' C O N CR ETE - STEEL CONSTRUCTION (DER EI S EN B ETO N B AU) . ' EAdlL.MORSCH FRANKLIN INSTITUTE LIBRARY PHILADELPHIA, PA. Digitized by tlie Internet Arcliive in 2015 https://archive.org/details/concretesteelconOOnnors Concrete-Steel Construction (DER EISENBETONBAU) BY PROFESSOR EMIL MORSCH Of the Zurich Polytechnic, Zurich, Switzerland AUTHORIZED TRANSLATION FROM THE THIRD (1908) GERMAN EDITION, REVISED AND ENLARGED BY E. P. GOODRICH Consulting Engineer NEW YORK THE ENGINEERING NEWS PUBLISHING COMPANY London: ARCHIBALD CONSTABLE AND COMPANY, Ltd. 1909 Copyright, 1909, BY THE ENGINEERING NEWS PUBLISHING COMPANY Entered at Stationers' Hall, London, E.C, 1909 J. F. TAPLEY Co., New York PREFACE TO SECOND EDITION In the absence of a uniform literature, and in view of the number of pro- fusely recommended systems, the first edition of this work, published l)y the firm of Wayss & Freytag in 1902, effected the purpose of familiarizing those interested in the scientific principles of reinforced concrete with all the experi- mental researches available at that time. The firm in question was impelled to publish it because systems based on wholly unscientific methods of calcula- tion, and offering no adequate security, were being pushed into recognition by systematic advertisement, so that the danger was imminent that reinforced con- crete would forfeit a large proportion of the confidence it already enjoyed, espe- cially if a few failures should occur. More than a year after the publication, in conneetion with the first edition, of information in regard to reinforced concrete, were published the "Leitsatze" (Recommendations) of the Verbands Deutscher Architekten und Ingenieur- Vereine, and of the Deutscher Beton Verein, as well as the "Regulations" (Bestimmungen) of the Prussian government, but they harmonized exactly with those of the first edition. The publication of the second edition had another purpose. The "Leitsatze" and the official "Regulations" had inspired wide- spread confidence in the new method of building, but even the best of direc- tions could not altogether obviate mistakes and failures, where the proper knowledge of the cooperative effects of the two materials — steel and concrete — was lacking. In addition to this, all directions presumed a knowledge of approved rules of construction, as the " Leitsatze " could not possibly be amplified into a book of instructions on reinforced concrete. This knowledge was, how- ever, very difficult to obtain from the class journals and other literature, because in these, all sorts of systems were simultaneously described, and conflicting opinions were also expressed. The active part taken by the firm of Wayss & Freytag, as well as the under- signed. Prof. E. Alorsch, in the compilation of the preliminary "Recommenda- tions," and the interest they manifested in making them final, caused them to bring out the present second edition, which represents a complete revision of the first edition, and facihtates the application of the " Leitsatze." The general portion deals with examples chiefly relating to the practical reinforcement of T-beams, columns, and arches, under the most widely varied loads. The succeeding, and most comprehensive part, treats of the theory of reinforced concrete, covers exhaustively the properties of materials, and then iii I8897 iv PREFACE TO SECOND EDITION applies the theory in the closest possible manner to the results of the tests. The author has avoided a repetition of useless theories on reinforced work, of which there is no lack. On the other hand, he has succeeded in showing by- means of tests that the methods of calculation given in the "Leitsatze'^ (which are identical with those published in the first edition) are well founded and useful. At the same time the actual distribution of stress in reinforced sections was thoroughly studied. The firm of Wayss & Freytag placed the whole of their experimental data (in great part hitherto unpublished) at the dis- posal of the author in the preparation of the work. In addition Bach gave the valuable results of the tests conducted for the reinforced concrete commission of the Jubilaumstiftung der Deutschen Industrie, published in the course of the current year, especially those relating to adhesion. The third portion, covering the uses of reinforced concrete, reviews the most important fields of its utilization. All the examples cited represent work done by the firm of Wayss & Freytag, and, for the most part, executed under the direction of the author in his capacity as director of the Technical Bureau of the before-mentioned firm, selected from their fifteen years' experience in reinforced concrete work. This limitation of the choice of examples is war- ranted, inasmuch as all the reinforced construction work completed by the firm in question during the past five years has been calculated in accordance with the methods recommended in the " Leitsatze," and in accord with the rules given in the theoretical and general sections of the book regarding construction work. The field of employment for reinforced concrete is constantly widening; there can therefore be no claim raised that it has been completely covered; only the most important features have been presented. But the operations of this single firm give an excellent idea of the versatility of the employment of rein- forced concrete. The firm is well aware that the material herewith presented is of service to their competitors, but believe that by a general deepening of knowledge of rein- forced concrete, they are rendering the most service to the subject. Wayss & Freytag. Neustadt, a d. Haardt, November, 1905, Professor E. Morsch, Zurich, November, 1905. PREFACE TO THIRD EDITION Owing to the quick sale of the second edition, at the request of the publishers and of the firm of Wayss & Freytag, the undersigned undertook the preparation of a third edition. Of the new experiments conducted by the firm in the interim, special attention must be called to those relating to shear in T-beams and those made upon continuous Ijeams. These experiments, in connection with the recently published results of the tests undertaken for the Reinforced Concrete Commission of the Jubilaumstif- tung der Deutschen Industrie, by the Testing Laboratory at Stuttgart, made possible a detailed treatment of the subject in question. Compared with the preceding editions it is here that the principal additions occur. In addition, the theo- retical chapters relating to flexure and bending with axial stress, were consider- ably extended. In the applications, the chapters on buildings, columns, and silos have likewise been enlarged. In the preface to the second edition, the grounds were given that led to the exclusive use of the w^ork of the firm of Wayss & Freytag. These reasons still apply in regard to the new edition, for most of the examples referred to in the applications were made under the author's direction, and he also furnished the firm with the suggestions for the new tests. The author has also collaborated, as a member of the Commission, in the program of tests conducted by the Testing Laboratory at Stuttgart. In view of the present general development of reinforced concrete the standpoint of this work may possibly be designated as one-sided. It may be answered that the present advance in the art is, in large part, due to the efforts of the firm of Wayss & Freytag, and that, on the other hand, no complete presentation of all of the applications of reinforced concrete are contemplated, because the scope of this work is much too limited. Professor E. Morsch. Zurich, November, 1907, V PUBLISHERS' NOTE Professor Morsch's Eisenbetonbau^' is probably the clearest exposition of European methods of reinforced concrete construction that has yet been published. It has for some years been a recognized standard in Europe and has also had a considerable demand in this country, but the comparatively limited usefulness of the German edition to American engineers prompted us to make arrangements with Professor Morsch for the rights of translation and publication of the book in the English language. In the original German edition there is no division into chapters, but for the sake of clearness and system, and in conformity with American custom, the translation has been divided into parts, (i) The Theory of Reinforced Concrete, and (2) The Applications of Reinforced Concrete, which have been subdivided into Chapters and an Appendix. On account of the impossibility of securing the original drawings and photo- graphs from which to make reproductions for illustration, it was necessary to import electros of the cuts used in the German book. Wherever possible, the wording of these has been translated into English and altered in the cut, but in many cases such alterations were impossible and the German lettering has been left. The measurements used in the German editions were in the metric system only; in the translation, the metric system has been retained, but the English equivalents are given wherever measurements are quoted, as well as in all tables. Furthermore, a table of metric and English equivalents has been included at the end of the book. It is hoped that the efforts of the publishers to make available to English- speaking engineers the contents of this valuable work will merit their approval and appreciation. The Engineering News Publishing Company, Book Department. New York, November, 1909. vi CONTENTS INTRODUCTION PAGE CHAPTER I. Slabs 4 T-Beams 8 Columns ii Arches 14 PART I THEORY OF REINFORCED CONCRETE CHAPTER n. Strength and Elasticity 15 Steel 16 Concrete 18 Strength and Elasticity of Concrete 19 Elasticity Test of Concrete 23 CHAPTER 111. Shear, Adhesion, Torsion 31 CHAPTER IV. Extensibility 50 CHAPTER V. Compression 59 CHAPTER VI. Simple Bending 74 Rectangular Sections — Slabs 76 Rectangular Sections, Double Reinforcement 87 CHAPTER VII. Actual Ultimate Bending Tests of Reinforced Slabs in their Relation to Theory 90 Bending Tests of Concrete Beams with Double Reinforce- ment 93 CHAPTER VIII. Bending with Axial Forces 119 Bending with Axial Tension 127 Graphical Methods of Calculation 130 Method of Computation for Stage 11a 137 CHAPTER IX. Effects of Shearing Forces 139 Formulas for Shearing and Adhesive Stresses 141 vii viii CONTENTS PAGE CHAPTER X. Experiments Concerning the Action of Shearing Forces ... 151 CHAPTER XL Stuttgart Experiments Concerning Shear, Continuous Members, etc 174 Deductions from the Experiments 181 Shearing Stresses in Beams of Variable Depth 190 Deformation 192 Computation of Forces and Moments 194 Experiments with Continuous T-Beams 199 PART n APPLICATIONS OF REINFORCED CONCRETE CHAPTER XIL Historical 204 Buildings 209 Stairs 232 Arches in Buildings 235 Spread Footings 241. Sunken Well Casings 244 Water-tight Cellars 247 Piles 250 CHAPTER XIIL Bridges 256 With Horizontal Members. Slab Culverts 256 Arch Construction 267 Reservoirs - 284 Silos 287 APPENDIX PRELIMINARY RECOMMENDATIONS (LEITSATZE) FOR THE DESIGN, CONSTRUCTION, AND TESTING OF REIN- FORCED CONCRETE STRUCTURES 317 L General. II. Building Preliminaries. III. Checking Plans. IV. Building Construction. a. Supervision of Work and Employees. h. Materials and Their Handling. 1. Reinforcement. 2. Cement. 3. Sand, Gravel, and other Aggregates. 4. Concrete. c. Forms and Supports. Time of Removal. d. Protection of Structural Parts. CONTENTS ix V. Inspection and Test of Work. a. Tests during Erection. b. Tests after Completion. c. Duties of the Contractor. VI. Exceptions. APPENDIX TO THE FOREGOING RECOMMENDATIONS RE- GARDING mp:thods of calculation to be used IN TESTING RJ:INF0RCED CONCRETE STRUCTURES... 321 I. Fundamental Assumptions. a. ILxternal Forces. 1. Loads. 2. Reactions, Moments, and Shears. b. Internal Forces. c. Safe Stresses. 11. Approximate Methods of Computations. a. Simple Bending. 1. Rectangular Sections. Slabs. ^ 2. T-B earns. b. Compression. III. Supports for Safe Loads . IV. Examples of the Method of Computation for a Few Simple Cases. a. Simple Bending. 1. Slabs. 2. T-Beams. 3. Continuous T-Beams. b. Compression, Columns. REGULATIONS OF THE ROYAL PRUSSIAN MINISTRY OF PUBLIC WORKS, FOR THE CONSTRUCTION OF REIN- FORCED CONCRETE BUILDING, MAY 24, 1907 333 I. General. a. Testing. b. Construction. c. Removal of Forms. 11. Recommendations for Statical Computations. a. Dead Load. * b. Determination of External Forces. c. Determination of Internal Forces. d. Permissible Stresses. III. Methods of Calculation, with Examples. a. Simple Bending b. Central Loading. c. Eccentric Loading. d. Examples CONCRETE-STEEL CONSTRUCTION (Der Eisenbetonbau) CHAPTER I INTRODUCTION Reinforced concrete (Eisenbeton) is the name given to all varieties of construction in which are combined cement-concrete and steel, in such manner that the two elements acting together, statically resist all external forces. In this connection it is to be understood that the concrete resists compres- sive stresses principally, while the steel resists tensile ones in large measure — that is, gives the concrete a higher tensile strength. In this type of construction many advantages and valuable properties result from the combination of these two quite dissimilar materials. Buildings erected in this manner combine the massiveness of concrete with the lightness of steel construction, and their wide distribution and daily growth in numbers is due to considerable economic advantages possessed by reinforced concrete over corresponding work in stone, wood or iron. Besides being cheaper in first cost than iron or wood, practically all maintenance charges can be eliminated in reinforced concrete, because of the rational manner in which use is made of the wearing qualities of the two elements. Another excellent property of reinforced-concrete work is its resist- ance to fire. Because of this quality, concrete has been employed for some time in building work, in the shape of partitions and stairways, and for the fire- proofing of steel beams and columns. Now, columns and beams are built of the same materials which were formerly used simply for fireproofing purposes, and in this way is secured a more uniform and cheaper fireproof construction. These several advantages, and the usefulness of reinforced concrete for the •structural parts of beams, columns, and floor slabs, arise from the following fundamental properties of concrete and steel in combination: I. Steel Covered with Concrete is most Perfectly Protected by it -against Corrosion. This is now a recognized fact, but it should be added that only with relatively rich mixtures, and with a plastic condition of the concrete (not ^arth-moist) can there be attained the intimate covering and adhesion neces- sary to give proper protection. If a leaner and drier mixture is employed, it 2 CONCRETE-STEEL CONSTRUCTION is necessary to wash the reinforcement with cement grout just before the deposit of the concrete, to obtain the desired adhesion and security against rust. As a proof of the existence of this property of protecting against rust, there may be cited the numerous reinforced-concrete reservoirs and sewers which have already stood for several decades and as yet show no signs of any corrosion of the reinforcement. Some examinations of twenty-year old sewers showed the steel absolutely uninjured and of the same color as when it left the rolling mill. Additional proofs are constantly being adduced by the repeated loading of structures, and through the demolition of old reservoirs and floors, in none of which has ever been disclosed any corrosion of properly covered reinforcement, even when of considerable age. Bauschinger gives the following report of some observations as to freedom from corrosion in several test specimens which had been broken in Ootober, 1887, and had lain in the open air till 1892: "From several slabs, the concrete covering the reinforcement was knocked away with a hammer. The chips broke only in small pieces where the concrete was struck, showing good adhesion between the steel and the concrete, and the exposed reinforcement was entirely free from rust, even close to fractured edges. "A tank was cracked and otherwise damaged through rough treatment during transportation, so that the reinforcement was partially exposed. Natu- rally, the portion longest exposed showed corrosion, and some rust was revealed when the concrete was removed adjacent to an old crack. However, when the metal was exposed under an unbroken, hard surface, no rust was revealed and the same adhesion was observed as in the slabs. "On July 23, 1892, several fragments of floor slabs 6 to 8 cm. (2.4 to 3.1 in.) thick, were examined. They had lain around the end of a sewer, and the pieces next the entrance were most of the time covered with water which often con- tained sewage. According to a statement of the owner, the pieces had been in place about four years, and had been purchased by him at the sale of the frag- ments of the tests made in 1887. They plainly showed the fractured ends from which the reinforcement stuck about 5 cm. (2 in.). On one piece which lay somewhat lower than the others, the reinforcement was scarcely i cm. (0.4 in.) beneath the upper surface. This upper layer was chiseled away, the concrete proving very hard and adhering firmly to the steel. The latter was absolutely rustless to within a distance of i cm. (0.4 in.) from the fractured edge." (See Beton und Eisen, No. IV, 1904, p. 193.) 2. The Adhesion between Embedded Steel and Cement Concrete is Considerable and about equal to the shearing strength of concrete. This adhesion can be demonstrated by direct experiment, but its presence is clearly shown by the great bending strength of reinforced concrete slabs as compared with those of plain concrete. This bending resistance, with reinforcement aggregating 1% of the cross-section, amounts to 178 kg/cm- (2532 lbs/in^) and increases to 247 kg/cm^ (3513 lbs/in^) with 1.45% of reinforcement; whereas the bending strength of a plain concrete slab of similar section amounts at most to 47 kg /cm- (668 lbs/in^). If adhesion were lacking, slabs with embedded steel would show smaller bending strength than similar slabs without reinforce- ment, because of the diminished net concrete section. For some time adhesive strength was assumed as 40 kg/cm^ (569 lbs/in^) as found by Bauschinger, and until lately its actual value was considered unim- INTRODUCTION 3 portant, since adhesion was never taken into account in making computations. However, this point is of great importance, and the anchorage of reinforcing rods should always be investigated. Other tests will be discussed later. With an adhesive strength of 35 kg/cm- (498 lbs/in^), the length to which a rod must be embedded in concrete so that its tensile strength (3600 kg /cm-, or 51,200 lbs/in^) is exceeded by the adhesion developed, will be, for a round rod of 10 mm. diameter, 26 cm. (s in.) (10.2 in.) 20 mm. < ( 52 cm. (f in ) (20.4 in.) 30 mm. 78 cm. in.) (30.6 in.) and it is seen that the transfer of stress from the concrete to the steel, or vice versa, may be considered as proportional for shorter lengths. Furthermore, as an additional precaution against slipping (which costs very little extra) the ends of all rods should be hooked. 3. The Coefficients of Linear Expansion by Heat of Steel and Concrete are Practically Identical. The coefficients were determined by Bonniceau {Annals des ponts ct chaiissees, 1863, p. 181) for 1° C. as 0.00001235 for steel rods, and 0.00001370 for Portland cement concrete. but it is to be understood that the coefficient for concrete is subject to small varia- tions from differences in the quality of the aggregate. Some experiments of Keller published in No. 24 of the Tonindustriezeitung, 1894, may be cited further. The concrete of the test specimens consisted of part gravel, of particles of 20 mm. (f in.) diameter, and part Rhine sand. The average coefficients of linear expansion for 1° C, between —16° and + 72° C, were as follows: Mixture t:o, coefficient 0.0000126 " 1:2, *' O.OOOOIOI 1:4, ^' 0.0000104 " 1:8, " 0.0000095 The coefficient for steel is usually assumed as 0.000012. Since the coefficients of expansion by heat are so nearly equal, the objection formerly made against reinforced concrete is therefore groundless — that the necessary adhesion which must exist between two such dissimilar materials as compose it, would be endangered by changes of temperature. In any case, the temperature of thoroughly encased steel cannot be far different from that of its concrete cover. Furthermore, being poor conductors, such bodies will 4 CONCRETE-STEEL CONSTRUCTION absorb very little heat, and this absorption will take place only very slowly and at points directly exposed to temperature effects. The concrete cover there- fore protects the reinforcement very effectively against temperature change. According to official fire tests, a failure of adhesion which would be danger- ous to strength does not take place even with large and sudden temperature changes (see "Das System Monier," 1887, by G. A. Wayss). With usual dif- ferences in temperature the variation in expansion is compensated by small internal stresses (Zeitschrift des O ester r. Arch- und Ingenienr-Vereins, 1897, No. 50). The variation in volume of concrete, due to its humidity, has the greatest influence upon the distribution of the stress between the steel and the concrete. Through experiments, especially those of the French Commission,* it has been determined that concrete which sets in air, shrinks; while that which sets under water expands. General, accurate figures for the different kinds of cement, and their different mixtures, cannot be given, although these phenomena are worthy of more attention on the part of designers than they have hitherto received. The several structural parts of reinforced concrete buildings are slabs, T- beams, columns and arches — the characteristics of each of which will first be briefly described. SLABS Slabs are the simplest reinforced-concrete constructions built to resist bend- ing stresses. It is well known that in a slab simply supported at each end and centrally loaded, the upper layers are subjected to compressive stresses, while the lower layers are acted upon by tensile ones. Since the tensile strength of concrete is much smaller than its compressive strength, the failure of such a concrete slab will take place through exceeding the ultimate tensile strength. It is the province of the added reinforcement to overcome this defect, and increase the resultant strength of the structure, by carrying the major part of the tensile stresses. The reinforcement must be designed so as to have its strength in a proper ratio to the compressive strength of the concrete. In slabs assumed as simply supported at the ends, the reinforcing rods should run parallel with the lines of action of the tensile stresses, and should lie as close to the bottom of the slab as is consistent with proper protection. With good mortar, small rods may be properly covered with 0.5 cm. (0.2 in.) of concrete; while slightly heavier material should have at least i cm. (0.4 in.) of covering, and still larger rods should be placed at greater distances above the bottom of the slab. Usually, in addition to these "carrying rods," others at right angles to them, called "distributing rods," are installed. They are primarily employed to keep the carrying rods properly spaced during the construction of the slab, and the two series are therefore wired together at points of intersection. Of course, the number and size of these distributing rods must depend upon * Commission du ciment arme. Experiences, rapports, etc., relatives a Femploi du beton arm^. Paris, H. Dunod et F. Pinat, 1907. INTRODUCTION 5 the conditions of loading and support. They also assist in distributing con- centrated loads over a larger carrying area of the slab. If the slabs are supported on four sides, the heavier carrying rods are laid in the direction of the shorter s})an, and the smaller distributing rods, perpendicular to it. The section of the carrying rods must vary as the span and the load to be carried. Their spacing should be from 5 to 15 cm. (2 to 6 in.), and it is to be noted that light rods, closely si)aced, carry more than larger rods with greater spacing. The criterion for calculating this spacing is the unit adhesive stress on the surface of the rods over the supports. The diameter of the distributing rods is usually 5 to 7 mm. (t% to I in.) and their spacing 10 to 40 cm. (4 to 16 ins.). The distributing rods have another important province in cases where conditions are such that stresses due to temperature change are set up in the slab at right angles to the carrying rods. In such cases the distributing rods take up the stresses and thereby prevent cracking. Sometimes a light system of reinforcement is installed near the upper surface of a slab. This is done where absolute freedom from cracking is necessary, and where large secondary stresses are to be expected, due to shrinkage or temperature change. Above, have been considered only slabs freely supported at their ends. In most constructions, however, a certain amount of restraint is experienced where slabs are supported in outside walls, and many slabs run continuously over girders of rolled beams or of reinforced concrete. Because of this restraint, due to continuity of structure, the moment at the middle of the slab span is reduced, but bending moments of opposite kind are produced over the supports, and because of this condition, reinforcement must be introduced near the tops of the slabs in the vicinity of the supports, so as to take up the tensile stresses Fig. 2. at those points. In this way is derived the type of bent rod, originally used by Monier, which corresponds (in its relation to the neutral axis) with the line of maximum moments. A single type of bent rods is usually not sufficient, since moving loads must be considered. More frequently, both a maximum and a minimum moment line is involved, to which the reinforcement must correspond. Frequently, too, it is necessary to employ continuous top rods, especially when a short span adjoins a long one. (See Fig. i.) Fig. 3 shows in detail the arrangement of reinforcement employed in the continuous slab of Fig. 2, consisting of four spans supported between I-beams. The dotted line in Fig. 4, which is drawn between the two maximum moment lines, represents the moments under conditions of perfect restraint at the ends 6 CONCRETE-STEEL CONSTRUCTION and a uniformly distributed load. Under such conditions the moment is 24 in the center, and 12 at the ends. Continuous reinforced concrete floors between I-beams are usually con- structed with slightly arched ceilings, the arches being formed by constructing haunches down to the lower flanges of the beams. The advantage of these Fig. 3 haunches is that for the moments near the supports (which exceed those at the centers) the concrete has been so increased in depth that no special increase in reinforcement is necessary. An increase in the section of concrete at the sup- ports is needed, if the slab thickness at the center of the span is so thin as just to resist the compression at that point. If this thickness were carried over the intermediate supports, the concrete would be over-stressed at those points. According to the theory of continuous beams with variable section, because ^^^^ Fig. 4. of the arch form of the slab, a slight reduction results in the moments at the centers of the spans, with a corresponding increase of those over the supports. Since ample reinforcement is generally provided at the latter points, the exact and detailed computation of moments may be omitted in most practical cases. In the same manner, floor slabs which run continuously over reinforced concrete girders must be reinforced. (See Fig. 5.) For want of accurate knowl- edge concerning the matter, no account is taken, in either case, of the torsional resistance exerted by the rolled steel or reinforced concrete beams. Thus, a somewhat larger factor of safety is secured. INTRODUCTION 7 In thin slabs up to about lo cm. (3.9 in.) thickness, the bending of the rods should be done with a slope of 1:3. In thicker and shorter slabs the slope can be steeper — 1:2 to 1:1^. It is evident, in this connection, that in all continuous slabs, without regard to an arrangement to fit the distribution of moments, so much reinforcement must be bent that the bent portion is able to carry the whole load of the central portion of the slab over into the ends, which act as canti- levers, even though the slab be cracked entirely through in the vicinity of the bends. This rule is easy to follow, and is the more important the less the amount Fig. 5. of straight reinforcement and the more the concrete is exposed to outside stresses from shrinkage and temperature change. Instead of finishing the ends of the straight and bent rods as hooks, it is evi- dent, under such circumstances, that the ends which lie next the centers of the slabs can remain straight and simply be anchored in the zone of compression of the concrete. The number of "systems" of reinforced concrete floors is large, and new ''.systems" are constantly being devised. In most cases, however, their new- ness does not include any improve- ments. As stated before, many systems are at fault in that no reinforcement is provided near the upper surface over the beams, as computations show necessary, reinforcement being used only near the bottom; while others employ a wrong distribution between the upper and lower systems of rods. One improvement in such floor systems aims at a seiDaration, as far as pos- sible, of the zones of tension and compression, without essentially increasing the total weight of the structure. This is accomplished by employing numerous smafl ribs separated by hollow blocks or grooves filled with light pumice con- crete. The reinforcement is placed in the lower parts of the ribs. (Fig. 6.) Fig. 6. 8 CONCRETE-STEEL CONSTRUCTION T=BEAMS If the hollow blocks above described, or the other light fiUing material, is omitted, the floor construction consists of T-beams of concrete with the steel enclosed by the stems of the T's. If the ribs are arranged further apart, and are built proportionately larger, then what was formerly the compression zone must now be treated, in ac- cordance with established rules, as a restrained reinforced con- crete slab between beams. Fig. 7. In this way is developed a construction in which the slabs and beams combine to form a statically effective T-section. It is also possible to design slabs and independent beams of proper strength and of simple rectangular sections, but it is clear that by making the slabs carry the compressive stresses, a considerable economy is practised. The stressing of the concrete slab in two directions at right angles to each other, is not at all hazardous, and occurs in numerous other types of construction. From a theo- retical standpoint, a slab strengthened with ribs is more economical of material than a slab of uniform thickness. At a certain span, the greater cost of instal- ling the ribs equals the saving in material, so that T-beams can first be built economically with spans of between 3 and 4 meters (10 to 13 ft.). Between the slabs and beams, naturally occur shearing stresses, for the transference of which most builders arrange special vertical reinforcing mem- bers called stirrups (Biigel) consisting of 6 to 10 mm. (J to f in.) round rods, or of thin, flat iron. These enclose the bottom reinforcing rods and thus pre- vent the formation in the concrete of the ribs, of possible longitudinal cracks Fig. 9. Fig. which might be caused by the hooked ends of the main reinforcing rods. The stirrups thus increase the adhesive strength, so that it equals at least that employed in calculations. As proved by tests which will be described later, stirrups with none but straight main reinforcing rods have only a small effect on the increase of the shearing strength of the ribs, so that their practical value consists in more securely connecting the slabs and beams, and producing a better distribution of the adhesion. So as to secure the best transfer of forces from one to the other, the connec- tion l>etween beams and slabs is variously designed, as illustrated in Figs. 8 to 10. By so doing, the advantage is also gained of strengthening the slabs where greatest moments occur. With this design, arched ceilings between reinforced concrete beams are produced. (Fig. 8.) As with flat slabs, a single low layer of reinforcement is not found satisfactory, especially if there is any restraint at the ends, or, if the beams are continuous, INTRODUCTION 9 over several supports. Similarly, at points of negative moment, steel must be introduced near the tops of the T-beams, or by carrying certain rods up and over the supports. Under certain load conditions, continuous top reinforcement may be neces- sary, especially with unequal spans. Furthermore, at the simply supported ends of the slabs of heavily-loaded T-beams, some of the lower reinforcement should be bent upwards (at an angle of about 45°) so as to take up the shearing stresses, or rather, the diagonal tensile stresses in the slabs, for which reinforcement must be provided. Since the moments decrease toward the ends of the slabs, not all of the rods are necessary close to the bottom in the vicinity of the supports, so that a part can advantageously be bent upward. The ribs are usually located underneath the slabs, but there are also cases in which they may be placed above them. One or the other arrangement will be employed, according to circumstances. Since the moments are partly positive and partly negative for restrained and continuous beams, no special advantages are gained with ribs located above the slabs. Fig. II. At the intermediate supports, where the greatest moments are found, the compression occurs along the lower edge of the beam. In order to lessen the unit stress, the beam section is increased at such points by means of a bracket or knee, producing a slightly arched effect. In cases where the ribs are located above the slabs it is possible to do without these knees, since the whole width of the slab between ribs serves as a zone of compression. The knees or brackets at the intermediate supports have the added advan- tage of considerably reducing the unit shearing stresses, partly because of the increased depth of beam, but principally because the compressive stresses along the lower edges of the beam at such i)oints act obliquely upward and thus equili- brate a part of the diagonal forces. (See Fig. 13.) Figs. II and 12 respectively, illustrate advantageous arrangements of rein- forcing rods for a simply supported and a continuous T-beam. Bending the rods upward at the intermediate support, and anchoring them in the adjoin- ing beam brings about an economy in their use in resisting the regular distri- bution of moments; and furthermore, this bent form increases the resistance of the stem of the T-beam against shearing forces. Viewed in this light, it is evidently to be recommended (with reference to better anchorage in the concrete of the ribs), that some at least of the upper steel which terminates near the 10 CONCRETE-STEEL CONSTRUCTION intermediate supports should be bent obliquely downward, as shown in Fig. 14. In that figure is also shown how the knees may be reinforced so as to increase their compressive strength. If the stem of the T-beam does not afford enough room to allow all the reinforcing rods to be placed side by side, they may be arranged in layers, in which case it is possible to place the bent rods on top, as shown in Fig. 11. This should be done only in case of necessity, since the rods are more effective statically when closer together than when they are arranged in two or more Fig. 12. — Reinforcement and moment lines for continuous beams of three spans. layers, because their centroid is then lower. With continuous beams over spans of varying lengths, if a long span is fully loaded it may be necessary to provide continuous top reinforcement in the adjacent shorter spans. The amount of restraint afforded continuous beams by intermediate reinforced concrete sup- ports or partition walls, is comparatively small and may well be neglected. The same is even truer in the case of the end supports, since positive restraint will occur only in the rarest instances, where special means have been adopted to provide it. At simply supported ends of T-beams care should be taken to run some of INTRODUCTION 11 the lower rods straight over the supports. The required number is to be deter- mined by the necessary adhesion. . With large spans the standard lengths of rods will not suffice, so that welding will be necessary. The weld should be located where the rod is not fully loaded, which, in general, is in a bend. If a room of given dimensions is to be floored, it is first divided into panels by main girders, with intermediate supports if neces- sary. These girders are then connected by simple slabs, or beams may be introduced between the girders so as to diminish the slab spans. In that case the slabs are supported Fig. 13. on all four sides, and require a correspondingly light reinforcement, especially in a direction parallel with their greatest dimension. The principal reinforce- ment is placed in the opposite direction, or perpendicular to the beams. When both girders and beams are employed, and the slabs are used as flanges of the girders, these slabs will be thrown into compression and their Fig. 14. — Reinforcement for an intermediate support of a continuous beam. stress must be added to their proper stresses from bending. For that reason it is recommended that only small widths of slabs be used in computing girders, and that the slabs be constructed with haunches, where slabs and girders meet. COLUMNS In columns, several varieties are to be distinguished. Some are reinforced with vertical round rods, others with rolled shapes which are made into rigid frames, and since 1902 the spiral reinforcement invented and patented by Considere has been employed. Further, in the first two varieties, the horizontal connections between the vertical pieces are of special importance in connection with the strength of the column. Instead of temporary wooden forms, rein- forced cylinders or cement blocks can also be used, the latter being especially appHcable to bridge piers. In building work, the concrete columns take the place of cast or wrought iron ones, and must be as small in diameter as possible. Consequently, the use of a permanent shell is out of the question. By the term "reinforced-concrete column" is usually understood one con- taining vertical round rods. Such a column is constructed in the following way: 12 CONCRETE-STEEL CONSTRUCTION so A concrete column of any section contains a certain number of vertical rods which are placed close to the surface. At certain points the rods are fastened together with horizontal wire tires. The whole reinforce- ment thus forms a skeleton, which encloses the concrete and prevents lateral bulging. The result is that even in long columns, ignoring the necessary safety against bend- ing, the strength of plain cubes will be attained. The latter is higher than that of prisms. The ties are placed from 20 to 40 cm. (8 to 16 ins.) apart. For a square column, the reinforcement usually consists of four rods located in the corners, with ties of 7 to 8 mm. (approximately \ to 3% in.) wire. With large dimensions, eight rods are used. (See Figs. 15 and 16.) The lower ends of the ver- tical reinforcing rods rest on a grid of flat bars, so that the load carried by the rods may be distributed over a larger area of concrete. This grid is usually placed in a separate concrete pedestal, which dis- tributes the column load over a larger surface of the foun- dation concrete proper, corres- ponding with the lesser allow- able unit stress of the latter. In columns which extend through several stories of a building, the sections diminish upward, and the rods have to be offset at each change of diameter. Further, rods have to be spliced, which can be pipe over the blunt ends. Flat-iron Fig. 15. — Base and section of a reinforced concrete column. short piece of done simply by slipping a (Fig. 17.) Greater resistance against bending is afforded, however, by lapping the INTRODUCTION 13 vertical rods from 50 to 80 cm. (20 to 30 ins. approximately) and by having their ends hooked. (See Fig. 18.) Naturally, the column section may be rectangular, hexagonal, octagonal, circular, etc., and the number of reinforcing rods can be increased in propor- tion to the load. With eccentric loading, they should all be placed on one side. The interiors of columns can also be made hollow by enclosing pipes in the con- crete. These can serve for rain leaders, or may contain gas or water mains. The diameter changes to correspond with the load to be carried, and with the factor of safety desired. It may run from 20 by 20 cm. (8 hy 8 ins.) to 70 by 70 cm. and more (28 by 28 ins.). The diameter of the rods may vary from 14 to 40 mm. (h in. to ins. approximately). i i Fig. 17. Fig. 18. Splicing of rods in concrete columns. Columns of spirally reinforced concrete designed by Considere have rela- tively light-strength longitudinal rods, while the greater part of the load is carried by a spiral wrapping which encloses the longitudinal rods and the concrete core within them. This spiral affords great resistance against the bulging of the concrete under load. The spirals should be covered by concrete, so that the best shape for such a column is round, octagonal, or hexagonal. The first pub- lication by Considere concerning his "beton frette," or hooped concrete, was in "Genie Civil," in November, 1902. His investigations on concrete cylinders with spiral reinforcement disclosed an efficiency 2.4 times greater for the rein- forcing material than when used simply as straight rods, and the strength of the concrete was increased to 800 kg/cm^ (11,400 lbs/in^), or about quadrupled. Practical applications are already quite numerous and are especially useful in cases where it is necessary that a very heavily loaded column should have a small diameter. 14 CONCRETE-STEEL CONSTRUCTION ARCHES The reinforcement for small arches can be determined in the same manner as for simple slabs. Since no bending moments act on an arch with a parab- olic profile and uniform loading, a system of lightly interwoven reinforcement near the soffit is usually sufficient. Usually, however, such simple reinforcement is not enough, a second layer near the upper surface extending from the abut- ments over the haunches being needed. In bridge arches which are subjected to variations of load, reinforcement is introduced throughout near both the upper and lower arch surfaces. Reinforced concrete arches have the advantage over arches of plain concrete that the reinforced arch can withstand tensile stresses as well as compressive ones. For short spans it is thus possible to secure reinforced arches, which make full use of the compressive strength of the concrete. Under such conditions, arches of much less thickness are secured than when non-reinforced concrete is used, the thickness of which for short spans must be made so great as to pre- vent the appearance of appreciable tensile stresses. In arches of larger span, properly designed to meet the conditions involved, tensile stresses do not occur, and the question of reinforcement lessens in impor- tance since it does not change the unit compressive stresses enough to compensate for its employment. With wide spans, the profile of the arch is of considerable importance, so that the unit compressive strength of the concrete shall not be exceeded; while, with short spans, and with the introduction of reinforcement, the form of the arch can be freely chosen within certain limits. Cases often occur in building work where the form of an arch must be selected for archi- tectural reasons not corresponding at all with the statical conditions, and only a reinforced arch can be employed. Just as in slabs, so in arches — lateral reinforcement is employed, which serves the same purpose as the distributing rods in slabs, and is similarly designated. The upper and lower systems of reinforcement in arches are held in the desired relative positions by means of wire ties. Besides round rods, rolled shapes are sometimes used in arches (Melan sys- tem). Then the arch is composed of a series of parallel ribs which are entirely embedded in concrete. In floor arches and other small structures, the ribs are T-bars, rails or wide-flanged I-beams, and are connected with each other only at the points of support. With larger spans and deeper ribs, the latter are built as lattice girders, and bars are run between them. These serve mainly as sup- ports for the arch forms. PART I CHAPTER II THEORY OF REINFORCED CONCRETE STRENGTH AND ELASTICITY In the early stages of the development of reinforced concrete, its l^uilders had at hand no recognized methods of calculation, and Monier and Franc^ois Coignet erected their work solely by practical instinct and experience. Of late, a real rivalry has developed in the production of new theories concerning reinforced concrete, and thejr authors have been anxious to explain the particular excel- lence inherent in a combination of steel and concrete, with reference to their combined statical action. Practice has here been far ahead of theory. The principal question in controversy has been whether the tensile strength of the concrete in bending should be considered. Among practical builders this ques- tion was really decided at the start, and decided against its inclusion, because absolutely no attention is paid to it and the steel is stressed to the maximum safe limit. The tensile strength of the concrete is entirely ignored. On this assumption was based the first method of theoretical computation of slabs, devised by Koenen (Government architect) in Berlin in 1886, and his method has been used by the majority ever since. Theoretical investigators, unfamiliar with the practical side of concrete con- struction, usually considered the tensile strength of the concrete, and some even went so far in the older methods as to assume the elasticity in tension and com- pression as equal. Later, the modulus of elasticity in tension was accepted as smaller than that for compression, and a parabola was assumed as the stress- strain curve. Finally, the stress curve for concrete in tension was found by Considere's investigations to be a straight line parallel with that of the steel. It is evident that with such assumptions, results are obtainable which appear extremely accurate to the several authors; but long formulas are not attractive to practical builders, and in this connection it is to be observed that the employ- ment of a parabola for the stress-strain curve is actually less accurate than the use of a straight line, because a certain amount of violence must ])e used if the stress-strain curve is forced into parabolic form. But, ignoring this point, such methods of calculation do not provide the desired degree af safety, and may even become actually dangerous if too small a percentage of reinforcement is used. It is not the object of this book to give a review of all proposed methods of calculation. This would be useless, and furthermore, the methods of checking designs contained in the "Vorlaufige Leitsatze fiir Eisenbetonbauten " (Tentative 15 16 CONCRETE-STEEL CONSTRUCTION Recommendations concerning Reinforced Concrete Construction) published in 1904 by the Verband Deutscher Architekten- und Ingenieurverein and the Deutscher Beton-Verein, and in the " Bestimmungen fiir die Ausfiihrung von Konstruktionen aus P^isenbeton bei Hochbauten" (Regulations for the Execution of Constructions in Reinforced Concrete in General Building Work), issued by the Prussian govern- ment, are identical with those contained in the first edition of this book (1902). The new requirements of the French Ministry of Public Works of October 20, 1906, for posts and telegraphs, also contain the same assumptions and methods of computation. Therefore, here will be discussed only the theory above described, which has been proved best by several years of trial and in a large number of constructions. Since the first edition, the results of numerous experiments have been secured which test the accuracy of these methods of calculation and especially explain the importance of shear in T-beams. Methods of calculation will therefore be found in close connection with the results of experiments. In no other subject is it more important to rely as completely on the results of tests, if disagreeable experiences are to be avoided, since the present knowledge concerning reinforced concrete is at best imper- fect and liable to surprises. Before turning to the methods of calculation, which are very simple, a review will be made of the strength and elastic prop- erties of steel and plain concrete, so that the formulas may be more susceptible of daily use. The properties of steel (wrought iron or steel) are well known to-day. In calculations relative to steel construction, the relation between stresses and strains is assumed, and the limiting ratio will never be exceeded in actual load- ing. Furthermore; the tensile strength is the same as the compressive strength, and the elastic behavior is the same under tensile and compressive stresses. As to the modulus of elasticity and the safe working stress, opinions do not differ materially. Usually, wrought iron in the form of rods is employed * for rein- forcement. In Table I some results are given for ordinary material from stock. In it, d represents the diameter of the machined test specimen, not of the rod from which the specimen was prepared. In special locations, such as arch bridges, the steel reinforcement can be STEEL (EISEN) Fig. 19. * In Europe. — Trans. THEORY OF REINFORCED CONCRETE 17 used in the form of rolled shapes or of lattice girders. The American expanded metal (Fig. 19) invented by Golding, made by stamping and bending sheet metal, has been highly recommended for the reinforcement of slabs. Any required strength can be obtained by change of thickness, and size of mesh. However, when iLsing expanded metal one does not have as easy a means of adapting the design to the variation of the moments, as with the use of round rods, so that expanded metal can be used only for simple slal)s. The lighter grades are used for ornamental plaster beams of various kinds. In stamping the meshes from the sheet, the material experiences a heavy stress, and, since the strength and abihty of ingot iron to stretch are damaged by stamping, the sheets must be annealed in order to remove this defect. Table I RESULTS OF TESTS ON ROUND IRON MADE BY THE TESTING LABORA- TORY OF THE ROYAL TECHNICAL HIGH SCHOOL, STUTTGART Stretch in Reduc- Diameter Elastic Limit Tensile Strength Modulus of Elasticity Length tion in of 10 d Area mm. in. kg/cm2 lbs/i.i2 kg/cm2 lbs/in 2 kg/cm2 lbs/in2 % % 10 0-39 2994 42590 4178 59430 2192000 31 180000 10 0-39 3026 43040 4182 59480 2143000 30480000 26.4 66.9 10 0-39 3104 44150 4123 58640 2140000 30440000 27.0 69. 1 10 0-39 3117 44330 4234 60220 2172000 30890000 24.8 66.9 10 0-39 3038 43210 4329 61570 71 .0 15 0-59 2710 38550 3810 54190 21 I 6000 30100000 27. 2 55-3 15 0-59 2725 38760 4146 58970 2150000 30580000 30-0 71.7 15 0-59 2627 37370 3870 55050 2140000 30440000 26.4 55-6 15 0-59 2938 41790 4124 58660 2133000 30340000 28.0 71.6 15 0-59 3277 46610 4610 65570 30-0 53-7 20 0.79 2650 37690 3940 56020 2184000 31060000 30-3 64-4 20 0.79 2166 30810 3790 53910 2165000 30790000 31.2 64.0 20 0.79 2681 38130 3991 56760 2161000 30740000 30.4 64-4 20 0.79 2627 37360 3845. 54690 2177000 30960000 31.2 63-6 In America various forms of reinforcement are employed, all of which are designed to prevent slipping of the rod in the concrete. In the Ransome rod (Fig. 20), this is secured by twisting the square steel bar; in the Johnson bar, Fig. 20. elevations on the surfaces of the rods are produced in the rolling; and the Thacher or knotted bar is provided with swellings, while maintaining a con- stant sectional area. These "knots" may well have the desired effect when 18 CONCRETE-STEEL CONSTRUCTION the rod is anchored in a large mass of concrete, but they will act in an opposite manner in the small stems of T-beams, especially at their bottom.s, where they will have a splitting effect and thus cause premature failure of bond. It will be shown later that the adhesion in the case of ordinary round rods with hooked ends is ample to transfer all actual stresses, and furthermore, the arrangement of the principal reinforcement may be so designed with respect to the shearing stresses that no occasion should arise to make up any deficiency through the use of those costly special bars. CONCRETE For reinforced concrete work only rich mixtures of fine-grained materials should be employed. Practically, only with rich, wet concrete, will the neces- sary adhesion and rust prevention be secured, because only then will the tamp- ing force enough grout against the reinforcement to completely coat it. This coating of grout adheres to the concrete in spite of cracks and even rupture between the concrete and the steel, and forms the real rust preventative, as can be demon- strated. When using drier and poorer concrete, it is important to coat the rein- forcement with cement grout immediately before depositing the concrete. The sand aggregate exerts a great effect in determining the quality of the concrete. With the cement it forms the mortar, ond on the strength of this mortar depends the strength of the concrete. The strength of the latter is usually somewhat greater than when no gravel is used. In the " Mitteilungen iil^er Druckelastizitat und Druckfestigkeit von Betonkorpern mit Verschiedenem Was- serzusatz" (Communication Concerning the Compressive Strength and Elasticity of Concrete Specimens with Different Admixtures of Water), Stuttgart, iqo6, pages II and 14, the following figures are given, which are of interest in this connection: The compressive strength of m.ortar taken from a i : 2J: 5 mixture, amounted, in 28 days 100 days to 294 kg/cm2 (4182 lbs/in2) ^32 kg/cm^ (4722 lbs/in^), while the strength of the corresponding earth-moist concrete of 1:2^:5 mixture was 225 kg/cm^ (3200 lbs/iii2) 321 kg/cm^ (4566 lbs/in2). Similarly, for a 1:4:8 mixture (pages 10 and 13), the results were Mortar 280 kg/cm^ (3982 lbs/in^) 258 Kg/cm- (2670 lbs/in^. Concrete 230 kg/cm^ (3271 lbs/in^) 254 kg/cm^ (2513 lbs/in2). The " Leitsatze " of the Verbandes Deutscher Architekten- und Ingenieurverein recommended that in the composition of concrete for reinforced work, the mor- tar contain sand of graded sizes of particles up to 7 mm. (0.28 in.), and be mixed not poorer than 1:3. Further, that the addition of gravel or stone chips in STRENGTH AND ELASTICITY OF CONCRETE 19 quantities up to that of the sand was permissible. The size of the gravel or stone should be l)et\veen 7 mm. (0.28 in.) and 25 mm. (o.q8 in.). Of cement, only the best Portland should l)e used corresponding at least with the "Normen"* since not enough experience has been obtained concerning other cements, es])ecially as to their action on reinforcement. Under certain conditions, ])umice may advantageously l)e used as the j)rin- cipal aggregate of concrete. On account of its smaller strength, j)umice con- crete can only be used for light slab construction, princi{)ally roofs, where, besides the advantage of its lighter weight, it also has that of insulating against temperature changes. Although pumice-concrete is ])rinci])ally used in arches, between steel beams, it can also be used in the slabs of reinforced floors, pro- vided the beams are made of gravel concrete. Pumice-concrete is usually made with river sand as i)art of the aggregate. STRENGTH AND ELASTICITY OF CONCRETE Compressive Strength. — The resistance which concrete offers to crushing is quite varial)le, and changes with the proportions of the mixture and with the properties of the sand, gravel, and broken stone, as well as with the tamping during making. The form and size of the test specimen also influences the apparent strength. The compressive strength per square centimeter decreases when the section of the specimen is enlarged. The apparent strength is espe- cially dependant upon the ratio of the height of the specimen to its base. When this ratio is small (as in mortar joints) the strength is considerable. But when the height is several times the diameter of the 1)ase, failure will occur along a diagonal plane, because the shearing strength has been exceeded, and the compressive strength, which is not involved, appears small when the break- ing load is divided by the area of section. The compressive strength of concrete cubes is called the "cubic strength" (Wurfelfestigkeit) of concrete, and is usually assumed as the allowable compressive strength in reinforced work, because in such constructions, diagonal shearing is prevented by the use of proper rein- forcement. As to the increase of strength with age, some very interesting tests are avail- able. They were made in connection with the erection of the bridge over the Danube at Munderkingen. With i part cement, 2J parts sand, and 5 parts pebbles, mixed wet, the test cubes 20 cm. (7.8 in.) on each edge developed the stresses shown in Table IL Table II LONG TIME COMPRESSIVE STRENGTH TESTS OF CONCRETE After 7 days an average compressive strength of 202 kg/cm" (2873 lbs/in^). After 28 days an average compressive strength of 254 kg/cm" (3613 lbs/in^). After 5 months an average compressive strength of 332 kg 'cm^ (4722 Ibs/in^j. After 2 years, 8 months an average compressive strength of 520 kg/cm^ (7396 lbs/in^). After 9 years an average compressive strength of 570 kg/cm^ (8107 lbs/in^). Lately, discussion has turned much to the question of earth-moist or plastic concrete. As plastic concrete is here understood it contains 50 per cent more * German standard. — Trans. 20 CONCRETE-STEEL CONSTRUCTION water than necessary, so that it can be placed in thicker layers and be brought to a proper consistency by a less number of blows of the tamper than can moist concrete. To solve the problem as to whether moist or plastic concrete was. the better, a large number of experiments were made at the Testing Labora- tory of the Technical High School of Stuttgart on the compressive strength and elasticity of different proportions. The results of the tests, published by Bach * are of considerable value. Even by these the question is not conclu- sively answered, since with exactly the same materials the above described specimens, which were made in Ehingen and in Biebrich, gave variously divergent results. While the specimens from Ehingen almost invariably gave substantially higher results for the plastic concrete, the specimens prepared in Biebrich showed a superiority for the moist concrete, but within two years the plastic concrete increased as much in strength as did the moist. The use of the moist concrete requires particularly expert workmanship and rigorous inspection, but even then involves the troubles incident to defective work. On the other hand, a considerable security is obtained with regard to the uniformity of the mass when plastic concrete, that is, such as has an excess of water, is used. In reinforced concrete work, plastic concrete is especially valuable, since tamp- ing is often almost impossible through several layers of reinforcement. Prismatic specimens, like Fig. 21, on which elasticity tests were made, gave the following compressive strengths, (each result is the average of three obser- vations; mixture, i cement to 3 gravel and sand; plastic): After 3 months, 172 kg/cm^ (2446 lbs/in^) After 2 years, 308 kg/cm^ (4381 lbs/in^) The strength of reinforced concrete buildings, therefore, increases with time,, so that one-fifth of the cubic strength at an age of twenty-eight days may well be assumed as the safe working stress. According to the " Leitsatze," under ordinary weather conditions, at an age of twenty-eight days the concrete should develop a compressive strength in 30 cm. (12 ins. approximately) cubes, of 180- 200 kg/cm^ (2560-2845 lbs/in^). If this strength is not developed with any particular sand when mixed in mortar proportions of i to 3, then more cement is to be added. Moreover, the i : 3 mortar mixture is to be considered the extreme limit, especially with regard to the securing of ample protection against rust. Tensile Strength. The results of tensile tests are more variable than those of compression. All the conditions which affect the apparent compres- sive strength, affect the tensile strength as well, and the shape and size of the test specimen is of even more importance. In the majority of cases, tensile tests are made on mortar specimens, that is, on bodies composed only of cement and sand, and are prepared only to afford a test of the cement. Few tests on regular concrete specimens have ever been made. The latter give lower results than do specimens made of mortar, as is * " Mitteilungen iibcr die Herstellung von Petonkorpern mit Verschiedenem Wasserzusatz, sowie iiber die Druckfestigkeit und Druckelastizitat derselben," Stuttgart, 1903. Konrad Witt- wer. (Report Concerning the Manufacture of Concrete Specimens with varying Percentages, of Water, together with their Compressive Strength and Elasticity.) The second edition (igo6) contains the experiments on specimens two years old. STRENGTH AND ELASTICITY OF CONCRETE 21 shown by the experiments made in connection with some elasticity tests at the Testing Laboratory in Stuttgart, the specimens for which are illustrated in Fig. 21. The results contained in Table III are averages of three tests of specimens made of Heidelberg cement and Rhine sand and gravel, mixed wet: Table III TENSILE STRENGTH OF CONCRETE Mixture 1:3 1:3 1:4 Age 3 months 2 years 3 months Tensile Strength 12.6 kg/cm^ (179 lbs/in-) 15.5 kg/cm^ (220 lbs/in^) 9.2 kg/cm^ (130 lbs/in^) "X l..._.Jt Even on similar specimens the results are quite variable, as is shown by the fact that the number 15.5 is the average of 8.8, 15.8, and 22.0. Elasticity of Concrete. — Just as it is impossible to assign a definite value to the breaking strength, so it is impossible to do so for the modulus of elasticity of concrete, since all the above mentioned points influence the elasticity as well as the strength. For this reason the results obtained by different observers cannot be compared, and therefore it is necessary to make special tests in practical cases or to select results made under comparable conditions. Experiments concerning the elastic deformation of Portland cement concrete under pressure have been made by Durand- Claye,* by Bauschinger, and by the committee on arches of the Oesterr. Ingenieur- und Architektenverein, etc.; but the most accurate and best known are those made by Bach. All former tests w^re defective in that they employed specimens of too small dimensions. Further, no distinction Fig. 21. was made between elastic and permanent deformations. This point was first brought out by Bach in his experiments for the Wiirtt. Ministerialabteilung fiir Strassen und Wasserbau, in 1895."}* His cylindrical specimens were 25 cm. (y.8 in.) in diameter, and i meter (39.4 in.) long. The shortening in a length of 75 cm. (29.5 ins.) was measured at two diametrically opposite points. The experiments were conducted as follows: A load corresponding to 8 kg/cm- (113.8 lbs/in^) was brought to bear on the specimen and then removed. This operation was repeated several times, until only pure elastic deformation resulted. The load was then increased to 16 kg/cm^ (226.6 lbs/in^), and the same process of loading and unloading repeated until the maximum permanent set for this load had been attained. In this manner the operation was continued, and with each increment the total deformation, elastic deformation, and permanent set were measured. Curves were drawn to represent the values thus obtained. A definite elastic limit was not disclosed by these curves; rather, from the start the shortening seemed to increase with the stress. In specimens made with Blaubeurer cement, a straight line can be substi- tuted for the stress-strain curve up to stresses of about 40 kg/cm^ (569 lbs/in^). * Annales des ponts et chaussees, 1888. f Zeitschrift des Vereins Deutscher Ingenieure, 1895-1897. 22 CONCRETE-STEEL CONSTRUCTION The deformation curves found by Bach are so regular that they may be represented by an exponential equation, the relation between the compression and the stress being such that E=a where E is the deformation in unit length, o the corresponding stress, and a and m are coefficients which depend upon the properties of the material. Sim- ilar relations have been deduced for sandstone, granite, cast iron, etc., for all materials in which no constant proportionality exists between the stresses and the strains, and in which the tensile and compressive elasticities differ considerably. The equations of Table IV * have been deduced for several different mixtures, but they are not correct for all brands of cement: Table IV EXPONENTIAL EQUATIONS OF STRESS-STRAIN CURVE OF CONCRETE I cement: 2i sand:!; gravel, ii= — o•^■^^ ("^^r ' coefficient for inches and lbs.) ^ ^ ' 298000 ^5676100 I cement: 25 sand:^ stone, £= o'^'^^, ( coefficient for inches and lbs.) 457000 ^9190500 ^ I ^ I X I cement: 3 sand, E= ■ o^'^^, (- coefficient for inches and lbs. I 315000 ^0520300 ^ I cement: li sand, E=- — -- — o^'^^, ( — coefficient for inches and lbs.) 356000 ^7567200 Considerable information concerning the compressive elasticity of much tamped concrete of different mixtures and degrees of humidity is to be found in the above mentioned " Mittelungen liber die Herstellung von Betonkorper," etc., of Bach, 1903 and 1906. The dearth of elastic tests on such concrete as is used in reinforced con- struction work, and the comparatively few tests which had been made on the elasticity of concrete in tension for the arch committee of the Osterr. Ingen-. und Arch.-Verein, by Grut and Nielsen, led to the making of some further tests on the elasticity of concrete in compression and tension, at the Testing Laboratory of the Royal Technical High School of Stuttgart. Specimens like those illustrated in Fig. 21, were made of Mannheimer Port- land cement and Rhine sand and gravel. The aggregate consisted of about 3 parts sand of o to 5 mm. (o to 0.2 in.) grains, and 2 parts of gravel of 5 to 20 mm. (0.2 to 0.78 in.) pebbles. The results are shown graphically in Figs. 22 to 25 inclusive, and are also given in Tables V to VII. The numbers are always the averages of three tests. Six specimens were prepared of each of the mixtures, 1:3, 1:4, and 1:7, with 8 per cent and 14 per cent of water, one-half being tested in compression and the other half in tension. The measured length was 350 mm. (13.8 ins.). The repetition of load was omitted, so that the experiments would not take so long and be so tedious, and to produce an equivalent result at each step, the load was maintained for three minutes. The age of the specimens was quite uniform, viz., 80 to 90 days. Consideration is here given only to the i : 3 and i : 4 mixtures, because the results obtained from the i : 7 mixture were of less value compared with the other two, and because such proportions are not used for reinforced concrete. * In metric units. — Trans. STRENGTH AND ELASTICITY OF CONCRETE ELASTICITY TESTS OF CONCRETE Table V 1:3 MIXTURE T K Unit Stress 8% Water 14% Water Vletric .g/cm2 lbs/in 2 Deformation in Millionths E Deformation in Mililonths E Metric English Metric English [61-3 871.9 255 240000 3413000 293 209000 2973000 49.0 697.0 198 247000 3513000 227 216000 3072000 36.8 523-4 143 257000 3655000 165 222000 3158000 30.6 435-2 117 261000 3712000 135 227000 3226000 c _o '5! 24-5 348-5 92 266000 3783000 104 235000 3342000 18.3 260.3 67 273000 3883000 76 241000 3428000 a S 15-3 217.6 55 278000 3954000 62 246000 3499000 0 12.2 173-5 43 284000 4039000 48 254000 3613000 9- 2 130.8 32 287000 4082000 36 260000 3698000 6.1 86.8 21 290000 41 25000 23 265000 3769000 . 3-0 0 42.7 0 10 300000 4267000 II 272000 3869000 r 1.6 22.8 6 267000 3798000 7 230000 3271000 c 3-1 44.0 13 238000 3385000 15 207000 2954000 .9 i • 4-6 65-4 20 230000 3271000 23 200000 2845000 6.2 88.1 28 221000 3143000 32 194000 2759000 7-7 109.4 38 203000 2887000 44 175000 2489000 I 9.2 130.8 47 196000 2788000 T ensile Streng th T ensile Streng th 12.6 (179.2 ) 10.5 (149-3) Table VI 1:4 MIXTURE. Unit Stress 8% Water 14% Water Metric English Deformation E Deformation in Millionths E in Millionths Metric English Metric English 61.3 871.9 290 21 1000 3001000 360 170000 2418000 49-0 697.0 225 218000 3101000 276 177000 2518000 36-7 522.0 163 225000 3 200000 198 185000 263 1 000 30.6 435-2 133 230000 3271000 160 I 91 000 2716000 c _o 24-5 348.5 104 235000 3342000 124 198000 2816000 cn QJ l-i ■ 18.3 260.3 76 241000 3428000 90 203000 2887000 a S 15-3 217.6 62 247000 3513000 73 210000 2987000 0 0 12.2 173-5 49 250000 3556000 58 215000 3058000 9-2 130.8 36 257000 3655000 42 219000 3 II 5000 6.1 86.8 23 265000 3769000 27 226000 3214000 - 3-0 0 42.7 0 II 273000 3883000 12 250COO 3556000 r .6 22.8 6 266000 3783000 6 250000 3556000 0 3-1 44-1 13 240000 3414000 14 221000 3143000 4.6 65-4 21 224000 3186000 22 200000 2845000 6.2 88.2 31 200000 2845000 32 I 94000 2759000 I 7.8 no. 9 41 190000 2702000 T ensile Streng th I ensile Streng th 9-2 (130-8) 8.8 (125 2) 24 CONCRETE-STEEL CONSTRUCTION When deformations are taken as ordinates and stresses as abscissas, the curves of Figs. 22 to 25 are obtained: 14% water 14% water Figs. 22-25. — Stress-strain curves for concrete. STRENGTH AND ELASTICITY OF CONCRETE 25 The deformation curves are quite regular in shape. The tensile strength of large concrete specimens is always considerably less than of octagonal mortar ones, since the latter can be compacted much better than can larger ones. Con- cerning the percentage of water used, it is to be noted that the specimens were molded in water-tight cast-iron forms; that the sand and gravel was not abso- lutely dry, and that the addition of 14 per cent of water (especially with the poorer mixtures) proved superabundant — a condition not reached in practice even with plastic concrete. Measurements of deformations cannot be carried as near to the ultimate strength as is to be desired, because of the danger of damaging the measuring instruments. Just as the ultimate strength of concrete increases with age, so does the modulus of elasticity. This can be seen from the experimental results of Table VII obtained on specimens two years old mixed 1:3, with 14 per cent of water. The results of tests on three-month old specimens are also given for comparison. Table VII ELASTICITY TESTS OF OLD CONCRETE Unit Stress Kg/cm2 lbs/in2 ' 86.0 1223. I 73-7 1048.2 61.3 871.9 1 49.0 697.0 XT. 36.8 523-4 a 30.6 435-2 6 0 24-5 348.5 u 18.3 260.3 12.2 173-5 . 6.1 0 0 ■ 1.6 22.8 3-1 44-1 4-6 65-4 c 6.2 88.2 0 "tr, < C 7-7 109.5 9.2 130.8 10.8 153-6 12.3 174.9 I 13-8 196.3 Three Months Old Deforma- tion in Millionths 293 227 165 135 104 76 48 23 7 15 23 32 44 Metric 209000 216000 222000 227000 235000 241000 254000 265000 230000 207000 200000 194000 175000 English 2973000 3072000 3158000 3229000 3342000 3428000 3613000 3769000 3271000 2944000 2845000 2759000 2489000 Tensile strength. 10.5 (149-3) Two Years Old Deforma- tion in Millionths E Metric English 334 257000 3655000 280 263000 3741000 229 268000 381 2000 180 272000 3869000 132 278000 3954000 109 280000 3983000 87 283000 4025000 64 286000 4068000 42 290000 41 25000 20 305000 4330000 4-7 340000 4836000 9-8 316000 4495000 14.8 31 1000 4423000 20.0 310000 4409000 25.0 308000 4381000 30-3 303000 4310000 35-5 303000 4310000 40.8 301000 4281000 46.2 298000 4239000 Remarks Tensile strength. 15.8 (224.7) The stress-strain curve for the two-year old concrete is shown in Fig. 26. Bending Strength of Concrete. — The tensile strength of rectangular con- crete beams calculated from actual bending tests carried to rupture, by means of Navier's formula, is always about twice the value obtained from direct tension tests. Several results of bending tests on plain concrete beams will first be given. 26 CONCRETE-STEEL CONSTRUCTION and then the theoretical explanation of this seeming contradiction will be dis- cussed. Fig. 26. — Stress-strain curves for concrete 3 months and 2 years old. Experiments by Hanisch and Spitzer. Not only were the bending strengths of the slabs determined, but the tensile and compressive strengths were also ascertained, of specimens carefully cut from the broken slabs. The mixture was 1:34, the clear span 1.50 meters (59 in.), the width of slab 60 cm. (23.6 in.), and the age 268 days. (See Table VIII.) The explanation of the seeming contradiction has to be sought in the phe- nomenon of the variation of the modulus of elasticity and its difference for ten- sion and compression. Consequendy, Navier's formula cannot be used except for comparative purposes, the computed extreme tensile stress given by it being too high. Prof. W. Ritter, of Zurich, has given, in Part I of his " Anwendung der 'Graphischen Statik," 1888, a graphical method of computing stresses which exceed the elastic limit, applicable to all materials of which the deformation dia- gram is curved, as is that of cast iron, the relative behavior of which is very similar to that of concrete. STRENGTH AND ELASTICITY OF CONCRETE 27 Table VIII COMPARISON OF COMPRESSIVE, TENSILE, AND BENDING STRENGTHS No Thickness Concentrated Live Load Dead Load Compressive Strength Tensile Strength Bending Strength, cm. in kg. lbs. kg. bs. Metric. Engl'h. Metric. Engl'h. Metric. Engl'h. I 2 3 4 5 6 7-8 8.0 lO.O lO.O 3- 0 4- 5 4-5 3-1 3-9 3-9 8oo 1400 1500 700 1 200 1 200 1764 3086 3307 1543 2644 2644 170 240 240 175 210 210 375 529 529 385 463 463 296 329 256 314 352 300 4210 4680 3641 4666 5007 4267 29 24 27 23 20 29 412 341 384 327 284 412 54-6 43-2 46. 1 49-1 46. 2 49-1 777 614 656 698 657 698 Average. . . . 308 4381 25 356 48.0 6S3 The stresses might also be computed with the help of the exponential equa- tion of the stress-strain curve, as was explained by Carling in the Zeitschrift des Oesterreich. Ingenieur- und Architektenverein, 1898. Assuming the elastic properties assigned to granite by Bach, Carling computes, with the help of the exponential law, the location of the neutral axis in a rectangular section, the corresponding maximum tensile and compressive stresses, and the relation between depth of beam and moment for assumed tensile stresses. But since the exponential law applies only to low stresses, it cannot be em.ployed in com- putations of conditions near rupture. In the same volume of the above mentioned Zeitschrift, Spitzer gave a method of calculation for beams of materials possessing a variable deforma- tion coefficient, which, while only approximate, is applicable to all beam shapes, and for which a knowledge only of the stress-strain curves for tension and com- pression is required. The simplest explanation of the high-bending strengths of concrete is found in the graphical method first above mentioned, which is as follows: If Navier's hypothesis is assumed, as is here done, in accordance with vv-hich plane sections before bending are supposed to remain so after flexure, then the deforma- tions are represented in Fig. 27 by the Hne DD' and the different stresses by the line EOE'. Since the ordinates are proportional to the deformation, the curve EOE' is none other than the experimentally determined stress-strain curve. Fig. 28 shows this line, which can also be taken to represent the stress- distribution in a beam of rectangular cross-section. The area within the curve above the neutral plane shows the total compressive stress, and the area below it is the tensile stress. Since no external horizontal forces act on the beam, in 28 CONCRETE-STEEL CONSTRUCTION every section the total compression must equal the total tension. That means that the areas OAB and OCD, above and below the neutral axis, must be equal. Abscissas above and below, which intersect the deformation curve so as to pro- duce equal areas, therefore indicate corresponding maximum tensile and com- pressive stresses. Each compressive stress corresponds with a perfectly definite tensile stress. If 5j and 5^ are the centroids of the areas OAB and OCD, then the moment of the internal stresses is equal to Dy=Zy, in which y is the dis- D Tensile Strength Fig. 28. tance between the centroids. This moment must be equal to that of the exter- nal forces. When a certain edge stress above or below is assumed, the moment can be expressed as a function of (as is also the case with the exponential law). If the ultimate tensile strength is assumed as the lower edge stress, the maximum possible moment for non-reinforced conditions will be obtained. If the deformation curve is extended beyond the ultimate tensile stress, as is done in Fig. 28, then are obtained the corresponding edge stresses given in Table IX for the specimens described on page 22 of a 1:3 mixture. STRENGTH AND ELASTICITY OF CONCRETE 29 Table IX CORRESPONDING EDGE STRESSES IN CONCRETE BEAMS Compression 3.5 Kg/cm2 5-3 7-2 9-4 20.8 26. 2 Tension 3.1 Kg/cm2 4.6 6.2 7.7 12.6 Further, there is obtained at the point of rupture, with 0^ = 12.6 and for unit width, D=Z = SAh; y = 0.6411 M = 5.4Xo.64X/^^=3.45^^ * From this moment, the edge stresses are found by Navier's formula to be tT=— ==-^2"~=3.45X6 = 2o.7 kg/cm2, whereas the actual stresses are 12.6 on the tension side and 26.2 on the compres- sion side. Three actual bending tests of the above mentioned mixture gave an average of 21.4 kg/cm^ for the bending stress computed by Navier's formula. This is in accord with the value expected from the deformation curve computation. In other words, it may be used as a partial check, since the bending tensile stress shown by it is entirely different from that secured in true tension tests. Specimens when about three months old were tested to rupture with a center load, and the bending strengths given in Table X were developed according to Navier's formula: Table X* COMPARISON OF BENDING AND TENSILE STRENGTHS Mixture 1:3 1:4 1:7 Per cent of water. . . . 8 14 8 14 8 14 Bending strength. . . 21.4 23-2 16. 1 16.7 13-3 12.8 Tensile strength 12.6 10.5 9.2 8.8 4.4 5.5 The specimens had a length of i meter (39.37 in.), a width of 15 cm. (6 ins.), and a height of 20 cm. (7.87 in.). They were mixed with Mannheimer Port land cement and Rhine sand and gravel. * Metric. — Trans. 30 CONCRETE-STEEL CONSTRUCTION The bending strength of concrete is often used in connection with the com- pressive strength as a test of the quality of the material, since tests of it are easier to make than tensile ones, which latter depend largely on the degree of accuracy with which the load is applied at the exact center of the specimen. So long as the fact is kept in mind that Navier's formula gives results good only for com- parative purposes, and that the actual tensile stresses are only about half those shown by it, that method can conveniently be used. CHAPTER III THEORY OF REINFORCED CONCRETE SHEAR, ADHESION, ETC. Shearing and Punching Strength of Concrete (Schub- und Scherfestig- keit). — The great importance played by shearing forces in reinforced concrete construction, and a study of the results of other tests, led to the making of the following series of experiments, partly by the writer and partly by the Testing Laboratory of the Royal Technical High School at Stuttgart. The experiments disclosed a marked difference between the qualities of shear and punching resistance Schubfestigkeit " and " Scherfestigkeit "). As is known, there exists in every section of a homogeneous beam loaded like those shown in Figs. 29 and 30, normal stresses o and shearing stresses t, Fig. 29. P'lG. J. . which combine to form two inclined mutually perpendicular principal stresses, so-called, viz.: and 2 \ 4 the directions of which are found from 2X tan 20; = a If it is understood that between any adjacent sections no external concen- trated forces act on the beam, the shearing stresses, which are to be computed OS according to the formula t=—^, occur in pairs, and at every point the horizon- tal shear r is equal to the vertical shear. If nothing but shearing stresses act in any two adjacent sections, so that (7 = 0 (as happens in a cylinder subject only to torsion), then any rectangle ABCD (Fig. 31) will be deformed by the pairs * Formed by differential portions of two adjacent sections, AD and BC. — Trans. 31 32 CONCRETE-STEEL CONSTRUCTION of stresses into a rhomboid, in which the diagonal AC has been lengthened, and BD shortened. The principal stresses are then o^^+z and o^^ = — r, and the angle a =45°. These values appear directly from the rectangular form of the figure ABCD. If there is also to be considered the influence of the lateral dilation, it is evident that for this dilation, due to the corresponding stresses on the material in the proper directions, \ m/ or with w = 4, the allowable stress t=o.8o a^, a value frequently employed in steel construction and one found by experiment. t D .1 4 D A Fig. 31, In distinction from the types of loading of Figs. 29 and 30 is that of Fig. 32. In the former, only shearing stresses (Schubspannungen) were supposed to act, that type being distinguished from other cases by the condition that the beam is subject only to flexure and consequently deflects. The other variety is the case of pure shear.* This differs from the foregoing, both spoken of as shear, in that no bending takes place and the external force is here theoretically applied only on a single section; while before, it was constant through several adjoining sections (or with a uniform load, varied only slightly from one to another). It is thus evident that pure shear is scarcely possible in practical work. Fig. 32. Fig. 33. F1G.34. The action of concrete amply justifies a distinction between beam shearing stresses and pure shearing or punching stresses (Schub- und Scherspannungen) , since they give entirely different surfaces of rupture and offer different resistances. To obtain a relation between the compressive, tensile, and punching strengths, one may imagine the resistance to shear to be similar to that offered by a series * Best illustrated by ibe action of a punch. — Trans. SHEAR 33 of small teeth, Fig. 33,* along the infinitesimal faces of which compressive and tensile forces act in oblique but mutually perpendicular directions. The hori- zontal components of these forces must balance among themselves, and the vertical components must equal the total shearing force S. Or, in other words, the shear ct in the vertical section of a tooth (Fig. 34) is the resultant of the two normal forces ba^ and aa^, and must pass through their point of inter- section, which determines the perpendicularity of the faces of the teeth. Because of the condition that a rupture of this series of teeth can occur only when the compressive stresses and the tensile stresses a, simultaneously reach their ultimate values, a definite shape is imposed upon the right triangle abc and a definite relation must exist between the compressive, tensile, and shearing strengths. In the triangle of forces c2 t^ = a^af+b^(7^^. The equation of the horizontal components gives bo^^b^ao^-, or, b^G^ = a^o^, which, in connection with the first equation, gives, c2 f=b^ o^ + a^ a^=a^ a^{a^+b^) from which The theoretical maximum pure shearing strength would therefore be the geometrical mean of the tensile and compressive strengths. In an absolutely homogeneous material with equal tensile and compressive strengths, t would equal a, or with regard to lateral dilation there is obtained In the case of actual tests of wrought iron and steel, the strength in pure shear equals 0.7 to 0.8 of the tensile strength, thus developing equally large shearing and torsional strengths (compare Bach, " Elastizitat und Festigkeit "). With concrete, however, of which the tensile strength is not as large as the com- pressive strength, tests show that the shearing strength is considerably larger than the tensile one, and close to the theoretical value t=-Va^ a^^. Experiments | concerning Pure Shear in Concrete with the Arrangement shown in Fig. 36. — The 18 by 18 cm. (7 by 7 in.) prismatic concrete specimens were fixed on one side in a Marten testing machine, with cast-iron plates above and below, so that the space between the two upper plates corresponded * Figures 32 to 43 are loaned by the "Schweizer Bauzeitung," where they were first pub- iished by the author. t These and the following described experiments were made by the author. 34 CONCRETE-STEEL CONSTRUCTION accurately with the width of the lower plate. When the load was applied on the non-reinforced specimens, a crack a first show^ed itself in the middle, run^ ning from top to bottom. This was doubtless caused by a bending of the speci- men. However, the load on the machine could yet be considerably increased, and only then did the load take full bearing on the edges of the plates, as is neces- sary in order to obtain the real shearing strength. I. Test on three concrete specimens, mixed 1:3, with 14 per cent of water 18 by 18 cm. (7 by 7 in.) in section, age 2 years, Fig. 35. Fig. 35. — Shear test. The bending crack a appeared at a load P = s tonnes (11,000 lbs.), but the load was increased to P = 40 t. (88,000 lbs.) when shearing along crack b took place. In the second specimen, the bending crack appeared atP = iot. (22,000 lbs.) and the shearing took place at P = 38t. (83,600 lbs.), while the third speci- men sheared at P = 5o t. (110,000 lbs.). On the assumption of an equal dis- tribution of P between the two sections to be sheared, the shearing strengths of the three specimens result as shown in Table XI. Table XI SHEARING STRENGTH ^=^^^-^'-^ ^g/^™' (^^79 lbs/in^) lo X lo ^=^^«=58.7 kg/cm^ (835 lbs/in^) ^= '0^^^ =77-2 kg/cm2 (1C98 lbs/in^) ioXlo Average 65.9 kg/cm^ (937 lbs/in^) Tests of three specimens of each kind, and of the same age and mixture, 74 cm. (29.12 in.) high and 18 by 18 cm. (7 by 7 in.) in section, like Fig. 21,. SHEAR 35 broken at the Testing Laboratory of the Technical W\gh School at Stuttgart, gave the following average values: Tensile strens:th 8.8 + 15.8 + 22.0 1,0 „ . . Oz=^ ^ = 15-5 kg/ cm- (220 lbs ni-), Compressive strength = = t,oS kg/cm^ (5405 lbs/in^). In accordance with the theory described above, the limit of shearing strength would be (Td=V 15.5X308 = 69 kg/cm- (981 lbs/in-), while the observed strength was 65.9 kg/cm^ (937 lbs/in-). 2. Test with 18 by 18 cm. (7 by 7 in.) con- crete prisms, months old, and 1:4 mixture with 14 per cent of water. The aggregate con- sisted of 3 parts sand of o to 5 mm. (o to 0.2 in.) grains, and 2 parts of gravel of 5 to 20 mim. (0.2 to 0.78 in.) pebbles, and was also of the same quality as the other specimens. The arrangement is illustrated in Fig. 36. Specimen i: Bending crack in the middle at P = i5 t. (33,000 lbs.); sheared at P = 25 t. ^5 (SSjOoo Hjs.). If a uniform distribution of stress (Dimensions in cm.) is assumed, the unit shearing strength will be f- . « : M i ^ *<.- — — >• '^TiyTs^^^-^ (549 lbs/in^). Specimen 2 gave / = 4i.7 kg/cm- (593 lbs/in^), Specimen 3 / = 31.0 kg/cm^ (441 lbs/in^). Tension and compression tests were not made in connection with these speci- mens. There exist, however, tests on concrete prisms like Fig. 21, 3 months old, of similar composition, of which the average of three strength tests were (72 = 8.8 kg/cm^ (125 lbs/in^), and 0,1 = 1 j 2 kg/ cm- (2446 lbs/in-), so that /=V 8.8X172=38.8 kg/cm2 (439 lbs/in2). The average of the three shearing tests is, 38.6 + 4'.7+3i-^' /=" = 37.1 kg/cm^ (528 lbs/in^). 3. Tests with reinforced concrete prisms. a. With straight rods only. 36 CONCRETE-STEEL CONSTRUCTION The experiments were performed on specimens of the same age, size, and mixture as the foregoing; but each specimen was reinforced with four rods lo mm. (4/10 in.) in diameter, near the upper and the lower surfaces, as illus- trated in Fig. 37. The rods were not connected by ties. They prevented a rupture of the specimen, reduced the size of the cracks, and allowed the load to be considerably increased after one shearing crack had appeared and until, and even after, the other crack had opened. r=i] Fig. 37. Specimen i. At P = i2 t. (26,400 lbs.) a fine, low, horizontal crack showed itself. At. P = iS (33,000 lbs.) a fine bending crack became visible in the center, and shearing took place on the left at P = 2o t (44,000 lbs.), ^ = 31.0 kg/cm^ (441 lbs/in^), on the right at P = 3o / (66,000 lbs.), /=46.3 kg/cm^ (659 lbs/in^), Average, / = 38.6kg/cm2 (550 lbs/in^). In spite of these cracks, the load was increased to P=42 t. (92,400 lbs.) where the sole resistance against shear was the sixteen rod sections which then held /e = -^-°^— = 3350 kg/cm2 (47,650 lbs/in2). :6Xl2x- 4 Specimen 2 showed shearing cracks, on the left at P = i8 t (39,600 lbs.), 1 = 2^.8 kg/cm^ (395 lbs/in^), on the right at F = 2j t (59,400 lbs.), /=4i.8 kg/cm^ (595 lbs/in^), Average, ^ = 34-8 kg/cm^ (495 lbs/in^). The load was increased to P = 4o t. (88,000 lbs.) at which point a horizontal crack appeared at the left end. For this load te = — — =3180 kg/cm2 (45,230 lbs/in2). i6X^-^Xi2 4 From Table I on page 17, the tensile strength of the reinforcement can be taken as 4200 kg/cm^ (59,740 lbs/in^), so that its shearing strength would be about 0.8X4200 = 3360 kg/cm^ (47,790 lbs/in^). The unequal shearing SHEAR 37 resistances on the left and right can be explained in the first arrangement, as due to an unequal distribution of the load P on the two plates. In the latter case, the arithmetical mean gives the correct value of the shearing strength. These tests show that the shearing cracks appeared in the reinforced prisms at practically the same load as in the non-reinforced ones, and conse(iuently that only after the shearing strength of the concrete is exceeded does that of the iron come into play, but then is developed to its full value. With this manner of loading for pure shear, a combination of the strength of the two materials thus seems impossible of attainment. In any case final rupture depends on the resistance of the steel. h. With some bent reinforcement. In the two following tests (Fig. 38), besides two straight reinforcing rods 10 mm. (4/10 in.) in diameter, three bent ones of the same diameter were used, n ^ — ^ QUO Fig. 38. and so designed as to cut the shearing planes at an angle. Otherwise, the size, shape, and mixture were as before. The age was six weeks. Specimen i. Shearing crack, on the right at P = i8 t (39,600 lbs.), t = 2j.S kg/cm^ (395 lbs/in^), on the left at P=-3o t (66,000 lbs.), 1 = 46.4 kg/cm^ (660 lbs/in^), Average, 37.1 kg/cm^ (528 lbs/in^). The load was increased to 35 t. (77,000 lbs.). When the area of a vertical section through the bent reinforcement along the plane of shear is taken into account, the area is increased 1.25 times, and the unit shearing stress is (4 + 6X1.25)- 3870 kg/cm2 (55050 lbs/in2). Specimen 2. Shearing crack, on the left at P = i6 t (35,200 lbs.), ^ = 24.7 kg/cm^ (351 lbs/in^), on the right at ^ = 25 t (55,000 lbs.), / = 38.7 kg/cm^ (551 lbs/in^). Average, 31.7 kg/cm^ (451 lbs/in^). The load was increased to P = 3o t (66,000 lbs.); /e = 33io kg/cm^ (47,080 lbs/in2). 38 CONCRETE-STEEL CONSTRUCTION Specimen 3. A bending crack appeared at P = i2 t (26,400 lbs.) ; shearing occurred at the left at P = i5 t (33,000 ibs.), / = 23.2 kg/cm^ (330 lbs/in^), at the right at ^ = 28 t (61,600 lbs.), / = 43.3 kg/cm^ (616 lbs/in^), Average, 33.3 kg/cm2 (473 lbs/in^). Fig. 39. Solid Cylinders. Fig. 40. Fig. 41. Fig. Hollow Cylinders. The load was increased to P = 32 t (70,400 lbs.); ^e = 3540 l^g/cm2 (50,350 lbs/in2). Consequently, the same observations apply to tests b as to tests a. TORSION 39 Torsion Experiments with Concrete Cylinders. — In a cylinder under- going a twist, without any axial forces at i)lay, no normal stresses exist within any section, only shearing stresses acting, and at each point the latter are equal along directions parallel and perj)endicular to the axis, so that all elements in the body are stressed, as is illustrated in Fig. 31, page 32. It has been shown by the shearing experiments that the resistance offered by concrete to shear is somewhat greater than its tensile strength. Consequently rupture of a cylinder subject to torsion must take place along a screw surface with a i)itch of 45° at right angles to the major dilation or the oblic^ue tensile stresses. (See Figs. 39-42.) These torsion experiments were made at the Testing Laboratory of the Royal Technical High School of Stuttgart. The mixture of the concrete was 1:4, and its age 2 to 3 months. a. Solid cylinder, 26 cm. (10.24 in.) in diameter. The length of the speci- men under test was 34 cm. (13.38 in.). (See Figs. 39 and 40.) The twisting moment was applied on the hexagonal heads. (See Table XIII.) Table XIII TORSIONAL STRENGTH OF SOLID CYLINDERS No. Torque M^i Torsional Strength according to the Formula "<= ■ 16 Age in Days. kg/cm in /lbs kg/ cm^ lb/in2 V \T VII VIII 01 500 66500 46000 59500 53300 57600 39800 51500 19-3 13-3 17.6 259 275 189 250 89 85 79 98 Average. . . | 17. i 243 b. Hollow cylinders of the same external dimensions, with inner diameters about do = is cm. (5.9 in.), gave the torsion moments shown in Table XIV. Table XIV TORSIONAL STRENGTH OF HOLLOW CYLINDERS No. Torque Mj. Torsional Strength, Age in Days. kg/cm in/lb kg/cm^ ibs/in2 XVI XVII XVIII 30000 24500 29000 26000 21 200 25100 9-4 7-9 9-3 134 112 132 54 55 52 Average .... 27830 24100 8-9 126 40 CONCRETE-STEEL CONSTRUCTION The tensile strength of some hollow cylinders of similar section and equal age provided with the corresponding heads, gave an average of ^72 = 8.0 kg/cm^ (113.8 lbs/in^), while the similar above described tensile specimens, like Fig. 21, gave 7.7 kg/cm^ (109.5 lbs/in^). The results found from the hollow cylinders agree quite satisfactorily with each other, while the above described theory for solid cylinders has not been confirmed. Aside from the greater age of the solid cylinders, the greater value of Td is explained on the ground that since the modulus of elasticity diminishes with increase of stress, the sections near the center carry a relatively large part of the load, as is shown by the formula 16 so that the load is reduced on the outer portion. The torsional strength of con- crete, therefore, bears the same relation to its tensile strength as do the bend- ing and tensile strengths. In this manner can be explained the high value of 1 7.1 kg/cm^ (243 lbs/in^), when compared with the tension test specimens of the same material and mixture which gave about 9 kg/cm^ (128 lbs/in^), when 3 months old. And with hollow cylinders, in which the rupture takes place along a screw surface with a 45° pitch and at right angles to the maximum tensile strains, the computed torsional stresses also correspond with the actual ones. It must be mentioned, however, that only through the use of extremely plastic concrete will this agreement be obtained, and only with wet concrete can the tamping be thoroughly effective, as is especially necessary with hollow cylinders. With regard to torsion investigations concerning spirally reinforced concrete hollow cylinders, see page 53, " The extensibility of concrete." Shearing Experiments with Slotted Concrete Beams. — These tests were conducted on specimens with slits molded along the neutral axis, so that with the method of loading shown in Fig. 43, the failure would take place by Fig. 43. a shearing of the connecting bridges at the ends. The tests were made at the Testing Laboratory at Stuttgart. At the ultimate load, the shearing stresses existing in the sections a-a, are calculated as follows: TORSION 41 The unit shear at any point x along the neutral plane is * P S 2 Jb' where S is the statical moment of the cross-section lying above the neutral axis in relation to it, and / is the moment of inertia of the whole section. Thus the total shear from o to — is 2 r-r^)- = - — - 2 2 J 2 ana the shearing strength in a-a is given by _P S I It must be explained that the side subject to tensile stresses had to be rein- forced, so that the weakest points in the body would be the bridges over the supports, and so that the specimen would not fail prematurely through tension. ^ . Al . . 1 \ay- 37$ V - - . ^. .m...-^.-..- 3.75_ 1^ 30 — — 1^ — 30 A A ■ ■ 1 1 1 -t — Fig. 44. — Slotted beam shear test. Further, the bodies were not supported accurately under the centers of the bridges, so that some bending was experienced at those points. This would produce a result equivalent to a partial reduction in the effective width of the bridges, as compared with the original arrangement. Example. Specimen "85," wet mixture, 1:3, age 105 days. Under a load P = i43o kg. (3146 lbs.), the crack hi appeared conspicuously through the whole bridge. At P = i620 kg. (3564 lbs.), a\ showed itself through the whole bridge, and m\ started in the edge. J'^y pl My C^My pbdy, but M = — or p = , so that stress = I bdy = o y I Jo I M ry r — j bdy. M= j Fdx=Fdx when x is very small, so that F means stress in differential Fdx fy length of beam. Total stress= I hdy. Divide by area over which total stress acts = ^ Jo Fdx fy F bdx to get unit stress. Unit stress = I bdy= — S, where 5 is static moment of section Ibdx J lb above neutral axis about that axis. — Trans. 42 CONCRETE-STEEL CONSTRUCTION At P = i77o kg. (3894 lbs.), a2 appeared. At P = 20oo kg. (4400 lbs.), W2 appeared. Under a load of P = 24io kg. (5302 lbs.), a wide crack formed at W3 and W2 widened considerably. The load could not be further increased. In Table XV the observed shearing strengths are given, together with the tensile and compressive strengths of the specimens illustrated in Fig. 21 (page 21). The results are, each, averages of three specimens. Table XV SHEARING, TENSILE AND COMPRESSING UNIT STRENGTHS Mixture. I 3 I 4 1 : 7 Per cent Water Unit Stress. 8% 14% 8% 14% 8% 14 % e \ B a: "c ■<, w B 0 s "s \ B 0 B Shear 512 30 427 31 441 28 398 26 370 19 270 Tension 12.6 179 149 9-2 131 8.8 125 4.4 63 5-5 78 280 398 T95 2770 220 3130 153 2180 127 I8I0 88 125 I The shearing strength for 1:4, here observed, of from 31 to 28 kg/cm^ (441 to 398 lbs/in^), is a little smaller than the one found by direct shear, of 37 kg/cm^ (526 lbs/in^). The reason probably lies in the not entirely rigorous methods of calculation used in connection with the slotted prisms, or else in that the solid end connections had an appreciable thickness, so that partially inclined cracks could occur from diagonal tension. In practice, the case of pure shear is very rare. Diagonal tensile stresses are always combined with shearing ones, and the former become of critical impor- tance long before the shear does, as the torsion experiments plainly show. Th's point will later be discussed more fully, in connection with shearing tests on beams. Adhesion or Sliding Resistance between Steel and Concrete. Experi- ments aiming at the determination of the amount of adhesion between concrete and embedded steel, or the resistance offered to sliding, can be carried out in various ways. The resistance experienced by an embedded rod when drawn out can be directly measured, or the adhesion may be ascertained by computation from bending tests. Consideration will later be given to a discussion of experi- ments of the latter kind concerning adhesion, which naturally are of the greatest importance in this subject. The published figures for directly ascertained adhesion, disclose many dif- ferences, produced by the variableness of concrete, the method of test, the nature of the surface of the rod, etc. For a long period the value of the unit adhesion of from 40 to 47 kg/cm^ (569 to 668 lbs/in^), determined by Bauschinger in his nivestigations for the A.-G. fiir Beton & Monierbau, was accepted. Among later experiments may be mentioned those of Tedesco,* with six-day old mortar * "Du Calcul des ouvrages en ciment avec ossature metallique," by MM. Ed. Coignet and N. De Tedesco, Paris, 1894. ADHESION 43 i IG. 45. prisms which gave an adhesion of 20 to 25 kg/cm^ (284 to 355 lbs/in^), and those of the " Service franjaise des phares et balises," * with 25 to 36 mm. (i to 1 1 in. approximately) round rods which were anchored ; for a length of 60 cm. (23.6 in.) with Portland cement into stone blocks. After setting for a month in the open f air, the rods were pulled out. The adhesion \\as found ; to vary with the diameter of the rod, and was between 20 \ and 48 kg/ cm- (284 and 682 lbs/in^), j The larger values correspond with thicker rods, and i with material possessing a higher elastic limit. With equal sections, the stress was quite constant, and was practically equal to the elastic limit of the steel involved. Thus, the adhesion between concrete and steel was broken when the sections of the rods began to diminish perceptibly. With a slow, regular withdrawal of the rods, almost as large a sliding resis- tance was disclosed, which varied between 39 and 71 kg/cm- (555 and loio lbs/in-) of surface of contact. { The variation in this sliding resistance may be explained by the fact that the surface of commercial rod iron is not a mathematical cylinder. Some experiments were made by the writer in 1904 on the " pressing through " of rods set in concrete, as is illustrated in Fig. 45. The cubes were 20 cm. (7.8 in.) on an edge; the concrete was mixed in the proportions of I to 4, with different percentages of water, and was four weeks old. The specimens did not crack, and it was shown that after the adhesion was overcome, there existed a considerable constant sliding resistance. A second series of tests was made upon exactly similar cubes with 20 mm. (| in. approx.) rods, and special precautions were taken to prevent cracking of the con- crete by embedding in it a 4.5 mm. (3/16 in. approx.) wire spiral with 3 cm. (1.2 in.) pitch and 10 cm, (3.9 in.) diameter. (Fig. 46.) The age of the specimens was four weeks. The results are as given in Table XVL Table XVI ADHESION TO ROUND RODS Per Cent of Water. Adhesion from Average of Four Specimens. Without Spiral. With Spiral. kg 'cm2 lbs /in 2 kg cm 2 lbs in 2 10 48.8 694 50.8 723 12.5 31.2 444 45-9 653 15 29. 1 414 54-0 768 * Annales des ponts et chaussees, 1898, III. 44 CONCRETE-STEEL CONSTRUCTION The percentage of water given is only nominal, since the sand and gravel were moist. The pressure of the testing machine was increased rather rapidly for the larger loads. The results approach closely the shearing strength of similar concrete speci- mens. The compressive stresses in the rods reached a maximum of 2140 kg/cm^ (30,440 lbs/in^), and consequently were below their observed elastic limit of from 2600 to 3200 kg/cm^ (36,980 to 45,520 lbs/in^). Although the non-reinforced concrete cubes were not cracked by the pressing through of the rods, their adhesive strength was smaller than was that of the ones containing spirals. The results of some American tests were pubHshed in Engineering News," 1904, No. 10. The adhesive strength of rods of different shapes was examined for a mortar mix of i to 3 and for various concrete mixtures. With cubes of cement mortar, 15 cm. (6 ins.) on an edge, the average values of Table XVII were obtained: Table XVII AMERICAN ADHESION TESTS Section. Unit Steel Stress Oe Adhesive Strength. kg/cm^ lbs/in2 kg/cm^ lbs/in2 Square, 1570 22330 30.2 430 Round, 1780 25320 35-8 509 Flat, 1270 18060 20.5 292 Square, 2430 34560 25.8 364 It is seen from this that the round rods developed a greater adhesive strength than the square ones, and considerably more than the flat iron. Some concrete prisms 20 by 20 cm. (7.8 in.) in section and 25 cm. (10 in.) high, contained square rods 25 by 25 mm. (i in. approx.), and developed adhe- sive strengths of 34 to 41 kg/cm^ (484 to 583 lbs/in^), or an average of 37.5 kg/cm^ (533 lbs/in^), which agrees well with the values found by Europeans. In a very careful and exhaustive manner. Bach carried out a series of tests on the sliding resistance of steel embedded in concrete,* for the investiga- tions of the Eisenbetonausschusses der Jubilaumsstiftung der Deutschen Indus- trie. His results shed new light on the subject of adhesion. The concrete specimens were made in the form of square prisms 22 cm. (8.7 ins.) on a side and with heights of 10, 15, 20, 25, and 30 cm. (4, 6, 8, 10, 12 in. approx.). The concrete was mixed in the proportions of 1:4, with Rhine sand and gravel, of which the aggregate contained 3 parts sand of o to 5 mm. (o to 0.2 in.) grains and 2 parts gravel of 5 to 15 mm. (0.2 to 0.6 in.) pebbles. Heidelberg Portland cement was used, so that the specimens were exactly like those described on pages 43 and 44. *"Versuche iiber den Gleitwiderstand einbetonierter Eisen," by C. v. Bach, Berlin, 1905, and also No. 22 of the "Mitteilungen iiber Forschungsarbeiten auf dem Gebiete des Ingenieur- wesens." ADHESION 45 The experiments included tests for the determination of the influence of the amount of water used, the quantity of sand, the influence of jarring the specimen before the concrete had set, and finally, time tests of specimens up to three months old. The following conclusions were deduced: That percentage of water was best with which it was just possible to manu- facture the specimens satisfactorily. With the proportions above described, this was 12 per cent. Within certain limits, the relative proportions of sand and gravel have no important influence on the resistance to sliding, so long as the percentage of water is proportionately small when small amounts of sand are used. The resistance to sliding will be increased by jarring the finished specimen before setting is completed, at least when the specimen stands on a wooden bottom, which gets jarred by being struck by other bodies. This increase is more important when small percentages of water are used, and is to be explained by the fact that, through the jarring, the grout which is necessary to a good bond will be enabled to collect around the reinforcement. The sliding resistance is considerably greater in tests conducted at high rates of speed than at slower ones where the loads act for longer periods at each step. Also, tests in which rods are " pushed through " are somewhat higher than when they are pulled through." In regard to the practical employment of these results, it is to be noted pri- marily that it is impossible to obtain, in actual work, the exact percentage of water above mentioned, on account of humidity of the various aggregates, but that it is necessary to rely almost entirely on experience and good practice. On the other hand, an excess of water does not then have the harmful effect that it does on test specimens molded in solid cast-iron forms, since the wooden molds absorb a part of the water and some more is lost through the cracks between the boards. Furthermore, in building construction, the fresh concrete will receive plenty of jarring from the forms, so that the highest value obtained from the experiments, in which the specimens were shaken as well as tamped, may be assumed as a proper working stress. A very important point, and one here brought out for the first time, is that for steel stresses far below the elastic limit, the unit adhesion diminishes with the length of rod embedded. The explanation of this phenomena is as follows: The tensile stress in the rod will decrease from the outside of the concrete to the inner end of the rod as the stress is transferred from its surface to the con- crete. Because of its elasticity, the rod will stretch under the tension, while the concrete will be throw^n into compression and will shorten. Consequently, even under small tensile stress, because of the changes of length in opposite directions in the two materials, a sliding effect will be produced along the rod, near its outer end, so that the tensile stress in the steel will not be uniformly distributed over the whole length of the rod embedded in the concrete. It will first be taken up by the adhesion at the outer end, and only after that is exceeded and a slight displacement takes place, will the distant parts of the concrete be stressed. It follows from this unequal distribution of stress, that the observed values of this stress are too small and that they should more properly be termed the " frictional resistance," as Bach has done. The shorter is the embedded 46 CONCRETE-STEEL CONSTRUCTION length of rod, the smaller are the tension and elongation, and the more nearly equally distributed will be the effect over the whole surface. When the rods are pushed through, there exist practically the same con- ditions, but in less degree, because then the steel and concrete are loaded in like kind. Even then a slight sliding will occur very early along the outer portions of the rod. This slight sliding explains the influence shown by the rate of appli- cation of the load. It is easily seen that with a high rate, the sliding does not have time to develop, and that the adhesive stress is then more uniformly dis- tributed over the embedded area of the rod. In Fig. 47 are given the principal results of Bach's tests. They refer entirely to 1:4 concrete prisms with 15 per cent of water. The earlier tests were con- jI, o fOO f^if ^00 Jl^ 500 >n^ Fig. 4/. — Results of adhesion experiments. ducted with applications of load for short periods — each step occupied one-half a minute (which is really long as compared with most experiments). The embedded lengths of the rods are plotted as abscissas, and the observed resis- tances to sliding, as ordinates. If the curves for adhesion on pushed and pulled rods under short load periods, and also the curve shewing the results for longer duration of load (from nothing up to no minutes) are extended to intersect the axis of ordinates, all three meet at practically the same point, which corresponds with an adhesive strength of 38 kg/ cm- (540 lbs/in^). At this value, which corresponds to a length / = o, the influences of the embedded length of rod, of premature sliding, of time, and the difference between pulling and pushing, all vanish. This value of 38 kg/cm^ (540 lbs/in^), happens to correspond with that found by the author on specimens of the same mixture and age for shearing strength, and also approaches closely that for the quickly oper- ated adhesion experiments. The low point of the middle curve at / = 2oo mm. may be explained by the fact that those specimens were first manufactured, and the operator had not yet acquired proper experience. In addition to the experiments on the slipping resistance of embedded round Tods, made at the Testing Laboratory in Stuttgart, a series was also conducted with Thacher * bars. The specimens were again prepared of the same mixture * Versuche mit einbetonierten Thatchereisen von D.-Ing., C. v. Bach, Berlin, 1907. ADHESION 47 of I part of Portland cement and 4 parts sand and gravel, with 15 per cent water. The height of the specimens was 20 cm. (7.9 in.), while the length of the side of the square base was, in some cases, 22 cm. (8.7 in.), some 16 cm. (6.3 in.), and some 10 cm. (3.9 in.), and the resistance to pulling out was found to vary with the diameter of the specimen, since all split when the Thacher rods were withdrawn. If the pull P is uniformly distributed over the embedded surface (O), the resistance to sliding for the several specimens was as given in Table XVIII. Table XVIII UNIT ADHESION FOR DIFFERENT LENGTHS OF EMBEDMENT Metric. English. Metric. English. Metric. English. Length of side ^*max. 22 cm. 58-5 8.7 in. 832 16 cm. 56-1 6.3 in. 799 10 cm. 33-4 3-9 in- 475 0 It is evident from the last figure that with a minimum thickness of specimen equal to 3.75 cm. (1.5 in.), and with lesser values, the splitting effect of the knots Fig. 48. is so great that greater adhesion cannot be expected than that of common round rods as they come from the mills. With greater thickness of concrete the splitting occurred when the elastic limit of the steel had been reached. Only those adhesion experiments in which the steel stress remains under the elastic limit, give a proper value of the adhesive strength to be used in the design of reinforced concrete structures, and consequendy all steel must be so arranged as to length, shape, and thickness that it will effect a safe transfer of stress to the concrete. Actual tensile stresses are usually small, however, and 48 CONCRETE-STEEL CONSTRUCTION an increase up to the elastic limit through overloading of beams is seldom to be feared. In singly reinforced slabs, the ends of the rods rest in large masses of concrete, so that a diminishing of the adhesion, because of premature cracking of the surrounding concrete, is not to be feared. In slabs, the amount of the embedding is less, but near the ends of beams, stirrups are introduced which surround the concrete to some extent, and so preserve its adhesive strength. In this connec- tion are here given the valuable results of the French Reinforced Concrete Com- mission's * experiments: Certain prisms with centrally located rods were manu- factured, in which only 2 to 2.5 cm. (0.8 so i.o in.) of concrete existed between the rod and the outside surface. Besides these, some were made without stirrups, as illustrated in Fig. 49. A second series had three flat iron stirrups 30 by 2 mm. (lA by ins.) as are used in the Hennebique system, and which enclosed the 30 mm. (ifg in.) diameter rods tightly. In the third series, open stirrups of the same flat iron were employed, which enclosed a larger mass of concrete and were separated from the rods by a space of about i cm. (| in.). The concrete was composed of 300 kg. (661 lbs.) of cement, 400 1 (14 cu.ft.) sand, and 800 1 (28 cu.ft.) gravel, with 8.8 per cent by weight of water, and were six months old at the time of the test. The resistances shown in Table XIX were developed against pulling the rods out of the concrete: Table XIX ADHESION IN THE PRESENCE OF STIRRUPS Specimen. Starting Resistance. Average Sliding Resistance. Stirrups. Figure. No. of kg/cm2 of kg /cm- of Specimens. Surface. lbs/in 2 Surface. lbs/in2 None 49a 2 ■ 7-2 102 8.1 115 19.9 283 14.2 202 Hennebique.. 49& 2 20.0 284 17.2 245 16.9 240 12.8 182 Open 49c 2 25-7 366 18.2 259 29.8 424 21.2 302 A repetition of these experiments with specimens three months old (in which the stirrups of flat iron were replaced with 9 mm. (JJ in.) rods, gave higher results, (see Table XX), each being the average of three specimens: Table XX ADHESION IN THE PRESENCE OF STIRRUPS Adhesion. Sliding Resistance. Specimen. kg/cm^ lbs/in2 kg/cm 2 lbs/in2 Figure 49a 24-7 8.8 125 Figure 496 26. 1 371 17.7 252 Figure 49c 31.2 457 30.0 284 * Commission du ciment arme. Experiences, rapports, etc., relatives a I'emploi du beton arme. Paris, 1907. ADHESION 49 It is seen that the open stirrups which surround the concrete have an advan- tage over those which enclose the rods more tightly, and that the adhesion then developed corresponds well with the Stuttgart results with rods in the centers of the specimens. Of interest are also the adhesion experiments of the French Commission on an old reinforced concrete beam, in which 5 mm. in.) steel wire, because of its somewhat crooked nature, developed a resistance to sliding of 80 to 92 kg/ cm- (1138 to 1308 lbs/ in-). These results are of practical impor- tance, since small rods are never absolutely straight. CHAPTER IV THEORY OF REINFORCED CONCRETE EXTENSIBILITY Extensibility of Reinforced Concrete. — Experiments with straight rein- forced concrete prisms, in which the extension is produced by axial tensile stresses, have several disadvantages. These consist primarily in the great trouble in securing an exactly central application of the tensile stress, and the fact that the force can be transferred to the reinforcement only through large adhesive stresses, so that the ends of the specimens crack prematurely. In the first edition of this book were given several theoretical investigations concerning combina- tions of steel and concrete under certain assumed conditions with regard to the elasticities of the two materials. Of much more value are tests in which the extensibility of reinforced concrete is ascertained through experiments on specimens subjected to flexure. Of such, the best known are Considere's. His first tests * were made on mortar prisms of square section 6 cm. (2.4 ins.) on a side, and 60 cm. (23.6 ins.) high, reinforced on the stretched side by round steel rods. The prisms were tested by fixing one end and applying at the other a bending moment in such manner that it w^as constant for all sections. The extension of the stretched side was then measured with each increase cf load. The mixture was 1:3, and the rein- forcement consisted of three round rods 4.25 mm. in. approx.) in diameter. For comparison, a few prisms had no reinforcement. With one prism, the bending moment w^as increased so that the tension side was stretched 2 mm. per meter (0.002 ft. per foot). Then a moment was applied 139,000 times, which was from 44 to 71 per cent of the first moment, and after each application the return to the initial condition was complete. These repeated applications gave extensions of from 0.545 to 1.25 mm. per m. (0.000545 to 0.00125 foot). Small strips 12 by 15 mm. (0.5 by 0.6 in.) in section were then cut from the prisms, and the bending moment again api)lied. The resulting strength was surprisingly high, almost equal to that of fresh specimens. From the com- parative tests of non-reinforced mortar prisms, it was found that the ultimate flexure was between o.i and 0.2 mm. (0.04 and 0.08 in.). It thus follow:" that in a reinforced concrete body the reinforcement gives the concrete the ability to bend to a considerably greater extent than when plain. Considere explains this as follows: As is known, in a metal rod subjected to tensile stress, the latter is at first distributed uniformly throughout the whole length; but with increase of stress the rod contracts at some point, and will * Genie Civil, 1899. 60 EXTENSIBILITY 51 then undergo considerable stretching. Thus, the total measured length may have increased only 20 per cent, while in the neighborhood of the point of ru))ture the actual stretch has been 10 to 15 times this amount. If it is suj)])c)S'j(l that the ])henomenon known as redaction in area " also applies to cement mortar, then the total elongation measured between the ends will give only an average value, and the mortar will, in reality, possess a very much greater ability to stretch than this value represents. In reinforced construction the concrete is attached to the steel, which latter possesses a much higher elastic limit than does the concrete. When undergoing stress, therefore, the steel will still tend to have the extension distributed uniform.ly over its whole length, at a stress at which the concrete tends to contract locally. But the adhesion makes it neces- sary for the concrete to follow the steel in its extensibility. It will therefore endure throughout its whole length the maximum possible deformation, and rupture will finally take place with an elongation (measured over all) which is considerably larger than if reinforcement were present. This explanation given by Considere is obvious, if the phenomenon of " reduction of area " really exists in concrete. In computations concerning these bending tests, Considere employed a method with reference to the relative distribution of stress between the concrete and steel, which made the concrete show no greater tensile strength than that developed by plain concrete prisms. This method was not entirely free from objections, and therefore Considlre subsequently made some true tension tests with reinforced concrete prisms.* Mortar prisms of square section, 47 mm. (1.85 ins.) on a side, symmetrically reinforced with four wires 4.4 mm. (3^ in. approx.) in diameter, were subjected to tension, and the stretch both in the reinforcement and the mortar was measured. They were always found prac- tically equal. From the known modulus of elasticity of the reinforcement, and the measured stretch of the steel, could be computed the proportion of the total tensile stress P, carried by the rein- forcement. The remainder, divided by the section of concrete, gave the unit tension in the mortar, to which its measured elongations corres- ponded. The observed law between stress and strain is shown in Fig. 50. The ordinates represent the total tensile stress on the prisms, while the abscissas give the corresponding stretch in the reinforcement. As long as the load does not exceed a certain value, Oa, the strains in- crease uniformly and are very small. They then increase suddenly, but soon again become uniform and are repre- sented by the flatter straight portion AB of the curve. From the measured stretch and the known area of the reinforcement, the part of the load carried * Genie Civil, 1899. 52 CONCRETE-STEEL CONSTRUCTION by it can be calculated. The curve for the steel is practically a straight line as long as the elastic limit is not exceeded. In the figure, this straight line is repre- sented by OF, which runs practically parallel with AB. For any stretch OP is then PN equal to the part of the load, PM, carried by the steel, NM " " concrete. It therefore follows from the curve that the concrete, in combination with the steel, is able to stretch considerably, but that after a certain elongation A, the stress on the concrete does not materially increase. The maximum stretch was 0.9 mm. (0.035 which corresponds with a steel stress of 1800 kg/cm^ (25,600 lbs/in^). (This is less than the first value of 2 mm. per meter, found by Considere). The lines CB, C'B', C"B", represent repeated loadings and unloadings. Considere's tests were repeated by the French Government Commission * with somewhat larger prisms of 1:2:4 concrete. Similar results were obtained, and it was further discovered that the extensibility of reinforced concrete which had set under water was greater than that which had set in air. Considere's tests very quickly became known, and were at once used by theorists in the formation of new methods of calculation, without waiting for con- firmatory experiments or even considering the limitations placed by Considere himself on the practical value of his results. In 1904 objections were raised to Considere's theory by both American and German experimenters, based on further tests made by them. The experiments of A. Kleinlogel, conducted in the Testing Laboratory of the Royal Technical High School at Stuttgart were published in Beton und Eisen," No. II, 1904, and also No. I of the " Forscherarbeiten aus dem Gebiete des Eisenbeton," Vienna, 1904. They comprised rectangular reinforced con- crete beams, 220 cm. (86.6 ins.) long and 15 by 30 cm. (5.9 by 11.8 ins.) in section. The mixture was i cement: i sand: 2 crushed limestone. For purposes of comparison some beams were made without reinforcement. The beams were supported at the ends and loaded with two symmetrically placed loads, i meter (39.4 ins.) apart. The stretch of the lowest concrete layer was measured on a length of 80 cm. (31.5 ins.), included within the central portion of a beam. In order to make the cracks more evident, the lower face and both sides of the beams were painted with a coat of whitewash. The six-months' old beams, which had been kept in damp sand, gave practically equal maximum extensions of the lower concrete layer for several different percentages of reinforcement. This amounted to between 0.148 and 0.196 mm. per meter (0.000148 to 0.000196 ft. per foot). Thus Considere's law was not confirmed, because the stretch of non-reinforced concrete was found about 0.143 '^n^- P^^ meter (0.000143 ft. per foot). (Accord- ing to Considere, it was o.i to 0.2 mm. per meter.) Kleinlogel's tests also furnished important information about adhesion, tO' which reference will be made later. * Beton und Eisen, No. V, 1903. EXTENSIBILITY 53 Because of the numerous objections raised concerning his hypothesis, Con- sidere repeated his experiments with larger specimens.* The concrete consisted of 400 kg. (880 lbs.) of Portland cement, 0.4 cubic meters (0.52 cu.yds.) of sand, and 0.8 cubic meters (1.04 cu.yds.) of crushed limestone. The beams were of rectangular section, 3 meters (9.84 ft.) long, 15 cm. (6 ins.) wide, and 20 cm. (7.8 ins.) high, and were reinforced on the lower side with two round rods 16 mm. (f in.), and three round rods 12 mm. (J in. approx.) in diameter. As in the before-mentioned experiments they were tested with symmetrically placed loads, 1.4 meters (55 ins.) apart, within which distance the moment was uni- form and no lateral forces acted. Of two specimens, one was kept under damp sand, and one under water for six months, at which age the specimens were tested. It was shown that the first beam stood a stretch between the layers A and B, of from 0.22 to 0.5 mm. (0.00866 ^ to 0.0196 in.), and the second (which had been kept under water), stood a similar stretch of from 0.56 to 1.07 mm. (0.022 to 0.04 in.). Fig. 51. A crack could not be found even ^ though the outer surface was coated with neat cement. The concrete between the layers A and B was sawed out and still showed the same strength as untouched concrete. Considere does not state (and such is the case with all his tests) whether he was able to cut away the section over the whole length of the beam in one piece, or whether in several. Of the experiments conducted by the Testing Laboratory of the Royal Tech- nical High School at Stuttgart, concerning the extensibility of reinforced con- crete, first will be discussed the Torsion Tests on Hollow Cylinders with Spiral Reinforcement. — Hollow cylinders of the same dimensions as those described on page 39 were provided with spiral reinforcement having a pitch of 45°, in the centers of the walls. They were so arranged that torsion tests would produce tension in the spirals. Cylinder IX, with five spirals of 7 mm. (| in. approx.) round iron, was tested when 60 days old. Under a torque if^ = 72,500 cm-kg (62,800 in-lbs), two cracks, a and b (Fig. 52), at right angles to the spirals, were observed. The torque was increased to 86,500 cm-kg (74,900 in-lbs), at which load the cracks opened considerably. * Beton und Eisen, No. Ill, 1905. 54 CONCRETE-STEEL CONSTRUCTION In Cylinder X, which corresponded exacdy with the other, a fine crack appeared with = 70,000 cm-kg (60,600 in-lbs). The torque was, however, increased to 120,000 cm-kg (104,000 in-lbs) when further parallel cracks appeared. If there is subtracted from the value for Cylinder IX at which the first cracks appeared, the torque = 54,560 cm-kg (47,200 in-lbs) carried by the non- reinforced hollow cylinders of equal age, there remains in Specimen IX the moment 1/"^ = 17,940 cm-kg (15,600 Fig. 53. in-lbs). This gives, in the circle of 21 cm. (8.27 in.) diameter in which the spirals lay, a total horizontal circumferential strength 5 = ^— =1710 kg (3762 lbs.), 10. s ^ half of which must be taken up at the moment of cracking by the reinforcement which lies at an angle of 45° to this theoretical stress, and half by the compres- sive resistance of the concrete acting at right angles to the direction of the rein- forcement. Consequently, from Fig. 53, Z=P=^a/ 2, and the stress in the five spirals is .?55^ kg/cm2 (8960 lbs/in2). 5X0.72^ 4 This stress may also be obtained from the torque 71/^ = 17,940, by a proper distribution of the inclined tensile stresses r over the section of the reinforcement. For cylinder X, the steel stress at the appearance of the first crack was found to be (7e = 540 kg/ cm2 (7680 lbs/in^). With Cylinder XI, with 10 spirals of 10 mm. (| in. approx.) round rods, otherwise like the foregoing, the first crack {a) appeared at = 125,000 cm-kg (108,200 in-lbs), with other cracks running in the same direction, and final rupture at 1/^ = 155,000 cm-kg (134,200 in-lbs). With the same suppositions as before, there is obtained for the steel stress at the appearance of the first crack in Cylinder XI, ^^, = 603 kg/cm2 (8580 lbs/in^); Xil, (7c = 56o " (7070 " ). EXTENSIBILITY 55 It is thus found that with the four hollow cylinders the first cracks in the concrete appeared at an extension which corresponded with an average steel stress of ^^^ 630+540+603 + 560 ^^3^ kg/cm^ (8290 lbs/in2). 4 The extension at this stress is -=0.27 mm. per meter (0.00027 ^t. per foot). 2160 If the shearing stresses at the appearance of the first crack and at rupture are computed from the formula Md 16 d the results of Table XXI are obtained: Table XXI SHEARING STRESSES AT FIRST CRACK AND AT RUPTURE Cylinder No. At the First Crack, Td- At Rupture. kg/cm^ lbs/in2 kg/cm^ lbs/in2 IX X XI XII 25.2 24.4 43-6 41.8 358 347 620 • 595 30.2 42.0 49-5 54-0 430 597 704 768 It may be concluded from this, that through a proper arrangement of the reinforcement, that is, by placing it in the direction of the maximum tensile stresses, the shearing strength of reinforced concrete can be increased over that of plain concrete. In specimens with weak reinforcement, the stress at rupture rose to the ulti- mate stress in the steel; while with heavier reinforcement, such a stress could not be reached because the adhesion on the thicker rods was not sufficiently strong at the ends. Bending Tests with Reinforced Beams of 15 by 30 cm. Section. These specimens had the same dimensions as those tested by Kleinlogel, but were made with i cement to 4 Rhine sand and gravel. They were constructed in December, 1902, and tested three months later at the Testing Laboratory at Stuttgart. They were consequently older than Kleinlogel's specimens. They were tested with two symmetrically placed loads, so that a constant moment (with no external forces acting), was obtained throughout the central portion of 80 cm. (31.5 ins.) between the loads. Besides the stretch of the steel, the shortening of the top concrete layer was also measured, and the deflection within the measured length was also ascertained for different loads. The stretch in the steel was measured between projecting lugs A A, which were clamped to 56 CONCRETE-STEEL CONSTRUCTION the reinforcement. In the ends of the beams, the two reinforcing rods were arranged as shown in Fig. 54, and several stirrups were provided to counteract the local effects of the forces P and the shearing and adhesive stresses. These were such that no cracks appeared between the supports and the loads P. The six specimens were severally reinforced with two 10 mm. (| in. approx.), with two 16 mm. (| in.), and with two 22 mm. (| in.) rods. Of these beams, three were used for the determination of the steel stretch, and three for the shortening of the top concrete layer, because the apparatus w^as so designed that both observations could not be made simultaneously. The tension face of each beam received a coat of whitewash to make the cracks easier of discovery. The first cracks z v/ere always noted next the lugs A, probably because at those points the zone of tension in the concrete was weak- ened. Afterward, the cracks m, n^, and no, appeared within the central portion. All, indeed, were so minute that they probably would not have been seen except for the coat of whitewash. From the stretch in the plane of the reinforcement, and the shortening of the top layer, the extensibility of the lowest layer could be computed. The tests gave the values shown in Table XXII, at which the cracks appeared within the measured length: Table XXII EXTENSIBILITY EXPERIMENTS Reinforcement. Stretch of the Steel. Stretch of Lowest Con- crete Layer. Number of Round Rods Diameter. Per Cent. . mm. in. mm/m ft/foot mm/m ft/foot 2 10 1 0.4 0.42 0.00042 0.50 0.00050 2 16 5 8 1 .0 0-33 0.00033 0.40 0.00040 2 22 I 1.9 0.30 0.00030 0.38 0.00038 This was about treble that of non-reinforced concrete. After the specimens were prepared, they w^re kept moist for a considerable time, but were tested in an air-dry condition. The difference between Considere's tests and those of other experimenters can be partially explained, since concrete which sets under water swells and therefore stands greater stretching than that which sets in air and decreases in volume. It is also to be noted that with each repetition of his experiments, Considere found smaller results. From 2 mm. they fell to EXTENSIBILITY 57 o.Q mm., and finally to 0.5 mm. per meter (from 0.0020 to 0.0005 P^^ foot). The latter figure does not differ much from the results on pages 53 to 55. These bending tests will be discussed again later, in connection with the subject of the exact location of the neutral axis and the distribution of stress in the section. Also, there will be given an independent explanation of the large extensibility observed by Considere and of the stress distribution between steel and concrete, shown in Fig. 50. A com|)lete statement is imi)ossible with- out having first discussed the theory of reinforced concrete. Similar experiments were carried out for the Reinforced Concrete Com- mission of the Jubiliiumsstiftung der Deutschen Industrie in the Testing Labora- tory at Stuttgart. In them, Bach * thoroughly investigated the appearance of the first crack in beams of which the material, proportions, and load distri- bution were similar to those illustrated in Fig. 54, and the outside of which was given a coat of whitewash. With increasing load on the under side of the beams, small damp spots first showed themselves. These spots grew in size as the load was augmented. With further increase, cracks appeared, always where a spot of water existed, but not all such spots developed into cracks. These phenomena^ which had been described by Turneaure, " Engineering News," 1904, p. 213, and also by R. Feret, "Etude experimentale du ciment arme," 1906, developed in beams which had been kept under water, and may be explained by their porosity in certain portions which were stretched by the tensile stresses and from which the moisture w^orked outward and so formed the spots of water on the surface. The cracks appeared on the sides of the beams at somewhat higher loads than on the bottom. It was further shown that the cracks usually commenced at the bottom corner, furthest from the reinforcement. In the section shown in Fig. 55 a crack existed at a load of 6000 to 6500 kg. (13,200 to 14,300 lbs.) at depths about as shown by the lines ah and cd, and advanced under a load of 7000 kg. (15,400 lbs.) to the positions of ai^i, Cidi. In the beams with a single reinforcing rod the cracks appeared somewhat later in the narrower beams than in the wider ones. The first corner crack was observed at a stretch of from 0.127- 0.176 mm. in a length of one meter (0.0001 27-0.0001 76 ft. per foot) for a beam 15 to 30 cm. (5.9 to 11.8 ins.) wide with a single reinforcing rod. The spots of moisture always appeared with a stretch of 0.08-0.10 mm. per meter (0.00008 to 0.00010 ft. per foot), depending on the distribution of the steel in the section. This is, however, the ultimate .stretch of plain concrete. The formation of cracks will be delayed if the reinforcement in the vicinity of the porous spots in the stretched concrete receives additional assistance. When the reinforce- ment was uniformly distributed over the whole width of the beam, the cracks were actually found after greater stretching, but were much smaller and cor- respondingly harder to discover. In heavily reinforced beams the extension * " Versuche mit Eisenbelon-Balken," Part I., No. 39 of the Mitteilungen iiber Forschungs arbeiten and No. 26 of the Zeitschrift des Vereins Deutscher Ingenieure, 1907. 58 CONCRETE-STEEL CONSTRUCTION at which the first crack appeared was correspondingly increased to 0.267 "Sni- per meter (0.000267 ft. per foot). The maximum stretch, which amounted to 0.324 mm. (0.000324 ft.) for beams stored in moist sand, and 0.367 mm. (0.000367 ft.) for those in water, was found in beams in which the reinforce- ment was in the form of a plate 7 mm. (J in. approx.) thick containing holes and extending across the full width of the beam. The influence on the beams of dry and wet storage was also investigated. Beams of 30 by 30 cm. (12 by 12 in. approx.) section, provided on the under :side with one round rod 26 mm. (i in. approx.) in diameter, stored in air, stretched 0.097 mm. (0.000097 ft.), but those stored under water stretched 0.205 mm. per meter (0.000205 ft. per foot) before the appearance of the first crack. Since non-reinforced concrete which has been stored under water or in a moist condition swells, reinforced concrete which sets under water must develop tensile stress in the steel, and corresponding compressive and bending stresses in the concrete. These compressive stresses are naturally greatest in the layers near the reinforcement, and it is clear that when a load is applied it first overcomes these compressive stresses in the concrete and hence the stretch up to the first crack is greater than when concrete originally in an unstressed state, is stretched. In dry concrete which has set in the air, a reduction in volume or shrinkage takes place, so that in this case in unstressed beams the steel is com- pressed and the concrete subjected to tension and bending. Here the first crack will appear under less load and shorter stretch. The experiments with T-beams, which will be described later, also contribute something concerning the extensibility of concrete. Methods of calculation will next be considered, and in connection with them will be given further experiments on reinforced concrete bodies, so that the methods of computation can be checked by them. CHAPTER V THEORY OF REINFORCED CONCRETE COMPRESSION Calculation of Reinforced Concrete Columns with Longitudinal Rods and Ties. — In homogeneous bodies, subject to axial compressive stress, it is assumed that the resulting strain takes place by a lessening of the distance between adjacent imaginary parallel planes perpendicular to the line of stress, and in such manner that the planes remain mutually parallel after the strain. This same assumption is also made in the calculation of strains in concrete columns with longitudinal reinforcement if. First, that portion of the axial stress borne by the concrete is assumed as uniformly distributed over the whole cross-section, and Second, the reinforcement has the same deformation as the concrete. If represents the area of the concrete cross-section, the total area of the reinforcement, cr^ and the corresponding stresses in the two materials when equally strained, the total load P will be In the design of any new column, either the experimentally determined stress- strain curve or the exponential law, £^=a'(7j,^, may be employed in selecting corresponding values of and a^, which can then be inserted in the general load formula. On the other hand it is only by the method of approximations, or of interpolation in tables, that it is possible to find the exact stresses in an existing column. In the "Leitsatze" of the Verbands Deutscher Architekten- und E Ingenieurvereine, the ratio -^=^ = 15, is assumed as constant, so that with Fh equal strains on steel and concrete, Ee and the column load is P=--oh{Fb + isFe). 59 60 CONCRETE-STEEL CONSTRUCTION As the safe stress on concrete is assumed at cr;, = 35 kg/cm^ (497 lbs/in^), it follows that the safe load on a reinforced concrete column is from which may be derived In given columns, the unit stresses will be Ee . The ratio is less than 15 within the limits of perfect elasticity, being Eb approximately 10, but the higher value was chosen in the " Leitsatze " so as to take account of conditions near rupture. One point with regard to reinforced concrete design deserves the greatest consideration. With homogeneous materials, the dimensions of pieces are usually determined from safe stresses which are definite fractions of the ultimate loads.* In reinforced concrete, however, the question arises whether the allow- able and assumed load distribution existing with safe stresses still continues near the point of rupture, or whether conditions change so that the real causes of rup- ture are different, just as is the case in computations with regard to allowable tension in long columns. Nothing but experiments can afford information about these important questions. Until 1905 compression tests of columns were very rare, although great responsibility is involved in their design and construction. A column with 4^ per cent of reinforcement was tested at the Technical High School in Charlottenburg. Its sectional dimensions were 25 by 25 cm. (10 by 10 in. approx.), and height 3.22 m. (127 in.). The reinforcement was 4 round rods, 30 mm'. (13^ in.) in diameter, which were connected at 50 cm. (20 in.) intervals by horizontal flat iron ties 3 mm. by 80 mm. (J by 3 in. approx.) ; the mixture was i : 4, age 3 months. The column \vas prepared with accurate compression surfaces, and failed in such manner that the four rods buckled simultaneously between two ties, and the concrete between them crushed. The breaking strength was 255 kg/cm^ (3627 lbs/in^). If the reinforcing of concrete columns with longitudinal steel and horizontal ties is so done as to secure at least as much strength in the long members as in test cubes, then it is necessary, in designing, only to consider the strength of short specimens (or a certain part of such strength), and the load is P = FbOb. The steel would then be entirely omitted from consideration, but at the same * Exceptions, however, exist. For instance, the computation of the flexure at the breaking load will fail. The method of calculation of beams of the Schwedler type of construction is inaccurate, wherein the load is assumed greater so as to include the necessary safety in regard to diagonal tension. Also the computation of retaining walls and chimneys as to overturning. COMPRESSION 61 time enough must be employed in the form of longitudinal reinforcement and ties, so that the breaking load of a reinforced concrete column will be equal to that of a small cube. It is evident that a certain minimum of steel is necessary. In the " Leitsiitze," longitudinal reinforcement not less than 0.8 per cent of the cross-section is prescribed. But the spacing of the ties also influences the breaking load of a column. Their effect is even greater than that of the longitudinal rods, as was proved by the latest experiments of the Reinforced Concrete Commission of the Jubil- aumsstiftung der Deutschen Industrie. These tests,* made in 1905, were conducted by Bach at the Testing Lab- oratory of the Royal Technical High School of Stuttgart, and involved concrete prisms 25 by 25 cm. (10 by 10 in. approx.) in section, and i meter (39.4 in.) long, mixed in the proportions of i part Portland cement and 4 parts of Rhine sand and gravel, with 15 per cent of water. Thus they were of the same composition as the specimens described on pages 44 to 46, used in the adhesion experiments. Part of the specimens were without reinforcement. The others each had 4 rods with 7 mm. (J in. approx.) ties arranged as shown in Fig. 56. Five varie- ties of reinforcement were employed, viz.: 15 mm. (f in.) rods and 25 cm. (10 in.) tie spacing, Fig. 57 15 mm. in.) rods and 12.5 cm. (5 in.) tie spacing, Fig. 58 15 mm. (I in.) rods and 6.25 cm. (2^ in.) tie spacing, Fig, 58 20 mm. (I in.) rods and 25 cm. (10 in.) tie spacing, Fig. 58 30 mm. (i^V in.) rods and 25 cm. (10 in.) tie spacing, Fig. 59 At the same time was ascertained the compressive strength of cubes, 30 cm. (12 in.) on each edge. The elasticity in compression was measured for two or three specimens of each kind, for stresses up to 113 kg/cm^ (1607 lbs/in^). It was disclosed that the shortening diminished not only with increased section of longitudinal steel, ^ - " r- ™= 1. ..>r..iir.riti>>k,. ^^,L,..la Fig. 56. but also with increasing numbers of ties, when the longitudinal rods were the same. In the same manner as described on page 21, for stress increments of about 16 kg/cm2 (228 lbs/in^) were measured, the total compression, the elastic deformation, and the permanent set. From these results, curves were determined similar to those for plain concrete. The influence of the ties on the elastic phe- nomena is shown in Table XXIII. * C. V. Bach " Druckversuche mit Eisenbetonkorpera," 1905. " Mitteilungen iiber For- schungsarbeiten," No. 29. 62 CON(^RETE-STEEL CONSTRUCTION Fig. 57. Fig. 58. Fig. 59. Test Specimens. Table XXIII ELASTICITY TEST OF COLUMNS Stresses Diameter of Rods Tie Spacing Shortening in Millionths of the Length kg/cm2 lbs/in2 mm. in. cm. in. Total Elastic Dif. Permanent Set 32.3 32.3 32-3 32-3 459 459 459 459 Plain concrete 133 114 no 106 7 5 2 4 126 15 15 15 f 5 8 5 8 25 12.5 6.25 ID 5 2i 109 108 102 64.6 64.6 64.6 64.6 919 919 919 919 Plain concrete 333 267 264 241 37 20 18 13 296 247 246 228 15 15 15 1 5 8 5 8 25 12.5 6.25 ID 5 2i 97.0 97.0 97.0 97.0 1380 1380 1380 1380 Plain concrete 709 488 473 421 164 63 58 42 545 425 415 379 15 15 15 1 5 8 f 25 12.5 6.25 ID 5 2h COMPRESSION 63 It is there shown that, even with known elastic data for plain concrete, it is impossible to determine the distribution of stress between the steel ana concrete with usual stresses, since the ties alter the elasticity of the reinforced concrete. They prevent lateral expansion of the concrete and thereby increase its com- pressive strength. The assumed ratio of the moduli of elasticity of steel and concrete is a close enough approximation. Actually it varies from i:ii to 1:13 at the highest stresses covered by the elasticity experiments. For practical purposes, the observed breaking strengths are more important. (See Table XXIV.) Table XXIV BREAKING STRENGTH OF COLUMNS Specimen about 3 Months Old Diameter of Rods Tie Spacing Breaking Strength Each Average kg^cm2 lb /in 2 %of Rein- forcement 15 15 15 20 Plain Test concrete 25 12.5 6.25 25 25 cubes 10 5 2^ 10 10 146 171 168 212 169 174 I 168 138 139 161 172 187 175 200 203 169 172 199 197 169 171 \ 185 184 / 141 168 177 205 170 190 175 2010 2390 2520 2920 2420 2700 2490 o 1. 14 1. 14 1. 14 2.04 4.60 The appearance at fracture is shown in Figs. 60 to 64. According to the *'Leitsatze" of the Verbande Deutscher Architekten- und Ingenieur-Vereine, the allowable loads for the prisms were as in Table XXV. Table XXV ALLOWABLE COLUMN LOADS With four 15 mm. (| in.) rods, P = 625X35+i5X 7.1X35=25602 kg. (56324 lbs.) With four 20 mm. (fin.) rods, P = 625X 35+ 15X 12.6X35= 28490 kg. (62678 lbs.) With four 30 mm. (i^ in.) rods, ^=625X35+ 15X 28.3X35 = 36732 kg. (83010 lbs.) These computed allowable loads are in the proportion of 168:187:241, while the actual loads on specimens with a 25 cm. (10 in.) tie spacing were as 168: 170: 190. It is seen from these values that an increase in the area of longitudinal rein- orcement does not produce an increase in the breaking strength to the extent which would be indicated by the formula 64 CONCRETE-STEEL CONSTRUCTION In experienced hands this formula may give rise to constructions which are not sufficiently safe. Some designers are careless with regard to this point, and, in order to produce columns of small diameter, increase the percentage of longitudinal reinforcement disproportionately. This gives such columns a calculated margin of safety which they do not possess. When the increase in resistance is computed for one kilogram of steel in the form of longitudinal rods and of ties, it is discovered that the steel used as ties is nearly twice as effective as the straight rods. The former must, therefore, be given proper attention m the design of columns. Further experiments with long columns, in which the top and bottom are broadened to guard against premature failure, would be very desirable, and are planned by the Reinforced Concrete Commission. Fig. 6o. Fig. 6i, Fig. 62. Fig. 63. Fig. 64. Results of Crushing Tests. It is recommended, until further tests are available, in reinforced concrete building construction designed in accordance with the formulas of the German ''Leitsatze," that 20 kg/cm^ (284 lbs/in^) be assumed for the columns of the upper floor, and that this stress be increased in the lower floors to the maximum safe limit. The longitudinal reinforcement should be from 0.8 to 2.0 per cent, and the tie spacing approximately 5 cm. (2 in.) less than the diameter of the column, but never over 35 cm. (13.8 in.). If the strength of test cubes is made the basis for assumed safe loading without at all considering the steel, then the stresses may range from 25 kg/cm^ (355 lbs/in-) in the top story to 45 or 50 kg/cm^ (64c to 710 lbs/in^) in the lower ones. It is hardly imaginable that all floors of a building will ever be simultaneously fully loaded, so that the columns of the lowest story are very seldom fully stressed, and consequently are most favored. The computation of the tie spacing solely from the buckling length of the longitudinal steel, too often calls for excessive spacings. Furthermore, in prac- COMPRESSION 65 tice, the actual spacings are not mathematically exact, and do not remain fixed during the tamping of the concrete. They should prevent lateral Imlging of the concrete, and must therefore possess ample strength to resist lateral failure. Flexure. — No tests of the breaking strength of reinforced concrete columns exist, com])arable with those for steel. It therefore becomes necessary, in design- ing in reinforced concrete, to em])loy data applicable to all homogeneous bodies, Tetmajor shows that Euler's formula applies to long, slender steel columns, if the compressive stress lies below the elastic limit w^hen failure commences. To large cross-sections and short lengths, this formula probably does not apply, because the compressive stress at the breaking point has exceeded the elastic limit. Under such circumstances E, the modulus of elasticity, is not a constant. In all materials such as concrete which ZOO Fig. 65. Fig. 66. •do not possess a constant modulus, it is necessary to ascertain (by calculation from experiments), a value for E which will correspond with the compressive stress at the moment of rupture, in such manner that it may be measured by the tangent of the angle of inclination of the stress-strain curve and be expressed by da In calculating the area of the cross-section involved in the moment of inertia /, the area of the steel must be multiplied by the ratio EJEf^. No change in the distribution of stress can take place with such procedure, since the area of the reinforcement is replaced by a concrete area E^IE^ times as large. Since the exponential law of stress-strain variation applies only to stresses up to about 40 kg/cm^ (568 lbs/in^), neither can that law be utiHzed for the derivation of a suitable breaking formula. 66 CONCRETE-STEEL CONSTRUCTION In the 1899 volume of the Schweizerische Bauzeitung, is a communication by Ritter giving another basis for a formula. In it the following equation is employed: (7 = X(l-e-1000.S)^ K represents the ultimate stress in the concrete, E the corresponding strain, and € = 2.71828, the base of the system of natural logarithms. If the locus of this equation is plotted, the curve will be found to agree as well as can be expected with the stress-strain curves found by experiment under varying circumstances. By differentiating g with respect to e, the modulus of elasticity E is obtained. £=^ = XlOOOC-1000^ =IOOO(A'-(7). If this value is inserted in Euler's formula, there results P=-j^EJ=—iooo{K—a)J; a therefore represents the initial breaking stress. If P is replaced by Fa, J by Fr^ , and 71^ by 10, there is obtained for the breaking stress K I +0.0001—^ Example. Let it be desired to obtain the breaking load on a column having a cross-section of 25 by 25 cm. (9.84 by 9.84 in.) reinforced with 4 rods 18 mm. (0.708 in.) in diameter, Ee / = 4m. (157.48 in.), ;g^ = io. ^ = 250 (3555), 7=^25^ + 10X4X0.9^X7: X 10^=42702 cm^, (=^9-8^ + ioX4Xo.354'X7rX3W = i025 in^); F = 252 + ioX4Xo.92x-/r = 727 cm^, ( = 9.82 + 10X4X0.354 X27: = iT3 in2); r2 = -^ = 58.7 cm2 (9.1 in^); t COMPRESSION 67 Ok 250 I +0.0001 X 400 78T -> = i97^kg/cm2, 3555 I +0.0001 X I57-48- 9.1 = 2802 lbs/in2). With a factor of safety against rupture of eight, a safe working compressive stress of 25 kg/cm^ (355 lbs/in^) can be assumed. If partially fixed ends are considered, a corresponding free length of j / should be employed, and closer result will be somewhat larger. A be obtained with might somewhat W = II. In the example above, a comparatively slender column has been assumed. It is evident from the result of the calculation that reinforced concrete columns differ very materially from iron ones, the risk of breakage being much greater in the latter. The superiority of reinforced concrete is due to the greater sectional area (as compared with steel) and the smaller unit stress. Or, in terms of the Euler formula, the moment of inertia / is increased in greater ratio than the modulus of elasticity is reduced, when compared with a steel column of equal carrying capacity. Consequently, with concrete, only in exceptional cases will there be required a special calculation of the safety against rupture by flexure. Fig. 67. Calculation of Reinforced Concrete Columns with Spiral Reinforcement (Beton Frette)* Considere's method of calculating spirally reinforced columns will here be followed. From theoretical considerations, the correctness of which has been fully established by experiment, Considere reached the conclusion that rein- forcement, if introduced in the form of a spiral, ensured an increase in the carrying capacity 2.4 times as great as would be obtained with the same amount of reinforcement in the shape of longitudinal rods. If Fb represents the area of the concrete core, k the ultimate stress of non- reinforced concrete, the cross-section of the longitudinal reinforcement, J J the cross section, of imaginary longitudinal rods, of which the weight is equal to that of the spirals in an equal length of column, Oe the elastic limit of the rein- * Patented in France, Germany, England, United States, etc. 68 CONCRETE-STEEL CONSTRUCTION forcement (which, for commercial material, may be assumed at 2400 kg/cm^ (34,140 lbs/in^)), then the ultimate load is given by the formula I.5^F6 + ^7,(/, + 2.4//). In this expression, it is supposed that the elastic limit of the reinforcement determines the carrying capacity of the column. The factor 1.5 is employed because an octagonal cross-section, together with other usual conditions, make the gross sectional area about equal to 1.5 times the central portion enclosed by the 'spiral. It thus equals 1.5^5 of the whole concrete section. Considere * proved that test specimens, prepared with the care possible in a laboratory, developed very considerable compressive strength. The owners Fig. 68. Fig. 69. tiG. 70. Fig. 71. Test Specimens. of the German rights under Considere's patents, deemed it advisable to institute experiments with specimens manufactured without special care, at a building site. As a consequence, in the specimens so prepared, the pitch of the spirals was rendered somewhat irregular by the ramming of the concrete, and some eccentricity of position was perceptible. In the earlier tests the longitudinal reinforcement had a sectional area of at least one per cent of that of the specimen, with a pitch in the spirals of one-seventh of the diameter of the column. The * Genie Civil., Nov., 1902, Beton und Eisen, No. V, 1902. COMPRESSION 69 later experiments were intended to show whether it is advisable to increase the pitch of the spirals appreciably beyond this ratio. The specimens had an octagonal section with a diameter of 27.5 cm. (10.8 in.) and a length of i.oo m. (39.37 in.), and were made of a mixture of i part Ijy volume of Heidelberg Portland cement, to 4 parts of Rhine sand and gravel, with 14 per cent (by volume) of water. They were between 5 and 6 months old when tested. Three specimens of each kind were broken. Their general dimen- sions are given in Figs. 68 to 71 and Table XXVI. Table XXVI DIMENSIONS OF CONSIDER?: COLUMN TESTS Spiral Longitudinal Reinforcement ISIO. Average of of Figure Thickness d Pitch s Diameter Specimen No. of Rods mm. in. mm. in. m.m in. I 4 68 None None None None II 3 69 5 3 ic 38 1.50 4 7 1 III 3 69 7 1 4 37 1 .46 4 7 1 4 IV 3 69 10 f 42 1.62 4 7 1 4 V 3 70 5 3 10 38 1-50 8 II 7 • 16 VI 3 70 7 i 37 1 .46 8 1 1 7 10 VII 3 70 10 3 8 43 1 . 70 8 II 1^ VIII 3 69 7 1 31 1 . 22 4 7 1 4 IX 3 69 10 3 8 40 1.58 4 7 1 X 3 69 12 41 1.62 4 7 XI 3 69 14 37 1.46 4 7 i XII' 3 70 7 1 i 40 1.58 8 5 XII" 3 70 10 *9 40 T.58 8 7 1 4 XII'" 3 70 14 16 40 1.58 8 10 f XIII' 3 71 7 1 4 80 3-16 8 7 i XIII" 3 71 10 f 80 3-16 8 10 t XIII'" 3 71 14 9 TTi 80 3.16 8 12 XIV' 3 71 7 1 120 4.72 8 10 1 XIV" 3 71 10 f 1 20 4-72 8 12 XIV'" 3 71 14 120 4.72 8 14 9 In some of the tests * the permanent and elastic deformations were both measured, but no definite law could be deduced from the results except that the reinforced columns displayed somewhat less deformation, or a greater mod- ulus of elasticity than those which were not reinforced, just as was noted with regard to the ordinary (simply longitudinally reinforced) concrete prisms already described. In all cases the load was noted at which the first cracks were observed, as well as the ultimate load. The cracks appeared first in the concrete layer out- side the spiral, and large fragments of that shell finally became detached. The types of failure are shown in Figs. 72 and 73. * The tests were made at the Testing Laboratory of the Royal Technical High School at Stuttgart. The results were published by the president, Bach, in " Druckversuche mit Eisen.- betonkorpen, Versuch. B." Berlin, 1905. Also in No. 29 of " Mitteilungen iiber Forschungn- arbeiten." 70 CONCRETE-STEEL CONSTRUCTION Table XXVII gives the results of the tests, together with the increase in strength developed by the reinforced specimens as compared with specimen I, which were without reinforcement. Table XXVII RESULTS OF CONSIDERE COLUMN TESTS No. Figure Unit Stress of First Crack Oi Increase over Non-reinforced Column Oi— 133. etc. Ultimate Unit Load Increase over Non-reinforced Column i33> etc. Ultimate Unit Load on Central Core kg/cm 2 lbs/in^ kg/cm2 lbs /in 2 kg/cm2 lbs/in2 kg/cm2 lbs/in^ kg/cm2 lbs./in2 I 68 133 1892 133 1892 II 69 159 2262 26 370 159 2262 26 370 230 3272 III 69 161 2290 28 398 178 2532 45 640 257 3656 IV 69 170 2418 37 516 240 3414 107 1522 347 4936 V 70 224 3186 91 1294 226 3215 93 1323 327 4651 VI 70 230 3272 97 1380 230 3271 97 1379 332 4722 VII 70 243 3442 110 1550 281 3997 148 2105 406 5775 VIII 69 196 2788 63 896 200 2845 67 953 289 4111 IX 69 170 2418 37 526 211 3001 78 1 109 305 4936 X 69 180 2560 47 668 256 3641 123 1749 370 5263 'XI 69 158 2247 25 355 246 3499 113 1607 355 5050 XII 70 163 2318 30 426 163 2318 30 426 236 3357 XII 70 164 2333 31 441 230 3271 97 1379 332 4722 XII 70 184 2617 51 725 302 4295 169 2403 436 6202 xiir 71 162 2304 29 412 162 2304 29 412 234 3328 XIII" 71 179 2546 46 654 181 2574 48 682 261 3713 XIIU" 71 186 2646 53 754 199 2830 66 938 298 4239 ' XIV ' 71 2205 22 313 155 2205 22 313 224 . 3185 XIV" 71 183 2603 50 711 183 2603 50 711 264 3755 XIV"' 71 207 2944 74 1052 207 2944 74 1052 299 4253 Table XXVIII gives the results of applying Considere's formula to specimen? V, VI, and VII, with ^^ = 133 (1888 lbs/in2) and (7, = 2400 (3414 lbs/in2). Table XXVIII COMPARISON OF RESULTS OF CONSIDERE COLUMN TESTS Concrete Core Area of Reinforcement Strength of Reinforcement, Ultimate Strength No. Area. Strength, 1.5X133^^ i.5Xi89oF^ Longitu- dinal (Spiral) Equivalent Longitu- dinal. 2400 (/„4-2.4/;) 34140" (/^4-2.4/;) Kg. Lbs. cm2 in2 kg. lbs. cm2 in2 cm2 in2 kg. lbs. Ob- served Com- puted Ob- served Com- puted V VI VII 452 442 432 70.1 68.5 67.0 90100 88200 86200 198220 194040 189640 7.60 7.60 7.60 1. 18 1. 18 1. 18 3-90 7.78 13-49 0.60 I . 21 2.08 40700 63000 95900 89540 138600 210980 130800 1 51 200 182100 142000 144000 176200 287760 332640 400620 312400 3 I 6800 387640 COMPRESSION 71 In spite of the defects in the specimens, the strength developed l^y them corresponded approximately with the results indicated by the formula, and exceeds them for the specimens with the least reinforcement. Considere suggests the following lessons from the other results. Pitch of the Spirals. Specimens XIII and XIV, in which the pitch was exaggerated (80 and 120 mm. =3.15 and 4.72 in.), gave mediocre results. Specimen XIII''', although showing an increase, did not develop the strength indicated by the formula. This circumstance appears to be ascribable to a wrong relationship between the diameters of the longitudinal rods and of the spiral reinforcement. This deficiency was not eliminated by a decrease in the pitch of the spirals.* Fig. 72. — Failure from shear. Fig. 73. — Spalling of the outer (Spiral broken.) concrete shell. Relationship Between the Spirals and the Longitudinal Rods. In specimens II, III, IV, VIII, IX, X, XI, and XII' the sectional area of the longitudinal rods was small, and the results were consequently indifferent; but the greater the total weight of spiral reinforcement, the higher were the results. On the whole, the tests seem to prove that when the spirals are increased in strength, their pitch must be decreased, and the cross-section or number of the longitudinal rods must be increased ; "j" for with increase in strength of spirals, * Specimens XIII and XIV gave practically identical results. With the spiral reinforce- ment diameter and pitcb as designed, longitudinal rods of larger diameter should apparently have been used in XIII" and XIV" to give results proportional to the corresponding tests of XII and XIV.— Trans. fin order to secure a consistent increase in supporting power. — Trans. 72 CONCRETE-STEEL CONSTRUCTION the concrete is in a condition to resist a heavier pressure and its tendency to •force its way out between the longitudinal rods also increases. In planning the programme of tests of hooped concrete, a direct comparison was sought with the column tests conducted for the Reinforced Concrete Com- mission of the Jubilaumsstiftung der Deutscher Industrie (page 6i) , by making the cross-section of the octagon equal a square area 25 by 25 cm. (10 by 10 in. approx.), and by so arranging the spirals and longitudinal rods that in speci- mens II, III, IV, V, VI, and VII the amount of steel in the spirals was equal to that of the ties in the columns reinforced with 4 rods 15 mm. (0.59 in.) in diameter, and with spacings of 25 cm. (9.8 in.) 12.5 cm. (4.9 in.) and 6.25 cm. (2.45 in.). While with the ordinary form of tie the increase in strength compared with non-reinforced concrete prisms (3 months old) amounted to 27 kg/cm^ (384 lbs/in^), 36 kg/cm2 (512 lbs/in^), and 64 kg/cm^ (910 lbs/in^); with the employ- ment of the same amount of steel in the form of spirals, and with 4 longitudinal rods only 7 mm. (0.28 in.) in diameter (5 to 6 months old) the increase was 26 kg/cm^ (370 lbs/in^), 45 kg/cm^ (640 lbs/in^), and 107 kg/cm^ (1520 lbs/in^); and with eight rods 11 mm. (0.43 in.) in diameter, it was 93 kg/cm^ (1320 lbs/in^), 97 kg/cm^ (1380 lbs/in^), 148 kg/cm^ (2100 lbs/in^). In the last instance it is to be noted that the eight rods of 11 mm. (0.43 in.) diameter, have almost exactly the same section as the four 15 mm. (0.59 in.) rods in the last column test, so that the advantage of the spirals over the hooped results is an increase of strength of 66 kg/cm^ (939 lbs/in^), 61 kg/cm^ (868 lbs/in^), and 84 kg/cm^ (1195 lbs/in^). In the prisms VIII, IX, X, and XI the spirals were so designed that the quan- tity of steel in them was equal to that of both the ties and longitudinal rods of the column tests (page 61), and also so that a pitch of about one-seventh of the column diameter was obtained. In addition, for practical reasons, the spirals were held in position by four longitudinal rods, 7 mm. (0.28 in.) in diameter. The columns with four rods 20 mm. (0.79 in.) in diameter, and a 25 cm. (9.8 in.) spac- ing of ties, then had almost exactly the same amount of reinforcement as did the spirally reinforced prism IX, and also as did the column with four rods 15 mm. (0.59 in.) in diameter with a tie spacing of 12.5 cm. (4.9 in.). The increase in strength of the ordinary longitudinally reinforced columns, as compared with the non-reinforced specimens, according to Table XXIV, on page 63, amounted to 27 36 64 29 49 kg/cm^ (384) (512) (910) (412) (796 lbs/in2), and in the case of prisms VTII, IX, X, IX (again) and XI, according to Table XXVII, on page 70, it amounted to 67 78 123 78 113 kg/cm2 (952) (iiio) (1745) (iiio) (1605 lbs/in2). If the ratio of the increase in strength shown by these two series of tests representing the two types of design, but employing for closer comparison only COMPRESSION 73 those columns of the one kind which had the spacing of 25 cm. (9.8 in.), is com- puted, there is obtained 67 Q 78 . 113 — = 2.48, — = 2.60, ■ — - = 2.21. 27 ^ ' 29 ^' 49 These results are in satisfactory agreement with the figure 2.4 assumed l)y Considere, which, therefore, expresses the superiority of reinforcement in the form of spirals over its value in the form of longitudinal rods. The results obtained from specimens XII disclose the importance of com- bining with any spiral reinforcement a longitudinal reinforcement of about the same proportions. CHAPTER VI THEORY OF REINFORCED CONCRETE SIMPLE BENDING In homogeneous bodies possessing a constant modulus of deformation, equations of flexure can be derived on the assumption that sections which were plane before bending will be plane after bending. The question at once arises to what extent this assumption will apply to reinforced concrete bodies. By experiments with homogeneous bodies of rectangular cross-section, the correctness of this assumption has been established within certain limits, but it owes its general acceptance to a demand for the greatest possible simplification of methods of calculation. Furthermore, it is known that this assumption of the conservation of plane sections is irreconcilable with the existence of shearing stresses, which generally tend to produce an S-shaped deformation of any right section. With equal reason, therefore, there can be assumed the conservation of plane sections in the flexure of reinforced concrete beams, and it is to be noted that the strength of the beam computed on page 29, on the basis of such plane sections, but constructed of a material possessing a variable modulus of defor- mation, coincides satisfactorily with that obtained by experiment. If, therefore, in Fig. 74, AB represents the cross-section of a reinforced con- crete beam, A'B' is the curve of strain and the corresponding stress curve is repre- sented by EOF. The latter is really a properly plotted stress-strain curve for concrete. The steel reinforcement must follow the deformation of the concrete. The upper reinforcement is consequently I f -^ shortened an amount CC'^ while the lower TJ3' £ layer of steel is stretched an amount DD' . Fig. 74. The corresponding steel stresses are pro- portional to these strains. The distribution of stress in reinforced concrete, shown in Fig. 74, will occur only under very moderate loading, because the elasticity of concrete in tension is soon overcome. This condition of stress is designated Stage 1. In calculat- ing stresses in this stage, the lines OE and OF may be regarded as straight. With increasing load, the full tensile strength of the concrete will be attained throughout the whole zone of tension, and if the great capacity of reinforced 74 SIMPLE BENDING 75 concrete to stretch, observed by Considere, is assumed provisionally, then the distribution of stress under such conditions will resemble Fig. 75. This may be designated Stage II. According to Considere's tests, this second stage does not extend beyond the stress in the concrete corresponding to the strain of the reinforcement at its elastic limit. A continued increase in the load causes the elastic limit of the steel to be exceeded, the tensile strength of the concrete is no longer a factor, and finally a break occurs through a failure in the tensile strength of the steel or in the compressive strength of the concrete. This last condition of loading, the breaking stage, is designated Stage III. It is evident that in an exact theoretical study of the breaking stage, great difficulties are encountered; since, then, the elastic conditions usually employed as the basis for calculations do not exist. With regard to Stage II, it must be noted that no certain dependence can be placed on the tensile strength of the concrete, partly because of irregularities in its composition, but especially because recent tests have shown that the stretch of concrete does not extend nearly to the strain of the steel at its elastic limit. Cracks in the concrete may therefore be expected in the latter part of Stage II. The early part of this condi- tion of loading (without tension cracks in the concrete) may be called Stage lla, while the latter part may be Stage Wh (with tensile cracks in the concrete and steel stress less than the elastic limit). The distribution of stress in these yig 75. Fig" 76. two sub-stages is shown in Figs. 75 and 76. The question arises, which stress condition is to be made the basis from which to derive methods of calculation for practical purposes. When consid- eration is given to the fact that the object of every static calculation is not so much to ascertain the exact stresses in any structure resulting from a given load, but rather to secure an adequate degree of safety for the structure, then it must be concluded that attention should be given to the examination of the supporting power of reinforced concrete construction subject to bending — in Stage III (that of failure). This, however, can hardly be accomplished theoretically. Stage I must be excluded from consideration because it has already been passed even with perfectly safe loads. Stage \\a, which has been recommended in several instances as a basis for calculations, should be excluded because of the uncertainty of Considere's tests, and also because the concrete has been shown to be subject to cracks, attributable, variously, to deficient manipulation, inter- ruptions during the concreting process, to effects of temperature change, or to excessively rapid drying. Further, no more exact information is obtainable from this stage concerning the necessary amount of steel than is obtainable from Stages l\h or III. Stage lib is thus shown to be the only stress condition readily available for theoretical treatment and the one which most clearly shows the required amount of reinforcement. Moreover, the method derived from it has the great advan- tage of simplicity, and it can be adapted to Stage III, so that safe stresses can be selected which bear a proper relation to the results of ultimate bending tests. 76 CONCRETE-STEEL CONSTRUCTION In making designs based on Stage 11^ instead of on the rupture stage, no greater error is committed than is made in every calculation of ordinary timber or steel construction, in which Navier's theory of flexure is almost always employed, even though it is not applicable at the point of rupture. There follow a few methods of calculating reinforced concrete structures subjected to bending stress. They have been developed on the basis of Stage lib. In them it is assumed that, after deformation, the strained steel sections remain in the same planes with the corresponding compressed concrete sections. The tensile strength of the concrete is consequently ignored. Rectangular Sections — Slabs I. With rectangular cross-sections the calculations can be made with the aid of the stress-strain curve exactly as has already been described for non- reinforced concrete beams. In Fig. 77 the line OE represents the variation in compressive stress. The branch for tensile stresses is omitted, and its place is taken by the tension sur- FiG. 77. face of the reinforcement. Calling the width of cross-section equal to unity, and assuming ob and Og definite safe stresses for the concrete and the steel respectively, then the stress surface for the concrete is definitely determined, and also the position of the reinforcement. The tension surface of the steel is a long, narrow rectangle. In simple flexure, there are present no exterior longi- tudinal force components, and consequently the tensile and compressive forces must balance in each section. Or, the area of the compression surface must be equal to the rectangle of the tensile stress. If the distance of the centroid of the reinforcement from the upper edge is called h, then fe can be expressed as a function of h, ob, and the moment M. The latter is equal to the area of the compression surface multiplied by the distance of its centroid from the rein- SIMPLE BENDING 77 forcement, and consequently fe will be a function of h'^; h may be called the useful height of section. Determining dimensions of members according to this method is easy, whereas complicated trial calculations are necessary to ascertain the stresses in an existing structure. This method, in connection with the stress-strain curves shown on page 24, gives the following results: With a 1:4 mixture with 14 per cent of water, if Fe represents the section of the reinforcement in square centimeters per meter width of slab (also square inches per foot width), and M is computed for the same width in centimeter-kilograms (inch-pounds) with (76=40 kg/cm^ (569 lbs/in^), cre = 1000 kg/cm^ (14,220 lbs/in^), £e = 2,160,000 kg/cm^ (30,720,000 lbs/in2). h=o.o^o'i\/ M centimeters per meter width of M; *(/7.=o.o3i2\/il/ inches per ft. width of M); Fe=o.o277\/i/ square centimeters per meter width; *(Fe=o.oo255V^M square inches per ft. width). The thickness d of the slab is to be taken 1.5 to 2.0 cm. (0.6 to 0.8 in.) greater than the calculated useful height h, the bottom face of the concrete being lowered to that extent. It is further to be noted that this method also allows account to be taken of the tensile strength of the concrete. 2. The same general method may be followed along strictly analytical lines, by employing the exponential law. The conservation of plane sections is ex- pressed in the nomenclature of Fig. 78, by Fig. 78. the proportion a~ h' whence £b = T Se. Also ee = -^. 0 rL Now, according to the exponential law e.b=a ob^, * These values are for unit stresses of 569 and 14,220 lbs. in concrete and steel respectively, and for a moment equal to M in-lbs. per foot width in English units, while the formula in metric units is for M kg-cm per meter width. — ^Teans. 78 CONCRETE-STEEL CONSTRUCTION whence wherein E represents the modulus of elasticity of steel. Now, b = h—a, from which (h—a) a ob^=a-^, , ha Gh"^ whence a = E The moment M is: Now, (7^:(T6^: :7^:a, so that and from differentiation a a' v=^ dv= o^~^d(7. Therefore, M= { a^-^dG[b-\ Jo ob^ \ o,^l (76"* Jo Ob-'^Jo = a 0 ob-rd'^ \ — Ob. m + i 2m + i After substituting the values of a and 6, given above, m (76"*+^ E m + i /i (^-^e + Ob'^E^' / (m+i) (76"* £ ^ \ SIMPLE BENDING 79 The area of steel, }e, for unit width of slab, is given by the equation a dv I r"b (7 a m a -da a m Ob m+l Oe m + I Ob' a Ob Oe If h from the equation for M is substituted herein, there results for unit width of slab Ob Oe — obE^ — ob"^ 2m + 1 « If values of ob and Oe are assumed, the equations given above may be employed Ob 1-17 E in designing slabs. In this case with a= , w=i.i7, Sb-- 230,000 230,000' 2,160,000, (76 = 40 kg/cm^, and ^7^=1000 kg/cm^, if Fe represents the section of the reinforcement for one meter breadth of slab, and further if M is calculated for the same breadth in centimeter-kilograms. /z=o.o363\/ M centimeters for M on meter width; *(/z=o.o2 78\/il/ inches for M on foot width); Fe=o.o324\/ilf square centimeters for M on meter width; *(Fe = o.oo298\/ M square inches for M on ft. width). The thickness of the slab is to be increased 1.5 to 2 cm. (0.6 to 0.8 in.) over the value of //, this increase being made below the centroid of the reinforcement. 3. While the two methods above de- scribed permit only of designing — which is most important for the engineer — the following method may be employed to investigate the stresses in completed or completely designed reinforced concrete slabs. It is contained in the " Leitsatze" ig. 79. of the Verbands Deutsche Architekten- und Ingenieurvereine, and the Deutschen ! ' ! ^ 1 1 ! 1 ^ * See foot-note, page 77. 80 CONCRETE-STEEL CONSTRUCTION Betonvereins, of 1904, and is also included in the Prussian Ministerial Regu- lations " of 1904 and 1907. Here, instead of the exponential law, is employed the proportionality between the tensile and compressive stresses of the concrete, while the tensile strength is again ignored. If a constant modulus of elasticity of concrete Eh is assumed, and the distance of the neutral plane from the top of the slab is called x, then (see Fig. 79), If h represents the assumed breadth of the section, and Fe the total area of the tension reinforcement in the same breadth, then Z=D Further, the strains are in the proportion Ob Ge ,j . whence 2 Ge F( hx Substituting this value in the proportion above, gives 2 Oe Fe \x=—:{h—x) bxEb whence _2Fe Ee b Eh (h-x). With this may be transformed into the quadratic equation c, . Fe 1 + 2 — n x = 2 — n fi, from which With the value of x from this equation, there may be found SIMPLE BENDING 81 and the maximum stress on the concrete _2D_ 2M and on the steel Z M The position of the neutral axis is determined by the condition that it must pass through the centroid of the effective stress surface, in which the area of the reinforcement has been replaced by a concrete area w-times larger than that of the steel. The neutral axis thus forms the lower lim-it of the compressed portion of the concrete section. This gives as the equation for the statical moment of the effective areas with respect to the neutral axis X bx n Fe(h—x) =o, from which follows the quadratic equation r, . Fe J A formula may also be deduced in terms of the unit stresses. From the proportion Oh . there follows h Ob n X = - oe-\-n Oh The moment M for breadth h is h obx h oh^ n / h obn \ ■e+nob)\ 3{oe+nob)/ {30e + 2nOb). _b_ 2{Oe h ob^ n 6{oe-\-n Ob) 82 CONCRETE-STEEL CONSTRUCTION The total area of steel for breadth b is (j;, X b re = , or b h oh^ n 2 ae{oe-\-nab) If the safe unit stresses adopted in the " Leitsatze " are employed, (7^=40 kg/cm2 (569 lbs/in^), (7^ = 1000 kg/ cm^ (14,220 lbs/in^), if also 6 = 100 cm. (39.4 in.) and ^ = 15, there results /j=o.039oV M centimeters; Fe=o.0293\/ M square centimeters; *(/^=o.o2993\/ M inch for one foot width of M)] *(Fe = 0.002696V M square inches for one foot width of M). When employing the safe stresses (75=40, and (7e = iooo, the area of reinforce- ment bears to h the ratio ^ 0.02Q^- Fe= —h=o.n ^oh (metric) ; 0.0390 ^ ^' 0.002696 re= — /j=o.oq/^ (hnglish). 0.02993 If this ratio is exceeded, the steel cannot be fully utilized, because then the concrete would be over-stressed in compression. Reinforcement of this character would therefore be impracticable. With variously assumed values of ob and Oe the distance x of the neutral axis from the upper edge of section may be expressed in terms of h, and with n = i^y and 6 = 100 cm., the values in Table XXIX are obtained. The figures in heavy type represent stresses adopted in the Leitsatze " and the Prussian ''Regulations." With (76=40, and (7^ = 1000 kg/cm^ (569 and 14,220 lbs/in^, respectively), the neutral axis is located at f of the height, and the arm of the couple formed by the tensile and compressive stresses is |//. These results are of great value in preliminary calculations and rough estimates, since with moderate concrete stresses these quantities do not vary much. If, for instance, a continuous roof slab is to be designed, of which the greatest moment is 70,000 cm. -kg. (60,630 in.-lbs.), there must first be determined the thickness and steel section at points of maximum moment (by means of Table XX, for instance) , and then the section of steel at the points of minimum moment by the formula F- ^ re — — =y. (7et^ * See foot-note, -page 77. SIMPLE BENDING 83 Table XXIX BEAM ELEMENTS FOR VARIOUS UNIT STRESSES Oc hl\/M* Fels/M* x/h. \ 3 / / kg/cm^ Ibs/in2 kg/cm 2 Ibs/in2 cm. for M per m. in. for A/ per ft. cm^ for M per m. in 2 for M per ft. 30 427 750 10673 0.0451 0.0348 0.0338 0.00314 0.375 0.875 X ^ 00 498 1 J 1067^ 0.0401 0.0^07 0.0^8=; 0.00354 0.412 0.863 40 c;6o 10673 0.0^6^ 0.0270 • / y 0.0430 0.00394 0.444 0.852 640 1067^ o.o;^4. 0.0256 0.0474 O.OOd^2 0.474 0.842 712 750 10673 0.0310 0.0238 0.01^17 0.00476 0.500 0-833 30 427 800 11376 0.0459 0-0353 0.0309 0.00284 0.360 0.880 1 ^ 0 J 4Q8 800 1 1 ^76 '■'-01 0. 0408 0 OXXA o.oo'?2=; 0.396 0. oOo 40 c;6q 800 1 1376 0.0367 0.0282 0.0^07 o„oo^67 0.429 0.857 A^ 640 800 11376 0.0339 0.0261 0.0436 0.00402 0.458 0-843 712 800 11376 0,0314 0.0241 0.0475 0.004"? 7 0.404 0.839 30 427 900 12798 0.0474 0-0364 0.0264 0.00243 °=333 0.009 J J 408 ty>^ 000 y 12798 0.0420 0.0^2^ 0.0301 0.00277 O.3O0 0.877 40 5q8 900 12798 0.0380 0. 0298 0.0^^7 0.00310 0.400 0.867 4^^ 640 900 12798 0.0348 0.0268 o.o-?7^ 0.00344 0.429 0-857 50 0 712 900 12798 0.0322 0.0248 0.0407 0.00^,74 ^•455 0.040 20 299 1000 14220 0.0685 0.0526 0.0158 0.00145 0.230 0.923 J 1000 14220 0.0568 0.0436 0.0193 0.00178 0.273 0. 909 30 427 1000 14220 0.0490 0.0376 0.0228 0.00210 0.310 0.896 35 498 1000 14220 0-0433 O-O333 0.0261 0.00241 0.344 0.885 40 569 1000 14220 0.0390 0.0299 0.0293 0.00270 0-375 0.875 45 640 1000 14220 0-0357 0.0274 0 . 03 24 0.00301 0.403 0 866 50 712 1000 14220 0-0330 0.0253 0.0354 0.00326 0.429 0-857 30 427 1200 17076 0.0519 0.0398 0.0177 0.00164 0.273 0.909 35 49S 1200 17076 0.0457 0.0357 0.0203 o.ooiSS 0.304 0.898 40 569 1200 17076 0.0410 0.0315 0.0228 0.00210 0-333 0.889 45 640 1200 17076 0.0375 0.0288 0-0253 0.00234 0.360 0.880 50 712 1200 17076 0.0345 0.0255 0.0277 0.00255 0-385 0.872 Exact calculations will give stresses slightly smaller than (7e = iooo kg/cm^ (14,220 lbs/in^), so that a somewhat greater factor of safety is secured. If an existing design is to be checked, the equations of page 80 must be employed, or Table XXX used. In the latter case it is only necessary to find the assumed section of reinforcement Fe in terms of the useful area (for instance, Fe = pLbh, or ^iid then the values of x, oi^, and Oe may be found immediately. If the reinforcement is taken at approximately 0.79 per cent of the useful cross-section, the stress of the extreme layer ob will be equal to that in a homo- M geneous section, i.e., oh = A value of* 0.75 per cent also approximates * M is measured in kg.-cm. in one column and in in. -lbs. in the other, but in each case the ■coefi&cients are computed for the same numerical value of M. — Trans. 84 CONCRETE-STEEL CONSTRUCTION the customary amount of reinforcement employed, so that oh may be computed in this simple manner with sufficient accuracy. Table XXX BEAM ELEMENTS FOR VARIOUS PERCENTAGES OF REINFORCEMENT X Ob Oe Oe Per Cent. T M/bh^ M/bh^ 0.0100 0 18 b 009 Tit 116 0.0095 0*410 c 6 cn 5 -"bo o . 90 0 . 0090 0 . 402 J ■ /-I / 128 0.85 0.0085 0.393 5-852 23.1 135 0.80 0.0080 0. ^84 5.968 24.0 0.75 0.0075 0-375 6.096 25.0 152 0.70 0.0070 0.365 6.236 26.1 163 0.65 0.0065 0.355 6.394 27-3 T74 0.60 0 . 0060 0.344 6.572 28.6 188 0-55 0.0055 0.332 6.774 30.2 204 0.50 0.0050 0.320 7.006 32.0 224 0.45 0.0045 0.306 7.278 34.0 247 0.40 0 . 0040 0. 292 7-597 36-4 277 0.35 0.0035 0. 276 7.985 39-4 315 0.30 0.0030 0.258 8.471 43-1 365 0.25 0.0025 0.239 9.096 47-8 435 0. 20 0.0020 0. 217 9-945 54-2 539 The values of h and Fe for various moments are contained in Table XXX. Commencing on page 88 are to be found some examples of computations in full which were given in the " Leitsatze." In the " Zentralblatt der Bauverwaltung " for 1886 is to be found an approxi- mate rule devised by Konen, which is frequently employed in determining the necessary section of reinforcement. It makes the inaccurate assumption that the neutral plane is at the center of the slab, and that the distance between the centroids of the compression and tension areas is J^/, so that the area of steel is given by the formula M re — T-;. Oe \ d The distance is correct in accordance with what is shown on page 83, if id=o.S'jsK or d=lh. This equation will usually hold for slabs of thicknesses, d=6 to 12 cm. (2.4. to 4.7 ins.) so that in such cases, approximate calculations can be made with; |J in place of \h. Concerning tests made with rectangular slabs, see page 90. SIMPLE BENDING 85 Table XXXI BEAM ELEMENTS FOR VARIOUS MOMENTS (76 = 40 kg/cm^ (569 lbs/in^), <7g = iooo kg/cm^ (14220 lbs/in^) M for Meter Width. h d F c Corre- sponding cm. for M in. for M' cm- for M in2 for M' M' per ft. width. cm. -kg in. -lbs. per meter per foot cm. in. per meter per foot in. -lbs. width. width. width. width. 1 0000 8661 3-90 1-54 5-0 1-97 2-93 O-I39 2640 1 1 000 9428 4.09 1.61 5-0 1.97 3-07 0.145 2904 1 2000 10394 4.27 1.68 5-5 2.17 3.20 0.151 3168 13000 1 1 260 4-44 1 .74 5-5 2.17 3-33 0.158 3432 14000 I 21 26 4.62 1. 81 6.0 2.36 3-46 0. 164 3696 15000 12992 4.78 1.87 6.0 2.36 3-58 0. 169 3960 16000 4-94 1-94 6.0 2.36 3-70 0-175 4224 17000 14724 5 -09 2.02 6-5 2.65 3.81 o.iSi 4488 18000 15590 5-24 2.08 6-5 2.65 3-93 0.186 4752 19000 16456 5.38 2.13 6.5 2.65 4-03 0. 191 5016 20000 I732I 5-52 2.17 6.5 2.65 4.14 0. 196 5280 22000 19054 5-72 2.25 7.0 2.76 4-30 0. 202 5808 24000 20786 6.04 2-38 7.0 2.76 4-53 0.215 6336 26000 22518 6.29 2.48 7-5 2-95 4.71 0.223 6864 28000 24251 6.53 2-57 8.0 3.15 4.91 0.233 7392 30000 25984 6-75 2.66 8.0 3.15 5.06 0.240 7920 32000 27716 6.98 2.76 8-5 5-35 5-22 0.247 8448 34000 29448 7. 20 2.84 8-5 3-35 5-39 0.255 8976 36000 31180 7.40 2.91 8-5 3-35 5-54 0. 262 9504 38000 32913 7.61 3.00 9.0 3-54 5-70 0. 270 10032 40000 34645 7.80 3-07 9-0 3-54 5-85 0. 277 10560 42000 36377 8.00 3.15 9.0 3-54 6.00 0.284 11088 44000 38109 8.19 3.23 9.5 3-74 6.13 0. 290 I1616 46000 39832 8.37 3-30 9-5 3-74 6.28 0.297 1 2144 48000 41574 8.56 3-37 10.0 3-94 6.42 0.304 12672 50000 43307 8-74 3-44 10.0 3-94 6-55 0.310 13200 55000 47637 9-15 3.60 10.5 4-13 6.86 0.324 14520 60000 51968 9-56 3-76 11.0 4-33 7.16 0.339 15840 65000 56298 9-94 3-91 II-5 4-54 7-45 0-352 17160 70000 60630 10.32 4.06 12.0 4.72 7-74 0.366 18480 75000 64959 10.68 4.19 12.0 4.72 8.01 0-379 19800 80000 69291 11 .05 4-34 12.5 4.92 8.29 0-392 21 I 20 85000 73620 11.38 4.46 12.5 4.92 8.53 0.403 22440 90000 77952 11 . 70 4.60 13.0 5.12 8-75 0.414 23760 95000 82282 12.04 4-74 13.5 5.72 9-03 0.427 25080 I 00000 86614 4.85 1^.5 5.72 9.27 0.438 26400 105000 90944 12.67 4-97 14.0 5-51 9-50 0.449 27720 I I 0000 94280 12.90 5.07 14.0 5-51 9.68 0 . 459 29040 1 1 5000 98611 13-23 5.21 14-5 5-71 9.92 0.469 30360 1 20000 103940 13-52 5.32 15.0 5-90 10. 14 0.479 31680 125000 108270 13.80 5-43 15-5 6.18 10.35 0.489 33000 130000 1 1 2600 14.05 5-53 15.5 6.18 10.54 0.498 34320 135000 116930 14.33 5-64 16.0 6.30 10.75 0.508 35640 140000 121260 14.60 5-75 16.0 6.30 10.95 0.518 36960 145000 I2559I 14.87 5-85 16.5 6.49 11.15 0.528 38280 86 CONCRETE STEEL CONSTRUCTION Table XXXI — Continued (76 = 40 kg/cm^ (56.9 lbs/in2), (7e = iooo kg/cm^ (14220 lbs/in^) M for Meter Width. h d A Corre- sponding cm. for M in. for M' cm2 for M in 2 for M' Ivl XL* width. cm. =k. in. -lbs. per meter per foot cm. in. per meter per foot in. -lbs. width. width. width. I 50000 129920 15-13 5-96 10.5 0.49 1^-35 O-530 39600 160000 138580 15.60 6. 14 17.0 0 . 09 1 1 . 70 0.554 42240 1 70000 147240 16. 10 6 Id. 18.0 7.09 12.07 0.571 44880 180000 155900 16.60 6. 54 18.5 7.29 12.45 0-589 47520 190000 164560 17.00 6.69 19.0 7.4« 12.75 0.603 50160 200000 173210 17.45 6.87 19-5 7 . oo 13.09 0.619 52800 210000 181870 17.87 7-04 20 . 0 7.»7 13-45 0.030 55440 220000 190540 18.30 7 . 21 20.5 8.07 13-74 0.649 58080 230000 199200 18.71 7. ^7 21.0 8.27 14.06 0.664 60720 240000 207860 19.12 7. =^3 / - JO 21-5 14-35 0. 070 63360 250000 216520 19-50 7.68 22,0 R AA 14-65 0 . 692 66000 260000 225180 19.89 7.83 22.5 0 . ou 14.95 0.707 00040 270000 2^^840 20. 26 7.98 / ■ y 23.0 9-05 15-23 0. 720 71280 280000 242510 20.64 8.1^ 'J 23.0 9-05 15-51 0.733 73920 290000 251170 21 .00 8.27 23-5 9-25 15-70 0. 742 76560 300000 259840 21.36 8.41 24.0 9-45 16.05 0.759 79200 320000 277160 22.06 8.69 24-5 9-65 10 . 50 0.704 Q ^ ,1 R^ 0440O 340000 294480 22. 74 8-95 25.0 9-84 17.08 0.807 89760 360000 311 800 2^ .AO 0 21 26.0 10.24 17.58 0-831 95040 380000 ^201 ^0 2 A. OA Q 47 26.5 10.43 iO . OU 0.053 100320 400000 346450 24.67 9.71 27.0 10.63 18.54 0.070 105600 420000 363770 25.27 9-95 2o.O 1 1 .02 Ib.99 0.095 I 10880 440000 381090 2^ 87 10.19 28.5 11.22 19.44 0.919 I16160 460000 398320 26.45 10.41 29.0 II .42 19.87 0.939 I 21440 480000 415740 27.02 10.64 29-5 II .61 20.30 0.959 I 26720 500000 433070 27-58 10.86 30.0 T T Rt 20 .72 0.979 132000 550000 476370 28.92 11-39 31-5 12. 40 21-73 I .027 145200 600000 519680 ^0.21 11.89 33-0 12.99 22.70 1-073 I 58400 650000 562980 31 -44 12.38 34-0 13-38 23.63 I. 117 I 71600 700000 606300 ■^2.64 12.85 35-0 13-7^ 24.52 I -159 I04000 750000 649590 33-76 13.29 36.5 14-37 25.39 I . 200 198000 800000 692910 34-88 13-73 37-5 14.76 26. 20 1.238 211 200 850000 736200 0 J ' y 3 14.. I 38.5 15.16 27.01 I . 276 224400 900000 779520 37-01 14.57 39-5 15-55 27.79 I-3I3 237600 950000 822820 38.01 14.96 40.5 15-94 2^^.55 1-349 250800 I 000000 866140 39.00 15-35 1"; 53 29.30 ^ -305 I I 00000 942800 40.90 16. 10 43-5 17.12 30.62 1.447 290400 1 200000 1039400 42.72 16.82 45-5 17.91 32.10 I. 517 316800 1300000 1 1 26000 44.46 17.50 47-5 18.70 33-39 1.578 343200 1400000 I 21 2600 46.14 18.17 49.0 19.29 34.65 1.637 369600 1 500000 1299200 47-77 18.78 50.5 19.88 35-86 1.695 396000 1600000 1385800 49-32 19.42 52.0 20.47 37.02 1.749 422400 SIMPLE BENDING 87 RECTANGULAR SECTION, DOUBLE REINFORCEMENT Fig. So. -J? Where reinforcement is placed within the zone of compression but is of such size as to be far subordinate in effect to the concrete, and with the assumption of a constant modulus of elasticity, calculations may be carried out according to the method which follows and which corresponds with process 3, last preceding. With the nomenclature of Fig. 80 there is obtained for simple flexure, from the equality of tensile and compressive stresses in the cross-section, the equation: f_ _t. > A 1 1 1 • Fe Oe = ~Ob X + FeOe'j 2 (I) wherein the small reduction in the area of the concrete by the steel section Fe is ignored. Further, the following relations must hold: Oh ^ Oe Eb ' Ee =x:{h—x) (2) Ob ^ Oe Eb' Ee X h x:{x-h'). Ob h-^j+Fe'Oe\h-h'). (3) (4) These four equations suffice for the determination of the four unknowns Ee <7e, o/, Of^, if the remaining quantities are known. From (2) and (3) with -Er=n there results ob{h—x)n , ^ Oe= > (5) , ob(x — h')n Oe = . X (6) When these values are inserted in (i) there results a quadratic equation from which X may be derived „ Fe-\-Fe 271 , 7 / 77 /\ x^ + 2xn 7 = -rQi Fe+h'Fe). 0 0 (7) The same value may be deduced from the condition that the neutral axis passes through the centroid of the effective section, in which the area of steel has 88 CONCRETE-STEEL CONSTRUCTION been replaced by an equivalent area of concrete n times larger, and the centroid at the same time lies on the lower edge of the compressed concrete zone. From the solution of equation (7) b J ' b With X determined, (t^ is obtained from equation (4) "^^"bx^ish-x) +6 Fe' n{x-h'){h-hy and Oe and are given by equations (5) and (6). If Fe=o in equations (7) and (8) there result the values given on page 80 for single reinforcement. Exactly as for single reinforcement, condensed formulas for quick designing may be developed, but they possess no practical value. Example. — A reinforced concrete slab 100 cm. (39.4 in.) wide is to resist a bending moment of 600,000 cm. -kg. (519,680 in. -lbs.) but a thickness of 30 cm. (7.62 in.) cannot be exceeded. Since, according to Table XXXI, a thickness of 33 cm. (8.38 in.) is necessary, a fiber stress cr^ more than 40 kg/cm^ (569 lbs/in^) will be developed if none but lower reinforcement is provided, and such as will make (Te = iooo kg/cm^ (14,223 lbs/in^). In fact, with ^^ = 28. 5 cm^ (4.42 in^) (7b=46.$ and (Tg = ioio kg/cm^ (661 and 14,365 lbs/in^, respectively). In order ^ f,^fQ(^^i..^ !-= b'1.00 ^, >f — I 5^ rr: FT- ^ • i^e\ Sr d-'jo^ --T-- d^SO \ ; 1 . .^:f U I . ..ill'. . . . U.^ ^.- Fig. 81. Fig. 82. to reduce Ob, upper reinforcement is introduced amounting to ^^ = 9.5 cm^ (1.47 in2), and there is then obtained with /^ = 27 cm. and h' = s cm. (10.63 and 1. 1 8 ins. respectively). (See Fig. 81.) 100 ^ 100^ 100 = 10.8 cm. (4.75 ins.); 6X600,000X10. 100X10.8^(3X27 -10.8) +6X9.5 Xi5(io.8-3)(27-3) = 39-7 kg/cm2 (565 lbs/in2); ^^^^._^(^^ 39-7X16.2 ^3 ^ (,,7,6 1bs/in2) X 10.8 OhiX—h') 39.7X7.8 . , cy „ ,. 9^ ae'=n-^ ^ = i5X ^^ ' ' =431 kg/cm2 (6130 lbs/m2). X 10.0 SIMPLE BENDING 89 If now the upper reinforcement FJ is combined with the lower so that only a singly reinforced slab is secured, with 7^6 = 28.5+9.5=38 cm^ (5.89 in^) (see Fig. 82), there is obtained ^^i5X38r_^^/- 2X100X^1, 100 L ^ ^ 15x38 J 12.74 cm. (5.02 ins.) so that 2ilf 2X600,000 9 / o 1U /• 2\ (76= ^= =4i-S kg/cm-^ (587 Ibs/m^), . L x\ 100X12.75X22.75 ^ ^ ' ' 0 X \ 11 — ' Fe M 600,000 . , , 9 / 00 11. /• 9\ By comparing the two examples, it is seen that the unit compressive stress is almost as low w^hen the tension reinforcement is increased by F/.* From the standpoint of safety alone, the author prefers the proceeding of the last example in many cases instead of employing a compression reinforcement, because the reduction of the steel stress ae means corresponding increase in safety, since experiment shows that the compressive strength of concrete in bending increases with the percentage of tension reinforcement. * As when designed otherwise . — Trans. CHAPTER VII THEORY OF REINFORCED CONCRETE ACTUAL ULTIMATE BENDING TESTS OF REINFORCED CONCRETE SLABS IN THEIR RELATION TO THEORY Tests of reinforced concrete slabs have been copiously discussed in the tech- nical magazines * and have been subjected to thorough theoretical analysis by Ostenfeld, v. Emperger and others, somewhat along the lines already indicated. It has been found that the compressive strength of concrete developed, in tests, increases with extra reinforcement, because a decrease in the ratio n seems to take place during the rupture stage. That is to say, the increase in compres- sive strength was such as might occur should the steel stress exceed the elastic limit. In other words, calculations with w = i5, in cases of small percentages of steel where it is fully stressed, do not give correct results, so that a lower value for n must be adopted, which will produce a correspondingly higher value for Oh. When an effective depth of \d is assumed, it is easy to find the relation between Fe and d which will lead to a minimum cost, but this condition is unattainable with usual costs of materials, because with it the safe compressive strength of the concrete is exceeded. From a commercial point of view, therefore, the safe compressive strength in bending is of great importance. In No. II, 1903, p. 94 of Beton und Eisen, v. Emperger called attention to the fact that the strength in compression of mass concrete derived from direct pressure tests of plain concrete cubes, should not be used to determine the safe compressive strength of reinforced concrete in bending; but rather, the actual computed compressive stresses derived from ultimate bending tests, deduced in the same manner as those used to determine theoretical dimensions. This method possesses the advantage of almost entirely eliminating the effects of arbitrary inaccurate assumptions which enter most methods of calculation. It can also be employed with any other method of computation. Wayss and Freytag conducted some experiments in accordance with v. Emperger's ideas. The concrete was mixed in the proportions of 1:4, the same as the tests already described, and when 13 months old the specimens were * G. A. Wayss, "Das System Monier," 1887; Sanders, "Beton und Eiscn," No. IV, 1902. Ostenfeld, Christophe, "Beton und Eisen," No. V, 1902; Johannsen-Moskau, "Beton und Eisen," No. I, 1904. 90 TESTS OF SLABS 91 tested at the Testing Laboratory of the Royal Technical High School in Stutt- gart. The sections of three slab-like j)ieces, which averaged about lo by 31 cm. (3.9 by 12.2 in.), (Fig. 84), were 2.20 meters (86.8 in.) long and the reinforce- ment consisted of five round bars 10 mm. (f in.) in diameter. The other three had sections of about 10 by 25 cm. (3.9 by 9.8 in.), (Fig. 85), and were rein- forced with 10 round l)ars of 10 mm. (f in.) diameter. As is shown in Fig. 83, a part of the reinforcement was l)cnt diagonally upward near the ends, to prevent a premature failure from shear. In the test, the speci- mens were sup])orted so as to have a clear span of 2 meters (78.7 ins.) and the P XQQQ "t- -3/^ -> <- Z#S-----¥ Fig. 84. Fig. 85. load was applied at two symmetrically located points 0.50 meter (19.7 ins.) apart, in a continuous operation until rupture was produced. Because of the high per- centage of reinforcement employed (from 1.4 to 3.3% of the section), in all speci- mens, the Ijreak occurred at the upper surface, through over-stressing the concrete in compression. This failure occurred in the vicinity of one of the loads and between the two points of their application. The appearance of the fracture is shown in Fig. 86. The stresses were calculated according to method 3, page 80, with n = i^, the weight of the specimen being taken into consideration as well as the measured loads. In the specimen 31 cm. (12.2 in.) wude with 1.4% of reinforcement, at the occurrence of the first crack the average load was P = 57okg. (1254 lbs.) for which (7e = i57o kg/cm2 (22,330 lbs/in2), (76=92.5 kg/cm2 (1315 lbs/in2). In the case of the slab 25.1 cm. (9.9 in.) wide, with 3.3 of reinforcement, the load averaged P = io8o kg. (2376 lbs.) and (7e = i47o kg/cm^ (20,900 lbs/in^), (75 = 158 kg/cm^ (2247 lbs/in^). Fig. 86. 92 CONCRETE-STEEL CONSTRUCTION Stage 116 was really the one involved, since the steel stress was still within the elastic limit. For the breaking load there was obtained in the same manner with P = from 1444 to 2060 kg. (3178 to 4534 lbs.) with 1.4% reinforcement: (7^ = 3800 kg/ cm^, (7^ = 224 kg/ cm^, ^^ = 4.2 cm. (54047 lbs/in^) (3186 lbs/in2) (1.65 in.) with 3.3% reinforcement: (75 = 2750 kg/cm^, (75 = 296 kg/cm^, ^^ = 5.7 cm. (391 13 lbs/in^) (4210) lbs/in^ (2.2 in.) From these experiments is shown the amount of increase in compressive strength of reinforced concrete with increase of reinforcement. As already stated, the cause is to be sought in the fact that with low percentages of reinforcement, or with a steel stress above the elastic limit, calculations with ^ = 15 do not give correct results. According to the ''Leitsatze," one-fifth of the observed ultimate strength may be taken as a safe working stress. On the basis of the foregoing tests, there results with 1.4% reinforcement, ab= --^=45 kg/cm^ (640 lbs/in^), (7e=— ^ = 760 kg/cm2 (10,809 lbs/in^); with 3.3% reinforcement, ab= -^ = 59 kg/cm^ (853 lbs/in^), ...2ZL°.55okg/cmM73..1bs/in2). In the last case, however, it is impossible to fully stress the steel, and the stress decreases, the higher the percentage becomes. It can safely be maintained that the correct values have been selected in the " Leitsatze " with cre = iooo kg/cm^ (14,220 lbs/in^) and 0.75% of reinforce- ment together with (75=40 kg/cm^ (569 lbs/in^). It may be advisable in certain cases, in the compressed lower edges of beams of variable depth, for example, to allow higher stresses.. In such cases, however, the steel stresses must be kept down (by using greater percentages of reinforce- ment) . In addition to this series of tests, another very similar series was conducted, in which the age of the specimens was only two months. At the same time six cubes of the same age, made with the same wet concrete, were prepared. TESTS OF SLABS 93 With w = i5, the calculated stresses at the time of the first tension cracks were as follows: with 1.4% of steel: (7^ = 1310 kg/cm^, ^7^ = 77 kg/cm^, (18,632 lbs/in2) (1095 lbs/iii2) with 3.3% of steel: cr^ = 1195 kg/cm^, (76 = 128 kg/cm^, (16,996 lbs/in2), (1821 lbs/in2) At rupture, with 1.4% of steel: (7^=3150 kg/cm^, (76 = 185 kg/cm^, (44800 lbs/ in^), (2631 lbs/ in2), with 3.3% of steel: (76 = 1970 kg/cm^, (76 = 2ii kg/cm^, (28000 lbs/ in^), (3000 lbs/ in^). Owing to the retention by the cast-iron moulds of too much moisture, the compressive strength of the cubes was only 139 kg/cm^ (1977 lbs/in^). BENDING TESTS OF CONCRETE BEAMS WITH DOUBLE REINFORCEMENT. Tests of concrete beams containing reinforcement against both tension and compression are comparatively rare. The existing material is described and analyzed in Nos. Ill and IV, 1903, of Beton und Eisen by v. Emperger. The conclusion is reached that an increase in the compressive strength can be secured by the introduction of steel into the compression zone only when such reinforce- ment is well anchored by a proper number of stirrups, so as to prevent buckling of the compression rods, which might otherwise cause premature failure. Usually there can be applied to the calculations of doubly reinforced slabs subject to bending, the same formulas as for single reinforcement, since in most cases the tensile strength of the reinforcement will determine the carrying capacity. It is recommended with regard to compression reinforcement of slabs and beams, that the same precautions be employed as in the case of heavily reinforced columns. This should be done at least until by further tests the accuracy of the ordinary methods of calculation has been demonstrated. As was shown in the examples, it is much better to increase the tension reinforcement than to add steel to resist compression. Where it becomes necessary to strengthen the compression zone of reinforced concrete beams because of restricted depth of member, it can be effected with the greatest certainty by the introduction of spirals placed side by side throughout the critical portions. This point will be further discussed in connection with the subject of continuous beams. 94 CONCRETE-STEEL CONSTRUCTION Method of Calculation According to Ritter In the 1899 volume of the Schweizerische Bauzeitung, W. Ritter published several methods of calculation, based on various assumptions, of which the one described in the following paragraphs has found universal recognition in Switzer- land. For the determination of the position of the neutral axis, the concrete is regarded as possessing tensile strength and the section of the reinforcement is replaced by an w-fold greater concrete area, Ritter then supposes the neutral axis to pass through the centroid of the imaginary areas. He computes the moment of inertia of the section and then calculates the compressive stress in the concrete according to the usual formulas. With regard to the necessary section of steel, the assumption is made that the concrete may crack in tension, but that even then the location of the neutral axis is unchanged and it therefore follows that M h — — ]ae 3 With the method of calculation recommended on page 87, the unit stress on the concrete is somewhat lower, especially with deficient reinforcement, and the steel stress correspondingly slightly higher than in the Ritter method, because the arm of the couple between the tensile and compressive forces is slightly smaller.* For ordinary percentages of reinforcement, the Ritter method can be replaced for all practical purposes by the old Konen method, because the neutral axis lies very little below the center of the slab. In case a safe compressive stress for the concrete is assumed, practical and serviceable results are obtainable. As an example, the ultimate stresses in the previously described slabs have been calculated according to Ritter, in order to determine permissible working stresses to be used with his method. For the specimens with 1.4% of reinforce- ment (Fig. 87) and n = 2o (according to the Swiss " Normen ") the distance of the neutral axis below the center of the slab is d 20X3-93X4 o / . s X = ^^-^ = 0.8 cm. (0.31 m.); 2 31X10 + 20X3.93 ^ = i(5-83 + 4-23) +20X3-93X3.22 = 3585.6 cm4 (86.1 in4). The breaking moment is M = iii,825 cm. -kg. (96,856 in. -lbs.), so that the compressive strength of the concrete amounts to <'i.=^^^^^5^ = i8o kg/cm2 (2559 lbs/in^)- * And the position of the neutral axis somewhat altered. — Trans. TESTS OF SLABS 95 It is 224 kg/cm^ (3186 lbs/in-) according to page 92. Therefore, if according to the German Leitsiitze 00= \o kg/cm- (569 lbs/in^) is accepted, then the safe working stress according to the Ritter method on the basis of this test will be 40X180 . . ^ —J^-=32 kg/cm- (455 Ibs/m-). According to the Swiss ''Normen," cr6 = 35 kg/cm^ (498 lbs/in^) is allowed. With lower percentages of reinforcement, the difference between the two methods H * • It . J/^ ^ Fig. 87. is somewhat greater. For instance, with 0.75% of reinforcement, a stress of 40 kg/cm^ (569 lbs/in^) calculated according to the German Leitsatze," would cor- respond with one of only 28.5 kg/cm^ (405 lbs/in^) according to the Swiss. ''Normen." It would seem, therefore, that their allowable working stress of 35 kg/cm^ (498 lbs/in^) for concrete in bending is somewhat too high. According to the method and tables of pages 83 to 86, the neutral axis falls slightly above the center of the slab, whereas with the method of calculation fol- lowed in Switzerland, it falls below the center of the slab. Position of Neutral Axis An excellent explanation concerning the position of the neutral axis was secured through some tests as to the elasticity of reinforced concrete conducted at the Testing Laboratory at Stuttgart. The specimen shown in Fig. 88 was tested in bending, by means of two sym- metrical loads. Thus, a constant moment was secured throughout the space Fig. 88. between the loads where was located the measured length. At each stage of the loading, the shortening of the upper concrete surface was measured, together with the lengthening of the lower layer of steel. Because of the constancy of the moment and the absence of cross stresses within the measured length, the assump- 96 CONCRETE-STEEL CONSTRUCTION tion of the conservation of plane sections during deformation was justified at least as long as no cracks appeared in the tension concrete. Experiments of other testing laboratories with measurements taken at differ- ent heights have not shown this conservation of plane section. The fact remains, however, that, could measurements be made closely adjacent to a concentrated load, it would doubtless be found that changes of length at different heights were not proportional to the distance from the neutral axis. That is, because of changes in shearing stress, neighboring sections formerly plane, become curved.* Meas- urements during stage lib, when isolated cracks were visible, showed no apparent irregularity compared with those of the previous stage. This was probably due Fig. 89. to the great measured length, 80 cm. (31.5 in.), so that the effect of the separate cracks was distributed throughout the whole length. In Figs. 89 to 91, the measured compression of the concrete layer most distant from the neutral axis, and the stretch of the steel are plotted to a convenient scale — the figures employed indicating millionths of the length. The points of corresponding strain are connected by straight lines f corresponding with the idea of the conservation of plane sections, so that the location of the neutral axis for any corresponding strains is given by the point of infersection of the connect- ing line with the vertical representing the cross-section. * Compare v. Bach " Biegeversuche mit Eisenbetonbalken," Berlin, 1907, pages 7 and 8. t The effect of the weight of the specimen on the bending moment has been taken into account. Although but small in itself, it was only after this was done that it was possible to secure a proper agreement with regard to the stress distribution in the section. TESTS OF SLABS 97 The figures are the average of three tests. It will be seen that the neutral axis is lower, the greater is the amount of reinforcement; but that in all three Fig. 9c. varieties of specimens it moved upward with increasing load. Its initial posi- tion, with zero strain, may be determined, if in each position of the neutral axis 98 CONCRETE-STEEL CONSTRUCTION the corresponding moment is plotted upon a perpendicular to the cross-section,, and this moment curve is prolonged to an intersection with the section line. The curve thus obtained therefore furnishes a picture of the relation between the bend- ing moment and the displacement of the neutral axis. It is shown in Figs. 89 to 91, as a dotted line. It will be seen that a Stage I, with a constant modulus of elasticity of the concrete for tensile and compressive stresses does not exist, but that with the least loading an elevation of the neutral axis results. With the light reinforcement of 0.4% (2 rods 10 mm. (f in.)) in diameter, the initial position coincides almost exactly with the center of the slab, whereas with the heavier reinforcement of 1% (2 rods 16 mm. (f in.) ) in diameter, it falls con- siderably below the center. In all three cases it coincides very closely with the calculated position given by the Swiss Requirements, with n = 2o. On the other hand, the highest (measured) position of the neutral axis corresponds closely with that calculated by the German " Leitsatze " with ^^ = 15. From the dotted line showing the moments it can be determined with cer- tainty that with increasing moments, the neutral axis would approach asymp- totically a finite position that would differ but slightly from that obtained by cal- culation, at least as long as Stage 11^, or the elastic limit of the steel it not exceeded. It can therefore be concluded that the observed positions of the neutral axis in sections with stress conditions intermediate between Stages \\a and \\h, coincide with the positions calculated according to the ''Leitsatze." The exact location of the neutral axis in the cross-section where cracks have developed will probably never be certainly demonstrated experimentally. With large measured lengths only an average position is obtained. Later, the calculation of the position of the neutral axis for Stage \\a will be considered on the basis of the observed stress distribution in the cross- section. The tests under discussion afford a very instructive insight into this stress dis- tribution during Stage II. Since, with the arrangement adopted for the experiments, sections must always remain plane within the measured length, from Figs. 89 to 91, the deformation of the concrete at any point can be determined, and, with the help of the stress- strain curve made previously for concrete of the same age and composition, the corresponding stresses may be obtained. Hence, for each section there can be plotted a curve showing horizontally the stress corresponding to each observed deformation across the section considered as axis of ordinates (Figs. 92 to 94) and thus obtain for the pressure zone a stress surface, the area of which is equal to the resultant compressive force D, which must pass through its centroid. M Since the bending moment M is known, the equation y^~^ gives the arm of the couple formed by D and the tensile force Z which, with simple bending, must be equal to the compressive force U. The tensile force Z is composed of two components, viz., the strength Ze of the steel which can be calculated from the measured stretch Eg of the steel and its previously determined modulus of elasticity (2,160,000 kg/ cm2 = 30,600,000 lbs/in^) and a tensile force Zb representing the resultant of all tensile stresses in the concrete below the neutral plane. From the known points of application of TESTS OF SLABS 99 Z and Zg, that of Zh can be located. The value of Z^ must be equal to the area of the tension-stress surface of the concrete, and it should traverse the centroid of that area. In Figs. 92 to 94 the tension-stress curves have been drawn as full lines only as far as the observed stretch of the concrete corresponds with elasticity tests. The further presumptive course of the line is shown dotted. When such a course is chosen for this line that: 1. The surface it bounds is equal to Z^, 2. Its centroid coincides with the computed position of Zt, and 3. The previously observed tensile strength of non-reinforced concrete is not materially exceeded; then it may be concluded that the assumed course of the line of stress coincides with its actual course. As may be gathered from Figs. 92 to 94, this coincidence is very satisfactory in view of the variable composition of the concrete. It also .applies to higher loads where isolated cracks have been noted. Table XXXII gives information concerning the quantities M, D, Z, Ze, and Zb. From the last two columns of figures it may be seen to what extent the calculated Zh corresponds with the assumed value from the tension-stress surface of the concrete. With regard to the high position of Zh in the specimens with heavy reinforce- ment, it may be noted that the cross-section of the reinforcement is to be deducted from the concrete surface. All quantities are based on a width of i cm. Table XXXII D from Z^ from Rein- force- ment. Moment, kg-cm. the Stress Strain Curves, kg. kg. M cm. Z-Z, kg. the Stress Strain Curves, kg. First Crack 1992 96 51.8= 12 20. 7 84 85 2826 134 87.1= 20 21 .0 T13 117 i \ 3659 180 ^33-6-= 30 20. 2 150 148 0 . M (U 1 4492 218 0.105X 2.16X 206.8= 47 20.6 171 165 -ods, iamet 5326 254 389.8= 88 20.9 166 171 * 6159 323 649.5=147 19.2 176 180 6992 388 857-8=195 18. I ^93 200 £ 0 2833 148 57-0= 33 19. I 115 98 4083 213 99.8= 58 19.2 155 140 -0 II 5333 269 157.8= 91 19.8 178 165 6583 339 0. 268X2. 16X < 247.4=143 19.4 196 190 7833 388 365. 2= 212 20. I 176 171 ^ E ^ S 9083 442 479-5=278 20.5 164 180 10333 512 585-0 = 338 20.3 174 181 3673 200 ■ 58.7= 65 18.4 135 100 £^ 5340 273 1 00 . 0 = 1 1 0 19-5 163 137 7007 343 156.0=171 20.4 172 163 Is, 22 eter= 8673 456 0.507X 2.16X , 224.7=245 19.0 211 191 * 10340 527 298.0=327 19.6 200 196 e 1 12007 60s 371.0=407 19.9 196 201 13675 685 442.1 = 485 20.0 200 199 TESTS OF SLABS 101 15 30 cm., with varying pe/ccnlages of remforceinent. 102 CONCRETE-STEEL CONSTRUCTION The less satisfactory coincidence in the case of the first loadings with heavy reinforcement may be explained as due to initial stresses in the concrete, because of shrinkage. The measured tensile strength of i : 4 concrete in the case of the specimens used to measure its elasticity, Fig. 21, was from 8.8 to 10. i kg/cm^ (125 to 143 lbs/in^). A somewhat greater tensile strength in bending in connec- tion with reinforcement is not surprising, for in that case every eccentric strain is excluded, and a single weak section can have but a slight influence on the results of the measurements. A slight error in D, with the uncertain elastic prop- erties of the concrete, is easily possible, and might produce a wide variation in the position and size of Z^. In Figs. 95 to 97, the results of the tests are shown graphically in the following manner: The moments (which were constant throughout the whole measured length) are plotted as abscissas. The maximum compressive stresses ob, computed from the observed shortening of the edge of the concrete and the known stress-strain curves, are shown as ordinates upward. Downward ordinates represent the steel stress (7e, calculated from the measured stretch and the modulus of elasticity £e = 2.i6Xio6 (30,600,000 English equivalent). In this way the curves shown by heavy lines were obtained. The points at which cracks were observed do not correspond above and below, because both curves are the average of three tests each, and because the contractions and extensions could not be measured simul- taneously on any specimen. The figures also show by light lines the computed stresses in the steel and concrete for corresponding moments, calculated by method 3, page 80, with w = 15 (corresponding with the ''Leitsatze "). In the same manner the broken lines show the results of the Ritter method or according to the Swiss ''Normen,'' with w = 20. The diagrams thus obtained are very instructive and exemplify in a striking manner the following deductions: T. First is to be noted from the sharp drop in the tension line for light rein- forcement, the well-known fact that with slab reinforcement below 0.75% (that adopted in the "Leitsatze") the safe working steel stress is determinative, while with larger percentages of reinforcement the stress in the concrete is the limiting factor in design. 2. The theoretical compressive stress in the concrete, computed according to the " Leitsatze," is larger than the observed stress under safe load. With heavy reinforcement, the calculated value corresponds almost exactly with that found by measurement. Computations according to the Swiss " Normen " give stresses smaller than those actually observed. In Stage 11^, after the occurrence of cracks, the Gb obtained according to the Leitsatze " corresponds satisfactorily with the observed value (obtained from the longitudinal measurements). 3. The theoretical steel stresses obtained by calculation are much greater than .'ire actually observed. This holds good, of course, only until the appear- ance of cracks. From that point, the steel stress in the cracked cross-sections will be much higher than in the other parts and will attain the values established by calculation. 4. The curve of tensile stress takes the same course as is shown in the Con- sidere experiment, Fig. 50, page 51. Table XXXII, on page 99, shows in 'figures the same thing in regard to the distribution of tensile stress Z between the forces 104 CONCRETE-STEEL CONSTRUCTION Ze and Zft. While Z and Zg increase with increase of moment, Zh^ except for slight variation, remains practically constant after once attaining its maximum value. As claimed by Considere, therefore, a proportional distribution of tensile stress between steel and concrete must be admitted, but with this difference from Considere's claim, that in the tests here described, thanks to the great care exercised, the tension cracks in the concrete were discovered much earlier. In spite of their existence, however, the distribution of stress remains the same, and the tensile stress Zh suffers no material decrease. How can this phenomenon be explained, if the ductility of concrete assumed by Considere fails us? According to the records of the tests, cracks first appeared at the pins A ; next, within the measured length (the cracks w); and finally the crack m. As the lateral forces within the measured length are nil, there occur during Stages I and Ha within this part no sHding stresses. As soon, however, as Stage \\h is entered, and a crack occurs in a cross-section, the reinforcement is subjected at that point to more severe stresses, and in the adjoining sections the adhesion or rather resist- ance to sliding must assume its full importance in the adjustment of stresses between the concrete and steel. If africtional resistance of 33 kg/cm^ (469 lbs/in^) is assumed, there is obtained for the specimen with 2 rods 16 mm. (| in.) in diam- eter a length of 15Z6 15X180 f ' \ ^ — -=8.1 cm. (3.2 m.), 2X3-14X1.6X33 207 which is necessary to restore in the concrete the stress to which it was originally subjected. Because of friction against the reinforcement, and of the tensile strength which still exists in the pieces lying between cracks, even cracked con- crete decreases to some extent the stretch of the reinforcement.* Through these causes is obtained an almost constant value of Z^, even after the occurrence of cracks, as would be obtained in conjunction with the phenomenon of ductility of concrete, which, however, in reahty does not exist. It cannot be asserted positively that Considere, in his tests, overlooked the cracks, but on the other hand it should be observed that from the specimens of the tests here described, pieces of concrete 20 to 40 cm. (8 to 16 ins.) in length between cracks could have been removed entirely, and they would have displayed their full tensile strength. The cracks were at first visible only beneath the rein- forcement, so that it does not appear impossible that the higher concrete layers might yet resist tensile stress. * By employment of stretch measurements with small units of measure, even the relative displacement of the concrete with regard to the steel can be noted. See Christophe, Beton und Eisen, No. V, 1902, p. 14. On the other hand, the use of too small units is the cause of many diverse results in otherwise scientific experiments. TESTS OF SLABS 105 Safety of the Concrete against Tension Cracks 5. Especially with light reinforcement, the tensile stress taken up by the con- crete relieves the steel to such an extent that its stretch remains considerably below the calculated figures. With more liberal reinforcement, this is not the case, but here the limit of compressive stress in the concrete, warrants no further increase in the size of the reinforcement. Consequently, when designing accord- ing to the ''Leitsatze," i.e., according to the conditions in Stage 11^, in all cases is obtained a factor of safety against cracking in rectangular slabs which amounts to 2.12 with 0.4% of reinforcement; 1.50 with 1.0% of reinforcement; 1.64 with 1.9% of reinforcement. Similar results are afforded by the experiments described on pages 92 and 93, in which the computed unit stresses at the appearance of the first crack, as com- pared with c»e = iooo and ^7^=40 kg/cm^ (14,220 and 569 lbs/in^), give the follow- ing factors of safety against cracking of the concrete: 2.3 with 13 months old specimens with 1.4% of reinforcement; 3.9 with 13 months old specimens with 3.3% of reinforcement; 1.9 with 2 months old specimens with 1.4% of reinforcement; 3.2 with 2 months old specimens with 3.3% of reinforcement. In this connection is to be noted other valuable material by Bach in the Zeitschrift des Vereins Deutscher Ingenieure, 1907. With regard to rectangular sections with such reinforcement as is usually employed in practice, it is shown that, with the approved method of calculation which ignores tension in concrete, a factor of safety is obtained of 1.2 to 1.4 against the first, extremely fine, almost imperceptible tension cracks. The heavily reinforced beams, however, {i and k of the quoted list) showed the first tension crack at a computed steel stress of 765 kg/cm^ (10,881 lbs/in^) for the 1.4% of reinforcement, with a corresponding concrete compressive stress of 45.2 kg/cm^ (643 lbs/in^). In this case the com- puted stress was i.i times the assumed safe one. In these cases the cracks were so fine that they could not be observed with the usual whitened concrete surface. A certain amount of practice was necessary to see them, thereby showing clearly that in the earhest experiments of this kind on similar specimens, much higher stresses actually existed when the cracks were first discovered. It is thus found from these experiments that the customary methods of calcu- lation according to the ^'Leitsatze " or the Prussian ''Regulations," provide an aver- age factor of safety against the appearance of the first tension crack of 1.2 to 1.5. Of course this applies primarily to rectangular sections. The application to T- beams will be considered later. 106 CONCRETE-STEEL CONSTRUCTION The new Prussian "Regulations" of May 24, 1907, in Sec. 15, Par. 3, et seq., require that all buildings which are exposed to the weather, humidity, smoke, gases, and similar harmful influences, besides being designed according to Stage 11^, shall also have the added condition imposed that no cracks shall appear in the concrete because of tensile stresses. The allowable tensile stress on concrete must also be restricted to | of that obtained by tension experiments, or to of the bending strength, if the tensile strength is exceeded by it. The prescribed method of calculation is identical with that of Ritter, already explained — that is, the moduli of elasticity in tension and compression are considered equal and con- stant, and the steel may be replaced by a concrete area n times larger. After computation of the location of the neutral axis, as the centroidal axis of this modified section, the stresses can be determined by the well-known equation vM 7 The value 1 5 is selected for n. There follow some examples of the Stuttgart experiments tested by this new and compHcated method of design.* The distance x of the centroid of the section shown in Fig. 98 from the middle is 15X2.36X13-5 ^0.75 cm. (0.295 in-): 20X30 + 15X2.36 7 = -JX2o(i5. 75^ + 14. 25^) + 15X2.36X12.752 = 51,092 cm^ (1226 in^). \^ 1 1 1 1 1 1 r A 1 \ • 7F ■ - t 1 1 1 1 1 1 _ JO »■ - 1 K >\ Fig. 98. K -ZO- ----A Fig. 99. so that the tensile stress on the concrete at the appearance of the first crack at a moment of ilf = 98,348 kg-cm (85,183 in-lbs) was 14.25X98,348 51,092 27.4 kg/cm^ (390 lbs/in^), As a matter of fact the tensile strength of concrete is only about 13 kg/cm- (185 lbs/in^). The foregoing example is of a beam with only 0.43% of reinforce- ment, while the following is for one with 1.4% (Fig. 99). In it x^ 15X7-81X13 = 2.1 cm. (0.827 20X30 + 15X7. 81 / = iX2o(i7. 13 + 12.93) + 15X7-81X10.92 = 61,558 cm4 (1477 in4). * See also " Postuvanschitz," Beton und Eisen, No. VI, 1907. TESTS OF SLABS 107 The bending moment at the appearance of the first crack was ^1/= 141,010 kg-cm (122.134 in-lbs), so that the computed tensile stress on the concrete was approximately 12.9X141,010 ^ 11 /• 2\ ^2 = =29.5 kg/cm- (420 lbs/m2). According to the Regulations," a safety factor of ij against tensile cracks is intended, but sight has been lost of the fact that in plain concrete beams of rectangular section, because of the variable value of E, the tensile strength in bending is practically twice that found in direct tension tests. It seems natural, and is proved by these experiments, that the introduction of steel on the tension side makes very little change in this condition. It is thus evident that the Prus- sian Ministerial ''Regulations" of 1907 actually provide a three-fold factor of safety against the appearance of the first crack, and in consequence the execution of reinforced concrete work is needlessly costly and difficult. Of som.ewhat more practical value is Labes' '" Vorliiufigen Bestimmungen fur das Entwerfen und die Ausfilhung von Ingenieurbauten im Bezirke der Eisen- bahndirektion Berlin " (No. 52 of the Zentralblatter der Bauverwaltung, 1906). 6i/ . In it the bending strength 0=—— is taken as the tensile strength of the con- bh^ Crete and a factor of safety of 2.5 to 1.3 required. The last value applies to side- walks and light foot-bridges, mangers, water-tanks, and structures subject to slight vibration. For n, a value of 10 is taken, since it produces lower stresses in Stages I and I la (strictly, the steel section should be multiplied by n — i, because of the space displaced by it in the concrete). The value ^^ = 15, which is given in the "Leitsatze" for computations accord- ing to Stage \lh, would not here apply, in view of the results of elasticity experi- ments. It is to be noted, however, that this method of calculation does not consider the existing stress under the maximum allowable load, but rather a con- dition of necessary safety based on stresses developed by much higher loads. It is clear that the value of n should be adapted to this later condition. For slabs, that is, rectangular sections, the factor of safety against tension cracks provided by the above-mentioned discussion is clearly superfluous. The increasd safety is secured through more concrete, which, however, at the same time is favorable to vibration. Furthermore, the distribution of the reinforcement tends to prevent the appearance of the first fine cracks. T=BEAMS In T-beams, subject to positive bending moments, the slab is ahvays made of a certain width, so as to act statically with the stem, with which it forms a T- shaped section. If, however, the bending moment is negative, as will be the case with beams anchored at the ends, or with those passing over a central support, and again ignoring the tensile strength of the concrete, the calculation should be made just as if no slab existed. That is, one should proceed in exctly the 108 CONCRETE-STEEL CONSTRUCTION same manner as indicated above for a rectangular cross-section, but with the difference that the zone of tension is found with its reinforcement in the upper part, and the compression zone in the lower portion of the cross-section, and covering a width equal only to that of the stem (Fig. loo). jy: -yi JiT H i / 1 / T j- 1 Fig. ioo. — Distribution of stress with negative bending moments. On the supposition that the reinforcement in the stem is uniformly distributed with regard to the effective slab breadth b, calculations for positive bending moments can be made as for a corresponding rectangular section, if the neutral Fio. loi. — Distribution of stress with positive bending moments. axis falls within the slab or coincides with its lower edge. In the latter case, with the nomenclature of Fig. loi, D = Z from which In reality, the neutral axis always falls in the vicinity of the lower edge of the slab. Whenever it falls somewhat below that point, as in Fig. 102, the TESTS OF SLABS 109 shaded portion of the stem there shown (in which insignificant compressive stresses act), can simply be ignored. Consequently, the centroid of compression will be only slightly shifted from one condition to the other. If it is considered that the lowest possible position of this centroid can be the mid-point of the slab section, the maximum usual value of Z will be given by the formula It is thus seen that, because of the small possible variation in the location of the centers of tension and compression in T-beams, it is possible to ascertain the tensile stress in the reinforcement with sufficient accuracy for all practical pur- poses without recourse to special theoretical formulas. mmm • o • i ^ Fie. 102. — Distribution of stress with positive bending moments when x>d. The stress in the concrete at the upper edge of the slab does not vary within such small limits as does the arm of the couple of Z and D. However, for cases in which the neutral axis does not fall within the slab, there may be used the maximum value 2Z bd ' ab or one may proceed according to the following more exact method. The neutral axis is supposed to lie within the stem, and at a distance x below the upper layer of the slab, h is the distance of the reinforcement from the same layer and its area is represented by Fe. The small compressive stresses in the shaded area of the stem are simply neglected. Then, on the supposition of a constant modulus of elasticity Eh of the compressed concrete, there is found, as for rectangular sections (Fig. 102), -:^=-:(/,-^), from which with Eb there follows _nab(Ji—x) 110 CONCRETE-STEEL CONSTRUCTION and further _ bx ab(x—d)(x—d) Oet e = Oh 0 . 2X2 Substituting herein the value of a^, gives nob{h—x) bx (7b(x—d)^ ■ r e = (7b ■ J X 2 2X from which 2nhFe+bd^ x = 2{nFe + bd) The distance of the center of compression or of the centroid of its trapezoid from the neutral plane, computed by the equation of moments, is d d^ y=x h 2 6{2X — d) In this equation is clearly to be recognized for x=d the value d d d 2 , y=x \-— =x =—d, 26 33 and for greater values of x d y=x . 2 If the center of compression is known, the compressive stress fi—x+y as well as the stress a^, and OeX Oe (2nhFe + bd^) (76 = n{h—x) n bd{2h—d) can be computed. The position of the neutral axis may also be obtained from the condition that it must pass through the centroid of the modified section consisting of the slab and the w-times increased area of the reinforcement. The value of x may be immediately derived from the moment equation of this area about the upper edge of the slab. From x, the computation of y is easily made, and then the well- vM known equation 0= can be employed. In that case J xM (h-x)M TESTS OF SLABS 111 Example 1. — A reinforced concrete beam 28 by 50 cm. (11 by 19.7 ins.) stem section, with a reinforcement of 5 round rods 28 mm. (ij ins. approx.) in diam- eter, and a slab 10 cm. (3.9 ins.) thick, with an effective width of 250 cm. (98.4 ins.) has a positive bending moment of 1,430,000 kg-cm (1,236,000 in-lbs). 6 = 250 cm. (98.4 ins.), (/ = iocm. (3.9 ins.), /z = 57cm. (22.4 ins.), F, = 3o.8 cm2 (4.77 in2), ^ = 15. The position of the neutral plane is calculated to be Then 2X15X57X30-84-250X102 x= r — — — 7^ — ^ = 1^.1 cm. (5.26 m.). 2(15X30.8 + 250X10) - ^ 10 100 , „ . . y = '3.i-Y + 6(,xi3.i-zo) =9-' (3-58 m.); D = Z= — ^>43o>ooo — ^^i^Q^^ 27,000 kg. (59,000 lbs.); 57-13. 1+9.1 ^, = '-22^ = 878 kg/cm2 (12,488 lbs/in2); 30.8 ob= 878X13-1 ly^g kg/cm2 (249 lbs/in2). 15(57-13-1) If the neutral plane had been assumed to coincide with the lower edge of the slab,, there would have resulted ^_^_i^43o^_^ 57-3-3 (7e = 864 kg/ cm2(i 2,289 lbs/in2); 2X26,600 ^ H /• 9X or^v.^ =^^-3 kg/ cm- (303 lbs/m2). 250 A iO Example 2. — The same beam is to have double the reinforcement and be sub- jected to double the moment. The slab, however, is to be 10 cm. (3.9 ins.) thick, then equals 61.6 cm2 (0.965 in2), and there results 2X15X57X61. 6 + 250X102 . x = , , — ; ' ^ — = 10.0 cm. (7.5 m.); 2X(i5X6i.6 + 25oXio) ^ ^' ^ ^' 100 . , ' s y = i9.o-5 + ^^^^^^ ^_^^^ = i4.6 cm. (5.75 m.); 2,860,000 o n N ^^^^ 57-19.0 + 19.6 ^54-370 kg. (119,800 lbs.); ^. = ^^ = 883 kg/cm2 (12,559 lbs/in2); ^^^_883>09:0 ^ kg/cm2 (418 lbs/in2). 15(57-19.0) 112 CONCRETE-STEEL CONSTRUCTION Example 3. — The same beam as in Example i is supposed to be made of concrete possessing a higher modulus of elasticity, so that w = io. Then there follows 2X10X57X30-8 + 250X102 . x= , — r — ; = 10.7 cm. (4.2 m.); 2(10X30.8 + 250X10) ' ^' ^^^"7-T + 6(2Xio.7-io) =7.2 cm. (2.83 m.); Z=D^-^^^^^^^^^26,.oo\,g. (58,700 lbs.); 57-10.7 + 7.2 (7e = ^^^ = 867 kg/cm2 (12,331 lbs/in^); 30.5 867X10.7 lU /• 2N oh = — ~ ^ = 10.5 kg/cm^ (277 Ibs/m^). 10(57 — 10.7) ^ -3 fc" V ' ' ^ From the three foregoing arithmetical examples, the following conclusions may be derived: When, in a given beam, a doubling of the reinforcement makes possible its carrying double the l)ending moment, the steel stress varies only to an insignificant extent, while the stress on the upper surface of the slab (when the thickness remains unchanged) increases, but to a less extent than the exterior forces. In the examples given, the increase is from 17.5 to 29.4 kg/cm^ in place of 17.5 to 35.0. This retarded increase in edge stress has its origin in the movement of the neutral plane to a greater depth. A similar effect on its position, and in consequence on the concrete stress, is caused by a decrease in the modulus of elasticity Eh (or an increase in n) in such manner that a T-beam of poor material will show a lower stress than one with a richer mixture and correspondingly higher modulus of elasticity Eh under other- wise similar conditions. The same phenomena also occur in rectangular sections, such as simple slabs. The decrease in stress occurs, however, much more slowly with decrease of than does the diminution in the corresponding compressive strength, so that there is no inducement to employ other than a good mixture. Attention is again called to the fact that the simplified formulas for the cal- culation of T-beams are obtained by the somewhat improper neglect of the insignificant compressive stresses in the stem, and by the acceptance of a con- stant modulus of elasticity Eh. As to the width of slab 5, the "Leitsatze" and the ''Regulations" both stipulate that it shall not be greater than //3, that is, each side no greater than //6. At the same time h should evidently not be greater than the beam spacing. Investi- gations concerning the effective width of slab have not been made, but in this connection a natural limit in the calculations is set when the shear in the two vertical sections of the slab equals that of the stem. More will be said with regard to this point in the chapter on shearing stresses. The permissible compressive stress in the concrete may be assumed as large TESTS OF SLABS 113 in T-beams as in those of rectangular sections. This maximum stress can be employed in very few cases, however, since too shallow and excessively reinforced beams would be obtained, which above all are uneconomical, and a cheaper, better construction is produced with deeper beams and with a stress in the top layer less than 40 kg/cm- (570 lbs/in-).* In this connection, some authorities claim it is of considerable practical importance that the permissible concrete stress in the slab be considered that stress which is found by including the effect of a possible tensile stress in the concrete at right angles to the beams due to the continuity of the slab. The allowable stress should not exceed a theoretical value Oz . ) t \ 0.88 1 1 1 1 — -i- • • • • • • • • • e I. According to the exact formula. With F/ = o, Fe=io, 34 mm. (ij^ inch approx.) rods =90.79 cm^ (14.07 in^), bo = 3,^ cm. (15.0 in.), ^ = 160 cm. (63.0 in.), d = 20 cm. (7.9 in.), h = i02 cm. (40.2 in.), the equation box^ + 2x[d(b-bo) -i-nFe\=d^(b-bo)+2nhFb, becomes 38:x;2 ^ 2:x:(2oX 1 22 -f 1 5 X90.79) = 20^ X 1 22 -f 2 X 1 5 X 102 X90.79, 38jc2 + 76o4X = 326,61 7. Thus -7604 + v'7604^ + 4X38X326,617 2X38 = 36.4 cm. (14.33 in.); 7 = ^160X36.4^-122 X 16.43) + 15X90.79X65.62 = 8,253,418 cm4 (83,153 in4), 118 CONCRETE-STEEL CONSTRUCTION so that 8,021,000X65.6 ^ o lU /• 9\ ^^=^5 — Qor. .tR ^956 kg/cm2 (13,598 lbs/m2), 8,021,000X36.4 . , „ . ^u I- 9\ 2. Computation omitting compressive stresses in the stem. Then 2nhFe+bd^ 2X15X102X90.70 + 160X202 ^ . v 2(15X90.79 + 160X20) ^37-5 cm. (14.76 m.), d 400 „ / . V y=x —=^7.5 — 10+-- - = 28.7 cm. (11.^ m.), ^ 2 6{2x-d) ^ 6(75-20) ^ \ 6 so that M 8,021,000 ^ lU /• 9\ <7e=7TT^ r-T = 7 , — ^^ = 946 kg/cm2 (13,455 lbs/m2), Fe{h-x+y) 90.79(102-37.5 + 28.7) ^' ^ ^'^^^ GeX 946X37.5 A 1 / 2 / n /• 2\ C76=— , r = -7 ^^-^-^ = 36.7 kg/cm2 (522 lbs/m2). n(h-x) 15(102-37.5) 3. Computation according to the simple approximate formulas. M 8,021,000 . ^ I 2 / A lU /• = 90.79(102 -10) =96° kg/cm2 (13,654 lbs/m2), Oe (2nhFe+bd^) 960 (2X15XT02X90.79 + 160X400) w bd{2h—d) 15 160X20(204 — 20) =37.2 kg/cm2 (529 lbs/in2). Although in these examples, which are solved for an actual case, the neutral axis falls below the under side of the slab, and bo is small compared with b, the two approximate methods 2 and 3 give differences in the stresses ae and at scarcely worth mentioning. Their practical value is thus shown. The formulas under number 2, included in the " Leitsatze " and the "Regulations," were first given in the original edition of this book in 1902. CHAPTER VIII THEORY OF REINFORCED CONCRETE BENDING WITH AXIAL FORCES If the resultant of the external forces intersects the cross-section, the normal components can be replaced by an axial force N and a moment M. If the mod- ulus of elasticity of the concrete is accepted as a constant for the calculations, and, further, as often happens, a compressive force N is involved, two cases are to be distinguished. Consideration will here be given only to rectangular cross- sections, since for irregular sections the graphical treatment given later is preferable. I. Only compressive stresses are supposed to act over the whole section. By the centroid O of Fig. io6 is understood the centroid of the section produced — 7- T — : i """1 " i ^ a i.^ i > 1 1 1 . ^ I \ i C ~ (5^ \ Fig. 106. M when to the concrete area is added that of the reinforcement multiplied by n = If for I cm. width of section fe represents the area of steel, so that fe^~^ ^^'^ F ' // = —-, the location of the centroid* is given by the formula J2 z The compressive stresses produced by the normal force AT" acting at the cen- troid are distributed uniformly over the entire concrete section, so that A" hd^niFe^F^y * Below the top laver. — Trans. 119 120 CONCRETE-STEEL CONSTRUCTION The moment M, with reference to the centroid of the modified section, pro- duces on the one side compressive stresses and on the other side tensile ones. In this case, however, the tensile stresses, since they represent only a decrease in the uniformly distributed compressive stress, are to be calculated as for a homo- geneous section, in which the area of reinforcement is to be replaced by a concrete one — times larger. It is thus necessary to calculate the moment of inertia / Eh in the formula vM from the expression J = -u^ + -(d- «)3 + n Feih -uY+ nF/ (u - h')"^. 3 3 Bending with axial compression is the usual stress condition in the sections of arches. In them the reinforcement is usually symmetrically arranged, so that the centroid of the whole section coincides with the axis of the arch and the calcu- lation assumes a fairly simple form. The area of the modified section is then and the moment of inertia is F = bd + 2 n Fe, J = —d^ + 2nFe(--cy. 12 \2 If values are assumed for F and /, the same conditions exist in the reinforced section as regards the rib, as for a homogeneous section. 2. The resultant is supposed to have such an eccentricity that tensile stresses exist on one side of the section. If these tensile stresses are in- significant, the calculation may be made exactly as in i. If, however, they are appreciable, a special modulus of elasticity for tension must be introduced into the calcu- lations. Usually, in order to obtain a proper safety factor, the tensile strength of the concrete is disregarded, SLb in simple flexure. In Fig. 107, O represents the centroid of the concrete section to which the moment M is referred, and x is the distance of the neutral axis from the com- pression edge of the section. Then Fig. 107. N=--bx + Fe'Oe'-Feae, (t) 2 M = ohh X (d ---)+Fe'aJe' + Fe(Jee (2) 23/ BENDING WITH AXIAL FORCES 121 Further, because of the conservation of plane sections, d d ^ e-\ X e-\ X Ee 2 2 Oe=-prrsb — - = nab — , (3) J^l) oc oc ^ e' \-x e' , Ee 2 2 ''^=E,''^~^=''''—^— These four equations suffice for the determination of the four unknowns, x, ohy (7e, Oq. If the external forces and given dimensions are used to calculate x the following equation of the third degree results, which can best be solved by trial. Then Nx. "'^^^ ' — d — ^ — rr^ '^+nF/(e'- +xj —nFe{e-\ X As a rule, in arches and columns, the reinforcement is symmetrically arranged, and there are obtained, from equations (i) to (4), with Fe = Fe and e'=e, the following relations: A^ = ^6^ + F.K-.7.), (5) M = o^~i~-''~\-\-eFe{aJ\o^, (6) 2 \2 3 d e-\ X 2 Ge = nOb -, (7) X e \-x 2 Ge^nob , (8) X while the equation for the solution of x takes the form -x^ (n- -—\+2X M n 4-' fi^ {M d + iNe^) = o, 6 \ 4 2 / 0 0 122 CONCRETE-STEEL CONSTRUCTION or This equation may be solved l^y any approximate method, or directly. If, as is known, there is assumed in the general cubic equation + cix^ -\-hx ■\-c^o, the new relation x = z — ^a, there is obtained a reduced cubic of the form z^-\- pz + q = o, from which, according to Cardani's formula, may be derived z=|/-k+v(k)-+(i/')' With the values of equation (9) the reduced cubic becomes ¥1 4nFe M N'^'^ [4 NJ b N 2 M . AfiFt i2nFee^ — y-=°' from which it follows that d M x=z + ----. 2 J\' z here represents the distance of the neutral axis from the point of application of the resultant normal force. When X is ascertained, the stresses may be found by inserting the value of x in equations (8), (7), (6), and (5), and Ob bx , nFe, 1 (2X- d) 2 X d e-\ X 2 Ge = n Ob . X d e [-X Oe—nob -o X BENDING WITH AXIAL FORCES 123 The process is somewhat complex, and is not simplified when the reinforcement on the compression side is left out of consideration, so that F/=o. In practical cases, especially when the amount of reinforcement must first be determined, a l»riefer method may be followed: Compute the edge stress as for a homxOgeneous cross-section without reinforcement, as in the case of a rectangle. Ob X , 6M (7z bd bd^' Then suppose all the tensile stress in the section carried by the reinforcement, the strength of which must therefore be Further (Fig. io8). so that and Z^-ib(7z{d-x). d 6M 2 bd"- d—x = Ozd^b 12M' 24M is obtained. Furthermore, approximately, Fig. 108. Fe' When the edge stress from the rib moment has been obtained, Z can be cal- culated as the area of the tension surface. Example. — Assume a rectangular section in which b = i cm. (0.4 in.) and d^ 90 cm. (35.4 in.) for which ilf = 30,000 cm-kg (25,984 in-lbs), A^ = 66okg. (1452 lbs.), Fe= 0.37 cm- (0.057 'm-)=Fe, e^e' = /[o cm. (15.7 in.), ^ = 15. According to (9) there is obtained \ 2 660 •?o,coo O.S7 + :x X 12 X^-^— X 15 660 ^ I 1^X0.^7/^0,000 ■6X^-^(^^^90 + 2X40- ) -o, 660 or ^3 + i.364.\;2 + 302 7.3Jt;- 242,} 73.65 =c of which the root, found by the method described alcove, is jc = 46.3 cm. (18.65 i^-)- 124 CONCRETE-STEEL CONSTRUCTION From this, according to equation (lo), is found (Jb^^— = 28.2 kg/cm^ (401 lbs/in^), and according to (7), * c7e-i5X28.2 ^'^"^^^~^^-3 ^354 kg/cm2 (5035 lbs/in^), 40.3 cT/ = i5X28.24^^4|+46^_^^g j,g^^^2 (33^6 lbs/in^). The approximate method would have given. 660 30,000X6 lu /• ox (76=- \-- = 2Q.6 kg/cm^ (421 Ibs/m^), 1X90 1X90X90 ^ t,/ / and so that (72=-- 14.9 kg/cm^ (212 lbs/in^), ^ 62^/3^,2 1X903X14.9' 1 / QAIK N Z = — = ^ = 224 kg. (3186 lbs.), 24 . M 24X30,000 ^ & /> 2 24 <7e=^— = about 600 kg/cm2 (8534 lbs/in2). The approximate method thus gives an almost identical result for the com- pressive stress Ob, but one that is too far at variance for the steel stress cr^. For arch ribs, the rib-moments are first to be ascertained by customary methods, and from these moments are then to be computed the axial force N and the moment M with reference to the centroid of any cross-section,* w ~F'^ir Mko N M Ou- w F ir from which, through addition and subtraction of these equations, (7o + (7u Mku—MkOj^ A = P = Pj 2 2 IT Oo — Oujj. Mko + Mku so that the stresses ab and ae can be computed exactly. * W = Section modulus of rib at point in question. i*' = Modifie(i area of section. -o o,/ 0,2 0,5 0,4^ c>,s o,e o,?' o,^^d + 1 Soxjuid^ — gojud^^ corresponding to equation (11), with the nomenclature there employed. The curves (see Fig. 109) are therefore the branches lying on the negative ordinate side of the axis of abscissas, which correspond with those on the positive side, — { 1 1 e' 1 jr. J 1 1 1 \) — V V Fig. III. and complete each a curve of the third degree. The negative branches have vertical asymptotes in common with the corresponding positive branches and for x=Q all curves for different values of /i pass through the point with the ordinate ,3 1 • 7 S under which condition, with ^c==o, the concrete is theoretically useless. Nd 90 ^ In addition, at this point, all the negative branches have a common tangent, the inclination of which is tana-— ^. After X is determined, all the stresses can be ascertained from the following formulas: — 2Nx Oh- hx^^2pbdn{2X—d) ' o.^2d—x X x—o.oSd Ge = 1 5^76- BENDING WITH AXIAL FORCES 129 In flexure with axial tension also, Z and can be calculated approximately by the formulas from which flexure with axial compression is computed. M 31 -7 S For values of smaller than ^ =0.3528, the axis falls outside the section A d 90 and the tensile force is then to be divided according to the law of the lever, if the condition that the concrete is to carry no tension is maintained. Tal)le XXX gives a comparison between results obtained by the exact and the approx- imate methods. Table XXX Kind. h d 71/ M e cm. in. cm. in. kg. -cm. in. -lbs. kg. lbs. Nd cm. in. cm2 in2 Pending with axial com- pression I I I 0.4 0.4 0.4 50 50 19. 19. 19. 7 7 7 7500 I 2500 15000 6496 10826 12991 500 500 300 1102 1 102 661 0.30 0.50 1 .00 21 21 21 8.3 8-3 8.3 0.15 0-15 0.30 0.023 0.023 0.046 Bending with axial tension I I I 0.4 0.4 0.4 16.5 12.0 27.0 6. 4- 10. 5 7 6 1520 713 4000 1316 617 3464 45-6 25.6 106.4 1005 564 2346 2.02 2.32 1 .40 6.92 5.00 11-35 2-7 2.0 4-5 0. 165 0.103 0. 290 0.026 0.016 0.045 Exact. Approximate. Kind. X Oh Oe Gh Oe cm. in. kg/cm- lbs/in 2 kg/cm 2 lbs/in 2 kg/cm2 lbs/in 2 kg/cm 2 lbs/in 2 Bending with axial com pression 0.003 0.003 0.006 35-8 23-3 19.8 14. 1 9-2 7-8 25-9 44-2 39-5 368 629 562 1 10 6^8 782 1565 9217 11122 28.0 40.0 42.0 598 569 597 296 IITO 1041 4210 15787 14806 Bendnig with axial tension 0..0100 0.0086 0.0107 4.45 3.20 6.75 1-75 1 . 26 2.66 22.8 23.1 20. 1 324 329 286 826 850 820 1 1 748 12090 IT663 30-7 27.6 29.0 437 393 413 982 997 923 13967 14180 13138 The last three cases represent practical examples of silos actually erected. The results of the calculations can be readily tested as to their accuracy by introducing numerical values in equations (5) and (6) and noting whether the equations are satisfied. 130 CONCRETE-STEEL CONSTRUCTION GRAPHICAL METHODS OF CALCULATION All the foregoing discussion applied primarily to rectangular or T-shaped sections. It is quite possible, however, that sections of other shapes may occur — circular, annular, etc. For such cases, the formulas to be deduced would be very complex at best. The following graphical methods are therefore recommended. They lead directly to the desired result in a simple manner and for any desired rorm of section. In the two methods given on pages 12 and 13, a treatise by Autenrieth of Stuttgart will be followed. {a) Simple Bending.— li no normal force acts on a reinforced concrete sec- tion, only a bending moment M existing; and on the assumption that sections of steel after deformation remain in the corresponding planes with the compressed concrete sections; and from the equahty of the tensile and compressive forces, — it follows that the neutral axis (which at the same time limits the compression zone of the concrete section) is the centroidal axis of the area consisting of the compressed part of the section and the w-fold increased steel section on the ten- sion side. (See also page 81.) This surface is known as the modified cross-section. It follows, moreover, that for this modified section, the stresses can be calculated according to Navier's bending formula vM since the quantities are identical with those of a homogeneous cross-section, wherein the area of the tensile steel is replaced by an w-fold greater concrete area. The stress on the steel is then V M (7e=n a = n—j-, where / is the moment of inertia of the imaginary area, computed for the neutral axis passing through the centroid. For a symmetrical, otherwise unrestricted cross-section, the axis of sym- metry of which falls in the plane of the forces, two force polygons I and //, with equal polar distances H, are to be drawn, starting from B and A, so as to make a line polygon (Fig. 112). The loads which form the line polygon BD for the zone of compression are to be made up from strips of the compressed area taken perpendicular to the axis of symmetry, and in the polygon AD for the zone of tension, of the w-fold increased area of the reinforcement. When reinforcement is found in the zone of compression, as may often occur, the polygon BD is to include the w-fold increased steel area, beside the strip in which * "Berechunung der Anker, welche zur Befestigung von Flatten an ebenen Flachen dienen." Zietschrift des Vereins Deutscher Ingenieure, 1887. The treatise does not relate directly to reinforced concrete, but the conditions are identical with those here discussed, so that the methods can be appHed, without change, to reinforced concrete. BENDING WITH AXIAL FORCES 131 it acts. Now it is known that the moment of a system of parallel forces with respect to a parallel straight line is equal to the }:)ortion of the straight line inter- i Fig. 112. cepted by the two external sides of a line poylgon, multiplied by the horizontal; H, of such line polygon. Thus, referring to Fig. 113 and applying it to Fig. 112, there is obtained for the line DqD, passing through the point of intersection of the two line polygons, moments of equal size for the right and left areas. In other words, the centroidal axis and the neutral axis both pass through the point of intersection of both line polygons. To determine the stresses according to the formula vM the moment of inertia / of the modified section about the centroidal line is required. Fig. 113. For irregular areas it must be determined graphically. According to the method given by Mohr,* the moment of inertia in this case is / = 2i7Xarea ADB, so that all the quantities required for the computation of the stress are known. * In Fig. 113, the moment of inertia of the forces is equal to the area enclosed by the line polygon, the axis of inertia and the first external side of the line polygon, multiplied by 2iJ, 132 CONCRETE-STEEL CONSTRUCTION Applied to a rectangular cross-section, a straight line is obtained for the line polygon starting from A, and a parabola for that starting from B. The com- putation of their point of intersection leads again to the equation of the second degree, already given. Further, with H = i and b = i, the distance and so that 2 _ (h—x)x^ xM Gh = -—r- = I x^ I , x\ -X- = - ), 2M Consequently, the same value is obtained as with & = i, in the formula previously given. {b) Bending with Axial Compression. First Method. The point of applica- tion of the normal force N is supposed to act at C on the axis of symmetry Fig. 114. — (According to Autenrieth.) (Fig. 114). In a similar manner as above described, the two line polygons AD and BDG are drawn, wherein the latter also includes some steel w^hich would ])e within the zone of compression.* In distinction from the case of pure flexure, the neutral axis is shifted from D{) to Gq. If the distance of any area element of the modified cross-section from the neutral axis through G'o i^ designated ^', the following conditions of equahty are obtained: N = i:aXdF = -K^dFXv. V (Equation for vertical component) * If the force N did not exist. — Trans. BENDING WITH AXIAL FORCES 133 Na^:::dFXov^'^:::dFXv^. (Moment equation about neutral axis) Through a combination of the two there results V V from which J' here designates the moment of inertia of the modified section and AI' its statical moment, with reference to the neutral axis sought. Both quantities can be represented graphically by using the curves previously drawn. Now /' = 2i7Xarea ABDGK, or designating the area by /, Further M' = HxKG = Hz, so that 2/ or z az ^ 2 This equation provides a method of locating the neutral axis. By laying off from the line AB (Fig. 114) the ordinate differences 2 between the curves AD and BG, the curve DqO, starting from Do is obtained, and the two shaded areas will be equal. If DqL is made of such size that the triangle Do^^^^area ABD (ZZ* = triangle DqOC, it follows that — (that is, the area of the triangle GoCO) is almost equal to the area /. It is actually | too large by the area enclosed between the arc and the chord DqO. The po- sition Go of the neutral axis, therefore, still requires a slight correction. If (Fig. 115) such a piece COO' is cut off from ! the triangle GqOC, starting from C, that yig. 115. its area equals that bounded by the arc and chord ODq^ then the neutral axis sought will pass through 0\ because the 2/ * Find z = j~:, by measurement. — Trans. 134 CONCRETE-STEEL CONSTRUCTION area / has lost only the strip GqGq O'O, so that the quantity ^ is equal to the new area /. After the position of the neutral axis has been determined in this manner, the normal stress o at any desired point in the section can finally be found. _ vN _vN_vN or also V N a V N a J' 2Hf ' For the tensile stress in the reinforcement, there is found nv N' Hz The distances v and a, as well as z, are to be determined from the corrected position of the neutral axis. Second Method.^ The following is a simpler method than that of Mohr, for ascertaining the unit stresses in a homogeneous section subjected to bending loads outside the section itself, and in which tension is excluded, such as may be adopted for reinforced concrete columns. If the point of application G of the normal force N lies on the axis of sym- metry of the section, at a distance a from the neutral axis, then, as before, r where J' is the moment of inertia of the modified section, and M' is the statical moment about the neutral axis being sought. In Fig. ii6, the Hne polygon A'B'A" is so drawn for the force polygon on the left, with a polar distance H, that the portion A'B' belongs to the n-ioldi steel section, while the portion B'A" is for the strips of the concrete section. If GK is the true position of the neutral axis, then the statical moment of the effective modified area, that is, of the w-fold increased steel areas and the concrete area lying to the right of the axis, is M'=Hz. Then, in the line polygon of the effective section CA' is the first external side, and the side through G is the last external side. * C. Guidi, ' 'Sul calcolo delle sezioni in beton armato." Cemento, 1906, No. i. BENDING WITH AXIAL FORCES 13.5 The moment of inertia of the effective modified section is /'-2i7Xarea A'B'GK; so that J' 2 X area A'B'GK and Fig. ii6. Now, — is also equal to the area of the triangle C'GK. Hence, necessarily CGK^ A'B'GK. This is the case when the two shaded areas are equal. To locate the point G it is necessary to draw from the point C, located under 136 CONCRETE-STEEL CONSTRUCTION C on the first external side of the line polygon, a straight line C'G, so that the shaded areas equal each other; that is, so that C'LG=^A'B'L. Since the area of the figures A'B'L is known, the point G can be easily and exactly determined, if the difference is computed which a slight displacement makes in the value of the shaded area, derived from first locating the point tentatively according to judgment. When the location of the axis has been fixed, the unit stresses may be com- puted with the aid of the formulas of the first method, vN and vN vN HdFXv M' Hz' V being the distance from the neutral axis to the extreme layer. The second method seems somewhat plainer than the first. If desired, the steel found within the compression side can be ignored, and then in the line polygon B'A'\ simply the steel section can be treated, as shown in Fig. 117. T 1 J A •1 1 \\ 1 1 W m Fig. 117. This second method is very simple when applied to the rectangular section there shown. The line polygon A'B' is simply a straight line, and B'A" is a parabola. It should be noted that the graphical methods under {a) and {h) can aLc be applied to those forms of reinforcement in which the dimensions in a direction BENDING WITH AXIAL FORCES 137 parallel to that in which the forces act are so large that the moment of inertia of the steel section must be included. The section of the reinforcement is then to ])e considered as a concrete area composed of narrow strips ??-times as wide in a direction parallel with the neutral axis, in order to construct the line polygon These larger sections of reinforcement are T, I and j^-bars such as are used, for instance, in the Melan system. METHOD OF COMPUTATION FOR STAGE Ila For the sake of completeness, and in order to gain some insight into the difficulties attending the exact analytical inquiry as to deformations, the following method of computation for rectangular sections for Stage Ila is given: From Figs. 92 to 94 on pages 100 and loi, it is seen that the curves of stress in Stage II for rectangular reinforced-concrete sections, can be closely approximated by two straight lines, one of which passes through the neutral axis of the section and is prolonged into the tension side until the tensile stress reaches a value equal to the tensile strength of the concrete, from which point it becomes parallel with the line representing the cross-section. With the nomenclature of Fig. 118, for simple flexure, D^Ze-\-Zb, or 77 , v 2 Zy- haz XOz a= — ; Ob whence Zb= i d—x — —^^^boz, so that since Oe^nob^— — 2Gb / X hob FenobOi—x) , /, xoz\, — X = ^ -\-[d—x boz. 2 X \ 2(76/ For a given value of the ratio Ob X may be determined as follows, bx (Ji—x) — = Fen '" "' + [d 2 X or 138 concrete-stp:el construction whence Fefl The location of the neutral axis is thus determined. Further, then, M = Feae\ (i ^\ 7 .7 s/2 d—x\ hoz ,o IN . h -\-h(7z{d—x) —x-\ a{^x-\-la), \ 3/ \3 2 / 2 • y =y +?Oo{d^x){~ + '^^ -^^aoxfi{2X + xd), whence 6Mx (7b = nFe{h -x) {6h - 2x) -^xhp(d-x) (3^? +x) -hp^x^{2 +/?) If the formulas are applied to the test specimen described on page 99 with 0.4% of reinforcement, and on the assumption that w = io, and for a breadth of 15 cm., the following values are obtained: With/? = J ^ = 15X3659 cm-kg With/9=^i ^ = 15X5326 cm-kg X = 11.4 cm. Ob ^28.0 kg/cm^ Oz = 9.3 kg/cm2 Oe = 376 kg/cm^ X = 9.02 cm. Ob =49.7 kg/cm2 Oz = 9.97 kg/cm2 Oe = 978 kg/cm^ measured measured X = 12.4 cm. Ob = 28.5 kg/cm^ Oz = 9.5 kg/cm2 Oe = 288 kg/cm2 X = 9.6 cm. Ob = 48.3 kg/cm2 Oz = 9.5 kg/cm2 Oe^ 842 kg/cm2 With the exception of o^ the results agree in a satisfactory manner. It is seen that the computed position of the neutral axis changes with an increase of bending moment, and consequently the decreasing ratio — =/9 affects its position. Ob The limiting value with /? = o corresponds with the computations for Stage lib. In a beam loaded in the customary manner, so that the bending moment increases towards the center, the locus of the neutral axes through the various sections rises toward the middle of the beam. At the instant when cracks appear, it will have reached a culminating point, which will be lower in proportion to the average position of the line, the higher are the stresses. CHAPTER IX THEORY OF REINFORCED CONCRETE EFFECTS OF SHEARING FORCES While in rectangular steel beams the shearing stresses play small part and need be computed only in exceptional cases, in reinforced concrete beams they are of considerable importance and must be considered in the arrangement of the reinforcement. In reinforced T-beams in which only straight rods are employed, when bending takes place (provided the reinforcement is strong enough), the break does not occur near the center of the beam through tensile Fig. 119. — Failure cracks in the vicinity of the points of support of a concrete T-beam reinforced with only straight round rods. stresses, but near the points of support where inclined cracks form, due to the shearing stresses or the diagonal principal ones generated by them. Such cracks are shown in Figs. 119 and 120. In homogeneous beams possessing a constant modulus of elasticity, the diagonal principal stresses, that is, the maximum values of the tensile and com- pressive stresses in any inclined elemental area, are given by the formulas 0 o 2 \ 4 140 CONCRETE-STEEL CONSTRUCTION and their direction by 2r tan 20: = — . a The expression ^ 4 represents the limiting value of the shearing stress. The elemental areas in which the tensile and compressive stresses act, and in which the shear is zero, Fig. I20. — Failure cracks in the vicinity of the points of support of a T-beam reinforced with straight Thatcher bars. make with those in which the maximum shearing stresses act, an angle of 45°. If, from point to point of a beam, the direction of the greatest (or least) principal stresses at those points be followed, two series of mutually perpendicular curves are traced, which are called trajectories of the principal stresses. In Fig. 121 is given a diagram of the trajectories of the principal stresses for a simple, freely supported, homogeneous beam of T-section. All curves cut the neutral axis at 45°, at which point g = o and a^ = TQ. If the tensile strength is less than the shearing strength, as is the case for concrete, then the break will occur in consequence of the tensile stress a^, and the real shearing strength will not be developed. However, it cannot be finally determined that the best form of reinforce- ment is that which will follow the direction of the trajectory of the principal tensile stress. This point becomes evident upon working out this idea, espe- cially with regard to continuous beams with variable loads, in which the dis- tribution of stress in a section is different in a reinforced from a homogeneous EFFECTS OF SHEARING FORCES 141 beam. The principal stresses are also intluenced by vertical pressures between the various separate concrete layers. At all points in a beam where 0^ = 0, as at the supports of simple ones and at the points of zero moment of continuous ones, (1=45°. At these points the reinforcement should be bent at a 45° angle if it is to conform to the conditions, so that it can take up to the best advantage the diagonal tension stresses, which are then equal to r. As the middle is approached a, however, becomes smaller than 45°, so that flatter bends are advisable down to 30°. In adopting Stage I as a basis of computation, the value of the shearing stress r is an approximation for the section with 0^ = 0. In the ^'Leitsatze" and the Regulations," it is required that the horizontal tensile stress a, of the concrete is to be wholly carried by the lower reinforcement, so that in the calculation of the shearing stresses the tension in the concrete is wholly ignored. The diagonal tensile stresses produced by the shearing stresses should be carried by stirrups and bent rods. Since actual structures and test specimens built in this way have proven satisfactory, the simple method of com- Fig. 121. — Stress trajectories in a homogeneous T-beam. puting the shearing stresses according to Stage lib will be adopted, and an explanation given of the various formulas, followed by a careful application to the experimental data at hand, in order to ascertain what factor of safety is provided against a failure in -the shearing strength. Finally, will be considered Considere's theory of the great extensibility of reinforced concrete, under the influence of which the " Leitsatze" was prepared, but which has been found untenable in practice and has received certain modifications. FORMULAS FOR SHEARING AND ADHESIVE STRESSES In the same way in which the tensile strength of concrete is ignored in bend- ing, the formulas for shear and adhesion will be derived on the assumption that the stresses a^, and are equally effective in cracked sections as in all others. Further, only plain reinforcement will be considered. I. Rectangular section with simple plain reinforcement on the tension side. The normal stresses are to be found for Stage lib according to method No. 3 of page 80. Let AB and A'B' be two adjacent sections between which on the plane CC are applied shearing stresses equal in amount to the difference between the normal stresses on and A'C Then TXbXdl= I bXdvXda. J V 142 CONCRETE-STEEL CONSTRUCTION It has already been shown (page 8i) that 2M Gh bih from which dab "dl hxih dM x\ dl 2Q bx h-- wherein Q represents the total of the external forces acting on one side of the section. From the diagram of Fig. 122, do — —dot. X X C ^ Q. so that B B' TbXdl Fig. 122. : r bXdv J V X- 2vQXdl bxmi-- xb 2(3 x^ih /vXdv, V zb Q{x^-v^ ) x^[h-fl Consequently, on the top layer where v=x, the shearing stress is zero and increases toward the neutral axis to Q h--\b The expressions for r b and tq may also be ol^tained, if in the regular formula for homogeneous sections, - QS EFFECTS OF SHEARING FORCES 143 is substituted the modified section consisting of the compressed concrete and the «-times increased steel area. For the computation of zq, the value 5 of the statical moment of the compressed concrete with reference to the zero axis is 2 and the moment of inertia is J^^bx^-\-nFe(h-xf, so that 2 T0 = It follows, however, from the quadratic ecjuation for the determination of that 2nh—x .0= ^ bih-- so that finally as before 3 According to the assumptions made for Stage 116, no normal stress acts in the concrete below the neutral axis, the whole tensile forces being taken by the reinforcement. With this supposition, the shearing stress tq is constant between the line 00' and the reinforcement. In that case it is evident that the shearing stress To is also ec^ual to the difference in the tensile stress between two adjacent sections of the reinforcement. Hence it follows that bToXdl=dZ; 3 dZ dM I Q dl dl I x\ x'' ' 3/ ' 3 so that _Q_ 3 144 CONCRETP]-STEEL CONSTRUCTION This value of bzo also represents the total effective adhesive stress on unit length of the circumference of the steel, and consequendy the adhesive stress ri is bzo total circumference of the reinforcement* Example. A reinforced concrete slab with ^^ = 6.79 cm^ (i.o5in2)=6 rods 12 mm. (J in. approx.) in diameter, has a span of 2.0 m. (6.56 ft.) and carries a load of 820 kg/m^ (168 lbs/ ft^). In it //=9cm. (3.54 in.). The distance of the neutral axis from the upper layer, computed according to formulas already given, is ^ = 3-38 cm. (1.33 in.). Further, for & = ioo cm. (39.4 in.), (2 = 820 kg. (1704 lbs.). Hence iooTo = -^-^^^ = io4 kg/cm2 (1479 lbs/in2), To = i.04 kg/cm^ (14-8 lbs/in^). Since the reinforcement per meter width consists of 6 rods 12 mm. in diameter, the total circumference is ^ = 6X3.14X1.2 = 22.6 cm. (8.9 in.), and the adhesive stress Ti=^=4.6 kg/cm2 (65.4 lbs/in2). In simple slabs the shearing and adhesive stresses are usually so small that their computation seems unnecessary. For the same reasons, stirrups in simple slabs are deemed superfluous. Of more importance are the shearing and adhesive stresses in 2. T-beams. It is evident that the expression in the last section which applies to rectangular sections, is also applicable to T-sections when the distance of the reinforcement from the centroid of compression (Fig. loi, page EFFECT« OF SHEARING FORCES 145 X io8) is substituted for // , and for b the width of the stem /;o is used. For 3 xd (Fig. 102, page 109), is to be used. Approximately, the somewhat too small value h may be 2 used, so that for the distance of the centroid of compression from the reinforce- ment, Q ^0- (which is slightly too large) is the shearing stress in the stem between the reinforcement and the neutral plane. ExafHple. For the freely supported T-beam of Example i, page 11 l, / = 5.5 m. (18.0 ft.), 5 = 3780 kg/m (2535 lbs/running ft.). Thus, (3 = 2.75X3780 = 10,395 kg. (22,869 lbs.), ^0*0 = ^^37^9^ = ^96 kg/cm, and the shearing stress in the stem is To = ^ = 7.o kg/cm2 (99.6 lbs/in2). If all five of the 28 mm. (ij in. approx.) rods were carried to the support, the adhesive stress at that point would be =4.5 kg/ cm^ (64.0 lbs/ in-). 5X3-14X2 The foregoing formulas for the shearing stress are deduced on the assumption that no tensile stresses act on the concrete below the neutral axis. It is to be noted, however, that in both Stage I and lla, where the concrete yet carries some tension, the compressive force D = — , where z is the distance between z Z and D. Furthermore, the horizontal shearing force at the level of the neutral 146 CONCRETE-STEEL CONSTRUCTION axis between two adjacent sections must carry the whole of D, so that, since the distance z is constant between two successive sections, it follows that dD Q hTQ=—r= — . dl z The difference which exists between the actual value of the shearing stress in the neutral layer compared with the assumption made on the basis of Stage 11^, will only be caused by the difference between the calculated lever arm between the centroids of tension and compression, and the actual distance. From the column giving the value of y in the table on page 99, it is plain that these values do not differ much in rectangular sections, and an examination of the stress distribution shown in Figs. 92 to 94 proves that the actual distance is somewhat X smaller than that computed from the expression h . In consequence, in rectangular sections, the value tq along the neutral layer is slightly greater than that given by computations on the basis of Stage 11^. In T-beams, when ignoring the effect of the tensile stresses in the concrete of the stem, the result- ant Z of the tensile stresses falls nearer the steel stress Z^, and the arm of the couple between tension and compression in the section will be somewhat larger according to the computations than in reality. It is then to be expected that in T-beams, because of the influence of the slab, the shearing stresses in the stem will actually be more closely given by the approximate formula ^ Q Later, the relation of shear to reinforcement will be taken up. An exact theoretical method, however, is seen to be very difficult of development, in view of the uncertainty of computations based on Stage l\a, when it is to be con- sidered that the stress distribution shown in Fig. 118 corresponds only approx- imately with fact. The adhesive stresses given by the formulas developed above for Stage l\h are too large when compared with the actual conditions at the appearance of the first tension crack, and for Stages I and Ila. According to the table on page 99, the value of Z^ increases up to the appearance of the first crack. Consequently, the increase of Z^, which is directly proportional to the adhesive stress, is slower with increase of external moment than that of D, on which the computation of ri is based. In T-beams, where the influence of Zj, is less, the agreement will be better between the computed Zh and the actual than in rectangular sections. The shearing force h^TQ, found along the neutral axis, also acts in large part along the planes aa' perpendicular to it, which form the connection between the stem and the slab (Fig. 123). The average value of the shearing stress at those points will be ^_ 5oTo h— hp 2d h EFFECTS OF SHEARING FORCES 147 In the planes aa' there is no lack of reinforcement, since the slab rods are there present in consideraljle numbers. Their shearing resistance, however, does not come into play in taking their share of the transfer of the shearing stresses, but rather, their better tensile qualities are active. If it be imagined w ^ a. a \ d i . • • 1 a' Fig. 123. that the left flange of the T-beam is cut away along the plane aa' , besides the shearing stresses r, others perpendicular to them and due to the bending of the slab, will be brought into action on the section. This bending will be resisted by the combination of one flange with the stem and the slab on its opposite side, so that tensile and compressive stresses normal to the plane are brought into play to counteract the bending which would be produced by the shearing stresses r, and the flange is held in its actual condition, under stress. (Fig. 124.) To these tensile stresses are added the bending tensile ones in the r r 1 1 : cc Fig. 124. — Distribution of shearing and Fig. 125. — Probable courses of the stress trajectories normal stresses along the section aa' . in a floor slab acting as a compression member. slab itself, due to the moment at the support. All the stresses described above produce tension and compression trajectories in the slab, which take somewhat the courses shown in Fig. 125. Thus, the slab reinforcement lies so as to be favorable to the production of a reduction in the principal tensile stresses, which are here less than those of shear, since the accompanying compressive stresses diminish their amount. If the tensile stresses are entirely annulled, the com- 148 CONCRETE-STEEL CONSTRUCTION pressive trajectories will become arch lines, which will be held in balance hy the tensile strength of the slab reinforcement. If the beams are close together the arches may overlap one another. If there act on the sides of an infinitesimal parallelopiped (Fig. 126) the pairs of shearing stresses t, and also the mutually perpendicular normal stresses and Oy, then the values of the principal stresses may be computed by the formulas and their direction by 2T tan 2a = . Ox — Oy Fig. 126. In the case in hand, since and Oy cannot be determined with certainty, an exact theoretical treatment of the question as to the distribution of the stresses in a flat plate is very difficult, and without checking by experiments (which are still rare) would be worthless. Moreover, it is evident that the round- ing or sloping of the joint between slabs and stems of beams is of considerable value in the transfer of the shearing stresses at those points. Although in practice this point is often ignored, and the shearing stresses r along the plane aa' are really excessive in many actual structures (although dangerous consequences have not yet de- veloped), it is invariably best to follow only accepted and safe methods. The ends of reinforcing rods should always be made with a hook so that sole dependence is not placed on friction or adhesion. For this purpose the shape of the hook is of importance. The form commonly employed, of a simple right-angle bend, is not very effective when surrounded only by a thin concrete slab, as is often the case at the ends of beams. In such cases the ends should rather be given a larger bend of as much as 90°. Considere, in the French section of the International Society for Testing Materials, reported a new form of the end hook, which should be immedi- ately adopted in practice. By bending the end into a half circle, through which a short straight piece may be fastened, the principle of rope friction is employed and a greater frictional resistance is produced on the inner side of the bend, since the hook will be pressed hard against the concrete. Some expriments by Considere led to the result that rods with ends bent into semicircles could be stressed to the elastic limit, while the adhesion of plain rods is between 13.4 and 24.3 kg/cm^ (191 and 346 lbs/in^). Fig. 127. — Form of hook, according to Considere. EFFECTS OF SHEARING FORCES 149 When the average unit resistance to sliding developed by these hooks is computed, it is found to be about 77.4 kg/ cm^ (1095 lbs/in^) of contact between the steel and concrete, or about three times that of plain rods. These hooks possess the further merit of not depending to any great extent upon the character of the concrete or the care given the work, since a rope- like friction is secured by the large curve of pressure. This pressure naturally should not be too large, since then a crushing of the concrete results. According to Considcre, the best results are secured by giving the semicircular bend a diameter about five times that of the rod. The Action of Stirrups In the special literature of this subject the opinion is generally advanced that vertical stirrups have the power of reducing the shearing (schub-) stresses in the concrete because of their shearing strength, that they are stressed in shear as well as in tension. In order to compute the distribution of the shearing stress between the concrete and the stirrups, the area of the latter is to be considered Fig. 128. Fig. 129. as mcreased n-fold. The weakness of this idea is proven by the following points: If a piece of length dh be imagined as cut from a stirrup stressed thus in shear (the section of which, for sake of simphcity, is assumed as square) (Fig. 128), then it can be in equilibrium under the action of the shearing stresses r^a^ acting at the ends of the section, only when another couple due to adhesion comes into action. Then rea^Xdh = TiaXdhXa + 2TiX-XdhX- 2 2 must follow, so that r^ = i.5ri. That is, the shear in a stirrup cannot exceed one and a half times the adhesive strength. Similarly, for circular stirrups 150 CONCRETE-STEEL CONSTRUCTION The normal stresses upon the sides of the stirrup sections are infinitely small quantities of the second order, and are not considered. Also, normal stresses within the section itself of a stirrup cannot assist in producing equilibrium, because then the bending stresses in two adjacent sections must l)e opposite in sign (Fig. 129) according to this theory. Round stirrups can thus be stressed in shear to a maximum which is scarcely more than their full adhesive strength, which latter is practically nothing compared with their observed efficiency. An allowable shear for purposes of computation, no larger than the allowable adhesion is worthless. The favorable influence of the stirrups in the following experiments can be explained only through their acting in tension. CHAPTER X THEORY OF REINFORCED CONCRETE EXPERIMENTS CONCERINQ THE ACTION OF SHEARING FORCES The following results were secured by the author near the end of 1906, from experiments on T-beams. The accuracy of the conclusions drawn from them can be checked by means of the now well-known experiments of the Eisenbetonkommission der Jubilaumstiftung der Deutschen Industrie, in the preparation of the outline of the program of which the author assisted as a member. Experiments by the Author. The experiments were not conducted on T-beams designed according to normal methods, but such dimensions were chosen as would cause failure by exceeding either the value ti of the adhesion, or tq, that of the shear in the rib. The two beams were joined by a continuous slab, so • 0,60 i i« X].SO 4. ■ uo > 1' 1 i « 0 j 1 i r : 1 M Fig. 130. — Section of test beams. that the load, which consisted of bars of iron and sacks of sand, could be uni- formly applied and not produce torsional stresses as is possible with a single beam and slab. Roofing felt was applied to the ribs over the supports, so as to reduce friction.* The slab was so strongly built that it would carry with safety the breaking loads. (See Fig. 130.) The small span of 2.70 mm. (8.86 ft.) was adopted, so that the relation between the reactions and the center moments would be out of proportion; in other words, so that failure would take place at the ends before the middle broke. The two beams of each specimen were similarly reinforced. The scheme of loading the twelve specimens involved three groups, in the first of which the load was uniformly distributed; in the second, two symmetrically placed con- * In the majority of cases the friction at the supports was eliminated by supporting one end from a windlass so that it was free to oscillate. 151 152 CONCRETE-STEEL CONSTRUCTION centrated loads were used; and in the third, the beam was broken by a single center load. The specimens were about three months old, the concrete was mixed in the proportions of one part Heidelberg Portland cement to 4J parts Rhine sand and gravel, of such size that there were 72 parts of sand of o to 7 mm. (o to ^ in.) grains, and 28 parts of pebbles of 7 to 20 mm. to J in.) diameter. The sides and bottoms of the beams were whitwashed so as better to reveal the cracks. Without such a white coating the first cracks could not be found till a much later period. The six beams of the first group for uniformly distributed loading, had the same quantity of steel in each beam, although variously distributed. Beam I. Three straight round bars 18 mm. (J in. approx.) in diameter with ends hooked; one-half of the beam without stirrups, and the other half with them. Beam II. The same as I, except that the ribs were twice as broad. Beam III. Three straight Thacher rods without hooks; one-half of the beam without stirrups, the other half with them. Beam IV. The same area of reinforcement as I and II, except that there was one rod of 18 mm. (J in. approx.) diameter, and three round rods of 15 mm. {yq in. approx.) diameter, the latter bent upward at an angle of 45° near the sup- ports, the straight rod hooked at the end, the whole beam without stirrups. Beam V. The same area of reinforcement in the form of two rods 16 mm. (f in.), and two rods 15 mm. in. approx.) in diameter, the latter bent in the form of a truss from the third points to the tops of the beams over the supports. Beam VI. Like IV, except with stirrups throughout the whole length of the beam, the lower straight rod without hooks. The loadings produced the following results in the several beams: Beam I (Fig. 131). Three straight round rods of 18 mm. (f in. approx.) diameter, with hooks at the ends. For a total load of 11.5 t. (12.68 tons) on both beams, the computed stresses according to the ^^Leitsatze" were (7^ = loookg/cm^ (14223 lbs/in^) on the steel, ^7^ = 17.8 kg/cm^ (253 lbs/ in^) com- pression in the concrete, To = 8.4kg/cm2 (119 lbs/in^) shear over the supports, and Ti=6.g kg/cm^ (98 lbs/in^) adhesion. Thus, with this otherwise permissible load, the shear in the ribs was excessive. At a load of 7.0 t. (7.7 tons) the first cracks appeared (fine tension ones near the center), corresponding to a computed stress of (7^ = 668 kg/cm^ (9501 lbs/in^). The computation, according to Stage I, with ^ = 15, gave (7^ = 22.7 kg/cm^ (321 lbs/ in-). With increasing loads the tension cracks became more numerous and larger, and on the end of the beam with stirrups, some followed the stress trajectories. When the load reached 15 t. (16.5 tons) there appeared on the left side, that is, the end without stirrups, a distinct inclined crack, which, starting from the top gradually extended to the steel. At this load l SHEAR AND CONTINUOUS MEMBER EXPERIMENTS 177 ■described on pages 152 to 159. At a load of 18 t. (19.8 tons), 10 = 10.3 kg/cm^ (147 lbs/in^). Fig. 164. — Side view, bottom, and cross-section of beams of T-section with straight reinforcing-rods and stirrups. ^ ^s. - ^ The second group, with 7 mm. in. approx.) stirrups, developed cracks like those illustrated in Fig. 163, the diagonal, almost straight ones at an angle of 45° ap- pearing last. In the beams supplied with Hennebique stirrups (Fig. 164), of the cracks between the load and a support, the lower parts followed the stirrups, while in their upper parts they took an inclined direction toward the load points. At failure, the diagonal v. -20-^ Fig. 165. — Side view, bottom, and cross-section of "beams of T-section with one straight and four bent rods without stirrups. cracks shown in the illustration near the support were present. Other beams of similar dimensions were tested, the princi- pal reinforcement of which consisted of one round rod 32 mm. (ij in.) in diameter, and four bent rods of 18 mm. (f in. approx.) diameter. The slope of the latter was some- what flatter than 45°. Three specimens were without stirrups 7 mm. (3^ in. approx.) closely clasping stirrups, spaced 9 cm ..^j- while six had (3i in.) in 178 CONCRETE-STEEL CONSTRUCTION the outer thirds. In half of the beams the lower rod was absolutely straight, while in the other half a right angle hook was provided. The directions of the ^ Fig. 1 66. — Side view, bottom, and cross-section of beams of T-section with one straight and four bent rods and with stirrups. cracks were like those of the beams with only straight rods and with stirrups, as shown in Figs. 165-167. For extra clearness, the results are collected in Table XXXIV, in which are also included the beams of 2 m. (6.4 ft.) span, without stirrups, with rods bent about 45° (Fig. 168). Concerning the table, it should be added: That in the beams with straight tension reinforcement, the more stirrups were provided, the later was the occurrence of the I Fig. 167. — Side view, bottom, cross-section, and end view of a beam of T-section with a striaght round rod 32 mm. (ij in. approx.) in diameter with hook, from bent round rods 18 mm. (y|- in. approx.) in diameter and with stirrups. first slip and failure. This point is explained by the condition that the tensile strength of the stirrups prevented the downward pressure of the reinforcing rods near the supports after the appearance of diagonal cracks. The more th^ SHEAR AND CONTINUOUS MEMBER EXPERIMENTS 179 I o i hr X X X w pq < OC o. o o « 00 I- in -r 00 o <^ C) M 0 00 'i- 00 0 \0 M lo o 00 O O O O O M O) 04 M 00 in M 4 O CO NO CO 00 On O M LO On OO 1^ -1- NO -t -)- '■'-J On CO -to ro 00 M CN4 ro 04 ro <^ M (N -f 01 CO M M NO m OO to NO O 00 O ^ M O M On 04 CNi r0 -i- O O On On m O ^<-^ w On O i-o 1^ m O ' M NO LT- O On O 0 0 mo O 'O r*^ O lO <-0 O »^ w NO O O ^ CO O - II si O •£■5 q ci M3 180 CONCRETE-STEEL CONSTRUCTION tension in the stirrups was augmented, the harder did they press the main rods upward against the concrete, and the harder did the latter act diagonally downward from above, in consequence of which a considerable frictional resist- ance was developed, so that failure took place only after the first slip had occurred. For similar reasons, in beams supplied with both bent rods and stirrups, a greater frictional resistance was developed than when stirrups were absent. The action of the hooks at the ends of the straight rods resulted in an increase of the load from 41 to 46.5 t. (45.1 to 51.2 tons). In Fig. 167 can be seen the result of the failure of the concrete because of the straightening of the hooks. Fig, 168. — Side view and bottom of a T-beam of 2 m. (6.56 ft.) span with one straight round rod 32 mm, (ij in. approx.) and bent rods 18 mm. in. approx.) in diameter without stirrups. If the tensile stresses are computed for the stirrups of the beams which had only straight rods, on the assumption that at the appearance of the diagonal crack, they must carry the whole shear over a length equal to the distance between the centroids of compression and tension, the values found in Table XXXV are obtained. TABLE XXXV Beam Illustrated in Figure. Stress in Stirrups at the First Slip. at Failure. kg/cm^ lbs/in2 kg/cm2 lbs/in2 163 164 3450 1340 49271 19059 4200 1750 59738 24891 The beam shown in Fig. 167 had its end so constructed that failure took place inside the loads by compression of the concrete at the top. In the remainder of the beams, with bent rods, it may be supposed that as soon as a slip took place in the straight rods, the nearest bent one was stressed more heavily, so that the breaking crack was formed between the two bent bars. The nearest bent bar then acted as the tension member of the beam. In consequence of the SHEAR AND CONTINUOUS MEMBER EXPERIMENTS 181 great stretch of the reinforcement, the zone of compression in the slab would become shallower until the top layer of concrete crushed. (See Fig. i66.) From a comparison of Figs. 162-167, seen that the diagonal cracks produced by any load, extend very high in beams with only straight rods and no stirrups and for which tq as estimated is practically equal to the tensile strength of the concrete. They occur later when stirrups or bent rods, or both combined, are present. The more steel is cut by a crack, the later will it appear. DEDUCTIONS FROM THE EXPERIMENTS So far as the relations between the foregoing experiments go, for simply supported T-beams, the following deductions can be drawn: 1. In reinforced concrete beams, near the supports neither a pure vertical shear exists nor one in a horizontal direction, but rather the action of the shearing forces develop inclined cracks in the vicinity of the points of support. At these cracks, the tensile strength of the concrete will be exceeded by the diagonal principal stress, and it depends upon the manner of loading, breadth of span, and arrangement of reinforcement, whether the failure will take place in the center because of high bending moment, or near the supports indirectly through heavy shearing forces. In general, the diagonal cracks follow the directions of the stress trajectories. By employing stirrups and variously arranged bent rods, the direction of the cracks is not materially altered, but the inclined cracks near the supports occur later, showing that this steel diminishes the diagonal tensile stresses in the concrete. The strength of the concrete in pure shear plays no part in producing security against indirect failure of the concrete from shear, and, moreover, the horizontal and vertical shearing stresses produced during the bending of a reinforced concrete beam are to be considered such that the elemental areas affected by them are not perpendicular to the direction of the main tension and compression. 2. In any beam in which a failure would take place at a support because of lateral forces, when only straight rods are employed, the supporting power will be increased through an arrangement of stirrups and bent rods. Their use seems particularly advisable, since without much increase of material a greater load is assured. It is very important that the straight rods do not shp at the supports, since both series of experiments showed a very favorable action by the hooks at the ends in the increase of ultimate load. In comparison with the usual right angle or blunt hook employed heretofore, the arrangement proposed by Considere, and shown in Fig. 127, is of great value, since it renders unnecessary the com- putation of the adhesive stress. From the tests made by the author, it is shown that the best results follow when the bending of rods is so done that they may carry the diagonal tensile stresses equal to tq which act at an angle of 45°, and also provide the necessary amount of steel along the under side to care for the moments. A reinforced concrete beam of constant depth can then be compared to a single or double intersection truss or one of higher order (Figs. 143 and 144), in which tension 182 CONCRETE-STEEL CONSTRUCTION and compression members slope toward the middle at an angle of 45°. From the outset, it may be concluded that the double system is better than the single, since then the reinforcement is distributed more uniformly through the concrete rib. A somewhat flatter slope of the bent rods appears of no value, but it may be recommended for constructive reasons in large spans on account of conditions. At the upper ends of the inclined parts of bent rods the force carried by the tension member must be resolved into the force at right angles to it in the compression member, and a force in the direction of the top chord. The latter component acts in the upper part of the bent rod itself, and it must have a straight portion ending with an effective hook, capable of transferring its stress to that of the concrete in the too chord.* Similarly the tension in the Fig. 169. inclined part of a bent rod at the lower bend must be resolved in the direction of the compression diagonal acting at that point, and of the lower chord, clearly showing that the simplest and most effective course is to bend upward in a curve of radius r, one of the rods rendered unnecessary by the reduction in the bending moment. However, when this is done the adhesion on the remainder of the reinforcement will be more severely taxed. As is shown by Beams IV and VI of the author's experiments, it is necessary to keep at a low value the compression of the concrete at the bend of the rod. If the pressure acting on a unit area of the projection of the bend be computed, as is customary with rivets, then there is found approximately, if d represents the diameter p dr=S^Oe7Z — , so that ^ GeTt d r = . Ap With (T^ = iooo kg/cm^ (14,223 lbs/in^) and ^ = 6okg/cm2 (853 lbs/in^), ^ = 136? approximately. If the rods are bent cold, it is easier to do so with an even greater radius. It is recommended that the reinforcing rods be bent up, as soon as they become useless because of reduction of the moments near the sup- ports, and be anchored in the zone of compression, as described above; because, according to some earlier experiments of Wayss and Freytag (see second edition of this book), when the lower rods have been provided simply with hooks, cracks occur very early, due to the sudden change of stress. * Of the imaginary truss. — Trans. SHEAR AND CONTINUOUS MEMBER EXPERIMENTS 183 3. Stirrups also increase the carrying power, since their tensile strength resists a failure at the end of a beam. According to the experiments of the Stuttgart testing laboratory, they increase the adhesion of straight rods in this same manner. However, if the principal reinforcing rods are arranged as above described, then the stirrups have only a subordinate statical function, and can be considered a further item of security, which becomes active when other structural elements fail. Stirrups are also useful through the center i)ortions of l)eams when the latter are unsymmetrically loaded and shearing stresses are produced at points which are not usually included in statical computations. Then the stirrups must act •as vertical tension members, as illustrated in Fig. 141, or if a later stage be considered, as reinforcement of the parts of the concrete rib between cracks. Moreover, an experiment of Schiile (Table VII, No. X, Eidgenossischen Materialpriifungsanstalt, Zurich) with a heavily reinforced, uniformly loaded, T-shaped beam (No. 13), the end of w^hich was reinforced against shearing stresses by proper bent rods, showed that with increase of load the characteristic diagonal cracks which finally produced failure appeared in the central portion which was reinforced against shear neither with stirrups or bent rods, and cut directly through the tension cracks which had formed earlier. When they are present, stirrups thus have a value throughout the middle of a beam similar to the one they possess at the ends. Evidently it is not wise to confine the stirrups simply to the ribs. At the same time they assure a connection between the rib and the floor slab in cases where splitting apart of the concrete might occur. Further, it is believed that a beam with stirrups throughout its whole length withstands dynamic action better than one without them. The computation of the stirrups can be made on the assumption that the area of stirrups cut by a section taken at an angle of 45° through the rib, carries all the lateral forces existing in that sec- tion. In the centers of the beams, where bent rods cannot be arranged, the stirrups must be designed for the whole shear, while near the supports the whole of the shearing stresses can be computed as carried by the bent rods, or a part by the stirrups also. When it is intended that a given security against failure be provided in all parts of a reinforced concrete beam, it is not proper to consider the distribu- tion of stresses under a safe working load as measuring this security. Then it is the resistance just before failure which is involved. Consequently, the diagonal reinforcement and the stirrups should not be computed in connection with the diagonal tension in the concrete, since they would already be overloaded at the moment of failure. In the central portions of beams the tensile strength of the concrete in a diagonal direction cannot be computed as active, when cracks have already been produced by the normal tensile stresses. If the tensile strength of the concrete is assumed as 8 kg/cm^ (114 lbs/in^), then precaution against shear at the supports is unnecessary if tq does not exceed 2 kg/cm^ (28 lbs/in^). This condition usually exists in rectangular sections, like slabs, but nevertheless it is usual to bend upward at a flat angle a part of the reinforcement. In slabs, stirrups are unnecessary, since failure occurs in the center under usual conditions, invariably Stage I still being present near the points of support even at failure, and if diagonal cracks should occur at such 184 CONCRETE-STEEL CONSTRUCTION points, the resistance offered by the concrete to the downward pressure of the reinforcement is much greater than is that of the small ribs of T-beams. Stirrups placed normal to the lower reinforcement are considered the most suitable. Inclined at an angle of 45°, they would seem to be able better to carry diagonal tensile stresses. In this position the stirrups would resist ten- sion, but there is difficulty in transferring their stress to the lower reinforcement. They tend to slip along the rods and push off the concrete cover around the rods in the ribs, as was observed in some early experiments made by Wayss & Fig. 170.— Bursting effect of loose diagonal Freytag (Fig. 170). A solid connection stirrups. between diagonal stirrups and the lower rods is troublesome to secure, and hardly practical. An erect position slightly within the angle of friction would be somewhat better than an exactly vertical one. 4. The necessity, in all designs, of considering the adhesive stresses on the lower reinforcement is clearly shown in the foregoing experiments. In all beams, w^here a sHpping of the straight rods was observed, failure did not immediately result, the remaining structural parts (stirrups and bent rods) often increasing in stress until stretched beyond their capacity. Especially will the nearest bent rod assume the function of the lower chord, in which case in a statical sense the beam becomes one of variable depth. The greater is the resistance to slipping, the greater is the carrying power. (Compare the beams of Fig. 147 with 148, and of 166 with 167). Thus it is necessary in all beams which act as if of constant depth, and in which all elements perform their desired function up to failure, that the straight rods should also possess proper security against slipping. The most effica- cious element appears to be a good form of end hook, somewhat like Fig. 127 and as additional security a correspondingly low adhesive stress Ti at the ends of straight rods. Concerning the permissible value of Ti, opinion has been divided. In the " Leitsatze," a value of 7.5 kg/cm^ (107 lbs/in^) is suggested, while the Prussian Regulations allow only 4.5 kg/cm^ (64 lbs/in^). In both cases, the formula, bzQ circumference of reinforcement ' is given, but in the corresponding example in the " Leitsatze," for the circum- ference of the reinforcement only that of the straight rods is considered, while according to the old Prussian regulations, all the steel can be figured. This fact was not considered by those who accuse the " Leitsatze " and the author of poor judgment. The point especially involved in the question concerning on which reinforce- ment the adhesion acts, follows immediately from the experiments, since if only the straight rods are active, a slip would be observed in them, producing a gradual redistribution of internal forces which would lead to failure. Rods SHEAR AND CONTINUOUS MEMBER EXPERIMENTS 185 which are bent at three points cannot slip. Stretching and local shifting of the diagonal rods with respect to the concrete will certainly take j)lace, but when they are anchored within the zone of compression, as shown in Fig. 169, any real slip will be prevented. If the hypothetical value of ri is sought with respect to the circumference of only the straight rods, based on the hypothesis of as perfect a supporting power as a beam having only bent rods — then ti must ec^ual 00. Naturally, this is impossible, and as soon as such a beam is no longer exactly straight, it should be computed as possessing a bent tension chord. According to the experiments on rectangular beams with straight rods, and on the basis of the formula ri=^, an excellent agreement is found with the values of direct adhesion experiments, but it must be noted that this formula is not strictly applicable when bent rods are present. In this case, two ways are open — either the formula ti=^^ is simply assumed and the correspond- ing values ascertained from the experiments (which, obviously, will not agree with those of direct adhesion experiments but will represent simply comparative values), just as was done in regard to the Navier bending formula to find the bending strength of concrete; or it may be assumed that in bending, the same adhesion will be developed as in direct experiments, and endeavor is then to be made to find a suitable formula for ri. Both courses lead to the same result, as far as practical design and security are considered, since that value is used in design which has just been derived by the same method from experiments. (See also page 97). In Table XXXVI, the values found experimentally of ri at the first slip, are again collected. Table XXXVI Beam. bn~a kg/cm^i lbs/in 2 2U kg/cm^ lbs/in^ Ti Considering all Rods kg/cm2 lbs/in2 Rods Variety of Reinforcement Ends Stirrups Wayss and Frey tag „ Stuttgart, Table No. XXXIII, Beam No Stuttgart, Fig. No ^ jlV 3 4 5 161 162 163 164 165 166 167 168 53-6 32.6 22.0 21 . 1 10. 1 19.8 23-3 II . I 12.8 •7 •7 .0 ■ 4 ■7 762 474 313 300 272 282 331 158 182 20Q 408 498 546 522 26.8 16-3 14.4 17-5 19.2 18.4 381 237 205 249 273 262 15.3 9-3 22.0 21 . 1 19. 1 19.8 23-3 II . I 12.8 14.7 8-7 10.6 II. 6 II . I 218 132 303 300 272 282 331 158 182 209 124 151 165 158 Bent Bent Straight Straight Straight Straight Straight Straight Straight Straight Bent Bent Bent Bent Hooked Straight Straight Straight Straight Straight Straight Straight Straight Straight Straight Straight Hooked Straight For the beams with bent rods, the value of ti in the first column agrees fairly with that of the experiments in which the rods were pushed through at 186 CONCRETE-STEEL CONSTRUCTION high speed. Since in this former case the shpping takes place slowly, the computed values are too high. A better agreement is observed in the value computed by Q the formula ri= in the next column, on the assumption of a trussed 2ZU condition of the beam, when compared with the value found from rectangular beams, and also from direct experiments at slow^ speed. T-beams without ibent rods gave somewhat lower results, since the downward pressure of the rods •was less. According to experiments of the Stuttgart Testing Laboratory* the sliding resistance of embedded lengths of lo to 30 cm. (4 to 12 in. approx.) had an average of 15.3 to 25.1 kg/cm^ (218 to 357 lbs/in^). From the third column it is seen how inconsistent it is to consider the circum- ference of the bent rods, in computing zi. Since the beams with bent reinforce- ment developed less adhesion than the beams containing only straight rods, the conclusion could be drawn that a lessening of the adhesion took place with bending, which cannot be the case. It is readily seen that with a permissible adhesive stress of 4.5 kg/cm^ (64 lbs/in^), and taking into account all rods, under these circumstances a factor of safety of only about two is secured against slip of the straight rods. According to the experiments, the permissible adhesive stress for slabs or beams without bent reinforcement, on the basis of a safety factor against slip of four, can be taken at about 5 kg/cm^ (71 lbs/in^), when failure is not to be feared from the shearing stress tq. If that stress is considered too high, a valua of 3.5 kg/cm^ (50 lbs/in^) should suffice. The security against diagonal tensile cracks and their damaging results is not increased by this means, however. If bent rods are employed in such amount that they can carry all the diagonal tensile stresses, then a factor of safety of four can be secured with or 7.5 kg/cm^ (107 lbs/in^) in connection with the formula tj =-^> 3.75 kg/cm^ (53 lbs/in^) in connection with the formula zi=^^~; 2 (J in which U represents the circumference of the straight rods which extend over the supports; and the hooked ends, which should always be used, will increase the security factor so that it is more than five. If the bent rods are not all so arranged as to resist the diagonal tensile stresses tq, then a part of the lateral forces can be carried by stirrups. For this part, ri is to be computed according to the first formula, while for the rest, the second formula is to be used, and a stress chosen between 3.75 and 5 kg/cm^ (53 and 71 lbs/in^), according to the amount of Q taken by the bent rods and the stirrups. * Versuche iiber den Gleitwiderstand einbetonierter Eisen, by Bach, Berlin, 1905, or also No. 22 of the Mitteilungen iiber Forschungsarbeiten auf dem Gebiete des Ingenieurwesens. SHEAR AND CONTINUOUS MEMBER EXPERIMENTS 187 It is recommended in this connection that the whole of the diagonal tension near the points of support be taken by the bent rods, even though stirrups be employed throughout the whole length of the beam, and effective hooks are placed on the ends of the straight rods. 5. With reference to the foregoing recommendation, the method can be followed which is shown in the sketches of T-beams of Figs. 171 and 172. In Fig. 171 a double intersection truss is assumed, in which the bent rods Si, S2, and S3 are designed to carry the whole of the diagonal tension tq. Accord- ing to the method of Fig. 142 and page 160, the forces ^i, S2, and Ss are repre- sented by the shaded areas shown at an angle of 45°, which areas are to be multi- ])lied by the breadth ^0 of the rib. The bending of the rods will reduce the resisting moment of the beam toward the ends, but the line of maximum moments can be used to ascertain whether a sufficient carrying power is present. It is thus easy to compare with the area representing the maximum moments the area showing the moment which the beam can carry at each point, so that the lower reinforce- ment may not be stressed beyond the allowable limit at the points of bend. Where the bent rods are discarded for resisting the moment, the method gives results favorable to the moment diagram and therefore on the safe side. The reentrant angles of the corresponding polygon should lie outside the curve bounding the area of maximum moments. So much straight reinforce- ment must be carried over the supports that zi will be between 7.5 and 3.75 kg/cm^ (107 and 54 lbs/in^). When the bending is done so as to correspond with the diagonal members of a double intersection truss, the forces Si, S2, and Ss can also be determined by a resolution of the lateral forces of the corresponding panels, as is illustrated in the lower diagram of Fig. 171. In a single intersection system S would equal 188 CONCRETE-STEEL CONSTRUCTION If computations are made with partial live load, as it is well to do in all cases, then the lateral forces are not zero at the center, and to care for them the bend- ing of the rods must be started so soon that not enough steel will remain to care for the moments. Although a certain arch-like dimunition of shearing stress takes place, it seems wise to provide some structural elements. As such, stirrups are most easily available, and they can be computed on the assumption that they carry all the lateral forces in a length equal to z. If e is the stirrup spacing, then the force in a single stirrup is z For general reasons already given, a certain stirrup spacing (20 to 30 cm. — 8 to 12 ins.) should not be exceeded, and they should be employed even where the rods are bent. As a further precautionary measure the inner bend may be somewhat flatter, or a greater rounding of its angles be made, as is shown by the dotted line on the right-hand half of the sketch. In Fig. 172 the bent parts are closer together, and a part of the lateral force the area for tq. If the tensile strength of a single stirrup is B, then the breadth of this strip is — . The bent rods cor- eho respond to equal parts of the ro-area, if the rods are all of the same size. The German " Ausschuss flir Eisenbeton" also includes in its program, experi- ments to make clear the action of shearing forces. Upon completion of these tests, opportunity will be given of proving the accuracy of the ideas and methods here set forth. 6. Security against cracking of T-beams. While it can be concluded, from bending tests of rectangular reinforced concrete beams, that sufficient security against the appearance of the first tension crack is provided when the methods of design contained in the " Leitsatze " are followed, as a matter of fact the same has not yet been etablished for beams of T-section. Here, the amount of SHEAR AND CONTINUOUS MEMBER EXPETRIMENTS 189 reinforcement is much greater in relation to the concrete subjected to tension, so that the tensile stresses in the concrete are not great enough to diminish those in the reinforcement so that tension cracks in the concrete will be avoided. Con- scquendy the security against cracking decreases with increase of the percentage of reinforcement in the ribs. According to the experiments cited in Table XXXII, page 172, in Beams I, III, IV, V, VI, with a reinforcement amounting to 2.18% of the area of the rib, the first crack was observed at a stress computed at approx- imately 700 kg/cm^ (9956 lbs/in^ (according to Stage II b). In Beam II, with 1.09% of reinforcement, the computed steel stress rose to 1200 kg/cm^ (17068 lbs/in^). The tension in the concrete, computed according to Stage I, with ^ = 15, is given in the several descriptions of the tests, and varied between 20 and 27 kg/cm^ (284 and 384 lbs/in^). The results of the experiments of the Stuttgart Testing Laboratory concerning the appearance of the first crack, are gathered in Table XXXVII. The amount of reinforcement in the T-sections is given as a percentage of the concrete area between the bottom of the beam and the top of the slab and of a breadth equal to that of the rib. Table XXXVII Beam Reinforce- ment % First Crack Observce between the Limits. Steel Stresses According to Stage II 6 Concrete Tensile Stresses Accord- ing to Stage I with m = 15 Figure No. Stirrups kg/cm 2 lbs/in2 kg/cm^ lbs/in 2 162 None 1.79 773-839 I 0994-1 1947 33-3-36-0 474-512 163 Used 1.79 747-812 10624-11549 52.3-35-^ 459-501 164 Used 1.79 642-728 9131-10354 27-7-31-4 394-447 165 None 1.82 725-808 10312 -11492 32.3-36.0 459-512 166 Used 1.82 680-744 9672 -10582 29.6-32.3 421-459 167 Used 1.82 699-762 9942-10838 30.5-33-3 434-474 168 Used 1.82 753-785 I1477-11165 33-2-34-6 472-492 The directly observed tensile strength of the concrete was about 13 kg/cm^ (185 lbs/in2). Similar stresses in the reinforcement, between 600 and 700 kg/cm^ (8534 and 9956 lbs/in^) w^ere observed by Schiile, and reported in the Mitteilungen der Eidgenossischen Materialpriifungsantalt, Zurich, No. X. The first tension cracks in the concrete, which are so fine that they cannot be discovered on rough, unpainted surfaces of beams, need give no anxiety unless the tensile strength of the concrete has been included in making calcula- tions. This should rarely be done, however. When, with the usual arrangement, the practically invisible cracks are exactly crossed by a needful amount of reinforce- ment, there is nothing to fear, because this condition is found in the greater number of well constructed reinforced concrete structures, many of which are subjected to very severe conditions. No danger of rust need be considered, since the covering which affords protection against it does not consist of the porous concrete but rather of the cement film immediately covering the steel. Further- more, the first cracks along the edges do not usually extend entirely to the reinforc- ing rods. 190 CONCRETE-STEEL CONSTRUCTION Nevertheless, if it is desired to design T-beams which will be wholly free from cracks, it is necessary to use broader concrete ribs or less reinforcement. In consequence, however, the design of many buildings with T-beams will be uneconomical, and it will be better to employ some form of arch construction. In usual building work, absolute freedom from rusting is generally unimportant, so that excessive care need not be exercised. In the use of reinforced concrete, it often happens that the depth of a beam is increased where the moment is greatest. Figs. 173-176 show the usual arrange- ments for positive and negative bending moments. The direction of the section to be made for purposes of computation cannot be assumed at random, since, in the neighborhood of the outside layers, the compressive stresses act parallel with them, and the steel stresses naturally act in the direction of the rods. In order to simplify the derivation of formulas for To, however, the section will be assumed as vertical, since otherwise it would involve lateral forces, and should also rigorously take acount of bending and of axial pressure. For all the cases shown in Figs. 173-176, when only vertical loads are assumed so that the increment of Z in the adjacent section is SHEARING STRESSES IN BEAMS OF VARIABLE DEPTH dZ zdM—Mdz or dZ, dl ldM_M dz z dl z^ dl' Further, in all cases, and h ToXdl = U TiXdl=dZ. In Figs. 175 and 176, _dl___dZ_ cos a cos a The arm 2 ot the couple between tension and compression may be assumed as equal to J h for all practical purposes, so that, with SHEARING STRESSES 191 there is obtained Q 7 b To = U Ti= — tan a. Z 5 2" This is a less value than if the height were constant, since the second quantity is subtracted from — . If the formula is written z bTo = Q-|- — tana Q-^-Z tan a Q-^Dinna 8 2 8 8 it is evident that in comparison with a beam of constant depth, instead of the total shear, a value found by diminishing it by | D tan a is to be used in com- puting To. If, in Figs. 173-174, the resultant pressure is assumed as inclined in the direc- tion of a line connecting the centroids of compression in adjacent sections, then z A Fig. 173. — Positive Bending Moment. Fig. 174. — Negative Bending Moment. ' X i ^ D Z Fig. 175. — Positive Bending Moment. Fig. 176. — Negative Bending Moment. since D must be its horizontal component, the subtractive quantity J D tan o: can be none other than its vertical component. Since the compressive stresses act on layers parallel to this direction, this deduction is entirely plausible. In Figs. 175-176, the resultant pressure may be assumed as acting in the direction of the line connecting its points of application, and the quantity represents the difference between the vertical components of the upwardly directed pressure and the downwardly inclined tension. In the equation for h tq is seen the beneficial influence of arching the inter- 192 CONCRETE-STEEL CONSTRUCTION mediate sections of a continuous reinforced concrete beam. If the moment in- creases with decrease of //, then the minus sign in the formula is to be changed to plus. DEFORMATION Concerning the experiments already described on page 105, it was said that the concrete, because of its tensile strength, relieved the stress in the reinforcement to some extent, and that in Stage 11^ this condition also existed to an even greater extent. Because of this action, the deflections of reinforced concrete struc- tures are generally very small. It is to be further noted that because of the fixed connection between all parts of a reinforced concrete structure, more structural parts contribute to the support of the loads than are usually considered in making computations. In researches concerning structural bridges, deformations are given a con- siderable importance at the present time, but without proper foundation it would seem, since the total deformation is the result of a large number of very small elastic deformations of the various parts and sections. Thus, it is impossible to trace in the computed total deformation the effect of one or more defects in a member, such as a poor rivet, etc., comprising, as it would, such a small part of the whole; but a fairly exact determination can and should be made of the whole structure, on the basis of known causes. The amount of deformation of any reinforced concrete construction is of even less value as a measure of its quality, since adequate determinations of the influence of shearing and adhesive stresses are wanting, and since the distribu- tion of load within the construction cannot be followed with mathematical exactness. If one cannot leave matters to the experience of the company execut- ing the work, nothing remains but to become familiar with the details of construc- tion and design in order to be sure of the necessary care during construction. When exact experimental knowledge as to the safety of the structure is to be brought into the question, the size of the actual stresses produced in steel and concrete should be known, and they can best be determined during an experiment by means of a suitable measuring device (such as a Rabut-Manet). In this in- stance the deformations are not determined indirectly. In Fig. 177 are given the deflection diagrams of the tests described on page 95 (Fig. 88). The constant bending moments at the centers are plotted as abscissas, and the resulting deformations as ordinates. The circles on each curve show the permissible loading according to the " Leitsiitze." The condition of the curves up to these points is practically rectilinear, but at the appearance of the first crack an upward turn takes place. That is, a sudden increase in the deflection occurs, so that the conclusion may be drawn that in reinforced concrete slabs, cracks actually exist at those points. For T-beams, the courses of the curves are similar. According to these diagrams, the action of a flexed reinforced concrete beam is such as to give a deflection diagram consisting of two straight lines with a transition curve between them. The first part starting from the origin cor- responds with stresses in Stage I, in which the concrete yet exerts tension. The DEFLECTION 193 broken line connecting the two parts corresponds with Stage Ua, in which the tensile strength of the concrete has reached its ultimate point, and finally the commencement of Stage 11^, where at several points the tensile strength has been exceeded and fine cracks appear. The further practically recti- linear character of the curve corresponds with Stage lib, with increasing cracks, during which the reinforcement is prevented from stretching indefinitely only by the practically constant condition of the resistance offered to its sliding, by its concrete covering. Consequently, here, the deflection is not proportional to the load. Fig. 177. — Deflection diagrams of beams of pp, 100, loi. A mathematical determination of the deflection must thus fit these several conditions. For Stage I, computations can be carried out according to the usual theories of flexure, but the expression for the moment of inertia must have the sections of reinforcement replaced by w-fold larger ones of concrete. The kind of deflection in Stage 11^ makes its computation impossible, since its determina- tion rests on the average stretch in the steel, the spacing of the tension cracks and the frictional resistance between the steel and concrete. Also, a method of computation for Stage lla, during which the stress condition (up to the appear- ance of the first crack) is quite accurately known, is equally shown to be out of the question, since the special formulas (p. 138) for :v and rr;, show that it is not possible to find a serviceable expression for the angle between the adjacent sections, when it is recognized that /? must first be approximately determined through deductions from experiments. 194 CONCRETP]-STEEL CONSTRUCTION COMPUTATION OF FORCES AND MOMENTS In the foregoing pages it has been shown how to compute the stresses in a section, produced by known moments and normal forces, and how structural parts should be designed. In the following paragraphs will be discussed the general case of the computation of lateral forces and moments. While it is usually sufficient to know simply the maximum moment which may occur at any point within the span of a steel beam of constant area (such as any rolled section), and in some cases also of plate girders; for an economical design of reinforced concrete beams, the inaximum moments of both kinds must be known for a large number of sections. Above all, it is necessary to know in which direction the moment acts, since the location of the reinforcement in the beam depends upon it. As has been shown, the lateral internal forces, such as the shearing stresses, play a considerable part in the design of the sections of reinforced concrete beams. A further condition which must be included in designing is that the permissible stresses of each of two different materials must not be exceeded. On the other hand, a much simpler form of cross-section is to be dealt with. For all statically determined reinforced concrete construction, the bending moments are to be computed from the exterior forces according to the rules of statics. The question arises, however, whether for statically indeterminate reinforced concrete construction, such as restrained and continuous beams, arches without hinges, etc., the stresses are to be determined in the same manner as for homogeneous materials. It was shown by Spitzer in the Zeitschrift des Osterreichischen Ingenieur- und Architektenvereins, in connection with the Purkersdorfer test of a Monier arch, that his method of calculation may be followed, as may also the elastic theory as applied to homogeneous materials, when the area of reinforcement is replaced by an w-fold greater area of concrete in all expressions for areas F and moments of inertia /. It is to be noted that in bending with axial pressure, as in an arch, the tensile strength of most sections is not involved, since the action of the moment and of the pressure are to be added in the same manner as for homo- geneous sections. A restrained or continuous T-beam in which no axial force acts will not be considered. So long as the angle of inclination between two adjacent sections is proportional to the bending moment, the methods of computation for restrained or continuous beams are to be followed. According to the deflection diagram, this proportionality between moment and deformation in rectangular sections continues up to the maximum permissible loading. But even when the propor- tionality ceases with higher loading, the stress distribution is not greatly altered, as the following simple cases show: As an extreme case it will be assumed that the angle of inclination between the adjacent sections is proportional to the third power of the moments (so that the deflection diagram of Fig. 177 would be a cubical parabola), and thus d(j)=^CMHx, wherein C represents some constant. THEORIES OF MOMENTS 195 For a beam fixed at both ends, and carrying a concentrated load P at the center, 2 It must happen that The solution gives the well-known moment 8 • ■M, P. / - Fig. 178. An unsymmetrical case will be selected of a beam in which one end is fixed and the other freely supported, Fig. 179, Mx^A x-^x^, A ^—j ) dx. Since the end of the beam at A cannot move vertically, 0= )xd(l), or 0=j'^^x(^Ax- x^ I dXf The equation derived from the integration of this expression is satisfied with A = ^gl, the known proper value on the basis of a frictionless support. 0 It may be concluded from these two cases that a relation differing from true proportionality between moment and deformation makes no appreciable change in the section, from that found by pure elasticity formulas. The propriety of computing continuous reinforced concrete beams as such, will be determined from the experiments hereafter described, made by the firm of Wayss & Freytag, 196 CONCRETE-STEEL CONSTRUCTION upon continuous reinforced beams. The assumptions on which the success of the test rested were, that the reinforcement should conform to the conditions of restraint and continuity and not be arranged simply according to some " System. " It is often forgotten that the elastic conditions in homogeneous materials hold only while strict propordonality lasts, and that thus with regard to the distribu- tion of stress at rupture, the same uncertainty exists as with reinforced concrete. Reinforced concrete beams and slabs can be computed by the formulas for continuous beams of constant section with as much reason as continuous structural Ends freely supported. Ends restrained. Fig. i8o. — Maximum moment lines for continuous beams of three spans. beams are designed on the assumption of a constant moment of inertia, when the area changes; that is, when the maximum moment is employed. Computations will actually be too favorable, if, for sake of simplicity, the slabs over the ribs and the ribs themselves over the intermediate columns, are considered as freely supported (although really continuous members), and the restraint of the beams at the columns is ignored However, the saving which exact computa- tions would make, could be only insignificant. The restraint of the end panels THEORIES OF MOMENTS 197 of slabs at the outside walls can only in rare cases be secured through structural measures, and is at best very uncertain, even though the same method is maintained at the ends and some reinforcement bent upward in the vicinity of the supports. i i — 1 I I i 1 1 Fig. i8i. The restraint afforded at the ends of T-beams which rest in walls is even less. If the beams are supported by wall columns, a certain amount of restraining action is produced, which may be included in computations under certain circumstances. The wall columns should always be built to resist bending. The girders shown in Fig. i8i support a reinforced concrete floor and are con- tinous beams of two spans freely supported at the ends. Those of Fig. 182 are to be designed as continuous beams of three spans with partial restraint of the 198 CONCRETE-STEEL CONSTRUCTION ends by the wall columns which are built in the outside walls, and support the ends of the beams. The computations for a uniformly distributed load in the simplest form, from the ordinates of the maximum moment line for continuous beams is given by Winkler, in Vol. I of " Vortrage iiber Briickenbau. " His tables are reproduced in the appendix, and are of considerable value in connection with the computa- tion of continuous beams. The maximum moment line for three spans is given in Fig. 1 80. Of considerable value, also, are the interpolation tables for the ready determination of the influence lines for moments and shears in continuous beams, prepared by Gustav Griot, Zurich. A further discussion of the computa- tion of such moments will not be given, since the various methods are to be found in text-books of the statics of building construction. For many reasons, the " remnant " stresses in reinforced concrete construc- tion are of considreable importance. It so happens that the concrete, especially on the tension side, undergoes, by the first loading, a certain permanent defor- mation in addition to the elastic one, which causes certain permanent stresses in both materials, because of the connection between the steel and the concrete. These permanent stresses can but slightly influence the ultimate load through repetition, since with repeated loading, the concrete correspondingly alters its coefficient of elasticity, which corresponds rather to the final deformation than the one produced by the first application of load. Thus, when it is stated that reinforced concrete beams which have been subjected to bending have a residual compression in the lower concrete, due to the reaction of the steel against the permanent stretch produced in the concrete by the first loading, it must also be considered that with repetition of load this compression in the concrete invaria- ably must first be overcome before tensile stresses appear therein; that, further, the concrete is not so readily extensible after the first time; and that the tensile stresses quickly increase, so that the final value is again approximately equal to the first one. Aside from this fact, the residual concrete stresses would have some importance for the designer if they exerted an influence upon the ultimate stresses in the loaded condition of a beam, since he would then have to employ in his designs a very much lower resistance to load (Stage 11^ with cracked concrete), so as to provide a proper factor of safety. However, these " remnant " concrete strains and stresses have little influence upon this condition. The careful determination of the actual stresses in the steel and concrete under permissible loads shows them to correspond with those now customary in steel structural work, but this is no reason why they should be advocated for reinforced concrete. In an earlier chapter it was shown in several cases that many of the methods adopted by custom in reinforced concrete design, give wholly erroneous results; also that regular steel construction was often superior when one could not employ "safe stresses." (In this connection may be cited, for instance, the experiments of Schiile of Zurich on I-beams, in which the top flange failed by lateral cracking.) Only because designers of reinforced concrete have kept in sight from the outset the safety of their structures, keeping well within the usual stresses under permissible loads wherever possible, has reinforced concrete reached its present development. The shrinkage of concrete produces secondary stresses in reinforced concrete 1 Fig. 1S3. — Continuous Test Beam I with haunches. Breaking load 34.4 t. (37. S tons). ' t i f } I I, * .1 t s ~T~~^ Llll 1 Fig. 184— Continuous Test Beam II without haunches but with spirals and shear rods. Breaking load 31.9 t. (35.1 tons.) % -A i " — ■ S.SS Fig. 185. Continuous Test Beam III, without haunches but with spirals; reinforcement same as Beam II except for shear rods and the substitution of a 20 mm. (} in. approx.) rod for one 14 mm. (} in. appros.) in diameter. Breaking load 25.4 t. (28.0 tons). To face page 100 CONTINUOUS T-BEAM EXPERIMENTS 199 structures. The amount of this shrinkage bears a definite relation to the pro- portions of the mixture. In long structures allowance must be made for expansion and contraction from temperature change, by means of expansion joints of suitable size (20 to 40 mm.) (J to in. approx.). In high buildings, the joints can be carried either through the centers of beams and columns or through the panels. The latter arrangement leads to a cantilever construction of the floor and beams. The expansion joint can be constructed without any open space, since it tends only to open because of the shrinkage of the concrete. Buildings provided with such joints actually remain free from cracks, while otherwise the danger of cracks occurring at undesirable points always exists. EXPERIMENTS WITH CONTINUOUS T=BEAMS Since experiments with properly constructed continuous T-beams were unknown, and because of the importance of the subject, the firm of Wayss and Freytag carried out tests of three beams of T-section in accordance with plans prepared by the author. The specimens are illustrated in Figs. 183-185. They consisted of continuous beams of two spans each 5.9 m. (19.4 ft.) wide, the ribs of the beams being 14 cm. (5.5 in.) broad and 25 cm. (9.8 in.) high, above which was a slab i m. (39.4 in.) wide and 10 cm. (3.9 in.) thick. For sake of stability, lateral ribs were built over the supports. The arrangement of the reinforce- ment was made in accordance with the moment line for uniformly distributed loading over two openings. The specimens were five weeks old at the time of the test. Beam I showed the usual arrangement of continuous reinforced concrete beams, with haunches at the center support. Since the dead load amounted to 325 kg m (218 lbs/ft) a five load of equal amount would produce a total of 650 kg/m (436 lbs. per running ft.) under which the reaction at the left support would be (2 = 650X6.15X1 = 1500 kg. (3300 lbs.), so that tq = 3.6 kg/cm^ (51 lbs/in^). Since the bent reinforcement was ample at that point to resist the diagonal tension, the adhesion could be computed according to the formula ti = ^ and this gave "1 = 2.85 kg/cm^ (41 lbs/in-), so that a 2Z U large factor of safety was secured at the ends. At 0.4/ the bending moment under the same load would be i\/=o.o7X65oX6. 152x100 = 172090 cm.-kg. (153385 in.-lbs.) and consequently, (7e = iooo kg/cm^ (14223 lbs/in2), and (T6 = i7-4 kg/cm^ (247 lbs/in^). Over the center pier the computed stresses were (Te = 990 kg/cm2 (14081 lbs/in^), and ^7^ = 45. 7 kg/cm^ (649 lbs/ in^). 200 CONCRETE-STEEL CONSTRUCTION At a total load of 6t. (6.6 tons) on both spans, the first cracks had already appeared in the region of the greatest positive moments, and at i4t. (15.4 tons) they showed themselves at the points of negative moment. With increase of load, somewhat later slightly diagonal cracks followed in the outer zones of positive and negative moment. Between these two regions at points about jl from the end, where the moment was zero, a space in the beam remained with no cracks up to the ultimate load. At those points, the principal stresses were of shear, which were carried by the bent rods, according to the computations made above. From the distribution of cracks, which are shown for only one span in Fig. 183, it is clear that a theoretically continuous reinforced concrete beam which is actually so constructed will act as such. Rupture resulted at a load of 34. 4t. (37.8 tons), from crushing of the concrete on the under side of the haunches. This occurred at a point where a stirrup had been displaced by the ramming of the concrete, so that the stirrup spacing was 24 cm. (9.4 in.) as that point, instead ot 15 cm. (5.9 in.). The two round rods of 10 mm. (fin.) diameter, were buckled in consequence, in the manner shown by the photograph in Fig. 186. A compression reinforcement thus shows Fig. 186. — Beam I, Cracks and Rupture in the vicinity of the intermediate support. itself of value, when it is prevented from buckling by closely placed stirrups, but otherwise it may be an actual detriment. Simultaneously, with the break at the center support, one also took place at 0.4I, since with the giving way of the haunch the positive moment was considerably increased so that failure naturally resulted at that point. For the ultimate load of 34. 4t. (37.8 tons), when the dead weight is also con- sidered, the computations give To = i7.3 kg/cm2 (246 lbs/in2), Ti = i3.7 kg/cm2 (195 lbs/in2). At 0.4I, according to Stage 116, there results (Te=48oo kg/cm^ (68,273 lbs/in^), ^7^ = 83. 5 kg/cm2 (1188 lbs/in2). CONTINUOUS T-BEAM EXPERIMENTS 201 When the steel stress is computed at the time of rupture with the arm of the couple at 32 cm. (12.6 in.), it is found to be 4450 kg/cm^ (63,294 lbs/in-). At the point where the concrete was crushed on the haunch, the moment of the continuous beam (computed for a constant section) equals 7.0 m-t. (25.3 ft. -tons). From this there results (7c = 3700 kg/cm^ (43,626 ll)s/in2), (76 = 171 kg/cm^ (2432 lbs/in^), so that the destruction of the concrete at this point is adequately explained. If it is further considered that the compression acted parallel to the under side, and that the stress computed for a vertical section is to be divided by cos^x, a compression on the concrete of 184 kg/cm^ (2574 lbs/in^) is obtained. The shearing stress computed for the section which failed, according to the formula Q iM bTo = - — ---^ tana, Z OS" amounts to 70 = 9.9 kg/cm^ (141 lbs/in^). The horizontal crack at the point of rupture should be noted. At 28. 6t. (31.5 tons) it was already visible and corresponded to a shearing of the concrete. At the front end of the haunch, the compression in the concrete acted horizon- tally, and because of the abrupt change of direction of the lower edge, the concrete of the haunch could only be affected by horizontal shearing stresses in a way to alter the [direction of the. — Trans.] compressive forces,, Since the change is abrupt, the principal stresses cannot develop at that point, and moreover the transfer may be supposed to take place through a sort of toothing, as in the case of plain shear (Fig. 33). The danger of this horizontal shear is the greater, the steeper is the haunch. Flat and especially rounded transitions are therefore preferable. Beam II had no haunches, the upper reinforcement at the center support being increased over that of Beam I so as to give equal carrying power, and the lower side was strengthened against compression by a spiral of 10 mm. (f in.) round steel, with a diameter of 10 cm. (3.9 in.) and a pitch of 3 cm. (1.2 in). Since the shearing stresses were greater because of the absence of a haunch, two extra shear-rods were introduced (Fig. 184). The distribution of cracks was just the same as in the foregoing beam, only the point of zero moment occurred somewhat nearer the center support (which is in accord with theory), because the larger section near the center pier of Beam I would move the point of zero moment further into the span. The stresses computed for the section with maximum positive moment at the load of 31.9 t. (35.1 tons) which produced rupture of the concrete on the lower side near the center support, were (je=425o kg/cm2 (60,449 lbs/in2). (7^ = 76.6 kg/cm2 (1089 lbs/in2). 202 CONCRETE-STEEL CONSTRUCTION Since failure occurred in this section at the same time with that of the concrete at the center support, Oe was relatively the same at the two points, just as in Beam I. When the bending moment and corresponding stresses are computed for the point of failure located about 0.3 m. (0.98 ft.) from the axis of the center support, there result at the instant of failure ^7^,= 33 15 kg/cm^ (47,150 lbs/in^), and (7^ = 311 kg/ cm^ (4423 lbs/ in^). Because of the 50 cm. (19.7 in.) breadth of the supports, a certain added security was attained at those points, so that these figures are rather too small than too large. The diagonal cracks cutting the shear rods, were apparent at a load of 16 t. (17.6 tons), corresponding to a shearing stress of to = i5.5 kg/cm^ (220 lbs/in^). There, the vertical pressures between the concrete layers produced by the reaction of the support, acted to delay the appearance of the first shear crack, as did Fig. 187. — Beam II, Cracks and Rupture in the vicinity of the intermediate support, also the shear rods and the stirrups. The adhesion must be computed at those points for the circumference of the upper reinforcement, and was 6.1 kg/cm^ (87 lbs/in2) failure, according to the formula '^^^'^Jf- Since in continuous beams, an abundance of steel is to be found over the intermediate supports, the adhesion there will always be found low in computed value. Beam III was constructed exactly like Beam II, except that the two 14 mm. in.) shear rods were omitted and the upper 14 mm. reinforcing rod was increased to 20 mm. (| in.). The distribution of cracks as shown in Fig. 185 was hke that of Beam II. Failure again occurred through rupture of the lower concrete near the center support, while failure also took place at the point of maximum positive moment at the same time that the ultimate carrying power was exceeded at the other point. With the failure of the concrete on the under side, indications also existed of a shearing of the concrete in a horizontal direc- tion just above the spiral (Fig. 188). Since the breaking load reached only 25.4 t. (27.9 tons), when compared with Beam II the utiHty of special shear rods is disclosed. These may be unnecessary when the main reinforcement can be bent at the desired points without reducing its area below that clearly necessary to resist the negative moments. CONTINUOUS T-BEAM EXPERIMENTS 203 Experiments with continuous beams are more difficult to execute than those on simple beams, since an inequality in the loading affects the results. The firm of Wayss & Freytag will repeat these experiments with somewhat heavier reinforcement near the ends, and also make two other beams without spirals or haunches. From the foregoing results, the effect of lack of haunches in continuous rein- forced concrete beams can be seen. Their economic value is shown in the follow- FiG. i88. — Beam III, Cracks and Rupture in the vicinity of the intermediate supports. ing table, which gives the amount of the main reinforcement including the spirals but omitting the stirrups: Beam I II III Ultimate Load 34-4 t. 31-9 25-4 Reinforcement 75-3 kg. 90.0 91.0 From these experiments, especially from Beam I, it may be concluded that reinforced concrete beams can be constructed as continuous members. It is believed that they should almost always be so designed when it is considered that all parts of a building of this type are monolithic. This also necessarily applies to floor slabs carried by reinforced concrete beams, which must withstand the negative moments over the latter, for since they act as the compression chords of the latter, they should not be allowed to become cracked on the upper side next the beams. PART II CHAPTER XII APPLICATIONS OF REINFORCED CONCRETE HISTORICAL Joseph Monier, who made the first practical use of it in 1868, is usually credited with being the inventor of reinforced concrete. However, traces of this method of construction are found at an earlier period. For instance, at the Paris Exposition of 1855, Lambot exhibited a boat made of reinforced concrete. At the International Exposition of 1867, besides the better known Monier, Francois Coignet was represented. Since i860 he had designed floors, arches and pipes, in the construction of which the fundamental principles of reinforced concrete construction are recognizable. To Monier, however, belongs the credit of having devoted himself to the new method of construction with perseverance and success. Originally, he was the owner of an important nursery in Paris. His first attempts were to make large plant tubs which would be more durable than those of wood, and more readily transportable than those of cement. He sought to attain his object by introducing iron rods of small diameter into the cement sides of the tubs, and extended this method of construction to the production of large water tanks. In 1867 he took out his first French patent, which he soon followed with a large number of others on reservoirs, floors, straight and arched beams in combination with floors, etc. In his patent drawings are already dis- closed all the elements which are to-day employed in the various construction details of the vairous systems. It is easy to understand how Monier's invention, but little understood and based as it was on an entirely empirical foundation, was destined to develop along entirely different lines in the hands of engineers. In 1884 the so-called Monier patents were purchased by the firms of Freitag and Heidschuch in Neustadt-on-the-Haardt, and of Martenstein and Josseaux in Offenbach-on-the-Main. The first mentioned firm acquired the rights for all of South Germany, with the exception of Frankfort-on-the-Main, and vicinity, which territory was reserved to themselves by the last mentioned firm. At the same time both parties acquired from Monier options for the whole of Germany, which, however, they rehnquished a year later to engineer Wayss. The latter, in connection with the above mentioned firms, conducted load tests in Berlin, the results of which were made pubHc in 1887, in the brochure " Das System Monier, Eisengerippe mit Zementumhullung," on the basis of which Wayss succeeded in introducing the Monier system into public and private edifices. 204 HISTORY 205 In that pamphlet, Wayss first expressed the decided opinion that the steel in reinforced concrete construction must be placed where the tensile stresses occur. He perceived that, owing to the extraordinary adhesion of cement concrete to iron, both elements must act together statically, and found his theory confirmed by his numerous tests. The Wayss experiments included not only strength tests of all kinds, but were also extended to include tests of protection against fire, and protection of the embedded steel against rust, as well as of the adhesion of iron to concrete. Examples were also given in the above mentioned pamphlet explaining the commercial praticability of the new method of construction compared with the older systems. The great capacity of the Monier slabs to resist shock had at that time already been demonstrated. The tests were witnessed by official representatives and private engineers and architects. Government Architect Koenen, now Director of the Actiengesellshaft fur Beton- und Monierbauten, in Berlin, was commissioned by Wayss to deduce methods of computation from these tests, which latter were published in the same pamphlet and also in the volume of the " Zentralblatter der Bauverwaltung " for 1886. Commencing at that time, a theoretical foundation was evolved, according to which the design of reinforced concrete work could be effected, and through these preliminary labors, this method of construction was extensively adopted in Germany and Austria. A turning point in its development was the Interna- tional Exposition in Paris, in 1900, and the report by Emperger, published at that time in regard to the position which the subject occupied. Because of the scientific investigation of reinforced concrete during the past few years, it has made rapid progress in Germany. It was specially pro- moted by the publication, in 1904, through the cooperation of experts and prac- tical men of the "Leitsatze" of the Verbands Deutscher Architekten- und Ingenieurvereine and the Deutschen Betonvereins, as well as by the " Regulations" of the Prussian Government, which abolished many restrictive rules, cleared the way, and inspired in the widest circles confidence in the new method of building. At the present time, in many countries, commissions are investigating the subject of reinforced concrete. The French Commission has already completed its labors and pubhshed its findings in a special report. The German Commit- tee on reinforced concrete has arranged an extensive programme, which is being executed in numerous testing laboratories. Besides this committee, there is the Reinforced Concrete Commission of the Jubilee of the Foundation of German Industries, which continues in existence, and some of the important results of the activities of which have been given in the foregoing matter. The Swiss Com- mission will complete its work during the next year, and make suggestions for the construction and design of reinforced concrete structures. In the United States reinforced concrete has been employed for a considerable time, but the wide difference among the systems employed, and the lack of method in their preparation, have prevented rational development. The Ran- som, Wilson, expanded metal, etc., systems, have found wide adoption in build- ings in that country. The Melan system was introduced into America by F. von Emperger, and came into extensive use in bridge building. 206 CONCRETE-STEEL CONSTRUCTION In addition to the Monier system, which was widely developed in France (the land of its origin), a large number of other systems arose there, of which only the names — Cottancin, Bordenave, Coignet, Bonna, Latrai, etc. — can be mentioned. The best known system is that of Hennebique, which, at the start, like those of his English and American co-laborers, aimed chiefly at security against fire. It is extensively employed in France, Belgium, and Italy. Henne- bique's ideas of construction were not altogether new, perhaps, being contained in part in the Monier patent specifications. Thus, are found, beams reinforced Fig. 189. — Monier's Reinforced Concrete Patent Drawings of 1878. with similar heavy round rods and wide stirrups, bent rods used in floors, beams,, etc. The first reinforced concrete beams in combination with slabs occurred as early as 1886, in connection with the erection of the library in Amsterdam. Since 1892, in chronological order, there followed Coignet, Sanders, Ransome, and then Hennebique. Of the better known systems for floor slabs and beams, the following may be enumerated: The Monier system (Fig. 190) employs numerous distributing rods at right angles to the supporting rods, the two being wired together at the points of intersection. Formerly, the distributing rods crossing the supporting Fig. 190. — Monier System. FiG. 191. — Hyatt System. ones were regarded as a means of preventing the concrete from slipping length- wise of the supporting rods. When it was realized, however, that the adhesion was sufficient for this purpose, the distributing rods were spaced further apart. The Monier system, or the ordinary grid of round rods spaced 6 to 10 cm. (2.4 to 4 in.) apart, found extensive employment in the construction of reservoirs of every description. Hyatt (Fig. 191) made the supporting rods of flat iron laid on edge, in which holes v/ere punched, through which were passed distrubuting rods, made of small round iron. The Ransome system. Fig. 192, which attained considerable importance in America, suppresses the distributing rods entirely, and uses for the supporting SYSTEMS 207 rods, spirally-twisted square iron, to prevent any slipping in the concrete. Other inventors, like Cottacin (Fig. 193) have woven supporting and distributing rods together, to form a rectangular network. For continuous floors with steel beams, bent rods in the form of flat iron, to which are riveted angle iron clips, are used in the Klett system (Fig. 195) Fig. 192, — Ransome System. Fig. 193. — Cottacin System. and the Wilson and Koenen systems (Fig. 194). In the two last-named systems, the slab is made heavier at the supports, as required by the greater moment of a continuous slab at those points. For varying loads, however, a single reinforce- ment must be regarded as inadequate. Fig. 194. Koenen System. Fig. 195. Klett System. The Matrai system incloses in the concrete a network of wires suspended, chain-like, between iron supports, crossing one another so as to form squares. In the Hennebique system (Figs. 196, 197), the ' reinforcement of slabs and beams consists of two series of rods. One series is straight and lie in the lower part of the concrete. Over the supports are top rods which are bent down near the center of the span and finally He close to the straight ones. Fig. 196. Slab of the Hennebique System. Fig. 197. Beam of the Hennet^ique System. Fig. 198. Flat-iron Stirrup. Hennebique rightly recognizes in the bent rods a preventive of shearing strains and uses them also with slabs and beams which are not restrained. He also uses flat iron stirrups in slabs and beams, of the form shown in Fig. 198. Hennebique is entitled to the credit of having introduced into construction work, on a large scale, reinforced concrete beams and columns, and of having developed new fields for reinforced concrete work. The systems of Klett and Wilson, used for floor slabs, in which the reinforce- ment consists exclusively of suspended, bent, flat iron, have their counterpart 208 CONCRETE-STEEL CONSTRUCTION in the beam construction of the Moller system (Fig. 199) used for the construc- tion of bridges of spans up to 20 m. (66 ft.). In this case, the ribs are reinforced on the lower side by suspended, bent, flat irons that are anchored over the points of support, and on which angle iron clips are riveted. The ribs have the fish belly shape, and follow exactly the line of the suspended rods; their depth decreases as they approach the points of support, while the thickness of the floor slab is increased at those points. The floor must take care of the longitudinal pull of the hanging flat irons, and is reinforced at right angles to their length by small I-beams or angle irons. Fig. 199. — T-beam Construction of the Moller System. The efforts made to manufaacture the structural parts of a reinforced con- crete structure at a special plant, and then transport them in finished state to the building, have produced a number of floor systems. The most successful in this respect are the hollow beams of the Siegwart system, which form a continuous floor when laid close together, and the beams of the Visintini system* which are used in the same manner, and consist of upper and lower transverse members with connecting webs. The theory of reinforced concrete construction has undergone many changes. The first suggestions of the prevailing methods of computation as given in the ''Leitsatze" are to be found in a communication from Coignet and de Tedesco in 1894, where, for instance, the quadratic equation is given as the means of determining the distance of the neutral layer from the upper surface of the slab. This publication is little known and consequently these formulas were independ- ently discovered by various authors, by Ritter of Zurich, for instance, in 1899, and Emperger. The centroid of compression was also first located in the center of the pressure zone, in place of at two-thirds of its height. In the first edition of the work of Christophe, Annale des Travaux publics de Belgique," 1899, the theory to-day contained in the "Leitsatze" is completely discussed, including the provision that the concrete can withstand no tension. Autenrieth of Stuttgart contributed to Vol. XXXI, 1887, of the Zeitschrift des Vereins Deutscher Ingenieure,^ ''a graphical method of calculating the anchors which fasten floors to plane surfaces." The methods there set forth are identical with those herein employed for the calculation of reinforced concrete * Beton und Eisen, No. Ill, 1903. t "Berechnung der Anker, welche zur Befestigung von Flatten an ebenen Flachen dienen.*' BUILDINGS 209 work, so that the processes indicated for simple bending and for flexure with axial pressure have been carried over, without change, to reinforced concrete work. This graphical method has been reproduced on page 130 et seep, and it is recommended for use in connection with complicated forms of cross-section. BUILDINGS In buildings, reinforced concrete may be used only for floors between steel beams, or in monolithic construction for beams and columns as well. Reinforced concrete floors between I-beams were fomerly constructed as Monier slabs, resting freely on the lower flanges of the beams, or in the form of flat plates carried continuously over the upper flanges. The present customary method of constructing continuous concrete floors between I-beams is shown u Fig. 200. — Holzer System. in Fig. 3. The lower flanges of the beams are wrapped with woven wire, so that they wiU take the ceihng finish which also protects the metal to some extent from the direct effects of fire in case of a conflagration. Among the systems using reinforced concrete between I-beams, the foUowing may be enumerated: Floors according to the Holzer system (Fig. 200). This belongs to the class of simple reinforced concrete slabs with free ends, and consists of a reinforce- ment of small beams 22 mm. (| in.) deep. The only value of this special form Fig. 201. — Zollner Cellular Floor. of section inheres in the possibility of erecting the floors without wooden forms, the small beams not possessing any increased carrying capacity in comparison with round rods of equal size. The support for the concrete is a cane mat, stiffened by round rods and suspended from the I-beams by wires. The mats are thus made to support the load of the floor during construction. The usual spanss are from i.om. to a maximum of 2.5 m. (3 to 8 ft. approximately). The Zollner ceUular floor system (Fig. 201) is suited to wider spans, about 4.5 to 7 m. (15 to 23 ft. approx.). With comparatively small dead weight, it 210 CONCRETE-STEEL CONSTRUCTION possesses considerable structural depth, and when set between the lower flanges of I-beams, requires but little filling. The great depth and low weight are obtained by the use of a series of hollow blocks, made from burned clay. The actual supporting structure of the floor consists of T-beams of concrete, arranged side by side, the stems of which carry the reinforcement. Instead of hollow blocks, a light cinder concrete may be used. It is also possible to construct a continuous cellular floor between I-beams or reinforced concrete ribs.* For sake of completeness there may also be mentioned the Monier arch exten- sively employed during the early periods of reinforced concrete work, but now replaced by more practical systems. The floor systems designed to be used with reinforced concrete members already in place, are very numerous and their enumeration would be too lenghty. Monolithic System of Reinforced Concrete. Of far greater importance than the various methods of floor-slab construction between steel beams, are the complete reinforced concrete buildings, in which all the load sustaining parts Fig. 202. — Warehouse for the Government Railway at Elberfeld in Opliden. (floors, beams and columns) are executed in reinforced concrete. This mate- rial is best adapted for long span, heavily loaded floors, and consequently, advanta- geously replaces the usual building materials for all factories, warehouses, etc. All parts are made in situ, so that the entire frame is of a perfectly rigid, mono- lithic character. The columns are rigidly joined with the beams, and the joint is given additional strength by the haunches under the beams. By this means, even with thin enclosing wafls, the stabiHty of the entire structure against lateral forces is considerably increased, as compared with those employing steel beams and columns. In the latter there is always a certain amount of joint-like mobility in the connections. Reinforced concrete beams may be supported directly on the outside walls, when of ordinary masonry, but in order to prevent settlement the walls must have a very solid foundation and should be laid in cement mortar. Or, wall columns may be carried up to receive the floor loads transmitted by * Beton und Eisen, No. Ill, 1903. BUILDINGS 211 Fig. 204. — Reinforced Concrete Construction of Floors and Columns in the Speyer Cotton Mill on the Rhine. 212 CONCRETE-STEEL CONSTRUCTION . the main beams. If the wall columns are then connected with special wall beams, which are made to support the floors at the outside walls, the load carry- ing reinforced concrete frame can be reared by itself, and completed independently of all w^all work. When relieved alike of the structural weight of floors and their live loads the masonry of the enclosing walls becomes simply a covering that serves merely to impart to the building the customary appearance. (Fig. 202). But when an outside wall is erected of such a variety, revealing nothing of the particular interior construction, it can be built much lighter than could otherwise be allowed or even than the building laws prescribe. It is then possible, and this is of the greatest importance in factory construction, to leave very large opeings for the admission of light. For this purpose, the whole height to the lower edge of the floor slab may be employed, since the ribs of the wall beams and window lintels can be carried above the floors just as well as below them. The wall columns are subjected to certain bending stresses from the load- ing oE the girders and because of their rigid connection with them, and in conse- quence are usually carried up of rectangular section, with the long side parallel to the length of the girder. For some time, steel columns and beams have not been considered fire-proof building material. Columns bend at 600 to 800° C. (iioo to 1400° F.). while beams break or push apart the outside walls, by their expansion. An instructive picture of the destruction to which a steel structure is exposed in case of a conflagration, is the view shown in Fig. 203, of the upper story of the Speyer woolen factory after a fire. The interior of the same story, rebuilt in rein- forced concrete, is also shown in Fig. 204. The concrete columns stand on the cast iron ones of the lower story, which were spared by the fire (Fig. 205). To insure better insulation and reduce the weight on the old columns as much as possible, the floor slabs were made of pumice-stone concrete, while Rhine gravel was used for the concrete in the beams and columns. In distinction from steel construction, buildings of reinforced concrete are fire-proof, for in them the metal does not play as important a part, and moreover it is effectively protected from the effects of the fire by the envelope of concrete. Concerning this, the following copy of a testimonial furnishes evidence. TESTIMONIAL The Wayss & Freytag Co., are hereby informed that the reinforced concrete construction carried out by them in 1901 in my factory at Neidenfels, consist- ing of floors, beams and columns, on the occasion of a fire on the 5-6 of Sept. prevented the spread of the flames to the factory rooms on the lower stories. The floor was found, by a load test made after the fire, to be completely BUILDINGS 213 secure at all points. With the exception of the finish coat on the floor, which was directly exposed to the fire and falhng debris, the concrete construction remained absolutely intact. I can therefore recommend the above mentioned construction to all whom it may concern, as absolutely fire-proof. (Signed) Julius Glatz. Neidenfels, Sept. 15, 1903. Rhine Palatinate. The foregoing statement of the firm of Julius Glatz, paper manufacturer, Neindenfels, describes the actual circumstances. The load test of a part of the floor in question, with 1800 kg. to the sq. meter (369 lbs/ft^) showed that the reinforced concrete construction, with the exception of the finish coat, had suffered no injury, but remained entirely intact. The Royal Fire Insurance Inspector. Neustadt-on-the-Hardt, Sept. 16, 1903. Characteristic occurrences of a similar nature, are recorded in Beton und Eisen," Nos. II and III, 1903. The superiority of the concrete protection of steel beams and columns, as a means of security against fire, compared with the terra cotta employed in America, is also emphasized. The earthquake and subsequent fire in San Francisco left unhurt the concrete protection of beams and columns, while the terra cotta covering fell away because of insufficient anchorage, and allowed the fire free access to the steel work. Concerning entire buildings of reinforced concrete, no experience can be gained, since the local building regulations did not allow the erection of entire structures in rein- forced concrete. See The San Francisco Earthquake and Fire, " by Himmel- wight; also " Deutsche Bauzeitung," 1907, No. 28. As shown in the accompanying illustrations (Figs. 206, 207) of a spinning factory in Finland, hangers for shafting, etc., can be attached at any point to the ceiling, or beams, girders, or columns. As to this point, preference is to he given to a ceiling with beams spaced 2 to 3.5 m. (6.5 to 11. 4 ft.) apart, compared with large paneled ceilings carried directly between girders. In the latter case, special stiffening beams must be provided between the columns, and haunches between the floor slabs and the girders are also to be recommended. (See Fig. 207.) So-called hanging columns may also be advantageously provided for holding shaft hangers, etc. In Fig. 208, showing the interior of the new building erected for the Speyer cotton mills, some of these members are illustrated. Vibration, even with high speed machinery, is hardly perceptible, and it is this freedom from susceptibility to shock and vibration that is one of the great advantages of reinforced concrete edifices. As a rule, the elastic deflection of a reinforced concrete beam can be placed at ^ to J that of an equally large steel one. With regard to vibration, however, the great weight in the reinforced-concrete floors and beams, rather than their stiffness, is the determining factor. The inclosing of the supporting structural members by the erection of fagade walls, does not allow of the utilization to the fullest extent of the advantages of 214 CONCRETE-STEEL CONSTRUCTION BUILDINGS 215 reinforced concrete. They are much better reahzed when the wall-beams and columns are left exposed and the panels thus left are closed with brick or thin concrete filling walls. In such cases, walls only a single brick in thickness on all stories will suffice. In such buildings, the wall beams must Ije made heavy enough to carry the masonry of the next story above. This consistent utiliza- tion of reinforced concrete work is as stable as when complete inclosing walls of masonry are employed, and allows a material saving in masonry work and foundations, permitting the best possible use to be made of the site — an important consideration where the cost of land is as great as in cities. In buildings erected without a cellar, the walls can be carried on transverse arches or reinforced concrete girders, extending between the separate foundations of the wall piers. An example of this type of construction is shown in Fig. 209, and a complete lay-out is also included, together with details of the reinforcement for the new building of the Daimler motor factory in Unterturkheim.* Fig. 210 shows a part of the plan of the ground floor and Fig. 211 is a section of the building which has a length of 131 m. (429 ft.) and a breadth of 46 m. (151 ft.). The columns throughout the building are spaced 5X5 meters apart (16.4 ft.), at which distance are also spaced the girders running across the building. Perpendicular to them are the beams, with a spacing of 2.5 m. (8.2 ft.), on which the floors are carried. Over the room on the front of the building are located two rows of girders 10 m. (32.8 ft.) long. In the outer walls all the beams are supported on reinforced concrete columns, and the floor panels are also there carried by beams of the same material, stretching between these wall columns. This arrangement creates rectangular panels in the outside walls, which, because of the great width of the building, are utilized as much as possible for windows. Except the brick curtain under each window, with concrete window sills and a small brick mullion in the center of each panel, no masonry exists in the outside waUs. The ribs of the wall beams are not placed beneath the floor slabs as is customary, but are immediately above them, so that the windows extend up to the under side of the floors. (See Fig. 214.) The second floor is provided with a number of openings, corresponding with the roof skylights, which openings are covered with glass and contribute to the better illumination of the ground floor. The concrete wall footings extend in the form of arches between the column foundations. In the front of the building, the intermediate piers are also executed as reinforced concrete columns, Ijecause in that case they have to carry the loads of the intermediate beams. The broken lines in Fig. 210 of the plan indicate expansion joints which traverse the entire structure, including the floors and beams. By them the structure is cut in two longitudinally, and also divided into Ave parts by four cross joints. These expansion joints, which are absolutely necessary in large buildings to pro- vide against cracks and injurious stresses, were introduced for the first time in this case, as far as known. It is evident the arrangement of such joints does not contribute to the simplification of construction, but rather complicates the design. The joints, which were constructed closed, later opened in part to a width of 6 mm. (J in.), thus affording the best proof of their necessity and of * Deutsche Bauzeitung, Cement Supplement, No. i, 1904. Fig. 209. — Office building of Wayss & Freytag, Neustodt-on-the-Hoardt. BUILDINGS 217 CONCRETE-STEEL CONSTRUCTION Fig. 213. — Factory of the Daimler Motor Co. in Unterturkheim near Stuttgart. BUILDINGS 219 their practical value. Because of the joint running longitudinally through the building, the auxiliary beam which would be cut by it, is rc})laced by two smaller ones, over which the floor slab is cantilcvered for 85 cm. (33.5 in.). Next the elevators the expansion joints make necessary the use of beam brackets having an angle less than 45°, and in the roof concrete, ridges with zinc coverings for the joints had to be provided, as shown in Fig. 211. The roof is covered with pitch strewn with gravel. The wall beams are constructed above the roof level, thus making a gravel stop unnecessary, and Fig. 214. are finished with a molding run in cement. The zinc protecting strip engages in a joint under the cornice. From Figs. 215 and 216 may be seen the details of the ground floor columns, together with all their reinforcement. The columns under the girders which are spaced 5 m. (16.4 ft.) apart, have a section of 32 X 32 cm. (13 in.), with a reinforcement of four round rods, 20 mm. (} in. approx.) in diameter, which rest at the bottom on a flat iron grid, and at intervals of 20 cm. (7.9 in.) are connected by 7 mm. in. approx.) round wire ties. The columns under the girders having the 10 m. (32.8 ft.) spacing, have a section of 40X40 cm. (15.7 in.), and are reinforced with four 20 mm. (f in. approx.) round rods at the corners, and four 18 mm. (^J in. approx.) rods between them. 220 CONCRETE-STEEL CONSTRUCTION The section of the wall columns, Fig. 217, was designed with regard to the window openings. The reinforcement consists of six rods, 16 mm. (f in.) in diameter. Figs. 215 and 216. — Ground floor interior columns. The live load for the second story was 600 kg/cm^ (123 Ib/ft^), and in the computations of floors, beams, and girders the most unfavorable distribution of this load had to be taken into consideration. In this connection it was assumed that the floor slabs would rest freely on the beams, that these would be supported BUILDINGS 221 freely on the girders, and that the latter would rest freely on the columns. Thus, all these structural meml)ers would be continuous beams with a larger or smaller number of spans, and with cantilever ends because of the presence of the transverse expansion joints. * ts H W j\ Fig. 217. — Ground floor wall columns. In Fig. 218 are shown the positive and negative moment curves for a girder of four spans with free ends. They were calculated according to the tabular values given by Winkler, and the influence of the end spans is only considered 222 CONCRETE-STEEL CONSTRUCTION with reference to the adjoining intermediate ones. The necessary reinforcement at top and bottom was detemined on the basis of these maximum curves, and was constructed as shown in Fig. 219. The necessary section of reinforcement at the top over the supports, was provided by long overlaps of the bent lower bars. This condition cannot be obtained from the small overlaps employed in certain systems." In the computations, the restraint of the beams at the walls was ignored, but since a part of the lower reinforcement was bent upward to care for the shearing forces, a partial restraint was created by this top reinforcement and its anchorage in the wall columns. The girders join the columns, and the beams join the girders with haunches, so that the allowable compressive stress of the concrete on the lower surface of BUILDINGS 223 the beam may not be exceeded at those points. The two round rods i8 mm. (xg in. approx.) in diameter which pass through the columns, and arc carried into the girders on each side, serve the same purpose. The reinforcement of th3 beams and the floor slabs was carried out on the same principle. The latter is shown in Fig. 220. In Fig. 221 the details are shown of the connection of the two intermediate beams with the wall beams. The 25 cm. (10 in. approx.) brick curtain walls originally planned were replaced by rein- forced concrete ones, 8 cm. (3 in. approx.) thick, which were considered equal in fire-proof quality to the brick curtains. The construction was com- pleted in three months, the daily rate for floors and beams being about 500 m^ (5381 ft^). This structure is considered a typical example, in which all the advantages of concrete con- struction were secured, so that its economic benefits were also obtained. The short period of I ^ 3r Fig. erection shows that with the necessary trained staff and the required equipment, reinforced concrete buildings may be erected with a speed of which no other type of building admits. 224 CONCRETE-STEEL CONSTRUCTION Figs. 222 and 223 show examples of the tasteful interior treatment of the Tietz Shop on the Bahnofplatz in Munich (Heilmann and Littmann, architects). The same rectangular reinforced concrete panel was employed on all fleers. The doubly reinforced slabs with spans of 5.15 m. (16.9 ft.) in both directions had haunches on all four sides at the beams, which were all of equal size. In the center of the building rose a large light shaft of elliptical section. The illustra- tions show the completed decorative interior in reinforced concrete, the light shaft with stairways and the counter space. An extremely simple architectural treatment of the interior of the building was intended. The entire under sides Fig. 222. — Interior view of the Tietz shop in Munich. of the floors and beam bottoms with their light moldings were made white, down to the two brown lines around the tops of the columns, making a very simple decorative treatment of the interior of the salesrooms. The reinforced concrete columns and parapets around the light well were covered with tasteful marble panels in different colors, and decorated with colored mosaic glass tile. Fig. 224 shows the placing of the reinforcement. The double reinforcement of the floor slabs was done with a beam spacing of 6.5X6.5 m. (21.3 ft.). It is possible, however, to divide the panels produced by a square column spacing into smaller squares by beams crossing each other, reinforcing the slabs as square panels partially restrained at the supports. Such an arrangement is shown in Fig. 225. The beams then rest freely on the inclos- ing walls. BUILDINGS 225 Fig. 224 — Placing reinforcement for Tietz shop. 226 CONCRETE-STEEL CONSTRUCTION Fig. 227 shows the floors of the immense Deimhardt wine cellar during con- struction in Coblenz. Since only a small height could be employed for the Fig. 225. — Storehouse in Ulm. Square floor panels and intersecting floor beams. structure, and at the same time large spans and considerable live loads (1000 and 1500 kg/m^ — 200 and 300 lbs/ft^) were required, intermediate beams were necessarily omitted and the floor panels were given a width of 6.1 m. (20 ft.) between. Fig. 226. — Printing house in Heibronn. girders. Because of the small head room, it was necessary to make the latter very wide. The floors were constructed according to the ZoUner cellular system, in which gutters were placed throughout the central parts of the panels. BUILDINGS 227 Fig. 227. — Erection of the three-story Deinhardt wine cellar, Coblenz. Fig. 228. — Cross-section of the Singer Manufacturing Co. factory in Wittenberg, 228 CONCRETE-STEEL CONSTRUCTION There may also be mentioned the immense factory now in course of construc- tion for the Singer Manufacturing Co., in Wittenberg, near Hamburg. It covers Fig. 229. — View of a floor in the Singer Manufacturing Co. factory showing widening of the beams where they intersect the girders. Columns of beton frette. Fig, 230 — Krefeld warehouse, with flat roof and monitor skylights. an area of over 5000 m^ (54,000 ft^ approx.) while the whole building with its four floors affords a working space of about 20,000 m^ (240,000 ft^ approx.). Teh BUILDINGS 229 Fig. 233. — Loft of the forge shop of the Daimler motor factory. 230 CONCRETE-STEEL CONSTRUCTION floors, which had to carry a heavy hve load, could have only shallow intermediate beams, because of the power transmission requirements, and for the same reasons they could not be very deep at the girders. The heavy compressive . Fig. 234. — Hall in Pfersee. Fig. 235 — Locomotive house in Haben Krefeld. The roof covering best adapted for reinforced concrete is of tar, laid with a grade of 2j%. The 10 cm. (4 in. approx.) layer of gravel on such a roof, in most Fig. 236. — Interior veiw of locomotive house in Krefeld. BUILDINGS 231 cases provides sufficient insulation against changes of temperature. A roof cover- ing with a double layer of felt is also employed, and has the advantage of lower cost, and allows a steeper pitch, up to about 15%. Sheet zinc is attacked by the cement concrete, and requires an intermediate layer of rooting ])a])er. In buildings of large area, some roof panels may be omitted, and subsecjuently covered with monitor skylights. (See Fig. 230.) For architectural or other reasons, high pitched roofs are also constructed. They may be covered with sheet metal or tile. (See Fig. 231.) Roofs are also constructed with long span girders. Fig. 234 shows the roof of a hall in Pfersee, with girders in the form of l)raced arches, while Figs. 235-237 show a locomotive roundhouse at Hafen Krefeld, Fig. 237. — Exterior view of locomotive house in Krefeld. in which the girders are in the form of arched beams. The design of the latter can be done through consideration of the elastic conditions produced by live load, and snow and wind pressures. The forms for the floors of reinforced concrete buildings several stories in height, are best so arranged that the side pieces of the beams form can be removed after the concrete has hardened sufficiently, w^hereas the supports and bottom pieces should be allowed to remain for a longer period (4 to 6 weeks). To facilitate the removal of the forms, all the supports rest on wedges. In determin- ing the period for removing the forms, besides the weather conditions during the period of setting, the question has to be considered as to whether the floor in question has to carry the forms for the stories above it. Spirally reinforced concrete is especially applicable to the columns of high 232 CONCRETE-STEEL CONSTRUCTION buildings. Above all, its usefulness is disclosed where heavily loaded columns would require large diameters, such as those shown in Fig. 229 in the new build- ing for the Singer Co., which are supplied with spirals. They were wound on hollow cylinders, with the coils close together, and could then be easily stretched longitudinally until the required pitch was secured. The last half- turn of the spiral should overlap, and the end of the rod be made into a hook carried to the center of the coil. Fig. 238 shows the steel skeleton of such a column. STAIRS Reinforced concrete stairs may be variously applied. In Fig. 239 is shown a staircase built according to the old method after the Monier system; the flights being supported by flat arches, braced against the structural parts of the land- ings. The steps are of concrete. Winding staircases in residences (Fig. 240) may also be erected in reinforced concrete, by arranging as the carrying structure, a winding slab of rein- forced concrete from 10 to 14 cm. (4 to 6 in. approx.) thick, supported in slots, 6 to 10 cm. (2J to 4 in. approx.) deep, cut in the masonry, on which slab the steps are formed. Winding stairs with side strings are also practical, as FlG.238.-Steel skeleton of a spirally shown in Fig. 241. reinforced column. Stairs with Straight flights can be arranged in different ways, according to conditions. Thus, Fig. 242 shows a stairway of reinforced concrete, in which, for lack of other support, the landing is suspended from the reinforced concrete beam overhead; while, in Fig. 243, an arrangement is exhibited in which brackets of reinforced concrete are employed, which bisect the angle of the inclosing walls, each forming an integral part of one tread, thus carrying the flights. Artificial stone stairs can also be constructed of concrete, each separate step forming a reinforced concrete beam, molded in advance, and set in place after having sufficient time to harden. Both ends may be supported, or in the case of flying staircases, one end may be set in a wall. Properly constructed stairways of reinforced concrete are just as safe as other parts. This cannot always be said of stairways built of stone or wood, in accordance with usual methods. In regard to security against fire, reinforced concrete stairs are superior not only to those of wood or stone, but also to those of iron. The little dependence which can be placed on stairs built of limestone is shown in an illustration in ''Beton und Eisen," No. II, 1903, p. 79. STAIRS 233 CONCRETE-STEEL CONSTRUCTION 234 The treads of reinforced concrete stairs can be finished with any desired covering, such as linoleum, oak, etc., so that they are adapted to the highest requirements. Fig 243 — Reinforced concrete stairs in the storehouse of the National Railway, Elberfeld in Opladen. ARCHES IN BUILDINGS 235 ARCHES IN BUILDINGS The archfcs which occur in buildings may serve the most varied ends. If they are merely decorative in character, they may be executed as thin ''Monier" arches, without forms, with woven wire on an iron framework. They are then more easily carried and more durable than when built of " Rabitz "* construction. By employing heavier T and I> iron framing, larger openings may be spanned without forms. This method has been employed for numerous vaulted church roofs, as shown in Fig. 245. The framing iron is there located within the arch ribs. In distinction from this may be mentioned the work shown in Fig. 246, of the vaulted ceiling of St. Joseph's Church in Wurzburg, for which the arch ribs, with a span of 20 m (65.6 ft.) were rammed in wooden forms and reinforced with round rods. In order to give the ribs the same appearance as the work below the spring of the arches, which was composed of genui::e shell limestone, crushed stone, and grit for the concrete, was prepared from the same material, and the ribs afterwards dressed by a stone cutter. The appearance of these arches fully met all requirements. The inverts between the ribs were made after the Holzer patents, small framing irons being employed :ind giving excellent results. The advantages of this type of arch are its security against cracking and its stability compared with a stone structure. In case of fire, such a vault will withstand the fall of the burned roof with perfect safety, and thus it forms an important pro- tction for the interior of the church. Manifestly all such barrel or groined arches can be constructed wholly on forms. Fig. 247 shows a fire-proof arrangement of a continuous barrel arch with transverse openings over the gymnasium of a school house in Munich. Among other uses of the reinforced concrete arch in building work may be mentioned an arch in which the profile which is required for architectural reasons, does not conform to that necessary from a statical point of view. Only in rein- forced concrete can such structures be executed — an example being shown in Fig. 249 — since tensile stresses can then be cared for and the line of pressure need not necessarily be located within the middle third. Reinforced concrete is particularly well adapted for the construction of domes. Through variation in the number and size of the reinforcing members placed in meridinal and parallel circles, it is possible to provide for all possible stresses. Since reinforced concrete will resist tensile stresses, almost any surface of revolu- tion can be used as the dome. Around its l^ase a tension ring, preferably of channel section, resists all the horiontal components of the meridinal stresses, so that the interior is entirely free from structural parts. If the crown of the dome is to be broken, so as to admit a skylight or cupola, the meridinal compressive stresses can be resisted by a concrete pressure ring with reinforcement. Domes (6 to 12 cm. thick — 2.4 to 4.7 in.) can be erected with the help of forms * Cement mortar on wire lath. — (Trans.) CONCRETE STEEL CONSTRUCTION Fig. 245. — Groined arches of reinforced concrete in the monastery chapel in Landau. ARCHES JN BUILDINGS 237 Fig. 246. — Arches in St. Joseph's Church in Wiirzburg. ■Fig. 247. — Fireproof Monier arches over the gynasium of the school in the Gatzingerplatz, Miinich. 238 CONCRETE-STEEL CONSTRUCTION Fig. 249 — Reinforced concrete semi-circular arches in the cupola of the new station in. Niirnberg. 239 or with shaping members in the directions of the meridinal and parallel circles, the trapezoidal spaces between irons being covered with woven wire, held in place by wires run through holes in the frames. The concrete is then made to Fig, 250 — Reinforced concrete cupola and lantern of the Army Museum in Munich, diameter 36 m. (52.5 ft.). cover the whole of the metal. The design of domes is best done graphically, just as in domes of the Schwedler type. Figs. 250-253 show a reinforced concrete dome of the Army Museum in Munich. In Fig. 250 is shown the exterior view of the whole dome. Including the 240 CONCRETE-STEEL CONSTRUCTION 9 m. (30 ft.) high lantern (187 ft.). ith its spire, the total height above grade is 57 m. Fig. 253 is a plan of the dome, and in Fig. 252 a section is shown. Fig. 251 shows, at a larger scale, the support for the base of the dome shown, both inner and outer shells were provided. The first, of a radius of 8.1 m. (26.6 ft.), carries only its own w^eight and is supported about i m. (3.3 ft.) lower than the outer dome, the surface of which is generated by radii of varying size. Each dome is supported on a footing ring, consisting of a D.N.P.* 14 cm. (5.5 in.) channel, set in the tambour, which had a thickness of from 38 to 51 cm. (15 to 20 in.). The concrete was from 5 to 6 cm. (2 to 2.4 in.) thick, and was strengthened by reinforcement arranged in the directions of the meridinal and parallel circles. In the outer dome the upper com- pression ring is made of an angle 50X50X7 mm. (2X2X1^ in. approx.), the ring upon which the lantern 4 m. (13. 1 ft.) in diameter rests, is a D.N.P. 12X6 cm. T (4.7X2.4 in.), while the remaining parallel circles are 8x4 cm. T's, (3.iXi.6in.) and the meridians As there Fig. 251. — Detail of base of cupola. Fig. 252. — Vertical section through the cupola of the Army Museum in Munich. are 9x4.5 cm. T's (3.5X1.8 in.). Between the meridinal and ring members was placed a 7 mm. (3^ in. approx.) round wire grid, with 10 cm. (3.9 in.) meshes. The inner dome is covered only with mortar on the outer side, the German standard. — (Trans.) SPREAD FOOTINGS 241 outer dome being covered with copper, fastened to wooden plugs which were placed in the dome along with the concrete. The copper is separated from the concrete by a coat of asphalt. 0 5 10*" 1 1 ^ i 1 i J 4 ^ Fig. 253. — Plan of cupola of the Army Museum. SPREAD FOOTINGS To prevent bad settlements, the areas of the foundations of a bulding must be proportioned to the carrying capacity of the soil. If the carrying capacity is small, a very large bearing area is required, which can be secured by the use of reinforced slabs possessing the necessary bending strength. In this way it is possible to reduce the pressure on the soil to 0.5 kg/cm^ (0.5 tons/ft^) or less, so that poor building areas and filled ground can be used for the foundations. Fig. 254 shows the bottom of a sewer in Wiesbaden, with footings 8 m. (26 ft.) deep. Since the soil could not be loaded more than 1.5 kg/cm^ (1.5 tons/ft^), the entire space between walls had to be called upon to resist the pressure. This was accomplished by the construction of a foundation slab, 45 cm. (17.7 in.) thick, extending from one wall to the other. Considered statically, the slab constituted a beam supported at both ends, and bent by the upward earth pressure. The reinforcement required per meter length of sewer was 10 rods 24 mm. (i^^ in.) in diameter, which, because of the curvature of the surface, had to be anchored into the concrete by means of stirrups 7 mm. (yq in. approx.) thick. As a sub- base for the sewer, a 15 cm. (6 in. approx.) layer of stone was laid in cement mortar. Spread footings should be employed when, for any reason, piling is imprac- ticable. In many instances, such footings are cheaper than piling. Fig. 255 shows a usual form of reinforced concrete footing, under a row of silo columns. In this instance it was necessary to distribute the concentrated column load uniformly longitudinally as well as laterally, and consequently CONCRETE STEEL CONSTRUCTION 242 reinforcement was needed in both directions. The longitudinal reinforcement served to distribute the concentrated loads over the space beween the columns. Fig. 254. — Reinforced concrete sewer base in Wiesbaden, Walls of buildings can be founded on reinforced concrete footings. For this purpose slabs of necessary width should be constructed along the line of the wall, projecting equally on both sides, as shown in Fig. 255. Such footings do not, Fig. 255. — Silo column foundation. however, distribute the load uniformly if eccentrically located with respect to the wall, as sometimes happens when the wall is built flush with the property line. In such a case a slab is required which extends under the whole building, or a SPREAD FOOTINGS 243 244 CONCRETE-STEEL CONSTRUCTION large part of it. If the centroid of the slab coincides with that of the building load, it is possible to distribute the load uniformly over the site, by means of a reinforced concrete foundation. In Fig. 256 is shown the upper and lower reinforcement of a footing slab, 50 cm. (20 in.) thick under a business block in Stuttgart. Upon the assumption of a uniform earth pressure of 0.7 kg/cm^ (0.7 tons/ft^) it was possible to compute approximately the positive and negative moments of the various panels and to arrange the reinforcement accordingly. Between the piers a stronger reinforce- ment is provided, so that the slab will act as a beam to distribute the concentrated load uniformly over its entire length, and transmit it to the line of reinforcement running perpendicular to the heavier material. At distances of about 50X50 cm. (20X20 in. approx.) the upper and lower reinforcement was tied together with stirrups. Similar footings as much as 75 cm. (30 in. approx.) thick were constructed by Wayss and Freytag for the large Elbhof block in Hamburg, and for various large silos. As a rule, in complicated ground plans, the statical relations are not entirely clear, so that calculations are made on somewhat unfavorable assumptions, and a certain excess of reinforcement should be employed. Sometimes the computa- tions cannot be based on a uniform soil pressure. In such cases the resultant of all the loads does not coincide with the centroid of the ground plan, or some of the loads are much heavier than others. Then varying pressures are to be reckoned under the several parts. Foundation slabs constructed in this manner permit of a saving of excavation, require less material, and solve the problem better than when the usual heavy concrete footings are employed, reinforced with grillages of rails or I-beams. They are adapted to the foundations of buildings, chimneys, fountains with heavy super- structures, monuments, etc., which are to be set on light soils. SUNKEN WELL CASINGS Sunken well casings of reinforced concrete were constructed according to the Monier system at an early period of its development. They serve either for obtaining water, in which case they are provided with holes in the walls, or as foundations for buildings, bridge piers, etc., and must be filled with concrete after being sunk. Compared with masonry well casings, they are capable of resist- ing heavier external pressures, and on account of the thinness of their walls they easily penetrate the soil, but their relatively lighter weight makes necessary a greater artificial load. Fig. 257 shows such a well being sunk by hand dredging for the municipal plant of Bamberg. The top is constructed in the shape of a dome. In Fig. 258 are shown the piers for the Kocher Bridge at Brockingen, erected on such sunken wells, details of reinforcement being also given. The casings, with a clear width of 1.5 m. (5 ft. approx.) or less, were constructed as reinforced concrete pipes, those of the larger diameter being constructed in situ between, forms. Over the well is laid a heavy slab of reinforced concrete, upon which the masonry rests. SUNKEN WELL CASINGS 245 An extensive water supply system was built in '1902 for the Pasinger Pai)er Mill (see Fig.' 259). The method of construction was as follows: First, a Fig. 257. — Wells for the municipal electric plant of Bamberg. I Fig. 258. — Well casing foundations for the Kocher Bridge at Brockingcn. circular excavation was made in the gravelly soil, with fairly steep banks down to elevation 0.0. Then the [ ] rings of the upper part of the well, fastened together Fig. 259. — Reinforced concrete well for the Pasingen paper mill. WATER-TIGHT CELLARS 247 vertically by round rods, were erected, commencing at the bottom, and outside them wooden planks were j)laced and driven down as the excavation was dee{)ened. At the same time the rings below the o.o level were placed, so that between grades o.o and — 3.3 m., the shaft was excavated under partially water-tight conditions with vertical sides. At this j)oint the construction of the casing proj)er was commenced at level 1.5,* this being above water level. Because of the heavy compressive stresses to which the casing would l)e subjected, it was constructed of reinforced concrete, with inwardly j^rojecting ribs reinforced with es])ecially heavy channels and round rods. The outside surface was finished smooth, and the lower edge was provided with a cutting ring. During the sinking oj)erati()n, sub-aqueous excavation was not employed because of practical considerations, the water being kept down and the material removed by mechanical api)liances. To drain the excavation, large centrifugal ])umps and electric motors were set up on steel elevators erected on the heavy ribs of the casing, so that the machinery could be easily raised, should the electric current fail. Upon the completion of the sinking of the casing, the concreting of the lining and of the reinforcing rings above level — 3.3 was undertaken, and at the same time a water-tight reinforced concrete floor was installed over about | of the area of the well, and a reinforced concrete wall carried from this floor above the level of the highest ground water. In this way a convenient entrance was provided to the well proper, as well as a perfectly water-tight spacious pump chamber, at a depth which lessened the suction on the pumps. Iron steps made pump pit and well easily accessible. Above ground, the well was inclosed by a small building which accommodates the power machinery. Concrete and reinforced concrete pneumatic caissons are discussed in a paper by K. E. Hilgard, in the Transactions of the American Society of Civil Engineers, Vol. 63, 1904. According to this paper, such caissons have been used in large numbers in Switzerland for foundations for bridge piers in the correction of tlie course of the Rhine, and as foundations of turbine pits. WATER=TIQHT CELLARS In all cases where the highest ground water level is above the bottom of a cellar which it is desired to maintain in a useful condition, a water-proofing of the walls and bottom is necessary. An excellent method of water-proofing the floor is to employ shallow inverted reinforced concrete arches, which span directly between the foundation walls in the form of cyhndrical arches, or are built as groined arches between the walls and the intermediate column foundations of the building. On these arches a water-tight cement finish is laid which really acts as the water-excluding medium, and consequently must be protected from injury by a concrete filHng over it. This filling concrete is leveled and finished with a wearing coat. At the walls, the arches are anchored and continued as a v/all coating of concrete with proper reinforcement, on which concrete the * Probably - 1.5.— (Trans.) 248 CONCRETE-STEEL CONSTRUCTION water-proofing is applied. Should the purpose for which the cellar is to be used so demand, the wall coat must also be protected from injury. In Fig. 260 is shown the woven wire reinforcement in a portion of the new building for the Daimler Motor Company. The cellar construction was first planned, so that the column foundations were to be walled and coated. A Fig. 260 — Woven wire reinforcement for a water-tight cellar in reconstruction of the Daimler motor factory in Unterturkheim. certain amount of continuity of construction beneath the foundations was effected by pumping cement grout into the gravelly soil immediately under the columns. The arches are inverted groined ones. In Fig. 260, the reinforced coating of water- proof cement for the column foundation is shown completed, while the metal grid for the floor arches is clearly visible. Fig. 261. — Reinforcement of the beams of a water-tight cellar. Another form of construction with floor arches 8 m. (26 ft.) wide abutting against reinforced concrete beams, is shown in Fig. 262. The necessity of water-proofing was discovered only at a late stage of construction, and groined arches were impracticable, because of the long rectangular spaces between columns. Wedge-shaped reinforced concrete beams were therefore constructed between the column foundations, and between the beams, flat, 8 m. (26 ft.) span, WATER-TIGHT CELLARS 249 barrel arches were built. In Fig. 261 the details of the reinforcement are shown, especially the arrangement of the beam reinforcement to withstand shearing stresses. The ground water level was 1.3 m. (4.2 ft.) above the cellar bottom. Fig. 262. — Section through beams and arches of a water-tight cellar. Similar proofing against ground water is often required in the boiler pits of steam heating systems. In this case, however, the water-proof cement coating must be made to resist the radiant heat of the boilers. Fig. 263. — Pump house of the Wolsum pulp mil In the pump-house of the pulp mill at Walsum, the bottom of the pump chamber had to be water-proofed against a high water level, 7 m. (23 ft.) above 250 CONCRETE-STEEL CONSTRUCTION the floor. Because of this heavy pressure two reinforced floor arches, one over the other, were employed, each with a water-proof layer, and both joined to the reinforced waU covering. The arrangement is shown in section in Fig. 263. PILES Spiralen ^10""" Reinforced concrete bearing piles and sheet piles have the advantage when compared with those of wood, that the concrete ones may be used above ground water level, whereby a considerable saving on the whole foundation may be effected. The reinforcing of the common square pile corresponds exactly with that of a reinforced column, the longitudinal rods being combined at the pointed end in an iron shoe. A good connection between the longitudinal rods by ties spaced not too far apart is very important, just as for columns. Further, so as to distribute the impact during driving, it is necessary to interpose some medium between the pile driver hammer and the pile, and also use a solid cap inclosing the head of the pile, to prevent its destruction. This variety of pile was intro- duced in excavation work by Hennebique. A detailed description of the subject was given by Diemling in " Beton und Eisen, " No. II, 1904. On account of its high compressive strength, spirally reinforced concrete is especially suitable for piles. The section is then usually circular or octagonal. Because of the excellent results which have been attained in Paris on Considere piles, which have been driven without any protecting device,* Wayss and Freytag acquired the exclusive rights to spirally reinforced piles and have secured excellent results in numerous instances. Fig. 264 shows the construction of one of the reinforced piles for the top of the 600 m. (2000 ft. approx.) long and 37 m. (121 ft.) wide, Ill-Hoch canal in Miilhausen in Alsace. The driving took place when the con- crete was only six weeks old. The load to be carried by each pile was 36 t. (39.6 tons); the reinforcement consisted of 8 longitudinal rods 14 mm. in.) in diameter and a spiral of 10 mm. (J in.) material Fig 264 — Reinforced ^ pitch of 6 cm. (2.4 in.) which was reduced to 3 concrete pile for the (^'^ head and the point. The wrought- top of the canal in iron point had four prongs, the upper ends of which Mulhausen in Alsace, were slightly offset and were punched. At the offset was a turn of the spiral and the holes provided a firm attachment between the pile shoe and the interior structure. The 1200 * See Deutsche Bauzeitung, 1906, Zementbeilage, No. 21. PILES 251 kg. (2645 lb.) hammer fell with a 1.3 to 1.5 m. (4.2 to 5 ft.) droj) upon an oak block. In Figs. 265-267 arc shown the fabrication of the reinforcement, the phicing of the concrete in the horizontal forms, and a stock of complete j^iles with some finished units of reinforcement. The latter were built by placing upon a l)ench the spiral, which had been formed upon a reel, the longitudinal rods were then put in place and wired to every second or third turn of the spiral. The concrete was mixed in proportions of 1:4^. The forms were so arranged that the side Fig. 265. — Factory in Metz. Fabrication of pile reinforcement. boards could be removed in one or two days, while the pile remained ui)on the bottom piece for about 8 days. For the foundations of the station in Cannstatt, ])iles from 5.5 to 10 m. (18 to 33 ft.) long were driven after hardening from 35 to 45 days. Their section and reinforcement were exactly like those of Fig. 264. The driving was done with a Mench and Hambrock steam pile driver. 'J'he 2750-kg. (6050 lb.) driving mechanism consisted of a ram with lifting ro])e. As a guide for the wooden block, a wrought-iron cap was used, which also j)revented a splitting of the concrete, because it completely inclosed the pile head. Between the pile and the block was a layer of sawdust to lessen the impact. Concrete piles provided with spiral reinforcement possess such impact resistance that the wooden block may be entirely omitted and the hammer allowed to strike directly on the concrete. It may crumble the concrete of the head somewhat, as is shown in Fig. 270, for which no cushion was used. This damaging of the pile top is without danger, Fig. 267. — Behrend Building, Kiel. Stock of finished piles and some reinforcement units. PILES 253 Fig. 269. — Business building in Metz. Driving piles. PILES 255 since this concrete must almost always l)e removed to build the reinforced concrete beam or column, so that the reinforcement for the new member can be joined to that of the pile, and then all concreted together. In a similar way, reinforced concrete j^iles can be spliced. In the foundations of the station in Cannstatt, the piles were driven with a hammer fall of about i m. (3.3 ft. approx.), the penetration being 4-5 mm. to ii^ )- The maximum perform- ance of the driver was 100 running m. per day (328 ft.). In computing the carrying capacity of the piles, the Brix formula is usually ^'^^^ 27o.-Station in Cannstatt. , , Head of pile which was driven without wherein h is the fall of the hammer; Q the weight of the hammer; g the weight of the pile; e the penetration of the pile under the last blow p is double the safe allowable load for the pile. The quantity e will naturally be the average of the last few blows. Fig. 269 shows a steam pile driver at work on a new business building in Metz. CHAPTER XIII APPLICATIONS OF REINFORCED CONCRETE BRIDGES (a) With Horizontal Members. Slab Culverts. — In the early stages of railroad construction, culverts roofed with natural stone were extensively em- ployed. With the advent of concrete and of cement pipe, arched conduits easily constructed in concrete or, for smaller openings, cement pipes were substituted. With the introduction of reinforced concrete, however, slab culverts again became Fig. 271. — Laufbach Bridge in Laupheim. Placing Reinforcement. useful. Since it is possible with the aid of reinforcement to make the concrete slab resist any bending stress, the span of the slab or the clear way of these culverts can be increased to about 6.5 (21 ft.), so that their field of usefulness has been greatly extended. The span might be still further increased, but beyond about 5m. (16 ft.), T-beams are cheaper than simple slabs. Slab culverts with reinforced concrete covers, are used over railroads as well as for streets. For instance, Wayss & Freytag constructed for the Gaildorf- Untergroningen Railroad a whole series of such culverts, and in the station at 256 BRIDGES 257 Soflingen, a foot tunnel of this same sort. • Figs. 271-273 show the construction of a slab culvert of 4 m. (13 ft.) clear span under the market ])lace of Laui)heim. The 35 cm. (13.7 in.) slab was calculated for a steam roller. It was l)uilt with 12 rods, 16 mm. (f in.) in diameter j)er m. width (39.37 in.), g being straight and 3 bent upwards at the ends. Its surface is crowned to discharge surface water. The test loading showed no measurable deilection. Numerous similar slabs under streets, up to spans of 6.5 m. (22 ft.), have been built. Fig. 272. — Laufbach Bridge in Laupheim. Cross-section. The abutments of such culverts are usually built of concrete, but use is often made of existing ones of masonry. The slab covers the abutment, thereby effecting a saving in wide spans through a saving of masonry. The slabs are usually constructed at the site, on forms, but have also been made in sections 80 to ICQ cm. (30 to 40 in. approx.) wide, and placed after hardening. This is necessary where no interruption to traffic can be allowed, one-half the street being first constructed, the traffic then diverted over that portion while the other half is completed. I I ( % oo ><- o so -^^ Fig. 273. — Laufbach Bridge in Laupheim. Longitudinal section showing thoroughfare. Under railroad embankments, the strength of the reinforced concrete slab can •always be suited to the load, by reducing the thickness towards the ends of the culverts. Walls over the ends of the culvert to retain the fill and shorten the length of the masonry work can advantageously be employed and anchored to the slab by the cross rods. Cantilevers. — Reinforced concrete slabs can be employed not only as mem- bers between two supports but also when secured at one end, the other projecting freely. This form of construction can be employed for instance for widening a street along a river. The sidewalk can then be allowed to project over the 258 CONCRETE-STEEL CONSTRUCTION river, as in Fig, 274. Naturally the reinforcement must then be placed near the upper surface, and be anchored in a block of concrete behind the wall of the em- bankment, the block being of sufficient size to prevent the overturning of the walk. Such an arrangement was built by Wayss & Freytag at Wildbad. In a similar manner the footwalks of old bridges may be arranged in order to widen the roadway. Reinforced concrete slabs can also be used to advantage as the flooring of steel footbridges and viaducts, and also for the construction of the flooring of the main thoroughfares of bridges in place of Zores* iron and buckleplates. The cost of such structures will be lessened by the employment of reinforced concrete and in all cases the construction will be simplified. The slabs for sidewalks are, in most cases, made in advance, and then laid. They may be < — zoo ->K.„.^..-. 70 ...^ ..»» Fig. 274. — Cantilevered sidewalk, Schramberg. made with a cement surface coat, or an asphalt wearing surface can be applied after laying. The reinforced concrete construction of main bridge thorough- fares consists of continuous reinforced slabs that are supported either between the longitudinal beams or usually between the cross beams, and are reinforced according to the maximum moment diagrams. T-Beam Bridges. — For larger spans, the rectangular slab section is uneconom- ical, and T-beams are usually employed for the carrying members of spans exceed- ing about 5 m. (16 ft.). The usual arrangement is to span the opening with several similar parallel girders and lay a floor slab between them. With ref- erence to the concentrated load of a steam roller, a girder spacing of between 1.3 and 1.6 m. (4.2 to 5.2 ft.) should be used, and for the same reason, and because of the unequal deflection of the girders, the floor slabs should have straight rods the full width both above and below, as well as bent ones, and numerous dis- tributing rods. Such a T-beam bridge vv^ith a clear span of 12.07 (39-6 ft.) (the Horn- bach Bridge at Zweibriicken) is shown in Figs. 275-276. The bridge is skew^ * Z-bars, Phoenix column shapes, and some other similar sections. — (Trans.) BRIDGES 259 and rests partly on pre-existing stone masonry abutments. The girders are connected by cross-beams, which serve to distribute more uniformly over several girders the concentrated loads and those of the brackets on which the reinforced Fig. 275. — Hornback Bridge near Zweibriicken. concrete slabs of the sidewalk are supported. As reinforcement, the girders have five straight round rods of 30 mm. (13% in.), and five bent ones of 28 mm. (ij in. approx.) diameter, the bending of the latter being arranged to care for Fig. 276. — Hornback Bridge near Zweibrucken. Cross and longitudinal sections. the diagonal tensile stresses produced by the shearing forces. Under the test load of a 20 t. (22 ton) steam roller, the girders were deflected only 0.3-0.4 mm. (0.015 260 CONCRETE-STEEL CONSTRUCTION T-beam bridges of this type, of spans up to i6 m. (52 ft.) are entirely prac- ticable, and in m^st cases cheaper than steel bridges. Special instances occur of 20 m. (66 ft.) spans, and unconnected girders also exist, such as are shown in Fig. 277. With longer spans, the girders become rather heavy, so that T- beam bridges possess little superiority over steel ones. In narrow bridges, up to 6 ra. (20 ft.) width, less depth is involved when only two parallel girders are employed, and the weight of the roadway is transferred to them by cross beams. Such small depth is important where railroads have to replace grade crossings by bridges. An example of such a case is shown in Figs. 278-281, of a bridge at Grimmelfingen, near Ulm. The girders with a clear span of 9 m. (30 ft.) have a rectangular section 70 cm. (27.6 in.) wide, and Fig. 277. — Bridge near Krapina, with open girders of 200 m. (66 ft.) span. extend up above the roadway, thus forming a low parapet, upon which only a small railing is necessary. The reinforced concrete slabs which carry the road- way and span the 1.533 (S ^^•) spaces between beams, are covered with a water- tight coating of layers of asphalt felt. Ready drainage is effected not only by a slope toward the girders from the center of the roadway, but also by a slope toward the abutments from the center of span. The roadway proper is con- structed of concrete. The static computation was made for a load consisting of a crowd of people of 450 kg/cm- (92 lbs/ft^) and a 6 t. (6.6 ton) wagon with a 1.5 t. (1.65 ton) wheel load. The first loading determined the girders, while the latter affected the deck slabs and the floor beams. The calculation and dimensioning of the deck was based on the assumptions of continuity throughout the several panels, and of free support of the beams. The reinforcement consisted of seven round rods of 7 mm. (yq in. approx.) diameter per meter width, above and below, throughout, 262 CONCRETE-STEEL CONSTRUCTION and of seven bent rods, lo mm. (f in. approx.) diameter, bent upward in the vicinity of the beams and carried over them. The cross beams were calculated without regard to possible end restraint. At the same time, however, the arrangement of the reinforcement provides some rigidity at their connections with the girders. Of the four rods (26 mm. — 13^ in. approx.) required at the centers of the beams, two are bent upward near the girders, at points where their area is not required in the lower chord. The two girders are not anchored at the abutments and consequently must be considered as freely supported beams of rectangular section. Of the ten Fig. 282. — Highway Bridge in Grimmelfingen near Ulm. Tested to 450 kg/m^ (92 lbs/ft^) with gravel; deflection 0.2 mm. (o.ooS in.). round rods, each of 30 mm. (if in. approx.) diameter required at the center, at special points six are bent upward at an angle of 45° to resist in the most effec- tive manner the shearing or diagonal tensile stresses. Naturally, the bending is done at points where the moment is sufficiently reduced. The bent ends are carried over the supports so that possible reverse moments may be resisted. The zone of compression of the girders is reinforced with three round rods 18 mm. {{^ in.) in diameter, connected together and anchored in the concrete of the girders by 7 mm. (J in. approx.) stirrups. They strengthen the upper side of the girder against compressive stresses. An important advantage of such reinforced concrete bridges over railways, is that they are not affected by the gases from the locomotives, which, in the case of busy stretches of track, and where difficult of access, cause active corrosion and high maintenance charges for steel structures. Fig. 283 gives an instruc- BRIDGES 263 tive illustration of the destruction wrought by smoke gases in unfavorable condi- tions. The piece there shown was removed early in 1907 from a steel longitu- dinal girder under the roadway of a bridge over a freight station erected in 1886. If the length of a horizontal reinforced concrete bridge is greater than 16 to I Fig. 283. — Rusting of a main girder by locomotive gases, ^ 20 m. (43 to 66 ft.), intermediate supports must be provided. They may consist of ordinary masonry intermediate piers, especially where such may have been left standing from a previous wooden bridge; but usually, however, are made of reinforced concrete in the shape of columns. It is best to place a separate support under each girder, and to connect the columns at the base by means of a common pedestal and a single foundation. Fig. 284. — Reinforced concrete bridge over a railroad cut. Continuous members with three spans are well adapted for bridging railroad cuts (Fig. 284), a considerable saving being effected in abutment masonry. In Fig. 285 are shown the entire design and details of a section through the thoroughfare of such a bridge over a ravine at Bad Tolz. In this case the abut- ments are carried down through the shifting soil to bedrock on tubular piles of 264 CONCRETE-STEEL CONSTRUCTION reinforced concrete. The reinforcement of the continuous bridge girders is designed exactly like those of buildings. Fig. 286 shows a view of this bridge. Fig. 285. — Bridge over a ravine near Bad Tolz. In Straight girder bridges of greater length an expansion joint must be provided about every fourth opening. This necessitates cutting one of the Fig. 286. — Bridge over a ravine near Bad Tolz. piers longitudinally, or constructing two piers close together and allowing the girders to project over them as cantilevers. Bracketed beam ends may often be employed to advantage in reinforced concrete bridge construction. Pains should BRIDGES 265 be taken to arrange the supports of reinforced concrete bridges so that the j)res- sures at those points are properly carried. In small bridges, sliding and tangen- tial tilting supports are suitable, roller bearings being useful only in exceptional cases. Practice has indeed not yet shown the absolute necessity of such devices for reinforced concrete construction, since at such points much smaller move- ments take place, and where girders are supported by columns such devices are of small moment, because of the elasticity of the column. Where necessary^ the secondary stresses so produced can be calculated. With beams of only a single span, dangerous though invisible conditions exist with rigid supports, since the bottom of the beam lengthens from the tensile stresses which appear Fig. 287. — Cover of the lU-Hochwasser Canal, Miilhausen. when the forms are removed, and the beam exerts a pressure on the abutments, which is bad for both beams and wall. In continuous beams, however, which rest on masonry center piers, such an arrangement, when j)roperly constructed, is advisable. All varieties of deck structures are identical in jjrinciple with T-beani bridges, especially those over streams and railroads. In those over streams it often happens favorably that existing river walls may be used as abutments for shear- free constructions. In covering railroad cuts, the principal advantage of rein- forced concrete is its capacity to withstand the locomotive gases. An extensive cover of this kind in Miilhausen, in Alsace, over the Ill-Hoch- wasser canal is shown in Fig. 287. The work illustrated includes a deck 36 m. (118 ft.) wide and 660 m. (2165 ft.) long. The beams, spaced 3 m. (9.8 ft.) apart, were continuous over spans of 11, 14 and 11 m. (36,46 and 36 ft.) with ends ARCHES 267 elastically restrained l)y reinforced concrete columns in the side walls. They were designed with that end in view and were computed for a live load of 500 kg/m- (102 lbs ft-). Since these reinforced concrete columns, which had also to resist the earth pressure, were inclosed in the wall construction, it was possible to construct the girders with less depth. The 7 m. (23 ft.) high, octagonal, intermediate supporting columns rested on the reinforced concrete piles described previously (Fig. 264). Between the columns, 5 m. (16 ft.) high reinforced concrete walls were erected, so that at high water it llowcd in three separate streams. These partitions afforded a stronger j)rotection for the columns and a considerable stiffening of the construction in a longitudinal direction. Four streets crossed the canal, and it was because of the high loads from street cars and heavy trucks that this s])ecially strong construction was re(|uircd. The largest piece of work of this description is that of the Vienna Municipal Railroad, which extends 2 km. (1.24 miles) with spans of 12.7 m. (41.6 ft.), and was constructed by G. A. Wayss & Co., of Vienna. (/;) Arch Construction. — In arched bridges, reinforced concrete can be em- ployed either for the arch alone, or the superstructure including the roadway, or in all structural parts. In small spans, the reinforcement in the arch enables the full compressive strength of the concrete to be utilized, since the tensile stresses are independently resisted. In medium spans of 40 to 50 m. (130 to 165 ft.), the employment of reinforced concrete as the arch material is less frequent, since in this case, provided a proper profile has been employed, no tensile stresses occur, because of the large dead load. On the other hand, reinforced concrete is better adapted for long spans. If the safe compressive stress in the arch is not to be exceeded, it is necessary to limit the weight of the superstructure, and this can be done by a suitable employment of reinforced concrete. In this way the dead load stresses will be greatly reduced, but the small edge stress may decrease to zero, or change to tension under unfavorable live loads, so that reinforcement is again necessary. In consequence, where it is desired in small and medium spans that no tensile stresses shall exist, or in other words where only concrete work is employed, the superstructure cannot be kept too light. A light superstructure is then justifiable only when demanded for architectural reasons or when it is necessary to impose as little weight as possible on the foundations. An example of a reinforced concrete bridge without special superstructure is shown in Fig. 288. The arch of 36 m. (118 ft.) span and 4.2 m. (13.8 ft.) rise, is 50 cm. (20 in.) thick at the crown and was computed as fixed at the ends.* The reinforcement consisted of ten 14 mm. in.) rods near both the top and the soffit of the arch, and at distances of about 50 cm. (20 in.), the two systems of rods were tied together by 7 mm. in. a])prox.) stirrups. At the spring- ings the arch rested with a widened foot upon the abutment concrete so as to secure the assumed, computed restraint. The arch is faced with ashlar masonry. The space between the asphalt waterproofed top of the arch and the roadway is filled with gravel, upon which rests the concrete foundation of the asphalt street surface. The arrangement of the arch centers and other details is also * See article by the author published in the " Schweizerischer Bauzeitung," 1908, Nos. 7 and 8; and also the separate brochure, " Berechnung von eingespannter Gewolben." 268 CONCRETE-STEEL CONSTRUCTION shown in the illustration. The maximum stresses amounted to 37.4 kg/cm^ (532 lbs/in^) compression, and o.i kg/cm^ (1.4 lbs/in^) tension, the reinforce- ment being thus only a safeguard in case an abutment should settle and cause an increase in the tensile stresses. During the last twenty years a large number of bridges similar to this have been erected by Wayss & Freytag. With regard to the application of reinforced concrete to the construction of the roadway and the superstructure over the arch, several arrangements are possible. I. The reinforced concrete slab which carries the roadway may rest on 40 to 60 cm. (16 to 24 in.) walls of concrete or masonry, which are combined with the arch and carry the loads to it. The arches can be constructed either re- strained or three hinged by using suitable material, entirely independent of the rein- FiG. 289. — Highway bridge near Hamburg. Longitudinal and horizontal sections. forced concrete construction of the roadway. The reinforced slabs of the latter correspond with the spandrel arches of the usual arrangement, but are more advantageous, because they may extend somewhat over the outside walls and because no horizontal shear is produced by spandrel walls. Consequently, a narrower arch is possible, with a saving in the abutments and piers. When spandrel walls of considerable height are used, it is wise to stiffen them with intermediate decks. A bridge of this arrangement of superstructure is illus- strated in Fig. 289. The arch of that bridge consists of mass concrete with- out reinforcement. Since the ground behind the abutments slopes up to the level of the roadway, the longitudinal walls with the reinforced concrete slab resting on them could be extended into the ground over the abutments, so that wing walls and embankments were unnecessary. In this manner a saving in cost could be accomplished. ARCHES 269 2. Over the arch, and at right angles to its face, cross walls can be erected to support the reinforced concrete construction of the roadway. These cross walls are usually j)laced at such a distance apart that a continuous reinforced concrete slab may l)e provided for the supi)ort of the roadway. The outside walls thus fall entirely beyond the arch, and the latter does not receive as much stiffening from the superstructure as in the last case, so that its form must be accurately designed and executed. Its superiority from a static point of view rests in the decreased load on the abutments and foundations. The arrangement with cross walls is very light and j)leasing in appearance. The walls may be constructed of masonry, or of concrete, with or without rein- forcement. The latter kind of construction permits a narrow width to be em- ployed and may be used when the arch consists of reinforced concrete. Fig. 290. — Highway bridge near Hamburg. Section through superstructure and thoroughfare. In Fig. 291 is illustrated a foot bridge over the canal near the Grosshesseloher railroad bridge. The pleasing appearance is secured without employing architec- tural adornment, but solely through the structural work, all conspicuous parts of which consist of reinforced concrete. A highway bridge of the same type is shown in ¥ig. 292. The bridge has a 32 m. (105 ft.) span, and was erected on the site of a dilapidated wooden one. In Figs. 293-294 are illustrated a foot bridge on the Metz-Vigy line. The axes of the ribs in these bridges were assumed as parabolas, so as to be able to apply advantageously available formulas for the immediate computation of the influence line for the load point moments of a restrained parabolic arch. Since usually the axes of restrained arches are assumed so as to coincide with the line of pressure for dead load, in the foregoing cases the assumption of a parabola was permissible, because the dead load is small and very nearly uniformly distributed over the span. It is to be especially noted that here the connection of the structure with the bank is effected by means of concrete walls which inclose a hollow space decked over with a reinforced concrete slab. In this way the earth pressure is diminished against the wing walls. Fig. 292. — Highway bridge in Gunzesried (Allgan). Span 32 m. (105 ft.). ARCHES 271 3. The cross walls in the foregoing arrangement may be replaced with trans- verse rows of columns which carry the T-beam construction of the roadway. In this way the weight of the superstructure over the arch will l^e reduced as Fig. 293. — Foot bridge over the Metz-Vigy Line. Test load. much as possible, this arrangement being adapted for large spans. The best example to date of this class, is the Isar bridge in Grunwald (Figs. 295-304), a short description of which will here be given.* Fig. 294. — Foot bridge over the Metz-Vigy Line. Longitudinal section. This bridge, which spans the Isar between Hollriegelsgreuth and Grunwald, was built by the reinforced concrete companies* of Munich after the designs of * See article by the author in the " Schweizerische Bauzeitung," 1904, XLIV, Nos, 23 and 24, also published separately. Figures 296 to 304 are borrowed from that publication, t Wayss & Freytag of Neustadt-on-the-Hardt, and Kallmann & Littmann of Munich. •272 CONCRETE-STEEL CONSTRUCTION ARCHES 273 the author. This highway bridge is of about 220 m. (722 ft.) length, with two arches of 70 m (229.6 ft.) span, each with a rise of 12.8 m. (42.0 ft.) over the beds of the river Isar and the power canal for the electric plant. These two main spans are finished on the right by a single, and on the left by four approach s})ans 8.5 m. (27.9 ft.) long, which are decked with a straight reinforced concrete structure. The two main spans were designed as three-hinged arches. In determin- ing the selection of this form of construction, besides its general excellence, one main condition also existed, that up to the date of the arrangement of this pro- ject, no positive data existed concerning the underlying conditions, so that much caution was necessary. Fig. 295. — Highway bridge over the Isar near Griinwald. The bridge was designed for a crowd of people of 400 kg/m^ (82 lbs/ft^), and a steam roller load of 20 t. (22 tons). As described by the author in the Zeitschrift fur Architektur and Ingenieurwesen," No. II, 1900, the arches were so proportioned as to form and thickness that at each section the maximum stresses in the upper and lower edges were equal to the one then considered permissible of 35 kg/cm^ (497 lbs/in^). At the crown, the arch thickness was 75 cm. (29.5 in.), at the springing 90 cm. (35.4 in.), and at joints V and VI* it reached 1.20 m. (47.2 in.), its greatest thickness. No tensile stresses appeared, but the comj^ression was reduced to 2.1 kg/cm^ (29.8 lbs/in^), with unfavorable loading. For each centimeter (0.4 in.) which the arch rib deflected, the edge stress varied about i kg/cm^ (14.2 lbs/in^), so that with a deflection of 4 to 5 cm. * Under the verticals counted from the abutments. — (Trans.) 274 CONCRETE-STEEL CONSTRUCTION (1.5 to 2 in. approx.), tensile stresses would exist in sections IV- VI. Since such a deflection did not appear impossible through uncertainty of construc- tion, or unequal lowering of the centers, the arch was provided with a rein- forcement, which obviously was impossible of computation, and was introduced only as an additional measure of security. This reinforcement consisted of nine round rods 28 mm. (ij in. approx.) in diameter, both above and below over the whole width of 8 m. (26.2 ft.). At distances of i m. (40 in.) the upper and lower steel was tied together by 7 mm. in. approx.) round iron stirrups. The hinges of cast steel w^re of the dimensions shown in Figs. 297-298, and their smoothed faces rested agahist squared reinforced artificial stones with a 4 mm. in.) layer of sheet lead between. The pressure over the bearing area was 100 kg/cm^ (1422 lbs/in^). The hinges were designed for a permissible Fig. 297. — The abutment hinge. Fig. 298. — The crown hinge. bending stress of 11 00 kg/cm^ (15,642 lbs/in-), and the two halves met on cylindrical surfaces of 250 and 200 mm. (9.84 and 7.87 in.) radii. Granite was first proposed for the hinge bearing blocks, but later, reinforced concrete blocks were selected because more economical, and after compression experiments made in Miinich had disclosed good results. The hinge-bearing blocks are under pressure over only a part of their area. Similarly loaded test specimens failed by cracking in the direction of stress, so that a reinforcement was neces- sary, running at right angles to the direction of stress and transverse to the pressure areas, in order to increase sufficiently the ultimate strength. In constructing the blocks, which were molded in perfect cast-iron forms with smooth surfaces, the reinforcement was uniformly distributed throughout the full height, since the cracks had appeared in the test specimen in the part between the pieces of reinforcement. The length of each block was 79 cm. (31 in.) and ARCHES 275 corresponded with the length of a hinge piece, so that a single piece rested on each block. At the abutment hinges the i)arts were so arranged that both pieces of stone rested on the centers, thus making impossible anything except a simultaneous shifting. The arch was concreted in separate sections, the size and order of which is shown in Fig. 296. In making this arrangement, care was taken concerning Fig. 299. — Detail of the forms of the Isar Bridge. the centering that the largest sections, 1-6, usually covered a whole panel, with a small space left free at the ends of the panels over the posts, and also a small space next the hinge stones. For the order of concreting the small sections, 7-14, it was specified that those next the hinges should be done last, so that only hght loads could come upon them, and thus prevent the development in them of a dangerous deformation. The last sections were those immediately adjacent to the hinge stones in the abutments. The arch was built with a mixture of i part Blaubeur Portland cement, 2 parts Isar sand, and 4 parts Isar gravel. 276 CONCRETE-STEEL CONSTRUCTION The forms for both main spans consisted of seven sections of the construc- tion shown in Fig. 296. By this arrangement, it was insured that the support- ing of the vertical loads from the concrete construction would be done most directly by the piles, so that the panels would be the only construction parts- subjected to bending stresses. In this way deformation of the scaffolding was reduced to a minimum, and to this end, as far as precautionary measures could be taken, the posts and braces were not allowed to bear directly against the wood of the sills, that is, so that the latter would be overstressed in a direction at right angles to the fibers; 13 to 15 kg/cm^ (185 to 213 lbs/in^) was assumed as a safe permissible stress on timber in that direction, and pieces of channel iron were employed to distribute the pressures of the posts and piles over the sills and cross-beams of the scaffolding. (Fig. 299.) Fig. 300. — Highway bridge over the Isar near Griinwald. View of the left abutment. Furthermore, sand boxes were employed, except for the first range from the abutment, where wedges were used. Compared with screw jacks, a saving in cost was effected; and further, the sand boxes offered the added advantage of a stabler support for the scaffolding, and, with sufficient caution and experience, as safe a form of centering was secured as with the average screw jack. All foundations of abutments and piers were carried down by pumping to the rock (a kind of marl), which could be loaded to 5 kg/cm^ (5 tons/ft^). The highest pier of the approach spans and the superstructure of the principal pier contained open spaces, limited in width by the reinforcement of the upper thoroughfare arches on these piers, and the condition that the floor beams were given sufficient bearing surface. The bridge floor had a breadth of 8 m. (26.2 ft.) between the side rails, of ARCHES 277 which 5 m. (16.4 ft.) was given to the roadway and 1.5 m. (4.9 ft.) on each side to a sidewalk. The floor sloped from the center pier in each direction on a 1% grade and was drained through the piers, over the abutments of each main span. The roadway was carried by a reinforced concrete construction, consisting of a reinforced slab 8.6 m. (28.2 ft.) wide and 20 cm. (7.9 in.) thick, which trans- ferred its load to five longitudinal beams 25X40 cm. (9.8X15.7 in.) in section^ which in turn were supported from the arch by reinforced concrete columns 4 m. (13. 1 ft.) apart. The slab and the beams were designed as continuous mem- bers, in which the unfavorable assumption was made that the slab was free to move on the beams and the latter on the columns. For the calculation of the roadway construction the wheel load of a steam roller was critical. The reinforcements of the slabs and beams are shown in Figs. 303 and 304. Over the Fig. 301. — View under the arch over the right-hand stream. columns, the section of each beam was increased by haunches, so that the com- pressive stress on the underside, because of a large negative end moment, did not exceed safe limits. These haunches also reduced the shearing stresses. The columns have a section of 40X40 cm. (15.7 in.) with the exception of those in plain sight at the sides of the bridge, which were given a T-section to improve their appearance, so that they had a breadth of 70 cm. (27.6 in.) on the outside. The reinforcement of the longest columns consisted of eight round rods 24 mm. (Jf in.) in diameter, while the succeeding rows had eight rods of 22 mm. (I in.), four rods of 24 mm. (jf in.), and four rods of 22 mm. (Jin.) diameter respectively; and the outside columns were reinforced with from eight rods 20 mm. (xf in.) to four rods 18 mm. (xJ in. approx.) in diameter. In all columns, a tie spacing of 35 cm. (13.8 in.) was used, for the 7 mm. (^ in. approx.) round wires employed. The column steel extended about 40 to 50 cm. (16 to 20 278 CONCRETE-STEEL CONSTRUCTION >|fEj:::::.ii::::::ft)::3^^^^ I k! 06'i Off'i >fi osii 06i- < wjOfi >H ■ 0?- > — •+-1 :3 O QJ / n 4 ■I / — / 71 ^ l_ I V I y I y ARCHES 279 in,) into the arch concrete, and under each row of columns, the arch was rein- forced laterally by four i6 mm. (f in.) round rods below and two similar rods above, so as better to distribute the concentrated loads of the columns over the whole arch width. The last sup})ort over the abutments was built as a rein- forced concrete wall, with openings, so as to provide the necessary lateral stability for the thoroughfare deck. Over the hinges at crown and abutments, expansion joints were provided in the deck construction, the joints being covered with sheet metal in the usual way. The reinforced concrete construction over the 8.5 m. (27.9 ft) wide approach spans consisted of a deck slab and girders. Since the girders had the same spacing as those over the main spans, the slab was built exactly like that one. The beams were designed and constructed like simple, freely supported mem- FiG. 303. — Reinforcement of the deck slab. bers, so as to simplify the reinforcement which consisted of five round rods 36 mm. (i^ in.) and one rod 24 mm. (}f in.) in diameter. (See Fig. 304). The architecture of the bridge is completely determined by its construction. With the exception of the center one, no pier is at all decorated. All concrete surfaces were left without manipulation, except one prominent ridge which was formed by a crack between the form boards. The railing was also constructed of concrete with open panels, without extra finish. The water-tight covering of the concrete slab which carried the thoroughfare consisted of a layer of asphalt, that of the arches being a water-tight cement coating. When the centers were struck, the concrete was about three months old, and the whole dead load, including the pavement, was in place, so that the abutment pressure had exactly the direction computed for it. The striking of the centers was so arranged that first, at a given signal, the sand boxes were opened under both middle sections beneath the center joint, and 1 1. (J pint) of sand allowed to run out. The opening was then closed, and a couple of blows struck upon the sand box which caused a settlement of a few millimeters. The same operation was repeated simultaneously on the next four rows of supports each side the center, and so on, to the third series, after which the process from the crown outward w^as repeated, and all except the last row was 280 CONCRETE-STEEL CONSTRUCTION lowered. A total of twenty-eight men with the requisite inspection force was necessary, each one being equipped with a wrench, an ax, a measuring vessel, and a mallet. Since the centering was itself strained elastically, only a very small deflection of the arch was observed. Since the deflection was not larger, the oak wedges next the abutments were also loosened. When a lowering of the centers of lo cm. (3.9 in.) under the crown had taken place, the deflection of the arch down to its final position was only 17 mm. (0.669 ii^O 5 the amount of the deflection of the arch measured to the center- ing was not uniform. This observed deflection measured to the centering amounted on the right span to 6.5 mm. (0.256 in.), and on the left one to 10 mm. (0.394 in.). Before and after the lowering, the width of the hinge opening Schnin a-b.-1:20. Fig. 304. — Thoroughfare construction between the piers on the left bank. between the stones was measured, but a diminution of not more than j\ mm. (0.004 ii^O could be observed. Shifting of the foundations could not be ascer- tained with certainty with the instrumental arrangements at hand. The computations for the deflection of the crown gave a satisfactory agreement with the measured amount, but also gave the conclusion that small deviations from the profile planned influenced the deflection considerably.* The hinge openings of the arch were filled with cement mortar, so as to pro- tect the steel hinges from rust. The mobility of the joint was maintained by a layer of asphalt, concreted into the center of the opening. Since the actual cost of the bridge was only about 260,000 M. ($62,000 approx.), it is demonstrated, as far as the Griinwald-Isar bridge is concerned, that arched bridges with a proper arrangement of reinforced concrete, and of large spans, can compete successfully with steel construction. * See the description in the " Schweizerischer Bauzeitung," 1904. ARCHES 281 In Fig. 305 is shown a smaller bridge with a hingeless arch and similar super- structure. The span is only 23 m. (75.4 ft.), the thickness of the arch at the crown being 35 cm. (13.7 in.), and at the springings 60 cm. (23.6 in.). Besides the usual round rod reinforcement, plate or lattice girders are of advantage for the reinforcement of arch bridges, according to the Melan system. A description follows of the highway bridge over the Mosel Road at Wasser- liesch, in which a shallow depth of construction was secured by arranging the reinforcement in the form of lattice girders. In this bridge, which replaced a grade crossing, not enough space existed between the clearance required and the arch profile to erect a scaffold of the usual variety; even an iron structure to support the sheeting would commonly occupy the whole of the clearance space, which cannot usually be spared. Such metal Fig. 305. — Nagold Bridge near Cain (Wiirtemberg). forms can only be employed to advantage when they may be used a large number of times in a more uniform structure. Since an arch of the Monier variety was in this case out of the question, it was necessary to build the reinforcement so that it formed a supporting arch structure, upon which the forms could be hung in such manner that the space below the arch would be entirely free from scaffolding. In Fig. 306 these steel supporting members are shown in detail. There are six members side by side in a distance of 0.9 m. (2.95 ft.), and they are secured against lateral tipping by a light horizontal connection. At the grades of the upper and lower layers was a network of longitudinal and cross wires 7 mm. in.) in diameter, so as better to tie together the concrete. The 3.5 cm. (1.38 in.) form boards were supported by 50X50X7 mm. (2X 2X1Q in. approx.) angle irons, bent into a curve, and hung from the arch rein- forcement by 15 mm. in. approx.) screw bolts at intervals of 80 to 100 cm. 282 CONCRETE-STEEL CONSTRUCTION H < rizontal ti< < > eme of Ho w m I o 6 ARCHES 283 (30 to 40 in.). When the forms were removed, the bolts were withdraw^n from the concrete matrix, and the holes so left were filled with cement mortar. The steel arch-supports with fixed ends consisted of four angles 50X50X7 mm. (2X2X1% in. approx.), which were held together at distances of 50 cm. (19.7 in.) by plates. The ends were supported in their exact position by pairs of wedges. The calculation of the arch supports was made on the assumption that half the arch would be placed at one time, although this unfavorable load- ing could be obviated by commencing the concreting at both springings and at the crown at the same time. In the foregoing type of construction, the concrete is loaded much less than in the usual reinforced arch, because the dead load is carried exclusively by the arch-supports, and the weight of the additional superstructure and of the live load is carried jointly by the concrete and the steel in proportion to their elastic deformations. A greater advantage of the above described construction is that the removal of the forms can be done earlier (after about eight days) and without special care, whenever the concrete has become so hard that it can be left free between the steel ribs with safety. In the usual reinforced concrete arch the scaffolding should not be removed inside of about four weeks. A bridge of the Melan type with three hinges is described with all details in " Beton und Eisen," No. Ill, 1903. In very large spans the girders, later to be concreted, give a guaranty for the maintenance of the proper form of the arch, which is of great importance, which can be secured with w^ooden centers only through considerable care. The girders always carry the whole or a greater part of the centering. When a wooden scaffold is employed in addition, it can be made much hghter than is otherwise necessary. A noticeable appHcation of the Melan system was also made in the Chauderon-Montbenon bridge in Laus- anne, where the high supporting scaffolding was omitted. Newer Methods of Arch Construction in Reinforced Concrete. — In the foregoing, it was always assumed that the carrying part of the arch was rectangular in section. The serviceable application of reinforced concrete to arch-like structures of other shapes has lately been made. There may be mentioned: (a) Reinforced arches of rectangular section, to which the throroughfare is attached by hanging columns. In this case the thoroughfare can also act as a tie so that all horizontal shear is taken from the abutments. (See ''Deutsche Bauzeitung, Zementbilage," Nos. 17 and 21, 1905). The best example of this kind is the railroad bridge over the Rhone at Chippis, with a span of 60 m. (197 ft.), in which the thoroughfare contains an expansion joint at the center of the span. (See " Schweizerisch Bauzeitung," 1907.) {b) According to a method of construction already much used in Switzer- land, by Maillart of Zurich, for reinforced concrete arch bridges, the side walls and the deck, which both consist of reinforced concrete, can be built in cantilever form decreasing in depth from the abutments to the crown. Since the combina- tion of the several parts of this section is perfect, it can be employed in its entirety for carrying stresses. This construction appears to be specially adapted only for three-hinged arches. See " Schweizerische Bauzeitung," October i, 1904. (c) The arch can also be constructed cf separate ribs of rectangular section 284 CONCRETE-STEEL CONSTRUCTION placed side by side, wherein spirally reinforced concrete affords extra strength. The thoroughfare is then supported by columns from the arch ribs. They are prevented from moving laterally by bulkheads or continuous slabs. If the latter are so built as to be flush with the upper layers of the ribs at the crown and with the lower layers near the abutments, this arrangement provides con- siderable resistance against deformations from normal stresses and from change of temperature. {d) Spirally reinforced concrete is especially applicable to bridges and to arches consisting of separate ribs of octagonal or rectangular section, and also to truss-like members in the form of beams or arches. Banded concrete is, however, not yet well known, so that designers do hardly more than experiment with it. See Considere, Essai a outrance du Pont dTvry," " Annales des Fonts et Chaus- sees," No. 3, 1903, and " Beton und Eisen," No. i, 1904. RESERVOIRS. The Monier system, with its network of wire, is well adapted to the con- struction of reservoirs of all kinds. When a serviceable method of computing slabs was found, the thickness of walls and amount of reinforcement was determined with a proper margin of safety. Even earlier, an extensive apphcation of reinforced concrete was made in the construction of various reservoirs for industrial purposes. Because of the satisfactory experience with these structures, reinforced concrete now occupies an even broader field in this line. In the catalogue issued by Wayss and Freytag, in 1895, are found several examples of this apphcation, explained by sketches. There may be mentioned: for paper mills, — bleaching cyHnders, drip boards, chloride holders, acid tanks, mixing vats, settling basins; for breweries, — barley-soaking vats, drying arches, icehouses; for tanneries, — tan pits, etc.; for pulp mills, — similar parts to those in a paper mill, and so on. If the tanks are circular, the horizontal reinforcement has to resist the tangen- tial stress, the vertical rods acting only as ties. Long walls of rectangular tanks are rigidly connected to the bottom, which is always constructed as a reinforced slab monolithic with the walls. In that case the largest stresses are in the vertical reinforcement. The water tightness is obtained by a water-proof cement coating on the inside. The cylinders are protected from the injurious influence of acids by a cover- ing of porcelain tile. The construction of bleaching cylinders with brick or mass concrete walls has proven unsatisfactory, since the heating through of the relatively thick wall takes too long. When the hot paper stock is introduced, the inside is warm and the outside cold, and cracking takes place. Cylinders of reinforced concrete do not possess this disadvantage, since with the thinner walls, a quicker equalization of temperature results, and furthermore, the reinforcement resists the stresses produced. Several constructions adapted to the needs of special industries will be described in detail. The largest water supply reservoirs may be built of reinfored concrete in various ways. Either the walls may be made of mass concrete or masonry, RESERVOIRS 285 and reinforced concrete used for roof and partitions; or bottom, walls and top may all be built of reinforced concrete. An example of the first kind, which is most often employed, is shown in Fig. 307. The top, when built in reinforced concrete, offers economic advantages over the usual arched construction only when the price of gravel and its transporta- tion cost is high. Fig. 307. — Water reservoir with reinforced concrete cover. Reservoirs entirely of reinforced concrete, up to about 300 cu.m. (79,250 gals.) capacity, are usually of hemispherical or cylindrical form, with a dome-shaped top. Fig. 308 shows a section through a hemispherical reservoir, such as are commonly builj: for small water sup- plies. With graphical methods of calculating domes, the various stresses to resist loads in the directions of the meridians and parallels may be found, and the reinforcement corres- pondingly determined. Cylindrical reservoirs with flat dome-shaped tops must be provided with a heavy tension ring to resist the horizontal shear of the dome and transfer it to the cylinder. Fig. 310 Fig. 308. — Water reservoir in hemispherical form, shows a water reservoir 7 m. (22.9 ft.) in diameter, reinforced with round rods and having I-beams as an inclosing ring, and with a top constructed to resist heavy street traffic. To construct of reinforced concrete the whole of a large reservoir of elongated rectangular plan is not always as economical as a well built reservoir of mass con- crete. Under certain circumstances the former allows, however, a better employment of the ground area available and is to be recommended with ex- pensive gravel and sand. In Fig. 311 is shown such a reservoir, of 4000 cu.m. (1,057,000 gals.) capacity, for Brussels. The side walls, which withstand a water head of 2 m. (6.6 ft.) were reinforced concrete 12 cm. (4.7 in.) thick, spanning between the top and the bottom. The beams in the top form rectangular panels, so that the deck slab was reinforced in two directions. Water towers are well adapted for construction in reinforced concrete, the substructure as well as the tank being of the same material. The structural 286 CONCRETE-STEEL CONSTRUCTION members may be placed on the outside and thus add to the architectural ap- pearance. Figs. 312-313 show a cyHndrical tank with dome-shaped bottom, supported on masonry walls. The shear produced by the arched form of the bottom was resisted by eight round rods 40 mm. (i^ in.) in diameter, forming a ring. Fig. 309. — Cylindrical water reservoir with dome-shaped top. J'lacheiserv In Fig. 314 is shown the reservoir of the Gross wartenberg water supply system. The tank overhangs the supports, because of the interior construction. The insulating wall is also constructed of reinforced concrete. The details of construc- tion, and all the arrangements are shown in the figure. Figs. 315-316 show a vertical section and general view of the water tower in Rixensart, in which the substructure consists of a pleasing reinforced concrete frame, the panels of which are filled with brickwork. As far as the tensile stresses involved are concerned, gas receiver frames can profitably be constructed of reinforced concrete. For the upright guides, special concrete piers must be built, both out- side and in connection with the receiver walls. See Fig. 317. In the cylindrical walls of such a tank, besides the circumferential stresses, bend- ing stresses in a vertical direction exist, developed because the cylindrical walls are prevented by their rigid connection with the bottom from assuming the deforma- tion corresponding to the circumferential stresses, that is, of increasing* in di- ameter. Since the cylinder is prevented to the greatest extent from enlarging near the base, vertical tensile stresses appear there on the inner side, while they act near the top on the outer side. These vertical stresses are cared for by Fig. 310. — Cylindrical water reservoir. Ten- sion ring of the dome- shaped top and con- nection with the cylin- drical sides. RESERVOIRS 287 vertical steel near the inner and outer faces of the cylindrical wall, which must be strongly connected with the bottom. Concerning an investigation into the method Fig. 311. — Water reservoir of 4000 (141,275 ft^) capacity for Brussels. Cross-section. of calculating these bending stresses by Reich, see " Beton und Eisen, 1907, No. 10. Silos. — Silos are bins for certain dry materials, such as grain, coal, cement, ore, broken stone, etc., in which, because of their shaft-like arrangement, the material which was received above, can be extracted when necessary from the lowest point of the bin. In this connection may be distinguished large silos 288 CONCRETE-STEEL CONSTRUCTION I^iG- 313- — Cylindrical tank with dome-shaped bottom. Detail of reinforcement. RESERVOIRS . 289 Fig. 314. — Grosswartenberg water tower. 290 CONCRETE-STEEL CONSTRUCTION without separate partitions, or such as are relatively large in area with respect to their height; and cellular silos, or silos consisting of compartments of rectangular,, or better, of square, round, or hexagonal section. Examples of the first variety, without interior division walls, are the ore bins for the Burbach smelters, the general arrangement and details of which are given in Figs. 318 to 320. The foundation of these bins consists of a continuous reinforced concrete slab 70 cm. (27.6 in.) thick, which distributes the load uni- formly upon the soil at a stress of 1.5 kg/cm^ (1.5 tons/ft^). Since the columns, were spaced 3.33 m. (10.9 ft.) apart in both directions, reinforced concrete beams- Fig. 315. — Water tower in Rixensart. Flg. 316. — Water tower in Rixensart. were built into the foundation slab in each direction under the rows of columns^ the square panels between being correspondingly reinforced. The 60X60 cm. (24X24 in.) openings in the funnel-shaped square panels in the bottoms of the bins were supplied with slide valves. Beams also extend in both directions over the rows of columns. The outside walls 6 m. ^23. 6 ft.) high, were anchored to the beams in the bottoms of the bins by ribs 25 cm. (9.8 in.) thick. Between them the outer wall acts as a continuous reinforced concrete slab. The top of the wall is stiffened by a somewhat thicker rib. Three raikoad trestles enter the bin on 25 cm. (9.8 in.) thick supporting walls, 6.66 m. (21.8 ft.) apart, carried by the lower columns. The trestle stringers are continuous reinforced concrete girders. Spaced 26.4 m. (87.2 ft.) apart are expansion joints through floors and walls.. All exposed edges of ribs, supporting walls and columns are protected against wear by channel and angle irons. SILOS 291 A smaller silo, without interior partitions, is shown in section and plan in Fig. 319. The coal in storage can be discharged directly into the boiler room through several openings. The walls carry the lateral pressure of the stored material horizontally to the columns which are tied together by the roof, and between the funnels by cross-beams, 45X45 cm. (17.7 in.) in section. In the silo for the Odenwalder copper wwks (Fig. 321) a trestle of the kind described above is built over the several pockets. The latter are of an elongated Fig. 317. — Gas holder of the Jagstfeld Railroad Station. rectangular form, and make a row along the length of the building serving for the storage of crushed porphyry of various sizes of particles. The building is described at greater length by the author in Beton und Eisen," No. i, 1903, with details, to which reference should be made. The malt silo of the Lowen brewery in Munich, contains cells 3.5X3.75 m. (11. 5X12. 3 ft.) in plan, 16.5 m. (54 ft.) high, with a capacity of 2200 hi. (6242 bushels). The points of intersection of the walls are supported by reinforced concrete columns, which carry the load to a continuous foundation slab, i m. (39.4 in.) thick, so that the foundation pressure is only 2.5 kg/cm^ (2.5 tons/ft^). 292 CONCRETE-STEEL CONSTRUCTION The outer walls show pleasing reinforced concrete ribs, the panels between which are filled with brickwork. Between the latter and the outside silo walls an Fig. 318. — Section and plan of the Burbach ore bins. insulating air space is left. The silos are emptied by screw conveyors located at the level of the bottoms of the bins, in the lateral passage (Fig. 322). While in usual structures the silo funnels are constructed simply as hanging pyramids, the large ones in the saw-dust bins of the pulp mill shown in Fig. 323 SILOS 293 are supported by a T-beam deck, and stiffening walls are erected upon this. The outside walls 8 m. (26.2 ft.) apart are also tied together by an anchor beam in the center of each panel between the cross walls. The principal reinforcement of the outside walls runs vertically between the two horiontal wall beams. Fig. 324 gives a view of these silos, which are erected on a high brick substructure. Arched beams are used for the roof girders. k tp9 4 Horizontal-section. Fig. 319. — Coal pocket in Kirn. Examples of cellular silos with very large pockets of 6.7X8.5 m. (21.9X27.9 ft.) ground plan, are those of Figs. 325 and 326, showing the coal pockets of the Volklingen smelters. In this case it was essential that the underside of the floor should be absolutely flat, making it necessary that the beams and girders extend above the slab. The silo rested on a continuous reinforced concrete foundation slab. Details of the girders are shown in Fig. 326. Reinforced concrete can doubtless be considered the best building material for silos; since, aside from its fire-proof qualities, the reinforcement of the walls affords at the same time the best anchorage for them. It so happens most advantageously that both functions of the horizontal reinforcement in the cell SILOS 295 walls are not necessary at the same time; the anchorage stresses being greatest with simultaneous filling of adjacent pockets, when, however, the bending stresses are nill; and with maximum bending stresses, as when a single j)ocket is filled, the anchorage stresses diminish one-half. As to the bending with axial tension, which occurs in this case, see page 127. The constructive excellence of reinforced concrete has further been proved, because, during the past year, in the construc- tion of a large silo foundation, an additional application has been found. The following is a single example of silo construction executed by Wayss and Freytag this year and last. Fig. 321. — Reinforced concrete silo for the Odenwald Copper Works at Rossdorf. Fig. 329 is a view of the reinforcement in the funnel of the silos of seven hexagonal pockets 16 m. (52.4 ft.) high, for the Alsen Portland cement works in Itzehoe. Since the walls of square cells are to be considered as fully restrained at the corners when adjacent pockets are filled, the moment at the centers of panels is — , and in the corners — . For this reason the walls should be twice as thick 24 12 at the corners as at the centers. In Fig. 327 is shown the reinforcement of the walls for a silo with 44 square cells of 4X4 m. (13. i ft.) ground plan for the Alsen Portland Cement Works at Itzehoe. It is seen that the wall thickness is doubled at the corners. The outside walls are covered with brickwork, with an insulating air space between it and the reinforced concrete walls, and its support '296 CONCRETE-STEEL CONSTRUCTION Fig. 322. — Lower Brewery, Silo, Munich. Fig. 324. — Sawdust bins of the Waldhof Pulp Mill. Half longitudinal section Cross-section. Longitudinal section. Fig. 323. — Sawdust bins of the Waldhof Pulp Mill. SILOS 297 SO 01 t VSO 085 i I 080 0/J XJL HI oes 065 Fig. 325. — Valklingen coal pockets. Fig. 326. — Valklingen coal pockets. Details of the rods in the girders in the bottoms. 298 CONCRETE-STEEL CONSTRUCTION is obtained at certain levels from projecting reinforced concrete ribs monolithic with the concrete wall. A view of two silos, each with seven hexagonal cells, is shown in Fig. 328. The bending moments in them are relatively less than in square pockets. Smelters require a special variety of silo, called an ore pocket, the cells of which are not arranged in two or more separately entered groups. In Figs. 330 and 331 are given a cross-section and a longitudinal section of such an ore pocket for the Mosel smelter at Maiziere, which is 178 m. (584 ft.) long. The sloping floor slabs were supported by very heavy cross-beams, upon which also rested the reinforced concrete columns supporting the three railroad trestles. Because of the great length of the construction, four expansion joints were installed. For Fig. 329. — Cement bins. Reinforcement of the funnels near the hexagonal cell bottoms. 300 CONCRETE-STEEL CONSTRUCTION Fig. 331.- Longitudinal section. SILOS 301 especially designed to resist shearing stresses, special bent rods being employed, while the upper and lower rods were not diverted. In the lower part, the longitudinal walls spanned directly from one girder to SILOS 303 the next, while in the upper, open part tney extended between vertical reinforcing beams which were anchored together by tie-beams at the tops. The column arrangement is not regular, being made to conform to the requirements of the necessary lateral passages which required a i)eculiar modification from that used in the usual sections of silos. The emptying of the pockets takes place at the lowest point and also at the center, through properly con- structed gates which were concreted into the outside walls and the sloping bottoms by angle iron frames. Three parallel railroad tracks ran over the whole length of the bins, two of i m. (39.37 in.) gauge from the mines, and one of standard gauge for the transportation of coke. The extent of this plant required for its construction about 500 t. (550 tons) of fabricated round rods. The time of construction was somewhat more than six months from the time of starting the foundations. In Fig. 334 is show^n the reinforcement of the lateral beams, and in Fig. 335 that of the sloping bin bottoms. The ore pockets in Dudelingen have a capacity of about 5000 cu.m. (176,500 cu.ft). The heavy cross walls which carry the sloping bottoms and the longitudinal walls, are rein- forced and constructed as somewhat overhanging. The beams of the ore crusher floor are stiffened against the heavy vibrations of traffic by special cross- beams. The roof, which is supported by free columns 7.3 m. (24 ft.) high, is arched, and is supplied w^ith tension members. To resist the wind pressure in a longitudinal direction along the roof, diagonal members are supplied. In this case, also, several expansion joints were installed, to effect which the necessary columns and cross walls were built double. 304 CONCRETE-STEEL CONSTRUCTION Fig. 336 shows a cross and a longitudinal section of the building, and Fig. 337 shows the placing of the reinforcement. Fig. 336. — Diidelingen ore pockets. Cross and longitudinal sections. The Getreide silo at Hafen in Genoa is shown in Fig. 338. The building consists in the main of an enlargement of the silos previously built by Hennebique. Fig. 337. — Diidelingen ore pockets. Placing reinforcement. Besides a large granary, divided into rooms, a new building was erected 65 m. (203 ft.) long, 40 m. (131 ft.) wide, and 30 m. (98 ft.) high, which contains SILOS 305 126 bins, 3X3 m. (9.8 ft.), and 3X5 m. (9.8X16.4 ft.) in plan. These increased the capacity from about 28,000 t. (30,800 tons) to 50,000 t. (55,000 tons). The whole building rests on reinforced concrete columns 90X90 cm. (35.4 in.) thick, which have a carrying capacity of 400 t. (440 tons). Under the silos are five loading tracks, the substructure for which, together with the loading platform, are of reinforced concrete. The total load of the silo and tracks was uniformly distributed by means of a continuous reinforced concrete slab upon ^ twenty -year-old fill of an inkt from the harbor, in such manner that the maxi- mum soil presure was only 1.7 kg/cm^ (1.7 tons/ft^). The construction of the slab with quasi-inverted arches was decided upon because of the necessity of having a flat top surface, so as to secure enough space for the loading tracks and platforms. Fig. 338. — Getreide grain isilo, Genoa. Cross-section. Furthermore, this arrangement enabled the loading tracks to be brought close to the rigidly set automatic scales connected to the mouths of the bottom funnels produced by concreted inclined surfaces on top of the horizontal bin bottoms. The full load of the bottom is hung upon the cross walls, which are correspond- ingly reinforced so as to form, in combination with the bottom and floor slabs, a continuous beam 15 m. (49.2 ft.) deep, with three spans each 8 m. (26.2 ft.) long. The construction of the large floor, which was divided into rooms, is clearly shown in the section, w^hile the succeeding illustrations give pictures of the method of erection of the building. During the period of construction of 200 working days, the following quan- tities of material were used : About 900 t. (990 tons) steel almost entirely of German manufacture. About 2,900,000 kg. (6,393,400 lbs.) cement from the mills of Flli. Palli Caroni Deaglio Casale. 306 CONCRETE-STEEL CONSTRUCTION About ii,ooo cu.m. (14,380 cu.yds.) sand, gravel and fine slag, largely from the shores of both rivers. The concrete was mixed by an electric impulse mixer with fixed drum. In the best months, from August to November, when the daily rate was about 60 cu.m. (78 cu.yds.) of finished concrete, the monthly performance averaged about 200,000 lire ($38,600). It is very important, in the design of silos, to know the lateral pressure exerted against the walls by the material in the pockets. In silos of large size, without cross walls, or those with very long rectangular cells, the computation is to be Fig. 339. — Getreide grain silo, Genoa. Reinforcement of the bottoms. made according to the usual formulas for earth pressure. Neglecting the friction against the walls, the total lateral pressure on the height h is P = ir/^2tan2(45°-^/2), and the pressure on a differential area at the height h is = tan2(45° -0/2). For several materials the quantities in Table XXXVII can be employed: Material. kg/m.^ lbs/It.^ kg/m^ lbs/ft' Gas coal 800-900 50-56 45 146 30 Cement 1400 87 40 305 62 Small slag 1600-1800 100-112 45 290 59 Malt - 530 33 22 240 49 Wheat 820 51 25 333 68 Minette (ore) . . 2000 125 45 343 70 Fig. 341. — ^View of the silo in Genoa with the colonnade toward the harbor, 308 CONCRETE-STEEL CONSTRUCTION In celled silos of considerable height, these figures give very heavy pressures; in the lower parts, and the lightening effect of the friction of the material against the walls may be considered. Two pubHcations exist agreeing in all essentials concerning the computation of the lateral pressures in silo cells, by Janssen in " Zeits- chrift des Vereins deutscher Ingenieure," 1895, p. 1046, and by Konen in the " Zentralblatt der Bauverwaltung," 1896 p. 446. The friction between the mate- rial and the side acts so that the lateral pressures can never exceed a certain frac- tion of /'max.- This fraction may be introduced and the weight of any layer will then become a function, in the computation of the frictional resistance developed. Fig. 342. In Fig. 342 let it be assumed that in a full cell, at a depth a layer of thickness d x\'$> cut out, then the following forces are active.* qF^ from above, where q is the special pressure in a vertical direction; Fydx^ the weight of the layer; (q-\-dq) F, the vertical resistance from below; pUdx, the horizontal pressure against an area Udx; pU t2in(j)idx, the frictional resistance of the walls at this horizontal pressure and acting upward; where F is the area of the cell (Trans.); and U is the circumference of the cell (Trans.). From the equating of the vertical components of the opposite forces, it follows that dq=dx( y—p^t2in In a material devoid of cohesion under a vertical pressure q, there is developed a lateral pressure. tan2(45-^/2), whence dq=dx(^-q tan2[45° -9^)/ 2] ~ tan . * See Konen " Zentralblatt der Bauverwaltung," 1896. SILOS 309 If ?n is inserted in place of the constant factor tan^ (45°— ^ A 1 ' t 1 Y z ^ K_ "}< \ — ~ < jyc > A \ ^ Fig. 2. B Fig. (&) With double reinforcement. According to the nomenclature of Fig. 2, the distance x of the neutral plane may be determined by the quadratic equation : x^ + 2xn — , ^-rihFe + h'Fe)\ 0 0 X having been ascertained, the compressive stress in the concrete will be 6Mx Ob- bx^iS^i - x) ^ 6F/n{x - h'){h-h') the tensile stress in the lower reinforcement is ob{h—x)n <7e = ; X and the compressive stress in the upper reinforcement is , Gb{x — h')n X / APPENDIX 325 2. T-Beams The effective width of slab h is to be taken as h ^ J/, in which / represents the distance between the supports of the beam, but h must not be greater than the beam spacing. Tw^o different cases exist: {a) x^d (see Fig. 3). X i )• Fig. 3a and 3^. The formulas given under A, 1 a apply in this case. Under certain circum- stances the shear in the rib and the adhesive stress on the reinforcement at the support must be calculated. These are -Co boih-x/sY a i \ A Circumference of the reinforcement {b) x>d (see Fig. 4). Ignoring the small compressive stress in the rib, there are found 2nhFe-{-hd'^ J/ Fig. 4. x = (7e = 2{nFe + hd) M and and d d^ y ^= . 4" 2 6{2x—dy Ob- OeX Fe{h—x-\-y) ^" n{h-x)' B. COMPRESSION The reinforcement of columns must aggregate at least 0.8% of the total area. The reinforcement subject to stress is to be secured against buckling by lateral ties (usually of round iron). The spacing of these ties should not exceed the diameter of the column. I. SUPPORTS FOR SAFE LOADS {a) Central Loading. — If Fb represents the area of the concrete member, the permissible load will be P = ob{Fb-^nFe), 326 CONCRETE-STEEL CONSTRUCTION where ^ = 15. Further, Fb+nFe „ , I'd re-] • n (b) Eccentric Loading (Bending with Axial Stress). — The computation can be made in the same manner as for a section of homogeneous material, provided that in all expressions representing the areas of sections and their moments of inertia, the area of reinforcement is to be computed at ^ = 15 times its actual value compared with the concrete section. If tension occurs, the reinforcement located on the tension side must be capable of carrying it. No danger of buckling exists, provided the supports have at least the following dimensions: Stress of the Concrete, kg 'cm 2 1 lbs /in2 Least Diameter of Round Column in Terms of its Length. Least Length of Short Side of Rectangular Column in Terms of its Length. 427 1/18 1/ 21 35 498 1/20 40 569 1/16 1/19 45 640 1/15 1/18 50 711 I./14 1/17 Since few experiments exist concerning the resistance to buckling, smaller dimensions than those given above should not be used. IIL EXAMPLES OF THE METHOD OF COMPUTATION FOR A FEW SIMPLE CASES A. SIMPLE BENDING 1. Slabs (a) Freely Supported Slabs with Single Reinforcement. Clear span 2.00 m. Thickness of slab o-i5 " Distance between supports 2.15m. The live load /> = iooo kg/m^, the dead load ^ = 0.15X2400 = 360 kg., and thus the total load ^ = 1360 kg/m^, and the moment for i m. breadth (see Figs. I and 5), M = i36oX^^^^-Xioo = 78,583 cm..-kg. APPENDIX 327 In a breadth of i m. were 9 lower rods of 10 mm. diameter, making F^. = 'j.oy cm2 For ^ = 13.5, ^ = 15, and 6 = 100, the distance x of the neutral axis below the top of the slab is cm. fi y>7 1 , ^1 mmm J^Jj§^^^$J^ 1 1 Stress in the concrete 2M Fig. s. 2X78,583 bx{Ii-xls) iooX4-39(i3-5 -4-39/3) Stress in the steel 29.7 kg/cm^. M 78,583 7:^-^923 kg/cm2. Fe{h-x/s) 7.07(13.5-4.39/3) The shear at the support is 7 = ^X1360X2.0=1360 kg., so that the shear- ing stress is V 1360 b{h-x/s) 100(13.5-4.39/3) = 1.13 kg/cm2, which is thus less than the permissible value of 4.5 kg/cm^. The adhesive stress of the above reinforcement at the supports is looX 1. 13 Circumference of the reinforcement 9X1. 0X3. 14 4.0 kg/cm^ (b) Freely Supported Slabs with Double Reinforcement. — The dimensions and loading of the slab are the same as in the preceding example, so that ^ = 78,583. Besides the lower reinforcement consisting of 9 rods, 10 mm. diameter, there is an upper layer of 6 rods of 10 mm. diameter. Then F/ = 4,7^, /i' = i.5 (see Fig. 2, p. 324). The distance x of the neutral layer below the upper surface of the slab is to be computed from the quadratic equation x^ + 2xn^^— = 2~{hFe + h'F/), or :>;2 + 2X:vXi5^^^^^^- = 2^(i3.5X7-07 + i. 5X4.71). 328 CONCRETE-STEEL CONSTRUCTION Whence or x = 4.o^ cm. Then the compressive stress on the concrete is 6Mx Ob bx^{T,h -x) + 6Fe'n{x-h') {h-h') 6X78,583X4.05 100X4-05^(3 X 13-5 -4-05) +6X4-71 X 15(4-05 -i.5)(i3.5- 1-5) The tensile stress in the lower reinforcement is = 26.25 kg/cm-. = 12.1 cm., ^^^^^,(^^^ 26.25(13.5-4-05)15 ^ 3 ^1^^, X 4.05 and the compressive stress in the upper reinforcement is ^^,^ ..(^-/0^ ^ 26.25(4.o5-i.5)i5 ^ g k X 4.05 ^ The distance between the centroids of tension and compression in this case will be M ^ 78,583 FeOe 7.07 X91I whence ^ ^360 , , 2 100X12. 1 100X12. 1 ^ ^' ' and the adhesive stress on the lower reinforcement at the supports is hTQ 100X1.13 1 / s T\ = — — — = ~ =4.0 kff/cm^ Lircumierence 01 remiorcement 9X1. 0X3. 14 2. T-Beams Simple, Freely Supported T-Beams. — Clear span 10.60 m., distance between mpports 11.00 m., live load 400 kg/m^. Load per running meter of beam: Live load, 400X1.7 =680 kg. Asphalt layer, 30X1.7= 51'' Dead load, 2400(0.25X0.50 + 1.7X0.10) =708 Uniform total load, approximately, 1440 kg/m^. M = ^- = i44oX-^Xioo = 2, 178,000 cm. -kg. APPENDIX 329 The reinforcement consists of 8 round rods, 28 mm. in diameter with F^ = 49.26 cm-. The distance of the neutral axis from the upper layer of the slab (see Fig. 4, p. 325) is to be computed by the formula: that and or 2nhFe+hd^ ^~'2{nFc + bdy ^^2Xi5X54X49.26 + i7oXio2_ ^ 2(i5X49-2() + i7oXio) d d^ 10 19.84-— +■ 10^ 2 ' 6{2X-d) 2 6(2X19.84-10) y = 15.40. d. o lo 1 aso /,.0S Then there results finally, M - lu 6o - -11.00 r 6 -0.25 Fig. 6a and 66. 2,178,000 (Te = Fe{h-x-^y) 49.26(54.0-19.84 + 15.40) = 892 kg/cm^, (76 = OeX 892X19.84 = 34.5 kg/cm2. n{]i-x) 15(54-19.84) At the supports the shear is greatest, being F = i44oX = 7632 kg., 2 so that the shearing stress in the concrete is V 7632 hQ{h-x + y) 25(54-19-84 + 15-4) 6.2 kg/cm^, and the adhesive stress at the supports, on the four round rods of 28 mm. diameter, which extend to them, is 25X6.2 , , ^ ^i=-v7-^ — ^^ = 4-4 kg/cm-. 4X3.14X2.8 The shearing stress reaches the maximum permissible value of 4.5 kg/cm^ at the point at which 7632X4-5 6.2 = 5540 kg., 330 CONCRETE-STEEL CONSTRUCTION that is, at a point ^^1^1 — 5540 .^^ ^^^^ p. ^ 1440 and the total diagonal tension Zi, which must be carried by the bent rods will be Zi = ^(6.2-4.5)XiX25 = 2i8o kg. V 2 ^1 Fig. 6c. Fig. 7. When, therefore, the four upper rods of 28 mm. diameter are bent upward through a distance of 1.45 m., they will carry a stress of only i7e = — TTT 7 = 09 Kg/cm'^.* 4X6.16 Continuous T-Beams Each running meter of the beam (Fig. 7) has to carry a dead load of ^ = 2000 kg., and a live load of p = 1,600 kg. The following moments therefore result: (a) At 0.4/ of the first span: Mg= +0.080X2000X6. 752X 100= + 728,960 cm. -kg. —Mp= —0.020X3600X6. 752X 100-- — 328,032 " -\-Mp= +0.100X3600X6.75^X100= +1,640,160 so that + 2,369,120 (b) Over the center supports: Mg= — 0.10000X2000X6. 752X 100= — 911,200 cm. -kg. —Mp= —0.11667X3600X6.752x100= — 1,913,575 " +Mp= +0.01667X3600X6.75^X100= + 273,415 so that Mmax= -2,824,775 " (c) In the center span: Mg= +0.025X2000X6.75^X100= + 227,800 cm.-kg. —lfp= — 0.050X3600X6.752X100= — 820,080 +Mp= +0.075X3600X6.752X100= + 1,230,120 i +Mmax= +1,457.920 ( -Mn,ax=- 592,280 SO that *The assumption of 4.5 kg/cm^ for the shearing stress in connection with the computation of the bent rods is not free from objection. See in this connection, p. 183 and 187. — (The Author). APPENDIX 331 These moments give the following stresses: (a) At 0.4/ of the first span: Since the girders are 4.5 m. apart, the permissible width of slab is ^=^/3=— J^ = 2.25 m. Fe = four round rods of 32 mm. diameter =32.17 cm-. // = 77 cm., ^/=i2 cm., /) = 225 cm. (see Fig. 8); and the distance x of the neutral axis from the top of the slab computed by the formula 2nhFe + bd''^ IS Further y=x ^ 2{nFe + bd)' 2X15X77X32.17 + 225X12^ 2(15X32.17 + 225X12) d d-^ 16.8 cm. 2 6{2x—dy and fmally ^ ^ 12 12^ y = i6.8 H ^ 2 6(2X16.8-12) II. 9 cm., M 2,369,120 FeQi—x+y) 32.17(77 — 16.8 + 11.9) = 1020 kg/cm^, oex 1020X16.8 , , ^ <76=— ^ r= / A qT="^9-o kg/cm^. n{h-x) 15(77-16.8) r ' — s OJO k-O.JJ i - - f I- Fig. 8. ffl Fig. 9 ^,i^z; N \ 1j)Q . - ♦ * The stress in the reinforcement can easily be reduced below 1000 kg/cm^, by replacing one 32 mm. rod by one of 34 mm. diameter. {b) Over an Intermediate Support. — Since the tensile strength of the concrete is ignored, the floor slab receives no consideration in connection with the nega- tive pier moment, so that in the computations only a rectangular section (see Fig. 9) of breadth ^ = 35 cm. is employed. _ 4X3-2^X- , 2X3-4-X- 2 Fe=^ \ + = 50-33 ^m2; 4 4 ^ = 35 cm., /^ = io7cm., n = \^. nFe\ , A , 2hh\ 332 so that CONCRETE-STEEL CONSTRUCTION x = 15X50.33 Further 35 :x: = 49.5 cm. 2M L ^ 15x50.33 J Ge 2X2,824,775 ^ 35X49-5 107 3 F.(//-x/3) 2,824,775 50.33 107 49-5 621 kg/cm^. (c) At the middle of the center span: + i/max= +1,457^920 Cm.-kg. 3X3.22X7: re = =24.13 cm^, 6 = 225 cm., /^ = 77 cm., ^/=i2cm. so that, as under (a). 2 X 15X24.13X77 + 225X122 :x; = r — — = 14-4 cm., 2(15X24.13 + 225X12) 12 :y = i4.4 + 12^ M Oe- Gh- 2 6(2X14-4 — 12) 1,457.920 FeQi-x+y) 24.13(77-14.4 + 9.8) GeX 833X14-4 01/ < — - = — - = i2.8 kg/cm^ n{h-x) 15(77-14.4) -Mmax= -592,280 Cm.-kg. iX3-4^X7r 9.8 cm. 833 kg/cm2; = 77 cm., F. nF 9.08 cm2, 6 = 35 cm. or x = 15X9-08 .N- 2x35x77 ] 15x9-08 J' ajj 35 L :x: = 20.9 cm.; 2M 2X502,280 , , 9 r- = 23.2 kg/cm^ Fig. 10. Gh = hxih—xj-x) I 20.9 ^ '^^ 35X20.9(77 M Ge = Fe{h-Xls) 592,280 , , „ -7-^ r = 932 kg/cm2. 9.08 77 20.9 The computation of the shearing stresses follows, as in example 2. APPENDIX 333 B. COMPRESSION, COLUMNS The intermediate supports of example 3 (ignoring the continuity) have to carry a load of P = 6. 75(2000 + 3600) =37,800 kg. The section is 35X35 cm., and 4 round rods of 24 mm. diameter, with F^ = 1 8. 1 cm-, are employed. Then 7^6 = 1225 cm^. Fe=i8. 1 cm2. P 37,800 , , „ ^^ = -E^-T~^ = 7- o =25.3 kg/cm2, Fb + nFe 1225 + 15X18.10 ^ ^ &/ ' (76=^(76 = 25.3X15=380 kg/cm-. REGULATIONS OF THE ROYAL PRUSSIAN MINISTRY OF PUBLIC WORKS, FOR THE CONSTRUCTION OF REIN- FORCED CONCRETE BUILDINGS. MAY 24, 1907 L GENERAL A. TESTING Sec. 1 1. The construction of buildings or their structural parts in reinforced con- crete is to be subject to special supervision by the building inspectors. For this reason, when application is made for a permit for a structure in whole or in part of reinforced concrete, drawings, statical calculations and specifications must be submitted in which the general plans and all important details are shown. In case the owner or contractor does not decide upon the use of reinforced concrete until the work is under way, the building inspectors must insist that the above described drawings for the reinforced concrete work be filed a sufficient period before commencing the construction. Under no circumstances is work to be begun before permission therefor has been granted. 2. The specifications must state the source and the kind of concrete aggregates to be used, their proportions, the amount of water, and the compressive strength which is to be developed by 30 cm. (11.8 in.) cubes, 28 days old, made at the building site, and of the materials stated. If required by the building inspectors, the compressive strength must be shown by test before commencing work. 3. The concrete must be mixed in proportions by weight; a bag of 57 kg. (125.4 lbs.), or a barrel of 170 kg. (374 lbs.) of cement being the unit of measure. 334 CONCRETE-STEEL CONSTRUCTION The aggregates may be either weighed, or measured in vessels the capacity of which has previously been so arranged that the weight corresponds with the proportions already determined. 4. The contract is to be signed by the owner, and by the general contractor and the special contractor who is to do the work. The building inspectors are to be notified of any change of contractors. Sec. 2 1. The quality of the concrete materials is to be certified by an official test- ing laboratory. As a rule, such certificates must not be more than a year old. 2. Only such Portland cement may be used as fulfils the Prussian require- ments. The certificates of its quality must give particulars as to its constancy of volume, time of set, fineness, as well as of its tensile and compressive strength. The constancy of volume and time of set must be independently tested by the builder. 3. Sand, gravel and other aggregates must be proper for the manufacture of concrete, and the special purpose intended. The size of the particles must be such that the placing of the concrete, and its tamping between the reinforcing rods and between them and the forms can be done with certainty and without displacing the steel. Sec. 3 1. The method of computation must provide at least as much security as that provided according to the Leitsatze," section II, and according to the methods of calculation with examples in section III of these Regulations. This must be demonstrated by the contractor, if required. 2. In the case of new types of construction, the building inspectors may condi- tion the permit on the results of preliminary test structures and loading exper- iments. The latter are to be carried to the point of failure. B. CONSTRUCTION Sec. 4 1. The building inspectors may have the quahty of the materials in use in the work tested by an official laboratory, or in any other manner deemed suitable by them, and can also have tested the strength of the concrete made therefrom. The strength tests can also be made at the building site by a concrete press, the accuracy of which has been certified by an official testing laboratory. 2. The specimens to be so tested must be cubes, 30 cm. (11.8 in.) on an edge. These specimens are to be dated, sealed for identification, and stored until they have properly hardened, in accordance with the instructions of the building officials. 3. The cement must be delivered in the original package at the point of consumption. APPENDIX 335 4. The concrete must be mixed in such manner that the quantities of the several constituents always agree with the specified proportions, and can always be readily determined. Where measuring vessels are employed, they are always to be filled in the same manner and to the some degree of compactness. Sec. 5 1. As a rule the manipulation of the concrete must commence immediately after it is mixed, and must be completed before it has begun to set. 2. In warm, dry weather, the concrete must not lie unused longer than one hour, and in cooler damp weather, longer than two hours. Concrete that is not immediately placed, must be protected from climatic influences, such as sun, wind and heavy rain, and must be re-mixed before use. 3. The manipulation of the concrete must always be continuous until the tamp- ing is complete. 4. The concrete must be laid in layers not more than 15 cm. (6 in. approx.) thick, and should be tamped as much as the amount of water present will permit. Rammers of suitable form and weight are to be used for the tamping. Sec. 6 1. Before placing, the reinforcement is to be cleaned of all loose rust, dirt and grease. Special care is to be exercised to see that the reinforcement is in proper position, that the rods are of correct form and are properly spaced and kept in place by necessary contrivances, and are completely enveloped in special fine con- crete. Where the reinforcement in the beams is in several layers, each one must be separately embedded. Beneath the reinforcement must be a layer of concrete at least 2 cm. (J in. approx.) thick in beams, and at least i cm. (| in. approx.) thick for slabs. 2. The forms and supports of the floors and beams must be strong enough to resist bending, and be solid enough to withstand the tamping. The forms are to be so devised that they can be removed without disturbing such supports as may be necessary until the concrete has properly set. As far as possible, only unspliced lumber is to be used for supports. If splices are unavoidable, the supports must be strongly and firmly connected at the joints. 3. Column forms must be so constructed that the depositing and compacting of the concrete can be done through one open side, which is closed as the work progresses, so that it may be closely inspected. 4. At least three days' notice must be given the building inspectors l^efore the completion of the forms and the proposed commencement of the concreting for each story. Sec. 7 1. As far as possible, the several concrete layers must be deposited on fresh material; but in every case, the top of the old layer must be roughened. 2. When w^ork is to be done on hardened concrete, the old toj) surface must first be roughened, swept clean, wetted, and coated with a thin cement grout before new material is deposited. 336 CONCRETE-STEEL CONSTRUCTION Sec. 8 In the construction of walls and columns in buildings several stories in height, work on th^ upper members can be continued only after the structural parts in the lower stories have become sufficiently hard. Sec. 9 1. During freezing weather, work can be carried on only under such condi- tions as will preclude the possibility of injury by the frost. Frozen materials must not be used. 2. On the advent of mild weather, after a prolonged freeze (Sec. ii), work can be continued only after permission so to do has been obtained from the building inspectors. Sec. 10 1. Until the concrete has properly hardened, the structural parts must be protected against the effects of frost and against premature drying out, as well as against vibration and loading. 2. The interval which must elapse between the completion of the tamping, and the removal of the forms and supports must depend upon the prevailing weather, the spacing of supports and the weight of the members. The side forms of beams, column forms, and floor slab forms must not be removed in less than eight days, and the supports of the beams in not less than three weeks. With large spacings of columns and areas of members, the interval must be extended to six weeks. 3. In many-storied buildings, the supports under the lower slabs and beams can be removed only when the hardening of the upper ones has so far advanced that they can support themselves. 4. When the tamping was completed but a short time before frost occurred, the removal of forms and supports is to be done with the greatest care. 5. Should freezing take place during the period of hardening, in view of the fact that the latter may have been retarded by the frost, the intervals mentioned in section 2 should be extended by at least the frozen period. 6. For the removal of forms and supports, special contrivances (wedges, sand boxes, etc.) must be employed to prevent shock. 7. The building inspectors must be given at least three days' notice of the removal of forms and supports. Sec. 11 A diary must be kept of the progress of the work, and it must always be open for inspection at the building site. Freezing weather is to be specially noted therein, together with the temperatures and the hour of their observation. APPENDIX 337 C. REMOVAL OF FORMS Sec, 12 1. The structural parts must be exposed at different points as required by the building inspectors, so that the character of the work may be determined. The right is reserved to determine by tests the hardness and strength of ques- tionable parts. 2. Should question arise as to the proportions and degree of hardness of any portion, test samples may be taken from the finished parts. 3. Where load tests are considered necessary, they are to be conducted accord- ing to the directions of the proper officials. The owner and the contractor will be given due notice, and those interested informed. Loading tests should be made not less than 45 days after the concrete has set, and be restricted to the portion deemed necessary by the building inspectors. 4. In load tests of slabs and beams, the following method is to be adopted: In loading a whole floor panel, if g is the dead load, and p the uniformly dis- tributed live load, the applied load is not to exceed 0.5 ^ + 1.5 p. With live loads of more than 1000 kg/m^ (205 lbs/ft^), simply the live load may be used. If only a strip of the floor is to be tested, the load is to be uniformly distributed over a space at the center of the slab, the length of which is equal to the span, and a third of the span in breadth, but never less than i meter (3.3 ft.). The applied load in this case is not to exceed a value of ^ + 2 p. The dead load is that of all the structural parts, including the weight of the flooring and slab, the live load being of the character specified in Sec. 16, No. 3. 5. In any loading tests of columns, unequal settlement of the structural parts and an overloading of the subsoil is to be avoided. II. RECOMMENDATIONS FOR STATICAL COMPUTATIONS A. DEAD LOAD Sec. 13 1. The weight of the concrete, including the reinforcement is to be assumed at 2400 kg/m3 (149 lbs/ft^), unless otherwise specifically stated. 2. With that of the slab is also to be included the weight of the material forming the finish, determined according to known units. B. DETERMINATION OF EXTERNAL FORCES Sec. 14 I. In members subjected to flexure, the applied moments and reactions are to be computed according to the usual rules for the kind of loading and manner of support of freely supported or continuous beams. 338 CONCRETE-STEEL CONSTRUCTION 2. In freely supported slabs, the clear span, plus the thickness of the slab at the center, is to be considered as the length in computations, and in continuous slabs the distance between centers of supports is to be employed. In beams, the length is to be considered as the free span increased by the width of the supports. 3. In slabs end beams, continuous over several spans, where the moments and reactions ccnnot be computed according to the rules for continuous members, under the assumptions of free supports at ends and intermediate points, or cannot be determined by experiment, the bending moments at the centers of spans are to be taken as four-fifths of the value for a slab resting freely on two supports. Over the supports, the negative moments are to be the same as the span moments when both sides are freely supported. Beams and slabs are to be computed as continuous according to these rules only when the supports are rigid and at the same level, or consist of reinforced concrete beams. In arranging the reinforce- ment, under all circumstances the possibility of the occurrence of negative moments is to be considered. 4. In beams, moments from end restraint are to be included in computations only when special structural conditions make true restraint possible. 5. Continuity is not to be assumed over more than three spans. With live loads of more than 1000 kg/m^ (205 lbs/ft^) the calculations are to be made for the most unfavorable arrangement of load. 6. In computations of T-beams, the width of each flange at the center of span is not to be assumed as more than one-sixth the length of the beam. 7. Rectangular slabs, freely supported on all sides, with double reinforcement and uniformly distributed load can be computed according to the formula, M = - — , when the long side a is less than one and a half times the breadth h. Special forms and distribution of reinforcing rods are to be employed to care for negative moments at supports. 8. Even when so computed, thickness of slabs and of the flanges of T-beams is never to be less than 8 cm. (3.1 in.). 9. In columns, the possibility of eccentric loading is to be considered. C. DETERMINATION OF INTERNAL FORCES Sec. 15 1. The modulus of elasticity of the reinforcement is to be assumed as fifteen times that of the concrete, unless otherwise specified. 2. The stresses in any section of a body under flexure are to be computed on the assumption that the strains are proportional to the distances from the neutral axis, and that the reinforcement carries all the tension. 3. In buildings or members exposed to the weather, to dampness, to smoke gases, and similar deleterious influences, it must be shown that cracks wiU not occur from the tensile stress to which the concrete is subjected. 4. Shearing stresses are to be considered, unless the form and arrangement of the members show their harmless nature. They must be carried by a proper APPENDIX 339 arrangement of the reinforcement, whenever the design of the structure is not made to care for them. 5. As far as possible, the reinforcement must be so constructed as to preclude its displacement in the concrete. Adhesive stresses should always be susceptible of mathematical determination. 6. Columns should always be computed with regard to buckling unless the length is less than eighteen times the least lateral dimension. The spacing of the longitudinal reinforcement is to be maintained by cross ties. The spacing of the ties must not exceed the least dimension of the column or be more than thirty times the diameter of the reinforcing rods. 7. In computing columns with regard to buckling, the Euler formula is to be employed. D. PERMISSIBLE STRESSES Sec. 16 1. In members subjected to flexure, the compressive stress of the concrete should be one-sixth of its ultimate value, and the compressive and tensile stresses in the reinforcement should not exceed 1000 kg/cm^ (14,223 lbs/in^). 2. If the tensile stress in the concrete must be considered as required in Sec. 15, No. 3, two-thirds of the tensile strength of the concrete as determined by experiment, may be allowed. When tension tests are wanting, the tensile stress is not to be assumed greater than one-tenth the ultimate compressive strength. 3. The following loading values are to be assumed: {a) In members subject to moderate vibration, as in the floors of dweflings, stores, warehouses,- -the actual dead and live loads. (h) In members subject to more violent vibration or to widely fluctuating loads, as in the floors of places of assembly, dance hafls, factories, storehouses — the dead load — plus the live load increased 50%. {c) In loads attended by violent impact, as in the roofs of ceflars under drive- ways and courtyards — the dead load plus the live load increased 100%. 4. In columns, the concrete should not be stressed to more than one-tenth its ultimate strength. In computing the reinforcement with regard to buckling, a factor of safety of five is to be employed. 5. The shearing stress in the concrete is not to exceed 4.5 kg/cm^ (64 lbs/in^.). If a greater shearing strength is possible, the permissible stress is not to be more than one-fifth of the ultimate strength. 6. The adhesive stress is not to exceed the permissible shearing stress. 340 CONCRETE-STEEL CONSTRUCTION III. METHODS OF CALCULATION, WITH EXAMPLES A. SIMPLE BENDING (a) Without reference to the tensile stress in the concrete. With single reinforcement of total area /, of a beam or slab of width b, if the ratio of the moduli of elasticity of the steel and the concrete is represented by n, then the distance of the neutral axis below the top is given by the equation of the statical moments of the elemental areas about this axis (see Fig. i). ~^=nfe{h—a—x) (i) is nfe (2) From the equality of the moments of the external and internal forces, it will follow that M = oi,^h(h — a—^^= ocfc (^h — z — ^ : (3) wherein indicates the maximum concrete compressive stress, and the average steel tension. From this there follows 2M Ob bxi h — a M Fig. I. (4> (5)' Under certain circumstances the following easily obtained equations are of value n{h—a)ai bx a}j =OeJe 2 (6> (7) In T-shaped sections, such as T-beams, the calculation does not differ from that above when the neutral axis falls within the slab or at its lower edge. If the neutral axis is in the stem, the small compression in the stem may be ignored. APPENDIX 341 Then (see Fig. 2) x-d Ou = Oo^\ h —a—x (Jc = n -Oo', (To +(7: bd = Ocje, (8) (9) (10) or, by introducing into equation (10) the values of cr,^ and from equations (8) and (9) \-nfe{h — a) x=- bd+nfe (II) r — 4 ' — Be- I Fig. 2. Since the distance of the centroid of the compression trapezoid below the top is d Oo-\-2au x—y-= 3 Oo^Ou ' there is obtained by introducing the value of from ecpation (8) d d? 2 ( (x-dy^' y=x Htt -K = -[^ + -7 2 o{2x—d) 3 \ 2x—a M Ge- . . . (12) . . . (13) . . . (14) n{h-a-x) ^^^^ When beams and slabs also contain top reinforcement, the following equa- tions may be employed: For the location of the neutral axis: bx'^ — —fe{x—a)-\-nfe{x—a)=nfe{h—a—x), (16) fe{h — a—x-\-y)' XOo from which = ^ - + -j, -j +-^l{n-i)f/a + nMh-a)\. . ( * , from equation (5) is Oe h — a — s{h-a) or when // 1/ , . this becomes -VMb^tVMb. , , (42) 346 CONCRETE-STEEL CONSTRUCTION The values of x, h — a, and Z^, found according to diis method for different values of and may be tabulated.* Such a table may also be used for T-beams, when the neutral axis coincides with the under side of the slab, or when such a location of the axis is made a condi- tion of the design. B. CENTRAL LOADING If F is the area of the concrete under pressure, and is the total area of rein- forcement, the permissible load will be so that P = {F + nfe)ob, (43) nP ^ ^ ,^ = nab=y^^^ (45) C. ECCENTRIC LOADING The calculations are to be made as for homogeneous materials, the area of reinforcement being replaced by an equivalent concrete area w-times larger in all expressions for areas of sections and moments of inertia. Tensile stresses which may occur are to be cared for by reinforcement. D. EXAMPLES « The maximum stresses in the steel and the concrete are to be ascertained for a freely supported dwelling floor of 2 m. span j. _ cm J ^ 1;^^^,,^^^.^ 2^nd TO cm. thick, reinforced with 5.02 cm^/m ^< 2 00 ^ p width (10 rods of 8 mm. diameter) placed 1.5 cm. y from the bottom of the slab to the centers of the rods. The dead weight of the floor per m^ is 0.10X2400 240 kg. Over which is placed 10 cm. of rolled cinders 60 kg. A wooden floor 3.3 cm. thick with stringers 20 kg. A finish 1.2 cm. thick 20 kg. Live load 250 kg. Total ^ 590 kg. * The original contains such a table with steel stresses varying from 14,223 to 11,379 lbs/in^, and concrete stresses from 640 to 284 lbs/in^ This table has not been translated because these values are so far below usual American practice that it would be of no practical value. —(Trans.) APPENDIX 347 Then 590X2. i-Xioo , M = ^ g = 32,500 kg.-cm.; ^^15X5-02 100 r / 2x100x8.5 1 VI +— ^ — I =2.0 cm. 15x5-02 J ^ 2X32,500 _ . , 2 ah= 't-^^ - = 20.8 kg/cm2; 100X2.9(8.5-0.97) ^ (Te = r = 86o kg/cm2. 5.02(8.5-0.97) The concrete compressive stress of 29.8 kg/cm^ is permissible if the concrete employed has an ultimate compressive strength of 6X29.8 = 178.8 kg/cm^. By using Table i, since 7^=5.02, there are found, 100X8.5 m= = 170 approx. ; 5.02 6.617X32,500 01/2 100X8.52 =29.8kg/cm2; (7c = 29.01 6X29.8 = 865 kg/cm^. 500 X 2.00 To ascertain the shear and adhesive stresses at the supports, V = ~ — ^ = 590 kg. must be found. Then the shearing stress is ro = -r-^ r = 7^^^ r=o.78 kg/cm^. h[li The adhesive stress is then in which 11 represents the circumference of the reinforcement. 100X0.78 - , 2 Ti = ;7 — = ^.io kg/cm-^. ^ 10X0.8X3. 14 Neither shearing or adhesive stresses reach the maximum permissible limits. 2. A simply supported T-beam, with single reinforcement is assumed, with a span of 2 m. The live load is 1000 kg/m^, for a factory. The necessary size of concrete and reinforcement is to be ascertained on the assumption that the concrete employed will develop a compressive strength of 180 kg/cm^. For the calculation of the dead load, the thickness of the slab will be tenta- tively assumed as 18 cm., so that the total span considered is 2 18 m. The dead weight of the slab per sq.m. is 0.18X2400= . . . 432 kg. The covering of cinders 20 cm. thick 120 kg. Cement finish 2.5 cm. thick 48 kg. Total 600 kg. 348 CONCRETE-STEEL CONSTRUCTION ^, 6oo + i.t;Xiooo 1 The-n M = ^ X 2.182x1 00 = 124, 700 kg. -cm. 8 Since ^7& = -^ = 3o and (7e = iooo kg/cm^ are the permissible stresses, accord- ing to equation (6), 15X30 -{h—a) =o.^i{h—a), 1000 -[-15X30 and then by equation (41), According to equation (i) fe is found to be o.^iN ' 100 I ^5-)o.3iX3o 24,700 17.3 cm. bx^ 100X0.312X 17.32 . ^ je = — 77 \^~t: — 7 cm^ 2n{h-a-x) 2X15(17.3-0.31X17-3) Nine round rods, 11 mm. in diameter, with a total area of 8.55 cm^ may be employed. The total thickness of slab on account of the covering for the steel must be increased to 19 cm. From Table II* for (7^= 1000, and (^5 = 30, there is found h — a^o.4g\^ 1247 = 17.3 cm. /e = 0.00228v 12,470,000 = 8 Cm2. The shear at the abutment is 7 = 600-^1.5X1000 = 2100 kg. The shearing stress is 2100 ^1/9 To = J ^=1.36 kg/cm2. .r.J.^ . Q-3IXI7 -3' ioo[i7.3 — The adhesion is 100X1.36 2 Ti =—- r-^ =4-38 kg/cm2. 9X1.1X3-14 3. The floor described under 2 is to be investigated as to the stresses which occur when the tensile strength of the concrete is taken into consideration. According to equation (27), with the concrete also acting in tension, i^i^Vi5X8.55Xi7.3 x= ^ = 10.02 cm., 100X19 + 15X8.55 * See Note page 346.— (Trans-^ APPENDIX 349 and according to equation (31), 124,700X10.02 ^ 1 00X10.02-^ looXcS.oh'^ ^ ^.^ ^ ° + ^-^ + 15X8.55X7.282 ig — 10.02 , , „ (762 = -X 19.4 = 17-4 kg/cm-; 10.02 I c;(i7.S — 10.02) , , „ Oe=- ^ ' ^X 19.4 = 21 1.4 kg/cm2. A tensile stress of 17.4 kg/cm^ is permissible when an ultimate tensile strength of -1X17.4 = 26.1 kg/cm^ is demonstrated by experiment. If such test is not feasible, the concrete employed must show an ultimate compressive strength of 10X17.4 = 174 kg/cm^. This ultimate compressive strength must reach 180 kg/cm^ because of the assumed permissible stress of 30 kg/cm^. To determine the shearing stress at the neutral axis, the distance z between the centroids of tension and compression must be found. This is obtainable from the condition that M=Dz, where D^—ob= ^^^^ ^9-4X 10.02 ^ 2 2 ' so that 124,700 z = — = 12.83 cm. 9720 Then ^^=7^o-^^^"^4kg/cm2. Because of the tensile strength of the concrete, the shear at the level of the reinforcement is somewhat less. In general VS wherein 5" is the statical moment of the section above* the level in question, f and / is the moment of inertia of the whole section. Thus, for the section of the level of the reinforcement, 5 = ioo(-^— -^^^-^ + 15X8.55X7.28 = 3698; so that ^ Mx 124,700X10.02 . / = — = =64,420; Oh 19.4 , 2100X3698 , , ^ To'=7 ^r^=i.2i kg cm2. 64,420 X 100 The adhesion is then , 100X1.21 , , p t/ = =4 kg/cm^. ' 9X1.1X3.14 ^' * Or below, as in the example. — (Trans.) t With reference to the neutral axis. — (Trans.) 350 CONCRETE- STEEL CONSTRUCTION t< 20 4. A reinforced concrete beam of 4 m. span, with dimensions as in the accompany- ing figure has an apphed moment of 120,000 kg. -cm. The maximum compressive stress in the concrete and the stresses in the ^ = 4ffO'^/m=4 5z^^'' reinforcement, ignoring the tensile stress in the concrete, are required. According to equation (17) Fig. 8. 14X1.51 + 15X4-52 20 ^^^ x4Xi.5i + i5X4.5 y^A(,,xx5i + x5X4.5^X33) = ix.35cm. Then, according to equation (19), 1 20,000 (76 = ^^^-^(33-3.78) + i4Xi.5xX^fx3o ^ ■'•■^•35 31.7 kg/cm2; ^^-^Sf X^''^ = ^5o kg/cm2; 21. 6< 01, o 3-^X350 = 908 kg/cm2. In computing the shearing stresses, the distance y\ from equation (24) is to be found, 20X11.37^ 3 14X8.372x1. 51 20X11.37^ 7.67 cm. + 14X8.37X1.51 Since the load per m. is 600 kg., 7 = 2X600 = 1200 kg., and 1200 -^0 = 20(21.65 + 7.67) 20X 2.05 = 2.05 kg/cm2; 4X1X3-14 At the upper reinforcement, since 3.27 kg/cm^ and 5 = 20 ^ •-^ + 15X1.51X8.35 = 780, ^ 120,000X11.^^ /= ^=42,970, 31-7 , 1200X780 , , 2 To = — = 1 .00 kg/ cm^, ° 20X42,970 ^ ^ APPENDIX 351 If the tensile stresses in the concrete are considered, according to equation (38), 20 X 36- ^ + 14(1.51X3+4.52X33) x = — — ■ ^ = 18.8 cm., 2oX3() + i4(i-5i +4.52) so that, according to equation (39), T 20,000X18.8 , . _ '''"' = 20X18.83 20X17.-' : ~ :r'^-^ ''S/cm^; o o '^bz=^^X2s.4 = 2i.4. kg/cm2; (76 = 15X^^^X21.4 = 265 kg/cm2. The shearing stresses at the level of the upper reinforcement, since 7 = 96,410, will be 1200 /i8.82- 15.82 i5Xi.5iXi5.8\ ^ , , o ro=— — + ^ ^ ^ =0.87 kg/cm2, 96,4io\ 2 20 / ' ^ and the adhesive stress 20X0.87 Ti= — = 2.^ kg/cm^. 3X0.8X3.14 ^ At the neutral axis, 1200 /i8.82 15X1. 51X15. 8\ , , 96,4io\ 2 20 / ° 5. A floor panel 3 m. wide and 4 m. long is to consist of a plain concrete slab, freely supported on all sides, with reinforcement in two directions parallel to the sides. Live and dead load amount to 600 kg/cm2. The necessary thick- ness of floor and amount of reinforcement is required. The applied moment, computed from the shortest span, is 600X3.1^X100 M= = 48,050 kg. -cm. The permissible stresses are (Tc = iooo and ^76=40 kg/cm^. Then, from Table II * ^ 148,050 ^ h-a = o.s9\^-^ =8.54 cm., /e=o.oo293\/^4,8o5,ooo = 6.42 cm2. The thickness of the slab should be increased to 10 cm. For the reinforce- ment in the direction of the shorter span, 10 round rods of 9 mm. diameter with a total area of 6.36 cm2/m. width, should be used. The longer rods can be somewhat fewer, about in the ratio of the breadth to the length of the slab. For them, 8 rods per m. width, of the same size, may be employed. * See note, page 346. — (Trans.) 352 CONCRETE-STEEL CONSTRUCTION 6. A T-beam of the dimensions shown in the accompanying figure is assumed with a span of 7.5 m. and a column spacing of 7.8 m., with a hve load of 500 kg/m, in a store. The reinforcement con- sists of 6 round rods of 2.5 cm. diameter, with a total area of 29.45 cm. The maxi- mum stresses in concrete and steel are to be determined. tso t / 1 1 [ a • • f Fig. 9. The dead load consists of The weight of the T-beam = (1.5 Xo.i +0.32X0.25) X 2400 .... 552 kg. The weight of the floor filHng, 6 cm. of rolled cinders ... 36 kg. The weight of the cement finish, 2 cm. thick 40 kg. The weight of the plaster ceiling 14 kg. Total per sq. meter 90 kg. Thus, for 1.5 m2, 1.5X90= i35 kg. The live load 500 kg. Total 1187 kg. or approximately, 1200 kg/m length of beam. Then i2ooX7-82Xioo ^ . M. = = 912,000 kg. -cm., and, according to equation (11), 150X102 2 15X29.45X36 150X10 + 15X29.45 and according to equation (13) >' = i2.o5-5 10^ 6(2X12.05— 10) consequently, according to equation (14) 912,600 12.05 cm. = 8.23 cm., 29.45(36-12.05 + 8.23) and according to equation (15) 12.05 =963 kg/cm2, Gh=- 15(36-12.05) The shear at the abutment is |r_ 7-5Xi2 oo X 963 =32.3 kg/cm2. = 4500 kg., APPENDIX 353 so that the shearing stress in the concrete is V ^0 = 4500 bi{h —a—x-\-y) 25(36 — 12.05+8.23) = 5.6 kg/cm2. The permissible stress is thus exceeded. It is consequently advisable to bend upward near the end two rods of the upper layer of reinforcement. The point at which such bending should take place is determined by the condition that at this point the shear V should be only 4500X4-5 5-6 3616 kg. This is found at a point ^^^^ — ^^^==0.74 meters from the abutment. The total tension Z to be taken by the bent portions of the rods is equal to the shear to be transferred to them, i.e., Z = ^t(5.6-4.5)iX25 = 72o kg. V2 The stress in the bent rods is therefore 720 ^^=-i— — = 73 kg/cm2. The adhesive stress on the four lower rods at the supports, amounts to 25X5-6 = 4-5 kg/cm2. u 4X2.5X3-14 If it is desired to ascertain the tension in the concrete, x must be determined from equation (32) 25X42^ , 125X10^ + 15X29.45X36 25X42 + 125X10 + 15X 29.45 and according to equation (13) 16.12 cm. 100 }/ = i6.i2-5+— - = 11.87 cm., 6(32.24-10) so that by equation (33a) [150X10X11. lf = 9i2,6oG= — ^^^(2X16.12-10) +^(6.123 + 25. + 15X29.45X19.^ from which ,~| Obd J 16^12' obd = 2S.4 kg/cm2, 2 ^ 88 . r f^~^ h-a-x\ Vbx nfe, ,,1 (a) P = --a, + nfeO,[--- __j = J , and from condition 2 (b) F{x-e)= Ob— + nfeob I — — — + ^ J 3 Ob [— + — (2^2 — 2^:x: + 2^2 + 7/2 — 2a//)l 3 ^ J Equating the value of obtained from these equations and reducing, there results T^x^ — ^x^ — (2e — h)x = 2a^ + h^ — (2a+e)h, 6nfe 2nfe or by substituting the values, ^ = 25; ^ = 15; /^ = 6.28; ^ = 2.5; h = 2^; a = 3; x3-7.5:x;2 + 452.i6x = 9734. The solution is most easily accomplished by trial and there is found with sufficient accuracy x = i6.^ cm. Then, by the aid of equation (a) /2c;Xi6.^ 1^X6.28 \ (76 = 20, 2 kg/cm^, and further, Oed=— ^ = 249 kg/cm2, 16.3 (7^2 = 249-^^ = 107 kg/cm2. 13-3 APPENDIX 359 Table B MAXIMUM MOMENTS IN CONTINUOUS BEAMS CONTINUOUS BEAMS OF TWO SPANS 1:1 X T Moments From ^ From p M Max. ( + M) Max. (-M) + o 0 0 0 O.I + 0.0325 0.03875 0.00625 O. 2 + 0-0550 0.06750 o.oi 250 + 0.0675 0.08625 0.01875 0.4 + 0.0700 0.09500 0.02500 + 0.0625 0.09375 0.03125 0.6 + 0.0450 0.08250 0.03750 0.7 + 0.0175 0.061 25 0.04375 0.75 0 0.04688 0.04688 0.8 — 0.0200 0.03000 0.05000 0.85 — 0.0425 0.01523 0.05773 0.9 — 0.0675 0.0061 I 0.07361 0-95 — 0.0950 0.00138 0.09638 I — 0. 1250 0 0.1 2500 pP Note. — In this and the following table, in cases in which the calculations include only quiescent loads, as for roofs, it is recommended that the positive moments at centers of spans be increased to at least — . In computations concerning par- tial live loads p in unfavorable positions, the deficient value due to ,.; found in the table is overbalanced, so that the tabular values can be employed as they stand. In beams with more than four spans, the end ones can be calculated like the first one of a beam of four spans, and the other spans in the longer beam like the second span of a continuous beam of four spans. 360 CONCRETE-STEEL CONSTRUCTION Table C MAXIMUM MOMENTS IN CONTINUOUS BEAMS CONTINUOUS BEAMS OF THREE SPANS 1:1:1 Moments X T From g From p M Max. (+M) Max. (-M) First opening o O.I 0.2 0.4 0-5 0.6 0.7 0.8 0.85 0.9 0.95 I Second opening 0 0.05 0.1 0-15 0.2 0.2764 0-3 0.4 0-5 0 + 0-035 + 0.060 + 0.075 + 0.080 + 0.075 + 0.060 + 0.035 0 — 0.02125 — 0.04500 —0.07125 — 0. 10000 — 0. 10000 — 0.07625 — 0.05500 — 0.03625 — 0.020 0 + 0.005 + 0.020 + 0-025 0 0.005 0.010 0.015 0.020 0.025 0.030 0.035 0.04022 0.04898 0.06542 0.08831 0.11667 0.11667 0.09033 0.06248 0.05678 0.050 0.050 0.050 0.050 0.050 + 0 0.040 0.070 0.090 0. 100 0. 100 0.000 0.070 0.04022 0.02773 0.02042 o.oi 706 0.01667 0.01667 0.01408 0.00748 0.02053 0.030 0.050 0-055 0.070 0.075 pP pP APPENDIX 361 Table D MAXIMUM MOMENTS IN CONTINUOUS BEAMS CONTINUOUS BEAMS OF FOUR SPANS 1:1:1:1 Moments X T From g From p M Max. {-M) Max. (+M) First opening o O.l 0.2 0-3 0.4 0.5 0.6 0.7 0.7857 0.8 0.85 0.9 0-95 I 0 Second opening 0 0.05 0.1 0-15 0.2 0. 2661 0-3 0.4 0.5 0.6 0.7 0.8 0.8053 0.85 0.9 0.95 1 .0 0 + 0.03429 + 0.05857 + 0.07286 + 0.07714 + 0.07143 + 0.05572 + 0.03000 0 -0.00571 — 0.02732 — 0.05143 — 0.07803 — 0. 10714 — 0 . 10714 —0.08160 -0.05857 — 0.03803 — 0.02000 0 + 0.00857 + 0.02714 + 0.03572 + 0.03429 + 0.02286 + 0.00143 0 —0.01303 — 0.03000 —0.04947 — 0.07143 0 0.00536 0.01071 0.01607 0.02143 0.02679 0.03214 0.03750 0.04209 0.04309 0.05216 0.06772 0.09197 0.12054 0. 1 2054 0.09323 0.0721 2 0.06340 0.05000 0.04882 0.04821 0.04643 0.04464 0.04286 0.04107 0.04027 0.04092 0.04754 0.06105 0.08120 0. 10714 + 0 0.03964 0.06929 0.08893 0-09857 0.09822 0.08786 0.06750 0.04209 0.03738 0.02484 0.01629 0.01393 0.01340 0.01340 0. 01 163 0.01455 0.02537 0.03000 0.04882 0.05678 0.07357 0.08036 0.07715 0.06393 0.04170 0.04092 0.03451 0.03105 0.03173 0.03571 pP pP 362 CONCRETE-STEEL CONSTRUCTION Table E CONVERSION TABLE, METRIC TO ENGLISH No. Kilograms to Averdupois Pounds. Tonnes to Tons of 2000 Pounds. Centimeters to Inches. Meters to Feet. oQU3,re Centimeters to Square Inches. Square Meters to Square Feet. I 2. 20462 I . 10231 0-39370 3-280833 0-155 10.76387 2 4.40924 2.20462 0.78740 6.561667 0.310 21-52773 3 6.61387 3-30693 1 .18110 9.842500 0.465 32. 29160 4 8.81849 4.40924 1.57480 13-123333 0.620 43-05547 5 II .02311 5-51156 I .96850 16.404167 0-775 53-81934 6 13-22773 6.61387 2.36220 19.685000 0.930 64.58320 7 15-43236 7.71618 2-75590 22.965833 1.085 75-34707 8 17.63698 8.81849 3.14960 26. 246667 1 . 240 86.11C94 9 19.84160 9.92080 3-54330 29-527500 1-395 96.87481 No. Cubic Meters to Cubic Yards. Hektoliters to Bushels. Kilograms per Square Centimeters to Pounds per Square Inch. Kilograms per Square Meter to Pounds per Square Foot. Kilograms per Cubic Meter to Pounds per Cubic Foot. I 1-30794 2.83774 14.22340 0. 20482 0.06243 • 2 2.61589 5-67548 28.44680 0.40963 0. 12486 3 3-92383 8-51323 42.67020 0.61445 0.18728 4 5-23177 11-35097 56.89359 0.81927 0. 24971 5 6-53971 14.18871 71 .11699 I .02408 0.31214 6 7.84766 17.02645 85-34039 I . 22890 0.37457 7 9-15560 19.86420 99-56379 1-43372 0.43700 8 10.46354 22. 70194 113. 78719 1.63854 0.49943 9 II. 77149 25-53968 128.01059 1-84335 0.56185 INDEX PAGE ^DHESION 42 proof of existence of 2 Adhesive forces, formulas for 145, 146 stresses developed 157, 161, 184, 186 Anchorage for ends of rods 148, 149 Arch bridges, examples 266-284 Arches 14 in buildings , 235 gEAMS, graphical methods of calculating 130-137 of variable depth, shear in 190, 191 Bending and tensile strengths, explanation of discrep- ancy between 26, 27, 28 relation of 27-29 of rods, effects of 181, 182 strength of concrete 25 tests to determine extensibility 55? 56 with axial compression, formula for 1 19-123 with axial stress, graphical methods for 132-137 with axial tension 127-129 with axial thrust 1 19-127 Beton Frette 67 Brackets 8, 9 Breaking strength of columns 63 Brick curtain walls 215 Buildings of reinforced concrete, examples of 210 /CANTILEVERS, examples 257 Cellars, Water-tight 247 Cement bins, examples 295-298 Coal pockets, examples 290 Coefficient of expansion, concrete 3 steel 3 363 364 INDEX PAGE Column tests 60, 6 1 Columns 11 breaking strength of 63 effect of ties on 62 flexure of 65 with longitudinal rods 59 with spiral reinforcement 67 Compression 59 Compressive strength of concrete 19, 20 Computation, suggested methods of 323, 333, 340 Concrete, bending strength of 25 compressive strength of 19 elasticity of 21 extensibility of 50 punching resistance of 31 shearing strength of 31 tensile strength of 20 torsional strength of 39? 40 Continuous bridges, examples 263-266 T-beams, experiments with 199-203 slabs 5, 6 Cooling towers, example of 3'^3~3^5 Corrosion, proof of prevention of 1,2 Cracking of T-beams 188-190 stage, methods of computation for 137, 138 Cracks, safety against tension 105, 106, 107 Curtain walls of brick 215 J^EFLECTION of beams 192, 193 of Isar-Grimwald arch liridge 280 Deformation 192, 193 Deformed bars, objections against i7ji8 Distributing rods 5 Domes 235 Double reinforcement, formulas for 87, 88 for slabs, with methods of figuring 94 tests of slabs with 93 JgFFECTIVE depth, alteration of, from cracks 171 Effect of humidity on extensibility of reinforced concrete 53~5^ Elasticity of concrete 21 Elastic limit of concrete 21 Examples of methods of computation 326-333 Expanded metal, objections against 17 Expansion by heat, coefficient of, concrete 3 coefficient of, steel 3 INDEX 365 PAGE Extensibility 50 from bending tests 55 j 56, 57 of reinforced concrete 51 5 52 Euler's formula not applicable to long columns 65 J^IREPROOF quality of reinforced concrete, example of 212 Flexure of columns 65 Floor systems 209 Footings 241 r^OVERNMENT (Prussian) regulations 233 Grain bins, examples 304-306 Graphical methods of calculating beams 130-137 Grimwald-Isar arch bridge 271-280 TJAUNCHES 6, 8, 9 in continuous beams, effect of 203 History 204, 205 Hooks, effects of 158, 161, 162, 164, 178, 182 Humidity, effect of on concrete 4 JINTERIOR treatment of reinforced concrete buildings 224 Isar-Grimwald arch bridge 271-280 J^NEES 8, 9, 10 J^EITSATZE 15,317,333 Longitudinal and spiral reinforcement for columns 72,73 rods, effects of 64 jyjELAN arch bridges 281-283 Methods of computation, suggested 323, 333, 340 Modulus of elasticity of concrete 21-25 Moments, computation of 194-197 in slabs 6 ^^EUTRAL axis, position of 95-98 ^^RE pockets, examples 290, 293, 298 piLE formula 255 Piles 250 Prussian government regulations for reinforced con- crete work 16 366 INDEX t PAGE Pumice concrete 19 Punching resistance of concrete 31 ■REGULATIONS of Prussian government 16 Reinforcing rods, arrangement in columns ... . . 12 arrangement of, in arches 11 arrangement of, in slabs 5, 6, 9, 10 Remant stresses 198 Reservoirs, examples 284-287 Rich mixture, necessity of 18 Roof construction 230 coverings 219-230 Rust, proof of prevention of 2 OAFE stresses, suggested 323-339 Safety against tension cracks 105-107 Sand-boxes for Isar Grimwald arch bridge 276 Scherfestigkeit 31? 32 Schubfestigkeit 31^32 Shearing forces, effects of 139, 141, 147, 148 experiments concerning, by author. . . . 151-173 experiments concerning at Stuttgart Testing Laboratory 173, 174 formulas for 141-146 theory of action of , 151 strength of concrete 3^^ 33. 34, 35> 3^, 37» 3^ from slotted beams 40,41,42 relation of tensile and compressive to.. 32, 33, 35, 42 stresses 31,32 in beams of variable depth 190, 191 in reinforced concrete 31, 32, 55 Shipping platform, example of 312 Shrinkage stresses 198 Simple bending 74 Silos, computation of stresses in 306-312 examples 287-312 Slab culverts, examples 256 Slabs 4 formulas for computation of 77-82 graphical computation of 76 tables for computation of S3-86 tests of simple 90 tests of, with double reinforcement 93 with double reinforcement, methods of figuring 94 Sliding resistance of rods in concrete 43, 44, 45, 46, 47 Slope of bent rods 7 INDEX 367 Spiral and longitudinal reinforcement for columns reinforcement for columns notes regarding for columns, how computed Spirally reinforced columns PAGE 73 231 71 67 47 74 7=^ 232 16 8 149^ 150 Stirrups action of effects of 155, 156, 158, 159, 164, 168, 175, 177, 178, 180, 181, 183, 184, 188 on adhesion 48, 49 tests concerning effect of 152 Stress distribution over section 98-104 Stresses at failure 172-179 in T-beams 11 2-1 1 3 for slabs, working 92 Sunken well casings 244 Suspension system of reinforcement, tests concerning. . 164, 165, 166, 168 Systems of construction 206-209 npABLES for beams subjected also to thrust T-beam bridges, examples T-beams continuous, experiments with cracking of diagrams for economical spacing of experiments concerning, by Stuttgart Labora- tory formulas for stresses in Tensile strength of concrete Tension bending with axial Tests of spirally reinforced columns Thacher rods, spHtting effect of Theory, development of Thrust, axial bending with Ties in columns, effect of ,. , Torsional strength of concrete Torsion tests on hollow cylinder Truss action of reinforcement . .;. /-J-l' theory of . J. 1 Tunnel lining, example of 125, 126, 127 258-266 8-107 1Q9-203 188-190 115 176 108-110, 114, 116, 117 112 20 127, 128, 129 68, 69 47 208 119-127 53-. 54, 55 ,^ 160. ; 16 1 ; , : /• 161, ''i83,'i8-;,'i«8^ 312 , . . 36S INDEX PAGE ■^^^ARIABLE depth, shear in beams of 190, 191 Tl/'ATERTTGHT cellars 247 Weld, location of 11 Well casings 244 Wet mixture, strength of 18-20 Working stresses for slabs 92 f GETTY CENTER LIBRARY 3 3125 00002 1424