A = iber 32 Price $1.00 
 
 EPRINT AND CIRCULAR SERIES 
 
 OF THE 
 
 NATIONAL RESEARCH 
 COUNCIL 
 
 MOMENTS AND STRESSES IN SLABS 
 
 By H. M. Westergaard 
 
 Assistant Professor of Theoretical and Applied Mechanics 
 University of Illinois 
 
 and 
 
 W. A. Slater 
 Engineer Physicist, U. S. Bureau of Standards 
 
 
 
 
 V 
 
 Reprinted from Proceedings of the American Concrete Institute, vol. 17, 1921 
 By permission of the American Concrete Institute
 
 MOMENTS AND STRESSES IN SLABS. 
 BY H. M. WESTERGAARD * AND W. A. Si- 
 
 I. INTRODUCTION. 
 
 1. The subject of the strength of flat slabs has received considerable 
 attention during the past ten years. In November, 1910, the floor of the 
 Deere and Webber Building J at Minneapolis was tested. This was the 
 first field test of a reinforced-concrete building floor in which strain meas- 
 urements in the reinforcement and in the concrete were taken at various 
 places in the building. Since that time many other tests have been made 
 and much study has been given to the analytical side of the problem. 
 
 While considerable work has been done on the correlation of the 
 analytical and the experimental results, it does not seem that the possibili- 
 ties of useful work in this direction have been exhausted. It is the purpose 
 of this paper to present information which correlates the results of tests of 
 a fairly large number of slab structures with the results of analysis, so that 
 the report may aid in the formulation of building regulations for slabs. 
 
 The field of this report may be divided into three parts: (a) analysis 
 of moments and stresses in slabs, (b) study of the relation between the 
 observed and the computed steel stresses in reinforced-concrete beams, made 
 for the purpose of assisting in the interpretation of slab tests, (c) a study 
 of the test results for flat slabs with a view of comparing x the moments of 
 the observed steel stresses with the bending moments indicated by the 
 analysis, and of estimating the factor of safety. 
 
 The mathematical analysis is the work of Mr. Westergaard. The 
 analysis of the beam tests to show the relation between the computed and 
 the observed stresses is the work of Mr. Slater. 
 
 2. ACKNOWLEDGMENT. The expense of the report has been borne jointly 
 by the American Concrete Institute and the United States Bureau of 
 Standards. 
 
 The Corrugated Bar Co., of Buffalo, N. Y., and A. R. Lord, of the 
 Lord Engineering Company, of Chicago, have furnished the results of a 
 number of tests which had not been published, or which had been published 
 only in part. 
 
 Acknowledgment is made to-M. C. Niehols, graduate student, and to J. 
 P. Lawlor and K. H. Siecke, seniors in engineering, in the University of 
 Illinois, for their assistance in working up the data of the tests. 
 
 * Assistant Professor of Theoretical and Applied Mechanics, University of Illinois, 
 t Engineer Physicist, U. S. Bureau of Standards. 
 
 t A. R. Lord, Test of a Flat Slab Floor in a Reinforced-concrete Building, 
 National Association of Cement Users, v. 6, 1910.
 
 II. ANALYSIS OF HOMOGENEOUS ELASTIC PLATES. 
 
 BY H. M. WESTERGAARD. 
 
 3. SCOPE OF THE ANALYSIS. A slab is sometimes analyzed by considering 
 it as divided into strips, each carrying a certain portion of the total load. 
 One may expect to obtain, by this method, an exact analysis of a structure 
 consisting of strips which cross one another and carry the loads as assumed. 
 This structure, however, is quite different from the slab. The degree of 
 approximation obtained may be judged by the resemblance or lack of 
 resemblance between the strip-structure and the actual slab. As the resem- 
 blance is not very close, the approximation, naturally, is not very satisfac- 
 tory. The ordinary theory of beams, too, is approximate, not exact, when 
 applied to actual beams. Assumptions are introduced in the beam theory: 
 for example, the plane cross-sections remain plane after the bending, and 
 the material is perfectly elastic. But the approximation in the beam theory 
 is much closer than in the strip analysis of plates. The explanation is 
 simple: the beam to which the beam theory applies exactly has a closer 
 resemblance to actual beams than the strip-structure has to slabs. It is 
 possible, however, to analyze slabs more exactly than can be done by the 
 strip method. If an analysis of slabs is to compare in exactness with beam 
 analysis, then it must be based on a structure which resembles actual slabs 
 more closely than does the strip structure. It is hardly possible at present 
 to cover by analysis the whole range of designs of reinforced-concrete slabs. 
 It is expedient, therefore, to confine this investigation to a single type. The 
 homogeneous slab of perfectly elastic material is selected ; homogeneous 
 slabs have a fairly close resemblance to other slabs, and exact methods 
 exist by which they may be analyzed. The selection of a homogeneous 
 elastic material agrees witli common practice in the investigation of 
 statically indeterminate structures. For example, the distribution of bend- 
 ing moments in a reinforced-concrete arch or frame is often determined by 
 replacing the structure by one of homogeneous material. The plan is then 
 to investigate distributions of moments in homogeneous slabs. These distri- 
 butions mav be used as a basis for the study of the experimental data. 
 
 Theoretical analysis is under the disadvantage that its processes are 
 often rather more remote from the actual phenomena which are studied, 
 than are the processes of direct physical tests. For this reason alone it 
 would be out of the question to rely on the results of theoretical analysis 
 only. There are, on the other hand, advantages of theoretical analysis 
 which fully warrant its extensive use in conjunction with physical tests. 
 One may appreciate these advantages by looking upon the theoretical analy- 
 sis as being, in a sense, a test in which the testing apparatus consists of 
 the principles of statics and geometry, expressed in equations, and in which 
 the structure tested is the structure assumed in the analysis. The equations 
 may be solved with any desired exactness, and the structure has dimensions
 
 MOMENTS AND STRESSES IN SLABS. 3 
 
 and properties exactly as assumed, and is not subject to the incidental 
 variations which so often have made it impossible to draw definite conclu- 
 sions from physical tests. Besides, by the analysis one may cover whole 
 ranges of variations of the structure or at least a great number of indi- 
 vidual structures, while a physical test can deal with only a limited number 
 of cases. For these reasons the analysis is of particular value as a basis of 
 comparison and as a method of establishing continuity between results of 
 separate, individual tests. 
 
 It will be seen from the historical summary which follows that the 
 problem of flexure of plates is one on which scientists have been at work 
 for more than a century. The methods, which have been developed by such 
 men as Navier, St. Venant, Kirchhoff, and Lord Kelvin, have found their 
 way into engineering literature in Europe. It has been possible, therefore, 
 to build the present report, in part, on the work of previous investigators. 
 The agreement between the results of different analyses of rectangular 
 slabs supported oh four sides, serves as evidence that the methods are 
 dependable. 
 
 It has been thought desirable to follow the method of presenting the 
 results first, and details of the processes afterwards. A historical sum- 
 mary, a statement of the limitations of the theory, and a derivation of the 
 fundamental equations are followed by the report of results. The results 
 deal with rectangular slabs supported on four edges, and with flat slabs 
 supported on round column capitals. Details of the analysis will be given 
 in the appendix A. 
 
 4. HISTOBICAL SUMMARY. The incentive to the earliest studies of the 
 flexure of plates appears to have been an interest in their vibrations, in 
 particular those producing sound, rather than an interest in the stresses 
 and strength. Euler, after having developed his theory of the flexure of 
 beams, attempted to explain the tone-producing vibrations of bells by 
 assuming a division into narrow strips (or rings), each of which would 
 act as a beam, 1 but this application of the strip method was not satisfac- 
 tory. Jacques Bernouilli (the younger), in a paper presented in 1788, 2 
 treated a square plate as if it consisted of two systems of crossing beams or 
 strips, and he attempted in this way to explain the results of Chladni's 
 experiments with vibrating plates, 3 in particular the so-called nodal figures. 
 As might be expected, the results of this theory did not agree very well 
 with the experimental data. In 1809 the French Institut, at the instiga- 
 tion of Napoleon, proposed as a prize subject a theoretical analysis of the 
 tones of a vibrating plate. Mile. Sophie Germain 4 made some unsuccessful 
 attempts to win this prize, but won it in 1815, when she arrived finally at a 
 
 1 Euler, De sono campanarum, Novi Commentarii Academiae Petropolitanae, v. 10, 
 1766. 
 
 2 Jacques Bernouilli, Esaai theoretique sur les vibrations des plaques elastiques 
 rectangulaires et libres, Nova Acta Academiae Scientarum Petropolitanae, v. 5, 
 1787 (printed 1789). 
 
 * E. F. F. Chladni, Entdeckungen iiber die Theorie des Klanges, Leipzig, 1787. 
 4 See Todhunter and Pearson, A History of the Theory of Elasticity, Cambridge, 
 1886, p. 147.
 
 4 MOMENTS AND STRESSES IN SLABS. 
 
 fairly satisfactory, though not faultless derivation of a fundamental equa- 
 tion for the flexural vibrations. But in the meantime, in 1811, Lagrange, 
 who was a member of the committee to pass on the papers, had indicated 
 in a letter this equation, which is known, therefore, as Lagrange's equation 
 for the flexure and the vibration of plates (with the term depending on the 
 motion omitted, it is the same as (11) in Art. 6). 
 
 In 1820, Navier, 5 in a paper presented before the French Academy, 
 solved Lagrange's equation for the case of a rectangular plate with simply 
 supported edges. By this solution one may compute the deflections and, 
 therefore, also the curvatures and the stresses at any point of a plate of 
 this kind, under any distributed uniform or non-uniform load. Navier's 
 solution could be applied only to plates of this particular shape and with 
 this type of support. Furthermore, a really acceptable derivation of 
 Lagrange's equation, a derivation based on the stresses and deformations 
 at all points of the plate, had not been found so far. Poisson, 6 in his famous 
 paper on elasticity, published in 1829, obtained such a proof. With it, he 
 derived a set of general boundary conditions (conditions of equilibrium 
 and of deformation at the edge of the plate), and was then able to obtain 
 solutions for circular plates, both for vibrations and for static flexure under 
 a load which is symmetrical with respect to the center. Poisson's theoret- 
 ical results were compared with results of tests, namely, with the experi- 
 mental values, found by Savart for the radii of the nodal circles of three 
 vibrating circular plates. A close agreement was found. 
 
 In a paper, published 1850, Kirchhoff 7 derived Lagrange's equation and 
 the corresponding boundary conditions by using the energy principle, or the 
 principle of least action. He found one boundary condition less than 
 Poisson, namely, four at each point instead of Poisson's five. This differ- 
 ence gave rise to some discussion, but finally, in 1867, Kelvin and Tait ' 
 showed that there was only an apparent discrepancy, due to an inter- 
 relation between two of Poisson's conditions. This conclusion, as well as 
 Kirchhoff's and Poisson's theories as a whole, applies, as might be expected, 
 with limitations which are analogous to the limitations of the ordinary 
 theory of beams. For example, the plate-theory ceases to apply when the 
 span becomes small compared with the thickness of the plate, but, of course, 
 in that case the structure has really ceased to be a plate in the ordinary 
 sense. The question of the exact nature of the limitations called for 
 further researches. Such were made by Bouissinesq. 9 His investigations 
 have established the applicability of Poisson's and Kirchhoff's theories to 
 
 * See Saint-Venants annotated edition of Clebsch's Theory of Elasticity, Paris, 
 1883. Note by Saint-Vcnant, pp. 740-752. 
 
 * S. D. Poisson, Memoire sur 1'equilibre et le mouvement des corps elastique, 
 Memoirs of the Paris Academy, v. 8, 1829, pp. 357-570. See Todhunter and Pearson, 
 History of the Theory of Elasticity, 1886, pp. 241, 272. 
 
 * G. Kirchhoff, Ueber das Gleichgewicht und die Bewegung einer elastischen 
 Scheibe, Crelles Journal, 1850, v. 40, pp. 51-88. 
 
 1 Kelvin and Tait, Natural Philosophy, ed. 1, 1867. See A. E. H. Love, Mathe- 
 matical Theory of Elasticity, ed. 1906. p. 438. 
 
 6 J. Boussinesq, fitude nouvelle sur 1'equilibre et le mouvement des corps solides 
 elastiq'ues dont certaines dimensions, sont tres-petites par rapport a d'autres, Journal 
 de Mathematiques, 1871, pp. 125-274. and 1879, pp. 329-344.
 
 MOMENTS AND STRESSES IN SLABS. 5 
 
 homogeneous elastic plates whose ratio of the thickness to the span is 
 neither very large nor very small, that is, plates whose dimensions are not 
 extreme. 
 
 With a theoretical foundation thus laid, the time was ready for efforts 
 to obtain numerical results by application of the theory, that is, by solution 
 of the general differential equation in specific cases of technical, or other- 
 wise scientific importance. There was due also a change of chief interest 
 in the problem from the question of vibrations to that of stresses and 
 strength, that is, the time had come for the structural, rather than the 
 acoustic problem to stand in the foreground. Lavoinne, 10 in 1872, tackled 
 the question of a plane boiler bottom supported by stay-bolts. The problem 
 is essentially the same as that of the flat slab (of homogeneous material) 
 supported directly on column capitals, without girders, and carrying a 
 uniform load. Lavoinne's solution is for the case in which Poisson's ratio 
 of lateral contraction is equal to zero, but, as will be shown later, a correc- 
 tion for this lateral effect may be made afterward without any difficulty. 
 Lavoinne, by the use of a double-infinite Fourier series, solves Lagrange's 
 equation for a uniformly loaded, infinitely large plate which is divided by 
 the supports into rectangular panels, and which has its supporting forces 
 uniformly distributed within small rectangular areas around the corners 
 of the panels. The scries for the load become divergent when the size of 
 the rectangles of the supporting forces becomes zero, that is, when the 
 supports are point-supports. The same problem was treated by Grashoff," 
 whose solution, however, is incorrect, since it disregards some of the 
 boundary conditions. G. IT. Bryan, 12 in 1890, made an analysis of the 
 buckling of a rectangular elastic plate, due to forces in its own plane. 
 Maurice Levy 13 showed how Lagrange's equation, when applied to rec- 
 tangular plates with various types of supports, may be integrated by a 
 single-infinite series depending on hyperbolic functions, instead of the 
 double-infinite Fourier series in Navier's solution. 
 
 In the meantime, a different path of investigation, namely, that of 
 semi-empirical methods, had been entered into by Galliot and by C. Bach. 
 Galliot" compared observed deflections of plates in lock-gates (under 
 hydraulic pressure) with the results of an approximate theory, which, in 
 this manner, he found applicable as a basis of design. Bach's 15 empirical 
 formulas are based on laboratory tests in connection with some very simple 
 theoretical considerations. An example will illustrate his method. He 
 found by test that the line of failure, the danger section, of a square plate, 
 
 10 Lavoinne, Sur la resistance des parois planes des chaudieres a vapeur, Annales 
 des Fonts et Chaussees, v. 3, 1872, pp. 276-303. 
 
 11 F. Grashof, Elasticitat und Fcstigkeit, ed. 1878, p. 351. 
 
 13 G. H. Bryan, On the stability of a plane plate under thrusts in its own plane, 
 London Math. Soc., v. 22, 1890, pp. 54-67. _ 
 
 18 Maurice Levy, Sur 1'equilibre elastique d'une plaque rectangulaire, Comptes 
 Rendus. v. 129, 1899, pp. S3S-539. 
 
 14 Galliot, fitude sur les portes d'ecluses en tole, Annales des Fonts et Chaussees, 
 1887, v. 14, pp. 704-756. 
 
 16 C. Bach, Versuche fiber die Widerstandsfahigkeit ebener Flatten, Zeitschr. d. 
 Ver. deutscher Ingenieure, v. 34, 1890, pp. 1041-1048, 1080-1086, 1103-1111,1139-1144.
 
 6 MOMENTS AND STRESSES IN SLABS. 
 
 simply supported along the edges, is along the diagonals. It happens that 
 the average bending moment per unit length across the diagonal can be 
 determined by a simple analysis based on elementary principles of statics 
 (the result is 1/24 id 2 where w is the load per unit-area, I the span). But 
 this analysis gives no information as to the distribution of the bending 
 moment. Bach, then, multiplies the average moment by an empirical con- 
 stant, found by comparison with the tests. The investigation included cases 
 of rectangular and circular plates, with distributed or concentrated loads. 
 Bach's formulas, because of their simplicity and sound empirical basis, 
 have been used rather extensively. A similar treatment of the problem 
 of the plane boiler-bottom, supported by stay-bolts, was added later. 16 The 
 analysis of flat slabs, which was indicated in 1914 by Nichols," may be 
 recognized as falling into the same category as Bach's analyses. Since its 
 first appearance, Nichol's analysis has been used by many as a basis of 
 comparison between results of tests and rules of design. 
 
 Since the close of the nineteenth century the investigators of the theo- 
 retical side of the question have been confronted with three definite tasks. 
 Analyses which would cover the extreme cases in which the plate is either 
 very thick or very thin were called for; new theoretical methods were 
 needed, for example, for the solution of Lagrange's equation; and numer- 
 ical results applying to specific cases had to be worked out. 
 
 Theories applying to plates of ordinary thickness, as well as to thick 
 plates with a short span, have been developed by Michell, 18 Love, 19 and 
 Dougal." The latter, when he applied the exact theory to plates of ordi- 
 nary thickness, found agreement, in a number of specific cases, with the 
 results derived by Lagrange's equation. Thin plates, whose deflections 
 have become so large compared with the thickness that the curving of the 
 cross-section must be considered, have been treated by A. Foppl.- 1 
 
 As an example of the development of methods, mention may be made 
 again of Levy's solution of hyperbolic functions. Dougall. in the paper 
 just quoted, used Bessel-functions, and obtained thereby some rapidly con- 
 verging solutions. Other solutions by various series have been contributed 
 by Hadamard," Lauricella," 3 Happel, 24 and Botasso. 25 The modern theory 
 
 16 C. Bach, Die Berechnung flacher, durch Anker oder Stehbolzen unterstiitzer 
 Kesselwandungen und die Ergebnisse der neuesten hierauf bezuglichen Versuche, 
 Zeitschr. d. Ver. deutscher Ingenieure, 1894, pp. 341-349. 
 
 17 T. R. Nichols, Statical limitations upon the steel requirement in reinforced- 
 concrete flat slab floors. Am. Soc. C. E., Trans., v. 77. 1914, pp. 1670-1681. 
 
 18 J. H. Michell, On the direct determination of stress in an elastic solid, with 
 application to the theory of plates, London Math. Soc. Proc., v. 31, 1899, pp. 100-124. 
 
 19 A. E. H. Love, Mathematical Theory of Elasticity, ed. 1906, pp. 434-465. 
 
 20 J. Dougall, An analytical theory of the equilibrium of an isotropic elastic plate, 
 Edinburgh Royal Soc. Trans, v. 41, 1903-4. pp. 129-227. 
 
 "A. Foppl, Technische Mechanik. v. 5, ed. 1918, pp. 132-144; also: A. and L. 
 Foppl, Drang und Zwan?, v. 1. 1920, pp. 216-232. 
 
 22 J. Hadamard, Stir le probleme d' analyse relatif a 1'equilibre des plaques 
 elastiqucs encastrces. Institut de France, Acad. des Sciences, Memoires presentes par 
 divers savants, v. 33, 1908, Xp. 4, 128 p.p. 
 
 21 G. Lauricella, Sur 1'integration de 1'cquation relative a 1'cquilibre des plaques 
 elastiques encastrees, Acta Mathematica, v. 32, 1909, pp. 201-256. 
 
 -* H. Happel, Ueber das Gleichgewicht rechteckiger Flatten. Gottinger Nachrichten, 
 Math. phys. Klasse, 1914, pp. 37-62 (rectangular plate with fixed edges and with a 
 concentrated load at the center). 
 
 25 Matteo Botasso, Sull'equilibrio delle piastre elastiche piane appoggiate lungo il 
 contorno, R. Accademia della scienze di Torino, Atti, v. 50, 1915, pp. 823-838.
 
 MOMENTS AND STRESSES IN SLABS. 7 
 
 of integral equations has opened the way for new solutions, by series which 
 may fit almost any type of plate (see, for example, Hadamard's and 
 Happel's \vorks, which were just quoted). A method of a different type is 
 Ritz's 28 approximate method, which was indicated in 1909, which may be 
 applied to any elastic structure, and which was applied by Ritz himself to 
 plate problems, and after him, by other writers, to water tanks, domes, etc., 
 and to plates. The method makes use of series of properly chosen functions, 
 each of which must satisfy the boundary conditions of the problem, and 
 each of which is introduced in the series with a variable coefficient which is 
 unknown beforehand. Then one determines a suitable, finite number of 
 these coefficients by the principle of energy-minimum. Ritz's method has 
 proved itself an effective addition to our analytical equipment. Another 
 approximate method is that of difference equations which was used by N. J. 
 Nielsen" in a work on stresses in plates. His results will be mentioned 
 later. By the method, the differentials of the differential equations are 
 replaced by finite differences, and the problem is then reduced to the solution 
 of a set of linear equations, in which, for example, the deflections at a finite 
 number of points enter as variables. The method is used, in fact, when 
 string curves for distributed loads (for example, in the investigation of 
 beams) are replaced by string polygons. 
 
 Investigations in the theory of plates, made with the purpose of obtain- 
 ing definite results in specific cases, have appeared in a fairly great number 
 during recent years. Estanave, 28 in a thesis in Paris, 1900, analyzed various 
 cases of the flexure of rectangular plates. Simic, 29 in 1908, gave an approxi- 
 mate solution for rectangular plates with simply supported edges. He 
 used a rather short series of polynomials. The results agree fairly well 
 with those found by later investigations. Hager, 30 in a work published 
 in 1911, applied trigonometric series, and used Ritz's method, in an investi- 
 gation of rectangular slabs. His results are incorrect, in so far as they 
 apply to homogeneous plates, because the torsional moments in the sections 
 parallel to the edges are not considered; the results may, however, have 
 some interest with reference to two-way-reinforced concrete slabs, which 
 have a reduced torsional resistance in these sections. The same criticism 
 applies to an investigation, first published in 1911, by Danusso. 31 . He 
 
 28 Walter Ritz, Ueber cine neue Methode zur Losung gewisser Variationsprobleme 
 der mathematischen Physik, Crelles Journal, v. 135, 1909, pp. 1-61. See also H. 
 Lorcnz, Technische Elastizitatslehre, 1913, p. 397. 
 
 27 N. J. Nielsen, Bestemmelse af Spaendinger i Plader ved Anvendelse af Dif- 
 ferensligninger, Copenhagen, 1920. 
 
 28 E. Estanave, Contribution a 1'etude de 1'equilibre elastique d'une plaque mince, 
 Paris, 1900. 
 
 29 Jovo Simic, Ein Beitrag zur Berechnung der rechteckigen Flatten, Zeitschr. des 
 oesterr. Ingenieur- und Architekten-Vereines, v. 60, 1908, pp. 709-714. Another paper 
 by Simic (Oesterr. Wochenschrift fur den offentlichen Baudienst, 1909), was criti- 
 cized by Mesnager (see the paper quoted later, of 1916, p. 417) on the ground that 
 torsional moments had not been duly considered. 
 
 30 Karl Hager, Berechnung ebener rechteckiger Flatten mittels trigonometrischer 
 Reihen, Munich and Berlin, 1911. For criticism, see Mesnager's paper of 1916, which 
 is quoted below, pp. 414-418. 
 
 31 See Arturo Danusso, Beitrag zur Berechnung der kreuzweise bewehrten Eisen- 
 betonplatten und deren Aufnahtnetrager. Prepared in German by Hugo von Bronneck 
 after the articles by Danusso in II Cemento, 1911, No. 1-10; Forscherarbeiten auf dem 
 Gebiete dcs Eisenbetons, v. 21, 1913, 114 pp.
 
 8 MOMENTS AND STRESSES IN SLABS. 
 
 replaces the rectangular slab, as Jacques Bernoulli! had done in 1789, by 
 two systems of crossing beams which are connected at the points of inter- 
 section, only he considers a finite, instead of an infinite number of such 
 beams. This structure, again, has no torsional resistance in the sections 
 parallel to the sides. A structure consisting of three closely spaced systems 
 of crossing beams in three different directions would, on the other hand, 
 have both torsional and bending resistance in all directions. Such a struc- 
 ture was used by Danusso in the analysis of a triangular plate, and his 
 results may be expected to be approximately correct in this special case, as 
 long as Poisson's ratio may be assumed equal to zero. 
 
 In a note issued in 1912 by the French Council on Bridges and Roads 12 
 some design formulas were presented, together with various analyses based 
 on the differential equation of flexure. The years 1913 to 1916 brought 
 forth a rather valuable collection of exact or approximately exact studies 
 of rectangular slabs supported on four sides. The authors referred to are 
 Hencky," Paschoud, 34 Leitz, 35 Xadai, 34 and Mesnager, 37 and they appear to 
 have worked entirely independently of each other. Their numerical results 
 agree, on the whole, very well. A treatment of flat slabs by Eddy, 18 pub- 
 lished in 1913, was, unfortunately, not free from faults. Incorrect boundary 
 conditions, inconsistencies in the consideration of the negative moments 
 across the rectangular belts, and the use of an abnormally high value of 
 Poisson's ratio, namely, one-half, led, naturally, to incorrect results. One 
 may also object to his use of the terms "true" and apparent" stresses and 
 bending moments in a manner which is contrary to common usage. 
 
 N". J. Nielsen's 27 investigation, published in 1920, was mentioned on 
 account of the use of difference equations. He proved the applicability of 
 the method by applying it to known cases, where the results found by 
 previous investigators had shown approximate agreement. Analyses of 
 rectangular slabs supported on four sides served this purpose. He then 
 analyzed the action of flat slabs with different loading arrangements, with 
 square or rectangular panels, stiff or flexible columns, etc., and he made 
 special studies of the stresses in exterior panels and corner panels. The 
 approximation obtained does not seem to be quite satisfactory in all the 
 
 12 Conceil General des Fonts et Chaussees, Calcul des hourdis en beton arme, 
 Annales des Fonts and Chaussees, 1912, VI, pp. 469-529. 
 
 33 H. Hencky, Ueber den Spannungszustand in rechteckigen ebener Flatten, 1913, 
 94 pp. (thesis in Darmstadt). 
 
 * 4 Maurice Paschoud, Sur I'applicatipn de la methode de Walter Ritz a 1'etude de 
 1'equilibre elastique d'une plaque carree mince, thesis in Paris, 1914, 56 pp. (See 
 Mesnager's paper, quoted later.) 
 
 ** H. Leitz, Die Berechnung der frei aufliegenden, rechteckingen Flatten, Forsch- 
 erarbeiten auf dem Gebiete des Eisenbetons, v. 23, 1914, 59 pp. He added later an 
 analysis of rectangular plates with fixed edges, see his paper: Die Berechnung der 
 eingespannten, rechteckigen Platte, Zeitschr. f. Math. u. Phys., v. 64, 1917, pp. 262-272. 
 
 " Arpad Nadai. Die Formanderungen und die Spannungen von rechteckigen 
 elastichen Flatten, Forschungsarbeiten auf dem Gebiete des Ingenieurwesens, v. 170- 
 171, 1915, 87 pp. Also, in a shorter presentation, in Zeitschr. d. Ver. deutscher 
 Ingenieure, 1914, .pp. 487-494, 540-550. 
 
 r Mesnager, Moments et fleches des plaques, rectangulaires minces, portant une 
 charge uniformement repartie, Annales des Fonts et Chaussees. 1916, IV, pp. 313-438. 
 
 ** H. T. Eddy, The theory of the flexure and strength of rectangular flat plates 
 applied to reinforced-concrete floor slabs, 1913.
 
 MOMENTS AND STRESSES IN SLABS. 9 
 
 cases. This deficiency might have been remedied by the use of a greater 
 number of terms, but thereby the complexity of the work would, of course, 
 have increased. Nielsen's analysis is the first in which approximately exact 
 methods of analysis are applied, on an extensive scale, to the flat-slab prob- 
 lem. His results will be quoted later, on various occasions. 
 
 The experimental work on steel plates had been continued, in the 
 meantime, by Bach." 1 Another contribution of the sort is due to Craw- 
 ford. 40 A test of a rubber model, designed to represent a flat-slab structure, 
 was made by Trelease tt for the Corrugated Bar Co. The experimental work 
 on concrete slabs is mentioned at other places in this report. 
 
 Among the treatises in which the slab-problem is dealt with extensively 
 may be mentioned those by Love, Foppl, and Lorenz. 42 
 
 5. LIMITATIONS OP THE THEORY. The properties of the plates dealt 
 with in the following analysis will now be defined. 
 
 a. In order to simplify the discussions the plates will be assumed to be 
 horizontal, the applied forces vertical. 
 
 6. The plates are medium-thick. A medium-thick plate is defined here 
 as one which is neither so thick in proportion to the span that an appre- 
 ciable portion of the energy of deformation is contributed by the vertical 
 stresses ( shears, tensions, and compressions ) , nor so thin that an appre- 
 ciable part of the energy is due to the stretchings and shortenings of the 
 middle plane when the plate is bent into a double-curved surface. All 
 plates and plate-like structures may be divided into four groups according 
 to thickness: thick plates, in which the vertical stresses are important; 
 medium-thick plates, to which the present analysis applied; thin plates, 
 whose resistance to transverse loads depends in part on the stretching of 
 the middle plane; and membranes, which are so thin that the transverse 
 resistance depends exclusively on the stretching. The membrane, of course, 
 is not a plate in the ordinary sense, any more than a suspended cable is a 
 beam. The thick and the thin plates require special theories, such as those 
 developed by Michell, Love, Dougall, and Foppl (see Art. 4). These extreme 
 cases are eliminated by definition. Plates of such proportions as are gener- 
 ally used in reinforced-concrete floor slabs may be classified as medium- 
 thick, and fall within the scope of the analysis. 
 
 c. The plates are homogeneous and of uniform thickness. Since the 
 vertical stresses do not contribute directly to the work of deformation or to 
 the deflections, it is sufficient to specify the elastic properties with regard 
 
 89 C. Bach. Versuche iiber die Formanderung tind die Widerstandsfaliigkeit ebeiier 
 Wandungen, Zeitschr. d. Ver. deutscher Ingenieure, 1908, pp. 1781-1789, 1876-1881. 
 
 40 W. J. Crawford, The elastic strength of flat plates, an experimental research, 
 Edinburgh Roy. Soc. Proc., v. 32, 1911-1912, pp. 348-389. Additional note by C. G. 
 Knott, pp. 390-392. 
 
 41 Corrugated Bar Co., Bulletin on flat slabs, Buffalo, 1912. 
 
 "A. E. H. Love, Mathematical Theory of Elasticity, ed. 1906, Chapter XXII. 
 
 A. Foppl, Technishe Mechanik, v. 3 and v. 5 (ed. 1919 and 1918). 
 
 A. and L. Foppl, Drang und Zwang, v. 1, 1920. 
 
 H. Lorenz, Technische Elasticitatslehre, 1913, Chapter VII. 
 
 See also: Todhunter and Pearson, History of the theory of elasticity, 1886-1893; 
 and: Encyclopedic der mathematischen Wissenchaften, Vol. IV, 25, 1907, pp. 181-190, 
 and 27, 1910, pp. 348-352.
 
 10 MOMENTS AND STRESSES IN SLABS. 
 
 to the horizontal strains, llooke's law is assumed to apply to the horizontal 
 strains, and the elastic properties, then, depend on two constants; namely, 
 E = modulus of elasticity for horizontal tensions and compres- 
 sions, and 
 
 K = Poisson's ratio of lateral horizontal contraction to longitu- 
 dinal horizontal elongation. 
 
 d. A straight line, drawn vertically through the plate before bending, 
 remains straight after bending. This assumption is consistent with the 
 preceding specifications that the plate is medium-thick, and is homogeneous 
 and of uniform thickness, and it is entirely analogous to the assumption in 
 the theory of beams that a plane cross-section before bending remains plane 
 after bending. It follows that the horizontal unit-stresses, tensions, com- 
 pressions, and shears, in vertical sections are distributed according to 
 straight-line diagrams, as the tensions and compressions in the cross-section 
 of a beam. 
 
 e. The zero-points in these diagrams for the horizontal stresses in ver- 
 tical sections are in the middle plane, which is therefore a neutral plane. 
 
 As to the question of the consistency of these assumed properties refer- 
 ence may be made to the theoretical works mentioned in the historical sum- 
 mary (Art. 4) . 
 
 6. THE EQUATIONS APPLYING TO A SMALL KECTANGULAB ELEMENT 
 OF THE SLAB. 
 
 The following notation is used: 
 
 x. y = horizontal rectangular coordinates (see Fig. 1). 
 z = vertical deflection, positive downward. 
 
 V = vertical shear per unit length in the section perpendicular to a; at 
 noint (x, tj) ; the positive direction of V is indicated by the 
 arrow in Fig. 1. 
 
 V . = same in section perpendicular to y. 
 
 A/ x = bending moment per unit length in the section perpendicular to a? 
 at point (x, y} ; M is positive when causing compression at the 
 top and tension at the bottom. 
 M =: same in section perpendicular to y. 
 
 M torsional moment per unit length in sections perpendicular to x 
 and ?/ at point (x, y} ; the positive directions are indicated by 
 the arrows in Fig. 1 ; that is, the torsional moment is considered 
 positive when it causes shortenings at the top along the diagonal 
 through the corner (#,?/) of the element. 
 K = modulus of elasticity of the material. 
 K =Poisson's ratio of lateral contraction to longitudinal elongation. 
 
 Concerning E and K, see the preceding article. 
 
 7 moment of inertia per unit longlh; / = ^ d 3 when d is the thick- 
 ness of the slab. 
 
 Fig. 1 shows a small rectangular element of the slab with the forces 
 and couples acting on it. The location of the element is denned by the 
 horizontal coordinates x and ;/ of the midpoint of the lower left-hand edge
 
 MOMENTS AND STRESSES IN SLABS. 
 
 11 
 
 in the figure. The dimensions are dx and dy in the x- and y-directions, and 
 the thickness of the slab in the ^-direction. The deflection at point (x, y) is 
 measured by ~, which is positive downward. 
 
 The loads are: first, the applied surface load w per unit-area, in the 
 ^-direction; that is, a total load of w dx dy; secondly, the internal vertical 
 
 y 
 
 K'j^M 
 
 * surface had in d/recfon z. 
 x ondy are ftorizenfa/ cvordinates. 
 
 cfx 
 
 FIG. 1. KECTANGULAB ELEMENT OF SLAB. 
 
 shears, bending moments and torsional moments which are listed in Table 1. 
 The values per unit-length are y y + x d x y etc., hence 
 
 the total values are y x dy, (F x + -=- x dx) dy, V y dx > etc - The 
 
 ox 
 
 vertical shears and the bending moments are of the same nature as the 
 vertical shears and the bending moments in beams. The torsional moments
 
 12 
 
 MOMENTS AND STRESSES IN SLABS. 
 
 M are resultants of the horizontal shears in the vertical faces. The values 
 of M for the lower and left-hand faces in Fig. 1 are equal on account of 
 the law of equality of shears in sections perpendicular to one another. 
 
 TABLE I. FORCES AND COUPLES IN FIG. 1. 
 TABLE I. 
 
 Face 
 
 Vertical shear 
 
 Bending moment 
 
 Torsional moment 
 
 Value 
 
 Direc- 
 tion 
 
 Value 
 
 Direc- 
 tion 
 
 Value 
 
 Direc- 
 tion 
 
 Left face 
 
 tidy 
 
 -z 
 
 ** 
 
 xz 
 
 M z dy 
 
 yz 
 
 Riqht face 
 
 (V^dxty 
 
 +z 
 
 fa<$*tt 
 
 zx 
 
 (M^dxjdy 
 
 zy 
 
 Lower face in F/q. 1. 
 
 Vyd* 
 
 -z 
 
 Mydx 
 
 yz 
 
 Mtdx 
 
 m 
 
 Upper face in F/g.l. 
 
 fa*$W* 
 
 *J* 
 
 (Myt^&dyjdx 
 
 zy 
 
 (M^dy^ 
 
 zx 
 
 Tlie directions indicated are the positive directions of the loads. 
 
 The forces shown in Fig. 1 must hold the element of the slab in equi- 
 librium. By equating the sum of the vertical components (in the z-direc- 
 tion) to zero and dividing by dx dy we find 
 
 <5x by , (1) 
 
 By equating to zero the sum of the moments about a line parallel to y 
 through the center of the element, and dividing by dx dy, we find 
 
 and by analogy, 
 
 Vl"L7 -rr 
 T^7 = VV 
 
 (2) 
 
 (3) 
 
 By differentiating (2) and (3) with respect to x and y, respectively, 
 and substituting in (1), we find 
 
 The equations (!) to (4) arc equations of equilibrium. It the slab is 
 bent, say, in the ^-direction only, so that the lines parallel to y remain 
 straight and parallel to j/, then the slab acts as a beam, V , M , and M 7 
 become zero, and the formulas (1) to (4) are reduced to the well-known 
 equations from beam theory: 
 
 X _j 
 
 i ' x ~~ x 
 dx ox 
 
 j 
 dx z 
 
 The terms containing M represent the effect of the torsional resistance.
 
 MOMENTS AND STRESSES IN SLABS. 
 
 13 
 
 The equations ( 1 ) to ( 4 ) were derived by the statical conditions of 
 equilibrium without reference to the deformations. That is, ( 1 ) to ( 4 ) 
 apply without reference to the particular elastic properties. Since they are 
 merely equations of equilibrium, they apply to non-homogeneous slabs, such 
 as reinforced-concrete slabs, and to slabs with reduced torsional resistance, 
 as well as to the homogeneous elastic plates. 
 
 We now consider the deformations and their relations to the loads. 
 The plate is again assumed to be homogeneous. Fig. 2 illustrates three 
 types of deformation: bending in the a?-direction shown in Fig. 2 (a) ; in 
 the t/-direction shown in Fig. 2(b) ; and torsion in the a?i/-directions shown 
 in Fig. 2(c). Any state of flexure of an element of the slab may be 
 resolved into component parts of these three types. The amounts of defor- 
 
 8 2 7, 
 
 mation are measured in Fig. 2 (a) and Fig. 2(b) by the curvatures _ 
 
 S 2 
 an( j ?. (as in beams), and in Fig. 2(c) by the rate of change of slope, 
 
 that is, by 
 
 8x 
 
 82 
 Z 
 
 8\8)/ 
 
 (a) (bl '"I 
 
 FIG. 2. DEFORMATIONS OF ELEMENT OF SLAB. 
 
 A bending moment M acting alone, produces the curvatures _ = 
 
 M X ' 8 2 rc M ^ X 
 
 ? in the or-direction, _ _ JT? in the y-direction, and no twist; 
 
 El Sy 2 El 
 
 these results may be taken directly from the theory of beams. The torsional 
 
 couples M produce a twist _ 
 
 Sx 
 
 which may be determined by intro- 
 
 ducing temporarily another system of coordinates, as' , y' , making angles of 
 45 with the system of on, y. The couples M are replaced by an equivalent 
 combination of couples consisting of the bending moments M M in the 
 
 in the i/'-direction. These bending moments 
 
 M z -(l+K) 8 2 z 
 _ ___ ~ 
 
 El 
 
 x -direction and M = M 
 
 y z 
 
 8 2 
 produce the curvatures 
 
 j ,2 ~ 
 OX 
 
 El El 8y' El 
 
 in terms of which the twist may be expressed by the transformation formula 
 
 _ 
 
 ~ 
 
 El
 
 14 MOMENTS AND STRESSES IN SLABS. 
 
 In the general case the bending moments M and M , and the tors 
 moment if are all present, and the resultant deformations are then 
 
 (5) 
 
 SxSy El (7) 
 
 Equations (5), (6), and (7) express the deformations in terms of the 
 moments. By solving them with respect to the moments, the moments are 
 found in terms of the deformations: 
 
 M El ( S Z Z rsS?Z\ 
 
 M * = Mf*(&?-Xfyz) ' (S) 
 
 ., El ( fftfz. S*z\ 
 ty-JTPW&f-Jf*) . (9) 
 
 ., El ( S?Z\ 
 and Mz=M;\{toSy) (10) 
 
 By substituting these values of the moments in equation (4), which is 
 a relation between the moments and the applied load, a direct relation is 
 found between the applied load * and the deformations, .;. The result is 
 Lagrange's equation for the flexure of plates, or, the "plate equation," 
 
 We may introduce Laplace's operator 
 
 = Jif + 6^ 
 
 ox 6y 
 
 which gives 
 
 ^ (Sx 4 &r<5y z Sy 4 
 Then Lagrange's equation (S), may be written in the simpler form 
 
 (12) 
 
 One may determine, in a similar manner, a direct relation between the 
 shear Y and the deflections z s by combination of the equations (2), (5), 
 (tf),and (7). One finds: 
 
 El ^AZ 
 
 r . El 
 
 Y y-~l-K* Sy ' ,14) 
 
 Calculations are sometimes made under the assumption that Poison's 
 ratio is equal to zero. Let M x< J/ yi M^ r x , T' y , and c, denote the 
 moments, shears, and deflections when Poisson's ratio is equal to zero, while 
 M'x . . . . , V\ . . . . , ~' are tne corresponding values, for the same load, 
 when Poisson's ratio has a value K which is different from zero. There are
 
 MOMENTS AND STRESSES IN SLABS. 15 
 
 certain relations between the two sets of values which apply when the 
 boundary of the area under consideration, as marked by the supports and 
 by the edges of the plate, is fixed, or consists of simply supported straight 
 edges, or consists of parts which are fixed and parts which are simply 
 supported along straight edges. These relations apply to the slabs dealt 
 with in this report, but they do not apply, in general, when the supports 
 are elastic, or when there are unsupported edges, or simply supported 
 curved edges. The relations may be verified by inspection of equations (11) 
 or (12), (8) and (9), (10), and (13) and ( 14) , respectively. The relations 
 are: 
 
 (16) 
 
 K/=K , 
 
 Since Poisson's ratio K varies according to the material used, it is 
 expedient to make the calculations of , j^j M ...... , etc., on the basis of 
 
 K = 0. The values ', M 1 M' ...... , etc., for any particular value of K 
 
 may then be determined afterward by formulas (15 to (18). 
 
 When K = Q, then the equations (12), (5) to (10), (1.3) and (14) 
 assume the simplified forma: 
 
 S*z\ ,, / S*z f 
 
 (20) 
 
 (21) 
 
 where Az =TTT -t -5? 
 
 o" oy ' OA^ oA~oy~ oy ' 
 
 Equations (19) to (22), in connection with equations (15) to (IS), 
 constitute a set of fundamental relations, by which plates may be analyzed. 
 The most difficult part of the problem lies in the solution of Lagrange's 
 equation, (19). This equation must be solved in each case witli due con- 
 sideration of the particular boundary conditions. 
 
 The reactions in a beam are expressed by the end shears, and end 
 moments. In a similar way, the reactions in a slab may be expressed in 
 terms of the shears, bending moments, and torsional moments at the edge. 
 The torsional moments along a straight simply supported outer edge may 
 be replaced by an equivalent vertical reaction by the method indicated by 
 Kelvin and Tait. The method is described fully in Xadai's work on rec- 
 tangular plates ( see the historical summary ) .
 
 16 MOMENTS AND STRESSES IN SLABS. 
 
 The above differential equations in rectangular coordinates have fur- 
 nished the larger number of the results which are indicated in the following 
 articles. Polar coordinates may be introduced instead of rectangular 
 coordinates by transformation of the above equations; they have been used 
 with advantage in the analysis of circular slabs (see for example Foppl's 
 treatment). Beside the method of differential equations two other methods 
 stand out as effective in analysis, and they have furnished some of the 
 results which are quoted and used in the following articles. These methods 
 are: Ritz's method, which is based on the energy variations for the whole 
 plate; and the method of difference equations, which was used by Nielsen 
 (see Art. 4) . 
 
 7. MOMENTS IN RECTANGULAR PLATES SUPPORTED ON FOUR SIDES. 
 
 The following notation is used : 
 a = longer span. 
 
 b = shorter span. 
 
 a = b/a = ratio of shorter span to longer span. 
 w = uniformly distributed load per unit-area. 
 A/,,^ positive moment per unit-width at the center of the panel, in the 
 
 direction of the short span. J/ bc is referred to as the "positive 
 
 moment in the short span." 
 M maximum positive moment per unit width in the direction of the 
 
 long span, or "maximum positive moment in the long span." 
 
 This maximum moment occurs somewhere on the center line 
 
 parallel to the long sides, but not necessarily at the center of the 
 
 panel (see the small diagrams at the top in Fig. 3). 
 M be = negative moment per unit-width at the center of the long edge, in 
 
 the direction of the short span, or "negative moment in the short 
 
 span." 
 Jl/ negative moment per unit-widtli at the center of the short edge, 
 
 in the direction of the long span, or "negative moment in the 
 
 long span." 
 M d , = moment per unit-width at the corner across a line through the 
 
 corner, making angles of 45 degrees with the sides (see the 
 
 sketches at the top of Fig. 3 ) . 
 
 Fig. 3 to Fig. 11 show results of analyses of rectangular slabs sup- 
 ported on four sides. The slabs are single panels. The edges are assumed 
 to remain undeflected in their original plane. The edges are either simply 
 supported or fixed, as indicated in titles of the figures. The load is uni- 
 formly distributed. 
 
 In Fig. 3 to Fig. 10 the abscissas represent the ratio, oc , of the 
 shorter span 6 to the longer span a. The right-hand edge of each diagram 
 corresponds to cc = 1, that is, to a square slab, while the left-hand edge 
 corresponds to cc = 0, or a = oo , that is, to an infinitely long slab 
 supported along the two parallel edges. The ordinates in Fig. 3 to Fig. 10 
 are coefficients, M/irlr, of moment per unit width. The diagrams (a), to
 
 MOMENTS AND STRESSES IN SLABS. 
 
 17 
 
 the left in Fig. 3 to Fig. 8, show moments coefficients calculated 
 by analysis, while the diagrams (b), to the right, consist of simplified 
 curves of approximately the same shape as the curves to the left. The 
 values indicated in the diagrams to the left in Fig. 3 to Fig. 8, are based on 
 a Poisson's ratio, K, equal to zero. The points marked by small squares 
 and triangles are based on results found by Nadai and by Hencky, respect- 
 ively.* The points marked by circles were determined in the present 
 investigation by independent calculations. In these calculations infinite 
 series were used which are based on Navier's and Levy's solutions! of 
 Lagrange's equation ((11), (12), or (19), in Art. 6). The series are 
 similar, but not identical, to those used by Nadai and Hencky. Each coeffi- 
 
 FIG. 
 
 0.2 0.4 0.6 0.8 1.0 
 
 Ratio of Short Span to LonqSpan b /a-a 
 
 (al 
 3. BENDING MOMENTS PER UNIT WIDTH IN RECTANGULAR SLABS 
 
 o o.z 0.4 0.6 0.8 ia 
 Ratio of Short Span fo LonqSpan Va-oc 
 
 WITH SIMPLY SUPPORTED EDGES. 
 
 Poisson's ratio equal to zero; (a) calculated values; (b) simplified curves. 
 
 cient indicated in Fig. 3 to Fig. 8 is the outcome of a rather large amount of 
 numerical work. The resiilts shown in Fig. 3 to Fig. 8 might be supple- 
 mented by coefficients which have been determined by Leitz, Mesnager, and 
 lSTielsen,t some of whose results will be quoted later. On the whole, the 
 results obtained in the different investigations are very consistent. 
 
 The results stated by Nadai and Hencky are based on a Poisson's ratio 
 K = 0.3, while the values given in Fig. 3 to Fig. 8 are for Poisson's ratio 
 equal to zero. Coefficients which apply when Poisson's ratio is zero may be 
 derived by formulas (16) in Art. 6, from the corresponding coefficients 
 which apply when Poisson's ratio has some other value. Nadai's and 
 
 * See Art. 4, footnotes 36 and 33, respectively. 
 
 t See Art. 4, footnotes 5 and 13; also, A. E. H. Love, Mathematical theory of 
 elasticity, 1906, p. 468. 
 
 } See Art. 4, footnotes 35, 37, and 27, respectively.
 
 18 
 
 MOMENTS AND STRESSES IN SLABS. 
 
 llencky's results were transformed in this manner, as will be shown by ait 
 example. Take the moments at the center of a simply supported slab with 
 6/a = O.G. Nadai, in hia Table 6, p. 38, indicates the values M' =0.1289 
 to (a/2) 2 for the short span, and M'^ = 0.0704w(a/2) 2 , for the long span. 
 Formulas (16), in Art. 6, then determine the corresponding values for 
 K : 
 
 M v - 
 
 M' - KM' 
 
 0.1289- 0.3X0.0704 
 
 wb* 
 
 1 - K* I 0.3 2 0.6 2 .2 2 
 
 and, in the same way, J/ x 0.0243u'6 2 These values of the momenta at 
 the center, for cc = 0.6, are indicated in Fig. 3 (a). The coefficients given 
 in Fig. 3 to Fig. 8 are for Poisson's ratio equal to zero. 
 
 
 
 s 
 
 
 
 
 
 
 
 
 
 
 
 \ 
 
 
 s*h 
 
 
 I 
 
 z 
 
 *h' 
 
 
 
 
 
 
 \ 
 
 H-0cc t +6cc* 
 
 
 
 
 
 
 \ 
 
 
 
 
 
 
 
 
 
 
 \ 
 
 
 
 
 
 
 
 
 
 
 
 \ 
 
 
 
 
 
 
 
 
 
 
 \ 
 
 
 
 
 
 
 
 
 
 
 
 \ 
 
 
 
 
 
 
 
 
 
 
 \ 
 
 \ 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 -A 
 
 H 
 
 <'& 
 
 ft 
 
 QJa 
 
 r) 
 
 
 \ 
 
 
 
 
 _ 
 
 
 
 -^ 
 
 
 
 
 
 
 
 
 
 
 
 10 
 
 06 
 
 0.2 0.4 06 
 
 of 5AJ/7* 5oon to Long Span 
 
 Ratio of Short Span to Lonq Span Va- a: 
 
 FIG. 4. BENDING MOMENTS PER UNIT WIDTH IN SIMPLE SPAN OF REC- 
 TANGULAR SLAU.S; Two PARALLEL EDGES FIXED AND Two EDGES 
 SIMPLY SUPPORTED. 
 
 Poisson's ratio equal to zero; (a) calculated values; (b) simplified curves. 
 
 Fig. 3 deals with rectangular slabs witli simply supported edges. 
 When a = oo , that is, cc = 0, the slab acts as a beam, and the moment 
 coefficient for the span b becomes one-eighth. When <x increases from zero 
 to one, then the moment coefficient A/ bc /wb 2 decreases from 0.1250 to 
 0.0369, and the corresponding coefficient for the center of the long span 
 increases from to 0.0369. The value 0.0369 applies to the square slab 
 (6 = a, a :=:!). For this case Hencky's analysis gives the coefficient 
 0.0365, Leitz's analysis of 1914 gives 0.0368, Mesnager's 0.0368, and Niel- 
 sen's 0.0366.* In Fig. 3 (a) there is a separate curve for the maximum 
 
 See the investigations quoted in Art. 4 in footnotes 33 (Hencky p 
 (Leitz, p. 27); 37 (Mesnager, p. 369); and 27 (Nielsen, p. 132). 
 
 34); 35
 
 MOMENTS AND STRESSES IN SLABS. 
 
 moment in the long span. This curve defines coefficients which are greater 
 than the corresponding values at the center of the long span; that is, the 
 maximum moment in the long span of a rectangular panel does not occur 
 at the center, except when the slab is nearly square. Moment diagrams for 
 the center line of the long span are shown in Fig. 11. Two cases are 
 represented, namely, oc = i and oc = 0. The coefficients 0.0257 and 
 0.0234, which are indicated in Fig. 3 (a), appear as maximum ordinates in 
 Fig. 11; and the coefficient 0.0174, for oc = y 2 , appears in both figures, as 
 applying to the moment at the center, in the direction of the long span. 
 
 A simply supported square or rectangular slab (simply supported on 
 four sides), when loaded, has a tendency to bend up at the corners. In 
 the slabs treated here the corners are assumed to be anchored, that is, the 
 supports provide for a concentrated downward reaction at each corner. 
 With this force acting, stresses and moments are set up at each corner: 
 there is a positive moment, M across the line which makes angles of 45 
 
 deg. with the sides, that is, across the diagonal in the square slab; there is 
 
 O.Z 04 06 08 1.0 
 
 Ratio of Short Span to Long Span b /a-a: 
 
 ft 
 
 02 0.4 06 
 
 Ratio of Short Span to Long Span %-fc 
 
 (bl 
 
 FIG. 5. POSITIVK BENDING MOMENTS PER UNIT WIDTH IN FIXED SPAN OF 
 RECTANGULAR SLAB; Two PARALLEL EDGES FIXED AND Two EDGES 
 SIMPLY SUPPORTED. 
 
 Poisson's ratio equal to zero; (a) calculated values; (b) simplified curves. 
 
 an equally large negative moment in the direction of this line, and there is 
 an equally large torsional moment in the sections parallel to the sides. 
 The presence of negative moments in the direction of the diagonal of a 
 square slab may be understood easily when one considers the curve of 
 deflections along the whole diagonal. This curve has a horizontal tangent 
 at the corner, because the deflected surface lias a horizontal tangential 
 plane at this point. The convex side of this elastic curve, therefore, is 
 upward; that is, the moment is negative. The following values of the 
 coefficients, M Ai /ivb 2 , in a square slab were determined : by Nadai's 
 analysis, 0.0479; by Mesnager (his paper, p. 369), 0.0464; and by the 
 present investigation, 0.0463. The concentrated downward reaction is equal 
 to twice the diagonal or torsional moment per unit width; that is, the 
 present analysis leads to a corner reaction equal to 2 X 0.0463wj6 2 = 
 0.0926w& 2 . Leitz indicates this reaction as 0.092w6 2 . 
 
 The average coefficient of moment across the diagonal in a simply sup-
 
 20 
 
 MOMENTS AND STRESSES IN SLABS. 
 
 ported square slab may be determined by simple statical principles.* It is 
 1/24 0.0417; that is, practically the average of the extreme values, 
 0.0463 and 0.0369, occurring at the corner and at the center, respectively. 
 The coefficient 1/24 has been used frequently as a basis of design. This 
 value, 1/24, may be justified on the ground that when the proportional 
 limit is exceeded, or when the material begins to yield, in a part of the 
 diagonal section, the stresses will be redistributed so that they become more 
 nearly uniformly distributed. 
 
 The curves in Fig. 3 (a) do not have equations which can be expressed 
 by simple algebraic formulas. It is possible, however, to indicate simple 
 
 
 
 
 
 ^ 
 
 -M* 
 
 jnb* 
 
 "~ I +0.30:" 
 
 ^ \ 
 
 
 
 -L 
 
 ng _ 
 
 ipar 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 *4 
 
 A 
 
 J^ 
 
 s 
 
 
 
 She 
 
 rfS 
 
 oan 
 
 l+O.Za:-' 
 
 \ 
 
 
 
 
 
 
 
 
 
 * - 
 
 --. 
 
 s 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 . 
 
 y* 
 
 pi* 
 
 b 
 
 \ 
 
 
 
 
 
 
 
 
 
 
 
 a 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 02 0.4 0.6 
 
 Ratio of 5horf Span fo Longman tfa- ex 
 
 OZ 04 OS 1.0 
 
 Ratio ofShorf Span to Long Span a~a: 
 
 la) .Ib) 
 
 FIG. 6. NEGATIVE BENDING MOMENTS PEE UNIT WIDTH IN FIXED SPAN OF 
 KECTANGTJLAB SLABS; Two PARALLEL EDGES FIXED AND Two EDGES 
 SIMPLY SUPPOBTED. 
 
 Poisson's ratio equal to zero; (a) calculated values; (b) simplified curves. 
 
 formulas which give nearly the same values as are found in Fig. 3 (a). 
 Such formulas, and the curves which represent them graphically, are indi- 
 cated in Fig. 3(b). The formulas and the curves give the coefficient I/ 24 
 for the square slab. The curve for Af dla l' e9 lower in Fig. 3(b) than in 
 Fig. 3 (a). The decrease for oc from 0.0678 in Fig. 3 (a) to 0.0625 in 
 Fig. 3(b) may be defended on the ground that in very long slabs the 
 stresses at the corner, due to M dla are local stresses upon which the 
 safety of the slab as a whole does not depend. And in the case of oc = 1 
 the probable redistribution of moments and stresses across the diagonal, 
 when the material begins to yield at one point, will account for the proposed 
 reduction of the coefficient, from 0.0463 to 1/24. In selecting the formulas 
 
 "Used by Bach in his plate theory, see the paper quoted in Art. 4, footnote 15.
 
 MOMENTS AND STRESSES IN SLABS. 
 
 21 
 
 indicated in Fig. 3(b) some weight was given to the desirability of having 
 simple formulas, upon which design computations might be based. 
 
 Fig. 4, Fig. 5, and Fig. 6 deal with rectangular slabs which have two 
 fixed opposite edges and two simply supported opposite edges. In a uni- 
 formly loaded single continuous row of simply supported panels each panel 
 acts, on account of the continuity, in the same way as the single panel 
 with two fixed and two simply supported edges. The torsional moments 
 and bending moments are zero at the corners in these slabs. As in Fig. 
 3 (a), separate curves are indicated in Fig. 4 (a) and Fig. 5 (a) for the 
 maximum moments in the long span; these curves lie, in part, above the 
 corresponding curves for the moment at the center. Certain individual 
 points, which were derived from Nadai's work (p. 62, Table 9, in his work), 
 lie at a distance from the curves drawn through the rest of the points. 
 There are three such points lying below the bottom curve in Fig. 4 (a), and 
 
 .06 
 
 .04 
 
 02 0.4 0.6 0.8 
 
 Ratio of Short Span to Long Span 
 
 02 0.4 0.6 0.8 1.0 
 
 Ratio of Short Span to Long Span %-a: 
 
 FIG. 7. POSITIVE BENDING MOMENTS PER UNIT WIDTH IN RECTANGULAR 
 SLABS WITH FIXED EDGES. 
 
 Poisson's ratio equal to zero; (a) calculated values; (b) simplified curves. 
 
 three points, belonging to the same cases, lying above the top curve in 
 Fig. 5 (a). In Fig. 6 (a) one point lies below the bottom curve. There is a 
 possibility of an error in these points. Fig. 6 (a) shows the peculiar result 
 that greater negative moments are produced when the long span is fixed 
 than when the short span is fixed. The simplified curves to the right in 
 Fig. 4, Fig. 5 and Fig. 6 follow rather closely the curves to the left. 
 
 Fig. 7 and Fig. 8 deal with slabs fixed on four sides. Unfortunately, 
 this case, on account of the greater difficulties involved, has been treated 
 less extensively than the preceding cases. Navier's and Levy's solutions 
 do not apply to these slabs. Ritz's method, which was applied to these 
 slabs, for example, by Nadai, leads to a fairly satisfactory analysis. The 
 curves in Fig. 7 (a) are drawn according to Hencky's results. For the 
 moments at the center of a square plate various writers have indicated 
 values, which lead to the following coefficients: Heneky, 0.0177; Nadai, 
 0.0177; Mesnager, 0.018; Leitz, 0.01S4; Nielsen, 0.0171; the present 
 investigation, by an approximate method, 0.0194. For the, negative moments 
 at the center of the edge of a square panel the same writers have indicated
 
 22 
 
 MOMENTS AND STRESSES IN SLABS. 
 
 the following coefficients:* Hencky, 0.0513; Nadai, 0.0487; Mes- 
 nager, 0.0474; Leitz, 0.0515; Nielsen, 0.0511; the present investiga- 
 tion, by an approximate method, 0.0493. The curve for M , in Fig. 
 7(b), and the line for M in Fig. 8(b), have been drawn according to an 
 estimate, and they may have to be revised later. 
 
 Fig. 9 contains a summary of all the simplified cur^s in Fig. 3 to Fig. 
 8. The curves for the negative moments are shown to the left, those for 
 the positive moments to the right. Table II gives a summary of the for- 
 mulas represented by these curves. A corresponding set of formulas, 
 applying exactly to an elliptic plate with fixed edges and with Poisson's 
 ratio equal to zero,f is indicated, for the purpose of comparison with the 
 other formulas, in the bottom line in the table. 
 
 2fc 
 
 M^'-^wb^ 
 
 0? 0.4 0.6 0.3 10 
 
 Ratio ofShorf Span -fa LongSpan tya-cc 
 
 FIG. 8. NEGATIVE BENDING MOMENTS PER UNIT WIDTH IN RECTANGULAR 
 SLABS WITH FIXED EDGES. 
 
 Poisson's ratio equal to zero; (a) calculated values; (b) simplified curves. 
 
 Fig. 10 (a) illustrates the influence of change in Poisson's ratio. Such 
 a change causes a redistribution of the moments in the slab. The case 
 dealt with is again that of the slab with simply supported edges. Two of 
 the curves in Fig. 3 (a) are reproduced, namely, the curve for the moment 
 in the short span at the center, and the curve for the moment at the corner, 
 in a section making angles of 45 degrees with the sides; that is, the curves 
 
 for M^ and M*. respectively'. These curves are marked X = 0; those 
 DC alag 
 
 for K = 0.3 are indicated in the figure. The computations of the changed 
 moment coefficients were made according to forrmilas (16) in Art. 6. They 
 apply when Poisson's ratio, K, is equal to zero. The corresponding curves 
 
 * See the investigations quoted in Art. 4 in footnotes 33 (Hencky, p. 53) ; 36 
 (Nadai, p. 86); 37 (Mesnager, p. 413); 35 (Leitz); 27 (Nielsen, p. 139). Leitz's 
 paper of 1917, unfortunately was not available to the writer. Leitz's results for the 
 square slab are quoted from Nielsen. 
 
 t See A. Foppl, Technische Mechanik, Vol. 5, ed. 1918, p. 106.
 
 MOMENTS AND STRESSES IN SLABS. 
 
 23 
 
 change from A = to K = 0.3 is seen to increase the moment at the center, 
 and to decrease the moment at the corner. In the square slab the moments 
 across the diagonal are redistributed; the point of maximum moment 
 across the diagonal is moved from the corner to the center. 
 
 The stresses at a point in a homogeneous slab are directly proportional 
 to the moments at the point. But the maximum stress at a point does not 
 necessarily define the "nearness of rupture" or "tendency to failure" at the 
 point. This tendency depends on the whole "state of stress" at the point, 
 or, in the slab, on the state of moments at the particular point. In a square 
 slab the moments at the center are equal in all directions, while at the 
 
 
 
 
 
 
 
 
 
 
 
 
 
 - 
 
 . 
 
 --- 
 
 Jfe 
 
 Ik 
 
 
 
 
 
 
 
 
 
 
 ^v 
 
 & 
 
 
 
 
 
 
 
 
 
 
 % 
 
 
 
 
 
 
 
 
 
 h 
 
 s 
 
 
 
 
 
 
 Othe 
 
 ^Hfe^ 
 
 **.., 
 
 L 
 
 v 
 
 
 
 
 
 ""--, 
 
 3 
 
 i 
 
 <3 
 
 ^ 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 ^ 
 
 
 
 
 Other Span hxed, M^ 
 
 
 
 -\ 
 
 
 
 1 1 1 1 
 Full 'Unei- 'Moments fy in theshaltryan 
 Dotted Lin&Mmnh Ma in ttx longer yon 
 w - uniformly distributed load, 
 a - lonq span, 
 b -short span 
 
 *:* 
 
 : : 
 
 
 o o* 0.4 0.6 o.a 1.0 
 Ratio of Short Span toLonqSpan b/a-cz 
 
 O.Z 0.4 0.6 08 1.0 
 
 Ratio of Short Span to Long Span h /a~<x. 
 
 P'IG. 
 
 9. SUMMARY OF APPROXIMATE CURVES IN FIG. 3 TO FIG. 8; (a) 
 NEGATIVE MOMENTS; (b) POSITIVE MOMENTS. 
 
 corner a positive moment across the diagonal is combined with an equally 
 large negative moment along the diagonal. Though the stresses may be 
 numerically larger at the center than at the corner, failure, nevertheless, 
 may be nearer at the corner, because here the positive and negative moments 
 are combined. Various theories concerning the tendency to failure have 
 been advanced. One is represented in Fig. 10 (b); it is the "shear and 
 strain" theory, which was originated by A. J. Becker,* and which was indi- 
 cated by him as applying to steel. Corresponding to a state of stress one 
 may compute an "equivalent stress,"f which is a simple tension, in one 
 direction only, which is as dangerous as the given compound state of stress. 
 
 * A. T. Becker. The Strength and Stiffness of Steel Under Biaxial Loading. 
 University of Illinois Eng. Exp. Sta. Bull. 85, 1916. 
 
 + See, for example, Journal of the Franklin Institute, v. 189, 1920, p. 635.
 
 24 
 
 MOMENTS AND STRESSES IN SLABS. 
 
 If the shear and strain theory applies, the equivalent stress at a given point 
 may be computed, as the larger of the following two quantities: one is the 
 modulus of elasticity times the greatest unit-elongation or unit-shortening 
 at the point in any direction; the other is the greatest shearing stress at 
 the point, divided by a certain constant, which, according to Becker's 
 results, is 0.6. An equivalent moment in a slab is a bending moment which 
 would produce the equivalent stress. According to Becker's results, the 
 equivalent moment at a point is computed, then, as the larger of the follow- 
 ing two quantities: one is El times the numerically largest curvature in 
 any vertical section at the point; the other is the largest torsional moment 
 
 Hff* 0656) 
 Sending Moments at 
 Comer across ttie 
 Diagonal 
 
 02 04 06 03 10 02 04 06 OS 
 
 Ratio of Short Span to Long Span ^/a-tz Ratio of Short Span to Long Span 1 : 
 
 (a) (b) 
 
 FIG. 10 (a). INFLUENCE OF VARIATION IN POISSON'S RATIO, K, ON THE 
 
 MOMENTS IN RECTANGULAR SLABS WITH SIMPLY SUPPORTED EDGES. 
 
 FIG. 10 (b). "EQUIVALENT MOMENTS," BASED ON THE "SHEAR AND STRAIN" 
 
 THEORY, IN RECTANGULAR SLABS WITH SIMPLY SUPPORTED EDGES. 
 
 in ai^y section at the point, divided by 0.6. Such equivalent moments are 
 indicated in Fig. 10 (b). The slabs are the same as in Fig. 10 (a), and the 
 curves refer to the center and to the corner. The methods of computation 
 are indicated in the figure. According to equations (15) and (20) in Art. 
 6, El times the curvature may be computed as ( 1 K-) times the moment 
 corresponding to Poisson's ratio equal to zero. M and M , as used in 
 (he notes in Fig. 10(b), are the moments corresponding to Poisson's ratio 
 equal to zero. The curves in Fig. 10 (b) explain why a square slab with 
 K = 0.3 may fail at the corners first, in spite of the fact that the stresses 
 are smaller at the corner than at the center. 
 
 Fig. 10 fa) and Fig. 10 (b) and the discussion in connection with these 
 figures Bhow how the results derived for the case in which Poisson's ratio is
 
 MOMENTS AND STRESSES IN SLABS. 
 
 25 
 
 zero may be interpreted and used when Poisson's ratio has any other value, 
 provided the law of failure of the material is known. Whether or not the 
 curves and formulas indicated in Fig. 3 to Fig. 9, in Fig. 11, and in Table 
 II may be applied as a basis for design of actual slabs, should be deter- 
 mined for the individual materials by comparison with experimental 
 results. 
 
 8. MOMENTS IN SQUARE INTERIOR PANELS OF UNIFORMLY LOADED FLAT 
 SLABS. 
 
 Notation : 
 
 I = span, measured from center to center of the columns. 
 
 c = diameter of the column capitals. 
 
 w = load per unit-area, uniformly distributed over all panels. 
 W = total panel load. 
 
 The slab under consideration is a girderless or "flat" slab, supported 
 directly on the column capitals, which are assumed to be round. Lines 
 connecting the centers of the columns divide the floor into square panels, 
 
 =3E3? 
 
 o .2b 
 
 6b .8b b 
 
 tzb 
 
 l.4b I6b 
 
 idb sb 
 
 Distance a/ong x- 
 
 @ 
 
 FIG. 11. MOMENTS ALONG THE CENTER LINE OF THE LONG SPAN IN REC- 
 TANGULAR SLABS WITH SIMPLY SUPPORTED EDGES. 
 
 with a column at each point of intersection. An interior panel is consid- 
 ered. It is surrounded on all sides by similar panels, all carrying the same 
 load. For the convenience of the analysis it may be assumed that there is 
 an infinite number of equal, square panels, all carrying the same load. The 
 slab is assumed to be fixed in the column capitals at the edge of each column 
 capital. A panel of this description, loaded as indicated here, will be 
 referred to as "a normal panel." On account of the symmetry, the column 
 capitals supporting the normal panel will not tend to rotate about a hori- 
 zontal axis, the tangents across the edges and center lines of the normal 
 panel will remain horizontal, and the torsional moments along the center 
 lines of the panel and along the parts of the edges between the column 
 capitals will be zero. 
 
 Fig. 12 shows sections for which it lias become customary to indicate 
 the moments. The terms column-head sections, mid-section, outer sections. 
 and inner section are in accordance with common practice. The moments 
 in these sections are taken in a direction perpendicular to the straight parts 
 pf the sections. Moment coefficients for tbese sections will be stated pres-
 
 26 MOMENTS AND STRESSES IN SLABS. 
 
 ently, but first some remarks will be made concerning the processes of the 
 analysis. 
 
 The present investigation is built, in part, on certain results which 
 were found by N. J. Nielsen * in his analysis of plates by the method of 
 difference equations. Nielsen analyzed various types of square interior 
 panels of uniformly loaded flat slabs: first, point-supported slabs, in which 
 tli column capitals and the columns have been reduced to point supports; 
 second, slabs in which the supporting forces are uniformly distributed within 
 squares with side 0.21 and with the centers at the centers of the column; 
 third and fourth, slabs supported on square column capitals with the sides 
 0.21 and OAl, respectively; fifth, a slab with dropped panels (areas of in- 
 creased thickness around the supports) ; and sixth, a slab in which there is no 
 bending resistance across the central parts of the edges of the panels, that 
 is, no bending resistance in parts of the mid-sections. Nielsen divided the 
 panel into elementary squares with side \ = 0.11. l/6i, 0.21, or 0.251. The 
 
 Column-head M/d-Secr/on CaJu/nn -head 
 Secf/on) \Secf/on 
 
 ] Outer ' Inner i Outer 
 \5ecfiffr Section S0:f/on\ 
 
 FIG. 12. MOMENT SECTIONS FOR FLAT-SLAB PANELS. 
 
 variables in the equations are the deflections at the corners of these squares; 
 one equation is indicated for each such corner. Since finite squares are used 
 instead of infinitesimal rectangular elements the method is approximate, 
 not exact, as applying to homogeneous slabs.f The smaller the value of \ 
 the closer is the approximation. The value \ = O.ll gives, on the whole, 
 rather satisfactory results. \ = 1/61 to 0.251 gives results, use of which 
 may be made in comparative studies of distributions of deflections and 
 moments in different slabs; such use of some of Nielsen's results will be 
 made later (in Art. 10). But the moment coefficients found with \ = 
 1/61 or more, hardly seem to be sufficiently exact when considered as inde- 
 pendent results applying to fundamental cases. For this reason the use of 
 Nielsen's results was limited here to those obtained with \ = O.ll. This 
 value of X was used by Nielsen only in the first two of the cases men- 
 
 * N. J. Nielsen, Bestemmebe af Spaendinger i Plader ved Anvendelse af Differen- 
 sligninger, 1920 (referred to in Art. 4, footnote 27). 
 
 t The difference equations apply exactly to a certain rib-structure in which the 
 bending deformations are concentrated at the points of intersection of the ribs, and 
 in which the torsional resistance is supplied by special structural elements which con- 
 nect one rib with another.
 
 MOMENTS AND STRESSES IN SLABS. 
 
 27 
 
 tioned, that is, in the analyses of the point-supported slab and of the slab 
 with the supporting forces uniformly distributed within small squares; 
 in the other cases he used \ > O.H. Those of Nielsen's results, with 
 X r= O.H, which apply to the point-supported slab, were represented graph- 
 
 TABLE IT. APPROXIMATE FORMULAS FOR BENDING MOMENTS PER UNIT 
 
 WIDTH IN RECTANGULAR SLABS AND ELLIPTIC SLABS 
 
 SUPPORTED ON THE PERIPHERY. 
 
 The Tormulas are represented qrophicatlLimfiq31orT<i9. 
 a -/onqerspan, b - shatter span ;ot- Obi Fbissons rot/o-O. 
 
 Momenta in span b. 
 
 /It center ' /?/" center 
 of edqe. of 5/ab. 
 
 Moments in span g 
 
 At center 
 
 of edqe 
 
 ~M ae 
 
 A/onq center 
 fine of slob. 
 
 Fburedqes 
 
 simply 
 
 supported. 
 
 wb 
 
 1+ZoC 3 
 
 Span & 
 fried; 
 
 -Span g_ 
 simple. 
 
 1-t-OAoC? 
 
 o 
 
 wb^ 
 60 
 
 Span g_ 
 
 f/xed; 
 
 Spanb. 
 
 simple. 
 
 ItO-Boc 
 
 All 
 edc/es fixed. 
 
 llipt/c j ~/ab 
 wiih fixededqe; 
 diameters 
 
 j> 
 
 f+jof+a* 
 
 -0C 
 
 ically for the purpose of the present investigation, and adjustments were 
 made, similar to those by which a string polygon is modified into a string 
 curve. By such adjustments of the curves for moments and deflections it 
 was possible to improve the approximation slightly. It would be possible 
 to analyze the point-supported slab by differential equations, and to obtain, 
 thereby, an increased degree of exactness, but in this study it seemed 
 desirable to make use of the available results. The degree of approximation 
 obtained by this use may be judged by comparing the moment coefficients 
 for square slabs supported on four sides, found by Xielsen with \ equal to 
 one-tenth of the side, as quoted in the preceding article, with the corre- 
 sponding moment coefficients found in other analyses.
 
 28 
 
 MOMENTS AND STRESSES IN SLABS. 
 
 The use of Nielsen's results in connection with results found in the 
 present investigation by means of the differential equations will now be 
 described. The point-supported slab is not important in itself, because 
 actual slabs do not have point supports. But the results found for the 
 point-supported slab may be used in the analysis of the normal panel, sup- 
 ported on round column capitals, in the same way as moment diagrams for 
 simple beams are used in the study of continuous beams or beams with 
 fixed ends. The diagrams for fixed beams can be found from the diagrams 
 for simple beams by adding the effects of the end moments. The simple 
 beam is considered in this connection as a "substitute structure" which 
 temporarily replaces the given fixed beam, and which is made to act like the 
 given beam by adding the end moments. In a similar way the point- 
 supported slab may be used as a substitute structure which temporarily 
 replaces the slab supported on column capitals, and which is made to act 
 like the original slab, that is, have the same deflections and moments at all 
 
 J 
 
 2 
 
 3 
 
 v 
 
 I 
 
 T 
 
 ^ 
 
 4- 
 
 _7 
 
 m & 
 
 
 
 
 - 
 
 I 
 
 T 
 
 7\ 
 
 FIG. 13. MOMENT SECTIONS REFERRED TO IN TABLE IV; THE HEAVY SHORT 
 LINES INDICATE THE POSITIONS AND DIRECTIONS OF THE SECTIONS. 
 
 points outside the circles marked by the edges of the column capitals, by 
 adding certain loads near the center of the column. The nature and 
 intensity of these loads must be such that the resultant deflections and 
 slopes at the circles marking the edges of the column capitals become zero. 
 The essential part of the load of this kind, added at each column, may be 
 termed a "ring load." It consists of a combination of an upward, force 
 uniformly distributed over the circumference of a small circle, drawn on 
 the slab, with an equally large downward force at the center of this circle. 
 The ring load may be considered as concentrated, just as a couple acting on 
 a beam may bo considered as concentrated at one point, for example, at the 
 end point if the end is fixed. A ring load of the proper intensity at each 
 point support, combined with a certain uniformly distributed bending 
 moment applied at the edge of the whole slab (which includes many panel?) 
 and, combined with the original uniformly distributed load and the reac-
 
 MOMENTS AND STRESSES IN SLABS. 
 
 29 
 
 TABLE III. PERCENTAGES OF SUM OF POSITIVE AND NEGATIVE MOMENTS 
 RESISTED IN SECTIONS SHOWN IN FIG. 12. 
 
 Results of analysis of a square interior panel of a uniformly loaded homogeneous flat slab. 
 The sum of positive and negative moments is approximately equal to 
 
 c=diameter of column capital; /=span; W= total panel load; Poisson's ratio =0. 
 
 
 c/l 
 
 Average. 
 
 0.15 
 
 0.20 
 
 0.25 
 
 0.30 
 
 Negative Moments 
 Positive Moments 
 
 Column head section 
 
 Across edge of capita! 
 Outside capital 
 
 31.8 
 16.5 
 48.3 
 17.0 
 65.3 
 20.9 
 13.8 
 34.7 
 
 37.1 
 11.3 
 48.4 
 16.7 
 65.1 
 20.9 
 14.0 
 34.9 
 
 40.2 
 S.I 
 48.3 
 16.6 
 64.9 
 20.8 
 14.3 
 35.1 
 
 42.7 
 5.7 
 48.4 
 16.3 
 64.7 
 20.7 
 14.6 
 35.3 
 
 20+80 c/l 
 2880 c/l 
 48 
 17 
 65 
 21 
 14 
 35 
 
 Total . ... 
 
 
 Total negative mome 
 Outer section 
 
 it 
 
 
 Inner section 
 
 
 Total positive momes 
 
 t 
 
 
 TABLE IV. COEFFICIENTS OF MOMENT PER UNIT WIDTH, IN- 
 SERTIONS SHOWN IN FIG. 13. 
 
 Results of analysis of a square interior panel of a uniformly loaded homogeneous flat slab. Poisson's 
 ratio =0. 
 
 The coefficients are values of the expression 
 
 M 
 
 They are found by multiplying the coefficients in Fig. 14 to 16 by the factors 8, 9.52, 10.31, 11 . 39, and 
 12.46 for c/J = 0, 0.15, 0.20, 0.25, and 0.30, respectively. 
 
 
 
 
 e/l- 
 
 
 
 Approximate 
 Coefficients 
 
 Section. 
 
 
 
 0.15 
 
 0.20 
 
 0.25 
 
 0.30 
 
 forc/J = 
 0.15 to 0.30 
 
 1 . .. . 
 
 
 2.120 
 
 1 854 
 
 1 609 
 
 1.424 
 
 -H C-J-+0 
 
 2 
 
 488 
 
 461 
 
 0.435 
 
 0.409 
 
 0.370 
 
 ^ c ' 
 0.5 1.5( -fV 
 
 3 
 
 255 
 
 0.270 
 
 0.275 
 
 0.282 
 
 0.289 
 
 \l 1 
 
 0.028 
 
 4 
 
 258 
 
 130 
 
 048 
 
 046 
 
 0.169 
 
 0.23 4.5(4 V 
 
 5... 
 
 015 
 
 015 
 
 015 
 
 0.016 
 
 0.015 
 
 V i J 
 
 +0.02 
 
 6... 
 
 474 
 
 474 
 
 0.466 
 
 0.461 
 
 0.447 
 
 0.46 
 
 7 
 
 0.331 
 
 0.341 
 
 0.340 
 
 0.343 
 
 0.343 
 
 0.34 
 
 8. 
 
 226 
 
 242 
 
 246 
 
 0.254 
 
 262 
 
 0.25 
 
 9 
 
 0.233 
 
 188 
 
 0.156 
 
 0.121 
 
 0.072 
 
 0.23 l.s(4Y 
 
 10 
 
 0.121 
 
 0.110 
 
 0.102 
 
 0.093 
 
 0.080 
 
 \lj 
 K-K(|V 
 
 
 
 
 
 
 
 \l /
 
 30 MOMENTS AND STRESSES IN SLABS. 
 
 lions at the point supports, will make the slab deflect in such a way that 
 the circles marking the edges of the column capitals practically become a 
 contour line along which the tangential planes are horizontal and coin- 
 ciding. By introducing certain supplementary loads, beside the ring loads, 
 the conditions of the edges of the column capitals may be satisfied with any 
 desired degree of approximation. Corrections by means of such supple- 
 mentary loads were omitted, because the degree of approximation obtained 
 without these loads appeared to be acceptable; besides, since the degree of 
 approximation is limited in one part of the problem by the use of the 
 approximate results found by difference equations, the gain by a further 
 increase of exactness in the part of the problem discussed here would be 
 only slight. 
 
 It remained^ then, to investigate the effects of the ring loads, to deter- 
 mine their intensity, and to make the proper additions to the moments in 
 the point-supported slab. The method used was that of differential equa- 
 tions. Lagrange's equation ((11), (12), or (19), in Art. 6) was solved 
 for the case of the ring loads by double-infinite series, and the moments at 
 definite points, produced by the ring loads, were computed by corresponding 
 double-infinite series. As in the preceding article, and for similar reasons, 
 Poisson's ratio was taken as zero (compare, in particular, the discussion 
 made in connection with Fig. 10 (a) ). Details of this analysis will be pre- 
 sented in Appendix A. 
 
 The results will now be described. Reference is made again to Fig. 12, 
 which shows the customary moment sections. Table III gives results 
 found for these sections. The moments are stated in per cent of the sum 
 of the numerical values of positive and negative moments in all the sections 
 in Fig. 12. Values are given for four sizes of the column capital. The per- 
 centages resisted in the different sections in Fig. 12 are seen to change 
 only slightly when c changes from 0.151 to 0.30Z, and to deviate only slightly 
 from the constant values given in the last column : namely, 48 per cent in 
 the column-head sections, 17 per cent in the mid-section, 21 per cent in the 
 outer sections, and 14 per cent in the inner section. 
 
 The total moments in the various sections depend upon the moments 
 per unit-width at the individual points of the slab. Fig. 13 indicates 
 points and sections at which the moments per unit-width are of particular 
 interest. Moment coefficients for these sections are stated in Table IV. 
 
 Fig. 14 shows diagrams of coefficients of moment per unit-width across 
 the edge and the center line of the panel. The coefficients are values of 
 Jf/wP. 
 
 Lavoinne,* in a paper published in 1872, derived the stresses in certain 
 sections of a uniformly loaded point-supported slab. He stated coefficients 
 of stresses; but moment coefficients MAc/l-, of the type used in Fig. 14, 
 may be found by dividing his stress coefficients by six. Thus, Lavoinne's 
 
 * Lavoinne, Sur la resistance des paroi? planes des chaudieres a vapenr, Annal 
 Fonts et Chaussees, 1872, pp. 276-295; the numerical coefficients are quoted frc 
 ?86. See the historical summary in Art. 4, footnote 10.
 
 MOMENTS AND STRESSES IN SLABS. 
 
 31 
 
 analysis gives the following moment coefficients: at the center of the slab, 
 0.17/6 = 0.0283, to be compared with 0.0283 in Fig. 14; at the center of 
 the edge along the edge, 0.34/6 0.0567, while Fig. 14 gives 0.0592; at 
 the center of the edge across the edge, 0.20/6 = 0.0333, while Fig. 14 
 gives 0.0319. Since Lavoinne stated only two decimal places in each 
 
 FlG. 14. COEFFICIENTS OF BENDING MOMENTS PER UNIT WIDTH IN A 
 SQUARE INTERIOR PANEL OF A UNIFORMLY LOADED FLAT SLAB WHEN 
 POISSON'S RATIO is ZERO; MOMENTS ACROSS THE EDGE AND THE 
 CENTER LINE. 
 
 coefficient, the agreement may be considered as fairly satisfactory. 
 Lavoinne's solution is by double-infinite trigonometric series. 
 
 Fig. 15 shows coefficients, M/wl~, of moment per unit-width along 
 the edge and the center line of the panel. Eacli of the curves must satisfy 
 a certain condition, which applies also to beams with fixed ends: the 
 positive and the negative area of each of the moment diagrams must be
 
 32 
 
 MOMENTS AND STRESSES IN SLABS. 
 
 Fi. 15. COEFFICIENTS OF BENDING MOMENTS PER UNIT WIDTH IN A 
 SQUARE INTERIOR PANEL OF A UNIFORMLY LOADED FLAT SLAB WHEN 
 POISSON'S RATIO is ZERO; MOMENTS ALONG THE EDGE AND THE CENTEB 
 LINE.
 
 MOMENTS AND 'STRESSES IN SLABS. 
 
 33 
 
 numerically equal. This condition applies to the curves in Fig. 15 because 
 the slope of the slab is zero at the edge of the column capital and at 
 the center, that is, at the points where the curves in the diagram end; the 
 application of the condition under these circumstances follows from equa- 
 
 FIG. 16. COEFFICIENTS OF BENDING MOMENTS PEE UNIT WIDTH IN A 
 SQUABE INTERIOR PANEL OF A UNIFORMLY LOADED FLAT SLAB WHEN 
 POISSON'S RATIO is ZERO; MOMENTS ACROSS AND ALONG THE 
 DIAGONALS. 
 
 tions (20) in Art. 6. An examination of the curves in Fig. 15 showed 
 equality of the positive and negative areas. 
 
 Fig. 16 shows coefficients of moment per unit-width acrosa and along 
 the diagonals. In drawing the ciirves for the momenta along the diagonal, 
 use was made of the condition that the positive and negative parts of each
 
 34 
 
 MOMENTS AND STRESSES IN SLABS. 
 
 moment diagram must be numerically equal. The diagonal momenta at the 
 edge of the column capital could not be determined with the degree of 
 exactness obtained elsewhere, because the torsional moments at these points, 
 in the point supported slab, in sections parallel to the panel edges, had not 
 been determined by the difference equations with the same degree of exact- 
 ness as other moments. The negative moment across the edge of the column 
 capital is approximately the same along the diagonal, as along the panel 
 edge, in fact, it is approximately constant all the way around the column 
 capital. Accordingly, the coefficients of end-moment along the diagonal, 
 stated in Fig. 16, were taken as equal to the corresponding negative 
 moment coefficients in Fig. 15, which apply at the panel edge. Certain 
 small discrepancies, which possibly may be explained by the deviations in 
 the negative diagonal moments, will be discussed in connection with the 
 next two figures. 
 
 Mi. 
 
 OZnl 3 
 
 Moments 
 
 FIG. 17. DIAGRAM SHOWING THE EQUILIBRIUM OF THE FORCES AND COUPLES 
 ACTING ON AN OCTANT AND A QUADRANT OF A SQUARE PANEL 
 (c = 0.30). 
 
 Fig. 17 shows forces and couples acting on two separate octants of a 
 normal panel with c = 0.30Z. For each octant the downward resultants of 
 the applied loads acting at the centroids of the area, and the upward resultant 
 of the vertical supporting forces or shears at the edge of the column capital 
 
 form a couple; these couples are: M , for one octant, M' , for 
 
 i -fn i +11 
 
 the other. In the plane vertical sections there are bending moments, jV/ m 
 M lv , and M yi in one octant, M' lir M' 1V , and A/' VI i" the other, but 
 on account of the symmetry there are no torsional moments and no vertical 
 shears in these sections. The moment Af y and M' are the resultant 
 moments in the curved sections at the edge of the column capital. In the 
 diagram to the right, in the figure, the different couples are represented as 
 vectors. Each vector is laid off parallel to the vertical plane of the couple 
 which it represents; the direction is that of the upper one of two horizontal 
 force* representing the couple. The couple vectors M and M',.,,
 
 MOMENTS AND STRESSES IN SLABS. 
 
 are shown resolved into the components i/ r and M U , M\ and M' n . 
 The couple vectors form two polygons, one for each octant, but with the 
 side Jf yi in common. Eacli octant is in equilibrium; therefore, if the 
 couples are represented correctly, each polygon should close. Fig. 17 shows 
 small gaps at the end points of the vectors M and M> These gaps 
 
 are a measure of the discrepancies which have entered, so far, into the 
 calculations and into the particular graphical representation in the figure. 
 The method of obtaining the particular vectors represented in the 
 vector diagram in Fig. 17 will now be described, and possible sources of the 
 gaps will be discussed. The couple M I , with the components M and 
 Afjj, is considered first. The two components could be computed exactly 
 if the point of application of the resultant shear were known exactly. 
 Afj and M u were computed under the assumption that the vertical shear 
 at the edge of the column capital is uniformly distributed, or, that the 
 resultant passes -through the centroid of the circular arc formed by the 
 
 Afr 
 
 c^ 
 
 C-QZOl 
 
 '.'j 
 
 Scale 
 
 ' 01 .0?Wl 
 
 FIG. 18. COUPLES ACTING ON AN OCTANT OF A SQUARE PANEL (c = 0.15; 
 
 0.20; 0.25). 
 
 section. The distribution of the shear at the edge of the column capital 
 is known to be approximately uniform, but if it is not entirely uniform, 
 the end point of M. may have to be moved slightly. It is possible that a 
 shifting of the end of M into its correct position would reduce the gap 
 
 were determined by 
 
 at the end of M, 
 
 The vectors M and 
 
 measuring areas in Fig. 14, of the diagrams of moments across the center 
 line and the edge. Af y was determined by a corresponding area in Fig. 
 16. M was computed under the assumption that the bending moment 
 across the edge of the column- capital is constant, and that the torsional 
 moment along the edge of the column capital is zero. If these assumptions 
 are correct, the direction of lf y will bisect the 45 deg. angle between the 
 panel edge and the diagonal. But the assumption of even distribution is 
 not more than approximately correct; as stated before, the diagonal 
 moment across the edge is not known with the degree of exactness obtained 
 elsewhere. By a slight change in the bending moments and by introducing 
 small torsional moments, If may be changed so as to eliminate the gap 
 between M and M In fact, the gap may be eliminated, practically, 
 
 by changing the direction of the couple jl/ y slightly without changing the 
 magnitude.
 
 36 MOMENTS AND STRESSES IN SLABS. 
 
 Fig. 18 shows vector polygons of the same kind as those shown in the 
 preceding figures. The diagrams apply to slabs with c/l = 0.15, 0.20, and 
 0.25. As in the preceding figure, gaps are left open between M and Al 
 
 The method used here in the study of the equilibrium of the resultant 
 couples acting upon an octant of the slab is analogous to that used by 
 J. R. Nichols * in his study of the moments in a quadrant of the slab. 
 In fact, the analysis represented in Fig. 17 and Fig. 18 may be looked upon 
 as Nichols's analysis applied to an octant of the slab. The sum of positive 
 and negative moments indicated in Fig. 17 is the total moment indicated 
 by Nichols's analysis. The approximate value which Nichols gave in the 
 discussion of his paper t 
 
 M = \Wl(l - \?-)* (23) 
 
 o O ( 
 
 is so nearly equal to the value which he derived originally that it may be 
 used instead; it is the value used in connection with Table 3 and stated 
 at the head of this table. 
 
 The moment coefficients for the column-head sections, mid-section, outer 
 sections, and inner section, as taken directly from the diagrams in Fig. 14, 
 Fig. 15, and Fig. 16, led to the gaps in the polygons in Fig. 17 and Fig. 18. 
 While a part of the discrepancy may be due to a slight error in the total 
 moment, and while it is possible that the main part of the discrepancy is 
 due to the unevenness in the distribution of moments at the edge of the 
 column capital, it \vas considered feasible to make adjustments by correct- 
 ing each bending moment in proportion to its size. That is, the percen- 
 tages of total moment indicated in Table 3 were left unchanged; they are 
 the original percentages based on areas measured in the diagrams in Fig. 
 14. Adjustments of the coefficients of moment per unit-width, stated in 
 Table 4, were introduced by a correction of the factors which are stated 
 at the head of the table, and which were used in transforming the coeffi- 
 cients M /wV, of the type used in the diagrams, into coefficients of the type 
 used in Table IV. 
 
 9. UNBALANCED LOADS ON FLAT SLABS. The load on one panel of a 
 flat-slab floor-structure has some influence on the stresses in the adjoining 
 panels. If a load which is originally uniformly distributed over all panels 
 is changed by removing or reducing the loads on some of the panels, the 
 stresses in the remaining panels will be increased in some sections and 
 decreased in others. The loads, by this change, become unbalanced. 
 Unbalanced loads cause the tangents across the panel edges to rotate; they 
 may produce bending moments in the columns or may cause the column 
 capitals to rotate about horizontal axes. Unbalanced loads on continuous 
 beams produce analogous effects; for example, the positive moments in one 
 span increase when the downward loads in the two adjacent spans are 
 removed. In the analysis of flat-slab structures the varying degree of 
 
 See Art. 4, footnote 17. 
 
 t J. R. Nichols, Discussion on reinforced-concrete flat-slab floors. Am. Soc. C. E., 
 v. 77, 1914, p. 1735.
 
 MOMENTS AND STRESSES IN SLABS. 
 
 37 
 
 stiffness of the columns must be taken into consideration; this stiffness of 
 the columns affects the stresses under unbalanced loads. 
 
 Fig. 19 and Fig. 20 show certain results of the study of unbalanced 
 loads. Further studies of the effects of these loads are made in connection 
 with Fig. 21 to Fig. 24. 
 
 Ratio of Loads 
 
 FIG. 19. BENDING MOMENTS IN OUTER SECTIONS DUE TO UNBALANCED 
 LOADS; DIFFERENT UNIFORM LOADS ON ALTERNATE Hows OF PANELS. 
 
 A flat-slab structure is considered in which each floor consists of a 
 large number of equal square panels. For the sake of convenience of 
 analysis the number of panels may be assumed to be infinite in all direc- 
 tions, as in the preceding article. All the columns are assumed to be alike. 
 Poisson's ratio is again assumed to be equal to zero. The small figures at 
 the bottoms of Fig. 19 and Fig. 20 show the loading arrangement on one 
 floor: each alternate row of panels carries the full uniform load, w per 
 unit-area, the other rows carry the reduced uniform load, w per unit-area.
 
 38 
 
 MOMENTS AND STRESSES IN SLABS. 
 
 The positive moments in the outer and inner sections shown in Fig. 19 and 
 Fig. 20 reach extreme conditions when w is as large as possible, and when 
 w is as small as possible, for example, when W Q Is equal to the dead load 
 only. The abscissas in Fig. 19 and Fig. 20 represent the ratio w /to of the 
 The left-hand edges correspond to w = 0, that is, every other row 
 
 50 
 
 loads. 
 
 A .6 
 
 Ratio of Load s ^/ 
 FIG. 20. BENDING MOMENTS IN INNER SECTIONS DUE TO UNBALANCED 
 LOADS; DIFFERENT UNIFORM LOADS ON ALTERNATE Rows OF PANELS. 
 
 of panels is entirely unloaded. The right-hand edges correspond to w =. to, 
 that is, uniform load tr on all panels as in the preceding article. The 
 ordinates in Fig. 19 and Fig. 20 represent percentages of the total moment 
 
 1 2 
 
 MO = - Wl (1 c) 2 ; M is the sum of the numerical values of the 
 
 8 o 
 
 moments in the outer sections, inner section, column-head sections, and mid- 
 section when the panel is loaded by w. The values indicated on the right-
 
 MOMENTS AND STRESSES IN SLABS. 
 
 39 
 
 hand edges, 21.0 in Fig. 19, and 14.0 in Fig. 20, are the percentages apply- 
 ing to the condition of uniform load over all panels. These two per- 
 centages are approximate values, taken from the last column in Table III 
 in the preceding article; they are nearly equal to the exact values, which 
 vary only slightly within the range of variation of c/l. The two upper 
 pencils of lines in Fig. 19 refer to the fully loaded panels, the panels loaded 
 by w; the two lower pencils refer to the panels which carry only the par- 
 tial load w o . Two extreme cases are represented, one by the two middle 
 pencils, the other by the two outer pencils. In one extreme case the col- 
 umns are perfectly rigid, and in the other the column capitals are perfectly 
 free to rotate about horizontal lines. The latter condition may be estab- 
 lished by introducing hinges in the columns directly below the column 
 capitals and directly above the slab. Actual slab-structures fall between 
 Hie two extreme conditions, which, therefore, are analyzed first. Methods 
 by which one may interpolate between the extreme cases will be indicated 
 later. 
 
 (a) (b) 
 
 FIG. 21. UNBALANCED LOADS PRODUCING MAXIMUM POSITIVE MOMENTS 
 
 IN THE SLAB; 
 
 (a) total applied load; 
 
 (b) component, + -^, of applied load. 
 
 (c) component, , of applied load; 
 
 According to Fig. 19, when w is zero, and when the columns are rigid, 
 the percentage of moment in the outer sections in the loaded panels ranges 
 from 20.3 for c =z OM to 23.5 for c 0.15L In the unloaded panels the 
 corresponding percentage is small, ranging from + 0.3 to 2.6, the latter 
 figure indicating a small negative moment. When the columns are free 
 to turn, the load on one panel has a marked influence on the moments in 
 the other panels. According to Fig. 19, when w is zero, the percentage 
 for the outer section ranges from 41.4 to 49.6 for the loaded panels, and 
 from 20.4 to 28.6 for the unloaded panels. The negative percentages 
 represent fairly large negative moments. The use of the diagram may be 
 illustrated by an example. Assume w QAw, c = 0.2l; then if = 
 
 121 2 
 
 - wl (I - - c) 2 = - wl* (1 - - . 0.2) 2 = 0.0939t0Z 3 . 
 
 The diagram gives 
 the following values of the moment in the outer sections: in the fully
 
 40 MOMENTS AND STRESSES IN SLABS. 
 
 loaded panels, 0.218M when the columns are rigid, 0.347M when the 
 column capitals are free to turn; in the partially loaded panels, 0.075Af 
 when the columns are rigid, 'o.053M w ^ en the column capitals are free 
 to turn. 
 
 Fig. 20, which refers to the inner section, is analogous to Fig. 19. 
 The percentage of moment varies in the same general manner as in Fig. 19. 
 
 The procedure of the analysis will now be discussed. The load consist- 
 ing of w and to on alternate rows of panels is denoted by w,w The 
 moments produced by this load are linear functions of w Q ; and when w 
 ia considered as a constant, they are linear functions of the ratio w o /w. 
 
 It follows that when the moments for the extreme values w =0 and 
 
 o 
 
 w =w are known, that is, the moments represented at the left and right 
 edges in Fig. 19 and Fig. 20 are known, then the moments for intermediate 
 values may be determined by linear interpolation, as represented by the 
 straight lines in the two figures. It remains, therefore, to make an analysis 
 for the load w,0, (or w and zero in alternate rows of panels) . This load, w,0, 
 may be resolved into two components by the scheme indicated in Fig. 21 
 and Fig. 23.* The load w,Q on each floor in Fig. 21 (a) is resolved into two 
 
 romponents: -f- , shown in Fig. 21 (b), uniformly distributed over all 
 2 
 
 panels; and -|- shown in Fig. 21 (c), consisting of the upward and 
 
 2* 2' 
 downward uniform loads t/;/2 on alternate rows of panels. The load 
 
 _ is anti-symmetrical with respect to the dividing edges, that is, the 
 
 2' 2 
 
 edges at which the load changes from -f- w '/2 to w/2. The structure 
 
 itself is symmetrical with respect to the vertical sections through these 
 
 edges. Consequently, under the load -}- _ the deflections at points 
 
 2' 2' 
 
 which are symmetrical with respect to the dividing edges are equal and 
 opposite; the dividing edges remain straight and undeflected; the moments 
 in sections which are symmetrical with respect to the dividing edges are 
 equal and opposite; and the moments at the dividing edges are zero. 
 
 When the column capitals are free to turn, and when their diameter 
 
 is gmall, the slab, under the load -f- will deflect within each row 
 
 2' 2' 
 
 of panels as if that particular row were separated from the rest of the 
 slab, and as if it were simply supported on girders at the two parallel 
 edges of the row. The moment per unit-width across the center line of 
 
 the row, accordingly, is -f- I" in the row loaded by -}- _. and - /' 
 
 89 O ' Q O 
 
 W 
 
 in the row loaded by _ Since the column capitals are in the regions 
 
 2' 
 near the points of inflection, a change in the size of the diameter of the 
 
 This scheme was used by Nielsen (Spaendinger i Plader, 1920, p. 192).
 
 MOMENTS AND STRESSES IN SLABS. 41 
 
 column capitals, even to the greatest size c = 0.31, has only a slight influ- 
 ence on the state of flexure of the slab under the load -|- as long 
 
 2' 2' 
 
 as the column capitals are free to turn. Minor local redistributions of the 
 moments occur near the edges of the column capitals as a result of this 
 change in the diameter of the column capital, but in the inner and outer 
 sections the influence of this change is negligible. That is, the values 
 
 .t 1 2 may be used without reference to the size of the column capital. 
 
 8 2 
 
 The method of calculation for the load w,Q, when the column capitals are 
 free to turn, may be shown by an example. Take c = 0.21. The corre- 
 sponding total moment is M = - wl 3 (1 - -. 0.2) 2 = 0.0939wZ 3 . The 
 
 8 3 
 
 W W 1 W I 
 
 moments in the outer sections due to the load + , ~ are =*= - I 2 - 
 
 wl 3 
 = = 0.333Af . The moment in the outer sections, due to the 
 
 w w 1 
 uniform load , , is - . 0.21 Af = 0.105A/ . The resultant moments in 
 
 t 
 
 the outer sections are then: in the loaded panels, 0.333A/ + 0.105A/ = 
 0.438Af o ; in the unloaded panels _ 0.333AT,, + 0.105A/ = _ 228A/ . 
 The percentages 43.S and 22.8 are shown at the left-hand edge in Fig. 19. 
 The other percentages represented at the left-hand edges in Fig. 19 and 
 Fig. 20 were calculated in the same manner. 
 
 The slab-structure with rigid columns and immovable column capitals 
 was analyzed as a statically indeterminate structure in which the turning 
 couples transferred from the slab through the column capital to the column 
 are introduced as statically indeterminate quantities. The structure with 
 column capitals free to turn is used in the analysis as a substitute structure. 
 In this substitute structure the column capitals turn under the influence 
 
 of the load _| _ but the slope of the column capitals may be 
 
 reduced to zero, and thus the substitute structure may be made to act like 
 the original structure, by applying a turning couple of the proper magni- 
 tude and direction at each column capital. The resultant moments in the 
 slab are found, then, by adding the moments produced by the turning 
 couples to those already existing. In order to find the magnitude of the 
 turning couples and the effects of them, a study was made of bending 
 moments and slopes produced by turning couples 21 of constant magni- 
 tude, applied at the column capitals. Results of this study are stated in 
 Table V(a). Tn order to derive these results the structure with column 
 capitals free to turn was replaced by a second substitute structure in which 
 the column capitals are removed altogether. This second substitute struc- 
 ture is the same point-supported slab that was used in the preceding article 
 in the study of the slab with the same load in all panels. The second sub- 
 stitute structure is loaded at the points of support by concentrated couples
 
 42 
 
 MOMENTS AND STRESSES IN SLABS. 
 
 21 and by certain additional concentrated loads which may be called ring 
 couples. A ring couple may be obtained by applying two equal and two 
 opposite ring loads, of the kind described in the preceding article, so close 
 together that the whole system of forces may be considered as a concen- 
 trated load. The concentrated couples acting alone do not produce uniform 
 slopes at the circles marking the edges of the column capitals; but uni- 
 formity of the slopes at these circles is restored by adding ring couples of 
 the proper intensity. The second substitute structure is thus made to act 
 like the first substitute structure, which is the slab with column capitals 
 which are free to turn relative to the columns. The influence of the ring 
 couples on the moments at the center line of the row is small; it is meas- 
 ured by the difference between the moments stated in the first column in 
 
 TABLE V (a). PENDING MOMENTS AND f LOPES DUE TO TURNING COUPLES 
 +21 APPLIED AT THE COLUMN CAPITALS. 
 
 The turning couples are clockwise and counter-clockwise in alternate rows of columns. 
 
 i is perpendicular to the rows, y is in the direction of the rows. The couples are in planes parallel to zt. 
 
 * = gpan; c = diameter of column capital; / = moment of inertia per unit-width; Poisson's ratio =0. 
 
 ell 
 
 
 
 0.15 
 
 0.20 
 
 0.25 
 
 0.30 
 
 Momenta-direction ( < f f *"** tO '" ' ' 
 
 0.886 
 1 081 
 
 0.898 
 1 073 
 
 0.907 
 1 066 
 
 0.920 
 1 057 
 
 0.936 
 1 046 
 
 per unit-width. | Average for width 1 
 
 1 
 
 1 
 
 1 
 
 1 
 
 1 
 
 Moments in v-direction { < J ^"^ * *' ' ' 
 
 0.294 
 248 
 
 0.282 
 240 
 
 0.273 
 233 
 
 0.260 
 224 
 
 0.244 
 213 
 
 per unit-width. | Average for length L ....... .. 
 
 
 
 
 
 0. 
 
 
 
 
 
 
 469; 
 
 472* 
 
 475? 
 
 478* 
 
 482* 
 
 
 531* 
 
 5281 
 
 0.525.' 
 
 522* 
 
 518* 
 
 Slope of column capital, in i-direction 
 
 
 1.081* 
 
 0.975* 
 
 
 0.834* 
 
 Approximate values of , calculated by the formula 
 i c 
 
 
 2EI 
 
 2E1 
 
 
 2EI 
 
 ' 2-0 82EI 
 
 
 1.037* 
 
 0.975* 
 
 
 . 854* 
 
 
 
 2EI 
 
 2EI 
 
 
 2EI 
 
 Table V(a) and the moments in the remaining columns. The moments and 
 slopes in Table V(a) were calculated by means of infinite series which are 
 solutions of Lagrange's differential equation; details of this analysis will 
 be given in the appendix. 
 
 The slope of the column capitals under the influence of the load 
 
 -f -, , or the load w, 0, is found to be, with close approximation, 
 
 wl 3 
 4&EI 
 
 ('"i(f) 1 )- 
 
 (24) 
 
 The values of the slope s of the column capitals under the influence of the 
 couples 21 are stated in Table V(a). The turning couples ~^ M c 
 which are necessary to make the resultant slope of the column capital
 
 MOMENTS AND STRESSES IN SLABS. 43 
 
 o 
 
 equal to zero is determined, then by the formula M C = 2 1. - (25) 
 
 8 
 
 M c is the resultant couple which is transferred through each column capital, 
 from the columns to the slab in the original structure, which is the struc- 
 ture with rigid columns and immovable column capitals. An example will 
 show the manner of computing M c and the resultant bending momenta 
 which are shown at the left-hand edges in Fig. 19 and Fig. 20. Tako 
 
 c = 0.2Z. Equation (24) gives S Q . 0.985. Table V(a) gives 
 
 9751 48EI 
 
 s = - . The couple transferred through each column capital is, then, 
 
 according to formula (25), M e = -= ^- .-- = 0.0842wZ 3 = 0.897A/ . 
 
 s 0.97o \2i 
 
 According to Table V(a) the moment in the inner section in the slab with 
 column capitals free to turn, produced by the turning couples ^ 21 is 
 =*= 0.525?. The corresponding moment produced by the turning couples 
 
 - 0.525Z . M c - 1 
 + M c is then + - - = + - . 0.525 . 0.897 M = + 0.236A/ . 
 
 I J-i 
 
 The signs, minus and plus, refer to the loaded and unloaded row of panels, 
 respectively. According to Fig. 20 the moments in the inner section in the 
 slab with column capitals free to turn, produced by the load w,0, are 
 0.403A/ in the loaded row of panels and 0.263Af in the unloaded row. 
 
 O 
 
 The resultant moments in the inner sections in the slab with rigid columns 
 are then: 
 
 in the loaded row of panels: 0.403A/ _ 0.236A/ = 0.167 M Q - 
 in the unloaded row of panels: _ 0.263M Q + 0.236A/ = -0.027 A/ . 
 The coefficients 0.167 and 0.027 are expressed as percentages and are 
 shown at the left-hand edge in Fig. 20. The remaining percentages belong- 
 ing to the two middle pencils in Fig. 19 and Fig. 20 were computed in a 
 similar manner. 
 
 Two extreme cases have been considered so far: one with perfectly 
 stiff columns and fixed column capitals; the other with columns which are 
 flexible or supplied with hinges at the ends, so as to allow the column 
 capitals to turn freely with the slab. Actual slab-structures have an inter- 
 mediate degree of rigidity of the column capitals. These structures can be 
 dealt with if a method of interpolation between the extreme cases can be 
 devised, so that one can say that a given case belongs, for example, 70 per 
 cent or 0.7 to one extreme case, and 30 per cent or 0.3 to the other. For 
 the purpose of the interpolation two definite ratios are introduced measur- 
 ing the degree of fixity of the column capitals and the degree of freedom of 
 the column capitals to rotate. These ratios are 
 
 k = fixity of the column capitals, 
 
 k' = 1 k = freedom of the column capitals to rotate. 
 These ratios are defined as follows: Let M denote the moment in a certain 
 section, A/, and A/ B the moments which would occur in the same section 
 if the column capitals were fixed and free to turn, respectively. The ratios
 
 44 
 
 MOMENTS AND STRESSES IN SLABS. 
 
 fc and k' are defined, then, as far as the particular section is concerned, by 
 the equations: 
 
 M**kM A +k'M Bt (26) 
 
 fc+fc'-l. (27) 
 
 For example, M = 130000 in. Ib, M A = 100000 in. Ib, and M B = 200000 
 in. Ib, gives k = 0.7, k' = 3 ; the column capitals may be said to be 70 
 per cent fixed and 30 per cent free to turn. M may be calculated by formula 
 (26) when M A , A/ B , k and k' are known. The limiting values of k and k' 
 are and 1; the combination k = 1, k' = represents the extreme case of 
 fixed capitals; the combination k = 0, k' = 1 represents the case of column 
 capitals which are free to turn. 
 
 The values of k and k' may be different at the different moment sec- 
 
 FKJ. 22. MOMENTS IN COLUMNS DUE TO UNBALANCED LOAD SHOWN IN 
 
 FIG. 21; 
 
 (a) columns without capitals; 
 (h) columns with capitals; 
 
 tions. But in certain important, rases, for example, in those shown in Fig. 
 21 and Fig. 23, /.- and k' are independent of the position of the moment 
 section; k and k', then, are constants belonging to this structure as a 
 whole. Let 
 
 M' ~ moment transferred from a column through the column capital 
 
 to the slab in the structure with intermediate rigidity of the 
 
 column capitals; 
 M moment transferred from a column through the column capital 
 
 to the slab in the structure with fixed column capitals. 
 Application of (2(5) to these moments gives 
 
 A/' c -W e (28) 
 
 where fr is the fixity k referring to the moments which are transferred 
 through the column capitals. Assume that a calculation by (28) leads to
 
 MOMENTS AND STRESSES IN SLABS. 
 
 45 
 
 the same value of k for all the column capitals within an area which 
 includes a large number of panels in botli directions. It may be shown 
 that A; in (26), under this condition, is the same for all sections and is 
 equal to the constant value k ; the moment M in any section may be 
 expressed as a linear function of the moments M' ', by substituting 
 M' _. k M , the moment M becomes a linear function of ke', according 
 to (26) and. (27) M is a linear function of k; the limits of k and k c 
 are the same, and 1 ; consequently, k = k as was to be proved. In 
 Fig. 21 and Fig. 23 M' and M are the same in all columns, except for 
 the direction clockwise or counter-clockwise; that is, in each slab, k is the 
 same at all column capitals and in all moment sections. The loading 
 arrangement in Fig. 21 produces the greatest possible moments in the outer 
 and inner sections; the arrangement in Fig. 23 produces large moments 
 in the columns. 
 
 1 
 
 L. 7 ,|, 7 -J. 7 I ^ > 
 
 (a) (b) (ci 
 
 FIG. 23. UNBALANCED LOADS PRODUCING LARGE MOMENTS IN COLUMNS; 
 
 (a) total applied load; 
 
 (b) component, + -^, of applied load; 
 
 (c) component, , of applied load. 
 
 
 
 The analysis is facilitated by resolving the loads ivQ shown in Fig. 
 
 21 (a) and Fig. 23 (a) in each case into two components: namely,^-, , 
 
 2t 
 
 shown in Fig. 21 (b) and Fig. 23(b), and !fl, - } shown in Fig. 21 (c) and 
 
 
 
 Fig. 23 (c). To make the analysis simple, the dimensions, including the 
 diameter of the columns, are assumed to be the same through several 
 stories. 
 
 The dimensions and the loads are denoted as in Fig. 21 and Fig. 23. 
 Fig. 22 and Fig. 24 show diagrams of the bending moments in the columns; 
 they show also the notation for these bending moments. The moments of 
 inertia of the sections are 
 
 T = moment of inertia of the cross section of the slab for the whole 
 panel width, 
 
 7 = moment of inertia of the columns. 
 
 A simplified case is considered first, in which the slab-structure is 
 replaced by a frame whose girders and columns have the same moments of 
 inertia, /' and J, as the slab structure; at the joints there are no column
 
 46 
 
 MOMENTS AND STRESSES IN SLABS. 
 
 capitals, but the connections between the four adjoining members are rigid. 
 Tin: moment diagrams of the columns are shown in Fig. 22 (a) and Fig. 
 24 (u). Analysis, for example, by the method of least work, by the method 
 of tlio substitute structure, or by the slope-deflection method, leads to the 
 following values: 
 
 In Case I, Fig. 21 and Fig. 22 (a), with 
 
 K 
 
 me finds 
 
 that is, 
 
 Jl 
 
 I'h' 
 
 .wl* d_ 1 \ 
 
 -24 I 1 I+K) 
 
 k - 1 - 
 
 1+K ' 
 
 1+K 
 
 (29) 
 (30) 
 (31) 
 
 to) (b) 
 
 Fin. 24.- MOMENTS IN COLUMNS DUE TO UNBALANCED LOAD, SHOWN IN 
 
 FIG. 23; 
 
 (a) columns without capitals; 
 
 (b) columns with capitals 
 
 In Case II, Fig. 23 and Fig. 24 (a), again with 
 
 Jl 
 
 K - 
 
 one finds 
 
 that is, 
 
 24 
 
 I'h" 
 1 - 
 
 (32) 
 (33)
 
 MOMENTS AND STRESSES IN SLABS. 47 
 
 In the slab-structure the bending moments are not uniformly distrib- 
 uted over the width of the sections. One may say, that the moment of 
 inertia /' of the section is not fully effective. The presence of the column 
 capitals, on the other hand, has the effect of reducing the clear spans and 
 increasing the rigidity of the structure. The slopes or angles of rotation 
 of the column capitals in Fig. 21 and Fig. 22 (b) must be equal or equal 
 and opposite. The moment diagram of the cylindrical part of the column 
 between the top of the slab and the bottom of the column capital above, 
 must have its centroid, therefore, at the center of the total distance meas- 
 ured between the centers of the slabs. In Fig. 2-1 (b) the point of inflection 
 is at the center of the cylindrical part of the column. The dimensions of 
 the moment diagrams in Fig. 22 (b) and Fig. 24 (b) have been computed 
 without considering the influence of the thickness of the slab upon tin; 
 stiffness of the column. This influence may be taken into account by 
 measuring h from the top of the slab to the bottom of the column capital 
 above, i from the bottom of the column capital to the bottom of the slab; 
 7t'= h -f- i is, accordingly, the clear distance from the top of one slab to 
 the bottom of the slab above. 
 
 The slope of the column at the capital must be equal to the angle of 
 rotation or slope of the capital. In terms of the moments in the columns, 
 the slopes, Q, of the columns at the capitals are 
 
 irwr in Fi S- 22 (b) ' - wr in Fig 24 (b) ' ( 34) 
 
 i fLJ O C<J 
 
 In terms of the moments 2X, transferred from the columns through the 
 capitals to the slab, and in terms of the applied load w,0 or _j_ ^ _ }-. f 
 
 ^- 
 
 the angle of rotation, or slope, of the column capitals in Fig. 21 to Fig. 24 
 may be expressed approximately as follows, as may be seen by comparison 
 with the last line in Table V(a) and with formula (24) : 
 
 * 
 
 2 -0.80 -1.02E7' ' 48-1.02#/' 
 
 By equating Q to Q> and substituting the values of X given on the diagrams 
 in Fig. 22 (b) and Fig. 24 (b), in terms of X', the following values; are 
 obtained : 
 
 In Case I, Fig. 21 and Fig. 22 (b), one finds 
 
 x ' ~ 24 !~" + / 1+3 i\ 2 "A "_~c\"_ : JT ;;; (30) 
 
 The fixity k of the column capitals is the ratio of this quantity to the. value 
 of the same quantity when J = oo that is, 
 
 k -1 - -JL A/ L_ (S7J 
 
 l+K' l+K'
 
 48 MOMENTS AND STRESSES IN SLABS. 
 
 where 
 
 (38) 
 
 0.80-1.02 I'h 
 
 In case II, Fig. 23 and Fig. 24 (b), one finds 
 
 Jl 
 x , wl 3 1. 021' h , 39) 
 
 " 24 1 j. (\ +1\ /I - \ JZ 
 
 3 + \ /i/\ F/ 0.80 -1.027'/i 
 
 The fixity k of the column capitals is determined as in Case I by comparing 
 X' with its value when J = oo one finds 
 
 A- = 1 k' = (40) 
 
 1 + 3K ' 1 + 3K ' 
 
 where 
 
 j _ _\ IL/__ j__ LZ. " (41) 
 
 0.80- 1.02 7'/t 
 
 When the fixity has been computed, the maximum moment X' in the cylin- 
 drical part of the column may be computed as 
 
 X' _^?0fc_ 
 
 24 
 
 Sample calculations of freedom of the column capitals to rotate, fixity 
 of the capitals, and moments in the columns, according to formulas (26) 
 to (42), are shown in Table V(b). In examples (1) the columns are rather 
 slender, in examples (2) and (3) they are comparatively stiff. The mate- 
 rial is assumed to be homogeneous. The dimensions i should be measured, 
 as stated before, as the vertical distance between the bottom of the column 
 capital and the bottom of the slab. In example ( 1 ) , Case I, the column 
 capitals are found to be about 28 per cent fixed, 72 per cent free to turn. 
 Fig. 19 and Fig. 20 show that with these degrees of fixity and freedom to 
 rotate, unbalanced loads will have a considerable influence on the bending 
 moments. In examples (3), Case I, the fixity is 90 per cent; that is, the 
 moments in the inner and outer sections of the loaded panels will not differ 
 greatly from the moments under uniform load. 
 
 It should be noted that those approximate values of fixity, freedom to 
 rotate, and moments in the column, which are computed by considering 
 the structure as a frame, do not differ greatly from the corresponding 
 values in the lower part of Table V(b), which are calculated more exactly. 
 One may conclude that other flat-slab structures may be analyzed, in most 
 cases with a satisfactory approximation, as far as the moments in the 
 columns are concerned, by methods applying to frames. For example, the
 
 MOMENTS AND STRESSES IN SLABS. 
 
 49 
 
 TABLE V(b). SAMPLE CALCULATIONS OF THE RIGIDITY OF COLUMN 
 
 CAPITALS. 
 
 T/re /rjcr/e/-/er/ /s ersstfnee/ fe 6e Aosnevf/reot/s. Tftf 
 /octets ore /tre//ictrtK/ //j /-/f.2**/ and f/a.JtSf^f. 7#f 
 of r/te co/umns ane ' 
 
 moment of inert/a of s/ot> />er /*ne/ nf/'t///if 
 moment of /nerf/a of co/t/mns; 
 fixity of coAumrt c<yo/7a/s ; 
 'freeeto/rr o/ co/t/nrn caprta/s to 
 
 //O. 
 
 fv 
 
 ofco/umr. 
 
 / Sfory 
 r/jett o 
 0/amfffr of co/umn. </ 
 
 O.ffOl 
 
 s/nucfufv tv/'/A c 
 
 6.Z3 
 
 Case I 
 
 o.r/s 
 
 0/37 
 0.4&3 
 
 O 096 
 O.904- 
 
 0459 
 
 0.03+ 
 
 ffo/aevt///7 co/um/i X-A'- 
 
 3. /7a/--S/ao structure 
 c 
 
 i 
 /r 
 
 aosz 
 
 0.31 
 O./02 
 
 O.Z2 
 
 0.041 
 07/1 
 
 0.32 
 0-662 
 
 O.Z2 
 0.462 
 
 0.32 
 
 0091 
 
 0-4/1 
 
 Case I 
 
 K = 
 
 0.7Z0 
 O.Z00 
 
 OL350 
 
 0.7 4/ 
 
 6.ZZ 
 
 0./S4 
 
 0.36Z 
 
 .69 
 
 0.09ft 
 0.904 
 
 0O98 
 O.902. 
 
 /ff.36 
 
 rtf. 24T6 
 
 0.4S/ 
 O. "49 
 0. 02.17 
 
 0.47/ 
 
 0.049 
 
 O.OZ27 
 
 0.0375' 
 
 0.948 
 O.O397 
 
 30./O 
 0032 
 0.968 
 O.O37/ 
 
 0033 
 O 967 
 O.O373
 
 50 MOMENTS AND STRESSES IN SLABS. 
 
 slope-deflection method * may be used to advantage when the design is less 
 uniform than assumed in Table V(b) ; for example, when the diameter of 
 the columns is different in the different stories. 
 
 So far, attention has been given mainly to the moments due to unbal- 
 anced loads in the outer and inner sections. The negative moments in the 
 column-head sections and mid-section are affected less by unbalanced loads 
 than are the positive moments. In a frame structure the maximum nega- 
 tive moment in the second-floor girder between the third and fourth span 
 occurs with the following combination of loaded and unloaded spans; W 
 is the total load on one span: 
 
 Loads on Span Xo. 
 123456 
 
 4th fl.x>r O W W O 
 
 3rd floor W O O W 
 
 2nd floor BOWWOW 
 
 1st floor W O O W 
 
 With span I, total story height h', and moments of inertia I' of the girders 
 and J of the columns as before, the negative moment in the second-floor 
 girder between the third and fourth span becomes, as may be verified by 
 the slope-deflection method, 
 
 _ _ / " . __T . . 
 
 (43) 
 
 where K = IJL as before (formula (29)). This moment is found to be 
 approximately equal to 
 
 TI7 II f\ A \ 
 
 (44) 
 
 When J = oo the moment becomes Wl/24. Or, the ratio, Q, of increase of the 
 negative moment by a change to columns of finite stiffness is approximately 
 
 Q = l + T 4^ = l+0.4fc', (45) 
 
 1 + K 
 
 where k' is the freedom of the column capitals to rotate under the loading 
 arrangement in Fig. 21. The ratio Q may be assumed to apply approxi- 
 mately to the negative moments in flat slabs. For example, k' = 0.72, as 
 given in the first column in Table 5(b), gives Q = 1.29, or 29 per cent 
 increase, while k' = 0.10 gives an increase of 4 per cent. 
 
 The case in which only a single panel is loaded is of no particular 
 importance as a condition for design; greater bending moments are pro- 
 duced by a uniform load on all panels or by loading in rows than by single- 
 panel loading. A number of tests have been made, however, with a single 
 panel loaded, and the case should be investigated for the purpose of the 
 study of these tests. 
 
 See Wilson, Richart, and Weiss, Analysis of Statically Indeterminate Structures, 
 by the Slope Deflection Method, Univ. of 111. Eng. Exp. Sta. Bull. 108, 1918; Hool 
 and Johnson, Concrete Engineers' Handbook, 1918, p. 411, p. 629.
 
 MOMENTS AND STRESSES IN SLABS. 51 
 
 When the columns are stiff, the single-panel load produces approxi- 
 mately the same effects in the panel as a "checker-board load," by which 
 the panels corresponding to the black and white fields of a checker-board 
 are loaded and unloaded respectively.* The checker-board load leads to a 
 simpler analysis than the single-panel load. The checker-board load W, 
 
 may be resolved into two components, one, . .. uniformly distributed 
 
 9 9 
 
 ~\A7 \\f 
 
 over all panels, the other, -f - , -~, positive and negative in alternate 
 
 
 
 panels. The first component produces one-half the moments derived in 
 Art. 8. The second component, on account of the anti-symmetry leaves the 
 panel edges straight; that is, if the column capitals are small, each panel 
 deflects as a single square panel which is simply supported on four sides 
 and loaded by + W/2 or W/2. 
 
 A study is made of the equilibrium of one-half of a loaded panel. This 
 half-panel is a rectangle with two sides of length I, containing the column- 
 head sections, mid-section, outer sections, and inner section, and two sides 
 of length 1/2. The moments are taken about the edge containing the mid- 
 section and column-head sections. The following results were found for the 
 point-supported slab with checker-board loading W, O, the moments being 
 
 expressed in terms of the total moment M = Wl: 
 
 Moment in inner section: 0.13A/ ; 
 
 outer sections: 0.13A/ o ; 
 
 mid-section: 0.09A/ ; 
 
 column-head section: 0.24A/ . 
 
 Total for the moment sections: 0.59A/ 
 
 Moments of vertical shears plus torsional moments 
 
 in the short sides of the rectangle: 0.41M 
 
 Total: \~WM~ 
 
 Moment of applied load W/2: _ 1.00M 
 
 Total moment about edge containing mid-section : 0. 
 
 According to these results 41 per cent of the total moment leaks out 
 by shear and torsion in the short edges of the rectangle. In deriving the 
 moments in the various sections use was made of the results, stated in 
 Art. 8, Table 3, obtaiued for a flat slab with all panels loaded; of the value 
 stated in Fig. 3 (a), of the moment at the center of a square slab simply 
 supported on four sides; and of the value of the moments and shears at 
 other points of a square slab simply supported on four sides, stated by 
 Leitz.f In slabs with column capitals the proportions of moments may be 
 assumed to be approximately the same as in the slabs with point supports, 
 provided the column capitals are not too large. Or, by substituting 
 
 1 *? r 
 
 M _ --Wl(l -)- one may use the expressions just stated for the 
 8 of 
 
 * See Nielsen, p. 196, where this case is invaitigated. 
 t See Nielsen, p. 133.
 
 52 MOMENTS AND STRESSES IN SLABS. 
 
 point-supported slab, as approximate expressions for moments in the slab 
 with column capitals. 
 
 10. MOMENTS IN WALL PANELS, CORNER PANELS, OBLONG INTERIOR 
 PANELS, AND PANELS WITHOUT BENDING KESISTANCE IN THE MID-SECTION. 
 Table VI contains approximate values of moments in panels of several 
 different types. The results of the computations are found in the column 
 next to the last in the table. The calculations are based on Nielsen's * 
 work, and they are of an approximate character. In the analysis of these 
 cases by difference equations Nielsen used a rather large value of the 
 side X of the elementary square, and he assumed point supports instead 
 of supports on column capitals. It was his scheme that the values deter- 
 mined in this manner should be used as a basis of comparison between the 
 different cases; this use has been made of his results in Table VI. The 
 exterior panels dealt with in Table VI are assumed to be simply supported 
 along the walls on rigid lintel beams. 
 
 An example will illustrate the method followed in computing Table VI. 
 Take sections D and E in the third case in Table VI, that is, the case of 
 two adjacent rows of wall panels with lintel beams. Nielsenf gives the 
 following coefficients M/wl- of moment per unit width when Poisson's ratio 
 is zero: at the center of D, a t = 0.0861; at the center of E, Cj = 0.0661; 
 midway between, 6j = 0.0725. The spaces between these points are equal to 
 the side \ = i/4 of the elementary square which was used in the difference 
 equations. The three coefficients are to be interpreted as the altitude of 
 three rectangles, placed together as in Fig. 25 at the left. The three rec- 
 tangles constitute the moment diagram in approximate form. A more 
 nearly correct diagram is obtained by drawing a smooth curve which has 
 the altitudes of the rectangles as average ordinates in the three intervals 
 covered by the rectangles. The curved diagram is now replaced, as shown 
 at the right in Fig. 25, by two rectangles whose altitudes are average 
 ordinates within the two intervals covered by the bases. The formulas 
 stated in Fig. 25 for these altitudes apply approximately, and they were 
 used in the calculation of the table. In this manner the following coeffi- 
 cients of moment per unit-width are found in sections D and E : 
 in the outer section D, 
 
 ai + bi Q! - ci 0.0861 + 0.0725 0.0861 - 0.0661 
 
 ~2~~ ~~15~ ~~2~~ ~T5~~ 6 ' 
 
 and in the inner section E, 
 
 61 + c, a. - c, 0.0725 + 0.0661 0.0861 - 0.0661 
 
 ~T ~T5~ ~T~ ~l5~ 
 
 The corresponding moments per unit width, O.OS06W and 0.0680W, where 
 W = icV, and the average value 0.0743W applying to section F, are stated 
 in Table VI. By applying the same method, with the side of the elementary 
 
 * N. J. Nielsen, Spaendinger i Plader, 1920; the five cases represented in Table 6 
 are dealt with in his work on pages 210, 189, 212, 217, and 205, respectively, 
 t Nielsen, p. 214.
 
 MOMENTS AND STRESSES IN SLABS. 
 
 53 
 
 TABLE VI. MOMENTS IN OBLONG PANELS, WALL PANELS, CORNER PANELS, 
 AND PANELS WITHOUT BENDING RESISTANCE IN THE MID-SECTION. 
 
 -* load per panel (uniform over a// panels.). M -B Wl(i-$ )* 
 ' A Wld-3 V*. Ma- 1 Wa(/~ )t Mt-AWtif-J 
 
 Types 
 of 
 panels 
 
 Sect/on 
 
 Length of 
 secf/on 
 
 Momenf.rln, 
 perunifwidfh 
 in Me Is en's 
 ana/ys/s 
 (point supports, 
 A=/^ or /) 
 
 Rat 10. a, 
 of/% 
 to the 
 Mnfor 
 normal 
 panel 
 
 Deduced 
 MomentM 
 
 f" Ma '(J> 
 
 Mf 'moment 
 innonTMl 
 panel] 
 
 L oca t ion 
 of 
 sections 
 
 Square interior 
 
 Normal 
 
 Col-head 
 Mid 
 Total ne^ 
 Outer 
 Inner 
 Totalpos 
 
 A 
 B 
 C 
 
 D 
 E 
 
 r 
 
 kl 
 zl 
 I 
 kl 
 il 
 I 
 
 -O.I067W 
 -0.0495W. 
 -0.0781 W 
 0.057$ W 
 
 6.0469W 
 
 1.00 
 I.OO 
 I.OO 
 1.00 
 I.OO 
 I.OO 
 
 -0.48M 
 -O.I 7 M 
 -0.65Mo 
 O.ZIMo 
 
 JC 
 
 JM1[ III] 1 M ! 1 1 ][ ! * ! 1 11 1 i ' j i JI 
 
 1 
 
 ' 1 ^ 7_^ 
 
 ^-L **- 1 f 
 3 'E |c F 
 
 Without bend- 
 Ing resistance 
 across mid 
 section 
 
 Cohheact 
 Mia 
 Outer 
 Inner 
 
 A 
 B 
 D 
 E 
 
 \l 
 kl 
 kl 
 
 
 1.23 
 0.00 
 1.10 
 129 
 
 -0.59Mo 
 0.00 
 O.Z3M 
 O./8M 
 
 Square exterior 
 
 Two 
 adjacent 
 rows of 
 wal/ 
 pane/s 
 with 
 lintel 
 beams 
 
 Col-head 
 Mid 
 Jotalneg. 
 Outer 
 inner 
 Tota/pos 
 Lintel ( 
 beams\ 
 Col-tead 
 Mid 
 
 Totalneg. 
 Outer 
 Inner 
 
 Tofa/pos 
 
 A 
 
 B 
 C 
 D 
 
 r 
 
 L 
 M 
 
 A' 
 
 B: 
 
 r> 
 
 u 
 
 V 
 
 E' 
 
 E; 
 
 F' 
 
 \ 
 
 il 
 kl 
 I 
 
 il 
 
 I 
 il 
 kl 
 
 n 
 i 
 
 -O.I 380 W 
 -0.0646 W 
 -0.101 3 W 
 0.0806W- 
 0.0680W 
 0.0743W 
 -O.035 W 
 0.01 76 W 
 -0.1133 W 
 
 - 0.0033 W 
 -0.0456W 
 0.060/W 
 O.O&Z W 
 0.003ZW 
 0.0269W 
 
 I.Z9 
 I.3O 
 I.3O 
 1.39 
 1.89 
 1.58 
 
 1.06 
 0.66 
 
 0.58 
 1.04 
 0.62 
 
 0.57 
 
 -0.22 M^ 
 
 -OMfH> 
 
 6.<55fil 
 
 -0.035WI 
 O.OI76WI 
 -0.5lr1o 
 -O.llMo 
 -O.OlMo 
 -O.38M 
 02ZM, 
 
 0.09Me 
 
 0.0 1 Mo 
 O.SOM* 
 
 
 III ! ', ! I! ' ' ' ME, ' : '[Ilfl 
 
 \ 
 
 - (. 
 
 -^c 
 
 ~rfc- 
 
 ki""uj ft ^ 
 
 li.. 
 
 .b J 
 
 Four 
 adjacent 
 corner 
 
 panels 
 with 
 lintel 
 beams 
 
 Co/.-head 
 Mid 
 
 Outer 
 Inner 
 
 A 
 B 
 B, 
 D 
 E 
 E 
 d 
 e 
 f 
 
 i 
 
 unit 
 unit 
 unit 
 unit 
 unit 
 unit 
 
 -0.1264 W 
 -0.0302 W 
 -0.0008 W 
 0.074OW 
 0.0446 W 
 0.0121 W 
 0.0833 W 
 0.044-7 W 
 
 0.066 W 
 -O.O66 W 
 
 1.18 
 O.6I 
 
 I.Z8 
 1.24 
 
 -O.57Mo 
 -aLOMo 
 -Q002Mo 
 
 o.nrio 
 
 0.04 Mo 
 
 f]df)w(?~ 7/^1 
 
 .t^rt/ffUr *Jc/ 
 
 
 
 Ehi-SH-g) 
 
 :^ A: 
 
 .4-ei 
 
 J 1 ., 
 
 
 Ob/ong 
 inter/or 
 panels 
 a '= I 5 
 
 Col-head 
 Mid 
 Totalnea 
 Outer 
 Inner 
 Totalpos 
 Cathead 
 
 Mid 
 Tota/n^ 
 Outer 
 Inner 
 Total pas 
 
 A 
 B 
 C 
 D 
 E 
 F 
 A' 
 B' 
 C' 
 D' 
 E 
 F 
 
 ka 
 ka 
 a 
 za 
 
 a 
 
 ib 
 
 Z b 
 tb 
 zb 
 b 
 
 -0.1248 wb z 
 -0.0314 Wb* 
 -0.0781 Wb* 
 00696 wb z 
 0.0242 wb* 
 0.0469 wb* 
 -0.10/2 wa* 
 -0.0608 wa* 
 -0.0810 wa* 
 0.0473 WO* 
 00407wa z 
 00440wa* 
 
 W8' 
 0.77' 
 1.00" 
 1.24" 
 0.65' 
 /.OO" 
 0.94' 
 1.20* 
 I.OO* 
 0.90* 
 1,16' 
 WO* 
 
 -0.52Mb 
 -0.13Mb 
 -0.65Mb 
 
 o.zeMi, 
 
 0.09Mb 
 0.35Mb 
 -0.43M, 
 
 -6.66 Ml 
 
 6. 1 6 Ma 
 
 i^ ^P -A 
 
 T r? 
 ]Wjj 
 
 * Ratio of M to tneM for norma/ pane/ w/fti 2*0 or I
 
 54 
 
 MOMENTS AND STRESSES IN SLABS. 
 
 square again equal to Z/4, to the normal interior panel, one finds the 
 moment in the outer section D equal to 0.0579 W. The ratio of increase for 
 the wall panel is calculated, then, as q = 0.0806/0.0579 = 1.39 (see fifth 
 column for q). If this ratio is correct, the moment in section D in the 
 wall panel, when the columns are replaced by point supports, may be com- 
 puted as M " qM q where M is the moment in section D in the normal 
 
 panel, that is, M = 1.39 '0.21 1 0.29 Z - When the point supports 
 
 w i 
 are replaced by columns with capitals, one may expect that the factor 
 
 1 1 r 1 
 
 is to be replaced by M' = - Wl (1 _ 1)2. the coefficient i occurs in 
 
 8 3 I 3 
 
 o 
 this expression instead of the usual because the clear span in this 
 
 o 
 case is I %c instead of the value I c found in the interior panels. 
 
 The moments in the lintel beam in sections L and M are computed 
 under the assumption that the beam is continuous, with supports opposite 
 the columns. 
 
 ^v 
 
 
 < 
 
 1 = 
 
 T 
 
 \ 
 
 X, 
 
 \ 
 
 r 
 
 -Q 
 
 
 ' ' 
 
 ~ - 
 
 g+b+d^c -T- 
 
 ' 2. is \b+c a-c 
 
 I 
 
 
 
 <j 
 
 
 z is 
 
 i 
 
 
 
 i 
 
 
 i i 
 
 FIG. 25. DIAGEAM INDICATING RELATIONS BETWEEN AVEBAGE OKDINATES. 
 
 The values stated for panels without bending resistance across the mid- 
 section were estimated on the basis of a similar case treated by Nielsen, in 
 which there is no bending resistance in four-fifths of the mid-section. The 
 proportions are nearly equal to those given by Nielsen. The side of the 
 elementary square used in this case was \ = 1/5. 
 
 The oblong interior panels have sides a and &, with a = 1.55. The 
 elementary squares used by Nielsen had the side \ = 6/4 = a/6- The 
 moments M in the short sections A', B', D', and E' were determined in the 
 manner described in sections D and E in the wall panels. The average 
 moment per unit-width in each of the sections A, B, D, and E was deter- 
 mined as the average of three values given by Nielsen, one for each ele- 
 mentary square adjacent to the section. The proportions of negative and 
 positive moments M found in this way are approximately the same as in 
 the normal panel. If they can be assumed to be exactly the same, the 
 total moments in the sections C, C', F, and F' become _065A/ 6 , 0.65A/ a , 
 0.35Af , and 0.353/ , respectively, as stated in the column for M. These 
 total moments are assumed to be distributed between the column-head sec- 
 tions and mid-sections, outer sections and inner sections according to the
 
 MOMENTS AND STRESSES IN SLABS. 
 
 55 
 
 proportions between the values of M n , For example, one finds under this 
 assumption in the column-head section: 
 
 - 0.1248**.. L ( _ 
 
 M - 
 
 0.52M, 
 
 - 0.0781u>& 2 a v -.~w b) - v,.u^ ft . 
 The remaining values of M were computed in the same manner. The 
 ratios of M to the corresponding M in the normal panel were computed 
 afterward. 
 
 The moments in the oblong panels are represented graphically in Fig. 
 26, where the abscissas are values of the ratio V*i of the spans,; Z, is the 
 span in the direction in which the moment is taken. The points represent- 
 ing the moments in Table VI lie on or near the two straight lines and the 
 
 -45 
 
 I* 
 
 \30 
 $ 
 
 V* 
 
 *1 
 
 ! /5 
 /* 
 
 ^lu&g&L 
 
 ms.tt% 
 
 VTufcl 
 
 _f^f^^ 
 
 r&&M^~ 
 
 oiz 
 
 0.4 
 
 i.o 
 
 O.6 O.8 
 
 Ratio of Spans, 
 
 FIG. 26. MOMENTS IN OBLONG PANELS. 
 
 iz 
 
 1.4 
 
 two parabolas which are shown, with their equations, in Fig. 26. These 
 equations may be used as formulas for the moments in the various moment 
 sections in oblong panels. They show, as might be expected, that the 
 moments are more nearly uniformly distributed over the short sections 
 than over the long sections. 
 
 Nielsen derived the following values of the reactions of the columns: 
 in the case of two adjacent rows of wall panels, R = 1.203W; in the case 
 of four adjacent corner panels, R = 1.313W. 
 
 Nadai * derived the following values of the moments per unit width 
 for a single square panel, which is simply supported on point supports at 
 the four corners; he assumed Poisson's ratio equal to 0.3; at the center of 
 the panel 3/ = 0.1115W, at the center of the edge in the direction of the 
 edge, M = 0.151TF, the latter being the maximum moment in the panej. 
 
 NAdai, p. 70.
 
 56 MOMENTS AND STRESSES IN SLABS. 
 
 III. RELATION BETWEEN OBSERVED AND COMPUTED TENSILE 
 STRESSES IN REINFORCED CONCRETE BEAMS. 
 
 BY W. A. SLATER. 
 
 11. GENERAL CONSIDERATIONS. It has been generally recognized that in 
 the tests which have been made on reinforced-concrete floors the measured 
 tensile stresses in the reinforcement do not account for all of the moments 
 which are applied to the slab. This has been especially apparent in the 
 cases in which the measured stresses were low. In the tests of flat slabs the 
 coefficient of the resisting moment of the measured steel stresses has been 
 found to increase as the measured stresses increased. This increase in the 
 coefficient indicates that, for the low loads at least, the tensile stress 
 accounts for only a portion of the applied bending moment. Table VII 
 quotes published results which show that for different loads, the difference 
 in the proportion of the total moment which is accounted for by the 
 measured tensile stresses is likely to be considerable. It is very desirable 
 that a means be found for determining how great this difference is, but the 
 slab tests cannot be used for this purpose since the coefficient of the bending 
 moment is the main thing that is in question in the slab tests. 
 
 To show the relation between the bending moment and the resisting 
 moment in beams a study of observed tensile stresses in 84 reinforced- 
 concrete beams tested at an age of 13 weeks has been made. These beams 
 were tested by the Technologic Branch of the United States Geological 
 Survey at St. Louis about 1905 to 1908, and have been reported* in Techno- 
 logic Paper No. 2 of the Bureau of Standards, by Humphrey and Losse. 
 All the test data of those beams here presented may be found in that paper. 
 The results of this study are shown in Fig. 31 and 32 as a relation between 
 the observed and the computed stresses, but it is evident that the relation 
 of the observed stress to the computed stress is the same as the relation of 
 the computed moment of the observed tensile stresses to the applied moment. 
 Test results from the same source and from other sources were used by 
 Professor Hatt in a study of the relation of the computed moment of the 
 measured tensile stresses in the reinforcement to the applied moment for 
 beams and slabs. Professor Hatt found that beam tests from different 
 sources gave different results, and the writer also has found this to be true. 
 It is shown, however, in Professor Hatt's paperf that there is a certain 
 degree of conformity between test results from a considerable variety of 
 sources, and the writer feels justified in using the beams of Technologic 
 Paper No. 2 for the purpose of determining a law which will serve as a 
 basis for the comparative study of the moment in slabs. 
 
 12. LIMITATIONS TO GENERAL APPLICABILITY OF THE TEST RESULTS 
 While it is recognized that there are limitations which must be observed in 
 the application of these results to other conditions than those under which 
 they were obtained, it is believed that, because of the wide range which the 
 
 See alsn Bulletins 329 and 344, U. S. Geological Survey. 
 
 t W. K. Hatt. Moment Coefficients for Flat Slab Design with Results of a Test 
 Proceedings American Concrete Institute, Vol. 14, p. 165 (1918).
 
 MOMENTS AND STRESSES IN SLABS. 
 
 57 
 
 tests cover, the limitations are less serious than would be those of any other 
 tests which might have been used for this purpose. 
 
 For the beam tests there are reasons for expecting a difference between 
 the observed stresses and the computed stresses. In a cracked beam the 
 stress at the cracks may approach the computed stress, but between the 
 cracks the concrete assists so greatly in carrying the stresses that the 
 average measured unit-deformation over the gage length is likely to be 
 considerably less than the maximum unit-deformation, especially at the 
 lower loads. It is possible also that even at the section where a crack 
 occurs a portion of the moment may be resisted by the tensile stresses in the 
 concrete. There probably are other reasons for the differences though it is 
 believed that these are the most important. 
 
 By taking measurements of deformation over a short gage length 
 within a longer gage length on the same reinforcing bar it has been shown* 
 that higher stresses exist at certain places than are indicated by the average 
 measured deformation. The fact that in the beams used as the basis of the 
 present study failure occurred at observed stresses which were somewhat 
 
 TABLE VII. UNCORRECTED MOMENT COEFFICIENTS FROM 
 PUBLISHED REPORTS. 
 
 (Sum of coefficients for positive moment and negative moment.) 
 
 Test. 
 
 Load, 
 Ib. per 
 
 Coef- 
 ficient. 
 
 Load, 
 Ib. per 
 
 Coef- 
 ficient. 
 
 Load, 
 Ib. per 
 
 Coef- 
 ficient. 
 
 Load, 
 Ib. per 
 
 Coef- 
 ficient. 
 
 Reference. 
 
 
 sq. ft. 
 
 
 sq. ft. 
 
 
 sq. ft. 
 
 
 sq. ft. 
 
 
 
 Shredded Wheat Factory 
 
 56 
 
 0363 
 
 120 
 
 0356 
 
 191 
 
 0439 
 
 
 
 Proc. A. C. I., 1914, 
 
 
 
 
 
 
 
 
 
 
 p. 404. 
 
 Worcester Test Slab 
 
 102 
 
 026 
 
 215 
 
 049 
 
 
 
 
 
 Univ. of Illinois Eng. 
 
 
 
 
 
 
 
 
 
 
 Exp. Sta. Bull. 84, 
 
 
 
 
 
 
 
 
 
 
 p. 103. 
 
 Purdue Test, "J" Slab.. 
 
 150 
 
 0.0148 
 
 300 
 
 0.0246 
 
 450 
 
 0.0400 
 
 600 
 
 0.0539 
 
 Proc. A. C. I., 1918, 
 
 
 
 
 
 
 
 
 
 
 p. 181. 
 
 lower than the known yield point of the steel is another indication that even 
 at the high loads the real stresses were greater than the observed stresses. 
 
 There is another limitation to the interpretation of test results ot 
 slabs, which the results of the study of the beam tests does not help to 
 remove. This limitation lies in the fact that in the measurement of tht 
 deformation in a slab at a position of rapidly changing stress the result 
 represents the average unit deformation over the entire gage length and 
 not at the end where the stress in the gage length is a maximum. In the 
 beams tested the moment was constant over the entire gage length. 
 
 13. TEST SPECIMENS AND METHODS OF TESTING. Technologic Paper 
 No. 2f of the Bureau of Standards describes the specimens and tho 
 methods of testing. Only the features which arc of tJio most importance in 
 connection with the present study arc repeated hero. 
 
 * F. R. McMillan, in unpubli.-lied data furnished to the writer, 
 t See also Bulletin 329, U. S. Geological Survey.
 
 58 
 
 MOMENTS AND STRESSES IN SLABS. 
 
 All the beams were 13 ft. long, 8 in. wide, and 11 in. deep. The 
 amount of reinforcement varied in the different beams from two one-half- 
 inch round to eight one-half-inch round bars. For the beams having four 
 bars or less all were placed in one layer. For the beams having five bars 
 or more the bars were placed in two layers. The vertical distance between 
 the centers of the bars of the two layers was l 1 /^ in. The distance from the 
 top of the beam to the center of the lover layer of bars was 10 in. The 
 ratio of reinforcement, based upon the depth to the center of gravity of 
 the reinforcement, varied in the different beams from 0.0049 for the beams 
 with two bars, to 0.0212 for the beams with eight bars. In Technologic 
 Paper No. 2 the percentage of reinforcement was based upon the depth to 
 the center of the lower layer of bars. On account of the difference in the 
 method of computing the ratios of reinforcement the ratios given here are 
 slightly greater for the beams having two layers of bars than the ratios 
 given in Technologic Paper No. 2. 
 
 All the concrete used in the beams was of a 1:2:4 mixture. Four 
 different aggregates were used. These were granite, gravel, limestone, and 
 
 TABLE VIII. STRENGTH MODULUS OF ELASTICITY AND VALUES OF n. 
 
 
 Compressive 
 Strength 
 !b. per sq. in. 
 
 Modulus of 
 Elasticity, 
 Ib. per sq. in. 
 
 Values of n. 
 
 
 4,200 
 
 4,430,000 
 
 6.8 
 
 
 4,000 
 
 4,620,000 
 
 6.5 
 
 Limestone Concrete .. ... .... 
 
 3,600 
 
 3,810,000 
 
 7.9 
 
 Cinder Concrete 
 
 2,200 
 
 1,820,000 
 
 16.6 
 
 
 
 
 
 NOTB. The term n represents the ratio of the modulus of elasticity of the steel to that of the concrete. 
 For this computation the modulus of elasticity of the steel is taken at 30,000,000 Ib. per sq. in. 
 
 cinders. The strengths and moduli of elasticity of the concretes from these 
 aggregates are given in Table VIII. It will be seen that while all of the 
 concrete had a good strength there was considerable range in the strengths. 
 
 All the bars used as reinforcement in these beams were tested and 
 they were classified according to the yield point. (In the earlier beams 
 the elastic limit was used as the basis of classification.) This insured 
 practically the same yield point for all the bars of any beam. The highest 
 and the lowest yield points for the beams quoted in this paper were 43470 
 and 36110 Ib. per sq. in. respectively. The unit used in the discussion in 
 this paper is the average for three beams of a kind. On this basis the 
 highest and lowest average yield points were 42900 and 36400 Ib. per sq. in. 
 respectively. The average yield point was 40200 Ib. per sq. in. The yield 
 points for all but two of the twenty-eight groups were within o 1 /^ per cent 
 of the average. 
 
 A span of 12 ft. was used in all the tests, and the loads were applied at 
 the one-third points of the span. The deformations were measured over a 
 gage length of 29.25 in. by means of extensometers which were attached to,
 
 MOMENTS AND STRESSES IN SLABS. 59 
 
 the concrete and which did not measure directly the deformations in the 
 steel. The arrangement of the extensometers gave the deformations in the 
 steel at the level of the bottom layer of reinforcement. These are the 
 deformations on which the studies in the Technologic Paper are based. 
 Since in the present study the depth of the beam was taken as the depth 
 to the center of gravity of the cross section of the reinforcement it became 
 necessary to reduce the deformations reported in the Technologic Paper to 
 the corresponding deformations at that depth. This modification affected 
 only the beams having more than one layer of bars. 
 
 14. METHOD OF ANALYZING RESULTS OF BEAM TESTS. The analysis of 
 the results of the beam tests consists in deriving an empirical equation for 
 the observed stress in terms of the computed stress and the percentage of 
 reinforcement. For the beams used in this study the load-strain diagrams 
 (in which values of H/bd? were plotted as ordinates and the unit- 
 deformations in the steel were plotted as abscissas) are made up of three 
 parts, ( 1 ) the part in which little or no cracking of the concrete had taken 
 place; (2) the part in which the concrete had cracked and the stress in the 
 reinforcement was below the yield point; and (3) the part in which the 
 stress in the steel was at or beyond the yield point. It was found that the 
 first two parts were nearly straight lines which, if projected, intersected 
 at a point which corresponds quite closely to the unit-deformation at which 
 a breaking down of the concrete in tension may be expected to occur. In 
 the study of the results of these beam tests empirical equations for these 
 two straight lines were determined. In the diagrams representing these 
 empirical equations the straight lines are connected by smooth curves, but 
 no attempt has been made to state an equation for the curved portion. The 
 part of the load-strain diagram for which the yield point of the steel has 
 been reached or passed has not been included in the study. 
 
 Fig. 27 gives typical load-strain diagrams* for certain of the gravel 
 concrete beams used in this study. In Fig. 29 there is a sketch of a load- 
 strain diagram with notation which will assist in making clear the manner 
 in which the analysis was carried out. The lines OA and BC represent the 
 straight lines which may be fitted to the two portions of the diagrams 
 below the yield point of the steel. The slopes of the line OA for all the 
 beams used were plotted, and from these points an equation for the slope 
 was derived. The plotted points and the equations of the lines which were 
 fitted to them are given in Fig. 28. Likewise the slopes of the lines BC for 
 all the beams used were plotted in Fig. 29 and the equation of the slope was 
 derived. The intercept OB of the lines BC for all the beams were plotted 
 in Fig. 30 and equations of the intercept were derived in like manner. The 
 height of the intercept OB might be used as a measure of the load at which 
 the breaking down of the concrete in tension occurred, but it is not entirely 
 satisfactory as a measure of this action of the beams, and Fig. 30, showing 
 these intercepts plotted as ordinates, is introduced only to indicate this 
 
 The tabulated data used in this figure are given in Tech. Paper No. 2, U. S. 
 Bureau of Standards, 1911.
 
 60 
 
 MOMENTS AND STRESSES IN SLABS. 
 
 Strain 
 
 Deflection 
 
 Neutral Axis 
 
 ~4 
 
 '100 
 
 <^ 
 
 ^ 
 
 ^ 
 
 M 
 ffl 
 
 100 
 
 
 300 
 200 
 100 
 
 Beam Na34d 
 
 ^7/77 
 
 3-k"rd rods-074% 
 
 i 
 
 
 
 Deflect/on 
 
 I 
 
 Q\5 
 
 ' 
 
 : 
 
 06 
 
 07 
 
 _L 
 
 01 
 
 oz 
 
 FIG. 27.- --TYPICAL LOAD-STRAIN DIAKKASUS FOR BEAM TESTS.
 
 MOMENTS AND STRESSES IN SLABS. 
 
 61 
 
 step in the derivation of the equations of the relation between the observed 
 and the computed stresses. The equation of the slopes of the lines OA, and 
 the fact that the lines pass through the origin give sufficient information 
 with which to determine the equation of the lines OA. The equation of the 
 slope of the lines BC, and the equations of the intercept OB of these lines 
 on the vertical axis, were stated in terms of observed and computed stresses, 
 and were solved simultaneously for the equations of the lines BC. The 
 graphical representation of the equations determined in this way for the 
 granite, gravel, and limestone concretes is given in Fig. 31. The equations 
 which apply to the beams of cinder concrete are somewhat different and 
 are shown in Fig. 32. In all the computations the value of / (the ratio of 
 the moment arm to the depth d) was taken as 0.86G. 
 
 kj 2500000 
 
 = zoooooo 
 
 fc 1500000 
 *^ 
 ^ 
 ^ 1000000 
 
 ^n 
 
 
 
 v-QmrirteCOt 
 o Gravel 
 Limestone 
 ^Cinder 
 
 x/gvfe 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 O i " 
 
 
 
 
 ^-* 
 
 o 
 
 '0000- 
 
 o 
 
 fJ^W; 
 & 
 
 Vl\ 
 
 ^~ ' 
 
 ' 
 
 "~~~ 
 
 .^ 
 
 I 
 
 
 
 ~~~~~ 
 
 
 
 X 
 
 "" 
 
 ' 
 
 V 
 
 
 
 
 
 
 .^ 
 
 oo+z 
 
 -i. 
 
 soooa 
 
 +~ 
 
 7 P 
 
 , 
 
 .. -* 
 
 4 
 
 r" 
 
 ~* 
 + 
 
 -T 
 
 ^ -t 
 M 
 ~e5d* 
 
 
 -^252 
 
 1 ij jUUOOO 
 
 L 
 FIG.. 28. Ri 
 
 LOAD-S' 
 
 
 
 
 
 
 
 
 
 
 
 ? .<aa? .## .# .-006 .010 .o/z .014 .0/6 .o/d 020 .022 
 Faf/o of/?einforcemenf,p 
 
 :LATION BETWEEN RATIO OF REINFORCEMENT AND SLOPE OF 
 CHAIN CURVE BELOW LOAD AT WHICH CONCRETE CRACKED. 
 
 15. EFFECT OF QUALITY OF CONCRETE AND AMOUNT OF REINFORCEMENT 
 ON RATE OF INCREASE OF TENSILE DEFORMATION. Fig. 28 and 29 were 
 prepared in order to study the action of the beam during the first two 
 stages of the test which have been referred to in Art. 14. In each figure the 
 ordinates of the plotted point represent slopes of a portion of the load-strain 
 diagrams for all the beams used in this study. These slopes may be inter- 
 preted as the rate of increase of deformation with load. Fig. 28 represents 
 the stage of the test below the cracking of the concrete. Fig. 29 represents 
 the stage of the test between the cracking of the concrete and the reaching 
 of the yield point of the reinforcement. 
 
 For stages of the test below the cracking of the concrete the rate of 
 increase of tensile deformation was affected in an important degree by the 
 quality of the concrete, while the effect of the amount of reinforcement on 
 the rate of increase of deformation was almost negligible. For stages of 
 the test above the cracking of the concrete the conditions were reversed;
 
 62 
 
 MOMENTS AND STRESSES IN SLABS. 
 
 the rate of increase of tensile deformation was affected in an important 
 degree by the amount of reinforcement, while the effect of the quality of 
 the concrete on the rate of increase in deformation was entirely negligible. 
 In Fig. 28 the magnitudes of the ordinates to the respective points are, 
 in general, in the order of the values of the modulus of elasticity of the 
 concretes . represented. The ordinates representing the cinder concrete, 
 which had a modulus of elasticity about half as great as that of the other 
 concretes (see Table VIII), are uniformly about half of those for the other 
 
 5500OO\ 
 
 OOZ 004 006 .006 OIO .0/Z .0/4 0/6 
 
 ffbtio of Reinforcement, p 
 
 0/8 020 022 
 
 FIG. 29. RELATION BETWEEN RATIO OF REINFORCEMENT AND SLOPE OF 
 LOAD-STRAIN DIAGRAM ABOVE LOAD AT WHICH CONCRETE CRACKED. 
 
 concretes. Even for the granite, gravel, and limestone concretes, whose 
 moduli of elasticity showed only slight differences, the magnitudes of the 
 average ordinates to the points take the same order as the values of the 
 modulus of elasticity. Although the compressive strengths occupy the same 
 order of magnitude as the moduli, it seems logical to attribute the effect 
 on the rate of deformation to the variation in the modulus of elasticity 
 rather than to the variation in the compressive strength. Whether the 
 important factor was the modulus of elasticity or the compressive strength, 
 the facts here pointed out justify the statement that the rate of tensile
 
 MOMENTS AND STRESSES IN SLABS. 
 
 6.3 
 
 deformation was affected in an important degree by the quality of the 
 concrete. 
 
 Since the abscissas in Fig. 28 are ratios of reinforcement the slopes of 
 the curves fitted to the plotted points in this figure will be a measure of the 
 effect of the amount of reinforcement on the rate of increase of tensile 
 deformation. Both average lines (that for stone and gravel concretes and 
 that for cinder concrete) have slopes which are very small in proportion 
 to the slope of the line in Fig. 29, whicli represents the conditions for the 
 stage of the test after the formation of cracks. This comparison justifies 
 the statement that the effect of the amount of reinforcement on the rate 
 of deformation was almost negligible for the stage of the test in which the 
 concrete was not generally cracked. 
 
 The greatly increased slope of the average line in Fig. 29 over the 
 slopes shown in Fig. 28 forms the basis of the statement that for stages 
 
 %Z50 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 i=f?o +( '0-002)3000 
 bd z , / 
 
 
 
 _* 
 
 _ ' 
 
 -* 
 
 -r 
 
 "' 
 
 -" 
 
 ox- 
 
 -{ ~ 
 
 
 + 
 
 
 
 
 
 -t- 
 
 I + 
 
 /* 
 
 
 
 
 + 
 
 
 
 
 
 z* 
 
 
 
 
 
 xGra 
 06 m 
 Lm 
 +Cin(. 
 
 nite Concrete. 
 'el ' 
 tsfone. 
 1er 
 
 
 
 
 
 
 
 
 
 
 
 ,8 
 
 -5 .002 .004 .006 .003 .O/O .0/2 .014 .0/6 .0/8 .O2O .022 
 
 ffistio of fcs/nforcemenf, p=-j 
 
 FIG. 30. RELATION BETWEEN RATIO OF REINFORCEMENT AND INTERCEPT 
 OB OF FIG. 29 ON LOAD Axis. 
 
 of the test above the cracking of the concrete the rate of tensile deformation 
 was affected in an important degree by the amount of reinforcement. To 
 appreciate fully the relative importance of this factor account must be 
 taken of the fact that in Fig. 29 the scale of ordinates is only one-tenth of 
 that used in Fig. 28. 
 
 In the arrangement of the points in Fig. 29 there is no regularity 
 which is dependent upon the strength or modulus of elasticity of the con- 
 crete. The points representing the cinder concrete are intermingled with 
 the points representing the other grades of concrete in such a way that one 
 average line represents all the results with as great accuracy as would be 
 possible with an independent line for each kind of concrete. This clearly 
 warrants the previous statement that for the stages of the test above which 
 cracks occurred the effect of the quality of the concrete on the rate of 
 tensile deformation was entirely negligible.
 
 64 
 
 MOMENTS AND STRESSES IN SLABS. 
 
 16. RELATION BETWEEN OBSERVED AND COMPUTED TENSILE STRESSES. 
 The significance and scope of the results of the study of the relation 
 between the observed and the computed stresses in the reinforcement may 
 best be visualized by reference to Fig. 31 and 32, which show graphically the 
 derived equations of the observed stress in terms of the computed stress 
 and the percentage of reinforcement. For the part of the test below which 
 the concrete is cracked these equations are, 
 
 0.52/ s 
 
 1 + 
 
 .021 
 P 
 
 1.04f 
 
 for the stone and gravel concretes and 
 
 for the cinder concrete. 
 
 (1) 
 
 (2) 
 
 For max. load J C -400OO^ 40000(02^ 
 
 10000 20000 3OOOO 40000 50000 6OOOO 7QOOO 
 
 Computed 5fresj, , /b per 54. in 
 
 FIG. 31. RELATION BETWEEN OBSERVED AND COMPUTED TENSILE STRESSES 
 IN REINFORCEMENT OF BEAMS TESTED AT ST. Louis. 
 
 Diagrams from derived equations for beams of stone concrete and of gravel 
 concrete's. 
 
 For the part of the test above which the concrete is generally cracked the 
 equations are, 
 
 1.04p/_ - 144 
 
 p - .002 
 1.04p/_ - 144 
 
 3600 for stone and gravel concretes and, 
 
 for cinder concrete. 
 
 p - .002 
 
 (3) 
 
 (4) 
 
 In these equations, / is the observed stress, / is the computed stress, and 
 p is the ratio of longitudinal reinforcement based upon the depth from the
 
 MOMENTS AND STRESSES IN SLABS. 
 
 65 
 
 compression surface of the beam to the center of gravity of the tension 
 reinforcement. 
 
 In Fig. 31 and 32 the lines representing the stresses in beams having 
 less than 0.5 per cent of reinforcement are dotted because these lines repre- 
 sent extrapolation below the lowest percentage of reinforcement used in any 
 of the beams tested. Since most of the slabs to whose study the results of 
 these beam tests may be applied have not more than 0.5 per cent of rein- 
 forcement it is important to consider whether the extrapolation is justifi- 
 able. The average lines fitted to the points in Fig. 28, 29 and 30 were 
 projected from 0.0049, the lowest ratio of reinforcement for any of the 
 beams tested, to the lower value of 0.002. This extrapolation covers a 
 small portion of the total range in the percentages of reinforcement repre- 
 
 4000C 
 
 for max. 
 
 /Mdf-4tXm-400G0(8Z+7p)-y&/^-r--. 
 ) I /- X//1/ / 
 
 "ffiZzoodT 
 
 o loooo zoom 30000 40000 50000 soooo TOOOO 
 Gompufed Stress, g, Ib. per sq/n 
 
 FIG. 32. RELATION BETWEEN OBSERVED AND COMPUTED TENSILE STRESSES 
 IN REINFORCEMENT OP BEAMS TESTED AT ST. Louis. 
 
 Diagrams prepared from derived equations for beams of cinder concrete. 
 
 sented by the beams which were tested, and the curves which were projected 
 are well denned by the experimental points. It seems, therefore, thstt there 
 is justification for this extrapolation. An inspection of Fig. 28, 29 and 30, 
 on which the curves of Fig. 31 and 32 are based, will assist in forming an 
 opinion as to whether the extension of the scope of the diagrams of Fig. 31 
 and 32 below 0.49 per cent of reinforcement is warranted by the test data. 
 Fig. 29, 31, and 32 indicate that for beams having only 0.2 per cent 
 of reinforcement when the concrete breaks down in tension the reinforce- 
 ment immediately would be stressed to failure. That is, when the concrete 
 breaks down in tension the slope of the load strain diagram becomes zero 
 for a beam with only 0.2 per cent of reinforcement. That this condition 
 ia approached as the amount of reinforcement becomes small is shown by
 
 66 
 
 MOMENTS AND STRESSES IN SLABS. 
 
 inspection of Fig. 29. The same thing is shown directly in the flatness 
 of the slope of the load stress diagrams for beams 336, 337 and 338 of 
 Fig. 27, which have the smallest amount of reinforcement of any of the 
 beams studied. That there should appear to be no difference in the amount 
 of reinforcement required to bring about this condition for the beams of 
 cinder concrete from that which was required for the beams of stone 
 concrete, may be due to a break in the mean line of Fig. 29 between 0.5 
 and 0.2 per cent of reinforcement. Whether such a break occurs is not 
 
 40000 
 
 ^30000 
 
 * Granite Concrete 
 
 o6ra/ef 
 
 Limestone 
 
 zoooc 
 
 10000 
 
 10000 20000 3000O 4OOOO 5OOOO 
 Computed Stress - Ib. per 53. //?. 
 
 FIG. 33. RELATION BETWEEN OBSERVED AND COMPUTED TENSILE STRESSES 
 IN REINFORCEMENT OF BEAMS TESTED AT ST. Louis. 
 
 Test results for beams of stone concrete and of gravel concrete compared with 
 results from derived equations. 
 
 known because no beams with less reinforcement than 0.49 per cent were 
 tested. 
 
 In order to make certain by a direct comparison, that equations ( 1 ) to 
 (4) represent the relation between the observed and the computed tensile 
 stresses, Fig. 33 and 34 have been prepared. Points showing the observed 
 and the computed unit deformations throughout the tests of representative 
 beams have been plotted, and for comparison with them the graphs of the 
 equations which represent the relation between the observed and the 
 computed stresses for the same beams are shown in the same figures. 
 Each point plotted in these figures represents the average load and the 
 average deformation for the three beams of its kind. Considering the
 
 MOMENTS AND STRESSES IN SLABS. 
 
 67 
 
 range of concretes and the range in the amounts of reinforcement used in 
 the beams represented, the agreement between the test results and the 
 empirical equations seems good. 
 
 In Fig. 31 and 32 the slopes of the lines which represent the stages of 
 the test in which the concrete had not cracked were approximately inversely 
 proportional to the values of the modulus of elasticity of the concrete. 
 The slopes (1.04/ 8 ) of all the lines for the cinder concrete beams were 
 just twice as great as the slopes (0.52/ 8 ) for the corresponding beams of 
 stone or gravel concrete. The values of n (the ratio of the modulus of 
 
 10000 
 
 20000 30000 
 
 5f/TS55, /, A 
 
 4000O 5QOOO 
 
 
 
 FIG. 34. RELATION BETWEEN OBSERVED AND COMPUTED TENSILE STRESSES 
 IN REINFORCEMENT OF BEAMS TESTED AT ST. Louis. 
 
 Test results for beams of cinder concrete compared with results from derived 
 equation. 
 
 elasticity of the steel to that of the concrete) were, on the average, 2.33 
 times as great for the cinder concrete as for the stone and the gravel 
 concretes. Assuming that the value of the slope may be taken as propor- 
 tional to the value of n an equation is found which gives values closelj 
 approximating the test results when the proper values of n are used in the 
 equation. The equation is 
 
 f\"7 m f 
 
 (5)
 
 68 MOMENTS AND STRESSES IN SLABS. 
 
 For all the beams of stone or gravel concrete reported in Technologic 
 Paper No. 2 the average of the unit-deformations in the reinforcement at 
 the time that the first crack was observed is 0.000113 and this corresponds 
 to a stress of 3390 Ib. per sq. in. For all the cinder concrete beams reported 
 in that paper, the average unit-deformation at the occurrence of the first 
 crack was 0.000179. This deformation corresponds to a stress of 5370 Ib. 
 per sq. in. The intersections of the two straight portions of the diagrams 
 of Fig. 31 for the stone and gravel concretes, lie at an observed tensile 
 stress of about 3200 Ib. per sq. in. In Fig. 32 the intersections for the 
 cinder concrete lie at a tensile stress of 6260 Ib. per sq. in. These values 
 are seen to correspond quite closely to the stresses at which the first cracks 
 were discovered. 
 
 For the stage of the test above the cracking of the concrete the only 
 difference between the equation which represents the relation between the 
 observed and the computed stresses for the stone and gravel concrete 
 ( equation ( 3 ) ) and the corresponding equation for the cinder concrete 
 beams (equation (4) ) is that in the former there is an additive term 
 ( 3600 Ib. per sq. in.) which is lacking in the equation for the cinder 
 concrete beams. It may seem unexpected that such a term as this should be 
 present, but that the difference expressed by the term is present is made 
 entirely clear by attempting to fit the equation for. the cinder concrete to 
 the results for the stone and the gravel concretes. 
 
 The intensity of the bond stresses between cracks will be affected by 
 variations in the modulus of elasticity of the concrete, and it may be per- 
 missible to assume that the variation in the additive term in equation (3) 
 is proportional to the variation in the value of n (the ratio of the modulus 
 of elasticity of the steel to that of the concrete). With this as an assump- 
 tion a more general equation which, for the beams under consideration, 
 represents quite accurately the relation between the observed tensile stress 
 and the computed stress after the concrete had cracked is 
 
 17. OBSERVED TENSILE STRESS AT MAXIMUM LOAD. It is desirable to 
 determine the relation between the maximum loads which the beams car- 
 ried and the stress in the steel which corresponds to the observed deforma- 
 tion (here termed observed stress) at those loads. On account of the 
 possibility that the steel had been stressed beyond the proportional limit 
 before reaching the maximum load it is not feasible to determine the stress 
 at the maximum load directly from the measured deformation. In order to 
 determine the desired relation, the straight lines BC of Fig. 29 were pro- 
 duced to the maximum load, and the unit-deformation given by this line at 
 the maximum load was used to determine the stress at that load. In this 
 way the ratios, q, of the stress at maximum load to the yield point were 
 determined and are given in Fig. 35. The equation which expresses the 
 average relation between q and the ratio of reinforcement, p, is 
 
 q-0.82 + 7p. (7)
 
 MOMENTS AND STRESSES IN SLABS. 
 
 69 
 
 The yield-point stress used in these computations was 40,000 Ib. per sq. in. 
 The observed tensile stress at the maximum load was generally slightly 
 less than the yield point. It is possible that the stress at a crack was 
 enough greater than the stress found from the deformations over the 
 entire gage length to bring the stress at the maximum load up to the yield 
 point. This possibility is further indicated by the fact, which is brought 
 out in equation (7), that the observed stress approached the yield point 
 more closely for the beams with a large percentage of reinforcement than 
 for the beams with a small percentage. The result expressed in equation 
 (7) should not be unexpected since the bond stresses between cracks would 
 have more influence in reducing the total deformations in beams in which 
 the amount of reinforcement is small than in those in which it is large. 
 
 .002 .004 .006 .006 .O/O .012 .014 .O/6 .0/8 -020 .022 
 
 FIG. 35. RELATION BETWEEN RATIO OF REINFORCEMENT AND RATIO OF 
 OBSERVED STRESS AT MAXIMUM LOAD TO YIELD-POINT STRESS. 
 
 Equation (7) has been introduced into the diagrams of Fig. 31 and 32 
 to show the observed and the computed stresses at which tension failure in 
 the reinforcement is likely to occur. In making this application of the 
 equation the yield point was assumed to be 40,000 Ib. per sq. in., approxi- 
 mately the average value found in the tests of the coupons taken from the 
 beams. No test data were available from which to show the relation 
 between the stress at maximum load and the yield-point stress for higher 
 or lower yield points. However, by assuming that for small differences in 
 yield point the loads carried would be proportional to the yield-point stress, 
 the dotted curves for yield points of 38,000 and 42,000 Ib. per sq. in. were 
 obtained. The error of these estimates becomes large for the beams with 
 small amounts of reinforcement, hence these additional curves were not 
 carried beyond the values for one-half per cent of reinforcement. 
 
 18. FACTOR OF SAFETY AGAINST TENSION FAILURE. The factor of safety 
 for a structure may be denned as the ratio found by dividing the working
 
 70 
 
 MOMENTS AND STRESSES IN SLABS. 
 
 load for the structure into the load which will cause failure of the structure. 
 There may be differences of opinion as to how the load which is to be used 
 for determining the factor of safety should be applied. For these tests 
 there was only one possible load which could be considered, the load which 
 was built up by uniform increments until failure was brought about, the 
 whole test requiring not more than a few hours. For the purpose of elimi- 
 nating from the study of the factor of safety the effect of slight variations 
 in the yield point of the steel, the maximum loads given in Technologic 
 Paper No. 2 were corrected to give a load which presumably would have 
 caused failure if the yield point of the steel had been 40,000 Ib. per sq. in. 
 The maximum loads reported were increased or decreased by an amount 
 which was proportional to the difference between the yield-point stress and 
 40,000 Ib. per sq. in. To these corrected maximum loads were added the 
 weights of the beams, and the resulting loads were used in the computation 
 of the factor of safety. The working load was taken as the load which 
 gives a computed tensile stress of 16,000 Ib. per sq. in. In these computa- 
 
 TABLE IX. FACTORS OF SAFETY AGAINST TENSION FAILURE. 
 
 Kind of Concrete. 
 
 Ratio of Reinforcement. 
 
 Average 
 Factor of 
 Safety. 
 
 0.0049 
 
 0.0074 
 
 2.83 
 2.92 
 2.62 
 2.85 
 
 2.80 
 
 0.0098 
 
 0.0133 
 
 0.0159 
 
 0.0186 
 
 0.0212 
 
 2.53 
 2.78 
 2.70 
 2.36 
 
 2.59 
 
 
 3.09 
 3.25 
 2.96 
 2.96 
 
 3.06 
 
 2.75 
 2.88 
 2.65 
 2.74 
 
 2.76 
 
 2.69 
 2.95 
 2.64 
 2.68 
 
 2.74 
 
 2.67 
 2.88 
 2.48 
 2.56 
 
 2.65 
 
 2.52 
 2.77 
 2.51 
 2.63 
 
 2.61 
 
 2.72 
 2.92 
 2.65 
 2.68 
 
 2.74 
 
 Gravel 
 
 Limestone 
 
 Cinder 
 
 Average 
 
 
 tions the value of ; (the ratio of the moment arm to the depth d) was 
 taken as 0.875. The factors of safety found in this way are shown as 
 plotted points in Fig. 36. Each point represents the average for three 
 beams. The values from which these points were plotted are given in 
 Table IX. 
 
 From the definition of the factor of safety it will be seen that the ratio 
 of the computed stress at the maximum load to the computed stress at the 
 working load will be the factor of safety. From this relation another 
 method of determining the factor of safety is afforded, since Fig. 31 and 32 
 show, more or less exactly, the values of the computed stress in tension at 
 the maximum load. These -stresses divided by 16,000 Ib. per sq. in. give the 
 values of factor of safety shown by the smooth curves in Fig. 36. The 
 irregularities have been smoothed out by the fact that the values shown 
 by these curves represent the intersections of mathematical curves. The 
 dotted portions of the two curves express the indications of Fig. 31 and 32 
 as to what the factor of safety would be for beams which have smaller 
 percentages of reinforcement than any of the beams which form the basis 
 of this study.
 
 MOMENTS AND STRESSES IN SLABS. 
 
 71 
 
 The agreement between the plotted points and the smooth curves is as 
 good as will generally be found from methods which are as nearly inde- 
 pendent as were these. Both sets of results are based upon the same 
 reasoning, but they are reached by different methods of using the test data. 
 
 The indication of the smooth curves is that the factor of safety for the 
 cinder concrete was less than that of the other concretes, but the plotted 
 values do not bear out this conclusion. It cannot be said, however, that the 
 arrangement of the points is independent of the kind of concrete. If an 
 attempt were made to draw a curve which fits the values shown for each 
 kind of concrete the curve for the gravel and that for the limestone con- 
 crete would be the highest and the lowest respectively, while the curves 
 for the granite concrete and the cinder concrete would be intermediate and 
 would practically coincide. This is somewhat surprising since both the 
 
 s 
 
 
 
 \ 
 
 \ 
 
 
 
 
 
 
 A 
 
 W 
 Yi 
 
 X 
 
 106 
 
 m 
 
 e/a 
 d 
 d 
 
 7/7 
 
 '^ 
 7/7< 
 
 > 
 r A<2 
 
 rtf 
 
 dc 
 
 e a 
 
 7S/ 
 ?in 
 ni 
 ve 
 
 7rc 
 
 '/bet 
 ress, 
 /, 40 
 
 te 
 / + 
 
 welc 
 
 iff? 
 
 /< 
 00 
 Lir 
 Cit 
 
 We 
 
 s,/3we<. 
 wolbpa 
 Olb.per 
 neston 
 ider 
 
 r&fes 
 
 ?A 
 "W 
 
 w 
 
 a 
 
 in. 
 in 
 
 
 
 
 
 \ 
 
 i 
 
 
 
 
 
 
 
 
 
 
 
 \ 
 
 \; 
 
 
 S/ 
 
 L or 
 
 e 
 
 
 
 
 
 
 
 
 H 
 
 \ 
 
 - . 
 \ 
 
 .0 
 
 - 
 
 - 
 
 
 
 
 o 
 
 
 \ 
 
 
 O 
 
 
 
 
 
 
 
 
 
 
 yr- 
 
 
 \ 
 
 
 
 * 
 
 
 
 
 
 
 
 
 
 
 
 '/Vc 
 
 ie, 
 
 r c 
 
 ~o. 
 
 icref 
 
 
 
 
 
 
 f 
 
 ~"0 002 D04 .006 008 0/0 .012 .014 .016 OI8 .OZ 
 Ratio of Re/nforcemenf p 
 
 FIG. 36. FACTOR OF SAFETY AGAINST TENSION FAILURE OF BEAMS TESTED 
 
 AT ST. Louis. 
 
 compressive strength and the modulus of elasticity were highest for the 
 granite concrete and lowest for the cinder concrete. This would indicate 
 that it is some other property of the beams than the strength of the con- 
 crete which determined the differences in the factor of safety developed. 
 
 It is of interest that a tendency toward a higher factor of safety for 
 the beams with a small amount of reinforcement than for the beams with 
 large amounts of reinforcement appears in this figure. Apparently, as a 
 criterion of the load-carrying capacity of the beams throughout the range 
 of the tests the yield point of the steel is much more nearly correct than is 
 the ultimate strength of the steel. It is quite clear that this figure gives no 
 basis for claiming a factor of safety of four for a beam having the usual 
 amount (say 0.7 per cent) of reinforcement of structural grade, if the 
 working stress in tension is 16,000 Ib. per sq. in. An estimate of the 
 factor of safety of these beams, based upon the ultimate strength of the 
 steel, would give from 3.25 to 4.0. These and many other tests have made
 
 72 MOMENTS AND STRESSES IN SLABS. 
 
 it clear that such a basis of estimating the factor of safety is wrong, and 
 yet occasionally a claim of such a factor of safety, arrived at in this 
 manner, is made. 
 
 19. SUMMARY, (a) It was found that the load-strain diagrams for the 
 beams could be represented quite closely by two straight lines which inter- 
 sect at a point which corresponds to the strain, in the tension side of the 
 beam, at which the concrete cracked in tension. Through a study of the 
 average slopes, and average intercepts of these lines, it was found to be 
 possible to state equations which give, with a considerable degree of 
 accuracy, the relation between the observed and the computed stresses in 
 the reinforcement of the beams. 
 
 (b) The relation between the observed and the computed stresses in 
 the reinforcement for the beams studied was found to be affected by the 
 variation in the quality of the concrete, the amount of reinforcement, and 
 the intensity of the computed stress. 
 
 (c) For stages of the test below the cracking of the concrete the 
 rate of increase of the tensile deformation was affected in an important 
 degree by the quality of the concrete, while the effect of the amount of 
 reinforcement on the rate of increase of tensile deformation was almost 
 negligible. 
 
 The rate of increase in the tensile deformation in the reinforcement 
 at this stage of the test was found to be approximately proportional to the 
 reciprocal of the modulus of elasticity of the concrete in compression as 
 determined by tests of control cylinders. 
 
 (d) For stages of the test above the cracking of the concrete the rate 
 of increase of the tensile deformation was affected in an important degree 
 by the amount of reinforcement, while the effect of the quality of the con- 
 crete on the rate of increase in deformation was entirely negligible. 
 
 The total amount of deformation, however, was found to be greater for 
 the beams of cinder concrete than for the beams having a greater com- 
 prcssive strength and modulus of elasticity. The difference in amount of 
 the deformation for the different concretes was constant for all percentages 
 of reinforcement and for all stages of the test between the cracking of the 
 concrete and the reaching of the maximum load, as far as the data of the 
 tests give a basis for judgment on this subject. 
 
 (e) The observed tensile stress in the reinforcement was less than the 
 computed stress for all loads up to and including the maximum load. The 
 difference was greater both proportionally and quantitatively for the beams 
 with small percentages of reinforcement than for beams with large per- 
 centages. This was true for all loads. Correspondingly for all percentages 
 of reinforcement the difference was greater for low loads than it was for 
 high loads. 
 
 (f) The indications were that with a reinforcement of not less than 
 0.2 per cent the strength of the reinforced beam would be the same as the 
 strength of an unreinforced beam. This holds for the cinder concrete 
 beams as well as for those made with concrete of a higher compressive
 
 MOMENTS AND STBESSES IN SLABS. 73 
 
 strength and higher modulus of elasticity. Since no beams with less than 
 0.49 per cent of reinforcement were tested this observation must be taken 
 as an indication and not as a fact established for the beams studied. 
 
 (g) The observed stress in the reinforcement at the maximum load 
 was found to be less than the yield point for all the beams studied. It was 
 found, however, that the ratio of the stress at maximum load to the yield 
 point was greater for the beams with large percentages of reinforcement 
 than for the beams with small percentages. 
 
 (h) The average factor of safety (see Art. 18 for definition) was found 
 to be 10 per cent greater than the ratio of the yield point of the reinforce- 
 ment to the working stress in tension which was used in determining the 
 working load. When averages for all the concretes are considered a con- 
 sistent decrease in the factor of safety with increase in percentage of rein- 
 forcement was found. The factor of safety was 18 per cent greater for 
 beams with 0.49 per cent of reinforcement than for beams with 2.12 per 
 cent of reinforcement. 
 
 IV. TESTS OF SLABS SUPPORTED ON FOUR SIDES. 
 BY H. M. WESTERGAABD. 
 
 20. TESTS OF SLABS SUPPORTED ON FOUR SIDES. Information as to the 
 strength of slabs supported on four sides was obtained by a series of tests 
 made during the years 1911 to 1914 in Stuttgart in Germany under the 
 direction of Bach and Graf,* and by a test made in 1920 at Waynesburg, 
 Ohio, for J. J. Whitacre, under the direction of W. A. Slater. 
 
 In Bach's and Graf's test 52 slabs, simply supported along the edges, 
 and 35 control strips, supported as beams, were loaded to failure. The 
 strength of the slabs was to be compared with the strength of the strips. A 
 record of the progress of the test was obtained by measuring the deflections 
 at a number of points and the slopes at the centers of the edges, and by 
 observing the development of cracks. Fig. 37 and Fig. 38 show typical 
 examples of the record made of the cracks; the numbers indicate the loads 
 in metric tons at which the particular cracks appeared. 
 
 Two mixtures of concrete were used, with the following properties: 
 
 Type A B 
 
 Mixture ". 1:2:3 1:3:4 
 
 Per cent of water 9.2 9.7 
 
 Prism strength in compression after 44-48 
 
 days, Ib. per sq. in 2,290 1,835 
 
 Initial modulus of elasticity in compression, 
 
 Ib. per sq. in 4,050,000 3,400,000 
 
 The 1: 3: 4 mixture was used in three slabs, which were designed to fail 
 
 * Reported by Bach and Graf in Deutschcr Ausschuss fur Eisenbeton, v. 30, 
 1915. See the Bibliography in Appendix C.
 
 74 
 
 MOMENTS AND STRESSES IN SLABS. 
 
 in compression (slabs g in Table II) and in the corresponding six control 
 strips ( 26 and 27 in Table X ) . The 1:2:3 mixture was used in all the 
 other slabs and strips. At the time of the test the age of the specimens 
 was from 40 to 54 days. The yield point of the steel was from 49,600 Ib. 
 per sq. in. to 75,200 Ib. per sq. in. The slabs were two-way reinforced, with 
 the bars parallel either to the sides or to the diagonals. 
 
 FIG. 37. TOP OF SQUARE SLAB OF 200 CM. SPAX, TESTED BY BACH AND GRAF. 
 
 Table X and Fig. 39 show certain results of the tests of the control 
 strips. In order to imitate the conditions of the two-way reinforced slabs 
 most of the strips were built with transverse bars either above or below 
 the longitudinal reinforcement (as indicated in one of the columns in 
 Table X). The transverse bars were found to hasten the development of
 
 MOMENTS AND STRESSES IN SLABS. 
 
 75 
 
 the first crack, but the table shows that these bars have only little influence 
 on the ultimate strength of the strips. The strips of typos 18 to 25 failed 
 by tension in the steel. The table gives the modulus of rupture of the 
 steel, that is, the steel stress computed by the ordinary theory of reinforced- 
 concrete with n = 15, for the observed maximum load (with the dead 
 
 FIG. 38. BOTTOM OF SQUARE SLAB OF 200 CM. SPAN, TESTED BY BACH AND 
 
 GRAF. 
 
 weight taken into consideration). In the strips that failed in tension the 
 ratio of the modulus of rupture of the steel to the yield point of the steel 
 was found to be approximately / If - 1.26 (1 lOp), where p is the 
 ratio of steel. The strips of types 26 and 27 failed in compression at an 
 average modulus of rupture of the concrete of 3490 Ib. per sq. in. (computed 
 with n 15), that is, 1.90 times the prism strength.
 
 76 
 
 MOMENTS AND STRESSES IN SLABS. 
 
 Table XI* and Fig. 40 show some of the results obtained from the tests 
 of 40 slabs out of the 52 which were tested. The surface of each of the 
 square slabs, a to g of the rectangular slabs h, and of the rectangular slabs 
 i, was divided into 50 cm. squares; that is, into 16, 24, and 32 squares, respect- 
 ively; equal loads were applied at the centers of these square's, each load 
 
 fe 
 3 
 O 
 
 H 
 
 O2 . 
 
 r-i I* 
 
 H - 
 
 < 
 
 O 
 
 11 
 
 BH 
 
 4 
 
 a* 
 
 M 
 
 en >*- 
 
 ** G^ 
 
 Xfi 
 
 o 
 
 1 
 
 ffc oo co 
 
 ^ 'A <A 
 
 Q) <U 
 
 
 
 O O 
 
 cq cq c>4 
 
 0' C5 
 
 co &> O 
 
 vfi sS> 
 O O 
 
 (to 
 
 K N- o 
 S 
 *>' oo' OQ 
 
 C> C\ 
 
 CM ro 
 CM CM 
 
 N 
 
 *- ! 
 
 o rv. 
 o" OS 
 
 Ci 
 
 o t- 
 
 o o 
 CD 06 
 
 - cs 
 
 acj c> 
 
 * Table XI is modeled after similar tables used by E. Suenson (Ingenioeren, 
 1916, p. 541) and by N. T. Nielsen Hngenioeren, 1920. p. 724), in the studies of the 
 same experimental material. Table XI was computed from the original data reported 
 by Bach and Graf; the results computed here agree approximately with those found 
 by Suenson and by Nielsen. Suenson compared the experimental coefficients of 
 moment in rectangular slabs with those found by an approximate theory. Nielsen 
 computed the stresses by using coefficients which he derived by the method of difference 
 equations.
 
 MOMENTS AND STRESSES IN SLABS. 
 
 77 
 
 distributed within a circle 9 cm. in diameter. Thus, the loads on each of 
 these slabs were nearly uniformly distributed. Of the remaining twelve 
 slabs, not included in Table XI, two were loaded by a concentrated load at 
 the center, seven by eight concentrated loads near the center, while three 
 slabs were double panels, continuous over a transverse beam, and carrying 
 nearly uniformly distributed loads. 
 
 If a square slab, simply supported on four sides, is loaded uniformly 
 by the total load W, the average moment across the diagonal becomes 
 1/24 W = 0.041 7 W. If the total load W is divided into 16 concentrated 
 equal loads applied at the centers of 16 squares into which the slab is 
 divided, the average moment across the diagonal becomes 3/64 W = 
 0.0469 W. When these 16 loads, instead of being concentrated, are distrib- 
 
 /-; 
 1.35 
 ^,.30 
 
 \I.ZO 
 l./S 
 rin 
 
 o 
 
 no transverse bars, 
 transverse bars above long/tud/na/ steel, 
 transverse barj below longitudinal steel. 
 Each point represents the average for 
 three or four strips 
 , i 
 
 
 
 
 
 
 *^-^ 
 
 ^-i 
 
 o 
 
 ~ -^^ 
 
 /^=f.?6(> 
 
 -**) 
 
 
 
 
 
 
 
 
 r^C, 
 
 ^ 
 
 ^**^**x 
 
 
 ^^^. 
 
 
 
 
 
 
 
 
 
 
 5T~-^ 
 
 .__. .OOQ .OO9 .OlO 
 
 Fatio of steel, p 
 
 FIG. 39. RATIOS OF MODULUS OF RUPTURE (COMPUTED ULTIMATE STEEL 
 STRESS, /, \ TO YIELD POINT / ix CONTROL STRIPS TESTED BY BACH 
 
 Jbsl > Jyi 
 
 AND GRAF. 
 
 uted uniformly within small circles, drawn at the centers of the squares, 
 with a diameter equal to 9/200 of the span, then the average moment per 
 unit-width across the diagonal, as may be verified by a simple statical 
 computation, becomes 0.0460W,* that is, 1.104 times the moment due to a 
 uniform load. An equivalent uniform load was derived, therefore, in Table 
 XI, for the square slabs, by multiplying the load applied at 16 points by 
 1.104 and by adding the dead loads, 1200 kg and 800 kg for the 12 cm and 
 8 cm slabs, respectively (that is, the dead weights within the supporting 
 edges). In the rectangular slabs the section of maximum stress is not 
 along the diagonal; for these slabs the equivalent uniform load was com- 
 puted as the sum of the applied load and the dead load. The following 
 coefficients of moments were used in Table XI: 0.0417, the coefficient of the 
 average moment across the diagonal, was used for all the square slabs; in 
 
 E. Suenson, in Ing-enioeren, 1916. p. 538, states this moment as - 
 
 0.0459 1C. 
 
 21. S W =
 
 78 
 
 MOMENTS AND STRESSES IN SLABS. 
 
 addition, the coefficients 0.0369 and 0.0463, applying to the center and to 
 the corner, respectively, see Fig. 3 (a) in Art. 7, were used for the square 
 slabs with non-uniform spacing of the steel (slabs d^ and d 2 ) ; finally, 
 0.0733 and 0.0964, coefficients of moment per unit-width, in the short span 
 at the center, taken from Fig. 3 (a) in Art. 7, were used for the rectangular 
 slabs h and i. Since the bending moments computed for the square slabs 
 are moments across the diagonal, the section modulus pjd? per unit-width 
 may be taken as the average for the two layers of steel if the steel is par- 
 allel to the sides; in the slabs, / / a , and g, with reinforcement parallel to 
 
 160 
 >55 
 f.& 
 
 >45 
 
 !+} 
 
 
 
 
 f] 
 
 
 
 
 
 
 
 ^c/77. slabs 
 *> cm. slabs 
 '; square slabs, ZOOcmxZooc 
 ecianqular slabs, 3X)cm*30C 
 octangular slabs, ?OOcm.x40L 
 
 - 
 
 
 
 
 
 
 
 
 
 
 o 
 
 + i 
 
 
 
 
 
 /- 
 
 Keck. 
 
 '/TJfcife 
 
 rsla 
 
 tf 
 
 tor 
 
 m. 
 vn. 
 
 
 
 
 
 
 i"*4 
 
 
 
 i;r 
 
 fc/7Z 
 
 
 
 
 * 
 
 s* 
 
 ^ 
 
 on-ur 
 
 iforn 
 
 ?^a7^ 
 
 ;inq 
 
 
 
 
 
 
 
 1.20 
 
 H5 
 HO 
 
 i'Ti 
 
 
 
 
 
 
 r 
 
 ~5arr 
 
 lesp 
 
 acin^ 
 
 T inh 
 
 wo tc, 
 
 lyers 
 
 
 
 
 
 
 
 
 
 J 
 
 +GZ 
 
 C/05< 
 
 ?rsp<. 
 
 . \ 
 Tcinq in u t 
 
 oper 
 
 layer 
 
 
 
 
 ^ 
 
 ----. 
 
 ^4 
 
 k^ 
 
 - -^ 
 
 ^ 
 
 L- 
 
 * 
 
 
 
 
 
 
 
 
 
 
 
 G 
 
 0- 
 
 <* 
 
 >^. 
 
 1 , 
 
 ^^ 
 
 -Con 
 
 "~>^ 
 
 troh 
 
 tripste-^ 
 (y 
 
 126(1 
 
 -lOp) 
 
 
 
 
 
 
 \5; 
 
 eel parallel 
 to diagonals 
 
 -~~^ 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 ClQSt. 
 
 '.rspc. 
 
 Tang 
 
 in Uf. 
 
 iperk 
 
 jyer 
 
 1.00 
 
 
 
 
 
 
 
 
 L 
 
 ' 5 
 
 fee/f. 
 
 >arali 
 
 el -to 
 
 d/agc 
 
 ma/5 
 
 *9 
 
 .001 .002 .003 .004 .005 006 007 .008 .009 .010 .Oil .OIZ .013 0/4 Qi 
 
 of s fee/, p 
 FIG. 40. RATIOS OF MODULUS OF RUPTURE f TO YIELD POINT 
 
 > /S) > 
 
 SLABS SUPPORTED ox FOUR SIDES, TESTED BY BACH AND GRAF. 
 
 d, AND rfj REPRESENT AVERAGES OF Two TESTS EACH, OTHER POINTS 
 
 AVERAGES OF THREE TESTS EACH. 
 
 / IN 
 
 JV> 
 
 POINTS 
 
 the diagonal, the section moduli in the two directions are practically equal 
 and the average value may be used again. In the rectangular slabs, h and i, 
 the section modulus by which the stresses are computed may be taken as 
 that defined by the bottom layer, which is in the direction of the short span. 
 As an example of the computations in Table XI the derivation of the 
 stress f in the slabs Oj may be shown : 
 
 M 0.0417 45700kg u 22 
 
 Ib.cm 2 
 
 = 70000 
 
 in 2 
 
 pjd* * 0.387cm 2 " kg.in 2 
 
 The stress f is the steel stress, computed by the ordinary theory of
 
 MOMENTS AND STRESSES IN SLABS. 
 
 79 
 
 reinforced-concrete with n = 15, developed under the observed maximum 
 load; that is, t is tlie modulus of rupture in bending. 
 
 The stresses f developed in the slab may be judged by comparison 
 with the yield point / of the steel and the modulus of rupture f (level- 
 
 oped in the strips. Such comparisons are made in the last three columns in 
 Table XI. The ratios fjf are represented graphically in Fig. 40. 
 
 The slabs e were designed to fail in compression. The stresses devel- 
 oped were: tension, 39,600 Ib. per sq. in.; compression, 3430 Ib. per sq. in..
 
 80 MOMENTS AND STRESSES IN SLABS. 
 
 which is 0.983 times the corresponding stress developed in the strips, and 
 1.87 times the prism strength of the concrete in compression. 
 
 Certain conclusions may be drawn from Table XI and from Fig. 40: 
 
 (a) The slabs show, on the whole, the same decrease of modulus of 
 rupture with an increasing ratio of steel as did the strips which were 
 tested as beams. 
 
 (b) The thinner slabs develop, on the whole, greater moduli of rupture 
 than the thicker slabs with the same span and reinforcement. This result 
 may be explained by the dish action which occurs when the deflections have 
 become appreciable compared with the thickness of the slab. The slabs 
 a, and a,, for example, which were 12 cm and 8 cm thick, respectively, 
 deflected about 6 cm at the center at the maximum load. By the double 
 curvature of a slab the vertical sections resisting the bending moments 
 assume an arc-shape instead of the original rectangular shape, and thus the 
 section modulus is increased. At a given deflection, this effect is compara- 
 tively greater in a thin slab than in a thick slab. The dish action of the 
 thin slab may be interpreted as a reversed dome action, in which the central 
 area is essentially in tension, while the outer area is essentially in com- 
 pression. The additional tensions and compressions explain the added 
 carrying capacity, beyond what may be expected on the basis of the coeffi- 
 cients of moment which were derived for the medium-thick stiff homo- 
 geneous elastic plates. 
 
 (c) The design with closer spacing of the bars in the upper than in 
 the lower layer of steel, so as to make the section moduli equal for the two 
 layers, does not appear to be advantageous. 
 
 (d) Reinforcement parallel to the diagonals appears to be less effective 
 than reinforcement parallel to the sides. If the corners had been prevented 
 from bending up by anchoring, the corners were observed to deflect 
 slightly upward, and if the steel along the diagonal had been bent up so 
 as to reinforce against negative moments at the corner, greater strength 
 might possibly have been developed with the same amount of steel. 
 
 (e) The slab has an ability to redistribute the stresses as the deflec- 
 tions increase, as the steel stresses approach or reach the yield point, and 
 as cracks develop. By the redistribution the large stresses become smaller 
 and the small stresses larger than would be predicted according to the dis- 
 tribution in the homogeneous elastic slabs for which the theory in Part II 
 was derived. The phenomenon of redistribution is well known from other 
 fields. For example, in a flat steel tension bar with a circular hole there 
 is, at small stresses, a relative concentration of stress at the edge of the 
 hole, but when the yield point has been reached the stresses may be prac- 
 tically uniformly distributed. Redistribution of stresses is a typical gen- 
 eral feature in statically indeterminate structures of ductile materials. 
 Thus, the property of the slab as a highly statically indeterminate structure 
 becomes important; it explains additional strength beyond what might 
 otherwise be expected. In the homogeneous elastic square slab with simple 
 supports on four sides and with Poisson's ratio equal to zero the coeffi- 
 cients of moment per unit-width across the diagonal are, according to Art.
 
 MOMENTS AND STRESSES IN SLABS &L 
 
 7: at the corner, 0.0463; at the center, 0.0369; average for the whole 
 diagonal, 0.0417. The stresses across the diagonal may be redistributed so 
 as to become nearly uniform; accordingly the average coefficient, 0.0417, 
 was applied to all the square slabs in Table XI. The maximum coefficient 
 0.0463 would have made the corresponding stresses and ratios of stresses 
 in Table XI and in Fig. 40, 1.11 times greater than the values shown. The 
 coefficients 0.0463 would have led to a less close agreement between the 
 slab strength and the strip strength than is found in Table XI and in Fig. 40. 
 The redistribution of stresses across the diagonal may explain the rather 
 large stresses computed for the corners of the slabs d^ and d t . These 
 stresses were computed by using the largest coefficient of moment in con- 
 nection with the smallest percentage of steel. Evidently the stresses have 
 transferred toward the center, where the spacing of the steel is closer, and 
 as a result of this redistribution the steel at the center appears to be more 
 effective, per pound weight, in resisting the utlimate loads, than the steel 
 near the edges. In the rectangular slabs the rather large moduli of rupture, 
 84,900 and 95,500 Ib. per sq. in., computed by the coefficients of moment for 
 the short span at the center, may be explained partly by a redistribution 
 of the stresses across the long center line, whereby the actual stresses at the 
 center are reduced, and partly by a transfer of stresses from the short span 
 into the long span. N. J. Nielsen,* by using moment coefficients found by 
 the method of difference equations, with Poisson's ratio equal to zero, deter- 
 mined ratios of slab strength to strip strength for the square slabs loaded 
 nearly uniformly, for the square slabs loaded by eight forces near the 
 center, for the square slabs loaded by one force at the center, and for the 
 rectangular plates. He found the ratio of slab strength to strip strength to 
 be fairly uniform for all the slabs, including the rectangular slabs, by 
 assuming different values of the moment of inertia for the two spans, 
 namely, such values that the maximum computed stresses become equal in 
 the two spans, with the steel in the two directions utilized fully. 
 
 A comparative study of computed and observed deflections of one of 
 the double panels ( two square panels, continuous over a transverse beam ) , 
 under a load equal to about one-fourth of the ultimate load, was made by 
 X. J. Nielsen,f who used the method of difference equations. By considering 
 the plate as made of homogeneous material with a modulus of elasticity of 
 4,110,000 Ib. per sq. in., and by taking the deflections of the transverse 
 beam as observed, he found the computed and the observed deflections at 
 the centers of the panels to be equal, and he found the deflection curves 
 and contour lines shown in Fig. 41. Near the transverse beam the observed 
 deflections are seen to be smaller than the computed deflections. Thi-s 
 difference may be due to the rather heavy reinforcement across the trans- 
 verse beam. 
 
 The test made in 1920 at Waynesburg, Ohio, for Mr. J. J. Whitacre, 
 throws further light on the question of the redistribution of stresses, as 
 compared with the stresses in homogeneous elastic slabs, and on the ques- 
 
 * Ingenioeren, 1920, p. 724. 
 
 tN. J. Nielsen, Spaemlinger i Plader, 1920, p. 74.
 
 82 
 
 MOMENTS' AND STRESSES IN SLABS. 
 
 tion of the ultimate strength of the slabs.* The test was made with a two- 
 way reinforoed-concrete and hollow tile Hour slab, in. thick, with 18 
 panels. Fig. 42 shows the plan of the floor. The tiles are G in. by 12 in. 
 by 12 in., open at the ends so as to allow the concrete to How in, filling up 
 a part of the tile. The tiles are separated by 4-in. concrete ribs in both 
 directions. Each rib is reinforced by a ^-in. round bar at the bottom, and, 
 in addition, in the part of the rib near the panel edges, by a ^-in. round 
 bar at the top. The yield point of the steel was 54,000 Ib. per sq. in. 
 
 Large negative moments were produced at the edges by loading all the 
 panels and the cantilevers adjacent to B, C, and E, at the same time. 
 Results of this part of the test are shown in Table XII. The applied load 
 on each panel was a nearly uniformly distributed load, consisting of four 
 piles of bricks with 18-in. aisles. Equivalent entirely uniformly distributed 
 applied loads were derived by multiplying the average applied loads 
 (within the areas defined by the clear spans) by the factors 0.91, 0.92, and 
 0.93 for the square, medium long, and longest panels, respectively; these 
 
 FIG. 41. COMPARATIVE STUDY BY N. J. NIELSEN OF OBSERVED DEFLECTIONS 
 (SHOWN BY DOTTED LINES) AND COMPUTED DEFLECTIONS (SHOWN BY 
 FULL LINES) IN A DOUBLE PANEL TESTED BY BACH AND GRAF. 
 
 factors were determined by an approximate theory. The equivalent loads 
 stated at the head of Table XII are found by adding the dead load, 50 Ib. 
 per sq. ft., to the equivalent uniform applied load. The observed stress, /, 
 at the center of the edges, was considered to be made up of an estimated 
 dead-load stress of 500 Ib. per sq. in., plus the increase of stress due to the 
 live load; this increase was found as the maximum ordinate of a smooth 
 curve plotted from strain-gage readings on several gage lines across the 
 
 * A detailed report on this test has not yet been published. Only certain aspects 
 which have a general bearing on the question of the moments and stresses in slabs are 
 discussed here.
 
 MOMENTS AND STRESSES IN SLABS. 
 
 83 
 
 particular edge. In Table XII the ratio of the corrected steel stress / 
 corresponding to the computed stress in a beam, to be observed stress, /, 
 has been determined by means of formulas (1) and (3) in Part III, as 
 though the material were solid stone or gravel concrete instead of concrete 
 and hollow tiles. Since the reinforcing bars are 16 in. apart, the ratio of 
 reinforcement at the center of the edge is, p = 0.00260. Formula ( 1 ) , 
 which applies at small stresses, before the concrete has cracked, gives 
 then, 
 
 /. 0.52 J_ 
 
 7" " i + 021 17 - 5 
 p 
 
 and formula (2), which applies after the cracking has begun, gives the 
 relation 
 
 54000 
 
 0.222, 
 
 by which the values were computed in Table XII. The computation of the 
 -^ i 1 i 1 
 
 f 
 
 , !Q 
 
 \ 
 
 
 
 
 
 J 
 
 A 
 
 B 
 v t 
 
 C 
 ^ j 
 
 D 
 
 
 
 -<- v- 
 
 F 
 <, . > 
 
 G 
 
 ..... ^ 
 
 h 
 \ > 
 
 1 
 
 j 
 
 J 
 
 K 
 1 
 
 , --.^ 
 L 
 
 M 
 
 > ^ 
 
 N 
 
 
 
 P 
 
 Q 
 
 K 
 
 FIG. 42. PLAN OF FLOOR SLAB TESTED AT WAYNESBUKG, OHIO ; THICKNESS, 
 
 6 IN. 
 
 coefficients of moment in Table XII may be shown by an example. For the 
 edge AB (in the first line of the table) the observed moment coefficient, 
 based on the observed stress, 45,000 Ib. per sq. in., is found to be 
 
 M 
 
 0.00260 0.920 4.7 Pin 2 45000 . 2 
 388 lb_- 15 2 ft 2 
 ft 2 
 
 Since the corrected stress is 1.422 times the observed stress, the correspond- 
 ing corrected moment coefficient becomes 
 
 1.422- 0.02735 0.0389.
 
 84 MOMENTS AND STRESSES IN SLABS. 
 
 TABLE XII. COEFFICIENTS OF NEGATIVE MOMENTS IN WAYNESBVRG TEST. 
 
 Test of reinforced concrete and hollo** fi/e floor slat; IB panels supported 
 on girdtrs as shown in f r i'ff.'-42<5trfsses and moments at-centej-s of 
 edges. All panels /oadfct. fauiya/ent trni/orm loads (incl dead load) in /b.per sq.ft. 
 
 a*Vi _- _ _ f_ r-t ** > * j _ 3 rf? l_f Ay ^/\ O *. /" n J T S\ T\ n^>t r* fr'u . _'* 
 
 
 / 
 
 Jf. 
 
 0^ 
 
 folio 
 
 CotfficiejiU of max. 
 
 ffj* 
 
 O ltftvrJ 
 
 Ralio 
 
 Coefficients of ma*. 
 
 
 
 
 tfrtts 
 
 A 
 
 nea. moment tf/wb 
 
 
 ttr, 
 
 JL 
 
 3' r / 
 
 
 
 
 f 
 
 
 Ot~r~J 
 
 Cor^cffJ 
 
 Throrj. 
 
 
 f 
 
 f 
 
 fce r >ed 
 
 c ' trftt " 
 
 TKrory 
 
 
 
 
 1b/in l 
 
 
 
 
 flf.fli) 
 
 
 Ik/in 1 - 
 
 
 
 
 /7y 8M 
 
 -C 
 
 fl 
 
 flB 
 
 45000 
 
 uzz 
 
 .0274 
 
 0)89 
 
 
 FIG 
 
 Sisoo 
 
 1.270 
 
 .031) 
 
 .0598 
 
 
 *<\ 
 
 & 
 
 Bfl 
 
 38000 
 
 / 64) 
 
 .02/8 
 
 .0358 
 
 
 
 
 
 
 
 
 ^ 
 
 B 
 
 &C 
 
 17000 
 
 5.597 
 
 .0098 
 
 .03* 
 
 
 BH 
 
 3(,500 
 
 1.70 / 
 
 .OZIO 
 
 .JO 3 57 
 
 
 1. 
 
 G 
 
 
 
 
 
 
 
 Gfl 
 
 50OOO 
 
 I.30Z 
 
 .0287 
 
 .0374 
 
 
 \ 
 
 G 
 
 GH 
 
 19fOO 
 
 1-589 
 
 .0217 
 
 .0360 
 
 
 CM 
 
 41000 
 
 1.55? 
 
 .02)f 
 
 03 6Z 
 
 
 ^ 
 
 H 
 
 HG 
 
 JJ500 
 
 1.589 
 
 .0211 
 
 .0360 
 
 
 HB 
 
 48500 
 
 1.356 
 
 .0278 
 
 .0371 
 
 
 * 
 
 H 
 
 HI 
 
 38000 
 
 1.643 
 
 .OUB 
 
 .0358 
 
 
 HH 
 
 48500 
 
 1-356 
 
 .0178 
 
 057Z 
 
 
 s. 
 
 fl 
 
 tin 
 
 44000 
 
 1.449 
 
 .0268 
 
 .0388 
 
 
 flG 
 
 55000 
 
 1.240 
 
 O)ZZ 
 
 .0400 
 
 
 ? 
 
 /v 
 
 fiM 
 
 41000 
 
 1559 
 
 .0235 
 
 .056% 
 
 
 NH 
 
 51500 
 
 1.270 
 
 0296 
 
 .0376 
 
 
 i 
 
 N 
 
 tfO 
 
 42500 
 
 /.495 
 
 -0244 
 
 .0.364 
 
 
 
 
 
 
 
 
 
 
 ffrerage coefficient- 
 
 .0363 
 
 .0417 
 
 flreraye coefficient: 
 
 .0?7 
 
 .0417 
 
 
 C 
 
 ce> 
 
 29000 
 
 2.084 
 
 .OZZl 
 
 .0463 
 
 
 
 
 
 
 
 
 ^ 
 
 C 
 
 CD 
 
 24500 
 
 2425 
 
 .0188 
 
 .0455 
 
 
 Cl 
 
 26000 
 
 1.500 
 
 .0/99 
 
 .0458 
 
 
 J$ 
 
 D 
 
 DC 
 
 53600 
 
 1.654 
 
 .0257 
 
 .0471 
 
 
 
 
 
 
 
 
 % 
 
 D 
 
 D 
 
 3F?00 
 
 1744 
 
 .0272 
 
 .0474 
 
 
 DJ 
 
 42500 
 
 1.493 
 
 .0)25 
 
 .0486 
 
 
 X 
 
 I 
 
 IH 
 
 38000 
 
 1.645 
 
 .0291 
 
 .0478 
 
 
 1C 
 
 Soooo 
 
 1.50% 
 
 .0383 
 
 .0499 
 
 
 <\ 
 
 
 I 
 
 IJ 
 
 59500 
 
 1.589 
 
 .0503 
 
 .0481 
 
 
 10 
 
 flfOO 
 
 1.270 
 
 .0395 
 
 .0501 
 
 
 
 J 
 
 J 1 
 
 38000 
 
 1-645 
 
 .0291 
 
 .0478 
 
 
 JD 
 
 51500 
 
 1.270 
 
 .0575 
 
 .0501 
 
 
 ^>' 
 
 J 
 
 JK 
 
 47000 
 
 1.371 
 
 .0360 
 
 .0494 
 
 
 JP 
 
 51500 
 
 1:270 
 
 .0595 
 
 .0501 
 
 
 | 
 
 
 
 ON 
 
 58000 
 
 1.64) 
 
 .0291 
 
 .0478 
 
 
 01 
 
 50000 
 
 1-501 
 
 .038) 
 
 .0499 
 
 
 ^ 
 
 
 
 OP 
 
 48500 
 
 1.356 
 
 .0371 
 
 .0496 
 
 
 
 
 
 
 
 
 r 
 
 P 
 
 PO 
 
 38000 
 
 1.643 
 
 .0291 
 
 .0473 
 
 
 PJ 
 
 41500 
 
 1.49) 
 
 .0325 
 
 .0481, 
 
 
 1 
 
 P 
 
 PQ 
 
 44500 
 
 1.4.5 (> 
 
 .0341 
 
 .0490 
 
 
 
 
 
 
 
 
 ^ 
 
 
 ffverage > forty Span: 
 
 .0478 
 
 .0417 
 
 /Jyej-aye., sAorfsfon.- 
 
 .0491 
 
 .OS7Z 
 
 
 
 
 ED 
 
 2000Q 
 
 2.922 
 
 .0179 
 
 .0522 
 
 
 
 
 
 
 
 
 ^ 
 
 E 
 
 EF 
 
 55500 
 
 1-834 
 
 .0300 
 
 .0549 
 
 
 EK 
 
 41000 
 
 1.539 
 
 .0367 
 
 .0564 
 
 
 ^ 
 
 F 
 
 FE 
 
 24500 
 
 2.425 
 
 .0219 
 
 .0531 
 
 
 FL 
 
 Slfoo 
 
 1.2.70 
 
 -04to 
 
 .0565 
 
 
 N 
 
 K 
 
 KJ 
 
 42500 
 
 1.495 
 
 .0380 
 
 .0567 
 
 
 K 
 
 54000 
 
 1.221 
 
 .0483 
 
 .0590 
 
 
 * 
 
 A 
 
 KL 
 
 42.500 
 
 1.493 
 
 .0366 
 
 .0567 
 
 
 HO 
 
 54000 
 
 1.222 
 
 .0493 
 
 .0590 
 
 
 
 
 L 
 
 LK 
 
 59500 
 
 I.f89 
 
 .0355 
 
 .0561 
 
 
 LF 
 
 f/foo 
 
 1-270 
 
 .04.60 
 
 .0585 
 
 
 v 
 
 L 
 
 
 
 
 
 
 
 Lf{ 
 
 54000 
 
 /.22Z 
 
 04S3 
 
 .0590 
 
 
 43 
 
 Q 
 
 QP 
 
 555oo 
 
 1.6*4 
 
 .0)00 
 
 .0549 
 
 
 QK 
 
 54000 
 
 I-ZU, 
 
 .0483 
 
 OS90 
 
 
 
 Q 
 
 Q% 
 
 48500 
 
 I55& 
 
 .0433 
 
 .0*79 
 
 
 
 
 
 
 
 
 
 
 R 
 
 RQ 
 
 f4000 
 
 1.222 
 
 .048) 
 
 OfffO 
 
 
 RL- 
 
 54000 
 
 l.2^^ 
 
 .O483 
 
 .osyo 
 
 
 
 
 fJrera 9 e . 7or>yJp0n 
 
 OfSj 
 
 0417 
 
 flfe rage , J ho r-f span ; 
 
 .0535 
 
 .0674
 
 MOMENTS AND STRESSES IN SLABS. 85 
 
 The theoretical values of the coefficients were taken from the approxi- 
 mate curves in Fig. 8(b) in Art. 7. These values are somewhat smaller 
 than the corresponding directly determined coefficients in Fig. 8 (a), but 
 they may be assumed to apply after the stresses across the edges have been 
 redistributed to some extent. That a further shifting of the stresses from 
 the short span to the long span takes place at increasing deformations, is 
 indicated by the fact that the oblong panels in Table XII have corrected 
 coefficients, based on observations, which are greater than the theoretical 
 values for the long span, and are smaller than the theoretical values for the 
 short span; but the sum of the corrected coefficients for the long span and 
 the short span is approximately equal to the sum of the corresponding 
 theoretical coefficients. It is probable that at increasing loads there is 
 also a transfer of stress from the edges to the central portion of each panel; 
 and that this transfer may partly explain the rather small values found 
 for some of the coefficients in Table XII. 
 
 In a later part of the test large loads were applied in panels H, J, and 
 K, while the loads on the surrounding panels were reduced. In the square 
 interior panel H, for example, the average applied load was increased to 
 1413 Ib. per sq. ft., giving, by the computation used in Table XII, an 
 equivalent uniform load of 50 + 0.91-1413 = 1336 Ib. per sq. ft. This 
 value is probably somewhat too large because of the unavoidable arch action 
 in the piles of brick; the four piles were joined together 12 ft. above the 
 slab and were continued as one pile up to the total height of almost 22 ft. 
 When the deflections increase the resultant pressure transmitted through 
 each of the four piles is thrown toward the corners of the panel, and the 
 equivalent uniform load becomes correspondingly smaller. Since the 
 amount of the reduction is not known, the value just stated, 1336 Ib. per 
 sq. in., will be used without reduction in the computation of stresses. Since 
 the adjacent panels were unloaded, the panel H may be considered in this 
 computation as a single square panel with the edges half fixed and half 
 simply supported. Accordingly, the average of the numerical values of the 
 moment coefficients at the center and at the edge in slabs with simply sup- 
 ported and with fixed edges is used, that is (see Fig. 3 (a), Fig. 7 (a), and 
 Fig. 8 (a)). 
 
 i (0.0369 + +0.0177 + 0.0487) = 0.0258. 
 
 4 
 
 By assuming this moment coefficient, and by assuming the same effective 
 depth and ratio of steel as in the calculation of Table XII, one finds the 
 "computed stress" in panel H under the maximum load equal to 
 
 , 0.0258 wV 0.0258 1336 15 2 
 
 > = -jjdr- = 0.00260- 0.92 -4.7P = 14600 lb ' per Sq ' m ' 
 This stress is 2.71 times the yield point stress of the steel, j y = 54,000 lb. 
 per sq. in., and 2.21 times the strip strength, f bs =66,200 lb. per sq. in., 
 as determined from Bach's and Graf's tests by the line in Fig. 39. In 
 estimating the significance of this result it should be noted that some arch 
 action in the piles of brick probably made the applied load not fully effect- 
 ive; that the material is different from ordinary reinforced-concrete; that
 
 86 MOMENTS AND STRESSES IN SLABS. 
 
 the moment coefficients may have been reduced by redistribution of the 
 stresses across the center line, across the diagonal and across the edge; 
 that this redistribution may have been aided by the deflections of the sup- 
 porting girders; and that the deflection at the center was so large, 1.4 in., 
 that the dish action or reversed dome action which is characteristic of thin 
 slabs, may have aided in carrying the load. It is not known what load 
 would have" produced failure. 
 
 The average loads, determined by dividing the total applied load by 
 the panel area, applied in panels J and K at the same time and under 
 similar conditions without producing failure, were 1184 Ib. per sq. ft. and 
 920 Ib. per sq. ft., respectively. 
 
 V. TESTS OF FLAT SLABS. 
 BY W. A. SLATER. 
 
 21. GENERAL DESCRIPTION. In the following pages are given the 
 results of tests of certain flat slabs. It has been necessary to make the 
 discussion of the results very brief and only sufficient statement on each 
 subject has been made to enable the reader to interpret the data given in 
 the diagrams and tables. 
 
 The study of the tests is based almost entirely upon tensile stresses 
 since it is impossible to know with sufficient accuracy for this purpose what 
 compressive stresses arg indicated by the compressive deformations and 
 because the amount of reinforcement in flat slabs is generally so small that 
 the tensile stresses will almost always be critical rather than the com- 
 pressive stresses. 
 
 The results for most of the tests have been published previously. Those 
 for the two Purdue tests, the Sanitary Can Building test, and the Shonk 
 Building test have not been published, and those of the International Hall 
 test* were published only in part. Because of the fact that the results of 
 the Purdue test had not been published, the reinforcing plans, the location 
 of gage lines, the measured deformations and the deflections are given in 
 Appendix B. Further data are given in Table XIII. 
 
 It had been expected to give the results for the Sanitary Can Building 
 test and the Shonk Building test as fully as for the Purdue tests, but this 
 has not been possible. The following statement, together with the data 
 given in Table XIII, will be sufficient to give significance to the moment 
 coefficients given in Fig. 45 for these tests. It is expected that in a later 
 publication of the Bureau of Standards the full data of these tests will be 
 included. 
 
 The tests of the Sanitary Can Building and the Shonk Building were 
 made by A. R. Lord, of the Lord Engineering Company, Chicago, 111. Prof. 
 W. K. Hatt, of Purdue University, Lafayette, Ind., was in touch with these 
 tests at the request of the Corrugated Bar Company. The report by Mr. 
 
 Trans. A. S. C. E., Vol. LXXVII, p. 1433 (1914).
 
 MOMENTS AND STRESSES IN SLABS. 87 
 
 Lord and that by Professor Hatt have been drawn upon for the data used 
 in preparing this paper. 
 
 Both buildings are located at Maywood, 111., a suburb of Chicago. 
 The floors of both are fiat slabs, having column capitals 5 ft. in diameter 
 and dropped panels 8 ft. square. In each building four panels were loaded 
 and in each case two of the loaded panels were wall panels and the other 
 two were the adjacent interior panels. The panel length in the direction 
 parallel to the wall and also perpendicular to the wall for the two interior 
 panels is 22 ft. for both buildings. For the wall panel, the panel length 
 perpendicular to the wall is 21 ft. 3 in. for the Sanitary Can Building and 
 20 ft. 7 in. for the Shonk Building. The Sanitary Can Building has 24-in. 
 octagonal interior columns and wall columns 20 by 45 in. rectangular in 
 cross section. The Shonk Building has 22-in. octagonal interior columns 
 and wall columns 21% in. rectangular in cross section. The Sanitary Can 
 Building has two-way reinforcement and the Shonk Building has four-way 
 reinforcement. There was some difference in distribution of reinforcement, 
 but in the two slabs the total area provided for negative moment was 
 about the same, and the total area for positive moment was about the same. 
 The area of reinforcement at the principal design sections and the meas- 
 ured depth d to the reinforcement are shown in Table XIII. 
 
 Each floor was designed for a live-load of 150 Ib. per sq. ft. The 
 maximum superimposed test load was about 400 Ib. per sq. ft. The dead 
 load brought the total test load up to about 535 Ib. per sq. ft. in each test. 
 The highest observed stresses in the reinforcement were about 24,000 Ib. 
 per sq. in. in both floors, and this high stress occurred in only a few places. 
 
 The misplacement of the reinforcement in the Bell St. Warehouse* 
 probably had an influence on the distribution of resisting moments between 
 positive and negative, which would not be expected in slabs of that type. 
 At least, this feature should be considered in studying the results of 
 the test. 
 
 The original data of the* test of the Western Newspaper Union Building 
 were not available, therefore, in determining the moment coefficients for this 
 test, it was assumed that the moment of the observed tensile stresses at any 
 load less than the maximum bore the same ratio to the resisting moment 
 reported for the test load of 913 per sq. ft.f that the sum of the observed 
 stress and the estimated dead load stress for the load under consideration 
 bore to the .sum of the observed stress for the maximum test load and the 
 estimated dead load stress. 
 
 22. CORRECTION OF MOMENT COEFFICIENTS. In reporting results of 
 flat slab tests it has been customary to state the moment of the observed 
 stresses in the reinforcement as a proportion of the product of the total 
 panel load and the span. It has been known from beam tests that the 
 
 Pacific N. W. Soc. Civ. Eng., Vol. IS (Jan. and Feb., 1916); Eng. Record, Vol. 
 73, p. 647; Eng. News-Record, April 19, 1917. 
 
 t Bulletin 106, Univ. of 111. Eng. Exper. Sta., p. 36. Also Proc. A. C. I.. Vol. 
 XIV, p. 192 (1918).
 
 88 
 
 MOMENTS AND STRESSES IN SLABS. 
 
 total moment of the observed stresses is less than the applied moment. 
 The ratio of the applied moment to the moment of the observed stress is 
 equal to the ratio of the computed stress to the observed stress. This may 
 be shown as follows: 
 
 M = KW I = Af s jd 
 A/, - KiW I 
 from which 3f K / s 
 
 and 
 where 
 
 7v = 
 
 K and K i are coefficients 
 KW I = applied moment 
 KI W I = moment of observed stress 
 fs = computed stress 
 /i = observed stress 
 Other terms have their usual significance. 
 
 J6r 
 
 J4 
 
 1 - Total Moment f numerical 
 
 sum of pas and neq.) 
 
 2 - Total tfeqafive Moment 
 
 3 - Neq. Moment; Col. Head5ecf ion 
 
 4 - Total flssfffa Moment 
 
 5-Fbs.Moment;OuterSedion 
 6 -Net). - ;Mid 
 \.l-Fbs. - > Inner 
 
 i 
 
 ZOOOO .30000 40000 50000 60000 ^0000 30000 40000 50000 60000 70000 
 
 Computed Average Tensi/e Stress, Ibpersq./n. 
 
 (a) (6) 
 
 FIG. 43. MOMENT COEFFICIENT FOR SLAB J; (a) UN CORRECTED; (6) 
 
 CORRECTED. 
 
 These equations show that for a beam the true moment coefficient K^ 
 may be determined by multiplying the moment coefficient of the observed 
 stress by the ratio of the computed stress to the observed stress. This ratio 
 is here termed the moment correction. Although it is recognized that the 
 behavior of a slab differs from that of a beam it seems reasonable to assume 
 that the relation between applied moments and the moment of the observed 
 tensile stresses in the reinforcement should be the same for a slab as for a 
 beam if the percentage of. reinforcement, the modulus of elasticity of the 
 concrete, the depth d and the depth of covering of the reinforcement are the 
 same for the slab as for the beam.
 
 MOMENTS AND STRESSES IN SLABS. 
 
 The beams tested by the U. S. Geological Survey, and reported in 
 Part III, afford a basis for determining the moment correction for a wide 
 range in the percentage of reinforcement and the modulus of elasticity of 
 the concrete, and to this extent the moment corrections found for these 
 beams will be useful for estimating from the observed stress in the slabs 
 the moment applied to the slabs. The fact that in this investigation only 
 one depth, d, and only a slight variation in the covering of the tension 
 reinforcement were used, limits the usefulness of this investigation as 
 applied to interpreting test results of flat slabs, but no other series of 
 tests is known which covers so wide a range of conditions, and, notwith- 
 standing these limitations, it seems reasonable to expect that, on the whole, 
 the application of the moment corrections from Fig. 31 and Fig. 32 to the 
 
 16 
 
 S 
 
 
 / - Tola I ' Momentfnumertcal \ 
 
 sum of pas. and neg) J 
 
 ' Z -Jbfat NsqafMsMament \ 
 
 ~3 - Neq. Moment, Caf head Section 
 
 4- ~~ 
 
 06 
 
 /f 
 
 30000 40000 50000 6000O 70000 30000 40000 50000 60000 70000 
 
 Computed Average Tenst/e Stress, /b. per 53. in. 
 (a) (6) 
 
 FIG. 44. MOMENT COEFFICIENTS FOR SLAB S; (a) UNCORRECTED; 
 
 CORRECTED. 
 
 (6) 
 
 moment coefficient, determined by means of the observed stress in the flat 
 
 slab, should give a fair idea of the true moment coefficients for these slabs. 
 
 23. MOMENT COEFFICIENTS. The moment coefficients have been stated 
 
 M 
 
 as values of the expression 
 
 in which M is the sum of the 
 
 m - a- r 
 
 positive and negative moments in the direction of either side of the panel, 
 W is the total panel load, c is the- diameter of the column capital and I is 
 the span in the direction in which moments are considered. This is a con- 
 venient form of expression and it has been found possible to state the 
 moments found by the analysis in terms of it witli a satisfactory degree of 
 
 See Art. 8.
 
 90 
 
 MOMENTS AND STRESSES IN SLABS. 
 
 The vise of the same form of expression in stating the test results 
 simplifies the comparison with the analytical result. In determining the 
 value of M from the tests for use in calculating these moment coefficients 
 the equation M Afjd was evaluated. Wherever it was possible the 
 moments were determined separately for the sections of positive and of 
 negative moment shown in Fig. 12, using the values A, f, and d, shown by 
 observation for these sections. In some cases it was necessary to use an 
 average value of fj for both sections of positive moment and another average 
 value for both sections of negative moment. In some cases measured values 
 22 
 
 ID 
 
 06 
 
 I - Total ' Moment (nu-- 
 
 merical sum of - 
 pas andneq.) - 
 
 2- Tofal Negative _ 
 Moment 
 
 3-Neq.Moment;Col. 
 
 - Head Section - 
 4-Totol Posffire - 
 
 - Moment 
 
 5 -Po5. Moment: 
 Outer Sect/on 
 
 6 -Neq. Moment; 
 Mid Section. 
 
 7- Pos. Moment; 
 Inner Section. 
 
 \ 
 
 \ 
 
 zo 
 
 .18 
 
 j 
 
 1* 
 
 j|./<? 
 
 s/ 
 
 
 
 
 
 
 J- 
 
 r 
 i 
 
 "a/. 
 
 > 
 
 : 
 
 7. 
 
 n 
 
 
 
 
 
 1- Total Moment(n 
 merical sumo) 
 pos. andneq.) 
 Z-Tofal Neoatin 
 Moment 
 3-NeqMoment,L 
 head Sect/or 
 4 -Total Positive 
 Moment 
 5 -Pos. Moment, 
 Outer5ection 
 6- Neg. Moment 
 Mid Section 
 7- Pos. Moment 
 ImerSectioi 
 
 
 
 
 
 
 
 \ 
 
 
 
 
 
 ^ 
 
 
 
 
 
 
 \ 
 
 
 
 
 
 V 
 
 
 
 
 
 \ 
 
 
 
 
 
 
 \ 
 
 
 
 
 
 
 'b/ 
 
 
 
 
 
 
 
 
 
 
 
 
 ^ 
 
 
 
 
 
 s 
 
 
 
 
 
 
 ^v 
 
 
 
 
 
 
 
 
 
 
 
 \ 
 
 > 
 
 >v 
 
 *? 
 
 
 
 
 
 
 
 
 
 
 Sj 
 
 *^v. 
 
 'y* 
 
 .04 
 & 
 n 
 
 
 
 
 
 
 
 
 
 h 
 
 
 
 
 
 
 f 
 
 ^ 
 
 f> j 
 
 
 
 
 > 
 
 \ 
 
 
 
 
 ( 
 
 / 
 
 
 tf 
 
 
 
 
 
 K 
 
 K 5 
 
 i 
 
 / 
 
 s 
 
 i^ 
 
 
 
 
 
 
 i 
 
 c 
 
 
 ^ 
 
 
 7 
 
 
 
 *^ 
 
 ^^ 
 
 TTtft 
 
 
 -=s 
 
 =^ 
 
 =3?f> 
 
 
 
 
 
 
 ZOOM 3000040000 ZOOM 3000040000 
 
 Computed Average 1eny/e5tres5, Ibpersqin. 
 
 () (6) (a) 
 
 FIG. 45. MOMENT COEFFICIENTS FOR SANITARY CAN 
 UNCORRECTED ; ( 6 ) CORRECTED. 
 
 FIG. 46. MOMENT COEFFICIENTS FOR SHONK BUILDING; (a.) UNCORRECTED 
 
 ( 6 ) CORRECTED. 
 
 Computed Aeraqe Tensile Stress Ibpersq/n. 
 
 BUILDING; (a) 
 
 of d were not available and it was necessary to use what appeared to be the 
 most probable values, taking into account the total thickness of the slab, 
 the size of reinforcing bars and the number of layers of reinforcement. It is 
 apparent that these uncertainties will. introduce corresponding uncertain- 
 ties into the moment coefficients, but the errors are believed to be no 
 greater than the errors which are inevitably involved in other experimental 
 work of an equal degree of complexity. 
 
 Generally the computations were made separately for the moments in 
 the directions of the two sides of the panel, and were combined for presenta- 
 tion in Fig. 43 to 48. In all the slabs for which the moment coefficients
 
 MOMENTS AND STRESSES IN SLABS. 
 
 91 
 
 are shown the panel lengths in the two directions were so nearly the same 
 that it did not seem desirable to show separately the coefficients for the 
 two directions. The deficiencies in the test data available, and the unknown 
 factors which affect the behavior of the slab, would introduce errors which 
 are larger than the difference between the moments in the two directions. 
 In order to determine experimentally the difference in moments in the two 
 directions, when the spans are so nearly equal, a large number of tests 
 would be required, and the deficiencies in the test data would have to be 
 
 /- Total Moment(nu- 
 
 pos.andnea.) 
 <?- Total Negative 
 Moment. 
 
 3- Neq.Moment;Col. 
 Head Section. 
 
 4- Tofa/ Positive 
 Moment 
 
 5 -Fbs. Moment; 
 
 OuferSection. 
 6-Neq.Moment- 
 
 Mid Section 
 7- fas. Moment: 
 
 Inner Section 
 
 FIG. 47.- 
 
 ZOOM 30000 40000 30000 
 
 Computed Average Tensi/eSfress /bpersq/n 
 (a) (0) 
 
 -MOMENT COEFFICIENTS FOR BELL STREET WAREHOUSE; 
 UNCORRECTED; (6) CORRECTED. 
 
 (a) 
 
 supplied. One test, that of the Larkin Building, was available, in which 
 the differences in the size of the panels was considerable; but in this test, 
 except in the lower loads, the number of panels loaded was not sufficient to 
 give corresponding conditions in the two directions, and the coefficients for 
 that slab are not presented. 
 
 The computed average tensile stress, shown as abscissas in Fig. 43 to 
 48, were determined from the equation 
 
 \ 
 
 Ajd 
 
 in which 
 
 W is the total panel load, live and dead, and 2 Ajd is thp sum of the
 
 92 
 
 MOMENTS AND STRESSES IN SLABS. 
 
 values of Ajd for the sections shown in Fig. 12. The values of Ajd for 
 these sections are given in Table XIII. 
 
 It was shown in Art. 16 that the relation between the observed stress 
 and the computed stress in the reinforcement is affected by variations in the 
 value of n (that is, of the modulus of elasticity of the concrete, since that 
 of the steel is practically constant). Equations (5) and (6), Art. 16, show 
 this effect below and above the load at which the concrete cracked. Corre- 
 spondingly the value of the corrected moment coefficient will be affected by 
 these variations in n. 
 
 Fig. 49 has been prepared to show the effect of a variation in the value 
 of n on the coefficients. Small circles indicate the coefficients for the value 
 of n, which was used in obtaining the corrected moment coefficients shown 
 in Fig. 43 to 48. This is the average value of n for the stone and the 
 
 JJ|M-/2" 
 ^ 
 ^ J 
 
 \a 
 
 I ^ 
 
 ^ &> 
 1 (K 
 
 . l-Joto/ MomeiJ 
 merical sum 
 p-os. and ne 
 2- Tote! NeGCi'i'/t 
 4- " Fbfif/re. 
 
 , 
 
 7; 
 
 / 
 
 J- 
 
 ..,r 
 
 r 
 
 
 ~ 
 
 ~f- 
 
 --4-- 
 
 
 ^, 
 
 L J I 
 
 
 
 
 ^ 
 
 / 
 
 
 
 
 
 
 J\ 
 
 ~? 
 
 
 
 
 T 
 
 > 
 
 -0- 
 
 
 * Z 
 
 
 
 / 
 
 L? 
 
 ,: 
 
 
 
 
 
 
 
 
 
 , 
 
 ^ 
 
 
 
 
 
 
 
 
 
 
 / 
 
 'S 
 
 
 
 
 
 
 
 
 ~-~. 
 
 , 
 
 
 
 <_ 
 
 
 
 
 
 
 
 
 
 
 
 
 
 4 
 
 
 
 
 J>- 
 
 
 a-" 
 
 4 
 
 
 
 
 
 
 
 
 
 s 
 
 K ^*" 
 
 
 
 
 
 
 
 
 
 
 
 
 a' 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 KXJOO ZOOM 3OOCO 4OOOO IOOOO ZOXO 3OOOO 4OOCO 
 
 Computed Oreraqe Tensi/e Jtnsss, /b.perjq.in. 
 
 (a) (6) 
 
 FIG. 48. MOMENT COEFFICIENTS FOR WESTERN NEWSPAPER UNION BUILDING ; 
 (a) UN CORRECTED; (6) CORRECTED. 
 
 gravel concretes of the beams used in the tests by the U. S. Geological 
 Survey. (See Table VIII, Part III). The smooth curves show the cor- 
 rected coefficients which have been found by using a range of values of A in 
 the determination of the corrections. 
 
 It will be seen that in order to bring the coefficients to the theoretical 
 value of % the value of n for the Purdue tests would be about 9 and that 
 for the Sanitary Can Building and the Shonk Building would be about 11.5. 
 While it cannot be stated that these are the correct values of n for the 
 concrete in these slabs, it seems reasonable to believe that they are more 
 nearly correct than the value of 7, which was used in computing all cor- 
 rected moment coefficients. The available data from the slab tests strengthen 
 this belief, but the reliability of the test data which bear on this subject 
 was not sufficient to justify introducing into the computation of the cor- 
 rected moment coefficients the experimental values of n, determined in the 
 tests of these structures.
 
 MOMENTS AND STRESSES IN SLABS. 
 
 93 
 
 It will be seen in Figs. 43 to 48 that the uncorrected moment coefficients 
 are all less than the theoretical value, %. For the lower computed stresses 
 the corrected coefficients are generally in excess of y s , but for the highest 
 computed stress, that is, for the highest load applied, the average coefficient, 
 0.111, is less than the theoretical coefficient. 
 
 The fact that for the lower computed stresses the corrected moment 
 coefficients generally were higher than the theoretical value, indicates that 
 too large a correction factor was used. No means of knowing how much 
 the factor used was in error is evident, but the fact that as the computed 
 stress increases the uncorrected and the corrected moment coefficients 
 approach each other in value seems to reduce the uncertainties as to the 
 correct values of the moment coefficients to a narrower margin than that 
 which has limited the usefulness of practically all the field tests that have 
 ever been made on reinforced-concrete slabs. On the average, the agreement 
 
 Z 4 68 IO IZ 14 
 
 Values of n 
 
 FIG. 49. EFFECT OF VARIATION IN MODULUS OF ELASTICITY ON VALUE OF 
 MOMENT COEFFICIENTS AT Low LOADS. 
 
 of the corrected coefficients for the higher computed stresses with the result 
 found by analysis in Part II is sufficiently close to warrant the belief that, 
 if all the sources of error in the measurement of deformations and in the 
 interpretation of test results could be removed, the analysis and the tests 
 would be in substantial agreement. It is pointed out in Art. 20, in discuss- 
 ing the results of the test of a slab in which there were beams on the 
 boundary lines of the panels, that although, for the lower loads, the test 
 results were in fair agreement with the analysis of that type of slab, there 
 was, for the higher loads, an accommodation of the slab to the conditions 
 imposed upon it, which made the slab capable of carrying a much greater 
 ultimate load than is accounted for by equating the applied bending moment 
 and the apparent resisting moment. On account of the lack of tests of flat 
 slabs carried to the point of failure, and in which the design was such as 
 to preclude failure from some other cause than bending, it cannot be stated 
 that, to the same extent, a similar source of additional strength is present 
 in flat slabs as was present in the Waynesburg slab, which was supported on
 
 94 MOMENTS AND STRESSES IN SLABS. 
 
 four sides. There are some indications, however, that there was greater 
 strength in the flat slabs than appears from the conclusion that the test 
 results and the analysis of moments are in fair agreement. Some indica- 
 tion of this greater strength is seen in the description of the Purdue tests, 
 Appendix B. 
 
 24. FACTOB OF SAFETY. Table XIII gives summarized data of all the 
 tests studied. The purpose of this table is to give the best estimate of what 
 the factor of safety against failure of the structure would have been if 
 sufficient load had been applied in each case to produce failure. For all 
 the tests reported, except the Purdue tests, the factor of safety is an esti- 
 mated quantity which gives the ratio of the estimated maximum load which 
 the slab would carry to the design load (sum of dead and live load), com- 
 puted as indicated in the table. Although the factor of safety is shown for 
 three moment coefficients, the design load is shown for only one coefficient. 
 The design load will be inversely proportional to the coefficient used in the 
 design. For the Purdue test slabs the factors of safety given are the ratios 
 of the load actually applied to the slab to the design load shown in the 
 table. The "average observed stress, /" of the table, therefore, has no sig- 
 nificance for those slabs, but they are given because of their value as mat- 
 ters of general information concerning the tests. The stresses for the 
 maximum load of 872 Ib. per sq. ft., on slab J, were not reported, and the 
 average observed stress given for that slab is, therefore, that for the load 
 of 664 Ib. per sq. ft. (live and dead), the highest load for which the meas- 
 ured stresses were reported. For all other tests the average observed 
 stress given in the table is that for the "maximum test load" given in the 
 table. The average observed stresses, as reported, were roughly weighted 
 to take account of the distribution of the gage lines over the sections of 
 maximum stress due to negative moment and positive moment. For the 
 Purdue tests this weighting was not necessary, since the stresses were meas- 
 ured in all the bars crossing these sections. 
 
 The "estimated dead load stress" of Table XIII is given by equation 
 (1) or equation (3) of Part III, in which y s is the value of 
 
 , I 
 
 Js 
 
 2 Ajd 
 
 In this equation W is taken as the total dead load of the panel. Since the 
 value of / was generally below the points in the curves of Fig. 31, which 
 represent the cracking of the concrete, equation ( 1 ) was generally used in 
 estimating the dead load stress rather than equation ( 2 ) . 
 
 In estimating the maximum load for slabs it was assumed that the 
 yield point of the steel was 40,000 Ib. per sq. in., and that failure would 
 have occurred at a computed average stress, which is the same as the com- 
 puted stress given in Fig. 31 at the intersections of the straight lines, of 
 equation ( 3 ) , part III, with the locus representing the equation / = 40,000 
 (0.82-|-7p) given in Fig. 31. The assumption of 40,000 Ib. per sq. in. as
 
 MOMENTS AND STRESSES IN SLABS. 
 
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 'xx don moment of
 
 96 MOMENTS AND STRESSES IN SLABS. 
 
 the yield point was made in order to bring all the tests to a common basis 
 for the purpose of comparison and in order to obtain a factor of safety not 
 higher than might be expected if reinforcement having that yield point 
 were used. Obviously reinforcement having a yield point of 55,000 Ib. per 
 sq. in. should be expected to give a higher factor of safety than that having 
 a yield point of 40,000 Ib. per sq. in., when the working stress in tension 
 is 16,000 Ib. per sq. in. in both cases. 
 
 The considerations in the two preceding paragraphs indicate that the 
 true factor of safety was probably higher than the values given in Table 
 XIII. On the other hand it must be recognized that the design load found 
 with the use of the measured depth d would be smaller and the factor of 
 safety somewhat larger if the depth d assumed in the design of the slabs 
 were used for the computations of factor of safety. This is due to the fact 
 that, on account of misplacement of the reinforcement, the measured depth 
 to the reinforcement is usually somewhat less than the depth used in design. 
 
 The average factor of safety for the structures reported in Table XIII, 
 
 estimated on the basis of a moment coefficient of 0.1067 Wl 1 1 - - -l 2* 
 
 \ 3 l) 
 
 is 3.23 and that for the moment coefficient of 0.09 Wl (\ _ ? 2\ 2* is 2.72. 
 
 \ 3 l) 
 
 It is to be noted that for the tests which were carried to destruction of the 
 slab, or nearly so (the two Purdue tests and Western Newspaper Union 
 Building test) the values were above these average values. 
 
 It has been established by tests that the maximum load on a simple 
 beam occurs at a tensile stress only slightly greater, say 10 per cent, for 
 steel of structural grade, than the yield point of the steel in tension and not 
 at the ultimate strength of the steel. This is true not only for reinforced 
 concrete beamsf but also for steel beams. $ 
 
 Kecognizing this fact, and at the same time using a working stress of 
 16,000 Ib. per sq. in: in steel of structural grade, whose yield point is 33,000 
 Ib. per sq. in., is, in fact, recognizing the sufficiency of a factor of safety 
 
 of about 2.25. Based upon the use of a moment of 0.09 Wl M _ ^ . \ 2 for 
 
 \ O i / 
 
 design, the slabs reported in Table XIII would, on the average, develop this 
 factor of safety of 2.25 even though they were to have failed at loads 
 approximately 15 per cent greater than the loads which were applied. To 
 one familiar with the behavior of the structures listed in Table XIII, or 
 with similar structures during and after the tests, it is obvious that they 
 would have carried at least that much additional load. 
 
 * These are the total moments recommended by, respectively, the Joint Committee 
 on Concrete and Reinforced Concrete and the American Concrete Institute Committee 
 on Reinforced Concrete and Building Laws. 
 
 t Art. 18 and Fig. 36; also A. N. Talbot Bull. 1. Univ. of 111. Eng. Exp. Sta. 
 (1904), p. 27; Turncaure and Maurer "Principles of Reinforced Concrete Construc- 
 tion," 2nd ed. (1909), p. 142. 
 
 t H. F. Moore Bull. 68, Univ. of 111. Eng. Exp. Sta. (1913), p. 14. 
 
 $ Standard Specifications for billet steel concrete reinforcement bars A 15-14. A. S 
 T. M. Standards. 1913. D. 148.
 
 MOMENTS AND STRESSES IN SLABS. 97 
 
 It seems certain, therefore, that for the flat slabs under discussion 
 the factor of safety against failure in bending, based on a total moment ol 
 
 0.09 Wl(l - - y) 2 , is at least as great as that which can be counted upon 
 
 in the most elementary flexural unit, the simple beam built of, or reinforced 
 with, steel of structural grade and designed with the usual working 
 
 stresses. Based upon a total moment of 0.125 Will _ ) * the factor of 
 
 \ u 1 1 
 
 safety is correspondingly greater. 
 
 An effort was made to obtain, from failures of flat slabs in service, 
 information as to what were the serious weaknesses in the design. A study 
 of a letter from Edward Godfrey* citing 29 failures of reinforced-concrete 
 buildings brings out the fact that all of the building floors there discussed, 
 including flat slabs among others, failed during construction or as a result 
 of a severe fire. While some of the failures during construction may have 
 been partly due to deficiencies in the design the lack of proper safeguards 
 in construction were so evident and so important that with the meager 
 data available any effort to analyze the failures would be long drawn out 
 and largely speculative. The fact that the majority of the failures referred 
 to occurred during winter weather or in structures which had no oppor- 
 tunity to cure properly in warm weather is in itself sufficient indication 
 that poor construction conditions contributed largely to the failure. To 
 attempt to guard against abusive lack of safeguards in construction by 
 severe requirements for design would be ineffective and prohibitively 
 extravagant and would itself encourage the omission of these safeguards. 
 There probably are cases in which deficiencies of design have caused 
 trouble in flat slab structures, but cases of this kind, with data of the 
 design and loading sufficient to be of value in the study of the factor of 
 safety, have not been found in the preparation of this paper. 
 
 25. SHEARING STRESSES. In Table XIII the maximum shearing stresses 
 on a vertical section at a distance d, from the edge of the column capital 
 for the tests summarized in that table are given. Those shearing stresses 
 were computed on the basis of a depth jd. The highest shearing stress 
 developed was in the case of the Western Newspaper Union Building. The 
 reportf does not indicate that the test developed any weakness in shear. 
 In order to develop a factor of safety as high in shear as the estimated 
 
 factor of safety in bending based upon the total moment, 0.1067 Will 
 ~ j J , the shearing stress in the section under question could have been at 
 
 O i j 
 
 least 87 Ib. per sq. in. at the design load. Using the factor of safety in 
 bending for the total moment of 0.09 Win _ ~ . ) the shearing stress for 
 
 the design load could have been at least 102 Ib. per sq. in. There were 
 some indications that the failure of slab J may have been due to shear. 
 
 * Edward Godfrey, "An open letter to W. A. Slater," Concrete, February, 1921. 
 t Univ. of 111. Eng. Exp. Sta. Bulletin 106.
 
 98 MOMENTS AND STRESSES IN SLABS. 
 
 The allowable shearing stresses at the design load which would give the 
 same factors of safety as those shown for the moments would have been 
 49 Ib. per sq. in. and 58 Ib. per sq. in. respectively. 
 
 The shearing stresses are not given for the Jersey City Dairy building 
 nor for the International Hall. This is because the loads reported in this 
 table were not such as to give uniform shear around the perimeter of the 
 column capital, and it is not known what the maximum shear was. 
 
 VI. SUMMARY. 
 
 The theoretical analysis deals with slabs of homogeneous perfectly 
 elastic material and of uniform thickness. Two types are considered in 
 particular: slabs supported on four sides and flat slabs supported on 
 columns with round capitals. Moment coefficients derived by principles of 
 equilibrium and continuity are shown in the diagrams and tables. 
 
 The close agreement between the moment coefficients, determined by 
 several investigators, in slabs supported on four sides, is an indication that 
 dependable methods are now available by which homogeneous elastic slabs 
 may be analyzed. 
 
 In the analysis of flat slabs the moment sections used in the Joint 
 Committee report of 1916 were found suitable for the purpose of stating the 
 resultant moments. In a square interior panel of a uniformly loaded 
 floor slab, with a large number of panels in all directions, the percentages 
 of the total moment (or sum of positive and negative moments) which 
 are resisted in the column-head sections, mid-section, outer sections, and 
 inner section are found to be nearly independent of the size of the column 
 capital. 
 
 A study is made of unbalanced loads, for example, loads in rows; 
 unbalanced loads are found to produce large moments in the slabs if the 
 columns are slender, and large moments in the columns if the columns are 
 rigid. Moment coefficients are stated also for various cases of oblong 
 panels, wall panels, and corner panels. 
 
 The tests of slabs supported on four sides indicate that when the 
 deformations increase, certain redistributions of moments and stresses take 
 place, with the result, in general, that the larger coefficients of moments 
 are reduced. The ultimate load is found to be, in general, larger, and in 
 some cases much larger, than would be estimated on the basis of the theo- 
 retical moment coefficients and the known strength of beams with the same 
 ratio of steel. 
 
 When the moments of resistance of the observed stresses in the rein- 
 forcement in flat slabs were multiplied by the ratio of the applied moment 
 in simple beams to the resisting moment of the observed stress, corrected 
 moments for the slabs were obtained which, in comparison with the results 
 of the analysis of flat slabs presented in Part II, were (a) much greater 
 for the lower loads than for the higher loads, (b) greater for the lower 
 loads than the theoretical moments and (c) slightly less for the higher 
 loads than the theoretical moments.
 
 MOMENTS AND STRESSES IN SLABS. 99 
 
 If the effect of the difference in the modulus of elasticity for the slabs 
 from that for the beams on which the comparison is based could be elimi- 
 nated it seems that the agreement between the analysis and the tests would 
 be fair. Such information as is available on the effect of the modulus of 
 elasticity on the results points in the direction stated. 
 
 The average value of the estimated factor of safety for the slabs 
 studied was 3.23 for the working loads based upon the moment coefficients 
 recommended by the Joint Committee on Concrete and Reinforced Concrete 
 and 2.72 for the working loads based on the coefficients recommended by the 
 American Concrete Institute. 
 
 APPENDIX A. 
 
 DETAILS OF THE ANALYSIS OF HOMOGENEOUS PLATES. 
 BY H. M. WESTERGA.ARD. 
 
 Al. Notes referring to Art. 7. Solutions of the Differential Equation of 
 Flexure for Slabs Supported on Four Sides, (a) Rectangular slabs 
 with simply supported edges.* 
 
 The analysis of the rectangular slab with simply supported edges is 
 simplified by assuming certain special values of the dimensions, of the 
 clastic constants, and of the load: namely, 
 
 a = "f = long span, in the direction of x; 
 
 b = oc TT = '_ = short span, in the direction of y; 
 
 M 
 w = . = load per unit-area; 
 
 7T 2 
 
 7 = 1; Poisson's ratio K = 0. 
 
 The origin of the coordinates x, y is at the center of the slab. 
 
 With these special values, one finds wb- = 1 ; that is, the moments 
 
 M 
 become equal to the coefficients of moment, -ii_ . Since El = 1 and K = 0, 
 
 wb 2 
 
 the moments or coefficients of moment become numerically equal to the 
 curvatures. The expediency of analyzing with a Poisson's ratio equal to 
 zero was discussed in Articles 6 and 8, where also methods of modifying 
 the results when Poisson's ratio has some other value were indicated. 
 (See equations (15) to (18) in Art. 6 and Fig. 10(b)). 
 
 In order to solve Lagrange's equation of flexure ( (11), (12), or (19) 
 in Art. 6), 
 
 where $ 4 5" g< 
 
 AA=r- 4 + 2 rj^-i + 7-4 
 dx* t>x &y by 
 
 * Navier's solution is used here (see, for example, A. E. H. Love, Mathematical 
 Theory of Elasticity, ed. 1906, p. 468). Levy's solution (Love, p. 469) was used by 
 Nadai in dealing with the same problem (see the historical summary in Art. 4, foot- 
 note 36).
 
 100 MOMENTS AND STRESSES IN SLABS. 
 
 8* 
 the term is expressed by a double-infinite Fourier series, as follows: 
 
 mtn 
 
 8 '6/3 m ~(~U 
 
 (m.n- 1,3,5,7 ) (47) 
 
 This expression applies at all points of the slab except at the edges. If 
 the load w had consisted of only one of the terms in (47), the solution of 
 Lagrange's equation would be: z equal to a similar term, which is equal to 
 a constant times the load at the particular point. With w equal to the 
 complete series ( 47 ) , the solution of ( 46 ) becomes : 
 
 CO5 (48) 
 
 This solution satisfies equation (46), as may be verified by substitution, 
 and it satisfies also the boundary conditions, that at the edges z - 0, 
 
 5>2 $2 
 
 - 0, and Z - 0. The deflections, therefore, are expressed correctly by 
 Sz 2 8j/ 2 
 
 equation (48). 
 
 By double differentiations of (48), one finds the bending moments 
 (according to (20) in Art. 6) to be: 
 
 M S'z I6tf * n -{-tfFm 
 
 '- Sx' = -fig. g . nlm'+fWy C 5 005 P n y (49) 
 
 and 
 
 (50) 
 and the torsional moment (according to (21) in Art. 6) to be 
 
 rl .,Q3 m , . \ al ^ D 
 
 Mr x -T-r = j 2123 -. , a i ,, z sin mx sinBny 
 
 6x5y TT i.3t,3"(m+J)n) (51) 
 
 The moment coefficient at the center of a square slab is found by 
 substituting x = y = and ft = 1 in (49) or (50), and it is 
 
 M _J6_ -H) m l a m 
 ~ 
 
 16 
 
 5+\f 3(9+9)* 5( 
 
 _J _ N 1 
 
 7(1+49)*) "J 
 
 .^0.2245-00369, (52) 
 
 as shown on the diagram in Fig. 3 (a). This calculation is typical; other 
 coefficients shown in Fig. 3 (a) were computed in the same way. It may be 
 noted that in (52) the terms of the double-infinite series are arranged in
 
 MOMENTS AND STRESSES IN SLABS. 101 
 
 groups with m -(- = 2, 4, 6, 8, 10 ......... , respectively, and thus the 
 
 double-infinite series is transformed into a single-infinite series. 
 
 The moment M a across the diagonal at the corner in a square slab 
 is numerically equal to the torsional moment M at the point, defined by 
 (51) ; that is, 
 
 (53) 
 
 The moment M at the corner of a rectangular slab across a line 
 making angles of 45 degrees with the sides, is determined by an expression 
 similar to (53), but containing the ratio fi of the long span to short span. 
 In the limiting case in which ft = oo (or, ex =0), this expression may 
 be reduced to the form, 
 
 (54) 
 which is the value shown at the left-hand edge in Fig. 3 (a). 
 
 (b) Infinitely long strip, extending from x to x = oo between 
 
 the simply supported edges y - ; fixed edge along the i/-axis. This 
 
 t 
 
 slab is a special rectangular slab with the spans a ** oo, b jf. Load 
 
 w--; v* = - . K = 0; El = 1. 
 4 4 
 
 The solution of Lagrange's equation, 
 
 is written in the form 
 
 z = z -\- 2, 
 
 where c, is the deflection at the point (x,y) when the support at the t/-axis 
 is removed, so as to make the edge deflect freely. The remainder z t is the 
 amount which is added when external forces applied at the free edge at the 
 y-axis make the deflections and the slopes at this line again equal to zero. 
 2, is the deflection of a simple beam with a span equal to TT , and may be 
 expressed as a polynomial in y, but may also be expressed by the Fourier 
 series 
 
 21 = cos y & cos3?/ -f B cosSy - ^ cos7y + ...... , 
 
 as may be verified by comparison with the expression for the load 
 
 w = - = cos y -- cos3?/ + - cos5y - cos7y + ....... 
 
 4 357 
 
 The deflection z 2 must satisfy the following conditions: at all points, 
 
 Sz 
 A A zz = 0; at the short edge, z 2 = zi and = 0; at the long edges, z 2 - 
 
 8 2 z 
 and =0. These conditions are satisfied by 
 
 
 , (-\fy- 
 
 (/-t-mxje N s co5 my
 
 102 
 
 MOMENTS AND STRESSES IN SLABS. 
 
 i 
 
 Fince = 0, the bending moment along the z-a\is becomes 
 
 (55) 
 
 When x = O, this moment becomes M a - 2 ~^T 
 
 m 3 
 
 = Ri'r 
 
 IT' . ' "* 1 
 
 wb* - , the corresponding momsnt coafficient becomes ~? = _ , as 
 4 u-'O 2 8 
 
 shown at left-hand edge in Fig. 6 (a). The series (55) converges rapidly. 
 Values of Max were computed for x = 05, 1.0, 2.0, 3.0, and 4.0; the greatest 
 positive value, 0.1339, which was found with x = 2.0, gives the coefficient 
 
 M 
 
 = 0.0173, as shown at the left-hand edge in Fig. 5 (a). 
 u-b 2 
 
 y y 
 
 5 imply supported 
 
 Fiu. Al. RECTANGULAR SLAB. 
 
 (c) The rectingulnr slab shown in Fig. A I. Two parallel edges are fixed 
 and tv.'o parallel edges simply supported Simple span = ""; fi*ed span = I. 
 
 Load w = ; when / >TT, then ivb 2 = ; when I 
 44 
 
 , then wb 2 = - I 2 ; 
 4 
 
 7=1; tf = 0. 
 
 The solution given here is essentially Levy's solution.* The procedure 
 is essentially the same as in the preceding case, (b), which is the special 
 case in which I = oo . 
 
 Lagrange's equation 
 
 AAz = 
 
 (56) 
 
 is solved by 
 
 z = zi + z-i (57) 
 
 where, as in case (b), z l is the simple-beam deflection, obtained when the 
 
 * See the historical summary, Art. 4, footnote 13, or A. E. H. Love, Mathe- 
 natical Theory of Elasticity, Ed. 1906, p. 469.
 
 MOMENTS AND STRESSES IN SLABS. 103 
 
 edges y and y' are removed, c, may be expressed either by a polynomial in 
 y or by the Fourier series 
 
 Zi = cosij - - cos'3y + cos5y - ...... (58) 
 
 The- remaining part c s of the deflection must satisfy the following condi- 
 tions: 
 
 AA 2 , = () at all points; (59) 
 
 8, 
 
 Z2 = 0, s =0 at the simply supported edges; (60) 
 
 5 
 
 s = 0, ~Y = at the fixed edges; (61) 
 
 Zi = z\ at the fixed edges. (62) 
 
 The conditions (59) and (60) may be satisfied, as may be verified by 
 differentiation, by an expression of the form 
 
 S (-D^r 
 z^Z -r-$ m co S my (63) 
 
 where 
 
 e+xe (6 4) 
 
 in which K m and k are cimstants. This solution will satisfy the condi- 
 tion (61) when 
 
 >+(ml-l) e - m (65) 
 
 and it will satisfy the condition (62) when 
 
 K = _ L_ 
 
 / + (+rnh- TO l)e (66) 
 
 The deflections, then, are defined completely by equations (57) (58), and 
 (63) to (66). 
 
 The bending moments in the x- and y- directions may be found by 
 double differentiations of (57), (58), and (63). One finds 
 
 S'z S z z, pf-i) 2 ^ SX 
 M* = - TT - - r~t = ^ j -r~r cos my 
 ^ J .. 3 
 
 + e cos my 
 
 and 
 
 * v J5v* = ~Jt z ~ JT* = T \~A ~y I r ' ' ( 68) 
 
 The series (67) and (68) converge rapidly. When these series are used in 
 connection with formulas (64) to (66) they are suitable for numerical
 
 104 MOMENTS AND STRESSES IN SLABS. 
 
 computations, and they were used in determining the points shown by 
 small circles in Fig. 4 (a), Fig. 5 (a), and Fig. 6 (a). 
 
 With i/=0 and 1= oo, formula (67) becomes the same as (55). 
 
 (d) Slabs with four ficced edges. 
 
 The approximate moment coefficients for square slabs, represented in 
 Fig. 7 and Fig. 8 by points marked with circles, were determined by approx- 
 imate expressions which contain trigonometric and exponential functions 
 of x and y; they are somewhat similar in form to those applied in the 
 preceding cases. Neither Navier's nor Levy's solution applies directly to 
 the slab with four fixed edges. Ritz's method, which was used, for example, 
 by Nadai and Paschoud in analyses of fixed slabs, is found to lead to suit- 
 able solutions of the problem.* 
 
 A2. Theory of Ring Loads, Concentrated Couples, and Ring Couples. 
 Certain concentrated loads, each consisting of a group of forces within a 
 small area, were introduced in Articles 8 and 9, where procedures of 
 analyses of flat slabs were outlined, and where the results of these analyses 
 were presented. The loads introduced are: the ring loads, which were 
 defined in Art. 8, and which are used in the analysis of the normal square 
 interior panel of a uniformly loaded flat slab; and the concentrated 
 couples and ring couples, which were defined in Art. 9, and which are used 
 in the study of unbalanced loads. 
 
 By the use of the concentrated loads in the analysis of flat slabs a 
 procedure is followed which has general applicability, and which was used 
 in one form in the preceding article (in cases (b) and (c) ) : 
 
 Lagrange's equation, 
 
 AA* = Lz__* 2 -, (12) 
 
 is solved, for the given loads and boundary conditions, by expressing the 
 deflection in the form 
 
 z = * + 2* m (69) 
 
 where z satisfies (12), without necessarily satisfying the boundary condi- 
 tions, while each function z satisfies the equation 
 
 AA2 m = 0, (70) 
 
 which is Lagrange's equation for w =0. 
 
 The deflection z in (69) may be, for example, the deflection of the 
 point-supported slab under the load w. Then, z may be the deflection due 
 to one of the concentrated loads, acting alone on the slab at a point of 
 support, with the surrounding supports removed. By introducing one such 
 concentrated load at each point of support or panel corner, and by adding 
 the deflections due to all of the loads, one forms the series (69) , which may 
 be an infinite series, by differentiation of which one may obtain corre- 
 sponding series for the moments. The concentrated loads must be so 
 selected that all of the loads, including the applied load w, will cause the 
 
 Sec the historical summary, Art 4, footnotes 26. 34, and 36.
 
 MOMENTS AND STRESSES IN SLABS. 105 
 
 point-supported slab to deflect, outside the circles marked by the edges of 
 the column capitals, exactly as the slab deflects which is supported on 
 column capitals and loaded by w. 
 
 In the theory of the concentrated loads, now to be presented, it is 
 assumed at first that only one concentrated load acts on the slab. Then, 
 groups of loads are considered. The slab is assumed to extend indefinitely 
 in all directions. Furthermore, let: 
 
 K = Poisson's ratio = 0, as before; 
 
 r = y x 2 _j_ yi = radius vector measured from the origin of the co- 
 ordinates. 
 
 (a) Ring loads. 
 
 Lagrange's equation 
 
 AAz = 0, (71) 
 
 for the case in which 10 =0, is satisfied at all points, except at the origin, by 
 
 z = Clr+c', (72) 
 
 where C and c' are constants. The deflected surface, according to this 
 equation, is a surface of revolution about an axis through the origin. A 
 load, concentrated at the origin, and producing the state of flexure defined 
 by (72), may be called, by definition, a ring load. The intensity of this 
 ring load is measured conveniently by CEI, where El is the usual stiffness 
 factor of the slab. Since C is a distance, the ring load CEI may be meas- 
 ured in Ib. in. 3 units. One finds by differentiation of (72) : 
 
 ^_ cy 
 
 (73) 
 *~ x ' ) *L* , C(x_'-_y_v; 6jz ZCxy 
 
 <5y z f 4 ' <5x<5y *"* (74) 
 
 that is 
 
 8 z z. <5 Z 2 
 
 that is -=~? + v- / Az = &nd AAz = . 
 
 AV Ay' 
 
 According to formulas (22) in Art. 6, the vertical shears are proportional 
 to the derivatives of &z; that is, the vertical shears are zero at all points 
 except the origin. The moments in the directions of x and y are defined by 
 the second derivatives in (74). The moment in the direction of radius 
 vector, or the radial moment, is 
 
 MJ-J T ~ O-/^/ /7K\ 
 
 2 . ( / O I 
 
 In a circular section with center at the origin and with radius r, there 
 is, then, a uniformly distributed radial moment, defined by (75), but no 
 torsional moment and no vertical shear. In the light of the state of flexure 
 in the circular section with center at the origin and radius r,, one may 
 explain the nature of the ring load. Assume that the material is removed 
 within the circle with radius r,, and that a radial moment, determined by 
 (75) is applied as an external load, uniformly distributed over the circum- 
 ference of the circle. Then the slab, under the influence of this load alone.
 
 106 MOMENTS AND STRESSES IN SLABS. 
 
 will deflect according to formula ( 72 ) , because thereby it satisfies all the 
 boundary conditions. Now assume that the circle with radius r, is not cut 
 out, but that instead some load within the circle produces at the circumfer- 
 ence of the circle the state of flexure that was assumed before as a result 
 of the external loads. On account of the identity of conditions at the cir- 
 cumference of the circle, the state of flexure outside the circle will remain 
 unchanged, as determined by equation (72). More than one kind of load 
 within the circle may produce this same effect: the load may consist of 
 upward and downward loads P, uniformly distributed over the circumfer- 
 ences or areas of two concentric circles; or it may consist of an upward 
 load P at the origin combined with a down load P which is uniformly dis- 
 tributed over the area or the circumference of a circle with center at the 
 origin and radius not larger than r,. But whether the load is made up in 
 one way or another, if the radius r l is small, the load may be considered as 
 one concentrated load; namely, the ring load whose magnitude is measured 
 by CEI, and whose effects are defined completely by equation ( 72 ) . 
 
 It may be noted that, according to equation (72), the deflection at the 
 origin is infinite. Since the origin lies always within the smallest circle 
 containing the whole load, the infinite deflection at the center has only 
 theoretical significance. When the ring load is applied at a point of sup- 
 port, then, in order to avoid the assumption of any infinite deflections, one 
 may conceive of the support as being distributed over the circumference 
 of a small circle whose center is at the original point of support and at a 
 fixed elevation. 
 
 The expressions (74) are well suited for computations of such series 
 as may be formed when a large number of ring loads are applied at the 
 same time. 
 
 (b) Concentrated couples. 
 
 Lagrange's equation ((71)) 
 
 for w = is satisfied at all points except at the origin by the solution 
 
 z = A x I . r + ax, (76) 
 
 where A and a are constants. By differentiation one finds: 
 
 ,77, 
 
 &z = A z , (79) 
 
 02 
 
 this value of A ? is proportional to the value of in the preceding case, 
 
 o 
 
 (a), which gave A^ = 0; it follows, therefore, in the present case, that 
 AA * = 0. 
 
 In the state of flexure just represented the ?/-axis remains undeflected. 
 A circle drawn on the slab, with center at the origin of coordinates, remains
 
 MOMENTS AND STRESSES IN SLABS. 107 
 
 plane, but rotates about the y-axis, through some angle. On account of the 
 anti-symmetry with respect to the y-axis, the resultant of the stresses 
 in the cylindrical section r = r, is a couple about the y-axis. Now r, 
 may be given any small value, that is, the couple must be transferred to the 
 slab at the origin as a concentrated couple. 
 
 The magnitude of the concentrated couple may be found by considering 
 the stresses in two sections parallel to the y-axis, on opposite sides of the 
 origin. By formulas (22), in Art. 6, one finds the vertical shear per unit- 
 width in a section parallel to the y-axis to be 
 
 The total vertical shear in this section becomes, then, 
 
 The total bending moment in a section parallel to the y-axis, with x posi- 
 tive, is equal to 
 
 while a negative x gives -)- 2^ AEI. The resultant of the stresses in the 
 two sections + x is then equal to the concentrated couple =4 TT AEI (80), 
 turning, when A. is positive, in the direction from z to x. 
 
 A number of concentrated couples may be dealt with by computing 
 series of the terms contained in equations (77) and (78). 
 
 (c) Ring couples. 
 
 The deflection due to two equal and opposite ring loads, close to the 
 origin and to one another, and with centers on the x-axia, may be expressed 
 as equal to a constant times the first partial derivative, with respect to x, of 
 the deflection which is due to a single ring load, and is expressed by equa- 
 tion (72). This derivative was given in equation (73). Thus, when B is 
 a constant, the function 
 
 Z = *k, (81) 
 
 r 2 
 
 is the deflection due to a ring couple which is applied at the origin in the 
 direction of x, and is measured in intensity by the quantity BEI. One 
 finds by differentiation of (81) : 
 
 x r 4 Sy~ r* (82) 
 
 7 ? = ~~^/ = ~^ r ( / ~ r 2 ! ' (83) 
 
 and Az = AAz =
 
 108 
 
 MOMENTS AND STRESSES IN SLABS. 
 
 (d) Other types of concentrated loads. 
 
 If the function z = F(x,y) satisfies Lagrange's equation A A * = 
 at all points except at the origin, which is a singular point, then also the 
 function 
 
 gm+n F 
 
 2 " 8x m 8y n 
 
 will satisfy the equation A A* = at all points except at the origin ; and 
 z, like the original function F, will define some concentrated load at the 
 origin. Other solutions may be formed by integration of F. Thus from the 
 fundamental solutions (72) and (76), for ring loads and concentrated 
 couples, and from the solution 
 
 rl.rdr 
 
 (85) 
 
 which defines a single concentrated force proportional to D,, at the origin, 
 one may derive an infinite number of solutions, each defining a correspond- 
 
 y 
 
 Y 
 
 R, 
 
 >- -t 
 
 ** 1 
 
 J 
 ^ - 
 
 ?m 
 >- - 
 
 , 
 
 ^ < 
 
 J 1 
 
 -z- 
 
 J; - 
 
 -z* 
 
 I - 
 
 ~ l - 
 
 -z* 
 
 ^z^ 
 
 
 
 
 
 N 
 
 , 
 
 
 
 1 
 
 
 
 
 FIG. A2. RING LOADS IN A Row. 
 
 ing concentrated load. It is an example of this procedure, that the deflec- 
 tions due to the ring couple were derived by differentiation of the expression 
 for the deflections due to the ring load. 
 
 (e) The co-action of a number of ring loads. 
 
 A number of equal ring loads, R^ 7? n , of intensity CEI, is 
 
 assumed to act in a row parallel to the #-axis as shown in Fig. A2. The 
 spacing is constant and equal to I. By using equations (73) and the rela- 
 tions = x 4. ,, one finds the change of slope between the two origins of 
 coordinates, o and a at a distance of I, to be 
 
 (86) 
 
 that is, the difference is expressed in terms of the coordinates of the first 
 and the last load in the row. 
 
 In Fig. A3 ring loads of intensity CEI are applied at the points marked 
 with small circles. The group of loads is symmetrical with respect to the 
 lines y = Q, a=' 1 /2l, and a? = 1 &l =*= y. According to (86) the change of 
 slope between the points a and 6 may be expressed as follows, in terms of
 
 MOMENTS AND STRESSES IN SLABS. 
 
 109 
 
 the coordinates of the extreme loads in the first quadrant, that is, the loads 
 on the line A : 
 
 S-'.^C AP- (87) 
 
 If the number of loads is very large, then the summation may be replaced 
 by an integration. By making use of the axes of symmetry, one finds then 
 
 r 
 
 x, <dy 2"C f *dy~ ydx 2.C. /" * r 08 irC, 
 
 This difference of slope may be reduced to zero if at the edge of the plate, 
 which is assumed to be at infinity, a certain additional load is applied; 
 
 FIG. A3. GKOUP OP RING LOADS. 
 
 namely, a bending moment, uniformly distributed over the edge, producing 
 a uniform bending moment in the slab in all directions, equal to 
 
 (89) 
 
 With this additional moment present, small circles, drawn in the central 
 portion of the slab, concentric with the ring loads, all with the same radius, 
 will remain co-planar; and they will be contour lines of the deflected 
 surface if one of them remains horizontal. On account of this relation to 
 the ring loads the moment M is assumed, in the analysis of the uniformly 
 loaded flat slab, to act with the group of ring loads. 
 
 (f) The co-action of concentrated couples and ring couples. 
 
 A concentrated couple 4 T AEI, and a ring couple BEI, whose effects 
 are determined by equations (76) and (81), respectively, are assumed to 
 act on the slab, at the origin, in the o^-plane. The circles r-= const, remain 
 plane under the influence of this combined load. The slope in the direction
 
 110 
 
 MOMENTS AND STRESSES IN SLABS. 
 
 of x varies, in general, from one point to another on the circle r = const. 
 The condition may be imposed, however, that all points of a certain circle, 
 
 for example, the circle r= *L which marks the edge of a column capital, 
 
 Jt 
 must have a common tangential plane. By comparing equations (77) and 
 
 (82) one finds that all elements of the deflected surface at the circle r = 
 
 f 
 
 I- 
 
 C-rB-p A r&T C rB-r-A r^n C -rBn A- 
 
 ~ :~c 
 
 r-X 
 
 cc 
 
 np 
 
 CD 
 
 (0) 
 
 5- 25 q'l 
 
 T)JD 
 
 X-~N 
 
 za 
 
 B-1 
 
 5- -25-1' b 
 
 =54- 
 
 2- 25-1'b 
 
 C Jfj 
 
 p* 
 
 5 V 
 
 ii : o'b 
 
 a 
 
 i-db 
 
 [5V 
 
 -i 
 
 - 1 - 
 
 1 
 
 -H 
 
 I '- 
 
 FIG. Bl.^ REINFORCING PLAN FOB SLAB J. 
 
 have a zero slope in the direction of y, and consequently have a common 
 tangential plane, when 
 
 B=4-- (90) 
 
 Concentrated couples =* ^TtAEI and ring couples =t B7 are used 
 in the analysis of flat slabs with alternate rows of panels unloaded. The
 
 MOMENTS AND STRESSES IN SLABS. 
 
 Ill 
 
 rows considered are parallel to the i/-axis. One concentrated load of each 
 kind is assumed at each column center. The double signs =t refer to 
 alternate rows of columns. Because of the loads at the surrounding sup- 
 ports, equation (90) expresses in this case only approximately the condi- 
 tion that there is a common tangential plane at all points of the circle 
 
 r = . The following formula, which* is a modified form of (90), and 
 
 
 
 which is a close approximation, takes into consideration the concentrated 
 
 
 
 @ 
 
 FIG. E2. I OCATION OF GAGE IINES FOR TOP OF FLAB S. 
 
 loads at the nine points definea by the coordinates y = I, 0, -f- I, and cc 
 = / (negative values of A and B) and x = (positive A and B) ;this 
 formula was derived by equating to one another the slopes in the direction 
 
 of x at the points /O, -\ and (- , 0\ : 
 
 (91) 
 
 Formula (91) was used in computing values of B when alternate rows of 
 flat slab panels are unloaded.
 
 112 MOMENTS AND STRESSES IN SLABS. 
 
 APPENDIX B. 
 
 TESTS OF SLABS AT PURDUE UNIVERSITY 
 BY W. A. SLATER. 
 
 Bl. Description of Tests. The tests referred to as the Purdue teata 
 were made for the Corrugated Bar. Co. under the direction of Prof. W. K. 
 
 flfeZE 
 
 Gaqe lines are shawoi 
 they mutt qfeor if seen 
 through the slab from obnv 
 
 FIG. B3. LOCATION OF GAGE LINES FOB BOTTOM OF SLAB J. 
 
 Hatt at Purdue University, Lafayette, Ind., on two test slabs, J and S, 
 each of which had four panels 16 ft. square. 
 
 The dimensions of the concrete in the two slabs were the same, but the
 
 MOMENTS AND STRESSES IN SLABS. 
 
 113 
 
 Goqe lines are shown 
 a5 they would appear if 
 
 seen through the column 
 from the outside 
 
 FIG. B4. LOCATION OF GAGE LINES ON COLUMNS AND MARGINAL BEAMS 
 
 OF SLAB J. 
 
 Unit Be*mutKn ardy^s to Scale Mealed J-awH oammoxmim 
 
 SmriySfpff^n 
 
 FIG. B5. LOAD STRAIN AND LOAD-STRESS DIAGRAMS FOR Top OF SLAB J.
 
 114 
 
 MOMENTS AND STRESSES IN SLABS. 
 
 -
 
 MOMENTS AND STRESSES IN SLABS. 115 
 
 amount of reinforcement for slab S was considerably less than that for slab J. 
 The latter fact will help to account for the smaller load which was carried 
 by slab S than by slab J, but another important consideration is the fact 
 that for slab S the average strength of the concrete control cylinders at 28 
 days was only 1215 Ib. per sq. in. while the strength of the control cylinders 
 for slab J was 2305 Ib. per sq. in. 
 
 The methods of making the test are similar to those which have been 
 described in reports of various tests on floors of buildings.* 
 
 The important dimensions of the slab are shown in Table XIII. The 
 amount and distribution of the reinforcement are shown in Fig. Bl and B9. 
 
 JO 10 JO .10 10 ?0 30 40 50 60 70 60 00 100 I/O 120 130 1.40 ISO 160 
 Deflection. Inches 
 
 Fio. B7. LOAD DEFLECTION DIAGRAMS FOR SLAB J. 
 
 The location of gage lines is shown in Fig. B:>, B3, B4, BIO, Bll, and B12. 
 The measured stresses in the reinforcement and deformation in the concrete 
 are shown in Fig. B5, BO, B13 and B14. The deflections are shown in Fig. 
 B7 and B8. Certain information concerning these tests has already been 
 published! and reference to the published report will supply certain results 
 of the test which are lacking in this paper. 
 
 In the tests of both slabs the load Avas applied as nearly uniformly as 
 possible. In order to afford access to the gage lines on the top surface of 
 the slabs, aisles were left in the loaded area. When the load was high 
 enough these aisles were bridged over and sufficient load was placed 
 immediately over them to give practically a uniform distribution of load. 
 
 * Univ. of 111. Eng. Exper. Sta. Bulletins 64 and 84. 
 
 t W K Halt "Moment Coefficients for Flat-slab Design with Results of a Test," 
 Froc. A. C. I. V. 14, p. 174 (1918).
 
 116 
 
 MOMENTS AND STRESSES IN SLABS. 
 
 B2. Loading of Slab J. In the test of slab J the highest load applied 
 uniformly over the entire slab was 595 Ib. per sq. ft. At this load the 
 measured stress in the reinforcement was at the yield point in gage lines 
 which crossed the mid-section of the slab (see Fig. 12 which shows location 
 of sections), and the highest deflection reported for any panel was 1.1 in. 
 After the load had been in place about two days longer the deflection had 
 increased to 1.25 in. At this stage of the test it is reported that there was 
 no evidence of crushing of the concrete. The entire load was removed from 
 the slab and about 40 days later a load of 803 Ib. per sq. ft. was applied 
 "over one panel, the overhang, and into the adjoining panel, etc."* Failure 
 occurred under this load by punching of the column* capital through the 
 
 0016 
 
 Legend 
 
 Section A- A inferior panel 
 B-B wall 
 
 FIG. B8. STRAIN DISTRIBUTION FOR REINFORCEMENT IN WALL PANEL AND 
 INTERIOR PANEL OF SLAB J. 
 
 dropped panel. The fracture had the angle of a diagonal tension failure. 
 Assuming the full live and dead load of an area 16 ft. square to have 
 been carried on the central column, the computed shearing stress on the 
 vertical section of depth jd, which lies at a distance d from the edge of the 
 column capital, was 233 Ib. per sq. in. Although the "failure of the slab 
 . . . began with a feathering of the concrete on the dropped panel at the 
 edge of the column capital"* this shearing stress is high enough that it 
 seems that diagonal tension may have been a factor in causing failure. 
 
 B3. Loading of Slab S. The maximum load applied to slab S was 
 450 Ib. per sq. ft. The official report of the test states that this load "was 
 attended by complete failure of the concrete in compression and the stretch- 
 ing of the steel to the yield point."* The load-strain diagrams, Fig. B13 
 and B14, show that the reinforcement generally was highly stressed both 
 at sections of negative moment and at sections of positive moment, and 
 
 Proceedings A. C. I., Vol. XIV, pp. 182 and 183 (1918).
 
 MOMENTS AND STRESSES IN SLABS. 
 
 117 
 
 photographs of the slabs show the crushing of the concrete around the 
 capital. However, the highest deflection reported was only 1.30 in. at the 
 center of a panel, and when the load was removed the deflection decreased 
 to 0.4 in. Relatively this deflection was small and the recovery was large 
 and the test does not afford a conclusive answer to the question as to what 
 load would have been required to cause collapse of the structure, or, in other 
 words, as to what was the factor of safety against destruction of life and 
 
 f- 
 
 t- 
 
 C 
 
 ^^ 5 
 
 -''; 
 
 'K 
 
 5 T" 
 
 -A rOT C T- 
 
 
 I 
 
 E- " 24-3 b 
 
 5-- 24 -5 b 
 
 
 
 2- - Z4-3 b 
 
 5- 24-3 b 
 
 -B-rA p-B-r c 
 
 .0-0 
 
 v> 
 
 '9'-* 
 
 SP- 
 
 (3) 
 
 t^ 
 
 2=%" 10 -8' 3 
 - W-3' b 
 
 5-- 24.'-3' b 
 
 f-T~ A~ T-B-^-C-1 ' 
 
 - Z4-3 b 
 
 >? if 1 
 
 10 ' 6 :'? 
 24-3 b 
 
 5- 24-3' b 
 
 10-8' 3 
 
 a-- 24-3 b 
 
 s-^io-s's 
 
 5- Z4-3' b 
 
 2- V 10- 6 
 ' 
 
 ' b 
 
 'T 
 
 t 
 
 J r 
 
 .S # 
 
 I 
 
 o'2b 
 
 
 
 
 
 
 ... 
 
 J_ 
 
 FIG. B9. REINFORCING PLANS FOB SLAB S. 
 
 property. In view of the large load carried by the Waynesburg slab after 
 the yield point of the negative reinforcement had been reached it does not 
 seem unreasonable to believe that this slab might have carried more load 
 without actual collapse. 
 
 B4. Moments in Wall Panels. In the design of slab J and slab S 
 provision for greater positive moments in the wall panels than in the 
 interior panels was made by using a larger area of reinforcement for the 
 positive moment in the wall panels than in the interior panels. The same
 
 118 
 
 MOMENTS AND STRESSES IN SLABS. 
 
 number of bars was used at the two positions, but for the wall panels square 
 bars were used and for the interior panels round bars were used. This 
 gives 27 per cent more reinforcement for the wall panel than for the interior 
 panel. Strains measured are shown in Fig. B8 and B16. For the lower 
 loads the stresses were almost equal in the two panels of slab J, but were 
 somewhat higher for the wall panel in slab S than for the interior panel. 
 For the higher loads the stresses were higher for the interior panels in 
 
 FIG. BIO. LOCATION OF GAGE LINES FOB Top OF SLAB S. 
 
 both cases. The indication from this test is that the allowance of 27 per 
 cent greater moment for wall panels than for interior panels was in excess 
 of the requirement for wall panels. The moments in the wall panels will be 
 dependent upon the moment of inertia of the wall columns and probably upon 
 the manner in which the negative reinforcement at the edge of the wall 
 panel is distributed, and for this reason the results in Fig. B8 and B16 
 should not be applied to other cases without taking into account the effect 
 of these features of the design.
 
 MOMENTS AND STRESSES IN SLABS. 
 
 119 
 
 APPENDIX C. 
 BIBLIOGRAPHY. 
 
 References are made in the following list, to published results and to 
 some unpublished results of tests on flat slabs or on slabs supported on 
 beams which lie on the edges of the panels, but which have no intermediate 
 beams. 
 
 In general the following sequence is used in references cited: Name or 
 designation of structure tested, city, brief characterization of type of rein- 
 forcement, number of panels loaded, reference to periodicals by number in 
 parentheses, date of publication. 
 
 NDTE--Z qxf. foBoO on N side 
 of this column txbn cap 
 
 f - Oaqe lines are tfom as they nou/d appear 
 if seen through ffe shb from,etwe. 
 
 FIG. Bll. LOCATION OF GAGE LINES FOR BOTTOM OF SLAB S. 
 
 -^^JT^VTPX^C* : ~^''ft=&MM y 3<g y 3?*$c " T 
 ...^^^^.^^^j.- -.f t-^-^^V^^S^^ -' 
 
 .._..., 
 
 
 
 Gaqe lines on inner face 
 
 
 
 
 
 NOTE. : Gaqe lines are shown 
 
 ssXL, 
 <3#r w 
 
 
 
 
 as they would appear if 
 
 
 
 
 
 seen through the column 
 
 
 
 
 
 from the ot/fyde. 
 
 
 
 | | 
 
 Fro. B12. LOCATION OF GAGE LINES ox COLUMN Z AND MARGINAL BEAKS 
 
 07 SLAB S.
 
 120 
 
 MOMENTS AND STRESSES IN SLABS. 
 
 The periodicals or institutions referred to in the bibliography are 
 designated by the following numbers: 
 
 ( 1 ) Proceedings National Association of Cement Users and of its 
 
 successor, the American Concrete Institute. 
 
 (2) University of Illinois, Engineering Experiment Station. 
 
 (3) Indiana Engineering Society. 
 
 (4) Proceedings Pacific Northwest Society of Civil Engineers. 
 
 Gaqe lines on steel 
 concrete 
 
 FIG. B13. LOAD STRAIN AND LOAD-STRESS DIAGRAMS FOR TOP OF SLAB S. 
 
 (5) Transactions American Society of Civil Engineers. 
 
 (6) Journal of the Engineering Institute of Canada. 
 
 (7) Bulletin on Flat Slabs by Corrugated Bar Co. 
 
 (8) Engineering and Contracting. 
 
 (9) Engineering News. 
 
 (10) Engineering Record. 
 
 (11) American Architect.
 
 MOMENTS AND STRESSES IN SLABS. 
 
 121
 
 122 
 
 MOMENTS AND STRESSES IN SLABS. 
 
 .10 .29 10 ?0 30 40 .X .60 W BO 90 ICC 110 12) 130 140 
 
 Deflection, Inches, 
 FIG. B15. LOAD DEFLECTION DIAGRAMS FOB SLAB S. 
 
 Legend 
 
 5ecttonA-A interior panel 
 B-B wall 
 
 ^TX?X*K7*^ 
 
 ,f .-/. - .. 
 
 FIG. B16. STRAIN DISTRIBUTION IN WALL PANEL AND INTERIOR PANEL 
 
 OF SLAB S.
 
 MOMENTS AND STRESSES IN SLABS. 123 
 
 LIST OF TESTS. 
 
 1. C. Bach and O. Graf: Versuche mit allseitig aufliegenden, quadra- 
 tischen and rechteckigen Eisenbetonplatten, Deutscher Ausschuss fiir Eisen- 
 beton, v. 30, Berlin, 1915, 309 pp. These laboratory testa were made in 
 Stuttgart, 1911 to 1914, under the direction of Bach and Graf. 52 slabs 
 supported on four sides and 35 control strips supported as beams were 
 tested to failure. The tests are reported in detail, without attempt, how- 
 ever, to explain or analyze the results. Analyses of the results have been 
 made later by Suenson and by Nielsen; see E. Suenson, Krydsarmerede 
 Jaernbetonpladers Styrke, Ingenioeren (Copenhagen), 1916, No. 76, 77, 
 and 78; N. J. Nielsen, Krydsarmerede Jaernbetonpladers Styrke, Inge- 
 nioeren, 1920, pp. 723-728. 
 
 2. Deere and Webber Building, Minneapolis, 1910, 4-way, 9 panels, 
 (1) 1910, (2) Bull. 64, 1911, (9) 12-22-1910, (8) 12-22-1910. 
 
 3. Test of Rubber Model Flat Slab, 1911, 9 panels, (7) 1912, (1) 
 1912, p. 219. 
 
 4. Powers Building, Minneapolis, 1911, 2-way, 4 panels, (1) 1912, p. 
 61, (9) 4-18-1912, (10) 4-20-1912. 
 
 5. Franks Building, Chicago, 1911, 4-way, 4 panels, (1) 1912, p. 160. 
 
 6. Barr Building Test Panel, St. Louis, 1911, 2-way supported on 
 beams, (1) 1912, p. 133. 
 
 7. St. Paul Bread Co. Bldg., 1912, 4-way, 1 panel, (5) 1914, p. 1376. 
 
 8. Larkin Building, Chicago, 1912, 4-way, 5 panels, (1) 1913, (10) 
 1-1913. 
 
 9. Northwestern Glass Company Building, Minneapolis, 1913, 4-way, 
 4 panels, (5) 1914, p. 1340. 
 
 10. Worcester (Mass.) Test Slab, 1913, (2) Bull. 84, 1916. 
 
 11. Shredded Wheat Factory, Niagara Falls, N. Y., 1913, 2-way, 9 
 panels, (1) 1914, (2) Bull. 84, 1916. 
 
 12. International Hall, Chicago, 1913, 4-way, 4 panels, (5) 1914, pp. 
 1433-1437. 
 
 13. Soo Line Terminal, Chicago, 1913, 4-way, 4 and 5 panels, (2) Bull. 
 84, 1916, (9) 8-16-1913. 
 
 14. Curtis Ledger Factory, Chicago, 1913, 2-way at columns, 4-way 
 elsewhere, 4 panels, (2) Bull. 84, 1916. 
 
 15. Schulze Baking Co. Building, Chicago, 1914, 4-way, 4 panels, (2) 
 Bull. 84, 1916. 
 
 16. Schwinn Building, 1914-15, long time test, 4-way, 1 panel, (1) 
 1917, p. 45. 
 
 17. Sears Roebuck Building, Seattle, 1915, 2-way, (4) Jan. and Feb., 
 1916. 
 
 18. Bell St. Warehouse, Seattle, 1915, 4-way, 4 panels, (4) 1916, (10) 
 5-13-16. 
 
 19. Eaton Factory. Toronto, Out., about 1016. 4-way, 4 panels, (6) 
 April, 1919.
 
 124 MOMENTS AND STRESSES IN SLABS. 
 
 20. S.-M.-I Slab, Purdue University (1917) circumferential reinf., 4 
 panels, (1) 1918. 
 
 21. Sanitary Can Building, Maywood, 111., 1917, 2-way, 4 panels, (1) 
 1917, p. 172, and 1921, p. 500. 
 
 22. Shonk Building, Maywood, 111., 1917, 4-way, 4 panels, (1) 1917, p. 
 172, and 1921, p. 500. 
 
 23. Western Newspaper Union Building, Chicago, 1917, 4-way, 4 panels, 
 (1) 1918, p. 291, (2) Bulletin 106, 1918. 
 
 24. Slabs J and S, Purdue Univ., 1917, 2-way, 4 panels each, (1) 1918, 
 p. 174, also 1921, p. 500. 
 
 25. Slab R, Purdue Univ., 1917, circumferential" reinf., 4 panels, (1) 
 1917, p. 172. 
 
 26. Arlington Building, Washington, D. C., 1918, 2-way tile and con- 
 crete supported on beams. (Under preparation as Tech. paper of U. S. 
 Bureau of Standards.) 
 
 27. Whitacre Test Slab, Waynesburg, Ohio, 1920, 2-way, tile and 
 concrete, supported on beams, 18 panels, (11) 8-11-20 and 3-16-21 also under 
 preparation as Tech. Paper U. S. Bureau of Standards. 
 
 28. Channon Building, Chicago, 1920, circumferential reinf. 4 panels, 
 (1) 1921, p. 500. 
 
 29. Jersey City Dairy Company's Building, Jersey City, N". J., 1913, 
 2-way, 1 panel, tested by Corrugated Bar Co., Buffalo, N. Y., not published.