Chapter 16 (cont..15)

For doing the analysis and design, we must first calculate the loads acting on any one slab. For slabs, we usually take the load on 1 m² area, and when we consider a strip of 1m width, this will be the load per meter length of the strip. But in our present case, we cannot take a 1m wide strip because the width ‘B’ of the tread slab is less than 1m. The area on which the load is to be calculated is shown in green colour in the fig.16.91 given below:

Fig.16.91
Plan showing area of the slab on which load is taken

Self wt. of tread slab:
The area of the green coloured portion = 1 x B = B m²
Thickness perpendicular to the plane of the paper = 't'm
So volume = Bt m³
Unit wt. of RCC = 25 KN/m³
Thus wt. of the green coloured portion = 25Bt kN
This is the self wt. on 1m length of the slab.
So we can write:

Eq.16.26
Self wt. of tread slab = 25Bt kN/m

Wt. of finishes:
This is obtained from the codes or data books. It is specified as load per 1m² of horizontal area. But we are having an area, which though is horizontal, have a lesser value (1 x B m²= Bm²) than 1m². So we have to modify the value that we obtain from the code or data book, by multiplying it with the actual area that we have. So, if we denote the value obtained from the data book as w_f,data then we can write:

Eq.16.27
Wt of finishes = w_f,data x B kN/m

Live load:
This is also obtained from the codes or data books. It is specified as load per 1m² of horizontal area. But we are having an area, which though is horizontal, have a lesser value than 1m². In the case of LL, the area that we have is still lesser than B m² because the LL will not be applied on the overlapping area. This is shown in the fig.16.92 below:

Fig.16.92
Lesser area on which LL act

Thus the area that we have is T m². So we modify the value that we obtain from the code or data book, by multiplying it with the actual area that we have. If we denote the value obtained from the data book as w_LL,data then we can write:

Eq.16.28
LL = w_LL,data x T kN/m

If we sum up the above three items, we will get the total load acting per 1m length of the slab. But we have to do one more step in this type of stairs. We have to consider another type of LL as specified by the loading code. Accordingly, each tread slab should be capable to resist a concentrated load of 1.3 kN acting at the free end as shown in the fig. below:

Fig.16.93
Concentrated load at the end of cantilever

These two Live loads can be considered separately. That is., we can assume that they do not act at the same time. So we have to consider two possible line diagrams as shown in the figs. below:

Fig.16.94
Uniformly distributed DL and LL

Fig.16.95
Uniformly distributed DL and concentrated LL

Out of the above two cases, the one which gives the maximum bending moment at the support must be considered for analysis and design.

The expression for calculating the bending moment caused by a concentrated load is different from that for a UDL.

BM at the support of a cantilever due to UDL = w l² /2

BM at the support of a cantilever due to concentrated load at free end = Wl

So we have to consider these loads separately. The procedure can be summarized as follows:

First we calculate the BM due to the self weights (DL) based on Eqs. 16.26 and 27

Multiplying this by the Load factor, we will get the factored BM. So  we can write:

Eq.16.29:

This is a final quantity, and will not change because we are not considering any other types of DL.

Now we calculate the BM due to the usual LL based on Eq.16.28. We will call it as ‘type I’ LL.

Multiplying this by the Load factor, we will get the factored BM. So  we can write:

Eq.16.30:

Next we calculate the BM due to the concentrated LL. We will call it as ‘type II’ LL.

Multiplying this by the Load factor, we will get the factored BM. So  we can write:

Eq.16.31:

Next we add the two LL moments  to the DL moment separately, to get the two possible Bending moments that can occur. So we can write:

Eq.16.32:

Sum₁= M_u,DL  + M_u,LL,I  
Sum₂= M_u,DL  + M_u,LL,II   

Out of the above two sums, the largest one is our required value.

The above steps can be represented by a simple flow chart as shown below:

Fig.16.96
Flow chart for final Bending moment

Thus we get the final factored bending moment, which can be used for the design.

In a similar way, the shear force at the support can also be calculated. The two possible alternative Live loads should be considered, and the one which gives the maximum effect should be taken for the design. For a cantilever member, the reaction at the support is simply equal to the total load on the member. So we can write:

Eq.16.33

Sum₁= (w_u,DL  + w_u,LL,I ) l

Sum₂= (w_u,DL  ) l+ w_u,LL,II 

Out of the above two sums, the largest one is our required value.

In the above equation,
w_u,DL = Load factor x (25t + w_f,Data)B Based on Eq.16.26 and 27
w_u,LL,I = Load factor x (w_LL,Data)T Based on Eq.16.28

w_u,LL,II = Load factor x 1.3 Based on Fig.16.93

So we are now in a position to design a typical tread slab of this type of stair.

For preliminary proportioning of the slab, value of ‘t’ can be assumed to be approximately equal to l /10. But it should be finalized only after the required checks.

In the next section, we will see a solved example based on the above discussion.

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