Chapter 17 (cont..4)- Torsionally restrained two way slabs

In the previous sections, we discussed about the analysis and design of Simply supported two way slabs. The method of analysis that we used is called the Rankine-Grashoff method. The values of α_x and α_y that are given in the table 27 of the code are known as the Rankine-Grashoff coefficients. Simply supported two way slabs are usually designed using this method. But there are other types of two-way slabs which are not simply supported. For example, the two-way slab may be continuous over one or more supports. Fig.17.1 that we saw earlier is the plan view of such an example. Another example is when the two-way slab is built monolithically into a supporting beam. Fig.17.10(a) below shows the sectional view of such an example. Yet another example is when a wall is present above the supporting wall as shown in the sectional view in fig.17.10(b). In these cases, the corners of the slab are not free to lift up, and so, Rankine-Grashoff theory is not applicable.

Fig.17.10
Examples of restraints applied at supports

Fig.17.11 below shows the view of a two way slab. The yellow arrows represent the uniformly distributed load acting on the slab. The red arrows represent the restraining force acting at the supports. This view is only a schematic representation. The restraining force may be due to a wall built above the support, or the slab being built monolithically into a beam, or the slab being continuous. In any case, a restraining force is acting at the supports of the slab, and so the corners are being prevented from lifting up. When such a restraint is applied to a slab subjected to uniformly distributed load, it deforms into the shape of a 'dish'.

Fig.17.11
View of a two way slab whose corners are held down

Let us divide the slab into strips. The fig.17.12 below shows the slab divided into a set of strips parallel to l_y.

Fig.17.12
Strips parallel to ly

Let us take one particular strip from this set. The second one from the longer support. It is shown in the fig.17.13 below:

Fig.17.13
Analysis of a single strip

We can see that, at the middle portion of the strip, the inner edge is at a lower level than the outer edge. And towards the ends of the strip, both the edges are at the same level. So the strip is twisted. In other words, the strip is experiencing torsion. This torsional effect is more towards the ends of the strip. When we take the whole set of red strips, the strips nearer to the two long walls experience this torsion to a greater extent than the strips near the middle of the whole slab. In the same way, in the other set of blue strips which are perpendicular to the red strips, the torsional effect will be more at the ends of those blue strips which are nearer to the short walls. So when we take the slab as a whole, we can see that the torsional effect will be more pronounced at the four corners of the slab.

Thus we can see that the slab is resisting the external loads applied on it, not only by bending, but also by torsion. But we have to provide special steel reinforcement to resist this torsion. Let us now see how this steel is to be provided:

Fig.17.14 below shows a corner of the slab separated from the rest. We can see that, due to the restraining effect, cracks will be formed on the top surface of the slab. These cracks are in a direction perpendicular to the diagonal of the slab.

Fig.17.14
Top surface at corner portion

We have to provide steel reinforcement at the top of the slab so that these cracks are intercepted. This means that we have to provide steel bars in a direction parallel to the diagonal of the slab. This is shown in the plan view given below in fig.17.16(a)

Now we will see the under side of the slab. That is., the bottom surface. This is shown in the fig.17.15 below:

Fig.17.15
Bottom surface at corner portion

At the bottom surface, cracks will be formed in a direction parallel to the diagonal of the slab. So here we have to provide steel bars in a direction perpendicular to the diagonal of the slab so that these cracks are intercepted. This is shown in the fig.17.16(b) below:

Fig.17.16
Arrangement of corner bars

There is another method to provide these corner bars. In this method bars are not provided parallel or perpendicular to the diagonal. Instead, they are provided parallel to the sides of the slab. This is shown in the fig.17.17 below:

Fig.17.17
Alternative arrangement for corner bars

From the fig.17.17(a), we can see that near the top surface, there are two sets of bars. One set consists of bars parallel to l_x and the other set consists of bars parallel to l_y. These two sets are provided in two layers. The bottom layer is just below the top layer. Both these sets act together to prevent the formation of the cracks at the top surface.

Similarly (fig.17.17(b)), near the bottom surface also, there are two sets of bars. One set consists of bars parallel to l_x and the other set consists of bars parallel to l_y. These two sets are provided in two layers. The top layer is just above the bottom layer. Both these sets act together to prevent the formation of the cracks at the bottom surface. Thus there will be a total of 4 layers.

It is better to provide 'U' shaped bars (section XX in fig.17.18) so that all the bars at the corners will act as a single unit, and also chances of displacement of the bars at the time of pouring concrete is minimum.

Fig.17.18
U-shaped bars at corner

In later sections, we will see the length required for these corner bars, and also the diameter and spacing.

In the next section we will discuss the methods for obtaining the bending moments in a restrained two way slab.

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