Slabs | Civil Engineering SSC JE (Technical) - Civil Engineering (CE) PDF Download

Chapter 3

 Slabs

Slabs are palte elements forming floors and roofs of buildings and carrying distributed loads primarily by flexure. Inclined slabs may be used as ramps for multistorey car parks. A staricase can be considered to be an inclined slab. A slab may be supported by beams or walls and may be used as the flange of a T-or L-beam.
Moreover, a slab may be simply supported, or continuous over one or more supports and is classified according to the manner of support.

(a) One-way slabs spanning in one directions. (b) Two-way slabs spanning in both directions. (c) Circular slabs. (d) Flat slabs resting directly on columns with no beams.

If the cross-sectional areas of the three basic structural elements: beam, slab and column are related to the amount of steel reinforcement provided, it will be seen that the percentage of steel is usually maximum in a column than in a beam and the least in a slab. The distinction between a beam and a slab can be made as follows:

(a) Slabs are analysed and designed as having a unit width, that is, 1 m wide strips.
(b) Compression reinforcement is used only in exceptional cases in a slab.
(c) Shear stresses are usually very low and shear reinforcement is never provided in slabs. It is preferred to increase the depth of a slab and hence reduce the shear stress rather than provide shear reinforcement.

One-way Slabs

One-way slabs are those in which the length is more than twice the breadth. A one-way slab can be simply supported or continuous i.e.,

where,l_y = length of longer span
l_x = length of shorter span

Design Steps

(a) Basic rules for design

1. Effective span The basic rules for the effective span of a slab are the same as given by beams. Thus, in the case of freely supported slab, the effective span is taken equal to supports, or the clear distance between the supports plus the effective depth of the slab whichever is less.

2. Control of deflection The same rules apply, as given for beams earlier.

3. Minimum reinforcement The mild steel reinforcement in either direction in slabs shall not be less than 0.15% of the total crosssectional area. However, this value can be reduced to 0.12% when high strength deformed bars or welded wire fabric are used.

4. Maximum diameter The diameter of the reinforcing bars shall not exceed one eight of the total thickness of the slab.

5. Spacing of bars The horizontal distance between parallel main reinforcement bars shall not more than three times the effective depth of a solid slab or 300 mm, whichnever is smaller. The horizontal distance between parallel reinforcement bars provided against shrinkage and temperature shall not be more than five times the effective depth of the slab or 450 mm whichever is smaller. The horizontal distance between two parallel main reinforcement bars shall usually be not less than the greatest of the following:

(i) The diameter of the bar if the diameters are equal.

(ii) The diameter of the larger bar if the diameter are unequal, and

(iii) 5 mm more than the nominal maximum size of coarse aggregate.

6. Cover to reinforcement As discussed in Beams.

7. Curtailment of tension reinforcement
(a) For curtailment, reinforcement shall extend beyond the point at which it is no longer required to resist flexure, for a distance equal to effective depth or 12 times the bar diameter whichever is greater, except at support or end of cantilever.
(b) At least one-third the positive moment reinforcement in simple members and one–fourth of the positive moment reinforcement in continuous members shall extend along the same face of the member into the support, to a length equal L_d/3.

(c) At simple supports, and at point of inflexion, positive moment tension reinforcement shall be limited to such a diameter that

NOTE:
The value of  in the expression may be increased by 30% when the ends of the reinforcement areconfined by a compressive reaction.
At least one-third of the total reinforcement provided for negative moment at the support shall extend beyond the point of inflextion, for a distance not less than the effecitve depth of the member or 12 φ or one-sixteenth of the clear span whichever is greater.

TWO-WAY SLABS
Simply Supported Slab (Corners ar e free to lift up): A two way slab which is simply supported at its edges, tends to split off it supports near the corners when loaded. Such a slab is the only truly simply supported slab.  The values of the B.M. used for the design of such slabs can be calculated as follows.

and 
 where M_x, M_y = maximum moments at mid span on strips of unity width and spans I_x and I_y, respectively.
I_x = length of shorter side I_y = length of longer side

α_x , α_y = moment coefficient (as given in clause D-2.1 of IS 456 : 2000) 

• The code requires that at least 50% of the tension reinforcement provided at mid span should extend to the supports. The remaining 50% should extend to within 0.1 I_x and 0.1 I_y of the support as appropriate

Restrained Slab

A slab may have its few or all edges restrained. The degree of restraints may vary depending whether it is continuous over its supports or cast monolithically with its supporting beams. A hogging or a negative bending moment will develop in the top face of the slab at the support sides. In these slabs the corners are prevented from lifting and provision is made for torsion. The maximum moments M_x and M_y are given by:

β_x and β_y are moment coefficients,
β_x depends both on type of panel and moment and (I_y/I_x), given in clause D-1.1 of IS 456 : 2000.
β_y depends only on type of panel and moment. 

• Slab are considered as divided in each direction into middle strips and edge strips as shown in Figure. The middle strip being three-quarters of the width and each edge strip one-eight of the width.

•  The maximum moments calculated in equation (i) and (ii) apply only to the middle strips and no redistribution shall be made.
• Tension reinforcement provided at mid-span in the middle strip shall extend in the lower part of the slab to within 0.25 l of a continuous edge, or 0.15l of a discontinuous edge. 
• Over the continuous edges of a middle strip, the tension reinforcement shall extend in the upper part of the slab a distance of 0.15 l from the support and at least 50 percent shall extend a distance of 0.3 l. 
• At a discontinuous edge, negative moments may arise. They depend on the degree of fixity at the edge of the slab but, in general, tension reinforcement equal to 50 per cent of that provided at mid-span extending 0.1l into the span will be sufficient.

• Torsion reinforcement shall be provided at any corner where the slab is simply supported on both edges meeting at the corner. It shall consists of top and bottom reinforcement, each with layers of bars placed parallel to the sides of the slab and extending from the edges a minimum distance of one-fifth of the shorter span. The area of reinforcement in each of these four layers shall be three-quarters of the area required for the maximum mid-span moment in the slab. 
• Torsion reinforcement equal to half that described above shall be provided at a corner contained by edges over only one of which the slab is continuous. 
• Torsion reinforcement need not be provided at any corner content by edges over both of which the slab is continuous.

Shear  Stress
(a) For Homogeneous beam

where, q = shear stress at any section
V = shear force at any section

 = Moment of area of section above the point of consideration
I = Moment of inertia of section = 

(b) For Reinforced concrete beam

(i) Shear stress above N.A

(ii) Shear stress below N.A

As per IS 456 : 2000
Nominal shear stress, 

The maximum shear stress

 obtained from elastic theory, is greater than the nominal shear stress(or Average shear stress) ζ , suggested by IS : 456 : 2000. 

• Design shear strength of concrete ( ζ _c) without shear reinforcement as per IS : 456 : 2000 ζ _c depends on
  (i) Grade of concrete (ii) Percentage of steel.

where,A_st = Area of steel B = Width of the Beam d = Effective depth of the beam

•  Maximum shear stress ( ζ _cmax) with shear reinforcement is

•  Minimum shear reinforcement (As per IS 456:2000)

 This is valid for both W.S.M and L.S.M

where, A_sv = Area of shear reinforcement
S_v = Spacing of shear reinforcement Where ζ_v exceeds ζ_s , shear reinforcement shall be provided in any of the following forms:

 (i) Vertical stirrups (ii) bent-up bars along with stirrups and (iii) inclined stirrups.

Where bent-up bars are provided, the contribution towards shear resistance shall not be more than half that of the total shear reinforcement.

Shear reinforcement shall be provided to carry a shear equal to V_u – ζ_c Bd.  

The strength of shear reinforcement Vvs shall be calculated as below:

(a) For vertical stirrups

 b) For inclined stirrups or a series of bars bentup at different cross-section:

(c) For single bar or single group parallel bars, all bent-up at the same cross-section.
V_us = 0.87 f_y A_sv sin α
Angle between the inclined stirrup or bent-up bar and axis of member not less than 45º 

• Spacing of shear reinforcement 

Maximum spacing is minimum of (i), (ii) and (iii)

(ii) 300 mm

(iii) 0.75d — For vertical stirups d — For inclined stirrups
where, d = effective depth of the section  

• Critical section for design shear

The above provisions are applicable for beams generally carrying uniformly distributed load or where the principal load is located faster than 2d from the face of support.

•   Vertical stirups:

Shear force 'V_s' will be resisted by shear reinforcement provided in 'd' length of the beam,

 V_s =   A_sv σ_sv →  For WSM

 where,A_sv = Cross-sectional area of stirrups S_v = Centre to centre spacing of stirrups

  V_su =  A_sw (0.87f_y) _→  For WSM

•  Inclined stirrups : or a series of bars bent-up at diferent cross-section:

 V_s =  A_sw. σ_sw. (sin α + cos α)  _→  For WSM

 V_su =  A_sw.(0.87f_y)(sin α + cos α) _→  For WSM