Slabs Civil Engineering (CE) Notes | EduRev

RCC & Prestressed Concrete

Civil Engineering (CE) : Slabs Civil Engineering (CE) Notes | EduRev

The document Slabs Civil Engineering (CE) Notes | EduRev is a part of the Civil Engineering (CE) Course RCC & Prestressed Concrete.
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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.,
Slabs Civil Engineering (CE) Notes | EduRev
where,ly = length of longer span
lx = length of shorter span

Design Steps

Slabs Civil Engineering (CE) Notes | EduRev
Slabs Civil Engineering (CE) Notes | EduRev

(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 Ld/3.

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

 Slabs Civil Engineering (CE) Notes | EduRev

NOTE:
The value of Slabs Civil Engineering (CE) Notes | EduRev 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 Slabs Civil Engineering (CE) Notes | EduRev
 where Mx, My = maximum moments at mid span on strips of unity width and spans Ix and Iy, respectively.
Ix = length of shorter side Iy = length of longer side

αx , αy = 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 Ix and 0.1 Iy 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 Mx and My are given by:
Slabs Civil Engineering (CE) Notes | EduRev

βx and βy are moment coefficients,
βx depends both on type of panel and moment and (Iy/Ix), given in clause D-1.1 of IS 456 : 2000.
βy 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.

 Slabs Civil Engineering (CE) Notes | EduRev

  •  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.

 Slabs Civil Engineering (CE) Notes | EduRev

  • 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.

 Slabs Civil Engineering (CE) Notes | EduRev

Shear  Stress
(a) For Homogeneous beam

 Slabs Civil Engineering (CE) Notes | EduRev
Slabs Civil Engineering (CE) Notes | EduRev
Slabs Civil Engineering (CE) Notes | EduRev

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

Slabs Civil Engineering (CE) Notes | EduRev = Moment of area of section above the point of consideration
I = Moment of inertia of section = 
Slabs Civil Engineering (CE) Notes | EduRev

(b) For Reinforced concrete beam

 Slabs Civil Engineering (CE) Notes | EduRev

(i) Shear stress above N.A

 Slabs Civil Engineering (CE) Notes | EduRev

(ii) Shear stress below N.A

 Slabs Civil Engineering (CE) Notes | EduRev

As per IS 456 : 2000
Nominal shear stress, 
Slabs Civil Engineering (CE) Notes | EduRev
The maximum shear stress
Slabs Civil Engineering (CE) Notes | EduRev
 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.

Slabs Civil Engineering (CE) Notes | EduRev
where,Ast = Area of steel B = Width of the Beam d = Effective depth of the beam

 Slabs Civil Engineering (CE) Notes | EduRev

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

 Slabs Civil Engineering (CE) Notes | EduRev

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

 Slabs Civil Engineering (CE) Notes | EduRev

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

 Slabs Civil Engineering (CE) Notes | EduRev

where, Asv = Area of shear reinforcement
Sv = Spacing of shear reinforcement Where ζv exceeds ζs , 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 Vu – ζc Bd.  

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

(a) For vertical stirrups

Slabs Civil Engineering (CE) Notes | EduRev

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

 Slabs Civil Engineering (CE) Notes | EduRev

(c) For single bar or single group parallel bars, all bent-up at the same cross-section.
Vus = 0.87 fy Asv 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)

Slabs Civil Engineering (CE) Notes | EduRev

(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

Slabs Civil Engineering (CE) Notes | EduRev

Slabs Civil Engineering (CE) Notes | EduRev

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:

 Slabs Civil Engineering (CE) Notes | EduRev

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

 VsSlabs Civil Engineering (CE) Notes | EduRev  Asv σsv →  For WSM

 where,Asv = Cross-sectional area of stirrups Sv = Centre to centre spacing of stirrups

  VsuSlabs Civil Engineering (CE) Notes | EduRev Asw (0.87fy) →  For WSM

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

 Vs =  Asw. σsw. (sin α + cos α)  Slabs Civil Engineering (CE) Notes | EduRev→  For WSM

 Vsu =  Asw.(0.87fy)(sin α + cos α) Slabs Civil Engineering (CE) Notes | EduRev→  For WSM

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