Lecture-X Two Way Slabs

BITS Pilani Hyderabad Campus

CE F311 Design of Concrete Structures Instructor: Mr. J S Kalyana Rama

Course Lecture-X Two Way Slabs

BITS Pilani, Hyderabad Campus

Introduction

Instructor: Mr. J S Kalyana Rama

When the aspect ratio of a floor panel is less than two, the contribution of the longer span in carrying the floor load becomes substantial. Since the floor load is transmitted in two directions, this type of slab is called a two-way slab, and the flexural reinforcement has to be designed in both the directions.

Two-way slab systems include two-way solid slabs supported by beams, flat plates, flat slabs, and waffle slabs (see Fig. 2).

A slab supported on beams on all sides of each floor panel is generally referred to as a two-way slab system.


Fig. 1 Difference in behaviour of one-way and two-way slabs (a) One-way slab (Ly/Lx > 2.0) (b) Two-way slab (Ly/Lx 2)

Introduction



Fig. 2 Types of two-way slabs (a) Two-way slab (b) Flat plate (c) Flat slab (d) Grid or waffle slab

Introduction



Consider a two-way slab as consisting of beam strips in each of the two directions, intersecting each other. Two such strips, L1 in the longer direction and S1 in the shorter direction, are shown in Fig. 3(a).

When acted upon by the applied load, each of the two strips acts similar to a beam, sharing the applied load and transferring it to their respective edge supports.

Thus, bending exists in both directions. To resist these moments, the slab must be reinforced in both directions by two layers of reinforcements that are perpendicular to each other.

Behaviour of Two-way Slabs



Consider the same slab with two sets of three strips in two perpendicular directions as shown in Fig. 3(b).

The behaviour of the outer strips S2, L2, S3, and L3 are different from that of the centre strips S1 and L1. These outer strips are bent as well as twisted.

This twisting results in torsional stresses and torsional moments, especially near the corners, which in turn will result in lifting up of the corners, unless restrained.




Fig. 3 Behaviour of two-way slabs (a) Single beam strip in each direction (b) Multiple beam strips in each direction




When the load is increased, hairline cracks start to appear from the point of maximum deflection as shown in Fig. 4(a).

The slab no longer has constant stiffness, as the cracked regions have lower stiffness than the other regions.

In rectangular slabs, the crack pattern may differ in the two directions and the slab may not remain isotropic.

Most of the slabs will have hairline cracks under service loading.




When the load is further increased, yielding of reinforcements takes place at the mid-span. The bending moments get redistributed to the non-yielding portions that still remain elastic.

As the load is increased further, this inelastic redistribution will continue until a large area of steel in the central portion of the slab yields and a mechanism is formed when the slab fails (see Fig. 4b).

The yield lines divide the slab into a series of trapezoidal or triangular elastic plates in the case of rectangular slabs and triangular plates in the case of square slabs (see Fig. 4b).




Fig. 4 Mode of failure of simply supported two-way slabs (a) Onset of yielding of bottom Reinforcement at point of maximum deflection (b) Bottom steel yielding along yield lines forming a mechanism




Slab thickness is the primary factor affecting the serviceability and shear strength. Fire resistance requirements may also govern both the cover and slab thickness.

The vertical deflection limits may be assumed to be satisfied in two-way slabs with short spans up to 3.5 m with mild steel reinforcement and for loading class up to 3 kN/m2, when the span to overall depth ratios are not greater than the following values(Clause 24.1 & 23.2):

1. Simply supported slabs - 35 2. Continuous slabs 40 For Fe 415 grade steel these values to be multiplied by 0.8

Minimum Thickness of Slabs



1. Linear elastic analysis for thin plates (classical plate theory, finite difference method)

2. Non-linear analysis (finite element analysis) 3. Plastic or limit analysis (lower bound like the strip method and upper

bound like the yield-line analysis method) 4. Idealized frame method of analysis (equivalent frame method) 5. Simplified methods of analysis (methods based on moment coefficients,

direct design method) 6. Combination of elastic theory and limit analysis

Analysis of Wall-supported Slabs



Structures have a tendency to transmit loads to the supporting systems along the shortest possible path. This tendency is seen in slabs as illustrated in Fig. 10, which shows the principal moment directions in slabs with ratios of sides 1.0 and 2.0 for simply supported and fixed boundary conditions.

In the central region in the longer direction of the longer slabs, the direction of the principal moments, is nearly perpendicular to the supports.

Thus, the load is mainly transferred in the shorter direction, implying that the moments in the shorter direction are greater than those in the longer direction.

Trajectories of Principal Moments in Two-way Slabs



Fig. 10 Trajectories of principal moments in beam-supported rectangular slabs (a) Ly/Lx = 1 (b) Ly/Lx = 2.0




The principal moment directions in the corner regions are inclined to the support, indicating twisting moments.

Approximate regions of negative bending moments in simply supported rectangular slabs under uniformly distributed slabs are shown in Fig. 11(a) and these regions for slabs with fixed ends is shown in Fig. 11(b).

These regions should be provided with reinforcements on both the upper and lower faces of the slabs.




Fig. 11 Regions of negative bending moments in rectangular slabs under uniformly distributed slabs (a) Simply supported slab (b) Slab with fixed ends




Hence, considering the two middle strips as shown in Fig. 3(a), the deflection at their common intersection point will be the same (as these two strips of beams belong to the same monolithic slab). The following equations as given in Clause D-2.1 of IS 456 are based on

their method:

Where

Where, wu is the factored UDL , L

x is the short span, and r is the is the aspect ratio of

spans= Ly/Lx. See Table 27 of IS 456 also.




Restrained slabs are defined as those that are cast integrally with beams and in which the corners are prevented from lifting and provision for torsion is made at simply supported corners.

They may be continuous or discontinuous at the edges.

The maximum design moments per unit width of such restrained slabs may be calculated using the following equations:

Mx = ax wu L2x My = ay wu L2x

here Mx and My are the moments on strips of unit width spanning Lx and Ly, respectively, ax and ay are the bending moment coefficients. wu is the uniformly distributed factored load on slab.

Table 26 of IS 456(see Table 4 of the book) is based on this method

Coefficients Based on Yield-line Theory



The equations may be used only when the following conditions are satisfied:

1. The characteristic dead and imposed loads on adjacent panels are approximately equal to the loads on the panel being considered.

2. The spans of adjacent panels in each direction are approximately equal.

3. The slabs are essentially subjected to uniform loads.




The following are the rules to be observed when the equations are applied to restrained slabs (continuous or discontinuous): 1. Slabs are considered as divided in each direction into middle strips

and edge strips as shown in Fig. 12.

2. The maximum design moments calculated as mentioned earlier apply only to the middle strips and no redistribution is allowed.

3. Reinforcement in the middle strips should be detailed in accordance with IS 456.

4. Corner reinforcements are provided to resist the torsional moments.




Fig. 12 Division of slab into middle and edge strips (a) For span Lx (b) For span Ly




Different types of support conditions



Bending moment coefficients for two way restrained slabs ( Table 26, IS 456-2000) Detailing requirements



When there is a series of continuous slabs in one or both directions, the negative moments obtained at a common support on the left- and right-hand sides, may not be equal and may differ significantly, as shown in Fig. 13.

This may be due to any one or more of the following reasons: 1. The two adjacent spans being unequal 2. The loading on one panel being different from that of the other 3. The boundary conditions in the two adjoining panels being different

Unbalanced Moments in Adjacent Spans



Fig. 13 Unbalanced moments in adjacent spans of a continuous slab

Unbalanced Moments in Adjacent Spans



1. Calculate the sum of the mid-span moment and average of the support moments (neglecting the signs) for each panel.

2. Treat the values found from Table 26 of IS 456 as fixed-end moments (FEMs).

3. Distribute the FEMs across the supports according to the relative stiffness of adjacent spans, giving new support moments.

4. Adjust the mid-span moment for each panel: this should be done in such a way that when it is added to the average of the support moments (neglecting signs) from step 3, the total should be equal to that from step 1.

Procedure to Balance Unbalanced Moments



If, for a given panel, the resulting support moments are significantly greater than the value obtained from Table 26 of IS 456 , the code suggests that the tension steel over the supports should be extended beyond the provisions of Clause D-1.5.

It also recommends the following procedure: 5. Take the span moment as parabolic between supports: the

maximum value is found from step 4.

6. Determine the points of contraflexure of the new support moments (from step 3) with the span moment (from step 5).




7. Extend half the support tension steel at each end to at least an effective depth or 12 times the bar diameter beyond the nearest point of contraflexure.

8. Extend the full area of support tension steel at each end to half the distance 6.6 from step 7.

Even though this procedure has been specified in the code, several engineers consider it logical to take the larger value of moment (M1 as shown in Fig. 13) as the design negative moment at the common continuous edge.




The magnitude of shear stresses are comparatively lesser in two-way slabs than in one-way slabs due to the two-way action. Hence, they will not govern the design.

The distribution of shear forces at the various edges of a two-way slab is not easy to determine. However, Clause 24.5 and Fig. 7 of the code (see also Fig. 1b) recommend a simple triangular distribution of load on the short edge and a trapezoidal distribution of load on the long edge.

This type of shear is often called a one-way shear, which is different from the two-way shear or punching shear, which also has to be checked in the case of flat slabs.

Shear Forces in Two-way Slabs



Fig. 15 Assumed distribution of loads for determination of shear force in two-way slabs

Shear Forces in Two-way Slabs



Extending plate-based code methods to beam-supported continuous slabs introduces a degree of approximation with respect to the support rigidity.

A major assumption in the plate-based methods is that a rectangular slab panel is rigidly supported on its four sides. If beams are provided along the column lines and if these beams are rigid, then the analysis and design of the slab may be considered in the same manner as wall-supported slabs.

The design and detailing of two-way slabs supported on stiff beams may be carried out similar to the design and detailing of two-way slabs supported on the walls.

Beam-supported Two-way Slabs



Fig. 16 Minimum depth of beam to ensure rigidity

Beam-supported Two-way Slabs



The different steps involved in the design of two-way slabs, considering them as beams of one metre width, are similar to the design of one-way slabs.

The bending moments are obtained using the coefficients provided in Tables 26 and 27 of the code.

The required thickness of the slab may be estimated initially based on the serviceability limit state criteria of deflection, using the suggested value of effective span to depth ratio given in Clause 24.1 of the code or by using Fig. 7 of the Book.

Design Procedure for Two-way Slabs



As the depth chosen based on the deflection criteria will normally be greater than that required for bending, the tension steel required will be less than that for the balanced steel and the section will be under-reinforced.

The reinforcements calculated for the bending moments occurring in the middle strip of the restrained discontinuous slabs have to be provided in the middle strip alone (see Fig. 12) and the edge strips require only minimum reinforcement.

After calculating the reinforcements, the assumed thickness should be checked for L/d limits imposed by the code, using modification factors given in Figs 4 and 5 of the IS code.

Design Procedure for Two-way Slabs



Simply Supported Two-way Slabs

When the slab is truly simply supported at the edges, there will not be any negative moments near the supports. However, there may be some unforeseen partial fixity.

To safeguard against partial fixity, either alternate bars are bent up or separate top steel is provided, with an area equal to 0.5 times of that provided at the bottom of the mid-span, with an extension of 0.1Lx or 0.1Ly from the face of the support (see Fig. 17).

Detailing of Reinforcements



Fig. 17 Detailing of wall-supported two-way slabs with bent-up bars

Detailing of Reinforcements



Torsional stresses and torsional moments are developed near the corners of a simply supported slab, which will result in the lifting up of the corners as shown in Fig. 18(a) unless the slab is restrained at corners.

This will result in cracking of the slab near the corners as shown in Fig. 18(b).

Hence, torsional reinforcements are to be provided in the corners of the slab to take care of these torsional moments. Two types of reinforcement are indicated in Figs 18(c) and (d).

Detailing of Torsional Reinforcement at Corners



Fig. 18 Torsional effects and torsional reinforcements (a) Lifting of corners due to

torsion (b) Potential crack pattern

(c) Torsion reinforcement using skewed bars

(d) (d) Torsion reinforcement using top and bottom bars




When restrained slabs are designed by using the moment coefficients given in Table 26 of IS 456 , they should be detailed as discussed below:

1. The tension steel calculated for the positive design moments (per unit width) at the short and long spans should be provided, as shown in Fig. 19, at the bottom of the mid-span in the middle strip in the short- and long-span directions, respectively.

2. The tension steel calculated for the negative design moments in the short and long spans at continuous supports should be provided at the top and uniformly distributed across the edge strips of the short and long spans, respectively (see Fig. 19).

Detailing for Restrained Two-way Slabs



3. At discontinuous edge, negative moments may arise due to partial fixity. Hence, to safeguard against such situations, 50 per cent of the bottom steel at the mid-span should be provided at these edges and such steel should extend over a length of 0.1Lx or 0.1Ly from the face of the support, as shown in Fig. 19.

4. Reinforcement in an edge strip parallel to the edge need not exceed the minimum area of tension reinforcement together with the recommendations for torsion given as per Clause D-1.7 of the code.




Fig. 19 Reinforcement detailing in restrained two-way slabs (a) Plan (b) Section AA Instructor: Mr. J S Kalyana Rama


Fig. 20 Simplified rules for curtailment of bars in two-way slabssection through middle strip (a) Using bent-up bars (b) Using straight bars




Torsion reinforcement should be provided at any corner where the slab is simply supported on both edges meeting at that corner.

The following points need to be noted (see Fig. 21): This torsion reinforcement should be provided at the top and bottom

in a mesh or grid pattern, each with layers of bars placed parallel to the sides of the slab and extending from the edges to a minimum distance of one-fifth of the shorter span

Torsion reinforcement equal to half that described in 1, should be provided at a corner where one edge of the slab is continuous and the other edge is discontinuous.

Torsion reinforcement need not be provided when both edges meeting at a corner are continuous.




Fig. 21 Detailing of torsional reinforcement in restrained slabs



Roof slabs are subjected to weathering action in addition to supporting the self-weight, occasional imposed load, and the weathering course.

It is important to provide adequate drainage facilities to avoid the problem of ponding and overloading and/ or leakage of roof slab.

Roofs in tropical areas are subjected to large variations of temperature. Strains due to thermal changes are sufficient to cause micro-cracks in concrete. it is advisable to provide sliding joints between the roof and the wall in wide roofs, using elastomeric bearing pads.

Parapet wall is important to hold down the corners of slabs.

Roof Slabs



Green roof of City Hall in Chicago, Illinois.

Green Roofs



To mitigate Heat-island Effect of cities green roofs are adopted. A green/living roof is partially/completely covered with vegetation planted

over a waterproofing membrane. It may also include additional layers such as a root barrier and drainage and

irrigation systems. Hence those loads (typically 1-5 kN/m2 ) should be considered.

Green Roofs



References

Reinforced Concrete Design- Krishna Raju Reinforced Concrete Design- Pillai and Menon IS 456:2000 Design of Reinforced Concrete Structures-N Subramanian IIT Madras Reading Material for RCC and Concrete


BITS Pilani Hyderabad Campus

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Lecture-X Two Way Slabs

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