THEORY AND ANALYSIS OF FLAT SLAB

Abstract

Flat slab construction is one of the modern type of construction now widely used throughout the world. The flat slab is a statically indeterminate structure so that the static equations of equilibrium arenot sufficient to determine the bending moment and shear in the slab. The problem can be solved only by consider-ation of the deflection of the slab and the relation between the deformation and stresses. Several solutions have been obtained thr-ough different mathematical approach for the maximum value of bend-ing moments and shears. Several field tests have been conducted on prototype structures and on models to study the distribution of moments in the column and middle strips and to compare with analy-tical results.t The available information on the above is widely scatt-ered and an attempt has been made here to bring in one place the various analytical approaches to the problem and the important test results and compare the same with the provisions of the code on which the design of flat slabs is based. The Indian Code of Practice and the Code of the American Concrete Institute are taken for purposes of comparison. The application of the yield line theory which is the ultimate load method for slabs has been indicated for flat slabs and ultimate load based on this theory has been calculated for a test slab. ThW usage of partial prestressing for the flat slabs over columns to avoid conjestion of reinforcement is briefly given. In conclusion some improvements in the provisions of the Indian Code of Practice for flat slabs have been suggested. Consider the interior half panel as shown in figure 1(a) with uniformly distributed load q . Since the straight boundaries of the slab are all lines of symmetry they are free from torsion and shear. All the shear and torsion must be carried around the cur-ved corner sections which follow the folumn capital. The free body as shown in figure 1(b), is subject to a downward load 1,11 acting at the centroid of the loaded area, an equal upward force WI acting on the curved quadrants, the total positive moment Mj acting on the middle section ab and the total negative moment M2 acting about the y axis on cdef. = L 2 c2 ) =_a_ (L2 ..1. C2 2 ) The upward shear W1 is assumed to be uniformly distributed around the quadrants cd and of and the resultant acts at a distance ctrr from the y axis. Moment of downward load Wl about y-y axis is q L2 L 2i Tr,C28q x 3 Ti ) 2 C qL3 qC3 8 12 Equilibrium of moments'about y axis gives I' C2 C . 14 . m + qL . ( L2 - ) = 1 2 7— 12 - 2 4 IT M1' + M2 r. Mo w cIL3( + C3 C 8 3L3 ) The parenthesis 'on the right Side!tan be:antrnyimately taken to be ( 1- 2 1. 3 L ' = qL3 2 M --6. ( 1 ..2/3 C/L) VIA - _ (1 2o g. /.2 8 " -66 L where W is total load.on panel The above approximation has been done for easier calculations with-out losing much of accuracy. A simpler and more accurate formula hasbeen given by C.P. Seise(15) which is =-( 1- 1.25 8 (2) which gives values less than 0.2% on the low side for values upto C/L = 0.3. The formula developed by Nichols are applicable thainly to panels with round columns but the same is applicable for panels with square and rectangular columns and capitals with slightly gre-ater error say upto the order of 2 to 3 percent. A general formula more accurate and simple than Nichols has been devised by. C.P.Seiss for slab with rectangular capitals on similar lines as Nichols. The assumptions made in the method are the same as 1 and 2 in Nichols analysis. Two different assumptions are made regarding the distribution of shear around the periphery of the capital. (1) The shear is uniformly distributed, (2) All the shear is concentrated at the corners of the column capital. (1) Uniformly Loaded Spear. Proceeding on the basis as shown in the figure (1), the general expression for static moment is M _LI 0 E , C2- Ca CZ / 4- --- , Cl where C1 and C2 are overall dimensions of the rectangular cap-ital in directions of span L1 and L2. For square capital [3, c c 3 8 " 2 a 17- r.,) j- The term in the parenthesis can be simplified and written as Mo = q (1 - 1.45 ) 8 Concentrated Shear The general expression is M = WL1 ( 1 - 2 el + '12 02 o 8 1,1 L2 L2 For square capital Mo =r 2 C 8 t_ - r )3] Approximately WL (1 1.95 E-6- ) This statics solution does not give any idea as to how this total moment is distributed between the positive and negative moment regions and the variation of moments along the slab width. (3) (4) Hence initial design of flat slabs were based an emphe- rical coefficient drawn from the results of tests conducted on flat slabs. Later theoretical slab analysis have developed considering the flat slab as an isotropic plate and several solutions have been arrived at, which are enumerated in the following chapter