra amount 
 required at the top face at the support may be made up by rods placed near the top face and 
 extending 3<3 s either way from the counterfort, or further if necessary for moment or bond. 
 Stirrups, if placed near the counterforts, should be placed in accordance with the shear 
 reinforcement in continuous beams or slabs. 
 
 When the vertical wall is in front of the center of the base, longitudinal steel is likely to 
 be required in the bottom face of the back slab between the center of the base and the vertical 
 wall, as in this portion the upward pressures may be greater than the downward pressures. 
 This is especially notable when the face wall is at the front edge of the base slab (see Plate IV). 
 
 The Counterfort. — The reinforcement of the counterfort must take the tension developed 
 in the direction AC, Fig. 23; the tension across BC due to the overturning 
 moment of the stem AB; the tension across AB due to the tendency of the 
 face slab to move forward away from the counterforts; the shear on BC due to 
 the earth thrust on AB; and the shear on AB due to the resultant downward 
 pressure on BC. 
 
 The counterfort is designed as a wedge-shaped cantilever beam fixed 
 to the base slab. Such an assumption requires that the base slab be rigidly 
 attached to the bottom of the counterfort over its entire length. The principal 
 reinforcing in the counterforts is the steel along the inclined face tying the 
 upper end of the face wall to the back edge of the base. Both ends of this steel must be 
 thoroughly anchored. This may be done by hooking, or by hooking to cross rods, provided 
 that in the latter case the bearing of the cross rods does not exceed the compressive work- 
 ing unit stress of the concrete. 
 
 Illustrative Problem. — Given a counterfort 20 ft. high, 16 in. thick, and 8 ft. wide at the base slab; required 
 the tensile steel necessary to prevent exceeding the unit stresses fc = 650, and fs = 16,000 lb. per sq. in. Moment 
 on the counterfort = 7,000,000 in.-lb. Allowing 0.2 ft. for steel covering, 
 
 M 7,000,000 
 
 hd"^ (16)(7.8)2(12)2 
 
 Entering Diagram 2, Sect. 7, page 360, at 50, for/s = 16,000, p = 0.0035. Slope of back face = tan -10.4 = about 
 22 deg. Entering Diagram 3, page 361, at the lower margin with 3.5, trace vertically upward to = 0, then 
 horizontally to /3t = 22 deg., then vertically upward to values of C, where 3.85 is found. Pointing this off 
 properly, the corrected value of p = 0.00385. 
 
 As = (16) (7.8)2 (12)2 (0.00385) = 5.4 sq. in. 
 
;. 13-8e] 
 
 RETAINING WALLS 
 
 599 
 
600 
 
 CONCRETE ENGINEERS' HANDBOOK 
 
 [Sec. 13-9 
 
 It will be found by considering various horizontal sections of the counterfort that it will 
 not be necessary to run all of this steel to the full height of the counterfort. Wherever any 
 steel is stopped because it is no longer needed for moment, it should be extended sufficiently to 
 develop the full strength by bonding, or should be adequately anchored. 
 
 Horizontal steel should extend from the counterfort into the front face to keep the latter 
 from being torn fron the counterfort. This steel should be hooked to the horizontal rods in the 
 slab, or should be bent either way into the slab to thoroughly attach it to the vertical slab. 
 
 Similar steel should be extended and anchored into the footing slab. 
 
 If the cross-section of the base of the counterfort is not large enough to carry in shear the 
 thrust against the wall, the thickness of the counterfort should be increased, or stirrups should 
 be extended from the base slab into the counterfort. 
 
 Typical designs are shown in Plates III and IV. 
 
 9. Special Types of Reinforced-concrete Walls. — Fig. 24 shows the cellular type of re- 
 inforced-concrete retaining wall. This wall is formed of two longitudinal curtain walls (A) 
 and (B), connected by transverse walls (C). The vertical space between the parallel longitudi- 
 
 Center Line of Side Track 
 
 r/3' To Center Line 
 cf Main Track 
 
 Fig. 24. 
 
 nal walls and the transverse walls is filled with earth. This type of wall gives a lower maxi- 
 mum soil pressure than either the cantilever or counterforted types, but under average condi- 
 tions its cost per linear foot is greater. Its use would seem to be restricted to poor soil with 
 no opportunity to drive piles and where the wall must be built as close as possible to a given 
 property line — conditions which often occur in railway work. 
 
 Fig. 25 shows a design of wall which gives a maximum soil pressure even less than the 
 cellular type and which costs approximately the same per linear foot of wall. This design 
 was employed by the C. M. & St. P. Ry. in a long retaining wall at Elgin, 111., and was chosen 
 in preference to the design shown in Fig. 24 on account of the much lower bearing pressure on 
 the soil. The wall consists of a footing (F) which supports a longitudinal wall (T) and the 
 cross walls (t/). The cross walls at the outer end support the girder (X), which spans from one 
 cross wall to another. The slabs (F) are supported at one end by the girder and at the other 
 end by the longitudinal wall. The great reduction in soil pressure is brought about mainly 
 by eliminating the weight of the earth directly above the footings, the space between the 
 cross walls (C7) being an open and empty space in this design. In this type of wall, and in the 
 cellular type as well, stability against sliding should be carefully investigated. 
 
Sec. 13-10] 
 
 RETAINING WALLS 
 
 601 
 
 10. Construction of Retaining Walls. 
 
 10a. Backfilling and Drainage. — Quite as much attention should be paid to the 
 earth filhng and to its drainage as to the design and construction of the retaining wall proper 
 or to the matter of providing a suitable foundation. If the earth is deposited in layers inclined 
 from the wall, the pressure will be small compared to that resulting if the layers are sloped 
 toward the wall. It is quite often the case that by depositing the backfilling so as not to 
 
 Center Line of S/cfe^ Track 
 K— - 11-6" 
 
 i3' to Center Line 
 ' of Main Track 
 
 Cross Seci-ion 
 
 Sectional Elevation 
 
 Fig. 25. 
 
 These fouf boards forming a 
 panel to be set at wor/ang p/ace 
 only to prevent spilling of concrete 
 
 Runwciy /or loacfect wheel barroiys 
 
 / Place boards -for crossover 
 [opposite kYorking place 
 \ ....--Punway "for empty 
 
 wheelbarroifS 
 
 Bracing of. Wall Forms .2-6 "C. to C 
 ^IG. 26 
 
 slide against the wall, a light wall may be made to stand where, under the same conditions, if 
 the earth is dumped so as to slide against the wall, even a heavy wall will fail. 
 
 When placing a backfill near a steep undisturbed slope of earth or rock, care should be 
 taken that the earth does not arch, or does not form a wedge. If the proper precautions are 
 not taken, a heavy lateral thrust will occur near the top of wall. 
 
 Water behind a wall is a frequent cause of failure. It adds to the weight of the backing 
 and softens the material so that the lateral thrust is increased. Also, undrained backfilling 
 will freeze and create lateral thrust due to the consequent expansion. To drain the backing, 
 weepers or weep holes should be left through the wall just above the footing. Tile, 3 or 4 in. in 
 
602 
 
 CONCRETE ENGINEERS' HANDBOOK 
 
 [Sec. 13-106 
 
 diameter is generally used and, in the North Central States, placed usually not more than 10 
 to 15 ft. apart. The tile should be connected with a longitudinal drain in front of the wall. 
 If the backing is retentive of water, a vertical layer of broken stone, coarse gravel, or cinders 
 should be placed next to the wall to act as a drain. The filling in front of the wall should also 
 be carefully drained. 
 
 106. Forms. — Formwork as applied to buildings is treated in detail in Sect. 2. 
 Since the method of constructing forms and the directions for their removal are very much the 
 same for different types of structures, very little need be said under this heading. 
 
 The lumber used for forms should have a nominal thickness of at least l}i in. before 
 surfacing and should be of a good quality of Douglas fir or Southern long-leaf yellow pine. 
 
 Fig. 27. 
 
 The lumber for face work should be dressed on one side and on both edges to a uniform thick- 
 ness and width. The lumber for backing and other rough work may be unsurfaced and of 
 an inferior grade of the kinds above mentioned. 
 
 Forms should be substantial and unyielding, and built so that the concrete will conform 
 to the dimensions shown on the designer's plans, and they should also be tight so as to prevent 
 the leakage of mortar. Forms may be either continuous or sectional, or a combination of 
 both, depending upon the economy of the work. The concrete in any given section should be 
 allowed to harden for 36 hr. before the forms are removed and, in freezing weather, extra care 
 must be taken to make sure that the concrete has had sufficient time to become thoroughly 
 set. Material once used for forms should be cleaned before being used again. 
 
 A design of a form for a reinforced-concrete cantilever wall is shown in Fig. 26. 
 
 Fig. 27 shows the method of constructing and reinforcing the counterforts of a retaining 
 wall at Buffalo, N. Y. 
 
SECTION 14 
 
 SLAB AND GIRDER BRIDGES 
 
 The methods of designing slabs, beams, and girders are explained at length in Sect. 7, and no attempt is 
 made in this section to treat of these methods in detail. Many things must be considered, however, in designing 
 bridges of this class, aside from the proportioning of simple and continuous beams, and such matters are given due 
 consideration. 
 
 The erection of forms and other operations in slab-and-girder bridge construction are essentially the same for 
 ordinary conditions as the corresponding operations in the construction of buildings. On this account, construc- 
 tional methods are referred to only incidentally under this heading. 
 
 The loadings to use in design are, of course, the same as for the floors in arch bridges of open-spandrel con- 
 struction (see Art. 6, Sect. 16). In fact, it should be noted that the framed structure which is supported by a 
 ribbed arch is virtually a trestle form of girder bridge and the loadings and general design are identical. 
 
 Impact may properly be neglected in arch-ring analysis but becomes important in bridge-floor design. An 
 increase of 25 % is usually made in the live load or live-load stresses for highway bridges and 50 % in those for rail- 
 road bridges. 
 
 From the standpoint of economy, slab bridges should in general be limited in span length to about 25 ft., and 
 ordinary girder bridges to about 50 ft. 
 
 SLAB BRIDGES 
 
 1. Slabs Under Concentrated Loading. 
 
 la. Illinois Tests. — From tests made on simply supported slabs at the Uni- 
 versity of Illinois, Prof. W. A. Slater has recommended that where the total width of slab is 
 greater than twice the span, the effective width e (Fig. 1) be assumed as 
 
 e = Hx + d 
 
 where x is the distance from the concentrated load to the nearest support and d is the width 
 at right angles to the span over which the load is applied. 
 
 0:0 0 
 
 1^ 
 
 Fig. 1. 
 
 O.Z 04 0.6 08 
 Ratio of total width to span 
 
 Fig. 2. 
 
 Tests showed the effective width to be but little influenced by the depth of the slab or by 
 the percentage of longitudinal reinforcement. Prof. Slater has recommended, however, that 
 the latter be limited to 1% "because of the possibility that in a beam with a large amount 
 of longitudinal reinforcement and a relatively small depth, failure may be caused by trans- 
 verse tension in the concrete and not by longitudinal steel stress." 
 
 The above formula refers to a total width of slab greater than twice the span. For a 
 slab whose total width is less than this, the effective width may be found from Fig. 2 (full line) 
 which shows the ratio of the effective width of the span as determined from the measured 
 steel stresses in the University of Illinois tests. 
 
 603 
 
604 
 
 CONCRETE ENGINEERS' HANDBOOK 
 
 [Sec. 14-16 
 
 16. Ohio Tests.^ — The object of the tests was to obtain, if possible, a sufficient 
 knowledge of the distribution of loads through and by concrete floor slabs to enable the de- 
 signer to rationally proportion the joists of a slab floor, and also the slab itself, to carry con- 
 centrated loads. 
 
 The following conclusions regarding the distribution of concentrated loads on a rein- 
 forced concrete slab, to the floor joists, seem to be warranted by these tests: 
 
 1. The percentage of reinforcement has little or no effect upon the load distribution to the joists, so long as 
 safe loads on the slab are not exceeded. 
 
 2. The amount of load distributed by the slab to other joists than the one immediately under the load, 
 increases with the thickness of the slab. 
 
 3. The outside joists should be designed for the same total live load as the intermediate joists. 
 
 4. The axle load of a truck may be considered as distributed uniformly over 12 ft. in width of roadway. 
 
 Fig. 3. — Standard design for slab bridges of 12-ft. span, Wisconsin Highway Commission. 
 
 In a slab of a certain span and indefinite width, there is some width symmetrical with the 
 load, beyond which a single concentrated load will have no effect. The stresses in this slab 
 will be a maximum under the load and will decrease in each direction from it. 
 
 The "effective width" of a slab is that width used in designing over which a single con- 
 centrated load may be considered as uniformly distributed on a line down the middle of the 
 slab parallel to the supports. 
 
 The tests of slabs seem to warrant the following conclusions: 
 
 1. The effective width" is affected very little by the percentage of transverse reinforcement (parallel to 
 supports). 
 
 2. The " effective width" decreases somewhat as the load increases. 
 
 3. The "effective width" in percentage of the span, decreases as the span increases. 
 
 4. The following formula will give a safe value of "effective width" where the total width of slab is greater 
 than IW + 4: ft. 
 
 e = 0.61 + 1.7 ft. 
 where e is effective width in feet and / is the span in feet. 
 
 1 From Bull. 28, State of Ohio, Highway Department. 
 
Sec. 14r-lc] 
 
 SLAB AND GIRDER BRIDGES 
 
 605 
 
 Ic. Tests by Goldbeck. — The results of these tests are shown by the dotted 
 
 curve in Fig. 2, 
 
 2. Slab Bridges of Single Span. — The floor of a slab bridge may be designed as any simply 
 supported slab except that shearing stresses may require very careful attention where the live 
 
 „ , / I'^Tyy. Ifods 'f'c. foe. SO ' lonq 
 
 Roadway <^ ,„aj.^ „ ;2'c./toc iw^-^'^. f fnr /Ms-za'cnc.-e' /o> 
 
 5fCfetya//r 
 
 ^""Tw. " rz'c.foc Bo'-e'lq. U' 
 ( fnr. » 3'c.-fDC. /9'-6'^lq . ) I 
 
 o'-o"- >)<- 
 
 '°Tw. /fodsii'c foe. 19^ 
 
 'long 
 ^'-€'13. 
 
 i Tw. \z''ctoc 5io''/ong- 
 Transverse Secrion on Centisr Lin^of Stream ^/^>-- 
 
 3 
 
 // - £" 
 
 Plan of Abutmenl- 
 
 C L of Bridge^ 
 
 Fig. 4. — Details of Christiana bridge, Town of Cross Plains, Dane County, Wis. 
 
 load is relatively large, as in railroad structures. The whole reinforcing system may be made 
 absolutely rigid by wiring the main reinforcing rods to the transverse spacing rods at the ends 
 
 rmished Poa dway 
 
 ^ l" Ifods izcfoe. 
 
 ffelnforcement 
 Elevation 
 
 ■f°'ffods E"c.ioc. Wing 
 
 "°ffocls 
 l£"cfoc. 
 
 Plan of 
 One Abutment 
 
 
 
 sz'-o' 
 
 rrUe Drains, S'ctoC. 
 ■■.-^"Cyrfods Bi'-8"long 
 
 ^~\---i'"'ffods.iS'l^ 
 
 Section Exterior 
 Elevation 
 
 [""Rods 
 is'c.toc. S^'l 
 
 Cross Section 
 
 Fig. 5. — Typical slab bridge, Illinois Highway Commission. Abutment wings of the cantilever type. Main wall 
 vertical slab supported at top by superstructure and at base by footing. 
 
 of the bent-up steel. Both the straight longitudinal rods and those bent up to provide for 
 diagonal tension should be hooked at the ends. 
 
 Practice varies in regard to the use of expansion joints between the slab floor and the 
 abutments. There are none provided in Figs. 3, 4 and 5, but such joints are placed at both 
 
606 
 
 CONCRETE ENGINEERS' HANDBOOK 
 
 [See. 14-2 
 
 abutments in Fig. 6. In the latter figure, vertical end expansion joints are provided at the 
 angle points between abutment wings and slab, making it necessary to cantilever the outer 
 portions of the slab width on account of insufficient abutment support. 
 
 In Figs. 3, 4 and 5 the main wall or vertical slab of the abutment is supported at the top 
 by the floor and at the bottom by the footing, and may be figured for earth pressure as a simple 
 slab with the main steel near the outer or stream face. Of course, the main wall should also 
 be designed to act as a column to support the superstructure. Since a joint of low friction is 
 
 Half Plan View 
 
 A^>7^p.■ 
 
 So proportion the wing 
 length that A: B •■C:D as l-i^- 
 The point £ yyhere the /y I slope 
 from end of wing intersects the 
 elevation of stream bed may be 
 varied to suit local conditions. 
 
 In streams whose channels 
 are nearly the span width, or 
 where there is a possibility of 
 cutting and channel widening, the 
 point E Should be tairen at abutment face 
 
 <--- 18 Clear Roadway > 
 
 > 4" Drains spaced not more 
 \ /than IS 'c. toe. Perforaivd 
 Cover 
 
 Expansion Joints of tar paper 
 or galvanized iron 
 
 Z4'-0'l - 
 
 Cross Sec-Mon of Slab 
 
 Fig. G.- 
 
 -Typical slab superstructure, Iowa Highway Commission. 
 
 not provided between floor and abutments, the unit tensile stress in the steel of the super- 
 structure is usually kept low (12,000 lb. per sq. in.) so as to provide properly for the additional 
 tension in the steel caused by the contraction of the bridge in cold weather. The wings may 
 be designed as self-supporting retaining walls of the cantilever type, using the methods ex- 
 plained in Sect. 13. Theoretically the maximum efficiency of the footing for the wing walls 
 can be obtained by placing the wing wall at about the outer middle-third point of the base, 
 but in many cases considerable saving in excavation may make it more desirable to shift the 
 footing a little toward the stream bed. Counterforted walls are advisable only for abutments 
 over 20 ft. in height, 
 
Sec. 14-3] 
 
 SLAB AND GIRDER BRIDGES 
 
 607 
 
 Figs. 6 and 7 show an unusual abutment design adopted by the engineers of the Iowa 
 Highway Commission for both slab and girder bridges. Expansion joints being provided at 
 each end of the superstructure, both the main portion and the wings were designed as self- 
 supporting retaining walls. The main portion, however, was not only analyzed in the ordinary 
 manner for pressure on the base, but was also analyzed taking into account the stability due 
 to the weight of the wings. The mean value of the maximum pressure at the toe of the foun- 
 dation by the two methods being found safe, and an analysis for sliding and overturning being 
 satisfactory, the dimensions shown were adopted. The horizontal rods designated as tension 
 bars were inserted in order to utilize the weight above mentioned. These rods are placed 
 in the main stem near the upper surface and extend continuously through the wings, with 
 splices at the center of the main portion. The effect of considering the wings as a part of the 
 
 Fig. 7. — Typical substructure for slab and girder bridges, Iowa Highway Commission. 
 
 abutment body is to shift the center of gravity of the entire mass farther from the stream face 
 and thus reduce the eccentricity of pressure on the foundation. The horizontal reinforcing 
 rods shown near the stream face of the abutment, and which are carried about 5 ft. into the 
 wings, were employed to counteract a tendency to the formation of vertical cracks on the 
 outside at the corner of wing and abutment. The vertical rods in the front face serve as a 
 framework upon which to build the horizontal rods and they also prevent any tendency toward 
 the formation of horizontal cracks in the stream face due to the clogging of an expansion joint. 
 
 3. Slab Bridges of Multiple Spans. — Slab bridges of multiple spans will be treated under 
 the four following headings : 
 
 Concrete pile trestles. Trestles with framed bents. 
 
 Pier trestles. Cantilever flat-slab construction. 
 
 3a. Concrete Pile Trestles. — Figs. 8 and 9 give the essential details of design 
 of the pile trestles built by the Illinois Central Railroad. They can be considered typical of 
 concrete pile trestles in general. These trestles replace similar wooden structures over swamps 
 
CONCRETE ENGINEERS' HANDBOOK 
 
 [Sec. 14r-Sa 
 
 I I 
 
 P'r; n-n n^n r;T; ht; r.-n 
 
 Bars 4-0 long f| 
 /'-J /ong 
 
 ~>|/2'|<- • 
 
 I J'lBors /j'-6'/oncf :. | ^ I | 
 
 SecTion on 
 Cen-ter Line of Track 
 
 10-^' Bars, B-e'/ory 
 
 Concrete Piles 
 
 ■A'^ 3-8 3-8" 3-8 "Xlk 
 
 7- ^ "Bars, 13-6" lonj ■ 
 -H^ ■ " " -211 Bars, 13'- 6" long 
 
 ■4'-3"-A 
 
 /"Bars 
 
 Section on 
 Center Line 
 of Track 
 
 Anchor Pier 
 
 Fig. 8. — Details of substructure, standard concrete pile trestle, Illinois Central R. R. 
 
Sec. 14-3a] 
 
 SLAB AND GIRDER BRIDGES 
 
 609 
 
 and shallow streams which may not be filled and where bridges on more permanent supports 
 would be extremely expensive because of their great length. The construction consists of pile 
 bents spaced generally from 16 to 20 ft. c. to c, and with a height above ground not greater 
 than the span. The piles are capped with reinforced-concrete girders which support the 
 floor slabs. 
 
 The piles and deck slabs are usually cast in a convenient yard, allowed to season from 60 
 to 90 days, and are then hauled to the bridge site. The lifting stirrups shown permit of the 
 slabs being set in place by a wrecking crane. The ballast and track are laid directly on the 
 slabs after the longitudinal and transverse joints (except at anchor bents) are filled with cement 
 
 
 
 D 
 
 
 □ 
 
 
 
 1 Mu<y L ine 
 
 
 
 rr 
 
 49 
 
 Cross Section 
 This concrete poured 
 
 after pre cast slabs 
 are in place 
 
 Expansion 
 Joint 
 
 \ 2 1, Extending slab stei 
 /feinforcement \ S a / to be greased 
 in slab ' ' - ^/^ 
 
 End of fence slab iv be 
 greased to present bonding 
 to concrete in post 
 Fence Post 
 
 Section through Grout surface 
 Typical Bent .--before placing slab 
 
 /"Sand 
 Joint 
 
 I -Iron 
 
 1". 9" Plate 
 
 "--4 >![<-;■• ,-J''''5p//fe's 
 
 -n£^--Post 
 Expansion Joint 
 at Railing 
 
 Expansion Joint 
 in Roadway 
 
 Fic 
 
 (Bent Cap 50) 
 
 10. — Details of pile trestle across the Miles River near Easton, Md. 
 
 mortar and after the floor surface is thoroughly waterproofed. The slabs are set on a bed of 
 grout on the pile caps. An anchor bent is used at suitable intervals to take up longitudinal 
 stresses due to tractive force and, by means of an expansion joint, to prevent any great accumu- 
 lation of movement of the deck due to temperature changes. 
 
 A concrete pile trestle for carrying a highway is shown in Fig. 10 ^ It was found economi- 
 cal to cast the piles, deck slabs, and railing slabs at Baltimore, 60 miles away, and transport 
 them to the site on scows. Expansion joints were located in the roadway slabs, curb, and 
 railing slabs at every fifth bent. 
 
 1 See also Eng. News, Feb. 5, 1914. 
 39 
 
 / 
 
610 
 
 CONCRETE ENGINEERS' HANDBOOK 
 
 [Sec. 14-36 
 
 36. Pier Trestles. — Thin concrete piers are preferable to pile bents when the 
 height of bridge above the ground line is greater than about 16 ft. Fig. 11 shows a typical 
 trestle of the solid bench-wall type built by the Illinois Central Railroad. 
 
 7 6 SB 4 3 e I ^^■-^.^ 
 
 ^ Base erf Rail ^ _^ 
 
 I S'hnj pij 
 
 Top of Pier 5B 
 li' Base of Bai l 
 
 fPgds, i'-9"!z"c foe. 
 
 Top of Pier 
 
 !<.. s'.o':i^'^^^ V j7'-o" ■■{■:■ 
 
 fffods. W'c. to c f/fods. e'c fo C. 
 
 Piers 2.3.4.6,& 7 
 
 Fig. 11. — Pier details, Illinois Central R. R. trestle over Kaskaskia River near New Athens, 111. 
 
 3c. Trestles with Framed Bents. — Slab bridges with framed bents forming 
 subways are used on at least fifteen railroads in this country. A design which may be considered 
 
 \<' - 'y'-z" - X - ij'-z" >j .. z' 
 
 ' % I ,° I i ! i T 
 
 ftAs'-z" 
 
 T 
 
 _LliLil: ! L|l! I ilili.i_l j .U III lij-L ij I ! ili I 
 
 Half Elevation- Curb Pier Half Elevation- Roadway Pier 
 
 U 6'-6" --J 
 
 Section B-B 
 
 Detail of Notch 
 for Curb Piers 
 
 Details for Sliding ^lev. /sse-.. 
 Plates , Center Pier Elevation, 
 
 C L. of Track 
 
 ■ h • — p^y Batter S " ' ' 
 
 ■^.r- 7'-^" >H e'-e" ^ 
 
 ' r /? Alternate rows 
 
 Batter S-IZ of st,rrvps ■ 
 
 Parapet Bars inverted- .... 
 
 11 
 
 
 
 
 A| 
 
 i— 
 
 
 
 Section A-A 
 
 Plan 
 
 Fig. 12. — Details of Mozart Street subway, Bloomingdale Road track elevation, Chicago, Milwaukee & St. Paul 
 
 Railway. 
 
 typical is shown in Fig. 12. The deck slabs may either be cast in place or cast at some central 
 yard and placed in a similar manner to the slabs for pile or pier trestles. In Fig. 12 the design 
 is shown for slabs to be cast in place.. 
 
Sec. 14-3c] 
 
 SLAB AND GIRDER BRIDGES 
 
 611 
 
 ^...S/idincf joints - 
 
 I "spaces between sfdbs 
 ■fWed with grout-. 
 
 Section Through 
 Sidewalk Girder 
 
 f'Bars./B c.toc. \ 
 y\ Alternate bars bent • 
 
 Section 
 through Slob 
 
 Inside Elevation ond Section 
 showing Reinforcement in 
 Sidewalk Girder and Bent 
 
 Inside Elevation 
 and Section of 
 Side Parapet a Bent ~i~ ''T' "^'-0'' 
 
 ^""Bars 5^6" long X 'W 
 
 J-L_. .i-LL XI. 
 
 Y ij' Bar. s'/ong Place bars in 
 ' ^ ■' position shown 
 
 before grouting 
 open space 
 
 J^'.Q' 
 
 Arrangement for 
 '^ Lifting Slab 
 
 Detail of End Slab 
 
 ! i"''Bars 
 
 7-^5 ^ -> 
 
 ■— , //'-//' 
 Half Longitudinal Section "I^'picaf Cross Section 
 
 oF Intermediate .Slab. for Tnd and Intermediate Slabs 
 Fig. 13. — Details of trestle apErroach to Cumberland River bridge, Illinois Central R. R. 
 
612 
 
 CONCRETE ENGINEERS' HANDBOOK 
 
 [Sec. 14-3d 
 
 A framed-bent trestle with continuous side girders to resist stresses due to traction is 
 shown in Fig. 13. The girder on one side acts simply as a tie and parapet, while the other with 
 a cantilever projection at the side acts also as a sidewalk. The slabs were cast in a yard at some 
 distance from the bridge site, loaded on flat cars, taken to the job, and swung into place with 
 derricks. Expansion is allowed for at both ends by providing a sliding joint between the 
 bridge superstructure and the abutments. The shallow 4-in. curbs at the ends of the slabs 
 were provided to prevent seepage from getting into the 1-in. grout joint between slabs. The 
 pier footings are reinforced longitudinally in top and bottom, and were figured as continuous T- 
 
 Section B-5 
 
 Fig. 14. — Cantilever flat-slab bridge on Mississippi River Boulevard, St. Paul, Minn. 
 
 beams uniformly loaded by the pressure on the soil. The cross girders at the top of the columns 
 support the deck slabs previously referred to, and are made continuous. 
 
 Sd. Cantilever Flat-slab Construction. — Fig. 14 is a flat-slab structure of the 
 Turner Mushroom type. The methods which may be used in the design of the roadway slab 
 are treated in Sect. 11. The hollow abutments should be noted. 
 
 Many cantilever flat-slab bridges are built in which the abutments are of the ordinary 
 reinforced-concrete type. The abutment walls are considered as held at the top by the super- 
 structure to which they are anchored by bending the vertical rods into the slab. 
 
 I 
 
Sec. 14-4] 
 
 SLAB AND GIRDER BRIDGES 
 
 613 
 
 SIMPLE GIRDER BRIDGES 
 
 4. Deck Girders. — The deck-girder type of construction usually proves more economical 
 than the through-girder type wherever sufficient headroom is available. The girders, of 
 course, should be relatively thin and deep for the greatest economy, and a curtain wall should 
 be provided between the girders at each end of span to retain the earth fill, thereby avoiding 
 complicated parapet walls on the abutments. 
 
 Standard details of deck-girder bridges designed by the engineers of the Iowa Highway 
 Commission are shown in Fig. 15. The floor slab was analyzed both as fixed and as continuous, 
 
 •i-9''^ -is"Spacinq •^^fS'^ Spacing ■><-Z4"$pacinq 
 
 K- Lonqifudinal girder tars ix> be \ I- 
 \y' secure/y hooked at ends over ^ \anchor bars 
 
 StirrdfS 
 
 \<I5> < 
 
 mil 
 
 £-;^"sVee/ P/ai^s ■ expansion Join f : "b ' < 
 a ..>l<-..b •->l<- C -d <^rrdereach ^/rde 
 
 Longitudinal Section through Girder 
 
 Handrail ( ''"^'Y/ J ''^ '''''' 
 ) Hor. S-f ears 
 
 ft. ■>\S [<■•/ 4" mepiro/es on each side of roadway 
 I I not more than /2 ff- apart. Perforated 
 
 ^^^"^ '^'^^ covers 
 f'"^/ .i'"'Bar5.S"c.foc. \ i" ^P^^ 
 
 1 1 1 1 1 1 
 
 Elevation and Section 
 of Post over Wing 
 
 z'-i-^'.^isi '-i" open ^xp Joint 
 Transverse Section through G/rder Elevation of Spindle Roi 
 
 Table of Dimensions and Girder Reinforcerneni" 
 
 Span 
 
 H 
 
 d 
 
 G 
 
 f 
 
 stirrup Spaces 
 
 a 
 
 b 
 
 c 
 
 Girder 
 
 Bars 
 
 !2" 
 
 18" 
 
 Z4'> 
 
 Top Row 
 
 BotlDm Row 
 
 34' 
 
 2'-Z" 
 
 
 ■}". 
 
 H" 
 
 10 
 
 6 
 
 z 
 
 z'- 3" 
 
 z'- 3" 
 
 2'- 0" 
 
 3 ' 1"" 
 
 
 26' 
 
 2'- 4'' 
 
 H" 
 
 3" 
 
 zi" 
 
 12 
 
 6 
 
 z 
 
 Z'-6" 
 
 2'-l" 
 
 z'-s" 
 
 3- 1'" 
 
 3'- 
 
 Id' 
 
 2-6' 
 
 H' 
 
 3" 
 
 ti" 
 
 12 
 
 C 
 
 3 
 
 2'- 5" 
 
 Z'-Z" 
 
 2'- 11' 
 
 3-1°" 
 
 /'- /'!' 
 B'- IB" 
 
 30' 
 
 2-9" 
 
 3" 
 
 3i" 
 
 ^i" 
 
 10 
 
 C 
 
 5 
 
 Z'-8" 
 
 Z'' 6" 
 
 2'-IO' 
 
 3 . ,"0 
 
 3-ir 
 
 32' 
 
 Z-IO'' 
 
 3" 
 
 H" 
 
 7// 
 
 10 
 
 C 
 
 6 
 
 Z'-7i' 
 
 z'-ei" 
 
 3'- 1" 
 
 3-li"° 
 
 3-ir 
 
 34' 
 
 3-1" 
 
 3" 
 
 4" 
 
 3" 
 
 10 
 
 10 
 
 4 
 
 2'- II" 
 
 z'-io" 
 
 3'- 3" 
 
 1 •li"" 
 
 3-li'"' 
 
 36' 
 
 3'- 3" 
 
 3" 
 
 4" 
 
 3" 
 
 10 
 
 10 
 
 S 
 
 3'-0" 
 
 Z'-ll" 
 
 f-io" 
 
 3 - li'" 
 
 
 38' 
 
 3' 5" 
 
 3" 
 
 4" 
 
 3" 
 
 10 
 
 10 
 
 6 
 
 3'- 4" 
 
 Z'-IO" 
 
 3'- 4" 
 
 
 S'ly" 
 
 40' 
 
 3- a" 
 
 4" 
 
 4" 
 
 4" 
 
 12 
 
 6 
 
 9 
 
 r-6« 
 
 ■s'-z" 
 
 3'- 10" 
 
 3-4"" 
 
 3-/r 
 
 Fig. 15. — Standard details of concrete deck-girder bridges, Iowa Highway Commission. 
 
 and was designed to resist maximum stresses caused by either method of analysis. The method 
 of fastening the girder steel to anchor rods should be noted. Expansion joints are provided 
 under the girder stems by means of sliding steel plates anchored into the body of both super- 
 structure and substructure. 
 
 Fig. 16 illustrates the type of deck-girder bridge adopted as standard by the Illinois 
 Highway Commission. The following description of the methods employed in providing for 
 expansion in girder bridges is given in the fourth report of the Commission: 
 
614 
 
 CONCRETE ENGINEERS' HANDBOOK 
 
 [Sec. 14-4 
 
 Two methods of providing for expansion in girder bridges have been used and both have proved satisfactory. 
 In one method, the wing walls of one abutment are entirely separated from the abutment wall proper, the latter 
 
 ^^^i "Felt Joint 
 Section A-A Steel Plate 
 
 Fig. 16. — Details of Embarrass River bridge, Cumberland Co., 111. 
 
 being free to move at the top with the expansion or contraction of the superstructure. The wing walls are designed 
 to be self-supporting. As girder spans designed by the Commission have so far been limited to 60 ft., the amount 
 
Sec. 14-4] 
 
 SLAB AND GIRDER BRIDGES 
 
 615 
 
 of movement either way from the normal is small and is taken up by deflection of the main wall or a slight rocking 
 of the wall on the footing. Earth pressure against the wall is of little importance in this connection as it but tends 
 to reduce the tension in the girder steel during expansion and to cause the abutment wall to follow the superstruc- 
 ture during contraction. It does not increase the stress in the compression area of the girder as the load is applied- 
 at the bottom of the girder, tending by this eccentricity of application to reverse the dead- and live-load stresses 
 in the girder. 
 
 This method has been found to be entirely successful, but is somewhat objectionable as a slight movement of 
 the wings due to earth pressure and unequal settlement sometimes causes the wing walls to move forward slightly 
 at the top, making a somewhat unsightly offset between the wing and abutment walls. This has never been more 
 than 2 or 3 in. for the highest walls, but as it is not understood by the ordinary observer, an impression of weakness 
 is sometimes caused. 
 
 The present method of providing for expansion is to design the abutments and wings in the ordinary way, 
 separating the superstructure completely from one of the abutments by a thick paper joint and supporting each 
 girder at the free end on a single cast-iron rocker of large diameter. The reaction is transmitted to the girder and 
 abutment from the rocker through planed structural-steel plates stiffened with I-beams when necessary. The 
 rocker surfaces in contact with the bearing plates are turned to insure perfect bearing on the plates. The diameter 
 of the rocker is made proportional to the load imposed per linear inch, in the same manner as is commonly used in 
 proportioning roller nests for steel bridges. The upper and lower plates are bedded in the concrete of the superstruc- 
 ture and abutment. The rocker is located in a pocket built in the abutment. This pocket is filled with a soft 
 ' asphalt to prevent the entrance of water or dirt and to protect the metal from corrosion. 
 
 The rocker method of providing for expansion has proved very satisfactory and is but little more expensive 
 than the other method, especially when it is considered that the wings may be tied to the main wall when rockers 
 are used and advantage taken of the mutual support thus obtained. 
 
 Crown of F/'n/shed /foadway 
 
 Section A-A Half Rear Elevation 
 
 I'Bars l2"c-foc. '■ ^'Bars /s'c.iuc. ^"'/\<.-4'-9"-A 
 Half Front Elevation 
 
 ± — 1 
 
 
 
 
 
 
 
 
 
 
 2 End 
 of Wing 
 
 Fig. 17. — Type of abutment used for girder bridges by the Illinois Highway Commission. Wings of cantilever 
 type. Main wall supported by wings as counterforts. 
 
 Fig. 17 shows a type of abutment adopted by the lUinois Highway Commission in cases 
 where the girders are supported on cast-iron rockers and the wings are nearly parallel to the 
 roadway or make an angle of more than 45 deg. with the face of the abutment. The wing 
 walls are considered to act as counterforts and the reinforcing steel in the main walls is hori- 
 zontal and placed near the stream face of the wall. 
 
 Fig. 18 gives the details of a girder bridge, the main portion of the abutments and the 
 wings of which were designed and figured in the same manner as the slab bridges of Figs. 3, 
 4 and 5. 
 
 In the structure shown in Figs. 19 to 23 inclusive, cross girders and stringers were provided 
 in addition to the longitudinal girders, this type of floor system being found to be economical 
 for wide bridges of long span. One end of each span is anchored to the pier and the other end 
 is allowed to expand and contract in a joint packed all around with in. of tar paper and 
 bearing on a pair of milled steel plates. Expansion joints were also made in the roadway 
 slab and railing. Cast-iron scuppers were placed in each curb on 25-ft. centers. For maximum 
 stresses in the cross girders, the sidewalks were assumed to be unloaded. 
 
Fig. 19. — North Samuels Avenue viaduct, Fort Worth, Texas. 
 
Sec. 14-5] 
 
 SLAB AND GIRDER BRIDGES 
 
 617 
 
 Reinforced-concrete deck-girder bridges for railroad traffic are rather unusual, but a few 
 such structures have been built. The superstructure in such bridges is usually supported by 
 plain-concrete piers and abutments. 
 
 6. Through Girders. — From the standpoint of economy, the through-girder bridge should 
 not be built except where insufficient headroom or other local conditions prevent the use of the 
 deck girder. 
 
 Plan of Reinforcement In Footing 
 
 Fig. 20. — Details of south abutment, North Samuels Avenue viaduct, Fort Worth, Texas. 
 
 The standard type of through-girder bridge, adopted by the Iowa Highway Commission 
 for locations where the deck type is impracticable, is shown in Fig. 24. The girders themselves 
 were designed as simple beams and the floor as a slab partially fixed, the point of contraflexure 
 being arbitrarily assumed at about 1 ft. from the edge of the girder. This is approximately in 
 accordance with tests made by the Illinois Highway Commission in 1907. 
 
 In the tests referred to, stress measurements were made on a through-girder span with an 
 18-ft. roadway loaded with crushed stone and pig iron. The full applied load was 418 tons, 
 which gave a distributed load per square foot of floor of 1450 lb. Deformation measurements 
 
618 
 
 CONCRETE ENGINEERS' HANDBOOK 
 
 [Sec. 14r-5 
 
 , gj/r -27-04 ■■• 
 
 ^/ey. 495-0 
 
 3-^'^- /eh" hops i/nder 
 6" /each girder seat 
 
 y after nafe 
 
 Elevation of Stream Pier 
 
 Section A-A 
 
 4 Bars bent to form 
 one stirrup 
 
 IT- i"' Bars, 4-0? 
 
 
 
 
 
 1 il i[ r \ 
 
 
 
 1 1 1 1 1 
 
 iSiiii 
 
 
 
 
 
 Detail of 
 Snear Bars 
 
 * n- Section B-B 
 
 Elevation of Pier 
 
 Fig. 21. — Details of typical piers, North Samuels Avenue viaduct, Fort Worth, Texas. 
 
Sec. 14-5] SLAB AND GIRDER BRIDGES 619 
 
 A Center Line of stringer and ra//inq 
 ^ r ^/^// .G ^ ; J^,^,A, 
 
 ■\- 30-0 face tv face oi^curb 
 
 7'- 6" H<- 
 
 1 K 
 
 fiecTion 6-6 
 
 Section FF 
 
 Detail Cff Copper 
 Expansion Joint 
 (to be used for side- 
 wQlk slab and curb) 
 
 Fig. 22. — Details of floor system, North Samuels Avenue viaduct, Fort "Worth, Texas. 
 
620 
 
 CONCRETE ENGINEERS' HANDBOOK 
 
 [Sec. 14- 
 
 Center L/ne of F/oorbeam and pier 
 
 /£'.$" >i ,.C.L.of F/oorbeam 
 
 C.L.ofF/.beam \ <^rrc( girder 
 
 . I^^i i i K i i i 
 
 — >]]<:-->l ;V/ ^ 
 
 ///'• K>,....-?../£^^^.. ^-^ '''' c. to c. piers 
 ' ? II-" 
 
 ^ Elevation of girder 
 
 Center line of -Diaphragm 
 
 1 
 
 ■>]/-//[<- 
 5ect.ion 
 
 VJ-i'^. 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 "A 
 
 Stringer under Roqdvyaj 
 
 6 ' 
 
 Siope /- J 
 
 <?ro^/- Expansion ^ , ^ 
 jQint Joint lined k[ 
 witty 
 
 Section A-A Section. B-B 
 
 r 
 
 tor paper Scuppers 
 
 (to be piaced midway 
 between fioorbeams 
 and 25 ft c.toc.) 
 
 .Chamfer i" Expansion Joint-jfc^// 
 
 7//^_/.// . ^1!^ tre/^tjt ofrai/in^ 
 '8-'. B 
 
 I Li'^'l'*"' A J<^/^ V* ipiufkyuin z>juu ■ ^^i"''Conduif,-\^^^^ Tar 
 
 if>7?. Str/njer •■'^ 
 
 Typical Post Typical Panel of Hand Railing Post carrying Lamp 
 
 Fig. 23. — Details of floor system and railing, North Samuels Avenue viaduct, Fort Worth, Texas. 
 
Sec. 14-5] 
 
 SLAB AND GIRDER BRIDGES 
 
 621 
 
 /S^-O" Roadway 
 
 I I F/oor may be movecf 
 
 . ' Sars, 6 "c. -foe. every S"^ and 
 \ S^'^ bar fo be bent up as shown 
 
 Bars 
 
 Bars, Zbof..4 top 
 
 1^ Theoretical Half Span 
 
 fe'-; i5'-6" ■-. 
 ■■ir-^-o>^-4^i'-&' 
 
 6/rder 
 
 F/oor 
 
 ( /-/'"Bars 
 
 30- 
 
 5' 
 
 Z-l"" 
 
 
 36'- 
 
 11" 
 
 3-1"" 
 
 
 3Z' 
 
 0" 
 
 Z-li"" 
 
 
 3Z' 
 
 0" 
 
 
 
 II' 
 
 3" 
 
 
 
 Zl' 
 
 0" 
 
 44-1"° 
 
 
 7' 
 
 0" 
 
 
 
 Zl' 
 
 6" 
 
 Zf-V" 
 
 
 zs' 
 
 6" 
 
 /3-i"° 
 
 
 3Z' 
 
 0" 
 
 
 
 
 
 Fig. 24. — Standard sections for through girder bridges of 30-ft. 
 
 Iowa Highway Commission. 
 
 corners Transverse Section 
 
 Fig. 25. — Details of superstructure of Funkelien bridge, Town of Christiana, Dane Co., Wis. 
 
 r-'T 
 I 
 
 ^'-o" 
 
 4 - a" Bars on \ 
 each end ofevery\ 
 other -floor bar 
 
 ^"Tar paf^ r Joint 
 
 ^"Tar paper K;* 
 Joint \: o. 
 
 --. Asphalt 
 / peneiration 
 
 I PC mm ) 
 
 Spacing for j"" Stirrups 
 
 t< - 8" ••■><•■•- IZ" ■■•J>1< Z4" >f'' 
 
 \ cfoc \ c-toc. I croc- I 
 
 -3'Ti/e Drains 
 6'c.toc. 
 
 Fig. 26. — Details of through girder bridge of 45-ft. span, Illinois Highway Commission. 
 
622 
 
 CONCRETE ENGINEERS' HANDBOOK 
 
 [Sec. 14-6 
 
 were made in the reinforcing steel of the suspended floor which indicated stress equivalent to 
 that theoretically resulting from a sinple span of 15.5 ft. In other words, since the roadway 
 was 18 ft. between girders, the point of contraflexure was apparently about 15 in. from each 
 girder face. 
 
 Figs. 25, 26 and 27 illustrate other design of through-girder superstructures. 
 
 ,Bafter 
 / "/fods, 'f'-ol /e'cfuc. 
 
 \ Batter ? 
 
 . j"/fods,s'-3" IE"ctOC 
 
 .- 2"ffod5, /8'identj f4"c -oc 
 
 I" Pod 5. yVctoc 
 
 il'tf0ds\ j |\ I ^"f?ods' I \l''ff0d5:3i'ct0c'\ I 
 ■■.■■3'-ii"-Xi'-^ r'-o' ■ — ->l/-e3<---- e'-o" —■>^i-6i< r'-o'' •>i/-'e4f— j'- //'--: 
 
 Fig. 27. — Cross-section of through girder bridge, C. B. & Q. R. R. 
 
 Fig. 28 shows a rather unusual type of through bridge on account of the fact that the 
 girder reinforcement is in the form of a truss of sufficient strength to carry the dead and con- 
 struction loads. The piers are simply columns braced between by either a vertical slab or by 
 struts. As no falsework is necessary, this type of construction is especially adapted for high- 
 way bridges over railroads and electric lines, or at locations where the soil is very soft. 
 
 ] D 
 
 
 n|i — 
 
 
 
 —y 
 
 
 
 
 
 
 r-/^ 1 
 
 
 
 1 — 
 
 u-tt 
 
 i ft 
 
 
 
 A 
 
 A 
 
 
 
 
 
 
 
 f 
 
 J 
 
 i 
 
 
 
 
 Section A-A 
 
 
 
 Fig. 28. — Details of bridge over Muddy Creek, Hamilton County, Ohio 
 
 CONTINUOUS-GIRDER BRIDGES 
 Monolithic Construction 
 
 6. Expansion Joints. — In order to prevent an accumulation of movement due to contrac- 
 tion and expansion, expansion joints should be provided at least every 100 ft. in length of the 
 structure. If this is not done, severe stresses are likely to occur in the end columns. 
 
 In long bridges an expansion joint is usually provided between the superstructure and the 
 
Sec. 14-7] 
 
 SLAB AND GIRDER BRIDGES 
 
 623 
 
 abutments for the reason that, if an abutment witli a heavy pressure of earth against it is 
 rigidly connected with a number of continuous spans, the expansion and contraction tend to 
 act in one direction only — that is, away from the abutment — the earth pressure back of the 
 abutment not allowing movement in the opposite direction. Such a condition would lead to 
 difficulties at the center of the bridge, or over the expansion piers next to the abutments, and 
 the abutments and piers would also be severely overstressed due to the continuous movement 
 in one direction. Each time an abutment would move slightly due to contraction, the earth 
 against it by reason of the heavy moving loads would fill in the small space left by the contrac- 
 tive movement, and when expansion again took place, the abutment would be restrained by 
 the earth so that enormous stresses might be developed. 
 
 7. Examples of Typical Bridges of the Continuous-girder Type. — A rather simple highway 
 trestle, applicable to comparatively low crossings, is shown in Fig. 29. The longitudinal beams 
 are continuous over three spans, an expansion joint occurring over every third pier. Inter- 
 mediate piers are made monolithic with the floor by means of rods from the columns and stir- 
 rups from the cross beams. Because of the indeterminate degree of fixity, the lightness of the 
 structure, and unknown construction factors, all members affected were designed both as fixed 
 
 Half Transverse Section Longitudinal Section on Center Line 
 
 Fig. 29. — Details of standard trestle spans, Engineering Department, State of Arizona. 
 
 beams and as beams freely supported. The slabs were designed as continuous, with equal 
 positive and negative steel throughout. The center longitudinal beam was designed as a 
 T-beam with a 36-in. flange. The pier web, or wall between supporting columns of bents, is 
 carried 2 ft. above high water. 
 
 A low trestle or viaduct type of construction is shown in Fig. 30. The slab-beam-and- 
 girder spans were selected since arches, it was thought, would not appear to advantage for such 
 low construction. The cost of the girder type was also found to be much less than for a series 
 of arches, due principally to decrease in the dead weight of the structure and to simplicity in 
 form work. Expansion joints occur about every 200 ft. and make the viaduct virtually a 
 number of independent structures, a double row of columns being provided at each joint. The 
 arched girders capping the column bents were designed as straight rectangular beams and no 
 account was taken of the possible arch action. The entire top surface of the roadway slab was 
 waterproofed with a layer of burlap and two layers of felt laid in hot asphalt. 
 
 Fig. 31 gives the details of a continuous-girder bridge to span a stream which is generally 
 dry, but which at flood times reaches over a wide area of bottom lands. This railway bridge 
 is also within the backward area of the White Rock reservoir of the Dallas water supply so 
 that ample provision had to be made for high-water conditions. The floor consists of a double 
 T-beam which is monolithic with the bents and the abutments. There is no expansion joint in 
 the structure since it was considered of sufficient strength to take all movement from end to 
 pnd. Both ends of the structure are open between girder supports with short trestles con- 
 
624 
 
 CONCRETE ENGINEERS' HANDBOOK 
 
 [Sec. 14-7 
 
 B Cameqie Tie M^S Ur— s'-Si" 
 
 Concrete Ballast '-^jA ^ \ 
 
 (8-Q Bars ,r^-,-gpnnediate 
 ) Hoops 6"c. to c. Bracket 
 
 '■/"stub, Id" A 
 
 ^oops'^^l^ 
 
 4-l"^T....\ 
 
 •3'm,,-P!.B^''K§''xiZ' 
 
 I — r 
 
 T — I 
 
 Expansion Joinl" 
 in Rocdway U W U 
 
 SJ-O' -■: " 
 
 <i''ivood Block. 
 
 . - Z '^""Tyr. Bars ^ ^^^^^ Cosh/on \ 
 
 M''"'{^-/f"Bars 
 
 u..._ — a< 7itf'^ 
 
 B-/'"'r^y. Bars \ 
 
 ^ Section A-A 
 
 Fig. 30. — Details of Gilbert Avenue viaduct, Cincinnati, Ohio. 
 
Sec. 14-8] 
 
 SLAB AND GIRDER BRIDGES 
 
 625 
 
 necting with the bridge which at some future time may be filled in if high water gives no trouble 
 at this point. The girders were designed for Cooper's E-30 loading with an impact allowance 
 of 100% of the live load. The ribbed abutments should be noted. 
 
 A continuous-girder bridge or trestle with no expansion joints and with abutments of the 
 ordinary type is shown in Fig. 32. The structural-steel core in the column bents is unusual 
 but undoubtedly adds greatly to the rigidity. 
 
 Fig. 33 gives the details of one of the typical bridges in the Chicago, Milwaukee & St. 
 Paul track depression work at Minneapolis. The girders are continuous from end to end with 
 expansion joints at the abutments — that is, the girders were considered continuous over the 
 two interior supports and simply supported at the ends. The moments and shears were 
 calculated by influence lines in accordance with the theory of continuous structures assuming 
 constant / and unyielding supports (see Art. 48a, Sect. 7). Corrections were made for >^-in. 
 
 End Elevotion Side Elevation 
 
 Fig. 31. — Details of railway bridge over White Rock Creek near Dallas, Texas. 
 
 settlement of supports for variable /. An unusual feature of the bridges is a curved shelf on 
 the outer face of the outside girder which is intended to act as a smoke shield by diverting the 
 smoke from the parapet as engines pass under the bridge. 
 
 8. Analysis of Stresses in Rigid Viaduct Structures. — A viaduct structure, composed of 
 one or more cross frames or bents, and two or more spans of longitudinal deck girders, is in 
 reality a rigid frame when girders and bents are rigidly attached to each other. Reinforced- 
 concrete structures of this kind will act as rigid frames between expansion joints, and should 
 be investigated as such. The general problem may be reduced to two problems for analysis: 
 the stresses caused in the frame as seen in longitudinal elevation (hereafter referred to as the 
 viaduct frame) ; and the stresses in the cross frame seen in a transverse section of the structure 
 (hereafter called the cross frame, or bent). These problems will be treated in the order here 
 given. 
 
 The viaduct frame should be designed to withstand: (1) the dead load of the frame; (2) 
 the vertical live load, plus impact; (3) the horizontal traction forces, or braking forces, plus 
 40 
 
626 
 
 CONCRETE ENGINEERS' HANDBOOK 
 
 [Sec. 14-8 
 
 
 
 . 1 
 
 
 ST 
 
 
 
 -J '"^^ 
 
 Vz'^^ 
 
 3d'' 8" — ^ > 
 
 , fBars S'c. foe. 
 
 /9-4!' -> 
 
 Side Girder ReirrTorcernent 
 
 Interior Girder Reinforcennen+ 
 
 AS.&py. Co's rr/an^/e Mesh 
 Sty/e No. 7-50 - /a "rndtfr 
 
 /?ound Corners, io /'^/Fad/us 
 jSymmefr/ca/ about 
 ^ Center L/ne except 
 
 3-2 , of drain \ for S/deyt^affr 
 
 . 2 Bar 
 
 /neacft side 
 of each span 
 
 ^^Fv?-^ ttC :_ 
 
 ■e-f' 
 
 4, <V4 
 
 <3-3-h.-/6''--M'-— 6- 7" J<- i6"--X V-/^ 
 
 Transverse Section 
 
 B 
 
 ^.S.&tr. Cos Tr/an^/e Mesir . 
 Siy/eAo 7- i8"yyidttT- mnd 
 spiraiiy aroc/nd cg/umn 
 
 Hoies 7^r ^'^/^ncfror Boits 
 
 ZPis.6K/8fK^' 
 
 
 
 
 
 1 
 
 
 
 
 
 
 ( 
 
 Section A-A \ 
 
 Cross Girder ^ 
 /^n^/es 
 
 Plan 
 
 Section B-B th. B4".B4".f^'' ' 
 
 Fig. 32. — Details of Mill Creek bridge, Village of Cazenovia, Richland Co., Wis. 
 
Sec. 14-8] 
 
 SLAB AND GIRDER BRIDGES 
 
 627 
 
 ■M , Bridge Seat ^ 
 
 y9" 
 
 8'' Z'^-->'f:-5'-5"-^- 
 
 3/ -6" 
 
 <- 
 
 Bearn "C 
 
 '>1< 
 
 IS'.S't: J 
 
 Sectional Elevation \<-6-9"-iA 
 
 Baiter4^:/B 
 
 SecTion B-B 
 
 « 4.5-0^ ^>|;c >| 
 
 B i'^^'^y^^ Paper, ''^ 
 
 1< ^'-^//->| S/770/f6'---l^ J/'-^''' -J 
 
 Shie/d ■ _ 
 
 Section A-A Elevation ^Smoke 
 
 Fig. 33. — Details of Bryant Avenue bridge over tracks of the C. M. & St. P. Ry., Minneapolis, Minn. 
 
628 
 
 CONCRETE ENGINEERS' HANDBOOK 
 
 [Sec. 14-8a 
 
 impact; (4) stresses due to changes in temperature. Any one of these cases may develop 
 large moments in both girders and columns. The reactions on the footings cannot be deter- 
 mined in an unsymmetrical frame, or in a symmetrical frame with unsymmetrical loading, 
 without consideration of the elastic distortion of the structure. The dead load would consist 
 of the estimated weight of the frame. The live load and its impact would be of the classes of 
 loading given for arches (see Art. 6, Sect. 16). 
 
 8a. Viaduct Frames. — Before proceeding with the analysis of the viaduct 
 frame, it is necessary to determine the conditions of support of the bases of the columns and 
 of the outer ends of the girders. The greatest moments in the columns due to unbalanced- 
 loads on adjacent spans will occur when the column bases are fixed; and the greatest column 
 stresses due to lateral, or tractive, forces will occur when the column bases may be considered 
 as hinged. Since the tractive forces produce very large column stresses, and since there is 
 usually great difficulty in securing a perfectly fixed column base, the discussion here given will 
 apply first to viaduct frames whose column bases are hinged, after which certain modifications 
 of the development will be suggested to care for the case of the perfectly fixed column base. 
 End girders will be assumed to have frictionless bearings in the expansion pockets, thus allow- 
 ing the maximum horizontal deformation of the frame to occur. 
 
 The extent to which the bent will act as a pair of columns in the viaduct frame should 
 also be determined. In Fig. 34 is shown a two-legged bent with slightly battered columns. 
 Girders attached at B and C will deflect in vertical planes; hence it is convenient to replace 
 
 AB and CD with columns lying in vertical planes, as A'B and 
 CD\ whose stiffness, or ability to restrain the points B and C, is 
 just equal to that of the legs AB and CD. The length of these 
 equivalent columns will be h, and their moments of inertia about 
 a horizontal axis lying in the plane of the bent will be Ia'b = Iab 
 cos d. When the bent is composed of more than two columns, 
 each column should be replaced by a vertical one, as above; and 
 the sum of the moments of inertia of all equivalent columns on 
 one side of the axis of symmetry should be considered as the 
 moment of inertia of a single resultant column A'B. This resultant column will be called 
 "the column" in the following treatment of viaduct frames. 
 
 It is immaterial whether or not the whole viaduct frame is split lengthwise into two parts 
 for analysis. The designer should arrange this matter to suit his convenience in computations. 
 
 Three cases of loading will be applied to 'the viaduct frame: (1) a moving vertical con- 
 centrated load; (2) a vertical symmetrical load, placed symmetrically on a single span; (3) 
 a horizontal load acting axially on the deck girder. It will be possible from these investiga- 
 tions to plot influence lines for live load, and to determine the effect of dead load on the span. 
 
 The following series of solutions will begin with the general case of the four-span frame. 
 From this case the other cases may be deduced, since each of the cases following is a special 
 form of the general case. All solutions will be made by the method of slope-deflections (see 
 Art. 2, Sect. 10). The nomenclature used is as follows: 
 
 M = moment. Subscripts will denote where this moment is taken — as, Mab = moment 
 
 at A in the member AB. 
 6 = change in slope of the tangent to the elastic curve at a given joint. Subscripts 
 
 will denote the joint in question. 
 di = lateral movement of the top of any first-tier column due to eccentric vertical, or 
 
 lateral, loads. 
 
 di = lateral movement of the top of any second-tier column. 
 h = column height. Subscript designates the column in question. 
 I = girder length. Subscript designates the girder in question. 
 
 n = T- Subscript designates the column in question. 
 
Sec. 14-86] 
 
 SLAB AND GIRDER BRIDGES 
 
 620 
 
 K 
 F _ 
 I 
 
 X = 
 
 for girder; or ^ for column 
 
 Subscript indicates member in question. 
 
 constant for symmetrical loading. See page 413 for values for different types of 
 loading. 
 
 moment effect at left end of girder due to load P on that girder, and in general 
 
 equals — ^ , in which a is the distance from the left end of the girder to P, and 
 
 a -\- h = I. Subscripts will denote the girder in question, 
 y = moment effect at right end of girder due to load P on that girder, and in general 
 Pa% 
 
 equals — jj-. Subscripts will denote the girder in question. 
 
 Sb. Four-span Viaduct Frame with Rigidly-connected Column Tie (Type I). — 
 The general case of the unsymmetrical frame shown in Fig. 35 will 
 be analyzed for the loads Pi and P2 and Q occurring separately. 
 
 Since the structure and its loads are for the general case un- 
 symmetrical, there will be a horizontal movement of the members 
 AB, and of the members FL, besides rotation at all of the joints. 
 Vertical components of this nearly horizontal motion will be neg- 
 lected in this analysis. 
 
 Referring to Arts. 2 and 3, Sect. 10, the equations for the 
 moment at the end of each may be written as follows: 
 
 N 
 
 K, 
 
 K2 
 
 K3 
 
 K4 
 
 
 
 
 
 
 
 Kg 6 
 
 H 
 
 Kj J 
 
 K, L 
 
 
 0 
 
 
 
 \ 
 
 k 
 
 s 
 
 Fig. 35. 
 
 Mfn 
 
 
 SEKi^idF - niM 
 
 Mle 
 
 
 2EKi3{2dL + Be - 
 
 3ni3(i2) 
 
 Mpg 
 
 
 2EK^{2dF + Og) 
 
 Mlt 
 
 
 SEKisidL - nisdi) 
 
 
 Ufa 
 
 
 2EIU{2dF + dA - 3/19^2) 
 
 Mlj 
 
 
 2EKs(2dL + dj) 
 
 
 Map 
 
 
 2EK^{2dA Of - ?>n^d2) 
 
 Mjl 
 
 
 2EK^{2dj + Ol) 
 
 
 Mab 
 
 
 2EKi{2dA + ds) - Xab 
 
 Mjs 
 
 
 dEKn(dj - ni7di) 
 
 
 Mba 
 
 
 2EKi{2dB + 0^) + Yba 
 
 Mjd 
 
 
 2EKi2(2ej + 60- 
 
 3ni2C?2) 
 
 Mbg 
 
 
 2EKio(2eB + dG- Sniod2) 
 
 Mjh 
 
 
 2EK7{2dj + Bh) 
 
 
 Mbc 
 
 
 2EK 2(268 + dc) - Xbc 
 
 Mhj 
 
 
 2EK'ji2dH + ej) 
 
 
 McB 
 
 
 2EK2(2dc + Ob) + Ycb 
 
 Mhr 
 
 
 SEKuidH - nidi) 
 
 
 McH 
 
 
 2EKii(2ec + dH - Snud2) 
 
 Mhc 
 
 
 2EKni2dH + dc - 
 
 3nii(i2) 
 
 McD 
 
 
 2EKz{2dc + Od) 
 
 Mhg 
 
 
 2EK,{2dH + Og) 
 
 
 Mdc 
 
 
 2EKz{2dD + dc) 
 
 Mgh 
 
 
 2EK6(2dG + Oh) 
 
 
 Mdj 
 
 
 2EKi2{2dD + dj - 3ni2d2) 
 
 Mgo 
 
 
 SEKi5(dG - niM 
 
 
 Mde 
 
 
 2EK4.{2dD + Be) 
 
 Mgb 
 
 
 2EKio{2dG + eB - 
 
 3nioc^2) 
 
 Med 
 
 
 2EK4{2dE + do) 
 
 Mgf 
 
 
 2EKr,[2dG + Of) 
 
 
 Mel 
 
 
 2EKiz{2dE + dL- 3ni3fi2) 
 
 
 
 
 
 Mob 
 
 Ma 
 
 Since from statics, any joint in a structure is in equilibrium, the sum of the moments about 
 that joint must equal zero. Thus, 
 
 Mfn + Mfg + Mfa = 0 
 
 The values of each moment (from above) may be substituted into this equation. 
 
 In liice manner the equations above for the members concurring 
 at any joint may be summed up and this sum set to zero. This will 
 result in ten equations, one for each of the joints A to L, inclusive, 
 which will involve twelve unknowns; that is, a 6 for each of these 
 joints, and in addition, c?2 and di. For solution it is necessary to write 
 as many equations as there are unknowns. One of the two additional 
 equations may be supplied from the following condition: 
 The sum of the moments at the top and bottom of all columns of one tier, plus the product 
 of the shear (Q), and the height of the tier (/ig), equals zero. Thus, referring to Fig. 36, 
 
 Fig. 36. 
 
630 
 
 CONCRETE ENGINEERS' HANDBOOK 
 
 [Sec. 14-86 
 
 Maf + Mbo + McH + Mdj + Mel + Mfa + Mgb + Mhc + ATjz) + ATle + Qh^ = Q 
 
 The second of the two equations may be written from the requirement that the sum of all hori- 
 zontal components of reactions at the bases to T, inclusive, must equal Q. Thus, 
 
 MpNnii + MGoni5 + MuRTiie + MjsUn + MltUu - Q = 0 
 
 These two equations, and the ten equations mentioned above, have been tabulated in Table 
 I, after having first been divided through by E. All terms having no variables have been put on 
 the right-hand side of the equations, and have been tabulated as three cases: (1) with the load 
 Pi on AB; (2) with the load P2 on BC; (3) with the horizontal load Q at E. Vertical loads 
 on other spans may be treated as in Cases (1) and (2). This arrangement allows any case 
 to be analyzed entirely separately. In either Case (1) or (2), the position of the load may 
 be altered by changing the distances a and b, hence changing the values of X and Y (see 
 page 629, Fig. 35). It was found convenient to divide the last two equations by a common 
 constant term. 
 
 After a table similar to Table I has been prepared, the known terms may be evaluated, 
 and the resulting simultaneous equations solved as in the illustrative problem under Type 
 VII., Art. 8h. 
 
 Table 1. 
 
 Variable Terms 
 CLeft hand side of equations) 
 
 Constant Terms 
 (Right hand side of equations) 
 
 
 
 
 ec 
 
 Go 
 
 
 Or 
 
 
 e„ 
 
 ej 
 
 
 d^ 
 
 d, 
 
 Case I 
 
 Casen 
 
 Casein 
 
 1 
 
 
 
 
 
 
 
 
 
 
 
 -6 Kan 9 
 
 
 
 
 
 2 
 
 
 
 
 
 
 2 Kg 
 
 
 
 
 
 
 
 
 
 
 3 
 
 eK. 
 
 4K,MKj+4Ka 
 
 
 
 
 
 2K,o 
 
 
 
 
 -SKionio 
 
 
 -\\ 
 
 
 1 
 
 4 
 
 
 
 
 
 
 
 
 
 
 
 -6K„n,i 
 
 
 £D 
 
 
 6 
 
 5 
 
 
 
 
 
 
 
 
 
 
 
 -6Kjn,2 
 
 
 C 
 
 1 1 1 1 
 
 load -Span BC 
 
 MM 
 
 Horizontal load ^ 
 
 M M 1 
 
 6 
 
 
 
 
 
 4Kit4K,3 
 
 
 
 
 
 
 -6K|3n,3 
 
 
 CO 
 
 7 
 
 
 
 
 
 
 
 
 
 
 4K^4Ko+3Kia 
 
 
 -3K„,n,6 
 
 
 8 
 
 
 
 
 
 
 
 
 
 +4^:^+3^,7 
 
 
 
 -3Knn,7 
 
 — £ — ^ 
 
 9 
 
 
 
 
 
 
 
 
 
 
 
 -6K,in,i 
 
 "3K,6n|6 
 
 — L^y — 
 
 
 10 
 
 
 
 
 
 
 ^Ks 
 
 
 
 
 
 -6K|onio 
 
 -3K,5n,5 
 
 — s_ 
 
 
 
 If 
 
 
 
 
 
 
 K|4ni4 
 
 
 
 
 
 
 -(|<«?i!4+l(itn,s+IC4ri|6 
 •fKiinS+KiiTiis') 
 
 Hi 
 
 -1- 
 
 ?{^[ 
 
 IE 
 
 
 
 K„ 
 
 
 K,3 
 
 K9 
 
 
 K„ 
 
 
 K,3 
 
 
 
 6 
 
 
 
 It sliould be noted both in the moment equations, and in Table I, that when the load is on 
 AB^ Xbc and Ycb both equal zero. 
 
 F 
 
 When the load is symmetrically placed on a member, X = F = Thus, if a load is 
 
 symmetrically placed on AB, Mas = 2EKi(2eA + Ob) - y Equation (2), Table I, equals 
 
 ^^^for Case I. Equation (3), Table I, equals^ — for Case I. The same scheme holds 
 
 F 
 
 for other members. For values of y for various symmetrical loadings, see page 413. 
 
 Case a {Fig. 37). — The solution for this case may be obtained from the general equations 
 of Table I, by substituting Ki = K13 = Kis = 0. 
 
 C D E A B C D E 
 
 2 
 
 10 
 
 6 6 
 
 15 
 
 3 
 
 II 
 
 H 7 
 
 Fig. 37. 
 
 / 
 
 2 
 
 9 
 
 10 
 
 5 G 
 
 6 H 
 
 14 
 
 15 
 
 
 h 
 
 C 
 
 
 
 Fig. 
 
 3 \ ^ ^ 
 
 II 
 
 It is important to note the variation if hn = 0, and a hinge is placed at J. Then JD 
 and JH would be hinged at J. In this special case Kn = 0; Mjd = Mjh =0; di = 0; 
 
Sec. 14-8c] SLAB AND GIRDER BRIDGES 
 
 631 
 
 Mdj = SEK12 {Od - niid^)', Mhj = SEKjdn. With these modifications a table Hke Table I 
 may then be constructed. 
 
 Case h (Fig. 38).— In this case = Ks = Kiz = Ku = = 0; Mjd = 0; Mdj = SEK^ 
 {Od — riiM. These variations from the general moment equations, pages 629 and 630, will 
 allow a new table, similar to Table I, to be made up. 
 
 Case c {Fig. 39). — The equations of Table I apply to this frame when = = Kg = 
 Kiz = Kii = K18 = 0. 
 
 2 
 
 J 
 
 10 
 
 11 
 
 G 6 
 
 H 7 
 
 15 
 
 16 
 
 1 
 
 7 
 
 ; 
 
 ? ~ 
 
 Fig 
 
 39. 
 
 I 
 
 2 
 
 
 3 
 
 
 
 9 
 
 10 
 
 
 II 
 
 
 12 
 
 S 6 
 
 6 
 
 H 
 
 7 
 
 
 J 
 
 14 
 
 1 
 
 15 
 
 16 
 
 . : 
 
 17 
 
 Fig. 40. 
 
 8c. Three-span Viaduct Frame with Rigidly-connected Column Tie (Type II). — 
 
 Equatiojis of moment at the ends of each member (Fig. 40) in the frame may be written as in 
 
 Type 1. 
 
 
 
 
 Mfn 
 
 = SEKnidF - nidi) 
 
 Mdj 
 
 = 2EKi2{2dD + dj - 
 
 3ni2C?2) 
 
 Mfg 
 
 = 2EK,{2dF + Bg) 
 
 Mjd 
 
 = 2EKi2{2dj + do - 
 
 3ni2C?2) 
 
 Mfa 
 
 = 2EK,{2dF + eA'- SnM 
 
 Mjs 
 
 = SEKuidj - nudi) 
 
 
 Maf 
 
 = 2EK,{2dA + dp - Sn,d2) 
 
 MjH 
 
 = 2EK^{2dj + Oh) 
 
 
 Mab 
 
 = 2EK,{2dA + Ob) - Xab 
 
 Mhj 
 
 = 2EK,{2dH + ej) 
 
 
 Mba 
 
 = 2EKi(2dB + Oa) + Yba 
 
 Mhr 
 
 = SEKuidH - niM 
 
 
 Mbo 
 
 = 2EKtoi2dB + do - Sniod2) 
 
 Mhc 
 
 = 2EKni2dH + do - 
 
 37211(^2) 
 
 Mbc 
 
 = 2EK2(2dB + do) - Xbc 
 
 Mhg 
 
 = 2EK,{2dH + Og) 
 
 
 Mcb 
 
 = 2EK2{2dc + Ob) + Ycb 
 
 Mgh 
 
 = 2EK^{2dG + en) 
 
 
 McH 
 
 = 2EKii{2dc + Oh - ^niM 
 
 Mgo 
 
 = SEKi,{dG - nM 
 
 
 McD 
 
 = 2EKsi2dc + Od) 
 
 Mgb 
 
 = 2EKioi2dG + 9b - 
 
 3nioc?2) 
 
 Mdc 
 
 = 2EK,{2dD + ec) 
 
 Mgf 
 
 = 2EK,{2dG + dp) 
 
 
 The general equations set up from these moment equations are the first eight of the follow- 
 ing. The ninth and tenth are found in the same manner as the eleventh and twelfth of the pre- 
 ceding type (see page 630). 
 (1) 2K,dA + (3Ki4 + 4^5 + ^K,)dF + 2K,dG - ^K^ndi - 3Kunidi = 0 
 
 Xab 
 
 E 
 
 (2) (4Xi + 4:Kg)dA + 2KidB + 2KgdF - QK^n, 
 
 (3) 2KidA + (4Ki + + 4:K:o)dB + 2X260 + 2KiodG - ^K,,n,od2 = - -|- 
 
 (4) 2K2eB + (4^2 + 4Z3 + 4:Kn)ec + 2X^60 + 2KixdH - QKiinnd2 = 0 
 
 (5) 2K,dc + (4^3 + 4X12) + 2Ki2dj - QKufindi = 0 
 
 (6) 2Ki2dD + 2K7eH + (3Xi7 + 4K7 + 4Xi2)^j - 6Ki2ni2C^2 - SK^nndi = 0 
 
 (7) 2Kiidc + 2^6% + (3Zi6 + 4^6 + 4K7 + 4Kii)0ff + 2K7dj - QKnnnd2 - 3Kuni^di=0 
 
 (8) 2Ki,dB + 2K,dF + (3Ki5 + 4X5 + 4^6 + 4Kio)0g + 2KedH - 6XionW2 - SKuUndi = 0 
 
 (9) KuUudF + Ki^ni^dG + Ki^niedH + KnUndj - {KuUu^ -f- Xunis^ -h Kunie^ + Knni-j'^)di 
 
 = 0 
 
 (10) KgdA + Kio05 + Kxidc + Ki25z> + -^90/^ + KioOg + i^ii^// + Kx2ej - 2d2{Kgng + 
 
 i^ionio + KiiMii +^^i2ni2) = 0 
 The right-hand side of the above equations is given for a vertical load Pi in span AB. For 
 a vertical load P2 on span BC, replace (— Yba/E) in equation (3) with (X^c/^) ; set equation 
 (4) equal to (— Ycb/E) instead of zero, and set equation (2) to zero. For a horizontal load at 
 D, set equation (9) equal to (Q/3E); set equation (10) equal to ( — Qh^/QE); and all other equa- 
 tions to zero. It will be noted that the modifications are in accordance with Cases I, II and III 
 of Table I. 
 
632 
 
 CONCRETE ENGINEERS' HANDBOOK 
 
 [Sec. 14r-Sd 
 
 When any horizontal member is loaded symmetrically, then for that member, X = Y = j- 
 
 F 
 
 Thus, if a load is placed symmetrically on AB, Mab = 2EKi{2dA +63) - j', Mba = 
 F 
 
 2EKi(2dB + ^a) +7' equation (2), page 631, would equal (F/ZE) ; and equation (3), page 631, 
 
 would equal ( —F/IE). Values oiF/l for various symmetrical loadings are given in the table on 
 page 413. 
 
 These ten equations may now be tabulated as in Table I, and may be solved by the same 
 method as that employed in a problem under Type VII (see Art. Sh). 
 
 Case a {Fig. 41). — This frame may be analyzed by letting K12 = Kt = Kn = 0, in the gen- 
 eral equations of Type II, page 631. 
 
 /5 
 0 
 
 Fig. 41 
 
 D 
 
 
 I 
 
 
 1 3 2L 
 
 3 22. 
 
 F 
 
 9 
 
 5 6 
 
 10 
 
 r 
 
 H 
 
 
 
 14 
 
 i 
 
 C 
 
 15 
 1 
 
 
 Fig. 4fi. 
 
 Suppose /ii6 = 0, and a hinge is placed at so that HC and HG are hinged. Then Kie = 
 0 = Mhc = Mhg = 0; Mgh - SEKgOg] Mch = SEKuidc - UiM^di = 0. These modifica- 
 tions would be made in the moment equations on page 631, and a new set of general equations 
 would be written. 
 
 Case b (Fig. 42).— For this case, Ke = K7 = K12 = Ki^ = Kn = 0] Mhc = 0; Mch = 
 SEKniOc — ^11^2). These values would be set into the moment equations on page 631 and the 
 resulting modifications would be made in the ten general equations. 
 
 00 / 
 
 c 
 
 2 
 
 10 
 
 6 
 
 3 .22 
 
 II 
 H 
 
 1 
 
 /■ 
 
 15 
 
 16 
 
 < 
 
 Fig. 43. 
 
 Fig. 44. 
 
 Case c {Fig. 43). — Equations (1) to (10), on page 631, may be used for this case by placing 
 in them K^ = Kt = K, = K12 = Ku = Kn = 0. 
 
 Sd. Two-span Viaduct Frame with Rigidly -connected Column Tie (Type III). — 
 
 The method of analysis does not differ from that of the preceding cases. The moment equations 
 are as follows (see Fig. 44) : 
 
 Mfn 
 
 
 SEKu{dF 
 
 — niidi) 
 
 
 Mfg 
 
 
 2EK,{2dF 
 
 + Og) 
 
 
 Mfa 
 
 
 2EK,{2dF 
 
 + - 
 
 2>n^d2) 
 
 Map 
 
 
 2EK,{2dA 
 
 + Bf - 
 
 2n^d'i) 
 
 Mab 
 
 
 2EKi{2dA 
 
 + dB) - 
 
 - Xab 
 
 Mba 
 
 
 2EKi{2dB 
 
 + Oa) + Yba 
 
 Mbg 
 
 
 2EKio{2dB + do - 
 
 3/110^2 
 
 Mbc 
 
 
 2EK2{2dB 
 
 + ec) 
 
 
 McB 
 
 
 2EK'i{2dc 
 
 + Bb) 
 
 
 Mch 
 
 = 2EKn{2ec + Oh - 
 
 3^11^2) 
 
 Mhc 
 
 = 2EKn{2dH + dc - 
 
 3nii(i2) 
 
 Mhr 
 
 = SEK,e{dH - niM 
 
 
 Mhg 
 
 = 2EK,{2dH + Og) 
 
 
 Mgh 
 
 = 2EKe{2dG + Oh) 
 
 
 Mgb 
 
 = 2EKio{2dG + Ob - 
 
 37110^^2) 
 
 Mgo 
 
 = SEKi,{eG - niM 
 
 
 Mgf 
 
 = 2EK^{2dG + dp) 
 
 
Sec. 14-86] 
 
 SLAB AND GIRDER BRIDGES 
 
 633 
 
 General equations: 
 
 (1) (4i^9 + 4:Ki)dA + 2K,eB + 2K,eF - QK^n^dz = Xab/E 
 
 (2) 2KidA + (4/^1 + 4K^o + ^K,)dB + 2K,ec + 2KicdG - QK,oniod2 = -Yab IE. 
 
 (3) 2^:205 + (4^2 + 4ivu)^c + 2K^xQh - QKuUudi = 0 
 
 (4) 2K9dA + (3i^i4 + 4^5 + 4:K,)dF + 2K50(? - SKi^nudi - QK,n,d2 = 0 
 
 (5) 2KiodB + 2/^:6^^^ + (4^5 + 4Xio + 4^6 + SKis)^ + 2^60^/ - 8X15^15(^1 - C)Kum,,d, = 0 
 
 (6) 2Kudc + 2K6^G + (4Kii + 4:K, + SKie)^// + ZK,,n,di - QK,,n,,d2 = 0 
 
 (7) K.miAdF + Ki5ni5^G + Ky.m^dH - {K,mx^ + X^nu^ + K,,n,^)d^ = 0 
 
 (8) Xg^A + i^io^5 + Kiidc + K,dF + i^io^G + KiiOh - {K^n9 + KwUio + Knnn)2d2 = 0 
 
 For a horizontal load at C, equation (7) would equal equation (8) would equal 
 
 and all others would equal zero. 
 
 See preceding case for directions for analysis with symmetrical loading on a horizontal 
 member. 
 
 Case a {Fig. 45). — Solution of this frame is accomplished by the above set of eight general 
 equations, by letting K 6 = ^ii = Kie = 0. 
 
 Be. One-span Viaduct Frame with Rigidly-connected Column Tie (Type IV). — 
 
 The general moment equations for this case (Fig. 46) are as follows: 
 
 Mfn 
 
 = SEKiiidF 
 
 — niidi) 
 
 
 Mba 
 
 = 2EK,{2dB 
 
 + dA) + Yab 
 
 Mfg 
 
 = 2EK,{2eF 
 
 + Og) 
 
 
 Mbg 
 
 = 2EF,,{2dB 
 
 + 60— 8n 10(^2) 
 
 Mfa 
 
 = 2EK,{2eF 
 
 + Oa - 
 
 8719(^2) 
 
 Mgb 
 
 = 2EKi,{2dG 
 
 + Ob — Sriiodi.) 
 
 Maf 
 
 = 2EK,(2dA 
 
 + Of - 
 
 8719(^2) 
 
 Mgf 
 
 = 2EK,{2dG 
 
 + Of) 
 
 Mab 
 
 = 2EKi{2dA 
 
 + Ob) - 
 
 Xab 
 
 Mgo 
 
 = 3EKi5{dG 
 
 - nidi) 
 
 0 
 
 Fig. 45. 
 
 A 
 
 B C 
 
 
 D E 
 
 
 2 
 
 
 
 n 
 
 6 
 
 n 
 
 Y\ 
 
 
 7 / 
 
 
 1 L 
 
 Fig. 46. 
 
 Fig. 
 
 General equations: 
 
 (1) (4X9 + 4X1)0^ + 2KieB + 2K,dF - QK,n,d2 = Xab/E 
 
 (2) 2KidA + (4Xi + 4Kic)05 + 2Kio^G - 6KionW2 = -Yba/E 
 
 (3) 2K,dA + (3i^i4 + 4K5 + 4:K,)dF + 2K,dG - ^Kimxdi - ^K,n,d2 = 0 
 
 (4) 2KiodB + 2K,dF + (4^5 + 4i^io + 8X15)^0 - ^Ki.mdi - QKion,od^ = 0 
 
 (5) KiiTiiidF + Ki^nudG - {Kunii^ + Kunis-)^! = 0 
 
 (6) K,dA + KioOb + KsdF + KiodG - (K,n, + Kionio)2d2 = 0 
 
 For a horizontal load at B equation (7) would equal {Q/3E); equation (8) would equal 
 i—Qhg/QE); and all others would equal zero. 
 
 For a load placed symmetrically on AB, replace the terms Xab and Yab by F/l in above 
 equations, without altering the present signs. Values of F/l for various symmetrical loads are 
 given on page 418. 
 
 8/. Four-span Viaduct Frame (Type V). — The general case here given (Fig. 47) 
 deals with a frame that is either symmetrical or unsymmetrical. A load on a single span will 
 cause a sidewise movement of the deck. The vertical component of this movement is insig- 
 nificant here, and will be neglected. 
 
 The following moment equations and general conditional equations will treat of a vertical 
 load Fi on AB; a vertical load P2 on BC] and a horizontal load Q at E. 
 
634 
 
 CONCRETE ENGINEERS' HANDBOOK 
 
 [Sec. 14-^ 
 
 Map = SEK,(dA - n,d) 
 
 Mab = 2EK,{2dA + ds) - Xab 
 Mb A = 2EKi{2dB + Oa) + Yba 
 Msg = SEK.idB - n,d) 
 
 Mbc = 2EK2{2dB + ec) - Xbc 
 
 McB = 2EK2{2dc + Ob) + Ycb 
 McH = SEK^idc - n-jd) 
 
 General equations {E is constant) for loads Pi and Q'. 
 
 (1) (3i^5 + 4.Ki)dA + 2K,dB - SK.nd = Xxb/E 
 
 (2) 2KrdA + (3^6 + 4Ki + 4X2)^5 + 2K2dc - SK.nd = 
 
 (3) 2K2dB + (3^7 + 4^2 + 4K3)^c + 2X3^ - SK^n7d = 
 
 (4) + (37^8 + 4^3 + 4:Ki)dD + 2X4^^ - ^Ksnd = 
 
 (5) 2K4dD + (4^4 + SK,)dE - 3A%n9(Z = 0 
 K^UidA + K^nedB + i^yny^c + K^n^do + K^^n-^dE 
 
 McD = 2EKs{2dc + 0d) 
 ikfi>c = 2EK^{2dD + M 
 Mzj^ = SEKsidD - nsd) 
 Mde = 2EK4{2dD + 0^) 
 M^D = 2EK4{2dE + ^i)) 
 ilf^^L = SEK^idE - ngd) 
 
 (6) 
 
 K,n,^)d = Q/3^; 
 
 When only Pi is acting, Q = 0. When only Q acts, Xab Yab = 0. When only P2 
 acts on BC, = Yba = 0; Q = 0] equation (2) equals {Xbc/E)] equation (3) equals 
 
 ( — Yqb/E). When a symmetrical load is placed on a member, X and Y for that member are re- 
 
 F F . . 
 
 placed by y of the load. Values of j for various symmetrical loads may be found on page 413. 
 
 4 £0 
 
 Fig. 48. 
 
 Fig. 49. 
 
 G " 
 
 Fig. 50. 
 
 Fig. 51. 
 
 Case a {Fig. 48). — A special case arises when = 0, or joint E rests on rollers. A solution 
 of this case may be reached by making Kg = 0 in the general equations of Type V. 
 
 Case b {Fig. 49). — This case is very common. It may be analyzed by placing Ks = 
 Kg = 0 in the general equations for Type V. 
 
 8^. Three-span Viaduct Frame (Type VI). — A solution of this frame (Fig. 50) 
 may be made from the equations given on page 633, by substituting into those equations 
 Ki = Kg = 0. It will be noted that Be drops out, and in accordance, equation (5) disap- 
 pears. In all other respects the solution is identical. 
 
 Sh. Two-span Viaduct Frame (Type VII). — General equations for this frame 
 (Fig. 51) may be set up by placing Kz = Ki = Ks = Kg = 0 in the general equations for 
 Type V. Hence, for the loads Pi and Q: 
 
 (1) (3^6 + 4:K,)dA + 2KidB - SK.nd = Xab/E 
 
 (2) 2KidA + (3^6 + 4Ki + 4:K2)dB + 2i^2^c - SK.nd = -Yba/E 
 
 (3) 2K2dB + (3X7 + 4:K,)dc - SK^nyd = 0 
 
 (4) K^n.dA + Ken.dB + K-.mdc - {K,n,^ + K^n,^ + K7m^)d = Q/SE 
 
 In these equations, when Pi acts alone, Q = 0. When Q acts alone, Pi = 0. When a 
 
 F 
 
 load is placed symmetrically on AB, Xab and Yba are replaced by j for that load (see page 413). 
 
 Illustrative Problem. — 'A viaduct frame of Type VII has two spans of 17.0 ft. each, with columns of 
 length 42.5 ft., 34.0 ft., and 29.75 ft. The moments of inertia are: /i = I2 = 80,000 in. /s = 100,000 in.*; Je = Ii 
 = 10,000 in.*. Determine horizontal reactions at F, G, and H, when a tractive force of 12,000 lb. acts along the 
 deck (as at C). From these data the following table of constants is made up: 
 
Sec. U-Sh] 
 
 SLAB AND GIRDER BRIDGES 
 
 635 
 
 Member 
 
 Length 
 (in.) 
 
 I (in.4) 
 
 T 
 
 1 
 
 3Kn 
 
 3Kn2 
 
 1 
 
 204 
 
 80,000 
 
 392 . 50 
 
 
 
 
 2 
 
 204 
 
 80,000 
 
 392 . 50 
 
 
 
 
 5 
 
 510 
 
 100,000 
 
 196.20 
 
 0.00196 
 
 1.154 
 
 0.002263 
 
 6 
 
 408 
 357 
 
 10,000 
 
 24.50 
 
 0.00245 
 
 0.180 
 
 0.004410 
 
 7 
 
 10,000 
 
 28.05 
 
 0.00281 
 
 0.236 
 
 0 . 000664 
 
 From this table of constants the four general equations were reduced to the following equations, in which 
 E = 3,000,000 lb. per sq.- in. The fourth equation has been multiplied by 3, which permits values to be drawn 
 directly from the above table. All equations were finally divided by 10. 
 
 Left-hand side of equations 
 
 Right-hand 
 side 
 
 Equation No. 
 
 OA 
 
 OB 
 
 oc 
 
 d 
 
 Constant 
 terms 
 
 1 
 
 2 
 3 
 4 
 
 215.86 
 78.50 
 
 0. 1154 
 
 78.50 
 321.35 
 78.50 
 0.0180 
 
 78.50 
 165.42 
 0.0236 
 
 -0.1154 
 -0.0180 
 -0.0236 
 -0.0003368 
 
 0 
 0 
 0 
 
 0 .0004 
 
 1' 
 2' 
 4' 
 
 1.0 
 1.0 
 1.0 
 
 0.364 
 
 4.08 
 
 0.156 
 
 1.0 
 
 0.2045 
 
 -0.000534 
 -0.000229 
 -0.00292 
 
 0 
 0 
 
 0.00347 
 
 2' - 1' = 5 
 - 4' = 6 
 
 
 3.716 
 3.924 
 
 1.0 
 
 0 . 7955 
 
 0.000305 
 0.002691 
 
 0 
 
 -0.00347 
 
 5' 
 6' 
 3' 
 
 
 1.0 
 1 .0 
 1.0 
 
 0.2693 
 0.2027 
 2.106 
 
 0.0000822 
 0.000685 
 -0.0003008 
 
 0 
 
 -0.000884 
 0 
 
 3' - 5' = 7 
 - 6' = 8 
 
 
 
 1.8367 
 1.9033 
 
 -0.0003830 
 -0.0009858 
 
 0 
 
 0 . 000884 
 
 7' 
 8' 
 
 
 
 1.0 
 1.0 
 
 -0.0002085 
 -0.0005170 
 
 0 
 
 0 . 000464 
 
 8' - 7' = 9 
 
 
 
 
 -0.0003085 
 
 0.000464 
 
 Solving equation (9) gives d = — 1.503 (inches lateral shifting). 
 
 ec = - 0.0003135 (rotation of C in radians). 
 
 Ob = + 0.0002078 (rotation of B in radians). 
 
 Oa = - 0.000876 (rotation of A in radians). 
 
 Map ^ E = SKsdA - SKsn^d = - (588.6) (0.0008786) + (1.154) (1.503) = 1.223 
 Mbg -i- E = SKedB - SKened = (73.5) (0.0002078) + (0.18) (1.503) = 0.2863 
 McH ^ E = SKjdc - ZKimd = - (84.15) (0.0003135) + (9.236) (1.503) = 0.3286 
 
 Map = 3,669,000 in.-lb. Hp = 7,190 lb. 
 
 Mbg = 858,900 in.-lb. Hg = 2,110 1b. 
 
 McH = 985,000 in.-lb. Hh = 2,760 lb. 
 
 Total to check = 12,060 lb. (error 0.5%) 
 
636 
 
 CONCRETE ENGINEERS' HANDBOOK 
 
 [Sec. 14-8^ 
 
 Case a (Fig. 52).— The solution may be made by using the general equations of Type VII, 
 making Kr = 0. 
 
 Case b (Fig. 53) —In this case = Kt = 0. Substituting these into the general equations 
 of Type VII, the solution is found at once. 
 
 Bi. One-span Frame, Unequal Columns (Type VIII). — The solution for this 
 frame (Fig. 54) may be obtained from the equations of Type VII, by making K2 = Kt = 0. 
 This will eliminate equation (3), and modify the other equations. 
 
 B 
 
 I 2 £2 
 
 ' \ 
 
 G 
 
 Fig. 52. 
 
 4 
 
 £S T 
 
 E 00 
 
 G 
 
 Fig. 53. 
 
 Fig. 54. 
 
 8;. Temperature Stresses. — Stresses caused by changes in temperature may 
 become very large, and require thorough irivestigation. They come about from the fact that 
 the members tend to change length, whence each causes a lateral displacement of a member at 
 right angles to it. 
 
 The following analysis will be made for a frame of Type I, in which the lower tier of columns 
 increase in length toward the left (see Fig. 55). We will note later that this was to cause all 
 values of A to be positive; and will then give the effect upon the resulting equations when A 
 
 is negative. The change of temperature is as- 
 sumed in the development to be positive. 
 Later, corrections will be noted for cases when 
 the change is negative. 
 
 If the rise in temperature is then for a 
 material having a coefficient of expansion of C t, 
 the increase in a length Z is + Cttl. Its value 
 is negative for a drop in temperature. This 
 increase in length should be computed and 
 tabulated for each member. 
 
 Suppose for the present that the column 
 LT is held vertically during a rise in tempera- 
 ture. Then the horizontal movement of any 
 joint to the left of L, will equal the elongation occurring between L and that joint. LT does 
 not really remain vertical, however, since L will move to the right an amount dependent 
 upon the elastic rigidity of the frame. Hence the horizontal movement of any joint to the 
 left of L, will be equal to di minus the elongation occurring between L and that joint. The 
 shifting of the frame causes flexure in the columns, which in turn develops horizontal reac- 
 tions at the bases. The sum of these horizontal reactions must equal zero. 
 The moment equations for the frame are as follows: 
 
 Maf 
 
 
 2EK,{2eA 
 
 + dp - 
 
 
 Mde 
 
 = 2EKi{2dD + Be - 
 
 3n4A8) 
 
 Mab 
 
 
 2EK,{2dA 
 
 + 6b - 
 
 3niA5) 
 
 Med 
 
 = 2EK^{2Be + Bd - 
 
 3n4A8) 
 
 Mba 
 
 
 2EKi{2dB 
 
 + eA - 
 
 3niA5) 
 
 Mel 
 
 = 2EKu{2Be + Bl - 
 
 3^13C^2) 
 
 Mbo 
 
 
 2EKioi2dB 
 
 + 60 + ^niodi) 
 
 Mle 
 
 = 2EKiz{2Bl + Be - 
 
 37113^2) 
 
 Mbc 
 
 
 2EK2{2dB 
 
 + ec - 
 
 3n2A6) 
 
 Mlj 
 
 = 2EKs{20l + Bj - 
 
 3n8A8) 
 
 McB 
 
 
 2EK2{2dc 
 
 + dB - 
 
 3n2A6) 
 
 Mlt 
 
 = SEKisiBL - nisdi) 
 
 
 Mch 
 
 
 2EKn(.2dc 
 
 + Bii - 
 
 - 3niid2) 
 
 MjL 
 
 = 2EKs{2Bj + Bl - 
 
 3n8A8) 
 
 McD 
 
 
 2EKz(2dc 
 
 -\-dD - 
 
 3M3A7) 
 
 MjD 
 
 = 2EKu(2Bj + Bd - 
 
 - 3ni2(i2) 
 
 Mdc 
 
 
 2EK3{2dD 
 
 ^Bc - 
 
 
 Mjs 
 
 = SEKn[Bj - nxi{di 
 
 -es)] 
 
 Mdj 
 
 
 2EKu{2eD + dj - 
 
 - 3^126^2) 
 
 Mjii 
 
 = 2EK^{2Bj + Bh - 
 
 - 3n7A7) 
 
Sec. 14-8i] 
 
 SLAB AND GIRDER BRIDGES 
 
 637 
 
 Mhj 
 
 = 2EK^{2dH 
 
 ■\-ej - 
 
 
 
 Mgf = 2EK,{2eG + dp - BnsAs) 
 
 Mhc 
 
 = 2EKn{2dH + dc - 
 
 - 3niic?2) 
 
 
 MiPG = 2EK,{2eF + dG - B^sAs) 
 
 Mhr 
 
 = SEKieidn 
 
 — nu{di 
 
 - 68 — 
 
 67)] 
 
 Mfa = 2EK,{2dF + 0a - Snad,) 
 
 Mhg 
 
 = 2EKe(2dH 
 
 + dG - 
 
 
 
 Mf,v = 3£'i^i4[0f — nii{di — es — 67 — 66 — 
 
 Mgh 
 
 = 2EKe{2dG 
 
 -\- Oh - 
 
 
 Mfmu 
 
 14 + MgoUx^ + MhrUig + MjsUn -\-MLTrii8 ■ 
 
 Mgb 
 
 = 2EKio{2dG 
 
 + eB - 
 
 ■ 3^10^2) 
 
 Map + Mfa + Mbg + Mgb + Mch + M//c + 
 
 
 
 
 
 
 Mdj + Mjz) + Mel + Ml£ 
 
 Mgo 
 
 = ^EKi,[dG ■ 
 
 
 — 68 - 
 
 ei - 66)] 
 
 
 General equations: 
 
 (1) (4i^i + 4i^9)0A + 2KxeB + 2K,dF- QK,n,d2 = GZiniAg 
 
 (2) 2Ki0A + (4Ki + 4^2 + 4Zio)0B + 2K2 dc+ 2KiodG - QKi^nxod^ = GXanaAe + GKiniAs 
 
 (3) 2K2dB + (4^2 + 4^3 + Kxi)ec + 2X3^ + 2KiieH - 6Knnud2 = 6K2n2A6 + QK^n^Aj 
 
 (4) 2K30C + (4^3 + 4K4 + 42^12)^0 + 2KidE + 2Ki2dj - QKi^nizdz = QKm^^A^ + GKsnaA? 
 
 (5) 2X4^ + (4^4 + 4.Kx^)dE + 2KxzdL - QKun,zd2 = 6K4n4A8 
 
 (6) 2KueE + (3i^i8 + 4^8 + 4:Kis)dL + 2^8^/ - SKi.nisdi - QKizUizd^ = GXsWsAs 
 
 (7) 2Ki20z) + (3Ki7 '+ 4^7 + 4Ks + 4Ki2)0j + 2^70// + 2K,dL - 2>K,,nudi - 
 
 6Ki2ni2d2 = QK^nrA^ + Gi^s^^sAg — 3Ki7ni768 
 
 (8) 2Kndc + (3Zi6 + 4^6 + 4^7 + 4:Ku)dH + 2K,dG + 2K7dj - ZK^mxdi - 
 
 6KiiniiC?2 = GKeneAe + 6K7n7A7 — BKieWieCe? + 63) 
 
 (9) 2i^io0fi + (3Ki5 + 4^5 + 4^6 + 4Xio)0G + 2^5^^ + 2K,eH - SK,,ni,d, - 
 
 QKioniod2 = GKsnsAs + GKeneAe — 3X15^15(66 + 67 + eg) 
 (10; 2Kc>dA + (3J^i4 + 4^5 + 4:K,)dF + 2X5^0 - 2>Ki^ni4i - QK,n^d2 = 6K5n5A5 - 
 
 3i?i4ni4 (65 + 66 + 67 + es) 
 
 (11) KiiniiOF + KuUi^dG + KiGTiiedH + KnUnOj + KisTIisOl - d{KuniA'^ + i^is^is^ + Ki6ni62 
 + Ki7ni72 + Kisnis^) = - [^68 2)14 -K^' + e7 Xn^ + ee Kn^ + 65(Ki4ni42)] 
 
 (12) QK^Oa + 6Kio0B + QKudc + 6i^i20z) + Gi^is^F + QK^dp + 6i^io% + QKudn + 
 
 6i^i20j + 6Ki30L - 2d(K,n, + i^ionio + Kunn + ^12^12 + Xis^u) = 0 
 
 The left-hand side of the above equations is identical to that of the equations in Table 
 I. All values on the right-hand side are known as soon as one assumes a given change in tem- 
 perature and computes the corresponding changes in length. They may be put into a separate 
 column, similar to Case I, Table I, for instance. Solution of this column may be made after 
 some one other case has been solved. Thus, in the problem on page 635, if a column for tempera- 
 ture change is added to the right-hand side, its solution could be made by following the pro- 
 cedure noted in the column headed Equation No." at the extreme left of the table. 
 
 In the foregoing equations, it was assumed that all values of A were positive. For instance, 
 before the rise in temperature, F and G were on the same level. After the temperature change 
 the final position of G is helow the final position F by an amount equal to + A5, since to attain 
 this sloped position the member FG would have to rotate in a positive direction about either F 
 or G. Suppose, however, that GO for a given case is longer than FN. Then the final position 
 of G would be above the final position of F by an amount equal to — A5. Hence, for such a 
 case, all terms involving A5 in equations (1) to (11), above, would become negative. The 
 following rule may therefore be stated: Beginning at the left end of the frame after distortion 
 from temperature change has taken place, if a normally horizontal member slopes downward 
 to the right, A for that member is positive, and its value is equal to the vertical displacement 
 of its right end. If the member slopes upward to the right, A for that member is negative, 
 and its value is equal to the vertical displacement of its right end. This relation between 
 the final positions of points holds for either a rise or fall of temperature. If all columns of the 
 structure have the same length, then all values of A are zero. 
 
 The value of e in the foregoing equations would become negative for a fall of temperature. 
 
638 
 
 CONCRETE ENGINEERS' HANDBOOK 
 
 [Sec. 14-8/c 
 
 8k. Effect of Fixed Bases. — Certain modifications made in the foregoing equa- 
 tions will permit them to be used for the analysis of frames of the type which they affect, but 
 with fixed, instead of hinged bases. Take, for instance, a portion of the frame of Type I 
 (see page 629). There is no rotation at (Fig. 56) when N is perfectly fixed, hence 
 
 Mfn = 2EKi4 (2dF - 3ni4i) 
 
 This would replace that for Mfn given on page 629. Equations precisely similar, except for 
 subscripts, may be written for the moment in the top of all other lower columns of the frame. 
 
 These values of moment would replace corresponding ones in the equations on 
 pages 629 and 630. Having made these changes in the moment equations, the 
 general conditional equations will be affected thereby, and must be revised ac- 
 cordingly. 
 
 After the conditional equations have been solved, the moments at the ends of 
 ■ each member may then be found. The moment at the base of the fixed column 
 
 N FN above, is 
 
 Fig. 56. Mnf = 2EKu{dA - ^riidi) 
 
 = y2{MFN + QEKunudr) 
 
 Having found the moment at the ends of each column, the point of inflection may readily 
 be obtained; and, supposing a hinge to be introduced at the point of inflection, the horizontal 
 reaction at this ''hinge" — or shear on the column — may be found as for a frame with hinged 
 bases. The vertical reactions may likewise be found as for hinged bases. 
 
 The same modification to take into account fixity of the base of supporting columns may 
 be applied to any other of the foregoing types of viaduct frame. 
 
 Bl. Viaduct Bent. — The cross-frame or bent, provides the lateral stiffness for 
 the viaduct structure, in addition to being the supporting unit. It should be designed to with- 
 stand: (1) the dead load of the entire structure; (2) the direct and flexural stresses set up in 
 the "columns" of the viaduct frame, due to live load, as determined in the foregoing discussion; 
 (3) lateral forces of wind, and centrifugal forces on curves; (4) lateral expansion; (5) moments 
 due to loads on floor girder; (6) moments due to overhanging floor beams carrying walks, etc. 
 
 For bents having a batter steeper than 1 to 6, the procedure of analysis for the first five 
 cases of loading given above is identical with that for the viaduct frame of similar dimensions. 
 Frames with excessive batter, however, require consideration of the vertical component of 
 lateral displacement due to lateral forces or to underbalanced loads (see assumptions, page 
 628). The relation is, of course, purely trigonometrical, so that the 
 method of attack is as already outlined, save for the adding of this 
 component of lateral movement to the lateral displacement of the ends 
 of horizontal members. 
 
 When the bent is symmetrical and carries lateral loads at the 
 joints, there is a point of inflection of moment in each horizontal mem- 
 ber at the point where it is intersected by the axis of symmetry. For 
 the analysis of such a frame see "Modern Framed Structures" by 
 Johnson, Bryan and Turneaure, Part II, Arts. 283-4. 
 
 The sixth case of loading — that of a known moment applied at a joint may be treated by 
 a slight modification to the foregoing equations. Fig. 57 shows a bent or cross-frame of Type 
 III, of viaduct frames, with a cantilever at A applying a known moment M'. Referring now 
 to page 632, the equations there given for Mab and Maf are equally true for this case. It will 
 be noted, however, in forming equation (1) on page 633, where no load is acting on AB, that the 
 sum of the moments in the A-ends of members meeting at A is zero. In this case where M' is 
 acting, equation (1) would now equal M', since the moments in the members must offset the 
 external moment at any joint. M' as shown in the figure is negative on the right-hand side of 
 the equation, since the moment required to resist it is clockwise. 
 
 / 
 
 2 
 
 9 
 
 10 
 
 F 5 
 
 G 6 
 
 14 
 
 i 
 
 IS 
 
 A 
 
 N 
 
 0 
 
 Fig. 
 
 57. 
 
Sec. 14-9] 
 
 SLAB AND GIRDER BRIDGES 
 
 639 
 
 CANTILEVER BRIDGES 
 
 A type of bridge which in appearance is a concrete arch, but which in reaHty is composed 
 of balanced cantilevers, is shown in Figs. 58 to 61 inclusive. A structure cf this type can be 
 made with longer spans than the ordinary girder and is suited to locations where the real arch 
 would be exceedingly costly on account of unsatisfactory foundation conditions. 
 
 9. Theory of Design. — A pier and the cantilever arms on each side compose a unit, the 
 arms being balanced for dead load and for full live load. The piers are designed for bending 
 
 F/n/shed sc/rface 
 
 of Roadway , "'^ 
 
 Typical Elevation of 
 Inside 35ft. Cantilever Beam 
 
 Short strc!i^t7t bars in 
 top and bottom at Jo/nt, ;s'^c toe 
 ->![<■• ,// /^/ternate bars bent. 
 
 Section 
 
 Fig. 58. — Details of Hopple Street viaduct, Cincinnati, Ohio. 
 
 due to the maximum eccentric load that can be applied, and considering the load on only one 
 of the cantilever arms at a time. The pier footings are designed so that the pressure on the 
 base of a pier due to this same eccentric loading will not cause an intensity greater than the 
 unit bearing value of the soil. 
 
 10. Examples of Cantilever Bridges. — The viaduct shown in Fig. 58 consists of twenty- 
 five skewed spans, each span comprising two curved cantilever arms supported on reinforced- 
 
 Cxpansion Joint 
 
 Detail of Expansion Joint 
 
 I Line of 
 V- Bridge 
 
 Fig. 59. 
 
 -Half-elevation and detail of expansion joint of arch-shaped cantilever bridge over Rouge River, Wayne 
 
 County. Mich. 
 
 concrete piers. A single cantilever arm occurs at each end of the viaduct. Each cantilever 
 arm comprises four curved ribs which were designed as cantilevers from the skewed piers. 
 The joint at the center of each span is shown in detail. In designing, this joint was considered 
 as transmitting only shear from one cantilever arm to another and not any bending or arch 
 action. Since each pier and its cantilever arms are symmetrical about the center line of the 
 pier, no bending exists in the pier due to dead load or to full live load on both cantilevers. 
 Piers, footings, and piling were designed to withstand the overturning effect produced by the 
 loading of a single cantilever with the full live load. 
 
 The expansion joint at the center of the structure, shown in Fig. 59 is entirely different 
 from that employed in the bridge just described. It should be noted, however, that the same 
 
640 
 
 CONCRETE ENGINEERS' HANDBOOK 
 
 [Sec. 14- 
 
 U //'■o^'' ••••>1 
 
 Section B-B 
 
 Section A-A 
 
 -^1 A k£4"->^^S^. 
 ->^^j"°Bars e"c.toC 
 both vrays 
 
 SJ" 6'o"c.foc. 
 
 One Layer 
 Tar Paper 
 
 £'-o"c.t>c. 
 
 -/"''rl. Bars- Bent at i po/nt ^-f'Tir.ffa/^ 
 
 J-0' 
 
 .AM 
 
 Cross Section D-D Post 
 
 Fig. 60, — Details of Runnymede Avenue bridge over West Fork Creisk, Cincinnati, Ohio. 
 
Sec. 14-10] 
 
 SLAB AND GIRDER BRIDGES 
 
 641 
 
 result is accomplished — that is, that the two abutting arms are permitted to move longitudin- 
 ally, but not laterally or vertically. The joint consists of two 3-ft. piers of 2^-in. steel shafting 
 each inserted in two 13-^-ft. pieces of 3-in. gas pipe embedded in the concrete. End joints 
 were made by placing sheets of three-ply tar paper on top of the abutments before the end can- 
 tilevers were poured, thus permitting a slight movement of the ends of the structure under 
 changes of load and temperature. There was no apparent deflection at any of the expansion 
 joints due to live load. 
 
 The joints at the ends of the bridge shown in Fig. 60 are of the same type as employed in 
 the viaduct spans of Fig. 58. The absence of such a joint at the center of the middle span is, 
 however, the principal feature. In spite of this continuity between piers, no account was 
 taken of continuous action upon supports and the bridge was designed in the same manner as 
 the cantilever viaduct previously referred to. In fact, it has been found that a joint at the 
 center of span is an unnecessary refinement in cantilever-bridge design, and might be omitted. 
 
 Perspective View 
 
 Field Splices 
 
 Field Splices 
 
 Fig. G1.- 
 
 Th/s crossbeam 
 
 occurs at the ^^^^^ ^^.^^^ 
 
 Quarter and Half Points ^ 
 
 Half Section at Quarter and Half Points of Span 
 -Details of Washington Street bridge, Norwalk, Conn. 
 
 The bridge was designed and constructed so as not to rest on the abutments at all, the abut- 
 ments being used merely to hold back the earthfill at each end and to serve as anchorage for the 
 end cantilevers. A structure of this type with end openings closed by earthfill has all the ap- 
 pearances of a real arch. In such a case abutments are not needed. 
 
 If a number of bridges similar to the bridge of Fig. 60 were placed end to end, the result 
 would be essentially a structure such as shown in Fig. 61. It should be noted that the Nor- 
 walk bridge is continuous in sections of maximum length of 100 ft. The structural-steel work 
 was designed to be self-supporting during erection and to carry the erection stresses of the 
 forms and the fluid concrete in the ribs, cross-girders, and sidewalk brackets. Although a 
 deflection of 3^ in. at the free ends of the cantilevers was anticipated, a deflection of only 
 to ^8 in. actually resulted due principally to the rigidity of the forms and to the fact that the 
 concrete was continuously setting during the process of placing. The combined steel and 
 concrete in the ribs was proportioned to carry the roadway slab, paving, and all live loads. 
 The trusses were proportioned to carry all shear not safely taken by the concrete but were not 
 proportioned to carry all the tension developed by the bending moment since extra horizontal 
 rods were embedded in the concrete adjacent to the top chords of the trasses, 
 41 
 
SECTION 15 
 
 CONCRETE FLOORS AND ABUTMENTS FOR STEEL BRIDGES^ 
 
 1. Concrete Floors on Steel Bridges. — Within recent years it has become the general 
 opinion among railroad engineers that a ballasted solid floor is the most satisfactory form of 
 floor for steel bridges. Perhaps the best type of such a floor is the reinforced-concrete slab 
 resting directly on the steel-floor members. 
 
 Figs. 1 and 2 show details of reinforced-concrete deck slabs for plate-girder spans. The 
 slab floors are seen to rest directly upon the top flange of the steel girders. Comparison of the 
 two designs is of value since they show a wide difference in the concrete details and in the ar- 
 rangement of the reinforcement. The slabs are usually made at some convenient location and 
 hoisted into place when suflficiently cured. Before adding the ballast, the upper surface of the 
 slabs is thoroughly waterproofed by painting with tar paint. Drain holes are placed in such 
 a position as to keep the drip clear of the steel members. 
 
 Fig. 1. — Reinforced-concrete deck for steel-girder spans on D. M. & N. Ry. 
 
 A reinforced-concrete floor for a through plate-girder bridge is shown in Fig. 3. The con- 
 crete of the floor slab is seen to extend up on the sides to form curbs, and these curbs extend 
 entirely around the gusset plates. Steel trough floors filled with concrete are also used in 
 through plate-girder bridges. 
 
 Steel I-beams encased in concrete and supporting a reinforced-concrete floor slab is the 
 most common type of highway bridge with steel-floor members. On account of the ease with 
 which forms may be constructed to hold the concrete, this bridge for short spans is sometimes 
 
 * For treatment of concrete piers for steel bridges see "Foundations of Bridges and Buildings," by Jacoby 
 and Davis, or "Structural Engineers' Handbook," by Ketchum. 
 
 643 
 
644 
 
 CONCRETE ENGINEERS' HANDBOOK 
 
 [Sec. 15-1 
 
 Half Section Single Track Slab Half Section • inside Slab 
 
 End View 
 
 Fig. 2. — Standard reinforced-concrete slab for deck girders of C. M. & St. P. Ry. 
 
 j" Pods, iz'c foe 
 
 ::T^f^yj:T4:ir-r4:t:^h ' 
 
 .1-4-1- \-\-\-..- Z' Drain Ho/es ■■. 4- j- 
 
 Sectional Plan 
 
 - /4-8" 
 
 Section A-A 
 
 Sheet Zinc 
 
 Part Longitudinal 
 Section of Floor 
 
 Fig. 3. — Reinforced-concrete floor for through plate-girder bridge, C. B. & Q. R. R. 
 
 '^"°Bar over each floor beam 
 j "° Bars • 3 per panel 
 
 / i"''Bars,9''civc.- 
 
 ^^^"°Bars.9"cioc 
 
 8-9' 
 
 _ , Lonqitudinal Section of Floor 
 
 18 Roadway • ^ 
 
 ■ Weephole yyith 
 perforated cover 
 
 '■55*1 
 
 19'- O"— 
 
 Section 
 
 Fig. 4, — Concrete slab floor for highway steel truss span, Iowa Highway Commission, 
 
Sec. 16-2] 
 
 CONCRETE FLOORS AND ABUTMENTS 
 
 645 
 
 used in preference to slab bridges of all concrete. The only disadvantage of this bridge is in 
 point of economy. 
 
 Timber floors for highway bridges are not in great favor at the present time. Since the 
 expense of maintaining wood floors is considerable, the engineers of a number of highway com- 
 missions design practically all-steel bridges with concrete floors covered by a wearing surface 
 of gravel or macadam. For exceedingly light traffic on country bridges, driving is sometimes 
 
 allowed directly on top of the floor slab, mak- 
 ing an allowance of at least 1 in. in the thick- 
 ness of the slab for wear and cutting trans- 
 verse grooves to prevent slipping. Figs. 4 
 and 5 show typical designs of reinforced- 
 concrete floors for steel-truss spans. 
 
 2. Abutments for Steel Bridges. — An 
 abutment in its simplest form is a retaining 
 wall terminating the approach embankment 
 to a bridge, and provided with a bridge seat 
 for the end of the first span to bear upon. 
 The discussion here given is limited to those 
 
 Fig. 5. 
 
 fVeep fro/e yy/th perforafecf cover 
 
 ,-3 nif 
 
 -Concrete slab floor for highway steel-truss span, 
 Iowa Highway Commission. 
 
 abutments of plain or reinforced concrete which receive a vertical downward bearing from 
 the bridge. Abutments for arches are treated in Arts. 4 and 39, Sect. 16. 
 
 3. Types of Abutments. — Bridge abutments of concrete may be classified according to 
 general form as follows: 
 
 1. Pier abutments. 
 
 2. Wing abutments. 
 
 3. Cellular abutments. 
 
 4. U-abutments. 
 
 5. T-abutments. 
 
 6. Buried pier abutments. 
 
 7. Skeleton and arched abutments. 
 
 Base of rail 
 
 The design and advantages of each form will be discussed separately. 
 
 4. Pier Abutments of Plain Concrete.— This is the simplest form of abutment (Fig. 6). 
 Since many of the other forms are elaborations of this one, its stability will be studied in detail. 
 
 The thrust P of the earth against the back is found from the method of equivalent surcharge 
 h!^ as described on page 581. The force F 
 is due to frost expansion, and depends 
 upon the depth to which the ground or 
 ballast may freeze. It can at best only be 
 estimated, on the basis of ice pressures. 
 
 The vertical load of the trusses or 
 girders with their live load, cause two 
 forces B bearing on the bridge seat. The 
 dimensions s, c?, and n depend upon the 
 structure supported. The intensity of B 
 depends upon the span and loading, and 
 is taken from the design of the super- L 
 structure. The force T may either be ^ 
 caused by the tractive effort of the train 
 on the bridge; by braking of the train; 
 or by temperature changes not wholly 
 adjusted by poorly operating expansion joints 
 design of the superstructure. 
 
 The best form of abutment is that which puts the resultant pressure very close to the center 
 of the base. This requirement is particularly desirable in yielding soil, since the vibratory 
 loads will nearly always cause settlement. Many abutments have tipped forward notice- 
 
 FiG. 6. — Plain concrete-pier abutment. 
 
 All of these forces are determined from the 
 
646 
 
 CONCRETE ENGINEERS' HANDBOOK 
 
 [Sec. 15-5 
 
 ably because the pressure caused a non-uniform pressure on the soil, and hence a non-uniform 
 settlement. 
 
 The back wall should have a thickness a at its top of at least 12 in., and more for railway 
 bridges. Owing to the uncertainty of actual freezing forces, the thickness c at the bottom of 
 the back wall may be taken as 0.4/ to 0.45/, the larger value for railway structures. These 
 thicknesses have given good results in practice for back walls not reinforced. 
 
 The length of the back wall should be such that when material spills around its ends at a 
 slope of 1}^'2 to 1, it should not strike the pedestal of the bearing shoe. The batter of the back 
 face of the back wall should be about 2 in 12. The height / is of course set by the superstruc- 
 ture. The length w of the bridge seat is governed by the overall dimensions of the structure 
 being supported. J. E. Grenier^ specifies a minimum thickness of the coping as 18 in. for 
 railway, and 12 in. for other bridges. He also specifies the distance d to be "at least 12 in. 
 greater than required for the bedplates of steel superstructures." 
 
 It is a general rule that, below the coping, the plain concrete abutment shall have a thick- 
 ness at any point of from 0.4 to 0.5 of the depth of the point below the base of rail. The thick- 
 ness m should be such that the compressive unit stress at the forward edge is within that allowed 
 for plain concrete. This stress is determined precisely as though the wall rested on the soil 
 at that plane. 
 
 The base slab should have a forward projection o sufficient to keep the forward soil pressure 
 within the allowable range. The thickness of such a projection will determine the thickness t 
 of the slab. It is found the same as that for the toe of a reinforced retaining wall. The whole 
 slab should preferably be reinforced. 
 
 The body of the wall should be well bonded by dowel rods to the base slab, so that it can 
 neither rock nor slide upon it. 
 
 5. Pier Abutments of Reinforced Concrete. — Pier abutments of reinforced concrete are 
 divided into two general groups: (1) buttressed, and (2) counterforted. Fig. 7a shows the 
 
 general form of a buttressed abutment. The back wall is designed 
 as a cantilever wall, while below the bridge seat, the back slab is 
 designed as a continuous slab spanning horizontally the space 
 between buttresses. The back slab must be well anchored to the 
 base slab to prevent rocking forward on the base; and ample steel 
 should be pr9vided at the junction of the upper portion of the 
 wall with the base slab to prevent failure in shear (sliding) on the 
 plane of the top face of the base slab. Reinforcing in the body of 
 the buttress is very light, and is necessary only to prevent cracks. 
 
 Fig. 7b shows a counterforted pier abutment. The back 
 wall, as before, is designed as a cantilever wall. The whole 
 structure below the bridge seat is designed as a counterfort retain- 
 ing wall, and the discussion of the design of that wall will apply directly here. 
 
 It is of course the best arrangement to place the counterfort or buttress immediately 
 beneath the bridge seat, and at such other points as is necessary to provide against the thrust 
 of the earth. 
 
 6. Wing Abutments. — If wing walls are extended out beyond the ends of the bridge seat 
 of a pier abutment, the structure is called a wing abutment. The wings may either be on a 
 line with the face of the abutment, or deflected backward from the face (Fig. 8). The straight 
 wing is used where dry crossings are made, as for instance, street or railroad crossings. They 
 usually extend to the foot of the supported embankment, and are caped about 2 ft. above the 
 surface of the slope. When the toe of the slope at the end of the wing wall requires protection 
 from stream flow, deflected wings are usually built. They are usually put at 30 deg. with the 
 face wall. 
 
 The wing walls are designed for earth thrust, as in a retaining wall. 
 
 Fig. 
 
 7. — Reinforced-concrete 
 pier abutments. 
 
 The best design is 
 
 1 Gbenier's, "General Specifications for Bridges." 
 
Sec 16-7] 
 
 CONCRETE FLOORS AND ABUTMENTS 
 
 647 
 
 one which gives the same intensity of pressure over the base as is developed under the body of 
 the abutment; and it is particularly desirable that the resultant pressure on the base of the wing 
 wall cuts at relatively the same point as that of the body of the wall. Expansion joints are 
 often placed at the junction of the wing with the body. Such joints should be lock-joints so 
 that uneven tipping or settling will not cause unsightly offsets to develop between the wings 
 and the body. Where no joints are provided, especial attention should be paid to the similarity 
 of pressure on the entire foundation, as noted above. 
 
 Because of the desire not to obstruct stream flow, reinforced-concrete wing abutments are 
 usually of the counterfort type rather than the buttress type. Such a wall is shown in Fig. 9. 
 The body of the wall is designed like the same form of pier abutment. The counterforts of 
 the wing walls may either be parallel to the track, or normal to the wings. 
 
 Fig. 8. — Plain concrete wing abutment. 
 
 7. Cellular Abutments. — The cellular abutment, like the cellular retaining wall, consists 
 of a box-shaped pocket buried in the fill, to increase stability against overturning. A modified 
 form consists of a pier abutment, with wings running normal to the face of the abutment, and 
 a tie wall across the outstanding ends of these wings. Such an abutment is more costly than 
 the pier form, and has not been in common use. 
 
 8. U-abutments. — A very common form of abutment, called the U-abutment because of 
 its shape, is shown in Fig. 10. It consists of the face wall with bridge seat, and two wings at 
 right angles to the face wall. The pocket thus formed has a floor formed by the footing slab. 
 Such a type of abutment is very useful at the end of a moderately high fill which has a long 
 slope in the direction of the track. This form of abutment is usually of plain concrete and is 
 cast in one piece. Its total length is generally about 13-^ times its height. 
 
 The constituent parts of the U-abutment are designed like a pier abutment or retaining 
 walls. The outside faces are either vertical or slightly battered. The inside faces are either 
 heavily battered or stepped, the latter being the more common. The fill is allowed to slope 
 from the top of the outer ends of the wing walls in all directions away from the track at about 
 IK to 1. The inner faces should be battered 2 in 12 for the upper 3 ft., to provide for frost 
 expansion. The whole abutment should be thoroughly drained. 
 
 9. T-abutments. — This form of abiitment is shown in Fig. 11. It is of practically the 
 same cost as the U-abutment. The front face and stem are usually of plain concrete and the 
 floor over the stem of reinforced concrete. The wall is secure against tipping forward. 
 
648 
 
 CONCRETE ENGINEERS' HANDBOOK 
 
 [Sec. 15-10 
 
 10. Buried-pier Abutments. — Usually when extremely high abutments are required, or 
 when the abutment extends to considerable depth for suitable foundation, the earth slope may 
 be allowed to spill freely around the abutment. For this purpose a tall pier may be used, with 
 short wing walls to protect the bridge seat from the earth slope. An abutment of this type is 
 called a buried-pier abutment. It should be designed for the vertical reaction of the span, 
 
 40-7' - - »l< - - 40-7"- 
 
 Elevation 
 
 Section L-Z 
 near 4 of track 
 
 Fig. 9. — Reinforced-concrete counterfort wing abutment. 
 
 and the lateral earth and traction forces. It is especially desirable that the resultant pressure 
 on the base should pass through its center, particularly when the subsoil is yielding. The 
 economy of this abutment makes it more desirable than the wing abutment for high embank- 
 ments. 
 
 11. Skeleton and Arched Abutments. — A large number of special forms of abutments 
 have been developed recently, the most notable from an economic standpoint being the skeleton 
 
Sec. 15-12] 
 
 CONCRETE FLOORS AND ABUTMENTS 
 
 649 
 
 and arched abutments. The skeleton abutment consists of a very heavy viaduct frame of two 
 or more spans, upon the outer end of which is placed the bridge seat. The arched abutment 
 differs only in that it is a series of small arches on high walls. The proper length of these 
 abutments is governed by the cost per additional foot of abutment and of superstructure. 
 
 The analysis of the skeleton abutment does not differ from that of a viaduct frame (see 
 Art. 8, Sect. 14). The arched abutment may be analyzed as a system of arches; or more easily 
 by an approximate solution of a rigid viaduct frame of similar proportions, having girders of a 
 section equal to that at the crown of the arches. This is of course approximate; but limiting 
 conditions may be assumed, remembering that if the girder section is assumed too small the 
 
 ■43-6" 
 
 Side Elevation 
 
 j<-.7£5T-I_7<.6'!.j 
 
 Half |<-y(7-5'-->l 
 
 ,,_if-n 1 r-r7^» Section | Half 
 
 '-^'^■:^..eL3fl^.^a'-3k''\^^- ■ A-A front Elevation 
 
 Plan 
 
 Fig. 10. — U-abutment. 
 
 ■40-0" 
 
 ■5-^-^ 
 
 Side Elevation 
 
 K le-o" >j 
 
 Gas pipe dram-^ 
 
 End ml! 
 
 B 
 
 Cross section 
 /\-B 
 
 Fig. 11, — T-abutment. 
 
 stresses in the columns or bents will be too large, and vice versa. The deck girders may be ana- 
 lyzed as though continuous, and then their under sides arched, for rigidity and for appearance. 
 Two very important considerations should be borne in mind, namely, tractive or braking 
 forces, and temperature variation. 
 
 12. Care in Constructing Abutments. — In abutments, as in retaining walls, special care 
 should be taken to obtain a structure that is impervious to water. Fills and embankments 
 accumulate water, which will seek outlet through the abutment. Unless the wall is of proper 
 density throughout, without reliance on a smooth face, disintegration along percolating planes 
 and later partial or total destruction is sure to take place. 
 
 For valuable material on the economic selection and design of abutments see J. H. Prior, 
 Proc. A. R. E. A., vol. 13 (1912), page 1085. 
 
SECTION 16 
 
 ARCHES 
 GENERAL DATA 
 
 1. Definitions. — The following are some of the common technical terms applied to the 
 various parts of an arch (see Figs. lA and IB). 
 
 Soffit. — The under or concave surface of an arch. 
 Back. — The upper or convex surface of an arch. 
 
 Skewback. — The surface upon which the end of the arch rests. This definition applies particularly to the 
 stone or brick arch since the surface mentioned is purely imaginary in the case of the concrete arch. The term, 
 however, is useful in concrete-arch analysis. 
 
 Springing Line. — The line in which the soffit meets pier or abutment — that is, the inner edge of the skewback. 
 
 Span. — The horizontal distance between springing lines measured parallel to the center line of roadway. 
 
 Intrados. — The line of intersection of the soffit with a vertical plane taken parallel to the center line of roadway. 
 
 Extrados. — The line of intersection of the hack with a vertical plane taken parallel to the center line of roadway. 
 
 Crown. — The highest part of the arch ring. 
 
 Rise. — The height of intrados at crown above level of springing lines. 
 
 Haunch. — The portion of the arch ring about midway between the springing line and crown. 
 Spandrel. — The space between the back of arch and the roadway. 
 
 £xfrarcfo$ 
 
 Fig. IB. 
 
 Arches are divided into right arches and skew arches, depending upon the angle made by 
 the springing lines with the center line of roadway. A right arch is one that makes this angle 
 exactly 90 deg. 
 
 2. Curve of the Intrados. — The form or general outline of an arch is the first consideration 
 in its design. According to the curve of the intrados, arches are usually divided into circular, 
 multi-centered, elliptical, and parabolic. If the intrados is a semicircle, the arch is a semi- 
 circular arch; and, if the intrados is less than a semicircle, it is a segmental arch. A multi- 
 centered arch is one in which the intrados is composed of several arcs of circles tangent to each 
 other. Semicircular and semi-elliptical arches are full centered — that is, they spring from 
 horizontal beds — while segmental and parabolic arches spring from inclined beds called skew- 
 backs (see Fig. IB). Multi-centered arches may have beds either inclined or horizontal. 
 
 651 
 
652 
 
 CONCRETE ENGINEERS' HANDBOOK 
 
 [Sec. 16-2a 
 
 The parabola may be modified for the sake of appearance by short circular curves at its ends, 
 made tangent to the parabola and to the vertical side of the pier or abutment. Minor curves 
 joining the arch soffit to the pier are not effective, however, and should not be considered as 
 part of the arch rise. 
 
 2a. Three-centered Curve. — A segmental arch cannot often be used to advan- 
 tage, for it seldom can be made to fit the line of pressure. The three-centered arch is perhaps 
 the most common for solid-spandrel construction and gives a pleasing and generally an econom- 
 ical design. The formula for the radius of a circular segment when the chord distance 
 (span) and mid-ordinate (rise) of the segment are known is as follows: 
 
 m chord) 2 + (mid-ordinate) 2 
 
 Radius 
 
 2 X mid-ordinate 
 
 P'ollowing are the formulas for the radii of a three-centered curve (see Fig. 2). 
 
 + y' 
 
 R 
 
 2y 
 
 2 FE cos d - AF sin 
 
 Fig. 2. 
 
 Fig. 3. 
 
 26. Semi-ellipse. — The multi-centered curve can be made to approximate an 
 ellipse. Entirely graphical methods of obtaining the semi-ellipse and corresponding approxi- 
 mate multi-centered curves are as follows: 
 
 Let AD and CD (Fig. 3) be the semi-major and semi-minor axes, respectively, of the 
 
 ellipse. With D as a center, draw circular arcs with radii AD and c n 
 
 CD. From points where a common radius intersects the two circu- 
 lar arcs, draw vertical and horizontal ordinates. 
 The intersection of these ordinates gives one point 
 on the ellipse. Other points may be found in a 
 similar manner. 
 
 Suppose now that a three-centered intrados is 
 required which approximates a true ellipse. The 
 form of the true ellipse is first drawn by the 
 method given above and is shown in Fig. 4 by the 
 full line. The approximate form, shown dotted, is 
 what is required. Assume any two equal distances 
 CB and AE more than one-half of the semi-minor 
 axis. Join BE and bisect the line BE at F. Through F draw a perpendicular to BE, inter- 
 secting the line CD at 0. The two points O and E will be centers of two circular arcs which 
 will form an approximate ellipse. By first selecting the position of the point E so that the 
 circular arc described from E as center will conform as closely as possible with the true 
 ellipse, satisfactory curves will easily be found. 
 
 The method of drawing an approximate ellipse using a five-centered curve will now be 
 explained. In order to have a check on the work, it is advisable to first draw the form of the 
 
 Fig. 4. 
 
 Fig. 5. 
 
Sec. 16-2cl 
 
 ARCHES 
 
 653 
 
 true ellipse by the method given above. Let AD and CD (Fig. 5) be the given semi-axes. Join 
 A and C, and through B draw a perpendicular to AC, determining E and 0, two of the centers. 
 From 0, with OC as radius, draw an arc CK as long as thought suitable, and join K with O. 
 Make KG equal to AE. Join E and G. At the center of EG draw a perpendicular to jEJG, 
 and note its intersection H with KO. From if, with radius HK, draw an arc to HE (extended) ; 
 and from E, with EA as radius, complete the curve. 
 
 2c. Parabola. — The equation of the parabola. Fig. 6, is as follows: 
 
 Divide the line OR into any number of convenient equal parts, and number the points of divi- 
 sion 1, 2, 3, etc., beginning at the point nearest O. Then to find the values of y, for the various 
 abscissas x, the numbers 1, 2, 3, etc., should be inserted in the above equation for values of Xy 
 and the total number, which in the illustration is 6, should be inserted for the value of a. 
 
 Fig. 6. • Fig. 7. 
 
 A very simple graphical method of drawing the parabola is to lay off on the vertical line 
 RS, Fig. 7, the same number of equal divisions as are made on the horizontal axis OR, and from 
 0 draw radiating lines to the various division points on the vertical axis RS. From the 
 various points on the horizontal line OR draw vertical lines intersecting the radiating lines 
 from 0. The points of intersection of these vertical lines with corresponding radiating lines 
 are points on the required parabolic curve. 
 
 3. Arrangement of Spandrels. — Arch spandrels may be entirely filled with earth, or they 
 may be left more or less open and the roadway supported on a series of transverse walls, or on 
 a complete superstructure of columns, girders, beams, and slabs. If, as is rarely the case, a 
 heavy or massive appearance is desired in open-spandrel construction, then side curtain walls 
 may be used and all spandrel openings closed. In the open-spandrel type, the arch ring may 
 be either solid or composed of two or more longitudinal ribs. 
 
 With filled spandrels, the filling material is held in place laterally by retaining walls which 
 rest upon the arch ring. These retaining walls may be of either the gravity or the reinforced 
 type, or they may consist of thin vertical slabs tied together by reinforced-concrete cross 
 walls. Solid fillings increase the weight of the superstructure and make necessary thicker arch 
 rings and larger foundations. Open-spandrel construction, on the other hand, requires a rela- 
 tively larger amount of form work. At the present cost of labor and materials in this country, 
 the filled type of arch spandrel is preferable from the standpoint of economy for all arches of 
 moderate rise with spans less than about 100 ft. and also for flat arches of greater span where 
 the ratio of rise to span is not more than one-tenth. Fortunately, a proper artistic appearance 
 is usually obtained in satisfying these economical requirements. 
 
 4. Piers and Abutments. — The springing lines, or springs of an arch, should be located as 
 near the foundation as conditions will permit. This will often make possible a less expensive 
 design for the abutments and, where piers are employed, will reduce the overturning effect 
 on the piers to a minimum. 
 
 In the case of long bridges with a series of arches, what are called abutment piers should 
 be placed at frequent intervals (usually every five or six spans) so as to act as an abutment in 
 
654 
 
 CONCRETE ENGINEERS' HANDBOOK 
 
 [Sec. 16-5 
 
 case of failure of one or more of the arches. This type of pier is made of sufficient thickness 
 to resist the pressure for either arch standing and the other arch removed, and for both arches 
 standing. The ordinary arch pier should be analyzed for one adjacent arch without live load 
 and the other adjacent arch with live load over the whole span. 
 
 Arch bridges of four and six spans do not present a desirable appearance. For esthetic 
 effect, an odd number of spans should be selected and the span lengths should decrease each 
 way from the center of bridge. 
 
 The depth of arch foundations and the shape of abutments and piers is dependent upon 
 local conditions, and in some difficult cases have to be chosen after thorough study. A certain 
 shape of abutment or pier is first assumed; and this is then reviewed to see that the load upon 
 the ground does not exceed the allowable and that it is well distributed. Great saving is 
 effected in some cases by the use of hollow, or ribbed, abutments and piers. 
 
 5. Depth of Filling at Crown. — In making a preliminary design for an earth-filled arch 
 bridge, it is necessary to know approximately the required crown thickness of the arch ring and 
 also the amount of earth filling over the crown. This must be known in order to determine the 
 remaining distance from the crown to the springing line — that is, the available rise for the 
 arch. For highway bridges, a depth of filling including the pavement of from 1 to 2 ft. will be 
 sufficient; but for railroad structures a minimum depth of from 2 to 3 ft. below the ties will be 
 needed in order to form a cushion for the ties, to distribute the load, and to absorb the shock 
 from passing trains. 
 
 6. Loads. — The dead weight of the arch ring itself and of the superimposed material con- 
 stitute usually the principal loads on an arch ring. With open-spandrel construction, the dead 
 loads act vertically upon the arch ring or arch rib through the transverse walls or columns, 
 and are hence definitely known. With filled spandrels, the pressure produced on the arch ring 
 by the earth filling is really inclined and the dead load cannot be so accurately determined. 
 
 On flat earth-filled arches, it is better to consider only vertical loads as acting on the arch 
 ring, for the conjugate horizontal forces are small and may be neglected. On earth-filled arches 
 with large rise, the horizontal thrusts become great, especially close to the springing lines, and 
 it may be advisable in some cases to take these horizontal components into account. The 
 omission of these horizontal thrusts, however, is always on the side of safety. 
 
 A common assumption for weight of earth fill where the actual value is unknown is 100 lb. 
 per cu. ft. When sand is used, its weight should be taken at 120 lb. Pavement is usually 
 assumed as 12 in. thick and as weighing 150 lb. per cu. ft. 
 
 The live load to be used in the investigation of an arch bridge should be the greatest that 
 comes or is liable to come upon the roadway. Each location should be studied and the live 
 load chosen to fit the requirements. For ordinary conditions a standard loading is commonly 
 employed. Wind pressure is considered only on light or exceptionally high structures. 
 
 In earth-filled bridges where there is sufficient thickness of filling to distribute the con- 
 centrated loads over a considerable area of arch ring, uniform live loads are used in the arch- 
 ring design. City highway bridges are generally designed for 50-ton electric cars and for such 
 bridges, with spans of 200 ft. or more, a uniform load of 1200 lb. per lin. ft. is usually taken on 
 each railway track together with a uniform load of 80 lb. per sq. ft. over the remaining area of 
 roadway and sidewalks. For spans between 100 and 200 ft., the loads are taken proportionally. 
 The loads specified above for city bridges may be reduced by about 20% to apply to the arch 
 rings of light country bridges. The load on each street railway track is generally assumed to 
 cover a width of 9 ft. 
 
 In addition to the above loads, city bridges and bridges on thoroughfares likely to be used 
 for heavy hauhng should be designed to carry 20-ton trucks, with axles about 10 ft. c. to c, 
 14 tons on rear axle and 6 tons on front axle; wheels about 5 ft. c. to c. 
 
 Because of the permanent character of concrete bridges it may be wise to provide a larger 
 margin for increase of loading than is above suggested, or than is usually allowed in steel-bridge 
 design. Fortunately, in the case of concrete-arch bridges a large increase can be provided for 
 
Sec. 16-6] 
 
 ARCHES 
 
 655 
 
 with only a slight increase of expense due, of course, to the controlling influence of the dead 
 load. 
 
 Following is an extract from the report of a Committee on Reinforced Concrete Highway- 
 Bridges and Culverts, American Concrete Institute, presented at the Annual Convention at 
 Chicago, Feb. 17, 1914: 
 
 Class "A" Bridges. — Main thoroughfares leading from large towns. In view of the extensive introduction 
 of the heavy motor trucks and traction engines, and the probable general use of such vehicles in the future, it is 
 recommended that bridges on main thoroughfares and other roads which are likely to be used for heavy hauling, 
 be designed to carry 20-ton trucks, with axles about 10 ft. c. to c, 14 tons on rear axle and 6 tons on fore axle; 
 wheels about 5 ft. c. to c. Outside of the large cities it is recommended that only one such vehicle be assumed 
 to be on the bridge at any one time; the likelihood of more than one being on the bridge, in a position to produce 
 maximum stresses at the same time, is so remote that this assumption is considered safe. It is advised that such 
 very heavy loads be considered as occupying only the ordinary width of the road, about 8 ft. in width and about 
 35 ft. in length. Congested traffic of heavily loaded wagons or motor trucks will rarely impose a load of more than 
 100 lb. per sq. ft. over a considerable area. The above-mentioned 20-ton truck gives a load of about 140 lb. per 
 sq. ft., on the area actually occupied, but it is considered extravagant to assume that a large bridge is covered with 
 •such heavy loads. One hundred pounds per square foot is thought ample to assume for the loading of spans more 
 than 60 ft. long in designing the trusses or main girders. It is thought to be safe to reduce this assumed load in the 
 case of longer spans, to the following amounts: 
 
 Length of Assumed load 
 
 span (ft.) (lb. per sq. ft.) 
 80 90 
 
 100 80 
 
 125 75 
 
 200 and over 70 
 
 with all intermediate spans in proportion. 
 
 The greatest load that is liable to be imposed on a bridge sidewalk occurs when there is some excitement in 
 the neighborhood which attracts a large crowd, and for which the bridge affords an especially good point of view. 
 In that case the crowd forms a compact mass against the railing, not more than 4 ft. deep, making a load seldom 
 exceeding 100 lb. per sq. ft. over a very considerable space. The remaining portion of the sidewalk may be covered 
 by a moving crowd which can scarcely weigh more than 40 lb. per sq. ft. It may be advisable, sometimes, to 
 so design sidewalk slabs, that if a street car or motor truck accidentally gets upon the sidewalk, it will not go 
 through. Such accidents are so rare, that it is thought safe to allow materials to be stressed somewhat beyond 
 the elastic limit in such cases. 
 
 Class "B" Bridges. — Although it is impossible to determine beforehand, especially in the newer parts of the 
 country, whether any given road is to be used for heavy traffic, it seems extravagant, at least in the cases of larger 
 spans, to design bridges to carry much heavier loads than can be expected to come upon them. It is recommended 
 that bridges of this class be designed to carry 15-ton trucks, with axles 10 ft. apart, 5 tons on the front and 10 tons 
 on the rear axle. This will allow for a considerable overloading of existing motor trucks. It is further recom- 
 mended that only one truck be assumed to be on the bridge at one time, in designing the floor system, that it be 
 assumed to cover a width of 8 ft. and a length of 35 ft. and that the remainder of the bridge be covered with a 
 load of about 90 lb. per sq. ft., for spans up to 60 ft. 
 
 For longer spans, the trusses and main girders should be designed for the following loads: 
 
 Length of Assumed load 
 
 span (ft.) (lb. per sq. ft.) 
 
 80 80 
 
 100 70 
 
 125 65 
 
 150 60 
 
 200 and over 65 
 
 with intermediate spans in proportion. 
 
 Sidewalks should be designed to carry the same loads as in the case of Class "A" bridges. 
 
 Special Bridges. — City bridges and bridges carrying traffic connected with mines, quarries, lumber regions, 
 mills, manufactories, etc., require special consideration and should, of course, be designed to carry any load which 
 can reasonably be expected to pass over them, bearing in mind the likelihood of heavy traction engines and motor 
 trucks coming into extensive use in the not distant future. 
 
 Bridges Carrying Electric Cars. — Electric traction is still in its infancy and nobody is able to forecast its futiu-e 
 development. It seems probable, however, that it will not be profitable to run cars weighing more than 50 tons 
 each, at a speed that would be permitted on any public road. If very high speeds are desired, the traction company 
 will doubtless be required to operate over its own right-of-way. It is recommended that bridges carrying either 
 
656 
 
 CONCRETE ENGINEERS' HANDBOOK 
 
 [Sec. 16-7 
 
 urban or interurban electric cars be designed to carry 50-ton cars on two trucks, spaced 30 ft. c. to c, each truck 
 having two axles spaced 7 ft. c. to c. The Committee sees no reason for changing the customary practice of assum- 
 ing that an axle load is distributed over three ties. 
 
 For railroad bridges, Cooper's Standard Loadings are generally specified, the particular 
 loading to be used depending upon the location of the line and the future traffic that may be 
 expected. As regards the arch ring in earth-filled arches, where the thickness of filling is suffi- 
 cient to distribute the concentrated loads over a considerable area, an equivalent uniform 
 loading per linear foot per track is generally substituted. A load of 700 lb. per sq. ft. is common 
 for railroad traffic on spans, say, over 80 ft. in length. A uniform load of 1000 lb. per sq. ft. 
 is frequently adopted for shorter spans. The impact of live loads is not usually considered 
 except for the floors in all arches of open-spandrel construction. Braking or tractive stresses 
 are important only for bridges on heavy grades. 
 
 A concentrated load should be assumed to be distributed downward through the fill on a 
 30-deg. slope with the vertical starting from the ends of the ties.^ An axle load is assumed 
 as distributed over three ties in the direction of the track. 
 
 7. Empirical Rules for Thickness of Arch Ring. — Various empirical formulas have been 
 developed for the trial thickness of arch ring at the crown and are an aid to the judgment. 
 
 F. F. Weld^ gives the following formula: 
 
 fi V 10 200 400 
 
 where 
 
 h = crown thickness in inches. 
 I = clear span in feet. 
 
 w = live load in pounds per square foot, uniformly distributed. , 
 w' = weight of dead load above the crown of the arch in pounds per square foot. 
 
 W. J. Douglas gives the following tabulated formulas for different highway spans, the 
 values of h being given in feet : 
 
 Under 20 ft. h = 0.03(6 + I)* 
 
 20 to 50 ft. h = 0.015(30 + Z)* 
 
 50 to 150 ft. h = 0.00010(11,000 + Z2)f 
 
 Over 150 ft. h = 0.016(75 + 1)% 
 
 * For railroad arches, add 25%. 
 t For railroad arches, add 20 % . 
 t For railroad arches, add 15%. 
 
 Joseph P. Schwada gives the following formula which is founded on a rational basis but 
 has been checked with results from actual designs: 
 
 ^ ~ 57.6(r 
 
 where 
 
 h = crown thickness in feet. 
 
 I = clear span in feet. 
 
 r = rise of intrados in feet. 
 
 F = depth of fill at crown in feet (not including track and ballast or pavement). 
 B = weight of track and ballast or pavement in pounds per square foot. 
 w = uniform live load in pounds per square foot. 
 
 The following paragraphs and Figs. 8 and 9 are from Mr. Schwada's article in Eng. News:^ 
 
 In Fig. 8 is shown the application of the formula for crown thickness to a series of railroad arches for ratios of 
 rise to span from H to H and for conditions as noted in the figure. For convenience the coefficients of stress for 
 certain spans are arranged in tabular form. Intermediate values can be determined and the crown thickness for 
 any other conditions easily obtained. 
 
 1 See teste by M. L. Engeb, Eng. Rec., Jan. 22, 1916, p. 106. 
 
 2 Eng. Rec., Nov. 4, 1905, p. 529. 
 
 » Eng. News, Nov. 9, 1916, p. 880. 
 
Sec. 16-7] 
 
 ARCHES 
 
 657 
 
 To adapt the formula for crown thickness to highway arches a slight change must be made in the value of 
 coefficients for short spans, because of the light loads involved and because of a desire to obtain thicknesses which 
 are practical. The coefficients for /c for spans up to about 60 ft. therefore differ from the coefficients for the same 
 
 "d-Wghf of ballast and track tSi^OOIh/sqft 
 r-Dirtfill-e.Oi 
 W' Uniform live load: Ej^uiva/ent to Coopers 
 E50 reduced through 3'af fill and Jjallast 
 6001b. per sq. in. 
 
 ZO 30 40 50 60 10 80 90 
 Span in feet 
 
 Fig. 8. 
 
 wo I/O 120 
 
 B - Weight of pavement =0 
 F - Fill on crown = ^.0' 
 w ^Uniform live load =W0*/5q ft. 
 fc =550lb.per sq. in. 
 
 £0 30 40 50 60 70 80 90 100 I/O \Z0 
 Span j'n feet 
 
 Fig. 9. 
 
 spans for railroad arches. For spans over 60 ft. the coefficients are the same. A complete table of values for 
 highway arches is given in Fig. 9, with an application to a series of highway arches for conditions shown in the 
 figure. ^2 
 
658 
 
 CONCRETE ENGINEERS' HANDBOOK 
 
 [Sec. 16-8 
 
 To illustrate the application of the formula to a highway arch assume: span, 90 ft.; rise, 12 ft.; weight of 
 pavement, 100 lb. per sq. ft.; fill, 1 ft.; live load, 300 lb. per sq. ft.; unit stress, fc, 550 lb. per sq. in. Then K = 
 0.71. Assume depth at crown to be 23 in. = 1.92 ft. According to the expression, 
 
 90 X 90 
 
 h = 
 
 [5 + 12.0 + 15.36 + 6 + 15] = 1.91 ft. = 23 in. 
 
 57.6 X 10.08 X 0.71 X 550 
 
 If the resulting thickness does not check the assumed thickness, another trial must be made. 
 
 In applying the formula for crown thickness it should be understood that fc in the expression represents an 
 approximate maximum value of stress which one may reasonably expect to obtain if the arch is designed according 
 to the rules for curvature and thickness followed in the design of the arches here considered. The maximum 
 value of stress for these arches varied from 575 lb. per sq. in. to 650 lb. per sq. in., with an average value of about 
 625 lb. per sq. in. A value of 550 lb. per sq. in. to 600 lb. per sq. in. for fc is suggested for use in the formula. 
 
 The empirical rules given above, should be used only for trial. The exact shape of the 
 arch ring and the thickness at different sections must be determined by analysis. 
 
 8. Approximate Formula for Best Shape of Arch Axis. — Victor H. Cochrane has derived 
 equations^ giving approximately the best shape of axis for both open-spandrel and filled-spandrel 
 arches. The equations are given in Art. 32 of this section. 
 
 9. Proper Thickness of Arch Ring in the Haunch for Given Thicknesses at Crown and 
 Springing. — Victor H. Cochrane has analyzed a series of typical arches^ so chosen as to be 
 applicable to any span length, rise-ratio (ratio of rise to span), thickness at crown and 
 springing, and manner of loading. The characteristics of these typical arches are given in 
 Art. 33 of this section. 
 
 10. Dead Loads and Their Action Lines. — After a trial arch ring has been assumed, the 
 dead loads may be determined. 
 
 Assume, for example, a railway earth-filled arch. The earth filling and ballast, ties, and 
 rails should be reduced to an equivalent height of masonry, as shown in Fig. 10. If the earth 
 filling is taken at 120 lb. per cu. ft. and the ballast, ties, and rails at 150 lb. per sq. ft. of roadway, 
 then since the earth filling reaches 1 ft. above the extrados at the crown, the vertical distance ah 
 120 
 
 should be laid off equal tOT^ = 0.8 ft. In a similar manner the distance he should be laid 
 
 off equal to = 1 ft 
 
 150 
 
 Points d and / should be determined for the loading at c, and similar 
 
 points should also be found for other places along 
 the arch ring, a sufficient number being taken to 
 fully determine the curved line /e. This line is 
 called the reduced-load contour. 
 
 The arch with its load should now be divided 
 by vertical lines into trapezoids, or what are nearly 
 so. For testing the trial arch by the approximate 
 method (presented in the following article), the 
 horizontal distance between springing lines may be 
 conveniently divided into divisions of equal hori- 
 zontal length. Fig. 10 shows eight divisions on 
 each side of the crown. 
 
 The next step is to determine the area and 
 center of gravity of each trapezoid. With the 
 area known, the load corresponding to each trapezoid is found by multiplying by the weight 
 of a cubic foot of concrete, the arch considered being included between two longitudinal vertical 
 planes 1 ft. apart. The center of gravity for each of the trapezoids may be found as follows: 
 Extend AB, Fig. 10, so that BE = CD, and in the opposite direction extend CD so that CF = 
 AB. The intersection of EF and the median GH is the center of gravity sought. 
 
 11. Approximate Method of Testing Trial Arch. — Since the dead load usually controls 
 the shape of the arch ring, it is desirable to test the trial arch for this loading, employing an 
 
 1 Proc. Engineers' Society of Western Pennsylvania, vol. 32, No. 8. 
 
Sec. 16-12] 
 
 ARCHES 
 
 659 
 
 approximate graphical method. In the method referred to, that form of arch in which the Une 
 of pressure and the arch axis most closely approach each other is considered to be the best that 
 can be designed. 
 
 There are two classes of theories of the stability of the masonry arch — the line-of-thrust 
 theories and the elastic theories. The line-of-thrust theories do not consider the elastic proper- 
 ties of the material and are usually employed for arches made up of wedge-shaped stones, 
 called voussoirs. For instance, stone arches are generally calculated by these theories and the 
 stability of the arch ring is considered as depending upon the friction and the reaction between 
 the several arch stones. Various assumptions are made in arriving at these different line-of- 
 thrust theories, and the results derived from them are not so accurate as where the elastic 
 properties of the material are employed. They may be made use of, however, in determining 
 the proper shape of the arch ring before applying the elastic theory to monolithic arches. 
 
 If an equilibrium polygon is passed through the centers of the arch ring at the crown and 
 springing, this line will be very near the true line of pressure. In the trial arch mentioned 
 above, if the equilibrium polygon and the arch axis 
 do not closely approach each other there will surely 
 be another shape of arch ring which will fulfill this 
 condition, and hence have probably lower maximum 
 stresses resulting from the live and dead loads. Thus, 
 the method of obtaining the best form of arch is to 
 find for what form the line of pressure for the dead 
 load and the arch axis most nearly coincide. It 
 should be clear that this method will aid greatly in 
 getting pretty near, at any rate, to the proper form 
 for the arch before applying the elastic theory. 
 
 In order to pass an equilibrium polygon through 
 the centers of the arch ring at the crown and spring- 
 ing, the load line should first be laid off as shown in 
 Fig. 11 and then any convenient pole 0' selected in 
 the horizontal through the point k in the load line, this 
 point representing the end of the load nearest the 
 crown. The next step is to draw the rays to the force 
 
 polygon and then construct the corresponding equilibrium polygon beginning at the center of 
 the arch ring at the skewback (point A). If now the first and last strings are prolonged to 
 an intersection at B, a vertical line is determined which contains the resultant of the loads 
 occurring over the left half of span. 
 
 The equilibrium polygon which we have constructed does not pass through the center 
 of the arch ring at the crown and is not the one required. We do know, however, that the first 
 string which passes through A must intersect the last string in the vertical through B and that 
 the last string must be horizontal due to symmetrical loading. Drawing a horizontal line 
 through the center of the arch ring at the crown determines the point C which is the intersection 
 of the first and last strings of the equilibrium polygon required. A line drawn through m of 
 the force diagram parallel to CA determines the true pole 0 on the horizontal through k. If 
 the force diagram is now completed for this pole, the required equilibrium polygon may be 
 easily drawn. The line of pressure is seen to follow the arch axis very closely and the trial 
 arch will be accepted for analysis. 
 
 In both preliminary and final analysis the arch ring should be laid out to a scale of 1 in. 
 = 3 ft., and care should be used in the drafting work so as to have the dimensions exact and 
 the lines sharp. 
 
 12. Use of Temporary Hinges in Arch Erection. — Temporary hinges were employed by 
 the late George S. Morison in the construction of arches; they have also been used in a number 
 of European bridges. In arches with a rise less than one-fourth the span, these hinges materially 
 
 Jrue pole 
 
660 
 
 CONCRETE ENGINEERS' HANDBOOK 
 
 [Sec. 16-13 
 
 decrease stresses in the arch due to the ehmination of all dead-load bending stresses (including 
 arch shortening produced by dead-load compressive stresses) in addition to stresses from shrink- 
 age or settlement. Concrete should be poured in joints to close hinges only after full dead load 
 is on the structure and the shortening and shrinkage stresses have taken place. 
 
 13. Use of Reinforcement in Concrete Arches. — It would seem from purely theoretical 
 considerations that but little could be gained by the use of reinforcement in a concrete arch 
 since the direct compression usually controls to such an extent that the allowable stress in the 
 concrete permits of but a small unit tensile stress in the steel. From a broader viewpoint, 
 however, it is clear that the steel adds greatly to the reliability of the construction and makes 
 possible a higher working stress in the concrete than could properly be employed in the design 
 of plain-concrete structures. Higher working stress produces a thinner arch ring, and conse- 
 quently less dead load and lighter abutments. Undoubtedly, a large saving may result from 
 this cause in the case of long-span arches. 
 
 A considerable portion of an arch ring is subject to both positive and negative moments, 
 and for this reason the reinforcement should be placed, for some distance at least, near both 
 upper and lower surfaces. The general practice is to carry both rows of steel throughout the 
 entire span thereby eliminating any possibility of failure due to an inadequate provision for 
 tensile stresses. On account of the heavy compressive stress in arch rings, the upper and lower 
 reinforcement should be tied together to prevent buckling. 
 
 The percentage of longitudinal steel in arch rings is to a certain extent arbitrary. An 
 amount of steel between }4, and 1%% of the ring at the crown seems to be good practice in the 
 ordinary full-barrel arch-ring design although the exact amount depends upon the loading and 
 the form of arch selected, and must be finally tested by computation. Transverse rods at 
 right angles to the longitudinal are generally used to prevent cracks in the concrete and to 
 distribute the loads laterally. Web reinforcement is not required in ordinary construction. 
 
 14. Classification of Arch Rings. — Arches may be classified as hinged or hingeless. A 
 hingeless arch is one having fixed ends, while a hinged arch may have a hinge at the crown, a 
 hinge at each end, or a hinge at each end and one at the crown. Arches of one and two hinges 
 are not used to any extent in masonry construction since the three-hinged arch offers the ad- 
 vantage of more definitely fixing the line of pressure throughout the ring and thus makes pos- 
 sible a saving of material. Hinges, however, are often an expensive detail and the three-hinged 
 arch is by no means so common as the concrete arch having fixed ends. Friction on hinges is 
 also an important consideration. 
 
 Three-hinged arches are treated in Arts. 43 to 47 inclusive. 
 
 ANALYSIS OF THE ARCH BY THE ELASTIC THEORY^ 
 
 15. Deflection of Curved Beams. — Deflection formulas for curved beams (in which the 
 radius of curvature is large as compared with the depth) are employed in the development of 
 arch theory. 
 
 Let AB^ Fig. 12, be any portion of a curved beam in its unstrained form and A'B the same 
 portion in its strained form, assuming the beam rigidly fixed at B. Let X — X and Y — Y 
 be rectangular axes with origin at A, and denote the components of A' as Aa; and A?/. AO, 
 tangent to the arch axis at A, moves through the angle k. Formulas below give the following 
 values: (1) angular change of AO, (2) component A.r of A', (3) component A?/ of A'. 
 
 ^ - x: s 
 
 1 Method of analysis as given in Turneaure and Maurer's " Principles of Reinforced Concrete Construc- 
 tion," 2nd Edition, pp. 335 to 344. 
 
Sec. 16-16] 
 
 ARCHES 
 
 661 
 
 Fig. 12. 
 
 The smaller the elementary lengths of beam considered, the more accurately will the above 
 formulas apply. In the derivation of the formulas the values of M and / have been regarded 
 as constant quantities for each particular elementary length considered. Since this is not 
 true in practice on account of each element having appreciable length, a close approximation to 
 the actual M and 1 for a given element may be obtained by taking the values of the bending 
 moment and moment of inertia at the mid-point 
 of s. Distances x and y should also be measured 
 to this point. As in simple beams, M is considered 
 positive when it tends to increase the compression 
 on the back of the arch. The minus sign is used 
 in formula (2) because the effect of a positive 
 value of M in any element causes an upward de- 
 flection — that is, a minus value of Ay, considering 
 only the effect of bending in the element in ques- 
 tion. 
 
 16. General Procedure in Arch Analysis. — A 
 
 concrete arch with fixed ends is statically indeter- 
 minate. There are, in all, six unknown quantities — three at each support (the vertical and 
 horizontal components of the reaction, and the bending moment; or, what is the same thing, 
 the magnitude, direction, and point of application of the reaction) — and it is possible to de- 
 termine only three unknowns by the principles of statics. The three additional equations may 
 be found from the following conditions: 
 
 The change in span of the arch = Ax = 0 
 
 The vertical displacement at one end relative to the other end = Ay = 0 
 
 The angle between the tangents to the arch axis at the two ends of the 
 arch remain unchanged, or Zfc = 0 
 
 These three conditions must be true since the arch is fixed at the abutments. 
 
 Instead of actually finding the components of the reactions and the moments at the sup- 
 ports as outlined above, it is simpler for sym- 
 metrical arches to take the origin of coordinates 
 at the crown and find the thrust, shear, and 
 moment at that point. With these three un- 
 knowns determined, each half of arch may then 
 be treated as statically determinate. ^ 
 
 The analysis of a symmetrical arch con- 
 sists in finding the thrust, shear, and bending 
 moment at the crown and at intermediate sec- 
 tions in the arch ring or arch rib, and then find- 
 ing the stresses resulting therefrom. (The 
 method of finding stresses on an arch section, 
 knowing moment and thrust, is explained in 
 Sect. 9.) The thrust is here taken to be the 
 normal component of the resultant force on 
 the section, and the shear is the component 
 at right angles to the normal. The bending 
 moment will be considered positive when it tends to increase the compression on the back of 
 the arch, this being the same convention as for beams. 
 
 A horizontal thrust is produced at the crown when the arch is loaded symmetrically. For 
 non-symmetrical loading, an inclined pressure acts at the crown, but its horizontal component 
 is called the horizontal thrust for that loading. Its vertical component is the shear at the crown. 
 
 1 For unsymmetrical arches, see Art. 28. 
 
662 
 
 CONCRETE ENGINEERS' HANDBOOK 
 
 [Sec. 16-17 
 
 Assume the arch as cut at the crown and consider each half to act as a cantilever sustaining 
 exactly the same forces as exist in the arch itself. 
 
 The external forces holding a semi-arch in equilibrium (Fig. 13) are the loads Fi, P2, etc., 
 the horizontal thrust He, the vertical shear 7c, and the reaction at the skewback Ra. 
 
 After He, Vc, and Mc have been computed, the line of pressure (accurately enough re- 
 presented by the equilibrium polygon) can be constructed by help of the force polygon. Fig. 
 13. The value of Mc definitely determines the point of application of He and makes the con- 
 struction of the exact line of pressure possible. (For a positive value of Me, the thrust He 
 acts above the arch axis.) From this line of pressure and the accompanying force polygon may 
 be obtained the thrusts, shears, and bending moments at intermediate points of the arch. The 
 force polygon gives directly the thrusts and shears, while the line of pressure makes possible 
 the determination of the bending moment at any section, the bending moment being equal to 
 the resultant pressure at the given point multiplied by the perpendicular distance from 
 the arch axis to the line of pressure. Usually the line of pressure is drawn to serve only as a 
 check on the computations, and the bending moments at the various points are determined 
 algebraically. 
 
 The line of pressure of an arch is a continuous curve, but differs very little from an equi- 
 librium polygon for the given loads. Fig. 13. In fact this curve becomes tangent to the equi- 
 librium polygon between the angle points. The greater the number of loads, the nearer the 
 polygon approaches the line of pressure. 
 
 With He, Ve, and Mc determined, all external forces are known except the reaction at the 
 skewback, and this is determined by the closing line of the force polygon. An equilibrium 
 polygon may then be constructed as already mentioned, the first side being in the line of Re 
 produced, the second parallel to the ray 5, and so on until the last side through q gives the 
 position of i^o- 
 
 17. Notation. — The following notation will be employed: 
 
 Let s = length of a division of the arch ring measured along the arch axis. 
 nh = number of divisions in one-half the arch. 
 
 I = span of arch axis. 
 Co = average unit compression in concrete of arch ring due to thrust. 
 tc = coefficient of linear temperature expansion. 
 tD = number of degrees rise or fall in temperature. 
 Ec = modulus of elasticity of concrete. 
 
 At the crown, let 
 
 He = horizontal thrust. 
 Vc == vertical shear. 
 Rc = resultant of Hg and Vc- 
 Mc — bending moment. 
 
 At any point on the arch axis, with coordinates x and y referred to the crown as origin, let 
 N = thrust (normal) on radial section. 
 S = shear on radial section. 
 
 R = resultant force on radial section, resultant of A'^ and S. 
 Xo = eccentricity of thrust on section, or distance of A'^ from the arch axis. 
 t = depth of section. 
 
 I = moment of inertia of section including steel = /c + nls- 
 A = area of section including steel = Ac -\- nAs. 
 Po = steel ratio for total steel at section. 
 
 d' = embedment of steel from either upper or lower surface. 
 M = moment = Nxq. 
 
 mL = moment at any point on left half of arch axis of all external loads (Pi, P2, etc.) between the point 
 and the crown. 
 
 mR = moment at any point on right half of arch axis of all external loads between the point and the crown. 
 m = moment at any point on either half of arch axis of all external loads (Pi, P2, etc.) between the point 
 and the crown. 
 
Sec. 16-18] 
 
 ARCHES 
 
 663 
 
 18. Formijlas for Thrust, Shear, and Moment. — When the arch ring is so divided into sec- 
 tions that I is a constant, then the following formulas apply: 
 
 Loading: 
 
 He = 
 Vc = 
 Mc = 
 
 nh'SimL + mR)y - X(mL + mR)'Ly 
 
 2[nS2/2 - 
 S(mL - mR)x 
 2 2x2 
 
 S(mL + thr) — 2He'Z,y 
 
 (4) 
 (5) 
 (6) 
 
 2nh 
 
 M = Mc + HcV + VcX - rriL (7) 
 M = Mc + HcV - VcX - ruR (8) 
 All values of mL, mR, x, and y should be substituted as positive. All summations refer to one-half of the 
 arch axis. Positive value of Vc indicates that the line of pressure slopes upward toward the left; a negative 
 value, downward toward the left. Positive value of Mc indicates that the thrust He acts above the arch axis. 
 Signs preceding terms Mc and VcX in formulas (7) and (8) depend upon the results of (5) and (6). 
 Temperature: 
 
 I tdDluhEe 
 
 2[ri/.S2/2_(Sy)2] 
 Hcl^y 
 
 (tD should be inserted as + for a rise; — for a drop.) 
 
 (9) 
 
 (10) 
 
 "A 
 
 M = Mc + Hey (11) 
 The value of to should be inserted as plus for a rise of temperature; minus ( — ) for a drop. Signs preceding 
 He in formulas (10) and (11) depend upon the result of formula (9). Sign preceding Mc in formula (11) depends 
 upon the result of formula (10). Thus for fall of temperature, thrust and moment are of opposite sign from those 
 for a rise. I = span of arch axis. 
 Rib shortening: 
 
 I Calnh 
 
 He = 
 Mr. = 
 
 S 2[71aS2/2 
 
 M = Mc + HcV 
 
 Values of moments and thrusts for rib shortening are of same sign as for temperature fall. 
 
 (12) 
 
 (13) 
 (14) 
 
 = span of arch axis. 
 
 I 1 / ^y-" • 
 
 I \/ Distance 01003 Arch Axis, 
 j ^>^i from Skewback 
 
 I / 
 
 / 
 
 Fig. 14. 
 
 19. Division of Arch Ring for Constant j. — The graphical method shown in Fig. 14 is 
 
 usually employed. AB is drawn to any con- 
 venient scale equal in length to one-half the arch 
 axis. The curve EF is then drawn through 
 points whose ordinates are the values / and 
 whose abscissas are the corresponding distances 
 along the arch axis from the skewback. (In 
 order to make the drawing clear, the ordinates 
 and corresponding abscissas which determine 
 the curve EF are not shown.) A length AB. is 
 then assumed, a perpendicular hC erected at its 
 center, and the lines AC and CU determined. 
 Starting from point i?, lines are drawn parallel 
 alternately to AC and CU^ as shown in Fig. 14. 
 Only three or four trials will usually be required 
 to divide the line AB into the desired number 
 
 of divisions. The base of each triangle thus formed corresponds to s and its altitude to /. 
 
 Since all the triangles are similar by construction, the term y is constant throughout. 
 
 A convenient modification of the above method is to draw a second curve E'F' below AB^ 
 using the same ordinates as for EF. AH is then assumed as before and the perpendicular CC 
 erected at its center. Starting with C, diagonals and verticals are drawn alternately making 
 the diagonals parallel to AC. This method offers the advantage of drawing all the diagonals 
 parallel to the same line. 
 
664 
 
 CONCRETE ENGINEERS' HANDBOOK 
 
 [Sec. 16-20 
 
 20. Loadings to Use in Computations. — Arches should be analyzed for live load over 
 three-eighths of the span, five-eighths of the span, the middle one-fourth of the span, and the 
 end three-eighths of the span. These uniform loadings are only approximations to the true 
 loadings which produce the maximum stresses. The exact loadings to cause maximum con- 
 ditions may be found by the use of influence lines. 
 
 21. Use of Influence Lines. — It is common practice in arch design to consider the live 
 load as extending over certain definite portions of the span and to assume that these loadings 
 produce the maximum effects. For example, it is often assumed that by loading either the 
 whole span or the half span the greatest possible stresses at any given section are obtained. In 
 general this assumption is only a very rough approximation, and considerable inaccuracy may 
 result from such a method of procedure. In fact, in the case of large and important structures, 
 the only satisfactory way to analyze for maximum stresses is by what might be called a unit- 
 load or influence-line method. By this method the arch is first analyzed for a load of unity 
 at the several load points and then influence lines^ are drawn for either moment and thrust or 
 for fiber stress. 
 
 The position of the live load on an arch to cause maximum stress at any given section can- 
 not be determined in advance in the common method of analysis. An investigation will show 
 that different loadings are required for sections similarly located in arches of different propor- 
 tions. The only accurate method, then, is to draw a proper number of influence lines as above 
 described. In arches continuously loaded no definite load points exist at which to place the 
 load of unity in influence-line analysis, but in arches of this class points may be chosen for this 
 purpose sufficiently close together to give any desired degree of accuracy. 
 
 22. Internal Temperature Investigations. — Comparatively few experiments have been 
 made which furnish data on the internal temperature range in concrete structures. Undoubt- 
 edly the most important are those completed under the direction of the Engineering Experiment 
 Station at Ames, Iowa, on two highway arch bridges of the earth-filled type. These experi- 
 ments^ are described in detail in Bulletin 30 of the Iowa State College of Agriculture and Me- 
 chanic Arts where a summary is also given of the other tests that have been made on inter- 
 nal temperature variation. 
 
 The writers of the bulletin conclude that "to render an arch structurally safe, provision 
 should be made (in the latitude where the bridge tests were conducted) for stresses induced 
 by a temperature variation of at least 40°F. each way from an assumed temperature of no stress. 
 Particular circumstances may demand that a greater variation be used for drop in temperature 
 to prevent the appearance of cracks. This will always remain largely a matter of judgment 
 with the designing engineer." 
 
 23. Shrinkage Stresses. — Shrinkage stresses are at present ignored in arch analysis, as the 
 shrinkage coefficients on actual arches have not been determined. 
 
 24. Deflection at Any Point. — The deflection at any point in an arch may be found by 
 formula (2), Art. 15, or 
 
 The arch should be assumed as cut at the point in question, and either portion of the arch may 
 be considered. The cantilever selected should be subjected to exactly the same forces as 
 exist in the arch itself. 
 
 If the deflection of the crown of a symmetrical arch is desired, the value of M due to load- 
 ing for any section of the left cantilever may be found from formula (7), Art. 18; or, substi- 
 tuting this value in the above equation, we have 
 
 Ay = - 
 
 s 
 
 ~EJ 
 
 Ay = - 
 
 EJ 
 
 1 See Art. 48a, Sect. 7; also Art. 34 of this section. 
 
 2 By Messrs. C. S. Nichols and C. B. McCullough, 
 
Sec. 16-25] 
 
 ARCHES 
 
 665 
 
 For temperature changes, formula (11) of Art. 18 may be substituted in place of formula (7), or 
 
 Ay = - ^ (Mc^x + Hc^^xy) 
 _ _ tctnlinh'Exy — ^xXy) 
 
 25. Method of Procedure in Arch-ring Design. — The main steps that need to be taken in 
 the design of an arch ring may be enumerated as follows: 
 
 1. Assume a thickness for the arch ring at the crown and at the springing, using empirical formulas, if desired, 
 as an aid to the judgment. 
 
 2. Lay out the curve assumed for the intrados. 
 
 3. Lay out a curve for the extrados to give as nearly as possible the assumed ring thickness at the springing. 
 
 4. Draw the arch axis between the extrados and intrados. 
 
 5. Divide the arch axis into an even number of divisions such that the ratio ^ is constant for all. 
 
 6. Compute the dead and live loads, and indicate these loads properly on the drawing. 
 
 7. Compute He, Vc, and Mc at the crown for the different conditions of loading. 
 
 8. Draw the force polygons for the different conditions of loading and the corresponding equilibrium polygons, 
 or lines of pressure. 
 
 9. Determine the thrusts, shears, bending moments, and eccentric distances at the centers of they divisions 
 
 of the arch ring for the different conditions of loading. 
 
 10. Compute the thrust and moment at the crown due to variation in temperature; also the moments on 
 the various sections, and the corresponding thrusts and shears by resolving the crown thrust into tangential and 
 radial components. 
 
 11. Where necessary, compute the thrust and moment at the crown, and the thrust, shear, and moment at 
 various sections due to rib shortening. 
 
 12. Combine the thrusts, shears, and moments due to the different conditions of loading with the thrusts, 
 shears, and moments due to temperature and rib shortening. (The results usually show that the shearing unit 
 stresses are very small and need not be considered.) 
 
 13. Compute the maximum stresses — compression in the concrete and tension in the steel — due to the thrusts 
 and moments. If the stresses are either too small or too large, the dimensions or even the shape of the arch ring 
 must be changed and the computations repeated. 
 
 26. Uncertainty as to Fixedness of Ends of Arch. — This uncertainty can be reduced or 
 entirely eliminated by taking the skewback for purposes of analysis at a plane where the ends 
 of the arch are virtually fixed. Whenever the abutments are of such a form that there is no 
 pronounced change of section at the springing lines, then the analysis should include the whole 
 structure down to the point where the distortion due to the live load on the arch will be in- 
 appreciable. In some cases this may be the very bottom of the abutment. 
 
 27. Skew Arches. — Skew arches may be treated exactly as right arches, the span being 
 taken parallel to the center line of roadway and not at right angles to the springing lines of the 
 arch. 
 
 28. Unsymmetrical Arches. — Unsymmetrical arches are sometimes desirable in the end 
 spans of a series of two or more arches in order to reduce material in abutments, and at the 
 same time, to provide ample waterway area over streams. Also, arches of this type are often 
 necessary under other conditions, as, for example, when a river in a deep ravine is bordered by 
 a railway requiring maximum clearance near the abutments. ^ 
 
 Formulas for unsymmetrical arches depend upon the location Section near \ 
 of the coordinate axes. 
 
 28a. Origin of Coordinates Between Divisional 
 Lengths. — In the analysis of unsymmetrical arches, the entire 
 
 arch ring should be divided into a sufficient number of —j 
 
 divisions to obtain the desired degree of accuracy. The origin y 
 of coordinates may then be taken at the center of any one of the Fig. 15. 
 
 sections occurring between the divisional lengths, but for con- 
 venience in scaling the values of x and i/, this origin should be placed at one of the sections 
 rear the crown. The X and Y axes should be drawn perpendicular and parallel respectively 
 
666 
 
 CONCRETE ENGINEERS' HANDBOOK 
 
 [Sec. 16-286 
 
 to this section so as to permit the thrust at the section to be determined directly without com- 
 position and resolution of forces. Fig. 15 shows how these axes should be drawn. 
 
 The flexure formulas for Ho, Vo, and Mo for unsymmetrical arches, considering the origin 
 of coordinates near the crown, are exceedingly complex and inconvenient for use in practice. 
 The best plan is to use formulas as given below and to solve simultaneously for the above values 
 after the numerical values of the coefficients are substituted. Following are the formulas to 
 be solved in this way: 
 
 Loading: 
 
 Ho^y^ + Vo(7:xLVL - 'Lxrvr) + ilfoSy - -Lmy = 0 
 
 Ho^'LxLyL — 'LxRyR) + FoSx^ + Mo(2a;L — Sxfl) — 'LmLxL + ^mRXR = 0 
 Ho'Sy + VoiXxL — I^xr) + nMo - Zm = 0 
 
 Ml = Mo + HoyL + Voxl — mh 
 
 Mr = ilfo + HoyR — Voxr — mR 
 
 All values of mL, mR, xL, xr, yh, and yR should be substituted as positive. The subscripts L and R refer to 
 summations to left and right of the chosen section respectively. No subscript indicates that summation is for 
 entire arch. Positive value of Mo indicates that the thrust Hq acts above the arch axis. Considering the chosen 
 section as nearly vertical, a positive value of Fo indicates that the line of pressure slopes upward toward the left; 
 a negative value, downward toward the left. Signs preceding terms Mo, Voxl, and YoXR in the last two formulas 
 depend upon the signs of Mo and Fo resulting from the three simultaneous equations. 
 
 Temperature: 
 
 Ho-Ly"^ + FoCSxLJ/L - '^xRyR) + MoSy - ^ • tctnlEc = 0 
 
 Hoil^xLyL - I^xrur) + Fo2a;2 + Mo(2xL - Sxfl) = 0 
 HoSy + FoCSxL - Sxfl) + nMo = 0 
 
 Ml = Mo + HoyL + Foxz, 
 
 Mr = Mo + HoyR — Voxr 
 
 The value of to should be inserted as plus ( + ) for a rise of temperature; minus ( — ) for a drop. Signs pre- 
 ceding terms Mo, HoyL, HoyR, VoxL, and Vqxr in the last two formulas depend upon the signs of Mo, Ho, and Fo 
 resulting from the three simultaneous equations. I = span or arch axis measured parallel to X axis. 
 
 Rib shortening: 
 
 Ca 
 
 Rib shortening causes the same efifect as a lowering of the temperature. Solving for to gives equivalent 
 temperature drop. 
 
 286. Origin of Coordinates at Crown. — Assuming that by ''crown section" 
 is meant a section at the highest point of the arch, the following formulas result, where q 
 
 equals (^jj ^ divided by ^jj ^• 
 
 Loading: 
 
 Hcil^y^L + l^VR^q) + Vcil^xLyL - ^XRyRq) + Md^yL + Sj/flg) - 2mL2/L - ZmRyRq = 0 
 Hci'^xLVL — ^XRyRq) + FcCZxL^ + 'LxR'^q) + Mc(SxL - 2xflg) - I^mLXL + ^mRXRq = 0 
 Hei^yL + ^yRq) + Vci^xL - ^XRq) + M^nL + URq) - l^mL - ^mRq = 0 
 
 Ml = Mc + HcyL + VcXL — mL 
 
 Mr = Mc + HcyR - VcXR - mR 
 
 Temperature: 
 
 HcC2:yL^ + S2/ft2Q) + Fc(SxL2/L - "S-XRyRq) + Mc^'LyL + ^yRq) - ^— ' = 0 
 
 HciZxLVL - ZxRyRq) + Vd^XL^ + Sxiz^g) + Mc(SxL - Sxflg) = 0 
 Hci'^yL + Sl/fls) + Vdl^xL - ZxRq) + MdnL + nRq) = 0 
 Ml = Mc + HcyL + V^xl 
 Mr = Mc + HcyR - VcXR 
 
 Rib shortening: 
 
 Ca 
 
 E7t. 
 
 All values of mL, mR, xl, xr, yL, and yR should be substituted as positive. 
 
Sec. 16-28c] 
 
 ARCHES 
 
 667 
 
 28c. Origin of Coordinates at Left Springing.- 
 
 dinates x and y should be measured. The directions Hi, Vi, 
 considered as positive in the formulas given below. Values 
 of y measured below the axis X — X should always be con- 
 sidered as negative. 
 
 -Fig. 16 shows how the coor- 
 and Ml are shown for values 
 
 Y 
 
 Loading: 
 
 Hi'Ly'^ - ViZxy - MiZy + Xmy = 
 HiZxy — ViXx^ - MiZx + Hmx = 
 Hi-2y - ViT^x - nMi + Sm = 0 
 
 in which 
 
 Then 
 
 ^ ab — cd 
 ^ ~ ae — cf 
 
 a = 'Ex'Zy — 
 b = XmxXy - 
 c = l^x'^'Sy - 
 
 Hi 
 
 Ml 
 
 n'Sxy 
 - 'ExXmy 
 I.x'Lxy 
 
 d = l.m'Zy 
 e = 2x22/2 
 / = n22/2 - 
 
 - n'Lmy 
 
 - -Lxylly 
 (22/)2 
 
 M = Ml + Fix - Hiy - m 
 All values of m and x should be substituted as positive. Values of y below the X — X axis should be taken as 
 negative. The summations refer to entire arch. Positive value of Mi indicates that the reaction acts to the left 
 of the arch axis at the springing. Positive values of Hi and Vi indicate that the reaction acts upward to the right. 
 Signs preceding terms M\, Vix, and Hiy in the last formula depend upon the signs of Mi, Vi, and Hi resulting from 
 the preceding equations. 
 Temperature: , 
 
 //i22/2 - 7i2x2/ - Mi2y = --tetDlEc 
 
 HiZxy 
 HiXy - 
 
 a 
 
 Hi = -' 
 
 ■ ViZx^ 
 Fi2x - 
 
 - Afi2x = 
 nMi = 0 
 
 cf 
 
 Vi = 
 
 k-Zx - Hu 
 
 Ml = 
 
 HiZy - ViZx 
 
 in which 
 
 k = 
 
 tctDlEc 
 
 Then 
 
 M = Ml + Fix - Hiy 
 
 The value of to should be inserted as plus ( + ) for a rise of temperature; minus ( — ) for a drop. Signs pre- 
 ceding terms Mi, Fi, and Hi in the last formula depend upon the signs of Mi, Fi, and Hi resulting from the pre- 
 ceding equations. I = span of arch axis measured horizontally; that is, parallel to X axis. 
 
 Rib shortening: 
 
 Ca 
 
 Ectc 
 
 tD 
 
 Rib shortening causes the same effect as a lowering of the 
 temperature. Solving for tD gives equivalent temperature 
 drop. 
 
 29. Arch Structure of Two Spans with Elastic 
 Pier.i — A structure of this type is shown in Fig. 17. 
 
 The points indicated as fixed may be the bottoms of the pier and abutments, or they may 
 be at intermediate sections, depending upon where the designer considers the structure fixed. 
 The horizontal section A- A where arches and pier join is the section considered. In Fig. 18 
 the top of the pier is shown in detail. 
 
 1 An arch structure of more than two spans and with elastic piers may readily be analyzed by the ellipse-of- 
 elasticity method explained in vol. Ill of Hool's "Reinforced Concrete Construction." 
 
668 
 
 CONCRETE ENGINEERS' HANDBOOK 
 
 [Sec. 16-29 
 
 What may be called the skewbacks of the arches are shown. The weight of the material 
 between the skewbacks and the section A-A need be considered only in finding the resultant 
 thrusts on the pier sections. The origin of coordinates x and y for each arch is taken at the 
 middle point C of the section A-A instead of at the center of the skewbacks. 
 
 The following notation is employed: 
 
 Let XL, VL = coordinates of any point on the axis of the left arch referred to the center of the section 
 
 A-A as origin. Values of yh should be considered plus when measured above the axis 
 X-X, and as negative when measured below that axis. 
 XR, VR = coordinates of any point on the axis of the right arch referred to the center of the section 
 A-A as origin. Values of yR should be considered plus when measured above the axis 
 X-X, and as negative when measured below that axis. 
 yp = depth of any point on the vertical axis of the pier below the section A-A. All values 
 positive. 
 
 mL, mR = moment at any point on axis of left arch and right arch respectively of all external loads 
 between the point in question and the top of the pier. 
 MLy Mr, Mp = moment at any point on axis of left arch, right arch, and pier respectively. 
 
 nL, nR, np = number of j divisions in the left arch, right arch, and pier respectively. 
 
 CL, cR, CP = values of j for left arch, right arch, and pier respectively. 
 
 Hi, Vi = horizontal and vertical components of the thrust from the left arch at the top of pier. 
 
 Ml = moment at section A-A due to thrust from left arch = vertical component of thrust 
 from left arch multiplied by the distance from the point C (the center of the section) to where 
 this thrust produced cuts the section A-A. 
 Hi, Vi = horizontal and vertical components of the thrust from the right arch at the top of pier. 
 Mi = moment at section A-A due to thrust from right arch. 
 H3 = resultant shear on section A-A = Hi — H2. 
 V3 = resultant thrust (normal) on section A-A = Vi + V2. 
 M3 = resultant moment on section A-A = Mi — M2. 
 
 The arrows in Fig. 18 indicate what are considered positive values of the quantities. 
 
 The following six equations for loading may be solved simultaneously for the values of 
 Hi, 7i, Ml, H2, Vi and M<,\ 
 
 Loading: 
 
 cUMiHyL — HiZyL^ -\- ViZxLyL - 'LmhyL) = - cR{Mii:yR - HilLyR^ + ViUxrvr - 1,mRyR) 
 MillXL — Hi'ZxLyL + Vi'LxL^ — l^mLxL = 0 
 MiXxR — Hi'LxRyR + V21:,xr'^ — ZmRXR = 0 
 
 CLinLMi — HiZyL -\- Vi'ExL — XmL) = — cRinRM-- — Hi'S^yR + ViZxR — "Lvir) 
 cp[(il/i - M2)Syp + {Hi - i72)S2/p2] ^ cL[Mi-ZyL - HilLyh^ + ViLxLVL - ^mLyL] 
 cp[np(Mi — M2) + (Hi — i72)S2/p] = - chinhMi - H1S2/L + FiSxL - Smi] 
 
 Bending Moment at any point: 
 
 Ml = Ml — HiyL + ViXL — mL 
 Mr = M2 — HiyR + V2XR — mp 
 Mp = (Ml - M2) + (Hi - H2)yp 
 Values of yL and yp should be considered plus when measured above the axis X-X, and as negative when 
 measured below that axis. The values of H3, V3, and Mz may be obtained from the following relations: 
 Hz = Hi - Hi Vz= Vi + Vi Mz = Ml - Mi 
 
 All values of mL and mp should be substituted as positive. 
 
 For arch bridges symmetrical about the center line of pier, the labor involved in solving 
 the simultaneous equations mentioned above will be greatly reduced. 
 
 All the simultaneous equations given above pertain to the unknown forces acting at the 
 section A-A. With these completely determined, however, the moment and thrust at any 
 section may be found in same manner as for the single symmetrical arch. Each of the three 
 members must be considered separately and each subjected to exactly the same force that is 
 found to act upon it at the top of pier in the monolithic structure. 
 
 When the two arch spans are equal, either arch may be analyzed for temperature and rib 
 shortening stresses in the same manner as for a single arch having immovable or fixed supports. 
 
Sec. 16-30] 
 
 ARCHES 
 
 669 
 
 COCHRANE'S FORMULAS AND DIAGRAMS FOR USE IN THE DESIGN OF SYMMET- 
 RICAL ARCHES IN ACCORDANCE WITH THE ELASTIC THEORY^ 
 
 30. Accuracy of Formulas and Diagrams. — The formulas and diagrams in this chapter 
 are of sufficient accuracy to use for the final design in many cases, and in practically all instances 
 they will afford a means of determining the form and dimensions of the arch with assurance that 
 the final analysis will show that but slight changes, if any, are required. 
 
 31. Difficulties and Uncertainties Involved in Applying the Elastic Theory. — There art 
 many reasons for concluding that the use of rigidly exact theoretical formulas for arch design 
 is wholly unwarranted, and that if by making certain reasonable practical assumptions we can 
 devise greatly shortened methods of design, we are quite justified in so doing. For, in the 
 first place, the elastic theory as applied to hingeless arches is based on the assumption that the 
 ends of the arch are rigidly attached to immovable abutments, so that a section of the arch at 
 the skewback or springing is subject neither to vertical or horizontal displacement nor to rota- 
 tion. This assumption is never in entire agreement with actual conditions, and is only admis- 
 sible as an approximation in cases where the piers or abutments will, by reason of their size 
 and the character of the foundations, be subject to but slight distortions or movements. There 
 are other approximate assumptions made in deriving the working formulas, such as that the 
 material is homogeneous, that the modulus of elasticity is constant, and that the effect of the 
 radius of curvature may be neglected. Furthermore, it is necessary to replace the definite 
 integrals of the three equations of condition by summations. But this is not all. Even if the 
 thrusts and moments at any section for any given condition of loading, etc., could be found 
 with certainty by the elastic theory the resultant stresses could not even then be exactly 
 determined, on account of the approximate character of the flexure formulas and the uncertain 
 tensile stresses in the concrete. Finally, it may be pointed out that the live load itself is subject 
 to much uncertainty, both as to its amount and its distribution, and that the effect of variations 
 in temperature, while in many cases quite severe, cannot in the present state of our knowledge 
 concerning the matter be determined with any great degree of certainty. Ihe effect of the 
 shrinkage of the concrete is even more uncertain. These i 
 
 and other like considerations justify the use of approxi- V-'--5x--t-''^^<4^ 
 
 mate short-cut methods such as those proposed in this y-X^M^^ L \ 
 
 chapter. z^^^^*^- H J^^^s^^ 
 
 32. Best Shape of Arch Axis. — Using the notation '^/^^^^ I ^ 
 as shown in Fig. 19; also H" ^ ' 
 
 r = rise ratio = y ' Fig. 19. 
 
 u = ratio of thickness at any point to thickness at the crown = ^. 
 
 to 
 
 te 
 
 Us = J- 
 to 
 
 then the equations derived giving approximately the best shape of arch axis for both open- 
 spandrel and filled-spandrel arches may be expressed as follows: 
 
 For open-spandrel arches: 
 
 tan d = (3 + 5r) 
 
 6 + or 
 
 1 Taken verbatim, for the greater part, from paper on the " Design of Symmetrical Hingeless Concrete 
 Arches " presented before The Engineers' Society of Western Pennsylvania by Victor H. Cochrank;. This 
 paper was published in the Nov., 1916, issue of the Proc, vol. 32, No 8, pp. 647-713. Diagrams on pp. 682- 
 685 inclusive were supplied by Mr. Cochrane and take the place f^f formulas presented in the original paper. 
 
670 
 
 CONCRETE ENGINEERS' HANDBOOK 
 
 [Sec. 16-33 
 
 For filled-spandrel arches: 
 
 4rl 
 
 tan d = 
 
 4r 
 
 1 + 3r 
 
 (1 + 7.5r) 
 
 O.Zb 0 50 075 
 Values ofv=%i 
 
 Fig. 20. 
 
 33. Variation in Thickness of Arch Ribs. — Mr. Cochrane made a number of complete 
 designs and investigated a number of designs found in technical literature, in order to determine 
 what the thicknesses of the arch should be at various points in the haunch to give the same fiber 
 stresses as at the crown and the springing; or in other words, to determine the variation in u 
 
 with respect to y = — for equal maximum stresses throughout the arch ring. Plotting the 
 s 
 
 values of v as abscissas with the values of u as ordinates, curves were obtained of which those 
 shown in the full lines OAB and OCD in Fig. 20 are typical. In each case it was found that 
 a certain portion of the haunch might, so far as the stresses were concerned, have been made 
 thinner than the crown section. It was also found in each case that the thinnest section required 
 is at about the quarter point, and that between this point and the springing the thickness should 
 increase almost uniformly. It is easy to see why this should be the case, since the moments 
 
 ^ due to rib shortening and temperature variations are 
 small near the quarter point and greatest at the crown 
 or springing. A slight increase of thickness was required 
 near the crown, as shown at X in OCD, in some arches 
 of large rise-ratio designed for a relatively heavy live 
 load. 
 
 An arch of this shape would be unsightly and diffi- 
 cult to build, and consequently about the best that can 
 be done is to make the arch rib of uniform thickness, or 
 but slightly increasing thickness, from the crown to about 
 the quarter point, and to increase the thickness almost 
 uniformly from this point to the springihg. The point at which the thickness should begin 
 to increase rapidly is nearer the springing in arches of large rise-ratio carrying heavy live loads 
 than in flat arches carrying light live loads. The dashed curve OFD in Fig. 20 represents an 
 arch of the former kind and the curve OEB an arch of the latter kind, these curves representing 
 the actual thicknesses to be used instead of the possible minimum thicknesses shown by OCD and 
 OAB. The flatter arches generally require a greater relative thickness at the springing than 
 do those with large rise-ratios. 
 
 These conclusions led to the selection of a number of typical curves showing the variation 
 of u with respect to v for various values of Us. These curves are shown in Diagram 1 and are 
 designated as Types ^3.25, A3, A2.75, -^2.5, A2.25, A2, Ai.75 and A1.5, the subscript in each case 
 denoting the value of Us for that type. These curves were chosen after a careful study of 
 existing well-designed arches, and it is believed that the arch thicknesses required in any given 
 case will be closely represented by some one of them; we have only to select the one best suited 
 to the given conditions. 
 
 Since in each case the typical arch has a greater thickness in the haunches than would be 
 required to keep the extreme fiber stresses within desired limits, we reach the important conclu- 
 sion that it is unnecessary to compute the stresses in any hut the crown and springing sections. 
 The writer has yet to find a single well-designed arch having thicknesses corresponding closely 
 to those represented by one of these typical curves, in which the stresses at the crown and 
 springing are not very nearly the maximum found anywhere in the arch, provided the arch- 
 shortening effect is considered. Perhaps the only good reason for computing the stresses any- 
 where except at these two sections is in order to determine whether the amount of the steel 
 reinforcement may be reduced near the quarter point. 
 
 Many arches have been built having a much greater relative thickness in the haunch than 
 
Sec. 16-33] 
 
 ARCHES 
 
 671 
 
 indicated by these diagrams, but such practice is not to be recommended. Not only do such 
 arches require more material than necessary, but on account of the thicker haunches have 
 greater arch-shortening and temperature stresses at the crown and springing sections than do 
 those having thicknesses corresponding to the diagrams. Thickening the haunch does not 
 
 Diagram 1 
 
 O 0.10 O.ZO 030 040 Q50 0.60 070 0.80 Q90 /.GO 
 ^a/ues of /?<7f/o v 
 Assumed thickness of typical arches. 
 
 seem to improve the appearance of the arch. It is therefore clearly advisable to make the 
 haunch as thin as practicable. 
 
 If the half axis is divided into ten equal sections the ratio of the depth of the arch at the 
 center of each section to the depth at the crown is given in the following table : 
 
672 CONCRETE ENGINEERS' HANDBOOK [Sec. 16-33 
 
 Table 1. — Thicknesses of Typical Arches 
 
 Value of 
 
 
 
 
 
 Values of u 
 
 = — for tvoe 
 
 
 
 
 
 
 
 
 
 
 
 to 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 r - — 
 
 s 
 
 Al.6 
 
 Al.75 
 
 A2 
 
 A.2,25 
 
 
 ^2.75 
 
 As 
 
 A3. 25 
 
 0 
 
 1 
 
 .000 
 
 1.000 
 
 1 
 
 .000 
 
 1 . 000 
 
 1 .000 
 
 1 
 
 .000 
 
 1 
 
 .000 
 
 1.000 
 
 0.05 
 
 1 
 
 007 
 
 1.006 
 
 1 
 
 .005 
 
 1 .004 
 
 1 . 003 
 
 1 
 
 002 
 
 1 
 
 .001 
 
 1 .000 
 
 0.15 
 
 1 
 
 .021 
 
 1.018 
 
 1 
 
 .015 
 
 1 . 012 
 
 1 .009 
 
 1 
 
 006 
 
 1 
 
 003 
 
 1.000 
 
 0.25 
 
 1 
 
 035 
 
 1.030 
 
 1 
 
 .025 
 
 1 . 020 
 
 1 .015 
 
 1 
 
 010 
 
 1 
 
 005 
 
 1.000 
 
 0.35 
 
 1 
 
 049 
 
 1.042 
 
 1 
 
 035 
 
 1.028 
 
 1.023 
 
 1 
 
 021 
 
 1 
 
 023 
 
 1.030 
 
 0.45 
 
 1 
 
 063 
 
 1.054 
 
 1 
 
 048 
 
 1.048 
 
 1.057 
 
 1 
 
 070 
 
 1 
 
 083 
 
 1.101 
 
 U . Oo 
 
 1 
 
 077 
 
 1 n79 
 
 1 . U/ z 
 
 1 
 
 085 
 
 1.105 
 
 1.133 
 
 1 
 
 165 
 
 1 
 
 193 
 
 1 OQ1 
 
 0.65 
 
 1 
 
 095 
 
 1.125 
 
 1 
 
 168 
 
 1.215 
 
 1.269 
 
 1 
 
 328 
 
 1 
 
 385 
 
 1.455 
 
 0.75 
 
 1 
 
 145 
 
 1.223 
 
 1 
 
 311 
 
 1.403 
 
 1.508 
 
 1 
 
 625 
 
 1 
 
 737 
 
 1.865 
 
 0.85 
 
 1 
 
 245 
 
 1.393 
 
 1 
 
 547 
 
 1.700 
 
 1.862 
 
 2 
 
 025 
 
 2 
 
 185 
 
 2.355 
 
 0.95 
 
 1 
 
 406 
 
 1.621 
 
 1 
 
 837 
 
 2.055 
 
 2.277 
 
 2 
 
 495 
 
 2 
 
 709 
 
 2.932 
 
 1.00 
 
 1 
 
 500 
 
 1.750 
 
 2 
 
 000 
 
 2.250 
 
 2.500 
 
 2 
 
 750 
 
 3 
 
 000 
 
 3.250 
 
 We can readily obtain a formula for the areas of the vertical faces of these typical arches. 
 This forrnula is 
 
 . 182.2 + 18.6Sus + 5ASus^ ^ 
 
 The value of s, the length of the half axis, may be determined by scaling from a drawing, or 
 it may be taken by interpolation from the following table: 
 
 Table 2. — Lengths of the Half Arch Axis s in Terms of the Span Length 
 
 Kinds of arches 
 
 Values of s for rise-ratio r = 
 
 0.10 
 
 0.15 
 
 0.20 
 
 0.25 
 
 0.30 
 
 Open-spandrel arches 
 
 Filled-spandrel arches 
 
 0.513Z 
 0.515Z 
 
 0.529Z 
 0.534i 
 
 0.551^ 
 0.559^ 
 
 0.577Z 
 
 0.607Z 
 
 Table 3 gives values of -j" (referred to as j in preceding chapters for the typical arches, the 
 half axis in each case divided into 10 parts). 7o = moment of inertia at the crown. 
 
 Table 3 
 
 Type 
 
 
 Value of Y 
 
 ^1.6 
 
 0 
 
 0769 s -V- 7o 
 
 Ai.78 
 
 0 
 
 0732 s -^ 7o 
 
 A, 
 
 0 
 
 0699 s /o 
 
 ^2.25 
 
 0 
 
 0672 s -^ /o 
 
 -^2. 5 
 
 ' 0 
 
 0647 s h 
 
 ^2.75 
 
 0 
 
 0626 s 7o 
 
 ^3 
 
 0 
 
 0606 s 7o 
 
 ^1.26 
 
 0 
 
 0586 s -i- 7o 
 
 The approximate value of s may be taken from Table 2 above. 
 
Sec. 16-34] ARCHES 673 
 
 The following table gives the location of the centers of divisions having a constant ratio 
 
 As 
 
 Y". These points will be referred to for convenience as "division centers," or "centers of 
 divisions." 
 
 Table 4. — ^Location of Centers of Divisions Having a Constant Ratio 
 
 As . . 
 
 -J- (10 divisions) 
 
 Point at center 
 of division. 
 
 
 
 
 Values oi v = 
 
 = — for type 
 s 
 
 
 
 No. from crown 
 
 
 Al.Tb 
 
 A 2 
 
 A2.25 
 
 A 2. 5 
 
 A2.75 
 
 A3 
 
 A3. 25 
 
 1 
 
 0.039 
 
 0.037 
 
 0.036 
 
 0.034 
 
 0.033 
 
 0.032 
 
 0.031 
 
 0.030 
 
 2 
 
 0.119 
 
 0.112 
 
 0.107 
 
 0.102 
 
 0.098 
 
 0.095 
 
 0.092 
 
 0.089 
 
 3 
 
 0.201 
 
 0.190 
 
 0.180 
 
 0.172 
 
 0.165 
 
 0.159 
 
 0.153 
 
 0.149 
 
 4 
 
 0.286 
 
 0.270 
 
 0.255 
 
 0.243 
 
 0.232 
 
 0.222 
 
 0.214 
 
 0.208 
 
 5 
 
 0.374 
 
 0.352 
 
 0.332 
 
 0.316 
 
 0.301 
 
 0.287 
 
 0.275 
 
 0.265 
 
 6 
 
 0.464 
 
 0.436 
 
 0.411 
 
 0.389 
 
 0.370 
 
 0.353 
 
 0.338 
 
 0.325 
 
 7 
 
 0.558 
 
 0.522 
 
 0.491 
 
 0.466 
 
 0.443 
 
 0.423 
 
 0.405 
 
 0.389 
 
 8 
 
 0.657 
 
 0.614 
 
 0.578 
 
 0.549 
 
 0.524 
 
 0.502 
 
 0.481 
 
 0.463 
 
 9 
 
 0.766 
 
 0.722 
 
 0.684 
 
 0.652 
 
 0.625 
 
 0.600 
 
 0.578 
 
 0.558 
 
 10 
 
 0.912 
 
 0.890 
 
 0.872 
 
 0.856 
 
 0.842 
 
 0.829 
 
 0.817 
 
 0.806 
 
 To find the distances from the crown to the division centers, multiply the above coefficients 
 by the length of the half axis. These coefficients apply to an axis of any shape. 
 
 Table 4 was computed on the basis of a 1 % crown reinforcement, but it may be used for 
 plain arch ribs, since the moments and thrusts are changed but little on account of a consider- 
 able change in the location of the division centers. 
 
 34. Influence -line Diagrams. — The formulas used in figuring the influence-line values of 
 Diagrams 2 to 13 inclusive were the same as given in Art. 18. 
 
 The use of these influence-line diagrams requires little explanation. The designer having 
 determined by means of the diagrams referred to in 
 Art. 35 (or otherwise) the approximate actual or rela- 
 tive thicknesses at crown and springing, selects the 
 typical arch best suited to the particular case, and 
 reads from the diagrams, by interpolation if necessary, 
 the values of the moments and thrusts at crown and 
 springing. Having these values he constructs influence 
 line diagrams to. any desired scale and reads off from 
 these the moments and thrusts due to the specified con- 
 centrated loads. If no concentrated loads are speci- 
 fied, or if uniform loads may be substituted for the 
 concentrated loads, the influence lines are not needed 
 and the diagrams of Art. 35 are used instead. 
 
 It will be noted that the maximum live-load mo- 
 ments at the crown and springing sections are determined by four typical arrangements of the 
 live load designated in Fig. 21 as loadings 1, 2, 3 and 4. Loadings 1 and 2 are for the maxi- 
 mum positive and negative moments at the crown respectively, and when combined cover the 
 entire span. Loadings 3 and 4 are for the maximum positive and negative moments at the 
 springing section, respectively, and when combined likewise cover the span. In the figure, k 
 represents the ratio of the length loaded to the span length. Loadings 1, 3 and 4 are contin- 
 uous, while loading 2 is discontinuous, 
 43 
 
 i 
 
 L0/1D/NG J 
 
 for Max ■* M. a-/ Crown 
 
 llllllllllllllll 
 
 LOADING 2 
 For Max -Al. a-f Crown 
 
 L 0/1 DING 3 
 For Max. + M dSprn^ 
 
 mil miiil \ 
 
 LOAD/A/G4 
 For Max - M. atSpr'n'^ 
 
 Fig. 21. — Typical loadings for maximum 
 moments at crown and springing. 
 
674 CONCRETE ENGINEERS' HANDBOOK [Sec. 16-34 
 
Sec. 16-34] 
 
 ARCHES 
 
 675 
 
Sec. 16-34] 
 
 ARCHES 
 
 077 
 
678 
 
 CONCRETE ENGINEERS' HANDBOOK 
 
 [Sec. 16-34 
 
Sec. 16-34] 
 
 ARCHES 
 
 679 
 
680 
 
 CONCRETE ENGINEERS' HANDBOOK 
 
 [Sec. 16-34 
 
 An inspection of the moment diagrams reveals an unexpectedly small variation in the value 
 of the ratio k for each loading, the variation being greater with respect to the rise than with 
 respect to the type of arch. It appears that using approximate average values for k, the maxi- 
 mum positive moment at the crown occurs when the live load covers one-fourth of the span at 
 the middle, and the maximum negative moment occurs when the load covers the two end three- 
 eighths. The maximum positive moment at the springing is produced by a load covering five- 
 eighths of the span beginning at the opposite end, and the maximum negative moment occurs 
 when the load covers the adjacent end three-eighths of the span. It may be stated further that 
 loading 1 and loading 4 combined produce loading 3, and that one-half of loading 2 is the same 
 as loading 4. 
 
 Another striking fact is that for any given type of arch the coefficients for moments vary 
 only slightly with respect to the rise-ratio. Hence we may say that the moments for any typical 
 arch of given span length are practically independent of the rise-ratio. 
 
 As the ratio of springing thickness to crown thickness increases, the arch axis remaining 
 the same, the moments at the springing increase and the moments at the crown decrease, as 
 might be expected. The change in the moments as the arch axis changes in form, the ratio Us 
 remaining the same, may be seen by comparing the diagrams for an open-spandrel arch with 
 those for a filled-spandrel arch having the same ratio u^. Table 5 presents a typical case. The 
 positive moments at crown and springing increase as the arch axis departs farther from the 
 parabolic form and becomes more nearly elliptical, while the negative moments at crown and 
 springing decrease. 
 
 Evidently the areas enclosed by a moment influence line above and below the line of zero 
 moments are a measure of the maximum moments due to a uniform live load on the span. 
 The algebraic sum of the positive and negative areas is a coefficient of the moments due to 
 live load over the entire span. It is evident, therefore, from an inspection of the diagrams that 
 the moments for live load over the whole span are generally much smaller numerically than the 
 maximum moments, and are always positive at crown and springing. 
 
 Another interesting fact is that for arches of the same type (that is, having the same ratio 
 Us), but having axes of different form, the arithmetical sum of the positive and negative moments 
 is practically the same. For example, take an open-spandrel and a filled-spandrel arch of the 
 s.ame rise-ratio r = 0.2 and the same thickness-ratio Us = 2.5 (Diagrams 4, 7, 9 and 12), and the 
 moments are as follows: 
 
 Table 5. — Comparison of Moments for Type Aa.s Open-Spandrel and 
 Filled-Spandrel Arches (r = 0.20) 
 
 Section 
 
 Kind of moment 
 
 Open-spandrel arch 
 
 Filled-spandrel arch 
 
 Crown 
 
 Max. + moment 
 
 Algebraic sum 
 
 Arithmetical sum 
 
 +0. 00474 
 -0.00378w;Z2 
 
 +0. 00096^^^2 
 0.00852t(;Z2 
 
 +0 . 00637w;^2 
 -0. 00302 w;Z2 
 
 +0.003357/;Z2 
 0.00939i(;Z2 
 
 Springing 
 
 Max. + moment 
 
 Max. — moment 
 
 Algebraic sum 
 
 Arithmetical sum 
 
 +0.02990^^;^2 
 -0.02330w;^2 
 
 +0.00660w;^2 
 0 . 05320w;Z''' 
 
 +0.04110w;Z2 
 -0.01750w;Z2 
 
 +0 . 02360w;^2 
 0.05860w;Z2 
 
 Thus while the maximum positive moments are more than one-third greater for the filled 
 spandrel arch than for the open-spandrel one, the arithmetical sums of the moments are only 
 
Sec. 16-35] 
 
 ARCHES 
 
 681 
 
 one-tenth greater. These two arch axes differ widely, and hence for two axes differing but little 
 we may assume that the arithmetical sum of the maximum positive and negative moments is 
 the same for each arch. This fact leads to an easily applied approximate method for correcting 
 the known moments for an assumed axis to suit an actual arch having its axis differing in form 
 from the assumed axis. This method is given in Art. 36. 
 
 The moments in any typical arch due to unit loads are proportional to the span length, 
 while the thrusts are the same for all span lengths. The shears at the crown are nearly inde- 
 pendent of the rise and are the same for all span lengths. The thrusts at the crown increase as 
 Us increases, and also as the arch axis approaches an elliptical form, and very approximately 
 inversely as the rise-ratio r. 
 
 35. Diagrams for Moments, Thrusts, and Average Stresses. — The notation used in 
 Diagrams 14 to 21 inclusive is as follows: 
 
 I 
 
 
 the span length of the arch axis in feet. 
 
 h 
 
 
 the rise of arch axis in feet. 
 
 r 
 
 
 the rise-ratio h/l. 
 
 y 
 
 
 the ordinate of the arch axis at any point the abscissa of which is cl. 
 
 Us 
 
 
 ratio of the thickness at springing to the thickness at crown. 
 
 Mc 
 
 
 moment at crown in foot-pounds. 
 
 To 
 
 
 thrust at crown in pounds. 
 
 Ms 
 
 
 moment at springing in foot-pounds. 
 
 Ts 
 
 
 thrust at springing in pounds. 
 
 Vs 
 
 
 approximate dead load vertical end reaction, or one-half dead weight of span in pounds. 
 
 Wc 
 
 
 weight of arch at the crown, plus average weight of arch superstructure at the crown in pounds per 
 
 
 
 lineal foot of span. 
 
 w 
 
 
 live load in pounds per lineal foot of span (not necessarily the same for all positions of the live load). 
 
 te 
 
 
 the coefficient of linear expansion due to temperature changes. 
 
 tD 
 
 
 change in temperature in degrees Fahrenheit. 
 
 E 
 
 
 modulus of elasticity of concrete in pounds per square foot. 
 
 lo 
 
 
 the moment of inertia of the arch rib at the crown in feet^. 
 
 fa 
 
 
 average direct stress throughout arch in pounds per square foot. 
 
 Sac 
 
 
 direct stress at crown section in pounds per square foot. 
 
 From what has been said the use of these diagrams will be readily understood. Following 
 is a summary of the steps required in designing an arch by the use of the diagrams. It is 
 assumed that the arch superstructure has been designed, and that the dead load per lineal 
 foot at the crown, exclusive of the arch rib, is known; also that the span and rise of the axis 
 have been fixed. 
 
 1. Assume a crown thickness in accordance with one of the empirical formulas in use, or by comparison 
 with a previous design, and compute the total dead load per lineal foot of span at the crown (= w^). 
 
 2. Assume the arch type, or the thickness-ratio Us. For open-spandrel arches this ratio will usually be from 
 1.5 to 2.5, and for fiUed-spandrel arches from 2 to 3.25. 
 
 3. Determine from the diagram the dead-load thrusts and the maximum positive and negative moments and 
 corresponding thrusts at the crown and springing due to live load, temperature variation and arch shortening, and 
 calculate the extreme fiber stresses due to the proper combinations of moments and thrusts. If the stresses so 
 found are too great or too small, change the thickness at the crown or at the springing, or both, and repeat the above 
 operation. The second trial will usually be sufficient. If the thicknesses originally assumed are not correct, the 
 fact will usually be revealed before the first set of calculations is completed. 
 
 4. Lay out the assumed arch axis as per Art. 32 and divide it into 10 or more equal parts, locating the center 
 of each division. At the points thus found lay off the lialf thickness, as given in Diagram 1 or in Table 1, above and 
 below the assumed axis, and through the points thus determined pass, as nearly as practicable through all the points, 
 segmental, three-centered, or five-centered curves. A set of railroad curves is useful for this purpose. 
 
 5. Compute the dead loads at the panel points or at suitable intervals and lay out an equilibrium polygon 
 passing through the crown and springing, the value of the horizontal thrust being first computed as for a three- 
 hinged arch and used as the pole distance for the force polygon (see Art. 11). 
 
 6. By trial alter the shape of the arch axis so that it will fit the dead-load equilibrium polygon as nearly as 
 practicable, lay out the arch thicknesses again, and determine the radii of the intradosal and extradosal curves. 
 
 7. If the actual axis departs considerably from the assumed axis, scale the dead-load thrusts from the force 
 polygon and correct the maximum live-load moments by the method given in Art. 36, revising the stresses and 
 changing the thicknesses if necessary. This last step will seldom be required. 
 
682 
 
 CONCRETE ENGINEERS' HANDBOOK 
 
 [Sec. 16-35 
 
684 
 
 CONCRETE ENGINEERS' HANDBOOK 
 
 [Sec. 16-35 
 
Sec. 16-35] 
 
 ARCHES 
 
 685 
 
686 
 
 CONCRETE ENGINEERS' HANDBOOK 
 
 [Sec. 16-36 
 
 36, Approximate Method of Correcting Maximiun Moments when Actual Arch Axis 
 Deviates from Assumed Axis. — It has been found that for any given typical arch — that is, 
 for an arch having a constant ratio Us — the resistance line for live load over the entire span 
 crosses the arch axis at about the same horizontal distance from the center line of the span, 
 regardless of the shape of the arch axis. Also it has been found that for two arch axes of the 
 same type and which differ but little, the arithmetical sum {Si) of the maximum positive and 
 negative moments is the same for each arch. Making use of these facts, the formulas given 
 below were obtained. 
 
 In Fig. 22, the line CABS represents the axis of an open-spandrel arch of type A 2. 5 and 
 rise-ratio = 0.2, while CDES represents the axis of a type A-i.^ filled-spandrel arch. The 
 
 Fig. 22. — Resistance lines for an open-spandrel and a filled spandrel arch; live load over whole span. 
 
 vertical scale is times the horizontal. The resistance lines for full live load are QABN 
 and ODEP, respectively, and it will be seen that the intersection points A and D are about 
 equidistant from the center line, as are also B and E. It appears, then, that if the full-load 
 resistance line for one axis, as CABS, is known, the resistance line for another axis, as CDES, 
 may be approximately determined by passing the parabola ODEP through the two points 
 D and E having the abscissas Xi and X2 the same as for the known intersection points A and 
 B, and the ordinates yi and yo, referred to C as origin. 
 
 Let d = the rise of the required resistance line for live load over the entire span (parabolic). 
 
 di = the corresponding known term for the assumed arch. 
 
 a = the vertical intercept between the arch axis and the required resistance line at the crown. 
 b = the vertical intercept at the springing. 
 Hi = horizontal thrust for known resistance line QABN. 
 Mp' = known maximum positive moment at crown or springing for the assumed axis CABS. 
 Mn = known maximum negative moment at crown or springing for the assumed axis CABS. 
 Afp and Mn = corresponding terms for the actual axis CDES. 
 
 arithmetical sum of the moments Mp' and Mn' — Mp' — Mn'. 
 
 Then 
 
 Mp 
 
 Hia~ + S 
 a 
 
 2 
 
 Hia^ - S: 
 
 2 
 
Sec. 16-36] 
 
 ARCHES 
 
 687 
 
 For the springing moments substitute b for a in these formulas. 
 
 These formulas will give close approximations to the true values of the moments even if the 
 axis used deviates considerably from the assumed axis. 
 
 Illustrative Problem. — The design will be that of the 132-ft. arch, Kansas River bridge at Lawrence, 
 Kan. This is a rather flat open-spandrel arch. Typical half sections of the span are shown in Fig. 23. Follow- 
 ing are the dimensions: 
 
 Z = 132 h = 16 :. r = 0.121 
 
 fo = 2.5 te = 5.63 Us = 2.25 
 
 The live loads are as follows: 
 
 HALr- SECTION AT HAUNCH HALF-SECTION AT CROIVN 
 
 Fig. 23. 
 
 Live Loads 
 
 Member loaded 
 
 Portion of structure loaded 
 
 Track 
 
 Remainder of 
 roadway (18 ft.) 
 
 Sidewalks 
 
 For roadway floor slabs, fascia-girders, canti- 
 lever beams, cross-girders, spandrel columns 
 and approach girder spans. 
 
 Electric railway loading 
 (assumed to occupy a 
 transverse width of 
 12 ft.). 
 
 20-ton road roller 
 or 
 
 200 lb. per sq. ft. 
 
 150 lb. per sq. ft. 
 
 
 Electric railway loading 
 (assumed to occupy a 
 transverse width of 
 12 ft.). 
 
 60 lb. per sq. ft. 
 
 50 lb. per sq. ft. 
 
 Or the equivalent uniform loads given below. 
 
 
 3500 lb. per lin. ft. span 
 
 50 lb. per sq. ft. 
 
 40 lb. per sq. ft. 
 
 J lie vviieei luaus are snuwii iii rig. ^•±. 
 
 The working stresses assumed are given on page 688. 
 
 37300 
 37500 
 
 37500 
 37SOO 
 
 37500 
 37SOO 
 
 37SOO 
 37SOO 
 
 37500 
 37500 
 
 
 
 9 9 
 
 O O Q O E'TC, Of? 
 
 
 
 ki 23' 
 
 \s\ 8' \s]^£' 
 
 /I continuous line of 150,000 /b. cans mfh ax/e hods as abovre. 2 ax/es on/y. 
 Liv^e Load for E/ecfr/c Ry. Track 
 
 t 
 
 ZO-Ton Road Roller 
 Fig. 24. 
 
688 
 
 CONCRETE ENGINEERS' HANDBOOK 
 Working Stresses in Pounds per Square Inch 
 
 [Sec. 16-36 
 
 Member 
 
 Kind of stress 
 
 Stress 
 
 Floor slabs 
 
 
 600 
 16,000 
 
 Tension in steel 
 
 
 Arch rings 
 
 Compression on concrete, including L. L., D. L., rib shortening and temperature (for 
 
 600 
 800 
 
 
 Es = 30,000,000 Ec = 2,000,000 n = 15 tc = coefficient of expansion = 0.0000055. 
 
 In order to employ the formulas it is necessary to use a uniform live load in lieu of the concentrated loads 
 shown in Fig. 24 for the electric railway track. It is difficult, if not impracticable, to work out equivalent uniform 
 loads for arches such as are in use for simple beam and truss spans. It will usually be sufficiently accurate for prac- 
 tical purposes to use the average weight per lineal foot of car. Hence we assume the uniform load as follows: 
 
 Average weight of car = ^ ^^'^^f^ _ 3,490 
 
 Uniform load on roadway. 
 Uniform load on sidewalks. 
 
 (30 
 
 43 
 12) X 60 
 10 X 50 
 
 = 1,080 
 = 500 
 
 Total 5,070 1b. 
 
 Then we have wl = 5,070 X 132 = 669,000 lb. 
 
 and wl^ = 5,070 X 132^ = 88,340,000 ft.-lb. 
 It will be assumed to begin with that rough preliminary calculations have been made from which it is found 
 that the crown thickness to may be assumed = 2.5, and the ratio Ug = 2.25. The first step is to compute the dead 
 load per lineal foot at the crown (wc), thus: 
 
 Weight of track, paving, and paving base on sand fill 4,000 
 
 Weight of concrete in floor slabs, fascia girders and side walls 3,170 
 
 Weight of cantilever brackets, per foot of span 1,300 
 
 Weight of sand fill 2,880 
 
 Weight of railings 800 
 
 Weight in conduit spaces 250 
 
 Weight of arch rib, 24 X 2.5 X 150 9,000 ' 
 
 Total Uc = 21,400 lb. 
 
 Dead Load. — From Diagram 14, Tc (or He) 
 
 = (1.135)(21,400)(132) = 3,210,000 lb. 
 Vs = (0.600) (21,400) (132) = 1,700,000 lb. 
 Ts = 3,630,000 lb. 
 Live Load, Max. + Mom. at Crown. — From Diagram 15, 
 Tc = (0.524) (669,000) = 351,000 1b. 
 Mc = (0.00445) (88,340,000) = + 393,000 ft.-lb. 
 Line Load, Max. — Mom. at Crown. — From Diagram 15, 
 Tc = (0.545) (669,000) = 365,000 1b. 
 Mc = ( - 0.00391) (88,340,000) = - 345,000 ft.-lb. 
 Live Load, Max. + Mom. at Springing. — From Diagram 16, 
 Ts = (0.746) (669,000) = 499,000 lb. 
 Tc = (0.775) (669,000) = 518,000 lb. 
 Ms = (0.0282) (88,340,000) = + 2,490,000 ft.-lb. 
 Live Load, Max. — Mom. at Springing. — From Diagram 16, 
 Ta = (0.431) (669,000) = 288,000 lb. 
 Tc = (0.295) (669,000) = 197,000 lb. 
 Ms = (- 0.024) (88,340,000) = - 2,120,000 ft.-lb. 
 The arch is reinforced with 24 lines of V/i-in. square bars at top and bottom. The centers of the bars are 
 from the face of the arch, or 1 ft. from the axis at the crown. Hence the moment of inertia 
 J. _ (24)(2.5)3 , (24)(2)a.56)(l)2(15) 
 
 12 ' 144 
 The equivalent area at the crown = 
 
 2.5 +[<2«lj»],24,= 67.8 s,.(t. 
 Fall 0/ Temperature of 40°. — 
 
 tctoE = (0.0000055) (40) (288,000,000) 
 - 63,400 lb. per sq. ft. 
 
 = 39.05 
 
Sec. 16-36] 
 
 ARCHES 
 
 689 
 
 From Diagram 19, 
 Tc = 
 
 Mc = 
 
 (29.9) (63,400) (39.05) 
 
 (16) (16) 
 (20.1) (289,000) (16) 
 
 = - 289,000 lb. 
 = + 929,000 ft.-lb. 
 
 100 
 
 Ts = (1.09 - 1.75 X 0.121) (289,000) = - 254,000 lb. 
 Ms = + 929,000 - (16) (289,000) = - 3,695,000 ft.-lb. 
 Average Stresses. — From Diagrams 20 and 21 
 For dead load, 
 
 ^3,210,000n 
 
 fa = (0.88) (- 
 
 67.8 
 
 0 = 41, 
 
 700 lb. 
 
 For L. L. producing max. + moment at crown, 
 
 /,-(0.87)(M) = 4500 1b. 
 ForL. L. producing max. — moment at crown, 
 
 /365,000\ 
 /a= (0.90)(-^) = 
 
 4800 lb. 
 
 ForL. L. producing max. + moment at springing 
 '518,000\ 
 
 6700 lb. 
 
 /a = (0.87) (-^^ 3 / 
 For L. L. producing max. — moment at springing, 
 
 (0.89) (1§|») = 2600 lb. 
 For temperature drop of 40°, 
 
 fa 
 
 (0.83)(™) = - 3500 1b. 
 
 67. 
 
 For each combination of loading, the arch-shortening thrusts and moments bear the same ratio to the thrusts 
 and moments due to a fall of 40° in temperature, as does the total average stress to the stress IcIdE ( = 63,400 lb.). 
 
 I. Summary for Maximum + Moment at Crown 
 (a) D. L., L. L. and Arch Shortening 
 
 
 Thrust 
 
 Moment 
 
 Average 
 stress 
 
 D. L 
 
 + 3,210,000 
 + 351,000 
 - 200,000 
 
 
 + 41,700 
 + 4,500 
 - 2,400 
 
 L. L 
 
 Arch S 
 
 Total 
 
 + 393,000 
 + 642,000 
 
 + 3,361,000 
 
 + 1,035,000 
 
 + 43,800 
 
 (&) D. L., L. L., Temperature Variation, and 
 Arch Shortening 
 
 
 Thrust 
 
 Moment 
 
 Average 
 stress 
 
 D. L. and L. L. . 
 Temperature. . . 
 Arch S 
 
 Total 
 
 + 3,561,000 
 
 - 289,000 
 
 - 185,000 
 
 + 393,000 
 + 929,000 
 + 593,000 
 
 + 46,200 
 
 - 3,500 
 
 - 2,200 
 
 + 3,087,000 
 
 + 1,915,000 
 
 + 40,500 
 
 II. Summary for Maximum — Moment at Crown 
 (a) D, L., L. L., and Arch Shortening 
 
 
 Thrust 
 
 Moment 
 
 Average 
 stress 
 
 D. L 
 
 + 3,210,000 
 + 365,000 
 - 201,000 
 
 
 + 41,700 
 + 4,800 
 - 2,400 
 
 L. L 
 
 -345,000 
 + 646,000 
 
 Arch S 
 
 Total 
 
 + 3,374,000 
 
 + 301,000 
 
 + 44,100 
 
 
 It appears from the above that there is no negative moment at the crown. 
 41 
 
690 
 
 CONCRETE ENGINEERS' HANDBOOK 
 
 [Sec. 16-36 
 
 III. Summary for Maximum + Moment at Springing 
 (a) D. L., L. L., and Arch Shortening 
 
 
 Thrust 
 
 Moment 
 
 Average 
 stress 
 
 D. L 
 
 + 3,630,000 
 + 499,000 
 - 184,000 
 
 
 + 41,700 
 + 6,700 
 - 2,500 
 
 L. L 
 
 Arch S 
 
 Total 
 
 + 2,490,000 
 -2,675,000 
 
 + 3,945,000 
 
 - 185,000 
 
 + 45,900 
 
 (6) D. L., L. L., Temperature Variation, and 
 Arch Shortening 
 
 
 Thrust 
 
 Moment 
 
 Average 
 stress 
 
 D. L. and L. L. . 
 Temperature. . . 
 Arch S 
 
 Total 
 
 + 4,129,000 
 + 254,000 
 - 197,000 
 
 + 2,490,000 
 + 3,695,000 
 -2,860,000 
 
 + 48,400 
 + 3,500 
 - 2,700 
 
 + 4,186,000 
 
 + 3,325,000 
 
 + 49,200 
 
 IV. Summary for Maximum — Moment at Springing 
 (a) D. L., L. L., and Arch Shortening 
 
 
 Thrust 
 
 Moment 
 
 Average 
 stress 
 
 D. L 
 
 + 3,630,000 
 + 288,000 
 - 168,000 
 
 
 + 41,700 
 + 2,600 
 - 2,300 
 
 L. L 
 
 Arch S 
 
 Total 
 
 -2,120,000 
 -2,450,000 
 
 + 3,750,000 
 
 -4,570,000 
 
 + 42,000 
 
 (6) D. L., L. L., Temperature Variation, and 
 Arch Shortening 
 
 
 Thrust 
 
 Moment 
 
 Average 
 stress 
 
 D. L. and L. L.. 
 Temperature. . . 
 Arch S 
 
 Total 
 
 + 3,918,000 
 
 - 254,000 
 
 - 155,000 
 
 -2,120,000 
 -3,695,000 
 -2,257,000 
 
 + 44,300 
 
 - 3,500 
 
 - 2,100 
 
 + 3,509,000 
 
 -8,072,000 
 
 + 38,700 
 
 The following table gives the approximate extreme fiber stresses in pounds per square 
 inch computed from the above thrusts and moments on the assumption thai the concrete 
 takes no tension : 
 
 Case 1(a) 
 
 Case 1(6) 
 
 Case IV(a) 
 
 Case IV(6) 
 
 580 
 
 750 
 
 420 
 
 730 
 
Sec. 16-37] 
 
 ARCHES 
 
 691 
 
 The stresses computed by this method will be found to be somewhat less than those figured 
 by the exact method, but if some arbitrary allowance should be made for the dead-load 
 moments the difference would be less. A good rule for figuring the arbitrary allowance for 
 dead-load moments (in case the arch axis is made to fit the dead-load equilibrium polygon) 
 is to assume an eccentricity of application of the dead-load thrust above or below the axis 
 equal to one-fortieth the depth of section. 
 
 DETAILS OF ARCH BRIDGES 
 
 37. Spandrel Details in Earth-filled Bridges. — The filling material in solid-spandrel 
 bridges is held in place laterally by retaining walls which rest upon the arch ring. These retain- 
 ing walls may be of either the gravity or the reinforced type, or they may consist of thin vertical 
 slabs tied together by reinforced-concrete cross walls. In the usual type of solid-spandrel 
 construction, the sidewalk rests upon the earth filling, which is the type shown in Figs. 25 A and 
 26. Where the counterforted type of spandrel wall is employed, sidewalks are sometimes 
 
 Fig. 25 a. — Details of Pine Street bridge, Lima, Ohio. 
 
 cantilevered beyond the faces of the arch ring, as illustrated in Fig. 27. The faces of spandrel 
 walls may be entirely plain, or panels of approximately a triangular shape may be formed, 
 either by indenting the portion above the arch ring or by nailing beveled strips to the form- 
 work. Brick and stone are used in some cases as a facing for arch rings and spandrel walls. 
 
 Figs. 25A and 25B show a portion of a flat-arch bridge designed for the city of Lima, Ohio. 
 The spandrel walls are of the reinforced cantilever type. 
 
 A bridge with gravity spandrel walls is shown in Fig. 26. The brick facing for the arch 
 ring and the cast concrete and brick belt courses should be noted. The spandrel walls rest 
 partly on the brick facing and partly on the concrete portion of the arch, and are keyed into 
 the concrete portion by means of a projection which fits into a 6 by 12-in. groove in the arch. 
 
 A counterforted type of spandrel wall is shown in Fig. 27. These walls are 12 in. thick 
 and are reinforced on both faces with a double system of rods. The counterforts occur at 
 about 9-ft. intervals, and cantilever brackets are placed at these counterforts to support the 
 sidewalks. The following description is taken from Engineering Record, Feb. 22, 1913. 
 
 The entire width of the arch ring between outside faces of spandrel walls is 35 ft., and the roadway above is 
 39 ft. wide, thus giving an overhang of 2 ft. on each side of the bridge between the curb lines. This 2-ft. overhang 
 constitutes the concrete gutter of the roadway and, as such, will be subject to heavy concentrated wheel loads com 
 
692 
 
 CONCRETE ENGINEERS' HANDBOOK 
 
 [Sec. 16-37 
 
 ing upon the cantilever section. It was, therefore, built as a heavily reinforced concrete beam. This beam is 
 2 ft. 9 in. wide, having a depth of 15 in. at the spandrel wall and 10 in. at the curb, and is reinforced with fourteen 
 ^i-in. rods, with additional reinforcement at the brackets. 
 
 The bridge is designed for trolley traffic, and provision is made for the trolley poles by anchoring sections of 
 10-in. cast-iron water pipe in the brackets of the piers and abutments. Drainage is provided by means of 6-in. 
 cast-iron drains in the gutters over the piers. 
 
 1! 'ii^-^".s"Laj Bolt 
 
 SecTion 
 
 Details of Railfng 
 
 Fig. 255. — Details of Pine Street bridge, Lima, Ohio. 
 
 Details of the arch bridge over the Olentangy River on King Avenue, Columbus, Ohio, 
 is shown in Figs. 28 A, 285, and 28C. The type of spandrel walls without pilasters over piers 
 should be noted. Since the space beneath each sidewalk is hollow, the inner wall was designed 
 as a slab with the principal steel placed horizontally between cross walls. The longitudinal 
 walls under the car track were employed to prevent the usual settlement of the track when 
 laid on a new fill. The ties were laid directly on top of these walls and earth filling was dumped 
 both sides of, and also between, the longitudinal walls. 
 
Sec. 16-37] 
 
 ARCHES 
 
 693 
 
 Side Elevation Section A-A 
 
 Fig. 27. — Counterforted spandrel wall, highway bridge at Ansonia, Conn. 
 
694 
 
 CONCRETE ENGINEERS' HANDBOOK 
 
 [Sec. 16-38 
 
 Drains should be placed on each side of the roadway of a concrete bridge at intervals of 
 30 to 40 ft. when the roadway is level and about every 100 ft. when on a grade. These drains 
 should have a diameter of not less than 3 in. The minimum area of a drain in square inches 
 may be computed by the formula 
 
 A 
 
 ^=200 
 
 where A = area of the surface drained in square feet. 
 
 One type of expansion joint in a simple cantilever wall is shown in Fig. 29. 
 
 Elevation 
 
 Fig. 28 a. — Elevation of west end of Olentangy River bridge on King Avenue, Columbus, Ohio. 
 
 Fig. 28B. — Details of west abutment, Olentangy River bridge on King Avenue, Columbus, Ohio. 
 
 38. Spandrel Details in Open-spandrel Bridges. — The general types of open-spandrel 
 bridges have been described in Art. 3. Figs. 30 to 35 inclusive will serve to illustrate details 
 of some of these types. 
 
 Fig. 30 shows a full-barreled arch reinforced with typical Melan trusses made up of 3 by 
 3 by ^{e-in. angles and 2}i by }i-m lattice bars. The floor system is carried on a series of 
 transverse spandrel walls and the floor slab is provided with expansion joints as shown. The 
 sidewalks leave an overhang of about 3 ft. and are supported on cantilever brackets. The 
 
Sec. 16-38] 
 
 ARCHES 
 
 695 
 
 e8-6 
 
 Half Plan of Pier 
 
 [j U U J U L 
 
 U u J J l] 
 
 Half Sec+ion at Pier 
 
 Fig. 28C, — Details of Olentangy River bridge on King Avenue, Columbus, Ohio. 
 
 Sheet Lead ■ ■ 
 j' " thick 
 
 Top of 
 
 batter of 
 spandrel wall 
 
 Inside Elevation 
 of Expansion Joint 
 
 • Sheet 
 asphalt ^ 
 or its 
 equivalent 
 
 ■■■ Bottom of 
 batter erf 
 spandrel wall 
 
 . Sheet Lead 
 / 'thick 
 
 Section A-A 
 
 i'.i" Soft wood 
 Strip 
 
 Sheet Lead _ 
 ^' thick- 
 
 
 Sheet 
 "^^ ' Asphalt 
 
 
 
 
 
 
 
 Section B-B 
 
 Fig. 29. — Details of expansion joint in highway arch bridge over Chattahoochee River at Columbus, Georgia* 
 
696 
 
 CONCRETE ENGINEERS' HANDBOOK 
 
 [Sec. 16-38 
 
 Longitudinal Section 
 
 8o'-o" ■> 
 
 Half Section through Crown of llO-foot Arch 
 
 Fig. 31. — Details of Penn Street viaduct, Reading, Pa. 
 
Sec. 16-38J 
 
 ARCHES 
 
 697 
 
 Expansion Joint 
 
 Bottom of file to be flush 
 *y/fh slab at high points 
 
 y^sbesfos 
 
 ^,.4 "Ha If Tile fz'long 
 
 Board 
 S" Sheet Iron --" 
 Downspout extend- 
 ing thru arch rinj 
 
 6 Strip of/lsbestos Board 
 
 -3" Drain Tile 
 
 ?"°fods, 6"c ioc. £'3"ionj 
 along expansion joint 
 
 Section throu'gh Expansion Joint 
 at Center Drain 
 
 '■■•■s"''f?ods /3'lonj 
 
 s''o/?od lO' lonq 
 8 
 
 Mole for Trolley Pole ^ 
 
 l"''lfo.ds 4'-9" lon^ 
 
 4Lo''.'.-^ '-I'"- Pods b^nf up 
 
 from bottom of 
 Section B-B cantilever 
 
 4'.6". 
 
 Section C-C 
 
 rrtx 
 
 ■ ■ 
 
 Section A-A 
 
 ,3 ' Tile Drain 
 
 6",^" Strip of 
 Asbestos Board 
 
 -r--^T-+-^-^4 
 
 ,rr, 
 
 Section E-E 
 Section D-D 
 
 Fig. 32^.— Details of the Dallas-Oak Cliff viaduct, Dallas, Texas. 
 
698 
 
 CONCRETE ENGINEERS' HANDBOOK 
 
 [Sec. 16-38 
 
Sec. 16-38] 
 
 ARCHES 
 
 699 
 
 Fig. 34. — Wisconsin approach to high wagon bridge at Winona, Minn. 
 
700 
 
 CONCRETE ENGINEERS' HANDBOOK 
 
 [Sec. 16-38 
 
 Fig. 35. — Bridge over ravine on Mississippi River Boulevard, St. Paul, Minn. 
 
Sec. 16-38] 
 
 ARCHES 
 
 701 
 
 Longitudinal Sectior; 
 
 FiQ. 37.— Elk Run bridge across Cedar River, W. C. F. & N. Ry, 
 
702 
 
 CONCRETE ENGINEERS' HANDBOOK 
 
 [Sec. 16-39 
 
 cantilever section of the sidewalk is cast in units 5 ft. long and laid in place. Tile conduits are 
 provided under each sidewalk for necessary wires, and a 4-in. gas pipe and 12-in. water main 
 are laid in specially designed reinforced-concrete troughs beneath the roadway. 
 
 Transverse spandrel walls with openings to save material are shown in Figs. 32A and 32B. 
 The method of carrying the sidewalk should be noted. 
 
 The curtain walls between columns in Fig. 33 were considered simply as a bracing system. 
 The columns were designed to carry all loads, but doubtless the curtain walls help to distribute 
 the total loading. In taking the loading for the arch rings, it was assumed that the loadiag 
 from columns was equally distributed over 12 ft. of arch ring instead of having two loads con- 
 centrated on an arch ring 16 ft. wide. 
 
 39. Piers and Abutments. — The resistance offered by piers to the passage of water varies 
 with the type of starling. Experiments show that the value in this respect of the different 
 shapes of piers is in the following order: first, elliptical horizontal sections; second, rectangular 
 body with starlings formed by two circular arcs, tangent to the sides and described on the sides 
 of an equilateral triangle; third, rectangular body with semicircular starlings; fourth, rectangular 
 body with triangular starlings, the angle at the nose being 90 deg. ; and fifth, rectangular body 
 without starlings. 
 
 The ordinary type of abutment in earth-filled arches is shown in Figs. 36 and 37. What 
 might be called a buttressed abutment is shown in detail in Fig. 2SB. Fig. 38 represents a 
 very wide abutment in which a network of rods has been employed to make the entire abutment 
 act as a unit. Hollow piers for ribbed-arch structures are shown in Figs. 31 and 33. 
 
 Fig. 38. — Abutment of causeway arch construction, Galveston, Texas. 
 
 40. Railing and Ornamental Details. — A spindle balustrade is the common type of railing. 
 In such railings the spindles or balusters are usually the only members which are not cast in 
 place. Expansion joints should be provided each side of the posts and also over the spandrel 
 joints. Railing and ornamental details of various kinds are shown in Figs. 25B, 2SC, 30 and 
 S2B. 
 
 CONSTRUCTION OF ARCHES 
 
 41. Arch-ring Construction. — Arch rings with span lengths less than about 90 ft. are 
 usually constructed in longitudinal ribs 3 or 4 ft. wide, or in fact of such a width that one entire 
 rib can be poured in approximately 1 day's time. In narrow arches the entire arch ring is 
 sometimes poured at one operation. This method of construction has been successfully used 
 for much greater spans than 90 ft. but, unless special care is taken to make the centering very 
 stiff, the construction of any one rib may defo'rm the arch center to such an extent as prac- 
 tically to strike the center under the completed ribs. Of course, the ribs should be poured 
 continuously from each abutment toward the crown so as to obtain a symmetrical loading on 
 the falsework and thus eliminate distortion of the centering so far as possible. 
 
Sec. 16-41] 
 
 ARCHES 
 
 703 
 
 For spans of 90 ft. or over it is usually preferable to construct an arch ring or arch rib by 
 what is known as the alternate block or voussoir method. The arch is constructed in transverse 
 blocks of such size that each block can be completed at one pouring, or with about a day's 
 work. Obviously this method reduces shrinkage stresses in the arch ring to a minimum. 
 
 For the best results the blocks should be poured in such order as to give a uniform settle- 
 ment of the centering, and also prevent the crown of the arch from rising as the lower arch 
 loads are placed. If blocks close to the crown section are not placed before the blocks at the 
 haunch and springing sections, the centering will rise at the crown and the placing of the crown 
 loads will be likely to cause cracks at the middle of the haunch. Even in the construction of 
 an arch by the longitudinal-rib method, a temporary loading of the crown is often necessary. 
 
 The order followed in the construction of the Philadelphia and Reading R. R. bridge across 
 the Delaware River at Yardley, Pa. — an earth-filled bridge with clear span of 90 ft. 9 in. — is 
 shown in Fig. 39, the sections being concreted in the alphabetical order shown. The section 
 D is the keying section, and the section E a haunching section (quite unusual construction) 
 which was added after the lower portion of the arch ring was completed. The part of the arch 
 ring close to the springing lines was placed monolithic with the piers and abutments, making 
 
 Fig. 39. Fig. 40. 
 
 what is called an umbrella form for the piers. In large arches this umbrella type of construction 
 is frequently adopted. The pier forms in such cases are more expensive, but this increase in 
 expense for the piers is usually more than offset by the saving in the falsework for the arch 
 ring. 
 
 Fig. 40 shows the method of constructing the Larimer Avenue bridge at Pittsburgh — a 
 bridge of the open-spandrel type, with two ribs, having a clear span of approximately 300 ft. 
 The blocks were placed in alphabetical order and later the keys between them were concreted 
 to make the closure. 
 
 In constructing an arch rib or arch ring by the alternate block method the individual 
 sections or block spaces are closed off at the ends by timber bulkheads. On the steepest slopes 
 of the lagging these bulkheads adjoining keying sections are held in place by temporary struts 
 between voussoirs. ^ top form is usually needed for the block sections near the piers and abut- 
 ments. This top form should be laid up as the concreting progresses. 
 
 If arch reinforcement for large arches is put in place in long lengths, the settlement and 
 deformation of the centering during the pouring of the concrete will cause buckling of the steel 
 which will prevent the reinforcement from lying in its theoretical position. For this reason 
 steel lengths should not exceed about 30 ft. and the splicing should occur in the keyways. An 
 effort should be made to stagger the splices of adjacent rods and to locate the splices where the 
 tension in the steel is a minimum. 
 
 The upper reinforcement in arch rings may be kept in position either by means of spacing 
 boards nailed to props or by the steel being wired directly to transverse timbers supported above 
 the surface of the finished concrete. In the first method mentioned, the props and spacing 
 boards must be knocked out as the concrete is brought up with likelihood of disturbing the steel. 
 
 When steel ribs (either rolled sections or built-up lattice girders) are used as arch reinforce- 
 ment, great care should be taken to fix the ribs in the proper position, and in this position they 
 
704 
 
 CONCRETE ENGINEERS' HANDBOOK 
 
 [Sec. 16-42 
 
 should be braced until the concrete is placed. The use of such ribs is known as the Melan 
 system. 
 
 Spandrel walls for earth-filled arches are either built on top of the arch ring, or include a 
 portion of the arch, the bottom inner edge of the spandrel retaining wall lapping a short dis- 
 tance over the completed arch ring. 
 
 42. Centering. — The bent type of timber falsework is the type of centering generally 
 employed in arch construction except where a deep gorge is to be spanned or where a large 
 clearance under the arch is necessary while the bridge is under erection. Timber arches, Howe 
 trusses and bowstring trusses are sometimes employed when it is impossible to use the bent 
 type of centering, but these forms are expensive to build, deform badly under loading, and have 
 but small salvage value. Before using any of these types, consideration should be given to the 
 use of steel centers. 
 
 Fig. 41. — Common form of timber centering for arches of low rise. 
 
 42a, Timber Centers. — simple and common form of timber centering for arches 
 of low rise is shown in Fig. 41. The lagging had not been placed at the time this photograph 
 was taken, but some of the joists for supporting the lagging were already in place. The joists, 
 of course, extended from abutment to abutment and were supported by transverse bents of 
 round timber resting on sills the full length of each bent. The falsework rested on the con- 
 crete floor of a spillway channel so that mud sills, piles, or specially constructed concrete 
 footings were not necessary. Wedges were placed at the bottom of the posts so that the center 
 might be lowered conveniently after the arch ring was completed and ready to bear its load. 
 
 The centering used in the Third Avenue bridge at Cedar Rapids, Iowa, is shown in Fig. 
 42. In a paper presented before the Western Society of Engineers, April 13, 1914, Barton J. 
 Sweatt described the construction of this centering as follows : 
 
 The falsework for supporting the arches consisted of pile bents, the first bent being 6 ft. 6 in. from the face of 
 piers and abutments, the second bent 12 ft. 6 in. from the first, and the intermediate bents were 14 ft. 6 in. centers. 
 Oak piles were used and as a rule were driven to bed rock, the spacing was 6 ft. 0 in. for the three outside piles and 
 
Sec. 16-42a] 
 
 ARCHES 
 
 705 
 
 8 ft. 0 in. for the intermediate. The caps used were 12 by 12-in. yellow pine, false caps 6 by 10 in., joists 4 by 14 
 in., spaced 24 in. on centers and the lagging was 2 by 8 in. The proper curve for the intrados was obtained by 
 the use of 2-in. strips out to the proper curve and tacked to the regular joists. Oak wedges were used between the 
 main and false caps. These wedges were placed in pairs and spaced about 4 ft. apart. Small wedges were used 
 under the ends of the joists to bring them to the proper height. 
 
 In constructing the centering, an allowance of IVz in. was made for camber and in. for settlement after the 
 centering was removed. The actual settlement of the crown after removing the centering was % in. 
 
 Z' Circle Strips. 
 Z\l0"-2'-0'c.fO 
 
 11 
 
 4'jr/4 "Jo Is ts ■ ?4 "c. io c\ \ 
 (f^of double) 
 
 2" Filler. ^ ^"B'x l6'-o" Yellow P/ne Shsefin^.. \ |.? 
 6"xl0'L.^^ 
 SjDriffs 
 
 I " Bolts 
 
 Oair Piles .■rf 
 driven to 
 rock 
 
 Haff Side View Secrion showing Half erf Center Bent 
 
 Fig. 42. — Centering for Third Avenue bridge. Cedar Rapids, Iowa. 
 
 All Plumb Posts, Caps, and Stringers - 6"k 8" Yellow Pine Timber 
 All Batter Posts- 4"x 6" Y.P Timber 
 Arch Pibs - £"xiz'' Y.P Timber 
 
 Arch Flooring- £''x lo" Y P Timber ^ 
 
 Other Braces - Z''x e" YP Timber X 
 
 vi- 
 
 FiG. 43. — Falsework and centering for Cleveland Avenue bridge, Kansas City, Mo. Note structural steel girder 
 carrying the center of the span to allow for flood. 
 
 An article in Cement Age, March, 1912, describes the construction of the centering shown 
 in Fig. 43 in the following manner: 
 
 The centering for the arch consisted of four pile bents of four piles each, and two center pile bents of five piles 
 each. These bents were capped, top of caps being elevation of spring line of arch, and four lines of 6 by 8-in. 
 stringers were placed continuous from abutment to abutment — the ends, at elevation of the spring line, bearing 4 
 
 45 
 
706 
 
 CONCRETE ENGINEERS' HANDBOOK 
 
 [Sec. 16-42a 
 
 in. on the concrete abutments. At completion the ends of stringers were bored out and the holes in the abutments 
 tilled with concrete. On these four lines of stringers were placed a set of pine wedges over each pile, there being 
 four and five sets of wedges to each bent, and 3 by 12-in. timber was laid on the wedges over each bent through the 
 width of the arch. On these 3 by 12-iH. timbers, the arch bents were erected, each having four and five 6 by 8-in. 
 vertical posts V-braced, with 6 by 8-in. caps set edgeways, top of caps 12 in. below intrados of arch. On these 
 caps were placed 2 by 12-in. ribs, dapped to take square bearings on caps. These ribs were placed 18 in. centers 
 across the arch and were cut from timber of sufficient dimensions to make them lap over alternate caps. The ribs 
 were covered with 2-in. planking to form the intrados of the arch. 
 
 Fig. 44. — Details of arcli centers for proposed renewal, C. R. R. Co. of N. J. 
 
 A very simple and accurate method of laying out the ribs was used which consisted of laying out the full-size 
 arch intrados radii on a level place near the bridge site. The timber for the ribs was, therefore, marked by a full- 
 size drawing. Probably the most interesting feature of this arch centering was the simple straight work giving 
 maximum strength and maximum safety in every respect at the lowest cost. The five steel floor beams of the old 
 bridge were utilized to make an 18-ft. clear opening of maximum height in the centering. This was economical, 
 as it saved one pile bent and one centering bent, but its main purpose was to allow drift to pass through in case of 
 high water during construction. 
 
 Figs. 44, 45, 46 and 47 show timber centers similar to the one just described. 
 
Sec. 16-42a] 
 
 ARCHES 
 
 707 
 
 A patented type of centering is shown in Fig. 48, known as the Luten arch centering. The 
 idea in this center is to dispense with the usual wedges employed in lowering the falsework. 
 The top part of the uprights consists of two thin members with major dimensions transverse to 
 each other. These are arranged in the form of a T-column, and wired together at frequent 
 intervals. Each member separately is made too light to carry its loading so that clipping the 
 wires permits each member to buckle, which lowers the center. 
 
 e" Beveled Filler 
 Stringers not notched 
 
 ■ 8-6 ;'->k 8-6' 
 
 Mud Sills if required 
 
 Elevation 
 
 All lumber to be Yellow Pine unless 
 othenvise stated All bolts, with 
 tnv standard C. I was tiers Lagging to be 
 matched and dressed to a oniform thickness 
 so as to form a smooth and true surface. 
 /II/ lagging to be z"x 4" tongue and grooye 
 
 Elev. 219.00 
 All sills to be laid On 
 solid surface blocked 
 where necessary 
 
 Wedge 
 ( White Oak) 
 
 .Zx4 Lagging 
 
 iritnninnninnririnnnnnnrinif 
 
 Section A- A 
 
 Fig. 45. — Details of arch centers for Center Street bridge, Phillipsburg, N. J. 
 
 W. W. Washburn in an article in Concrete-cement Age, August, 1914, writes as follows in 
 regard to the centering shown in Fig. 49: 
 
 Since Buffalo Bayou is a navigable stream, it was necessary to leave an opening in the arch centering for 
 the passage of tugs and boats during construction. This opening was 24 ft. vertically above average water level 
 and 38 ft. wide. Protection piles on each side of the passageway were necessary, consequently the total span of 
 the opening in the arch centering was 49 ft. To carry the load over this opening nine 30-in., 200-lb. I-beams were 
 used. These I-beams will be used in the construction of other bridges. 
 
 Longitudinal bracing was arranged so as to counter, as much as possible, the "bucking up" tendency of the 
 centering at the crown as the arch was concreted. On account of the arch being skewed, special study was given 
 
708 
 
 CONCRETE ENGINEERS' HANDBOOK 
 
 [Sec. 16-42a 
 
Sec. 16-42a] 
 
 ARCHES 
 
 709 
 
 during the placing of braces, etc., so as to take care of all side thrusts. Struts were placed diagonally between 
 bents at right angles. Spacing of centering piles was such that no pile received a load of over 12 tons. 
 
 Timber centering for a bridge over railroad tracks at Fall River, Mass., is shown in Fig. 50 
 and described in Engineering Record, April 26, 1913, as follows: 
 
 In building the arch centering it was necessary to provide for certain requirements that necessitated a design 
 similar to the one shown in the accompanying drawing. 
 
 These requirements called for an overhead clearance of 18 ft. at a point 11 ft. 3 in. from the property lino, a 
 clear span of 40 ft. between inside vertical posts, and a minimum clearance of 15 ft. from the top of rail to the under 
 
 Fig. 47. — Falsework for bridge over Big Muddy River, I. C. R. R. 
 
 2" Facing 
 
 5>K n'-9" »•>}<•••• 13'- ii' 
 
 Leve l L ine -a ^1 
 
 Section parallel to Roadway 
 {96' Span) 
 
 Half Cross Section ^ 
 perpendicular to 
 Center Line of Road 
 
 Fig. 48. — Centers of Luten Design for Cicott Street bridge over Wabash River, Logansport, Ind. 
 
 side of the truss. The work had to be conducted without interruption of train service. The uprights supporting 
 the centering were carried on concrete mud sills and 2-in. tongue-and-groove lagging was used over the entire 
 arch ring. The centering was designed for a deflection equal to one eight-hundredth of the span. 
 
 Figs. 51A and 515 show the type of arch centering used in constructing a three-ribbed 
 arch on the Mississippi River Boulevard, St. Paul, Minn. Boxing for the ribs is also shown. 
 
 Posts in timber centering have sometimes been placed approximately normal to the arch 
 soffit, but such instances are quite rare. Since specially constructed footings are necessary 
 
710 
 
 CONCRETE ENGINEERS' HANDBOOK 
 
 [Sec. 16-42a 
 
 for inclined members, this form of center may be used economically only when rock or other 
 suitable foundation lies near the ground surface. 
 
 Sand boxes have been used to a very limited extent in this country in place of wedges for 
 the striking or lowering of arch centers. These boxes have given satisfaction in most instances, 
 but great care must be taken to keep the sand dry while the arch ring is being constructed. 
 This type of lowering device is expensive, but the extra first cost may be offset in large arches 
 by the high cost of striking wooden wedges. 
 
 Fig. 49. — Details of arch centering and supports for 110-ft. span of San Jacinto bridge, Houston, Texas. Note 
 
 size of opening for navigation. 
 
 z" Sheathing 
 
 
 
 < - 4o'-o"- 3. 
 
 
 
 
 
 Clear Span 82- 6 ' 
 
 
 
 
 
 
 
 
 
 ■■■ ■ •■• Concre-K Mud Sills ' _ 
 
 Fig. 50. — Highway bridge at South Park, Fall River, Mass., over tracks and right-of-way of the N. Y., N. H. & 
 
 H. R. R. 
 
 The chief disadvantage of using sand boxes lies in the fact that the sand will compress as the 
 weight on the centering increases. The amount of this compressibility is considerable, greatly 
 increasing deflection unless the sand is put under an initial compression, which is seldom feasible. 
 
 A sand box used in the main arch of the Edmondson Avenue bridge, Baltimore, is shown in 
 Fig. 52. The center was lowered by allowing the sand to run out through a 1-in. circular hole 
 in the oak bottom of the steel-plate cylinder. This hole was closed by a wooden plug while 
 
Sec. 16-42a] 
 
 ARCHES 
 
 711 
 
 Fig. 51B. — Rib boxing for bridge over ravine on Mississippi River Boulevard, St. Paul, Minn. 
 
 }6"~ Diameter Oak Cy I'm den 
 
 y±'^'-'-Felf Paper 
 
 Cylinder 
 
 Oafr Cylinder* 
 
 Fig. 52. — Sand box. 
 
712 
 
 CONCRETE ENGINEERS' HANDBOOK 
 
 [Sec. l"6-42fe 
 
 the centering supported its load. A disadvantage in the use of sand boxes lies in the fact that 
 the centering cannot be raised before the arch ring is poured in order to adjust the top members 
 to the curve of the arch intrados. 
 
 In the design of large arch centers an uncertainty exists regarding the pressures from vous- 
 soirs placed on the steepest portions of the lagging. Either of two assumptions are usually- 
 made as to the forces acting on the centering due to the weight of such voussoirs. In the 
 common method of design, the assumption is made that the centering sustains only the radial 
 components of the voussoir weights, the tangential components being resisted by temporary 
 struts between voussoirs so as to be transferred to the abutments. The more accurate method 
 is to assume that tangential pressures (in addition to the radial pressures) act on the centering 
 which, from any voussoir, may be as great as the product of the radial component and the coeffi- 
 cient of friction between the voussoirs and lagging. The original tangential component is 
 then reduced by this amount. 
 
 Since a timber center is only a temporary structure and has a high salvage value, great 
 accuracy in the design of the separate members is not necessary. The method of design need 
 only be such that the size of each member is well on the safe side. Then, too, rigidity is quite 
 as important as strength, so that all things considered, close figuring is out of the question. 
 Obviously the weight of centering may be omitted except for high arches. For the method of 
 designing lagging, joists, and posts see Arts. 65 and 66, Sect. 2. 
 
 As a rule, only hardwood should be used for caps and sills, although long-leaf pine may be 
 sufficiently hard in many cases. Wedges, however, should be made of hardwood without 
 exception. It is always advisable to reduce the number of joints in side-grain compression to 
 a minimum on account of the low bearing value of timber across the grain. Steel distributing 
 plates are of advantage in this connection. 
 
 Care should be taken to prevent lateral displacement of vertical posts due to radial pres- 
 sure from the arch ring. This may be avoided either by proper longitudinal bracing or by 
 notching out the joists and shimming them tight against the caps. 
 
 Many practical notes on the design and erection of falsework may be found in Sect. 7 
 of the "American Civil Engineers' Pocket Book." 
 
 In striking arch centers, wedges should be lowered gradually beginning at the crown and 
 working toward the springing lines. The lowering should be done symmetrically with respect 
 to the center of the arch ring. In a series of arches, centers between abutments or abutment 
 piers should be struck simultaneously. As a rule, centers should not be struck from arches in 
 less than 28 days under favorable weather conditions, and it is desirable that a longer period 
 should elapse if possible. 
 
 426. Steel Centers. — Steel centers of the arch-trussed type should receive con- 
 sideration when arches are to be built in series or where the character of the stream or crossing 
 renders timber and pile falsework impossible or expensive. Undoubtedly the cost of a steel 
 center is usually high, but if it can be used a number of times, as in a large series of arches, it 
 may not prove any more costly than timber. 
 
 It is generally recognized that there are some well-defined advantages in using three- 
 hinged arch centers. In the first place, the crown deflection using steel centers is usually much 
 less than that obtained by employing timber falsework. Furthermore, it is possible to compute 
 the deflection of each point of a steel center with some degree of accuracy while, in the case of 
 a wooden center, the probable settlement at each bent is pretty much a matter of guesswork. 
 Steel centers also have the additional advantages of allowing an obstructed opening for railroad 
 or other traffic and of eliminating danger from flood and ice in the construction of arches over 
 streams. The advantage of allowing the deflection to be quite accurately computed makes it 
 possible to give the centers a preliminary camber so that when the concrete is in place and the 
 centering withdrawn, the arch ring will assume its true position. 
 
 One disadvantage of using steel in arch centering lies in the fact that it is materially affected 
 by temperature changes. For this reason, in constructing large arches, only the alternate block 
 method should be employed. 
 
Sec. 16-426] 
 
 ARCHES 
 
 713 
 
 The steel centering used in constructing the three-span earth-filled arch structure which 
 carries Atherton Avenue in the City of Pittsburgh across the four tracks of the Pennsylvania 
 Railroad is shown in Fig. 53. This centering, fabricated by the Blaw Steel Cons. Co., consisted 
 of steel arch trusses spaced 5 ft. 5}i in. on centers. The trusses carried timbers and lagging, 
 and were supported on framed trestle bents placed close to the pier facfes. Sufficient trusses 
 were at first erected to concrete one-half the width of each arch ring, then the centers were shifted 
 transversely to themselves and the concrete placed for the second half of the structure. The 
 method used in construction and the details of the arch centering are described in Engineering 
 and Contracting, Feb. 19, 1913, as follows: 
 
 A six-post bent was erected on the footing shelf of the pier, the idea being to have a bent-post under each arch 
 rib. On the bent caps over each post was placed a block and double wedge and on these supports a 12 by 12-in. 
 plate on which rested the ribs. Between each pair of ribs a dolly was fastened to the bent-caps. 
 
 The shifting of the center to construct the second half of the arch was accomplished as follows: Jacks were 
 set up on the bent-caps alongside the dollies, and a strain taken on them until the wedges were loosened sufficiently 
 to be easily removed. The jacks were then lowered until the weight of the centers rested on the dollies. To pre- 
 vent the lagging and cross-timbers from being lifted off the ribs by sticking to the soffit of the arch ring, one end of 
 the center was lowered ahead of the other so as to give a stripping action in freeing the lagging. When lowered 
 into the dollies the whole center was shifted sidewise rubbing on the dollies, until it rested on the six-post bents under 
 the second half of the arch. The jacks were then placed on the caps of the second bents and the center raised and 
 the blocks and wedges inserted. A steamboat ratchet was used to pull the center on the dollies. Four men 
 working 8 hr. shifted a center. Incidentally the tie rods connecting the opposite ends of the ribs were found, when 
 planked across, to provide a most convenient bridge for the workmen engaged in shifting and adjusting the centers. 
 
 The lateral thrust on the centers due to their skewed position was taken care of by suitable lateral bracing 
 of the ribs. In anticipation of the center rising at the crown in concreting from the haunches upward, the ribs were 
 anchored back to the pier masonry. The joining carried by the ribs consisted of cross-timbers over which were 
 notched stringers with curved top edges. The stringers were spaced IV/i in. apart and were lagged with ^^-in. 
 boards. The bearings of the stringer ends against the piers were formed by wedges. 
 
 Steel centering employed in the construction of the South Eighth Street Viaduct, Allen- 
 town, Pa. (a two-ribbed arch structure of nine 120-ft. spans) is shown in Fig. 54. The Engi- 
 neering News, April 17, 1913, describes this centering as follows: 
 
714 
 
 CONCRETE ENGINEERS' HANDBOOK 
 
 [Sec. 16-426 
 
 For the nine 120-f t. arohes three full sets of steel arch centers were used, using each set for three of the arches. 
 Each set of centers consisted of two independent-trussed arches of the outHnes shown in Fig. 54, each arch support- 
 ing one of the twin concrete arch ribs and being itself made up of two steel arch ribs interbraced with steel struts. 
 Across the upper chords of these steel ribs, which were curved to the curve of the concrete arch, was bolted the 
 wooden lagging on which the concrete was deposited. The twin centering arches were held together by a timber 
 cross-beam and diagonal steel rods. 
 
 The arch trusses were fabricated in six sections and riveted on the ground into semi-arches, which were lifted 
 by derricks into place, to be bolted at the base to the supporting columns. At the crown it was riveted solidly in 
 the bottom chord, but bolted through slotted holes at the upper chord, to insure the stress passing through the 
 lower chord. 
 
 Fig. 55. — Arch rib forms on steel centers for Brooklyn-Brighton viaduct, Cleveland. 
 
 The centers were supported on inclined steel columns which footed on concrete steps purposely projected 
 from the main section of the pier and cut off after the centers were struck. The base plates of the columns rest on 
 cast-iron wedges which in turn rest on I-beam grillages, footing on the aforementioned concrete projection. Be- 
 tween the column base and the projection, 10-ton screw jacks are interposed to aid in the alignment and leveHng 
 of the centers; they are allowed to remain in place, though the load passes directly to the wedges which are used 
 for striking centers. A U-shaped clamp, made of a 1-in. bolt (not shown in the drawing) is passed around each 
 pair of wedges to prevent any possible lateral motion. A similar bolt is used for the same purpose higher up on the 
 main column. 
 
Sec. 16-43] 
 
 ARCHES 
 
 715 
 
 The unique feature in the steel centering used in constructing the Tunkhannock Creek 
 Viaduct on the relocation of the Delaware, Lackawanna, and Western Railroad was an adjust- 
 able panel at the crown of the steel arch trusses. 
 
 Arch rib forms on steel centers are shown in Fig. 55. These forms were used in constructing 
 the Brooklyn-Brighton viaduct, Cleveland. 
 
 THREE-HINGED ARCHES 
 
 43. General Discussion. — An arch with three hinges is statically determinate and conse- 
 quently can be analyzed much more readily for a given loading than is possible in the case of a 
 fixed-ended or solid arch. Furthermore, three-hinged arches do not need to be analyzed for 
 temperature changes, the hinges allowing contraction and expansion of the ribs without causing 
 any stress throughout the arch. Obviously this statement does not take into account the effect 
 which results from friction on the hinges, but such effect is usually considered to be negligible. 
 Whether or not hinge friction is likely to cause appreciable error in the analysis of three-hinged 
 arches is still a matter, however, in regard to which there seems to be a decided difference of 
 opinion. 
 
 Three-hinged arches are especially adapted to sites where abutments and piers must be 
 founded on compressible soil or on piles. The hinges permit of considerable settlement without 
 failure of the arch or without causing the huge cracks which are sure to develop in a fixed-ended 
 structure under like conditions. Of course, a solid arch may be designed on the assumption 
 that the abutments are yielding, but this is rarely done and 
 such computations in any event could not take into account 
 such settlement as might come from an unexpected source. 
 
 Hinges in arch-bridge construction are likely to be an 
 expensive detail, especially in short-span structures. The 
 claim is made, however, that in arches of large span, the 
 saving in concrete as compared with the fixed-ended type 
 much more than pays for the hinges. 
 
 44. Methods of Analysis. — Consider first the general 
 case of an unsymmetrical three-hinged arch subjected to a 
 number of vertical concentrated loads. By referring to 
 Fig. 56, it is seen that there are four unknown quantities — 
 namely, the horizontal and vertical components of each reaction — and four independent 
 equations are necessary to solve for these unknowns. We have the following three equations 
 from statics: 
 
 HV = algebraic sum of the vertical components = 0. 
 
 ZH = algebraic sum of the horizontal components = 0. 
 
 XM = algebraic sum of moments of all the forces about any point = 0. 
 
 The additional equation may be obtained from the fact that the bending moment is zero at the 
 crown hinge. Thus we have the following four equations with respect to the arch of Fig. 56. 
 
 Fa + - SP = 0 
 Ha - Hb =0 
 
 Taking moments about the left hinge 
 
 Heb - VbI + ^Pa = 0 
 Since the moment at the crown hinge is zero 
 
 VaIi - Ha! - So'iP {h - a) = 0 
 These four equations may be solved simultaneously to obtain the horizontal and vertical com- 
 ponents of the two reactions. 
 
716 
 
 CONCRETE ENGINEERS' HANDBOOK 
 
 [Sec. 16-44 
 
 The calculations may be simplified by resolving each reaction into a vertical force and a 
 force in the direction of the closing chord (Fig. 57). The four equations in this case are as 
 follows (since Ha = Hb from Fig. 56) : 
 
 Fi + 72 - 2P = 0 
 i^i - //2 = 0 
 - V2I + 2Pa = 0 
 Vih - So''P (Zi - a) -//ir - 0 
 
 or 
 
 Vih - 2o'^P(Zi - a) - Hac = 0 
 
 (The values of V have not been considered in the first equation as they are equal and opposite 
 in direction.) With the components of either reaction determined by these equations, the line 
 of thrust may be drawn throughout the arch as described in Art. 16 for the arch with fixed ends. 
 
 It should be noted that the values of Vi and V2 may be obtained from the above equations 
 (or by using SAf = 0 at both points A and B) in the following form: 
 
 Vi =1 SF(Z - a) (1) 
 
 I 
 
 V2 = \ ^Pa 
 
 (2) 
 
 These forces are thus identical with the reactions of a simple beam of the same span and similarly 
 loaded. 
 
 Fig. 57. 
 
 (3) 
 
 The bending moment at any point K (Fig. 57) may be expressed as follows* 
 
 M = VxX - llQ^'Pix - a) - Hav' 
 = Mk - HaV 
 
 where Mr is the bending moment at the point K of a similarly loaded beam. At the crown 
 hinge, letting Mc denote the moment of the vertical forces about the point C, we have 
 
 M = Mc - Hac = 0 
 
 or 
 
 Ms 
 
 c 
 
 (4) 
 
 Equations (1) to (4) inclusive are the formulas commonly employed in the analysis of three- 
 hinged arches — supplemented, of course, with the force and equilibrium polygons as in the case 
 of arches with fixed ends. 
 
 For symmetrical arches, Hi and H2 are horizontal and the line of thrust need be drawn for 
 onlj^ one-half the arch when the loading is symmetrical about the crown hinge. In such a 
 case of loading, the thrust at the crown hinge is horizontal and the line of thrust may be deter- 
 mined by trial in the manner described in Art. 11. This trial method gives exact results 
 when applied to a three-hinged symmetrical arch on account of there being two known points 
 (hinge points) on the line of thrust for each half of arch. 
 
Sec. 16-45] 
 
 ARCHES 
 
 717 
 
 The computations for uniform live loading are extremely simple and should be made 
 separately from those for dead load or concentrated live loads. For full loading, with the 
 crown hinge at mid-span, formula (4) gives 
 
 H.=l-^ (5) 
 
 where w is the uniform load per foot. The following equation, determined by substituting 
 in formula (3) gives the bending moment at any point (coordinates x and y) : 
 
 M = -^wx{l -x) - -~-y (6) 
 
 (For an arch of parabolic form, M = 0, and only axial stress occurs throughout the arch for 
 full uniform loading.) With only one-half of the span loaded 
 
 or one-half that due to full loading. The bending moment at any point in the loaded half 
 equals 
 
 1 1 7/;/2 
 
 M = ^ wx{Zl -4:x) -^'—-y (8) 
 
 and in the unloaded half 
 
 16 c 
 
 TIT 1 7 1 '^^^ 
 
 (In equations (8) and (9), the value of x is measured from that end of the arch which is nearer 
 to the point in question.) 
 
 A three-hinged arch is commonly analyzed for (1) dead and uniform live load over the 
 entire span, (2) for dead and uniform live load over the right half of span, and (3) for dead and 
 uniform live load over the left half of span. Full loading gives maximum stresses for the sections 
 near the hinges, while the half-span loadings give the greatest stresses near the quarter points 
 of the span. The usual method of design is to locate the hinges at the proper points and to 
 draw the force lines representing the load concentrations. These loads can be determined quite 
 accurately by making a complete design ^ 
 
 of the spandrels prior to the arch design , . — r=y:^Tr^jC^ , . - ^ ^ u ,' 
 
 and by approximatmg the weight of the /d?^'^'^'^ ^^'^^^^^ 
 
 arch ring — the arch ring, however, need '^^^^'^^^^^^ 
 
 not be drawn. The lines of thrust for ^- ^^ ^ ^""rhx 
 
 the three conditions of loading stated / Pb— ^ ...,.7t^^ ^ — M ^ 
 
 above are then drawn as shown in Fig. * 
 58. With the lines of thrust known, it 
 
 then becomes possible to determine the correct thickness of the arch at any point and decide 
 upon a suitable arch ring which, of course, should not differ appreciably in weight or position 
 from the arch ring previously assumed or else a second analysis should be made. 
 
 For a load of unity at the point L2 in Fig. 59, the direction of one of the reaction lines is 
 given by the line connecting the two hinges to one side of the load. The direction of the other 
 reaction is then known. It is thus an easy matter to construct influence lines similar to those of 
 Art. 34 and determine the exact maximum loadings. Method of constructing influence lines 
 is explained in Art. 48a, Sect. 7. 
 
 45. Common Type of Hinges. — The most common form of arch hinge consists of a struc- 
 tural or cast-steel pin bearing on two steel castings. Hinges of this type are shown in Figs. 
 60 to 63 inclusive. 
 
 46. Methods of Construction.— Three distinct methods of construction have been employed 
 in the erection of three-hinged arches: (1) casting the concrete ribs in forms on the ground and 
 then hoisting them into place; (2) erecting structural-steel reinforcement to be employed in 
 
718 
 
 CONCRETE ENGINEERS' HANDBOOK 
 
 [Sec. 16- 
 
 Thrust at section K 
 for unif had at *' 
 
 for un/t /oacf atL^.^^. 
 
 \-7 
 
 Shear at section K 
 for unit load at Lz- 
 
 Fig. 59. 
 
 K-— /?-j'''-->t-s-'- //-^'''■■->t<--- ii'-6''->r'--" 1 1 '-6 "■—>¥■—■• ii'-e'- 
 
 Z6-6 C-fOC-^. 
 
 Half Sec-Hon A-A 
 
 Anchor Bolts 
 
 
 
 n 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 Plan or 
 
 Side Elevation 
 Cost Steel Hinges 
 
 Fig. 60. — Details of Fourth Street bridge, Paducah, Ky. 
 
Sec. 16-47] 
 
 ARCHES 
 
 719 
 
 the arch ribs and using this reiiiforconunit to support the weight of the forms and phistic 
 concrete during construction; and (3) employing the usual type of centering and casting the 
 ribs in place. The first method is the one usually followed. Method No. 2 is of advantage 
 when a stream to be spanned is subject to sudden freshets and a minimum of falsework is 
 required. Method No. 3 is necessary only under unusual conditions. 
 
 The cheapest type of the three-hinged arch and also the type that is lightest and best 
 adapted to the use of hinges is one of detached ribs supporting spandrel columns. Such a type 
 of arch lends itself readily to the unit method of construction should this form of erection 
 be desired, and also eliminates the necessity for waterproofing which is a serious problem in the 
 case of a solid filled arch. 
 
 Abutmeni- Hinge Cast Steet Rib Hinge 
 
 Fig. G1. — Pratt Street bridge over Jones' Falls, Baltimore, Md. 
 
 47. Details of Design. — Figs. 60 to 63 inclusive give typical details of three-hinged arches. 
 
 The arch shown in Fig. 60 is founded on Ohio River mud, Raymond concrete piles being 
 used for the foundations. The reason for the use of the cast hinges in this case is thus apparent, 
 as settlement of foundations was anticipated. No appreciable settlem^ent, however, has ever 
 taken place. 
 
 The two halves of each rib of the bridge shown in Fig. 61 were designed to be erected simul- 
 taneously, without falsework, by derricks on opposite sides of the stream, and to be self-support- 
 ing as soon as the crown-hinge connection was made. Temporary sway bracing was provided 
 to insure lateral stability while the forms were being built and filled with concrete. 
 
720 
 
 CONCRETE ENGINEERS' HANDBOOK 
 
 [Sec. 16-47 
 
 Fig. 62. — Bridge over the Vermillion River at Wakeman, Ohio. 
 
 A 
 
 Hinge at Abutmem- Abutment 
 FirG. 63. — Details of Moffett Creek arch, Columbia Highway, Oregon. 
 
Sec. 16-47] 
 
 ARCHES 
 
 721 
 
 The three-hinged arch construction with cantilever ends, shown in Fig. 62, is unusual, 
 but was found to be more economical for an arch of this type and dimensions than a hingeless 
 structure. The cantilever ends were rendered necessary on account of the fact that there 
 were no stable foundations for abutments at the top of the fill at the ends of the bridge. The 
 cantilever ends decreased the dead-load thrust on the center hinges about one-quarter and 
 decreased considerably the angle which the resultant thrust on the lower hinges made with the 
 vertical, thus decreasing the size of abutment required. The cost of the hinges necessary for 
 the three-hinged arch was very considerably less than the cost of the additional steel reinforcing 
 in the arch ring required to take care of the additional bending moments in the hingeless arch. 
 No connection whatever was needed between the ends of the cantilevers and the abutments on 
 account of the extremely small amount of vertical motion at these points. A common type of 
 centering was used in constructing the arch ribs in place. The cast-steel hinges were entirely 
 encased in concrete after the centers were struck, a K-in. plate of sheet lead having been placed 
 at the center of each hinge to allow the necessary motion of the arch rib under live load and 
 temperature stresses. In this way all possibility of corrosion of the steel hinges was avoided. 
 
 46 
 
SECTION 17 
 HYDRAULIC STRUCTURES 
 
 DAMS 
 
 By a. G. Hillberg^ 
 
 A dam is built for the purpose of holding back a certain volume of water and usually to 
 raise the water level at a given point, so as to create a fall or head. Low dams designed 
 to permit the water to flow over them are termed weirs, while higher structures are called 
 spillways. Dams designed to hold back the water and not intended to assist in passing the 
 discharge are classified as bulkheads. 
 
 Dams, as a rule, can be divided into two main groups : Fixed and movable. The former 
 consist of masonry, earth, rock, steel, timber, or combinations of these materials and are de- 
 signed as immovable; while the second group contains movable structures, such as gates, stop 
 logs, flashboards, needles, rollers, etc., designed to be removed as required. 
 
 Often dams are combinations of these two groups as, for instance, at Keokuk, Iowa, where 
 11-ft. steel gates have been placed on top of a 32-ft. spillway dam of mass concrete. 
 
 1. Preliminary Studies. — Before the actual designing of a dam can be undertaken, a 
 great deal of knowledge of its purpose and the site must be at hand. As a rule, the geological 
 formation is the deciding factor as to what type to adopt, while very often the hydrographical 
 and topographical conditions decide the height of the structure. 
 
 la. Locating. — When investigating a river with the view of finding a suitable 
 dam site, the engineer, naturally, first looks for the narrow canyons. The reason for this is the 
 desire to get as short a structure as possible and also because the rock formation in such a narrow 
 place is usually firm and hard. However, the writer once investigated such a narrow box-can- 
 yon where the sides consisted of good, hard, blue limestone and thought the site to be excellent, 
 while borings afterward showed the canyon to be a volcanic split in the rock, probably 1500 
 ft. deep below the river bed. 
 
 When storage is required, a narrow box-canyon, while offering good dam sites, may not 
 give the desired reservoir capacity. It is, therefore, often necessary to investigate a number 
 of possible dam sites, as the one apparently best suited may not have any rock foundation nor 
 give the necessary storage capacity. A site, which at first does not seem suitable, might 
 after all be the most favorable because the formation is solid bedrock located not far below 
 the surface and the basin upstream such as to give the desired capacity with a structure of low 
 height. 
 
 lb. Geological Investigations. — These investigations must be very thorough 
 and the capital spent on careful explorations will be saved many times over during the con- 
 struction. The first matter of importance is to know whether or not bedrock can be reached. 
 If so, then it is important to know whether it is stratified, decomposed, or homogeneous; whether 
 hard or soft; or whether the formation is primeval, sedimentary, or volcanic. 
 
 Such data can be obtained only through core borings with diamond drills, and it is advisable 
 to test the holes by means of air or water under pressure. In one case the exact locations of 
 fissures were determined by having two gaskets on the pipe, so that air under pressure could 
 
 ' Hydraulic Engineer, New York City. 
 
 723 
 
724 
 
 CONCRETE ENGINEERS' HANDBOOK 
 
 [Sec. 17-lc 
 
 be applied to any predetermined portion of the hole.^ Should it be found that the forma- 
 tion is stratified or otherwise permeable, it is sometimes necessary to grout it. 
 
 In soft foundations, wash borings must be used and a record kept of the materials pene- 
 trated. Even here the main point is whether the formation is permeable or not. As a rule, 
 clay or hard-pan offer good foundations, while glacial deposits are always treacherous. 
 
 Hard-pan is not always to be depended upon since, if saturated with water under high 
 pressure, it will often disintegrate into yellow clay and gravel. ^ 
 
 As soon as the necessary data have been obtained, a subsurface map should be made. 
 From this map it can be decided what type of dam is most suitable and if any precautions have 
 to be taken in improving the foundations. 
 
 Ic. Selecting a Suitable Type of Dam. — As a rule, an engineer will first think 
 of a gravity-section dam, but if bedrock cannot be reached this type is out of the question. It 
 is also usually out of the question if no suitable materials for concrete are available at the site, 
 because hauling materials is expensive. For such conditions dams of reinforced concrete are 
 well worth investigating. However, the type of dam depends directly on the foundation 
 conditions, which can be divided into two groups: Rock foundations and soft foundations. 
 
 Rock Formation. — A type of dam always suitable in rock formations is the gravity-section. 
 However, if the site is narrow, an arched dam as a rule shows a considerable saving in material. 
 A still greater saving can sometimes be obtained by a reinforced-concrete dam. 
 
 If no concrete materials are available, an earthen dam might be the solution, but, in order 
 to make it tight, there must be both clay and sand or gravel available, as at least 20% of clay 
 is necessary to make it tight. 
 
 In many instances rock-fill dams have been used, but, in order to make them tight, a water- 
 tight membrane must be provided on the upstream side. This can be made of gravel, sand, 
 fine sand and clay graded so that they will form an impervious stratum. Reinforced con- 
 crete has also been used, but as a rule it is cheaper to support such a deck on concrete buttresses 
 rather than on a rock fill. To place a core wall of concrete near the center of a rock-fill dam 
 is not good practice, as the upstream portion of the rock fill is then submerged. 
 
 If good timber is available and the structure to be built is not too high, a rock-fill crib 
 dam will often be worth investigating. 
 
 Steel dams are structurally all right, but their maintenance is high. 
 
 Soft Foundations. — In soft foundations, gravity-section dams are out of the question, but 
 sometimes engineers have put low gravity-section structures on piling. The most suitable dams 
 for such conditions are earthen dams; reinforced-concrete structures placed on continuous 
 foundation mattresses; rock-fill timber cribs; or framed timber dams. 
 
 Id. Height of Structure. — Several conditions influence the height of a dam. 
 If it is to be used for storing water for irrigation, the reservoir capacity must be sufficient to 
 regulate the natural flow so that a certain area can be irrigated; if built in connection with 
 a hydro-electric power plant, the head created is just as important as the storage or the regu- 
 lation of the river discharge; if a low diversion dam, it must be high enough to divert water 
 into the pipe line, flume, or ditch at all stages ; if built for flood protection, it must have suffi- 
 cient capacity to retain all discharge in excess of the maximum carrying capacity of the river 
 channel below the dam, etc. 
 
 Very often the interference of the backwater with structures of immovable character, 
 such as operating mills, other power plants, buildings, railroads, etc., will limit the height of 
 a dam. 
 
 The top of the bulkhead section of a dam should always be higher than the maximum 
 water level, as wave action will occur on the surface of the reservoir. Such wave action is a 
 function of the exposed length of surface and, according to Stevenson 
 
 H = 1.5VT + (2.5 - VT) (ft.) 
 
 1 C. W. Smith: "Construction of Dams." 
 
 2 " Failure of Stony River Dam." Eng. Rec, Jan., 1914, p. 115. 
 
Sec. 17-le] HYDRAULIC STRUCTURES 725 
 
 where H = height of wave in feet; and I, length of exposed water surface in miles, measured 
 along a line perpendicular to the dam. 
 
 D. C. Henny gives the following formula: 
 
 H = 0.075 (F - 8.5) (ft.) 
 
 where V = velocity of wind in miles per hour. For dams designed by the U. S. Reclamation 
 Service, the following wave heights have been used : 
 
 Wind velocity, miles per hour. Wave height, feet. 
 
 35 2.00 
 
 40 2.50 
 
 44 2 . 50 
 
 48 3.50 
 
 50 3 . 50 
 
 56 4.00 
 
 75 5.00 
 
 The top of the dam should, of course, be somewhat higher than the wave height, especially 
 if it is an earthen dam, which might be severely eroded if water splashes over it. 
 
 le. Hydrographic Investigations. — For whatever purpose a dam is built an 
 intimate knowledge of the river and its drainage basin is required. The best possible data 
 to work on are actual observations on the river itself, such as are kept and published by the U. S. 
 Geological Survey and by the various State engineering offices. 
 
 If no such records are available at the point where the dam is to be built, records at other 
 points, preferably one above and one below the site, can be made use of and a rating curve 
 established. Should no records at all be available on the stream itself, the best method is to 
 compare it with adjacent streams on which records have been kept. 
 
 In many cases it is necessary to resort to runoff calculations, basing them on the rainfall. 
 Such computations are always very unreliable and should be made and used with the greatest 
 caution. The best available method is that developed by Vermeule.^ 
 
 E = F(15.50 + 0.16i^) (in.) 
 
 where E = yearly losses due to evaporation; F = (0.05T — 1.48), where T = mean yearly 
 temperature in degrees Fahrenheit; and R = yearly rainfall in inches. Vermeule found by 
 investigating the rivers in the East, especially those in New Jersey, that the monthly relation 
 between evaporation (including absorption by crops, etc.) and rainfall was as follows: 
 
 December e=0.42 + 0.10r June e = 2.50 + 0.25r 
 
 January e = 0.27 + O.lOr July e = 3.00 + 0.30r 
 
 February e = 0.30 + O.lOr August e = 2.62 + 0.25r 
 
 March e = 0.48 + O.lOr September e = 1.63 + 0.20r 
 
 April e = 0.87 + O.lOr October e = 0.88 + 0.12r 
 
 May e = 1.87 + 0.20r November e = 0.66 + O.lOr 
 
 Each of these monthly evaporations is to be multiplied by f = (0.05^ — 1.48), where t = 
 average monthly temperature in degrees Fahrenheit. 
 
 Where gage records are available they should be worked up as shown in Fig. 1, giving the 
 rating curve, the gage heights, and the duration per year of each gage height. The longer the 
 record, the better the average will be. 
 
 It is of great importance to determine, as correctly as possible, the maximum discharge 
 that can possibly occur. As the records given might not cover a sufficiently long period of time 
 to embrace such a maximum discharge, a cautious designer should increase the given maximum 
 
 New Jersey Geological Survey, 1894. 
 
726 
 
 CONCRETE ENGINEERS' HANDBOOK 
 
 [Sec. 17-1 / 
 
 from 20 up to 50%, and even then have sufficient freeboard to safeguard the bulkhead portion 
 from being overtopped. 
 
 1/. Capacity of Reservoir. — As soon as the dam has been located definitely and 
 the reservoir surveyed, a capacity diagram should be made as shown in Fig. 2. Such a diagram 
 
 is obtained by determining the areas enclosed by each contour, calculating the mean areas 
 between any two contours and multiplying by the distance iDetween them. A table should also 
 be made giving the areas and capacities. For convenience the unit measure is 1 acre-foot which 
 equals 43,560 cu. ft. Such a diagram is approximate only, as it is based on the assumption that 
 
 Capacities In acre-feet 
 
 Fig. 2. 
 
 the water level is horizontal; while, as a matter of fact, it is a modified parabola tangent to the 
 horizontal at the dam, and, at the upper end, to the slope of the water surface of the river feed- 
 ing into the reservoir. 
 
 The line thus formed is called the backwater curve and its exact shape cannot be calculated. 
 However, several methods have been devised whereby a fairly intelligent guess can be made. 
 
Sec. 17-1/] 
 
 HYDRAULIC STRUCTURES 
 
 727 
 
 If the dam were put in a channel with vertical sides or nearly so, the shape would be a para- 
 bola and the length I of its horizontal projection 
 
 where h is the height of the new water level above the natural slope S of the water surface at 
 the dam before it was built (Fig. 3). 
 
 However, as the shape of a reservoir basin is such that the respective cross-sections are 
 not uniform, the best method to apply is that of trial and error. Therefore, select first cross- 
 sections at suitable intervals, h, h, h, . . . In, and determine the area A, the wetted perimeter 
 
 Fig. 3. 
 
 p, the hydraulic radius R and the velocity v at different elevations for each one of them. Then 
 assume a value of n in Kutter's formula, basing it on the prevailing n at high and low water 
 in the river. This value is variable because the frictional resistance is greater for low stages 
 than for high, not only because of the increase in R but because of the decrease in velocity as 
 well. In many cases, especially if the dam is high, it is advisable to make a curve showing the 
 variation in n and extending it to cover stages as high as will obtain after the dam is built. 
 
 As the water level at the dam is known, the computation is started by guessing at the 
 slope Si in the first section Ix (Fig. 4). This will give a certain elevation at the first cross-section 
 so that A, R and v can be calculated {v = Q/A, Q being the discharge). Calculating the 
 
 Fig. 4. 
 
 average values of Am, Pm, Rm and Vm for the section and inserting them in the Chezy formula 
 
 C = ^ 
 
 ' Vrs 
 
 a value of Ci is determined. 
 
 By doing the same in the Kutter formula 
 
 , 0.00281 , 1.811 
 41.65 H ^ 1 — - 
 
728 
 
 CONCRETE ENGINEERS' HANDBOOK 
 
 [Sec. 17-2 
 
 a value of C2 is obtained. Of course, if *Si were correct, both values would have been the same. 
 By guessing at *Si and correcting Am, Vm, Rm, and Vm, values are finrally obtained that will make 
 
 Ci = C2 
 
 This is the correct value of Si, and it determines the height of the backwater curve at the 
 cross-section in question. 
 
 The next section is then taken and dealt with in the same way, and so on until finally a 
 point In is reached, where the slope coincides with the natural slope S of the water surface. 
 
 As all backwater computations are based on a given volume of discharge Q, it is obvious 
 that in order to find the limits of the influence of the water backed up, at least two sets of cal- 
 culations must be made, one for Qmaz and one for Qmin- 
 
 The inflow in a reservoir is 
 
 I = D + E + S 
 
 where / = inflow ; D, outflow ; E, evaporation and seepage ; and S, storage, which can be posi- 
 tive, negative or zero. 
 
 It is difficult to measure the inflow, as all of it might not be surface water, while the dis- 
 charge D can be measured directly and E calculated from evaporation records obtained from 
 pans and seepage records of similar reservoirs. 
 
 2. Design of Foundation. — When designing a dam, the substructure is just as important 
 as the superstructure, as a structure is no safer than its weakest part. If bedrock of good 
 quality is available, a gravity-section dam can, as a rule, be placed directly upon it, provided 
 the upper strata are removed. This is done not only to remove disintegrated portions, but 
 also to obtain a rugged surface with which concrete will give a good bond. 
 
 2a. Grouting. — Should the rock be disintegrated or fissured, it is often neces- 
 sary to grout it. This is done by drilling into it and pressing grout into the fissures through 
 the drill holes. The consistency of such grout must depend upon the size of the fissures, which 
 generally are so thin that neat cement and water must be used. Should any larger fissure 
 obtain, sand is mixed with the cement in various proportions and the grouting pressure is reduced 
 so as not to force the cement out of the mixture. 
 
 26. Cut-off Walls. — In fissured rock, cutoff walls of concrete are often used instead 
 of grouting, the underlying idea being to establish a long seam offering excessive resistance to 
 seepage. Of course, structures placed on such foundations must be calculated to resist a con- 
 siderable uplift. 
 
 2c. Caissons. — If the rock is so disintegrated that it consists of boulders loosely 
 cemented together between which a rapid percolation takes place, it might be found necessary to 
 use pneumatic caissons, as was done at Hale's Bar, near Chattanooga, Tenn.^. Several dams 
 in Europe have been placed on such foundations. 
 
 In soft foundations it has also been found expedient to use open caissons as foundation 
 for the core wall in earthen dams. The main difficulty in using caissons, whether pneumatic 
 or open, lies in the difficulty in making water-tight the joints between them. 
 
 2d. Pilings. — In soft foundation, piling is often used for improving the ground. 
 However, a portion of these piles must be placed in an inclined position parallel to the resultant 
 of all the forces acting on the structure above the plane of loading of the piles. Otherwise, 
 there will be no resistance to the horizontal component of the hydrostatic pressure and the dam 
 might move in a downstream direction. 
 
 2e. Sheet Piling. — In soft foundations, where it is impossible to reach bedrock 
 by any of the means given above, sheet pilings are often used. These can be of wood, 
 steel or reinforced concrete, but one thing applies to them all: they must be strong enough to 
 withstand the bending moment of the forces acting on their upstream side while transmitting 
 the load to the materials on the downstream side. As is obvious a considerable force must bci 
 
 1 "Foundations for the Hale's Bar Dam." Eng. Rec, Feb., 1913, p. 178, 
 
Sec. 17-3] 
 
 HYDRAULIC STRUCTURES 
 
 729 
 
 resisted on the downstream side at the top of such sheet piling, and it is, therefore, customary 
 to imbed the top firmly in the masonry. 
 
 One thing applies to all details appurtenant to the improving of foundations, whether 
 firm or soft : the improved portion must extend deeply enough so that the weight of the materials 
 including the weight of the structure itself as a surcharge on the downstream side more than 
 balances the forces on the upstream side of an arbitrary plane generally located at the upstream 
 side of the dam or through the center of the cutoff wall, if such is made use of. The forces on 
 the upstream side of such a plane are due to the full hydrostatic pressure only, if the foundation 
 is bedrock. If it is a soft foundation, the active pressure of the saturated soil must be added to 
 the water pressure, while on the downstream side the weight of the structure, acting as a sur- 
 charge, naturally increases the passive, or resisting, earth pressure. 
 
 3. Design of Dams of Gravity Section. — This type of dam resists the hydrostatic pressure 
 through its great weight, which is sufficient to form, with the horizontal component of the water 
 pressure, a resultant passing the plane between masonry and foundation at a point located within 
 the middle third. As masonry cannot be depended upon to resist more than a small amount 
 of tension, the design must be investigated at different elevations to insure against tensile stresses. 
 
 3a. Hydrostatic Pressure. — In the analysis it is customary to consider a strip 
 of the structure 1 ft. wide. As the 
 weight of 1 cu. ft. of water is w — assumed 
 at 62.5 lb., which is somewhat too high 
 — the intensity of pressure per square foot 
 at any depth is 
 
 p — wh (lb.) 
 
 where h = depth in feet below surface of 
 water. 
 
 The total pressure P on a vertical 
 plane extending from the water surface 
 to a depth H increases proportionally from zero at the top to wH at the bottom (Fig. 5) and 
 is expressed by 
 
 H wIP_ 
 2 
 
 wHX-^ = 
 
 (lb.) 
 
 Should the plane be inclined an angle a with the horizontal (Fig. 6), the maximum 
 
 intensity of the pressure remains the same, wH, while the length increases to H X 
 
 P = wH X 
 
 H 
 
 so that 
 
 (lb.) 
 
 2 sin o; 2 sin ot 
 
 If the plane is submerged, the pressure will be the difference of the total pressure P minus 
 the pressure -p corresponding to the upper part of the pressure triangle (Fig. 7), or 
 
 wH"^ wh^ _ w 
 1 
 
 Pi 
 
 For inclined surfaces multiply by 
 
 2 2^ 
 as shown above. 
 
 (lb.) 
 
 sm a 
 
 The resultant force of the hydrostatic pressure is always located through the center of 
 gravity of the pressure diagram and normal to the plane in question. When the pressure 
 
 diagram is a triangle, the resultant is located a distance above the lowest point (Figs. 5 and 
 
 while if it is a trapezoid (Fig. 7), the distance above the lower side is 
 
 H 
 
 ^ H - h wH + 2wh 
 y 3 ^ wH + wh 
 
 h ^ H + 2h 
 ^ H + h 
 
 (ft.) 
 
 36. Profiles of Dams. — The simplest form of a dam is a triangle with its upstream 
 side vertical (Fig. 8). 
 
730 . CONCRETE ENGINEERS' HANDBOOK 
 
 The overturning moment is 
 
 while the resisting moment is 
 
 PH ^ wH^ V ^ = ^ 
 3 2 3 6 
 
 2X _ mHX y 2X _ mHX^ 
 ^ ^ "2^3 3 
 
 [Sec. 17-36 
 
 (ft.-lb.) 
 (ft.-lb.) 
 
 where m = weight of 1 eu. ft. of masonry. 
 
 Dams are designed with a factor of safety against overturning of at least two : Therefore 
 
 Resisting Moment = 2{0veriurning Moment) 
 
 or 
 
 and the length of base will be 
 
 ~3~ ~ 
 
 X = H 
 
 (ft. 
 
 Fig. 10. 
 
 Fig. 11. 
 
 That for a factor of safety of two the resultant cuts the base at the outer edge of the 
 middle third can be proved as follows (Fig. 8) : 
 
 P'.W = 
 
 XH 
 3 '3' 
 
 but P=^,ndW = ^ 
 
 so that 
 
 (ft.) 
 
 The triangular shape of dam as an economical type was first pointed out by Edward 
 Wegmann.i He recommended that the water level for safety in the calculation be assumed as 
 extending to the top of the dam and that the superelevation (s) of the masonry above the 
 actual future water level and the top width (a) of the dam crest, each be made one-tenth of the 
 height of the dam, limiting the former to a maximum of 10 ft. and the latter to a minimum 
 of 5 ft. (Fig. 9). 
 
 He also recommended that the upstream side be kept vertical until the unit stresses for 
 "reservoir empty" approach the permissible maximum. Then the side should be sloped, 
 so as to keep the stresses at or somewhat below this stress. 
 
 As the limiting pressure will be reached sooner in the downstream than in the upstream 
 face, it is obvious that a similar slope must be started at a higher elevation on that side (Fig. 10). 
 
 Wm. P. Creager^ found that the economical top width of a dam is a function of the acting 
 forces (hydrostatic pressure, ice pressure, uplift, etc.) and that it should vary from 10 to 17% 
 of the height. However, as the height varies from nothing to a maximum and again to nothing 
 along the length of a dam, Creager's method would give a variable top width in addition to 
 irregularities in the face of the dam. An average section must, therefore, be adopted if this 
 method is used. 
 
 1 Weqmann: "The Design and Construction of Dams," 1st Ed., 1888. 
 
 2 Creager: "The Economical Top Width of Non-overflow Dams," Trans., Am. Soc. C. E., vol. 61, Nov., 1915. 
 
Sec. 17-3c] 
 
 HYDRAULIC STRUCTURES 
 
 731 
 
 In designing a dam it is advisable to start with a triangular profile and in that way obtain 
 a tentative section. The top width is then determined from practical considerations. As such 
 added weight at the top tends to place the resultant of the structure nearer the upstream face, 
 it is obvious that the profile can be somewhat reduced, and its downstream face given a slope 
 asymptotic to the face of the triangular profile (Fig 11). Thus a saving is effected as is 
 shown shaded in the figure. It is on this saving in material Creager has based his theory. 
 
 3c. Uplift. — Should a dam be founded on pervious material, uplift will result. 
 Its magnitude is a function of the permeability of the foundation and can vary from full hydro- 
 static pressure all along the base to nothing. Experiments at the Oester Dam in Germany 
 showed that in various sections the following pressures existed: 
 
 Section I — fall head at heel to one-half at toe. 
 
 Section II — full head at heel to one-fourth at toe. 
 
 Section III — three-fourths at heel to one-fourth at toe. 
 
 Section IV — full head at heel to zero at toe. 
 At the Neye Dam in Germany similar observations were made. This dam has its founda- 
 tion 26 ft. below the surface of the 
 
 rock, or about twice that at Oester. 
 Here 57% of the full pressure was 
 observed at the heel and 32% near 
 the toe. 
 
 In America it is customary to as- 
 sume two-thirds of the full hydro- 
 static pressure at the heel, diminish- 
 ing in a straight line to zero at the 
 toe (Fig. 12a). However, a more 
 correct way would be to assume full 
 pressure at the heel due to the water 
 
 on the upstream side and full pressure at the toe due to the water on the downstream side 
 (Fig. 126). 
 
 A still better way is to provide drainage immediately back of the cutoff wall and to use an 
 uplift corresponding to the elevation of the water surface below the dam under the entire base 
 excepting the portion in front of the drains on which full uplift is acting (Fig. 12c). 
 
 Depending upon which of the above assumptions is used, the pressures and moments 
 referred to the toe of the section will be as follows: 
 
 Force in pounds 
 1 
 
 l< 
 
 Ui + U2 = w{Ht + hv) 
 
 Fig. 12. 
 
 Fig. 12a 
 Fig. 126 
 Fig. 12c 
 
 U 
 
 U 
 
 wHb 
 
 XH + h)b 
 
 U = 
 
 M 
 
 M 
 
 M 
 
 Moment in foot-pounds 
 
 = 1(^+2)^^ 
 = ^[Ht{b-l 
 
 2 
 
 Some engineers apply the uplift theory to every horizontal joint in a dam and use then the 
 assumption shown in Fig. I2a for these joints. However, if care is taken and the upstream 
 
 side made as water-tight as possible and, in addition, vertical 
 drains, or weepers, are provided a few feet inside the face and con- 
 nected to a drainage gallery, such uplift can be eliminated entirely. 
 
 In order to get as impervious a surface as possible and at 
 the same time to provide as ample a drainage as possible, it has 
 been proposed to build in front of gravity section dams cellular 
 facings of reinforced concrete. This facing is anchored to the masonry, and at the bottom 
 the cells are piped through the base of the dam. The cells should be interconnected at certain 
 intervals from the bottom up (Fig. 13). 
 
 Fig. 13. 
 
732 
 
 CONCRETE ENGINEERS' HANDBOOK 
 
 [Sec. n-u 
 
 Zd. Wind Pressure. — No wind pressure needs to be considered, as when the 
 reservoir is full its action can only be on the downstream side, thus decreasing the unit stresses 
 in the masonry. When the reservoir is empty, such pressure on the upstream side will be con- 
 siderably less than the future hydrostatic pressure and, therefore, need not be considered. 
 If it acts on the downstream side, it will throw the line of pressure backward approximately 
 
 SmH + 6^ 
 
 where 6 = wind pressure in pounds per square foot; p, specific gravity of masonry (generally 
 2.5); m, unit weight of masonry per cubic foot; and H, height of structure above plane in con- 
 sideration. However, the influence of wind on the stability is so small as to make it negligible. 
 
 3e. Ice Thrust. — It is a much disputed point whether ice pressure should be 
 considered or not. Many engineers say that, as the dam is located in a narrow place, arching 
 stresses between the banks will relieve the dam of the ice pressure. However, several failures 
 are directly traceable to such pressure, but then, as in the case of Waldron, 111.,^ at Minneapolis, ^ 
 and at Thomaston, Conn., ^ the sheet of ice was cor.fined and the fluctuations in water level intro- 
 duced toggle joint effects in the ice. 
 
 Reputable engineers, acting conservatively, have recommended and used the following 
 ice pressures:^ 
 
 Pounds 
 per lin. ft. 
 
 St. Maurice, Que 50,000 
 
 Wachusett, Mass 47,000 
 
 Olive Bridge, N. Y 47,000 
 
 Kensico, N. Y 47,000 
 
 Croton Falls, N. Y 30,000 
 
 Cross River, N. Y 24,000 
 
 Keokuk, Iowa, on piers only 20,000 
 
 Such pressures are, as a rule, applied at the water surface, but as the sheet of ice is as- 
 sumed to be about 2 ft. thick, the actual point of application is about 1 ft. below the surface. 
 If, therefore, a joint is considered located at a depth h below the water surface and an ice pres- 
 sure / is assumed acting at a depth of 1 ft. below the surface, the moment will be 
 
 M = I{h - 1) (ft.-lb.) 
 
 The influence of the ice pressure on the profile of the dam decreases with the depth, necessi- 
 tating a material increase of the upper portions only, if the force is to be sustained by gravity 
 action. However, because of this decreasing influence, it is often advisable to reinforce the 
 upstream face, so as to enable the section to withstand the ice pressure by cantilever action. 
 
 3/. Initial Stress. — As the construction of a large dam covers several seasons, 
 the temperatures under which masonry is laid varies considerably. As yet no rules have been 
 laid down for taking into account in the design the effect of such stresses. 
 
 Zg. Temperature Stresses. — Seasonal changes in the temperature and, to a some- 
 what smaller extent, daily changes will cause the masonry to expand or contract. It is, therefore, 
 impossible to avoid cracking, especially in the upper and thinner portion. In order to limit 
 this cracking to a minimum and to locate such cracks where they cannot have any harmful 
 influence, it is customary to provide cracks or expansion joints at certain intervals. In order 
 to make them water-tight the masonry is often dovetailed together, or strips of copper flashing 
 used. 
 
 1 Eng. News, Apr. 23, 1908. 
 
 2 Eng. News, May 11, 1899. 
 
 3 Flinn: "Water Works Handbook," p. 119. 
 * Smith: "Construction of Dams," p. 117. 
 
Sec. 17-3/1 1 
 
 HYDRAULIC STRUCTURES 
 
 738 
 
 Such expansion joints should not be located closer than 20 ft. and not more than 50 ft. 
 apart. If an abrupt change in the foundation occurs, an expansion joint should be located 
 there as cracking cannot be prevented at such points. 
 
 Zh. Stresses in Masonry and on Foundation. — After the resultant of all the forces 
 acting on a section of a dam located above a certain plane has been determined as to magnitude 
 and location, it is necessary to calculate the unit stresses in the joint, so as to insure against 
 overloading. If in Fig. 14, the resultant R (consisting of the components V and H) cuts the 
 joint a distance I from its center, it is possible, without changing the conditions, to imagine two 
 forces equal to V and opposite in direction placed in the center of the joint. One of these 
 forces can then be combined with the component V to form a couple VI, and the other force V 
 gives an equal loading of the joint. The resultant unit pressures will then be a combination 
 of the corresponding pressures due to this central load and the couple, which for the central 
 load V is 
 
 V 
 
 Pi = — (Ib.per sq. in.) 
 
 where a is the area of the joint and equal to bd, or b, if the width cZ is 1 ft. 
 For the couple the maximum unit stresses are 
 
 (Ib.per sq. in.) 
 
 where S = section modulus for the area of the joint referred to an axis through its center, or 
 ^ if the joint is 1 ft. wide. Therefore, 
 
 P -P +P - ^ + Vl _V 6Vl _V 
 P - P, ± P, - - ± - ~± - ~ 
 
 V ytjivr pressurt 
 
 (Ib.per sq. in.) 
 
 Fig. 14. 
 
 Fig. 15. 
 
 Fig. 16. 
 
 Of course, as a negative stress would mean tension and, as such stresses are not permitted, 
 such outcome of the calculation proves that the length b of the joint is too short, or, in other 
 words, the resultant falls outside the middle third. 
 
 Prof. Rankine argued that the pressures near the faces are tangential to them and that, 
 therefore, in considering the pressures as above — viz.^ normal to a given horizontal joint — the 
 maximums are not obtained. M. Bouvier recommended that the pressure be calculated on 
 a plane normal to the resultant (Fig. 15). If the resultant R cuts the horizontal joint at a 
 distance I from its center and under an angle a. with the vertical, the inclined joint will also make 
 an angle <x with the horizontal. Consequently, the values in the above equations will be R 
 instead of F, b cos a instead of 6, and I cos a. instead of Z, so that 
 
 6 cos a \ b I 
 
 (lb. per sq. in.) 
 
 In Europe it is customary to figure the stresses on joints normal to both faces and bent 
 at an angle inside the dam (Fig. 16). The bend is located at a point where the unit stresses on 
 the projected horizontal joints are equal. 
 
734 
 
 CONCRETE ENGINEERS' HANDBOOK 
 
 [Sec. 17-3i 
 
 Several dam profiles (Rankine's, etc.) show a small amount of tension in the downstream 
 face when horizontal joints in the upper part are analyzed for the condition "reservoir empty" 
 and using the formulas given above. So does the triangular profile, because the added portion 
 at the top draws the pressure line upstream of the middle third. It is extremely doubtful that 
 tensional stresses actually develop in the joint, so long as the resultant remains within the 
 limits of the masonry. If the joints were open, tension would be out of the question and a 
 corresponding increase in the maximum compressive stress would be the result. In view of the 
 slight ability of concrete and especially cyclopean masonry to resist tensional stresses, it is ad- 
 visable to consider the joints as being open. If then the resultant R (Fig. 16A) cuts a joint 
 
 at a point located a distance c from the edge of the masonry, and c < the pressure dia- 
 
 gram would be a triangle with the base 3c. 
 of the joint is 1 ft. 
 
 If, furthermore, the width 
 
 P X3c 
 
 P = 
 
 27 
 3c 
 
 (lb. per sq. in.) 
 
 Fig. 16A. 
 
 where V is the vertical component of the resultant R. 
 
 The maximum permissible compressive unit stress should not exceed 
 300 lb. per sq. in., corresponding to 21.6 tons per sq. ft. If the stress is 
 more, it will be necessary to increase the length of the joint — that is, increase the area and 
 the section modulus, thus reducing the maximum stress. 
 
 3i. Shearing Stresses. — Each joint should also be investigated for shearing 
 stress. As the horizontal component is P, the unit shear s in the horizontal joint would be 
 
 p 
 
 s = - (lb. per sq. in.) 
 
 where a is the area in square inches. 
 
 If a joint such as d (Fig. 17) is considered, it is obvious that the shearing force is com- 
 posed of the hydrostatic pressure P cos j3 and a portion of the weight of the masonry = 
 W sin (3, so that the total shearing force is 
 
 S ^ P con (i + W sin ,3 
 
 Several joints must be investigated by the trial and error method 
 until that giving the maximum stress is found. 
 
 Two dams built by French engineers in Africa failed at points 
 located one- third down from the top because of excessive diagonal 
 shear. The permissible unit shearing stresses should not exceed 100 
 lb. per sq. in. and many engineers do not like to exceed 70 lb. 
 Often the joint between rock and masonry is investigated for fric- 
 tion, neglecting the adhesion, but this is not necessary, as (1) such 
 adhesion exists, and (2) the roughness of the rock surface gives a 
 good mechanical bond with the concrete. 
 
 Zj. Final Calculation. — After it has been decided what assumptions are to be 
 made as to the forces acting on the dam, a tentative triangular section calculated, and the top 
 width and the super elevation determined, it is necessary to investigate the design at a number 
 of different elevations, so as to be certain that the unit stresses do not exceed the permissible 
 adopted for the design, and that, on the other hand, the actual stresses are not so low as to make 
 the design uneconomical. The aim is to keep the pressure lines inside the middle third in such a 
 way that for the condition "reservoir empty" the line falls closely to the upstream limit of this 
 middle third and for "reservoir full" as closely to the downstream limit as possible. The ideal 
 dam would have these pressure lines coinciding with the middle third limits, but this cannot, for 
 practical reasons, be accomphshed at the top of the structure. Often the pressure line for 
 "reservoir empty" will fall slightly outside and upstream of the middle third limit, thus causing 
 
 Fig. 17. 
 
Sec. n-sj] 
 
 HYDRAULIC STRUCTURES 
 
 735 
 
 a slight tension in the downstream face. In one case a dam was on purpose designed so as to 
 lean upstream, or against the water pressure (Hauser Lake, Mont.)- 
 
 As a rule, both sides of a dam are kept parallel until a point is reached where the resultant 
 of the forces passes through the outer limit of the middle third (Fig. 18a) or 
 
 ' 6*3 
 
 If now, as recommended by Wegmann, the water surface is assumed as extending to the top 
 of the dam, P = and W = mad, which inserted in the 
 equation above gives 
 
 jvd^ 
 2mad 
 
 2d 
 
 or d 
 
 (ft.) 
 
 If an ice pressure / is to be taken into consideration, it 
 is impossible for the water surface to reach the top of the 
 dam. In this case moment equations are the more con- 
 venient to use, which, established in respect to the down- 
 stream edge (Fig. 186) with a factor of safety of two, would read 
 
 Fig. 18. 
 
 mad X 
 
 mda^ 
 
 In this equation it is best to consider the top width a as the variable, unless the dam is rein- 
 forced when the usual reinforced-concrete formula for cantilevers can be added to the right side 
 of the equation. 
 
 If uplift of %h diminishing to zero is assumed in the joint, the corresponding moment must 
 be added to the left side of the equation (see Art. 3c). 
 
 5425 
 5405 
 5385 
 5365 
 5345 
 5325 
 5305 
 5285 
 5265 
 
 ^5 
 
 ^dOOO 
 74,600 
 158^20 
 Z80,00d 
 
 634,690 
 868,000 ^950 
 
 1,138,630 
 1,446,700 
 
 RESEIRVOIR EMPTY 
 
 Cmv/fyline 
 
 5.00 5.00 
 
 7.53 
 
 11.47 
 16.00 
 
 433,650 20.4842.85 
 
 38.50 
 
 16.00 
 2520 
 34.00 
 
 25.005167 
 60.50 
 34.0069.33 
 
 78.17 
 
 Infensify 
 of pressure 
 
 140.000 H9.50 
 
 1192,200 
 
 3,990,000 m.70 
 9,520,000^ 
 18,785,200 
 
 32,784pO0m60 
 
 52 ,500,000 mo 
 
 78,934,300 fmo 
 113,094,800 
 
 mo 
 
 +19.50 
 -2.60 
 -3.70 
 ■8090 -3.10 
 
 mo 
 
 -2.90 
 -2.50 
 ■2.20 
 -2.00 
 ■m-1.90 
 
 RESEIRVOIR FULL 
 
 70^ 
 
 34,530 
 175,780 
 
 282,030 31.67 
 413/80 
 56933045.00 
 750.78051.67 
 957,030 58.33 
 
 If 
 
 35,160 
 446,000 
 1732.730 
 
 25.00 4M000 
 
 15,841,000 
 25,623,850 
 
 55,823,6002.03 
 
 Gmrihiine 
 
 2.0559.25 
 68Z0 
 17.10 
 
 Irrfensrfy 
 of pressure 
 
 375 
 
 14.27 
 18.30' 
 2250 
 
 50.0026.67 
 30.75 
 
 3957 
 
 +4.90 m.io 
 
 10.00 mo 
 
 neo 
 
 ^.20 
 
 mio+m 17.90 
 
 24.40 
 
 ■70.10 
 
 ■m 31.00 
 
 +5.2O^109.8L 37.40 
 e.40 (150.60 44.00 
 35.13 +3.10 m.70 50.50 
 
 +290 miO 57.00 
 
 11.40 
 
 Fig. 19. 
 
 In this way each joint is analyzed keeping in mind that all the forces acting on the dam at 
 or above the joint in question must be included. 
 
 A drawing should be prepared showing the profile of the dam, the limits of the middle thirds 
 and the pressure Hues of "reservoir empty" and "reservoir full." A table should also be pro- 
 vided giving in detail the various weights, moments, stresses, factors of safety, etc., as shown in 
 Fig. 19. 
 
 Every construction joint in the dam offers chances for defective work. Laitance, dirt, 
 etc., tend to establish open joints through which water might percolate. Such joints, of course, 
 
! 
 
 736 CONCRETE ENGINEERS' HANDBOOK [Sec. 17-4 
 
 do not offer the proper resistance to shearing, and joints should, therefore, never be made 
 horizontal. They should be stepped and located at different elevations in adjacent sections. 
 Rocks should be left projecting halfway up to enable the next course of masonry to get a 
 good hold. Before starting this next course the surface should always be scrubbed thoroughly 
 with steel brushes applying water and neat cement. 
 
 In Europe it is customary to place the courses in such a way that the top surfaces of them 
 are always normal to the faces of the dam and, in many cases, the courses are broken and 
 the outer part placed normal to the pressure line "reservoir full" (Fig. 20). 
 
 In many cases dams designed as gravity-section structures have been arched in plan 
 (Roosevelt, Arrowrock, Tallulah Falls). The reason for this is that in such dams temperature 
 stresses are counteracted by arch action in the structure. However, at Tallulah Falls when 
 the dam contracts it pulls away from the rock abutments, proving the necessity of providing 
 
 sections at the abutments where concrete should be placed in cold 
 
 weather so as to insure a tight joint under maximum contraction. This 
 course was followed by the U. S. Reclamation Service when building 
 the Arrowrock Dam. 
 
 4. Design of Arched Dams. — If the dam site is narrow and good 
 rock abutments are available, it is likely that an arched dam will prove 
 economical. Since, however, an arch is longer than a straight line, the 
 economy over a gravity-section dam depends upon whether the arch, 
 with its smaller cross-section, greater length, more expensive formwork, 
 greater accuracy in instrument work, sometimes richer mix of con- 
 crete and possible reinforcement requirements, will prove the less; 
 expensive. 
 
 Arched dams can be divided into three classes: 
 
 Buttressed AYches. — These are generally built of reinforced concrete and termed multiple- 
 arch dams (see under " Reinforced-concrete Dams", Art. 5/). 
 
 Constant-radius dams, which have a constant length of radius of either the upstream 
 side, the center, or the downstream side. 
 
 Constant-angle dams, which have a constant subtended angle, but a variable radius being 
 a function of the cord lengths at the different elevations. 
 
 4a. Constant -radius Dams. — Dams of this type are also called "true arches" 
 as they are built mainly on the principle that the pressure line of a uniform load acting on the 
 periphery of a circle has a circular line polygon. Consequently, as soon as the upstream 
 radius Ru has been determined upon, the stresses in the arch rings, assumed to be 1 ft. high, 
 can be calculated by the formula 
 
 S = PRu (lb.) 
 
 where P — water pressure in pounds per square foot measured to the center of the height o^ 
 the arch ring or wH, where w is the weight of water per cubic foot and H the depth in feel 
 of the center of the arch ring below the water surface. 
 
 As the pressure line coincides with the center of the circular arch, a correction in the 
 value of S in the above formula is required. 
 
 5, :S = Ru: ?^^±^ or = ^^^^ (lb.) 
 
 where Rd is the radius of the downstream side or Rd = Ru — T, where T is the thickness o 
 the arch ring in feet. 
 
 If the arch ring is h in. high and t in. wide, its area is ht sq. in. The permissible com 
 pressive unit stress is ^ lb. per sq. in., so that the total resistance is ^ 
 
 Q = qht (lb. 
 
Sec. 17-4a] 
 
 HYDRAULIC STRUCTURES 
 
 737 
 
 As now Q ^ Si, obviously Si ^ qht 
 
 or the required thickness t, if all other qualifications are given, 
 
 ^ qh 
 
 A correction must also be made for the vertical pressures due to the weight of the dam, 
 which will increase the stress to Si + AS. 
 
 For a dam with a vertical downstream face and a battered upstream side (Fig. 21a) 
 
 Ru \ 7) 
 
 AS = 
 
 1 + 
 
 3/i \ ' Ru + Rd/ 
 
 and for a dam with a battered downstream face and a vertical up- 
 stream side (Fig. 216) 
 
 AS 
 
 3m ( 
 
 1 + 
 
 R. 
 
 Ru + Rc 
 
 where p is the specific weight of masonry or 
 
 weight 
 "62^' 
 
 and 
 
 the re- 
 
 
 
 
 Rm 
 
 
 
 m 
 
 -f 
 
 (o) 
 
 Fig 
 
 For concrete - is from 0.16 to 0.22 or as 
 
 ciprocal of Poisson's ratio, 
 an average n = 5.^ 
 
 An arched dam is, as a rule, reinforced in both faces, in the upstream face so as to take 
 up tensile stresses due to cantilever action and in the downstream face so as to insure against 
 cracks due to tension caused by vertical beam action. The action of the dam, because of its re- 
 sisting moment when considered as a vertical beam 1 ft. wide, and because of the support 
 at the bottom, which is fixed, is to reduce the loading of the intermediate arch rings and to 
 increase it correspondingly in the upper rings. 
 
 By applying these formulas the thickness of the arch can be determined at as many ele- 
 vations as desired and the cross-section obtained. However, the formula is correct only for 
 an arch of circular form with fixed ends, while an arched dam is also fixed along its bottom. 
 
 Often the lower part is so short and the cross-section so thick that 
 horizontal beam action will result. It is, therefore, necessary to 
 investigate the tentative section for a variety of different condi- 
 tions: First, as a gravity section, until tension develops; second, 
 as a cantilever section, until a deformation occurs sufficient to 
 develop arching stresses; third, as a beam tending to equalize 
 such arching stresses, decreasing them at the bottom and in 
 creasing them at the top; and fourth as an arch.^ 
 
 Approximate methods for finding what proportion of the 
 TO' 80° 90' iQSi^ is taken by the arch and by the cantilever respectively have 
 been discussed by Harrison and Woodard.^ Shirreffs in his dis- 
 cussion of that paper develops the following formula for the 
 
 0 10° £0" 30° 40° 50° 60° 
 
 Values of ^ 
 Fig. 22. 
 
 crown deflection of an arch dam: 
 
 Dc = 
 
 C + PiRu"" 
 
 Et 
 
 where 
 
 and, 
 
 Pi =PX 
 
 2 sin 
 
 (1 — cos (p) + ^(cos 2(p — 1) 
 
 Zip 
 
 + (1 — cos <p) 
 
 -h cos (p — 4i 
 sin (p 
 
 1 Bellet: "Barrages en Marconnerie." 
 
 2 "For a full discussion of these stresses see Morrison and Brodie: " Masonry Dam Design," 
 
 3 Lake Cheesmau Dam and Reservoir," Trans. Am. Soc. C. E., vol. 53, p. 89. 
 47 
 
738 
 
 CONCRETE ENGINEERS' HANDBOOK 
 
 [Sec. 17-46 
 
 and is a factor which takes the curved beam action into consideration and can be found directly 
 from Fig. 22. E is the modulus of elasticity and t, the thickness of the arch ring. 
 
 This formula and curve, however, do not take the initial stresses in the dam structure into 
 consideration, and, therefore, before applying it for finding arch deflections it is necessary to 
 determine how much of this load is carried by the arch due to initial stresses, because only 
 after deducting this part of the load does the remainder divide up between arch and cantilever 
 action. 
 
 By initial stresses are meant stresses principally due to the weight of the structure and 
 to the water pressure. Therefore, -these stresses reach their maximum values at or near the 
 foundation, and are zero at the crest. 
 
 A number of arched dams of very slender dimensions have been built in Australia by the 
 English engineer Wade. The following table has been taken from Wegmann, "The Design 
 and Construction of Dams." 
 
 
 Max. 
 
 Top 
 
 Base 
 
 Radius of 
 
 Max. pressure, 
 
 Top above 
 
 Locality 
 
 height, 
 
 width, 
 
 width. 
 
 curve. 
 
 tons per square 
 
 water surface, 
 
 
 feet 
 
 feet 
 
 feet 
 
 feet 
 
 foot 
 
 feet 
 
 Parramatta 
 
 52 
 
 0 
 
 4 
 
 8 
 
 15 
 
 0 
 
 160 
 
 15 
 
 2.0 
 
 Lithgow No. 1 
 
 35 
 
 0 
 
 3 
 
 5 
 
 10 
 
 9 
 
 100 
 
 10 
 
 3.5 
 
 Parkes 
 
 33 
 
 5 
 
 3 
 
 0 
 
 13 
 
 5 
 
 300 
 
 24 
 
 5.0 
 
 Cootamundra 
 
 .46 
 
 0 
 
 3 
 
 0 
 
 13 
 
 0 
 
 250 
 
 25 
 
 1.0 
 
 Picton 
 
 28 
 
 0 
 
 7 
 
 0 
 
 13 
 
 6 
 
 120 
 
 12 
 
 10.0 
 
 Tam worth 
 
 61 
 
 0 
 
 3 
 
 0 
 
 21 
 
 5 
 
 250 
 
 20 
 
 2.0 
 
 Wellington 
 
 48 
 
 0 
 
 3 
 
 0 
 
 10 
 
 0 
 
 150 
 
 20 
 
 2.0 
 
 Mudgee 
 
 50 
 
 0 
 
 3 
 
 0 
 
 18 
 
 0 
 
 253 
 
 20 
 
 1.0 
 
 WoUongong 
 
 42 
 
 0 
 
 3 
 
 5 
 
 11 
 
 6 
 
 200 
 
 20 
 
 1.0 
 
 Katcomba 
 
 25 
 
 0 
 
 3 
 
 0 
 
 20 
 
 3 
 
 220 
 
 15 
 
 1.0 
 
 Lithgow No. 2 
 
 87 
 
 0 
 
 3 
 
 0 
 
 24 
 
 0 
 
 100 
 
 10 
 
 3.0 
 
 Medlow 
 
 65 
 
 0 
 
 3 
 
 5 
 
 9 
 
 0 
 
 60 
 
 12 
 
 3.0 
 
 Queen Charlotte Vale 
 
 32 
 
 0 
 
 3 
 
 0 
 
 8 
 
 6 
 
 90 
 
 10 
 
 2.0 
 
 Most of these dams are reinforced to take care of cantilever and temperature stresses. 
 
 The largest arched dams in the United States are located at Pathfinder and Shoshone, both 
 in Wyoming. They are of almost identical design, with a radius of 150 ft. figured to the center 
 of the top width. The upstream slope is 0.15 :1 and the downstream 0.25 :1 in both cases. 
 The former has a top width of 14 ft., a bottom width of 94 ft. and a total height of 210 ft.; 
 the latter a top width of 10 ft., a bottom width of 108 ft. and a total height of 328.4 ft. The 
 lower 85 ft. do not taper but are of a uniform thickness of 108 ft. Both dams are built by the 
 U. S. Reclamation Service. 
 
 46. Constant-angle Dams. — If a circular arch of given chord length is investi- 
 gated for economy, it will be found that the most economical shape has a subtended angle 
 2(p = 133.5 deg. It can also be shown that for variations between 120 and 150 deg. the increase 
 in area is negligible.^ As for any elevation the chord C is known, the corresponding mean 
 radius is (Fig. 23). i 
 
 Rm - ^ i 
 
 2 sin ^ I 
 
 The volume in a given section is equal to the area multiplied by the length of the mean arc, 
 
 V = area X Rm X 2(p 
 
 where ip is given in terms of tt. 
 
 1 Lars Jorgensen: "The Constant Angle Arch Dam," Trans. Am. Soc. C. E., vol. 78, p. 685, 1914. 
 
Sec. 17-5] 
 
 HYDRAULIC STRUCTURES 
 
 739 
 
 Poisson's ratio for lateral strains is taken into consideration in determining the relative 
 arch action in a dam of this type, a value of one-fifth being adopted, and the initial stresses 
 induced axially by the weight of the dam on the foundation, together with the water load, being 
 utilized to help support the latter. (See above under "Constant-radius Dams.") However, 
 dams designated in accordance with this method are liable to have an overhang, necessitating 
 a thickening of its lower part. This is, as a rule, accomplished by increasing the downstream 
 radius Ra for these lower arch rings, while keeping the 
 former radius Ru. The result is an arch thicker at the 
 crown than at the haunches. 
 
 When the tentative cross-section has been deter- 
 mined it must be analyzed in the same way as that 
 of a constant-radius dam. 
 
 Several dams of this type have been built in the 
 West and in Alaska. One constructed for the Alaska- 
 Gastineau Mining Co. near Juneau, Alaska, is 168 ft. 
 high. Another dam, which* ultimately will be 305 ft. 
 high, has been built for the Pacific Gas & Electric Co. 
 at Lake Spaulding, Cal. The maximum stress in this 
 dam is 23.8 tons per sq. ft. 
 
 5. Design of Reinforced-concrete Dams. — Rein- 
 forced-concrete dams possess many theoretical advan- 
 tages and they are by far the most satisfactory dam, so far as the type goes. The few 
 failures on record of such dams have never been due to fault of the dam proper; they have 
 always been traced to faulty foundation. Overconfidence in this type has led engineers to 
 neglect the facts that a secure foundation, sufficiently deep cut-off walls, adequate resistance 
 against sliding in soft ground, etc., are just as important details as the design and construc- 
 tion of the superstructure itself. 
 
 Because of the comparatively light weight of the structure, it is customary to support the 
 apron or water-tight membrane on a series of triangular buttresses in such a way that its inclina- 
 
 FiG. 2c 
 
 Fig. 24. 
 
 Fid. 25. 
 
 tion is about 45 deg. (Figs. 24 and 25). Obviously, the horizontal component of the water 
 pressure thus equals the vertical and the resultant has an inclination of approximately 45 deg. 
 From Fig. 26 it is evident that if the buttresses are of the shape indicated, the resultant of the 
 water pressure cuts the base in the outer edge of the middle third. The weight of the structure 
 draws the resultant somewhat inside this point. The corresponding unit pressures on the 
 foundation are thus greater at the downstream edge than at the upstream. 
 
 If soft foundation prevails, it is desirable that it be loaded as uniformly as possible. The 
 buttresses are then increased in length so that the resultant will cut it near the center of the 
 base. If the upstream slope of a bulkhead dam is 1 : 1 and the downstream batter of the 
 
740 
 
 CONCRETE ENGINEERS' HANDBOOK 
 
 [Sec. n-5a 
 
 IVater surface 
 
 buttresses 4 : 1, or, in other words, the length of the base is Y^E, where E is the height of the 
 dam, experience has shown that the resultant of water and concrete cuts the base almost exactly 
 in the center. n addition, the buttresses are placed on spread footings on a continuous founda- 
 tion mattress, so that the load will be distributed evenly over the whole base. 
 
 A dam of this type can be bulkhead (Fig. 24) or spillway section (Fig. 25). It can be 
 placed on any kind of foundation, and in 1909 a dam 22 ft. high was built on baled hay at 
 Anadarko, Okla. The top strata consisted of a pecuhar waxy clay or marl under which a 
 pocket of floating quicksand was found. To control the quicksand baled hay was laid down 
 on the surface of the sand pocket and on this was distributed a number of perforated pipes, 
 which were again covered with more hay. The pipes led to a sump outside the dam, from 
 
 which, during construction, the water was constantly 
 pumped. On the sand and hay foundation a contin- 
 uous foundation mattress was bull!; and the dam and 
 power house placed upon it. Although several severe 
 floods have passed over this structure, so far no de- 
 fects have developed. 
 
 In 1908 a dam was constructed near Douglas, 
 Wyo., 135 ft. high above the water hne, of which 80 
 ft. in the center are founded on clay and hard-pan. 
 The unit load under the mattress is 5.2 tons per sq. ft. 
 
 5a. Cut-off Walls. — When built on 
 soft foundations, cutoff walls of reinforced concrete 
 (sometimes extended by sheet pilings of steel, wood, 
 or reinforced concrete) are built at the upstream heel of the dam. The required depth of 
 such cutoffs can be calculated by means of the usual earth and hydrostatic pressure theories 
 after the necessary data have been obtained through test pits or borings. However, as a 
 rule, if such cutoffs extend below the ground level or bottom of the superstructure to a depth 
 one-half the height of the superstructure, they are safe. 
 
 As such cutoffs must be strong enough to resist the unequal pressures acting on them, they 
 must be reinforced and the superstructures designed for taking up a considerable reaction from 
 them. 
 
 56. Foundation Mattress. — In soft ground the buttresses are placed on a con- 
 tinuous foundation mattress, which must be reinforced in order to distribute the load from the 
 buttresses uniformly over the foundation. Weep holes are often 
 provided so as to prevent uplift from leakage, which possibly might jfr jfr 
 
 I I 
 
 fiTMltlllifl^ 
 
 Fig. 26. 
 
 Spread 
 ■ footing 
 
 Fig. 27. 
 
 find its way around the cutoff. 
 
 When the buttresses are placed on this mattress on widely , 
 spread footings, a certain amount of inverted arch action will take J y^nl^^ 
 place, thus reducing the reinforcement materially. The concrete ^ ^ 
 mixture in the mattresses is as a rule 1:3:5. The pressures on the 
 foundation are calculated by the formulas used for figuring the 
 stresses in a gravity-section dam (see Art. 3). 
 
 6c. Buttresses. — On top of the mattress or placed directly on the rock, if 
 such can be reached, are the buttresses. If built on a mattress or soft rock, such buttresses 
 are given a wide base by stepping them. Such widening should, if not reinforced, be made in 
 such a way that lines drawn at 45 deg. and intersecting between the buttresses at the underside 
 of the mattress will lie entirely within the steps (Fig. 27). If this is not done, the steps must 
 be reinforced. 
 
 Because of the reinforcement, the buttresses in some dams have been designed to take 
 tensile stresses in their upper parts (acting as cantilevers). In such cases the usual reinforced- 
 concrete beam formulas are used. Further down, where the buttresses are longer, no tensile 
 stresses are permitted and at the base the resultant must be well within the middle third. 
 
Sec. 17-5(1] 
 
 HYDRAULIC STRUCTURES 
 
 741 
 
 iforcemenf 
 
 ■-Reinforcement 
 
 6' to £4" 
 
 Vis- 
 
 Fig. 28. 
 
 The most dangerous stresses in a buttress are due to shear. As the apron or water-tight 
 membrane placed on the upstream side carries the load to the buttresses, it is obvious that 
 each buttress must withstand the load on one entire span. As the spacing, as a rule, is about 
 18 ft. these forces are considerable. So far no satisfactory method has been developed whereby 
 the exact lines of shear can be determined, and the designer, therefore, must be satisfied by 
 figuring the shearing stresses on horizontal joints, checking himself roughly to be certain that 
 the areas parallel to the water pressure are large enough. 
 
 As the shear increases with the depth, the buttresses become thicker. In many cases the 
 top portions have been made 12 in. thick, but 
 this is somewhat too small and 16 in. should 
 be adopted as the minimum. 
 
 In high dams the bottom thickness has 
 been as great as 72 in., but this, of course, is 
 ,a function of the length of the buttress. 
 
 In no case should the permissible unit 
 shear exfceed 100 lb. per sq. in. and even then, 
 some reinforcement should be used. In walls 
 not reinforced the stress should be limited to 
 70 lb. per sq. in. 
 
 Sometimes when the buttresses are very long, as for instance in high overflow dams, their 
 weight will draw the resultant downstream of the center. There. is no objection to this, pro- 
 vided the foundation is good and able to resist the unbalanced pressure. However, if the 
 foundation is soft and an equal loading is desired under the entire base, the portions of the 
 buttresses next to the apron are thickened sufficiently to place the resultant in the center. 
 
 In order to form a seat for the slabs of the apron, the buttresses at their upstream edge are 
 shaped as shown in Figs 28 and 29. The width of this seat varies from 6 to 24 in., depending 
 upon the thickness t of the slab. At the buttress the thickness of the 
 haunch must at least be t in order to develop the same resistance to shear- 
 ing as the slab itself. 
 
 The concrete mix in buttresses is generally 1:3:5, but sometimes a 
 richer mix is used. 
 
 hd. Bracing. — As buttresses are very high as compared 
 with their width, it is often necessary to brace one against the other. This is done by means 
 of rectangular beams as shown in Fig. 24. 
 
 5e. Apron. — The apron consists of slabs (Ambursen type) or arches (multiple- 
 arch type). Its sole function is to act as a water-tight membrane and to transfer the hydro- 
 static pressure to the buttresses. Slabs are designed as uniformly loaded and the usual formulas 
 apply. As a rule the moment is 
 
 M =-^^ (ft.-lb.) 
 
 where W is the load in pounds on a strip 1 ft. wide. As the unit load is wh lb. per sq. ft. and 
 the length of the loaded strip I ft., W = whl and 
 
 M = = 7.8125 hl^ 
 
 ■ o 
 
 M = 93.75 hl^ 
 
 (ft.-lb.) 
 (in.-lb.) 
 
 The reason for using this formula is that the buttresses are built first and the slabs 
 afterward and in such a way that no continuity of action can be depended upon (Fig. 28). 
 
 As these slabs are submerged, and as it is difficult to make them absolutely water-tight, 
 some engineers believe there is danger that in time the reinforcement will corrode. In order to 
 
742 
 
 CONCRETE ENGINEERS' HANDBOOK 
 
 [Sec. 17-5e 
 
Sec. 17-5/] 
 
 HYDRAULIC STRUCTURES 
 
 743 
 
 overcome this objection, slabs have sometimes been built as semi-arches, as shown in Fig. 29, 
 but the haunches must be designed for arch action and not like those shown in the figure. As 
 a rule, the concrete mix for these slabs or arches is 1 : 2 : 4. 
 
 After the proper slab thickness, the buttresses, etc., have been calculated, one entire panel 
 of the dam is analyzed for safety of design and 
 the corresponding unit stresses determined (Fig. 
 30). 
 
 Some practical details of design are shown in 
 Fig. 31. It is also customary to provide a ser- 
 vice bridge inside the dam, as shown in Fig. 30, so 
 that the underside of the slabs and their joints 
 with the buttresses can be inspected without 
 difficulty as often as desired. 
 
 The advantages of the hollow reinforced ^ 
 dam have led a number of designers to develop 
 various types. Many of them have never been 
 built like those developed by Ransome (Fig. 32) 
 and Morton (Fig. 33). The Ransome dam con- 
 sists of buttresses placed at a certain angle so that 
 they intersect. The slabs are thus of a variable 
 span. This is a very strong structure, but the cost 
 of forms excessive. In the Morton dam the apron 
 is supported on inclined columns braced by inter- 
 mediate beams. The disadvantages are that these 
 columns are liable to bend, that the load is con- 
 centrated on the column foundations, that the 
 form cost is high and that the placing of re- 
 inforcement and concrete difficult. Fig. 
 
 The Edge dam, used in a somewhat modi- 
 fied form in the reconstruction of the Austin Dam in Texas, offers several structural advan- 
 tages (Fig. 34). The apron is divided into square panels reinforced both ways, which gives 
 it great structural strength and the buttresses are thoroughly braced by intermediate walls 
 supporting the deck. However, it is not always possible to take advantage of the structur- 
 ally slender thickness of the apron, as a certain thickness against leakage is required. Further- 
 more, the form cost is excessive and the saving in concrete slight. 
 
 Plan 
 32. — Ransome dam. 
 
 Fig. 33.— Morton dam. Fig. 34.— Edge dam. 
 
 5/. Multiple -arch Dams. — To overcome what he considered objectional features 
 of the Ambursen dam — viz., the reinforced-concrete slabs and the many buttresses — Eastwood 
 developed a type consisting of a series of circular arches of long spans. The first dam built 
 
744 
 
 CONCRETE ENGINEERS' HANDBOOK 
 
 [Sec. 17-5/ 
 
Sec. 17-6] 
 
 HYDRAULIC STRUCTURES 
 
 745 
 
 of this type was for the Hume-Bennett Lumber Co. in CaUfornia in 1908. It consists of 12 
 circular arches of 50-ft. span supported on 13 buttresses.^ 
 
 A later dam of this type was condemned by the California Railroad Commission because 
 of insufficient bracing of the buttresses. The advantage of the slab type is, that should one 
 series of slabs collapse, the adjacent buttresses will prevent further damage to the structure, 
 while if a series of arches collapses the whole structure will fall because of the absence of a 
 counterforce at the buttress. 
 
 In some recent dams designed by Jorgensen^ the buttresses have been braced so that they 
 can, by cantilever action, sustain the arch reaction should the arches in the adjacent panel be 
 missing (Fig. 35). Quoting Jorgensen, the most economical spacing of the buttresses lies 
 between 30 and 50 ft. For low dams the lower limit would be the best and for high dams the 
 upper. 
 
 It should be noted that for arches with a large subtended angle and placed in an inclined 
 position, the crown lies at a considerably higher elevation than the haunches and that the load 
 varies due to the variation in hydrostatic pressure. 
 
 For designing the thickness of the arches, the methods given under ''Arched Dams" (Art. 4) 
 can be used. However, if the subtended angle is large and the inclination considerable, it is more 
 convenient to use graphic statics, drawing one pressure line for the weight and another for the 
 hydrostatic pressure and then combining them (Fig. 36). 
 
 6. Earthen Dams With Concrete Core Wall. — The design of earthen dams is purely a 
 matter of experience. The ruling factor is preventing water from traversing the dam. This 
 is effected by providing an impermeable core of clay, puddle, concrete, masonry, steel plates 
 protected with concrete or asphalt coatings, and, especially in smaller dams, sheet pilings of 
 lumber. It is absolutely necessary if an earthen dam is decided upon, that the necessary 
 materials are available at or near the site, as the quantities are so voluminous that it is impos- 
 sible to haul them any great distance. 
 
 1 Wegmann: "The Design and Construction of Dams," 1911 Ed., p. 436. 
 
 2 Labs Jorgensen: " Multiple-aroh Dams on Rush Creek, CaUfornia." Trans. Am. Sbe. C. E., March, 1917. 
 
746 
 
 CONCRETE ENGINEERS' HANDBOOK 
 
 [Sec. 17-6 
 
 An earthen dam may consist of : 
 
 1. A homogeneous bank of earth. 
 
 2. A bank of earth having a puddle core. 
 
 3. A bank of earth having a masonry core wail. 
 
 4. A bank of earth having a puddle placed on the water slope. 
 
 The first method can be- used only when the required quantity of earth or gravel, containing 
 enough clay to make it water-tight, can be obtained at a reasonable cost. 
 
 In cases where this would prove too expensive, a central core is provided of clayey earth 
 or gravel, ordinary earth being used in the other portions of the dam. In order to prevent leak- 
 age under the dam this core should be extended down in a trench to an impervious stratum, 
 or at least far enough down to make the path of such leakage as long as possible. 
 
 In the third plan, masonry is substituted for the puddle core. This plan is adopted when 
 no claye}'' earth can be obtained at a reasonable cost. As a rule it is more expensive than plans 
 1 and 2, but has great advantages as regards safety. 
 
 As regards plan 4, it is open to two objections: The puddle is injured if the slope settles, 
 which nearly always happens, and cracks will occur in the water line due to alternate wetting 
 and drying. 
 
 In some cases concrete paving, plain or reinforced, has been used on the upstream side 
 of an earthern dam for making the dam water-tight However, such paving is apt to slide 
 and should be heavily anchored down. Even then, settlements in the slope will cause cracking, 
 and leakage is bound to take place. 
 
 The theory of earth-dam design is very simple: The upstream portion and the core wall 
 serve to reduce the leakage to such an extent that the hydraulic gradient falls inside the dam 
 body, and the downstream portion, through its weight, will hold the saturated mass in position 
 and prevent it from sliding in a downstream direction. It has often been noted that earthen 
 dams have a tendency to fail through the caving of the downstream slope, which proves that the 
 hydraulic gradient did not fall inside the dam but intersected the slope some distance above 
 the foundation. 
 
 In order to broaden the base and to bring the downstream slope out as far as possible on 
 high dams, it is customary to step the slope (so-called berms) at elevations 20 to 30 ft. apart, 
 which obviously means a saving in materials. Such berms are usually paved and provided 
 with gutters on the inside. 
 
 In order to confine the hydraulic gradient to the embankment, the Pierson Engineering 
 Corporation, when building a series of earthen dams in Spain, ^ placed drainage dykes of broken 
 rock near the downstream side of each dam at the toe of a somewhat pervious section, and pro- 
 vided outlets for any leakage collected by these dykes. 
 
 When sheet pilings are used below a core wall and the foundation is very soft, it is often 
 necessary to increase the downstream portion of the dam, in order to create a surcharge suffi- 
 cient to counterbalance any unbalanced forces on the sheet piling due to hydrostatic pressure. 
 
 Experience has shown what general dimensions are most suitable for earthen dams. They 
 are, of course, to a certain extent a function of the materials used and the height of the struc- 
 ture, additional strength being given to high dams:^ 
 
 The upper portion of the upstream slope should be protected against erosion by waves 
 and burrowing animals by a riprap, 15 to 20 in. thick, or concrete paving, 5 to 8 in. thick, ex, 
 
 1 Eng. Rec, Aug. 29, 1914, p. 250. 
 
 2 Wegmann: "The Design and Construction of Dams." 
 
 Top width 
 
 Superelevation above high water 
 
 Upstream slope 
 
 Downstream slope 
 
 .10 to 30 ft. 
 . . 5 to 25 ft. 
 ..l:2:tol:3 
 l:lHto 1:2K 
 
Sec. 17-6] 
 
 HYDRAULIC STRUCTURES 
 
 747 
 
 tending some distance below and 5 to 15 ft. above the water line. The top of the dam, the 
 downstream slope and the upstream slope above the paving are generally covered with good 
 soil and sodded. 
 
 A fairly satisfactory formula for the top thickness is given by Lyndon:^ 
 
 r = 5+0.2i/ (ft.) 
 
 where H = height of dam in feet. 
 
 As the earth fill is bound to settle it is customary to increase the profile from one-fifteenth 
 to one-twentieth of its height (Fig. 36A). This shrinkage is, of course, a function of the quahty 
 of the materials and the degree to which they have been rammed. As the height varies from 
 zero to a maximum, this method will make the crest of the dam arched in the longitudinal 
 section. 
 
 ; , Surcharge^ H to^H 
 
 Fig. 36^. 
 
 Core walls are never designed to resist the whole water pressure as they serve merely as a 
 water-tight membrane and are backed by the downstream portion of the dam. They should 
 extend from 1 to 2 ft. above the highest water level in the reservoir and have a top width of 
 from 2.5 to 6 ft. The sides are battered from 16: 1 to 20: 1, so that the thickness at the natural 
 ground surface is from one-sixth to one-seventh of the height. Instead of battering the faces, 
 the increase in thickness can be obtained by offsets. 
 
 Parker^ gives a method, whereby the stability of core walls can be determined: Draw 
 in Fig. 37 on the water side a line ah making with the horizontal an angle </> equal to the angle of 
 
 Fig. 37. 
 
 repose of the saturated earth (usually = 20 to 23 deg.). Then draw on the downstream side 
 a line cd, making with the horizontal an angle O equal to the angle of repose of the rammed 
 earth (generally 9 = 45 to 55 deg.). 
 
 Calculate the areas located above these lines. For 0 = 20 deg. and a 1:3 slope of the 
 upstream side, the area ahc becomes very nearly Y/^h}) and with 6 = 45 deg. and a 1:2 slope 
 of the downstream side the area cde is J-^/i^. The weight of rammed earth can be taken at 
 132 lb. per cu. ft. and that of saturated at 160. The thrust on the core wall would thus be 
 
 y^h} (160) - (132) = 76/i2 (lb. per lin. ft.) 
 
 If the thickness of the core wall is x ft., its ultimate resistance to shear, when of concrete, 
 is about 30,000 x lb. per lin. ft. If a factor of safety / (usually / = 2) is used, 
 
 30,000 X = 7QhH or x = about ^ • (ft.) 
 
 1 Lamar Lyndon: "Hydro-electric Power," vol. I, p. 279. 
 'Philip A. Morley Parker: "The Control of Water," p. 318. 
 
748 
 
 CONCRETE ENGINEERS' HANDBOOK 
 
 [Sec. 17-7 
 
 Unless h is great this formula will give results somewhat smaller than experience has shown 
 as requisite to stop percolation through an ordinary mix of concrete. Especially for reinforced 
 concrete, it is necessary to use a very dense mix or to coat the upstream surface with an imper- 
 vious material. 
 
 HerscheP gives the following practical rules for first-class work: 4 to 5 ft. thick at bottom 
 of trench enlarged to 8 ft. at natural surface and tapering to 4 ft. at top of core wall. However, 
 for smaller dams these dimensions are entirely too clumsy. Herschel states also that a wall 
 2 ft. thick throughout is sufficient to stop percolation. 
 
 When building earthen dams the materials should be deposited in layers from 8 to 12 in. 
 thick and each course should be well packed. This is best accomplished by steam rollers, of 
 which the first one has grooved rollers and the other smooth ones. Especially in dams without 
 core walls, every layer should be somewhat inclined toward the water side of the dam and 
 arched upward, so that, later on, when the materials settle, no pockets are formed. Such 
 pockets, if impervious, will retain water and the hydrostatic pressure will be carried out to the 
 downstream slope, which then will cave. It has happened that dams, saturated in this way, 
 after having been in successful operation for several years have failed in an upstream direction, 
 because of the outer hydrostatic pressure being removed through a lowering of the water surface. 
 
 Hollow, cellular core walls of reinforced concrete extending to the top of the dam have 
 been proposed because they offer a convenient way of providing a continuous overflow along 
 the crest of the dam, thus preventing overtopping. 
 
 7. Passing the Discharge.— Openings must be provided in, at, or near a dam at such an 
 elevation, that when the water in the reservoir rises above a certain level, it will escape through 
 one or more of these openings, thus preventing the bulkhead portion from being overtopped. 
 Such relief must be designed to discharge the maximum flood observed or anticipated at the dam 
 site. If the reservoir has a large surface, the rise in water level to an elevation required to 
 give the necessary head on the spillway is sometimes considerable, and, if in addition the 
 floods are of short duration, a reduction can be made in the spillway requirements. 
 
 7a. Form of Spillway. — There are several ways of passing floods at dams. 
 Pipes laid at or near the base of the structure, thus acting under a great hydraulic head, are 
 sometimes used, but, as a rule, in conjunction with other types of spillways. Overflow spillways 
 are of many designs. Sometimes they are located apart from the main dam and the water 
 discharged into a gulley, canal, or channel conveying it back into the river a safe distance below 
 the dam; in other cases they are located in the dam itself, part of which has been lowered for the 
 purpose. 
 
 When spillways are located in places where attendance is possible, such as for power 
 plants with the station at the dam, they are often outfitted with some type of movable dam, 
 generally gates, as it provides means for keeping the water level at a higher elevation, thus 
 increasing not only the storage but also the head. Another advantage of this system is that 
 the spillway can be shortened materially as the discharge openings can be made very deep, with- 
 out necessitating a sacrifice in the storage capacity. Consequently, such dams, as built at 
 McCalls Ferry and for the Ozark Power Company on the White River in Missouri, are not 
 suitable. They are designed to resist and to pass the maximum floods, 16 and 12,5 ft., respec- 
 tively, and, because of the lack of movable dams or gates, the water level, after the flood has 
 passed, soon falls to the crest of the dam. Flashboards about 5 ft. high are now used at both 
 places, but such provisions are more or less to be considered as makeshifts. 
 
 When spillways are located in places where they cannot be under constant surveillance, 
 they must be given ample length, so that when the water level rises a comparatively large 
 discharge can be accommodated, thus preventing a rapid rise which might endanger the main 
 structure. In order to obtain a great length of spillway in as short a distance as possible, they 
 are sometimes arched or zig-zagged in plan. However, because of the interference at the cor- 
 ners of such broken lines the effective length is shorter than the actual total length. Sometimes, 
 
 1 Proc. I. C. E., vol. 132, p. 255. 
 
Sec. 17-76] 
 
 HYDRAULIC STRUCTURES 
 
 749 
 
 for high heads the water flows over the whole structure in such a way that the effective length 
 is the straight line between the end points. 
 
 76. Discharge Capacity. — The discharging capacity of a spillway depends upon 
 its shape. The general formula used reads: 
 
 Q = Vs tMlhV2gh or Q = H l^lV^g (hf (sec.-ft.) 
 
 where Q = discharge in second-feet; I, length of the spillway; h, head acting on it; g = 32.16; 
 and fi an empirical coefficient depending upon the shape of the spillway, and giver, in Fig. 38. 
 If the velocity of approach Vo is considerable, its influence on the discharge must be taken 
 
 Vo' 
 
 into account. This is done by transforming it into the corresponding velocity head k = 
 
 and introducing it in the equation, which thus becomes: 
 
 Q = y3f^lV2g[{h +kf^ 
 
 2sf 
 (sec.-ft.) 
 
 fj.'0.75-085 
 
 u,=a80-685 
 1^=060-0.65 
 
 0.75-0.65 
 
 Fig. 38. 
 
 Should the crest of the dam be very wide d> h (Fig. 39) with sharp corners, the discharge is: 
 
 Q = 0.351 V2g (h + kf' or fx = 0.525 
 and 
 
 e =mh + k) 
 
 and for rounded corners 
 
 Q = 0A0lV2g(h + k)^^ or ^ = 0.60 
 
 Francis found that for vertical, sharp-crested, rectangular weirs with complete contractions 
 and free overfall 
 
 Q = 3.33 
 
 where n = number of lateral contractions, 0, 1, or 2. His constant 3.33 corresponds to /x = 
 0.62 in the above given general formula. As a rule the end contractions can be neglected 
 when a spillway is considered so that the formula becomes: 
 
 Q = 3.33Z/i'^ 
 
 and the influence of the velocity of approach Vo is introduced as given above. 
 
 Because of the lack of experimental data on weirs with h <2 ft., engineers are using for 
 higher heads, formulas developed for smaller heads. However, it has been shown that the 
 Francis formula gives reasonably accurate results for heads up to 5 ft. 
 
 7c. — Profiles of Spillways. — Furthermore, spillways are seldom built in the 
 shape of sharp-crested weirs. They are, as a rule, given an ogee shape, so as to prevent a 
 vacuum below the falling sheet of water, the so-called nappe. 
 
 It is, therefore, customary to design the shape of the nappe for the maximum head acting 
 
750 
 
 CONCRETE ENGINEERS' HANDBOOK 
 
 [Sec. 17-7c 
 
 on the spillway using the general form as found by Bazin for sharp-crested weirs, and then to 
 form the concrete work accordingly. It is d,dvisable to let the concrete on the downstream 
 side of the apex of the curve for the lowex nappe encroach somewhat on the water, so as to be 
 certain that a vacuum cannot form (Fig. 40A). If the abscissae are x, the ordinates for the upper 
 
 +Y 
 
 +x 
 
 -0.1 
 -0.2 
 
 Nappe for slope oo I ( Vertical mir)- 
 Nappe for slope I Z 
 
 06-0.5-0.4-0.3-0,2-0,1 -I 0.1 0.2 0.3 04 Q5 06 0.7 0.3 0,9 LO I.I 1.2 1.3 1,4 
 
 Fig. 40. 
 
 nappe ?/„ and for the lower yi and the head Bazin found the following values for verticle sharp- 
 crested weirs: 
 
 X 
 
 u 
 
 yi 
 
 h 
 
 h 
 
 h 
 
 -3.00 
 
 0.997 
 
 
 -1.00 
 
 0.963 
 
 
 0.00 
 
 0.851 
 
 0.000 
 
 0.05 
 
 
 0.059 
 
 0.10 
 
 0.826 
 
 0.085 
 
 0.15 
 
 
 0.101 
 
 0 20 
 
 0.795 
 
 0.109 
 
 0.25 
 
 0.782 
 
 0.112 
 
 0.30 
 
 0.762 
 
 0.111 
 
 0.35 
 
 
 0.106 
 
 0.40 
 
 0.724 
 
 0.097 
 
 0.45 
 
 
 0.085 
 
 0.50 
 
 0.680 
 
 0.071 
 
 0 55 
 
 
 0.054 
 
 0.60 
 
 0.627 
 
 0.035 
 
 0.65 
 
 
 0.013 
 
 0.70 
 
 0.569 
 
 -0.009 
 
 1.40 
 
 -0.020 
 
 
 Consequently, if the head is 10 ft., = - 3.00 or = - 30ft. and ~^ = 0.997 or = 9.97 
 
 ft., etc. It is obvious that the lower nappe rises from origin to a height 0.1 12A in the distance 
 0.25/1 and that the shape is approximately an ellipse with the major axis horizontal and equal 
 to }y^h and the minor axis vertical and equal to }-^h. 
 
^ Sec. 17-7c] HYDRAULIC STRUCTURES 751 
 
 Boussinesque found that on the ordinate 0.25h the flow was practically horizontal. ^ From 
 f this observation can be deducted the velocity in the upper film of the nappe Vu = 0.475 \/ 2gh 
 and that in the lower film vi = 0.946 \/ 2gh. Assuming that the pressure increase on this 
 ordinate is in a direct proportion to the depth, the average velocity takes place one-third from 
 I the bottom of the nappe. 
 
 For a sharp-crested weir the discharge per linear foot is 
 
 Q = VsfJ^h V2^ with /X = 0.62 
 
 On the ordinate 0.25/i the thickness t of the nappe is 
 
 t = (0.782 -0.112)/i = 0.67/i (see Bazin's table) 
 
 so that the mean velocity is 
 
 and 
 
 Fig. 40A. 
 
 If it is assumed that the water jet on the downstream side of the ordinate 0.25/i follows 
 the laws for a heavy body thrown horizontally in a vacuum, then 
 
 X = Vmt and V = 2' 
 
 or 
 
 ^ Comptes rendus de L' Academie des Sciences, July-Oct., 1887. 
 
752 
 
 CONCRETE ENGINEERS' HANDBOOK 
 
 [Sec. n-7c 
 
 Introducing in this parabolic expression the value found above, or 
 
 2v„ 
 
 = 1.54/i it becomes y 
 
 1.54/1 
 
 and = {1.54:h)y 
 
 By plotting this parabola the thickness of the nappe at any point can be found by con- 
 structing graphically the tangential velocity Vx, keeping in mind that the horizontal velocity 
 remains constant Vm = O.Q2\/2gh (Fig. 40 A). As the discharge Q also remains constant, 
 the thickness tx of the nappe is expressed by 
 
 Q 
 
 Of this thickness ~ is plotted downward on the normal to the parabola in the point in 
 
 By combining the upper and the lower points respectively, the 
 
 2tx 
 
 question and upward. 
 o 
 
 shape of the nappe is found. 
 
 A parabola can be calculated, which approximates the shape of the underside of the nappe. 
 As the stability of overflow dams is calculated in the same manner as that for gravity-section 
 
 dams, reinforced-concrete dams, etc., it is not always possible to 
 keep this parabolic shape of the spillway all the way to the toe. 
 Generally a tangent is drawn to the parabola at such an angle 
 that its intersection with the base provides the required width 
 (Fig. 41). 
 
 From the above it is obvious that the true head for figuring 
 the discharge is larger than the head on the crest of the spill- 
 way. This true head can always be found if the elevation of 
 the crest of the spillway and the surface of the water perpen- 
 dicularly above it are known. If this thickness t of the nappe is 
 known, the true head h is 
 
 h = ^ = — ^ 
 
 0.7.82 - 0.112 0.67 
 
 Fig. 41. 
 
 1.49^ 
 
 which is the head to be used in the discharge formula. If the discharge is very small in com- 
 parison with that for which the spillway was figured, the Francis formula must be used with t 
 instead of h. 
 
 It sometimes happens, especially when ogee curves are fitted to spillways in arched dams, 
 that the crest thickness is insufficient to accommodate the curvature. 
 In such cases the crest of the spillway is cantilevered over the upstream 
 face as shown in Fig. 42. 
 
 Reinforced-concrete dams have decks sloping about 1 : 1, so that the 
 velocity of approach is increased gradually. Therefore, the oblique pres- 
 sure on the nappe, due to the part of the discharge coming from the body 
 of water located below the elevation of the crest, is at a smaller angle with 
 the horizontal, than for a vertical face of the dam. In consequence, the 
 curvature of the underside of the nappe is not so pronounced. 
 
 Bazin observed the flow over a sharp-crested weir inclined down- 
 stream on a slope of 1 :2 (Fig. 40). The coordinates observed are as 
 follows: 
 
 Fig. 4 2— Canti- 
 levered spillway. 
 
Sec. ll-7d] 
 
 HYDRAULIC STRUCTURES 
 
 753 
 
 X 
 
 l/u 
 
 yi 
 
 h' 
 
 h 
 
 h 
 
 0.00 
 
 0.730 
 
 0.000 
 
 0. 10 
 
 0.700 
 
 0.011 
 
 0.20 
 
 0.666 
 
 0.005 
 
 0.30 
 
 0.630 
 
 — 0.014 
 
 0.40 
 
 0.585 
 
 — 0.044 
 
 0.50 
 
 0.535 
 
 — 0.083 
 
 0.60 
 
 0.480 
 
 — 0. 130 
 
 0. 70 
 
 0.418 
 
 
 0.80 
 
 0.350 
 
 
 0.90 
 
 0.276 
 
 
 1.00 
 
 0.196 
 
 
 1.10 
 
 0.109 
 
 
 1.20 
 
 0.009 
 
 
 1.30 
 
 -0.098 
 
 
 For the same head h Bazin found that the indined weir gave nearly 13% greater discharge. 
 
 Knowing that for a slope 1 : co , or the horizontal, the water flows horizontally until the 
 crest (or in this case the end of the channel) has been reached and that then the shape of the 
 nappe follows the laws for a heavy body thrown horizontally (see under Boussinesque) that 
 for slopes of 1 : 2 and oo : 1, or the vertical, the shapes are as given above, it is comparatively 
 easy to determine approximately by interpolation the shape of the nappe for any slope. 
 However, in order to be safe, the concrete lines should be designed so as to encroach upon the 
 lower nappe for the maximum head on the spillway, as otherwise a vacuum will form 
 
 Horton^ in figuring the discharge over ogee curves uses the formula 
 
 Q = Clh}'' 
 
 where h is measured from the crest of the curve. A correction is introduced in C, which is 
 expressed 
 
 C= [3.62 - 0.16(*S - 1)] /i''"" 
 
 where S is the slope of the approach to the crest, or 
 
 ^ _ horizon tal run 
 vertical rise 
 
 This is obviously the best formula to use for dams with inclined water surface like re- 
 inforced-con Crete structures. 
 
 If thus ^ is 1 vertical to 2 horizontal and h = 4.0 ft., C = 3.71. 
 
 Actual experiments have shown that C is 3.74 which proves that the formula gives some- 
 what conservative values. It is to be noted that the portion upstream of the ordinate 0.25/i 
 (see Bazin's method) must be at least 3 ft. in width for this increase in C to take place. 
 
 7<i. Overflow Dams. — ^When a dam acts as an overflow and its height is com- 
 paratively large, the energy of the water when reaching its toe is sometimes considerable. 
 The force P. on a plane normal to the nappe is: 
 
 where v is the velocity ; Q, the quantity; w, the unit weight per cubic foot of water = 62.5; 
 and g, gravity = 32.16. 
 
 1 Water Supply Paper 200, p. 131. 
 48 
 
754 
 
 CONCRETE ENGINEERS' HANDBOOK 
 
 (Sec. Vt-le 
 
 If the water below the dam is of sufficient depth to act as a shock absorber, it is customary 
 to provide the spillway at its bottom with a sweep, so that the water is discharged from it in 
 a horizontal direction against the water below the dam (Fig. 41). 
 
 Should the water level below the dam be too low to act as a cushion, it is sometimes nec- 
 essary, as was done at Gatun, Panama Canal, to place concrete blocks in the path of the 
 water, thus breaking up its force by eddie formations. 
 
 Another method is by providing stilling pools, which often are formed by a low secondary 
 dam placed some distance below the main dam. 
 
 Often aprons are constructed to protect the river bed from erosion. Wegmann recom- 
 mends that if L is the length of such an apron in the downstream direction and the height 
 of the crest of the spillway above the top of the apron 
 
 L =2H 
 
 with the intention that the standing wave will occur in this distance. However, in many 
 instances this is not the case and in India L is made much longer. Often 
 
 L = 3 to 4// 
 
 and sometimes extended by a riprap 1.5H in length. 
 
 Rehbocki suggests that for the maximum head on the spillway hmax the length be made 
 
 L = 1.5H + 6hma. to 2H + Shma. 
 
 7e. Sluices. — Dams are generally provided with pipes laid through their base, 
 so-called sluices, to enable the drawing down of the water level below the spillway crest. Often 
 such sluices are designed with the intention of removing possible silt deposits, but their efficiency 
 is doubtful. Should a flood occur, greater than that for which the spillway is designed, such 
 sluices are very useful as their discharge capacity is great. 
 
 Such sluices should have their valves or gates placed in an open shaft, and stop logs should 
 be provided at their upstream side, so that the valves can be inspected or removed for repairs. 
 Coarse screens are sometimes placed in the conduit to prevent water-logged timbers or other 
 large objects from entering. Bell mouths should be formed in the concrete in order to make 
 the losses due to entry as small as possible. If h is the head measured from the center of the 
 opening to the water surface, or the difference in elevation between the upstream and down- 
 stream water surfaces, if the downstream opening of the conduit is submerged, and if I is the 
 length of the conduit the losses in the conduit are 
 
 ?;2 y2 p ,.2 ,.2 p 
 
 2g 2g a 2g 2g ^ a ' 
 
 where the first term is the losses due to the velocity of the flow; the second, due to entry; and 
 the third, the frictional resistance in the conduit. 
 The entry coefficient 
 
 so that fx varies from 0.06 for a pipe with a bell-mouthed entry to 0.50 for a pipe projecting into 
 the reservoir. 
 
 The coefficient C depends upon the frictional resistance in the conduit, p is the wetted 
 perimeter and a is the net area of the conduit. For round pipes 
 
 P ^ wd X 4c _4 
 a ird^ d 
 
 1 "Handbuch der Igenieurwissenschaften. " "Der Wasserbau," p. 37. 
 
Sec. 17-7/] 
 
 HYDRAULIC STRUCTURES 
 
 755 
 
 and for rectangular conduits t ft. high and 6 ft. wide 
 
 P _ 2(t + b) 
 a tb 
 
 A good trial value of C is 0.0075. 
 
 Generally the quantity Q is given and it is desired to find the area of the conduit or the 
 diameter d of a pipe sufficiently large to discharge this quantity under a given head ; thus 
 
 introducing 
 
 At = 0.5, C = 0.0075, and v = —, where a = ^ 
 
 a 4 
 
 and solving 
 
 = / iM + omi] 
 
 hw^g \ / 
 
 In this equation the only variable is d which is easiest found by trial. A good first trial 
 value is 
 
 (r) (9) (h) 
 
 Area A in each case meas ured on section "A- A" 
 Ka) Standard rnouttjpiec€- ir=0.82li^-.-Q=Av=Q82AiEgh.<,e) Re-entrant tube-ir'012iEgh- Q'^Av=0.7EAi^. 
 
 (b) Sharp edged orFios --v-^037^h - Q-0MAv-=d.6ZAiZgPi. (f) Conical diverging tube - Q-0 95AifFqh. 
 
 (c) Streamline contour-\H)96iMi--Q= Av =036 A IZgh. (g) ^nturi adjutage Angle 0= 5°to 8° Q= I.SAllEgh 
 
 (d) Borda'5 mouthpiece-vO$9/^--Q'054Av=0.53AiEgh. (h) Conical converging tube - Angle Q^S'tolO'- Q-OSSA^Egh. 
 
 Fig. 43. 
 
 In reinforced-concrete dams the conduit is, as a rule, very short and serves merely as a 
 setting for the sluice gate. It is continued through the dam in the open channel formed by the 
 buttresses. In such a case the conduit can be considered as a mouthpiece and the velocity 
 determined directly from Fig. 43. ^ 
 
 7/. Siphonic Spillways. — In places where the discharge to be handled is com- 
 paratively small, siphonic spillways can be used to advantage.^ As such devices generally are 
 designed for automatic operation, it is obvious that a close regulation of the water surface 
 will be obtained. As, furthermore, the operating head is the difference in elevation between the 
 water surfaces above and below less the hydraulic losses in the siphon, the discharging ca- 
 pacity per linear foot is considerably in excess of that of an overflow spillway. 
 
 1 Slocum: "Elements of Hydraulics," Ed. II, p. 69. 
 
 2 Hillberg: "Spillways of the Siphonic Type," Eng. Rec, May 3, 1913, p. 488. 
 
756 
 
 CONCRETE ENGINEERS' HANDBOOK 
 
 [Sec. 17-7/ 
 
 In the design three details are of importance: 
 
 1. The upper part must be so made that as soon as the water rises above the level to be 
 maintained, the siphon intake is sealed to the air and is kept sealed until the water level has 
 been drawn down again to normal. The air openings must then be large enough to admit 
 quickly sufficient air to break the siphonic action. Both of these features can be secured by 
 having long and sharp edges on the intake to the siphon at the normal water level. 
 
 2. The lower end of the siphon must be submerged deeply enough to secure a constant 
 seal from the beginning of the siphonic action. The upper edge of this opening must be as 
 sharp as possible to permit an easy escape of any air carried out by the water. 
 
 3. The cross-sectional area should be as large as possible at the intake to reduce losses due 
 to entry. Back of the intake provisions should be made for an efficient absorption of the 
 enclosed air. This is obtained by building a channel around the opening, so that in the be- || 
 ginning water will flow into it from all sides forming a spray. The neck of the siphon is gen- 
 erally curved, so that the water pressure on any entrained pocket of air will move it down- 
 ward flattening it so that the friction at the contact surface will tear off layers of air until i| 
 all of it has been carried out. A gradual narrowing of the cross-sectional area up to this point ' 
 is desirable because the increase in velocity head will lessen the static pressure thus creating 
 an overpressure on the upper part of such air pockets. Below this point no enlargement of 
 the area is permissible as the corresponding decrease in velocity will release a certain portion 
 
 of the entrained air, which, especially if the siphon is curved, will collect and cause interrup- 
 tion of the siphonic action taking the form of pulsations. From this it is obvious that the 
 best practice is to incline the lower leg, as released air would thus quicker reach the concrete 
 wall and be again entrained in the water. 
 
 In a siphon the sum of all losses must equal the difference in elevation E between the 
 water surfaces or 
 
 £ = |(l + + ^ 
 
 where v = velocity; g, 32.16 or gravity; /j., coefficient for loss due to entry (probable maximum 
 0.50); X, coefficient for loss due to friction in upstream leg (probable maximum 0.10); C = 
 coefficient for loss due to friction in downstream leg (varies from 0.005 to 0.009); p, its wetted 
 perimeter; a, its area; I, its length; and k, coefficient for loss due to bends or curves (probable 
 maximum 0.25). 
 
 This equation is applicable for all conditions up to E = 33.9 ft., which is the suction limit. 
 Should E be greater than this, the losses must be kept equal to or smaller than 33.9 ft. by taper- 
 ing the downstream leg in such a way that its smallest cross-section is at or near the lowest 
 point. As the maximum velocity will take place at this section, it is obvious that a certain 
 amount of head is still in the form of pressure in the sections above. This can be formulated 
 in the following: Calculate a conduit with a hydraulic gradient so that the sum of all losses is 
 33.9 ft. or less. One siphon at Gibswil, Switzerland (Fig. 44) operates under an elevation 
 difference of 52.48 ft. 
 
 Experiments made in Switzerland on the operation of siphons shows that the general 
 formula 
 
 Q = av ^ afx\^2gh (sec. -ft.) 
 
 can be used with a coefficient /x varying from 0.55 for smaller heads to 0.70 for higher. It is 
 always convenient to use this formula for trial computations, but to check the final design with 
 the more correct theoretical formula. It is to be remembered, however, that hmax = 33.9 ft. 
 
 K the discharge is too voluminous to be handled in one conduit, several are used and 
 placed side by side. 
 
 The materials used for building smaller siphons are steel pipes with reinforced-concrete 
 hoods or reinforced concrete is used throughout. Larger siphons are always built of concrete 
 
Sec. 17-7/1 
 
 HYDRAULIC STRUCTURES 
 
 757 
 
 328' High wQferTivel-^ 
 
758 
 
 CONCRETE ENGINEERS' HANDBOOK 
 
 [Sec. 17-8 
 
 which can be either mass or reinforced. One siphon built of reinforced concrete in Switzerland 
 is shown in Fig. 45. 
 
 Because of the suction the exterior load on the siphon walls equals the full atmospheric 
 pressure of 14.7 lb. per sq. in. or 2120 lb. per sq. ft. On the underside this load can be reduced 
 because of the weight of the water and the structure itself. 
 
 The biggest siphon so far proposed is designed for the Dunning's Dam of the Scranton, Pa. 
 water supply (Figs. 46 A and 465). It consists of five conduits 4 ft. high and 6 ft. wide and 
 
 Requinsd. SH 6'/6'xf-6'-e'fa 
 
 
 
 ' ill jll , i 
 
 • "Twisted ban ffchc. '^i//' Section "B-B" --f Twisted bars 8"cfoc 
 
 pBars, twisted 4'chc. 
 every other bar benf 
 as shown. 
 
 •'»[ li'Mi"/7r • J J '^t':<M'" Twisted 
 .'♦c^Hi Twiltfd. %^Mbars4'Joc. Pgi 
 tars 6 c toe L every other bm \^"\ 
 L fbentasshom.f'J 
 
 Section "B- 
 
 i'° Twisted bars S'ctoc. 
 
 bars.: 
 
 Section''C-C" 
 
 Fig. 4G5. — Details of siphon for Dunning's dam, Scranton Gas & Water Co. 
 
 its total discharge capacity is 3750 sec. -ft. Its overall width is 38 ft. and the length 90 ft. The 
 maximum stress in the concrete is 543 lb. and in the steel 13,750 lb. per sq. in. 
 
 8. Movable Dams.^ — Movable dams are used in places where a wide opening is required 
 to accommodate the flood discharge. Where such dams are used for river regulation they 
 are generally placed straight across the channel and erected on a substructure which is nothing 
 more than a low sill. The movable parts are so made that they can be laid down on this sill, 
 
 1 The best textbook on movable dams is " Handbuch der Ingenieurwissenschaften, Part III' 
 bau," Chapter III "Die Beweglichen Wehre," by Prof. K. E. Hilgard, 
 
 ' 'der Wasser- 
 
Sec. n-Sa] 
 
 HYDRAULIC STRUCTURES 
 
 759 
 
 removed entirely to the shore or hoisted between piers, so as to leave an unobstructed channel 
 of practically the same width as that before the placing of the dam. 
 
 Movable crests are often used in combination with spillways for dams in places where it 
 is desired to maintain as nearly constant as possible the elevation of the water surface above 
 the dam. 
 
 All types of movable dams can be divided into three groups: (1) Requiring operating 
 machinery; (2) operating under hydrostatic pressure differences; and (3) automatically 
 operating. 
 
 8a. Requiring Operating Machinery. — These dams can be operated manually, 
 electrically, hydraulically, or in any other mechanical way. The operating machinery is often 
 mounted on a traveler or a barge so that it can be moved along the dam and used at any desired 
 point. The principal types are: Stop logs, needle dams, A-frame dams, curtains, flashboards, 
 gates, wicket gates, bridge dams, taintor gates and rolling dams. 
 
 86. Operating under Hydrostatic Pressure Differences. — These dams are 
 generally so designed that they are operated by the opening or closing of valves in conduits 
 connecting an inclosed chamber under the gate with the high water above and the low water 
 below the dam. The usual types are: Bear traps, drum dams and butterfly dams. 
 
 8c. Automatically Operating. — For close regulation of water levels, automati- 
 cally operating devices have come to the front during the past 10 years. The first to be built 
 was in Connecticut as early as in 1902.^ They were made of oak and about 3.5 ft. high and 
 6 ft. wide. Each gate operated on the principle, that when the water level reached its top, 
 the resultant of the hydrostatic pressure fell above the point of support. In order to prevent 
 the gate from opening with an accelerated speed, the shaft was provided with toothed cams at 
 the ends, shifting the point of support upward on the gate when opening. In 1914 gates of 
 similar design and from 5 to 15 ft. high and 18.5 ft. wide were installed at Austin, Tex.^ They 
 consist of steel frames with wooden covering. Their lower part is filled with concrete to place 
 the center of gravity below the point of support. However, such gates have not been a success 
 because of the difficulty of designing a cam or a bascule which will prevent the gate from gaining 
 in speed when opening. Because of this the impact, when the gate reaches the open position, 
 is, by larger gates, sufficient to destroy them. Practically the only effective way of making 
 gates automatically operating is by providing them with counterweights, so designed that their 
 moments increase in proportion to the increase in moment due to hydrostatic and impact pres- 
 sures on the gate leaf. Among such balanced gates are those with overhead rolling counter- 
 weight, underhung counterweight, counterweights suspended at ends of levers, and those with 
 variable counterweights. 
 
 9. Fish Ladders. — To permit fish to pass dams in search of spawning grounds and of food, 
 the laws of many States require that fishways be provided. Such ways or ladders consist of 
 a number of compartments arranged in steps and separated by cross-partitions or baffles. 
 The construction varies depending upon the habits of the fish living in the stream in question, 
 but they can be divided into two main groups: Jump ladders and swim ladders. Combina- 
 tions of both are sometimes built if the prevailing conditions should so require. All fishways 
 must be so designed that the outlet is below low water level and so located as to have an unob- 
 structed discharge of water in order to attract the fish. The intake or upstream end should be 
 not less than 1 ft. lower than the crest of the dam. The slope of the bottom should never be 
 more than 1:4 and if possible be 1: 10 or even less. The width should not be less than 4 ft. 
 and often it is up to 10 ft. No compartment should be shorter than 4 ft. or in depth less than 
 2.5 ft. Plenty of light should be admitted or the fish will not use it. However, to protect the 
 fish from birds and human beings, fishways should be covered by gratings so built as to facili- 
 tate inspection and cleaning. There should be no regulating gates at the intake, necessitating 
 attendance. Fishways are built of wood, steel, masonry, or reinforced concrete. 
 
 1 Eng. Rec, Mar. 8, 1902, p. 222. 
 
 2 Lamar Lyndon: "Hydro-electric Power," vol. 1, p. 285. 
 
760 
 
 CONCRETE ENGINEERS' HANDBOOK 
 
 [Sec. 17-10 
 
 RESERVOIRS 
 
 10. General Types. — There are two general types of reservoirs — open reservoirs and covered 
 reservoirs. Open reservoirs vary in size from great catchment basins closed by dams or em- 
 bankments, either of earth, or of masonry, or both, to smaller containers constructed largely 
 or wholly of masonry, such as sewage treatment basins. Fig. 47 shows a 5,000,000-gal. open 
 circular basin constructed by the city of Duluth, Minn. 
 
 Fig. 47. — Duluth circular reservoir. Hoop rein- Fig. 48. — Covered equalizing and storage reservoir 
 forcement for wall was supported by notched steel for Hibbing water-works, 
 
 plate brackets riveted to vertical channels. 
 
 Covered reservoirs are of relatively limited size and usually are constructed of concrete 
 masonry in floors, roof, and walls, with earth covering on roof and against walls (see Figs. 48 
 and 49). Covered reservoirs prevent freezing or disagreeable warming of water, as well as 
 pollution from outside sources and organic growths which. require sunlight.^ 
 
 11. Quality of Concrete for Reservoir Masonry. — Concrete for reservoirs has density and 
 correlatively, impermeability as basic requisites. To this end, the selection of materials, the 
 
 1 For discussion see Ellms: "Water Supply;" Flinn, Weston and Bogert: "Water Works Handbook;" 
 Hazbn: " American Civil Engineers Pocket Book;" and others. 
 
Sec. 17-12] 
 
 HYDRAULIC STRUCTURES 
 
 761 
 
 proportioning, the mixing, the placing and, particularly, the quantity of water employed 
 should be subject to rigid regulation. Percolating and entrant water is the most active disinte- 
 grating agent to which concrete is normally subject. The use of arbitrary proportions and 
 careless methods of manipulation in such concretes is therefore not only poor engineering, but 
 a direct courting of trouble. The number of specifications and textbooks permitting and ad- 
 vocating such practices is at present regrettably large, either in ignorance of, or regardless of, 
 the definite authoritative and generally available knowledge as to their danger and incorrectness. 
 
 12. Open Basins with Embankment Walls. — With soil of proper character, open reser- 
 voirs may be constructed without masonry. It is usually preferable, however, after forming 
 dense banks from excavated or other materials, to cast upon them and upon the bottom, 
 
 Section thru secondary beams 
 
 \lf.5-0'lg-'"^ 
 
 f/£"c.fo<y 
 
 Elighfh Ave^ 
 18' SuDcly Pipe .14 'S cour and OT pipe 
 
 IS 
 
 la H H — 
 
 B IS 
 SI SI 
 
 — H 
 
 — H 
 
 B SI 
 IS SI 
 
 B SI la El B s - 
 
 Reservoir 
 B H SI H H B- 
 capacify Z000,000qal5 ■ 
 H H Sf H H M - 
 
 ,)^.„...4'-IO"- ^■'■■■3'-9" 
 
 Fig. 49. — Covered water-works reservoir at Regina, Sask. 
 
 slabs of concrete, reinforced or not, according to the stability and uniformity of the foundation 
 and embankments. 
 
 For embankment walls, concrete of rather stiff consistency is cast directly on a sand coat 
 over a layer of puddled clay. This clay layer should be from 10 to 24 in. in thickness; and the 
 slope about 3 : 1 and never steeper than 2:1. Reinforcing mesh or rods may be embedded 
 in embankment slabs if individual circumstances of location or material indicate this procedure 
 as advisable.! Concrete core walls may be used with earth embankment, with or without 
 masonry facing. In all cases, whether or not concrete facings are used, embankments should 
 be well settled and compacted and well worked into a thoroughly stripped and scarified subsoil. 
 
 13. Concrete Floors for Reservoirs. — Concrete floors for reservoirs are laid in one or more 
 separate layers, either as a continuous slab or superposed slabs, or in rectangular blocks with 
 
 1 See Eng. News, vol. 74, p. 267, 1915. 
 
762 
 
 CONCRETE ENGINEERS' HANDBOOK 
 
 [Sec. 17-14 
 
 closely abutting joints. As before stated, the concrete should be as impervious as possible; 
 and to prevent leakage, continuous construction of a monolithic slab is advantageous, inasmuch 
 as the often considerable leakage at joints is thereby prevented. The strength of the concrete 
 should be such as to support the weight of water over slight inequalities of bottom when the 
 
 reservoir is full; and at least sufficient to 
 withstand upward pressure of ground water 
 with the reservoir empty. A thickness of 
 from 6 to 12 in., depending upon subsoil con- 
 ditions, is usually sufficient. 
 
 When concrete floors are divided into 
 panels, the blocks of from 15 to 60 ft. square 
 are connected at the edges by some sort of 
 lock-joint. This is to provide for flexibility 
 during settlement, and for contracton. 
 There is usually a beam of concrete laid 
 along and below the joint. Details of such 
 joints are shown in Fig. 50. If a tongue- 
 and-groove joint is used, the tongue should 
 be V-shaped, otherwise it will be broken 
 off. 
 
 Pipes and other fixtures passing through 
 the bottom should be flanged and well 
 water-proofed. 
 
 14. Groined and Flat Floors. — Inverted groins have become a familiar type of reservoir 
 bottom in view of the advantages offered by footings adapted to distribute column pressures 
 formed by the thicker portions, with channels for water flow and cleansing offered by the con- 
 tinuous inverts. This type of floor is generally used with covered reservoirs, and it may be 
 
 Fig. 51. — Construction view of groined reservoir floor being placed in two layers. 
 
 laid in one or two layers, as desired. In Fig, 51 is shown in process of construction such a floor, 
 placed in two layers, the lower layer being run flat. 
 
 A modification of the groined floor is shown in Fig. 52. This type offers the additional 
 constructional advantage of permitting continuous placing of plastic concrete. This design was 
 
Sec. 17-151 
 
 HYDRAULIC STRUCTURES 
 
 763 
 
 Groined floor 
 
 Modrfied fkri- floor 
 
 Fig. 52. — Modification of groined floor. 
 
 developed by the John F. Casey Co. in their repairs to the Filtered Water Reservoir at the Divi- 
 sion Pumping Station, Cleveland, Ohio.^ 
 
 Flat floors may be used in either open or covered reservoirs. In large open reservoirs they 
 are customary as groined slabs are more difficult of construction and not advantageous 
 where no piers are to be supported, all necessary water channels being readily formed by sloping 
 slabs. A notable example of large, open concrete 
 reservoir construction of this type is the Hillview 
 Reservoir of the Catskill Water Supply for the 
 City of New York.^ 
 
 15. Concrete Walls for Open Reservoirs. — 
 Concrete walls for reservoir sides are designed as 
 retaining walls, except that reservoir walls must 
 withstand earth thrust from outside when the 
 
 reservoir is empty, and water pressure from within when the reservoir is full. The resistance 
 of outside earth embankments against water pressure is sometimes deducted, but this procedure 
 requires that to bear upon the earth, the wall may undergo no deflection; i.e., the earth does not 
 shrink away from the outer face of the wall but is constant in bearing. Various means are 
 used to prevent such loss of contact between wall and bank. The wall may be battered, or 
 better, stepped slightly. The portion of the fill next to the wall is sometimes puddled, but 
 where this is done, provision against excessive thrust must be made in the design, or the wall 
 must be braced. 
 
 When the embankment on one side of a wall is caused to resist pressure from the opposite 
 side of the wall, there is a tendency to force out a prism of earth. The largest thrust which the 
 earth will resist is called the -passive thrust. 
 
 In Fig. 53 let AB be a given wall with an embankment whose earth surface is Abfn tending 
 
 to resist a pressure acting against the wall 
 from the left. The friction between wall and 
 earth will act downward, since the earth will 
 tend to move upward. The reaction of the 
 earth P will thus be directed upward making 
 an angle above the normal to the wall of Z, 
 but never greater than 4> (see Art. 16, Sect. 
 13) . Lay off on the surface line points a, b, c, 
 etc. arbitrarily. Draw lines from these points 
 to B, thus bounding a series of prisms of 
 length 1 ft. perpendicular to the drawing. 
 Compute the weights of these prisms and to 
 any convenient scale lay them off in order, 
 on the line BW which makes an angle </> with 
 Draw through one of these points a line, as twd, which makes an 
 angle Z to the right of a normal to BW. Then through each point on BW draw a line parallel 
 to this direction tWd, to an intersection with corresponding ray through B, as We to Be at Se', 
 Wf to Bf at Sf', etc. Through the points S draw a smooth curve. (This curve will have 
 breaks at rays connecting B with points of change of slope.) Draw a tangent v to the curve 
 parallel to BW, and through the point of tangency draw a ray from B, as BSgg. This line will 
 represent the plane of rupture of the embankment if the passive thrust is exceeded. The 
 distance WgSg between BW and the tangent v, measured parallel to the direction tWd and to 
 the same scale as that used in laying off the weights on BW, is the magnitude of the passive 
 thrust. 
 
 the horizontal as Wa, Wb, Wc, etc. 
 
 1 See Eng. Rec, Dec. 9. 1916, p. 702. 
 
 2 See Proc. Am. Cone. Inst., 1915. See also Taylor and Lyndon on pavement of circular reservoir at 
 Austin, Tex., Proc. Am. Cone. Inst., 1916, p. 143. 
 
764 
 
 CONCRETE ENGINEERS' HANDBOOK 
 
 [Sec. 17-16 
 
 The point of application of P, the reaction of the earth against the wall, will be one-third 
 of AB above B. 
 
 Proof of the construction may be at once seen by revolving the force triangle BSgWg to 
 such a position that BWg lies on AB. Bwg is then seen to be the weight of the block of earth 
 AbfgB lying above the plane of rupture Bg; BSg is the resultant pressure on the plane Bg and 
 making the internal friction angle <^ thereto ; and as previously noted, WgSg denotes the passive 
 thrust, at an angle (90 deg. + Z) to the wall. 
 
 The moment of the passive thrust is added to the moment of resistance, taken about the 
 third point toward the earth, when the reservoir is full; and the moment of the active thrust is 
 taken as the attacking moment about the third point toward the water, when the reservoir is 
 empty. 
 
 16. Partition and Outside "Walls. — Walls subdividing the basin for purposes of filhng or 
 emptying are usually of concrete. They consist either of a cantilever wall or a double-counter- 
 forted wall with the stem at the center of the base. In the latter type the double-counter- 
 fort acts as a wedge-shaped beam with both faces inclined, and is reinforced to act in either direc- 
 tion. The stem of the cantilever wall may be designed in the same manner. The base slab 
 is designed to take the resultant pressure from either side. 
 
 Vertical joints in concrete walls and partitions should be made at intervals not to exceed 
 60 ft. The tongue-and-groove type of keyway with a folded metal strip, as in slabs, makes 
 a satisfactory joint. It should be thoroughly waterproofed. 
 
 Concrete in such walls should be well spaded next to the forms to obtain a smooth surface. 
 The whole material should be well tamped. The thinner the wall, the more essential is proper 
 selection and proportioning of materials, adequate mixing, and careful 
 placing, 
 
 17. Provision for Ice. — All walls for reservoirs in which ice is 
 likely to form, should be battered at the water line to heave the ice 
 
 , as it expands. Paved embankments usually give little trouble in this 
 respect. 
 
 18. Covers or Roofs for Reservoirs and Basins. — The purpose of 
 a roof may be (1) to prevent contamination or organic growths, (2) to 
 prevent freezing, or (3) to maintain lower water temperature in warm 
 climates. Three types of construction are used: beam-and-slab, flat- 
 slab, and groined- (elliptical) arch construction, support being afforded 
 by walls, by columns, or piers extending to the floor. The first two 
 types are the same as those for floors. The last requires separate 
 analysis. 
 
 19. Groined-arch Construction. — Experience shows that due 
 either to temperature or to settlement, cracks are likely to form at construction joints at the 
 crown a — a' (Fig. 54); and when such cracks have formed, little or no arch action prevails. 
 The nature of failures indicates that arch action is not sufficient to be considered in the design. 
 The design, therefore, should be made for shear and cantilever moment, and may be made as 
 for footing slabs. It is advisable to reinforce the ''umbrella" at the upper face over the 
 column, and at the lower face of the crown. Due to the increased cost of formwork and 
 difficulty of construction, this type of roof is not often as economical as the usual flat-slab, or 
 other types of floor construction. 
 
 .20. Construction Details of Columns and Roof. — Footings for walls and columns are laid 
 below the pavement or else the pavement is thickened at bearing points. To prevent undue 
 settlement pressures on footings should closely approximate those on floors of basins. If 
 separate footings are provided, expansion joints should be made in the pavement about the 
 column. Roof slab should be reinforced against temperature changes and shrinkage by adding 
 0.4% of steel. Many reservoir roofs have been built continuous with the side walls, but if 
 
 Fig. 54. 
 
Sec. 17-21] 
 
 HYDRAULIC STRUCTURES 
 
 765 
 
 the width of structure is considerable, or if material temperature change is anticipated, expansion 
 joints should be provided between roof and walls. 
 
 STANDPIPES AND SMALL TANKS 
 
 A standpipe is a cylindrical tank resting directly upon its foundation, and usually, though 
 not always, having a height greater than its diameter. It may be used for the storage of fluids 
 other than water, though the latter use is the most common. 
 
 Standpipes of concrete require particular care in construction. Not only must the con- 
 crete be of sufficient strength to withstand, without outside support, bursting from pressure 
 of water or cracking from temperature changes, but further the concrete must be so continuous 
 and of such strength, density, and finish that percolation at work planes or other points will 
 not take place. 
 
 21. Analysis of Stresses in Standpipes. — Since the pressure of a fluid varies in intensity 
 with the depth, the unit pressure at a depth y is 
 
 Py = wy 
 
 Suppose a circumferential strip 1 ft. wide to be cut from the shell at a depth y (Fig. 55). 
 The total tension on any diameter is 
 
 T = wr 
 
 where py is in pounds per square foot; r is the inner radius of the tank in feet; and T the pull 
 on one side of the tank per foot of height. 
 
 
 
 
 
 
 
 
 
 
 
 
 
 .Y 
 
 
 Fig. 55. 
 
 Fig. 53. 
 
 In a standpipe of concrete the tensile resistance is necessarily low, since at a given tensile 
 unit stress the concrete will crack, permitting seepage under high heads. Tests indicate that 
 concrete of good quality will crack minutely at an elongation of 0.00016 to 0.0006 inches per inch. 
 The value 0.00010 corresponds to a unit stress of 200 lb. per sq. in. in the concrete, and (when 
 bars are present) 3000 lb. per sq. in. in the steel. Stresses above these values require full tensile 
 resistance of the steel. Where high heads are used, low stresses should be employed; but in 
 tanks having a head of 30 ft. or less, steel stresses of 8000 to 10,000 lb. per sq. in. have proven 
 successful, since the cracks do not open enough to cause appreciable seepage at these low heads. 
 
 European practice employs a very rich mix for use in standpipes, varying from 1 cement: 
 
 graded aggregate, to 1 cement: 23^^ graded aggregate, these values varying for tanks from 
 100 ft. high to 30 ft. or less in height. 
 
 22. Restraint at Base. — The deformation of a standpipe is not wholly due to the elon- 
 gation of the hoops under stress. At the base the rigidity of the bottom prevents hoop expansions 
 and introduces a restraining moment on a vertical element. If the side of the tank were of 
 uniform thickness and a hemogeneous material were employed, the deformation of the rings 
 would vary as the depth of water. But in the design of a concrete standpipe with steel rein- 
 forcement, the hoop elongation will be limited to that elongation which corresponds to the 
 adopted working unit stress in the steel. Thus in Fig. 56, the portion BC of the exaggerated 
 
766 
 
 CONCRETE ENGINEERS' HANDBOOK 
 
 [Sec. 17-22 
 
 deformation is a constant for any depth between B and C. The deformation to AB may be 
 a straight Une or not; however, it is the gradual increase of deformation up to that correspond- 
 ing to the working hoop stress. 
 
 The restraint of the bottom is such that at D no hoop stress exists, but rather a fixity, 
 similar to a vertical beam fixed at the lower end. This restraint decreases so that at some point] 
 C only hoop stress, and likewise hoop deformation, prevails. It is evident that at this point] 
 C, the tangent to the deformation curve is again vertical, thus parallel to the tangent to the] 
 curve at the bottom. 
 
 Suppose the lower portion of the shell of the tank or standpipe to be made up of vertical' 
 beams restrained at their lower end, and having a length sufficient to include the deformation 
 
 CD, Fig. 56. Then (Fig. 57) the water pressure varies from its 
 upper end with w{h — I) to its lower end with wh. Let the beam 
 be acted upon at its upper end by a moment M', sufficient to 
 maintain the slope at that end parallel to that at the base. Then 
 the slope at C is zero with respect to that at D. The deforma- 
 tion A at C must be equal to that caused by ring working stress 
 only. 
 
 Now imagine the elastic hoops existing between the levels C 
 and D to restrain the deformation of the beam somewhat. Let 
 the deformation of the beam thus modified be that shown by 
 CD, Fig. 57. The hoops stresses between the levels C and D will 
 thus vary as abscissas to the deformation curve CD. Before we may again consider the ver- 
 tical beams separately, the effect produced by these hoop stresses should be used to modify 
 the water pressure on this beam. The deformation area DcC may be converted to a cor- 
 responding stress area, and subtracted from the pressure trapezoid obDc by placing cD upon 
 ah. The loading left to the restrained cantilever beam to be resisted by beam action is 
 therefore represented by the area hfecD. 
 
 Since the actual form of the curve hfec is still unknown, the resultant loading is likewise 
 unknown. So far as concerns the deflection A of the end C a very close approximation is ap- 
 parent by using the triangular loading bcD. It should be noted that this triangle exceeds the 
 actual loading at e (curve hfce) but is less than that at f; but since the upper portion of the beam 
 contributes more than half of the deflection A, the approximation becomes, if anything, slightly 
 on the safe side. For the loading adopted, and for a homogenous beam. 
 
 M = 
 
 24 
 
 A = 
 
 SOEI 
 
 (1) 
 
 From the theory of deflections of reinforced-concrete beams ^ the deflection A of the beam 
 CD becomes, instead of that just given for a homogeneous beam, 
 
 1 62.5 W n ^ 1 hP n 
 SOEs ' 24 hd^ ' a 368.8£/. ' a 
 
 A = 
 
 (2) 
 
 in which 
 
 h = depth of water at bottom of tank, in inches. 
 / = length of beam element CD, in inches. 
 d = effective depth of beam element CD, in inches. 
 Es = modulus of elasticity of steel, pounds per square inch. 
 n/a = numerical coefficient dependent upon p and n, in which n is recommended by 
 Turneaure and Maurer to be 8 or 10 (see Fig. 58). 
 A = deflection of end C, in inches. 
 
 I 
 
 Referring again to Figs. 56 and 57, it will be noted that A must equal the deflection, or 
 change in radius due to hoop tension. This may be expressed in terms of the steel working 
 stress fa, the radius r in inches, and E^. 
 
 1 See Art. 28, Sect. 7. 
 
Sec. 17-23] 
 
 HYDRAULIC STRUCTURES 
 
 767 
 
 Since this must equal the beam deflection, equating (2) and (3) gives 
 
 ^4 ^ 4425/'.rd3 
 
 (3) 
 
 whence 
 
 I = 8.16 
 
 (in.) 
 
 (4) 
 
 This is the value of I which will give the desired deflection A. Having thus found I, the moments 
 of its loading may be determined. Thus, with all linear dimensions in inches, M in inch- 
 pounds is from (1) and (4), 
 
 M = 3.612 
 
 fsrd% 
 
 In using this formula it should be noted that/^ is the unit stress in the hoops. M' 
 the relation 
 
 (5) 
 
 is given by 
 (6) 
 
 qbilO 
 
 5 100 
 u 
 
 f 90 
 
 8 80 
 
 3 60 
 
 The point of inflection is at 0.37Z above the bottom. At this point the reinforcement 
 may swing from the inner to the outer face. Although only a 
 third of the steel is needed at the outer face, the remaining 
 steel should be carried up sufficiently to provide ample bond. 
 
 The moment above the point C may be assumed to vary 
 as a straight line from the value M' at C to zero at the top. 
 This is not strictly true but is sufficiently close to provide a 
 means of cutting some of the steel near the outer face if 
 desired. 
 
 23. Shear at Base. — The shear at the base may be seen 
 from Fig. 57 to be equal to the triangular loading on the 
 beam. Thus, the shear per foot of circumference becomes 
 
 V = 0.2l7hl 
 
 where h and I are in inches. 
 
 05 
 
 Percentage of remforcemeni 
 Fig. 58. 
 
 Illustrative Problem. — Let h = 40 ft., diameter = 20 ft., thickness of shell = 10 in. U 
 per sq. in. for hoop stress. Assume 7i = 10 and p = 1.5%. 
 
 From formulas (2) and (3) in Art. 286, Sect. 7, or Fig. 58, page 7G7, - = 86. At the base 
 
 A = 8000 lb. 
 
 M 
 
 3 pOOO + 120 X (8.5 )3 x 480 
 
 207,000 in.-lb. 
 
 Assuming i = 0.83 (now n = 15) and noting from Diagram 2 on page 360 that when p 
 fc = 650, the moment value of 1 sq. in. of steel thus stressed is, for d = 8.5 in., 
 M = fsjd = (10,500) (0.83) (8.5) = 74,100 in.-lb. 
 207,000 
 
 1.5%, f, = 10,500 when 
 
 Checking, 
 
 74,100 
 As _ 2.8 
 bd 
 
 2.81 sq. in. steel 
 
 = 0.028, nearly 3 % 
 
 It will be necessary to taper the wall at the base. 
 
 207,000 
 d^ = = 131.7 
 
 (8.5) (12) 
 
 Solving for the effective depth required for p = 1.5% (/C = 131) 
 d = 11.5 in. or 14 in. total 
 
 (131) (12) 
 
 As = 12 X 11.5 X 0.015 = 2.07 s^. in. per ft. of shell. 
 Use ^i-in. sq. bars spaced 4>'2 in. on centers. 
 
768 
 
 CONCRETE ENGINEERS' HANDBOOK 
 
 [Sec, 17-24 
 
 The point of reversed moment is I above the base, or 
 I 
 
 = 8.16^. 
 
 8000 X 120 X 8.5^ 
 
 480 X 86 
 
 = 109.4 in. 
 
 Point of inflection is at 0.37Z = 40.5 in. above the base. The moment at the distance I is 
 
 V y Y 
 
 M' = - ■ 207,000 in.-lb. 
 3 
 
 69,000 
 
 69,000 in.-lb. 
 
 K = 
 
 12 X 8.52 
 
 This will require 0.6% of steel, or 
 
 As = 12 X 8.5 X 0.006 = 0.61 sq. in. 
 
 Every third bar of those needed at the bottom will suflBce. 
 The unit shear at the base is 
 
 0.217;iZ (0.217) (480) (109.4) 
 
 12 jd 
 
 (12) (0.83) (11.5) 
 
 = 99.5 lb. per sq. in. 
 
 ..>|/^"1<. Stirrups are thus required, for a distance of I — x from the bottom where 
 
 Fig, 59. 
 
 109.4 
 99.5 
 
 X = 43.9 in. 
 
 65.5 in. 
 
 The stirrups should be anchored to vertical 
 
 Inclined rods, figured as inclined stirrups in a beam, may be used, 
 rods near the outer face (see Fig. 59). 
 
 24. Small Tanks. — Tanks and reservoirs of small size are very readily constructed of 
 concrete, particularly when they may be built as monoliths. Their uses are manifold and 
 encouraging, in so far as the use of concrete in larger constructions to meet difficult situations 
 is concerned. In paper mills, concrete tanks for sulphite processing have proven satisfactory, 
 with a useful life of from 4 to 10 years. With proper grading 
 and choosing of aggregates, concrete resistant to acids and acid 
 fumes has been produced, this resistance being directly propor- 
 tional to the proportion of silicious materials and inversely pro- 
 portional to the cement exposed to contact with the attacking 
 fluid or gas. Concrete tanks are in successful use for various oils, 
 even including those of organic derivation; for tanning solutions; 
 for various fermenting substances, such as sauerkraut; and when 
 coated with mastic, for solutions incident to the refining of copper. 
 Among these latter may be mentioned those erected in Chile by 
 
 the Chile Exploration Co., which have successfully withstood earthquakes as well as chemical 
 attack. 
 
 The sides of small circular tanks are designed for hoop stress in the same manner as are 
 the sides of standpipes. In tanks under 15 ft. deep the restraint of the base is somewhat 
 different than for the foregoing case of deeper tanks. The two cases following give the statical 
 limitations of the secondary stresses caused by restraint. 
 
 (a) Walls Continuous with Base. — The moment at the base (Fig. 60a) is 
 
 M = 0.0QQ7wh^ 
 M' = 0.029Swh^ 
 V = 0.4rf 
 
 y = 0.553/i 
 
 (6) When the Base Is Separated by a Joint or hy Cracking (Fig. 606).- 
 
 M' = 0.0M2wh^ 
 V = 0.333w;/i2 
 
 y = 0.423/1 
 
Sec. 17-25] 
 
 HYDRAULIC STRUCTURES 
 
 769 
 
 In the above formulas, w is the weight per cubic foot of the hquid, and h is the height of 
 the tank in feet. All moments are in foot-pounds. 
 
 Tanks not circular in cross-section should be designed to resist moment in a horizontal 
 plane (see chapter on ''Deep Grain Bins," Sect. 18). 
 
 25. Construction Details of Tanks and Standpipes. — The piping for these structures 
 should be so placed that they will not interfere with expansion and contraction. Pipes should 
 be protected from frost. 
 
 It is held^ that thick walls are an advantage over thin walls because of the expense of the 
 taper of the latter; that some seepage is likely to occur; and that in cold climates little ice will 
 be formed, and no protective housing will be necessary if the structure is properly covered. 
 
 )<--■ j't>" >f s'^o"-- - t--o' - -■■■>! 
 
 Fig. 61. — Standpipe at Fulton, N. Y., having disconnected base and flexible seal. 
 
 Care should be taken to thoroughly splice the joints in the steel hooping. Mechanical 
 bond should be provided, preferably by bolted clamps. 
 
 Fig. 6P shows the detail of a tank at Fulton, N. Y., having a sliding joint at the junction 
 of the base and sides. This was patented by Wm. Mueser. It is claimed that there is no 
 lateral restraint of the sides at their base, and that therefore the secondary stresses at that 
 section are eliminated. This can be true only when the joint is perfectly frictionless. Some 
 friction does exist, though not large; and the moment in the sides due to this restraint should 
 be provided for by vertical rods near the outer face. 
 
 Fig. 623 shows the details of a low standpipe at Penetanguishene, Ont. Its capacity is 
 300,000 gal. The wall is thicker than required for strength, to prevent the formation of a 
 heavy ice crust. The connection between wall and bottom was analyzed for restraint by an 
 application of Grashof's theory. 
 
 1 W. J. Douglas: Trans. Am. Soc. C. E., vol. 74, p. 394, 1911. 
 
 2 Eng. Rec, Jan. 10, 1914, p. 43. 
 
 8 Eng. & Cont'g., Jan. 22, 1913, p. 110. 
 49 
 
770 
 
 CONCRETE ENGINEERS' HANDBOOK 
 
 [Sec. 17 
 
Sec. 17-26] 
 
 HYDRAULIC STRUCTURES 
 
 771 
 
 Fig. 63^ shows the details of a small standpipe of 143^^ ft. inside diameter and 40 ft. high, 
 built at Merrimack, N. H. The standpipe has a capacity of 50,000 gal. The concrete was 
 poured in one continuous operation which lasted 393^ hr, 
 
 26. Precautions in Construction. — The construction of standpipes of concrete has not 
 been thus far wholly satisfactory. It is a relatively simple matter to design against water 
 pressure alone, but true monolithic construction, without leaks at junction of base and walls 
 or at work planes, or cracks from secondary stresses, is less easy. 
 
 In addition to the procedures for producing dense concrete before referred to and treated 
 in detail in Sect. 2, care should be taken to so place the concrete as to secure thorough com- 
 pacting in the forms and close contacting with forms and with steel. To this end, forms in 
 relatively shallow lifts are advantageous for puddling and compacting, together with ex- 
 pulsion of entrained air, which can be further facilitated by tapping forms with mauls or 
 mallets, or better yet, with air hammers. 
 
 Although shallow lifts are advantageous, provision must be made in their design for rapid 
 addition of higher lifts, since the best construction, with avoidance of work planes, requires 
 continuous deposition, even though day and night work is entailed, preferably without per- 
 mitting any portion to take even initial set before new concrete is placed upon it. Excess 
 water is a potent source of trouble in constructions of this character, as through its use, laitance 
 in bands and pockets is encouraged. 
 
 Where continuous placement of concrete is impossible for one reason or another, vertical 
 diaphragms of sheet metal may be embedded in the concrete as work joints. Sheet copper, 
 with asphalt or asphalt mastic filler, can be made to give water-tight construction, but the 
 sightliness of the standpipe is usually impaired by the irregular belt of friable and easily weath- 
 ered material at each division. 
 
 ELEVATED TANKS 
 
 Elevated tanks may be classed in groups according to the type of floor employed: (a) 
 suspended, (h) flat, (c) beam-and-slab, (d) dam, or (e) double-dome. 
 
 The suspended bottpm was employed in the Middleboro tank^ (see Fig, 64). It was 
 designed in a manner similar to the same type of steel tank. For the anal3^sis of such a bottom, 
 see Ketchum's "Structural Engineers' Handbook," page 366. 
 
 Only small tanks have been constructed with flat circular floors. Such construction follows 
 that for flat slab floors. Floors of beams and slab are also designed after similar building floors. 
 
 When the floors restrain the sides of the tank, the negative moment at the base of the wall 
 may be provided for in a manner similar to that emploj^ed for standpipes. 
 
 Dome floors are common in Europe, and some have been built in this country. The 
 analysis of the double-dome floor follows. Tanks with only the curved dome have been built. 
 It is reported that trouble has been experienced due to restraint between walls and dome, 
 and to the lateral thrust of the dome. This difficulty may be eliminated by using the same 
 unit stress in the steel in the ring of the dome that is used in the hoops of the sides. Then 
 the dome ring will expand under load an amount equal to that of the sides and there will be 
 no restraint between the two. The analysis of a single dome will be that for a similar portion 
 in the following analysis. 
 
 27. Analysis of Stresses. — The roof A (Fig. 65) is a simple dome, and may be designed 
 as a thin spherical segment. In this case, all stresses are assumed to be tangential to the 
 spherical surface. 
 
 Assuming a uniformly distributed load of w lb. per sq. ft. on a horizontal projection, which 
 will include the weight of the dome since it is relatively flat, the vertical component per foot of 
 
 rim is w If the tangent to the shell at a vertical diameter makes the angle with the horizon- 
 
 1 Eng. News, Dec. 23, 1915. 
 
 2 Eng. & Contg., vol. 44, p. 473, Dec. 22, 1915. 
 
772 
 
 CONCRETE ENGINEERS' HANDBOOK 
 
 [Sec. 17-27 
 
 1, 3-i"Rodl5 wif-h iurnhuckles-^ .w 
 
 '^1 
 
 Lighi weight manhole with perforafed 
 cover and ^4" clear opening 
 
 These rods extend fo El S6I. 
 
 These rods iv El e65;. 
 These rods fa EIS69-1; 
 
 These rodstoEI.E73 
 
 Plan of Radial Refnf. in bottom of Tank 
 
 w-'z^io"--^ m 
 
 Elevatfon of Balcony 
 Developed on" 21-8" Radius 
 
 ^"°15"c. fo c. 108 in circunrtfrence 
 
 f^lZ''c.foc:4^Hc 
 
 Junction of Roof & Wall 
 
 aM\ J -45^^ 
 
 ''■■24 Circular opening 
 
 Sectional Elevation 
 
 Belting Zoi^rs^ 
 
 I yy j"<>o"c. to c £64 in circumfrence . 
 Base of Tov^er 
 
 Fig. 64. — Details of tower tank at Middleboro, Mass. 
 
Sec. 17-27] 
 
 HYDRAULIC STRUCTURES 
 
 773 
 
 tal, the outward pressure Ti per foot of rim is j 
 
 WD 
 
 7 (since Vi Ti = tan <t>'). The thrust 
 
 4 tan 0 
 
 tangent to the dome at its rim will be the resultant of these two forces, or \/T^^ +1^1^. 
 Taking a value of fc in pounds per square inch and letting t — thickness of shell, 
 
 12 tfc 
 
 1 
 
 t = j^Vt.^ + v.^ 
 
 It will be found that for diameters D less than 20 ft. the controlling factors in determining the 
 thickness will be practicability of building the thin shell, and provision for accidental punching 
 shear. Fairly satisfactory proportions for large diameters, remember- 
 ing these limitations, will be obtainable by assuming a low concrete 
 unit stress, say 100 lb. per sq. in. Metal fabric or lath should be 
 used as an added provision against possible concentrations. Vertical 
 shear at the base ring should be computed. 
 
 The thrust Ti should be resisted by a ring of reinforcement at 
 the top of the wall. The tension S in this ring will be 
 
 S = 
 
 TiD 
 
 Tan <(>' may be found from the relation 
 
 tan - = 
 
 FiQ. 65. 
 
 The steel required for the hoop will be 
 
 As 
 
 TiD 
 
 The tensile unit stress in the steel should not exceed about 8000 lb. per sq. in. 
 
 4 ft. Adopted load of 150 lb. per sq. ft. on horizontal projection. 
 
 Illustrative Problem. — D = 40 ft., r' 
 including dead load. Let fs = 5000. 
 
 y _ '^^ _ ^ 
 
 1500 lb. per ft. 
 
 Tan 
 
 - ) whence — = 11"18' 
 5 2 
 
 22''36' 
 
 tan <p' 
 1 
 
 tan 
 1500 _ 
 0.416 ~ 
 
 / = 0.416. 
 3600 lb. per ft. 
 1 
 
 12/c 
 3600 X 40 
 
 As = 
 
 2 
 
 72,000 
 5000 
 
 1 12 X 100 
 72,000 lb. 
 
 (3920) = 3.3, say 3>^ in. 
 
 14.5 sq. in., say 14 1-in. sq. rods. 
 
 A beam may be cast around the top of the tank in which half of the steel may be encased. The remainder may 
 be put in the dome itself near the rim. 
 
 The shell of the tank B (Fig. 65) is designed like the sides of a standpipe. Reinforcement 
 consists of rings, and sufficient vertical steel to support them. 
 
 The conical portion C carries a vertical load of V2 per foot of upper rim, which is equal 
 to the dead weight of the tank shell and the roof. In addition it carries a normal water 
 pressure of wh lb. per sq. ft. at the upper rim, and one of w {h + a) lb. per sq. ft. at its 
 
774 
 
 CONCRETE ENGINEERS' HANDBOOK 
 
 [Sec. 17-28 
 
 lower rim. The sides of the cone are assumed to slope at 45 deg. The shear per lineal foot 
 of the bottom rim is equal to the weight of the structure above this level plus the weight 
 of water above the conical portion, divided by the perimeter of the rim. Since the slope of 
 the cone is 45 deg., the thrust has the same value per lineal foot as F3. Likewise the thrust 
 at the upper rim, T3, is equal to the shear V2 at that level, or the weight of structure above it. 
 Two rings may be used, one to resist Tz, and one to resist T^, with reinforcement in the slab to 
 span between these rings; or, rings of steel rods may be distributed somewhat, through the coni- 
 cal side, to aid^the resistance of the sloping rods spanning between rings. This will be modified 
 with respect to the action of T4 when the dome thrust T4 is determined. Vertical shear at 
 the base should be computed and provided for. 
 
 The dome of the floor has a vertical shear at its perimeter equal to its weight plus the weight 
 of water directly above it. This shear is the vertical component of the thrust at the base ring 
 acting tangent to the same surface, or at the angle </>. Hence, 
 
 T5 = F4 cot (t> 
 
 This is the ring thrust acting at the base of the dome. 
 
 Fig. 66. Fig. 67. 
 
 Now the ring stress actually to be resisted is that produced by the difference between 
 Ti and T5. If they are made equal, then the reaction of the tank is vertical only at this point, 
 and no reinforcement need be placed for ring tension. 
 
 Fig. 66 gives the proportions of this type of floor when the thrust Ti balances the thrust T^. 
 
 28. Supporting Tower. — The tower carrying an elevated tank may be either a cylindrical 
 shell or a group of columns. For stresses due to vertical load and wind see the design of deep 
 grain bins or silos, Art. 4, Sect. 18. 
 
 A tower formed of a group of columns is shown in Fig. 67, acted upon by a total wind pres- 
 sure of W lb. at a height of a in. Let the distance from the axis of the group to the column far- 
 thest out be X in. Then the unit stress at this distarce x due to W will be 
 
 ^ Wax 
 ' Ic + cA 
 
 assuming little or no tension to exist on the section under the action of W and G. The number 
 of columns is denoted by c. This unit stress must be multiplied by the area A of the column 
 to obtain its load. Other columns will carry a similar stress when the wind is in another 
 direction. 
 
 Investigation of compressive stress due to wind should be made on the leeward side of the 
 tank at its base and at other sections when shear is vital. 
 
Sec. 17-29] 
 
 HYDRAULIC STRUCTURES 
 
 775 
 
 Bracing in reinforced-concrete towers consists usually of horizontal struts only, designed 
 to resist moment. The analysis of stresses in a bent of the tower may be made by the use of 
 the method of slope-deflections (see Sect. 10). 
 
 Fig. 68^ shows an elevated tank with double-dome bottom, built at Kitchener, Ont. It 
 forms an excellent example of this type of structure. 
 
 Fia. 68. — Tank for water-works at Kitchener, Ont. 
 CULVERTS 
 
 29. General Considerations. — The term culvert is usually applied to structures built to 
 carry surface water or small streams through highway or railroad embankments. When the 
 area of waterway required is comparatively small, a pipe culvert is usually the most economical. 
 For the larger openings either the box or arch form should be employed depending upon the avail- 
 
 1 Eng. News, vol. 69, p. 309, Feb. 13, 1913. 
 
776 
 
 CONCRETE ENGINEERS' HANDBOOK 
 
 [Sec. 17-30 
 
 able headroom, the depth of fill, the condition of the foundation, and whether or not an artistic 
 arch design is especially desirable. 
 
 The ordinary type of arch culvert and the box culvert without a load-supporting floor 
 (called open-box culvert) are in reality small bridges, and it is sometimes a question of how large 
 such structures may become before they should be considered strictly in the bridge class. No 
 arbitrary division is adhered to in the following articles except that a culvert is considered a 
 structure which can be completely and economically standardized, based on a given area of 
 waterway and height of embankment. 
 
 30. Factors in Culvert Design. 
 
 30a. Culvert Efficiency. — A culvert to be efficient in the amount of water it 
 can discharge should have its headwalls or wings arranged so as to facilitate the flow, and its 
 bed should be considerably inclined for those cases where the channel below the culvert will 
 permit the water to flow away freely. If any well-defined stream bed exists, the bed of the 
 culvert should have the same inclination as that of the stream, as otherwise either the outlet 
 or inlet end will clog depending upon whether the slope of the culvert is greater or less respec- 
 tively than the slope of the stream bed. 
 
 Any projections in the culvert bed should be avoided as they will retard the water and 
 diminish efficiency. It is also important that culverts be placed across roadways in the di- 
 rection of the stream flow since, if this is not done, clogging and subsequently washouts will be 
 likely to occur. 
 
 A culvert will discharge twice as much under a head of 4 ft. as under a head of 1 ft., but 
 water should not be allowed to dam up in this manner unless the culvert is well constructed 
 through a water-tight embankment. 
 
 306. Waterway Required. — Assuming an efficient culvert design, the area of 
 waterway required depends principally upon the maximum rate of rainfall, the area and shape 
 of the watershed, the kind and condition of the soil throughout this watershed area, and the 
 character and inclination of both drainage surface and stream bed. A number of empirical 
 formulas have been proposed by which to calculate the required culvert opening, but obviously 
 a problem of this kind does not admit of an exact mathematical solution and the desired size 
 of culvert should be determined by direct observation whenever that is possible. 
 
 In a new country an empirical formula is often the only method by which the required 
 area of waterway can be determined. Talbot's formula is the one most generally employed 
 and is as follows : 
 
 A = C^V' 
 
 where A = area of waterway in square feet, a = drainage area in acres, and C is a coefficient 
 which varies from 1 to ^■i in the following manner : 
 
 For steep and rocky ground, C varies from %tol. For rolling agricultural country subject to floods at times of 
 melting of snow, and with the length of valley three or four items its width, C is about ys; and if the stream is longer 
 in proportion to the area, decrease C. In districts not affected by accumulated snow, and where the length of the 
 valley is several times the width, H, M. or even less, may be used. C should be increased for steep side slopes, 
 especially if the upper part of the valley has a much greater fall than the channel at the culvert, i 
 
 The proper area of waterway can best be determined by knowing the dimensions of 
 existing openings on the same stream and by careful observation of the stream and the amount 
 of water which it carries at flood times. This amount of water can be determined by measuring 
 a cross-section of the stream at some narrow place near the culvert site. 
 
 30c. Length of Culverts. — The length of a culvert should depend upon the width 
 of roadway and the depth of fill on top of the culvert. The slope of an earth fill can generally be 
 taken as 1^ :1 — that is, for every 1 ft. in height, the horizontal distance is 1}^ ft. 
 
 In highway construction the roadway should never be narrowed at a culvert since such a 
 practice is dangerous and the construction unsightly. 
 
 1 Selected papers of the Civil Engineers' Club of the University of Illinois, No. 2, p. 14. 
 
Sec. 17-30fil 
 
 HYDRAULIC STRUCTURES 
 
 777 
 
 SOd. Design of Ends. — The arrangement of the headwalls or wings may be 
 substantially the same for all arch and box culverts. The arrangement should be such that 
 the embankment is protected and the flow of water facihtated. Wing walls may be built 
 parallel with or at right angles to the axis of the culvert, or they may be so placed as to make 
 an angle (usually 30 or 45 deg.) with this axis. 
 
 For the shorter spans (including spans for pipe culverts), wings parallel with the roadway 
 are generally used for low fills and, in highway construction at least, these headwalls are carried 
 up above the grade line to provide a low guard rail. This type of end, however, is not eco- 
 nomical for the larger spans since the straight wings under such conditions need to be made of 
 considerable length and height to retain the fill efficiently. A low guard rail may be formed with 
 flared wings by raising and coping both head and wingwalls. The top of the wings, of course, 
 should have a slope consistent with the slope of the earth fill. 
 
 Flared wings, especially at the upstream end, are the best for hydraulic reasons and, 
 when used, the culvert is less likely to become choked than when either of the other two forms 
 of wingwalls are employed. Straight wings — namely, wings parallel to the axis of the culvert — 
 are of advantage in railroad construction when an extension of the culvert is likely to be made 
 in the near future to accommodate another track. 
 
 It is common practice to design wing and headwalls of sufficient length to keep the culvert 
 opening clear when the earth is assumed to fall around the ends on a 13^^ : 1 slope. In some 
 cases a steeper slope could be assumed, but some soils take even flatter slopes than the standard. 
 
 S^- 8" -> 
 
 Fig. 69. — Reinforced-concrete pipe. 
 
 Box and arch culverts are built both with and without a floor, but in almost every case 
 the smooth waterway that can be obtained by using a concrete floor will greatly increase the 
 capacity of the culvert. A floor, if properly constructed, will also prevent any danger from 
 erosion of the stream bed and undermining of the foundation. The floor at the ends of the 
 culvert should be provided with an apron or baffle wall at its outer edge, and this wall should 
 in all cases be carried as low as the bottom of the footings. If especially desirable, the floor 
 should extend out to the end of the wingwalls. 
 
 31. Pipe Culverts. — One or more lines of pipe with suitable headwalls to protect the em- 
 bankment is the simplest form of culvert. The pipe may be of burned clay, cast iron, plain 
 concrete, or reinforced concrete; but, oji account of frequent breakages, there seems to be a 
 tendency at the present time to discontinue the use of vitrified and cast-iron pipe and also 
 pipes of plain concrete. All kinds of pipe culverts have the same type of concrete headwalls, 
 consequently the following articles treat only of pipe culverts of reinforced concrete. 
 
 Since it is desirable to make as few openings as possible through an embankment, water 
 is usually conducted along the side of the roadway until at least a 15-in. diameter of pipe is 
 required. In the following articles a pipe of at least this size is assumed. 
 
 Reinforced culvert pipes are usually made in from 4- to 8-ft. lengths, and with bell and 
 
778 
 
 CONCRETE ENGINEERS' HANDBOOK 
 
 [Sec. 17-31a 
 
 spigot joints. The largest diameter of pipe yet made is 72 in. The pipes usually have a 
 hoop reinforcement which is near the interior surface at the top and bottom of the pipe, and 
 near the exterior surface at the sides (Fig. 69), Pipe with a double line of reinforcing is also 
 used, as shown in Fig. 70. Longitudinal reinforcement is provided in pipes of the longer 
 lengths due to the likelihood of beam action if settlement takes place. 
 
 31a. Pressure in Trenches. — The most elaborate and thorough investigation 
 of the subject of pressure on pipes in ditches was made a few years ago at the Iowa State College 
 of Agriculture and Mechanic Arts, the results of which were published in Bulletin No. Sl^ 
 
 f°B'ars 
 /z'c.toc. EncI yiew 
 
 for diameters be/ow 
 » 30"-36" 
 
 '37-42" , X=JB" 
 
 Double //ne 
 
 of Peinforcement. 
 
 Section 
 
 ^re MesTr 
 
 or 
 
 : Expanded 
 Metal 
 equfya/enffo 
 ^.S.&lY.Co's 
 4-Mest7 shoyyn 
 '//7 t^b/e 
 
 £4", 
 42 ".F^ 
 
 Table of Dimensions 
 for Circular Pipe 
 
 Diameter 
 
 TTiickness of 
 
 Required Steel 
 
 
 Shell = T 
 
 /Area -for each Jine 
 
 IS" 
 /d" 
 84" 
 
 30" 
 36" 
 42" 
 
 2.^5" 
 2.50" 
 3.00" 
 3.50" 
 4.00" 
 4.50" 
 
 A5.& yV-Gy. Sq. in. 7 
 
 Sfy/e6*.058perJB 
 " 5-.077 " " 
 0 4'.I02 " " 
 42 '.f 5/ " " 
 " 23:170 " " 
 « 32-.225 " " 
 
 Fig. 70. — Standard dimensions for concrete pipe culverts with concrete head-walls, Iowa Highway Commission. 
 
 of the Engineering Experiment Station. The investigation was made with special reference 
 to drain tile and sewer pipe, but the results apply equally well to culverts laid in trenches. 
 The following tables and other data were taken from the bulletin above mentioned. The 
 correctness and reliability of the theory which was developed by reason of this investigation 
 were demonstrated with remarkable closeness by actual weighings of loads on pipes, the pipes 
 ranging from 12 to 36 in. in internal diameter placed in ditches from 0 to 19 ft. in depth. 
 
 1 Written by Prop. Anson Marston and Mr. A. O. Anderson. 
 
Sec. 17-31a] 
 
 HYDRAULIC STRUCTURES 
 
 779 
 
 Table i.— Approximate Maximum Loads in Pouni>6 Per Linear Foot, on Pipes in 
 Ditches from Common Ditch-Fillinq Materials 
 
 H = height of 
 fill above top 
 of pipe 
 
 B = breadth of ditch a little below top of pipe 
 
 1 ft. 
 
 1 2 ft. 
 
 3 ft. 
 
 4 ft. 
 
 5 ft. 
 
 1 ft. 
 
 2 ft. 
 
 3 ft. 
 
 4 ft. 
 
 5 ft. 
 
 Partly compacted damp top soil 90 lb. per ou. ft. 
 
 Saturated top soil 110 lb. per cu. ft. 
 
 2 ft. 
 
 130 
 
 310 
 
 490 
 
 670 
 
 830 
 
 170 
 
 380 
 
 600 
 
 820 
 
 1,020 
 
 4 ft. 
 
 200 
 
 530 
 
 880 
 
 1,230 
 
 1,580 
 
 260 
 
 670 
 
 1,090 
 
 1,510 
 
 1,950 
 
 6 ft. 
 
 230 
 
 690 
 
 1,190 
 
 1,700 
 
 2,230 
 
 310 
 
 870 
 
 1,500 
 
 2,140 
 
 2,780 
 
 8 ft. 
 
 250 
 
 800 
 
 1,430 
 
 2,120 
 
 2,790 
 
 340 
 
 1,030 
 
 1,830 
 
 2,660 
 
 3,510 
 
 10 ft. 
 
 260 
 
 880 
 
 1,640 
 
 2,450 
 
 3,290 
 
 350 
 
 1,150 
 
 2,100 
 
 3,120 
 
 4,150 
 
 
 Dry sand, 100 lb. per cu. ft. 
 
 
 Saturated sand, 120 lb. per cu. 
 
 ft. 
 
 2 ft. 
 
 150 
 
 340 
 
 550 
 
 740 
 
 930 
 
 180 
 
 410 
 
 650 
 
 890 
 
 1,110 
 
 4 ft. 
 
 220 
 
 590 
 
 970 
 
 1,360 
 
 1,750 
 
 270 
 
 710 
 
 1,170 
 
 1,640 
 
 2,100 
 
 6 ft. 
 
 260 
 
 760 
 
 1,320 
 
 1,890 
 
 2,480 
 
 310 
 
 910 
 
 1,590 
 
 2,270 
 
 2,970 
 
 8 ft. 
 
 280 
 
 890 
 
 1,590 
 
 2,350 
 
 3,100 
 
 340 
 
 1,070 
 
 1,910 
 
 2,820 
 
 3,720 
 
 10 ft. 
 
 290 
 
 980 
 
 1,820 
 
 2,720 
 
 3,650 
 
 350 
 
 1,180 
 
 2,180 
 
 3,260 
 
 4,380 
 
 12 ft. 
 
 300 
 
 1,040 
 
 2,000 
 
 3,050 
 
 4,150 
 
 360 
 
 1,250 
 
 2,400 
 
 3,650 
 
 4,980 
 
 14 ft. 
 
 300 
 
 1,090 
 
 2,140 
 
 3,320 
 
 4,580 
 
 360 
 
 1,310 
 
 2,570 
 
 3,990 
 
 5,490 
 
 16 ft. 
 
 300 
 
 1,130 
 
 2,260 
 
 3,550 
 
 4,950 
 
 360 
 
 1,350 
 
 2,710 
 
 4,260 
 
 5,940 
 
 18 ft. 
 
 300 
 
 1,150 
 
 2,350 
 
 3,740 
 
 5,280 
 
 360 
 
 1,380 
 
 2,820 
 
 4,490 
 
 6,330 
 
 20 ft. 
 
 300 
 
 1,170 
 
 2,420 
 
 3,920 
 
 5,550 
 
 360 
 
 1,400 
 
 2,910 
 
 4,700 
 
 6,660 
 
 22 ft. 
 
 300 
 
 1,180 
 
 2,480 
 
 4,060 
 
 5,800 
 
 360 
 
 1,420 
 
 2,980 
 
 4,880 
 
 6,960 
 
 24 ft. 
 
 300 
 
 1,190 
 
 2,540 
 
 4,180 
 
 6,030 
 
 360 
 
 1,430 
 
 3,050 
 
 5,010 
 
 7,230 
 
 26 ft. 
 
 300 
 
 1,200 
 
 2,570 
 
 4,290 
 
 6,210 
 
 360 
 
 1,440 
 
 3,090 
 
 5,150 
 
 7,460 
 
 28 ft. 
 
 300 
 
 1,200 
 
 2,600 
 
 4,370 
 
 6,390 
 
 360 
 
 1,440 
 
 3,120 
 
 5,240 
 
 7,670 
 
 30 ft. 
 
 300 
 
 1,200 
 
 2,630 
 
 4,450 
 
 6,530 
 
 360 
 
 1,440 
 
 3,150 
 
 5,340 
 
 7,830 
 
 Infinity 
 
 300 
 
 1,210 
 
 2,730 
 
 4,850 
 
 7,580 
 
 360 
 
 1,450 
 
 3,270 
 
 5,820 
 
 9,090 
 
 Partly compacted damp yellow clay, 100 lb. per cu. ft. 
 
 Saturated yellow clay, 130 lb. per cu. ft. 
 
 2 ft. 
 
 160 
 
 350 
 
 550 
 
 750 
 
 930 
 
 210 
 
 470 
 
 730 
 
 1,000 
 
 1,240 
 
 4 ft. 
 
 250 
 
 620 
 
 1,010 
 
 1,400 
 
 1,800 
 
 340 
 
 840 
 
 1,330 
 
 1,870 
 
 2,370 
 
 6 ft. 
 
 300 
 
 830 
 
 1,400 
 
 1,990 
 
 2,580 
 
 430 
 
 1,140 
 
 1,900 
 
 2,630 
 
 3,410 
 
 8 ft. 
 
 330 
 
 990 
 
 1,720 
 
 2,500 
 
 3,250 
 
 490 
 
 1,380 
 
 2,360 
 
 3,360 
 
 4,400 
 
 10 ft. 
 
 350 
 
 1,110 
 
 2,000 
 
 2,920 
 
 3,880 
 
 520 
 
 1,570 
 
 2,760 
 
 3,980 
 
 5,270 
 
 12 ft. 
 
 360 
 
 1,200 
 
 2,220 
 
 3,320 
 
 4,450 
 
 540 
 
 1,730 
 
 3,100 
 
 4,560 
 
 6,050 
 
 14 ft. 
 
 370 
 
 1,280 
 
 2,410 
 
 3,650 
 
 4,950 
 
 560 
 
 1,850 
 
 3,410 
 
 5,050 
 
 6,760 
 
 16 ft. 
 
 370 
 
 1,330 
 
 2,570 
 
 3,950 
 
 5,400 
 
 570 
 
 1,940 
 
 3,660 
 
 5,510 
 
 7,440 
 
 18 ft. 
 
 380 
 
 1,390 
 
 2,710 
 
 4,210 
 
 5,810 
 
 570 
 
 2,020 
 
 3,880 
 
 5,930 
 
 8,060 
 
 20 ft. 
 
 380 
 
 1,410 
 
 2,830 
 
 4,450 
 
 6,180 
 
 580 
 
 2,090 
 
 4,070 
 
 6,280 
 
 8,610 
 
 22 ft. 
 
 380 
 
 1,430 
 
 2,920 
 
 4,640 
 
 6,500 
 
 580 
 
 2,140 
 
 4,240 
 
 6,610 
 
 9,130 
 
 24 ft. 
 
 380 
 
 1,450 
 
 3,000 
 
 4,820 
 
 6,800 
 
 580 
 
 2,180 
 
 4,380 
 
 6,910 
 
 9,590 
 
 26 ft. 
 
 380 
 
 1,470 
 
 3,060 
 
 4,980 
 
 7,080 
 
 580 
 
 2,210 
 
 4,500 
 
 7,160 
 
 10,010 
 
 28 ft. 
 
 380 
 
 1,480 
 
 3,120 
 
 5,100 
 
 7,310 
 
 580 
 
 2,240 
 
 4,610 
 
 7,380 
 
 10,430 
 
 30 ft. 
 
 380 
 
 1,490 
 
 3,170 
 
 5,230 
 
 7,530 
 
 580 
 
 2,260 
 
 4,700 
 
 7,590 
 
 10,780 
 
 Infinity 
 
 380 
 
 1,520 
 
 3,410 
 
 6,060 
 
 9.480 
 
 580 
 
 2,340 
 
 5,270 
 
 9,360 
 
 14,620 
 
 The width of the ditch above the pipe makes practically no difference in the load on the pipe, which is just as 
 great for a vertical ditch as for one several times as wide at the top but of the same width a little below the top'of 
 the pipe. 
 
 In ditches of proportions customary in actual work, the diameter of the pipe used in any particular ditch of a 
 fixed given width makes practically no difference in the load on the pipe. A 12-in. pipe will have to carry the same 
 load as an 18-in. pipe, if both are placed in ditches 2 ft. wide under other similar conditions. 
 
780 
 
 CONCRETE ENGINEERS' HANDBOOK 
 
 [Sec. 17-31a 
 
 In case a wide ditch is necessary for constructive reasons, the load on the pipe can be diminished greatly, in 
 firm soil, by stopping the wide ditch a few inches above the top of the pipe and digging in the bottom the narrowest 
 ditch practicable to receive the pipe, making bell holes at the side for the sewer pipe, if necessary. 
 
 Grades or fills built over the surfaces of completed ditches, and piles of sand, gravel, and other materials having 
 internal friction, operate to increase the loads on pipes in ditches to the same extent as an equal added height of 
 ditch filling, for a breadth of ditch equal to that at a little below the top of the pipe. 
 
 A superload is any load applied to the upper surface of the ditch filling, except loads from fills or heaps of 
 granular materials. A long superload is one extending a considerable length along a ditch, as compared with its 
 depth and breadth, and may be caused by piles of paving brick, lumber, etc., over the ditch. Long superloads 
 on completed ditches cause increases in the loads on pipes in ditches by percentages of the superload which decrease 
 as depth increases, and safe values for which can be computed by Table 2. Table 2 has been closely checked by 
 actual weighings of the increase in loads on pipes in ditches due to super loads. 
 
 Table 2. — Approximate Safe Values of C to Use in Formula L 
 
 CLi 
 
 L = loads per unit of length, on pipes in ditches, due to Li. 
 Li = long superloads on ditches, per unit of length. 
 
 H 
 B 
 
 Sand and damp top 
 soil 
 
 Saturated top soil 
 
 Damp yellow clay 
 
 Saturated yellow 
 clay 
 
 0.0 
 
 1.00 
 
 1.00 
 
 1.00 
 
 1.00 
 
 0.5 
 
 0.85 
 
 0.86 
 
 0.88 
 
 0.89 
 
 1.0 
 
 0.72 
 
 t).75 
 
 0.77 
 
 0.80 
 
 1.5 
 
 0.61 
 
 0.64 
 
 0.67 
 
 0.72 
 
 2.0 
 
 0.52 
 
 0.55 
 
 0.59 
 
 0.64 
 
 2.5 
 
 0.44 
 
 0.48 
 
 0.52 
 
 0.57 
 
 3.0 
 
 0.37 
 
 0.41 
 
 0.45 
 
 0.51 
 
 4.0 
 
 0.27 
 
 0.31 
 
 0.35 
 
 0.41 
 
 5.0 
 
 0.19 
 
 0.23 
 
 0.27 
 
 0.33 
 
 6.0 
 
 0.14 
 
 0.17 
 
 0.20 
 
 0.26 
 
 8.0 
 
 0.07 
 
 0.09 
 
 0.12 
 
 0.17 
 
 10.0 
 
 0.04 
 
 0.05 
 
 0.07 
 
 0.11 
 
 A short superload is one extending a short distance along a ditch as compared with the breadth and depth, 
 and may come from the wheels of wagons, traction engines, steam road rollers, etc. Short superloads, on com- 
 pleted ditches, cause increases in the loads on pipes in ditches by percentages of the superload which decrease 
 as the depth increases, and safe values which can be estimated, but not very reliably, by Table 3. Table 3 has 
 not been checked by actual weighings of increase of loads on pipes in ditches. 
 
 Table 3. — Approximate Safe Values for C to Use in Formula L = CLi 
 
 L — loads per unit of length, on pipes in ditches directly under Li, due to Li. L\ = short superloads on 
 ditches, per unit of length, of length A along ditch. 
 
 
 Sand and damp 
 top soil 
 
 Saturated top soil 
 
 Damp yellow clay 
 
 Saturated yellow 
 clay 
 
 H 
 
 
 
 Ka 
 
 = K 
 
 Ka = 
 
 
 \Ka- 
 
 = K 
 
 Ka = 
 
 
 
 = K 
 
 Ka = 
 
 
 Ka- 
 
 = K 
 
 B 
 
 A = 
 
 A 
 
 
 A 
 
 
 A 
 
 
 A 
 
 
 A 
 
 
 A 
 
 
 A 
 
 
 
 B 
 
 B 
 10 
 
 B 
 
 B 
 10 
 
 B 
 
 B 
 10 
 
 B 
 
 B 
 10 
 
 B 
 
 B 
 10 
 
 B 
 
 B 
 10 
 
 B 
 
 B 
 10 
 
 B 
 
 B 
 10 
 
 0.0 
 
 1.00 
 
 1.00 
 
 1.00 
 
 1.00 
 
 1.00 
 
 1.00 
 
 1.00 
 
 1.00 
 
 1.00 
 
 1.00 
 
 1.00 
 
 1.00 
 
 1.00 
 
 1.00 
 
 1.00 
 
 1.10 
 
 0.5 
 
 0.77 
 
 0.32 
 
 0.70 
 
 0.12 
 
 0.78 
 
 0.33 
 
 0.71 
 
 0.13 
 
 0.79 
 
 0.34 
 
 0.72 
 
 0.13 
 
 0.81 
 
 0.34 
 
 0.74 
 
 0.13 
 
 1.0 
 
 0.59 
 
 0.11 
 
 0.49 
 
 0.02 
 
 0.61 
 
 0.11 
 
 0.51 
 
 0.02 
 
 0.63 
 
 0.11 
 
 0.52 
 
 0.02 
 
 0.66 
 
 0.12 
 
 0.55 
 
 0.02 
 
 1.5 
 
 0.46 
 
 0.03 
 
 0.34 
 
 
 0.48 
 
 0.04 
 
 0.36 
 
 
 0.51 
 
 0.04 
 
 0.38 
 
 
 0.54 
 
 0.04 
 
 0.40 
 
 
 2.0 
 
 0.35 
 
 0.01 
 
 0.24 
 
 
 0.38 
 
 0.01 
 
 0.26 
 
 
 0.40 
 
 0.01 
 
 0.27 
 
 
 0.44 
 
 0.01 
 
 0.30 
 
 
 2.5 
 
 0.27 
 
 
 0. 17 
 
 
 0.29 
 
 
 0.18 
 
 
 0.32 
 
 
 0.20 
 
 
 0.35 
 
 
 0.22 
 
 
 3.0 
 
 0.21 
 
 
 0. 12 
 
 
 0.23 
 
 
 0.13 
 
 
 0.25 
 
 
 0. 14 
 
 
 0.29 
 
 
 0.16 
 
 
 4.0 
 
 0.12 
 
 
 0.06 
 
 
 0.14 
 
 
 0.07 
 
 
 0.16 
 
 
 0.08 
 
 
 0.19 
 
 
 0.09 
 
 
 5.0 
 
 0.07 
 
 
 0.03 
 
 
 0.09 
 
 
 0.03 
 
 
 0. 10 
 
 
 0.04 
 
 
 0.13 
 
 
 0.05 
 
 
 6.0 
 
 0.04 
 
 
 0.01 
 
 
 0.05 
 
 
 0.02 
 
 
 0.06 
 
 
 0.02 
 
 
 0.08 
 
 
 0.03 
 
 
 8.0 
 
 0.02 
 
 
 
 0.02 
 
 
 
 0.03 
 
 
 0.01 
 
 
 0.04 
 
 
 0.01 
 
 
 10.0 
 
 0,01 
 
 
 
 
 0.01 
 
 
 
 
 0.01 
 
 
 
 0.02 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
Sec. 17-316] 
 
 HYDRAULIC STRUCTURES 
 
 781 
 
 H = height of fill in ditch, above top of pipe. 
 
 B — breadth of ditch, a little below top of pipe. 
 
 K = ratio of lateral pressure to vertical in the ditch filling. 
 
 Ka = ratio of longitudinal pressure to vertical in the ditch filling. 
 
 Values of C for Ka = 0 are given in Table 2. 
 
 The formula L = CLi holds true only directly under Li. Beyond Li in either direction, the intensity of load 
 on the pipe diminishes rapidly. 
 
 Calculations made from Table 3 are not very reliable since we are usually uncertain as to the proper value to 
 take for Ka, and there is great need of a series of tests of the actual loads on pipes caused by short superloads, but 
 such tests would be very difficult to make and test results are not available. 
 
 In the meantime, Table 3 will be of some value to engineers of good judgment in assisting them to make rea- 
 sonable safe allowances for the probable effect on the loads on pipes in ditches from heavy concentrated loads on 
 wagon wheels, traction engines, and road rollers. 
 
 316. Strength of Pipe. — The theoretical analysis of stresses in culvert pipe is 
 that of thin elastic rings and is similar to the general method employed for arches. The dif- 
 ference in the intensity of the load at the crown and at the extremities of the horizontal diameter, 
 due to the difference in the depths of the earth, is considered negligible, and the pressure and its 
 distribution on the lower half of the ring is assumed to be the same as that on the upper half. 
 
 Theory gives the following values of the bending moment at the top and bottom sections of 
 a pipe: 
 
 I. For single concentrated load (top and bottom), M = 0.159Pd. 
 
 II. For total uniformly distributed load over entire horizontal projection (top and bottom), 
 M = 0m25Wd. 
 
 III. For a uniformly distributed load over the top fourth of the circumference and with 
 the pipe supported on its bottom quarter circumference, M = 0.0845TFd. Where 
 
 d = diameter of pipe. 
 
 P = concentrated load at top. 
 
 W = total uniformly distributed load above horizontal diameter. 
 
 M = bending moment in pipe in a unit length. 
 The bending moments at the ends of the horizontal diameters under the above conditions of 
 loading are: 
 
 I. M = -OmiPd 
 II. M = -0m25Wd 
 III. M = -0.077Wd 
 
 The above moments will be reduced for any lateral restraint or lateral pressure. In fact for 
 equal uniform horizontal and uniform vertical forces (which may be considered equivalent to a 
 uniform radial pressure) the moments due to the lateral forces have equal but opposite signs 
 to those given for Case II above, and it can be proved that the total moments at all points are 
 zero. It is not good practice, however, to rely on any lateral restraint or pressure in the analy- 
 sis of the strength of pipes. Mathematical analysis shows that the weight of pipes causes only 
 five-eighths as much bending moment at the lowest point of the pipe as does an equal weight 
 of earth. 
 
 Since the exact load and the nature of its distribution over pipe surface is usually uncertain, 
 the probable range of bending moments under actual conditions of construction is all that 
 laboratory tests can be expected to furnish. In a series of tests at the University of Illinois,^ 
 reinforced-concrete rings and circular pipe (48-in. internal diameter and 4 in. thick) were tested 
 for concentrated loads at the top and bottom of the vertical diameter (Case I above), and for 
 uniformly distributed loads above and below the entire horizontal diameter (Case II above). 
 This latter loading was obtained by placing the pipe in a tight box so as to be entirely encircled 
 with sand and then applying a load to the top surface of the sand. The reinforcement for 
 most of the rings tested consisted of 3^^-in. corrugated rods placed near the intrados at top and 
 bottom and near the extrados at the sides. The rings were only 24 in. long, but the pipe 
 
 1 See Bulletin 22 of the Engineering Experiment Station of the University of Illinois, written by Prof. A. N 
 Talbot. 
 
782 
 
 CONCRETE ENGINEERS' HANDBOOK 
 
 [Sec. 17-31C 
 
 sections were from 102 to 104 in. in length with the usual bell and pigot points. To allow the 
 circumferential reinforcement in the pipes to be circular in shape, the pipe cross-sections (Fi-g. 69. 
 were made with the vertical diameter 4 in. longer than the horizontal diameter, thus bringing 
 the reinforcement at the points of tension in the loaded pipe. Using the yield point of the 
 steel in the common formula for the bending resistance of a reinforced section, there was found 
 
 Fig. 71. — Standard design for 24-in. circular culvert, Iowa Highway Commission. 
 
 a close agreement between the theoretical and experimental values for the strength of pipe 
 under these two methods of loading. 
 
 Marston and Anderson, in the investigation referred to in the preceding article, came to 
 the conclusion that "the typical field bedding and loading of pipes in ditches are such that their 
 effect on the pipe can be reproduced with practical exactness in laboratory tests by bedding 
 
 the pipes in sand during the tests for 90 deg. of the circumference at the bottom and also for 
 90 deg. at the top." This method of loading is Case III above. In the Iowa investigation 
 the weight of the pipe as well as that of the backfilling was taken into consideration. 
 
 31c. Circular Culverts Cast in Place. — A cast-in-place culvert with circular 
 opening is shown in Fig. 71. Fig 72 shows an adjustable, collapsible wood form which can 
 be very economically used for culverts of this type. 
 
Sec. 17-32] 
 
 HYDRAULIC STRUCTURES 
 
 783 
 
 The method of constructing and using this form is described as follows in a booklet entitled 
 "Small Concrete Bridges and Culverts," published by the Universal Portland Cement Co.: 
 
 This form is constructed of two by four's beveled and strung on wires, as shown in Fig. 72. The number of 
 staves to be used, varying with the size of the culvert, are placed side by side with a wire drawn through each end 
 of the stave as shown. The form is then rolled around a circular head size of the proposed culvert and wire bands are 
 tied tightly around it on the outside. Wedges are then driven as shown in Fig. 72 to hold the staves firmly in posi- 
 tion. After the culvert has been built the wedges are removed and the circular heads knocked in; the staves will 
 
 then collapse and are easily removed. This form can be used over and over again and Mr. Gearhart states that its 
 cost should not exceed $15 or $20. 
 
 32. Box Culverts. — The box type of culvert is especially adapted to locations where the 
 headroom is limited and, when planned for such locations, has a great advantage over the arch. 
 A culvert of this type, for example, can be built with less excavation and less disturbance to 
 the embankment and will give a greater distribution of load upon the foundation than the 
 
 ^very third rod 
 
 ^Corr Bars, /p'c toc.fop and 
 bo*tvm slabs 
 
 Longitudinal Section 
 Fig. 74. — Standard culvert of 6-ft. span, Hampden R. R. 
 
 ordinary form of arch culvert. Also, the formwork for this type is much simpler and the cost 
 of construction correspondingly lower, except perhaps in some cases where the number and 
 size of culverts to be constructed warrants the use of commercial arch forms of collapsible 
 steel. Box culverts are not always employed only under shallow fills, as is evident from Fig. 
 76. 
 
784 
 
 CONCRETE ENGINEERS' HANDBOOK 
 
 [Sec. 17-32 
 
 Botrom Level 
 
 pi \'''- 3'-o' or more depending 
 ^ — |j ^ on local conditions 
 
 I'-o" Longitudinal Section 
 
 4 <.^.— ^> <.... e'-o"—- • 
 
 K - 14-10" -->! 
 
 ,l>'Pods,l4' (Z8' cuts t) "V 
 
 X Cross Section 
 
 i Waterway 69^ sq.ft.) ^ /^'- T"!' 
 
 End Elevation 
 
 Fig. 75. — Double 6 X 6-ft. box culvert, A. T. & S. Fe. Ry. system. 
 
Sec. 17-32a] 
 
 HYDRAULIC STRUCTURES 
 
 785 
 
 32a. Forms of Box Culverts. — There are two distinct forms of box culverts 
 which may be distinguished by the terms open-box and closed-box. In the open-box (Fig. 73) 
 the side walls have dependent footings, while in the closed-box (called simply box) the load 
 is supported by the floor (Fig. 74). Double-box culverts (Fig. 75 ) are generally used where the 
 span is equal to, or greater than, twice the height, as the use of a single box beyond these pro- 
 portions greatly increases the cost. 
 
 326. Loading. — For small box culverts built in trenches the load on the top 
 slab may be approximately estimated by means of the tables in Art. 31a. For large box cul- 
 verts and for all culverts not built by trench construction, no allowance should be made for 
 the arching action of the material, which means that such culverts should be proportioned to 
 carry the entire weight of the fill above the cover slab. The lateral pressure of the earth 
 (including live-load surcharge) is usually assumed as that due to a fluid weighing about 30 
 lb. per cu. ft. — that is, a weight about one-fourth the weight of earth. It is obvious that the 
 pressure due to live load does not spread out through the filling at the ordinary angle of repose 
 of the material, but has a side slope, or line of zero stress, much more nearly vertical. It is 
 frequently assumed that the live load is carried down at a slope of horizontal:! vertical. 
 In railroad embankments this slope is taken from the ends of the ties. 
 
 An allowance is usually made for impact of the live load in the design of railroad culverts. 
 Some designers allow 50% for impact on all banks up to 40 ft. high. A more conservative 
 plan often followed is to allow 100% for fills of less than 2 ft., 75% for fills between 2 ft. and 
 4 ft., and 50% for all fills over 4 ft. 
 
 32c. Design of Cross-section. — The top slab of a box culvert is partially fixed, 
 but it is the general practice to design this slab as simply supported and to reinforce at the cor- 
 ners against negative bending. Negative reinforcement, however, is not always provided (see 
 Fig. 75), in which case the sides and top act as simple beams and more or less cracking occurs 
 on the outside near the corners. The walls or sides of a box culvert are usually designed 
 somewhat empirically, but are always provided with sufficient strength to support the lateral 
 earth pressure, neglecting any outward bending due to the bending of the cover slab. In 
 open-box construction, cross struts are often used to assist in holding the footings against 
 the pressures on the walls and also to provide bearing area in addition to that furnished by 
 wall footings (Fig. 77). The struts are designed as beams with a span equal to the width of 
 the culvert, and the struts are so spaced and proportioned as to obtain a uniform soil pressure 
 throughout. When a bottom slab is employed and assumed to support the load, its thickness 
 is made the same as that of the cover slab since the load for both slabs is substantially the 
 same. Longitudinal reinforcement should be provided on account of the possibility of beam 
 action due to unequal settlement. The amount of this reinforcement should depend upon the 
 character of the foundation. Where the foundations are very bad, it is the practice of some 
 engineers to figure the culvert as a whole to act as a beam, considering the length of beam as 
 12 times its depth. In long culverts under railroad tracks the load decreases beyond the ends 
 of the ties and the cross-section should decrease whenever a material saving is found to result. 
 
 In Fig. 77, which illustrates a standard box culvert adopted by the Chicago, Milwaukee & 
 St. Paul Railway, the cover is designed as a simply supported slab with span equal to clear 
 span when fillets are used and to clear span plus one-half the maximum cover thickness (but 
 not to exceed 1 ft.) when no fillets are provided. Stirrups and bent rods are employed to take 
 care of two-thirds of the shear when the shear exceeds 40 lb., bent rods being also used to care 
 for any negative moments which may develop due to connection with side walls. Longi- 
 tudinal steel is employed with a sectional area of about H 5 o of the area of the entire concrete 
 section. The side walls, cross struts, and footings are proportioned in the same manner as 
 above described. Keyways are formed in top of footings and side walls so as to offer shearing 
 resistance to movement of side walls due to lateral earth pressure. For fills up to 40 ft., the 
 load on the footings is assumed to include the live load and dead load of culvert, and the fill 
 directly above the culvert for the width overall including footings. For fills over 40 ft. high, 
 
 50 
 
786 
 
 CONCRETE ENGINEERS' HANDBOOK 
 
 [Sec. 17-32C 
 
 the total weight resting on the footings is considered as 623^ % of the culvert weight plus 62^^ % 
 of the fill above the footings. The live load is disregarded and, for ease in computation, the 
 weight of culvert is taken as equal to the fill it displaces. Wing walls over 8 ft. in height or 
 making an angle of less than 60 deg. with the headwall are made self-supporting cantilevers 
 with a joint at connection with the barrel. Wingwalls for all other conditions are made con- 
 tinuous with the main part of the culvert. The section of the culvert coming under the track 
 is made approximately equal in length to the theoretical spread of the live load, which is equal 
 to the distance out to out of ties (8 ft. for single track. 21 ft. for double track) plus the height 
 
 ^'^ Vert Pods 7-6 '\ 
 f Dowels. 2^- 
 
 4 
 
 -Base of hy/ ra/f 
 
 V V 
 
 I i 1 1 1 
 
 10- 
 
 -..J,L...'.>L...,._: 
 
 iDowe/s 
 
 ■'- -.-> 
 
 s/:-8 
 
 Sectional Elevation at Center Lin? of Culvert 
 
 I Break 
 
 <s'-w-A St/rrvpa. 
 
 Plan 
 
 i"Verf..7-e 
 
 'Dowels. 3' l;':^; 
 
 ■ ■ p4 l.f-. ^-jJ — : ^m d 
 
 j-->ji./:^/'>ik- y" 
 
 Section dt 
 Center Line of TracK, 
 
 Fig. 77— Single box culvert, C. M. & St. P. Ry. 
 
 of culvert plus one-half the height of fill above the cover slab. The invert is paved with a 
 concrete slab in all spaces between struts so as to form a continuous concrete invert. 
 
 Accurate analysis may be made of the moments in box culverts when such culverts are 
 reinforced so as to act monolithically. The following formulas and diagrams are given so 
 that the moments from such analysis may be readily determined. 
 
 The open-box culvert may well be divided into two classes for analysis, each claas de- 
 pending on the material upon which the footings rest. If the structure rests upon rock, 
 cemented gravel, or some other material having a high allowable bearing stress, the plane of 
 the base of each side wall will be held at a practically constant slope; that is, the side may be 
 considered fixed at the bottom. On the other hand, if the allowable bearing pressure of the 
 supporting material is low, the plane of the base will become inclined until the moment at the 
 
Sec. 17-32c] 
 
 HYDRAULIC STRUCTURES 
 
 787 
 
 footing has been reduced to a value that will be resisted by the moment of the upward pressure 
 on the base. Since in such a case the resisting moment is small, an altogether different dis- 
 tribution of moment will occur throughout the structure. It will be slightly on the safe side 
 to regard the sides as hinged at their base. 
 The following nomenclature will be used: 
 
 h = span of frame (c. to c. of side walls). 
 
 h = height of frame (c. to c. of top and bottom slabs). 
 
 R = = ratio of span to height. 
 
 . Ic = moment of inertia of cross-section of top or bottom slab (considered alike) for 
 one unit of length of the barrel. 
 Id = moment of inertia of cross-section of either side wall (considered alike) for 
 one unit of length of the barrel. 
 Ic 
 
 <S = = ratio of the moment of inertia of the horizontal slabs to that of the sides. 
 
 w = uniform load per unit area on top or bottom slab. 
 p = uniform load per unit area on side walls. 
 P = concentrated load. 
 He = thrust at the center of the top slab. 
 
 Vc = zero for symmetrical loading, as is the case in the analysis of culverts. 
 Mc = moment at the center of the top slab. 
 Md and 
 
 Mb = moment at center of side walls and center of bottom slab, respectively. 
 Ma and 
 
 Mb = moment at the upper and lower corners, respectively. 
 Type I. The Closed Frame J— 
 
 1 Assume the rectangular culvert of Fig. 78 to be rigidly fixed at the center of the bottom slab. With this con- 
 dition assumed, a unit length of the culvert may be treated in the same manner as a symmetrical arch with fixed 
 ends (see Art. 18, Sect. 16) and the following expressions result: 
 
 C ds Cds r ds r ds 
 
 Mil- (MY 
 
 in which 
 
 TO = moment at any point on either half of culvert of all external loads between point and crown section — 
 all values of m to be substituted in equations as positive. 
 
 ^ = integration referred to one-half of culvert only. 
 X, y = coordinates of any point. 
 
 From these equations were developed the formulas for Types I and II given above. 
 The bending moment at any point may be expressed as follows: 
 
 Mc = Mc + Hey — m 
 Mr = Mc + Hey — m 
 
788 
 
 Case 1. 
 
 Case 2. 
 
 Case 3. 
 
 CONCRETE ENGINEERS' HANDBOOK 
 
 R^ 
 
 [Sec. 17-32C 
 
 hit 
 
 B 
 
 wb=P 
 
 SA/tl 
 
 ,iiiiiini|M 
 
 ttitmttlir 
 
 wb 
 
 
 
 A 
 
 
 
 
 < b- 
 
 
 -> 
 
 
 
 
 
 
 
 
 
 Jl 
 
 
 
 r?/pe //. — O-pen Frame ivith Fixed Walls. 
 Case 1. — 
 
 £ 
 
 2 ^Ic 
 
 8 3i2 + *S 
 
 Ma = Mc 
 
 M, 
 
 Ph 
 
 Mc + Hch - — = Ma + i^c/i 
 
 He = 0 
 
 Mc MB 2i\R+S) 
 
 Md = Ma =M 
 
 Mc 
 
 He = ^ 
 
 Md = Mc + ^ 
 Mb = Mb = Mc 
 
 He 
 
 Mc 
 
 Ma 
 M, 
 
 P/ 3R^ \ 
 8\S + 2r) 
 Pb/ S -\- R \ 
 4 \S + 2r) 
 
 Mc j- 
 
 4 
 
 Ma + Hch 
 
 Case 2.- 
 
 wb 
 
 UIIIIU 
 
 -I. 
 
 1 See footnote on p. 789. 
 
 b-T-H 
 
 ^wb( R^ \ 
 4 \2R + Sj 
 ivb^ /2^-f3^\ 
 24 \2R + S ) 
 
 Mc 
 
 Ma = Mc - 
 
 Mb = Mc 
 
 wb^ 
 
 wb^ 
 
 Hch = Ma + Hch. 
 
Sec. 17-32c] 
 
 HYDRAULIC STRUCTURES 
 
 789 
 
 Case 3 — 
 
 
 « b- 
 
 .X 
 
 
 
 
 
 
 
 
 
 Mc = 
 
 
 
 
 
 
 
 rr ^Ph ( SR + 
 
 He 
 
 In sides M(max) occurs at from top. 
 
 M(max) = Mc + 
 
 T^T/pe ///. — Open Frame Hinged at the Base. ^ 
 Case 1. — 
 
 He' 
 2p 
 
 8 + 3i?/ 
 
 Mc=^ - Hh 
 4 
 
 Case 2- 
 
 I'c' ; 
 b- rH 
 
 Ma 
 
 H = 
 
 Mc = 
 Ma = 
 
 Hh 
 
 \2S + 3RJ 
 
 wbl R 
 
 ^ - Hh 
 
 -Hh 
 
 Case 3. — 
 
 Ph 
 
 ^^^^ 
 
 -Id 
 
 ph 
 
 3pA / >S + 2/e \ 
 4 \2*5 + Sr) 
 
 ^ _ ph 
 
 2 
 
 M (max) for the sides = 
 
 H 
 
 Mc = Ma 
 
 Hh 
 
 i/i V S \ 
 4 \2S + 3i?/ 
 
 2p 
 
 ^ at max. moment 
 
 1 The analysis of this type of frame follows that for a two-hinged arch. Let M' be the bending moment, at any 
 section, of the vertical forces only, as for a simple horizontal beam. Then for any point there results the total mo- 
 ment M = M' — Hy in which y is measured from the horizontal connecting the hinges, and H is the thrust acting 
 along this line and upon the hinges. From "Curved Beams" (see Art. 15, Sect. 16) the increase in span is, with E 
 constant, 
 
 ds 
 
 4f 
 
 '1 
 
 The span, however, is not permitted to increase, due to the resistance offered by H. 
 Substituting for M its value given above. 
 
 JiM' 
 
 from which 
 
 H = 
 
 - Hy)y:-^^ 
 
 Since the thrust only is statically indeterminate, its solution for any case permits statical solution of moment by 
 the equation for M given above. 
 
Sec. 17-32c] 
 
 HYDRAULIC STRUCTURES 
 
 793 
 
794 
 
 CONCRETE ENGINEERS' HANDBOOK 
 
 [Sec. n-S2d 
 
 Diagrams. — The statically indeterminate functions He and Mc for Types I and II, and 
 H for Type III may be read from the following graphs. The use of the diagrams will be found 
 to aid greatly the repetition of solutions caused by having to assume at the outset some values 
 of Ic and Id. These true values cannot be obtained until a tentative design is made, based 
 
 Ic 
 
 upon the stresses in the frame having an adopted stiffness ratio = S. In usual cases 
 one may well begin the design by assuming S = 4. The approximate sections required in sides, 
 
 top and bottom may be obtained from the moments under this assumption. The values Ic 
 and Id may then be calculated from these approximate sections, thus determining a new and 
 much closer value of S. The whole solution is then repeated until the required sections check 
 the assumed value of S. 
 
 32d. Construction. — The inside forms for small box culverts are usually arranged 
 to be collapsible or at least so that the frames against which the lagging is placed may be easily 
 
 Fig. 79. — Forms for 2-ft. box culvert. 
 
 knocked out after the concrete has properly set. In Fig. 79 the inner forms consist of boards 
 placed against frames made of three pieces of 2 by 4-in. and one piece of 2 by 6-in. joists, notched 
 as shown. The 2 by 6-in. piece at the top is not nailed in order that it may be readily knocked 
 out after the concrete has hardened. The removal of this piece allows the 2 by 4-in. joists to 
 
Sec. n-^2d] 
 
 HYDRAULIC STRUCTURES 
 
 795 
 
 be withdrawn, thus releasing the boards. Where the soil is hard and compact, the outer forms 
 may be omitted and the concrete deposited directly in the trench made to exact size of culvert 
 cross-section. Another type of small box culvert form is shown in Fig. 80. 
 
 Fig. 80. — Forms for small box culverts, Iowa Highway Commission. 
 
 Fig. 81.— Box-culvert forms, C. M. & St. P. Ry. 
 
 /^Bars,^'^c. foe. , t'Sars. 3^c. ioC. 
 
 Fig. 82.— Box-culvert forms, C. B. & Q. R. R. 
 
 The forms for large box culverts need no special consideration as they are similar to the 
 forms for walls and floor slabs in ordinary construction. Standard form details adopted by 
 two prominent railroads are shown in Figs. 81 and 82. 
 
796 
 
 CONCRETE ENGINEERS' HANDBOOK 
 
 [Sec. 17-33 
 
 Collap,sible steel forms are now being used to a considerable extent in culvert construction, 
 and in most cases with success. The Whalen steel form is show^n in Fig. 83. It is built in 
 but one size, yet by its unit method of construction, it readily lends itself to the building of 
 various-sized culverts. Although built in arched top sections, a proper assemblage of these 
 sections makes it possible to build most any size of flat top culvert. 
 
 If running water is encountered which cannot be temporarily diverted or dammed, the water 
 in the case of small culverts should be carried in a new trench around one side of the back 
 forms. In the case of the larger structures the trench excavated for the culvert should be ar- 
 ranged to flume the stream through between the abutments. 
 
 Fig. 83. — Whalen steel form. 
 
 33. Arch Culverts. — The arch type of culvert should be employed where an artistic design 
 is especially desirable, and should also be used in all cases where the fill to be supported is ex- 
 cessively high and the foundations suitable. High fills over box culverts necessitate a slab 
 of considerable thickness and the arch becomes the more economical under such conditions 
 because of the fact that an increase in fill does not produce a correponding increase in ring 
 stress. Arch culverts of reinforced concrete are not usually designed for spans less than about 
 8 ft., as plain concrete seems to answer the purpose for such small structures. 
 
 33a. Design of Cross-section. — In the ordinary type of arch culvert represented 
 in Fig. 84, the arch ring may be analyzed in the same manner as described for arch bridges in 
 Art. 16, Sect. 16. A uniform live load only is considered and this is placed over the whole 
 span. Although steel is used, the line of pressure is usually kept everywhere within the middle 
 third. In determining the dead load on the arch no allowance is made for the arch action of 
 the fill, but the horizontal components of the earth pressure are taken into account. Longitu- 
 dinal reinforcement is needed to prevent objectionable cracks caused by the shrinkage of concrete 
 
Sec. 17-33M 
 
 HYDRAULIC STRUCTURES 
 
 797 
 
 in hardening and the contraction due to a lowering of the temperature, and also to distribute 
 the load. 
 
 Inverts are employed in the designs shown in Figs. 85 and 86 in order to provide additional 
 bearing area and thus reduce the large abutments which would otherwise be needed in order 
 to bring the pressure on the soil to a safe value. An invert also tends to prevent any possibility 
 of water undermining the structure and the foundation need not always be carried to the same 
 depth as when the abutments aie independent of each other. 
 
 Fig. 84. — Standard design for 10-ft. arch culvert, State of Missouri Highway Department. (Actual dimensions 
 and shape of foundations governed by conditions of soil at location of site.) 
 
 Section A-A Section B-B ^\"--^" 
 
 
 ^ p" 
 
 \" - 
 
 1 ^.-^ 
 
 
 
 
 
 
 \</o Spaces @ 18"- ->| 
 
 ,-Z-/"Bars 
 
 M 
 
 |m If 
 
 i 
 
 
 3'-3'- 
 
 <-'•-->[ '']<-- e-e"^'- !<-'-'•> 
 <- /e'-o-- > 
 
 <- l6'-0'- ► 
 
 /- 
 
 Sectional Elevation ^ 
 Fig. 85. — Standard arch culvert for fills up to 40 ft. high, C. M. & St. P. Ry. 
 
 When the arch ring and invert are thoroughly tied together so as to act as a monolith (a 
 rare case except in masonry sewer design), the entire culvert may be analyzed after the manner 
 of the arch. For the analysis of this form of arch see Art. 33 on the design of conduits. 
 
 335. Forms. — The centers and forms for the arch culvert are similar to those 
 required for the arch bridge of small span. Where inverts are employed, the concrete for the 
 
798 
 
 CONCRETE ENGINEERS' HANDBOOK 
 
 [Sec. 17-33& 
 
 End Section End Elevation 
 
 Maxiwum Fill over Croivr? ■ 90^f.~ 
 
 Section of Barrel 
 Fig. 86. — Double-barrel culvert, D. L. «&; W. R. R 
 
 ^^4"x 6f4'-0"c.-foc. to 
 Fig. 87. — Centering for arch culvert, C. M. & St. P. Ry. 
 
 Bo+tom View of Cen-tiering 
 
 Fig. 88. — Centering for arch culverts. 
 
Sec. 17-34] 
 
 HYDRAULIC STRUCTURES 
 
 799 
 
 footings and floor in the stream bed should be deposited first and then the centers for the arch 
 erected on this concrete. Fig. 87 shows the type of arch culvert centers and forms used by the 
 Chicago, Milwaukee & St. Paul Railway. Other types of centers are shown in Figs. 88 and 89. 
 
 Two fhicJrnesses of 3' planir 
 tv form ring 
 
 .^Driff Bo/ts 
 
 Fig. 89. — Details of centering, Canoe Creek culvert, near Cloe, Pa., B. R. & P. Ry. 
 
 CONDUITS AND SEWERS 
 
 Sewers and pipe lines fall into two groups, since the former is acted upon by external pres- 
 sures only, and the latter primarily withstands internal pressure with the possible addition of 
 external pressures due to intermittent support, or to burying. 
 
 34. Stresses Due to Internal Pressure. — Let the internal pressure on a pipe be P lb. per 
 sq. in. Then the total tension at any longitudinal section through one side of the pipe, for 
 1 in. of length, is FR (see Fig. 90). This is resisted by A/s = p^/'s- From this 
 
 PR 
 
 Values of fs have ranged from 8000 to 16,000 lb. per sq. in. with good results in all cases. 
 Pressures up to 100 lb. per sq. in. do not seem to force seepage through the minute cracks. It 
 is important that the steel used be of high bonding capacity to distribute 
 the cracks, hence fabric, or for larger pipes, deformed bars with thorough 
 splicing should be used. 
 
 35. External Earth Pressure on Circular Pipe. — For earth pressure on 
 pipe in ditches, see Art. 31a. 
 
 36. Large Conduits and Sewers not Circular. — Definite conditions of flow 
 have for hydraulic reasons developed various types of conduit and sewer sec- 
 tions. Such sections consist of two parts: (1) the invert, or floor of the conduit; and (2) the 
 arch, or walls and roof of the structure. The invert may be parabolic, elliptical, semicircular 
 or a catenary, while the arch may be semicircular, parabolic, elliptical or segmental. A dis- 
 cussion of the hydraulic advantages of various sections will be found in Metcalf and Eddy's 
 
 American Sewerage Practice," Vol. I, Chap. XL 
 
 When the sewer or conduit rests directly upon rock, the invert is merely a pavement and 
 may be made relatively thin. The arch may then be analyzed by the methods given for the 
 elastic arch, considering it to have fixed ends (Case I, Fig. 91). 
 
 The pressures acting upon the arch may be found as follows. The rib is divided into con- 
 venient subdivisions, as described in the analysis of the elastic arch. The intensity of vertical 
 pressure, say for division 3, Fig. 91, is computed as described in Art. 31a and multiplied by the 
 horizontal projection of division 13, to obtain Viz. The intensity of horizontal pressure is 
 Ce times the intensity of vertical pressure (see Diagram 1, "Retaining Walls," page 577), 
 and this is multiplied by the vertical projection of division 13 to obtain Hn. Applying His 
 and Vis at the centroid of division 13 and completing the parallelogram, P13 may be found. 
 The other loads are then found in a similar manner. 
 
 When the structure rests on yielding material (Case II, Fig. 91), the section may be con- 
 sidered fixed at BB and cut at the crown as for the usual arch analysis. The rib is subdivided 
 
800 
 
 CONCRETE ENGINEERS' HANDBOOK 
 
 [Sec. 17-3(3 
 
 from the crown around to BB, and treated precisely as an arch. It is assumed that there are 
 vertical forces acting upward on the invert equal in amount to the total downward vertical 
 forces, and uniformly distributed over the invert. The oblique resultant force on block 16 is 
 
 Fig. 91. 
 
 due to the combining of the upward vertical force with the earth pressure acting on the left 
 side of the block. After having computed He, by the usual arch analysis, the force polygon 
 may be drawn, and the closing line will be Hb- Rays may then be drawn, since the pole 0 
 
 1 
 
 is now located and the pressure line drawn on the conduit section. The shears and moments 
 may be computed as for arch analysis. Shear will usually be large near the outer ends of the 
 invert, as is shown in Fig. 91. A typical method of placing reinforcement in horseshoe sec- 
 tions is shown in Fig. 92. 
 
Sec. 17-36] 
 
 HYDRAULIC STRUCTURES 
 
 801 
 
 Note - On hard bofhm no reinfbrcemenf required 
 Fig. 94. 
 
 51 
 
802 
 
 CONCRETE ENGINEERS' HANDBOOK 
 
 [Sec. 17-37 
 
 37. Construction. — Some large sewer sections are shown in Figs. 93 and 94, Fig. 93 
 shows a typical section of the Monroe St. sewer in Chicago. Fig. 94 shows a horseshoe section 
 in Evanston, 111. A 1 : 13-^ : 4^^ mix was used in the latter case. A large number of typical 
 sections are to be found in Metcalf and Eddy's ''American Sewerage Practice," Vol. 1, Chap. XII, 
 together with descriptions of each. 
 
 38. Longitudinal Reinforcement. — Temperature and shrinkage stresses require about 0.4 % 
 of longitudinal reinforcement in pipes and conduits. 
 
 Pipes resting at intervals on saddles are required to carry their own weight and should be 
 reinforced longitudinally to resist the flexure set up. The bending moment may be assumed 
 
 at support and center of span to be where I is the span in feet, and w is the load per foot 
 
 „3' Bl ■i'LcmiM/ncr/ Aars) A 
 
 Section D-D Section "C-C" 
 CHICAGO, BURUNGTON, & QUINCY R.R MICHIGAN CENTRAL R. R. 
 
 Fig. 95. — Three designs of reinforced-concrete pipe for railway culverts. 
 
 which the pipe is obliged to carry. Knowing the bending moment, and the dimensions of the 
 pipe, the steel requirements and unit stresses in concrete and steel maybe taken from Diagram 2, 
 Sect. 18, page 816. 
 
 Illustrative Problem. — Inside diameter of pipe 8 ft., thickness of shell 4 in. Find spacing of saddles for 
 longitudinal shearing stress of 35 lb. per sq. ir., and provide proper flexural reinforcement. 
 Weight of pipe = 8.3 X ir X H X 150 = 650 lb. per ft. 
 Weight of water = 16 X ir X 62.5 3140 lb. per ft. 
 
 Total, 3790 lb. per ft. 
 
 V 
 
 From Art. 15, Sect. 18, page 818,^= 3.15t'. 
 
 22,050 
 
 Allowable F = 50 X 4 X 3.15 X 35 = 22,050 lb. Clear spacing of saddles = g^^^ = 5.82 ft., say 6.8 ft. on 
 centers. 
 
 M 175,350 
 
 Bending moment = M2 (3790) (6.8)2(12) = 175,300 in.-lb. = (2500) (4) ^ ^^^^'■""g Diagram 2, 
 
 page 816, it is seen that at this extremely low value, the steel and concrete unit stresses are very low if a small amount 
 of longitudinal steel is in place. Use 0.4 % for temperature and flexure. 
 
Sec. 17-39] 
 
 HYDRAULIC STRUCTURES 
 
 803 
 
 39. Examples of Reinforced-concrete Pipe. — Figs. 95 and 96 show several types of rein- 
 forced pre-cast pipes. Most of the pipe lines laid are of pre-cast units. For culverts and sewers 
 the pipe may be oval or circular. For water under pressure the pipe should be circular, and 
 the joints carefully made. Two types of joints are common, bell and spigot, and sleeve joints, 
 both of which are shown in the figure. 
 
 ■d'-s'- 
 
 6-0' ■ 
 
 f . ' k o ' . ''! " 6: ' « i; 
 
 i<- --32, f'^Rods, 4"c.foc. >l 
 
 Longitudinal section Won vertical axis 
 
 Longitudinal section "C" on ^ point's A 
 Fig. 96. — Reinforced-concrete culvert pipe, Chicago & Northwestern Ry. 
 
 40. Forms for Sewers. — Because of desirability to repeat the use of form material, an 
 arrangement of collapsible form units is best. Steel forms are advantageous for this reason, 
 especially for heavy work. 
 
 Usually the sewer or conduit is cast in stages, the invert being poured first, and if bricked, 
 the pavement next laid. The arch centering and the steel for the arch are then placed. The 
 wall sections are next set and the arch poured and finished. The joint with the floor should 
 be well tied with dowel rods. The centering should be arranged so that it may be easily 
 
 "struck," or loosened for removal. Fig. 97 shows a simple arrangement for circular sections. 
 The same general principle is also used for arch forms both in wood and steel. 
 
 The backfill should cover the sewer before the centering, or core, is struck. General 
 practice {Eng. Rec, vol. 58, p. 664) indicates that under favorable circumstances forms may be 
 struck in 36 to 48 hr. for sewers smaller than 10 ft., and 72 hr. for larger sewers. Temperature 
 and humidity will greatly affect the setting of concrete, and should be considered as modifying 
 the above rule. 
 
I 
 
SECTION 18 
 
 MISCELLANEOUS STRUCTURES 
 
 DEEP GRAIN BINS OR SILOS 
 
 Deep bins will be considered of such proportions that the plane of rupture drawn from the 
 bottom of one side will not pass out at the free surface of the material retained. Analysis, 
 therefore, cannot be made according to the theories of retaining walls. 
 
 1. Action of Grain in Deep Bins. — A study of the action of grain in a deep bin shows that 
 the grain forms a dome from one side to the other, which is supported at its perimeter by the 
 friction between the grain and the side of the bin. This dome supports a portion of the grain 
 above it, the remainder being carried by further dome action to the sides. Compression thus 
 exists upon the horizontal sections of the wall, which varies in some ndanner with the grain 
 and the depth. The lateral radial pressure of the grain likewise varies in some manner with 
 the depth and the grain. 
 
 2. Janssen's Formulas for Pressure in Deep Bins.^ — 
 
 <t> = angle of repose of the filling. 
 (j>' = angle of friction of the filling on the bin walls. 
 u = tan (f) = coefficient of friction of filling on filling. 
 u' = tan 0' = coefficient of friction of filling on the bin walls. 
 w = weight of filling in pounds per cubic foot. 
 V = vertical pressure of the filling in pounds per square foot. 
 L = lateral pressure of the filling in pounds per square foot. 
 L = kV OT 
 
 K - y 
 
 A = area of bin in square feet, 
 U = circumference of bin in feet. 
 
 R = Yj = hydraulic radius of bin. 
 
 h = depth of granular material at any point. 
 Then 
 
 V = ^^j^l ~ ^ number whose common log is ^ gQ^^g ^ J 
 L = kV 
 
 Values of u' and k for different grains and bin materials are given in Tables 1 and 2. 
 
 Illustrative Problem. — Given wheat weighing 50 lb. per cu. ft.; u' = 0.444; k = 0.5; depth of material 
 50 ft.; diameter of bin 12 ft. Required vertical and horizontal pressures at base. 
 
 ^ A 36 
 
 ^ = c7 = r2 = 3 
 
 ku'h _ (0.5) (0.444) (50) ^ 
 2.303ie (2.303) (3) 
 
 The number whose common logarithm is 1.605 is 40.3 and 1 40.3 = 0.025 
 
 L = kV = (0.5) (660) = 330 lb. per sq. ft. 
 1 For derivation of formulas see Ketchum's "Structural Engineers' Handbook," 1st Ed., p. 319. 
 
 805 
 
806 
 
 CONCRETE ENGINEERS' HANDBOOK 
 
 [Sec. 18-2 
 
 Table 1.^ — Weights and Coefficients of Friction of Various Kinds of Grains 
 
 ON Bin Walls (Airy) 
 
 
 Weight of 
 
 a cubic 
 foot loosely 
 filled into 
 measure 
 (pounds) 
 
 
 Coefl&cients of friction 
 
 Grain on 
 grain u 
 (tan <p) 
 
 Grain on 
 
 rough 
 board u' 
 (tan <f>') 
 
 Grain on 
 smooth 
 board u' 
 (tan <(>') 
 
 Grain on 
 iron u' 
 (tan 4>') 
 
 Grain on 
 cement u' 
 (tan <l>') 
 
 Wheat . . . 
 
 49 
 
 0.466 
 
 0.412 
 
 0.361 
 
 0.414 
 
 0.444 
 
 Barley . . . 
 
 39 
 
 0.507 
 
 0.424 
 
 0.325 
 
 0.376 
 
 0.452 
 
 Oats 
 
 28 
 
 0.532 
 
 0.450 
 
 0.369 
 
 0.412 
 
 0.466 
 
 Corn 
 
 44 
 
 0.521 
 
 0.344 
 
 0.308 
 
 0.374 
 
 0.423 
 
 Beans. . . . 
 
 46 
 
 0.616 
 
 0.435 
 
 0.322 
 
 0.366 
 
 0.442 
 
 Peas 
 
 50 
 
 0.472 
 
 0.287 
 
 0.268 
 
 0.263 
 
 0.296 
 
 Tares. . . . 
 
 49 
 
 0.554 
 
 0.424 
 
 0.359 
 
 0.364 
 
 0.394 
 
 Flaxseed. 
 
 41 
 
 0.456 
 
 0.407 
 
 0.308 
 
 0.339 
 
 0.414 
 
 Table 2.^ — Values of = ^ for Wheat and Other Grains in Different Bins (Pleisner) 
 
 Bins 
 
 k = L/V 
 
 Wheat 
 
 Rye 
 
 Rape 
 
 Flax-seed 
 
 Cribbed bin 
 
 Ringed cribbed bin 
 
 Small plank bin 
 
 Large plank bin 
 
 Reinforced-concrete bin. 
 
 0 . 4 to 0 . 5 
 0.4to0.5 
 0 . 34 to 0 . 46 
 0.3 
 0.3 to 0.35 
 
 0.23 to 0.32 
 0 .3 to 0 .34 
 0.3 to 0 . 45 
 0.23 to 0.28 
 0.3 
 
 0 . 5 to 0 . 6 
 
 0 . 5 to 0 . 6 
 
 The formula for L has been used in plotting Diagram 1, for wheat retained in a reinforced- 
 concrete bin. The above problem could have been solved directly from this diagram. 
 
 3. Conclusions from Tests. — Prof. Ketchum has drawn the following valuable conclu- 
 sions ^ from tests* made by him and other experimenters upon model and full-sized grain 
 bins: 
 
 1. The pressure of grain on bin walls and bottoms follows a law (which for convenience 
 will be called the law of ''semi-fluids"), which is entirely different from the law of the pressure 
 of fluids. 
 
 2. The lateral pressure of grain on bin walls is less than the vertical pressure (0.3 to 0.6 
 of the vertical pressure, depending on the grain, etc.), and increases very little after a depth 
 of 2}'i to 3 times the width or diameter of the bin is reached. 
 
 1 From Ketchum's "Walls, Bins and Grain Elevators," p. 327. 
 
 2 From Ketchum's "Structural Engineers' Handbook," p. 321. 
 
 3 Ketchum's "Structural Engineers' Handbook," p. 325. 
 
 * Ketchum's "Walls, Bins and Grain Elevators," Chap. XVII. 
 
Sec. 18-3] 
 
 MISCELLANEOUS STRUCTURES 
 
 807 
 
 3. The ratio of lateral to vertical pressures k is not a constant, but varies with different 
 grains and bins. The value of k can only be determined by experiment. 
 
 4. The pressure of moving grain is verj^ slightly greater than the pressure of grain at rest 
 (maximum variation for ordinary conditions is, probably, 10%). 
 
 5. Discharge gates in bins should be located at or near the center of the bin. 
 
 6. If the discharge gates are located in the sides of the bins, the lateral pressure due 
 to moving grain is decreased near the discharge gate and is materially increased on the side op- 
 posite the gate (for common conditions this increased pressure may be 2 to 4 times the 
 lateral pressure of grain at rest). 
 
 7. Tie rods decrease the flow but do not materially affect the pressure. 
 
 Diagram 1 
 
 Lat-eral pressure In lb. per f t. 
 
 8. The maximum lateral pressures occur immediately after filling, and are slightly greater 
 in a bin filled rapidly than in a bin filled slowly. Maximum lateral pressures occur in deep bins 
 during filling. 
 
 9. The calculated pressures by either Janssen's or Airy's formulas agree very closely with 
 actual pressures. 
 
 10. The unit pressures determined on small surfaces agree very closely with unit pressures 
 on large surfaces. 
 
 11. Grain bins designed by the fluid theory are in many cases unsafe as no provision is 
 made for the side walls to carry the weight of the grain, and the walls are crippled. 
 
 12. Calculation of the strength of wooden bins that have been in successful operation shows 
 that the fluid theory is untenable, while steel bins designed according to the fluid theory have 
 failed by crippling the side plates. 
 
808 
 
 CONCRETE ENGINEERS' HANDBOOK 
 
 [Sec. 18-4 
 
 Experiments by Willis Whitedi and by Prof. Ketchum with wheat "have shown that the 
 flow from an orifice is independei t of the head and varies as the cube of the diameter of the 
 orifice." 
 
 4. Design of Walls. 
 
 4a. Vertical Load Carried by Walls. — Prof. Ketchum^ has shown that the 
 vertical load of grain carried by 1 ft. of circumference of the wall at a depth y will be very 
 approximately 
 
 Pu' = wR (approx.) 
 
 where P is the total side pressure on a vertical strip y high and 1 ft. wide. The unit stress 
 thus obtained must be added to that caused by the dead load of the structure. 
 
 46. Wind Stresses on a Horizontal Section. — The wind stresses in a deep bin are 
 very small as a rule, and it is common practice to permit the concrete to take tension on the 
 windward side up to about 200 lb. per sq. in., assuming the tensile stress to be low, the unit 
 stress in the concrete in tension and compression will be 
 
 21 
 
 Assuming a wind pressure of 30 lb. per sq. ft. of vertical projected area on a height of h ft. of 
 the bin, 
 
 h + nh 
 
 in which D = outside diameter or greatest dimension in feet, and Ic and 7s the moments of 
 inertia of the concrete and steel sections, respectively, about a diameter, expressed in inches*. 
 The steel unit stress is nfc approximately. For a round bin of one cell, isolated, use two-thirds 
 of the stress thus obtained. 
 
 If the tensile unit stress in the concrete exceeds 200 lb. per sq. in., the section should be 
 analyzed as for a chimney of similar dimensions. 
 
 4c. Thickness of Walls. — The stresses obtained from vertical and wind loads 
 are combined to obtain the maximum unit stress in the concrete. Common practice uses the 
 same working stresses in this structure as in buildings. The thickness of wall is then pro- 
 portioned to provide for these unit stresses. The same thickness of wall is usually maintained 
 for the full height to permit the use of moving forms. 
 
 4(i. Horizontal Reinforcement. — Circular Sections. — The lateral pressure tends 
 to burst the circular bin along some vertical plane. Reinforcement against this is provided in 
 the same manner as that for circular tanks. The area of steel As per foot of height, due to the 
 lateral pressure of P lb. per sq. ft., is 
 
 Prs 
 
 ^ 
 
 ^--^g Mr 
 
 f. 
 
 where rs is the radius in feet of the steel hooping. The amount of 
 steel would be proportional to the abscissas to the pressure curve 
 (see Diagram 1, page 807). 
 
 4e. Rectangular Sections. — There is a tendency for 
 ^ p ^ fl^^ sides of the polygonal bin to bulge, and the angle made by 
 
 two adjacent forces to open out (see Fig. 1). The moments, 
 determined by the method of slope deflections (see Sect. 10) may be written readily in the 
 following forms: 
 
 Moment at any corner = Ma = — — (TX-W^) — " 
 
 1 Proc. Eng. Soc. of West. Penn., April, 1901. 
 
 2 Structural Engineers' Handbook," p. 324, 
 
Sec. 18-4/] 
 
 MISCELLANEOUS STRUCTURES 
 
 809 
 
 V2[ l+b J 
 
 Ma = -bi + n 
 
 Ma j2 
 
 M. =^ - Ma 
 
 Md = ^ Ma 
 
 in which Ma is chosen from the proper case above. For square cells, 
 
 The moments given above must be provided for by horizontal bars running across the 
 inside of the corner and crossing to the outer face at the point of inflection. When the bins 
 are grouped, the moments may be caused from pressure on either side of an intermediate wall. 
 
 Sufficient reinforcement should be provided in the walls ac and bd to take the pull of the 
 wall ab or cd caused by the pressure p2. The tension in ac or bd is '^V'^h and in ab or cd is 
 
 The pressure pi and p2 may be found from the foregoing formulas and Diagram 1, page 807, 
 since the pressure on the side of a rectangular bin may be computed by computing the pressure 
 
 for a circular or square bin having the same hydraulic radius, R = jj- 
 
 4/. Hexagonal Bins. — In groups of circular bins the interspaces are irregular 
 in shape and do not hold as much as do the main cells. The hexagonal bin removes this 
 difficulty. Moments at corners and sides : 
 
 M corner = " K2?>&^ 
 Mside = y24.Vb^ 
 
 in which b is the length of any side. 
 
 5. Construction. — The foundation for a group of deep bins is usually a mat, unless the 
 foundation is unyielding. Pressures on the soil should be examined when half of the group is 
 loaded, and wind is acting. 
 
 Round cells are commonly placed in such a manner that at the points of tangency the wall 
 is of one thickness, so that the ring reinforcement from one cell laps over that of the adjacent 
 one. This, with the vertical steel passing through the link thus made, provides bonding between 
 cells. 
 
 Polygonal bins are arranged with walls in common, reinforced to take the pressure from 
 either side. The steel may well be arranged as for a continuous slab, the two systems crossing 
 each other at right angles at the corners. 
 
 Construction is usually continuous, the forms being jacked up on the vertical reinforce- 
 ment, or on vertical rods for the purpose, imbedded in the walls. Metal forms are commonly 
 used, since a smooth surface may be obtained. It is customary, when using moving forms, to 
 make the walls the same thickness from top to bottom. 
 
 The concrete bins of the Great Northern Ry.'s grain elevator at West Superior, Wis.,^ 
 
 1 Eng. News, Aug. 4, 1910. 
 
 When Ii = h 
 When pi = p2 
 
 For square cells, b = I and pi = 
 At the center of the sides. 
 
810 
 
 CONCRETE ENGINEERS' HANDBOOK 
 
 [Sec. 18-5 
 
 is shown in Fig. 2. The walls are of 1 : 2 : 4 concrete. The stresses used in the design were: 
 steel in tension, 16,000; concrete in compression, 600; bond, 100 lb. persq. in.;7i = 12. The 
 bins are 110 ft. high. The capacity of the 72 cells with interspaces is about 2,400,000 bu. 
 
 Fig. 3 is a typical section of rectangular bins in the elevator of the F. C. Ayres Mercantile 
 Co., Denver, Colo. A detailed description will be found in Ketchum's "Walls, Bins and Grain 
 Elevators," 2d Ed., page 454. 
 
 [<— to'-a'- ■>{<■- ^o'-3'■ - — >{<• - ^(^'-3'- 
 
 Fia. 2. — Concrete bins of Great Northern Ry's grain elevator, West Superior, Wis. 
 
 Fig. 3. — Rectangular bins in elevator of F. C. Ayres Mercantile Co., Denver, Colo. 
 
 SHALLOW BINS 
 
 If the plane of rupture, commencing from the bottom of one side of a bin, strikes the free 
 surface of the retained material, the bin is said to be shallow. Such bins are commonly used for 
 the storage of coal, coke, sand, ashes, etc. Pressures against the various sides are determined 
 by the methods commonly employed to determine earth pressures against walls. Two condi- 
 tions of loading will be given here : (1) level-full and (2) heaped bins. Bins with full sloped sides 
 and with partially vertical sides will be analyzed. 
 
Sec. 18-6] 
 
 MISCELLANEOUS STRUCTURES 
 
 811 
 
 6. Sloped Sides— Level Full (Case I). — Let ABC, Fig. 4, represent a bin with sloped sides. 
 On a vertical plane through B the pressure may at once be determined as for a vertical retaining 
 wall holding the prism BDC (see Rebhann's construction, Art. 16, Sect. 13). The triangle of 
 pressure aBD indicates the total pressure acting on the plane BD, the resultant being Pi, 
 acting liBD above B. Let Pi be produced to meet the weight 
 W of the prism ABD applied at the center of gravity O of the 
 triangle ABD. The resultant thus obtained is P2. This may 
 be resolved into the forces P3 and P4. P3 is then the resultant 
 normal pressure acting on AB. If the plane AB were smooth, 
 P4 would represent a component acting against DB; but since 
 the plane AB is rough, P4 becomes a thrust down the plane 
 against B, if the angle between the forces P3 and P4 is equal 
 to, or less than, the angle of friction between the plane and 
 the material. 
 
 Since P3 is the resultant pressure on AB, it passes through the centroid of the pressure tri- 
 angle AbB, and its magnitude is equal to the area AbB. The unit pressure bB may thus readily 
 be determined. 
 
 7. Partly Vertical Sides — Level Full (Case II). — So far as concerns the pressure on the side 
 
 AB (Fig. 5), it will be seen to carry the same load as a 
 similar portion of the sloping side AB, Fig. 4. 
 
 Locate the point A'. As before, determine Pi and W 
 (the latter applied at 0, the center of gravity of A'BD), 
 and find their resultant P2. P3 is the component of P2 
 normal to A'B, and represents the resultant of normal 
 pressure for the plane A'B. Its magnitude is equal to the 
 area of BA'b. The unit pressures Bb and Ac may thus be 
 found, defining the trapezoid AcbB, which represents the 
 total pressure acting on, and normal to, AB. 
 The pressure on the vertical side HA is represented by the pressure triangle HAd, which is 
 equal to the corresponding portion of the triangle BDa, and is therefore the pressure against a 
 vertical retaining wall of the height HA, and supporting the same material. 
 
 8. Sloped Sides — Fill Heaped to Angle of Repose (Case III). — No method has been given in 
 the discussion of retaining walls for 
 finding the thrust with a negative 
 surcharge. The following is an 
 application of Rebhann's method. 
 
 The pressures on a vertical 
 plane through B, as BD (Fig. 6a), 
 should balance either side. As- 
 sume that the thrust on BD acts 
 horizontally. Lay off BC making 
 the angle <f> (angle of internal 
 friction) with the horizontal BT, 
 and extending to DCC. Draw 
 DD' making the angle </> with BD 
 
 (hence normal to BC). With D'C as the diameter draw arc D'MC, and to it draw the 
 tangent BM from B. Make BN = BM, and draw NQ parallel to DD'. Make RN = QN. 
 The area of the triangle RQN times the weight per cubic foot of the material in the fill equals 
 the thrust Pi (Fig. 66) against BD. 
 
 Let Pi be extended to meet W\ the weight of the prism ABD. Their resultant is P2, whose 
 component normal to AB is P3. The true position of P3 should be }iAB from B. Its magni- 
 tude equals the area of the triangle ABa] and knowing the length AB, the unit pressure Ba 
 may be found. 
 
812 
 
 CONCRETE ENGINEERS' HANDBOOK 
 
 [Sec. 18-9 
 
 9. Partly Vertical Sides — Fill Heaped (Case IV) . — As in Case III, the pressure Pi against 
 BD is found, as in Fig. 6a. Pi is then combined with W, the weight of a prism A'DB, to form 
 P2 (Fig. 7). The component normal to A'B is P3, and represents the area of the pressure 
 triangle A'Ba. Knowing this area, and the length A'B, the unit pressures Ba and Ac may be 
 determined. The total pressure on -45 is given by the trapezoid AcBa. 
 
 Fig. 7. Fig. 8. 
 
 In some bins a flat portion is put into the bottom as in Fig. 8. In this case the vertical 
 plane may be extended to meet the extended plane A' ABB' at B'. The pressure Pi on DB' 
 is then found as in Case III, and combined with W, whence the normal pressure P3 is found. 
 This is the total normal pressure that would act on the plane A'B', and hence represents in 
 magnitude the area A'B'b'. The unit pressure B'b' could be found, and likewise Ac and Ba. 
 The actual normal pressure on A 5 is shown by the trapezoid AcBa. 
 
 10. Thrust due to P4. — The thrust P4 or the portion of P4 actually acting on plane AB, 
 should be provided for in the design of the slab AB and the supports at A and B. In Figs. 4 
 and 66 the thrust acting at B is equal to P4. In Fig. 7 the thrust at B is equal to P4 times the 
 ratio oi AB to A'B; and similarly in Fig. 8 the thrust at B is equal to P4 times the ratio oi AB 
 to A'B'. 
 
 If the span of the slab AB is in the direction AB, the stresses in it should be determined by 
 methods of combined flexure and direct stress. If the span is normal to the drawing, however, 
 simple bending takes place. 
 
 11. Data for Bin Design. — Table 1 gives the weights and angle of repose of several materials 
 commonly stored in shallow bins. For data on sand, earth, rock, etc., see table on page 575. 
 
 Table 1.^ — Weight and Angle of Repose of Coal, Coke, Ashes and Ore 
 
 Material 
 
 Weight (lb. 
 per cu. ft.) 
 
 Angle of re- 
 pose <i> (degree) 
 
 Authority 
 
 
 50 
 
 35 
 
 Link Belt Machinery Co. 
 
 Bituminous coal 
 
 47 
 
 35 
 
 Link Belt Engineering Co. 
 
 Bituminous coal 
 
 47 to 56 
 
 
 Cambria Steel 
 
 
 52 
 
 27 
 
 Link Belt Machinery Co. 
 
 
 52.1 
 
 27 
 
 Link Belt Engineering Co. 
 
 Anthracite coal fine 
 
 
 27 
 
 K. A. Muellenhoff 
 
 
 52 to 56 
 
 Cambria Steel 
 
 
 
 45 
 
 Wellman-Seaver-Morgan Co. 
 
 Slaked coal 
 
 53 
 
 37>^ 
 
 Gilbert and Barth 
 
 Coke 
 
 23 to 32 
 
 
 Cambria Steel 
 
 Ashes 
 
 40 
 
 40 
 
 Link Belt Machinery Co. 
 
 Ashes, soft coal 
 
 40 to 45 
 
 
 Cambria Steel 
 
 Ore, soft iron 
 
 
 35 
 
 Wellman-Seaver-Morgan Co. 
 
 
 
 1 From Ketchum's "Structural Engineers' Handbook," p. 311. 
 
Sec. 18-12] MISCELLANEOUS STRUCTURES 813 
 
 Table 2 gives some approximate friction angles for various materials against different 
 bin linings. 
 
 Table 2.^ — Angles of Friction of Different Materials on Bin Walls 
 
 
 Steel plate </>' 
 
 Wood cribbed 
 
 Concrete 0' 
 
 Material 
 
 in degrees 
 
 <j>' in degrees 
 
 in degrees 
 
 Bituminous coal 
 
 18 
 
 35 
 
 35 
 
 Anthracite coal 
 
 16 
 
 25 
 
 27 
 
 
 31 
 
 40 
 
 40 
 
 Coke 
 
 25 
 
 40 
 
 40 
 
 Sand 
 
 18 
 
 30 
 
 30 
 
 Fig. 9. — Plan and section of Duquesne Light Co.'s coal-storage pit at Pittsburgh, 
 
 12. Submerged Storage for Coal. — Coal stored in large quantities is commonly stored in 
 water, to prevent combustion. Bins or pits filled with water and coal should be designed to 
 resist the water pressure, as though full of water. Pits are then designed like open reservoirs, 
 except that the pavement for the pit must be heavier than that for the reservoir, to withstand 
 the dumping of the coal from the trestle overhead (see Fig. 9). 
 
 Ashes are frequently stored in pits of this nature, and very often water is run in (see Fig. 
 
 10). 
 
 When clam-shells or other forms of grab bucket are used to handle the material in the pits, 
 rails are commonly embedded in the pavement about 3 ft. apart, with the heads protruding 
 about }^ in. above the concrete, to protect it from the jaws of the bucket. This detail may be 
 noted in Fig. 10. 
 
 1 From Kbtchum's "Structural Engineers' Handbook," p. 312. 
 
814 
 
 CONCRETE ENGINEERS' HANDBOOK 
 
 [Sec. 18-12 
 
 i of track , (~> -Valve box 
 
 tofpt 
 
 -Cross 
 gi'-der 
 
 t of track 
 
 --H 
 
 Cross section of pit 
 
 Section thru end wall 
 
 Fig. 10. 
 
 Fig, 
 
Sec. 18-13] 
 
 MISCELLANEOUS STRUCTURES 
 
 815 
 
 13. Dock Pockets. — Storage pockets in docks for ore, and coal, etc., are usually unsymniet- 
 rical. The commonest form is that in Fig. 11. The pressure upon the face AB may be found 
 as for the vertical plane DB, Fig. 5. Thus P is the resultant of the pressures on AB, represented 
 by triangle ABb, and equals the area of triangle QNR times the weight per cubic foot of the 
 material retained. 
 
 Fig. 12. — Details of superstructure of Minneapolis, St. Paul & Sault Ste. Marie R. R. dock, 
 
 The ore dock of the Minneapolis, St. Paul & Sault Ste. Marie R. R. at Ashland, Wis.,i is 
 shown in Fig. 12. The front of the pockets are circular in plan, causing the wall stresses due to 
 the thrust of the ore to be tensile. This feature is patented by Max Toltz, who was the designer 
 of the dock. The concrete in the substructure was 1:3:5 while that in the superstructure was 
 1:2:4 mix. The horizontal thrust on the bin walls at a depth h was computed from p = 30/i, 
 
 » Eng. News, vol. 76, Aug. 10, 1916, p. 243. 
 
816 
 
 CONCRETE ENGINEERS' HANDBOOK 
 
 [Sec. 18-14 
 
 CHIMNEYS 
 
 The analysis of stresses in reinforced-concrete chimneys involves stresses due to (1) dead 
 load, (2) wind, and (3) temperature. 
 
 14. Dead-load Stresses. — The dead-load stress in the concrete may be written 
 
 Up 
 
 and 
 
 in which h is the height of chimney in feet above the section, and // is the steel unit stress in 
 compression. For prehminary estimate /c = 1.04/i (approx.). 
 
 0.5 
 
 Diagram 2 
 
 Percentage of longitudinal steel 
 1.0 1.5 20 25 3.0 35 4.0 
 
 05 
 
 1.0 1.5 ZO 2.5 3.0 3.5 4.0 
 Percentage of longitudinal steel 
 
 16. Stresses on Annular Sections in Flexure. — The following notation will be used (see 
 sketch, Diagram 2): 
 
 R = radius of center of chimney wall (inches) = also radius of reinforcing steel (inches). 
 t = thickness of wall (inches). 
 
 p = % of longitudinal reinforcement, based on cross-sectional area ( = A, Ac). 
 kR = perpendicular distance from center of chimney to neutral axis (inches). 
 
Sec. 18-15] MISCELLANEOUS STRUCTURES 817 
 
 The moment of the tension area of steel (arc NT A) about the neutral axis is 
 
 Sir r- h ~\ 
 
 2nptR''- r '2 (sin d - k)d9 = 2nptRA (1 - /c^)^^ + k sin'^k +^\ (1) 
 
 Jtt - sin-•^• L ^ J 
 
 The moment of the compression area, whose transformed stress width may be expressed by 
 [ 1 + (n — l)p]t, about the neutral axis is similarly 
 
 TT — sin~'A; 
 
 2RH[l + (n - l)p]J z (sin d - k)dd = 2Rh[1 + {n - l)p] \ {I - k^Y'^ + 
 
 2 L 
 
 A;sin-'A;-^J (2) 
 
 Since these two moments balance about the neutral axis, they may be equated, from whence 
 is derived the expression 
 
 [(1 _ k2y,i + /csin-i/c - ^1 
 
 P = p- ^- T . (3) 
 
 (1 - /c2)H + k s'm-'k - ^ - kirn] 
 
 Equation (3) has been plotted at the top of Diagram 2, assuming n = 15. 
 The moment of inertia of the transformed section may be written 
 
 /TT - sin-'A: Stt 
 ^ (sin d - 2A;sin0 + k'')d6 + 2R^pnt C 2 (sin^-d - 2k 
 
 2 J IT — sin"'A; 
 
 sin d + k^)dd = RH | (1 - p) [^(1 + 2A;2) - sin-^A;) - 3fc(l - k^Y^'A^ + irpn{\ + 2k'') | (4) 
 
 Let Cs = distance (inches) from the neutral axis to the extreme fiber of steel in tension 
 (on the transformed section). 
 Cs' = distance (inches) from the neutral axis to the extreme fiber of steel in compression 
 (on the transformed section), 
 and Ce = distance (inches) from the neutral axis to the extreme fiber of concrete in com- 
 pression. 
 
 Then Cs = Rn(l + k); c/ = Rn(l - /c); Cc = R{1 - k) (5) 
 
 If M is the bending moment in inch-pounds on any given cross-section due to external loads, then 
 from mechanics, 
 
 f,=Ml' f-^Mf = (6) 
 
 in which /«' = fiber stress of steel in compression, and fs and fc the same as in beam analysis. 
 By substituting equations (4) and (5) into equation (6) the curves of Diagram 1 were plotted, 
 assuming n = 15. This diagram does not include the stresses due to dead load, but gives only 
 those stresses due to bending. 
 
 It may be noted that jc relates to the compression in the concrete at the center of the chim- 
 ney wall, and not at the outer face (see sketch. Diagram 2). The maximum compression in 
 the concrete may be given by 
 
 /c(max) 
 
 The fractional term in the brackets is the increase over/c to obtain /c (max). The % increase 
 in concrete compression stress over/c to obtain /c (max) is plotted on Diagram 2(a) for various 
 values of t/R. The application of these curves relates to the stress due to flexure only. 
 52 
 
818 CONCRETE ENGINEERS' HANDBOOK [Sec. 18-16 
 
 Longitudinal Shear at Neutral Axis. — The intensity of longitudinal shearing stress at the 
 neutral axis may be given by 
 
 ^ = ^ (8) 
 
 for an annular section, in which V is the total transverse shear on the section and Mr equals 
 the sum of the moments of the tension and compression areas as given in equations (1) and (2). 
 Substituting into equation (8) there results 
 
 Rt 
 
 2 I (1 - p)[^(l + 2fc2) - - Ska - k^y^j +7rpn(l +2/c2) 
 
 (1 - - k^y^ + k sm-% - + '^Ma - k^y^ + k sin'^ k] 
 
 When n = 15 the term in the brackets becomes 3.15 for the usual ranges of k, whence 
 
 ^ = 3.15Z; (9) 
 
 From this equation the longitudinal shearing stress v may be computed. 
 
 16. Wind Stresses in Chimneys of Reinforced Concrete. — The action of wind pressure 
 alone upon a chimney is similar to the action of a distributed load on a cantilever beam of 
 annular section. The force of the wind is usually assumed, for a cylindrical surface, to be 
 two-thirds that on a plane surface. Thus, at 30 lb. per sq. ft., the pressure per foot of height of 
 chimney would be 20D, or 40JS. 
 
 The stresses caused by the wind may be determined by referring to Art. 15. A problem 
 will best illustrate the procedure. 
 
 Illustrative Problem. — Required % of vertical steel, and thickness of shell at the base of a chimney 200 ft. 
 high and 8 ft. mean diameter, such that the steel stress for wind shall not exceed 12,000 lb. per sq. in., and the con- 
 crete stress for wind shall not exceed 400 lb. per sq. in. 
 
 Moment on section = ilf = 20 X 8 X 200 X 12 X 100 = 38,400,000 in.-lb. 
 M 
 
 = 1575 for fs = 12,000 and fc = 400 (from Diagram 2) 
 
 _ 38.400,000 
 * ~ (4 X 12)2 X 1575 " ^^-^ 
 
 The vertical steel required is 3.65%. Should a smaller percentage be desirable, the steel stress would govern. 
 Thus with 3% vertical steel, M/R^t = 1320 for fs = 12,000. Then 
 
 38,400,000 
 ^ = (48)2 X 1320 
 
 This serves to illustrate the efifect of a governing stress when balanced stresses are not used. In the last instance 
 fc = 370 lb. per sq. in. 
 
 17. Chimney with No Vertical Reinforcement. — Short stacks, and the upper part of tall 
 stacks, may not have sufficient moment due to wind to cause tension on the windward side. 
 Assuming a wind pressure of 20 lb. per sq. ft. on a vertical projection, and noting that the 
 boundary of the kern of a thin hollow circular section is R/2 from the center, 
 
 y = 23.6 Rt (ft.) 
 
 in which y is the height of chimney in feet, above the section on which the resultant cuts the 
 kernal boundary, and R and t are the mean radius and thickness of shell, respectively, also in 
 feet. The compressive unit stress in the concrete, with no steel, is 
 
 W 
 
 fc = 2.08 ^ 
 
 where W is the weight above the section, and A is the sectional area in square inches. 
 
 The presence of vertical steel will affect the stresses, hence the formulas of Art. 15 will 
 be found useful for determining the flexural stresses. 
 
Sec. 18-18] 
 
 MISCELLANEOUS STRUCTURES 
 
 819 
 
 It should be noted that chimneys without vertical steel are subject to severe temperature 
 stresses if used under conditions where inside temperatures are those of flue gases, or under 
 conditions which give a temperature drop of 100°F. or more through the shell. 
 
 18. Longitudinal Shear in Chimneys. — Because the shell of concrete chimneys is relatively 
 thin, the unit shear on a longitudinal section requires consideration. Equation (9), Art. 15, 
 gives a means of solution of the longitudinal shearing stresses. The transverse shear 
 V = 2yR X 20 = 4:0yR, in which y is the height of chimney above the section considered 
 and R is in feet. From equation (9), 
 
 y = 0.079vt X 12 = O.MSvt. 
 When V = 40, the following table will give the relation of t to h: 
 
 t 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 10 
 
 11 
 
 12 
 
 h 
 
 151.2 
 
 189.0 
 
 226.8 
 
 264.6 
 
 302.4 
 
 340.2 
 
 378.0 
 
 415.8 
 
 453.6 
 
 When V = SO: 
 
 t 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 10 
 
 11 
 
 12 
 
 h 
 
 113.4 
 
 141.8 
 
 170.1 
 
 198.5 
 
 226.8 
 
 255.2 
 
 283.5 
 
 311.9 
 
 340.2 
 
 The above heights will give the limits for the given shearing unit stresses and thicknesses, on 
 the basis of a wind pressure of ^^(30) lb. per sq. ft. of projected area. 
 
 19. Temperature Stresses in Chimneys. — Flue gases have a temperature sufficient to 
 commonly give the chimney shell a temperature of 400° to 500°F. at the inner face near the 
 flue, and seldom exceed 700°F.i At a point three-quarters of the height above the base it is 
 found that the temperatures have not decreased more than 10 to 20% of the flue-level tem- 
 peratures. 
 
 The fact that the inner face of the shell tends to expand laterally and vertically causes 
 compression in the concrete and tension in the steel, in both directions. Assuming a constant 
 modulus of elasticity for concrete in compression for the above temperature range, and assuming 
 also a straight-line temperature gradient through the shell, Turneaure and Maurer have built 
 up a theory for the estimation of these temperature stresses. Applications of this analysis 
 yield a very high value of compression in the concrete vertically, and a moderate value (average 
 about 400 to 500 lb. per sq. in.) in a lateral direction. Appearance of large cracks in a lateral, 
 as well as vertical, direction, particularly near the top of the inner lining if one is present, bears 
 out the fact that large stresses do exist. 
 
 Prevention of temperature cracks cannot alone be made by heavier reinforcement. The 
 custom has been to extend a clay lining (in some instances though perhaps not in all cases war- 
 ranted, a fire-clay lining) from the flue-line to one-third the chimney height, having an air 
 space between the lining and the outer shell of 2 to 6 in. It is becoming evident, in the light 
 of past experience, that the lining should extend at least two-thirds the height of the chimney. 
 It is essential, also, that the air space between the lining and shell be provided with vents so 
 that a good circulation may be obtained. This is as important as any other feature of design. 
 
 Because of the meagre information concerning the properties of concrete under high tem- 
 peratures, it is not possible to build up a close theory of the stresses due to temperature. The 
 
 ' Report on reinforced-concrete chimneys to Assoc. Am. Port. Cem. Mfrs. by Sanford E. Thompson (1910). 
 
820 
 
 CONCRETE ENGINEERS' HANDBOOK 
 
 [Sec. 18-20 
 
 discussion of Turneaure and Maurer^ will offer a valuable guide to the estimation of the impor- 
 tance of these stresses, in the light of temperature data at present available. 
 
 20. Chimney Construction. — The shell is usually made with a smooth exterior in recently 
 constructed chimneys, and changes in thickness to obtain lighter sections at the top are made 
 by stepping the inner face. The forms used are usually steel either cylindrical or tapering, 
 and are jacked up on the vertical reinforcement (see construction of deep grain bins or silos, 
 Art. 5.), so that pouring is nearly continuous. Moderately dry 1 : 3 concrete is generally used, 
 and is well tamped. The mix should not be so dry that upon tamping, moisture is not brought 
 to the surface readily. Care should be taken to have a maximum silica (sand) surface on the 
 inside. The steel used may well be deformed, to distribute the temperature cracks. 
 
 21. Bases for Chimneys. — Chimneys are often placed on yielding soil, using a large base 
 slab. The common forms of bases are squares, circles, and octagons. Conditions of pressure 
 under these slabs maybe grouped in two cases: (1) resultant within the kern, (2) resultant out- 
 side the kern. 
 
 Kern sections for the forms of bases named are shown hatched in Fig. 13. 
 The section moduli, S, for these figures are given below: 
 
 Figure 
 
 Square 
 
 Circle 
 
 Octagon 
 
 S about 
 axis X-X 
 
 63 (about X'-X' 
 6~ 0.11863) 
 
 (approx.) 
 
 0.6906/2^ 
 
 When the resultant pressure lies within the kern of the base figure, the soil pressure at 
 the edge may be found by 
 
 W 
 
 Fig. 14. 
 
 Fig. 15. 
 
 A table of allowable soil pressures is given in Art. 1, Sect. 12. 
 
 The design of the base slab for moment and shear may be found in the design of footings, 
 Art. 7, Sect. 12. 
 
 1 "Principles of Reinforced Concrete Construction," 2d Ed., p. 413. 
 
Sec. 18-21] 
 
 MISCELLANEOUS STRUCTURES 
 
 821 
 
 Square Bases. — When tension exists between the foundation and the windward toe, there 
 is what might be considered a neutral axis and only part of the foundation is under pressure 
 (see Figs. 14 and 15). The following formulas result: 
 
 For direction of wind parallel to a side of the square (Fig. 14) 
 
 X = Sxq — ~ 
 
 pi 
 
 0-4") 
 
 w 
 
 62 
 
 Diagram 3 
 
 0 J 2 .3 .4 .5 .6 ,7 .8 .9 1.0 
 Values cff 1< 
 
 For direction of wind parallel to a long diameter of the square (Fig. 15) 
 
 1 
 
 Xo 
 
 r 
 
 Pi 
 
 7 3 _L 
 
 6 - 6A; + /c3 
 2- k W 
 
 2 — k W 
 If the quantity be denoted by L, then pi = L -r-' 
 
 Diagram 3 gives values of k and L for various values of — • Dotted lines with arrows indicate 
 
 r 
 
 how to obtain the values of k and L for a given value of 
 
 Xo 
 
SECTION 19 
 
 ESTIMATING 
 
 By Leslie H. Allen^ 
 ESTIMATING UNIT COSTS 
 
 1. Division of the Work. — Reinforced-concrete work may be considered under the follow- 
 ing four main divisions: 
 
 1. The concrete itself. 
 
 2. The forms or falsework. 
 
 3. The steel reinforcement. 
 
 4. The finish of the exposed surfaces. 
 
 Each of these divisions should be considered separately when making an estimate. 
 
 In the prices mentioned in this chapter, the cost of labor is based upon the rates paid in 
 the large cities at the present time (namely: carpenters 65 cts. per hr., laborers 35 cts. per hr., 
 carpenter foremen $7 per day) and include the overhead expense of superintendent and time- 
 keepers in charge of the work. 
 
 2. Estimating Unit Cost of Concrete. — With regard to the concrete itself, we shall take 
 into account: (a) the cement, sand, stone, and water; (b) the labor of unloading, mixing, and 
 placing of these materials; and (c) the plant necessary to accomplish this end. 
 
 2a. Materials. — The cost of cement in carload lots is, as a rule, about $1.50 
 per bbl. at the cement mill. If the cement is delivered in paper bags, there is an extra charge 
 of 10 cts. per bbl. for bags. If delivered in cloth, the extra charge is 40 cts., but this 40 cts. is 
 refunded by the mills if the cloth bags are returned in good condition. If furnished in wooden 
 barrels, there is a charge of 40 cts., and the barrel is not returnable. It is usually considered 
 the most economical to buy cement in cloth bags and return the bags when empty. The 
 freight on a barrel of cement on a haul of say 500 miles would be about 40 cts., so that the total 
 cost of a barrel of cement in cloth bags after returning and crediting the bags would be $1.90 
 per bbl. The cost of testing cement is from 3 to 5 cts. per bbl. It is customary among con- 
 tractors and engineers to have the whole shipment of cement tested at a testing laboratory, 
 and from $5 to $6 per carload is about the usual charge. The cost of unloading cement and 
 placing it in a storehouse close to the track is about 5 cts. per bbl. If the railroad tra cks 
 do not run to the site of the construction work, there must also be added the cost of teaming, 
 which would amount on a distance of 1 mile to about 5 cts. per bbl. In addition to this must 
 be figured the cost of handling and returning empty sacks, the freight on same, and the loss 
 of a few damaged or torn bags. This is usually estimated at about 3 cts. per bbl. Tabulating 
 the above, the cost of cement per bbl. ready for use would appear as follows: 
 
 Cement , $1.50 
 
 Freight 0.40 
 
 Cloth sacks 0.40 
 
 Total cost of cement f .o.b. cars at job $2 . 30 per bbl. 
 
 Deduct credit for empty sacks 0 . 40 
 
 $1.90 
 
 Add cost of testing 0.03 
 
 Add cost of unloading 0.05 
 
 Add cost of teaming, if any 0.05 
 
 Add cost of bundling and returning empty bags, and loss on same 0.03 
 
 Net price of cement ready for use in concrete $2 . 06 per bbl. 
 
 ' With Aberthaw Construction Co., Boston, Mass. 
 
 823 
 
824 
 
 CONCRETE ENGINEERS' HANDBOOK 
 
 [Sec. 19-2& 
 
 It is usual to obtain quotations from the cement companies for cement for jobs on which 
 estimates are being made. These quotations always include the freight and the bags, and to 
 arrive at the net cost it is necessary to deduct for the bags and add for the supplementary items, 
 according to the above list. 
 
 Sand usually costs about 50 cts. per cu. yd. to dig and load on teams or cars. If it has to 
 be screened or washed, it will cost from 60 to 70 cts. per cu. yd. Teaming or freight will vary 
 according to the length of haul, but will usually bring the cost of sand, ready for use, up to 
 $1.30 per cu. yd., f.o.b. the job. If it comes by rail, there should be added to this 15 cts. per 
 cu. yd. for unloading from cars. 
 
 Crushed stone can be bought at from $1.00 to $1.25 per ton at the crusher, to which must be 
 added the cost of teaming or freight, which will vary according to the length of haul. On a 
 haul of moderate length it is usual to pay from 30 to 50 cts. per ton, so that the cost of crushed 
 stone (f.o.b. the job ready for use) generally varies between $1.30 and $1.75 per ton. To this 
 should be added about 25 cts. per ton if it has to be unloaded from railroad cars. If gravel of 
 suitable size and quality is available for use, it can generally be obtained for $1.50 per cu. yd., 
 a considerable saving on the price of crushed stone. In comparing the price of gravel and 
 crushed stone, 1 cu. ft. of crushed trap rock or granite may be considered as weighing 100 lb. 
 
 Large bridges and other structures are sometimes built in places that are very difficult of 
 access and in consequence the cost of teaming materials may be much greater than above 
 mentioned. In some cases it may be found necessary to set up a crushing plant for the supply 
 of stone. The contractor, however, always avoids this, if possible, as the cost of operating a 
 small temporary plant is always greater than that of running a large permanent plant, and it 
 pays to buy crushed stone from such a plant, even if rock from the excavations is available. 
 
 26. Labor. — The labor of mixing and placing concrete varies considerably, 
 according to the conditions of the job and the nature of the work. It is obvious that to mix 
 and place concrete in heavy bridge abutments and concrete dams would cost a great deal less 
 than to place concrete in floor slabs 3 or 4 in. thick, in arch ribs, or in beam and column forms, 
 as the former would not require so much spading and spreading. 
 
 Assuming a well laid-out job and a machine mixer taking 4 bags to the batch, the cost of 
 mixing concrete should be from 70 to 80 cts. per cu. yd. The operations will consist of loading 
 wheelbarrows with sand and stone and wheeling them up to the mixer and charging same, bring- 
 ing cement from the cement shed and putting it into the mixer, and the work of an engineer 
 in running the mixer and discharging the concrete into wheelbarrows. 
 
 The cost of placing concrete will include the wheeling and dumping of the concrete in place, 
 and the spreading and spading of the concrete in the forms. This should cost about 90 cts. 
 per cu. yd. in average work. In columns and thin walls, where there is a lot of spading and where 
 care has to be used to get a good surface on the concrete, this cost would be about doubled. 
 On heavy masses of concrete, such as dams and thick retaining walls, these prices can be con- 
 siderably reduced, especially if the plant is well laid out and the equipment is good. Concrete 
 has been mixed and placed by mixer and derrick, or tracks and cars, for as low as 50 cts. per 
 cu. yd. 
 
 The following is an approximate schedule of labor prices for mixing and placing concrete: 
 
 Mixing and placing in footings $1 .75 per cu. yd. 
 
 Mixing and placing in floor slabs not exceeding 4^ in. thick. ... 2.00 per cu. yd. 
 
 Mixing and placing in floor slabs exceeding 5 in. thick 1.25 per cu. yd. 
 
 Mixing and placing in columns and thin walls 2 . 00 per cu. yd. 
 
 Mixing and placing in walls exceeding 18 in. in thickness 1.25 per cu. yd. 
 
 Mixing and placing in dams and thick retaining walls 1.00 per cu. yd. 
 
 On some jobs it is possible to unload sand and gravel direct from railroad cars to the wheel- 
 barrows which charge the mixer. In such cases the materials would be handled once instead 
 of twice before going into the mixer, and a saving of about 25 cts. per cu. yd. would result. 
 
Sec. 19-2c] 
 
 ESTIMATING 
 
 825 
 
 It is usual to specify that large stones may be embedded Jn massive concrete work to reduce 
 the cost of same. These stones are generally placed not less than 6 in. apart and are kept at 
 least 12 in. away from the face of the work. Some specifications will allow stones that one man 
 can handle; others will allow any stone that the derricks can lift. It will be found that from 20 
 to 50% of the volume of a massive pier can be composed of large stone in this way. The cost 
 of placing these stones, or "plums" as they are commonly called, should not exceed $1.25 per 
 cu. yd. If the rock has first to be excavated for the purpose, the cost of the rock excavation must 
 be added. 
 
 2c. Plant. — The cost of tools and plant and supplies on a job varies a good deal 
 according to the nature and size of the work. But, assuming a building job containing 5000 
 cu. yd. of concrete work carried out by a contractor of ability, it will usually be found that the 
 cost of plant, tools, and supplies, temporary buildings, office, cement shed, etc., will amount to 
 about $6000. Of this amount, about $1000 would be spent on labor in setting up the plant and 
 dismantling it; $300 for freight; $2000 for small tools and depreciation of mixer, hoisting 
 engines and large tools; and $2700 for coal or power, small tools, supplies and sundries. The 
 writer's practice is to estimate $1.25 for every yard of concrete on building jobs containing 
 between 4000 and 10,000 yd. of concrete. On larger jobs than this, the proportion would be 
 smaller, probably about 0.85 to $1.00 and, on smaller jobs than those containing 3000 yd., the 
 proportion would be higher — from $1.40 to $2.00 per cu. yd. On jobs containing less than 600 
 cu. yd. of concrete, machine mixing is not usually economical, and in that case it will be 
 necessary to estimate for mixing by hand, which will cost about $2 per cu. yd. more than the 
 prices given above for labor instead of including a charge for plant. 
 
 As a general rule, the plant required in bridge construction is more costly than that for 
 building construction. This is particularly the case where cableways are used. It is best to 
 calculate separately the cost of labor and depreciation on each item of plant in each case and add 
 the cost for supplies, fuel, and small tools. In general, however, it- will be found that the total 
 cost of plant for bridges, as in buildings, will vary pretty closely with the yardage of concrete. 
 In the writer's practice he has found that figures of $7000 for plant on a 5000-yd. job, $8000 on a 
 6000-yd. job, $6000 on a 4000-yd. job, and so on, check up quite closely with actual costs. 
 These figures, of course, do not include plant for excavating, drilling, etc. 
 
 2d. Summary. — The conditions on construction work do not approach those 
 of laboratory work, and there is always a considerable waste of cement, sand, and stone. It 
 has been found in practice, that, when estimating, it is not safe to allow less than the following 
 amounts of cement for different proportions of mix: 
 
 1:1^:3 mix 2 . 00 bbl. per cu. yd. 
 
 1:2:4 mix 1 . 66 bbl. per cu. yd. 
 
 1 : 23^ : 5 mix 1 . 40 bbl. per cu. yd. 
 
 1 : 3 : 6 mix 1 .20 bbl. per cu. yd. 
 
 The amount of sand and stone required varies considerably according to the percentage of 
 voids. This variation cannot be taken into account in the usual methods of estimating — as it 
 can only be ascertained by careful tests and varies from time to time, even when the source of 
 supply is the same — and therefore it is usual to allow 3-^ cu. yd. of sand per cu. yd. of 
 concrete, and 1 cu. yd. of crushed stone — ^figuring crushed stone to weigh 100 lb. per cu. ft. 
 
 The cost of 1 cu. yd. of concrete on a job containing 5000 cu. yd. of reinforced-concrete 
 work in floors and columns, etc., may be estimated as follows: 
 
 Cement 1.66 bbl. at $2.06 $3.42 
 
 Sand 0.5 yd. at $1.30 0.65 
 
 Stone 1.35 tons at $1.60 2.16 
 
 Labor 1.65 
 
 Plant 1.25 
 
 Total $9.13 per cu. yd. 
 
826 
 
 CONCRETE ENGINEERS' HANDBOOK 
 
 [Sec. 19-3a 
 
 The following illustrates the method of estimating the cost of concrete on a large typical 
 bridge job: 
 
 Abutments and piers — 1 : 2}4 - 5 mix: 
 
 Cement 1 . 4 bbl., @ $2 . 10 net $2 . 94 
 
 Sand 0.5 cu. yd.,@ 1.30 0.65 
 
 Crushed stone 1 . 35 tons, @1.60 2.16 
 
 Labor, mixing and placing 1 .00 
 
 Plant 1.30 
 
 Total $8.05 per cu. yd. 
 
 Abutments and piers — 1 : 2)4 ' 5 mix — with 30% of large stones: 
 
 7 cu. yd. concrete as above, @ $8 . 05 $56 . 35 
 
 3 cu. yd. placing large stones, @ 1 . 50 4 . 50 
 
 Arch ribs and deck slabs — 1 : 2 : 4 mix : 
 
 Cement 1 . 66 bbl., @ $2 . 10 
 
 Sand 0.5 cu. yd., @ 1.30 
 
 Crushed stone 1.35 tons, @ 1 . 60 . 
 
 $60 
 
 .85 
 
 6 
 
 .08 
 
 $3 
 
 .49 
 
 0 
 
 ,65 
 
 2. 
 
 16 
 
 1 
 
 .30 
 
 1 
 
 30 
 
 Total $8 . 90 per cu. yd. 
 
 The above costs do not include forms, steel, or finishing of surfaces. 
 3. Estimating Unit Cost of Forms. 
 
 3a. Considerations Involved. — Forms for building work should be measured by 
 the square foot of surface (measuring all sides that touch the concrete) and priced according 
 to the labor involved in erecting, studding, bracing and stripping. Strictly speaking, formwork 
 is the labor of supporting wet concrete, and the material — that is, the lumber used — is only 
 incidental to this labor; therefore it is not correct to take off the amount of lumber used and 
 price it according to the number of board feet thus estimated. It is the practice of some firms 
 to estimate forms by the latter method but such a practice is misleading, not only on the theo- 
 retical ground taken above, but on the practical ground that no two contractors would use the 
 same amount of lumber in erecting a given piece of formwork. Besides, it is not possible to 
 determine beforehand just how much lumber will be used for the same, since careful drawings 
 of the forms are not usually available. In the writer's practice, he has always found that even 
 though enough lumber to complete the work is ordered when a job is started it is always neces- 
 sary, on account of loss and waste of lumber, to order a great deal more before the job is finished. 
 The forms for a building consist principally of: 
 
 Forms to floors. Forms to walls. 
 
 Forms to beams and girders. Forms to footings. 
 
 Forms to columns. 
 
 Each of these should be measured separately and priced at separate and different rates. 
 
 The most uncertain and difficult item to estimate in bridge and similar construction is the 
 formwork. It is best to estimate by the square foot of surface contact, as in building work, 
 and to this add for the staging required for long arch spans. 
 
Sec. ld-36] 
 
 ESTIMATING 
 
 827 
 
 36. Materials. — The writer's experience, based on the cost accounts of many 
 reinforced-concrete buildings, large and small, shows that a good rule for estimating the cost 
 of lumber, nails, oil, and wire used in the construction of forms to a factory building, is to 
 reckon from $3.50 to $5.50 per 100 sq. ft. for these items, a good average being $4.00 per 100 
 sq. ft. This figure is higher than that used by a good many estimators at the present day, 
 but is based on actual experience of costs kept on a large number of jobs, and the writer is 
 constantly proving that this figure is correct. 
 
 In estimating bridge form work (the lumber, nails, oil, etc.) $5 per 100 sq. ft. should be 
 a sufficient allowance. 
 
 3c. Labor. — The cost of forms to floor slabs in buildings will take into account: 
 
 1. Making up panels. 
 
 2. Setting up posts and bracing same. 
 
 3. Putting girts and ledgers on tops of the posts. 
 
 4. Laying the panels on the same. 
 
 5. Stripping the centering after the concrete has set. 
 
 The labor of making up form panels will average about 43^ cts. per sq. ft., and as these are gener- 
 ally used 3 or 4 times, l}^ ct. per sq. ft. on the whole area is a good figure to use in esti- 
 mating. The labor of erecting studs and bracing will average about 3 cts. per sq. ft., and the 
 cost of putting on joists and girts and laying down panels will also cost about 3 cts. per sq. ft., 
 giving a total labor cost of erecting forms of 7)4 cts. per sq. ft. Add to this 1}4 ct. per sq. ft. for 
 stripping, and we get a labor cost of 8^ cts. per sq. ft. as a total. On a small and irregular build- 
 ing, of course this cost will be much higher, but for a plain, large factory a low cost such as this 
 can often be reached. If the height from floor to ceiling is over 16 ft., this price should be 
 increased, as the studs will have to be spliced and extra bracing will have to be put in. Two 
 or three cents a square foot should therefore be added in such a case. 
 
 The cost of beam and girder forms includes the cost of the following operations: 
 
 1. Making beam bottoms and beam sides. 
 
 2. Erecting same on the posts and joists which support the floor slab. 
 
 3. Stripping. 
 
 The cost of making up will be about 6 cts. per sq. ft., and as beam bottoms are left in longer 
 than the slab forms, it is not safe to figure on using these more than twice, giving an average 
 cost of 3 cts. per sq. ft. for making beam forms. Erecting the same will cost about 6 cts. per 
 sq. ft., and stripping 13^ ct. per sq. ft., giving a total cost of 10)^ cts. per sq. ft. for beam and 
 girder forms. If beams and girders are haunched at the ends, 50 cts. more should be allowed 
 each time for the haunching. 
 
 The labor of forming columns may be subdivided into: 
 
 1. Making up panels. 4. Bolting. 
 
 2. Erecting panels and placing yokes. 5. Stripping. 
 
 3. Plumbing. 
 
 The cost of making up panels, which are usually of 13^-in. stock, will be found to average 
 between 5 and 6 cts. per sq. ft. Allowing that these will be used 3 times, the cost per 
 square foot of formwork would be about 2 cts. The erecting, plumbing, and bolting will be 
 about 9 cts. per sq. ft., and the cost of stripping about 13^ cts. per sq. ft., giving a total labor 
 cost of 12)4 cts. per sq. ft. for labor on column forms. 
 
 Columns less than 8 ft. high cost a good deal more per square foot than higher columns, 
 owing to the fact that there is just as much time and labor spent in plumbing, erecting, and 
 bolting up as if the columns were twice as high with twice the amount of surface area. 
 
 In a similar way, the cost of erecting wall forms and footing forms nay be found. 
 
 No general rules or instructions can be given for estimating bridge and other formwork, 
 except to say that the labor should be estimated at not less than 13 cts. per sq. ft. 
 
828 CONCRETE ENGINEERS' HANDBOOK [Sec. 
 
 Zd. Summary. — The costs of forms per square foot to a reinforced-concrete 
 building may be tabulated as follows: 
 
 Forms to floor slabs: 
 
 Lumber, nails, oil, etc SO. 04 
 
 Labor making panels 0.015 
 
 Labor erecting studs and bracing 0 . 03 
 
 Labor laying panels 0 . 03 
 
 Labor stripping 0 . 0125 
 
 Total $0.1275 
 
 Forms to beams and girders: 
 
 Lumber, nails and oil $0.04 
 
 Labor making 0 . 03 
 
 Labor erecting 0 . 06 
 
 Labor stripping 0 . 015 
 
 Total „ $0,145 
 
 Forms to columns: 
 
 Lumber, nails, oil, etc $0.04 
 
 Labor making panels 0 . 02 
 
 Labor erecting, plumbing, and bolting 0.09 
 
 Labor stripping 0.015 
 
 Total $0,165 
 
 Forms to footings: 
 
 Lumber, nails, and oil $0 . 04 
 
 Labor making and erecting 0 . 08 
 
 Labor stripping 0.015 
 
 Total.... $0,135 
 
 Forms to walls: 
 
 Lumber, nails, oil, etc $0.04 
 
 Labor making 0 . 03 
 
 Labor erecting and plumbing 0 . 08 
 
 Labor stripping 0.015 
 
 Total $0,165 
 
 4. Estimating Unit Cost of Steel Reinforcement. — The cost of steel reinforcement, if 
 ordered in time to wait for delivery direct from the mill, will average about $3.00 per 100 lb. 
 The freight rate on a haul of about 500 miles will be about 18 cts., giving a total of $3.18 per 
 100 lb. for steel bars, f.o.b. cars at the job. At the time of publication of this book (Feb. 
 1918), deliveries are so very slow that the universal practice is to buy steel only from stock at a 
 delivered price of about $4.00 per 100 lb. The cost of steel bars varies according to the size 
 of the bar. Bars of from 1}4 to H in. diameter are taken at the lowest rate, which is called 
 the base price. Small bars take a higher rate as follows: 
 
Sec. 19-5] 
 
 ESTIMATING 
 
 829 
 
 From mill 
 
 From local 
 
 warehouse 
 
 % in. and i^'le in. 
 }i in. and %q in.. 
 
 base plus 5 cts. 
 base plus 10 cts. 
 base plus 20 cts. 
 base plus 25 cts. 
 .base plus 35 cts. 
 base plus 50 cts. 
 
 base plus 10 cts. 
 base plus 15 cts. 
 base plus 30 cts. 
 base plus 40 cts. 
 base plus 55 cts. 
 base plus 75 cts. 
 
 This differential is an important factor in design as well as in estimating. For example, assume 
 a floor 200 by 100 ft. having a slab 6 in. thick and an area of steel per square foot of 0.462 sq. 
 in. The total weight of steel required (allowing for laps) would be 33,301 lb. 
 
 Cost of H-in. square bars 6)^ in. on centers, 33,301 lb. @ $4.15 per 1001b. = $1381.95. 
 
 Cost of ^-in. round bars 3 in. on centers, 33,301 lb. '@ $4.40 per 100 lb. = $1465.20. 
 
 Thus there is a difference of $83.25 in favor of J^-in. bars if taken from stock deliveries. 
 It is usual in estimating on a building in normal times to figure on taking the steel for the foot- 
 ings and the lower part of the building, enough for the first 5 or 6 weeks work, out of local 
 warehouse stock and buy the rest direct from the mill. 
 
 The cost of unloading steel and piling it on the job is about 50 cts. per ton, and, if it has 
 to be teamed from the freight yards to the site of the work, this will cost 60 cts. per ton and 
 upward. 
 
 The cost of bending and placing steel in a building will vary according to the amount of 
 work that is done. Thus, placing steel bars ^^-in. diameter in a floor slab will cost about $6 
 per ton. If the bars have to be bent up at the end, $3 per ton should be added. Bending and 
 placing steel bars and stirrups in beams will cost from $8 to $10 per ton. Wiring up and placing 
 steel bars in columns and placing hoops around them will cost from $10 to $12 per ton. Placing 
 steel of ^-in. and %-in. diameter in walls will cost from $15 to $20 per ton. 
 
 These prices are sufficient to include the cost of wire, tools for bending, etc.; $12 per ton 
 is a good average price for labor on steel reinforcement all through the job. 
 
 The cost of steel reinforcement in a reinforced-concrete building may be estimated as 
 follows: 
 
 Steel from warehouse stock @$4.00 per 100 lb $80.00 per ton 
 
 Unloading, teaming, and piling 1.10 per ton 
 
 Labor, bending and placing 12.00 per ton 
 
 Tools, wire, and sundries 0 . 75 per ton 
 
 5. Estimating Unit Cost of Surface Finish. — One hundred square feet of granolithic finish 
 laid 1 in. thick in the proportion of 1 of cement, 1 of sand, and 1 of fine crushed stone, will require 
 1 bbl. of cement, 4 cu. ft. of sand, and 4 cu. ft. of crushed stone. This may therefore be esti- 
 mated as follows: 
 
 Total 
 
 $94 . 10 per ton 
 
 1 bbl. cement @$2 .06 per bbl. . . . 
 
 4 cu. ft. sand @$1 .35 per cu. yd, 
 
 400 lb. fine crushed stone. . . . @$2 .25 per ton. . . 
 
 $2.06 
 0.20 
 0.45 
 
 Labor mixing and placing 
 
 Finishers' time trowelling surface 
 
 $2.71 
 1.00 
 1.40 
 
 Total cost per 100 sq. ft. 
 
 $5.11 
 
830 
 
 CONCRETE ENGINEERS' HANDBOOK 
 
 [Sec. 19-6 
 
 If the surface of the concrete has to be cleaned off with acid or sand blasting, this will cost from 
 2 to 3 cts. per sq. ft. additional. 
 
 The exterior surfaces of concrete columns and beams are frequently rubbed smooth with 
 carborundum stone, using a little water and cement. ^ The cost of this work, including hanging 
 swing stages for the finishers, will be from 4 to 6 cts. per sq. ft. 
 
 For ornamental effect, external surfaces are sometimes picked with a pointed tool or 
 crandall hammer. The cost of this runs between 6 and 10 cts. per sq. ft. 
 
 ESTIMATING QUANTITIES 
 
 6. Systematic Procedure Advisable. — The operation of estimating quantities is that of 
 calculating (from plans supplied) quantities of labor and material which go to make up the 
 completed building. This is usually called taking off or scaling. This should be done quite 
 independently of the pricing or the arithmetical work of extending the quantities to obtain the 
 totals of the quantities of work. The secret of accurate, speedy taking off is to be found in a 
 systematic way of going about the work. No printed forms, tables, or special rules for taking 
 off will insure against error, but the surest way of making an accurate estimate is to have a good 
 system to work on and a clear and easily followed way of setting down the items. A good 
 method is to use plain squared paper 8 by 10}^ in., ruled in nine columns. In the first 
 column is placed a description of the items measured; the next four columns are for the 
 number, length, width, and height of the members of the building; the next two are for 
 arithmetical calculations and totals; and the last two for unit and total price. It is very im- 
 portant to keep length, breadth, and height in the same order in every item. Each can then 
 be readily identified. In taking off a reinforced-concrete building, start with the structural 
 members in the order in which they are built; that is, first take concrete footings, then columns, 
 then floor slabs, then beams and girders, then curtain walls and partitions, then cornice, and 
 then stairs and landings. Take all the concrete first, one item at a time and complete it. When 
 all the concrete is taken off, proceed to take off the forms, and after that take off the reinforce- 
 ment, and then the finish to the surfaces. After that take off excavation, windows and doors, 
 roofing and other incidental items necessary to complete the cost of the building. 
 
 In putting down the dimensions, it is well to put a note identifying each item, thus: 
 Concrete columns: 
 
 Basement, (mark A) 5 X 1>^ X 13^ X 10 
 
 (mark B) 4 X 1>^ X 1^^ X 10 
 
 (mark C) 7 X IH X IH X 10 
 
 First floor, (mark A) 5X13^X13^X10 
 
 (mark B) 4 X 13^ X 1^ X 10 
 
 (mark C) 7 X 13^ X 13^ X 10 
 
 It takes a little more time to do this, but it is well worth the labor, and any item can be readily 
 identified afterward. Also if an item is left out in error, it can be more easily detected. It 
 is good practice to put all dimensions in feet and fractions. Some estimators work in feet and 
 inches and some in feet and decimals. There seems to be the least chance for error in using 
 fractions, but this is a matter of individual judgment. 
 
 7. Riiles for Measurement of Concrete Work. — The following rules should govern the 
 measurement of concrete work: 
 
 All concrete should be measured by the cubic foot or cubic yard, and in all cases forms 
 should be measured separately. All concrete should be measured net as placed or poured in 
 the structure or building, and an excess measurement of concrete should never be taken to 
 pay for the cost of forms or extra labor in placing. All openings and voids in concrete should 
 be deducted, but no deduction should be made for steel reinforcement, I-beams, bolts, etc., 
 
 1 See Art. 526, Sect. 2. 
 
Sec. 19-8] 
 
 ESTIMATING 
 
 831 
 
 embedded in the concrete, except where such have a sectional area of more than 1 sq. ft. No 
 deduction should be made for chamfered, beveled, or splayed angles to columns, beams, and 
 other work. 
 
 For beams and girders it is usual to show on the plan the depth of concrete from the top 
 of the slab. Thus, if the quantity of concrete for the slab has been taken right through, it 
 will be necessary to consider only the extra concrete below the slab in taking off beam and 
 girder quantities. For example, in a floor 6 in. thick having 12 by 30-in. girders, the concrete 
 to take off for the girders should be considered as 1 ft. wide by 2 ft. deep, since the other 6 in. 
 is included in the slab. 
 
 Each class of concrete having a different proportion of cement, sand, or aggregate should 
 be measured and described separately. Concrete in the different members of a building or 
 structure should be measured and described separately according to the accessibility, location, 
 or purpose of the work; concrete in floor slabs should be measured and priced separately from 
 columns or walls, and so on. Concrete with large stones and rocks embedded in same (cyclo- 
 pean masonry) should be measured as one item and described according to the richness of the 
 mix and the percentage of rock in same. Concrete in stairs should be measured by the cubic 
 foot, and it is usual to include surface finish with same in this case, as it forms such a small 
 item in the cost. 
 
 8. Estimating Amount of Formwork. — Forms should be measured in square feet, taking 
 the area of the surface of the concrete which is actually touched by the forms or falsework. 
 Forms should in all cases be measured and described as a separate item and never included 
 with the concrete. No deduction should be made in measurement of surface of concrete sup- 
 ported by forms because of forms being taken down and re-used 2 or 3 times in the course of 
 construction. 
 
 It is not necessary to consider struts, posts, bracing, bolts, wire ties, oiling, cleaning, and 
 repairing forms, as these should be covered by the price put on the square foot measurements. 
 
 Forms to the different parts of a building should be measured and described separately 
 according to their nature; that is, forms to floor slabs, walls, columns, footings, etc., should be 
 separated from each other. No allowance need be made for angle fillets or bevels to beams 
 and columns, etc., but curved moldings should be measured and described separately. 
 
 No deduction in measurement of forms should be made for openings having an area of less 
 than 25 sq. ft. as the labor in forming same is often greater than the cost of the omitted area. 
 No deduction should be made in floor forms for heads of columns, or in column and girder forms 
 for ends of girders, cross beams, etc. No allowance should be made for pockets in column 
 forms for clearing out rubbish. 
 
 Fig. 1. 
 
 The correct measurement of column forms is the girth of the four sides, or circumference, 
 multiplied by the height from the floor surface to the under side of floor slab above. Forms 
 to octagonal, hexagonal, and circular cojumns should be measured and priced separately from 
 forms to square columns. Caps and bases to columns and other ornamental work should be 
 enumerated and fully described by sketches in the estimate with overall dimensions. 
 
 The correct measurement of beam forms is the net length between columns multiplied by 
 the sum of the breadth (5) and twice the depth below the slab {d), except for beams at edge of 
 floor or around openings, which shall have the thickness of the floor (0 added to the sum of the 
 breadth and twice the depth (see Fig. 1). 
 
832 
 
 CONCRETE ENGINEERS' HANDBOOK 
 
 [Sec. 19-9 
 
 Wall forms should be measured for both sides of concrete walls. 
 
 Forms to the upper side of sloping slabs such as saw-tooth roofs should be measured when- 
 ever the slope of such slab with the horizontal exceeds an angle of 25 deg. 
 
 Moldings in formwork should be measured by the linear foot. Forms to circular work 
 should always be measured separately from forms to straight work. 
 
 No measurement or allowance should be made for construction joints in slabs, beams, etc., 
 to stop the day's concreting, but construction joints or expansion joints in dams and other 
 large masses of concrete should be measured by the square foot as they occur. 
 
 Forms to cornices should be measured by the linear foot and the girth stated. Plain 
 forms to back of cornice should be measured separately. Forms to window sills, copings, and 
 similar work should be measured by the linear foot. Forms to the underside of stairs should 
 be measured by the superficial foot, and forms to the front edge by the linear foot. Forms to 
 the ends of steps should be measured by number. 
 
 9. Estimating Amount of Steel. — Reinforcing bars should be measured by the linear foot 
 and reduced to weight in pounds for pricing. The net weight placed in the building should be 
 taken and no allowance made for waste and cutting, or wire ties and spacers, etc., but laps should 
 be allowed for as called for by the plans or by the necessities of the design. Deformed bars 
 should be measured separately from plain. 
 
 The cost of bending and placing in columns and beams is greater than in slabs, but as the 
 difference is not great it is not usual to make any distinctions but to take olf the whole of the 
 steel together, except in special cases. 
 
 Pipe sleeves, turnbuckles, clamps, threaded ends, nuts, forgings, and other special items 
 should be measured separately by number and size, and allowed for in addition to the weight. 
 Wire cloth, expanded metal, and other steel fabrics sold in sheets are measured and described 
 by the square foot. The size of mesh and weight per square foot of steel will govern the price, 
 and should be stated. All laps should be measured and allowed for. 
 
 10. Estimating Amount of Surface Finish. — Finish of concrete surfaces should be measured 
 by the square foot. Finish should always be measured and described separately. No measure- 
 ment or allowance should be made for going over concrete work after removal of forms, and 
 patching up voids and stone pockets, removing fins, etc., as this is part of the labor incidental 
 to placing the concrete and the cost will depend upon the care used in spading the concrete 
 into the forms. 
 
 Granolithic finish should be measured by the square foot and should include all labor and 
 materials for the thickness specified. Finish laid integral with the slab should be measured 
 separately from finish laid after the slab has set. No allowance should be made for protection 
 of finish with sawdust, sand, or covering in to protect from weather. Grooved surfaces, gutters, 
 curbing, etc., should be measured separately from plain granolithic and should be measured 
 by the square foot or linear foot, as the case may require. 
 
 Putting on cement wash, rubbing with carborundum, scrubbing with wire brushes, tooling, 
 and picking, are other surface labors that should each be separately measured and priced. 
 The price should include the use of swing stages, tools, and materials required. 
 
APPENDIX A 
 
 STANDARD SPECIFICATIONS AND TESTS FOR PORTLAND CEMENT^ 
 
 These specifications are the result of several years' work of a special committee representing a United States 
 Government Departmental Committee, the Board of Direction of the American Society of Civil Engineers, and 
 Committee C-1 on Cement of the American Society for Testing Materials in cooperation with Committee C-1. 
 
 Specifications 
 
 1. Definition. — Portland cement is the product obtained by finely pulverizing clinker produced by calcining 
 to incipient fusion, an intimate and properly proportioned mixture of argillaceous and calcareous materials, with no 
 additions subsequent to calcination excepting water and calcined or uncalcined gypsum. 
 
 I. Chemical Properties 
 
 2. Chemical Limits. — The following limits shall not be exceeded 
 
 Loss on ignition, % 
 
 Insoluble residue, % 
 
 Sulphuric anhydride (S03),% 
 
 Magnesia (MgO), % 
 
 II. Physical Properties 
 
 3. Specific Gravity. — The specific gravity of cement shall be not less than 3.10 (3.07 for white Portland 
 cement). Should the test of cement as received fall below this requirement a second test may be made upon an 
 ignited sample. The specific gravity test will not be made unless specifically ordered. 
 
 4. Fineness. — The residue on a standard No. 200 sieve shall not exceed 22 % by weight. 
 
 5. Soundness. — A pat of neat cement shall remain firm and hard, and show no signs of distortion, cracking, 
 checking, or disintegration in the steam test for soundness. 
 
 6. Time of Setting. — The cement shall not develop initial set in less than 45 min. when the Vicat needle is used 
 or 60 min. when the Gillmore needle is used. Final set shall be attained within 10 hr. 
 
 7. Tensile Strength. — The average tensile strength in pounds per square inch of not less than three standard 
 mortar briquettes (see Sec. 51) composed of 1 part cement and 3 parts standard sand, by weight, shall be equal to or 
 higher than the following: 
 
 Age at test, 
 days 
 
 Storage of briquettes 
 
 Tensile strength, 
 lb. per sq. in. 
 
 7 
 
 28 
 
 1 day in moist air, 6 days in water 
 1 day in moist air, 27 days in water 
 
 200 
 300 
 
 8. The average tensile strength of standard mortar at 28 days shall be higher than the strength at 7 days. 
 
 III. Packages, Marking and Storage 
 
 9. Packages and Marking. — The cement shall be delivered in suitable bags or barrels with the brand and 
 name of the manufacturer plainly marked thereon, unless shipped in bulk. A bag shall contain 94 lb. net. A barrel 
 shall contain 376 lb. net. 
 
 4.00 
 0.85 
 2.00 
 5.00 
 
 1 These specifications and tests were adopted by letter ballot of the American Society for Testing Materials 
 on Sept. 1, 1916, and became effective Jan. 1, 1917. 
 
 53 8 33 
 
834 
 
 CONCRETE ENGINEERS' HANDBOOK 
 
 10. Sto/age. — The cement shall be stored in such a manner as to permit easy access for proper inspection and 
 identification of each shipment, and in a suitable weather-tight building which will protect the cement from 
 dampness. 
 
 IV. Inspection 
 
 11. Inspection. — Every facility shall be provided the purchaser foi; careful sampling and inspection at either 
 the mill or at the site of the work, as may be specified by the purchaser. At least 10 days from the time of sampling 
 shall be allowed for the completion of the 7-day test, and at least 31 days shall be allowed for the completion of the 
 28-day test. The cement shall be tested in accordance with the methods hereinafter prescribed. The 28-day test 
 shall be waived only when specifically so ordered. 
 
 V. Rejection 
 
 12. Rejection. — The cement may be rejected if it fails to meet any of the requirements of these specifications. 
 
 13. Cement shall not be rejected on account of failure to meet the fineness requirement if upon retest after 
 drying at 100°C. for 1 hr. it meets this requirement. 
 
 14. Cement failing to meet the test for soundness in steam may be accepted if it passes a retest using a new 
 sample at any time within 28 days thereafter. 
 
 15. Packages varying more than 5% from the specified weight may be rejected: and if the average weight of 
 packages in any shipment, as shown by weighing 50 packages taken at random, is less than that specified, the entire 
 shipment may be rejected. 
 
 Tests 
 
 VI. Sampling 
 
 16. Number of Samples. — Tests may be made on individual or composite samples as may be ordered. Each 
 test sample should weigh at least 8 lb. 
 
 17. (a) Individual Sample. — If sampled in cars one test sample shall be taken from each 50 bbl. or fraction 
 thereof. If sampled in bins one sample shall be taken from each 100 bbl. 
 
 (6) Composite Sample. — If sampled in cars one sample shall be taken from one sack in each 40 sacks (or 1 bbl. 
 in each 10 bbl.) and combined to form one test sample. If sampled in bins or warehouses one test sample shall repre- 
 sent not more than 200 bbl. 
 
 18. Method of Sampling. — Cement may be sampled at the mill by any of the following methods that may be 
 practicable, as ordered: 
 
 (o) From the Conveyor Delivering to the Bin. — At least 8 lb. of cement shall be taken from approximately each 
 100 bbl. passing over the conveyor. 
 
 (b) From Filled Bins by Means of Proper Sampling Tubes. — Tubes inserted vertically may be used for sampling 
 cement to a maximum depth of 10 ft. Tubes inserted horizontally may be used where the construction of the bin 
 permits. Samples shall be taken from points well distributed over the face of the bin. 
 
 (c) From Filled Bins at Points of Discharge. — Sufficient cement shall be drawn from the discharge openings to 
 obtain samples representative of the cement contained in the bin, as determined by the appearance at the discharge 
 openings of indicators placed on the surface of the cement directly above these openings before drawing of the 
 cement is started. 
 
 19. Treatment of Sample. — Samples preferably shall be shipped and stored in air-tight containers. Samples 
 shall be passed through a sieve having 20 meshes per linear inch in order to thoroughly mix the sample, break 
 up lumps and remove foreign materials. 
 
 VII. Chemical Analysis 
 Loss on Ignition 
 
 20. Method. — One gram of cement shall be heated in a weighed covered platinum crucible, of 20 to 25-c.c. 
 capacity, as follows, using either method (a) or (6) as ordered: 
 
 (a) The crucible shall be placed in a hole in an asbestos board, clamped horizontally so that about three-fifths 
 of the crucible projects below, and blasted at a full red heat for 15 min. with an inclined flame; the loss in weight 
 shall be checked by a second blasting for 5 min. Care shall be taken to wipe off particles of asbestos that may ad- 
 here to the crucible when withdrawn from the hole in the board. Greater neatness and shortening of the time of 
 heating are secured by making a hole to fit the crucible in a circular disc of sheet platinum and placing this disc over 
 a somewhat larger hole in an asbestos board. 
 
 (6) The crucible shall be placed in a muffle at any temperature between 900 and 1000°C. for 15 min. and the 
 loss in weight shall be checked by a second heating for 5 min. 
 
 21. Permissible Variation. — A permissible variation of 0.25 will be allowed, and all results in excess of the speci- 
 fied limit but within this permissible variation shall be reported as 4 %. - 
 
 Insoluble Residue 
 
 22. Method. — To a 1-gram sample of cement shall be added 10 c.c. of water and 5 c.c. of concentrated hydro- 
 chloric acid ; the liquid shall be warmed until effervescence ceases. The solution shall be diluted to 50 c.c. and di- 
 
APPENDIX A 
 
 835 
 
 gested, on a steam bath or hot plate until it is evident that decomposition of the cement is complete. The residue 
 shall be filtered, washed with cold water, and the filter paper and contents digested in about 30 c.o. of a 5 % solu- 
 tion of sodium carbonate, the liquid being held at a temperature just short of boiling for 15 min. The remaining 
 residue shall be filtered, washed with cold water, then with a few drops of hot hydrochloric acid, 1 : 9, and finally 
 with hot water, and then ignited at a red heat and weighed as the insoluble residue. 
 
 23. Permissible Variation. — A permissible variation of 0.15 will be allowed, and all results in excess of the 
 specified limit but within this permissible variation shall be reported as 0.85%. 
 
 Sulphuric Anhydride 
 
 24. Method. — One gram of the cement shall be dissolved in 5 c.c. of concentrated hydrochloric acid diluted 
 with 5 c.c. of water, with gentle warming; when solution is complete 40 c.c. of water shall be added, the solution fil- 
 tered, and the residue washed thoroughly with water. The solution shall be diluted to 250 c.c, heated to boiling 
 and 10 c.c. of a hot 10% solution of barium chloride shall be added slowly, drop by drop, from a pipette and the 
 boiling continued until the precipitate is well formed. The solution shall be digested on the steam bath until the 
 precipitate has settled. The precipitate shall be filtered, washed, and the paper and contents placed in a weighed 
 platinum crucible and the paper slowly charred and consumed without flaming. The barium sulphate shall then 
 be ignited and weighed. The weight obtained multiplied by 34.3 gives the percentage of sulphuric anhydride. The 
 acid filtrate obtained in the determination of the insoluble residue may be used for the estimation of sulphuric anhy- 
 dride instead of using a separate sample. 
 
 25. Permissible Variation. — A permissible variation of 0.10 will be allowed, and all results in excess of the 
 specified limit but within this permissible variation shall be reported as 2.00%. 
 
 Magnesia 
 
 26. Method. — To 0.5 gram of the cement in an evaporating dish shall be added 10 c.c. of water to prevent 
 lumping and then 10 c.c. of concentrated hydrochloric acid. The liquid shall be gently heated and agitated until 
 attack is complete. The solution shall then be evaporated to complete dryness on a steam or water bath. To 
 hasten dehydration the residue may be heated to 150 or even 200°C. for to 1 hr. The residue shall be treated 
 with 10 c.c. of concentrated hydrochloric acid diluted with an equal amount of water. The dish shall be covered 
 and the solution digested for 10 min. on a steam bath or water bath. The diluted solution shall be filtered and the 
 separated silica washed thoroughly with water. i Five cubic centimeters of concentrated hydrochloric acid and suf- 
 ficient bromine water to precipitate any manganese which may be present, shall be added to the filtrate (about 250 
 c.c). This shall be made alkaline with ammonium hydroxide, boiled until there is but a faint odor of ammonia, and 
 the precipitated iron and aluminum hydroxides, after settling, shall be washed with hot water, once by decantation 
 and slightly on the filter. Setting aside the filtrate, the precipitate shall be transferred by a jet of hot water to the 
 precipitating vessel and dissolved in 10 c.c of hot hydrochloric acid. The paper shall be extracted with acid, the 
 solution and washings being added to the main solution. The aluminum and iron shall then be reprecipitated at 
 boiling heat by ammonium hydroxide and bromine water in a volume of about 100 c.c, and the second precipitate 
 shall be collected and washed on the filter used in the first instance if this is still intact. To the combined filtrates 
 from the hydroxides of iron and aluminum, reduced in volume if need be, 1 c.c. of ammonium hydroxide shall be 
 added, the solution brought to boiling, 25 c.c. of a saturated solution of boiling ammonium oxalate added, and the 
 boiling continued until the precipitated calcium oxalate has assumed a well-defined granular form. The precipi- 
 tate after 1 hr. shall be filtered and washed, then with the filter shall be placed wet in a platinum crucible, and the 
 paper burned off over a small flame of a Bunsen burner; after ignition it shall be redissolved in hydrochloric acid and 
 the solution diluted to 100 c.c. Ammonia shall be added in slight excess, and the liquid boiled. The lime shall then 
 be reprecipitated by ammonium oxalate, allowed to stand until settled, filtered and washed. The combined fil- 
 trates from the calcium precipitates shall be acidified with hydrochloric acid, concentrated on the steam bath to 
 about 150 c.c, and made slightly alkaline with ammonium hydroxide, boiled and filtered (to remove a little alumi- 
 num and iron and perhaps calcium). When cool, 10 c.c. of saturated solution of sodium-ammonium-hydrogen 
 phosphate shall be added with constant stirring. When the crystallin ammonium-magnesium orthophosphate has 
 formed, ammonia shall be added in moderate excess. The solution shall be set aside for several hours in a cool place, 
 filtered and washed with water containing 2.5% of NH3. The precipitate shall be dissolved in a small quantity of 
 hot hydrochloric acid, the solution diluted to about 100 c.c, 1 c.c. of a saturated solution of sodium-ammonium- 
 hydrogen phosphate added, and ammonia drop by drop, with constan stirring, until the precipitate is again 
 formed as described and the ammonia is in moderate excess. The precipitate shall then be allowed to stand about 
 2 hr., filtered and washed as before. The paper and contents shall be placed in a weighed platinum crucible, the 
 paper slowly charred, and the resulting carbon carefully burned off. The precipitate shall then be ignited to con- 
 stant weight over a Meker burner, or a blast not strong enough to soften or melt the pyrophosphate. The weight of 
 magnesium pyrophosphate obtained multiplied by 72.5 gives the percentage of magnesia. The precipitate so 
 obtained always contains some calcium and usually small quantities of iron, aluminum, and manganese as 
 phosphates. 
 
 27. Permissible Variation. — A permissible variation of 0.4 will be allowed, and all results in excess of the 
 specified limit but within this permissible variation shall be reported as 5.00 % . 
 
 1 Since this procedure does not involve the determination of silica, a second evaporation is unnecessary. 
 
836 
 
 CONCRETE ENGINEERS' HANDBOOK 
 
 VIII. Determination of Specific Gravity 
 
 28. Apparatus. — The determination of specific gravity shall be made with a standardized Le Chatelier appa- 
 ratus which conforms to the requirements illustrated in Fig. 1. This apparatus is standardized by the United States 
 
 Bureau of Standards. Kerosene free from water, or 
 
 k- -- Scm ^ 
 
 |/./5c/77-;--^ 
 
 Capacity 
 of Bulk 
 approx. 
 ZSOcc ... 
 
 shall be used in 
 
 benzine not lighter than 62°B6. 
 making this determination. 
 
 29. Method.— The flask shall be filled with 
 either of these liquids to a point on the stem between 
 zero and 1 c.c. and 64 grams of cement, of the same 
 temperature as the liquid, shall be slowly introduced, 
 taking care that the cement does not adhere to the 
 inside of the flask above the liquid and to free the 
 cement from air by rolling the flask in an inclined 
 position. After all the cement is introduced, the 
 level of the liquid will rise to some division of the 
 graduated neck; the difference between readings is the 
 volume displaced by 64 grams of the cement. 
 
 The specific gravity shall then be obtained 
 from the formula 
 
 Specific gravity 
 
 Weight of cement (grams) 
 
 Displaced volume (c.c.) 
 
 30. The flask, during the operation, shall be 
 kept immersed in water, in order to avoid variations 
 in the temperature of the liquid in the flask, which 
 shall not exceed 0.5°C. The results of repeated tests 
 should agree within 0.01. 
 
 31. The determination of specific gravity shall 
 be made on the cement as received; if it falls below 
 3.10, a second determination shall be made after 
 igniting the sample as described in Sect. 20. 
 
 IX. Determination of Fineness 
 
 Have two 9.1 cc 
 Graduations extend 
 above I and 
 
 below 0 Mark- 
 
 32. Apparatus. — Wire cloth for standard sieves 
 
 for cement shall be woven (not twilled) from brass, 
 bronze, or other suitable wire, and mounted without 
 distortion on frames not less than in. below the top 
 of the frame. The sieve frames shall be circular, ap- 
 proximately 8 in. in diameter, and may be provided 
 with a pan and cover. 
 
 33. A standard No. 200 sieve is one having 
 nominally an 0.0029-in. opening and 200 wires per 
 inch standardized by the U. S. Bureau of Standards, 
 and conforming to the following requirements: 
 
 The No. 200 sieve should have 200 wires per 
 inch, and the number of wires in any whole inch shall 
 not be outside the limits of 192 to 208. No opening 
 between adjacent parallel wires shall be more than 
 0.0050 in. in width. The diameter of the wire should 
 be 0.0021 in. and the average diameter shall not be 
 outsiae the limits 0.0010 to 0.0023 in. The value of 
 the sieve as determined by sieving tests made in con- 
 formity with the standard specification for these 
 tests on a standardized cement which gives a residue 
 of 25 to 20 % . on the No. 200 sieve, or on other 
 similarly graded material, shall not show a variation of more than 1.5% above or below the standards maintained 
 at the Bureau of Standards. 
 
 34. Method. — The test shall be made with 50 grams of cement. The sieve shall be thoroughly clean and dry. 
 The cement shall be placed on the No. 200 sieve, with pan and cover attached, if desired, and shall be held in one 
 hand in a slightly inclined position so that the sample will be well distributed over the sieve, at the same time gently 
 striking the side about 150 times per minute against the palm of the other hand on the up stroke. The sieve shall 
 be turned every 25 strokes about one-sixth of a revolution in the same direction. The operation shall continue 
 until not more than 0.05 gram passes through in 1 min. of continuous sieving. The fineness shall be determined 
 from the weight of the residue on the sieve expressed as a percentage of the weight of the original sample. 
 
 j«- 9 cm 
 
 Fig. 1. — LeChatelier apparatus. 
 
APPENDIX A 
 
 837 
 
 35. Mechanical sieving devices may be used, but the cement shall not be rejected if it meets the fineness 
 requirement when tested by the hand method described in Sect. 34. 
 
 36. Permissible Variation. — A permissible variation of 1 will be allowed, and all results in excess of the speci- 
 fied limit but within this permissible variation shall be reported as 22 %. 
 
 X. Mixing Cement Pastes and Mortars 
 
 37. Method. — The quantity of dry material to be mixed at one time shall not exceed 1000 grams nor be less 
 than 500 grams. The proportions of cement or cement and sand shall be stated by weight in grams of the dry ma- 
 terials; the quantity of water shall be expressed in cubic centimeters (I c.c. of water = 1 gram). The dry materials 
 shall be weighed, placed upon a non-absorbent surface, thoroughly mixed dry if sand is used, and a crater formed in 
 the center, into which the proper percentage of clean water shall be poured; the material on the outer edge shall be 
 turned into the crater by the aid of a trowel. After an interval of H min. for the absorption of the water the opera- 
 tion shall be completed by continuous, vigorous mixing, squeezing and kneading with the hands for at least 1 min.i 
 
 38. The temperature of the room and the mixing water shall be maintained as nearly as practicable at 21°C. 
 (70°F.). 
 
 Fig. 2. — Vicat apparatus. 
 
 XI. Normal Consistency 
 
 39. Apparatus. — The Vicat apparatus consists of a frame A (Fig. 2) bearing a movable rod B, weighing 300 
 grams, one end C being 1 cm. in diameter for a distance of 6 cm., the other having a removable needle D, 1 mm. in 
 diameter, 6 cm. long. The rod is reversible, and can be held in any desired position by a screw E, and has midway 
 between the ends a mark F which moves under a scale (graduated to millimeters) attached to the frame A. The 
 paste is held in a conical, hard-rubber ring G, 7 cm. in diameter at the base, 4 cm. high, resting on a glass plate H 
 about 10 cm. square. 
 
 1 In order to secure uniformity in the results of tests for the time of setting and tensile strength the manner of 
 mixing above described should be carefully followed. At least one minute is necessary to obtain the desired plastic- 
 ity which is not appreciably affected by continuing the mixing for several minutes. The exact time necessary is 
 dependent upon the personal equation of the operator. The error in mixing should be on the side of over mixing. 
 During the operation of mixing, the hands should be protected by rubber gloves. 
 
838 
 
 CONCRETE ENGINEERS' HANDBOOK 
 
 40. Method. — In making the determination, 500 grams of cement, with a measured quantity of water, shall be 
 kneaded into a paste, as described in Sect. 37, and quickly formed into a ball with the hands, completing the opera- 
 tion by tossing it six times from one hand to the other, maintained about 6 in. apart; the ball resting in the palm of 
 one hand shall be pressed into the larger end of the rubber ring held in the other hand, completely filling the ring 
 with paste; the excess at the larger end shall then be removed by a single movement of the palm of the hand; the 
 ring shall then be placed on its larger end on a glass plate and the excess paste at the smaller end sliced off at the top 
 of the ring by a single oblique stroke of a trowel held at a slight angle with the top of the ring. During these opera- 
 tions care shall be taken not to compress the paste. The paste confined in the ring, resting on the plate, shall be 
 placed under the rod, the larger end of which shall be brought in contact with the surface of the paste; the scale shall 
 be then read, and the rod quickly released. The paste shall be of normal consistency when the rod settles to a point 
 10 mm. below the original surface in \2 min. after being released. The apparatus shall be free from all vibrations 
 during the test. Trial pastes shall be made with varying percentages of water until the normal consistency is 
 obtained. The amount of water required shall be expressed in percentage by weight of the dry cement. 
 
 41. The consistency of standard mortar shall depend on the amount of water required to produce a paste of 
 normal consistency from the same sample of cement. Having determined the normal consistency of the sample, the 
 consistency of standard mortar made from the same sample shall be as indicated in Table I, the values being in per- 
 centage of the combined dry weights of the cement and standard sand. 
 
 Table I. — Percentage of Water for Standard Mortars 
 
 Percentage of water 
 for neat cement 
 paste of normal 
 consistency 
 
 Percentage of water 
 for one cement, three 
 standard Ottawa 
 sand 
 
 Percentage of water 
 for neat cement 
 paste of normal 
 consistency 
 
 Percentage of water 
 for one cement, 
 three standard Ottawa 
 sand 
 
 15 
 
 9.0 
 
 23 
 
 10.3 
 
 16 
 
 9.2 
 
 24 
 
 10.5 
 
 17 
 
 9.3 
 
 25 
 
 10.7 
 
 18 
 
 9.5 
 
 26 
 
 10.8 
 
 19 
 
 9.7 
 
 27 
 
 11.0 
 
 20 
 
 9.8 
 
 28 
 
 11.2 
 
 21 
 
 10.0 
 
 29 
 
 11.3 
 
 22 
 
 10.2 
 
 30 
 
 11.5 
 
 XII. Determination of Soundness^ 
 
 42. Apparatus. — A steam apparatus, which can be maintained at a temperature between 98 arid 100°C., or 
 one similar to that shown in Fig. 3, is recommended. The capacity of this apparatus may be increased by using 
 a rack for holding the pats in a vertical or inclined position. 
 
 43. Method. — A pat from cement paste of normal consistency about 3 in. in diameter, in. thick at the cen- 
 ter, and tapering to a thin edge, shall be made on clean glass plates about 4 in. square, and stored in moist air for 24 
 hr. In molding the pat, the cement paste shall first be flattened on the glass and the pat then formed by drawing 
 the trowel from the outer edge toward the center. 
 
 44. The pat shall then be placed in an atmosphere of steam at a temperature between 98 and 100°C. upon a 
 suitable support 1 in. above boiling water for 5 hr. 
 
 45. Should the pat leave the plate, distortion may be detected best with a straight-edge applied to the surface 
 which was in contact with the plate. 
 
 XIII. Determination of Time of Setting 
 
 46. The following are alternate methods, either of which may be used as ordered: 
 
 47. Vicat Apparatus. — The time of setting shall be determined with the Vicat apparatus described in Sect. 
 39 (see Fig. 2). 
 
 48. Vicat Method. — A paste of normal consistency shall be molded in the hard-rubber ring G as described in 
 Sect. 40, and placed under the rod B, the smaller end of which shall then be carefully brought into contact with the 
 surface of the paste, and the rod quickly released. The initial set shall be said to have occurred when the needle 
 
 1 Unsoundness is usually manifested by change in volume which causes distortion, cracking, checking or 
 disintegration. 
 
 Pats improperly made or exposed to drying may develop what are known as shrinkage cracks within the 
 first 24 hours and are not an indication of unsoundness. These conditions are illustrated in Fig. 4. 
 
 The failure of the pats to remain on the glass or the cracking of the glass to which the pats are attached 
 does not necessarily indicate unsoundness. 
 
APPENDIX A 
 
 839 
 
 =1 
 
 
 
 
 -1- 
 
 
 C 
 
840 
 
 CONCRETE ENGINEERS' HANDBOOK 
 
APPENDIX A 
 
 841 
 
 ceases to pass a point 5 mm. above the glass plate in \i min. after being released; and the final set, when the needle 
 does not sink visibly into the paste. The test pieces shall be kept in moist air during the test. This may be accom- 
 plished by placing them on a rack over water contained in a pan and covered by a damp cloth, kept from contact 
 with them by means of a wire screen; or they may be stored in a moist closet. Care shall be taken to keep the 
 needle clean, as the collection of cement on the sides of the needle retards the penetration, while cement on the point 
 may increase the penetration. The time of setting is affected not only by the percentage and temperature of the 
 water used and the amount of kneading the paste receives, but by the temperature and humidity of the air, and its 
 determination is therefore only approximate. 
 
 49. Gillmore Needles. — The time of setting shall be determined by the Gillmore needles. The Gillmore 
 needles should preferably be mounted as shown in Fig. 56. 
 
 (a) Pat with top surface flattened for determining time of setting by Gilmore method. 
 
 '^/////////yy/y//////////////////////////////////^^^ 
 
 (b) Gillmore Needles. 
 Fig. 5. 
 
 60. Gillmore Method. — The time of setting shall be determined as follows: A pat of neat cement paste about 
 3 in. in diameter and l-i in. in thickness with a flat top Fig. 5a, mixed to a normal consistency, shall be kept in 
 moist air at a temperature maintained as nearly as practicable. at 21°C. (70°F.). The cement shall be considered to 
 have acquired its initial set when the pat will bear, without appreciable indentation, the Gillmore needle H2 in. in 
 diameter, loaded to weigh H lb. The final set has been acquired when the pat will bear without appreciable inden- 
 tation, the Gillmore needle l^i in. in diameter, loaded to weigh 1 lb. In making the test, the needles shall be held 
 in a vertical position, and applied lightly to the surface of the pat. 
 
 XIV. Tension Tests 
 
 61. Form of Test Piece. — The form of test piece shown in Fig. 6 shall be used. The molds shall be made of 
 non-corroding metal and have sufficient material in the sides to prevent spreading during molding. Gang molds 
 when used shall be of the type shown in Fig. 7. Molds shall be wiped with an oily cloth before using. 
 
 62. Standard Sand. — The sand to be used shall be natural sand from Ottawa, 111., screened to pass a No. 20 
 sieve and retained on a No. 30 sieve. This sand may be obtained from the Ottawa Silica Co., at a cost of 2 cts. per 
 lb., f.o.b. cars, Ottawa, 111. 
 
842 
 
 CONCRETE ENGINEERS' HANDBOOK 
 
 63. This sand, having passed the No. 20 sieve, shall be considered standard when not more than 5 grains pass 
 the No. 30 sieve after 1 min. continuous sieving of a .500-grain sample. 
 
 64. The sieves shall conform to the following specifications: 
 
 The No. 20 sieve shall have between 19.5 and 20.5 wires per whole inch of the warp wires and between 19 and 
 21 wires per whole inch of the shoot wires. The diameter of the wire should be 0.0165 in. and the average diameter 
 shall not be outside the limits of 0.0160 and 0.0170 in. 
 
 Fig. 6. — Details for briquette. 
 
 Fig. 7. — Gang mold. 
 
 The No. 30 sieve shall have bteween 29.5 and 30.5 wires per whole inch of the warp wires and between 28.5 
 and 31.5 wires per whole inch of the shoot wires. The diameter of the wire should be 0.0110 in. and the average 
 diameter shall not be outside the limits 0.0105 to 0.0115 in. 
 
 65. Molding. — Immediately after mixing, the standard mortar shall be placed in the molds, pressed in firmly 
 with the thumbs and smoothed off with a trowel without ramming. Additional mortar shall be heaped above the 
 mold and smoothed off with a trowel ; the trowel shall be drawn over the mold in such a manner as to exert a moder- 
 ate pressure on the material. The mold shall then be turned over and the operation of heaping, thumbing and 
 smoothing off repeated. 
 
APPENDIX A 
 
 843 
 
 66. Testing. — Tests shall be made with any standard machine. The briquettes shall be tested as soon as they 
 are removed from the water. The bearing surfaces of the clips and briquettes shall be free from grains of sand or 
 dirt. The briquettes shall be carefully centered and the load applied continuously at the rate of 600 lb. per min. 
 
 67. Testing machines should be frequently calibrated in order to determine their accuracy. 
 
 58. Faulty Briquettes. — Briquettes that are manifestly faulty, or which give strengths differing more than 
 15% from the average value of all test pieces made from the same sample and broken at the same period, shall not 
 be considered in determining the tensile strength. 
 
 XV. Storage of Test Pieces 
 
 69. Apparatus. — The moist closet may consist of a soapstone, slate or concrete box, or a wooden box lined with 
 metal. If a wooden box is used, the interior should be covered with felt or broad wicking kept wet. The bottom of 
 the moist closet should be covered with water. The interior of the closet should be provided with non-absorbent 
 shelves on which to place the test pieces, the shelves being so arranged that they may be withdrawn readily. 
 
 60. Methods. — Unless otherwise specified, all test pieces, immediately after molding, shall be placed in the 
 moist closet for from 20 to 24 hr. 
 
 61. The briquettes shall be kept in molds on glass plates in the moist closet for at least 20 hr. After 24 hr. in 
 moist air the briquettes shall be immersed in clean water in storage tanks of non-corroding material. 
 
 62. The air and water shall be maintained as nearly as practicable at a temperature of 21*^0. (70**F.). 
 
APPENDIX B 
 
 WORKING STRESSES 1 
 
 1. General Assumptions. — The following working stresses are recommended for static loads. Proper allow- 
 ances for vibration and impact are to be added to live loads where necessary to produce an equivalent static load 
 before applying the unit stresses in proportioning parts. 
 
 In selecting the permissible working stress on concrete, the designer should be guided by the working stresses 
 usually allowed for other materials of construction, so that all structures of the same class composed of different 
 materials may have approximately the same degree of safety. 
 
 The following recommendations as to allowable stresses are given in the form of percentages of the ultimate 
 strength of the particular concrete which is to be used; this ultimate strength is that developed at an age of 28 
 days, in cylinders 8 in. in diameter and 16 in. long, of the consistency described, 2 made and stored under laboratory 
 conditions. In the absence of definite knowledge in advance of construction as to just what strength may be 
 expected, the Committee submits the following values as those which should be obtained with materials and work- 
 manship in accordance with the recommendations of this report. 
 
 Although occasional tests may show higher results than those here given, the Committee recommends that 
 these values should be the maximum used in design. 
 
 Table of Compressive Strengths of Different Mixtures of Concrete 
 (In Pounds per Square Inch) 
 
 Aggregate 
 
 1 : 3* 
 
 1 : 4U* 
 
 1 : G* 
 
 1 -.IVi* 
 
 1 : 9* 
 
 Granite, trap rock 
 
 3,300 
 
 2,800 
 
 2,200 
 
 1,800 
 
 1,400 
 
 Gravel, hard limestone and hard 
 
 
 
 
 
 
 
 3,000 
 
 2,500 
 
 2,000 
 
 1,000 
 
 1,300 
 
 Soft limestone and sandstone. . . . 
 
 2,200 
 
 1,800 
 
 1,500 
 
 1,200 
 
 1,000 
 
 
 800 
 
 700 
 
 600 
 
 500 
 
 400 
 
 Note. — For variations in the moduli of elasticity see Sect. 8. 
 
 * Combined volume fine and coarse aggregate measured separately. 
 
 2. Bearing. — When compression is applied to a surface of concrete of at least twice the loaded area, a stress 
 of 35% of the compressive strength may be allowed in the area actually under load. 
 
 3. Axial Compression. — (a) For concentric compression on a plain concrete pier, the length of which does 
 not exceed 4 diameters, or on a column reinforced with longitudinal bars only, the length of which does not exceed 
 12 diameters, 22.5% of the compressive strength may be allowed. 
 
 (6) Columns with longitudinal reinforcement to the extent of not less than 1 % and not more than 4 % 
 and with lateral ties of not less than \i in. in diameter, 12 in. apart, nor more than 16 diameters of the longitudinal 
 bar: the unit stress recommended for (a). 
 
 (c) Columns reinforced with not less than 1% and not more than 4% . of longitudinal bars and with 
 circular hoops or spirals not less than 1 % of the volume of the concrete and as hereinafter specified :3 a unit 
 stress 55% higher than given for (a), provided the ratio of unsupported length of column to diameter of the 
 hooped core is not more than 10. 
 
 4. Compression in Extreme Fiber. — The extreme fiber stress of a beam, calculated on the assumption of a 
 constant modulus of elasticity for concrete under working stresses may be allowed to reach 32.5% of the compres- 
 sive strength. Adjacent to the support of continuous beams, stresses 15% higher may be used. 
 
 5. Shear and Diagonal Tension. — In calculations on beams in which the maximum shearing stress in a sec- 
 tion is used as the means of measuring the resistance to diagonal tension stress, the following allowable values for 
 the maximum vertical shearing stress in concrete, calculated by the method given in formula (22)^ are recommended: 
 
 1 From Final Report of the Special Committee on Concrete and Reinforced Concrete of the American Society 
 of Civil Engineers, presented before the Society, Jan. 17, 1917. 
 
 2 The materials should be mixed wet enough to produce a concrete of such a consistency as will flow sluggishly 
 into the forms and about the metal reinforcement when used, and which, at the same time, can be conveyed from 
 the mixer to the forms without separation of the coarse aggregate from the mortar. The quantity of water is of 
 the greatest importance in securing concrete of maximum strength and density; too much water is as objectionable 
 as too little. 
 
 3 Sec Art. 7, Sect. 8. 
 
 T 
 
846 
 
 CONCRETE ENGINEERS' HANDBOOK 
 
 (a) For beams with horizontal bare only and without web reinforcement, 2% of the compressive 
 strength. 
 
 (6) For beams with web reinforcement consisting of vertical stirrups looped about the longitudinal reinforcing 
 bars in the tension side of the beam and spaced horizontally not more than one-half the depth of the beam; or for 
 beams in which longitudinal bars are bent up at an angle of not more than 45 deg. or less than 20 deg. with the 
 axis of the beam, and the points of bending are spaced horizontally not more than three-quarters of the depth of the 
 beam apart, not to exceed of the compressive strength. 
 
 (c) For a combination of bent bars and vertical stirrups looped about the reinforcing bars in the tension 
 side of the beam and spaced horizontally not more than one-half of the depth of the beam, 5% of the compressive 
 strength. 
 
 id) For beams with web reinforcement (either vertical or inclined) securely attached to the longitudinal 
 bars in the tension side of the beam in such a way as to prevent slipping of bar past the stirrup, and spaced hori- 
 zontally not more than one-half of the depth of the beam in case of vertical stirrups and not more than three- 
 fourths of the depth of the beam in the case of inclined members, either with longitudinal bars bent up or not, 
 6% of the compressive strength. 
 
 The web reinforcement in case any is used should be proportioned by using two-thirds of the external vertical 
 shear in formulas (24) i or (25). 2 The effect of longitudinal bars bent up at an angle of from 20 to 45 deg. with the 
 axis of the beam, may be taken at sections of the beam in which the bent-up bars contribute to diagonal tension 
 resistance, as defined under Chap. VII, Sect. 8, as reducing the shearing stresses to be otherwise provided for. 
 The amount of reduction of the shearing stress by means of bent-up bars will depend upon their capacity, but in 
 no case should be taken as greater than 4J4% of the compressive strength of the concrete over the effective cross- 
 section of the beam (formula 22). 3 The limit of tensile stress in the bent-up portion of the bar calculated by 
 formula (25), 2 using in this formula an amount of total shear corresponding to the reduction in shearing stress 
 assumed for the bent-up bars, may be taken as specified for the working stress of steel, but in the calculations 
 the stress in the bar due to its part as longitudinal reinforcement of the beam should be considered. The stresses 
 in stirrups and inclined members when combined with bent-up bars are to be determined by finding the amount 
 of the total shear which may be allowed by reason of the bent-up bars, and subtracting this shear from the total 
 external vertical shear. Two-thirds of the remainder will be the shear to be carried by the stirrups, using formulas 
 (24)1 or (25)2. 
 
 Where punching shear occurs, provided the diagonal tension requirements are met, a shearing stress of 6% 
 of the compressive strength may be allowed. 
 
 6. Bond. — The bond stress between concrete and plain reinforcing bars may be assumed at 4% of the com- 
 pressive strength, or 2 % in the case of drawn wire. In the best types of deformed bar, the bond stress may be 
 increased, but not to exceed 5 % of the compressive strength of the concrete. 
 
 7. Reinforcement. — The tensile or compressive stress in steel should not exceed 16,000 lb. per sq. in. 
 
 In structural steel members, the working stresses adopted by the American Railway Engineering Association 
 are recommended. 
 
 8. Modulus of Elasticity. — The value of the modulus of elasticity of concrete has a wide range, depending 
 on the materials used, the age, the range of stresses between which it is considered, as well as other conditions. 
 It is recommended that, in computations for the position of the neutral axis, and for the resisting moment of beams, 
 and for compression of concrete in columns, it be assumed as: 
 
 (a) One-fortieth that of steel, when the strength of the concrete is taken as not more than 800 lb. per sq. in. 
 
 (b) One-fifteenth that of steel, when the strength of the concrete is taken as greater than 800 lb. per sq. in. 
 and less than 2200 lb. per sq. in. 
 
 (c) One-twelfth that of steel, when the strength of the concrete is taken as greater than 2200 lb. per sq. in 
 and less than 2900 lb. per sq. in., and 
 
 {d) One-tenth that of steel, when the strength of the concrete is taken as greater than 2900 lb. per sq. in. 
 
 Although not rigorously accurate, these assumptions will give safe results. For the deflection of beams 
 which are free to move longitudinally at the supports, in using formulas for deflection which do not take into account 
 the tensile strength developed in the concrete, a modulus of one-eighth of that of steel is recommended. 
 
 1 Vertical web reinforcement, 
 
 jd 
 
 2 Bars bent up at angles between 20 and 45 deg. with the horizontal and web members inclined at 45 deg., 
 
 A jd 
 
 (See Standard Notation, Appendix D.) 
 V 
 
 ~ bjd' 
 
APPENDIX C 
 
 RULINGS PERTAINING TO FLAT-SLAB DESIGN 
 
 Rulings have been adopted by the Cities of Pittsburgh and Chicago, by a Special Committee of the American. 
 Society of Civil Engineers, and by the American Concrete Institute, for the regulation of the design of fiat-slab 
 floors. These rulings are given below. Rulings have been adopted by other cities which are of merit, but those 
 of Pittsburgh and Chicago have been most widely used and properly applied have given satisfactory results. 
 
 The ruling of the American Concrete Institute is in many respects similar to that of the City of Chicago, 
 while the Report of the Special Committee of the American Society is similar as to method but more conservative 
 as to reinforcement, and in the light of all the test data available would seem to be in some respects ultra conservative. 
 
 For the method of application and a comparison of designs, the reader is referred to Art, 20, Sect, 11, page 
 
 487. 
 
 RULING ON THE DESIGN OF CANTILEVER FLAT-SLAB CONSTRUCTION IN 
 THE CITY OF PITTSBURGH 
 
 General. — The design and construction of reinforced-concrete flat slabs shall be carried out strictly in accord- 
 ance with all the provisions of the ordinance "Authorizing and Regulating the Use of Concrete and Reinforced 
 Concrete in the City of Pittsburgh," with special reference to Sect. 2, heading "Special Systems not Covered by 
 this Ordinance." 
 
 Reason for Ruling. — Since there has been a number and are likely to be more applications for building per- 
 mits, in which it is proposed to employ the flat-slab construction, and since there have been proposed several differ- 
 ent proprietary systems, it has become advisable to set forth a ruling with which each one must conform, and which 
 will state the interpretation of the ordinance and all its clauses which might relate to flat-slab construction as 
 made by this Bureau. 
 
 The ruling is expected to provide uniformity of the requirements among the several systems and equity 
 between them in preparing designs, and to so regulate their design and construction that they will be conducted 
 under correct principles and in conformity with the spirit of the ordinance. 
 
 Statement. — Inasmuch as the above-mentioned clause of the ordinance specified that the tests shall be made 
 to a breaking load in order to determine a factor of safety of 4, for loads for which the structure is planned, a state- 
 ment should be made to explain the reason for placing the ruling in its following form. 
 
 On account of the practical impossibility of so conducting a breaking load test that it would provide a basis 
 for a complete analysis which will prove the above required factor for safety and at the same time serve as a con- 
 vincing guide in the proper analysis for safe design and construction, as well as omitting to require a severe, 
 expensive and improper test which would not afford sufficient or consistent information, it has been formally 
 decided that: 
 
 First. — The tests shall have been conducted in such a manner as to demonstrate what are the actual strains 
 in the materials at the time working conditions are imposed upon the structure. This should also demonstrate 
 clearly the effect of an uneven or partial distribution of design loading, and that such loads will not cause the strains 
 in the materials to become substantially greater than those allowed for ordinary working conditions, and that an 
 unusual load or uncertainties of manufacture and installation of building materials will not result in weakness 
 or unsatisfactory result. 
 
 Second. — The test should be conducted in such a manner as to bring to light the relation which exists between 
 the strength of construction designed according to the provisions of the ordinance, and according to the provisions 
 of this ruling, and to show that all proper considerations have been provided for, therein. 
 
 Further, it is being understood that it is the intention of the ordinance to provide for safety of building con- 
 struction and equity among the different systems, it is decided that the requirements for all special systems as to 
 strains in the materials shall be the same as for ordinary construction as provided in the ordinance. 
 
 In view of the above reasoning, reports of strain-gage tests made by disinterested laboratories and testing 
 engineers from which it is possible to derive formulas, have been accepted and studied as they were submitted by 
 applicants. Inasmuch as the tests herein submitted have been conducted upon structures varying somewhat 
 from those designed and built under this ruling, in the matter of strength, and as there are matters to be deter- 
 mined under this ruling, which have so far not been determined under this ruling, which have so far not been de- 
 termined by test, it is sufficient that tests be conducted to determine: 
 
 First, the bending strains to which columns are subject within the interior of the structure as well as at the 
 
 walls. 
 
 Second, the proper percentage of increase in strength which should be provided in outer floor panels or wall 
 columns over that for interior construction, and for the effect upon the structure of uneven or partial distribution 
 of floor loading, such as placing the load between two rows of columns stretching clear across the structure than on 
 
 847 
 
848 
 
 CONCRETE ENGINEERS' HANDBOOK 
 
 interior or exterior panels. If it should be disclosed by such test that any provision of this ruling is too severe or 
 not severe enough, the ruling will be promptly revised only in so far as thus indicated. 
 
 Finally, it is herein determined that all systems admitted under this ruling shall have passed a strain-gage 
 test to demonstrate its abihty to conform to the requirements of this ruling before any further construction work 
 can be passed to a permit. 
 
 If it should be decided by all parties interested to conduct jointly one complete loading test which will also 
 be in all respects satisfactory to this Bureau, this test would be considered sufficient to allow all systems to be con- 
 structed without further test. 
 
 It being noted that the only additional information necessary will concern especially the points of uncertainty 
 described immediately preceding. And that additional tests will not be requested by this Bureau of those whose 
 systems are now approved. Tests by those whose systems are now approved will only be considered necessary 
 whenever these requirements are objected to by applicants as being too severe in the above-mentioned points. 
 
 Flat slabs as understood by this ruling shall consist of reinforced-concrete columns with enlarged capitals 
 on which is supported a flat reinforced slab floor with or without plates or depressed panels at the column cap. 
 The construction may be such as to admit the use of hollow panels in the ceiling or smooth ceiling with depressed 
 panel in the floor at the column cap. 
 
 The column capital shall be defined as the gradual flaring out of the column without any marked offset in 
 the concrete. 
 
 The depressed panel shall be defined as a square, rectangular or approximately circular depression around 
 the column capital extending below the adjacent slab. 
 
 The panel length shall be defined as the distance center to center of column of the side of a square panel, 
 and the panel length and breadth of a rectangular panel as the distance center to center of column in the long and 
 short directions respectively. The span length being defined as the distance in the clear from edge to edge of 
 column cap where it intersects either the floor slab or the depressed panel, measured on the length or breadth of 
 the panel as the case may be. 
 
 Requirements 
 
 Stresses. — All unit stresses shall be as specified in the ordinance governing the use of concrete and reinforced 
 concrete. The resisting moment and coincident stresses shall be computed under the assumption set forth in 
 the ordinance. 
 
 W U 
 
 Moments. — The negative bending moment at the support shall be taken — jj— in which W equals the total 
 
 load on one panel exclusive of any load within the area of the column capital, and L' is the clear span between 
 column capitals measured along the side of the panel. 
 
 WL 
 
 The positive moment at the center of the panel shall be taken as -jg- in which W is the total load on a panel 
 and L the distance center to center of columns measured along the side of the panel. 
 
 Resisting Sections. — The negative moment at the support shall be considered as acting on a vertical section 
 passing through the slab along the periphery of the column capital. The compressive stress in the concrete on 
 this section shall be calculated by the ordinary straight line assumptions of stress distribution, by the formulas 
 given in the ordinance, taking the periphery of the column capital, as the width of the section and the depth from 
 the lower face of the concrete adjoining the column capital to the center of gravity of the slab steel as the depth 
 of the section. 
 
 The area of slab steel resisting the negative moment of — — at the support shall be taken as the total section 
 of all slab rods cutting a conical critical section starting at the periphery of the column capital and flaring outwards 
 at a 45-deg. angle with the vertical. The spacing of rods thus determined for the width of the critical section shall 
 be maintained for the full width of the bands. 
 
 The positive moment at the center shall be resisted by the steel and concrete cut by a vertical critical section 
 through the slab having its center at the column center and its diameter equal to the main dimension of the sides 
 of the panel. 
 
 Drop Construction. — The thickness of the slab adjacent to the column capital may be increased, if necessary, 
 by means of a depending concrete drop panel centered on the column center. Where this drop panel is used the 
 resisting nioment^f the slab at the periphery of the drop shall be not less than that calculated from the formula 
 
 (lY 2L' "~ which W and L' are as defined above and X is the distance between the edge of the column 
 
 capital and circle of area equal to that of the drop used. This drop panel may be diminished in thickness at greater 
 distance from the column capital if desired provided the resisting moment at any section shall not fall below the 
 value determined by the above formula applied to that particular section. 
 
 Columns. — The columns shall be calculated for the unbalanced moment of the live floor load when the entire 
 area in one side of a fine through the column center is considered as loaded and the area on the other side unloaded. 
 The live-load reaction producing this moment shall be considered as uniformly distributed along the periphery 
 of the column capital. 
 
 Distribution of Slab Steel. — The computed area of steel per unit width of band determined from the moment 
 at the support shall be maintained the same for the entire width of each band. Width of bands shall be sufficient 
 to cause the whole area of the panel to be covered by reinforcing bars. The total steel area as determined from the 
 
APPENDIX C 
 
 849 
 
 computations may be distributed equally between all the bands or somewhat more than one-half may be placed 
 in the direct bands. In no case, however, shall the steel area of direct bands exceed that in diagonal bands by 
 more than one-third for square panels. 
 
 Rectangular Panels. — The slab thicknesses and the steel area in the various bands shall be determined in 
 rectangular panels by computations based on a square panel of the same dimensions. The steel area in the long 
 direct band shall be that required in the same band if situated on the edge of a square panel of the long dimensions 
 in size. The steel area in the short direct band shall be that required in the same band if situated on the edge of 
 a square panel of the short dimension in size. 
 
 The steel area in the diagonal bands shall be that required in the same band if situated diagonally in a square 
 panel whose size will be the average of the long and short dimensions. 
 
 W'L' 
 
 Profile of Column Capital. — The profile of the column capital shall satisfy the moment of at any con- 
 
 centric vertical section through it and the slab, the value of TF' and L' being taken for the particular section 
 considered. 
 
 Moments in Wall Panels. — The bending moment to be resisted by any band of reinforcing extending into 
 an exterior panel shall be increased by 20% if concrete column supports are present at the wall, and by 40% if 
 the slab rests on a brick wall at its exterior edge. 
 
 Concentrated Loads. — Girders and beams shall be provided where necessary to carry concentrated loads 
 in excess of the safe capacity of the floor slab. Such girders and beams shall be calculated to carry the full con- 
 centrated load. * 
 
 Openings Cut in Floors. — Girders and beams shall be provided on all sides of the opening wherever there are 
 openings or holes in the slab. They shall be calculated to carry the reaction of the floor on all sides of the openings. 
 
 RULING COVERING DESIGN OF FLAT-SLAB CONSTRUCTION IN THE 
 
 CITY OF CHICAGO 
 
 (For ruling as amended Jan. 1, 1918, see page 851.) 
 
 1. Definitions. — Flat slabs as understood by this ruling are reinforced-concrete slabs supported directly 
 on reinforced columns with or without plates or capitals at the top, the whole coastruction being hingeless and mono- 
 lithic without any visible beams or girders. The construction may be such as to admit the use of hollow panels 
 in the ceiling or smooth ceiling with depressed panels in the floor. 
 
 2. The column capital shall be defined as the gradual flaring out of the top of the column without any marked 
 
 offset. 
 
 3. The drop panel shall be defined as a square or rectangular depression around the column capital extending 
 below the slab adjacent to it. 
 
 4. The panel length shall be defined as the distance center to center of columns of the side of a square panel, 
 or the average distance center to center of columns of the long and short sides of a rectangular panel. 
 
 6. Columns. — The least dimensions of any concrete column shall be not less than one-twelfth the panel 
 length, or one-twelfth the clear height of the column. 
 
 6. Slab Thickness. — The minimum total thickness of the slab in inches shall be determined by the formula: 
 
 t = 0.023L\/w 
 
 where 
 
 t = total thickness of slab in inches. 
 L = panel length in feet. 
 
 w = total live and dead load in pounds per square foot. 
 
 7. In no case shall the slab thickness be less than one-thirty-second of the panel length for floors, and one- 
 fortieth of the panel length for roofs, and also not less than 6 in. 
 
 8. Column Capital. — The diameter of the column capital shall be measured where its vertical thickness is at 
 least VA in., and shall be at least 0.225 of the panel length. 
 
 9. The slope of the column capital shall nowhere make an angle with the vertical of more than 45 deg. Spe- 
 cial attention shall be given to the design of the column capital in considering eccentric loads, and the effect of wind 
 upon the structure. 
 
 10. Drop Panel. — The depth of the drop shall be determined by computing it as a beam, using the negative 
 bending moment specified elsewhere in this ruling. The width and length shall be determined by the allowable 
 unit shearing stresses on the perimeter, given below. 
 
 11. Shearing Stresses. — The allowable unit punching shear on the perimeter of the column capital shall be 
 three-fiftieths of the ultimate compressive strength of the concrete as given in Sect. 546 of the building ordinance. 
 The allowable unit shear on the perimeter of the drop panel shall be three one-hundredths of the ultimate compres- 
 sive strength of the concrete. In computing shearing stress for the purpose of determining the resistance to diag- 
 onal tension the method specified by the ordinance shall be used. 
 
 12. Panel Strips. — For the purpose of establishing the bending moments and the resisting moments of a 
 square panel, the panel shall be divided into strips known as strip A and strip B. Strip A shall include ihe rein- 
 forcement and slab in a width extending from the center line of the columns for a distance each side of this center 
 line equal to one-fourth of the panel length. Strip B shall include the reinforcement and slab in the half width re- 
 
 54 
 
850 
 
 CONCRETE ENGINEERS' HANDBOOK 
 
 maining in the center of the panel. At right angles to these strips, the panel shall be divided into similar strips A 
 and B, having the same widths and relations to the center line of the columns as the above strips. These strips 
 shall be for designing purposes only, and are not intended as the boundary lines of any bands of steel used. 
 
 13. These strips shall apply to the system of reinforcement in which the reinforcing bars are placed parallel 
 and at right angles to the center line of the columns, hereinafter known as the two-way system, and also to the sys- 
 tem of reinforcement in which the reinforcing bars are placed parallel, at right angles to and diagonal to the center 
 line of the columns hereinafter known as the four-way system. 
 
 Bending Moment Coefficients, Interior Panel, Two-way System 
 
 14. The negative bending moment taken at a cross-section of each strip A at the edge of a column capital or 
 over it, shall be taken as The positive bending moment taken at a cross-section of each strip A, midway 
 between column centers shall be taken as WL^/30. The positive bending moment taken at a cross-section of each 
 strip B in the middle of the panel shall be taken at WL'^/60. The negative bending moment taken at a cross-sec- 
 tion of each strip B on the center line of the columns shall be taken at WL^/QO, In the formulas hereinabove given 
 
 W = total live and dead load per lineal foot of each strip. 
 L = panel length in feet. 
 
 Bending Moment Coefficients, Interior Panel, Four-way System 
 
 15. The negative bending moment taken at a cross-section of each strip A at the edge of the column capital or 
 over it, shall be taken as WL^/ 15. The positive bending moment taken at a cross-section of each strip A, midway 
 between column centers shall be taken as WL^/4:0. The positive bending moment taken at a cross-section of each 
 strip B in the middle of the panel shall be taken as WL^/60, The negative bending moment taken at a cross-section 
 of each strip B on the center line of the column shall be taken at WL'^/GO. 
 
 Bending Moment Coefficients, Wall Panels 
 
 16. Wherever the coefficients Ms, Ho, Mo or \io appear in the moments given for interior panels in either the 
 two-way or the four- way systems, the coefficients M2, }/25, Vzz and Ybo respectively shall be used in the moments 
 for wall panels supported on concrete columns and girders. 
 
 17. When brick walls are used partly to support wall panels, these walls shall be stiffened by pilasters or piers 
 as directed by the Commissioner of Buildings. Wherever the coefficients Hs, J^o, Vio or Y^o appear in the mo- 
 ments given for interior panels in either the two-way or the four-way systems, the coefficients Mo, }^o, Vii and Mq 
 respectively shall be used in the moments for such panels resting on brick walls. 
 
 Point of Inflection 
 
 18. For the purpose of making the calculations of the bending moment at the sections away from the column 
 capital, the point of inflection shall be considered as being one-quarter the distance center to center of columns, both 
 crosswise and diagonally, from the center of the column. 
 
 Tensile Stress in Steel and Compressive Stress in Concrete 
 
 19. The tensile stress in steel and the compressive stress in the concrete to resist the bending moment shall 
 be calculated on the basis of the reinforcement and slab in the width included in a given strip, and according to the 
 assumptions and requirements given in Sects. 545 to 548 inclusive of the building ordinance. 
 
 20. The steel shall be considered as being concentrated at the center of gravity of all the bands of steel in a 
 given strip. 
 
 21. For the four-way system of reinforcement the amount of steel to resist the negative bending moment ovei 
 the support in each strip A shall be taken as the sum of the areas of steel in one cross band and one diagonal band 
 The amount of steel to resist the positive bending moment of each strip B shall be considered as the area of the steel 
 in a diagonal band. The amount of steel to resist the positive bending moment in each strip A shall be considered 
 as the area of the steel in a cross-band, and the amount of steel to resist the negative moment in each strip B shall be 
 the steel included in the width of strip B. 
 
 22. For the two-way system of reinforcement the amount of steel to resist the bending moment in any strip 
 shall be considered as the area of steel included in the width of the strip. 
 
 23. In both systems of reinforcement the compressive stress in the concrete in any strip shall be calculated by 
 taking the area of steel considered for each strip, and applying it in a beam formula based on the principles of Sect. 
 548 of the building ordinance. 
 
 24. When the length of a panel does not exceed the breadth by more than 5 %, all computations shall be made 
 on the basis of a square with sides equal to the mean of the length and breadth. In no rectangular panel shall the 
 length exceed four-thirds the breadth. 
 
 25. For panels with length more than 5 % in excess of the breadth, the slab shall first be designed for a bending 
 moment based on an assumed square panel with sides equal to the mean of the length and breadth of the rectangu- 
 lar panel. 
 
APPENDIX C 
 
 851 
 
 26. For the four-way system of reinforcement the amount of steel found for the positive moment of each 
 strip B by designing in this manner shall be that used in the diagonal band. For the positive moment in each 
 strip A, the required amount of steel in the cross-band shall be obtained by multiplying the steel used in the design 
 of the assumed square panel by the cube of the ratio found by dividing the length or breadth of the rectangular panel 
 by the side of the assumed square panel, for the long and short sides of the panel respectively. The compressive 
 stresses shall be calculated on the basis of a width equal to one-half of the side of the assumed square panel, and on 
 the assumptions used in the calculations of compressive stresses in square panels. In no case shall the amount of 
 steel in the short side be less than two-thirds of that required for the long side. 
 
 27. For the two-way system of reinforcement, the amount of steel found for the positive and negative moment 
 of each strip B by designing in this manner shall be obtained by multiplying the steel used in the design of the 
 assumed square panel by the cube of the ratio found by dividing the length or breadth of the rectangular panel by 
 the side of the assumed square panel, for the short and long side of the panel respectively. The mfethod of obtaining 
 the amount of steel required for each strip A, shall be the same as that given above for the four-way system. 
 
 28. Walls and Openings. — Girders or beams shall be constructed under walls, and around openings and to 
 carry concentrated loads. 
 
 29. Computations. — Complete computations of interior and wall panels and such other portions of the build- 
 ing as may be required by the Commissioner of Buildings shall be left in the office of the Commissioner of Buildings 
 when plans are presented for approval. 
 
 30. Placing of Steel. — In order that the slab bars shall be maintained in the position shown in the design dur- 
 ing the work of pouring the slab, spacers and supports shall be provided satisfactory to the Commissioner of Build- 
 ings. All bars shall be secured in place at intersections by wire or other metal fastenings. In no case shall the 
 spacing of the bars exceed 9 in. The steel to resist the negative moment in each strip B shall extend one-fourth of 
 the panel length beyond the center line of the columns in both directions. 
 
 31. All splices in bars shall be made over the column head. The length of the splice beyond the center line 
 of the column in both directions shall be at least 2 ft. nor less than that necessary for the full development of the 
 strength of the bar as limited by the unit bond stresses given by the ordinance. The splicing of adjacent bars shall 
 be avoided as far as possible. 
 
 32. Slab bars which are lapped over the column, the sectional area of both being included in the calculations 
 for negative moment, shall extend not less than 0.25 of the panel length for cross-bands, and 0.35 of the panel 
 length for diagonal bands, beyond the column center. 
 
 33. Test of Workmanship. — The Commissioner of Buildings or his representative may choose any two adja- 
 cent panels in the building for the purpose of ascertaining the character of workmanship. The test shall not be 
 made sooner than the time required for the cement to set thoroughly, nor less than 6 weeks after the concrete had 
 been poured. 
 
 34. All deflections under test load shall be taken at the center of the slab, and shall be measured from the nor- 
 mal unloaded position of the slab. The two panels selected shall be uniformly loaded over their entire area with a 
 load equal to the dead load plus twice the live load, thus obtaining twice the total design load. The load shall remain 
 in place not less than 24 hr. If the total deflection in the center of the panel under the test load does not exceed one 
 eight-hundredth of the panel length, the slab may be placarded to carry the full design live load. If it exceeds this 
 amount of deflection, and recovers not less than 80% of the total deflection within 7 days after the load is removed, 
 the slab may be placarded to carry the full design live load. If the deflection exceeds the allowable amount above 
 specified, and the recovery is less than 80% in 7 days after the removal of the test load, other tests shall be made on 
 the same or other panels, the results of which will determine the amount of live load the slabs will be permitted to 
 carry. 
 
 35. General. — The design and execution of the work shall conform to the provisions of the Chicago building 
 ordinances, and to correct principles of construction. 
 
 CHICAGO REINFORCED -CONCRETE FLAT-SLAB RULING AMENDED 
 
 (New ordinance adopted Jan. 1, 1918, while this book was in press) 
 
 1. Definitions. — Flat slabs as understood by this ruling are reinforced-concrete slabs, supported directly on 
 reinforced columns with or without plates or capitals at the top, the whole construction being hingeless and mono- 
 lithic without any visible beams or girders. The construction may be such as to admit the use of hollow panels in 
 the ceiling or smooth ceiling with depressed panels in the floor. 
 
 2. The column capital shall be defined as the gradual flaring out of the top of the column without any marked 
 
 offset. 
 
 3. The drop panel shall be defined as a square or rectangular depression around the column capital extending 
 below the slab adjacent to it. 
 
 4. The panel length shall be defined as the distance c. to c. of columns of the side of a square panel, or the 
 average distance c. to c. of columns of the long and short sides of a rectangular panel. 
 
 6. Columns. — The least dimension of any concrete column shall be not less than one-twelfth the panel length, 
 nor one-twelfth the clear height of the column. 
 
 6. Slab Thickness. — The minimum total thickness of the slab in inches shall be determined by the formula: 
 Vi 
 
 t = W /44 ( = square root of W divided by 44), where t = total thickness of slab in inches, W = total live load and 
 dead load in pounds on the panel, measured c. to c. of columns. 
 
852 
 
 CONCRETE ENGINEERS' HANDBOOK 
 
 7. In no case shall the thickness be less than of the panel length (I//32) for floors, nor >io of the panel 
 length (L/40) for roofs {L being the distance c, to c. of columns). 
 
 8. In no case shall the thickness of slab be less than 6 in. for floors or roofs. 
 
 9. Column Capital. — When used the diameter of the column capital shall be measured where its vertical 
 thickness is at least in. and shall be at least 0.225 of the panel length. 
 
 The slope of the column capital shall nowhere make an angle with the vertical of more than 45 deg. Special 
 attention shall be given to the design of the column capital in considering eccentric loads, and the effect of wind 
 upon the structure. 
 
 10. Drop Panel. — When used, the drop panel shall be square or circular for square panels and rectangular or 
 elliptical for oblong panels. 
 
 11. The length of the drop shall not be less than one-third of the panel length (L/3) if square, and not less than 
 one-third of the long or short side of the panel respectively, if rectangular, 
 
 12. The depth of the drop panel shall be determined by computing it as a beam, using the negative moment 
 over the column capital specified elsewhere in this ruling. 
 
 13. In no case, however, shall the dimensions of the drop panel be less than required for punching shear along 
 its perimeter, using the allowable unit shearing stresses specified below. 
 
 14. Shearing Stresses. — The allowable unit punching shear on the perimeter of the column capital shall be 
 %o of the ultimate compressive strength of the concrete as given in Sect. 533 of the building ordinance. The 
 allowable unit shear on the perimeter of the drop panel shall be 0.03 of the ultimate compressive strength of the 
 concrete. In computing shearing stress for the purpose of determining the resistance to diagonal tension the 
 method specified by the ordinance shall be used. 
 
 16. Panel Strips. — For the purpose of establishing the bending moments and the resisting moments of a square 
 panel, the panel shall be divided into strips known as strip A and strip B. Strip A shall include the reinforcement 
 and slab in a width extending from the center line of the columns for a distance each side of this center line equal to 
 one-quarter of the panel length. Strip B shall include the reinforcement and slab in the half width remaining in the 
 center of the panel. At right angles to these strips, the panel shall be divided into similar strips A and B, having 
 the same widths and relations to the center line of the columns as the above strips. These strips shall be for design- 
 ing purposes only, and are not intended as the boundary lines of any bands of steel used. 
 
 16. These strips shall apply to the system of reinforcement in which the reinforcing bars are placed parallel 
 and at right angles to the center line of the columns, hereinafter known as the two-way system, and also to the system 
 of reinforcement in which the reinforcing bars are placed parallel, at right angles to and diagonal to the center line 
 of the columns hereinafter known as the four-way system. 
 
 17. Any other system of reinforcement in which the reinforcing bars are placed in circular, concentric rings and 
 radial bars, or systems with steel rods arranged in any manner whatsoever, shall comply with the requirements of 
 either the two-way or the four-way system herein specified. 
 
 18. Bending Moment Coefficients, Interior Panel, Two-way System. — In panels where standard drops and 
 column capitals are used as above specified, the negative bending moment, taken at a cross-section of each strip A at 
 the edge of the column capital or over it, shall be taken as PFZ//30. 
 
 19. The positive bending moment taken at a cross-section of each strip A midway between column centers 
 shall be taken as WL/m. 
 
 20. The positive bending moment taken at a cross-section of each strip B in the middle of the panel shall be 
 taken as WL/ 120. 
 
 21. The negative bending moment taken at a cross-section of each strip B on the center line of the columns 
 shall be taken as TFL/120, 
 
 22. In the formulas hereinabove given W = total live and dead load on the whole panel in pounds, L = panel 
 length, c. to c. of columns. 
 
 23. Bending Moment CoeflScients, Interior Panel, Four-way System. — In panels where standard drops and 
 column capitals are used as above specified, the negative bending moment, taken at a cross-section of each strip A 
 at the edge of column capital or over it, shall be taken as WL /30. 
 
 24. The positive bending moment, taken at a cross-section of each strip A, midway between column centers, 
 shall be taken as WL/80. 
 
 25. The positive bending moment, taken at a cross-section of each strip B, taken in the middle of the panel, 
 shall be taken as PFL/120. 
 
 26. The negative bending moment, taken at a cross-section of each strip B on the center line of the columns, 
 shall be taken as WL/120. 
 
 27. Bending Moment Coefficients, Wall Panels. — Where wall panels with standard drops and capitals are 
 carried by columns and girders built in walls, as in skeleton construction, the same coefficients shall be used as for 
 an interior panel, except as follows: The positive bending moments on strips A and B midway between wall and 
 first line of columns shall be increased 25%, 
 
 28. Where wall panels are carried on new brick walls, these shall be laid in Portland cement mortar and shall 
 be stiffened with pilasters as follows: If a 16-in. wall is used, it shall have a 4-in. pilaster. If a 12-in. wall is used, it 
 shall have an 8-in. pilaster. The length of pilasters shall be not less than the diameter of the column, nor less than 
 one-eighth of the distance between pilasters. The pilasters shall be located opposite the columns as nearly as prac- 
 ticable, and shall be corbeled out 4 in. at the top, starting at the level of the base of the column capital. Not less 
 than 8-in. bearing shall be provided for the slab, the full length of wall. 
 
APPENDIX C 
 
 853 
 
 The coefficients of bending moments required for these panels shall be the same as those for the interior panels 
 except as provided herewith: The positive bending moments on strips A and B midway between the wall and first 
 line of columns shall be increased 50%. 
 
 29. Where wall panels are supported on old brick walls, there shall be columns with standard drops and capi- 
 tals built against the wall, which shall be tied to the same in an approved manner, and at least an 8-in. bearing 
 provided for the slab, the full length. Where this is impracticable, there shall be built a beam on the underside of 
 slab adjacent to the wall between columns, strong enough to carry 25% of the panel load. 
 
 The coefficients of bending moments for the two cases of slab support herein described shall be the same as 
 those specified in Sect. 27 and Sect. 28 for skeleton and wall bearing condition, respectively. 
 
 30. Nothing specified above shall be construed as applying to a case of slabs merely resting on walls or ledges, 
 without any condition of restraint. These shall be figured as in ordinary beam-and-girder construction specified in 
 the ordinances. 
 
 31. Bending Moment Coefficients, Wall and Interior Columns. — Wall columns in skeleton construction shall 
 be designed to resist a bending moment of WL/GO at floors and WL/30 at roof. The amount of steel required for 
 this moment shall be independent of that required to carry the direct load. It shall be placed as near the surfaces 
 of the column as practicable on the tension sides, and the rods shall be continuous in crossing from one side to 
 another. The length of rods below the base of the capital and above the floor line shall be sufficient to develop 
 their strength through bond, but not less than 40 diameters, nor less than one-third the clear height between the 
 floor line and the base of the column capital. 
 
 32. The interior columns must be analyzed for the worst condition of unbalanced loading. It is the intention 
 of this ruling to cover ordinary cases of eccentric loads on the columns by the requirement of Sect. 5. Where the 
 minimum size of column therein specified is found insufficient, however, the effect of the resulting bending moment 
 shall be properly divided between the adjoining slab and the columns above and below according to best principles 
 of engineering, and the columns enlarged sufficiently to carry the load safely. 
 
 33. Bending Moment Coefficients, Panels Without Drops, or Capitals, or Both. — In square panels where no 
 column capital or no depressions are used, the sum total of positive and negative bending moments shall be^qual 
 to that computed by the following formula 
 
 B.M. = (WL/8) (1.53 - 4^- + 4.18A;3) 
 where B.M. — Numerical sum of positive and negative bending moments, regardless of algebraic signs; 
 W = Total live and dead load on the whole panel; 
 L = Length of side of a square panel, c. to c. of columns; 
 k = Ratio of the radius of the column or column capital to panel length, L. 
 This total bending moment shall be divided between the positive and the negative moments in the same pro- 
 portion as in the typical square panels for two-way or four-way systems specified above for interior and wall panels 
 respectively. 
 
 34. Point of Inflectiori. — For the purpose of making the calculations of the bending moment at the sections 
 away from the column capital, the point of inflection shall be considered as being one-quarter the distance c. to c. 
 of columns, both crosswise and diagonally, from the center of the column. 
 
 35. Tensile Stress in Steel and Compressive Stress in Concrete. — The tensile stress in steel and the compres- 
 sive stress in the concrete to resist the bending moment shall be calculated on the basis of the reinforcement and slab 
 in the width included in a given strip, and according to the assumptions and requirements given in Sects. 532 to 535 
 inclusive of the building ordinance. The steel shall be considered as being concentrated at the center of gravity of 
 all the bands of steel in a given strip. 
 
 36. For the four-way system of reinforcement the amount of steel to resist the negative bending moment over 
 the support in each strip A shall be taken as the sum of the areas of steel in one cross band and one diagonal band. 
 The amount of steel to resist the positive bending moment of each strip B shall be considered as the area of the 
 steel in a diagonal band. The amount of steel to resist the positive bending moment in each strip A shall be con- 
 sidered as the area of the steel in a cross band, and the amount of steel to resist the negative moment in each strip 
 B shall be the steel included in the width of strip B. 
 
 37. For the two-way system of reinforcement the amount of steel to resist the bending moment in any strip 
 'shall be considered as the area of steel included in the width of the strip. 
 
 38. In both systems of reinforcement the compressive stress in the concrete in any strip shall be calculated by 
 taking the area of steel considered for each strip and applying it in a beam formula based on the principles of Sect. 
 535 of the building ordinance. 
 
 39. Where drop panels are used, the width of beam assumed to resist the compressive stresses over the column 
 capital shall be the width of the drop. 
 
 40. The width of beam, where no drop panels are used, shall be the width of steel bands. Where this is found 
 insufficient, the area shall be increased by introducing compression steel in the bottom of slab. 
 
 41. Rectangular Panels. — When the length of panel in either two-way or four-way system does not exceed the 
 breadth by more than 5%, all computations shall be based on a square panel whose side equals the mean of the 
 length and breadth, and the steel equally distributed among the strips according to the coefficients above specified. 
 
 42. In no rectangular panel shall the length exceed the breadth by more than one-third of the latter. 
 
 43. Rectangular Panels, Four-way System. — In the four-way system of reinforcement, where length exceeds 
 breadth by more than 5 %, the amount of steel required in strip A, long direction, both positive and negative, shall be 
 
854 
 
 CONCRETE ENGINEERS' HANDBOOK 
 
 the same as that required for the same strip in a square panel whose length is equal to the long side of the rectangu- 
 lar panel. 
 
 44. The amount of steel, strip A, short direction, positive and negative, shall be the same as that required for 
 the same strip in a square panel, whose length is equal to the short side of the rectangular panel. 
 
 46. The amount of steel in strip B, positive and negative, shall be the same as that required for similar strip in 
 a square panel whose length is equal to the mean of the long and the short side of the rectangular panel. 
 
 46. In no case shall the amount of steel in the short side be less than two-thirds of that required for the long 
 
 side. 
 
 47. Rectangular Panels, Two-way System. — In the two-way system of reinforcement the amount of steel 
 required for the positive and the negative moment of each strip A shall be determined in the same manner as indi- 
 cated for the four-way system above. 
 
 48. The amount of steel in strip B, positive and negative, running in short direction, shall be equal to that 
 required for the same strip in a square panel whose length equals the long side of the rectangular panel. 
 
 49. The amount of steel in strip B, long direction, positive and negative, shall be equal to that required for 
 the same strip in a square panel, whose length equals the short side of the rectangular panel. 
 
 50. In no case shall the amount of steel in strip B, long direction, be less than two-thirds of that in the short 
 direction. 
 
 61. Walls and Openings. — Girders and beams shall be constructed under walls, around openings and to carry 
 concentrated loads. 
 
 62. Spandrel Beams. — The spandrel beams or girders shall, in addition to their own weight and the weight of 
 the spandrel wall, be assumed to carry 20% of the wall panel load uniformly distributed upon them. 
 
 63. Placing of Steel. — In order that the slab bars shall be maintained in the position shown in the design dur- 
 ing the work of pouring the slab, spacers and supports shall be provided satisfactory to the Commissioner of Build- 
 ings. All bars shall be secured in place at intersections by wire or other metal fastenings. In no case shall the 
 spacing of the bars exceed 9 in. The steel to resist the negative moment in each strip B shall extend one-quarter of 
 the panel length beyond the center line of the columns in both directions. 
 
 54. Splices in bars may be made wherever convenient, but preferably at points of minimum stress. The 
 length of splice beyond the center point, in each direction, shall not be less than 40 diameters of the bars, nor less 
 than 2 ft. The splicing of adjacent bars shall be avoided as far as possible. 
 
 65. Slab bars which are lapped over the column, the sectional area of both being included in the calculations 
 for negative moment, shall extend not less than 0.25 of the panel length for cross bands and 0.35 of the panel length 
 for diagonal bands, beyond the column center. 
 
 56. Computations. — Complete computations of interior and wall panels and such other portions of the build- 
 ing as may be required by the Commissioner of Buildings shall be left in the office of the Commissioner of Buildings 
 when plans are presented for approval. 
 
 57. Test of Workmanship. — The Commissioner of Buildings or his representative may choose any two adja- . 
 cent panels in the building for the purpose of ascertaining the character of workmanship. The test shall not be 
 made sooner than the time required for the cement to set thoroughly, nor less than 6 weeks after the concrete had 
 been poured. 
 
 68. All deflections under test load shall be taken at the center of the slab, and shall be measured from the nor- 
 mal unloaded position of the slab. The two panels selected shall be uniformly loaded over their entire area with a 
 load equal to the dead load plus twice the live load, thus obtaining twice the total design load. The load shall 
 remain in place not less than 24 hours. If the total deflection in the center of the panel under the test load does not 
 exceed ^oo of the panel length, the slab may be placarded to carry the full design live load. If it exceeds this 
 amount of deflection, and recovers not less than 80% of the total deflection within 7 days after the load is 
 removed, the slab may be placarded to carry the full design live load. If the deflection exceeds the allowable 
 amount above specified, and the recovery is less than 80% in 7 days after the removal of the test load, other 
 tests shall be made on the same or other panels, the results of which will determine the amount of live load the slabs 
 will be permitted to carry. 
 
 59. General. — The design and the execution of the work shall conform to the general provisions and the 
 spirit of the Chicago Building Ordinances in points not covered by this Ruling and to the best engineering practice 
 in general. 
 
 60. Enforcement. — This Ruling shall go into effect on and after Jan. 1, 1918. All previous rulings on flat 
 slabs are hereby rescinded. 
 
 FINAL REPORT OF SPECIAL COMMITTEE OF THE AMERICAN SOCIETY OF 
 
 CIVIL ENGINEERS 
 PART PERTAINING TO FLAT-SLAB DESIGN 
 
 The continuous flat slab reinforced in two or more directions and built monolithically with the supporting 
 columns (without beams or girders) is a type of construction which is now extensively used and which has recog- 
 nized advantages for certain types of structures as, for example, warehouses in which large, open floor space is 
 desired. In its construction, there is excellent opportunity for inspecting the position of the reinforcement. The 
 conditions attending depositing and placing of concrete are favorable to securing uniformity and soundness in 
 the concrete. The recommendations in the following paragraphs relate to flat slabs extending over several rows 
 of panels in each direction. Necessarily the treatment is more or less empirical. 
 The coefficients and moments given relate to uniformily distributed loads. 
 
APPENDIX C 
 
 855 
 
 (a) Column Capital. — It is usual in flat-slab construction to enlarge the supporting columns at their top, 
 thus forming column capitals. The size and shape of the column capital affect the strength of the structure in 
 several ways. The moment of the external forces which the slab is called upon to resist is dependent upon the size 
 of the capital; the section of the slab immediately above the upper periphery of the capital carries the highest 
 amount of punching shear; and the bending moment developed in the column by an eccentric or unbalanced load- 
 ing of the slab is greatest at the under surface of the slab. Generally, the horizontal section of the column capital 
 should be round or square with rounded corners. In oblong panels the section may be oval or oblong, with dimen- 
 sions proportional to the panel dimensions. For computation purposes, the diameter of the column capital will 
 be considered to be measured where its vertical thickness is at least lli in., provided the slope of the capital below 
 this point nowhere makes an angle with the vertical of more than 45 deg. In case a cap is placed above the column 
 capital, the part of this cap within a cone made by extending the lines of the column capital upward at the slope of 
 45 deg. to the bottom of the slab or dropped panel may be considered as part of the column capital in determining 
 the diameter for design purposes. Without attempting to limit the size of the column capital for special cases, 
 it is recommended that the diameter of the column capital (or its dimension parallel to the edge of the panel) 
 generally be made not less than one-fifth of the dimension of the panel from center to center of adjacent columns. 
 A diameter equal to 0.225 of the panel length has been used quite widely and acceptably. For heavy loads or 
 large panels, especial attention should be given to designing and reinforcing the column capital with respect to 
 compressive stresses and bending moments. In the case of heavy loads or large panels, and where the conditions 
 of the panel loading or variations in panel length or other conditions cause high bending stresses in 1;he column, 
 and also for column capitals smaller than the size herein recommended, especial attention should be given to design- 
 ing and reinforcing the column capital with respect to compression and to rigidity of connection to floor slab. 
 
 (6) Dropped Panel. — In one type of construction the slab is thickened throughout an area surrounding the 
 column capital. The square or oblong of thickened slab thus formed is called a dropped panel or a drop. The 
 thickness and the width of the dropped panel may be governed by the amount of resisting moment to be provided 
 (the compressive stress in the concrete being dependent upon both thickness and width), or its thickness may be 
 governed by the resistance to shear^required at the edge of the column capital and its width by the allowable com- 
 pressive stresses and shearing stresses in the thinner portion of the slab adjacent to the dropped panel. Generally, 
 however, it is recommended that the width of the dropped panel be at least four-tenths of the corresponding side 
 of the panel as measured from center to center of columns, and that the off'set in thickness be not more than five- 
 tenths of the thickness of the slab outside the dropped panel. 
 
 (c) Slab Thickness. — In the design of a slab, the resistance to bending and to shearing forces will largely 
 govern the thickness, and, in the case of large panels with light loads, resistance to deflection may be a controlling 
 factor. The following formulas for minimum thicknesses are recommended as general rules of design when the 
 diameter of the column capital is not less than one-fifth of the dimension of the panel from center to center of 
 adjacent columns, the larger dimension being used in the case of oblong panels. For notation, let 
 
 t = total thickness of slab, in inches. 
 L = panel length, in feet. 
 
 w = sum of live load and dead load, in pounds per square foot. 
 Then, for a slab without dropped panels. 
 
 minimum t 
 for a slab with dropped panels, 
 
 minimum t 
 
 0.024 L\/w + IH 
 
 Position ofresuH-an) 
 oF shear on quarter 
 peripheries of fwo 
 coturnn capihjis 
 
 .7° 
 
 '^Cerrfer of gravity 
 of load on half 
 panel 
 
 Fig. 1. 
 
 0.02 L^w + 1 
 
 for a dropped panel whose width is four-tenths of the panel length, 
 
 minimum t = 0.03 L\/ w + IH 
 
 In no case should the slab thickness be made less than 6 in., nor should 
 the thickness of a floor slab be made less than one thirty-second of the panel 
 length, nor the thickness of a roof slab less than one-fortieth of the panel 
 length. 
 
 (d) Bending and Resisting Moments in Slabs. — If a vertical section of a slab be taken across a panel along a line 
 midway between columns, and if another section be taken along an edge of the panel parallel to the first section, 
 but skirting the part of the periphery of the column capitals at the two corners of the panels, the moment of the 
 couple formed by the external load on the half panel, exclusive of that over the column capital (sum of dead and 
 live loads) and the resultant of the external shear or reaction at the support at the two column capitals (see Fig. 1), 
 may be found by ordinary static analysis. It will be noted that the edges of the area here considered are along 
 lines of zero shear, except around the column capitals. This moment of the external forces acting on the half 
 panel will be resisted by the numerical sum of (a) the moment of the internal stresses at the section of the panel 
 midway between columns (positive resisting moment) and (b) the moment of the internal stresses at the section 
 referred to at the end of the panel (negative resisting moment). In the curved portion of the end section (that 
 skirting the column), the stresses considered are the components which act parallel to the normal stresses on the 
 straight portion of the section. Analysis shows that, for a uniformly distributed load, and round columns, and 
 square panels, the numerical sum of the positive moment and the negative moment at the two sections named is 
 given quite closely by the equation 
 
 ilf« = Hwl (Z - Hey 
 
856 
 
 CONCRETE ENGINEERS' HANDBOOK 
 
 In this formula and in those which follow relating to oblong panels, 
 w = sum of the live and dead loads per unit of area. 
 I = side of a square panel measured from center to center of columns. 
 h = one side of the oblong panel measured from center to center of columns. 
 h = other side of oblong panel measured in the same way. 
 c = diameter of the column capital. 
 Mx = numerical sum of positive moment and negative moment in one direction. 
 My = numerical sum of positive moment and negative moment in the other direction, i 
 
 For oblong panels, the equation for the numerical sum of the positive moment and the negative moment at the 
 two sections named becomes 
 
 My =^ wli (I2 - |c) ^ 
 
 where Mx is the numerical sum of the positive moment and the negative moment for the sections parallel to the 
 dimension, h, and My is the numerical sum of the positive moment and the negative moment for the sections parallel 
 to the dimension, h. 
 
 What proportion of the total resistance exists as positive moment and what as negative moment is not readily 
 determined. The amount of the positive moment and that of the negative moment may be expected to vary some- 
 what with the design of the slab. It seems proper, however, to make the division of total resisting moment in the 
 ratio of three-eighths for the positive moment to five-eighths for the negative moment. 
 
 With reference to variations in stress along the sections, it is evident from conditions of flexure that the resist- 
 ing moment is not distributed uniformly along either the section of positive moment or that of negative moment. As 
 the law of the distribution is not known definitely, it will be necessary to make an empirical apportionment along the 
 sections; and it will be considered sufficiently accurate generally to divide the sections into two parts and to use an 
 average value over each part of the panel section. 
 
 The relatively large breadth of structure in a flat slab makes the effect of local variations in the concrete less 
 than would be the case for narrow members like beams. The tensile resistance of the concrete is less affected by 
 cracks. Measurements of deformations in buildings under heavy load indicate the presence of considerable tensile 
 resistance in the concrete, and the presence of this tensile resistance acts to decrease the intensity of the com- 
 pressive stresses. It is believed that the use of moment coefficients somewhat less than those given in a preceding 
 paragraph as derived by analysis is warranted, the calculations of resisting moment and stresses in concrete and re- 
 inforcement being made according to the assumptions specified in this report and no change being made in the 
 ^ ^ ^ ^, ^ values of the working stresses ordinarily used. Accordingly, the values of 
 
 CtAimn heiod. HkJ-SecHon . Mum n-head , , . , , , , , , , , 
 
 Section p C D ^ Section the moments which are recommended for use are somewhat less than those 
 
 derived by analysis. The values given may be used when the column 
 capitals are round, oval, square, or oblong. 
 
 (e) Names for Moment Sections. — For convenience, that portion of 
 the section across a panel along a line midway between columns which lies 
 within the middle two quarters of the width of the panel (HI, Fig. 2) will be 
 called the inner section, and that portion in the two outer quarters of the 
 ! " '°" \ width of the panel (GH and IJ, Fig. 2) will be called the outer sections. 
 
 I 1 Of the section which follows a panel edge from column capital to column 
 
 1<_...^/ . — >/ ^t...^/^...-^ capital and which includes the quarter peripheries of the edges of two 
 
 ^ column capitals, that portion within the middle two quarters of the panel 
 / width (CD, Fig. 2) will be called the mid-section, and the two remaining 
 
 ~Fig~'2 portions (ABC and DEF, Fig. 2), each having a projected width equal 
 
 to one-fourth of the panel width, will be called the column-head sections. 
 (/) Positive Moment. — For a square interior panel, it is recommended that the positive moment for a section 
 
 in the middle of a panel extending across its width be taken as 2^ wl (j — |-c^ . Of this moment, at least 25% 
 
 should be provided for in the inner section; in the two outer sections of the panel at least 55% of the specified 
 moment should be provided for in slabs not having dropped panels, and at least 60 % in slabs having dropped 
 panels, except that in calculations to determine necessary thickness of slab away from the dropped panel at least 
 70 % of the positive moment should be considered as acting in the two outer sections. 
 
 (g) Negative Moment. — For a square interior panel, it is recommended that the negative moment for a section 
 which follows a panel edge from column capital to column capital and which includes the quarter peripheries of the 
 edges of the two column capitals (the section altogether forming the projected width of the panel) be taken as 
 1 / 2 \2 
 
 Y^wl [I — g-c j . Of this negative moment, at least 20 % should be provided for in the mid-section and at least 
 
 65 % in the two column-head sections of the panel, except that in slabs having dropped panels at least 80 % of the 
 specified negative moment should be provided for in the two column-head sections of the panel. 
 
 1 See paper and closure, "Statical Limitations upon the Steel Requirement in Reinforced Concrete Flat 
 Slab Floors," by John R. Nichols, Jun. Am. Soc. C. E., Trans. Am. Soc. C. E., vol. 77. 
 
 \ 
 
APPENDIX C 
 
 857 
 
 (^i) Moments for Oblong Panels. — When the length of a panel does not exceed the breadth by more than 5 % , 
 computation may be made on the basis of a square panel with sides equal to the mean of the length and the breadth. 
 When the long side of an interior oblong panel exceeds the short side by more than one-twentieth and by not 
 
 more than one-third of the short side, it is recommended that the positive moment be taken as -^wh (h — 5-' 
 
 on a section parallel to the dimension, h, and -Kn^h (h — o"<^) a section parallel to the dimenison, h; and that 
 
 the negative moment be taken as y^wh (h — o a section at the edge of the panel corresponding to the dimen- 
 
 sion, Z-2, and j^wZi — ^cj at a section in the other direction. The limitations of the apportionment of moment 
 between inner section and outer section and between mid-section and column-head sections may be the same as for 
 square panels. 
 
 (i) Wall Panels. — The coefficient of negative moment at the first row of columns away from the wall should be 
 increased 20% over that required for interior panels, and likewise the coefficient of positive moment at the section 
 half way to the wall should be increased by 20%. If girders are not provided along the wall, or the slab does not 
 project as a cantilever beyond the column line, the reinforcement parallel to the wall for the negative moment in the 
 column-head section and for the positive moment in the outer section should be increased by 20%. If the wall is 
 carried by the slab, this concentrated load should be provided for in the design of the slab. The coefficient of nega- 
 tive moments at the wall to take bending in the direction perpendicular to the wall line may be determined by the 
 conditions of restraint and fixedness as found from the relative stiffness of columns and slab, but in no case should 
 it be taken as less than one-half of that for interior panels. 
 
 0) Reinforcement. — In the calculation of moments, all the reinforcing bars which cross the section under con- 
 sideration and which fulfill the requirements given under paragraph (Z) of this chapter may be used. For a column- 
 head section, reinforcing bars parallel to the straight portion of the section do not contribute to the negative resisting 
 moment for the column-head section in question. In the case of four-way reinforcement, the sectional area of the 
 diagonal bars multiplied by the sine of the angle between the diagonal of the panel and the straight portion of the 
 section under consideration may be taken to act as reinforcement in a rectangular direction. 
 
 (k) Point of Inflection. — For the purpose of making calculations of moments at sections away from the sections 
 of negative moment to act integrally for a width equal to the width of the column-head section. 
 
 (o) Provision for Diagonal Tension and Shear. — In calculations for the shearing stress which is to be used as 
 the means of measuring the resistance to diagonal tension stress, it is recommended that the total vertical shear 
 on two column-head sections constituting a width equal to one-half the lateral dimension of the panel, for use in 
 the formula for determining critical shearing stresses, be considered to be one-fourth of the total dead and live 
 loads on a panel for a slab of uniform thickness, and to be three-tenths of the sum of the dead and live loads on a 
 panel for a slab with dropped panels. The formula for shearing unit stress given in the Appendix to this report 
 
 0.25 IF 0.30 PF 
 
 may then be written v = -^-^j^ior slabs of uniform thickness, and v = '^j^ ■ for slabs with dropped panels, 
 
 where W is the sum of the dead and live loads on a panel, b is half the lateral dimension of the panel measured 
 from center to center of columns, and jd is the lever arm of the resisting couple at the section. 
 
 The calculation of what is commonly called punching shear may be made on the assumption of a uniform 
 distribution over the section of the slab around the periphery of the column capital and also of a uniform distribu- 
 tion over the section of the slab around the periphery of the dropped panel, using in each case an amount of vertical 
 shear greater by 25 % than the total vertical shear on the section under consideration. 
 
 The values of working stresses should be those recommended for diagonal tension and shear in Appendix B. 
 
 (p> Walls and Openings. — Girders or beams should be constructed to carry walls and other concentrated 
 loads which are in excess of the working capacity of the slab. Beams should also be provided in case openings in 
 the floor reduce the working strength of the slab below the required carrying capacity. 
 
 (g) Unusual Panels. — The coefficients, apportionments, and thicknesses recommended are for slabs which 
 have several rows of panels in each direction, and in which the size of the panels is approximately the same. For 
 structures having a width of one, two, or three panels, and also for slabs having panels of markedly different sizes, 
 an analysis should be made of the moments developed in both slab and columns, and the values given herein modi- 
 fied accordingly. Slabs with paneled ceiling or with depressed paneling in the floor are to be considered as coming 
 under the recommendations herein given. 
 
 (r) Bending Moments in Columns. — Provision should be made in both wall columns and interior columns for 
 the bending moment which will be developed by unequally loaded panels, eccentric loading, or uneven spacing of 
 columns. The amount of moment to be taken by a column will depend upon the relative stiffness of columns and 
 slab, and computations may be made by rational methods, such as the principle of least work, or of slope and de- 
 flection. Generally, the larger part of the unequalized negative moment will be transmitted to the columns, and 
 the column should be designed to resist this bending moment. Especial attention should be given to wall columns 
 and corner columns. 
 
 2 
 
858 
 
 CONCRETE ENGINEERS' HANDBOOK 
 
 STANDARD BUILDING REGULATIONS FOR THE USE OF REINFORCED CONCRETE, 
 AMERICAN CONCRETE INSTITUTE, 1917, PART PERTAINING TO FLAT-SLAB 
 
 FLOORS 
 
 Continuous flat-slab floors, reinforced with steel rods or mesh and supported on spaced columns in orderly 
 arrangement, shall conform to the following requirements: 
 (a) Notation and Nomenclature. — In the formulas let: 
 
 w = total dead and live load in pounds per square foot of floors. 
 
 li = span in feet center to center of columns parallel to sections on which moments are considered. 
 h = span in feet center to center of columns perpendicular to sections at which moments are considered. 
 c = average diameter of column capital in feet at point where its thickness is l^-i in. 
 Q =■ distance from center line of the capital to the center of gravity of the periphery of the half capital 
 divided by V2C. For round capitals q may be considered as two-thirds and for square capitals as 
 three-quarters. 
 t = total slab thickness in inches. 
 
 L = average span in feet center to center of columns, but not less than 0.9 of the greater span. 
 The column-head section, mid-section, outer section, and inner section, are located and dimensioned as 
 shown in Fig. 3. Corresponding moments shall be figured on similar sections at right angles to those shown in 
 Fig. 3. 
 
 head sedio" 
 
 "" sect ion 
 
 Mid, section ^!l9!(p°j!J'T!r>.~ 
 
 head section 
 
 # • IH 
 
 Fig. 3. 
 
 L > V": .... / 
 
 r ~''W'less than aSL drop H 
 
 (o) Drop construction 
 I 
 
 (b) Cap construction 
 
 .-Not over J of slab thickness 
 
 j^..„ tiPl°yer_pf>L-Ceiling panel _ . 
 
 Cc> PaneHed ceiling consfrnxjtion 
 Fig. 4. 
 
 (b) Structural Variations. — Flat-slab floors may be built with or without caps, drops or paneled ceilings. 
 These terms are illustrated in Fig. 4. 
 
 Where caps are employed they shall be considered a part of the columns and the column capital dimension 
 c shall be found by extending the lines of the capital below to an intersection with the plane of the under surface 
 of the slab as indicated in Fig. 46. The cap shall be large enough to enclose this extension of the capital lines. 
 
 The column capital profile shall not fall at any point inside an inverted cone drawn, as shown in Fig. 4a, 
 from the periphery of the designed capital of diameter c and with a base angle of 45 deg. The diameter of the de- 
 signed capital c shall be taken where the vertical thickness of the column capital is at least 1}^ in. 
 
 The drop, where used, shall not be less than 0.3 of L in width. 
 
 Where paneled ceilings are used the paneling shall not exceed one-third of the slab thickness in depth and 
 the dimension of the paneling shall not exceed 0.6 of the paneled dimension (see Fig. 4c). 
 
 (c) Slab Thickness. — The slab thickness shall not be less than t = 0.02L\/ w -\- I io.. 
 
 In no case shall the slab thickness be less that L for the floor slabs nor less than — L for the roof slabs. 
 
 (d) Design Moments. — The numerical sum of the positive and negative moments in foot-pounds shall not 
 be less than 0.09 wh (h — gc)2. Of this total amount not less than 40% shall be resisted in the column-head sec- 
 tions. Where a drop is used not less than 50 % shall be resisted in the column-head sections. 
 
 Of the total amount not less than 10% shall be resisted in the mid-section. 
 Of the total amount not less than 18% shall be resisted in the outer sections. 
 Of the total amount not less than 12 % shall be resisted on the inner sections. 
 
 (e) Exterior Panels. — The negative moments at the first interior row of columns and the positive moments at 
 the center of the exterior panel on sections parallel to the wall, shall be increased 20 % over those specified above 
 for interior panels. If girders are not provided along the column line, the reinforcement parallel to the wall for 
 
APPENDIX C 
 
 859 
 
 negative moment in the column-head section and for positive moment in the outer section adjacent to the wall, 
 shall be altered in accordance with the change in the value of c. The negative moment on sections at the wall and 
 parallel thereto should be determined by the conditions of restraint, but must never be taken less than 50% of 
 those for the interior panels. 
 
 (/) Reinforcement. — In the calculation of moments all the reinforcing bars which cross the section under 
 consideration and which fulfil the requirements given under "Arrangement of Reinforcement" may be used. For 
 a column-head section reinforcing bars parallel to the straight portion of the section do not contribute to the negative 
 resisting moment for the column-head section in question. The sectional area of bars, crossing the section at an 
 angle multiplied by the sine of the angle between those bars and the straight portion of the section under considera- 
 tion may be taken to act as reinforcement in a rectangular direction. 
 
 (g) Point of Inflection. — For the purpose of making calculations of moment at sections away from the sec- 
 tions of negative moment and positive moment already specified, the point of inflection shall be taken at a distance 
 from center line of columns equal to Hih — qc) -f- J-^gc. This becomes %{h + c) where capital is circular. For 
 slabs having drop panels the coefficient of li should be used instead of Ya. 
 
 (h) Arrangement of Reinforcement. — The design should include adequate provision for securing the reinforce- 
 ment in place so as to take not only the maximum moments but the moments of intermediate sections. If bars 
 are extended beyond the column capital and are used to take the bending moment on the opposite side of the column, 
 they must extend to the point of inflection. Bars in diagonal bands used as reinforcement for negative moment 
 should extend on each side of the line drawn through the column center at right angles to the direction of the band 
 a distance equal to 0.35 of the panel length, and bars in the diagonal bands used as reinforcement for positive 
 moment, should extend on each side of the diagonal through the center of the panel a distance equal to 0.35 of the 
 panel length. Bars spliced by lapping and counted as only one bar in tension shall be lapped not less than 80 
 diameters if splice is made at point of maximum stress and not more than 50 % of the rods shall be so spliced at 
 any point in any single band or in any single region of tensile stress. Continuous bars should not all be bent up 
 at the same point of their length, but the zone in which this bending occurs should extend on each side of the 
 assumed point of inflection. 
 
 (i) Tensile and Compressive Stresses. — The usual method of calculating the tensile and compressive stresses 
 in the concrete and in the reinforcement, based on the assumptions for internal stresses, should be followed. In 
 the case of the drop panel, the section of the slab and drop panel may be considered to act integrally for a width 
 equal to the width of the column-head section. Within the column-head section the allowable compression may 
 be increased by 10% over that prescribed in Sect. 41.i 
 
 1 Section 41. — Reinforced-concrete structures shall be so designed that the stresses, figured in accordance with 
 these regulations, in pounds per square inch, shall not exceed the following: 
 
 Extreme fiber stress in concrete in compression — 37}^ % of the minimum compressive strength given in the 
 table below. Adjacent to the support of continuous members, 41 %, provided the member frames into a mass of 
 concrete projecting at least 50 % of the least dimension of the member on all sides of the compression area of the 
 member. 
 
 Concrete in direct compression — 25% of the minimum compressive strength given in the table. 
 
 Shearing stress in concrete when main steel is not bent and when steel is not provided to resist diagonal ten- 
 sion — 2% of the minimum compressive strength given in the table. 
 
 Punching shear in concrete, 7}^ % of the minimum compressive strength given in the table. 
 
 Shearing stress in concrete when steel to assist in resisting diagonal tension is provided — % of the minimum 
 compressive strength given in the table providing that sufficient web reinforcement is supplied to carry the stresses 
 in excess of the value allowed for the unreinforced concrete; and providing further, that this web reinforcement 
 extends from top to bottom of beam and is adequately anchored to the horizontal reinforcement. If main rein- 
 forcing bars are bent up and anchored, they may be considered as part of the web reinforcement. 
 
 Bond stress between concrete and plain reinforcing bars — 4% of the compressive strength. 
 
 Bond stress between concrete and approved deformed bars — 5% of the compressive strength. 
 
 Bearing upon a surface of concrete at least twice the loaded area — 50% of the compressive strength of the 
 concrete. 
 
 Tensile stress in steel — 16,000 lb. per sq. in., except that for steel having an elastic limit of at least 50,000 lb., a 
 working stress of 18,000 lb. per sq. in. will be allowed. 
 
 Tablk of Strengths of Different Mixtures of Concrete 
 (In Pounds per Square Inch) 
 
 Aggregate 
 
 1 : 3 
 
 1 :m 
 
 1 : 6 
 
 1 : 7H 
 
 1 :9 
 
 
 3,300 
 
 2,800 
 
 2,200 
 
 1,800 
 
 1,400 
 
 Gravel, hard limestone, hard 
 
 
 
 
 
 
 sandstone and approved slag . . 
 
 3,000 
 
 2,500 
 
 2,000 
 
 1,600 
 
 1,300 
 
 Soft limestone and sandstone .... 
 
 2,200 
 
 1,800 
 
 1,500 
 
 1,200 
 
 1,000 
 
 Cinders 
 
 800 
 
 700 
 
 600 
 
 500 
 
 400 
 
860 
 
 CONCRETE ENGINEERS' HANDBOOK 
 
 0) Provision for Diagonal Tension and Shear. — In calculations for the shearing stress which is to be used as the 
 means of measuring the resistance to diagonal tension stress, it shall be assumed that the total vertical shear on a 
 column-head section constituting a width equal to one-half of the lateral dimension of the panel, for use in deter- 
 mining critical shearing stresses, shall be considered to be one-fourth of the total dead and live load on a panel for a 
 slab of uniform thickness, and to be 0.3 of the sum of the dead and live loads on a panel for a slab with drop panels. 
 
 The formula for shearing unit stress shall hev = ' . , — for slabs of uniform thickness and v = - for slabs with 
 
 oja bjd 
 
 drop panels, where W is the sum of the dead and live load on a panel, b is half the lateral dimension of the panel 
 
 measured from center to center of columns, and is the lever arm of the resisting couple at the section. 
 
 The calculation for punching shear shall be made on the assumption of a uniform distribution over the section 
 of the slab around the periphery of the column capital and also of a uniform distribution over the section of the slab 
 around the periphery of the drop panel, using in each case an amount of vertical shear greater by 50 % than the 
 total vertical shear on the section under consideration. 
 
 The values of working stresses should be those recommended for diagonal tension and shear in Sect. 41. 
 
 (k) Walls and Openings. — Girders or beams shall be constructed to carry walls and other concentrated loads 
 which are in excess of the working capacity of the slab. Beams should also be provided in case openings in the floor 
 reduce the working strength of the slab below the reqmred carrying capacity. 
 
 (0 Unusual Panels. — The coefficients, steel distribution, and thicknesses recommended are for slabs which have 
 three or more rows of panels in each direction and in which the sizes of the panels are approximately the same. For 
 structures having a width of one or two panels and also for slabs having panels of markedly different size, an analy- 
 sis should be made of the moments developed in both slab and columns and the values given herein modified 
 accordingly. 
 
 (m) Bending Moments in Columns. — Provision shall be made in both wall columns and interior columns for the 
 bending moment which will be developed by unequally loaded panels, eccentric loading, or uneven spacing of 
 columns. The amount of moment to be taken by a column will depend on the relative stiffness of columns and slab, 
 and computations may be made by rational methods such as the principle of least work or of slope and deflection. 
 Generally the largest part of the unequalled negative moment will be transmitted to the columns and the columns 
 should be designed, and reinforced where necessary, with these conditions in mind. 
 
 The resistance of any column to bending in a direction parallel to h shall not be less than 0 . 022 wih {h — qz) 2, in 
 which wi is the designed live load per square foot. In determining the resistance to be provided in exterior columns 
 in a direction perpendicular to the wall the full dead and live load w shall be used in the above formula in place of 
 w\. The moment in such exterior column may be reduced by the balancing moment of the weight of the structure 
 which projects beyond the supporting wall-column center line. 
 
 Where the column extends through the story above, the resisting moment shall be divided between the upper 
 and the lower columns in proportion to their stiffness. The calculations of combined stresses due to bending and 
 direct load shall not exceed by more than 50 % the stresses allowed for direct load. 
 
APPENDIX D 
 STANDARD NOTATION 
 
 (tt) Rectangular Beams. 
 
 fs = tensile unit stress in steel. 
 fc = compressive unit stress in concrete. 
 Es = modulus of elasticity of steel. 
 Ec = modulus of elasticity of concrete. 
 Es 
 
 "^^^ 
 
 M = moment of resistance, or bending moment in general. 
 As = steel area. 
 
 b — breadth of beam. 
 
 d = depth of beam to center of steel. 
 
 k = ratio of depth of neutral axis to depth, d. 
 
 z = depth below top to resultant of the compressive stresses. 
 
 j — ratio of lever arm of resisting couple to depth, d. 
 jd = d — z = arm of resisting couple. 
 
 p = steel ratio = 
 
 {b)T -beams. 
 
 b = width of flange. 
 b' = width of stem. 
 t = thickness of flange. 
 
 (c) Beams Reinforced for Compression. 
 
 A' = area of compressive steel. 
 
 p' = steel ratio for compressive steel. 
 // = compressive unit stress in steel. 
 
 C = total compressive stress in concrete. 
 C = total compressive stress in steel 
 
 d' = depth of center of compressive steel, 
 r = depth to resultant of C and C. 
 
 (d) Shear, Bond and Web Reinforcement. 
 
 V = total shear. 
 
 V = total shear producing stress in reinforcement. 
 V = shearing unit stress. 
 
 u = bond stress per unit area of bar. 
 o = circumference or perimeter of bar. 
 So = sum of the perimeters of all bars. 
 T = total stress in single reinforcing member, 
 s = horizontal spacing of reinforcing members. 
 
 (e) Columns. 
 
 A = total net area. 
 As = area of longitudinal steel. 
 Ac — area of concrete. 
 
 P = total safe load. 
 
 861 
 
APPENDIX E 
 
 CONCRETE BARGES AND SHIPS 
 
 EXTRACT FROM REPORT OF THE JOINT COMMITTEE OF THE AMERICAN 
 CONCRETE INSTITUTE AND PORTLAND CEMENT ASSOCIATION 
 
 The idea of building ships or other floating vessels of reinforced concrete is not new. For many years and in 
 several different countries, attempts have been made from time to time to build small boats and barges of reinforced 
 concrete. From the information at hand, apparently some of these attempts have been successful and the vessels 
 thus built have been in use for considerable periods. However, no definite information regarding boats built 
 prior to the war is at hand which would assist your Committee in solving the general problem of the concrete ship. 
 
 Since the beginning of the war, however, owing to the loss of the world's tonnage due to submarine sinkings, 
 the attention of many naval architects, concrete engineers and others has been drawn to the question of concrete 
 ships to replace those sunk. The scarcity of steel, wood, and labor, and the length of time necessary to construct 
 ships of steel or wood directs attention generally to the substitution of reinforced concrete. Norway appears to 
 have taken the lead in this work and two different companies are already building ships of concrete. The Porsgrund 
 Cement Works, whose Vice President and General Manager, Jens Hauland, has recently been in this country, 
 has already launched one or more ships of 100-tons cargo capacity and is reported to have under construction a 
 ship of 1000-tons cargo capacity. The general design of these ships follows generally that of a framed steel ship, 
 and your Committee has been informed by Mr. Hauland that the weight compares very favorably with that of a 
 steel ship. No definite information is available, however, by which this statement can be verified. 
 
 The Fougner Steel Concrete Shipbuilding Company of Christiania, Norway, has been building vessels since 
 June, 1916. About eighteen in all have been launched up to the present time. Several others are under con- 
 struction. The " Namsenfjord" about 400 tons displacement launched some time ago has made a round trip 
 between Norway and England. No detailed information is available at the present time regarding these vessels 
 that would throw light on the general problem. 
 
 On this side of the Atlantic, at least two ships are under construction. At San Francisco, a large ship about 
 5000-tons capacity is being built and, from information at hand, will be shortly launched. Your Committee 
 learns that this ship is 320 ft. long, 44 ft. 6 in. beam and 30 ft. 0 in. deep. At 24-ft. draft the displacement is 
 said to be 8000 tons. The weight of the hull is said to be 2200 tons. At Montreal, Canada, a small concrete ship 
 which will have about 300 tons carrying capacity has already been launched and is now being equipped. This 
 vessel is being constructed by the Atlas Construction Co., Ltd., of which Chas. M. Morssen is President. Your 
 Committee is indebted to Mr. Morssen for considerable data relating to this ship and it takes this opportunity of 
 acknowledging the very courteous treatment shown its representatives by Mr. Morssen and the freedom with which 
 he discussed the details of the design and construction. This ship is 126 ft. 0 in. long, between perpendiculars, 22 ft. 
 6 in. beam, with a depth of 12 ft. 6 in. The displacement is about 650 tons. The ship will be self-propelled and 
 is now being equipped with boilers and engines. She will shortly make her first trip. No estimate of cost is at 
 present available. 
 
 With the exception of the two ships noted above, little information has been gained from the ships now under 
 construction which will assist in the solution of the concrete ship problem. 
 
 The present report will confine itself to a general statement of the several elements which make up the concrete 
 ship problem and a discussion of the information obtained from the tentative design of a concrete ship. 
 
 In order to make any advance toward the solution of the concrete ship problem, information must be obtained 
 concerning several points regarding which only meager information is now available. These points taken together 
 constitute the concrete ship problem. 
 
 1. The Relation Between Carrying Capacity and Displacement. — The displacement of a ship is the weight of 
 the water she displaces, and is therefore, the sum of the weight of the ship itself and its cargo capacity expressed in 
 tons. The cargo capacity will hereafter be spoken of as the "dead weight" — following the usual practice with 
 naval architects. Wherever displacement, dead weight or weight of ship is spoken of, it will be in terms of long 
 tons (2240 Ib^.y^which is the usual practice. 
 
 It is apparent that the efficiency of a ship as a cargo carrier depends upon the relationship between dead weight 
 and displacem,ent. Expressed in terms of %, in the average cargo ship built of steel, the dead weight is from 70 to 
 75 % of the displacement — taking into account as weight of ship all spars, fittings, deck houses, anchors and chains, 
 auxilia^3J^ engipes and tanks, but not boilers, engines or coal. In a wooden ship, the dead weight is from 60 to 65% 
 - W' 5s*^'v- 863 
 
864 
 
 CONCRETE ENGINEERS' HANDBOOK 
 
 of the displacement. It is quite evident that from the difference in weight of materials, it will be difficult to design 
 a ship of concrete that will give a relationship between dead weight and displacement approaching that of steel. 
 However, if ships are to be built of concrete for commercial use, the weight of the ship must be such as to provide 
 a reasonable dead weight or cargo capacity for the displacement. 
 
 2. Transverse Strength. — The stresses in the transverse members of a ship are, in still water, functions of the 
 draft and the stiffness, and may be computed by mathematical processes, although the computations are long and 
 laborious. When the material is reinforced concrete, the problem becomes much more complicated. Experience 
 has shown, however, that numerous elements other than draft effect the transverse strength of a ship, such as the 
 effect of rolling in a sea way, impact with docks or other ships and stresses incident to going into dry-dock. The 
 transverse members of cargo ships of today are, therefore, not designed to withstand computed stresses, but are 
 designed in accordance with various rules which embody the result of long experience in the construction and use 
 of ships. It should be noted in this connection that granting of insurance depends on compliance with these rules. 
 
 Steel ships are of two different types (a) framed ships in which transverse ribs or frames are spaced from 18 to 
 24 in. on centers, the plating being rivetted to these ribs without intermediate longitudinal members, excepting 
 in the bottom, and (6) longitudinally framed ships (Isherwood) in which heavy frames are spaced from 10 to 15 ft. 
 on centers with intermediate longitudinals to which the plating is rivetted. 
 
 From a comparison with the ordinary steel ship design, it would appear to be not difficult to design transverse 
 members of reinforced concrete of equivalent strength to steel members — the question of strength only being 
 considered. 
 
 3. Longitudinal Strength. — A ship must be able to meet conditions which are unlike any to which land structures 
 are subject. 
 
 In determining the longitudinal strength of a ship, it is customary to assume two conditions. Under the first 
 condition, the ship is assumed to be suspended between two wave crests, the length between crests being equal to 
 the length of the ship between perpendiculars, the height of the wave being equal to one-twentieth of that length. 
 In this case, the ship as a whole, is acting as a simple beam supported at the ends. This condition is termed "sag- 
 ging. " Under the second condition the ship is assumed to be supported amidships on one crest of the same wave. 
 Under this condition, the ship as a whole acts as a cantilever. This condition is termed "hogging." It is ap- 
 parent therefore, that when a ship is riding the waves both the deck and the bottom of the ship will be required to 
 withstand tensile and compressive stresses alternately — the maximum tensile stress following the maximum com- 
 pressive stress at very short intervals. In a steel ship the entire cross-sectional area of the midship section acts 
 to resist these stresses, taking into account, in determining the moment of inertia, all of the continuous members 
 such as continuous scantlings and deck, side and bottom plates. In the concrete ship, equivalent strength must 
 be provided. In the case of the concrete ship, however, only the steel reinforcement can be relied upon to take 
 tensile stresses. The concrete, assisted by the steel, will take the compressive stresses. 
 
 The effect of the rapid change of the character of the stress in either the deck or the bottom is much more serious 
 in the case of concrete ship than in the steel ship for the reason that owing to the low tensile stress of concrete, 
 cracks will occur at the extreme fiber under relatively low tensile stresses in the steel. These cracks, if any, alter- 
 nately opening and closing, may tend to cause disintegration, with resulting leaks or impairment of the 
 reinforcement. 
 
 At the present time, little information is available as to the effect of such reversal of stress, and but little can 
 be hoped for until an actual trial has been made of a concrete ship in a sea. 
 
 4. Elasticity. — There is an almost unanimous opinion among naval architects and seafaring men generally 
 that a concrete ship will be so inelastic that she will tear herself to pieces in a sea. While it is doubtless true that in 
 a concrete ship there will not be the same readjustment of stresses as in a steel ship when subject to the action of a 
 heavy sea, experience with reinforced concrete structures generally has shown that such structures have consider- 
 able elasticity and there is ample reason for the hope that reinforced concrete will prove a suitable material for 
 ship building purposes. 
 
 6. Effects of Sea Water on Concrete and Reinforcing Steel. — Until very recently little information has been 
 available as to the effect of sea water on concrete. The recent work of the Bureau of Standards, acting in co- 
 operation with the Portland Cement Association, has thrown considerable light on what may be expected from the 
 action of sea water. The result of their investigation tends to show that inferior concrete or concrete of which 
 the surface skin has been impaired suffers serious disintegration when in contact with sea water. Inasmuch as in 
 most instances the structures examined, which form the basis of the report of the Bureau of Standards, were built 
 without thought as to the action of sea water (it being assumed that there would be no deleterious action) there is 
 every reason to hope that where the effect of sea water is appreciated, and where steps are taken in the way of 
 selected materials and adequate workmanship, assuring a good mix and a satisfactory surface skin, the concrete 
 will not so deteriorate. 
 
 With regard to the effect of sea water on the reinforcing steel, there is some question as to whether a thin layer 
 of concrete can be relied upon to protect the steel from corrosion. To provide a thick protective layer of concrete 
 outside of the reinforcing steel is practically out of the question, owing to the large increase in weight. If the 
 reinforcement, therefore, is to be maintained in perfect condition, the steel must be protected by galvanization 
 and by increasing the efficiency of the protective concrete by means of additional care in materials and workman- 
 ship and by a surface coating of a waterproofing character. 
 
 6. Relative Cost. — Just at the present time, the demand for tonnage is so great that any ship of reasonable 
 capacity that can be used for carrying cargo will prove financially successful. The relative costs of ships of concrete 
 
APPENDIX E 
 
 865 
 
 and steel, or concrete and wood, is not therefore as important a consideration as it will be after the war when condi- 
 tions again approach the normal. However, it is necessary to have an adequate idea of the probable cost of a 
 concrete ship as well as a comparison with the cost of steel and wooden ships. 
 
 7. Speed of Construction. — Speed of construction is of vital importance in the ship building program today 
 owing to the immediate requirements for tonnage. If it shall be found that the concrete ship can be constructed 
 in much less time than a steel or wood ship, this fact will contribute largely to the success of the concrete ship. 
 
 Although there are some questions regarding the concrete ship which can only be answered by actual experiment, 
 the studies which your Committee has made point to the commercial success of the concrete ship. 
 
 Your Committee suggests that specifications for a concrete vessel should embody the following principles: 
 
 (a) Both cement and aggregates should be selected with great care to insure a concrete of maximum efficiency. 
 
 (b) The concrete should be placed in one continuous operation to insure monolithic construction. The concrete 
 mixture should be such as will develop a crushing strength in excess of 3000 lb. per sq. in. when tested in standard 
 cylinders at 28 days. A concrete consisting of one part Portland cement, one part sand and two parts H-in. 
 aggregate may be expected to give such a concrete. The mixture and workmanship in placing must be such as will 
 assure impermeability. 
 
 (c) The reinforcing steel should be in the form of deformed bars and should be galvanized. 
 
 (d) In parts of the vessel where cracks in the concrete would tend to cause leaks, the stress in the steel should 
 be kept low (preferable less than 12,000 lb.). 
 
 (e) Some form of elastic waterproofing coating should be applied to the hull below the deck. 
 
 65 
 
INDEX 
 
 Aberthaw Construction Co., 108, 110, 120, 122, 123, 
 
 131, 256, 823 
 Abrasive resistance of mortars, 255 
 Absorptive properties of concrete, 262 
 Abutments for steel bridges, 645-649 
 
 buried-pier, 648 
 
 care in constructing, 649 
 
 cellular, 647 
 
 pier abutments, 645 
 
 skeleton and arched, 648 
 
 T-abutments, 647 
 
 U-abutments, 647 
 
 wing abutments, 646 
 Abutments in arch bridges, 702 
 
 of arches, 653 
 Acids, effect on concrete, 257 
 Adhesive strength of mortars and concretes, 246 
 Adjustable beam saddles, 145 
 Aggregates, 12-31 
 
 blast-furnace slag, 17 
 
 cinders, 17 
 
 classification, 12 
 
 coarse, 12, 13 
 requirements, 20 
 
 colored, 92 
 
 cost affected by grading, 23 
 crushed stone, 19 
 density and strength, 22 
 
 effect on strength of mortar and concrete, 218-222 
 fine, 12, 13 
 
 materials suitable for, 17 
 
 requirements, 19 
 for concrete products manufacture, 152 
 grading of mixtures, 22 
 granite, 13 
 gravel, 16 
 impurities, 21 
 limestone, 15 
 mechanical analysis, 23 
 metamorphic rocks, 16 
 
 mica, clay, and loam, effect on strength, 224 
 mineral character, effect on strength of concrete, 
 219 
 
 N. Y. Public Service Commission test, 27-30 
 physical characteristics, 67 
 plums, 21 
 preheating, 77 
 preparation of, 171-173 
 quality of sands, 19 
 sand, 18 
 sandstone, 15 
 screenings, 19 
 sea sand, 19 
 sedimentary rocks, 14 
 shape and size of, 221 
 particles of sand, 19 
 
 Aggregates, size and gradation of particles, 22 
 specific gravity, tests for, 25 
 
 specifications of N. Y. Public Service Commission, 
 30 
 
 standard sand, 19 
 
 test of sand used in N. Y. City, 23 
 
 testing, 219 
 on the job, 68 
 
 tests of, 27-30 
 
 trap rock or diabase, 14 
 
 voids in, 25-27 
 Air bubbles on concrete surface, 167 
 
 entrained, 75 
 Akme flat-slab system, 467 
 
 computations for, 488 
 
 tables, 505, 506 
 Alaska-Gastineau Mining Co., 739 
 Alkali, action on concrete, 257 
 Allen, L. H., 823 
 Ambursen dam, 743 
 
 type of dam apron, 741 
 American bars, 44 
 
 American Concrete Institute, computations for Corr- 
 plate floors, 489 
 loading tests for floors, 483-486 
 paper on form design, 123 
 
 recommendations on design of exterior columns, 
 470 
 
 on natural and steam curing of concrete, 156, 158 
 report on bridges, 655 
 
 on concrete ships, 863 
 
 on design of flat-slab floors, 487 
 ruling on flat slab floors, 495 
 
 on flat-slab design, 858 
 specifications for concrete pavement, 24 
 
 for concrete stone, 168 
 American Concrete Steel Co., 479 
 American Society for Testing Materials, 10, 27 
 field testings of concrete, 79 
 specifications for reinforcement bars, 38, 41 
 
 for strength of mortar, 248 
 American Society of Civil Engineers, 28 
 on remixing of concrete, 232 
 rulings on flat-slab design, 854 
 
 See also Joint Committee on Concrete and Rein- 
 forced Concrete. 
 American Steel and Wire Co., 47 
 American System of Reinforcing, 44, 48, 58 
 American wire clamp, 119 
 Ames gage, 266 
 
 Analysis of aggregates, mechanical, 23 
 of beams, 314 
 
 of the arch by elastic theory, 660-668 
 Anderson, A. O., 778, 782 
 Anti-freezing mixtures, 78 
 Aprons of dams, 741 
 
 867 
 
868 
 
 INDEX 
 
 Arch bridges, details of, 691-702 
 
 piers and abutments, 702 
 
 railing and ornamental details, 702 
 
 spandrel details, 691, 694 
 culverts, 796-799 
 ring construction, 702 
 
 thickness of, 656, 658 
 Arched abutments for bridges, 648 
 
 dams, design of, 736 
 Arches, 651-721 
 
 analysis by elastic theory, 660-668 
 arch ring, thickness of, 656, 658 
 arrangement of spandrels, 653 
 axis, shape of, 658 
 classification of arch rings, 660 
 
 Cochrane's formulas for designing symmetrical 
 
 arches, 669-691 
 construction of, 702-715 
 curve of the intrados, 651 
 dead loads and their action lines, 658 
 definitions of terms, 651 
 designing an arch ring, 665 
 details of arch bridges, 691-702 
 filling at crown, depth of, 654 
 formulas for thrust, shear, and moment, 663 
 internal temperature investigations, 664 
 line-of-thrust theories, 659 
 loads, 654 
 parabola, 653 
 piers and abutments, 653 
 reinforcement of concrete, 660 
 
 report on bridges, of American Concrete Institute, 
 655 
 
 semi-ellipse, 652 
 
 temporary hinges, use in erection, 659 
 testing trial arch, 658 
 
 thickness of arch ring, rules for, 656, 658 
 three-centered curve, 652 
 three-hinged, 715-721 
 
 See also Elastic theory of stability of the arch. 
 Areas of rods, table, 354 
 Arrow Rock Dam, 236 
 
 tests on concrete, 261 
 Association of American Steel Manufacturers, steel 
 
 specifications, 37, 40 
 Atlas Construction Co., Ltd., 863 
 Austin dam, 743 
 Autoclave test, 11 
 Axial compression, 845 
 Ayres (F. C.) Mercantile Co., 810 
 
 Bach, Prof., 475 
 
 Backfilling of retaining wall, 601 
 Backwater curve of a dam, 726 
 Bailey, F. S., 290 
 
 Bank-run gravel, proportioning, 70 
 
 Bankers for concrete stone manufacture, 156 
 
 Bar bender, 140 
 
 Bar-chairs, 145 
 
 Barges, concrete, 863-865 
 
 Barney, Prof., 225 
 
 Barrett Co., 517 
 
 Barrows for concrete, 193 
 
 Bars, bent, for web reinforcement, 296 
 
 Bars, reinforcement, 36-45 
 
 Bars, reinforcement, American bars, 44 
 corrugated bars, 43 
 deformed bars, 42 
 diamond bars, 42 
 factors affecting cost, 42 
 Havermeyer bars, 43 
 inland bars, 44 
 rib bars, 44 
 size extras, 42 
 
 steel specifications, 37, 38, 40, 41 
 Barton Spider Web flat-slab system, 461 
 Basement floors, 438 
 
 walls, 545 
 Bates, R. H., 87 
 Bazin, 750, 753 
 
 Beam-and-girder construction, compared with flat- 
 slab, 457 
 
 moments at columns in, 416-425 
 
 monoUthic, 439-456 
 
 forms, 103 
 Beam concentrations, 331 
 
 spacers, 143 
 
 tests on reinforced concrete, 269 
 Beams and slabs, 273-370 
 
 designing tables, 341-359 
 diagrams, 360-370 
 rectangular beams, 273-307 
 
 shear and moment in restrained and continuous 
 beams, 318-341 
 
 slabs, 306, 307 
 
 special beams, 314-317 
 
 T-beams, 307-313 
 
 tables, 354-359 
 Beams, continuous and restrained, shear and moment 
 in, 318-341 
 
 deflection formulas for curved, 660 
 
 in flat-slab floors, 497 
 Beams, rectangular, 273-307 
 
 Beard and Schuler's charts, 350 
 
 bending points of horizontal reinforcement, 297 
 
 bent bars and vertical stirrups for web reinforce- 
 ment, 296 
 
 bond stress, 284 
 
 deflection of rectangular beams, 304 
 
 depth of concrete below rods, 300 
 
 designing, 341-344 i 
 
 diagrams, 360, 366-369 
 
 distribution of stress in homogeneous beams, 273 
 economical proportions, 302 
 flexure formulas, 276-280 
 
 formulas for percentages of steel in reinforced 
 
 beams, 304 
 Leffler's chart for designing, 348, 370 
 lengths of beams, 280 
 moment and diagonal-tension tests, 281 
 placing stirrups from the moment diagram, 291 
 plain concrete beams, 275 
 
 purpose and location of steel reinforcement, 275 
 
 ratio of length to depth of beam, 301 
 
 shear reinforcement, 286 
 
 shearing stresses, 280 
 
 steel in compressive side of beam, 302 
 
 strength in moment and shear, 301 
 
 strengthening against failure in diagonal tension. 
 
INDEX 
 
 869 
 
 Beams, rectangular, tables, 354-356, 359 
 
 tensile stress lines, 275 
 
 theory of flexure, 275 
 
 transverse spacing of reinforcement, 300 
 
 vertical stirrups, 289 
 
 web reinforcement, 285 
 Beams, reinforcement of, table of rods and area for, 357 
 
 restrained, shear and moment in, 318-341 
 Beams, special, 314-317 
 
 analysis, 314 
 
 designing double-reinforced, 317 
 moment of inertia, of complex sections, 317 
 neutral axis, location of, 315, 316 
 of complex sections, 314 
 resisting moment of beams, 316 
 Beams, T-beams, 307-313 
 analysis, 315 
 
 bonding of web and flange, 308 
 conditions met in design of, 310 
 continuous, at the supports, design of, 311 
 deflection formulas, 306, 313 
 designing, 345, 352 
 
 by Beard and Schuler's chart, 350 
 
 for shear, 310 
 economical considerations, 310 
 flange width, 308 
 flexure formulas, 308 
 
 formulas for percentages of steel in reinforced 
 
 beams, 313 
 Leffler's chart for designing, 349, 370 
 proportions, 310 
 
 steel in double-reinforced, formulas, 313 
 
 tables for designing, 364, 365 
 for formulas, 359 
 
 tests, 307 
 Beams, wedge-shaped, 314 
 
 diagrams, 361 
 Beard, R. S., 304, 309, 403 
 
 Beard and Schuler's charts for designing beams and 
 
 slabs, 350 
 Bearing capacity of soils, 557 
 table, 583 
 
 walls, 532 
 Belt conveyors, 177 
 Bending and direct stress, 385-409 
 Bending and placing reinforcement, 139-146 
 
 bar bender, 140 
 
 care necessary, 141 
 
 hand devices, 139 
 
 placing of reinforcement, 142 
 
 power-operated benders, 141 
 
 slab reinforcement, 142 
 
 types of bends, 139 
 
 Universal bar bender, 140 
 Bending-moment curves for beams, 335 
 
 moments in slabs, diagrams, 362, 363 
 Berger Mfg. Co., 57 
 Berry, Prof, H. C, 254 
 
 Billet-steel reinforcement bars, specifications, 37, 38 
 Bins, deep grain, 805-810 
 shallow, 810-815 
 
 designing, data for, 812 
 
 friction angles of various materials, 813 
 
 pressure on sides, 811, 812 
 
 submerged storage for coal, 813 
 
 Bins, shallow, weights and angle of repose of materials, 
 812 
 
 See also Grain bins. 
 Bitumens applied to concrete, 88 
 Blast-furnace slag, 6 
 
 as an aggregate, 17 
 
 proportioning in concrete, 71 
 Blaw Steel Construction Co., 110, 136, 713 
 Block, concrete, 146-169 
 Bond between concrete and steel, 265 
 
 stress of steel beams, 284 
 
 stresses, in column footings, 561 
 Bonding set and new concrete, 75 
 Boom plants, 203 
 Boussinesque, 751, 753 
 Bouvier, M., 733 
 Box culverts, 783-796 
 Brackets, column, 530 
 Bradley Knitting Co., 571 
 Brick, concrete, 146-169 
 
 exterior wall supports for floors, 498 
 
 parapet walls, 522 
 
 veneer, 539 
 Bridges, arch, details of, 691-702 
 
 arches, 651-721 
 
 cantilever, 639-641 
 
 concrete floors and abutments for, 643-649 
 
 continuous-girder, 622-638 
 
 girder, 613-622 
 
 slab, 603-612 
 
 See also Arches. 
 Brown Hoisting Machinery Co., 57 
 Brushing concrete surfaces, 92 
 
 stone surfaces, 165 
 Bucket, clam-shell, 176 
 
 unloaders and conveyors, 177 
 Buckets for concrete, 194 
 
 for concrete products manufacture, 154 
 Building Codes, Rochester, and Seattle, 431 
 
 frames, moments in, 411-430 
 
 tests, 480 
 Buildings, 431-555 
 
 columns, 527-532 
 
 construction methods and safeguards, 506 
 
 contraction and expansion, provision for, 554 
 
 elevator shafts, 552-554 
 
 floors, concrete, 431-512 
 
 roofs, 512-527 
 
 stairs, 549-552 
 
 steel-frame construction, 511 
 
 unit construction, 508-511 
 
 walls and partitions, 532-549 
 Buried-pier abutments, 648 
 Bush Stores, 182 
 
 Buttresses of reinforced-concrete dams, 740 
 
 Cableways for transporting concrete, 195 
 
 Caissons for dams, 728 
 
 Calcium chloride, effect on concrete, 236 
 
 quicklime, 2 
 California R. R. Commission, 745 
 Candlot, 233 
 
 Cantilever bridges, 639-641 
 
 flat-slab construction, 463, 481 
 of bridges, 612 
 
870 
 
 INDEX 
 
 Cantilever footings, 568 
 
 walls of reinforced concrete, 587-592 
 Carton molds for concrete tests, 81 
 Carts, concrete, 193 
 Casey, John F., Co., 763 
 Casting concrete in sand, 161 
 Catskill Reservoir, N. Y., 763 
 Cellular abutments, 647 
 Cement, 1-62 
 
 bulk, use of, 12 
 
 chemical analysis, 11 
 
 common lime, 1 
 
 consistency, 9 
 
 containers for, 11 
 
 fineness, 8 
 
 grappier, 2 
 
 grouting, 88 
 
 gypsum plasters, 1 
 
 handling in sacks, 177 
 
 hydraulic and non-hydraulic, 1 
 
 hydraulic lime, 2 
 
 manufacture, 6-8 
 
 natural cement, 3 
 
 non-hydraulic, 1 
 
 Portland and natural cement, 4 
 
 puzzolan, 2 
 
 rock, 6 
 
 sacks, bundling and storage, 178 
 sand, 4 
 seasoning, 12 
 slag, 2 
 
 soundness, 10 
 specifications for, 11 
 specific gravity, 11 
 
 storage and conveying for concrete products, 151 
 
 storing, 11, 173 
 
 strength, 10 
 
 testing, 8 
 
 time of setting, 9 
 
 tufa, 4 
 
 weight, 12 
 
 See also Portland cement. 
 Cement mortar, properties, 215-263 
 aggregates, 218-222 
 
 compressive and tensile strengths compared, 227 
 consistency, relation to strength, 225 
 contraction and expansion, 252-254 
 curing conditions, effect on strength, 234 
 durability, 254-259 
 effect of freezing, 236 
 
 of method of placing, on strength, 231 
 of mica clay, and loam in aggregates, 224 
 of size of sand on strength of, 221 
 elastic properties, 250-252 
 laboratory tests of strength, 215 
 method of mixing, effect on strength, 230 
 permeability, and absorptive properties, 262 
 porosity, 261 
 regaging, 232 
 
 relation between density and strength, 222 
 rise of temperature in setting, 259 
 salts, effect of, 236 • 
 strength, 215-250 
 
 compared with neat cement, 217 
 weight, 263 
 
 Centering arches, 704 
 Centering fabrics, 55 
 
 See also Self-centering fabrics. 
 Chair lock, 146 
 
 pinchers, 146 
 
 spacers, 145 
 Chalk, for cement manufacture, 7 
 Chambers, S. H., 258 
 Chanelath, 56 
 Chapman, C. M., 82, 238 
 Charging hoppers, 190 
 Charts, Beard and Schuler's, 350 
 
 Leffler's comprehensive beam, 348, 370 
 Checking materials on the job, 69 
 Chemical analysis of cements, 11 
 
 hardeners for concrete floors, 210 
 Chezy formula, 727 
 
 Chicago Building code, 460, 462, 465, 483-486 
 cube mixer, 230 
 
 flat-slab design rulings, 847, 849 
 Chicago, Milwaukee & St. Paul Railway, box cul- 
 vert, 785 
 
 bridges, 625 
 Chicago ruling for flat-slab construction, 494 
 
 for flat-slab panels, 501-503 
 Chile Exploration Co., 768 
 Chimneys, reinforced-concrete, 816-821 
 
 bases, 820 
 
 construction, 820 
 
 dead-load stresses, 816 
 
 eccentric load, 407 
 
 longitudinal shear, 819 
 
 stresses on annular sections in flexure, 816 
 
 temperature stresses, 819 
 
 wind stresses, 818 
 
 without vertical reinforcement, 818 
 China-wood oil, on concrete floors, 210 
 Churn drills, 171 
 
 Chutes for transporting concrete, 195 
 Cinder-concrete insulation for roofs, 514 
 
 concrete, strength of, 248 
 
 fill for roofs, 514 
 Cinders, as an aggregate, 17 
 
 proportioning in concrete, 71 
 Circumferential flat-slab system, 461, 471 
 Clam-shell buckets, 176 
 Clay cement in a sandstone, 18 
 
 for cement manufacture, 6 
 
 in aggregates, 21, 224 
 Cleveland Building Code, 462, 477, 478 
 
 Railway Co., 476 
 
 reservoir, 763 
 Cline, A. E., 158 
 Clinton Wire Cloth Co., 46 
 Coal bins, see Bins, shallow. 
 
 submerged storage for, 813 
 Coarse aggregates, 12, 20 
 Cochrane, V. H., 658 
 
 Cochrane's formulas for designing arches, 669-691 
 Coefl&cient of expansion of steel, 37 
 Cold, effect on concrete floors, 209 
 Cold-weather concreting, 77 
 Colored aggregates, 92 
 Colors, adding to concrete, 93 
 for concrete floors, 206 
 
INDEX 
 
 871 
 
 Colors of concrete products, 164 
 
 Columbia Univ., tests, 250 
 
 Column footings, see Footings, concrete. 
 
 forms, 96-103 
 
 heads, forms for, 108 
 
 moment, 486 
 
 reinforcement, table of rods and area for^ 357 
 Columns, concrete, 371-384, 527-532 
 
 bracket loads, stress in columns supporting, 381 
 
 brackets, 530 
 
 combined stresses in, 426 
 
 definition by Joint Committee, 371 
 
 design, details of, 527 
 
 diagram for designing, 374, 382 
 
 formula for bending stress, 371 
 
 hooped and longitudinal reinforcement, 372 
 
 in flat-slab construction, 497 
 
 load tables, 374-381 
 
 loading, 530 
 
 long, reduction formula, 381 
 longitudinally reinforced, 371 
 maximum combined stresses, 426 
 modulus of elasticity, 371 
 moments at, 415, 416-425, 486 
 plain, 371 
 
 reduction formula, 381 
 
 reinforced with structural-steel shapes, 372 
 strength of, 229 
 
 stresses for hollow and solid, 407-409 
 supporting bracket loads, stress in, 381 
 types, 371 
 
 working stresses, 373 
 Combined column footings, 565 
 
 stresses in columns, maximum, 426 
 Common lime, 1 ' 
 Composite wood and concrete piles, 573 
 Compression of beams, steel reinforcement for, 302 
 Compressive strength of cement, 10 
 
 of mortar and concrete, 227, 845 
 Concrete Appliances Co., Inc., 23 
 Concrete barges and ships, 863-865 
 
 bonding set and new, 75 
 
 cinder, strength of, 248 
 
 columns, 371-384, 527-532 
 
 cost of, 823-826 
 
 curbing, 212 
 
 depositing, 73-78 
 
 electrolysis in, 258 
 Concrete Engineering Co., 138 
 Concrete, field tests of, 78-82 
 
 finishing surfaces, 90 
 
 floors, 205-210, 431-512 
 
 and abutments, for steel bridges, 643-649 
 forms for, 93-139 
 
 freezing, 78 
 
 grading of sands for, 23 
 hardening rate in warm weather, 77 
 impervious, rules for, 89 
 machine vs. handmixing, 186 
 Concrete materials, 1-62 
 aggregates, 12-31 
 cement, 1-12 
 reinforcements, 36-62 
 water, 31-36 
 
 See also Handling concrete materials. 
 
 Concrete, mixing, 72, 186 
 
 parapet walls, 522 
 
 piles, 572 
 
 placing, 73-78 
 Concrete, plain, properties of, 215-263 
 
 contraction and expansion, 252-254 
 
 durabihty, 254-259 
 
 elastic properties, 250-252 
 
 permeability and absorptive properties, 262 
 
 porosity, 261 
 
 protection of embedded steel from corrosion, 262 
 rise of temperature in setting, 259 
 strength, 215-250 
 weight, 263 
 
 Concrete products, see Concrete stone, manufacture 
 and use of. 
 
 proportioning, 63-72 
 
 quality for reservoir masonry, 760 
 
 reinforced, see Reinforced concrete. 
 
 remixed and retempered, 76 
 
 removing entrained air, 75 
 
 roadways, 213-214 
 
 roof surfaces, 516 
 
 sea water, action on, 256, 864 
 
 ships and barges, 863-865 
 
 sidewalks, 210-212 
 
 spading, puddling, and tamping, 75 
 
 specifications to prevent excess of water, 36 
 Concrete Steel Co., N. Y., 143, 145, 465 
 Concrete Steel Engineering Co., 42, 43 
 Concrete Steel Products Co., Chicago, 463, 480 
 Concrete stone, manufacture and use of, 146-169 
 
 aggregates, kind and quality, 152 
 
 agitation after mixing, 153 
 
 air bubbles, 167 
 
 bankers, 156 
 
 brushing, 165 
 
 buckets and hoppers, 154 
 
 casting in sand, 161 
 
 cement storage and conveying, 151 
 
 colors, 164 
 
 combination molds, 161 
 commercial molds, 149 
 consistency, 148 
 crazing, 167 v 
 curing, 156 
 
 development of industry, 146 
 
 dry-tamp method, 147 
 
 efflorescence, 167 
 
 face design in standard units, 162 
 
 facing materials, 163 
 
 gang molds for wet-cast products, 151 
 
 materials, 151 
 
 methods of manufacture, 147 
 mixing, 152 
 molds, 159 
 mosaics, 167 
 natural curing, 156 
 operation of machines, 150 
 pallets, 155 
 placing, 154 
 • plaster molds, 159 
 pressure method, 148 
 rubbing surface, 166 
 sand molds, 161 
 
872 
 
 INDEX 
 
 Concrete stone, manufacture of, specifications of 
 American Concrete Inst., 168 
 
 spraying, 165 
 
 steam curing, 157 
 
 surfaces, 162 
 
 tamping, 150 
 
 tooling, 166 
 
 waste molds, 161 
 
 wet-cast method, 148 
 
 wheelbarrows, 155 
 
 wood molds, 159 
 Concrete, strength of, 64, 215-250 
 
 transporting, 72 
 
 waterproofing, 82-90 
 
 weakness, cause of, 64 
 
 work, measurement of, 830 
 
 working stresses, 845 
 Concreting in hot and cold weather, 76-78 
 
 effect of weather, 76 
 
 preheating aggregates and water, 77 
 Concreting plant, 181-203 
 
 balancing the plant, 181 
 
 cost, 181 
 
 machine vs. handmixing, 186 
 
 spouting plants, 203 
 
 time of mixer operations, 189 
 
 transporting and placing of concrete, 193-203 
 
 types of mixers, 187 
 
 typical plants, 181 
 Condensation on roof slabs, prevention of, 513 
 Condron, T. L., 483 
 Condron Co., 467, 488 
 Conductivity of concretes, 254 
 Conduits and sewers, 799-803 
 
 construction, 802 
 
 external earth pressure, 799 
 
 forms for sewers, 803 
 
 longitudinal reinforcement, 802 
 
 non-circular, 799 
 
 reinforced pipe, examples, 803 
 
 stresses due to internal pressure, 799 
 Consistency of concrete stone, 148 
 
 of mortar and concrete, 225 
 Consolidated Expanded Metal Cos., 51, 60 
 Constant-angle dams, 738 
 Construction, methods, 63-169, 506 
 
 bending and placing reinforcement, 139-146 
 
 field tests of concrete, 78-82 
 
 finishing concrete surfaces, 90-93 
 
 flat-slab, 457-508 
 
 floor, monolithic beam-and-girder, 439-456 
 forms, 93-139 
 hollow-tile, 447 
 
 manufacture and use of concrete stone, block, and 
 
 brick, 146-169 
 mixing, transporting, and placing concrete, 72-78 
 proportioning concrete, 63-72 
 waterproofing concrete, 82-90 
 See also Forms. 
 Construction of arches, 702-715 
 arch-ring construction, 702 
 centering, 704 
 sand boxes, 710 
 steel centers, 712 
 three-hinged arches, 717 
 
 Construction of arches, timber centers, 704 
 Construction plant, 171-203 
 
 concreting plant, 181-203 
 
 handling and storage of materials, 173-179 
 
 preparation of concrete aggregates, 171-173 
 
 typical installation, 179 
 Containers for cement, 11 
 
 Continuous beams, shear and moment in, 318-341 
 Continuous-girder bridges, 622-638 
 
 analysis of stresses, 625 
 
 effect of fixed bases, 638 
 
 examples of types, 623 
 
 expansion joints, 622 
 
 four-span viaduct frame, 629, 633 
 
 monolithic construction, 622 
 
 one-span frame, unequal columns, 636 
 
 one-span viaduct frame, 633 
 
 temperature stresses, 636 
 
 three-span viaduct frame, 631, 634 
 
 two-span viaduct frame, 632, 634 
 
 viaduct bent, 638 
 frames, 628 
 Continuous slab spacers, 145 
 Contraction in buildings, provision for, 554 
 
 of cement mortar and concrete, 252-254 
 coefficient of expansion, 252 
 Conveyances for concrete materials, 174 
 Cooper's Standard Loadings, for railroad bridges, 656 
 Core drill test specimens, 79 
 Cornell University, mortar tests, 222 
 
 tests of strengths of concrete, 229 
 on expansion of concretes, 252 
 Corr-mesh, 56 
 Corr-plate floors, 470 
 
 computations for, 489 
 
 tables, 503-505 
 Corr reinforcing system, 60 
 Corr-X-metal, 52 
 
 Corrosion of steel reinforcement, protection by concrete, 
 262 
 
 Corrugated Bar Co., 43, 52, 56, 60, 470, 480, 481, 489 
 Corrugated bars, 43 
 
 sheets, dovetailed, 57 
 Costs of concrete, estimating, 823-826 
 
 concreting plant, 181 
 
 conveying concrete materials, 174 
 
 forms, 94, 826-828 
 
 steel reinforcement, 828 
 
 surface finish, 829 
 Coulomb's wedge of maximum earth pressure, 577 
 Counterforted walls, design of, 592-600 
 
 back floor slab, 593 
 
 cantilever toe slab, 596 
 
 methods of reinforcing, 596 
 
 thickness and spacing of counterforts, 592 
 
 vertical or face wall, 593 
 Cracks in concrete surfaces, 167 
 Cramp & Co., 182 
 Crazing of concrete surfaces, 167 
 Creager, W, P., 730 
 Cream of lime, 1 
 Crushed limestone, 172 . 
 
 stone, 19 
 
 aggregate, preparation of, 171-173 
 screening, grading, and washing, 172 
 
INDEX 
 
 873 
 
 Crusher-run stone, proportioning, 71 
 Crushers, stone, 172 
 Culverts, 775-799 
 
 arch, 796-799 
 
 box, 783-796 
 
 circular, cast in place, 782 
 construction of box culverts, 794 
 design of cross-section of arch culverts, 796 
 of box culverts, 785 
 
 of ends, 777 
 diagrams of box culverts, 794 
 efficiency, 776 
 factors in design, 776 
 forms for arch culverts, 797 
 
 for box culverts, 795 
 length, 776 
 
 loading of box culverts, 785 
 
 loads on pipes in ditches, table, 779 
 
 pipe, 777 
 
 pressure in trenches, 778 
 
 strength of pipe, 781 
 
 waterway required, 776 
 Cumming's reinforcing system, 58 
 Curbing, concrete, 212 
 
 Curing in concrete products manufacture, 156 
 Curry tyer, 143 
 Curtain walls, 533 
 
 Dams, 723-759 
 apron, 741 
 arched, design, 736 
 automatically operating, 759 
 backwater curve, 726 
 bracing, 741 
 buttresses, 740 
 caissons, 728 
 capacity of reservoir, 726 
 constant-angle, 738 
 constant-radius, 736 
 cut-off walls, 728, 740 
 design of foundation, 728 
 discharge capacity of spillway, 749 
 Dunning's dam, 758 
 earthen, with concrete core wall, 745 
 Edge dam, 743 
 final calculation, 734 
 fish ladders, 759 
 foundation mattress, 740 
 Gatun dam, 754 
 geological investigations, 723 
 gravity section, design of, 729 
 grouting, 728 
 height of structure, 724 
 hollow reinforced, 743 
 hydrographic inivestigations, 725 
 hydrostatic pressure, 729 
 ice thrust, 732 
 initial stress, 732 
 Kensico dam, 236 
 locating, 723 
 Morton dam, 743 
 movable, 758 
 multiple arch, 743 
 operation of movable dams, 759 
 overflow dams, 753 
 passing the discharge, 748 
 
 Dams, pilings, 728 
 
 preliminary studies, 723 
 profiles, 729 
 
 of spillways, 749 
 Ransome dam, 743 
 
 reinforced-concrete, design of, 739-745 
 
 selecting suitable type, 724 
 
 shearing stresses, 734 
 
 sheet piling, 728 
 
 siphonic spillways, 755 
 
 sluices, 754 
 
 spillway, form of, 748 
 
 stresses in masonry and on foundation, 733 
 
 temperature stresses, 732 
 
 uplift, 731 
 
 wind pressure, 732 
 Davis, R. P., 222 
 Day's work planes, 34 
 Dead loads of arches, 658 
 
 of floors, 432 
 
 stresses on chimneys, 816 
 Decanville Automobile Co., 571 
 Deck girders for bridges, 613 
 Deere Webber Building Co., 479, 480 
 Defects in concrete floors, 207-210 
 Deflection formulas for curved beams, 660 
 
 of rectangular beams, 304 
 
 of T-beams, 306, 313 
 Deformed bars for reinforcement, 42 
 Delaware, Lackawanna, and Western R. R. bridge, 715 
 Density of aggregates, 22 
 
 of concrete, maximum, 68 
 
 of mortars and concretes, 222 
 Depositing concrete in forms, 73, 74 
 
 through water, 76 
 Designing columns, tables and diagrams, 374-382 
 
 double-reinforced beams, 317 
 
 floor construction, problem, 440 
 
 forms for concrete, 120 
 
 drafting-room methods, 123 
 
 formulas, 125 
 
 tables and diagrams, 124 
 
 hollow-tile floors, problem, 453 
 
 retaining walls, 584, 587-600 
 
 T-beams, 310 
 
 tables and diagrams for beams, 341-370 
 Beard and Schuler's charts, 350 
 diagrams, 360-370 
 
 effler's comprehensive beam chart, 348, 370 
 
 tables, 354-359 
 Deslaurier's Column Mold Co., 110 
 Detroit river tunnel, 232 
 Diabase, 14 
 
 Diagonal tensile stress in beams, 281 
 tension in column footings, 562 
 
 Diagrams for beams and slabs, 341-370 
 
 for designing symmetrical arches, 669-691 
 
 for stress, bending and direct, 388-402, 404, 405 
 
 Diamond bars, 42 
 
 Direct stress, 385-409 
 
 Dock pockets, 815 
 
 Dolomitic quicklime, 2 
 
 Door openings, 543 
 
 Douglas, W. J., 656 
 
 Dovetailed corrugated sheets, 57 
 
 Dow, A. W., 248 
 
874 
 
 INDEX 
 
 Drainage of retaining wall, 601 
 
 of roofs, 518 
 Drills for quarrying, 171 
 Drum, J. L., 479 
 Drum mixers, 187 
 
 Dry-tamp method of making concrete stone, 147 
 Duchemin's formula for wind pressure, 513 
 Dunning's dam, 758 
 
 Durability of cement mortar and concrete, 254-259 
 
 abrasive resistance, 255 
 
 action of acids, oils, and sewage, 257 
 of alkali, 257 
 of sea water, 256 
 
 electrolysis in concrete, 258 
 
 fire-resistance properties, 254 
 
 manure, effect of, 259 
 
 weathering qualities, 255 
 Dusting of concrete floors, 206 
 
 Earth pressure, 575 
 
 coefficients of internal friction, 576 
 
 Coulomb's wedge of maximum pressure, 577 
 
 Rankine's formula, 576 
 
 results of theories, interpretation of, 580 
 Earthern dams, 745 
 Easel-chairs, 145 
 Eastwood, 743 
 Easy chairs, 143 
 Econo expanded metal, 53 
 Edge, W. S., 457 
 Edge dam, 743 
 
 EflBorescence of concrete surface, 167 
 Elastic properties of cement mortar and concrete, 250- 
 252 
 
 modulus of elasticity, 251 
 stress-strain curves, 250 
 yield point, 251 
 Elastic theory of stability of the arch, 659, 660-668 
 analysis, procedure, 661 
 arch-ring design, 665 
 
 arch structure of two spans with elastic pier, 667 
 Cochrane's formulas for designing arches, 669-691 
 correcting maximum moments when axis deviates, 
 686 
 
 deflection at any point, 664 
 
 deflection of curved beams, 660 
 
 diagrams, for moments, thrusts, and average 
 
 stresses, 681 
 difficulties in applying, 669 
 division of arch ring, 663 
 
 formulas for thrust, shear, and moment, 663 
 influence-line diagrams, 673 
 internal temperature investigations, 664 
 loadings, 664 
 
 origin of coordinates at crown, etc, 665-667 
 
 shape of arch axis, 669 
 
 shrinkage stresses, 664 
 
 skew arches, 665 
 
 unsymmetrical arches, 665 
 
 use of influence lines, 664 
 
 variation in thickness of arch ribs, 670 
 Electric Welding Co., 58, 146 
 Electrolysis in concrete, 258 
 Elephant Butte Dam, 191 
 Elevated tanks, 771-775 
 Elevator shafts, 552-554 
 
 Elevator, pent houses, 553 
 
 pits, 552 
 Emperger columns, 529 
 
 Empirical rules for thickness of arch ring, 656 
 Entrained air, removal of, 75 
 Estimating, 823-832 
 
 concrete, cost of, 823-826 
 
 costs, 823-830 
 
 forms, unit cost of, 826-828 
 
 formwork, 831 
 
 labor, cost of, 824, 827 
 
 materials, cost, 823, 827 
 
 measurement of concrete work, 830 
 
 plant, cost of, 825 
 
 quantities, 830-832 
 
 steel, amount of, 832 
 
 steel reinforcement, cost of, 828 
 
 surface finish, amount of, 832 
 cost of, 829 
 
 unit costs, 823 
 Expanded metal, 50-54 
 
 Corr-X-metal, 52 
 
 econo, 53 
 
 GF, 53 
 
 Kahn mesh, 52 
 
 steel Crete, 51 
 Expanded Metal Cos., 546 
 Expansion in buildings, provision for, 554 
 
 joints, 86 
 
 of cement mortar and concrete, 252-254 
 moisture changes, 253 
 
 Facing materials of concrete stone, 162 
 
 Factor of limita,tion of retaining walls, 584 
 
 Feret, R., 243, 244 
 
 Fiber stresses of forms, 124 
 
 Field tests of concrete, 78-82 
 
 carton molds, 81 
 
 core drill specimens, 79 
 
 field-molded specimens, 79 
 
 limitations, 78 
 
 methods of American Society for Testing materials, 
 79 
 
 pre-use tests, 82 
 
 transverse tests on beam specimens, 79 
 Fine aggregates, 12, 17, 19 
 Fineness of cement, 8 
 Finishing concrete surfaces, 90 
 
 addition of colors to concrete, 93 
 
 brushing, 92 
 
 colored aggregates, 92 
 
 plaster finishes, 93 
 
 removing form marks, 90 
 
 rubbing, 91 
 
 sand-blasting, 92 
 
 specification for rubbed surface, 93 
 tooling, 90 
 
 Fire protection, depth of concrete below rods, 300 
 
 resistance properties of concrete, 254 
 
 -resisting windows, 541 
 Fish ladders in dams, 759 
 Flange width of T-beams, 308 
 Flat-slab bridges, 612 
 Flat-slab construction, 457-508 
 
 advantages over beam-and-girder, 457 
 
 Akme system, 467, 488 
 
INDEX 
 
 875 
 
 Flat-slab construction, Akme system, tables, 505, 506 
 Barton Spider Web system, 461 
 beams in floors, 497 
 brick exterior wall supports, 498 
 buildings to which adapted, 458 
 cantilever system, 463 
 Chicago ruling, tables, 501-503 
 columns, 497 
 
 computations on Chicago ruling, 494 
 
 on Pittsburgh ruling, 492 
 
 on ruling of American Concrete Inst., 495 
 Corr-plate floors, 470, 489 
 
 tables, 503-505 
 designing, methods, 459, 487 
 
 rulings pertaining to, 847-860 
 foundations, 571 
 
 four-way system, 461-466, 480, 494, 495 
 loading tests, 480 
 
 methods and safeguards, 506 
 moments in columns in, 425 
 mushroom system, 465 
 Pittsburgh ruling, tables, 498-501 
 roof design, 496 
 simplex system, 465 
 
 systems of designing and reinforcing, 461 
 
 tables for flat-slab panels, 498-506 
 of moments, 483, 484, 485 
 
 tests, discussion of, 482 
 
 three-way system, 476 
 
 Watson system, 466 
 
 See also Floors, concrete. 
 Flat Slab Patents Co., 461, 465 
 Fleming, Prof. B. P., 254 
 
 Flexure formulas for reinforced-concrete beams, 276- 
 280 
 
 assumptions in calculations, 276 
 for T-beams, 308 
 ultimate loads, 279 
 
 working and ultimate loads compared, 280 
 
 working loads, straight-line theory, 276 
 Flexure, theory of, 275 
 Floors, concrete, 205-210, 431-508 
 
 attaching shaft-hangers and sprinkler pipes, 434 
 
 basement floors, 438 
 
 bedding machinery, 437 
 
 causes of defects, 207 
 
 chemical hardeners, 210 
 
 coatings and paints, 210 
 
 dead load, 432 
 
 defects, 207-210 
 
 designing monolithic beam-and-girder construc- 
 tion, problem, 440 
 dusting, 206 
 
 economic considerations, 432 
 effect of excess water, 35 
 
 of freezing, 209 
 flat-slab construction, 457-508 
 
 rulings, 847 
 for reservoirs, 762 
 for steel bridges, 643-649 
 grinding, surface, 206 
 hardeners and surface compounds, 207 
 hollow-tile construction, 447 
 loads, 431 
 
 making good floors and surfaces, 206 
 
 monolithic beam-and-girder construction, 439-456 
 
 Floors,, concrete, openings, 433 
 paints, 210 
 patents, 479 
 pigments, 206 
 problems, 205 
 remedies for defects, 209 
 slabs, support of, 307 
 surface finishes, 206 
 surfaces, 93, 432 
 tests, 438, 480 
 tiles, 433 
 
 top coat separating from base, 209 
 types, 431 
 use of oils, 210 
 waterproof, 438 
 wood surfaces, 432 
 See also Flat-slab construction. 
 Footings, concrete, 558 
 bond stresses, 561 
 cantilever footings, 568 
 column, 559 
 combined, 565 
 diagonal tension, 562 
 examples, 571 
 
 four-way reinforcement, 562 
 single, 559 
 
 stepped and sloping, 562 
 wall, 559 
 
 width, in flexure computations, 561 
 Form marks, removal of, 90 
 Forms for concrete, 93-139 
 
 beam-and-girder forms, 103 
 
 Blaw light wall forms, 137 
 
 calculating floor forms, 121 
 
 clamp of Sterling Wheelbarrow Co., 100 
 of Universal Form Clamp Co., 102 
 
 column forms, 96-103 
 heads, 108 
 
 construction notes, 138 
 
 cost, 94, 826-828 
 
 curved pier forms, 119 
 
 depositing concrete in, 73, 74 
 
 design of, 120 
 
 drafting-room methods of design, 123 
 estimating amount of, 831 
 
 unit cost of, 826-828 
 examples of design, 96 
 for retaining walls, 602 
 Gemco adjustable steel shoring, 107 
 
 column clamp, 99, 101 
 Hodges system, 105 
 hydraulic steel column forms, 137 
 K. & W. clamp, 100 
 lumber for, 94 
 Meyer steelforms, 138 
 New England column clamp, 99 
 number of sets in building work, 96 
 pier, 103, 111 
 
 pressure of concrete against, 121 
 removal of, 95 
 
 Ross self-lock adjustable shore, 107 
 slab forms, 104 
 steel, 135 
 
 steel floredomes, 137 
 
 systematizing form-work on buildings, 131 
 tables and diagrams for designing, 124 
 
876 
 
 INDEX 
 
 Forms for concrete, time required before removing, 96 
 
 values to use in design, 121 
 
 wall and pier forms, 103, 111 
 
 wire clamps, 119 
 
 Wiscoforms, 138 
 Formulas, flexure, for reinforced concrete beams, 276- 
 280 
 
 for T-beams, 308 
 for designing forms, 125 
 for designing symmetrical arches, 669-691 
 for percentages of steel in reinforced beams, 304, 
 313 
 
 Foster Armstrong Co., 183 
 
 Fougner, Steel Concrete Ship-building Co., 863 
 Foundations, 557-574 
 
 advantage in using reinforced concrete, 558 
 
 bearing capacity of soils, 557 
 
 cantilever footings, 568 
 
 combined column footings, 565 
 
 column footings, 559 
 
 examples of column footings, 571 
 
 piles, 557, 572 
 
 plain concrete footings, 558 
 
 pressure on the soil, 558 
 
 raft foundations, 570 
 
 wall footings, 559 
 Four-way flat-slab construction, 461-166, 480, 494, 495 
 Framed-bent trestles, 610 
 Francis, 749 
 
 Freezing of concrete, 78, 236 
 
 French, A. W., 342 
 
 Fuller, W. F., 24, 68, 243, 263 
 
 GF expanded metal, 53 
 Gage, steel wire, 45 
 Gaging, use of oils, 240 
 
 use of sea water, 238 
 Gang molds for wet-cast products, 151 
 Gardner, Prof. Harry, 238 
 Garver, N. B., 122 
 Gatun dam, 754 
 Gearhart, 783 
 
 Gemco adjustable steel shoring, 107 
 Gemco Mfg. Co., 99, 101, 108 
 General Electric Co., 139 
 General Fireproofing Co., 53, 56, 138 
 Geological investigations for dam sites, 723 
 Gibb, H. M., 587 
 Gillmore needles, 841 
 Girder bridges, 613-622 
 deck girders, 613 
 through girders, 617 
 
 forms, 103 
 Glue molds, 160 
 Goldbeck, A. T., 253, 254 
 
 tests by, 605 
 Goodwin, R. E., 232 
 Grading crushed stone, 172 
 
 of mixtures in aggregates, 22 
 Grain bins, 805-810 
 
 conclusions from tests, 806 
 
 construction, 809 
 
 design of walls, 808 
 
 hexagonal bins, '809 
 
 horizontal reinforcement, 808 
 
 Janssen's pressure formulas, 805 
 
 Grain bins, load of walls, 808 
 
 rectangular section, 808 
 
 thickness of walls, 808 
 
 wind stresses on horizontal section, 808 
 Granite, 13 
 Granolithic, 206 
 
 finish of floors, 432 
 Graphical determination of stresses, 406 
 Grappier cement, 2 
 Grashof's theory, 769 
 Gravel, 16 
 
 bank-run, proportioning, 70 
 
 screening, 172 
 
 voids in, 26 
 
 washing, 173 
 Gravity mixers, 188 
 
 Great Northern Ry. grain elevator, 809 
 Grinding, surface, 206 
 
 Groined-arch construction for reservoirs, 762, 764 
 Grout, cement, 88 
 Grouting, 728 
 Guy-line plants, 203 
 Gypsum plasters, 1 
 
 Hain, J. C, 241 
 
 Hand vs. machine mixing of concrete, 186 
 Handling concrete materials, 173-179 
 
 belt conveyors, 177 
 
 bucket unloaders and conveyors, 177 
 
 bundling empty cement sacks, 178 
 
 cement in sacks, 177 
 
 clam-shell buckets, 176 
 
 conveyance economies, 174 
 
 sand, storage and care of, 174 
 
 shoveling from car to ground, 174 
 
 shovels, size and type, 175 
 
 stone, care of, 174 
 
 typical installation, 179 
 
 unloading economies, 175 
 
 water, storage and handling, 179 
 Hardeners for concrete floors, 207 
 
 chemical, 210 
 Hardman, R. C, 140 
 Harrison, 737 
 Hatt, T. K., 482 
 Hauland, J., 863 
 Havermeyer bars, 43 
 Heat conductivity of concrete, 254 
 Heating aggregates, 77 
 Hennebique reinforcing system, 60 
 Henny, D. C, 725 
 Hering, R, 258 
 Herschel, 748 
 High calcium quicklime, 2 
 Hillberg, A. G., 723 
 Hinged arches, 715-721 
 Hinges, use in arch erection, 659 
 Hodges, Jesse E., 104 
 Hodges adjustable shores, 105 
 Hoists for handling concrete, 200 
 Hollow-tile construction, 447 
 
 floors, designing problem, 453 
 
 table, 456 
 insulation, 515 
 weights, 452 
 Hooker, D. E., 481 
 
INDEX 
 
 877 
 
 Hooke's law, 275 
 
 Hooped reinforcement of columns, 372 
 Hoppers for concrete products manufacture, 154 
 Horton, 753 
 
 Hume-Bennett Lumber Co., 745 
 Hy-chairs, 145 
 Hydrated lime, 2 
 
 effect on concrete, 238 
 Hydraulic cements, 1 
 
 lime, 2 
 
 Hydraulic Pressed Steel Co., Ill, 137 
 Hydraulic structures, 723-803 
 
 conduits and sewers, 799-803 
 
 culverts, 775-799 
 
 dams, 723-759 
 
 elevated tanks, 771-775 
 
 reservoirs, 760-765 
 
 standpipes and small tanks, 765-771 
 Hydrographic investigations, for dams, 725 
 Hydrostatic pressure, in dams, 729 
 Hy-rib construction, 56, 546 
 
 Ice pressures on dams, 732 
 Igneous rock, 13 
 
 Illinois Central R. R. trestles, 607, 610 
 Illinois Dept. of Factory Investigation, 96 
 Illinois Highway Commission, deck-girder bridges, 
 613, 615 
 
 through-girder bridges, 617 
 Impervious concrete, rules for, 89 
 Impurities in aggregates, 21 
 Inertia, moment of, of beams, 317 
 
 of continuous beams, 339 
 Influence lines, 324 
 
 of arches, 664 
 diagrams, 673 
 Inland bar, 44 
 Inland Steel Co., 44 
 Insulation for roof slabs, 513 
 Insurance Engineering Station, 263 
 Integral waterproofing compounds, 87 
 International Association for Testing Materials, 256 
 Iowa Highway Commission, abutment design, 607 
 
 deck-girder bridges, 613 
 
 through-girder bridge, 617 
 Iowa State college, 664 
 
 tests of pressure on pipes, 778 
 Iron, powdered, as a surface hardener, 207 
 
 pyrites in an aggregate, 22 
 
 Jackson, J. H., 164 
 
 Janssen's formulas for pressure in deep bins, 805 
 Johnson, E. F., 427 
 Johnson, H. C, 65 
 Johnson, S. E., 236 
 
 Joint Committee on Concrete and Reinforced Concrete, 
 statement on: acids and oils, effect on con- 
 crete, 257 
 
 analysis of aggregates, 24 
 
 bands on columns, 528 
 
 bent bars for web reinforcement, 297 
 
 concrete barges and ships, 863 
 
 concrete in sea water, 257 
 
 corrosion of metal reinforcement in concrete, 262 
 
 definition of column, 371 
 
 depth of concrete below rods, 300 
 
 fire protection by concrete, 254 
 
 Joint Committee on Concrete and Reinforced Concrete, 
 statement on: flange width of T-beams, 308 
 flexure formulas for reinforced-concrete beams, 
 276 
 
 floor slabs, support of, 307 
 
 forces to be resisted by reinforced concrete, 273 
 permeability and waterproofing of concrete, 262 
 positive and negative moments of beams, 318 
 regaging, 232 
 
 reinforcing bars for compression in beams, 302 
 shearing strength of concrete, 301 
 spacing of reinforcement, 300 
 
 of vertical stirrups, 290 
 span length for beams and slabs, 318 
 supports of continuous beams to resist bending 
 
 moment, 339, 340 
 tests for flat-slab floors, 483-486 
 working stresses for concrete columns, 373 
 of steel, 37 
 Joints on concrete roadways, 214 
 Jorgensen, 745 
 Jumper, 171 
 
 K. & W. Clamp Co., 100 
 
 Kahn mesh, 52 
 
 Kahn reinforcing system, 57 
 
 Kardong Bros., 141 
 
 Kaufman, G., 258 
 
 Keator & Co., Edward O., 104 
 
 Kensico Dam, 236 
 
 tests on concrete, 261 
 Ketchum, Prof., 806, 808 
 Kinney, W. M., 158 
 Kutter's formula, 727 
 
 L-frame, 429 
 
 Labor, estimating costs, 824, 827 
 Laboratory tests of mortar and concrete, 215 
 Lackawanna R. R. Co., 11 
 Lackawanna Steel Co., 60 
 Laitance, 74, 75, 86 
 deposits, 34 
 
 effect on hardness of concrete, 241 
 Lang building, Haverhill, Mass., 535 
 Lauter Piano Co., 479 
 
 Leflfler's comprehensive beam chart, 348, 370 
 Leonard Co., Chicago, 481 
 Liberty Silk warehouses, 438 
 Lime, 1 
 
 hydrated, 2 
 
 hydraulic, 2 
 
 putty, 1 
 Limestone, crushed, 172 
 
 for cement manufacture, 6 
 Line-of-thrust theories, of stability of arches, 659 
 Lines, influence, see Influence lines. 
 Linseed oil on concrete floors, 210 
 Loading tests of flat-slab construction, 480 
 Loads, concentrated, figuring beams for, 324 
 
 floor, 431 
 
 of arches, 654 
 
 Cooper's Standard Loadings, 656 
 
 uniform, maximum moments, 330 
 moments in beams for, 323 
 moving, figuring spans for, 329 
 Loam in aggregates, 224 
 
878 
 
 INDEX 
 
 Lock-woven steel fabric, 49 
 
 Longitudinal reinforcement of columns, 371, 372 
 
 Lord, A. R., 481, 483, 487 
 
 Lorente, M. J., 344 
 
 Los Angeles Building Ordinance, 381 
 
 Low-charging mixers, 191 
 
 Lumber for forms, 94 
 
 Luten arch centering, 707 
 
 truss, 60 
 Lyndon, L., 747 
 
 McCuUough, Earnest, 234 
 
 McDaniel, A. B., 122, 235 
 
 Machine rock drills, 171 
 
 Machine vs. hand-mixing of concrete, 186 
 
 Machinery, bedding, in concrete floors, 437 
 
 McNeilly, Prof., 24 
 
 Magnesian quicklime, 2 
 
 Maney, G. A., 305, 427 
 
 Maney's method of determining deflection of beams, 
 305 
 
 Manufacture and use of concrete stone, 146-169 
 Manufacture of natural cement, 8 
 
 of Portland cement, 6-7 
 Manure, effect on concrete, 259 
 Marion Malleable Iron Works, 111, 119 
 Marl, 6 
 
 Marston, A., 778, 782 
 
 Massachusetts Institute of Technology, 244 
 Materials, 1-62 
 
 aggregates, 12-31 
 
 cement, 1-12 
 
 concrete, handling and storage, 173-179 
 reinforcement, 36-62 
 water, 31-36 
 Maurer, 819 
 
 Maurer's method of determining deflection of beams, 
 305 
 
 Maximum density tests of concrete, 68 
 
 Mayhew, A. B., 260 
 
 Measurement of concrete work, 830 
 
 of materials for concrete, 70 
 Measuring materials for concrete mixer, 192 
 Mechanical analysis of aggregates, 23 
 
 proportioning concrete by, 68 
 Melan system, 704 
 Melick, C. A., 427 
 Membranous waterproofing, 88 
 Metal Building Materials Co., 145 
 Metal lath and plaster partitions, 545 
 Metamorphic rocks, 16 
 Methods of construction, 63-169, 506 
 Meyer steelforms, 138 
 Mica in aggregates, 21, 224 
 Mills, A. P., 215, 222, 248, 255 
 
 Minneapolis, St. Paul & Sault Ste. Marie R. R., ore 
 
 dock, 815 
 Mixers for concrete, types, 187 
 
 for manufacture of concrete products, 152 
 Mixing concrete, 72, 186-193 
 
 discharge of the mixer, 192 
 
 drum speeds, 190 
 
 loading the mixer, 190 
 
 low-charging mixers, 191 
 
 measuring materials, 192 
 
 Mixing concrete, power loaders, 191 
 
 time of operations, 189 
 Mixing concretes and mortars, effect on strength, 230 
 Modulus of elasticity of concrete, 846 
 
 of mortars and concretes, 252 
 
 of steel, 37 
 Modulus of rupture, 242 
 
 Moisture, effect on mortars and concretes, 234, 253 
 
 on voids in sand, 26 
 Molding concrete and mortar, 216, 231 
 Molds for concrete stone, 149 
 
 for wet-cast products, 151 
 
 glue, for concrete products, 160 
 
 of paraffined paper, 81 
 
 wood and plaster, for concrete products, 159 
 Moments in continuous slabs, 306 
 Moments in rigid building frames, 411-430 
 
 analysis, application of method, 414 
 
 conception of rigidity, 415 
 
 columns in flat-slab construction, moments in, 425 
 criteria for combined stresses in columns, 426 
 diagrams, 421-424 
 
 exterior columns, moments at, 420-425 
 interior columns, moments at, 416-420 
 L-frames, 429 
 laws, 411 
 
 method of analysis, 411 
 
 roof frames, 428 
 
 special equations, 414 
 
 wind stresses, 427 
 Moments, of arches, diagrams for, 681 
 
 of columns, 416-425, 486 
 
 positive and negative, 318 
 
 theorem of three, 318 
 
 See also Shear and moment in beams. 
 Monolithic beam-and-girder construction, 439-456 
 Morison, G. S., 659 
 Morrow, D. W., 476, 477, 478 
 Morssen, C. M., 863 
 Mortar, cement, properties of, 215-263 
 
 See also Cement mortar, Portland cement. 
 Morton dam, 743 
 Mosaics, 167 
 Movable dams, 758 
 Mueser, W., 769 
 Multiple-arch dams, 743 
 Mushroom flat-slab system, 465 
 
 Natural cement, 3 
 
 compared with Portland. 4 
 manufacture, 8 
 testing, 8 
 
 Natural cement mortar, strength, 247 
 
 Natural curing in concrete products manufacture, 156 
 
 National Association of Cement Users, 258, 481 
 
 National Concrete Co., 60 
 
 Neat, mortar, and concrete strength, 217 
 
 Negative moments, 318 
 
 in continuous slabs, 306 
 Neutral axis of reinforced-concrete beams, 315, 316 
 Newberry, S. B., 254 
 New England Column Clamp Co., 99 
 New York City, Board of Water Supply, test for 
 sand, 23 
 
 Building Code, 465 
 
INDEX 
 
 879 
 
 New York Public Service Commission, specifications for 
 concrete aggregates, 30 
 tests of concrete, 79 
 
 of concrete aggregates, 27-30 
 
 of strength of concrete, 232 
 Non-hydraulic cements, 1 
 Non-staining cements, 2 
 Norcross patent, 465, 479 
 North Western Expanded Metal Co., 53, 56 
 Norton, Prof. C. L., 255, 263 
 Norway, concrete shipbuilding in, 863 
 Notation, standard, 861 
 
 Ohio tests of slabs under concentrated loading, 604 
 Oils, effect on concrete, 257 
 
 use of, on concrete floors, 210 
 
 used in gaging, 210 
 Ore docks, 815 
 Ottawa Silica Co., 19 
 Oursler, John, 165 
 Overflow dams, 753 
 Ozark Power Co., 748 
 
 Pacific Gas and Electric Co., 739 
 Page, L. W., 240 
 Page Special Process fabric, 49 
 Page Woven Wire Fence Co., 49 
 Paints for concrete floors, 210 
 Pallets, 155 
 
 Parabola of an arch, 653 
 
 Paraffine, used on defective concrete, 87 
 
 Parapet walls, on roofs, 521 
 
 Parker, P. A. M., 747 
 
 Partitions, 545 
 
 Patents for floor construction, 479 
 . Pathfinder, Wyo., dam at, 738 
 Paul, C. H., 260 
 Pavements, vault-light, 212, 545 
 Pennsylvania Railroad Co., 574 
 
 bridge in Pittsburgh, 713 
 Pent houses, 553 
 Percussion drills, 172 
 Perimeters of rods, table, 354 
 Permeability of mortar and concrete, 262 
 Perrine, H., 249 
 Pervious concretes, 86 
 
 Philadelphia and Reading R. R. bridge, 703 
 Pier abutments for bridges, 645 
 
 forms, 103, 111 
 
 trestles, 610 
 Piers and abutments of arch bridges, 702 
 
 concrete, 371 
 
 of arches, 653 
 Pierson Engineering Corporation, 746 
 Pigments for concrete floors, 206 
 Pile trestles, 607 
 Piles, concrete, 572 
 
 cast piles, 574 
 
 composite, 573 
 
 Pedestal, 573 
 
 Raymond, 572 
 
 Simplex, 573 
 Piles in foundations, 557 
 Pilings for dams, 728 
 Pin-connected system, 60 
 Pipe culverts, 777 
 Pipes, pressure in trenches on, 778 
 
 Pittsburgh Building Code, 465, 478 
 
 ruling for flat-slab construction, 492, 847 
 for flat-slab panels, 498-501 
 Placing concrete, 73-78 
 Placing of reinforcement, 139, 142-146 
 
 beam spacers, 143 
 
 easy chairs, 143 
 
 securo locking spacer, 143 
 
 supporting reinforcing bars, 143 
 
 ty-chairs, 145 
 
 tying slab and wall rods, 142 
 
 various devices, 145, 146 
 Plant, cost of, 825 
 Plaster finishes of concrete, 93 
 
 molds, 159 
 
 of Paris, 1, 7 
 
 partitions, 546 
 Plums, 21 
 
 Pneumatic mixers, 188 
 Poisson's ratio, 737, 739 
 Porosity in concretes, 83 
 
 of mortar and concrete, 262 
 Porsgrund Cement Works, 863 
 Portland cement, 2 
 
 adulterants, 4 
 
 chemical action with water, effect of weather, 76 
 
 analysis, 834 
 
 properties, 833 
 color, 93 
 
 compared with natural, 4 
 composition, 4 
 constitution, 5 
 
 determination of soundness, 838 
 
 of time of setting, 838 
 fineness, 836 
 
 hardening rate in concrete, 77 
 in proportioning concretes, 64, 65 
 inspection, 834 
 manufacture, 6, 7 
 
 mixing cement pastes and mortars, 837 
 
 mortars, regaging, 233 
 
 normal consistency, 837 
 
 packages, marking, and storage, 833 
 
 percentage of water, 838 
 
 physical properties, 833 
 
 rejection, 834 
 
 sampling, 834 
 
 setting and hardening, 5 
 
 slag or steel, 3 
 
 specific gravity, 11, 836 
 
 standard specifications and tests, 833-843 
 
 storage of test pieces, 843 
 
 strength, 10 
 
 strength compared to natural cement, 247 
 
 table of colors, 164 
 
 tension tests, 841 
 
 testing, 8 
 
 time of setting, 9 
 
 weight, 12 
 
 Portland Cement Association, report on concrete ships, 
 863 
 
 Positive moments, 318 
 Power loaders, for concrete mixers, 191 
 Pressure method of making concrete stone, 147 
 of concrete against forms, 121 
 
880 
 
 INDEX 
 
 Pressure on pipes in trenches, 778 
 Pre-use tests of materials, 82 
 Prior, J. H., 649 
 Proportioning concrete, 63-72 
 
 arbitrary, 65 
 
 bank-run gravel, 70 
 
 blast-furnace slag and cinders, 71 
 
 cause of weakness, 64 
 
 checking materials on the job, 69 
 
 crusher-run stone, 71 
 
 experiments of H. C. Johnson, table, 65, 66 
 
 for high-strength concretes, 64 
 
 maximum density tests, 68 
 
 measurement of materials, 70 
 
 mechanical analysis, 68 
 
 physical characteristics of aggregates, 67 
 
 properties of concrete and constituent materials, 63 
 
 theory, 63 
 
 unit, 65 
 
 void determinations, 65 
 
 void theory, 64 
 
 water, 71 
 Puddling concrete, 75 
 Pull-out tests, 265, 266 
 Pulver, H. E., 236 
 
 Punching shear, on column footing, 559 
 Purdue University, tests, 230 
 Putty, lime, 1 
 Puzzolan cement, 2 
 
 Quantities, estimating, 830-832 
 Quarrying aggregates, 171 
 Quicklimes, 1, 2 
 
 Raft foundations, 570 
 Railroad bridges, loads, 656 
 
 See also Arches. 
 Rail-steel reinforcement bars, specifications, 40, 41 
 Rankine, Prof., 733 
 
 Rankine's formula for resultant active earth pressure, 576 
 
 Ransome dam, 743 
 
 Ransome unit system, 508, 509 
 
 Raymond piles, 572 
 
 Rebhaun's construction, 580 
 
 Rectangular beams, 273-307 
 
 See also Beams, rectangular. 
 Reed, S. A., 255 
 
 Regaging concrete and mortars, 232 
 Rehbock, 754 
 
 Reinforced concrete, advantage in foundations, 558 
 advantages, 265 
 beam tests, 269 
 beams and slabs, 273-370 
 behavior under tension, 271 
 bond between concrete and steel, 265 
 building frames, moments in, 411-430 
 chimneys, 816-821 
 conduits, 803 
 dams, 739-745 
 
 effect of concrete setting under pressure, 269 
 length of embedment of bars, 270 
 properties, 265-272 
 pull-out tests, 266 
 
 ratio of the moduli of elasticity, 270 
 shrinkage and temperature stresses, 271 
 
 Reinforced concrete slabs, 306 
 walls, designing, 587 
 special types, 600 
 weight, 272 
 
 work, estimating, 823-832 
 Reinforcement, bending and placing, 139-146 
 cost of, 828 
 
 formulas for steel in rectangular beams, 303, 304 
 formulas for steel for T-beams, 313 
 of columns, 371, 372 
 of fiat-slab floors, 461 
 placing, 142-146 
 
 steel for compressive side of beams, 302 
 transverse spacing, 300 
 web, 285 
 Reinforcement materials, 36-62 
 bars, 36-45 
 
 expanded metal, 50-54 
 factors of cost of bars, 42 
 quality of steel, 37 
 rib metal, 55 
 
 self-centering fabrics, 55-57 
 
 steel specifications for bars, 37, 38, 40, 41 
 
 surface, 37 
 
 systems for beams, girders, and columns, 57-62 
 
 types of reinforcement, 36 
 
 wire fabric, 45-50 
 Reinforcing counterforted wall, 596 
 Reinforcing systems, for beams, etc., 57-62 
 
 Corr, 60 
 
 Cummings, 58 
 
 Hennebique, 60 
 
 Kahn, 57 
 
 Luten truss, 60 
 
 pin-connected, 60 
 
 shop fabricated, 60 
 
 unit, 58 
 
 Xpantruss, 60 
 Remixed concrete, 76 
 Removal of forms from concrete, 95 
 Reservoir capacity, 726 
 Reservoirs, 760-765 
 
 concrete floors for, 761 
 
 concrete walls for open, 763 
 
 construction details of columns and roof, 764 
 
 covers or roofs, 764 
 
 groined and flat floors, 762 
 
 groined-arch construction, 764 
 
 open basins with embankment walls, 761 
 
 partition and outside walls, 764 
 
 provision for ice, 764 
 
 quality of concrete for masonry, 760 
 
 types, 760 
 
 Resisting moment of reinforced-concrete beams, 316 
 Restrained beams, 318-341 
 
 See also Beams, continuous, and restrained. 
 Retaining walls, 575-602 
 
 backfilling and drainage, 601 
 
 base slab of cantilever walls, 590 
 
 bearing capacity of soils, table, 583 
 
 coeflBcients of friction, table, 582 
 
 construction of, 601 
 
 design of cantilever or T-walls of reinforced con- 
 crete, 587-592 
 of counterforted walls, 592-600 
 
INDEX 
 
 881 
 
 Retaining walls, design of cantilever or T-walls of plain 
 concrete wall?, 584 
 earth pressure, 575 
 equivalent surcharge, 580 
 expansion joints of cantilever walls, 592 
 factor of limitation, 584 
 
 of safety, 584 
 forms for, 602 
 
 live load on top of fill, 580, 581 
 pressure distribution, 581 
 reinforcing counterforted wall, 596 
 stability, 581 
 
 stem of cantilever walls, 589 
 
 Trautwine's table, 586 
 
 types, 584 
 
 of reinforced-concrete walls, 600 
 
 unit pressures, 583 
 Retempered concrete, 76 
 Retopping concrete floors, 209 
 Rib bars, 44 
 
 metal, 55 
 Ribplex, 57 
 Richardson, C, 248 
 Richart, F. E., 412, 413 
 
 Rigidity of reinforced-concrete building frames, 411 
 Roadways, concrete, 213, 214 
 Robinson, H. St. G., 246 
 Rochester Building Code, 431 
 Rock, igneous, 13 
 
 metamorphic, 16 
 
 sedimentary, 14 
 
 trap, 14 
 Rods, table of areas, etc., 354 
 Roebling Construction Co., 546 
 Roebling slab floor, 512 
 Roman, F. L., 225 
 
 Roof design, in flat-slab construction, 496 
 
 frames, stresses in, 428 
 Roofs, 512-527 
 
 concrete roof surfaces, 516 
 
 condensation on roof slabs, 513 
 
 drainage, 518 
 
 insulation, types of, 514 
 
 loading, 512 
 
 parapet walls, 521 
 
 sawtooth construction, 526 
 
 separate roof coverings, 517 
 
 structural design, 512 
 
 wind pressure, 513 
 Roos Co., H. W., 107 
 Roos self-lock adjustable shore, 107 
 Rotary drills, 172 
 Rubbing concrete surfaces, 91, 166 
 
 specification, 93 
 
 S-M-I flat-slab construction system, 471 
 
 Sabin, L. C, 164, 233, 234, 248 
 
 Salts, effect on concrete, 236 
 
 Sanborn, T., 240 
 
 Sand as an aggregate, 18, 19 
 
 -blasting concrete surfaces, 92 
 
 boxes for lowering arch centers, 710 
 
 cement, 4 
 
 derivation, 220 
 
 effect of size, on strength of mortar, 221 
 56 
 
 Sand, grading for concrete, 23 
 mechanical test, 23 
 molding, 161 
 
 requirements as aggregate, 59 
 screening, 172 
 
 selection of, as aggregate, 20 
 
 standard, 19 
 
 storage and care of, 174 
 
 testing on the job, 69 
 
 voids in, 25-27 
 
 washing, 173 
 Sandstone, 15 
 Sawtooth roofs, 526 
 Schilling, Adolph, 165 
 Schuler's charts for beams and slabs, 350 
 Schwada, J. P., 656 
 Schwendener, K. D., 298 
 Scofield, Prof., 230 
 Scranton, Pa., Dunning's dam, 758 
 Screening crushed stone, 172 
 
 sand and gravel, 172 
 Screenings, 19 
 
 Sea water, effect on concrete and steel, 256, 864 
 
 on mortars and cements, 238 
 Seabury, G. T., 261 
 Seasoning cement, 12 
 Seattle Building Code, 431 
 Securo locking spacer, 143 
 Sedimentary rocks, 14 
 Self-centering fabrics, 55-57 
 
 chanelath, 56 
 
 Corr-mesh, 56 
 
 dovetailed corrugated sheets, 57 
 
 hy-rib, 56 
 
 ribplex, 57 
 
 self-sentering, 56 
 Semi-ellipse arch, 652 
 Setting of cement, 9 
 Sewage, effect on concrete, 257 
 Sewell, Capt. J. S., 255 
 Sewers, see Conduits and sewers. 
 Shaft-hangers, attaching to floors, 434 
 Shales, for cement manufacture, 6 
 Shallow bins, 810-815 
 Shear and moment in beams, 318-341 
 
 beam concentrations, 331 
 
 bending up of bars, curves, 335 
 
 fixed and moving concentrated loads, 324 
 
 influence lines, 324 
 
 maximum moments from uniform loads, 330 
 moving uniform loads, 329 
 
 negative moment at ends of continuous beams, 334 
 
 positive and negative moments, 318 
 
 span length for beams and slabs, 318 
 
 theorem of three moments, 318 
 
 uniform load over all spans, 323 
 
 varying moment of inertia of continuous beams, 339 
 Shearing strength of mortars and concretes, 243 
 
 stresses of reinforced-concrete beams, 280 
 Sheet piling for dams, 728 
 Sheflfield Scientific School, tests, 225 
 Sherwin, R. A., 123, 131 
 Ships, concrete, 863-865 
 
 carrying capacity and displacement, 863 
 
 cost, 864 
 
882 
 
 INDEX 
 
 Ships, concrete, efifect of sea water, 864 
 
 elasticity, 864 
 
 longitudinal strength, 864 
 
 specifications, 865 * 
 
 speed of construction, 865 
 
 transverse strength, 864 
 Shirreffs, 737 
 
 Shop fabricated reinforcement system, 60 
 Shores, Hodges adjustable, 105 
 
 Roos self-lock adjustable, 107 
 Shoshone, Wyo., dam at, 738 
 Shovel for handling materials, 175 
 Shrinkage cracks in concrete, 84 
 prevention, 85 
 
 of reinforced concrete, 271 
 Shunk, Francis R., 121 
 Sidewalk lights, 212, 545 
 Sidewalks, concrete, 210-212 
 Sieves for sand analysis, 23 
 Silos, 805-810 
 Silt in an aggregate, 21 
 Simplex flat-slab construction system, 465 
 
 piles, 573 
 
 Singer Manufacturing Co., concrete storage arrange- 
 ments, 179 
 Single column footings, 559 
 Siphonic spillways of dams, 755 
 Skeleton abutments for bridges, 648 
 Slab and girder bridges, 603-641 
 
 cantilever bridges, 639-641 
 
 continuous-girder bridges, 622-638 
 
 girder bridges, 613-622 
 Slab bridges, 603-612 
 
 abutment design of Iowa Highway Commission, 
 607 
 
 cantilever flat-slab construction, 612 
 
 concrete pile trestles, 607 
 
 Illinois tests, 603 
 
 multiple spans, 607 
 
 Ohio tests, 604 
 
 pier trestles, 610 
 
 single span, 605 
 
 slabs under concentrated loading, 603 
 
 tests by Goldbeck, 605 
 
 trestles with framed bents, 610 
 Slab forms, 104 
 
 reinforcement, bending, 142 
 Slabs, cross-reinforcement, 307 
 
 designing, 344, 351 
 
 diagrams of bending moments, 362, 363 
 
 floor, support of, 307 
 
 in reinforced concrete, 273-370 
 
 moments in continuous, 306 
 
 provision for negative moment, 306 
 
 reinforced-concrete, 306 
 
 tables for spacing of rods in, 358 
 
 See also Beams and slabs, Flat-slab construction. 
 Slag, blast-furnace, 6 
 
 as an aggregate, 17 
 proportioning in concrete, 71 
 
 cement, 2 
 Slater, W. A., 481, 603 
 Sloping column footings, 562 
 Sluices of dams, 754 
 Smith, A., 427 
 
 Smulaki, E., 471, 482 
 
 Smulski flat-slab construction, 471 
 
 Soils, bearing capacity of, 557 
 
 table, 583 
 Soundness of cement, 10 
 Southwick, L. T. B., 225 
 Spacing of reinforcement, 300 
 Spackman, H. S., 238 
 Spading concrete, 75 
 Span length for beams and slabs, 318 
 Spandrel details in arch bridges, 691, 694 
 Spandrels of an arch, 653 
 Spaulding, R. E., 339 
 Specific gravity of aggregates, 25 
 
 cement, 11 
 Specifications for aggregates, 30 
 
 cement, 11 
 
 concrete stone, 168 
 vessels, 865 
 
 Portland cement, 833-843 
 
 production of a rubbed surface, 93 
 
 steel reinforcement bars, 37, 38, 40, 41 
 
 testing strength of cements, 10 
 Spillway of dams, 748 
 Spillways, siphonic, 755 
 Spiral Mushroom System, 479 
 SpofiFord, Prof. C. M., 244 
 Spouting plants, 203 
 Spouts for transporting concrete, 195 
 Spraying concrete stone, 165 
 Sprinkler pipes, attaching to floors, 434 
 Stability of the arch, see Elastic theory of stability of 
 
 the arch. 
 Stairs, 549-552 
 
 design, 549 
 
 details, 551 
 
 supporting, 550 
 Standard sand, 19 
 
 Standpipes and small tanks, 765-771 
 
 construction details, 781 
 
 precautions, 771 
 
 restraint at base, 765 
 
 shear at base, 767 
 
 small tanks, 768 
 
 stresses, analysis, 765 
 Staple chairs, 146 
 Statical moment, 273, 274 
 
 Steam curing in concrete products manufacture, 157 
 
 effect on hardening cement mortar, 235 
 Steel bridges, concrete floors and abutments for, 643- 
 649 
 
 centers of arches, 712 
 floredomes, 137 
 forms, 135 
 
 frame construction, 511 
 Steel reinforcement, cost of, 828 
 effect of sea water on, 865 
 estimating amount of, 832 
 protection from corrosion by concrete, 262 
 quality, 37 
 
 specifications for bars, 37, 38, 40, 41 
 wire gage, 45 
 
 See also Reinforced concrete. 
 Steelcrete, 51 
 
 Stepped column footings, 562 
 
INDEX 
 
 883 
 
 Sterling Wheelbarrow Co., 100 
 Stevenson, 724 
 Stirrups, vertical, 289 
 Stone, as an aggregate, 20 
 
 care of, for concrete, 174 
 
 concrete, 146-169 
 
 crusher-run, proportioning, 71 
 
 crushers, 172 
 Stone & Webster Engineering Corporation, 139 
 Storage of cement, 11, 151 
 
 of concrete materials, 173-179 
 
 pockets in docks, 815 
 Strehan, G. E., 249 
 
 Strength of beam, for moment and shear, 301 
 
 of cement, 10 
 Strength of mortar and concrete, 215-250 
 
 adhesive strength, 246 
 
 aggregates, effect on strength, 218-222 
 
 cinder concrete, 248 
 
 columns, strength of, 229 
 
 compared with neat cement, 217 
 
 compressive and tensile compared, 227 
 
 consistency, relation to strength, 225 
 
 curing by steam, effect of, 236 
 
 curing conditions, effect on strength, 234 
 
 depositing under water, effect of, 232 
 
 elements, 64 
 
 freezing, effect of, 236 
 
 hydrated lime and waterproofing compounds, 
 
 effect of, 238 
 laboratory tests, 215 
 laitance, effect of, 241 
 method of mixing, effect, 230 
 method of placing, effect, 231 
 mica, clay, and loam in aggregates, effect of, 224 
 modulus of rupture, 243 
 moisture, effect of, 234 
 
 natural and Portland cements compared, 247 
 
 oils, effect of, 240 
 
 rate of increase, 241 
 
 regaging, effect, 232 
 
 relation to density, 222 
 
 retrogression, 241 
 
 salts, effect of, 236 
 
 sea water used in gaging, effect of, 238 
 shearing strength, 243 
 transverse strength, 242 
 working stresses, 250 
 Stress, bending and direct, 385-409 
 
 analytical determination of, in rectangular sections, 
 387 
 
 diagrams of compression over whole section, 388- 
 393 
 
 diagrams of tension over part of section, 395-402, 
 404, 405 
 
 graphical determination of stresses, 406 
 hollow circular sections, 407 
 
 rectangular sections, determination of stress in, 
 
 387, 406 
 solid circular sections, 409 
 
 tension over part of section, diagram, 395-402, 404, 
 
 405 
 theory, 385 
 
 See alao Moments in rigid building frames. 
 Stresses in building frames, wind, 427 
 
 Stresses in columns, maximum, 426 
 
 in dams, 732-734 
 
 in reinforcing beams, 273 
 
 in rigid viaduct structures, 625 
 
 of arches, diagrams for, 681 
 
 on chimneys, 816 
 
 working, 845 
 
 See also Moments. 
 Structural-steel columns, 372 
 
 Structural Materials Laboratory, St. Louis, see U. S. 
 
 Bureau of Standards; U. S. Geological Survey. 
 Submerged storage for coal, 815 
 Surfaces of concrete, 162 
 
 estimating amount of finish, 832 
 cost of finish, 829 
 
 finishes, 206 
 Surfacing concrete floors, 93 
 Suspended ceilings, 516 
 Sweatt, B. J., 704 
 Systems, reinforcing, 37 
 
 for beams, girders, and columns, 57-62 
 
 T-abutments, 647 
 T-beams, 307-313 
 
 See also Beams, T-beams. 
 T-walls of reinforced concrete, 587-592 
 Tables, designing, for beams and slabs, 341-370 
 Talbot, Prof. A. N., 245 
 Tamping concrete, 75 
 
 stone, 150 
 Tanks, elevated, 771-775 
 
 analysis of stresses, 771 
 
 supporting tower, 774 
 
 small, see Standpipes and small tanks. 
 Tar and gravel roof, 517 
 Taylor, A. 238, 240 
 Taylor, T. W., 175 
 
 Temperature, effect on concreting, 76 
 
 effect on water-tightness of concrete, 86 
 
 of concrete structures, 664 
 Temperature stresses in concrete chimneys, 819 
 
 of reinforced concrete, 271 
 
 on dams, 732 
 
 on girder bridges, 636 
 Tensile strength of cement, 10 
 
 of mortar and concrete, 227 
 Tension in reinforced concrete, 394 
 
 shear and diagonal, 845 
 Terra-cotta tile partitions, 548 
 
 veneer for walls, 540 
 Testing materials on the job, 69 
 
 of cement, 8 
 Tests, autoclave, 11 
 
 field, of concrete, 78-82 
 
 for Portland cement, 833-843 
 
 of aggregates, 23, 27-30 
 
 of flat-slab floors, 480 
 
 of materials, pre-use, 82 
 
 See also American Society for Testing Materials 
 Theorem of three moments, 318 
 Theory of flexure for homogeneous beams, 275 
 
 of proportioning concrete, 63 
 
 of stability of the arch, 659-668 
 
 of stress, bending and direct, 385 
 Thompson, E. J„ 165 
 
884 
 
 INDEX 
 
 Thompson, S. E., 225, 238, 341, 481 
 Three-hinged arches, 715-721 
 
 details of design, 719 
 
 method of analysis, 715 
 
 methods of construction, 717 
 
 types of hinges, 717 
 Three-way fiat-slab construction, 461, 476 
 Through-girder bridge, 617 
 Thrusts, of arches, diagrams for, 681 
 Tile, hollow, weights, 452 
 Tiles as floor surface, 433 
 Timber centers for arches, 704 
 Time required before removing forms, 96 
 Toltz, Max, 815 
 Tooling concrete stone, 166 
 
 concrete surfaces, 90 
 Tower plants, 203 
 
 tanks, 771-775 
 Transporting concrete, 72, 193-203 
 
 barrows, 193 
 
 buckets, 194 
 
 cableways and buckets, 195 
 carts, 193 
 hoists, 200 
 
 regulating flow in spouting plants, 203 
 spouting plants, 203 
 spouts or chutes, 195 
 Trap rock, 14 
 
 Trautwine's table for retaining-wall design, 586 
 *Trelease, F. J., 481 
 Tremie, depositing concrete by, 76 
 Trestles, concrete pile, 607 
 pier, 610 
 
 with framed bents, 610 
 Triangle-mesh wire fabric, 47 
 Trough mixers, 187 
 
 Trussed Concrete Steel Co., 44, 52, 55, 56, 137 
 Tufa cement, 4 
 
 Tung oil on concrete floors, 210 
 Turneaure, 819 
 
 Turneaure's method of determining deflection of beams, 
 305 
 
 Turner, C. A. P., 465, 479, 481 
 Turner Construction Co., 182, 435 
 Turner Mushroom flat-slab bridge, 612 
 Two-way flat-slab construction, 461, 467, 470 
 Ty-chairs, 145 
 
 Tying slab and wall rods, 142 
 
 U-abutments for steel bridges, 647 
 Underwater concrete, depositing, 76 
 Unit-bilt construction, 508, 527 
 construction, 508-511 
 
 advantages, 508 
 
 method, 508 
 
 Ransome unit system, 508, 509 
 Unit Construction Co., 508, 526 
 
 costs of concrete work, 823 
 
 of proportioning concrete, 65 
 
 reinforcing system, 58 
 
 Wall Construction Co., 119 
 
 wire fabric, 48 
 United Shoe Machinery Co., 435, 555 
 U. S. Bureau of Standards, air-analyzer perfected at, 9 
 
 curve for proportioning concrete, 68 
 
 U. S. Bureau of Standards, steel wiTe gage, 45 
 
 tests, on action of sea water on concrete, 256 
 cinder concrete, 250 
 
 compressive and tensile strengths of concrete, 
 227 
 
 concrete, 217, 220 
 
 electrolysis in concrete, 258 
 
 effect of curing conditions, 234 
 
 effect of waterproofing compound on strength of 
 
 mortars, 238 
 hardening concrete by steam, 235 
 mixing methods, 231 
 permeability of mortars, 262 
 sands and aggregates, 223, 224 
 stress-strain of concrete, 250 
 weight of concrete, 263 
 yield point of mortars, 251 
 U, S. Dept. of Agriculture, tests on expansion of concrete 
 
 253 
 
 with oils in gaging concrete, 240 
 U. S. Geological Survey, tests on strength of mortar 
 
 and concrete, 217, 241 
 U. S. Navy Dept., 256 
 
 U. S. Reclamation Service, 260, 725, 736, 738 
 Universal bar bender, 140 
 cone nuts. 111 
 
 Form Clamp Co., 102, 119, 143 
 
 Portland Cement Co., 783 
 
 sand tester, 82 
 
 wire clamp, 119 
 University of Illinois, tests on bond for concrete and 
 steel, 265, 266 
 
 bond stresses in column footings, 561 
 
 column footings, 558 
 
 loads on pipe in trenches, 781 
 
 pressure of wet concrete, 122 
 
 shearing strength of concrete, 245 
 
 slabs under concentrated loading, 603 
 
 strength of concrete columns, 230 
 
 temperature and strength of concrete, 235 
 
 wind stresses in building frames, 427 
 University of Iowa, 254 
 University of Kansas, 238 
 University of Michigan, tests, 253 
 University of Pennsylvania, 254 
 University of Wisconsin, slab forms used at, 104 
 
 tests on bond of concrete and steel, 265, 266 
 detection of cracks in beams, 271 
 effect of salts on concrete, 236 
 strength of concrete columns, 230 
 Unloading materials, economies, 175 
 Uplift of dams, 731 
 
 Vault-light pavements, 212 
 Vermeule, 725 
 Vertical stirrups, 289 
 Vessels, concrete, 863 
 Viaduct bent, 638 
 
 frames, 628 
 
 structures, 625 
 Vicat method of determining time of setting, 838 
 
 needle apparatus, 9 
 Void theory of proportioning, 64 
 Voids, determination of, 65 
 
 effect of moisture on, 26 
 
INDEX 
 
 885 
 
 Voids, in aggregates, 25-27 
 
 in concrete caused by excess water, 34 
 
 Wade, 738 
 
 Wall and pier forms, 103, 111 
 
 footings, 559 
 
 scuppers, 438 
 Walls and partitions, 532-549 
 
 basement walls, 545 
 
 bearing walls, 532 
 
 brick and other veneer, 539 
 
 curtain walls, 533 
 
 details of Lang building, Haverhill, Mass., 535 
 
 door openings, 543 
 
 partitions, 545 
 
 window-openings, 540 
 Walls, cantilever, design of, 587-592 
 
 counterforted, design of, 592-600 
 
 retaining, 575-602 
 Wallace Supplies Mfg. Co., 140 
 Washburn, W. W., 707 
 Washing crushed stone, 172 
 
 sand and gravel, 173 
 Water, action on concrete, 82 
 
 amount used in mixing concrete. 72 
 
 percentage for standard mortars, 838 
 
 preheating, 77 
 
 proportioning in concrete, 71 
 storage and handling in concrete work, 179 
 Water for concrete, 31-36 
 
 bonding new concrete to old, 35 
 cold weather, concreting in, 35 
 day's work planes, 34 
 effect of excess on concrete floors, 35 
 
 on fluxing of cement, 32 
 
 on lubrication of mixture, 33 
 excess, prevention of, 36 
 functions, 31 
 laitance deposits, 34 
 
 quantity, effect on strength of concrete, 32 
 
 space occupied in concrete, 33 
 
 tests for acidity, 31 
 
 voids caused by excess, 34 
 
 waterproof concrete, 35 
 Waterloo Construction Co., 141 
 Waterproof floors, 438 
 
 Waterproofing compounds, effect on strength of con 
 
 Crete, 238 
 Waterproofing concrete, 82-90 
 cement grouting, 88 
 
 effect of temperature and atmosphere, 86 
 
 integral waterproofing compounds, 87 
 
 membranous waterproofing, 88 
 
 numbers of ply for various heads of water, 89 
 
 pervious concretes and laitance, 86 
 
 porosity in concretes, 83 
 
 protection of waterproofing, 89 
 
 rendering defective structures impervious, 87 
 
 resistance to water action, 82 
 
 rules for making concrete impervious, 89 
 
 shrinkage cracks, 84 
 
 water penetration, 83 
 Watertown Arsenal tests, 229, 248, 249, 259 
 Watson, W. J., «fe Co., 466 
 Watson flat-slab system, 466 
 Weakness of concrete, 64 
 
 Weather, effect on concrete, 76 
 Weathering qualities of concrete, 255 
 Web reinforcement of beams, 285 
 
 bonding of web and flange of T-beam, 308 
 Wedge-shaped beams, 314 
 Wegmann, E., 730, 735, 754 
 Weight of cement, 12 
 
 of mortar and concrete, 263 
 
 of reinforced concrete, 272 
 Weights of rods, table, 354 
 Weiss, C, 412 
 Weld. F. F., 656 
 Welded wire fabric, 46 
 Welland canal concrete tests, 79 
 Wellman, G. A., 225 
 Westinghouse Church Kerr & Co., 82 
 Wet-cast method of making concrete stone, 148 
 
 agitation after mixing, 153 
 
 gang molds for, 151 
 Wheelbarrows for concrete products manufacture, 155 
 Wheeler, Gen. E. S., 246 
 Whipple, Harvey, 146 
 White, Prof. A. H., 253 
 Whited, W., 808 
 Wig, R. J., 87 
 Wight & Co., W. W., 49 
 Willis, W. N., 225 
 Wilson, W. M., 412, 427 
 Wind pressure, formula, 513 
 on dams, 732 
 
 stresses in chimneys, 818 
 in deep bins, 808 
 in building frames, 427 
 Window openings, 540 
 Wing abutments for bridges, 646 
 Wire fabric, 45-50 
 
 lock-woven, 49 
 
 trian^le-mesh, 47 
 
 unit, 48 
 
 welded, 46 
 
 Wisco reinforcing mesh, 50 
 Wire form clamps, 119 
 Wire glass, 541 
 Wisco reinforcing mesh, 50 
 Wiscoforms, 138 
 Witherow Steel Co.. 50, 138 
 Withey, Prof., 475 
 Wolf, A. M., 520 
 Wolfe, W. S., 316, 406 
 Wood molds. 159 
 
 Woodard, 737 / ■ ' ' 
 
 Wooden floors, 432 / 
 Woolson, Prof. I. H., 254 ' ' ' >' 
 
 Worcester, J. R., 558 
 
 Working stresses, axial sompr'ifesiorj, ,i?45 , 
 
 bond, 846 . ; ; 
 
 for concrete columns, 373 ' ' ^ \, >' 
 
 for steel, 37 
 
 modulus of elasticity, 846 
 of concrete, 845 , j > 
 
 reinforcement, 846 , ^ \^ ^' \ ' 
 
 shear and diagonal tension, 845 ' ' ^ \ ^ 
 
 Xpantruss reinforcing system, 60 
 
 Yield point of mortars and concretes, 25] 
 
I 
 
K 
 
 3 3125 00000 4578 
 
 (