Design of RCC slabs

UNIT 5. DESIGN OF SLABS 5.1 GENERAL In reinforced concrete construction, slabs are used to provide flat, useful surfaces. A reinforced concrete slab is a broad, flat plate, usually horizontal, with top and bottom surfaces parallel or nearly so. It may be supported by reinforced concrete breams (and is usually cast monolithically with such beams), by masonry or reinforced concrete walls, by structural steel members, directly by columns, or continuously by the ground. The most common type of structural element used to cover floors and roofs of buildings are reinforced concrete slabs of different types. One-way slabs are those supported on the two opposite sides so that the loads are carried along one direction only. A common example of one way slab is the verandah slab spanning in the shorter direction with main reinforcements and distribution reinforcements in the transverse direction (Fig. 1a). Two-way slabs are supported on all four sides with such dimensions such that the loads are carried to the supports along both directions. Two-way slabs are common in the floors of multistorey buildings (Fig. 1b). Cantilevered slabs are generally used for chajjas over windows and in balconies projecting from the buildings. In T-beam-slab floors, the slab is continuous over T-beams and designed as a continuous slab with positive moments at mid span and negative moments over supports. Intermediate beams, as shown in Fig. 1c, may also be provided. If the ratio of length to width of one slab panel is larger than about 2, most of the load is carried in the short direction to the supporting beams and one-way action is obtained in effect, even though supports are provided on all sides. Concrete slabs in some cases may be carried directly by columns, as shown in Fig. 1d, without the use of beams or girders. Such slabs are described as flat plates and are commonly used where spans are not large and loads not particularly heavy. Flat slab construction, shown in Fig. 1e, is also beamless but incorporates a thickened slab region in the vicinity of the column and often employs flared column tops. Both are devices to reduce stresses due to shear and negative bending around the columns. They are referred to as drop panels and column capitals, respectively. Closely related to the flat plate slab in the two-way joist, also known as a grid or waffle slab is shown in Fig. 1f. To reduce the dead load of solid-slab construction, voids are formed in a rectilinear pattern through use of metal or fibreglass form inserts. Usually inserts are omitted near the columns, so a solid slab is formed to resist moments and shears better in these areas.

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Fig. 1: Types of structural slabs Reinforced concrete slabs of the types shown in Fig. 1 are usually designed for loads assumed to be uniformly distributed over one entire slab panel, bounded by supporting beams or column centrelines. Minor concentrated loads can be accommodated through two-way action of the reinforcement (two-way flexural steel for two-way slab system or one-way flexural steel plus lateral distribution steel for one-way systems). Heavy concentrated loads generally require supporting beams.

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5.2 DESIGN OF ONE-WAY SLABS Reinforced concrete slabs supported on two opposite sides with their longer dimension exceeding two times the shorter dimension are referred to as one-way slabs. The structural action of a one-way slab may be visualized in terms of the deformed shape of the loaded surface. Fig. 2 shows a rectangular slab, simply supported along its two opposite long edges and free of any support along the two opposite short edges. If a uniformly distributed load is applied to the surface, the deflected shape will be as shown by the solid lines. Curvatures, and consequently bending moments, are the same in all strips spanning in the short direction between supported edges, whereas there is no curvature, hence no bending moment, in the long strips parallel to the supported edges. The surface is approximately cylindrical.

Fig. 2: Deflected shape of uniformly loaded one-way slab For purposes of analysis and design, a unit strip of such a slab cut out at right angles to the supporting beams, as shown in Fig. 3, may be considered as a rectangular beam of unit width, with a depth ‘d’ equal to the thickness of the slab and a span ‘lx’ equal to the distance between supported edges. This strip can be analyzed by the methods that were used for rectangular beams, the bending moment being computed for the strip of unit width. The load per unit area on the slab becomes the load per unit length on the slab strip. Since, all of the load on the slab must be transmitted to the two supporting beams, it follows that all of the reinforcement should be place at right angles to these beams, with exception of any bars that may be placed in the other direction to control shrinkage and temperature cracking. A oneway slab, thus, consists of a set of rectangular beams side by side.

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Fig. 3: Unit strip basis for flexural design One way reinforced concrete slabs supporting floor or roof loads are generally designed as beams of unit width. For a given type of support condition, the span/depth ratio applicable for beams in IS: 456 are also valid for slabs. Since, the percentage of reinforcements in slabs is generally low in the range of 0.3 to 0.5 percent, a span/depth ratio of 25 to 30 is more appropriate by considering the modification factor Kt (1.2 to 1.4 for Fe-415 steel). Normally the thickness of slabs is so chosen that the shear can be resisted by concrete alone without any extra shear reinforcements. The shear enhancement factor (k) specified in Cl. 40.2.1.1 of IS: 456 code varying from 1 to 1.3 depending upon the thickness of slab will considerably increase the permissible shear stress in slabs when multiplied with the values of shear stress given in Table 19 of IS: 456-2000. In slabs, shear reinforcements may be allowed if the thickness is 200 mm or more but in no case the maximum shear stress in slabs due to ultimate load exceed one half of that given in Table 20 of IS: 456-2000. In the case of slabs the depth selected is usually greater than the minimum depth for the balanced section and hence the steel required may be calculated by the formula given in IS: 456-2000 or by the use of SP-16 charts and tables. The designed slab should be checked for shear stress and deflection control. 5.3 TWO-WAY SLABS Reinforced concrete slabs supported on all the four sides with their effective span in the longer direction not exceeding two times the effective span in the shorter direction are designed as two way-slabs. Two-way slabs deform with significant curvatures in two orthogonal directions with moments developed in the principal directions as shown in Fig. 4. The bending moments are maximum at the centre of the slab and the larger moment invariably develops along the short span.

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Fig. 4: Two-way slab action with moment and deflection diagram The bending moment in the slab depend upon the following parameters. i) The short and long span (Lx and Ly) ii) Edge conditions at the support (Fixed, free, continuous etc.) iii) Magnitude and type of load on the slab (uniformly distributed, concentrated etc.) 5.3.1 Simply supported slabs When a slab simply supported on all the four sides is subjected to transverse loads, the bending of the slab in the two principal direction causes the corners to curl and lift up as shown in Fig. 5 due to non-uniform variation of load transmitted to the supports. Simply supported slabs which do not have adequate provision to resist torsion at corners and to prevent the corners from lifting, the maximum moments per unit width are specified in the IS: 456-2000 code and computed by the following equations,

Fig. 5: Torsion effects in two way slabs

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Where, Mx and My are the design moments in x and y directions, w is uniformly distributed load on slab, Lx and Ly are the short and long span dimensions of the simply supported slab. The values of x and y are compiled in Table 27 of IS: 456 -2000 for simply supported slabs. These coefficients are due to Rankine-Groshoff theory in which slab is divided into a series of orthogonal beam strips and the load is apportioned to the short and long strips such that there is compatibility of deflection at the junction of strips. Cl. D.2.1.1 of IS: 456 code specifies that at least 50% of the tension reinforcement provided at mid span should extend to the supports. The remaining 50% should extend to within 0.1Lx or 0.1Ly of the support, as appropriate. 5.3.2 Two-way restrained slabs Restrained slabs are referred to as slabs whose corners are prevented from lifting. They may be supported on continuous or discontinuous edges. All the four edges of the two-way slabs are assumed to be supported rigidly against vertical translation. The design moments in restrained slabs are easily evaluated using the moment coefficients recommended in IS: 456 code and are given in Table 26 of IS: 456-2000. These moment coefficients are based on yield line theory with the following assumptions: a) The reinforcement for positive moment is uniformly distributed over the middle strip extending over 75% of the span. b) Edge strips cover a width equal to (Lx/8) or (Ly/8) as shown in Fig. 6. c) Minimum reinforcements prescribed for slabs should be provided in edge strips. d) Torsion reinforcement is provided at corners where the slab is simply supported on both edges meeting at the corner. The reinforcement comprising three quarters of the area required for the maximum mid span moment in the slab is provided in each of the four layers in the form of a mesh extending to a minimum distance of 1/5th of the shorter span. As shown in Fig. 6, full torsional steel is provided at corner A where the slab is discontinuous on both the edges meeting at that corner. At corner B where the slab is discontinuous on only one edge meeting at that corner, 50% of the full torsional steel is provided. At corner C, as the slab is continuous on both edges meeting at the corner, torsional steel is not required (Fig. 6a).

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Fig. 6: Middle and Edge strips in Two-way slabs

Fig. 6a: Provision of Torsional steel

5.3.3 Span to depth ratio In the case of two-way slabs, the magnitude of moments will be smaller than one-way slabs since the load is distributed in two principal directions. Consequently, the percentage reinforcement being small, the modification factor for tension steel Kt is higher resulting in higher values of maximum permissible span/depth ratios. Hence, the following span/overall depth ratios have been recommended in IS: 456 code Cl. 24.1 for two-way slabs with shorter span up to 3.5 m, using Fe-415 HYSD bars. a) Simply supported slabs = 28 b) Continuous slabs = 32 5.3.3 Deflection control The deflection of two-way slabs is controlled by span/depth ratio similar to the case of oneway slabs and beams. In two-way slabs the shorter span and the percentage of steel in that direction have to be considered for computations of modification factors. 5.4 PROBLEMS

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