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BOUGHT WITH THE INCOME 
 FROM THE 
 
 SAGE ENDOWMENT FUND 
 
 THE GIFT OF 
 
 Henrg m. Sage 
 
 1891 
 
 ^'^i^q-u, c^m/i 
 
 ii/rlt 
 

 3 1924 004 106 955 
 
The original of tiiis book is in 
 tine Cornell University Library. 
 
 There are no known copyright restrictions in 
 the United States on the use of the text. 
 
 http://www.archive.org/details/cu31924004106955 
 
C'-,': 
 
 
 
 'LB 
 
 l^ 
 
 
 O'REILLY ESTATE BUILDING, ST. LOUIS, MO. 
 
 A. B. Groves, Architect Murch Bros. Construction Co., Contractors 
 
 G. S. Bergendahl, M. AM., Soc. C. E., Engineer, St. Louis Representative, Mushroom System 
 
THE THEORY OF THE FLEXURE 
 AND STRENGTH 
 
 OF 
 
 RECTANGULAR FLAT 
 PLATES 
 
 APPLIED TO 
 
 REINFORCED CONCRETE 
 FLOOR SLABS 
 
 BY 
 
 HENRY T. EDDY, C. E., Ph. D., Sc. D. 
 
 PROFESSOR OF MATHEMATICS AND MECHANICS, COLLEGE OF ENGINEERING, 
 
 AND DEAN OF THE GRADUATE SCHOOL, EMERITUS, UNIVERSITY OF MINNESOTA. 
 
 CHAIRMAN OF BOARD OF EXAMINERS FOR THE MUSHROOM SYSTEM 
 
 APPOINTED BY THE COMMISSIONER OF BUILDINGS 
 
 CHICAGO, JUNE 15, 1912 
 
 PRICE ^5.00 
 
 ROGERS & COMPANY, MINNEAPOLIS 
 1913 
 
COPYRIGHT 1913 
 BY 
 
 HENRY T.EDDY 
 
 ALL RIGHTS KESERVED 
 
PREFACE 
 
 In reviewing the history of every type of structural work, we find 
 the designing engineer influenced in his first attempts at any new 
 type of structure by his knowledge of the practical forms of con- 
 struction with which he is already familiar. In fact until very 
 recent times precedent was the engineer's sole guide. It was to be 
 expected therefore that the pioneers in concrete-steel construction 
 should follow aiid closely imitate timber and structural steel con- 
 struction. In following this type, the idea has been to build up the 
 structure as a whole by assembling and joining together a number 
 of independent elements or units : whereas concrete, with or without 
 reinforcement, is a kind of material that is best suited by its nature 
 to construction in monolithic form. But when the attempt has been 
 made to treat such structures theoretically, preconceived ideas 
 have led to the effort to treat them by analysing them into separate 
 members and computing the strength of these arbitrarily selected 
 units, assumed to act independently as they do in steel structures. 
 Such treatment has led to errors of as much as two or three hundred 
 percent in the computation of slabs with two way reinforcement 
 supported on four sides, and to errors of four hundred percent in 
 case of continuous fiat slab construction such as occurs in the mush- 
 room system. 
 
 When we consider the fact that fire losses in Canada and the 
 United States amount each year to half a billion dollars, and that 
 the question of commercial economy determines whether buildings 
 shall be built of fireproof and incombustible materials such as rein- 
 forced concrete, or of infiammable materials such as are used in timber 
 construction, it is at once evident how important it is to the general 
 public to be able to determine on theoretically correct principles 
 whether safe fireproof buildings can be built at approximately the 
 same or less cost than combustible ones. In case of any uncertainty 
 on this question, the designer is compelled for safety to employ 
 materials in such lavish amounts as to render cost prohibitive. 
 
 The failure of engineers and mathematicians generally to apply 
 the mathematical theory of elasticity to the new material concrete- 
 steel, has led to considerable controversy between practical con- 
 structors of experience, and theoretical engineers without such 
 experience. 
 
IV PREFACE 
 
 Marsh in his treatise on Reinforced Concrete, edition of 1905, 
 Part V, p. 209, makes the following remarks upon this subject: — 
 
 When properly combined with metal, concrete appears to gain 
 properties which do not exist in the material when by itself, and 
 although much has been done by various experimenters in recent 
 years to increase our knowledge on the subject of the elastic be- 
 haviour of reinforced concrete, we are still very far from having 
 a true perception of the characteristics of the composite material. 
 
 It may be that we are wrong from the commencement in 
 attempting to treat it after the manner of structural ironwork, and 
 that although the proper allowances for the elastic properties of the 
 dual material is an advancement on the empirical formulae at first 
 employed, and used by many constructors at the present time, yet 
 we may be entirely wrong in our method of treatment. 
 
 The molecular theory, i. e. the prevention of molecular defor- 
 mation bjr supplying resistances of the reverse kind to the stresses 
 on small particles, may prove to be the true method of treatment for 
 a composite material such as concrete and metal. This theory is 
 the basis of the Cottacin construction which certainly produces good 
 results and very light structures, and M. Considere's latest researches 
 on the subject of hooped concrete are somewhat on these lines." 
 
 In this statement. Marsh undoubtedly has in mind the great 
 discrepancy between the results of tests and of computations of 
 multiple way reinforcement. 
 
 The empirical formulas of Hennebique, a pioneer in this field 
 and one of its most extensive investigators, give numerical results 
 at variance with the usual published theories for two way reinforce- 
 ment, while the empirical formulas published by Turner in 1908 
 exhibit an equally radical divergence. 
 
 It has remained apparently for Dr. Eddy to discover the reason 
 for this great discrepancy by a rigid application of the mathematical 
 theory of elasticity to the problem presented by multiple way rein- 
 forcement. 
 
 The clearing up of the mathematical difficulties with which the 
 theoretical engineer has heretofore struggled in dealing with 
 reinforced concrete, will lead to its more general adoption by the 
 elimination of uncertainty in design; and will lead to the adoption 
 of those types which are safest to erect, and those which possess a 
 degree of toughness, due to their monohthic construction, which is 
 lacking in types that merely imitate older forms of timber and steel 
 construction. 
 
 The record of the Mushroom system in the successful construc- 
 tion of between one and two thousand buildings, without accident 
 to the workmen, and without failure to make good the guarantee 
 
of test capacity, can be accounted for only on the ground that 
 scientifically designed multiple way reinforcement is inherently safer 
 to erect, and more reliable at all stages of construction than other 
 types. The theory given in detail which accounts for its economy 
 and safety, should, we believe be of interest to the profession at 
 large, and should prove of value to the practicing engineer in check- 
 ing designs for his clients. 
 
 The endorsement of this theory by the undersigned does not 
 constitute a license to use the patented type, though the extension 
 of the theory to older types which are not patented will undoubtedly 
 lead to the closest competition with it along legitimate lines. Any 
 loss to the patentee arising from such competition, will, in the writer's 
 judgment, be more than counterbalanced from a commercial stand- 
 point, by the increased safety and corresponding popularity of 
 reinforced concrete as a material of construction. 
 
 The invention covered by the claims of the broad patent 
 just mentioned includes much more than merely the standard mush- 
 room system. This is, however, the one form of all others 
 of the patent, which the writer prefers by reason of certain practi- 
 cal advantages which it possesses. Its arrangement of parts whereby 
 a continous multiple way reinforced flat slab is supported by a 
 large cantilever column head integral with the column and em- 
 bedded in the slab so as to resist tensile stresses both radial 
 and circumferential in zones near the top of the slab over 
 and around the columns, and in the zones near the bottom of the 
 slab toward the center of the panels, will be fully discussed in succeed- 
 ing pages. This discussion, which deals with the mushroom sys- 
 tem primarily, is intended as an advance chapter or two of a more 
 comprehensive treatise on Concrete-Steel Construction, in which 
 it is intended to treat somewhat fully concrete columns, beam and 
 slab construction, wall panels, etc., as well as flat slabs, and to 
 introduce the results of experimental work now under way to 
 determine the value of Poisson's ratio for different combinations 
 of steel and concrete. 
 
 This treatise will then represent the joint efforts of a pro- 
 fessional mathematician accustomed to treating these problems, 
 and a professional builder and designer of reinforced concrete with 
 many years of practical experience behind him. 
 
 The price charged for this booklet will be credited in return 
 for it, on the larger treatise which the authors intend to complete 
 as soon as the magnitude of the task will permit. 
 
 C. A. P. TURNER. 
 
CONTENTS 
 
 Section 
 
 
 Page 
 
 1. 
 
 Flat Slab Floors. The Mushroom System 
 
 
 1- 4 
 
 2. 
 
 Notation 
 
 
 5- 7 
 
 3. 
 
 True and Apparent Bending Moments 
 
 
 8-10 
 
 4. 
 
 Poisson's Ratio 
 
 
 11-12 
 
 5. 
 
 General Differential Equation of Moments 
 
 
 13-15 
 
 6. 
 
 General Differential Equation of Deflections 
 
 
 16-17 
 
 7. 
 
 Solution of the Differential Equation in Case of 
 
 Uni- 
 
 
 
 form Slab Supported on Columns 
 
 
 17-20 
 
 8. 
 
 Solution for Side Belts 
 
 
 21-24 
 
 9. 
 
 Practical Formulas for Stresses in Side Belts . . . . 
 
 
 24-30 
 
 10. 
 
 Practical Formulas for Stresses in Column Heads . 
 
 
 31-35 
 
 11. 
 
 Practical Formulas for Stresses in the Middle Area of 
 
 
 
 Panel 
 
 
 36^0 
 
 12. Deflections at Mid Span of the Side and Diagonal Belts. 41-43 
 
 13. Proportionate Deflections at Mid Span and Center of 
 
 Panel 43-46 
 
 14. Radial and Ring Rods, and Shear around Cap 47-55 
 
 15. Standard Mushroom System and other Systems . ... 55-61 
 
 16. Specimen Computation of Slab 62-70 
 
 17. Comparative Test to Destruction of Two Slabs 71-93 
 
 18. Suggestions as to Construction, Finish, Etc 94-100 
 
 Appendix. Specifications for Floors 101-104 
 
TISCHERS CREEK BRIDGE, DULUTH, :MIXX. 
 Spans are 26 feet longtitudinally 
 This type is built with spans up to 50'0" 
 
 View of Reinforcement in Place 
 
 TISCHERS CREEK BRIDGE, DULUTH, MINN. 
 
 Designed by C. A P. Turner Geo. H. Lounsbury, Contractor 
 
FLAT SLAB FLOORS 
 
 1 . The superiority of flat slab floors supported directly on columns, 
 over other forms of construction when looked at from the stand- 
 point of lower cost, better lighting, greater neatness of appearance, 
 and increased safety and rapidity of construction, is so generally, or 
 rather so universally conceded as to render any reliable information 
 relative to the scientific computation of stresses in this type of con- 
 struction of great interest. Heidenreich, in his Engineer's Pocket 
 Book on Reinforced Concrete, page 89, classifies this type as floors 
 without beams and girders — "Mushroom System." 
 
 Since "mushroom," as applied to concrete, is an arbitrary or 
 fanciful term, and indeed, almost a contradictory one, a word of 
 explanation as to its origin may be of interest. The term was 
 originated by C. A. P. Turner, of Minneapolis, and applied to his 
 flat plate construction, more particularly because of the fancied 
 resemblance to the mushroom, of the column and column head 
 reinforcement of that particular form of his flat plate construction 
 which he seemed to prefer by reason of certain practical advantages. 
 Another fancied resemblance is the rapidity of erection, comparable 
 to the over-night growth of the mushroom. Here the resemblance 
 ceases, since the construction, once erected, is enduring and per- 
 manent. 
 
 The Mushroom System is a continuous flat plate of concrete 
 supported directly on columns, and reinforced in such a maner that 
 circular and radial tensile stresses concentric with the column are 
 provided for by metal reinforcement in the tension zone above the 
 columns, and similar provision is made for tensile stresses in the 
 lower portion of the slab concentric with the center of the panel, 
 diagonally between the columns. Since all forces in a plane may be 
 resolved into equivalent components along any pair of axes at right 
 angles to each other, it is possible to provide reinforcement to resist 
 any horizontal tensile stresses in the slab by various arrangements 
 of intersecting belts of rods at zones where these stresses occur. 
 All arrangements of this kind are by no means equally effective. 
 A system of wide reinforcing belts from column to column com- 
 bined with a system of radial and ring rods to constitute a large, 
 substantial cantilever mushroom head at the top of each column 
 
Story COu 
 
 No 
 
 s 
 
 a 
 
 -I 
 
 a 
 o 
 
 IE 
 
 Kt-06ETuaiMa Spiw ^l 
 
 Fig. 1. Vertical Section of Standard Mushroom Head showing posi- 
 tion of Radial and Ring Rods, and Slab Rods, Vertical and Hori- 
 zontal Sections of Spirallj' Hooped Column, with Plain Bar Hoop 
 Collar Band, Vertical Reinforcing Rods and Elbow Rods. 
 
 Fig. 2. Plan of Reinforcement in Standard Mushroom System. 
 Radial and Ring Rods, Collar Band and Slab Rods. Diameter of 
 
 Head = sf = 7/l6(a+6). 
 
STANDARD MUSHROOM SYSTEM 3 
 
 provides a very effective and economical arrangement for controlling 
 the distribution of the stresses in the slab, and furnishes the resis- 
 tance necessary to support these stresses by placing the steel where it 
 is most needed. It not only has the same kind of advantage that the 
 continuous cantilever beam has over the simple girder for long 
 spans, but combines with it the kind of superiority that the dome has 
 over the simple arch by reason of circumferential stresses called into 
 play, which adds greatly to the carrying capacity of the slab. 
 
 In the standard mushroom type, which is quite fully discussed 
 in this paper, the heavy frame work, concentric with the colunrn, 
 supports the slab reinforcement at a fixed elevation, furnishes a high 
 degree of resistance to shear, and secures a high degree of safety 
 during construction. It extends as a cantilever approximately one 
 fourth of the way to the next column as shown in Figs. 1 and 2 on 
 page 2. Arranged upon the radial rods of this frame rest two or 
 more large hoops and upon these rest the wide spreading belts of rods 
 which extend both directly and diagonally from column to column. 
 Over the columns these belts lie near the upper surface of the slab, but 
 they run near the lower surface as they approach points midway 
 between columns. 
 
 The cantilever slab thus formed, not only has the same advant- 
 ages for this form of construction that the cantilever construction 
 has for long span bridges, but it causes the slab to have greater 
 stiffness and gives it greater resistance to shear in the neighborhood 
 of the columns; it removes the locus of zero bending moment to a 
 much greater distance from the column than would otherwise be the 
 case, thus dimininishing the area of that part of the slab which tends 
 to become concave on its upper face and enlarging the convex area. 
 
 The cantilever frame-work further, not only moves the locus of 
 zero bending outward from the column, but it also fixes the locus of 
 zero bending moment at a known position so that it does not vary 
 with increase and decrease of the load or change of the load from one 
 span to an adjacent span as would be the case were the mass of 
 metal in the frame and its stiffness largely reduced. This is ac- 
 complished as follows: 
 
 The locus of zero bending moments is fixed by the dip of the 
 reinforceing rods as they leave the upper surface of the slab near 
 the edge of mushroom and pass below the neutral surface to a level 
 near the bottom of the slab. Such change of tensile resistance in 
 the slab necessarily localizes at these points the zero bending mo- 
 ments. 
 
4 SLAB ACTION BIBLIOGRAPHY 
 
 In addition to the advantages just mentioned, which are 
 of so self-evident a character as to be readily appreciated even 
 by the layman, there is another of such an obscure and apparently 
 inexplicable a nature that it was for years denied as incredible and 
 regarded as non-existent by practical builders, and engineers as 
 well, unless they had opportunity to be convinced of its reality 
 by experiment. I refer here to the additional strength and stiff- 
 ness which is imparted to a belt of rods in a given direction in a 
 slab by another belt at right angles to the first belt, or at various 
 angles with it. This should be designated as slab action proper 
 in distinction from cantilever action. It depends for its amount 
 upon the value of Poisson's ratio of lateral contraction to direct 
 elongation in the slab, and is the basis of the so called circum- 
 ferential stresses, which make the strength and stiffness of such 
 reinforced flat slabs much greater than they are estimated to be 
 when these are neglected, as they usually have been. This mis- 
 taken view has in the past constituted the most serious obstacle 
 to the adoption of this form of structure, and has been the ground 
 of conscientious opposition to its introduction on the part of con- 
 sulting engineers. It is the object of this investigation to remove 
 if possible all reasonable uncertainty as to the rational theory of 
 this form of structure. 
 
 The following partial bibliography of this subject may be useful 
 to those unfamiliar with what has been done in this field. 
 
 Concrete Steel Construction, (305pp) 
 
 By C. A. P. Turner, M. Am. Soo. C. E. 
 816 Phoenix Bldg., Minneapolis, 1909. 
 
 Reinforced Concrete Construction, (259pp) 
 
 By Turneaure and Maurer, University of Wisconsin 
 Wiley, N. Y., 1907. 
 
 Concrete, Plain and Reinforced, (483pp) 
 By Taylor and Thompson, 
 WUey, N. Y., 1911. 
 
 Trans. Am. Soc. C. E. 
 Vol. LVI. June 1906. 
 
 Engineering News: — 
 Oct. 4, 1906, p. 361. 
 Feb. 18, 1909, p 176. 
 Dec. 23, 1909, p. 694. 
 
 Engineering Record: — 
 
 March 28, 1908, p 374. 
 May 2, 1908, p 575. 
 Oct. 10, 1909, p 411. 
 April 3, 1909, p 408. 
 April 10 ,1909, p 492. 
 
NOTATION 
 
 2. All lengths and areas are measured in inches, and all weights 
 in pounds. 
 
 A = area of cross section of steel reinforcement per unit ■width of 
 
 slab, in case it be assumed to be replaced by a uniform sheet 
 
 of equal weight, 
 .■ii = area of cross section of all the rods in one side belt. 
 A-2 = area of cross section of all the rods in one diagonal belt. 
 a = one half the longer side of a panel from center to center of 
 
 columns. 
 b = one half the shorter side of a panel. 
 B = the shortest distance along one side of a panel from the edge 
 
 of a column cap to the edge of the next cap. 
 Cj and 0-2 are constants depending on the relative lengths of the sides 
 
 of any panel, which reduce to unity for any square panel. 
 Di = the deflection of the middle of the longer side of the panel 
 
 below the edge of the cap. 
 D2 = the deflection of the center of the panel below the edge of the 
 
 cap. 
 
 d = the effective thickness of the slab at any point, being the 
 vertical distance from the center of action of the reinforce- 
 ment to the compressed surface of the concrete. 
 
 di = the A^ertical distance from the center of the rods in the side 
 belt at mid span to the top surface of the concrete. 
 
 d2 = the distance at the center of the panel from the center of the 
 rods in the second or upper diagonal belt to the top of the 
 concrete. 
 
 d-i = the distance at the edge of the cap from the center of the third 
 belt of rods from the top, to the compressed surface of the 
 concrete. 
 
 E or E^ = Young's modulus for steel = 3 x 10'^ 
 
 Ea = Young's modulus for concrete. 
 
 Ci = elongation in steel parallel to long side belt. 
 
 6-2 = elongation in steel parallel to short side belt. 
 
 ei = elongation in steel parallel to diagonal belt. 
 
NOTATION 
 
 F = modulus of elastic resistance to shearing. 
 
 /a = Ee = intensity of actual stress in steel. 
 
 /o = intensity of stress in concrete. 
 
 g = 7/I6 (a+b) = the diameter of the mushroom head and width 
 
 of belts. 
 h = the total actual thickness of concrete slab. 
 id = vertical distance from center of tension of steel to neutral 
 
 surface of slab. 
 jd = vertical distance from center of tension in steel to center of 
 
 compression in concrete. 
 kd = vertical distance from neutral surface to compressed surface 
 
 of concrete, hence i + k = 1. 
 K = Poisson's ratio of lateral contraction to longitudinal stretch- 
 ing for reinforced concrete slabs. 
 Li = 2a = long side of panel between column centers. 
 I12 = 2b = short side of panel between column centers. 
 I = distance from collar band at top of column to edge of cap. 
 mi = true moment of resistance of the tensile stresses in steel parallel 
 
 to the long side per unit of width of slab. 
 7112 = true moment of resistance of steel parallel to short side per 
 
 unit of width. 
 
 mi and m2 = apparent moments per unit of width of forces applied 
 
 parallel to the long and short sides respectively, 
 n = the apparent moments per unit of width of the equal 
 
 twisting couples parallel to either side. 
 ■pi = intensity of the forces applied parallel to the long side. 
 
 P2 = ditto for short side. 
 
 J) = intensity of stress in extreme fiber of radial rods. 
 
 q = load on slab in pounds per square inch. 
 
 Ri and R2 = the radii of curvature of vertical sections of the slab 
 
 parallel to the long and short sides respectively. 
 Si and S2 = the vertical shearing stresses per unit of width of slab 
 
 respectively perpendicular to the long and short sides 
 
 of the slab. 
 
 ■s = the intensity of vertical shearing stress in radial 
 
 rods. 
 
 t = either of the equal horizontal tangential or shearing 
 
 stresses parallel to the sides of the panel. 
 
NOTATION. SQUARE PANEL 
 
 t = the thickness of a radial rod. 
 
 u and V = deformations parallel to the long and short sides re- 
 spectively. 
 
 V = total vertical shearing stress in radial rod. 
 
 xy z = horizontal and vertical coordinates parallel to sides 
 
 of panel. 
 
 = difference of two vertical coordinates. 
 = deflections of radial rods. 
 = sign of partial differential. 
 
 2i and Z2 
 
 8 
 
 8z 
 
 8x 
 
 partial differential coefficient of z with respect to x. 
 
 Fig. 3. Plan of Reinforcement Mushroom System. 
 Square Panel, g = kL (as drawn). 
 Line of Ultimate Weakness. 
 
8 TETJB AND APPAKENT STRESSES 
 
 3. As preliminary to a general investigation of the rational 
 analysis of the flat slab, it seems desirable in the first place to 
 make a brief exposition of the relationship between the true bend- 
 ing moments and the apparent bending moments in the flat slab as 
 follows: 
 
 The fundamental equations of extensional stress and strain in 
 thin flat plates and slabs, established a generation ago and accepted 
 by Grashof*and by all authorities on the subject since then, maybe 
 written in the forms : 
 
 Eei = Vi— KV2 I (1) 
 
 Ee2 = P2 — Kpi j 
 
 (1 - K^)p, ^ E(e, + Keo) \ .^^ 
 
 {1-K')p2 = £{€.+ Ke^) i 
 
 in which pi and p2 are the external applied or apparent stresses per 
 unit of area of cross section of the plate, or of the reinforced slab, 
 which act parallel to the axes of x and y respectively if these latter 
 lie in the neutral plane of the slab; and ei and €-2 are extensometer 
 elongations of plate or slab reinforcement per unit of length parallel 
 to X and y respectively. E is Young's modulus of elasticity, and K 
 is Poisson's ratio of lateral contraction to linear elongation. Any 
 piece of material which is subjected to stress, and is of such shape 
 that more than one of its dimensions is considerable, as compared 
 with its remaining dimension, must have its stresses and strains 
 considered with reference to lateral contraction. This is the case 
 in plates and slabs, as it is not in case of rods and beams. 
 
 In the above equations -E'ei andEe-i are the true stresses per square 
 inch of section of reinforcement acting along lines parallel to x and y 
 respectively, whatever pi and p2 may be. These latter are the cause 
 of true stresses, but are not themselves the values of the true stresses, 
 as they are in case of rods, etc., where one dimension onlj' is large. 
 
 These equations show that the elongation ei in the direction of 
 X is not dependent alone upon the tension pi apphed in that direc- 
 tion, for it is diminished by any tension acting along y, but is in- 
 creased by any compression acting along y. It thus appears that 
 any tension pa along y assists the piece in resisting elongation along 
 X and makes it able to endure safely a larger applied stress pi with 
 the same degree of safety, i. e., -nith the same percentages of elonga- 
 tion or true stress. But it is also equally true that any compression 
 of amount p2 reduces the safe value of pi which may be applied to 
 
 *Theorie der ElasUeitat und Fcstgkeit, F. Grashof Berlin 1S7S. 
 
TRUE AND APPABEKT MOMENTS 9 
 
 it. These principles are not in accordance with those which hold 
 in ordinary computations for rods and bars, whose lateral dimensions 
 are small compared with their lengths, and whose lateral stresses 
 are negligible. This divergence between the true stresses as shown 
 by actual deformations, and the apparent or applied stresses, is a 
 fruitful source of error in the attempted computation of slabs. 
 
 Equations (1) in their present form apply to simple extensional 
 or compressive stresses and strains but may be extended to apply 
 to bending of slabs in the following manner : 
 
 Take A as the cross section of the reinforcement per unit of 
 width of slab when the actual reinforcement is regarded as distrib- 
 uted into a thin sheet of uniform thickness, and let jd be the vertical 
 distance from the center of the reinforcement to the center of com- 
 pressional resistance of the concrete regarded as a fraction j of d, 
 d being the distance from the center of the steel to the top of the 
 slab. Then 
 
 mi = Api jd, and m2 = Ap2 jd, (2) 
 
 are the apparent bending moments per unit of width of slab, of the 
 applied apparent stresses pi and p2, tending when positive, to cause 
 lines which before bending are straight and parallel to x and y re- 
 spectively, to become concave upwards. 
 
 Again mi = Eei Ajd, and OT2 = Ee2 Aid, (3) 
 
 are the true bending moments of the actual resistance stresses in 
 the reinforcement per unit of width of slab, as shown by extenso- 
 meter strains in the steel parallel to the axes of x and y respectively. 
 Multiply equations (1) thru by Ajd and substitute the values 
 given in equations (2) and (3), from which we obtain the following 
 relations between the true and apparent bending moments in the slab. 
 
 mi = mi — Km2\ (4) 
 
 m2 = m2 — Kniij 
 
 (1 — K^)mi = mi+ Km2 \ (^^a) 
 
 (1 — K^)m.2 = m2 + Kmi / 
 
 These equations bring out in a striking manner the essential diver- 
 gence of the correct theory of slab action from that of beam action 
 in which latter case we have the well known equations 
 
 mi = mi, and m2 = m2 
 
 i. e., in beams the moment of the applied forces is equal to the 
 moment of the internal resistance, which is not true in slabs. 
 
10 TRUE AND APPARENT MOMENTS 
 
 All attempts to base computations of the deflection of slabs 
 upon beam action are therefore necessarily erroneous. Such com- 
 putations are inapplicable and misleading, hence deflections and 
 stresses in slabs cannot be correctly computed by any form of 
 simple or compound beam theory. 
 
 Equations (4) show: 
 
 1st That at points where mi and m2 are of the same sign, (as 
 for example in the convex part of the mushroom near the columns 
 and also near the center of the panel) the true bending moments 
 nil and 1712, which determine the actual stresses in the reinforcement 
 are less than the apparent bending moments, which latter have been 
 ordinarily assumed, according to the beam theory, to determine 
 those stresses. 
 
 2nd That the compressive stresses in the concrete around the 
 column cap are determined on the same principles as the tensile 
 stresses and are consequently reduced in accordance with the value 
 of iiL by a considerable percentage below values corresponding to 
 mi and m2 of the beam theory. 
 
 3rd That at points where mi and m2 have different signs, as 
 they have for example in the middle part of the span at the side of 
 the panel directly between mushroom heads, the values of the true 
 bending moments are larger than the apparent moments as found 
 by the beam theory. 
 
 4th One deduction from this (which is also confirmed by 
 extensometer tests) is, that in slabs having equal side and diagonal 
 belts of reinforcing rods the greatest actual extensions and true 
 stresses in the steel occur at the mid points of those reinforcing rods 
 which run directly between the mushroom heads parallel to the 
 sides of the panel, and do not occur at the center of the panel where 
 mi and m2 have their greatest values. Further, the true stresses in 
 the reinforcement are not so large at the edge of the column caps as 
 at the points just indicated. Neither of these conclusions is in 
 accordance Avith the beam theory as implied in ordinary formulas 
 such as have been frequently adopted in practice in computing slabs. 
 
 5th In making any statement or specification respecting the 
 bending moments at any point of a slab, it is essential to state which 
 bending moments are contemplated, the true bending moments or 
 the apparent moments, with the understanding that the true bend- 
 ing moments only are to be used in determining cross sections and 
 stresses of steel. Any statement omitting this distinction is ambig- 
 uous, and any requirement seeking to proportion cross sections of 
 steel to apparent stresses and apparent moments is incorrect. 
 
POISSON S RATIO 11 
 
 4. Poisson's ratio K plays an important role in the theory of 
 flat slabs and plates, as is evident from equations (1) and (4). Few 
 attempts have been made to determine K by directly measuring the 
 amoimt of the lateral contraction accompanying the elongation of 
 test specimens, and, were such measurements made, the relative 
 dimensions of the cross section of the specimen would need to be 
 considered as affecting in a very complicated way the true value of 
 K to be derived from observation. Reliable determinations of K 
 usually depend upon observations of Young's modulus of elasticity 
 E and' the shearing modulus of elasticity F. 
 
 It is proven in the general theory of the deformation of isotropic 
 elastic solids that all the elastic properties of any such solid are 
 determined without excess or defect by its values of E and F, and 
 that Poisson's ratio is a function of E and F expressed by the equation 
 
 K + l = lE/F (5) 
 
 There is evidence to show that for concrete K is approximately 
 0.1*. For steel it is known that K = 0.3 nearly. 
 
 Now it is evident that a horizontal slab of reinforced concrete, 
 in which the reinforcement consists of rods, differs from one in which 
 its reinforcement is considered to be a simple uniform sheet of metal 
 in this, that the former has much less shearing rigidity in resisting 
 horizontal forces than the latter, for in it all stresses transmitted 
 from one band or belt of rods to any other belt crossing it are trans- 
 mitted thru concrete only, as is not the case if the reinforcement 
 consists of a continuous sheet. It is evident therefore that the value 
 of K which must be employed in applying the foregoing equations to 
 reinforced concrete slabs must exceed 0.3, the value required in case 
 the reinforcement is a sheet of steel. 
 
 This analysis of the conditions affecting the value of K for a 
 reinforced flat slab differs radically from assuming at ramdon that 
 because K = 0.3 for steel alone and if = 0.1 possibly, for concrete 
 alone, that therefore some intermediate value of K may be correct 
 for these two materials combined in a slab. Such an assumption 
 is merely a blind guess and has no rational basis. 
 
 As already partly stated, the view here put forth is this : Since 
 in any homogeneous, isotropic, elastic material the experimental 
 values of E and F perfectly define all its elastic properties, and since 
 we are evidently at liberty to assume our flat slab as sufficiently fine 
 grained in its structure to act nearly like a slab constructed of some 
 sort of homogeneous materials, it will be possible to determine 
 
 * Tumeaure and ^laurer's Reinforced Concrete Construction 2nd Ed. 1907, p. 210. 
 
12 POISSON S RATIO 
 
 certain mean values of E and F which will define its elastic proper- 
 ties. It is moreover evident that in a slab, where two kinds of elastic 
 solids are combined as they are here, the mean value of F for the 
 combination is much more affected by the concrete than is E, which 
 latter may be taken as that applying to the steel alone, and, conse- 
 quently as unchanged by the combination. It is otherwise, however, 
 with F, because the arrangement of the combination is such as to 
 require the assumption of a value of F lying somewhere between 
 that for steel and that for concrete. Since the latter value is much 
 less than the former, the mean value of F is smaller than for steel 
 alone. 
 
 This reasoning and other independent theoretical and kinemat- 
 ical considerations have led to the same conclusion, viz: that the 
 correct value of K for the slab is larger than 0.3. 
 
 Assuming E = 30,000,000, we may compute corresponding 
 values of K and F from (5) as follows : — 
 
 If X = 0.1 , F = 13,600,000 
 If X = 0.3 , F = 11,600,000 
 If if = 0.5 , F = 10,000,000 
 
 Were a perfectly complete and accurate mathematical theory 
 of the fiat slab at our disposal, we might consider every experimental 
 test of the deflection of such a slab, and every extensometer measure- 
 ment of its reinforcing rods as an experiment for determining the 
 numerical value of K, since deflections and extensions would then 
 all be known functions of K. Having brought such a rational 
 theory to a somewhat satisfactory degree of perfection, the writer 
 finds that, in the light of all known tests of slabs, the value that best 
 
 satisfies all conditions is K = 0.5. (6) 
 
 It is possible that this value of the constant K for slabs may need 
 some slight modifications hereafter, but for the present this may be 
 regarded as substantially correct for mushroom slabs. It may be 
 found necessary to assume a somewhat different value for other forms 
 of structure, as for example, beam and girder construction. That, 
 however, must be determined later. Moreover it must be said that 
 this value of K applies to tests made upon slabs from 2 to 4 months 
 old, and under loads which have been applied to such relatively soft 
 concrete as this for a period of usually not longer than one or two 
 days, and of an intensity such as to cause a maximum stress in the 
 steel of from 10,000 to 16,000 lbs. per square inch. Less loads on 
 better cured concrete, or longer time under load, may show con- 
 siderable deviation from this value of K. 
 
EQUILIBRIUM OF SLAB ELEMENT 13 
 
 How important a factor K is in slab theory is evident on con- 
 sidering equations (4) which show that in a square panel uniformly 
 loaded the true moments as shown by the elongations of the rein- 
 forcing rods at the center of the panel and over the centers of the 
 columns are only one half the corresponding apparent moments 
 derived from considering the moments required to hold the applied 
 forces in equilibrium, this being on the assumption of course that 
 K = 0.5. 
 
 5. In order to derive the general differential equation of shears 
 and moments in any rectangular panel in an extended horizontal 
 plate or slab, take the axes of x and y in the neutral plane of the 
 plate and parallel respectively to the longer and shorter sides of the 
 panel with the origin at its center before flexure occurs, and assume 
 that they remain fixed with reference to the points of support of the 
 panel. Then during flexure the center of the panel and all other 
 points of the slab or plate not in contact with the fixed points of 
 support will attain some deflection z, of amount to be determined 
 later. Take z positive downwards. 
 
 Then bxby is the horizontal area of an element of the slab 
 bounded by vertical planes, and if d be the effective thickness of the 
 slab or plate, the areas of the sides of this element which are respec- 
 tively perpendicular to x and y are dby and dbx, while dbxby is the 
 volume of the element. 
 
 We proceed to obtain the equations of equilibrium of this ele- 
 ment of the slab as follows : — 
 
 Let Si and S2 be the total vertical shearing stresses per unit of 
 width of slab for sections perpendicular to x and y respectively. In 
 case these shears are variable, as they are in a continuously loaded 
 slab, they respectively contribute elementary forces tending to move 
 the element vertically, of the following amounts : 
 
 5 Si 5 S2 
 
 bybx, and bxby 
 
 b X by 
 
 Assume that the slab carries a uniformly distributed load of q pounds 
 per square unit of area. Then the load upon the elementary area 
 bxby is qbxby, and the equation of equilibrium of the vertical forces 
 acting on the element reduces to this: 
 
 5 Si 5 S2 
 
 + — +q = (7) 
 
 b X by 
 
14 EQUILIBRIUM OP SLAB ELEMENT 
 
 in which Si and S2 are taken as positive when they are such as would 
 be produced in the slab by the loading q in case it were supported at 
 the origin only. 
 
 Let mi and m2 be the apparent moments per unit of width of 
 slab of the applied forces which tend to bend those lines in the slab 
 which before bending are parallel to x and y respectively. Take 
 them as positive when they tend to make those lines respectively 
 concave upwards. These are the moments obtained by multiplying 
 the total applied tension per unit of width of slab by the vertical 
 distance jd from the center of the reinforcement of the slab to the 
 center of compression in the concrete as given in (2) . These moments 
 are not identical in a slab with the true resisting moments mi and m2 
 in the same directions, which latter are the moments obtained by 
 multiplying jd by the actual tension in the steel per imit of width of 
 slab, which last is to be correctly computed by taking the product 
 of the area of steel per unit of width and its elongation multiplied 
 by E its modulus of elasticity as shown in (3) . 
 
 Again, let n be the twisting moment per unit of width of ver- 
 tical section of slab cut by planes perpendicular to either x or y, and 
 acting about either x or y, which moment n is regarded as due to the 
 variation of the vertical shearing stress Si when y varies, and to the 
 variation of S2 when x varies. The moment n is held in equilibrium 
 by horizontal shearing stresses in these same sections, which are 
 opposite in sign above and below the neutral surface. Let t be the 
 total horizontal shearing stress per unit of width of slab in the rein- 
 forcement on one side of the neutral plane, then: 
 
 n = t A j d (8) 
 
 At any point xy this horizontal shearing stress t must be the same 
 for the section perpendicular to x, as for the section perpendicular 
 to y, because in every state of stress the tangential components are 
 equal and of opposite sign on any two planes mutually at right 
 angles. Consequently the moment n is the same about x as about 
 y, as has been assumed in (8). 
 
 It is implicitly assumed in (2) and (3) that the concrete on the 
 same side of the neutral plane as the reinforcement is ineffective 
 and that its resistance is negligible, so that on that side the resistance 
 of the reinforcement alone counts. This condition actually occurs 
 only after a state of quite considerable stress obtains, and of itself 
 affords a sufficient reason why the formulas based on it fail of accu- 
 rately representing deflections and elongations at small loads and 
 low stress. 
 
DIFFERENTIAL EQUATION OF MOMENTS 15 
 
 The elementary couples acting on the vertical faces of the 
 element which are in equilibrium with those arising from the shear- 
 ing stresses are : — 
 
 (- 
 
 8 mi 
 
 d X 
 
 + 
 
 dn 
 8 y 
 
 -) 
 
 8x8y 
 
 about y, 
 
 (- 
 
 8 m2 
 d y 
 
 + 
 
 8 n 
 8 X 
 
 ) 
 
 8x8y 
 
 about x; 
 
 while those arising from the shears themselves are : — 
 
 Si 5a; 8y and S2 8x 8y. 
 
 Consequently the equations of equilibrium of the couples acting on 
 the element reduce to the following: 
 
 8 
 
 mi 
 
 8 
 
 x 
 
 8 
 
 m2 
 
 + + Si 
 
 8 
 
 n 
 
 8 
 
 y 
 
 8 
 
 n 
 
 ,(9) 
 
 + + S2 = 
 
 8 y 8 X 
 
 Differentiate equations (9) with respect to x and y respectively 
 and substitute in (7), and we obtain 
 
 S ^ mi 8^ n 8^ m2 
 
 + 2-— 7- + -—r = q (10) 
 
 8 x^ 8x 8y 8 y^ 
 
 which is a general differential equation of the apparent moments 
 of the applied forces which exist in a uniformly loaded slab in 
 terms of rectangular coordinates. From it the differential equa- 
 tion of the deflections may be derived as follows: — 
 
 6. To obtain the general differential equation of the deflec- 
 tions of a slab, note that from geometrical considerations such as 
 are familiar in the theory of beams we have 
 
 Ri ei = i d = R2 €2 (11) 
 
 in which Ri and R2 are the radii of curvature of sections of the 
 neutral surface by vertical planes parallel to x and y respectively; 
 and id is the distance from the center of the reinforcement to the 
 
16 MOMENTS AND CURVATURES IN SLAB 
 
 neutral surface. In equations (la) replace pi and p2 by values 
 ^iven in (2), and ei and 62 by values taken from (11) and we have: — 
 
 (1 — X^)mi = EA ij<f ( — + — ) 
 
 \Ri R2/ \ 
 
 (12) 
 
 \«2 rJ 
 
 (1 — K^)m2 = E A ijd 
 
 But from the theory of curvature 
 
 1 , 8^ z 1 , 8^ z 
 
 — = ± , and — = ± ... . (13) 
 
 Ri 8 X2 R2 8 y 
 
 Also write for brevity I = A ij (f (14) 
 
 Then we have from (12), (13) and (14): 
 
 52 . S2 
 
 (1 - K')m, = ± 
 
 / 8^ z 8^ z \ 
 EI I + K I 
 
 \8x' 8y' ; 
 
 \8y' 8x' ) 
 
 I 
 
 (15) 
 
 . S2 . s2 
 
 (1 — K^)m2 = JLEI 
 
 By the fundamental equations of elasticity we also have 
 / 8u 8v\ 
 
 ' = ''" = ''W+^) <■'' 
 
 in which F is the shearing modulus, 63 is the horizontal shearing 
 deformation of the reinforcement for two vertical planes one imit 
 apart horizontally, and 
 
 _L 62 ^ 8z 
 
 u = J_id — , V = _ I d — (17) 
 
 5a; 8y 
 
 are the deformations along x and y respectively, due to the vertical 
 distance id of the reinforcement from the neutral surface. 
 
 From (16) by help of (17) we have 
 , 8^ z 
 
 t = -2^^^^^ (18) 
 
 oxoy 
 
5* 
 
 z 
 
 8' z 
 
 + 
 
 d' z 
 
 5 
 
 x^ 
 
 5 2/* 
 
 DIFFERENTIAL EQUATION OF DEFLECTIONS 17 
 
 In (18) replace F by its value obtained from (5), and then sub- 
 stitute the resulting value of t in (8) : — 
 we then have 
 
 EI 8^ z 
 
 n = (19) 
 
 1 + K bxhy 
 
 From (15) and (19) obtain values of the second differential 
 coefficients of the moments appearing in (10), which on being intro- 
 duced into (10), transform that equation into the required general 
 differential equation of deflections as follows: — 
 
 (1 - K') 
 
 q (20) 
 
 EI 
 
 which is a partial differential equation of the fourth order that must 
 be satisfied by the coordinates x y z oi the neutral surface of any 
 uniform plate or slab initially flat, when deflected by the applica- 
 tion of a uniformly distributed load of intensity q, and supported in 
 any manner whatever. 
 
 It may be shown that any deviations from strict accuracy by 
 reason of local stretching of the neutral surface (here neglected) are 
 small compared with corresponding deviations in beam theory. 
 
 7. The solution of the general differential equation of deflec- 
 tions (20) for the case of a horizontal slab carrying a uniformly 
 distributed load and supported on rows of columns placed in rec- 
 tangular array and having the points of support all on the same 
 level, will now be considered. 
 
 The integration or solution of (20) would, since it is a partial 
 differential equation, introduce arbitrary functions of the independent 
 variables x and y whose forms would need to be so determined as to 
 cause the solution to satisfy the conditions imposed by the position 
 and character of the supports at certain points, or along certain 
 lines. It would be possible to expand these functions in terms of 
 ascending whole powers and products of x and y, and, in case the 
 supports are symmetrically situated with respect to the axes, the 
 expansions will contain no odd powers of x or y, because the value 
 of z must remain unchanged by changes of sign of either x or y, or 
 both x and y. Any form of polynomial expansion which satisfies 
 (20), and also all the conditions of any given case, must be the correct 
 solution for that case, for, the solution of any given case must be 
 unique. 
 
18 GENERAL EQUATION OP DEFLECTIONS 
 
 Instead therefore of carrying thru the tedious analytical devel- 
 opment involved in solving (20) mathematically and then applying 
 it to the case we are treating, we shall at once write down the form 
 of solution that applies to the case in hand and verify the fact that 
 it satisfies (2) and all the required geometrical conditions. It will 
 therefore be the solution sought for, which might also have been 
 obtained by the somewhat intricate analytical processes involved 
 in the intregation of such differential equations as (20). 
 
 Assuming at first that the slab is unlimited in extent and uni- 
 form thruout in the distribution of its reinforcement and loading, 
 and that the parallel rows of supporting columns divide the slab 
 into equal rectangular panels, we shall find a solution in which every 
 panel is deformed precisely in the same manner as every other. 
 Modifications made later will render it possible to take account of 
 variations and irregularities in the distribution and arrangement of 
 the reinforcement, and to estimate to some extent at least the effect 
 of loading only one or more panels. 
 
 Let 2a be the length and 26 be the breadth of a panel; then the 
 equation of its neutral surface, referred to axes parallel to its sides 
 and to an origin fixed in space at the center of the neutral surface of 
 the panel before deflection, is: — 
 
 48 EIz = g(l - K^) [{a^ — x^f + (6^ — y'f] (21) 
 
 This is the correct solution of (20) not only because it satisfies 
 (20), as it will be found to do by trial, (and just as many other func- 
 tions of X and y do also) but it also satisfies all the other conditions 
 required by the case proposed, viz. : 
 
 1st z = Q when both x = ^a and y = + h; 
 because there must be no deflection at these points of support which 
 are on the same level as the origin. 
 
 2nd & / rfx = 0, when x = 0, and also when x = ^a; as well as 
 dz/dy = Q, when y = 0, and also when 2/ = dl 6; because straight 
 lines drawn in space to touch the slab across its edges, and across 
 its mid sections parallel to those edges, must all be horizontal by 
 reason of the symmetry of the slab on each side of its edges and mid 
 sections. That these conditions hold is evident from the following 
 equations derived from (20) : 
 
 5 z (1— X') 
 
 q X {x^ — a^) 
 
 8x 12 E I 
 8z (l—K^) 
 5y 12 EI 
 
 qy(y'~ b') 
 
 .(22) 
 
TRUE STRESSES AND MOMENTS. LINES OF CONTUA-FLEXURE 
 
 19 
 
 It is of interest to note that the sections of this surface made 
 by all vertical planes parallel to the axes of y, i. e., by x = constant, 
 are precisely the same except in position, since their equations differ 
 by a constant only. The same is true of sections parallel to x. It 
 thus appears, that, in a square panel where a = b, the surface may 
 be regarded as a ruled surface described by using the two of these 
 curves on a pair of parallel sides of the panel as directrices and a 
 third one of these curves as a ruler sliding on the first two in such a 
 manner as to remain parallel to the other pair of parallel sides. 
 
 The deflections at the center of the panel and middles of the 
 
 sides are: 
 
 Atx = = y, 4:SE I z = q (l-X') (a* + b*) 
 
 Atx = +a,y = 0, ^8 E I z = q (l-K^) 6* 
 Atx = 0,y = ±b, 4&EI z = q {l-K^) a* 
 
 so that in a square panel the center deflection is twice the mid edge 
 
 deflection. 
 
 Differentiating equations (22) we have by help of (11), (13), 
 (14) and (3) : 
 
 id 5^ 2 
 
 ei = — = + i d 7 — 
 
 Rx - dx^ 
 
 id 8^ z 
 
 ^2 
 
 (1-A-^) 
 12E Aj d 
 
 q {3x^ — a^) 
 
 8x' 
 
 ■q{3/~b') 
 
 12 E Aj d 
 
 ,(23) 
 
 a-K') 
 
 nil = : — Q (3a;^ — a^) 
 
 ..2 j,2x 
 
 12 
 
 nu = 
 
 12 
 
 g(3t/^-6^) 
 
 . (23a) 
 
 in which the ambiguous signs are to be so taken that mi and TO2 in 
 (15) will be positive at x = = j/, and negative at a; = it a and 
 y=±b. 
 
 From (23) it appears that extensions vanish and contra-flexure 
 occurs at lines lying in vertical planes whose equations are 
 
 = if a^3 and 
 
 = ±i&V3. 
 
 y = j-z 
 
 (24) 
 
 It thus appears that the slab is subdivided by these lines (24) 
 drawn parallel to the edges into a pattern which consists of a rect- 
 angle occupying the middle part of each panel, of a size f aV3 by 
 
20 APPARENT MOMENTS AND SHEARS 
 
 f &V^3, i- e., of the same relative dimensions as the panel itself, and 
 bounded by lines (24), which rectangle is concave upward thruout. 
 On all four sides of this central rectangle are rectangles of saddle 
 shaped curvature directly between the central rectangles of adjoin- 
 ing panels, while each point of support is situated in a rectangle 
 which is convex upward over its entire area, of dimensions 
 2a(l— i ^^3) by 26(1 — i V^3). 
 
 From (22) we obtain the equation 
 
 5^ z / bxby = (25), 
 
 hence by (18) and (19) it follows that 
 
 t = = n, (26), 
 
 from which it appears that there is no horizontal shear in the steel, 
 and no twisting moment in vertical planes perpendicular to x or y. 
 This would be otherwise evident from considerations of symmetry. 
 It will be shown that this is not true of all other vertical planes. 
 
 Again from (15) and (23) we have 
 
 mi = i g [3/ - a^ + i:(3/ - &^) ] ,^„. 
 
 m, = h q imx' --a')+3y'~bn ^ ^ 
 
 in which we have omitted the sign ^l as superfluous. 
 
 From (9) by help of (26) and (27), we have 
 
 5mi 5m2 
 
 — Si = =^qx, and — Sa = = ^qy (28) 
 
 ox by 
 
 from which it appears that any strip of the panel parallel to x or y, 
 and one unit wide exerts a shear at its ends such as it would if it were 
 an isolated beam loaded uniformly with an intensity of \q per unit 
 of length. According to this, a total shear oi q a h, which is one 
 fourth of the total load carried by the panel, appears at each edge of 
 the panel, this total shear on each edge being uniformly distributed 
 along it. 
 
 It is seen therefore that the form of solution which we are 
 investigating implicitly assumes that at each edge of the panel there 
 is some auxiliary form of structure that will bear the shears coming 
 to it from each side and at the same time assume the curvatures and 
 deflections contemplated in (21). This will immediately engage 
 our further attention. 
 
SIDE BELTS 21 
 
 8. In order to investigate more fully the deflections, stresses- 
 and strains in tlie side belts of any panel directly between the mush- 
 room heads, let us consider the results just reached somewhat more 
 fully. 
 
 The conclusion drawn from (28) was, that a panel with rein- 
 forcement distributed with perfect uniformity thruout would require 
 to be supported by a narrow auxiliary girder extending from column 
 to column along each side, and of such resisting moment as to take 
 on, under its load, the precise curvature required by the neutral 
 surface in (21), which cur^'ature must be produced by a uniformly 
 distributed load oi 2 q a b, one half of it coming from each of the two 
 panels beside it. 
 
 It seems then, that up to this point, we have in reality been 
 treating the theory of the continuous uniform slab with specially 
 designed continuous beams supporting its edges, without as yet 
 investigating those beams in detail. But since no such beams in 
 fact exist under the flat slab, it is clear that the side belts of the slab 
 lying directly between the extended heads of the columns must 
 discharge the functions which would be discharged by the auxiliary 
 beams just spoken of. Such functions must necessarily be added 
 to those alread}^ discharged by those belts in supporting the loading 
 which rests directly upon them. In order that this may occur in a 
 mamier readily amenable to analysis, the extended stiffened head- 
 ings of the columns which constitute the mushrooms should in 
 general be approximately of the diameter required to support the 
 ends of a belt of reinforcing rods forming a flat beam which fills the 
 width along the edge of two adjacent panels between the two lines 
 of contra-flexure on each side of that edge, as given in (24). 
 
 This requires that the mushroom head should have a width of 
 at least (1 — | Vz) = -^'-3 of the width of the slab between col- 
 umns. For reasons that will appear later, it is current practice to 
 make these heads not less than A = .437 of this width. 
 
 The lines of contra-flexure in (24) have a flxity of position, (in a 
 flat slab constructed with mushroom heads of this size and stiffness,) 
 mider single panel loads, that does not exist in a uniform slab, or 
 where the headings ai-e not so stiff. It may be readily shown by 
 ^lohr's theorem respecting deflection curves as second moment 
 polygons, that where there are large sudden changes in the magni- 
 tude of the moment of inertia I, such as exist in this case at the lines 
 of contra-flexure at the edges of the mushroom, the lines of contra- 
 flexure remain fixed. But in systems where the diameter of the head 
 is smaller than given above, or its stiffness is much reduced, these 
 
22 SIDE BELTS 
 
 lines may be removed to greater distances from the center in loaded 
 panels surrounded by those not loaded than when all are loaded, 
 thereby increasing the deflections and stresses in a single loaded 
 panel over that of a uniformly loaded slab of many panels. 
 
 The lines of contra-flexure in (24) separate the slab into areas 
 which are largely independent of each other, since no bending 
 moments are propogated from one to another. The only forces 
 crossing these lines of section are the total vertical and horizontal 
 shearing stresses. The horizontal shears (which are imimportant so 
 far as deflections go) will be considered later so far as may be necessary, 
 but the vertical shears found by (28) are of prime importance. Let 
 us then consider one of these side belts. 
 
 In any extended slab with its panels all loaded uniformly 
 thruout, the vertical shear must vanish at all points along sections 
 made by vertical planes thru the centers of columns at each side of 
 any panel, as appears by reason of symmetry of loads. Let the 
 edges of the side belts be situated at some given distances, say xi and 
 2/1 on each side of the centers of all the panels, where Xi and yi are 
 not necessarily the values of x and y given in (24), altho those 
 values are also included in this supposition. Then by (28) there is 
 a uniformly distributed vertical shear of intensity i q yi along the 
 edge of the belt at y = y^, even tho the reinforcement in the side 
 belt may be greater than that in the central rectangle, for the devia- 
 tions caused by the irregularity of its distribution may be regarded 
 as unimportant and practicably negligible. 
 
 It may then be assumed that any side belt parallel to x must 
 carry, in addition to that already provided for in (21), a total loading 
 of q 2/1 per unit of length, uniformly distributed along the two edges 
 that are parallel to x. Now since the width of this belt is 2(6 — 2/1)1 
 the load already provided for in (21) is |g per unit of area, or q(b-yi) 
 per unit of length parallel to x, which added to that arising from the 
 shears just mentioned makes a sum total of g 6 per unit of length of 
 belt, which it will be noticed is independent of the width of the belt. 
 In other words, any such belt must support a load of one fourth of 
 the total load on the two panels of which it forms a part, or one half 
 of all that lies between the panel center lines which are parallel to 
 it on either side. This in effect transfers the entire loading of the 
 slab to tl^e side belts by the agency of the shearing stresses. It does 
 this in such a way that one half of the total loading of the entire slab 
 is carried by one set of side belts, and the other half by a second set 
 which crosses the first at right angles. 
 
DEFLECTION OF SIDE BELTS '24 
 
 In those parts of the slab area where these sets of belts cross, 
 forming the heading of the columns, the loading is superposed also. 
 
 The preceeding im-estigation of the shears at the edges of siile 
 belts ai^d their loading is independent of their width and of the posi- 
 tion of the lines of contra-flexure, but their mdth will be assumed in 
 what follows to be determined by the position of those lines as shown 
 in (,24) on accomit of the independence of action of belts of their 
 width, as previousl.y explained, where it was shown that no bending 
 moments are propogated across those lines. 
 
 The question now ai-ises, how the vertical shears at the edges 
 of the side belts are distributed across their -n-idth and carried by 
 them. Since bj' symmetrj" of loading, etc., there is no vertical 
 shear at the edge of the panel where y = b, the shear must diminish 
 from each edge of a belt to zero at that line. If it be assumed to 
 diminish uniformly, that is equivalent in its action to a uniformly 
 distributed load on the belt, which may be assumed in computation 
 to replace the shears at the edges. Whether it mil be so distributed 
 or not depends upon the stiffness of the mushroom head and the 
 smallness of its flexure. Extensometer measurements on the rods 
 of the side belt of the floor slab of the St. Paul Bread Company 
 Building by Prof. Wm. H. Kavanaugh show bej^ond question that 
 in the mushroom system the load is so distributed. Other exten- 
 someter measurements to which the wTiter has access also show that 
 in systems in which the heading of the coluimi is not so stiff as this 
 the distribution of loading cannot be taken as miiform over the side 
 belts. 
 
 Xow the belt parallel to .r was shown to carry a load per unit of 
 length of q h and to have a width 2{b — i/i), in general, or a width 
 -b(l — 3V^3) for the belt between the lines of contraflexure ; hence 
 the intensity of the loading on this belt is q h/2{h — j/i), instead of 
 q, as it would be in a miiformly loaded panel duly supported at its 
 edges by beams from cohunn to colmnn. Let 'ZA. designate the 
 area of the effecti^-e right cross section of the steel in the entire width 
 of a side belt regarded as forming a single sheet of metal of the width 
 of the belt; then ^A/2{h — ^i) is the effective right cross section 
 per unit of width of belt, and we may write (14) in the form 
 
 7 = U-rf-3A/2(6 — 2/1) (29) 
 
 We shall consider admissible ^-alues of —.4 later. 
 
 Since the deflection of the side belts may be taken independently 
 of the rest of the slab, let those values for the intensity of loading 
 and the moment of inertia (29) be introduced into (21). 
 
24 STRESSES IN SIDE BELTS 
 
 We then obtain an expression for the law governing the deflec- 
 tion of that part of the side belts parallel to x which lies between the 
 mushroom heads, and is bounded by hnes of contra-flexure, viz : 
 
 . = ^^~^'^^^ [ (a^ - x'f + (6^ - if ] (30) 
 
 with a corresponding equation for the side belts parallel to y, which 
 may be obtained by replacing g & in (30) by q a. Call this second 
 equation (31). Now (30) and (31) would hold thruout the entire 
 length of these belts from column to colunrn were they entirely 
 separate from each other and from the diagonal belts where they 
 cross each other. It will be necessary later to obtain the equation 
 which holds true where these belts cross and combine with each other. 
 
 9. Practical formulas for the stresses in the steel and concrete 
 of side belts between the lines of contra-flexure will now be obtained 
 from (30) and (31). 
 
 In order to do this, consider the summation in (30) expressing 
 the effective cross section of the steel in the mid area of the side belt 
 regarded as forming a single uniform sheet, that mid area being 
 bounded on all sides by lines of contra-flexure. 
 
 It is to be noticed that the factor (1 — K^) of (30) takes into 
 account the fact that the lattice of rods forming the reinforcement is 
 less effective than the same amount of metal in the form of a sheet, 
 the only question left being this: Will the great irregularity of 
 distribution of the reinforcement in this area cause it to act differ- 
 ently to any noticeable extent from the manner in which the same 
 amount of metal would act were it possible to distribute it uniformly 
 over the entire area? There are strong reasons which go to sustain 
 the view that this irregularity of distribution is negligible in the 
 standard mushroom slab, at least for loads less than those that 
 stress the steel below the yield point, or do not stress the concrete 
 for too long a time while it is imperfectly cured. On examining a 
 diagram of the reinforcing rods of a slab made with square panels 
 of such proportions that the width of the belts is one half the distance 
 between columns, then the pattern previously mentioned into which 
 it would be divided by these belts will be seen to consist of equal 
 squares whose edges are equal to the width of the belts, with one 
 central square in each panel concave upwards, and one half of each 
 of the saddle shaped squares which border it, also lying within the 
 same panel, and one quarter of each of the four convex squares at the 
 
MEAN REINFORCEMENT OF SIDE BELT 25 
 
 head of each of the columns at the corners of the panel, also lying 
 within the same panel, see Fig. 3, page 7. 
 
 Each side square will be found in this case to have double (or 
 two belt) reinforcement over one half or its area, single belt rein- 
 forcement over a diamond occupying one fourth of its mid area, and 
 triple reinforcement over four triangular areas along its sides which 
 together cover one fourth of the square. This gives a mean value 
 of 2.-1 = 2 Ai in which Ai is the total right cross section of the rods 
 in the side belt. 
 
 The belts in the standard mushroom are, however, not so wide 
 as this, since that system simply requires that the edges of the side 
 and diagonal belts intersect in a single point. Fig. 2, instead of forming 
 four areas of triple reinforcement on the sides. This makes the 
 width of the singly reinforced diamond sufficient to just reach across 
 the side belt. In this practical case we find that very approximately 
 
 SA = 1.5.-li (32) 
 
 in which, as before, A^ is the total right cross section of the side belt 
 in square inches. It is evidently impossible for this single side belt 
 of rods which crosses the diamond, to elongate without a correspond- 
 ing equal elongation of the double reinforcement on all its sides, or 
 at least it is impossible for readjustments to take place in any short 
 time such as will make these direct deformations within the diamond 
 larger than those in the areas along side of it, or before somewhat 
 more permanent deformations have taken place in the concrete. 
 
 In cases where the column heads are smaller than the standard, 
 and the side belts still narrower, not only may SA become much 
 less than 1.5 Ai but the belt become so weakened near the central 
 diamond as to render it very questionable whether the irregularity 
 of distribution of steel in the area considered may be safely disre- 
 garded. Diminution of the size of the heading thus not only dim- 
 inishes cantilever action, but reduces the effective resistance of the 
 reinforcing steel. Not much diminution of the size of head would 
 be required to reduce the value of SA to an amount as small as Ai. 
 
 Introducing the estimate given in (32) for the standard mush- 
 room into (30) we derive by (23), (23a) and (3), for that part of the 
 side belt parallel to x between x = + | a ^3 and x = — J a S^3, 
 
 S' 2 , (1 — K^)q b 
 
 f^ = Eei = +Eid - = + {3x^ ~ a^) 
 
 8 x^ ~' 18 jd A, 
 
 Ml = 1.5 Aij df,= ± q b (3a;' — a') 
 
 12 
 
 (33) 
 
26 TRUE STRESSES IN SIDE BELT 
 
 in which Mi is the total true moment of resistance of the side belt, 
 fa is the true stress per square unit of the reinforcement in the side 
 belt, and 1.5 ^li is the effective right cross section of the reinforce- 
 ment. This is independent of y as before noted, showing that the 
 values of /a and ei are the same for one rod as for another, but they 
 attain their greatest values at the mid length where x = 0. If units 
 be pounds and inches, and we assume j = 0.91 for the very small 
 percentage of reinforcement of the standard mushroom system, then 
 by (33) and (6) the practical formulas for design are: 
 
 /s 
 
 3qa^b W L 
 
 4 X 18 X 0.91 di Ai 175 di A^ 
 W L 
 
 Ml = 1.5 Ai j di f. 
 
 128 
 
 (34) 
 
 in which /s is the true stress in the steel, and Mi is the true bending 
 moment of the effective cross section 1.5 Ai of the steel in the entire 
 belt as shown by the elongation (at mid span) of the rods in a side 
 belt of length L, where L is either 2a or 26, and W = 4 q ab is the 
 total load on the panel in pounds, where di is the vertical distance 
 from the center of the rods in the single belt at mid span to the top 
 surface of the slab. 
 
 While the values obtained from (34) are conservative iorj = 0.91, 
 corresponding to a percentage of reinforcement for one belt of less 
 than 0.25%, (34) should be regarded merely in the light of a speci- 
 men equation for that percentage, and any slab where the percent- 
 age differs materially from that assumed value should be submitted 
 to separate computation in the same manner. 
 
 Values of j are given for beams by Turnearue and Maurer in 
 their "Reinforced Concrete Construction," page 57, for different 
 percentages of reinforcement on the straight line theory, which 
 latter is now accepted usage. As already stated, standard mush- 
 room design makes the percentage of reinforcement for warehouse 
 floors where the panels are, say 20' x 20', as low as 0.25% or less, at 
 the middle of the side belts, reckoned on the beam theory. But m 
 heavier and larger construction it may reach 0.33%. 
 
 We have taken the mean available steel in the belt as 1.5 A i, 
 hence the mean slab reinforcement will not be less than 1.5 x 0.23 = 
 0.4% in the side belt areas between lines of contra-flexure. 
 
 In case we assume the ratio of E, for steel to E, for concrete to 
 be 15, as is often prescribed, we find the above stated value of j as a 
 
THEORETICAL AND EXPERIMENTAL STRESSES COMPARED 27 
 
 good mean value, which will be less in cases where the percentage 
 of steel is greater. The small percentage of steel and great relative 
 thickness of concrete is one of the distinguishing features of the 
 standard mushroom design. 
 
 ^Ye may write (,34) in the form : 
 
 TT' L W L 
 
 ^s = > and M[ =Audf, = (34o) 
 
 175 rfi Ai 192 
 
 hi which Ml is the true bending moment of the actual cross section 
 Ai at mid belt. We have •WTitten this modification of (34), not for 
 use in design, but merely for the purpose of instituting a comparison 
 with empirical formulas obtained by Mr. Turner to express the 
 results of numerous tests made by him. On pages 26 and 2S of his 
 "Concrete Steel Construction" he has given equations expressing 
 the values of stresses and moments in mushroom slabs which in our 
 notation may be written as follows: — 
 
 TT' L W L W L 
 
 Ml = Ai j d U = , aud f. = = (35) 
 
 200 ■ ' 200 X 0.S5 d Ai 170 d A^ 
 
 in which he has assumed 0.S5 as a mean value of j. 
 
 It is seen that equations (34a), obtained from rational theory 
 alone, are in practical agreement with (35), which were deduced 
 from experimental tests of mushroom slabs, where the numerical 
 coefiicient mtroduced is entirely empirical. 
 
 As will be seen later, (34) is the equation which ultimately con- 
 trols the design of the slab reinforcement; so that the agreement of 
 these two entirely independent methods of establishing this funda- 
 mental equation cannot but be regarded with great satisfaction as 
 affording a secure basis for designs that may be safely guaranteed 
 by the constructor, as has been the custom in constructing standard 
 mushroom slalis. 
 
 The slab theorj- here put forth diverges so radically from the 
 results of beam theory that we introduce here the following compar- 
 ati^•e computation of the smallest values of true bending moment 
 and stress in steel, which can be obtained by beam theory for the 
 side belt parallel to .r. as follows: — 
 
 That part of the side belt between the lines of contra-fiexure is 
 simply supported at its ends by shearing stresses, and so may be 
 taken to be a simple beam resting on supports at these end lines. 
 
28 STRESSES BY BEAM THEORY AND BY TEST 
 
 Hence the true stress /g and the true bending moment M' at the 
 middle of this simple uniformly loaded beam may be computed from 
 the equation, 
 
 M' = Aiid/, = ITf'L' : (36) 
 
 in which M' is the total moment of resistance. 
 
 Ai is the total right cross section of the reinforcement, W' is the 
 total uniformly distributed load, and L' is the length of the beam. 
 The length of the simple beam in that case is evidently the distance 
 along X between Hues of contra-flexure, viz, L = f aV3 = 3 -^ V3, 
 where L is the edge of the panel, and the total load at most will be 
 that already proven to be carried by the side belt viz, q h per unit of 
 length, or a total for a span L' of W' = qh L' = iqab sfs = ^W Vz 
 where TF = 4 ga6 is the total load on the panel, hence 
 
 W L ^ ^ 
 
 M' = Ai jdfs = (36a) 
 
 48 
 
 It thus appears that according to simple beam theory the true 
 stress, or the cross section of steel required in the belt, is four times 
 that obtained by slab theory as shown by (34a). Since (34a) is in 
 good accord with experimental tests, this comparison justifies the 
 statements made near the beginning of this paper respecting the 
 inapplicability of beam theory to the computation of slab design. 
 
 The floor of the St. Paul Bread Co. Building, previously men- 
 tioned,is a rough slab 6" thick, and has panels 16' x 15', with ten 
 3/8" round rod reinforcement in each belt, built for a design load of 100 
 pounds per square foot; constructed in winter and frozen, the final 
 test was not made until the end of its first summer after unusually 
 complete curing, such as might make the value of K given in (6) 
 not entirely applicable. In one long side belt, extensometer measure- 
 ments were made at the mid span on three rods, (1) a middle rod, 
 (2) an intermediate rod and (3) an outside rod of the belt, with the 
 following results for the given live load in pounds per square foot: 
 
 Live Loads 
 
 108. 4# 
 
 316. 8# 
 
 416. 8# 
 
 /s = Ee, (1) 
 (2) 
 (3) 
 
 7650 
 7080 
 7320 
 
 15000 
 14190 
 13920 
 
 17940 
 16470 
 17160 
 
 Average 
 
 7350 
 
 14370 
 
 17200 
 
 /. by (34) 
 
 5000 
 
 14440 
 
 19000 
 
STRESSES AT YIELD POINT 29 
 
 The observed results are seen to be in excellent agreement with 
 those computed from (34) for the heavier loads, while any disagree- 
 ment is on the safe side. Agreement is not expected for light loads. 
 
 The accuracy and applicability of (34) and preceeding formulas 
 is dependent on the fixity of the lines of contra-flexure (24) which 
 were previously stated to be practically immovable because of the 
 sudden large change of the moment of resistance of the slab at those 
 lines. That fact may be put in a more definite and convincing 
 form than has been done so far. Consider for a moment that form 
 of continuous cantilever bridge where there are joints between the 
 cantilevers over the successive piers (which are in the form of a letter 
 T) and the intermediate short spans which connect the extremities 
 of the cantilevers. At such joints the resisting moments vanish, and 
 they form in a sense artificially fixed points of contra-flexure. The 
 same thing approximately occurs at the edge of the mushroom, 
 because there the reinforcing steel rapidly dips down from a level 
 above the neutral plane to one below it, and the sign of the moment 
 of resistance changes thru zero at that edge. 
 
 Furthermore, it may be proper to state in this connection that 
 the foregoing theory has been developed in consonance with the 
 general principles of elasticity, and that somewhat different condi- 
 tions and relations are thought to exist when the steel at the middle 
 of the side belts reaches its yield point, as it does in advance of the 
 rest of the reinforcement. As the yield point is reached equations 
 (34) no longer hold; for, as will be seen more clearly later, the single 
 belt of reinforcing steel, which crosses the circumference of an ap- 
 proximately circular area of radius L/2 about the center of each 
 column, will everywhere reach the yield point at practically the same 
 instant, and if loaded much beyond this will develop a continuous 
 line of weakness there. The equations that hold in this case will be 
 approximately those due to the actual cross section A i of the belt, in 
 place of (34) , which contain the effective cross section, viz : 
 
 W L 
 
 r 
 
 3 qa^ b 
 
 s 
 
 4x 
 Mi = A 
 
 12 X 0.91 di 4i 
 W L 
 
 117 di Ai 
 
 128 
 
 (37) 
 
 which may be regarded as expressing the relations that exist at the 
 limit of the elastic strength of the slab and the beginning of perma- 
 nent deformation, tho not necessarily of collapse. 
 
30 STRESSES IN CONCRETE 
 
 The percentage of reinforcement in standard mushroom slabs 
 is small enough to make their elastic properties depend upon 
 the resistance of the steel. The stresses in the concrete may then be 
 be computed from those in the steel, but many uncertainties attend 
 any such computation. It is usage, fixed by the ordinances of the 
 building codes of most cities to require the application of the so 
 called "straight line theory" in such computations, not because that 
 will give results which will be verified by extensometer tests of com- 
 pressions in the concrete, for it will not, but because it is definite 
 and on the side of safety. Furthermore it is usually prescribed 
 that the ratio of the modulus of elasticity of steel divided by that 
 of concrete shall be assumed to be 15, where the moduli are unknown 
 by actual test of the materials. This is usually far from a correct 
 value. The consequence is that the results of computation of the 
 stresses in concrete are highly artificial in character, and should not 
 be expected to be in agreement with extensometer tests. With this 
 understanding the computed stress in the concrete at the middle of 
 the side belt will be found as follows: — 
 
 Let id be the distance from the center of the steel to the neutral 
 plane. (It happens to be more convenient in this investigation to 
 use this distance id here and in our previous formulas than to intro- 
 duce the distance from the neutral axis of the slab to the compressed 
 surface of the concrete, as is done by many writers, under the desig- 
 nation k d. These quantities are so related that i -\- k = 1 ) . 
 
 Then, as is well known from the geometry of the flexure of 
 reinforced concrete beams, in case tension of concrete is disregarded, 
 
 k E, 
 
 /c = — • — /s (38) 
 
 i E, 
 
 where the subscripts c and s refer to concrete and to steel 
 respectively. 
 
 Applying (38) to the greatest computed stress f^ = 19000 in 
 the St. Paul Bread Go's Building, gives a computed stress /„ = 492; 
 but taking the greatest observed stress /^ = 17940 gives /c = 465 lbs. 
 per sq. inch, as the greatest computed compressive stress in the 
 concrete at the middle of the side belt, if i = 0.72. 
 
 The tensile stress across the middle of the side belt at the 
 extreme fiber of its upper surface is fixed by the curvature of the 
 vertical sections of the slab in planes that cut the side belt at right 
 angles. As stated previously all such planes make cross sections of 
 the side belt that are identical in shape. That is a consequence of 
 
DEFLECTIONS IN COLUMN HEAD AREAS 31 
 
 the conclusion reached previously, that all the rods in the side belt 
 are subjected to equal tensions. The curvature of these sections 
 is controlled by the stiffness of the mushroom heads, which is so 
 great as to make the curvature very small. No considerable tensile 
 cross stresses are consequently to be apprehended; but in case the 
 stiffness of the head were to be decreased, stresses might arise such 
 as to develop longitudinal cracks over the middle rod of the side belts. 
 
 10. In order to obtain practical formulas for the deflections and 
 stresses in the steel thruout the areas at and near the tops of the 
 columns where all the belts cross each other, and lying between lines 
 of contra-flexure, we shall have recourse to (30) and (31) which are 
 here superimposed on each other, and combined together. Were 
 there no steel here in addition to the side belts, that superposition 
 could be correctly effected by writing a value of z whose numerator 
 would be the sum of the numerators of (30) and (31), for that would 
 superpose the loads of the two side belts, and thus place the total 
 required loading upon this area as previously explained; and then by 
 writing for a denominator the sum of the denominators of (30) and 
 (31), for that would superpose and combine the resistance of all the 
 steel in both belts. But such a result would leave out of account the' 
 reinforcement arising from the diagonal rods, and the radial and 
 ring rods, which should also be reckoned in as furnishing part of the 
 resistance. 
 
 Supposing this additional steel to be distributed in this area in 
 the same manner as is that of the side belts, a supposition which is 
 very close to the fact, we may write 
 
 (1 - K^) g (g + &) 3 
 z = -— [(a -- X ) + (6 —y)\.. (39) 
 
 in which 2A is the cross section of the total reinforcement in this 
 area regarded as forming a uniform sheet, i and j stand for mean 
 values that have to be determined by the percentage of reinforce- 
 ment and its position, while d is the mean distance of the center of 
 action of the steel above the lower compressed surface of the con- 
 crete at the point xy. 
 
 We may conservatively assume in the standard mushroom 
 that the center of action of the steel is at the center of the third layer 
 of rods from the top, as will appear more clearly later. This defines 
 d, which we shall consequently designate by ^3. 
 
 It remains therefore to estimate the amount of the total rein- 
 forcement SA, and then find mean values of i and j- 
 
32 MEAN REINFORCEMENT OF HEAD 
 
 In case of reinforcing rods which are all of them continuous 
 over a head without laps, the percentage of reinforcement falls only 
 slightly below 4 times that at the middle of a side belt; but on the 
 other hand were none of them continuous for more than one panel 
 and each lap reached beyond the center of the column to the edge 
 of the mushroom, the percentage of reinforcement would not be less 
 than 7 times that at the middle of a side belt, and to this must be 
 added that due to the steel in the radial and ring rods. Thus the 
 percentage of reinforcement here may be varied not only by reason 
 of the larger or smaller number of laps over each mushroom, but by 
 reason of the length of the laps, from perhaps 3.75 to 7 times that 
 at the middle of a side belt. For standard mushroom construction 
 using long rods, it may be taken conservatively as a 4.25 times that 
 at the middle of a side belt. 
 
 It is impossible to make an estimate that will be accurate for 
 all cases, but commonly the 8 radial rods of a 20' x 20' panel are 
 equivalent in amount to a single 1-1 /S" round rod, or a 1" square 
 bar circumscribing the area under consideration, that is to 4 square 
 inches of additional reinforcement to be distributed in the width of 
 a single side belt. 
 
 The two rings rod, of which the larger is commonly 7/8" round, 
 and the smaller 5/8" round, may be taken to increase the reinforce- 
 ment of this area by at least one square inch of cross section, giving 
 all told some five square inches of cross section additional, equiva- 
 lent forty-five 3/8" round rods, or twenty-one l/2" rods. It thus 
 appears that the increased reinforcement from this source reaches 
 from 2 to 4 times Ai, and we may safely assume a mean total rein- 
 forcement over this area of 
 
 S.4 = 7.5 ^1 (40) 
 
 of which the center of action may be pretty accurately stated to be 
 at the middle of the third layer of reinforcement rods from the top. 
 
 In the standard design of mushroom floors for warehouses with 
 panels about 20' x 20', the mean percentage of reinforcement for a 
 single belt Ai being about 0.23%, may be taken by (40) for a rein- 
 forcement 7.5 Ai as 
 
 7.5 X 0.23 -I- = 1.75% The corresponding value of j is 0.83, 
 and we shall have 
 
 iS 4 = 0.83x7.5 Ai = 6^1 (41) 
 
 As previously stated, these equations (containing estimated mean 
 numerical values) are given as a specimen computation for the purpose 
 
STRESSES AT EDGE OF CAP 
 
 33 
 
 of making comparisons. In actual design, computations like these 
 should be made which introduce the exact values appearing in the 
 design under consideration. 
 
 We now derive from (39) and (40) by the help of (23) the follow- 
 ing equations for this area where the belts all cross : — 
 
 5^ 2 ±(1 — K"") q{a + b) 
 
 f, = Eei = ±Eids 
 
 dx' 
 
 QOjdsAi 
 
 {3x^ — a') 
 
 Ml = 7.5Add^fs 
 
 (1 - K ') 
 12 
 
 ■qia + b) (3a;^ — a^) 
 
 .(42) 
 
 in which j and rf;; are less than in (33) and (34), as has been stated 
 previously. 
 
 Apply (42) to find the stresses at the edge of the column cap 
 on the long side Li. 
 
 Let 5 = 2a; be the shortest distance along the middle of the 
 side belt parallel to x between the edges of the caps of two adjacent 
 columns, and introduce the values j = 0.83, K = 0.5, and W = iqab, 
 then; 
 
 W Li (Li + L2) (3B^/Lf —1) 
 
 /s = 
 
 800 rfg 4l L2 
 
 Ml = 7.5 Aud^f, 
 
 128 L, 
 
 1) 
 
 (43) 
 
 in which 7.5 A-i is the effective cross section of the steel in this area, 
 and Ml is the true resisting moment of the steel derived from the 
 elongation, and ^3 is as stated after (39). 
 
 Take the case of a square panel, and assume the diameter of 
 the column cap to be 0.2Li, then B = O.8L1 and (43) reduce to: 
 
 /s 
 
 W Li 
 
 435 da 4i 
 
 Ml = 7.5Audsfs 
 
 W Lr 
 70 
 
 (44) 
 
 It will be readily seen that if d^ in (44) is more than 0.4 of the vertical 
 distance designated by rfi in (34), (as it must be) then the stress /g 
 in (34) at the middle of the side belt exceeds /g in (43) at the edge 
 
34 GREATEST STRESSES OVER COLUMN 
 
 of the cap. But this does not prove that the stress in the concrete 
 at the edge of the cap is less than that at the middle of the side belt, 
 for, the value of i in (37) at the middle of the side belt is about 2/3 
 and at the edge of the cap about 1/2, as will be seen by consulting 
 Turneaure and Maurer, page 57, for values of i corresponding to the 
 values of j at these points. Hence, using these values of i, if primes 
 be used to designate the stress at the edge of the cap, we have by 
 
 (38), /;//e =2/;//, ^ (45) 
 
 from which it is seen that the stress fs at the edge of the cap must 
 be only half that in the side belt in order that the corresponding 
 stresses in the concrete may be equal. But ordinarily 2/s >/s, and 
 so/c >/c. The stress in the concrete at the edge of the cap will be 
 computed from that of the steel found in (44) by using (38), in which 
 if we put i = K = I, -we have /J = /J /15, as the computed value 
 of the stress at the edge of the cap. 
 
 Tests have seemed to show that much higher compressive 
 stresses may be safely permitted in the concrete around column caps 
 where there is compression in two directions, than in the extreme 
 fiber of a beam where compression takes place in one direction only. 
 A like principle applied to the extreme fiber at the middle of the 
 side belt where tension exists at right angles to the compression 
 would show that there only a low value should be permitted in 
 compression. 
 
 In order to compare the greatest stresses in the steel across the 
 mushroom with that at the middle of the side belts in a square panel 
 let B = Li = L2 in (43), then the stress in a section thru the column 
 center along the edges of the panel over the mushroom area is found 
 from the following equations: 
 
 W L, 
 
 f» = 
 
 200 ^3 Ai 
 Ml = 7.5 Aijd3n = 
 
 W Li 
 32 
 
 (46) 
 
 which are to be compared with (34), from which it appears that if 
 ^3 in (46) is more than 7 /8 of dj in (34), the stress in the steel across 
 the mushroom is less than at the center of the side belts. In any 
 case these stresses are so nearly equal that the inadvisability of 
 decreasing the steel in the mushroom head below standards indicated 
 above is evident. However, some of the steel at the edge of the 
 mushroom especially the outer hoop is at such level in this right 
 
STRESSES OVER CAP 35 
 
 section of the head as possibly to assist the concrete in bearing com- 
 pressive stresses. Such a large portion of this section, moreover, 
 falls within the cap, that no question of its stability and safety need 
 arise, in case the collar band of the column is sufficient to resist the 
 comparatively small tensions of the radial rods. 
 
 It will be noticed that in order to make /^ and f^ as small as 
 possible in this area ^3 must be made as large as possible, i. e., the 
 steel at the edge of the cap must be raised as near the top of the slab 
 as possible. Neglect of this is to invite failure and weakness such 
 as has overtaken certain imitators of the mushroom system. 
 
 A final remark is here in place respecting the values of j and ^3 
 in this area. The stresses /^ and /„ diminish very rapidly towards 
 the lines of contra-flexure, where they vanish, and the fact that the 
 steel also rapidly increases its distance from the top of the slab at 
 the same time might be regarded at first thought as requiring some 
 modification of the assumptions we have made as to the values of 
 j and dz, which are approximately correct at the edge of the cap 
 where the steel is placed as near the top surface as due covering will 
 permit. But the fact is this: the only consideration of importance 
 is the one respecting the position of the steel in that part of this 
 area where the moments and stresses are large. The effect of the 
 position of the steel near the lines of contra-flexure is negligible, and 
 the fact that the amount of reinforcement may be somewhat smaller 
 near these lines than elsewhere may also be neglected, so that the 
 mean effective reinforcement previously estimated is likely to be an 
 underestimate rather than the reverse. Further, the fact that the 
 slab is practically clamped horizontally either at the edge of the cap 
 or the edge of the superposed column, instead of at its center as 
 assumed in our formulas, renders the results given thus far slightly 
 too large. 
 
 Good average values of the size of steel used in the standard 
 mushroom system of medium span would make the radial rods 
 9/8" round, the outer ring rod 7/ 8" round, the inner ring rod 
 5/8" and the belt rods 3^ S" round. The importance of having 
 the belt rods small is that for a given thickness of slab the smaller 
 these rods are the larger is d in both (34) and (43) and consequently 
 the smaller is/s and Ai. 
 
 11. In attempting to consider the stresses in the diagonal rods 
 of the central rectangle between the side belts of a panel, it will be 
 noticed, as stated before, that no true bending moments are propo- 
 gated across the vertical planes or lines of contra-flexure (24) which 
 
36 DEFLECTIONS IN CENTRAL AREA OF PANEL 
 
 bound it, and since the vertical shearing stresses at these lines are 
 uniformly distributed along them, as already shown, (28), there are 
 no true twisting moments in these planes. The curvatures of this 
 rectangle will consequently depend upon its own loading and the 
 resistance of its own moment of inertia, regarded as uniformly dis- 
 tributed, independently of that of other parts of the slab. 
 
 Hence (21) may be correctly applied to this area, regardless of 
 the values which / (and q) may assume elsewhere, provided only 
 that the values of / in other areas may be assumed to have constant 
 values thruout those areas, and, further, that those areas are sym- 
 metrically disposed, so that all central rectangles have one and the 
 same given value of / thruout, all side belts also have one given 
 value of /, and the mushroom heads have a given value also, each of 
 these three sorts of areas being independent. The truth of this 
 proposition has been heretofore tacitly assumed in applying (21) 
 to these latter areas as has been done. 
 
 It will be seen however, that the values of z obtained from such 
 diverse equations express deflections of any point xy on the supposi- 
 tion that all the areas considered have the same value of /; but these 
 separate equations, each with its own peculiar value of /, can be 
 used separately to find the difference of level Zi — Z2 between any 
 two points Xi Hi and X2 y2 which lie in an area where / may be regarded 
 as constant. We shall return to this point when we come to the 
 derivation of practical deflection formulas. 
 
 For convenience in computing stresses in the rods of the diago- 
 nal belt, let the direction of the coordinates be changed so that in 
 square panels they will lie along the diagonals which make angles 
 of 45° with those used thus far. In (21) let 
 
 X = lV2{x' + y'), J/ = K2(a;' — 2/'), then 
 
 ' = J'^.^iv! t«' - «'(^" + y'"^ + ^'V' + 5(^V)i .... (47) 
 
 in which the panel is square and the axes of x' and y' He along its 
 diagonals, while the value of ^A/g is the effective cross section per 
 unit of width of all the reinforcement in this area regarded as a 
 single uniform sheet of metal, and g = 7/8 a, is the width of a 
 diagonal belt, and is equal to the diameter of the mushroom head. 
 In rectangular panels g = 7/l6 (a -|- b). 
 
ELONGATIONS AND SHEARS IN CENTRAL AREA 37 
 
 From (34) we have 
 
 5z (1 — K^) q g 
 
 r'^ 9AF--w2yJ ^'(^''+3;/'')-2a^^'] (48) 
 
 ox 24: E ij d zA 
 
 8'z dh {l~K')qg 
 
 ei=e2 = —id—-=—id—- = [2a^— 3(x'Hw'^](49) 
 
 dx' by^ 24Ejd^A 
 
 8'z il-K')qgx'y' 
 
 and — ■ = (50) 
 
 dx by 4Eijd''1,A 
 
 These expressions satisfy (20) as they should, for (20) is inde- 
 pendent of the directions of the rectangular axes x and y. 
 
 From (49) it appears that ei = = /s, on the circumference of 
 the circle x^-{- y'^ = f a^, which passes thru the points where the 
 lines of contra-flexure intersect. 
 
 By (19), which holds for any rectangular axes, and by (50), 
 we find 
 
 n' = \0-~K) qx' y' (26)' 
 
 From (26) ' it appears that in sections by all vertical planes 
 parallel to the diagonals, the twisting increases uniformly with the 
 distance from the diagonal. 
 
 Hence by (9) we have 
 
 / 5mi 5n 
 
 g — y , 1 ,- I = \qx' 
 
 \bx by / 
 
 / SmJ 5n ' \ 
 
 \by' bx' J 
 
 .(28)' 
 
 It thus appears that the same law holds for vertical shearing 
 stresses on planes parallel to the diagonals, as holds in (28) for planes 
 parallel to the edges of the panel. 
 
 In standard mushroom designs the edges of the diagonal belts 
 intersect on or very near to the edges of the side belts. That makes 
 the middle half of the central square to be covered by double belting, 
 and the remainder of it by single belting, so that 2 A = 1.5^2 or 
 perhaps 1.6 A2, and the mean value of A, the reinforcement per 
 unit of width of slab here, is to be found by dividing this by the 
 width of a belt, which is 7/8 a. We should then find A= 1.5 A2/ 
 7/8 a = 1.7 A2/a. But this mean value of A is not its mean effect- 
 ive value for this area, because the reinforcement is so disposed as 
 
38 MEAN REINFORCEMENT STRESS AT PANEL CENTER 
 
 to furnish the larger values of I in the central diamond just where 
 the largest true applied moments and stresses occur. The mean 
 value of A in the central diamond is 2A2 / 7/8 a = 2.3A2 / a. The 
 mean effective value lies between these two extremes, probably 
 nearer the latter than the former. A similar question was discussed 
 in connection with (40) and (41). We shall assume as the mean 
 effective reinforcement in this central rectangle, 
 
 A = 2A2/ a, and / = 2A2 ij dl/a 
 or in case of rectangular panels 
 
 I = 4.A2ijdl/(a + b) (51) 
 
 In case of rectangular panels the term 2a^ in (49) should be replaced 
 by a^ + &^ as a mean value to make it depend the dimensions of the 
 panel symmetrically, as it must. Making these substitutions in 
 (49) we have at a; = = y the center of the panel. 
 
 Tf (Li +L2) (Lf + ii) CiTFLi 
 
 L = Ee = 
 
 1024 Li L2 A2 j da 256 A2 j dz 
 
 W{L, + L2) {Ll + Ll) Ci W L, 
 
 .(52) 
 
 Ml = 2A2Jd2f, = 
 
 512 Ll L2 128 
 
 where Ci = UL^/Ls + 1) (1 + L^/h^. Take i = 0.89. 
 
 If 1 > L2 /-'^i> 0.75 then l<ci< 1.042, hence Ci varies less than 5% 
 while L2/L1 varies by 25% between its extreme permissible values. 
 Ci may ordinarily be taken as unity, or may be found with sufficient 
 precision by interpolation between the values just given. 
 
 The steps by which these equations (52) were deduced may not 
 seem conclusive, since they are not rigorous. They need be only 
 good, working approximations for the purpose for which they will be 
 here used, viz, to show that the stresses at the center of the panel 
 are less than those at the mid span of the side belts in case Ax = A2. 
 
 The value of da in (52) is less than di in (34), but always more 
 than 90% of it. We may define da as the vertical distance from the 
 center of the second and upper of the two diagonal belts to the top sur- 
 face of the concrete. We may assume da = 0.9di and j = 0.89 in 
 (52), and then we may compare these stresses for a square panel as 
 follows : — 
 
 175 
 
 ^ =205^ ^'^^ 
 
 where /J refers to the center of the panel. Even were the smaller 
 
LINE OF ULTIMATE WEAKNESS 39 
 
 value for the mean reinforcement, 1.7 A2/ a, used in deriving (52) 
 and (53), the stress given by these equations would not exceed that 
 given by (34). The compressive stress /„ in the concrete at the 
 center of the panel may readily reach a dangerous value in case the 
 forms are removed too soon. It should therefore be carefully con- 
 sidered in each case. Here, we have an approximate value ofi = 2/3 
 and (38) then becomes f^ = /s / 30 with no possible assistance from 
 steel reinforcement since that is all on the bottom of the slab. An 
 estimate that the elastic stress in the steel at the center of the panel 
 does not much exceed 80% of that at the middle of the side belt 
 cannot be far from the truth. 
 
 While this is undoubtedly the fact, it will appear on further 
 consideration that local stresses and strains which exist at incipient 
 failure are of such magnitude as to make the weakest points of the 
 diagonal belts to lie ultimately not at the center, but, instead, just 
 outside the diamond where they cross each other. 
 
 Take the standard case where the central diamond reaches just 
 across to the side belts. For square panels imagine a circle to be 
 drawn concentric with each column of radius L/2. Any circle at a 
 column will be tangent to the edges of four diagonal belts across the 
 tops of the four columns adjacent to it, and then the octagon cir- 
 cumscribing it, whose sides cut at right angles all the belts that cross 
 this column head, intersects but a single belt of rods as every point 
 of its perimeter. It is evident that, so far as reinforcement is con- 
 cerned, such a line or section cuts less steel per unit of perimeter 
 than any other regular figure concentric with the column and that 
 the reinforcement is entirely symmetrically disposed about the 
 column center, so that in case of equal diagonal and side belts, it 
 would be impossible from their geometry to distinguish the one from 
 the other by anything inside the octagon. That fact would make it 
 inherently probably that the stresses and strains of the rods where 
 they cross any one side of this octagon should be approximately the 
 same ultimately as in those that cross any other side, whether they 
 be rods in a diagonal belt or in a side belt. And what will be at- 
 tempted to be shown immediately is that ultimately the stresses 
 and strains in these several belts approach equality. If that should 
 be established, it will follow from the conclusion already reached as 
 to the excess of the stresses and strains of the side belt over those at 
 the center of the panel, that ultimately those at the edges of the 
 ocatgon exceed those in the same rods at the center of the panel. 
 
 The qualification implied above in affirming that this is what 
 will occur ultimately, is for the purpose of conveying the idea that 
 
40 LOCAL STRESSES AND STRAINS 
 
 this is the approximate distribution of stresses and strains which 
 will take place when the slab is sufficiently loaded to bring the steel 
 at the middle of the side belt to the yield point. At less stress than 
 this there is so much lag in the distribution of the effect of loading 
 that it penetrates to the various parts of the slab unequally. 
 
 Taking up now the deferred proof that the diagonal rods where 
 they cross the edge of the octagon are subject ultimately to the same 
 local stresses and strains as the direct rods of the side belts; note 
 that these diagonal rods lie in a triangular area between two side 
 belts, which latter experience equal elongations ei in directions at 
 right angles to each other. The edges of the triangle in which the 
 single layer of diagonal rods lie are continuous with the side belts 
 and necessarily experience the same elongations, which are propo- 
 gated from the side belts into the triangle by the agency of horizontal 
 shears on its edges. Such equal elongations at right angles imply 
 the same elongation in every direction in the triangle, as appears 
 from the fundamental properties of equal principal stresses and 
 strains. Hence we have the same elongations along the diagonal 
 rods as along the rods of the side belts at the edges. The existence 
 of an ultimate stress and strain in the diagonal belt equal to that in 
 the side belt would require that the cross sections A2 and Ai of the 
 two belts should be equal, altho so far as the elastic value of fg at 
 the center of the panel is concerned A2 might be less than Ai, as has 
 been already shown in (52) and (53). The relationships of stress, 
 load, etc., for this ultimate condition, have been already given in (37). 
 
 Besides the stresses and strains in the diagonal belts, just in- 
 vestigated, those due to the local stretching (arising from the deflec- 
 tions themselves) exert their greatest effect on the rods of the diagonal 
 and side belts just in the region of the line of weakest section, and 
 partly because of that fact. While these local stresses may not exceed 
 10% in addition to those already present, their existence should 
 prevent any thought of taking 24 larger than A 1 in (37) when deriv- 
 ing the ultimate stresses at the yield point. Similar results may be 
 formulated to cover cases where g is greater or less than 7/l6 L. 
 
 It is perhaps desirable at this point to consider a little more at 
 length the matter of local stretching in a slab. It is impossible for a 
 continuous flat floor slab to undergo the deflections which we are 
 treating, consisting of convexities, concavities, etc., without local 
 stretching to allow this to occur. A floor slab of many panels does 
 not undergo any change of its total linear dimensions which would 
 account for these corrugations. A continuous beam under flexure 
 would have its extremities drawn toward each other. But not so 
 
ACTUAL DEFLECTIONS IN SIDE BELTS 41 
 
 to any such extent with a slab. Such contractions are resisted by 
 local circumferential strains which result in true stresses. An 
 investigation of such stresses leads to the conclusion just stated that 
 in general they cannot exceed 10% of the ordinary stresses due to 
 slab bending when they are left out of the consideration. For this 
 reason a single panel alone will not function precisely in the same 
 way as a panel in a floor of many panels. 
 
 12. Actual deflections are distances which any given points 
 of a slab sink down by reason of the application of a given load, and 
 their theoretical values are to be computed by help of the formulas 
 which have been developed for z in the various areas into which the 
 panel has been divided. 
 
 We shall now make a slight modification in our definition of the 
 level of the origin of coordinates, and shall take it at the upper or 
 lower plane surface of the flat slab before flexure, in which surface 
 the axes of x and y are assumed to lie. It is of no consequence 
 whether it be the upper or the lower surface which is assumed, the 
 equations will be the same in either case. The reason for this new 
 definition of the position of the origin is this: Each kind of partial 
 area into which the slab has been supposed to be subdivided has its 
 neutral surface at a different depth in the slab, and so it does not 
 furnish a single suitable level from which to reckon deflections, as 
 does the upper or lower surface of the slab. None of the equations 
 which have been derived in this paper will undergo any modification 
 by reason of this change of definition. It has been assumed that 
 each kind of area has a separate value of I which remains constant 
 thruout, so that the neutral surfaces of different areas do not join 
 at their edges. As previously explained this is of no consequence 
 mechanically by reason of the zero true moments that exist at these 
 edges. The modification just introduced avoids the geometrical 
 perplexities arising from this discontinuity of neutral surfaces. 
 
 Deflections in the side belt area between the lines of contra- 
 flexure (24) are to be found from (30), or (31), and (32). To find 
 the deflection or difference of level in the mid side belt between 
 a; = 0, 2/ = 6, and x = \ a^2>, y = h, substitute these values in (30), 
 take i = 0.71, j = 0.91, k = 0.5 and subtract the value z at the 
 second point from that at the first point, which gives the following 
 value of the deflection of the one point below the other : 
 
 W Ll 
 10.7 X 10'° d? Ai 
 
42 ACTUAL DEFLECTIONS IN CENTRAL AREA, IN HEAD 
 
 in which hi is the vertical distance from the center of the single belt 
 of rods at the mid span of the side belt to the effective top of the 
 slab, considering the strip fill or other concrete finish at its effective 
 value. 
 
 In the same manner take the difference of level in the central 
 
 rectangle bounded by the lines of contraflexure between the center 
 
 point at a: = 0, y =0 and the corner x = ^ a V3, y = ^bVshy using (21) 
 
 and (51) and introducing the values i = 2/3, j = 0.89, etc., and 
 
 C2 = l/4(Li/L2 + 1) (1 + Lt/Lt), then: 
 
 C2W Ll 
 
 A22 = 7^^ (55) 
 
 6.56 X 10^° dl A2 
 
 in which A2 is the cross section of one diagonal belt and /i2 is the 
 vertical distance from the center of the upper or second diagonal 
 belt to the effective upper surface of the panel at its center. 
 
 On evaluating C2 above, we find 
 when 1>L2/Li >0.75 
 then 1> C2 >0.77 
 
 hence we may with sufficient accuracy for practical purposes assume 
 C2 = L2/L1 (56) 
 
 Deflections in the mushroom area between lines of contraflexure 
 (24) are to be derived from (39) (40) and (41) by introducing i = \, 
 j = 0.83, k = 0.5 and 1,A = 7.5 A^. Assuming the diameter of 
 the cap to be 0.2Li we have, at its edge where x = 0.8a, y = b, from 
 (39) 
 
 W Ll (L1/L2 + 1) /36\2 
 
 19.1 X 10'° dl A 
 
 W36\2 
 
 -(— ) (57) 
 
 1 Vioo/ 
 
 The value of z at the edge of the mushroom area, where x = ^ a Vz, 
 y = b, is to be obtained from (57) by replacing the last factor by 
 4/9; and the deflection between the edge of the cap and the edge of 
 the mushroom obtained by taking the difference of these quantities 
 is as follows: 
 
 WLUL1/L2 + 1) 
 60xlO'°d^4i 
 
 ^''= ..■■,.10.2. -- (58) 
 
 in which h-j is the vertical distance of the center of the third layer of 
 reinforcing rods over the edge of the cap above the bottom surface 
 of the slab. 
 
TOTAL DEFLECTIONS BELOW EDGE OP CAP 43 
 
 Similar expressions may be obtained for tlie values of z and A2 
 on the side parallel to ij, where x = a &i ij = 0.86, and y = \ 6^3, 
 by exchanging L^ and L2 in (57) and (58) . 
 
 Take half the sum of (57) and the corresponding values so ob- 
 tained aX X = a, y = 0.86, as the value of z at the edge of the cap 
 where it is intersected by the diagonal of the panel, viz. 
 
 ir (Li + Lo) {L\ + L\) / 36 •. 
 
 • (59) 
 
 38.2 X 10'° Li Lo dl A 
 
 2)/36_y 
 
 1 Vioo/ 
 
 and subtract this from the value of z on the diagonal at the corner 
 of the mushroom area where x = \ aV^S, y = 3 6^3 and we have 
 
 C2W L\ 
 
 A z, = ~~^~ (60) 
 
 12.5 X 10'° 4 Ai 
 
 as the deflection along the diagonal between the edge of the cap and 
 the intersection of the lines of contraflexure, in which C2 and h^ are 
 as previously defined. 
 
 Let Di = A^i + AzsX ^q^-^ 
 
 and D2 = Az2 + Azi,] 
 in which Di is the deflection of the mid point of the side belt below 
 the edge of the cap, and D2 is the deflection of center of the panel 
 below the edge of the cap. 
 
 The proportionate deflections of these points are obtained by 
 dividing by the spans, viz: Di/Li and D2/ ^ L\ + L\ 
 
 r2 
 
 13. Estimated proportionate deflections may be obtained from 
 (61) under such circumstances as to convey reliable information 
 respecting what may be reasonably expected. Let h = the total 
 thickness of the slab. The limiting values of the thickness of 
 standard mushroom construction are expressed as follows: 
 
 Li/20>/i>Li/35, (62) 
 
 and assuming that the reinforcing rods are l/2" rounds with l/2" 
 covering of concrete we shall have from the definitions of di, (^2 and 
 rfs, already given 
 
 h = di + 0.75 = ch + 1.25 = dg + 1-75 (63) 
 
 Substituting these in (62) etc. we have 
 
 Li /20 — 0.75 > di > Li /35 — 0.75 ] 
 
 Li/20 — 1.25 > da > A/35 — 1.25 \ ... .(64) 
 
 Li/20 — 1.7& > dg > Li/35 — 1.75 
 
44 
 
 PROPORTIONATE DEFLECTIONS 
 
 If it be assumed that we are dealing with medium sized panels 
 about 20' X 20' (64), may be written in the form: — 
 
 (1 — 0.062) Li/20 > rfi > (1 — 0.02) Li/35 
 (1 — 0.1) Li/20 >d2> (1 — 0.036) Li/35 
 (1 — 0.15) Li/20 > ^3 > (1 — 0.05) Li/35 
 
 or, 
 
 0.94 di 
 > — 
 
 20 ii 
 
 0.98 
 35 
 
 
 0.90 d2 
 > — 
 
 20 Li 
 
 0.964 
 > 
 
 35 
 
 
 0.85 dz 
 
 > — 
 
 20 Li 
 
 0.95 
 > 
 
 35 
 
 .(65) 
 
 In (54), (55), (58) and (60) replace W Lj by its value given in (34), 
 viz, 175 di Ai fs, and we have 
 
 A Z2 = 
 
 A zs = 
 A Zi = 
 
 LlL 
 
 6.11 X 10* di 
 
 3.75 X 10^ dl A2 
 
 d^L\{U/L2 + 1)/, 
 
 34.3 X 10* dl 
 
 C2d,L\U 
 
 7.14 X 10* dl 
 
 (66) 
 
 in which /^ is the greatest stress in the steel, i. e., at the mid side 
 belt, employed here to express deflections instead of expressing them 
 in terms of panel load as was done previously. 
 
RELATIVE DEFLECTIONS AT MID SPAN 
 
 45 
 
 Introduce into (66) the numerical ^'alucs given in (65) which 
 will then express limiting values of deflection for medium spans. 
 For simplicit}^ let Li = L2 then: 
 
 •2S7 > --: > 170 
 
 162 > > 106 
 
 urAz2 
 
 660 > > 451 
 
 lO'Azs 
 
 275 > > 188 
 
 10°AZ4 
 
 .(67) 
 
 By (61) we have the proportionate deflection of the side and diagonal 
 belts as follows: — 
 
 L287 66OJ 10° Li Ll70 45lJ 10° 
 
 Ll62 275jl0°%2 Ll^2 L1O6 ISSj 10° \2 
 
 Is 
 
 200 X 10° 
 
 < < 
 
 < < 
 
 ,/; 
 
 141.4x10° ii ^2 
 If /, = 16000, 
 
 If f, = 24000, 
 
 123.4 x 10° 
 95.9x10° 
 
 1 £>, 1 
 
 — < 1 < — 
 
 SS4 Li ^ 2 600 
 
 1 D2 1 
 < < 
 
 590 Li % 2 400 
 
 If /, = 32000, 
 
 1 D. 1 
 
 — < < — 
 
 442 Li ^ 2 300 
 
 .(68) 
 
 .(69) 
 
 Larger spans then 20 , or smaller steel than 1 2" round, or L2<Li 
 will reduce the above values somewhat, while smaller spans or 
 
46 THEORETICAL AND EXPERIMENTAL DEFLECTIONS COMPARED 
 
 larger steel will increase these values, all of which can in each case 
 be submitted to calculation by the methods here developed. 
 
 To recur at this point to the expression for the deflection D2 in 
 terms of the panel load W by help of (55), (60) and (61) 
 
 D. = 
 
 By (65) we find 
 
 CoW L^ 
 
 10'° A 
 
 - + (70) 
 
 1 16.5Q dl 12.5 dU 
 
 90 da 96.4 
 
 — > — > 
 
 85 ds 95 
 
 and using this inequality to eliminate ^3 from (70) we find after 
 reduction 
 
 C2 W L\ C2W L\ 
 
 < D2 < 
 
 4.46 X 10'° rfi ^1 4.33xl0'°d|^i 
 
 from which we may write as a mean value 
 _ C2W L\ 
 ^' ^ Ia X 10" dl A, ^^^^ 
 
 The empirical deflection formula given on page 29 of Turner's Con- 
 crete Steel Construction, when written in these units, is 
 
 W L\ 
 4.84 X 10'° di Ai 
 
 Do = (72) 
 
 This is identical with (71) when C2 = 0.909, and diverges from it 
 slightly for other admissible values of €2- The practical agreement 
 of (71) and (72) affords a second confirmation of the theoretical 
 deductions made thus far, and this taken in conjunction with the 
 practical identity of formulas (34) and (35), the theoretical and 
 empirical expressions for the maximum tensile stresses in the rein- 
 forcement, furnishes what on the theory of probabilities may be 
 regarded as so strong a probability of the general trustworthiness 
 of the entire theory as to exclude any rational suppositition to the 
 contrary. 
 
 The various formulas for stresses and for deflections which have 
 been developed in this paper have been obtained under the express 
 proviso that the panel under consideration was assumed to be one 
 of a practically unlimited number of equal panels constituting a 
 continuous slab, all of which are loaded uniformly and equally. The 
 
MUSHROOM PANELS INDEPENDENT 47 
 
 question at once arises as to the amount and kind of deviations from 
 these formulas which will occur by reason either of discontinuity of 
 slab or loading, such as occurs at the outside panels of a slab or at 
 panels surrounded partly or entirely by others not loaded. The 
 answer to this question depends very largely upon the construction 
 of the fiat slab itself. 
 
 In the standard mushroom construction it has been found that 
 the stresses and deflections of any panel are almost entirely inde- 
 pendent of those in surrounding panels. This is due to the fact 
 that the mushroom head is an integral part of the supporting column 
 in such a manner that it is impossible for it to tilt appreciably over 
 the column under the action of any eccentric or unequal loading of 
 panels near it. When single panels have been loaded with test 
 loads, no appreciable deflections have been discoverable in sur- 
 rounding panels, and no greater stresses and deflections have been 
 discovered than were to be expected in case surrounding panels were 
 loaded also. Future careful investigation of this may reveal 
 measureable effects of this kind, but they must be small. 
 
 A Hke statement cannot be made of other systems of flat slab 
 construction where the reinforcement over the top of the column is 
 not an integral part of the column reinforcement itself. Tests on 
 these systems have shown clearly the effects of the tipping of the 
 part of the slab on the top of the column, and lack of stiffness of 
 head, in the increase of the deflection of the single loaded panel over 
 the deflection to be expected in case of multiple loaded panels, and 
 especially in the disturbance of the equality of the stress in the other- 
 wise equal stresses in the rods of the side belts. Such distrubance, 
 by increasing the stress in part of these rods, would necessitate larger 
 reinforcement in the side belts of such systems than would be re- 
 quired in mushroom slabs. The great stiffness of the mushroom 
 head is also of prime importance in taking care of accidental and 
 unusual strains Hable to occur in the removal of forms from under 
 insufficiently cured slabs. 
 
 14. In considering the design of the ring rods and radial 
 cantilever rods of the mushroom head, it should be borne in mind 
 that they occupy a position in such close proximity to the level 
 of the neutral surface as to prevent them from being subjected 
 to severe tensile or compressive stresses by reason of the bending 
 of the slab as a whole. Their principal function as slab mem- 
 bers is to resist shearing stresses and the bending stresses due to 
 
48 VERTICAL SHEAK AROUND COLUMN CAPS 
 
 local bending. Their total longitudinal stresses are too small in 
 comparison to require consideration. 
 
 Let a cylindrical surface be imagined to be drawn concentric 
 with a column to intersect the slab, then the total vertical shearing 
 stress which is distributed on the surface of intersection is equal to 
 the total panel load W diminished by the amount of that part of the 
 panel load lying inside the cylinder. If the cylinder be not large, 
 the total shear may be taken as approximately equal to W itself. 
 
 It is evident that the smaller the diameter may be that is 
 assumed for this cylinder, the greater will be the intensity of the 
 vertical shear on its surface and that for two reasons : First, because 
 the totel load thus carried to the column will be greater the smaller 
 the diameter, and second because the surface over which the total 
 shear will be distributed decreases with its diameter. 
 
 The result of this is that the dangerous section for shear is the 
 cylindrical surface at the edge of the cap. For cylinders smaller 
 than this the increased vertical thickness of the cap diminishes the 
 intensity of the shear. We proceed therefore to consider the manner 
 in which the total vertical shearing stress of approximately TT^ in 
 amoimt is distributed in the material of the cylindrical surface at the 
 edge of the cap. 
 
 In a beam or slab the horizontal shearing stresses due to bending 
 reach a maximum at the neutral surface. It is a fimdamental con- 
 dition of equilibrium that shearing stresses on planes at right angles 
 shall be equal, and it is this condition that determines the distribu- 
 tion of the vertical shears, which are at right angles to the horizontal 
 shears resulting from bending the slab as a whole. From this we 
 have the well known fact that the vertical shear varies from zero 
 at the upper and lower surfaces to a maximum at the neutral surface, 
 and this is necessarily the manner in which the total shear is dis- 
 tributed at the edge of the cap. The top belt of rods will be sub- 
 jected to comparatively small shearing stresses, and successive 
 layers of rods will be under larger and larger shearing stresses by 
 reason of their greater nearness to the neutral surface, while the 
 total shear borne by the radial rods near the neutral surface will be 
 much larger than that upon the others. The shearing stress in 
 the concrete will need to be considered also. 
 
 It is to be noticed that all the steel of the belts and mushroom 
 head act together without the necessity of supposing large com- 
 pressive stresses in the concrete to transmit vertical forces, because 
 the belts of reinforcement rest directly upon each other, and these 
 in turn upon the ring rods and radial rods, all in metallic contact 
 
VERTICAL SHEAR IN RADIAL RODS, ETC. 49 
 
 with each other, in the mushroom head, and so they transmit and 
 adjust the distribution of stresses within the system to a very large 
 extent independently of the concrete. 
 
 We can then safely assign moderate values of the shearing 
 stress to each of the elements that constitute the slab at the edge 
 of the cap, with the assurance that they will each play a part in 
 general accordance with the distribution which has been already 
 explained. 
 
 The mushroom is constructed of great strength and stiffness 
 not merely to effect the results which have appeared previously in 
 the course of the investigation but also to ensure the stability of the 
 slab in case of unexpected or accidental stresses due to the too early 
 removal of the forms, before the slab is well cured, at a time when 
 the only load to which it is subjected is due to the weight of the 
 structure itself. 
 
 The working load to be assumed in designing the mushroom 
 may be taken as the dead load of a single slab plus the design load, 
 provided sufficiently low values of the shearing stresses be assumed 
 in the cross sections of steel and concrete at the edge of the cap 
 for the support of this working load, as follows : 
 
 For slabs having a thickness oi h = L/35 a mean working 
 shearing stress of 2000 lbs. per square inch at the right cross section 
 of each reinforcing rod which crosses the edge of the cap, a mean 
 shearing stress of 40 lbs. per square inch in the vertical cylindrical 
 section of the concrete at the edge of the cap, and 8000 lbs. per 
 square inch of right cross section of each radial rod. 
 
 For slabs having a thickness of /i = L/20 the intensities just 
 given may be safely increased by 50 per cent for reasons that will 
 be explained later. For slabs of intermediate thickness increase 
 the intensities proportionately. 
 
 These values are sufficiently low to enable the structure to sup- 
 port itself before the concrete is very thoroughly cured, and the 
 head so designed will be found after it is well cured to be so pro- 
 portioned as to carry safely a test load of double the live and dead 
 loads for which it was designed. 
 
 In this connection it seems desirable to investigate what takes 
 place in case of overloading and incipient failure of an insufficiently 
 cured slab, or one unduly weakened by thawing of partially frozen 
 concrete. Suppose that under such circumstances a shearing crack 
 were formed extending completely thru the head at the edge of the 
 ■cap, and we wish to investigate the stresses and behavior of the rods 
 
50 STRESSES IN RADIAL RODS 
 
 that cross the crack at which shearing deformation has begun to take 
 place. Designate the position of the crack by X. 
 
 The total vertical shearing stress on a radial rod at X is the 
 sum of two parts found as follows: First, the vertical reaction at 
 the top of a column is made up of the vertical reaction of the con- 
 crete core of the column and the reactions of its vertical reinforcing 
 rods. Call the vertical reaction of one of these rods Vi. The rod 
 is bent over radially and Vi expresses also the amount of the vertical 
 shear in that rod where it starts out radially from the column. 
 Between this point and X for a distance which measures usually 
 from 9 to 12 inches, the rod experiences the supporting pressure of 
 the concrete in the cap under it to a total amount which we will 
 designate by F2. The total shear in the radial rod at X will then 
 amount to 
 
 F = Fi+ 72 (73) 
 
 provided we neglect the weight of that small part of the actual load 
 of the slab which lies directly over this piece of the rod and may 
 be regarded as resting upon it. This portion of the radial rod of 
 length I is a cantilever fixed at one end in the top of the column, and 
 carrying a load V at the other end with a supporting pressure under- 
 neath of total amount V2 whose intensity is greatest at X and gradu- 
 ually decreases along I from X to the fixed end. The rod has a 
 point of contraflexure and zero moment at X. The portion of the 
 rod outside the crack has a fixed point in the slab at the place where 
 it supports the inner ring rod, at a distance from X which should 
 not exceed I as just defined. Similar conditions hold for this length; 
 i. e. there will be a totol shear in the radial rod at a point just inside 
 the inner ring, rod due to its total shear outside this ring rod and to 
 the vertical loading imparted to it by the ring rod itself. To this 
 must be added the downward pressure of the concrete between the 
 inner ring rod and X. All these, together, constitute the total 
 shear — V at X, in equilibrium with the reaction -|- V already ob- 
 tained at that point. 
 
 We shall discuss separately the action of Vi and V2 upon a radial 
 rod. A load Vi at the end of a cantilever of length I causes a 
 
 deflection of amount 2i = i Fi f /EI (74) 
 
 in which 7= 7r<*/64 where < = the thickness of the rod. 
 
 Alsoyi = siA , A=irf/4: 
 
 in which Sj = the mean shearing stress per square unit of cross sec- 
 tion and A is the cross section of the rod. Hence 
 
 si = SziEf/lQf (75) 
 
STRESSES IN RADIAL RODS 51 
 
 which shows that so far as Vi is concerned, for any given disphice- 
 ment Zi the shearing stress carried per square unit of rod will be pro- 
 portional to the square of its diameter, and up to its permissible 
 limiting shearing resistaiice, each unit of section of such a rod will 
 be effective in proportion to the square of its diameter. For econ- 
 omical construction, this will require the radial rods to be few and 
 large, rather than numerous and small. The bending moment is 
 greatest at the distance / from X and amounts to Vi I. The stress 
 in the extreme fiber due to the bending moment Vi 1 in the rod is 
 Pi = Vilt 27 = 8si //< (76) 
 
 This equation shows that the stress in the extireme fiber is so very 
 large at the fixed end of the rod compared with the shear at A' that 
 so far as T'l is concerned the rod will suffer permanent deformation 
 by bending long before there is any danger of its shearing. T'j is so 
 large compared with T'o that this conclusion will not be altered when 
 we come to consider the combined action of To. 
 
 Incipient failure of this kind ^\-ill therefore cause distortion and 
 sag without collapse. In case such sag as occurs in this case is 
 detected miderneath the head around the cap, the slab should be 
 blocked up at once and the concrete picked out at all parts showing 
 facture. This should then be refilled with a stronger concrete 
 Avhich will set rapidly. Such repair should not weaken the slab. 
 
 Whenever the intensity with which a radial rod presses upon 
 the concrete at the edge of a crack at A' passes the compressive 
 strength /„ of the concrete, it must begin to yield. Ati this instant 
 we shall have a pressure of the concrete against the rod which gradu- 
 ally diminishes as we pass along the rod from A' to the distance I, 
 where it becomes >!ero. We shall assume that the pressure dimin- 
 ishes miiformlj' -with this distance. This may not be precisely cor- 
 rect, but cannot be much in error. If the shear T'j at A' is the sole 
 cause of this pressure, then ^'2 = 2 tifc, and ^ Vn I = ^ t I- f^ 
 is the bending moment in the rod at the distance /, due to V2 at A 
 and the pressure distributed along /. 
 
 It will be found that these produce a deflection 
 
 3o = 3/e/* 20EI = 0.3fV2/EI (77) 
 
 a unit shear of 
 
 s., = v., /A = z.Ef/'-i.Sf (78) 
 
 and a stress on the extreme fiber at a distance I amounting to 
 
 p, = v., tl/2,1 = I6S2 / / (79) 
 
SHEAR IN OTJTER RING RODS 
 
 It thus appears that the equations expressing the action of V2 
 are precisely similar to those for Vi, differing only in their numerical 
 coefficients, and consequently all the statements as to the resistance 
 of the radial rods under the action of Fi hold for the action of 
 T'l and V2 together in the case of given initial deformations, 
 
 2l = Z2 Sit X. 
 
 While the preceding investigation has, in order to make ideas 
 explicit, ostensibly assumed a crack at X and an initial small shear- 
 ing deformation at X, the investigation applies equally well to the 
 elastic shearing deformation of the concrete at the dangerous 
 section in -which case the total shearing stress will consist of an addi- 
 tional componenent due to the resistance of the concrete, which 
 however may for additional safety be neglected. If the assumed 
 deformation be confined within limits so small that the concrete 
 is able to endure it without cracking then the preceding investiga- 
 tion may properly be applied to it. It is right here that the thick- 
 ness of the radial rods is able to render its most effective service, 
 for it appears from (75) and (78) that any permissible intensity of 
 shear may be developed in the radial rods by making them of suit- 
 able thickness, even tho the deflection be kept within the elastic 
 hmits of the concrete. 
 
 As already stated we must not overlook the fact that the major 
 stresses here are those under the head of Fj, which are due to the 
 direct metallic contacts of the steel rods resting one upon 
 another, where large stresses are transmitted and pass independ- 
 ently of the concrete except for the distortions of the steel which 
 meet resistance, and the secondary reactions such as have been 
 treated in a single aspect while investigating the action of V2. 
 
 It is due to this fact that large shearing stresses may be safely 
 borne by the slab at and near the edge of the cap, which the concrete 
 mostly escapes, it merely furnishing some lateral stiffening to the 
 steel. On this principle the outer ring rod should have a cross 
 section not much less than one half that of the radial rods on which 
 it rests. For, this arrangement provides for the transferal to the 
 radial rods of all the shear the ring rod is able to carry, it being in 
 double shear compared with the radial rod it rests on. 
 
 It is impossible to determine the cross section of the inner ring 
 rod, with the same definiteness as that of the radial rods, but 
 that is unimportant. Its position has already been fixed as not 
 more than I from the edge of the cap, where I is the distance from 
 the top hoop or collar band of the column to the edge of the cap. 
 
STRESSES IN CONCRSTB OF HEAD 53 
 
 The vertical shearing stresses may be regarded as sufficiently 
 resisted outside the mushroom by the concrete alone. The critical 
 cylindrical surface separating those areas where the shear may be 
 assumed to be safely carried by concrete alone, from those areas 
 where the steel may be relied on to carry as much of the shear as 
 may be required, should evidently be taken somewhat inside the 
 outer ring rod, but just where is of no particular consequence. 
 
 The supposition of the existence of a crack at X, either actual 
 or potential, on which our computation of the stresses in the radial 
 rods has been based, is sufficiently satisfactory so far as the rods 
 themselves are concerned ; but it seems desirable to consider in more 
 detail the phenomena attending the development of the stresses in 
 the concrete at and near the edge of the cap, especially in soft con- 
 crete when the limit of its compressive resistance is reached in this 
 region. 
 
 The horizontal compressive resistance of the concrete at the 
 lower surface of the slab is that alreadj' treated in (38) , and it is our 
 present object to consider how that is to be combined with the 
 vertical supporting pressures under the radial rods, and with the 
 horizontal and vertical shears in the slab due to bending. These 
 latter are greatest in the neutral surface, as has been previously 
 stated, and according the general theory of stresses are equivalent 
 to, and may be replaced by, a compression and a tension in the mate- 
 rial respectively at 45° with the vertical (and mutually at right 
 angles) of the same intensity as the shear. It is evident that the 
 combination and resultant of these three compressive stresses 
 would form the dangerous element in the stress, since the single 
 tensile element would be relatively unimportant, and it would find 
 assistance in its resistance from the steel running in a direction thru 
 the concrete such as to afford it substantial support. This direction 
 is that of the straight lines on the surface of a right cone whose 
 vertex is above the center of the column and whose slope is 1 to 1 . 
 
 Consider now two of the elements of the compression in the 
 concrete around the cap, viz, the horizontal compression which is a 
 maximum at the lower surface and zero at the neutral surface, and 
 that due to shear which is parallel to the sides of a right cone with 
 vertex downward, whose sides have an upward and outward slope 
 of 1 to 1, while its intensity is so distributed that it is zero at the 
 bottom of the slab and greatest at the neutral surface. It appears 
 consequently that the lines of greatest compression in the concrete 
 due to the combination of these two elements of compression would 
 
54 COMPRESSION ON CONCRETE OP HEAD 
 
 lie in vertical planes on a bowl or saucer-shaped surface that is hori- 
 zontal at the edge of the cap and inclined at a slope of 45° at the 
 neutral surface ; and if the concrete were to crush under these stresses 
 alone, the surface of fracture would have the shape indicated in- 
 stead of that of the cylindrical surface previously assumed. This 
 change would not, however, materially affect the computations we 
 have made of stresses in steel ; it merely serves to fix more definitely 
 the position of the points of contra-flexure of the radial rods. 
 
 But there is still one further element or component of the total 
 compression in the concrete to be considered and combined with 
 those just treated in order to arrive at the resultant or total com- 
 pression. This componenent is that due to the concentrated press- 
 ures underneath each of the radial rods. These rods are at some 
 distance apart circumferentially and so do not exert a pressure that 
 is uniformly distributed circumferentially. Any concentrated stress, 
 such as that in the concrete supporting a rod, diffuses itself in the 
 material in such a manner that its intensity rapidly diminishes with 
 the distance from the surface of the rod, in accordance the same law 
 as exists in case of centers of attraction. Since the supporting com- 
 pression under the rods is vertical, we can imagine the lines of great- 
 est compression in the concrete, when this component is combined 
 with those already mentioned, to lie in vertical planes on a bowl or 
 saucer-shaped surface which has as many indentations or scollops 
 around its edge as there are radial rods, at which indentations the 
 slope of the sides is such more nearly vertical than a slope of 45°. 
 At such parts of the surface the intensity is also more severe, and 
 especially is this the case if the slab is thin so that the concentrated 
 pressure has small opportunity to distribute itself by radiating into 
 a considerable body of material before it reaches the bottom of the 
 slab. It thus comes about that thick slabs are enabled to carry 
 safely larger intensities of shearing stress around the cap than can 
 thin slabs, which is in accordance with and in justification of the 
 statements already made as to permissible shears around the cap. 
 
 The resulting surface of fracture due to shear and compression 
 a,round the cap would be of irregular conical shape starting from 
 the edge of the cap and extending thru the entire thickness of the 
 slab, were this not interfered with in the upper part of the slab by 
 the mat of reinforcing rods, which are so tenacious as to tear to 
 pieces and fracture the upper surface to a considerable distance in 
 all directions whenever any such fracture occurs around the column. 
 
 Nevertheless such fracture as here described does not under 
 
RADIAL AND RING RODS PREVENT FRACTURE OF HEAD 55 
 
 any ordinary circumstances result in a dangerous collapse of the 
 slab, or one that cannot be repaired without much difficulty, for, the 
 radial rods and the reinforcing rods will at most have suffered some 
 individual deformation by bending and are still far from being 
 broken. This will become evident later where an experimental 
 attempt to load a full-sized slab to failure is described in detail, and 
 full account of the results reached is explained and illustrated. 
 
 It is stated on good authority that in experience with many 
 hundreds of buildings constructed on this system, no case of shear 
 failure or even of incipient shear failure or fracture has occurred in a 
 well cured slab near the column and while a few cases of incipient 
 failure have occurred in floors where forms were prematurely re- 
 moved, no injury or fatality has resulted therefrom to any person. 
 
 It appears that the line of weakest section in the cured slab of 
 the standard mushroom type is that discussed previously in obtain- 
 ing (37) and shown in Fig. 3 page 7. This is brought out later by a test 
 to destruction of a fairly well cured slab. The line of weakest sec- 
 tion in a partly cured slab is on the other hand not definitely fixed, 
 but may be and sometimes is, shearing weakness near the column as 
 has been discussed and pointed out. Provision against such weak- 
 ness or carelessness is a safeguard which, while costing a small 
 amount in the matter of steel, is an insurance against serious acci- 
 dent well worth the investment involved. It is secured by making 
 the radial and ring rods sufficiently stiff and strong. 
 
 15. This section will be devoted to a consideration of the 
 mushroom system, and to several more or less similar flat slab 
 systems, in order to comment on the modifications in mechanical 
 action that are produced by the particular modifications of the 
 arrangement of the reinforcement in these systems. 
 
 Fig. 1, page 2 represents the section of a standard mushroom 
 head by a vertical plane thru the axis of the column. In this the 
 elbow rods are shown, the vertical portions of which are embedded for 
 such distances as may be necessary in the columns or are them- 
 selves column rods. One of these is represented separately at the 
 right side of the Fig. They are confined just under the elbow at 
 the top of the column by a steel neck band, and are bent over at 
 the elbow to extend radially into the slab. This bent over portion 
 is formed to scale as to length and slopes in accordance with the 
 size and thickness of the slab in which it is to be used, in such a 
 way that when the ring rods and four layers of slab rods rest 
 upon it and are tied in place, the top of the upper layer will be 
 
56 STANDARD MUSHKOOM SYSTEM 
 
 0.75 inch below the top of the slab at a distance of the thickness 
 of the slab outside the edge of the cap, and at the same time the 
 extremities of the radial rods will be 0.5 inch above the bottom of 
 the slab. In order to accomplish this, the radial portions of these 
 rods must be nearly horizontal over the cap, and have a suitable 
 slope outside the cap as shown in Fig. 1. 
 
 Fig. 3, page 7, shows the ground plan of the reinforcement of the 
 mushroom slab when the panel is square so that Li = Z/2 = 2a 
 = 26. In this Fig. the diameter of the mushroom head is assumed 
 to be of the extreme size g = L/2, a size which would increase the 
 cantilever beyond that in usual practice to an extent not adopted 
 except in the case of very unusual intensity of loading. It will 
 be observed that the areas where the reinforcement consists of a 
 single belt or layer are thereby rendered small, and the slab action 
 due to the mutual lateral action of belts which cross each other 
 exists over nearly the whole slab. 
 
 In Fig. 2, the dimensions of the rectangular sides are so taken 
 that L1/L2 = 0.75, which is assumed to be the limiting or smallest 
 value of that ratio for constructional purposes. Further, the 
 diameter of the mushroom is made as small as will permit the rein- 
 forcing belts to cover the entire panel, viz. g = 7 (a + 6)/l6. For 
 example if Lj = 20, and L2 = 15, we have g = 7.65 + . This may 
 be considered to represent standard practice, where the edges of 
 the diagonal belts intersect on the edges of the side belts. This 
 was the case assumed for treatment in deriving the formulas of the 
 preceeding investigations. Those formulas could be modified to 
 apply to larger values of g, by taking lines of contra-flexure at the 
 edges of the head nearer the panel center than given by (24), and 
 by taking larger values of the effective cross section of steel than 
 those employed in (32), (40) and (51). 
 
 Now it is evident that systems similar to this may differ from 
 it in several ways: — 
 
 1st. The design of the frame-work at the top of the column 
 may be different from this without any change in the belts of re- 
 inforcing rods. It is hardly possible for any other form of frame- 
 work to be substituted for this which will exhibit the same rigidity 
 of connection between it and the column as do the elbow rods 
 embedded in the column and bent over radially in the slab so as 
 to make the column and slab integral with each other by means 
 of this common reinforcement. Any reduction of the stiffness of 
 connection between column and frame-work of head results in in- 
 creased tipping of the head under eccentric loading of the slab. 
 
OTHER SYSTEMS 
 
 57 
 
 Fig. 4. 
 
 Eccentric loading is any loading of one panel differently from 
 another. Tipping of the head increases some deflections at the 
 expense of others, aiid increased stresses in some of the reinforcing 
 rods at the expense of others, and so requires some additional 
 reinforcement. Such a frame-work is illustrated in Fig. 4, which 
 merely rests upon the top of the column without the support of 
 metallic connection with the vertical column rods. It consequentlj' 
 affords less resistance to tipping under eccentric loads than when 
 stiffened by such metallic connection. 
 
 2nd. The ground plan of the reinforcing belts may remain un- 
 changed but part only of the belt rods maj^ be carried at the top 
 of the slab over the column head, while the rest of them are carried 
 thru under the head at the bottom of the slab. This modification 
 of design, when a sufficient number of rods go over the head to 
 resist the negative bending moments there, is very uneconomical 
 of steel, because in the case where they all go over the head, it is 
 the fact that altho the mean tension of the steel is not so great as 
 at mid span, nevertheless, by reason of the overlapping of the belts 
 in crossing, the stresses in the rods at the top reach a value not 
 much less than at mid span, and cannot be safely diminished in 
 number. It thus appears that the rods carried thru on the bottom 
 are largelj' superfluous. Of these two mats of rods at top and 
 bottom, one of them is necessarily in tension and the other in com- 
 pression. But it is a mistake to use steel to resist compression 
 when concrete can be better used for this purpose. The lower mat 
 is superfluous for this reason. 
 
58 
 
 SMALL HEAD. TOP AND BOTTOM BELTS 
 
 r 
 
 Aiiumad line of I 
 
 • niaximum bending i 
 
 \ \ y I II momant . 
 
 ^^ZS^ ilneontical lint \ 
 i .. . /J ofinptctlon •. 
 
 ' Assumed line 
 
 of inflection 
 
 '\ Column tmad^ 
 
 1 W In pounds per square foot , , , . . i i 
 
 I 1 I 1 1 1 u 1 1 1 iTi 1 1 1 I 1 1 1 1 11 1 
 
 q In pounds per linear 
 
 Fig. 5 
 
 There is still another and, if possible, more serious objection 
 to this arrangement of rods to form a mat or double layer of rods 
 at the top and at the bottom of the slab near the columns. This 
 is because they are too far removed from each other in the slab 
 for the elongations of the steel in one mat to be resisted by lateral 
 contractions in the other. The reinforcement does not therefore 
 conspire to produce the slab action expressed by Poisson's ratio, 
 which requires that the interacting steel concerned should lie approxi- 
 mately in the same zone or level. 
 
 This arrangement is illustrated in Fig. 5, copied from Taylor 
 and Thompson's Concrete Plain and Reinforced, p. 484. In this 
 design the size of the head is small enough to reduce the width of 
 the belts so greatly that not only are the areas where we have a 
 single layer of rods on the plan much enlarged, but we find that 
 nowhere do more than two layers lie in metallic contact with each 
 other, and the areas where even this occurs are limited to one 
 relatively small square over each column, and one of equal size 
 at the middle of each panel. The remaining areas are subject to 
 the law of single rod reinforcement, where we must assume lateral 
 action to be such as greatly to diminish K for the combination, a 
 fact very injurious to the efficiency of the reinforcement. This 
 as has been said, is due partly to the smallness of the head and 
 partly to the separation of the layers between the top and the 
 bottom of the slab. 
 
BAD EFFECT OF ANY SHARP BEND OR ELBOW IN A ROD 
 
 59 
 
 3rd. Another modification of design without change of ground 
 plan is that where the rods that are carried over the head at the 
 top of the slab are given a sudden steep dip at the line of contra- 
 fiexure to cai-ry them to the bottom of the slab at that line. This 
 is also illustrated in Fig. 5. Such sudden bends or kinks anj'- 
 where in the rods may give rise to very serious fractures because 
 of straightenmg out under tension, especially when the forms are 
 removed. Such bends give rise to great differences of stress in the 
 extreme fibers of the rods, thus diminishing their resistaiice also. 
 All sudden bends in rods embedded in concrete should be sedulously 
 avoided as tending very effectivelj' to crack the concrete, whether 
 the rods ai"e part of the belts or in the frame-work of the head, as 
 sho^ii in Fig. 3, in which are many such angles said elbows unsup- 
 ported except by concrete, and therefore objectionable. 
 
 It seems fair to conclude that the cracks showii in the plan 
 of the floor of the Deere & Webber Company Building, INIinnea- 
 pohs, tested by j\lr. Arthur R. Lord, and occuring along the edges 
 of some of the loaded panels at the upper surface, where none usually 
 appear, were due to the elbows in the frame work of the head, like 
 that in Fig. 4, in conjunction with the comparatively small resis- 
 tance to bending in a vertical plane offered by the rods forming this 
 projecting elbow. 
 
 In the mushroom head the only bend permitted is that at the 
 elbow of the radial rods where a strong steel neck band prevents 
 any such bad effect as has just been pointed out. 
 
 Fig. 6 
 
60 TWO WAY REINFORCEMENT 
 
 4th. We may notice a form of design in which the diagonal 
 belts are omitted and the entire panel is covered by rods parallel 
 to the sides of the panel. This, while apparently very different in 
 ground plan from those just considered does not differ from it 
 materially in principle. It is clear that the lattice pattern of the 
 web in this case is in many parts of the panel not woven so close 
 as where diagonals exist, while in other parts of the mesh the num- 
 ber of layers in contact with each other has been decreased. Experi- 
 mental results do not as yet enable us to determine with certainty 
 whether Poisson's ratio for this combination is as great as for the 
 mushroom. Upon that depends in part the relative efficiency 
 of the two arrangements. A form of this design is seen in Fig. 6. 
 
 The maximum deflections at the center of a loaded panel of 
 the system of Fig. 6, would occur when the panels touching its 
 four sides were also loaded. In this particular it differs from a 
 loaded panel in a mushroom slab which would theoretically have 
 its deflection shghtly decreased by loading surrounding panels, 
 tho this is too insignificant to have been observed as yet. 
 
 Deflections shown by tests of this system of two way reinforce- 
 ment are wholly inconsistent with simple beam theory, and can 
 only be explained on the basis of slab theory. Nevertheless, some 
 of its advocates attempt to design its reinforcement and com- 
 pute its strength on the basis of beam theory, which actual de- 
 flections show to be untenable. Such attempts should be entirely 
 abandoned as erroneous and misleading. 
 
 All considerations which have been discussed under the three 
 previous counts are to be taken as applying equally to this plan 
 of arranging the reinforcing rods, especially as to carrying of 
 part of the belts thru on the bottom surface at columns. 
 
 5th. Another element of design is the relative number of 
 rods in the side and diagonal belts. We have previously adduced 
 reasons to show that in a square panel the same number of rods is 
 required ultimately in the diagonal belts as in the side belts, tho 
 for stresses less than the yield point of the steel, it would be pos- 
 sible to diminish the number of rods in the diagonal belts some- 
 what. Equation (34) shows that for equal stresses in the steel 
 of the side belts the number of rods should have the same ratio 
 as the lengths of the sides. 
 
 A different rule from this has been erroneously proposed, 
 viz., that the ratio of the number of rods in the side belts should 
 be equal to the ratio of the cubes of their lengths. The only foun- 
 dation for this rule is that according to the beam strip theory as 
 
RELATIVE CROSS SECTION OP BELTS 61 
 
 developed in Marsh's Reinforced Concrete, p. 283, a rectangular 
 plate carried by a level rigid support around its perimeter, would 
 divide the load per unit of area which is carried by two unit-wide 
 rectangular strips that cross each other, as the fourth power of their 
 lengths, and hence would carry to the edges of the rectangle loads 
 proportional to the cubes of the lengths of those edges. Were this 
 so, the case of a horizontal rigid support around the entire peri- 
 meter of the panel is wholly different from support on columns 
 at the corners, and such a rule would be wholly inapplicable there- 
 fore to a floor slab so supported. This rule was, however, evidently 
 adopted in the design of the Larkin Building, Chicago, as shown 
 by a photograph of its reinforcement in place before the concrete 
 was poured, to which the writer has access and published in Cement 
 Era for February, 1913. The very exhaustive tests of this build- 
 ing made by the Concrete Steel Products Company of Chicago, 
 and published in the Cement Era, for January 1913, show that this 
 ratio of rods caused the stresses for the larger loads to be more 
 than twice as great at the middle of the short side belts as at the 
 middle of the long side belts. This was assuredly an uneconomical 
 distribution of steel, since correct design would require these stresses 
 to be equal, when in fact one exceeded the other by 120 to 140 per 
 cent. This discrepancy would be largely rectified by making the 
 number of rods directly proportional to the lengths of the sides, 
 as required by (34). 
 
 It also appears that the diameter of the mushroom head and 
 the width of belts of slab rods in the Larkin Building is less than the 
 limiting size in the standard mushroom system, viz. g = 7(o-|-6)/l6. 
 This makes the intersection of the diagonal belts fall nearer the 
 center of the panel than the edges of the side belts. The very 
 considerable effect of a very inconsiderable change of this width 
 has been mentioned on p. 25. The result would be that the steel 
 would for this reason be far less effective, and its resistance 
 would be more nearly in accordance with (37) than with (34), a loss 
 of perhaps 25 to 30% in its effectiveness. 
 
62 SPECIMEN COMPUTATION OF THIN SLAB 
 
 16. This section will be devoted to a specimen computation 
 applying several of the preceeding formulas to a floor slab of practi- 
 cally the same dimensions and reinforcement as one or two recently 
 designed and now under construction (1913). 
 
 Long Side L^ = 28' X 12 = 336". 
 
 Short side L2 = 25' 10" = 310". 
 
 Thickness of rough slab, h = 10" = L/33.6. 
 
 By (56) C2 = Li/L^ = 0.9 nearly. 
 
 Diameter of head 3 = 7 (Li + L2)/32 = 141". 
 
 Diameter of cap L^—B = 0.2Li = 67". B = O.8L1 = 268.8". 
 
 Each belt has 25 — 7/l6" round rods. 
 
 Cross section of each belt, A = 25 x 0.15+ = 3.76 sq. inches. 
 
 Depth of center of mid side belt with | inch concrete cover- 
 ing, di = 10 — 0.5 — 0.2 = 9.3". 
 
 Depth of center of second layer slab rods at panel center, 
 ^2 = 10 — 0.5 — 0.64 = 8.86" 
 
 Depth of bottom surface below third layer of slab rods at edge 
 of cap with M" covering, dg = 10 — 0.75 — 1.1 = 8.15". 
 
 Design load per square foot = 150 lbs. 
 
 Dead load per square foot = 130 lbs. 
 
 Panel load, Tf = 280 x 28 x 25 5/6 = 202,550 lbs. 
 
 A maximum tension is found in the slab rods at the middle of 
 the long side belt, and is to be computed from (34) as follows: 
 
 202550 X 336 
 
 L= = 11,120 lbs. per sq. inch (80) 
 
 175 X 9.3 X 3.76 
 
 Any other loading within elastic limits of the steel would 
 produce proportionate stresses. 
 
 The tension in the steel at the center of the panel is com- 
 puted by (52), as follows: 
 
 1.02 X 202550 x 336 
 
 /3= = 9,145 lbs. per sq. in.. (81) 
 
 256 X 3.76 x 0.89 x 8.86 
 
 The radial tension at the edge of the cap is by (43) , 
 
 202550 x 336 x 646 (3 x 0.64 — 1) 
 
 /s = = 5320 lbs. per sq.in . (82) 
 
 800x8.15x3.76x310 
 
COMPUTED STRESSES IN THIN SLAB 63 
 
 The circumferential tension at the vertical section thru the 
 center of the column at the end of the long side may be computed 
 by placing B = Lj in (43) , and we obtain, 
 
 202550 X 336 x 646 x 2 
 
 /s= = 11,570 lbs. per sq. in. . (83) 
 
 800 X 8.15 X 3.76 x 310 
 
 as the mean computed intensity of stress in each of these rods, 
 regardless of its distance from the center of the column. This 
 stress may be reduced by increasing the number of laps over the 
 head. The result in (83) is, however, an over-estimate of the 
 tension across the top of the head because the head is integral 
 with the cap of the column where compressions in the concrete are 
 no longer confined merely to the thickness of the slab but take 
 in a much greater depth of concrete in the cap. This in effect 
 puts the neutral surface at a lower level throughout the cap and 
 by thus increasing the lever arm of the reinforcement reduces its 
 tension and deformation. This will react upon the rest of the 
 reinforcement in such a manner as practically to make the stresses 
 smaller than given by (83) because the mean lever arm will have 
 increased. In fact the greatest stress in these rods will be that 
 given by (80), instead of (83). 
 
 The compression in the concrete lengthwise of the longer 
 side belt at its middle is to be computed from (38) and (80) as 
 follows: By taking the percentage of belt reinforcement at 0.3%, 
 the corresponding value of i = 0.72, and Es/E^ = 15: 
 
 0.28x11120 
 
 /e= = 288 lbs. per sq. in (84) 
 
 0.72 X 15 
 
 The compression at the center of the panel where the per- 
 centage of slab reinforcement may be conservatively assumed at 
 0.6% and i = 0.66 may be computed thus: 
 
 9145 
 
 /o= = 305 lbs. per sq. in (85) 
 
 2 X 75 
 
 The compression at the edge of the cap lengthwise of the 
 side belt is uncertain in the absence of exact information as to the 
 laps in the slab rods over the head. Assume that one-half the rods 
 are lapped over each head, and that we take six belts as the reinforce- 
 ment of the slab, the percentage then is 1.8% and i = J, then, 
 
 5,320 
 
 /„ = = 355 lbs. per sq. in (86) 
 
 15 
 
64 COMPUTED DEFLECTION OF THIN SLAB 
 
 For reasons already given in discussing the circumferential 
 tensions in the head, it appears that any computation of the cir- 
 cumferential compressions in the concrete on the basis of (38) would 
 be incorrect and subject to large errors of as much possibly as 
 50%. That this is the fact appears evident when we consider the 
 large mass of concrete in the cap which must be actually diminished 
 in lateral dimensions before the slab which is integral with it can 
 be subjected to true stresses of equal intensity, and consider also 
 that near the edges of the head the radial rods and the outer ring 
 rods approach the lower surface sufficiently to afford reinforcement 
 to resist compression. It is consequently unnecessary to look 
 further than (86) in computing the greatest compression in the 
 concrete. 
 
 As previously stated, computations based on (38) are highly 
 artificial and arbitrary in their character, since they assume the 
 straight line theory as well as an arbitrary value of the ratio of 
 Young's moduli for steel and concrete. Furthermore, concrete 
 in compression in both circumferential and radial directions at the 
 same time, as it is at the edge of the cap, is known to resist with 
 safety compressive stresses of greater intensity than when in simple 
 compression in one direction. 
 
 If a test load of twice the design load, viz., in this case of 300 
 lbs. per square foot, be placed upon the slab, the deflections which 
 will be produced by the addition of this total load of 217,000 lbs. 
 may be computed as follows: 
 
 217000 X 336^ 
 
 By (54), A 2i = = 0.237 . . (87) 
 
 10.7 X 10^° X 9.3^ X 3.76 
 
 0.89 X 217000 X 336^ 
 
 By (55), AZ2= ; = 0.378 .... (88) 
 
 6.56 X 10^° X 8.86^x3.76 
 
 217000 x 336* x 2.084 
 60 x 10^° x 8.15^ X 3.76 
 
 By (58), A Z3 = ; = 0.115. . . . (89) 
 
 0.89 X 217000 X 336* 
 
 By (60), A 24 = ~ ; = 0.235 .... (90) 
 
 12.5 X 10'° x8.15^x 3.76 
 
PROPORTIONATE DEFLECTIONS 65 
 
 By (61), Z>i = 0.352, and D2 = 0.613 (91) 
 
 -Di 1 -D2 1 
 
 = , and = (92) 
 
 Li 960 Vi^2 ^ j^2 745 
 
 Any loading differing from this would produce deflections 
 proportionate to its intensity. 
 
 In this specimen floor slab, which is near the limit of least 
 thickness permissible in the standard mushroom system, viz., 
 d = Li/Z5, it is clear that the design load brings stresses to bear 
 upon its reinforcement which are very moderate in their intensity 
 indeed. It is also evident that were the slab to be loaded with a 
 test load of such amount that the total load sustained would be 
 twice the dead load of the slab itself plus twice the design or live 
 load, viz. 560 lbs. per square foot, none of tjie steel would be stressed 
 up to the yield point, and the first failure would take place by 
 cracking the concrete, tho the steel would still prevent sudden 
 failure and collapse. Altho the slab is relatively so thin the de- 
 flections are also very small for so large a span. 
 
 It has not yet been so generally recognized as it should be 
 that a thin construction, such as a flat slab is, should not be ex- 
 pected to show so small proportionate deflections as is required 
 in girders. 
 
 The observed results of quite a number of tests of mushroom 
 slab floors are to be found on pp. 32 and 44 of Turner's Concrete 
 Steel Construction. These are there compared with results com- 
 puted according to Turner's empirical formula, which translated 
 into our present notation has been reproduced in equation (72). 
 The observed and computed results show a very close agreement. 
 The results given by (72) are in close agreement, as has been seen, 
 with those derived from (61). 
 
 Some of these test slabs present peculiarities of reinforce- 
 ment such as need to be individually considered in order to make 
 exact computations of their deflections. It is thought that the 
 specimen computation already given will afford sufficiently guidance 
 in the methods to be employed. 
 
 Having considered the stresses and deflections of a slab which 
 is near the minimum thickness for the standard mushroom system, 
 viz. -Li/35, it will be instructive to consider a specimen or two 
 near the maximum thicloiess Li/20. 
 
66 
 
 STIFFNESS OF MUSHROOM SLAB BRIDGES 
 
 
 
 
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 Tischera Creek Bridge, Duluth 
 
 Test of Tischers Creek Bridge with 30 ton construction cars, each loaded with 20 tons of rails 
 Deflection leas than one twenty thousandth part of the span 
 
SPECIMEN COMPUTATION OF THICK SLAB 67 
 
 Take for example the bridge over Tischer's Creek, Duluth, 
 shown in the cuts on page vii and page 66. It is supported on three 
 rows of columns crossing the gorge, at a distance apart of 27 feet 
 center to center of columns, the two street car tracks being over 
 the side belt that lies along the center line of the bridge lengthwise. 
 Each of these rows consist of six columns lengthwise of the bridge, 
 at a distance apart of 26 feet from center to center, so that 
 
 Li = 27 X 12 = 324" 
 
 L2 = 26 X 12 = 312" 
 The size of the mushroom heads and width of the belts is 12 feet, 
 which is in excess of 7 (Lj + L2)/32 = 139 I/8" = 11.6', thus giv- 
 ing great stiffness. The object to be obtained by maximum thick- 
 ness and large head is to secure great stiffness and so reduce vib- 
 rations as well as decrease deflections. There are twenty 9/l6 
 inch roimd slab rods in each belt, or a total cross section in each 
 belt of ^1 = 5 square inches of metal. The slab is 15" deep at 
 its thinnest part at the gutter on each side of the roadway, and 
 the steel is kept down to that level throughout the slab, altho at 
 the crown of the roadway under the tracks and over the center 
 row of columns the slab is 5" thicker, or 20", with the same thick- 
 ness over the side rows of columns where the sidewalks are. The 
 mean thickness is somewhat in excess of L2/2O. This makes 
 di = 19" for the short side belts, di = 17" for the long side belts 
 and ^3 = 14" approximately for the heads. The design load per 
 square foot = 150 poimds. The dead load of the slab per square 
 foot = 300 pounds. Hence Tf = 450 x 26 x 27 = 315,900 pounds. 
 The effective cross section of slab steel is so great by reason of large 
 heads that instead of (34) we may take 
 
 W L 
 
 /s= (34)' 
 
 200 di Ai 
 
 For the long side belt this gives f^ = 6,033 pounds per square inch. 
 The total load imposed on the slab might be made six times as great 
 without causing the steel to reach its yield point, and the live 
 load might become 900 pounds per square foot without causing /g 
 to exceed 16,000 poimds. 
 
 This slab was tested as shown in the cut, page 66, by running 
 two construction cars loaded with 20 tons of rails each over the 
 bridge at the same time along one track of the short side belt 26 
 feet long. Weight of each car = 60,000 pounds. Weight of rails 
 40,000 pounds. Total weight of train = 200,000 pounds extend- 
 ing over several spans. The deflections was too small to be dis- 
 covered by observations with level and rod. It is useless to attempt 
 
68 
 
 COMPUTATION OF THICK SLAB 
 
 to compute the deflection of this slab under the test load because 
 the four steel rails of the railway tracks across the bridge were so 
 fastened to the steel cross ties which were embedded in the con- 
 crete as to make the rails a part of the reinforcement of the slab. 
 They furnish a cross section of reinforcement equal perhaps to 
 7Ai, which would effectually bar the appUcation of our deflection 
 formulas and reduce deflections to very small quantities. 
 
 In so thick a slab as this the action of any contemplated load 
 is widely distributed by the slab itself, and such loads, as well as 
 all shocks and vibrations are largely dissipated or absorbed by the 
 body of slab itself without causing observable local stresses as they 
 do in steel structures. 
 
 VIEW OF REINFORCING STEEL 
 
 Flat Slab Bridge, Denver, Colo. 
 
 Spars 43 ft. 6 in. 
 
 Carries Heavy Interurbau Cars 
 
COMPUTED STRESSES AND DEFLECTIONS 69 
 
 The Curtis Street bridge, Denver, Colorado, is one of four 
 bridges across Cherry Creek, shown by the cut on page 68, con- 
 structed on the mushroom system It has three rows of three 
 columns each crossing the stream, the middle column of each row 
 in mid stream with spans of 42 feet between columns centers length- 
 wise of the bridge, thus obstructing the waterway as little as pos- 
 sible. It has a width of 28 feet between column centers. The 
 slab is 17 inches thick at the gutters, 26.5 inches at the sidewalks 
 outside the gutters, and 21" over the center row of columns. The 
 sidewalk is stiffened with fourteen 3/8" round rods lengthwise 
 just below its top surface as supplementary reinforcement, and 
 there is an outside parapet giving added stiffness. There are 
 also three stiffening rods 24" apart across the bridge midway 
 between columns. There are three ring rods, and the width of the 
 belts is 16'. This is in excess of 7 (Lj + L2)/32 = 183.75" = 15 5/l6'. 
 The heads are exceptionally stiff each having twelve 1 3-8" round 
 radial rods. Each belt has twenty-six 5/8" round rods, hence 
 Ai = 26 X 0.3 = 8 square inches nearly. 
 
 Li = 42 X 12 = 504" , La = 28 x 12 = 336". 
 
 The dead load per square foot = 300 pounds. 
 
 The design load per square foot = 150 pounds. 
 
 IF = 450 X 42 X 28 = 529,200 pounds. 
 
 rfi = 20" for long side belt. 
 
 Compute the stress in the steel by (34) modified to (34)' by 
 reason of exceptional stiffness, and we obtain /^ = 13,320 pounds. 
 
 Compute the central deflection due to a test load of 100 pounds 
 per square foot. Let ds = 16". Then in (71) L2/Li = 2/3: hence 
 C2= 3/4, and we have D2 = 0.125". This is probably considerably 
 in excess of the correct deflection, since the slab is stiffer than the 
 one considered in equation (71), which was derived for 20 foot spans. 
 More correct values are to be computed from (54), (58) and (61). 
 Moreover for such comparatively light stresses in the concrete, 
 the deflections, as we have seen previously fall short of those com- 
 puted by the formula, which agrees with experiment for stresses 
 nearer the yield point of the steel. D2 = 0.125" is less than 
 one four-thousandth of the span, and the deflection under the 
 working load would undoubtedly be less than one sixth-thousandth 
 of the span. 
 
70 WORKING STRESSES AND FACTOR OP SAFETY 
 
 A word is here in place respecting working stresses and the 
 factor of safety in the reinforcement of slabs, to the effect that 
 the same values of these quantities in slabs affords a greater degree 
 of security than in ordinary structural steel construction, and that 
 occurs for several reasons: 
 
 1st. Steel rods such as are used in slabs have a higher yield point 
 by perhaps 25% than the steel of other structural members. Fur- 
 thermore, it is quite possible and desirable to use a higher carbon 
 steel for these rods than the mild steel necessarily used in structural 
 work, where it must be manipulated in such ways that high carbon 
 steel cannot be used. But in these rods which suffer no usage 
 tending to impair their condition, there is good reason to use a steel 
 of higher yield point and greater ultimate strength. This yield 
 point may readily be 70% greater than that of ordinary mild steel 
 for structural purposes. 
 
 2nd. Rods embedded in concrete do not yield as do bare 
 single rods in a testing machine or elsewhere by the formation of 
 a neck and drawing out at that point. The concrete embedment 
 prevents that. 
 
 3rd. In a reinforcement consisting of multiple parallel rods 
 acting together, no single rod can become overstrained and yield to 
 any appreciable extent before bringing into play adjacent rods. 
 This makes the construction tough, and not liable to sudden col- 
 lapse, as well as obviates concentration of stresses thus ensuring 
 a high degree of security. 
 
COMPARATIVE TEST OF TWO SLABS 71 
 
 17. This section will be devoted to a detailed consideration 
 of a test to destruction of two slabs, 12' x 12' between column 
 centers, constructed for experimental purposes. The tests were 
 made by Professor Wm. H. Kavanaugh, in November and December, 
 1912, and the results he obtained, together with a mathematical 
 discussion based upon them, will be here given. One slab was 
 constructed in accordance with the plans and specifications of the 
 U. S. Patent No. 698,542 issued to 0. W. Norcross for a slab for 
 flooring of buildings, and the other was a Turner Mushroom slab 
 under U. S. Patent No. 1,003,384. The test serves to bring out in 
 a striking maimer not only how two slabs, which present a super- 
 ficial resemblance in the plan of arrangement of reinforcement, 
 differ from an experimental and practical standpoint, but it also 
 makes evident their radical divergence of action mechanically and 
 mathematically. 
 
 That two slabs of the same span, thickness and amount of 
 reinforcement should on test show that one of them was more than 
 twenty times as stiff, and more than five times as strong as the 
 other, and that the failure of the weaker one was a sudden and 
 complete collapse, with little or no warning to the inexperienced 
 eye, while the other gave way by slowly pulling apart little by 
 little, thus gradually getting out of shape without any final break 
 down, are phenomena that deserve the close attention of the de- 
 signer, and are of the highest interest scientifically as well as practi- 
 cally. The enormous differences in the deflections and in the 
 stresses in the reinforcement as shown by extensomoter measure- 
 ments, and in the character of the failure in respect of safety and 
 its relation to the line or zone of weakest section, as well as in the 
 difference of design loads and breaking loads amounting to 500%, 
 all illustrate what scientific design will accomplish and what results 
 are possible by an ingenious arrangement of the reinforcement. 
 
 These slabs were each of the same thickness, viz 6", and were sup- 
 ported by columns placed at the corners of a square 12' x 12' from 
 center to center of columns. The slabs projected 2' to 3' beyond 
 the centers of the colunms on each side, and had precisely the same 
 number and size of reinforcing rods in each belt, viz eleven 3/8 
 inch round rods. The concrete was of a 1 : 2 : 4 mix, and while 
 only about four weeks old at the time of the test, it had been poured 
 warm and kept warm by steam heat under such unusually favorable 
 conditions as to have become well cured at the time of the test. 
 The steel used showed by test a stress at yield point of 51,000 to 
 55,000 pounds per square inch, and an ultimate strength of 76,000 
 
72 BEAM THEORY, VERSUS SLAB THEORY 
 
 to 80,000 pounds, with an elongation of twenty to twenty-five per 
 cent. 
 
 The first slab was made in accordance with the specifications 
 of the Norcross patent already referred to except that belts of rods 
 were substituted for the netting mentioned by the patentee. This 
 design was selected as one of the two for this comparative test, 
 not because it is a good design, or one that any engineer would 
 to-day care to employ, but because it exhibits, according to the 
 express intention of the patentee, simple tension on its lower surface, 
 everywhere between columns, and simple compression everywhere 
 on its upper surface between columns; this being in direct contrast 
 to the other design, which is arranged not only to resist direct ten- 
 sions over the supports, which the first does not, but also to resist 
 circumferential stresses both around the supports and around the 
 panel centers, as any truly continuous flat slab must. 
 
 This test may then be viewed in the light of an experimental 
 demonstration of the difference between a reinforced flat slab con- 
 structed in accordance with the beam theory and one constructed 
 in accordance with correct slab theory, where true and apparent 
 moments differ radically as shown at the beginning of this investi- 
 gation, but are wholly contradictory to any form of simple or con- 
 tinous beam theory. This test may be regarded as settling once for 
 all the question of applying simple beam theory to a cantilever flat 
 slab, reinforced throughout practically its entire area with a lattice of 
 rods crossing each other and in contact. It shows that it is impos- 
 sible to compute the deflections of such a slab by beam theory. 
 Furthermore this impossibility makes it certain that the stresses 
 in such a slab cannot be computed by beam theory, for to do this is 
 to commit an inconsistency such as has heretofore too often been 
 committed, but one which should hereafter be carefully avoided. 
 
THE NORCROSS TEST SLAB 
 
 73 
 
 Norcross in his patent already referred to describes his con- 
 struction as consisting "essentially, of a panel of concrete having 
 metallic network encased therein, so as to radiate from the posts 
 
 on which the floor rests The posts are first erected, and a 
 
 temporary staging built up level with the tops of posts. Strips of 
 wire netting are then laid loosely in place on top of the staging .... 
 The concrete is then spread upon or moulded in place on the staging 
 to enclose the metallic network. In practice I have sometimes 
 laid the concrete in layers of different quality, the lower layer of 
 the floor which encloses the wire being laid with the best concrete 
 
 available If the forces acting upon a section of flooring 
 
 supported between two posts be analyzed it will be found that the 
 tendency of the floor section to sag between its supports will cause 
 the lower layers of the flooring to be under tension while the upper 
 layers of the flooring will be under compression, these stresses being, 
 of course, the greatest at the top and bottom layers, respectively." 
 
 Fig. 7. Reinforcement of Norcross Slab 
 
74 
 
 THE NORCKOSS TEST 
 
 Fig. 8. Norcross Slab Carrying Load 3 
 ^■-i,- .L S'-Q- J, S-/1,- 
 
 Col. Cap P/ofe 20^20^i 
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 Fig. 9. Norcross Slab 
 
LOADS ON MUSHROOM SLAB 
 
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76 
 
 NORCROSS SLAB 
 
 The number and arrangement of the reinforcing rods in the 
 Norcross experimental slab, (eleven 3/8" round rods in each side 
 and diagonal belt) is clearlyshown inthe viewof Oct. 31st, Fig. 7, which 
 shows the forms ready for pouring the concrete. Steel plates 
 20" X 20" X 0.5" carry the rods and rest on the tops of the columns, 
 which last in this case consisted of steel pipes about 5|" in dia- 
 meter filled with concrete and embedded at their lower ends in large 
 concrete blocks. A vertical central bolt in the concrete at the 
 upper end of each pipe permitted the plates to be firmly secured to 
 the tops of the columns. The view of Nov. 30th, Fig. 8, clearly 
 shows the manner of placing the pig iron on the slab for load 3. 
 This slab is 16' x 16'. The loading at first covered an area having 
 the form of a Greek cross whose central square was five feet on a 
 side with arms 5' 6" long, as represented in accompanying diagram 
 of loaded areas A, B, C, D, E, Fig. 9, and of amounts shown in 
 Table 1. 
 
 Fig. 10. Collapse of Norcross Slab 
 
 When 10,000 pounds had been piled on the central part of the 
 slab in addition to load No. 4, of 66,812 pounds, the slab suddenly 
 failed. In anticipation of such failure timber blocking had been 
 placed under the slab to prevent its falling more than possibly ten 
 or twelve inches. 
 
NORCROSS TEST 
 
 77 
 
 Fig. 11. Collapse of Norcross Slab 
 
 The two views of Dec. 2d, Fig. 10 and Fig. 11, show the con- 
 dition of the slab after removing part of the final loading in order 
 to render the nature of the failure visible. Careful extensometer 
 measurements of the elongations of the steel rods at the middle 
 of the side and diagonal belts were made under the action of loads 
 1, 2, 3 and 4, and also similar extensometer measurements in the 
 concrete both on the top and the bottom of the slab along the center 
 line of the side and diagonal belts near those edges of two of the 
 steel plates which were nearest the center of the belts. Besides 
 these, certain other measurements of the concrete were made at 
 right angles to the diagonals. Deflections were also measured 
 under these loads at the middle of the diagonal belt and of two of 
 the side belts at V, W, X, Y, Z. 
 
 These measurements all show beyond question that the side 
 and diagonal belts act like simple beams in this form of construction, 
 since the stresses in the steel and concrete on the under side of 
 the slab in the direction of the rods is invariably tensile, while the 
 stresses in the same directions on top of the slab are always com- 
 pressive. It was the avowed intention of Norcross to reinforce 
 the slab in this manner since he regarded the upper part of the slab 
 as being subjected everywhere to compression and the lower part 
 to tension only, as stated in his specifications as already quoted. 
 
78 COMPUTATION OP THE TEST 
 
 The following computation, Table 2, shows a good approxi- 
 mate agreement of the results of this test with the beam theory of 
 flexure, assuming for simplicity that the stiff steel supporting plate 
 and interlacing of the ends of the belts diminishes the effective 
 span of the side belts by 12", and the diagonals in the same pro- 
 portion, and further assuming that the loading was all apphed at 
 the middle of the side and diagonal belts. 
 
 The extensometer measurements made were for a length of 
 8", consequently the stress in the steel per square inch would be 
 computed thus: 
 
 fs = l/8 (elongation in 8") x 30,000,000; (l)i 
 
 and, this being known from observation, it will be possible to com- 
 pute the load W carried by the beam in which the given elongation 
 occurs, as follows: 
 
 The bending moment due to a concentrated load W at the mid- 
 dle of a beam of length L is M = i W L, (2)i 
 
 and the equal moment of resistance of the reinforcement by which 
 
 it is held in equilibrium is M = A j d f^ (3)i 
 
 in which A is the total cross section of the steel in the belt = 
 11 x 0.11 = 1.215 sq. in., and the distance from the center of 
 the steel to the center of compressive resistance of the concrete 
 is assumed to be, j d = 0.9 x 5.75 
 
 when d = 5.75 is taken as the distance from the center of action 
 of the steel to the top of the slab. 
 
 Hence TF = 4 A j d f./L (4)i 
 
 is the load required to cause the stress /g in the steel. In the side 
 belts we assume the span L to be 132", and in the diagonals 132 V2. 
 
 In Table 2, which follows, it will be noticed that loading No. 1 
 is too small to develop sufficient elongations or deflections to 
 overcome the initial compressions in the concrete in which the 
 reinforcement is embedded, so that the load carried by the steel is 
 only about one half of the actual load, the other half being evidently 
 carried by the concrete in which it is embedded. This is in com- 
 plete accord with other similar experiments. But in case of loads 
 No. 2 and No. 3, where the steel is stressed close to the yield 
 point, the sum of the loads as shown by the stresses in the steel 
 is very close to the total actual load. It is assumed that these 
 total actual loads are carried by the various belts in the same pro- 
 portion as the computed loads, since there is no other way of 
 dividing the total load between the belts. This may be stated 
 mathematically, as follows: 
 
LOADS AND DEFLECTIONS OP NOKCROSS SLAB 
 
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80 NOKCROSS SLAB ON THE BEAM THEOBY 
 
 Let Wi = the computed load on a side belt, 
 and W2 = the computed load on a diagonal belt. 
 Let wi = the actual load on a side belt, 
 and W2 = the actual load on a diagonal belt. 
 Then 4PFi + 2W2 = total computed load on slab, 
 and 4TFi + 2W2 = total actual load on slab. 
 
 4 TFi' + 2 W2 Wi W'2 
 
 Then = = — (5)2 
 
 4.W1 + 2W2 Wi W2 
 
 from which Wi and W2 can be computed, Wi , W2 and 4:Wi + 4TF2' 
 being already known. 
 
 The stresses in the steel under load No. 4, are so far beyond 
 the yield point as to make computation useless. Having found 
 the actual distribution of loading Wi and W2 the center deflections 
 of the belts have been computed by simple beam theory from the 
 formula. 
 
 W' L^ 
 
 D2 = (6)1 
 
 4:8E Aij (f 
 
 in which i d = the distance from the steel to the neutral axis and 
 the value of j has been assumed to be 0.69; W' is the actual load on 
 the belt and L is its span as previously stated. 
 
 It appears from Table 2, that the effect of the reinforcement 
 is accounted for to a reasonably close approximation by consider- 
 ing the belts to act as a combination of simple beams, at least with- 
 in the range of loading near the yield point of the steel. 
 
 It appears that the steel reached its yield point under a total 
 load on the slab of from 15 to 18 tons and final collapse occured under 
 a total load of a little over twice the latter amount not distributed 
 uniformly but piled more in the general form of a pyramid. 
 
 It was observed that the application of the relatively small 
 loading on the corner areas F, G, H, I, had a very injurious effect 
 upon the slab, tending to break it across the tops of the columns. 
 
 The results of the test may be summarized in the Norcross 
 system as follows: 
 
 1st. This slab is of the simple beam type, and the test shows 
 no cantilever action and no circumferential slab action. 
 
 2nd. The narrow belts running diagonally leave large areas 
 without reinforcement, and there is consequently no provision for 
 resisting circumferential tensions as required in slab action. 
 
 3rd. The concrete showed compressive stresses on the upper 
 surface of the slab in the direction of all the reinforcing rods. 
 
SUMMARY OF NOBCROSS TEST 81 
 
 4th. The concrete showed tension at the bottom surface in 
 the direction of all the reinforcing rods, in agreement with Norcross' 
 own analysis. 
 
 5th. This slab deflected 1.6" under 33 tons and then broke 
 down completely under 38 tons. 
 
 6th. The first crack appeared under a load of 15 tons and 
 deflection of 0.7". 
 
 7th. The slab, not being reinforced on the top surface over 
 the columns, inevitably cracks at a column when the slab is loaded 
 around the column. 
 
 8th. At failure the steel had passed its yield point. The 
 percentage of reinforcement in the diagonal belt if we regard the 
 belt as about 18" wide is very nearly 1%, but since a width of 
 concrete somewhat greater than that may be assumed to act with 
 this steel, the percentage of reinforcement is somewhat less than 
 1%. Similarily, the side belts of width 36" have a reinforcement 
 less than 0.5%. The full strength of the steel in both belts was 
 developed by the concrete, which fact demonstrates that the con- 
 crete was of high grade and well cured. The steel was also of 
 good standard quality, and the test was therefore in every way 
 fair to the Norcross slab, since it was so loaded as to cause the 
 stresses in the side and diagonal belts to be practically equal, thus 
 using the steel most economically. The slab failed because the 
 steel yielded near the middle of the spans, thus causing the concrete 
 above the steel to crack and break. 
 
 The second slab was made according to the Turner Mush- 
 room System, under the patent already referr ed to. 
 
 Since all forces in a plane may be resolved into components 
 along any pair of axes at right angles to each other it is possible 
 to provide reinforcement to resist any horizontal tensile stresses 
 in the slab by various arrangements of intersecting belts of rods at 
 zones where these stresses occur. The combination of such belts with 
 radial and ring rods to constitute a large and substantial canti- 
 lever mushroom head at the top of each column affords a very 
 effective and economical arrangement for controlling the distribution 
 of the stresses in the slab, and it places the reinforcement where 
 it is most needed. It not only has the same kind of advantage 
 that the continuous cantilever beam has over the simple girder 
 for long spans, but combines with it the kind of superiority that the 
 dome has over the simple arch by reason of circumferential stresses 
 called into play, which greatly adds to the carrying capacity of the 
 slab. 
 
82 
 
 REINFORCEMENT 01" MUSHROOM TEST SLAB 
 
 Fig. 12. Reinforcement of Mushroom Slab 
 
 Column Rode 8^ /{* ' 
 
 Fig. 13. Mushroom Slab 
 
 Diom S~h 
 
THE MUSHROOM SLAB TEST 
 
 83 
 
 The mushroom test slab was six inches thick, and was sup- 
 ported on four 18" by 18" square reinforced concrete columns 
 distance 12' from center to center. These had square capitals, 
 42" X 42". The slab was appromimately 18' x 18', and the dia- 
 meter of the outer ring rod of the Mushroom was 66", while the 
 inner ring was 42". These were supported on eight l-l/S" round 
 radial column rods. 
 
 Fig. 14. Mushroom Slab, Load 4. 
 
 This will be clearly understood from the view dated October 
 31st, Fig. 12, which shows the reinforcement and forms ready for 
 pouring the concrete. The remaining views are explained by their 
 accompanying legends. 
 
 The diagram of loaded areas for the mushroom slab Fig. 13, is 
 like that already given for the Norcross slab in every particular 
 except that the size of the mushroom slab being 18' x 18', while the 
 Norcross slab was 16' x 16', the arms of the Greek cross in the 
 mushroom slab are each 5' 6" long and 5' wide. 
 
84 
 
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86 
 
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 Fig. 15. Mushroom Slab, Load 7. 
 
 Fig. 16. MuBhroom Slab, Load 9. 
 
COMPUTATION OF MUSHROOM TEST 87 
 
 The accompanying Table 3, exhibits the loads per square foot 
 of each of the subsidiary areas shown in the diagram as also the 
 total loads on each of those areas. The view of Dec. 3, Fig. 14, 
 shows load 4, and that of Dec. 13, Fig. 15, load 7, while that of 
 Dec. 16, Fig. 16, shows load 9. 
 
 Elongations of steel were measured by Berry extensometers 
 in two of the side belts and in one of the diagonal belts until the 
 yield point of the steel was reached at load No. 8. Deflections 
 were also measured. In Table 4, these will be considered so far as 
 they relate to the middle points of the belts. Loads 8, 9, 10, are 
 of great interest as exhibiting the behavior of the slab under ex- 
 cessive loads, showing, as they do, yielding and large permanent 
 deformation without dangerous collapse. 
 
 By (52) the uniformly distributed load per square foot of 
 panel area when the stress in the diagonal belt is /^ is found for a 
 square panel from the expression 
 
 256 j dz A 
 
 144g = w = TT'/144 = f, (52a) 
 
 144 L 
 
 which applied to this slab gives us 
 
 256 X 0.89 X 5.125 X 1.215 
 
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 144 X 144 
 
 The values of this uniformily distributed load w is tabulated 
 in table 4, for each of the observed values of the /g in the diagonal 
 belts. The values of iv so computed tend to become identical, 
 in case of the heavier loads, with the loads per square foot on the 
 central area C, as might reasonably be expected, w being the uniformly 
 distributed load which is equivalent so far as the stress on the dia- 
 gonal belt is concerned to the action of the actual loads which are 
 not imiformly distributed. 
 
 How compute by (54), (55), (58), (60) and (61), the deflections 
 at the mid side belt and at center of the panel, due to a uniform load. 
 These results are given in Table 4, and accord closely with those 
 actually observed, as they should, because the irregularity of dis- 
 tribution does not produce deflections that differ much from the 
 equivalent uniform load as computed above. 
 
 In these computations it is assumed that di = 5.5", ^2 = 
 5.125", da = 4" 
 
88 defijECTions of mushroom slab 
 
 The double set of values under loads 4 and 5 is due to the 
 fact that readings were had under load 4, immediately after the 
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 to the first set. 
 
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LOADS AND DEFLECTIONS OF MUSHROOM SLAB 
 
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90 DISCUSSION OF DEFLECTIONS OF THE TWO SLABS 
 
 It will be seen from Tables 1 and 3, that the first three loads 
 were practically the same for both slabs. In the Norcross slab 
 load 3, of 18 tons, stressed the steel up to the yield point, but in 
 the mushroom slab the stress was so small, (being in fact less than 
 ten per cent of the former) as probably not to remove all the com- 
 pression from the concrete in which it was embedded. Indeed the 
 load on the latter slab became five times as much, 90 tons, before 
 its steel approached the yield point, at which time it was carrying 
 about twice the load which caused the complete failure of the 
 Norcross slab. 
 
 Moreover the deflection of the Norcross slab under load 
 3, was twenty-two times that of the mushroom slab under the 
 same load. This result is in full accord with slab theory which shows 
 that the central deflection of a continuous diagonal beam with fixed 
 ends uniformly loaded with one sixth of the total load on the slab 
 and having the same thickness and reinforcement as the diagonal 
 belt, would have more than six times the central deflection of the 
 slab, while the stress in its steel would be three or four times as 
 much. This gives a measure of the effect of slab action. 
 
 By the phrase "slab action" we designate the increased strength 
 and stiffness of the slab by reason of its resistance to circumferential 
 stresses around the columns and around the center of the panel. 
 
 Furthermore, if this continuous beam be compared with a simple 
 beam uniformly loaded and having the same reinforcement, the 
 latter would have five times the defiection of the continuous beam, 
 or thirty times that of the slab, while the stress in the steel would 
 be one and one-half times that in the continuous beam, and six or 
 seven times that in the slab. This last exhibits the effect of canti- 
 lever action combined with slab action. 
 
 The apparent discrepancy between the observed ratio of de- 
 flections in these two slabs of 22 and the just computed deflections 
 of 30, is to be accounted for by the fact that the computation 
 assumed equal spans, whereas the Norcross span was assumed 
 to be diminished from 144" to 132" by the colimin plate. A re- 
 duction of the span of this amount will change the computed de- 
 flections in the ratio of 144^ : 132^ : : 30 : 23 which is in practical 
 agreement with the observed result of 22. 
 
SrMMARY OF TEST OP MUSHROOM SLAB 91 
 
 By the phrase "cantilever action" we designate the increased 
 strength and stiffness which is due to the continuitj'- of the beam 
 or slab at its supports so that it is convex upwards at such points. 
 
 While the concentration of the loading toward the middle of 
 the panel, such as was the case in this test, may prevent any pre- 
 cise agreement of these numerical estimates based on uniform 
 loadmg with the results of the tests, they cause the general agree- 
 ment shown in the tables and tend strongly to sustain our confi- 
 dence in the validity of the analysis from which these concordant 
 approximate estimates are obtained. 
 
 The amazing difference in the strength and stiffness of these 
 two slabs, which contain practically the same amount of concrete 
 and steel, is due to the difference of principle of their construction, 
 which may be summarized for the mushroom sjstem by consider- 
 ing its slab action and its cantilever action under the following 
 counts, viz: 
 
 1st. Circumferential slab stresses are most economically and 
 effectively provided for by the ring rods around the colunm heads. 
 
 2nd. The size of the mushroom heads is such as to make the 
 belts so wide as to provide reinforcement over the entire area of 
 the slab, thus securing slab action in the central part of the panel 
 where the belts lie near the lower surface. 
 
 3rd. The reinforcing belts cover a wide zone at the top of 
 the slab over the colunms and mushroom head, which thus provides 
 resistance to tension, and ensures effective cantilever and slab action. 
 
 4th. Concrete is thus stressed in compression at the bottom 
 of the slab for a wide zone around the columns. 
 
 5th. Under a load equal to the breaking load of the Xorcross 
 slab, amounting to thirty-eight tons, the mushroom slab deflected 
 at first only l/8", but after exposure to rain and great changes of 
 temperature for seven days had somewhat softened the concrete 
 the deflection increased to 1/4". 
 
 6th. The first crack appeared underneath the edge of the 
 slab across the side belt under load No. 5, of fifty-six tons, -^-ith a 
 center deflection of 0.4" and an average deflection at the middle 
 of side belts of 0.25". 
 
 7th. No cracks appeared on the upper side of slab at the 
 edge, nor were any seen elsewhere, until load No. 7, of 90 tons was 
 applied, when the yield point of the steel was e^-idently nearly or 
 quite reached, giving a center deflection of 1/2". 
 
92 
 
 FAILURE OF MUSHROOM SLAB 
 
 Fig. 18. Failure of Mushroom Slab. 
 
 Fig. 19. Failure of Mushroom Slab. Load Removed. 
 
FAILURE OF MUSHROOM TEST SLAB 93 
 
 8th. The slab carried its final load of over 120 tons for twenty- 
 four hours without giving way. It demonstrated the impossibility 
 of its sudden failure by gradually yielding until it reached a final 
 deflection of some nine inches, as seen in the views of Dec. 17th 
 and 24th, Figs. 18 and 19. 
 
 9th. While the slab steel in each belt was the same as in the 
 Norcross slab, the crossing of the belts increased the percentage 
 of slab reinforcement so much above that of the simple belt rein- 
 forcement that stress in the steel did not pass the yield point and 
 the failure was largely due to the giving way of the concrete around 
 the cap, but partly to some yielding at the line of weakest ultimate 
 resistance, both of which statements are confirmed by the view of 
 Dec. 24th, Fig. 19, where the removal of the loading permits the 
 irregular circular line previously mentioned to be made out at a 
 distance from the center of each column of somewhat less than L/2. 
 
 Less steel is required in this system than in the Nor- 
 cross slab for the same limiting stresses. Since the steel in this 
 slab did not pass the yield point any greater percentage of reinforce- 
 ment would be useless and would not increase the strength of the 
 slab. It has been found that good practice requires a percentage 
 of steel dependent in the following manner upon the thickness 
 of the slab: 
 
 If rf = L/35 the belt reinforcement = 0.2% 
 
 If d = L/24 the belt remforcement = 0.3% 
 
 If rf = L/20 the belt reinforcement = 0.4%, 
 
 Comparision of the steel in the test slabs: Norcross. Mushroom. 
 
 Size of slab 16' x 16' 18.4' x 17.8' 
 
 Area of slab 256 sq. ft. 328 sq. ft. 
 
 Length of 3/8" rods in the slab 1188 ft. 1450 ft. 
 
 Weight of 3/8" rods in the slab 446 lbs. 544 lbs. 
 
 Weight of Plates or Heads in the slab. . 268 lbs. 435 lbs. 
 
 Total weight of steel in the slab 714 lbs. 979 lbs. 
 
 Weight of steel per square foot of slab.. 2.8 lbs. 3 lbs. 
 
 Area of Panel 12 x 12 ft 144 sq. ft. 144 sq. ft. 
 
 Length of slab rods per panel 638 ft. 638 ft. 
 
 Weight of slab rods per panel 239 lbs. 239 lbs. 
 
 Weight in plates or heads per panel. . . 67 lbs. 109 lbs. 
 
 Total weight of steel per panel 306 lbs. 348 lbs. 
 
 Weight of steel per square foot of panel. 2 l/8 lbs. 2 5/l2 lbs. 
 
SUGGESTIONS REGARDING THE CONSTRUCTION 
 
 AND FINISH OF FLOOR SLABS 
 
 By C. A. P. TURNER 
 
 18. The Execution oe Work: Construction work of any kind 
 involves a great responsibility, not only on the part of the designer, 
 but also on the part of those in charge of the work, and that re- 
 sponsibility is for the safety of those erecting the work. 
 
 Perhaps the construction of no type of building is so free from 
 hazard and risk to the hves of those erecting it as reinforced con- 
 crete construction when scientifically designed and intelligently 
 executed. 
 
 During the last ten or twelve years, the manufacturers of Port- 
 land Cement, have through improvements in methods of manu- 
 facture and great reduction in cost, placed this material on the 
 market at such reasonable rates that it has given a remarkable 
 impetus to the construction of concrete work in all lines. Since, as a 
 material of construction, it has but recently come into general use, 
 it is not surprising that a large part of the engineering and archi- 
 tectural profession have not yet become so familiar with its char- 
 acteristics, but that designs lacking in conservatism from a scientific 
 standpoint have been frequently made, and this combined with 
 the execution of the work by unskilled contractors, has resulted in a 
 number of instances in needless sacrifice of life and large property 
 losses, such as a more thorough knowledge and study of the char- 
 acteristics of the material should entirely prevent. 
 
 It would be neglect of duty to fail even in this short discussion 
 to call attention pointedly to those properties and characteristics 
 of concrete which must be known and appreciated by the engineer 
 and constructor in order that he may avoid the serious disasters into 
 which those ignorant or forgetful of them have been too frequently 
 led. 
 
 The Hardening of Concrete: Concrete may be defined as. 
 an artificial conglomerate stone in which the coarse aggregate or 
 space-filler is held together by the cement matrix. The cement 
 should conform to the Standard Specifications for Cement, recom- 
 mended by the American Society for Testing Materials. 
 
HARDENING OF CONCRETE 95 
 
 The contractor and architect should, at least, see to it that the 
 cement is finely ground, and that it meets the requirements of the 
 boiling test. This last may be readily made by forming pats 
 of the cement of 3^ to 4 inches in diameter on a piece of glass, knead- 
 ing them thoroughly with just enough moisture to make them plastic, 
 so that they will hold their shape without flowing, and taper to a 
 thin edge. Store the pats under a moist cloth at a temperature of 
 sixty-five to seventy-five degrees Fahr. for a period of 24 hours. 
 Then place the pats in a kettle or pan of cold water, and after raising 
 the temperature of the water to the boiling point, continue boiling 
 for a period of four hours. If the pats do not then show cracks, 
 and if they harden without cracking or disintegrating, the con- 
 structor may be satisfied that the cement is suitable for use in the 
 work. Coarse grinding reduces the sand-carrying capacity of the 
 cement, and its consequent efficiency. 
 
 The fimction assigned to the concrete element in the combina- 
 tion of reinforced concrete is to resist compressive stresses in bend- 
 ing; but when first mixed the concrete is nothing more than mud, 
 and in order for it to become the hard, rigid material necessary to 
 fulfill its fimction in the finished work it must evidently pass in the 
 process of hardening thru all stages and varying degrees of hardness 
 from mud and partly cured cement to the final stage of hard, rigid 
 material. This curing or hardening being a chemical process, does 
 not occur in any fixed period of time, save and except the temper- 
 ature conditions are absolutely constant. Hence the time at which 
 forms may be safely removed is not to be reckoned by a given number 
 of days, but rather it must be determined by the degree of hardness 
 attained by the cement. In other words, during warm summer 
 weather, concrete may become reasonably well cured in twelve or 
 fifteen daj's. If the weather, however, is rainy and chilly, it may 
 not become cured in a month. In the cold, frostj' weather of the 
 spring and autumn, unless warm water is used in the mix, the con- 
 crete may require two or three months to become thoroughly cured, 
 while by heating the mixing water, whenever the temperature is 
 below 50 degrees Fahr., the concrete will harden approximately as 
 it does during the more favorable season. 
 
 Concrete which has been chilled by the use of ice cold water, 
 or that has become chilled within the first day or two of the time it 
 is cast, has this peculiarity, that it is very difficult indeed for the 
 most expert to determine when it is in such condition that it will 
 retain its shape after the removal of the forms. Once having been 
 chilled in the early stages, it goes through consecutive stages of 
 
POURING CONCRETE 
 
 sweating with temperature changes, and during these periods it 
 sometimes happens that the concrete diminishes in compressive 
 strength, and if the props are removed it sags and gets out of shape. 
 Such deformation will generally result in checks and fine cracks, 
 though there may not be any serious diminition of the ultimate 
 strength. These checks may be prevented as explained above by 
 the simple method of heating the mixing water whenever the tem- 
 perature has dropped below 50 degrees Fahr. In colder weather, 
 that is below the freezing point, not only must the water be heated, 
 but as a rule the sand and stone too, also a little salt may be ad- 
 vantageously used. The work must then be properly housed and 
 kept warm for at least three weeks subsequent to pouring. 
 
 Use of Salt in Cold Weathek : We have mentioned the use of 
 salt in cold weather. The action of salt is two-fold: It retards the 
 setting and thus enables us to use water heated to a higher temper- 
 ature than we could use without salt. It also lowers the freezing 
 point. Should the concrete then be frozen at the subsequent sweat- 
 ing period which occurs with a rise in temperature, the salt retains 
 the necessary moisture for crystallization because of its affiiiity for 
 moisture, thus preventing the softened concrete from drying out 
 and disintegrating through lack of moisture to enable it to crystalize 
 and harden properly. The amount of salt to be used is about a 
 cup to the sack of cement with the temperature from 18 to 20 de- 
 grees Fahr. If the temperature is below this, increase the amount 
 of salt, and when working below zero Fahr., use not less than two 
 cups of salt to the bag of cement. 
 
 PouRTNG Conceete: Bad work frequently results from im- 
 proper pouring, or casting of the work. In filling the forms, the 
 lowest portion of the forms should be filled first. A column should 
 be filled from the center and not from the side of the cap. Filling 
 from the center will insure a clean smooth face when the forms are 
 removed. Filling from the side will frequently give a bad surface 
 because the mortar will flow into the center of the column through 
 the hooping, leaving the coarse aggregate with voids imfilled at the 
 outside. As more concrete is then poured in, the voids between the 
 core and the out side portion will become filled, and the soft mor- 
 tar will not be able to flow back to completely fill the voids between 
 the hooping and the casing. Where the spacing of the hooping is 
 wide, this is not so important, but it becomes very important where 
 the spiral used has close spacing. It is better to cast the column and 
 mushroom frame complete, continuing to pour the concrete over 
 the center of the column so that it always flows from the column 
 
TEST FOR HARDNESS. LAP 97 
 
 into the Mushroom slab rather than the reverse. All splices must 
 be made in a vertical plane, in a beam preferably at the middle of 
 the span, and in a slab at a center line of a panel. 
 
 Test foe Hardness in Warm Weather: We have pointed 
 out that the criterion governing the safe removal of forms is the 
 hardness or rigidity of the concrete. A test of hardness in concrete 
 not frozen may be made by driving a common eight-penny nail 
 into it; the nail should double up before penetrating more than 
 half an inch. The concrete should further be hard enough to 
 break Hke stone in Imocking off a piece wdth the hammer. 
 Noting the indentation under a blow with the hammer, gives a 
 fair idea of its condition to those having experience. 
 
 Subcentering, as provided in the appended specification, is a 
 desirable method of preventing deformation, where the use of the 
 forms is desired for upper stories before the concrete is fully cured. 
 
 Test for Hardness in Cold Weather: Concrete freshly 
 mixed and frozen hard will not only sustain itself but carry a large 
 load in addition, imtil it thaws out and softens, when collapse in 
 whole or in part is inevitable. Partly cured concrete if frozen, 
 sweats and softens with a rise in temperature, hence in cold weather 
 there is danger of mistaking partlj^ cured concrete made rigid by 
 frost for thoroughly cured material. In fact the only test that can 
 be depended upon with certainty in cold, frosty weather, is to dig out 
 a piece of concrete, place a sample on a stove or hot radiator, and 
 note whether, as the frost is thawed out of it, it sweats and softens. 
 This gives the builder and engineer a perfectly conclusive test of the 
 condition of the concrete as to whether it is cured or merely stiffened 
 up by frost. 
 
 Lap of Reinforcement over Supports: Thoroughly tying 
 the work together by ample lap in the reinforcement is a prime 
 requisite for safety in any form or type of construction. This 
 general precaution insures toughness, and prevents instantaneous 
 collapse, should the workman exercise bad judgment in premature 
 removal of forms. 
 
 Responsibility of the Engineer: The steps which it is 
 possible for the engineer to take in securing safe construction are 
 Umited in the first place to the production of a conservative design, 
 and one which will present toughness, so that its failure imder over- 
 load or under premature removal of the forms will be slow and 
 gradual. This he can do, and this we beheve he is morally bound 
 to do. On the other hand, he cannot design reinforced concrete 
 
CRACKS IN CONCRETE 
 
 work which will hold its shape without permanent deformation, un- 
 less it is properly supported until the concrete has had time under 
 proper conditions to become thoroughly cured. 
 
 Concrete in setting shrinks, and sometimes cracks by reason of 
 this shrinkage, particularly when it hardens rapidly, as it does in 
 hot weather. This shrinkage sets up certain stresses in the concrete, 
 which, combined with temperature changes, occasionally manifest 
 themselves by subsequent cracks in the work. Such checks or 
 cracks do not of necessity indicate weakness, providing the concrete 
 is hard and rigid, since the steel is intended to take the tensile 
 stresses and the concrete the compressive. Such checks sometimes 
 cause an unwarranted lack of confidence in the safety and stability 
 of the work arising from the common lack of famiharity with the 
 characteristics of the material. For example, the owner of a frame 
 building would never imagine it to be unsafe because he found a few 
 season checks in the timber. He is sufficiently familiar with the 
 seasoning of timber to understand how these checks occur, and that 
 in most instances they do not mean a loss of strength, since, as the 
 timber hardens by thoroughly drying out, it becomes stronger, as a 
 rule, to an amount in excess of any slight weakness which might be 
 developed by ordinary season cracks or checks. So in concrete, 
 when the general public becomes more familiar with its character- 
 istics they will regard as far less important than they now do, checks 
 which are produced by temperature and shrinkage stresses, or 
 possibly by slight unequal settlement of supports. 
 
 Pbopee and Improper Methods op Floor Finish: In con- 
 crete work there are a number of small defects which occur through 
 failure to properly manipulate the material, for which the designer 
 of the engineering part of the work is frequently censured improperly. 
 For example, cases have occurred where a good splice was not 
 secured owing to the fact that in very hot weather the stone aggre- 
 gate became heated in the sun and was not properly cooled down 
 before mixing the concrete, and so the water dried out too quickly, 
 while the heat in the stone caused the cement to set so rapidly that 
 a good splice to the previous work could not be made. 
 
 The worst trouble, however, which has been observed, is that 
 resulting from poor surface finish of floors. Improper methods in 
 common practice are of two different kinds. One is the attempt 
 to finish the work approximately at the time it is cast, making the 
 surface finish integral with the slab. The difficulty with this method 
 of finishing lies in the fact that as soon as the columns are cast in 
 the story above, unequal moisture conditions are produced around 
 
FLOOR FINISH 99 
 
 the foot of the column owing to the excess of moisture in the column; 
 thus the concrete in the surface of the slab around and near the foot 
 of the column is expanded by the excess moisture, and it ultimately 
 shrinks, and leaves a series of spider web cracks as it dries out. 
 This will occur to a greater or less extent depending on the humidity 
 of the surrounding atmosphere during the curing or drying out of the 
 floor. If the weather is dry these checks will be verj^ pronounced 
 indeed, though they will not be very deep. If it is rainy and damp, 
 and the floor is kept soaked all the time, they may be nearly or quite 
 lacking. 
 
 Another objection to this method of finish is that unusual pre- 
 cautions must be taken to protect the floor before the centering can 
 be placed for a story above, and regardless of the method used to 
 protect it the floor usually becames scarred and deeply scratched before 
 the work is complete, leaving a surface difficult to satisfactorily repair. 
 
 Another method which leads to bad results is the following: 
 The rough slab is cast, and the centering removed in due time, the 
 slab cleaned and the finish coat applied in a sloppy or plastic form, 
 flowed in place, screeded to approximate surface, and then allowed 
 to partly set, so that the finishers can get on the floor and trowel it 
 down. A floor finished in this manner looks well when the work is 
 new. It does not wear well but dusts badly, pits and rapidly grows 
 rough and ragged under trucking. 
 
 The correct method of applying floor finish is as follows: 
 The finish coat should be not lessthanl and l/4 inches to 1 and l/2 
 inches in thickness. It should be applied after the rough slab has been 
 fairly well cured. The surface of the rough slab should be thoroughly 
 cleaned of dirt and laitance and thoroughly soaked with water. 
 Then the floor finish, a mixture preferably of one part of cement to 
 one and one-half sand (the sand a silicious sand with grains from 
 1/8 inch down, if such can be secured), should be thoroughly mixed 
 with just enough water to make an extremely stiff paste, one which 
 will hold its form if squeezed in the hand, but one which will not run 
 or flow, and will need a fair amount of tamping to bring the moisture 
 to the surface. This concrete, so mixed, should be applied to the 
 rough slab in blocks of from four to five feet square, first grouting 
 the rough slab with a neat cement grout, then tamp until the moist- 
 ure is brought to the surface, level up and trowel immediately. 
 The cement finish should not be mixed more rapidly than it can be 
 applied, so that the cement will not be killed by taking a partial set 
 before troweling, which is what occurs where the finish is applied 
 sloppy, and the workmen wait for it to partly harden before they can 
 
100 STRIP FILL FLOORS 
 
 get on it to trowel. A finish applied as just stated will stand severe 
 usage and last for several years without showing appreciable evi- 
 dence of pitting, dusting, or undue wear. 
 
 The addition of ground iron ore, to the amount of twenty 
 pounds to the barrel of cement, appears to improve the finish and 
 give it a more pleasing color. 
 
 Checks in cement finish have no relation whatever, as a rule, 
 to the strength of the work. They will invariably occur in the 
 cement finish where the finish coat is too thin. When it is less than 
 1 inch or 3/4 inch at one part of the floor with 1 l/4 inches or 1 l/2 
 inches at another, the surface will invariably check and crack badly if 
 applied at a sloppy consistency and allowed to partly cure before it 
 is polished down. We know of no type of construction where there 
 has not been much trouble with finished surfaces in such buildings 
 as have come under our observation. But experience has shown us 
 that these troubles are needless, and can be avoided by the proper 
 handling and application of the finishing coat. 
 
 It is difficult indeed to re-educate those who profess to be 
 cement finishers, whose experience has been largely in sidewalk 
 finish, or work of that character, to appreciate the necessity for a 
 different method of executing work in a building; but when this has 
 been accomplished the owner will have the use of a floor finish free 
 from the unpleasant defects above pointed out. 
 
 Strips and Strip Fill for Wood Floors: The proper time 
 for the application of the strips and fill is immediately after the 
 rough slab has become sufficiently hardened to work upon it, for the 
 reason that at this time the strips may be spiked to the partially 
 hardened concrete and wedged up or lined up to the desired level 
 ■without difficulty. Then the strip fill can be put in with the same 
 rig that is used to cast the floor slab. 
 
 The writer prefers the strip fill of the same mixture as the slab 
 except where the loads are so light that increased strength and 
 stiffness are of no importance. Then a one to three and one-half, 
 four, or even five, mix will answer the purpose. No natural cement or 
 lime should be used in the mixture, since when it is used, trouble 
 almost invariably follows, caused by its extremely slow hardening 
 and its retention of moisture until hardening takes place. This 
 moisture frequently swells and expands the flooring to such an 
 extent that it springs away from the fastenings, thereby necessi- 
 tating the entire relaying of the floors. Conservative practice ac- 
 cordingly is to use Portland Cement alone, which will dry out far 
 quicker than any natural cement or brown lime. 
 
APPENDIX 
 
 STANDARD SPECIFICATION FOR REINFORCED 
 CONCRETE FLOORS 
 
 By C. A. P. TURNER, Consulting Engineer 
 Minneapolis, Minn. 
 
 Reinforcement. Reinforcement shall be of sizes of bars shown on the accom- 
 panying plans and details which form a part of this specification. 
 
 All reinforcing metal shall be of medium open hearth or Bessemer steel, 
 meeting the requirements of the Manufacturers' Standard Specifications, in 
 composition, ultimate strength, ductility and elastic limit, and the required 
 bending basis. Hard grade may be used for slab rods only. 
 
 Bending. Bending shall preferably be done cold. If the column rods are 
 heated and blacksmith work is done, care must be exercised that the steel is not 
 burned in the operation, otherwise it will be condemned by the engineer. 
 
 Cement. Cement shall be of good quality of Portland Cement, of a brand 
 which has been upon the market and successfully used for at least four years, 
 meeting the requirements of the specification adopted by the American Society 
 for Testing Materials. 
 
 The contractor shall give the owner the opportunity to test all cement de- 
 livered, and shaU furnish the use of testing machine for this purpose. 
 
 The cement shall be delivered in good condition and properly protected 
 under suitable cover after delivery on the premises so that it may not be damaged 
 by moisture. 
 
 Sand. Sand used in the concrete work shall be clean and coarse, meeting 
 the requirements and approval of the engineer and architect. 
 
 Stone. Stone used shall be sound, hard stone, free from lumps of clay and 
 other soft unsatisfactory material, or hard smelter slag may be used. In size it 
 shall be crushed to pass a 1-inch ring, for slabs and columns, and shall be screened 
 free from dirt and dost. 
 
 Concrete. All concrete shall be mixed in a standard batch machine to the 
 consistency of brick mortar, so that it will flow slowly and require only puddling 
 around the reiaforcement. 
 
 Concrete shall be thoroughly mLxed in the following proportions: one part 
 cement, meeting the requirements of the standard specifications; two parts 
 clean, coarse sand free from clay, loam or other impurities; and four parts crushed 
 stone or clean gravel. 
 
 The concrete shall be poured in the low portions of the forms first. That is, 
 it shall be poured directly into the column boxes, beam boxes, etc., before it is 
 
102 STANDARD SPECIFICATIONS 
 
 poured on the slab. It shall be so placed that it will be forced to flow as little 
 as possible to get to the required position, since by flowing, the cement is readily 
 separated from the mixture. 
 
 Splices. Splices in beams or slabs are to be made in a vertical plane, prefer- 
 ably in the center of the panel or beam. 
 
 Proportions. Each sack of cement shall be considered equivalent to one 
 cubic foot in volume, and the mixture of the cement, sand and stone used in the 
 concrete shall be proportioned by volume on this basis and as hereinafter specified. 
 
 Concrete for footings, columns, beams and rough slabs throughout shall 
 consist of a mixture of one cement, two sand and four of crushed stone. 
 
 For the retaining walls, the concrete mixture shall be one cement, three sand 
 and five parts of stone. 
 
 Concrete in which the cement has attained its initial set shall not be used on 
 the work. Concrete, however, which has slopped out of the mixer, if cleaned 
 up within a short time, not over every half hour, may be put back in the mixer, 
 and after being thoroughly mixed again with water may be used on the work. 
 
 Forms. All forms for the reinforced concrete shall be substantially made 
 and true to line. Any irregularities due to defective workmanship in this re- 
 spect, shall be made good as directed by the architect, by dressing down the 
 finished work, or removal and properly replacing it in case that it cannot be 
 satisfactorOy done. 
 
 A fair quality of lumber, preferably 1x6 square edge fencing shall be used 
 for the slab forms. This lumber shall be dressed on the side next to the concrete 
 except where plaster is specified by the architect for office finish, in which case 
 the rough side of the boarding shall be placed upwards, next to the concrete. 
 
 Column Forms. Column forms shall be made up with plank not less than 
 Ij inches thick and stayed at intervals not more than 18 inches vertically be- 
 tween bands or straps and shall fit closely at the comer joints, or the forms may 
 be made of sheet metal. 
 
 Removal of the Forms. Forms shall not be removed imder the most 
 favorable conditions, prior to two weeks' time, and under less favorable con- 
 ditions where the temperature is lower than 50 " until the concrete is hard and 
 rigid. 
 
 The superintendent will keep in mind the fact that it is not the number of 
 days time which has elapsed since placing the concrete which shall determine the 
 earliest removal of the forms, but rather how rapidly the concrete has thoroughly 
 cured and hardened and that the concrete may be readily stiffened up by cold 
 and frost which, when it thaws, will sweat and fail to maintain 'the desired form. 
 
 Sub-Centering. Where a series of floors are cast one above the other, sub- 
 centering of substantial posts about 10 feet centers shall be kept in place until 
 there are at least two supportmg slabs that are well cured and hard so that the 
 concrete may not be overstained in the early stages of hardening. 
 
 Placing and Inspection of Reinforcement. Before Commencing the 
 Concrete Work, the reinforcement shall be properly placed and inspected by 
 the architect or the engineer representing the owner, and not until after this 
 inspection and approval may the work of casting the floor proceed. 
 
 The floor slab rods shall be wired together to hold them in the position as 
 shown on the plans. Special attention being given to placing the rods in belts 
 of the width of the mushroom frame and fairly uniform spacing, although this is 
 
STANDARD SPECIFICATIONS 103 
 
 of less importance than keeping to the general distribution through the full width 
 of the belts of reinforcement. 
 
 In placing the floor slab rods, all those running from column to column 
 directly on one side of a panel shall be placed first, then those running at right 
 angles, next all those in one diagonal belt, and then those in the other diagonal. 
 
 Where a belt of slab rods runs parallel to a wall place one rod at bottom on 
 forms. Then see that belts normal and diagonally are placed, following 
 up with slab rods parallel to the wall on the top of normal and diagonal belts. 
 
 In wiring the rods together it is desirable to use No. 16 soft annealed wire, 
 taking a piece, say a yard long, fastening an intersection, then carry the wire 
 diagonally to the next interaection, taking a half hitch and proceed until this 
 piece is used up and making the end fast. Then start with a new piece and 
 proceed as before. 
 
 Two lines of ties, crossing and norma] to the intersecting belts at the center 
 will hold these rods in position very nicely. 
 
 A similar tie across the parallel belts, and a suitable number of fastenings 
 around the mushroom head are required to hold the bars in position. 
 
 Floor Finish. The finish coat on the rough slab shall not be less than 1 inch- 
 thick, and the rough slab shall be prepared for its reception as follows : 
 
 The slab shall be thoroughly scrubbed with a steel brush and water, and then 
 after it has been thoroughly cleaned from dirt and laitance it shall be kept wet 
 for at least six hours. The surface shall then be coated with neat cement grout 
 and the finish coat applied. 
 
 The finish coat shall consist of a mixture of one cement to one and one-half 
 clean, coarse sand. The finish coat shall be mixed with just enough water to 
 make a very stiff paste and not enough to make it soft and sloppy. It shall be 
 tamped in place and troweled to a smooth finish . 
 
 Mixing the material wet and sloppy renders it necessary to wait until the 
 material hardens somewhat before it is possible to polish it down. In allowing 
 it to partlj' harden the finisher is then obliged to break up the surface of partly 
 hardened cement which results in a finished surface that will dust badly, pit 
 readUj- and wear rough under subsequent use, so that this method should not 
 be employed. 
 
 This finish coat is to be blocked off in squares along the center fine of 
 columns, and joints shall be made in this coat between panel joints at five to 
 six foot intervals. 
 
 Conduits. Before casting the concrete, the concrete contractor shall see 
 that the electric contractor has placed the necessary conduits for the wires. These 
 shall be kept above the reinforcement wherever they come in the center of a 
 panel, the idea beiag to have these conduit pipes above the steel and dip down 
 into the socket at the junction, or to use a special deep socket which would be 
 prefered by the engineer. 
 
 These conduit pipes should be carried below the level of the reinforce- 
 ment around the mushroom heads where the reinforcement is of necessity near 
 the top of the slab. 
 
 Depositing Concrete in Warm Weather. When the concrete is deposited 
 in temperatures above 70 ° Fahr., the slab shall be thoroughly wet down twice 
 a day for two days after it has been cast. Any preUminary shrinkage cracks 
 which occur on the surface of the slab due to too rapid drying shall be 
 immediately filled with liquid cement grout. 
 
104 STANDARD SPECIFICATIONS 
 
 Any concrete work indicating tiiat it has not been thoroughly mixed in the 
 required proportions shall be dug out and replaced as directed by the engineer 
 and architect. 
 
 Placing Concrete in Cold Weather. Where the temperature is below 45 ° 
 Fahr., the water shall be heated to a temperature of at least 110 ". Where the 
 temperature is below 30 " Fahr., artifical heat shall be used to assist in curing 
 the concrete, and this must be continued until such a time as the slab is thorough- 
 ly cured and dry throughout. 
 
 Pouring Concrete. In the mushroom system concrete shall be poured 
 over the center of the column until the column is filled. Then the pouring 
 shall be continued imtil the mushroom and mushroom frame is filled up so that 
 the concrete will flow from the column toward the center of the slab and not 
 from the center of the slab toward the column. In this way solid concrete 
 without joints and planes of imperfect bond will be secured aroimd and in the 
 vicinity of column heads, where it is most needed. 
 
 Test. No test shall be made until the concrete is thoroughly cured, is dry, 
 hard and rigid throughout. Ninety days of good drying weather at a tempera- 
 ture above 60 ° Fahr., either natural or artificial, shall be the criterion as to when 
 the test of double the working capacity can be reasonably made. 
 
 General. It is the general intent of this specification to require first class 
 work in all particulars, and work unsatisfactory to the engineer and architect 
 representing the owners shall be made good by the contractor as they direct. 
 
 PRINTED BY 
 HEYWOOD 
 
 MINNEAPOLIS 
 
LIST OF ONE HUNDRED BUILDINGS SELECTED FROM 
 MORE THAN A THOUSAND DESIGNED ON THE 
 
 MUSHROOM SYSTEM 
 
 1906 Johnson-Bovey Go's. Bldg Minneapolis, Minn. 
 
 1906 Hoffman Building Milwaukee, Wis. 
 
 1907 Bostwiek Braun Bldg Toledo, Ohio 
 
 1907 Lindeke Warner Bldg St. Paul, Minn. 
 
 1907 Hamm Brewery Bldg " " " 
 
 1907 Smythe Building Wichita, Kans. 
 
 1907 Forman Ford Bldg Minneapolis, Minn. 
 
 1907 Grellet Collins Bldg Philadelphia, Pa. 
 
 1907 Parsons Scoville Bldg Evansville, Ind. 
 
 1907 Bom BuUding Chicago, 111. 
 
 1908 South Dakota State Capitol Pierre, S. D. 
 
 1908 Merchants Ice & Cold Storage Bldg Cincinnati, Ohio 
 
 1908 St. Mary's Hospital Kansas City, Mo. 
 
 1908 John Deere Plow Co Omaha, Nebr. 
 
 1908 Minn. State Prison Bldgs ... (6) Stillwater, Minn. 
 
 1908 Ripley Apartments Tacoma, Wash. 
 
 1908 Velie Motor Bldg Moline,' 111. 
 
 1908 Park Grant Morris Bldg Fargo, N. D. 
 
 1909 Con P. Curran Bldg St. Louis, Mo. 
 
 1909 Manchester Biscuit Go's. Bldg Fargo, N. D. 
 
 1909 Blue Line Transfer & Storage Bldg Des Moines, la. 
 
 1909 Cutler Hardware Bldg Waterloo, la. 
 
 1909 Mass. Cotton Mills Boston, Mass. 
 
 1909 McMillan Packing Go St. Paul, Minn. 
 
 1909 Vancouver Ice and Gold Storage Go Vancouver, Bldg. 
 
 1909 Omaha Fireproof Storage Bldg Omaha, Nebr. 
 
 1909 J. I. Case Bldg Oklahoma City, Okla. 
 
 1909 Tibbs Hutchings & Co Minneapolis, Minn. 
 
 1909 Snead Mfg. Bldg Louisville, Ky. 
 
 1909 New England Sanitary Bakery Bldg Decatur, 111. 
 
 1909 International Harvester Bldg Milwaukee, Wis. 
 
 1909 Congress Candy Co Grand Forks, N. D. 
 
 1910 Y. M. C. A. Bldg Winnipeg, Man. 
 
 1910 West Publishing Go's. Bldg St. Paul, Minn. 
 
 1910 Beatrice Creamery Bldg Lincoln, Nebr. 
 
 1910 Iten Biscuit Co Omaha, Nebr. 
 
 1910 Turner Moving & Storage Bldg Denver, Colo. 
 
 1910 Congress Realty Go's. Bldg Portland, Me. 
 
 1910 Sniders & Abrahams Bldg Melbourne, Australia 
 
 1910 Strong & Warner Bldg St. Paul, Minn. 
 
 1910 Lexington High School Bldg St. Paul, Minn. 
 
 1910 Weicker Transfer & Storage Bldg Denver, Colo. 
 
 1910 ChehalKs County Court House Montesano, Wash. 
 
 1910 Missouri Glass Go's. Bldg St. Louis, Mo. 
 
 1910 Industrial Bldg Newark, N. J. 
 
 1910 Revel & Wagner Bldg Little Rock, Ark. 
 
 1910 Jobst Bethard Bldg Peoria, 111. 
 
 1910 International Harvester (Keystone Works) . . . Sterling, 111. 
 
1910 Patterson Hotel Bismarck, N. D. 
 
 1910 O'Neil Bldg Akron, Ohio 
 
 1911 Lindsay Bldg Winnipeg, Man. 
 
 1911 King George Hotel Saskatoon, Sask. 
 
 1911 Northern Cold Storage Bldg Duluth, Minn. 
 
 1911 Leighton Supply Co Fort Dodge, la. 
 
 1911 Kinsey Bldg Toledo, Ohio 
 
 1911 Lozier Motor Bldg Detroit, Mich. 
 
 1911 MuUin Warehouse Bldg Cedar Rapids, la. 
 
 1911 Griggs Cooper & Co St. Paul, Minn. 
 
 1911 Swift Canadian Co's. Bldgs Vancouver, B. C. 
 
 1911 MoKenzie Bldg Brandon, Man. 
 
 1911 Swift Canadian Co's. Bldg Fort WiUiam, Ont. 
 
 1911 Commerce Bldg St. Paul, Minn. 
 
 1911 Experimental Eng. Bldg. Univ. of Minn Minneapolis, Minn. 
 
 1911 St. Paul Bread Co's. Bldg St. Paul, Minn. 
 
 1911 Rust Parker Martin Bldg Duluth, Minn. 
 
 1912 Woodward Wight Co. Ltd. Bldg New Orleans, La. 
 
 1912 Internationa] Harvester Co's. Bldg Fort William, Ont. 
 
 1912 H. W. Johns-ManviUe Bldgs. ... (3) Findeme, N. J. 
 
 1912 Cooledge Bldg Atlanta, Ga. 
 
 1912 Lawrence Leather Co's. Bldg Lawrence, Mass. 
 
 1912 Sears, Roebuck & Co Dallas, Texas 
 
 1912 Vmeburg Bldg Montreal, Quebec 
 
 1912 Imperial Tobacco Co Montreal, Quebec 
 
 1912 Richards Pinhom Bldg Denver, Col. 
 
 1912 Kinney & Levan Co. Bldg Cleveland, Ohio 
 
 1912 Standard Oil Co. Bldgs ... (2) Cleveland, Ohio 
 
 1912 Silver Sunshine Bldgs ... (2) Cleveland, Ohio 
 
 1912 Commercial Improvement Co's. Bldg Columbus, O. 
 
 1912 Moore Department Store Bldg Memphis, Tenn. 
 
 1912 Main Eng. Bldg. Univ. of Minn Minneapohs, Minn. 
 
 1912 Honeyman Hardware Bldg Portland, Ore. 
 
 1912 RevUlon Wholesale Hardware Bldg Edmonton, Alta. 
 
 1912 Calgary Furniture Co's. Bldg Calgary, Alta. 
 
 1912 Willoughby Sumner Bldg Saskatoon, Sask. 
 
 1912 U. S. Post Office Minneapolis, Minn. 
 
 1912 Motor Mart Bldg Sioux City, la. 
 
 1912 Fmch Van Slyke & McConville Bldg St. Paul, Minn. 
 
 1912 Hudson Bay Co's. Warehouse Winnipeg, Man. 
 
 1912 Snell Bldg Moose Jaw, Sask. 
 
 1913 Y. M. C. A. Bldg Vancouver, B. C. 
 
 1913 Reynolds Tobacco Factory Bldg Winston Salem, N. C. 
 
 1913 Ford Motor Bldg Memphis, Tenn. 
 
 1913 Ford Motor Bldg Los Angeles, Cal. 
 
 1913 G. Sommers & Co. Bldg St. Paul, Minn. 
 
 1913 Knickerbocker Bldg Los Angeles, Cal. 
 
 1913 Trinity Auditorium Bldg Los Angeles, Cal. 
 
 1913 U. S. Alumium Co's. Bldg Pittsburg, Pa. 
 
 1913 Gordon Fergusen Co's. Bldg St. Paul, Mum. 
 
 1913 S. H. Kress & Co's. Bldg Houston, Tex. 
 
 'Los Muchachos" Bldg San Juan, Porto Rico 
 
 1913 
 
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