Chap 17 (cont..5)-Coefficients for Bending moments in restrained two way slabs

Now we will discuss how to determine the bending moment in each of the directions X and Y in a restrained two way slab. For this, let us consider the middle strip in each direction as shown below.

Fig.17.19
Middle strips in restrained two way slabs

Note that this fig. is different from the earlier fig.17.5 that we saw for a simply supported two-way slab. Because, here the ends of the strips are restrained or in other words tied down to the supports.

As in the case of simply supported slab we saw before, we can determine the maximum bending moments at the center portion of each of these restrained strips also. Let us denote them as M_x and M_y. If we use the factored loads, we will get the factored bending moments, and from those we can determine the steel required in each direction. But the bending moments that we obtain at the center portion of these restrained strips require some corrections to be applied to them. So we will call them M_x,1 and M_y,1 at this stage. The subscript '1' indicates that they are the first values in each direction, before applying any corrections. We will change this subscript as we proceed with the application of the various corrections.

Now we will discuss the first correction that has to be applied: The values M_x,1 and M_y,1 are obtained by considering the action of individual strips. But a slab consists of several such strips, and there will be interaction between adjacent strips. Consider a square slab subjected to uniformly distributed load of w/m². As it is square, the load share in each direction will be the same and is equal to w/2. So the maximum bending moment at the center of each strip will be

But if we analyse a square elastic plate subjected to uniformly distributed loads, by using the 'theory of bending of elastic plates', we will find that the maximum bending moment at it's center will be 0.048wl². This is lesser than 0.0625wl². This is because, when we analyse a plate as a whole, the interaction between the strips will also be taken into account, and so we will get a lesser value. So we have to apply a correction to M_x,1 and M_y,1. We will later see how this correction can be applied while designing a slab. In the mean time we will call the corrected value as M_x,2 and M_y,2.

Let us now discuss the next correction that has to be applied: M_x,1 and M_y,1 are the maximum bending moments at the center portion of the strips. These strips are like individual beams. If we increase the loads on the strips, a point will be reached where the steel at the center portion in the strips will begin to yield. If the load is increased further, the strips will fail. But in an actual two way slab, when the steel in the center portion fail, the failure of the slab as a whole will not occur. This is because the steel in the surrounding portion, will be able to take up the load. If the load is increased further, this surrounding steel will also fail. This continues until the steel fails at a considerable area in the center of the slab. So we can see that we need not provide steel for Mx1 and My1, as the failure of the slab will not occur even if they are exceeded. Thus we have to apply a correction here also. Let us denote the values obtained after applying both the above two corrections as M_x,3 and M_y,3.

The last correction that has to be applied is related to the torsion reinforcement that we saw in fig.17.17. In restrained slabs we have to provide these bars at the corners to prevent cracks. In such slabs which have adequate reinforcement for torsion, the bending moments at the center portion will be lesser than Mx1 and My1. Because the slab is resisting the load not only by bending, but also by torsion. The corner reinforcements have the effect of reducing the deflection and curvature at the center portion of the slab. This can be further explained based on the view in fig.17.20 below:

Fig.17.20
Interaction between perpendicular strips

The red strip in the above fig. is the same one which we saw earlier in fig.17.13 The blue strip is the middle strip in the perpendicular set of strips which are parallel to the short side. The red strip is under torsion. So torsional moments are induced in the red strip. These torsional moments try to twist the strip back to it's original shape. This is shown by the green arrow. Thus the torsional moment will reduce the deflection of the blue strip. This type of interaction between perpendicular strips will occur on the entire area of the slab.

So we have to apply a correction for this also. When this above correction is also applied, we can denote the values as M_x,4 and M_y,4. But now, all the required corrections are applied and we no longer need to use the subscripts 1,2 etc., and so we will denote the final values as M_x and M_y. , and the factored values will be denoted as M_u,x and M_u,y.

We have seen the various corrections that have to be applied. But we have not seen the methods for applying them. In design practice, these corrections are not applied separately. We use some 'Moment coefficients' α_x and α_y. These are similar to the Rankine-Grashoff coefficients that we saw in the case of simply supported two-way slabs. When we use these coefficients, the resulting bending moments will be the required final values M_x and M_y, with all the necessary corrections applied.

M_x = α_x w l_x² and
M_y = α_y w l_x²

If we use factored load w_u in the above equations, we will get the factored bending moments as shown below:

M_u,x = α_x w_u l_x² and
M_u,y = α_y w_u l_x²

Note that both the bending moments are in terms of the shorter side l_x

When there is fixity or continuity at any of the four supports of the slab, hogging (negative) moments will occur at those supports. So we need separate coefficients for those negative moments also. Thus we will need four coefficients: α_x⁺, α_x^-, α_y⁺ and α_y^-. The fig.17.21 shows the application of the four coefficients:

Fig.17.21
Four coefficients for bending moments

In the next section we will see some basic background details before we obtain the actual four coefficients.

PREVIOUS      CONTENTS       NEXT

             Copyright ©2015 limitstatelessons.blogspot.com- All Rights Reserved