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)
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.
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.
where,A_{st} = Area of steel B = Width of the Beam d = Effective depth of the beam
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
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
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.
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
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