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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 ‡ at Minneapolis was tested. This was the first field test of a reinforced-concrete building floor in which strain measurements 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 possibilities 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 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. Nichols, 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.
† Engineer Physicist, U. S. Bureau of Standards.
A. R. Lord, Test of a Flat Slab Floor in a Reinforced-concrete Building, National Association of Cement Users, v. 6, 1910.
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 resemblance is not very close, the approximation, naturally, is not very satisfactory. 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 with common practice in the investigation of statically indeterminate structures. For example, the distribution of bending 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 distributions may 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 analysis 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
and properties exactly as assumed, and is not subject to the incidental variations which so often have made it impossible to draw definite conclusions from physical tests. Besides, by the analysis one may cover whole ranges of variations of the structure or at least a great number of individual 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 on 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 summary, 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. HISTORICAL 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,' but this application of the strip method was not satisfactory. Jacques Bernouilli (the younger), in a paper presented in 1788,' 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, 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 instigation of Napoleon, proposed as a prize subject a theoretical analysis of the tones of a vibrating plate. Mlle. Sophie Germain 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 théorétique sur les vibrations des plaques élastiques rectangulaires et libres, Nova Acta Academiae Scientarum Petropolitanae, v. 5, 1787 (printed 1789). E. F. F. Chladni, Entdeckungen über die Theorie des Klanges, Leipzig, 1787. See Todhunter and Pearson, A History of the Theory of Elasticity, Cambridge, 1886, p. 147.
fairly satisfactory, though not faultless derivation of a fundamental equation 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, 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, 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 theoretical results were compared with results of tests, namely, with the experimental 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' 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 difference gave rise to some discussion, but finally, in 1867, Kelvin and Tait showed that there was only an apparent discrepancy, due to an interrelation 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. 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-Venant, pp. 740-752. S. D. Poisson, Mémoire sur l'équilibre et le mouvement des corps élastique, Memoirs of the Paris Academy, v. 8, 1829, pp. 357-570. History of the Theory of Elasticity, 1886, pp. 241, 272. G. Kirchhoff, Ueber das Gleichgewicht und die
Scheibe, Crelles Journal, 1850, v. 40, pp. 51-88.
Kelvin and Tait, Natural Philosophy, ed. 1, 1867.
matical Theory of Elasticity, ed. 1906, p. 438.
See Todhunter and Pearson,
Bewegung einer elastischen
See A. E. H. Love, Mathe
J. Boussinesq, Etude nouvelle sur l'equilibre et le mouvement des corps solides élastiques dont certaines dimensions, sont très-petites par rapport a d'autres, Journal de Mathématiques, 1871, pp. 125-274, and 1879, pp. 329-344.
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 otherwise 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," 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 correction 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 series 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. H. Bryan," in 1890, made an analysis of the buckling of a rectangular elastic plate, due to forces in its own plane. Maurice Lévy" showed how Lagrange's equation, when applied to rectangular 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 résistance des parois planes des chaudières a vapeur, Annales des Ponts et Chaussées, v. 3, 1872, pp. 276-303.
"F. Grashof, Elasticität und Festigkeit, ed. 1878, p. 351.
12 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.
Maurice Lévy, Sur l'équilibre élastique d'une plaque rectangulaire, Comptes Rendus, v. 129, 1899, pp. 535-539.
14 Galliot, Etude sur les portes d'écluses en tôle, Annales des Ponts et Chaussées, 1887, v. 14, pp. 704-756.
15 C. Bach, Versuche über die Widerstandsfähigkeit ebener Platten, Zeitschr. d. Ver. deutscher Ingenieure, v. 34, 1890, pp. 1041-1048, 1080-1086, 1103-1111, 1139-1144.