Chapter 16.10 - Loads on Transverse stairs

In the previous section we saw an example which gives the basic details about transverse stairs. In this section we will discuss more details. The fig.16.51 below shows the sectional elevation of the stair in the example that we saw.

Fig.16.51
Sectional elevation of stair

In the above figs., the waist slab is resting on the top of the stringer beams. Another arrangement possible is that in which the beams are projecting above the stair. The view and sectional elevation of this arrangement are shown in the figs.16.52 and 16.53, below:

Fig.16.52
View of stairs with projecting stringer beams.

Fig.16.53
Sectional elevation of stairs with projecting stringer beams

In the above fig. we cannot expect flange action because the flange will be in tension. So we must design it as rectangular beams.

Yet another arrangement that is possible, is to use masonry walls instead of beams. This can be used if adequate foundation can be provided for the walls, and also if the height of walls can be kept within permissible limits.

In all the above methods, the waist slab will be spanning in a direction, perpendicular to the direction of travel, and so it is a transverse stair.

Loads on a Transverse Stair

Let us now analyse the loads acting on this transverse stair.
In the case of a longitudinal stair, we determined the load on a horizontal projection having an area of 1 m², as shown earlier in fig.16.16. But for a transverse stair, we use the loads on a 1 m² area, on the inclined slab itself. This is shown in the fig.16.54 below:

Fig.16.54
1 m² area on inclined surface

The method of using such a load is used for the 'analysis and design of the stair' will be explained later. At present let us determine the magnitude of this load. It has four items as given below:
1) Self wt. of waist slab: This can be calculated based on the sectional view in fig.16.55 given below:

Fig.16.55
Self wt. of waist slab

• The area of the portion coloured in green in the above fig. is 1 x t m². (‘t’ is in m). Note that the '1m' here is the same 1m marked perpendicular to the red strip in the previous fig.16.54
• The length of the block (perpendicular to the plane of the paper) is 1m. This '1m' is the same 1m marked parallel to the red strip in the fig.16.54

So volume = 1 x t x 1 = t m³.
Thus the wt. of this block having an area of 1m² = t x 25 = 25t kN. where 25 is the unit wt of concrete in kN/m³.
Thus we can write:
Eq.16.20: Self wt. of waist slab = 25t kN/m²

2) Self wt. of steps:
• Area of cross section of 1 step = [0.5 x R(Rise) x T(Tread)] (∵area of the triangle = 0.5 x Base x Altitude)
• Length of the triangular prism perpendicular to the plane of the paper = 1m. This '1m' is the same 1m marked parallel to the red strip in the fig.16.54
• So volume = 0.5 x R x T x 1 =0.5RT
• Thus wt. of one step with a length of 1m perpendicular to the plane of the paper = 0.5RTγs  Where γs   is the unit wt. of the material of the step. (For brick masonry, γs = 20 kN/m3)

Now we have to calculate the no. of such steps with in the length of 1m shown in fig.16.55. For that, we will use the fig.16.56 given below:

Fig.16.56
No. of steps in 1m along the slope

From the fig., it can be easily seen that the no. of steps within the two red dotted lines will be equal to 1m divided by 'B' where B = √(R² + T²) (∵ B is the hypotenuse of the triangle with base T and altitude R)

   
So we can write
Eq.16.21
The total wt. of all the steps within the 1m² area =

It may be noted that 1/B will be a 'whole number + a fraction' in most cases. The fractional part indicates that 'parts of steps' will also be coming in the 1m² area.

The next item is the ‘weight of finishes’. We will see the details about it in the next section.

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Chapter 16.2 - Loads coming on the stairs

In the previous section we saw the support details and the calculation of 't'. Now we will discuss about the loads acting on the stair slabs. We know that the slabs are designed as beams of width 1m. We consider a 1 m wide strip from support to support. The same procedure is adopted for the stair slabs also as shown in the fig. below:
Fig.16.14
1m wide strip on stair slab

The width of the beam is 1m and total depth is equal to the thickness of the slab 't'. To calculate the bending moments and shear forces acting on the beam, we need to know the load acting on the beam per unit length. So another 1m is taken along the length of the strip. Thus we will get a 1 x1m square area as shown in the fig. below:
Fig.16.15
1 x1m square area on the strip

So if we have the load acting per ‘meter square area’ on the stair, we have the load acting per ‘meter length’ of the beam. This is a square area which is inclined at an angle to the horizontal. For convenience, we take the load on a 1 x1m square on the horizontal surface as shown below:

Fig.16.16
1 x1m square area on horizontal plane

We can see that the width remains the same. But the '1m length on the sloping slab' does not project exactly into '1m on the horizontal surface'. The horizontal projection will be less than 1m. This can be further illustrated using the elevation view as shown below:
Fig.16.17
Horizontal projection is always less than 1m

Let us now see how this shortage of length can be remedied while calculating the 'self wt of the slab'. From the fig.16.17 above it is clear that if we want the 'self wt of the slab' on 1m sq of horizontal area, we must take more than 1m along the slope. We must calculate this new increased length that we have to measure along the slope. For that, we will use the fig.16.18 given below:
Fig.16.18
Increased length for getting exact 1m projection

The blue colored triangle at the bottom portion, represent the step with Rise R and Tread T. We rotate this triangle through 180 degrees and then bring it under the top surface of the slab (the triangle shown within the red region). The size and shape of the triangle is not altered. Now, from similar triangles we can write

From the triangle pqr we get

From this we get
Eq.16.2

• The square root of (R² + T²) will be always greater than T. So the value of pq will always be greater than 1.
• The weight of the concrete block colored in red will give us the self wt of the waist slab on a horizontal area of 1m².
• This block has a length of pq and a thickness of 't'. It's width is perpendicular to the plane of the paper, and is equal to 1m. (∵  we are considering 1m wide strip for slab design).
• So the volume of the block is pq x t x 1. Multiplying this by the wt per cu.m (weight density) of reinforced concrete (25kN/m³ ), we will get the wt. of the block. So substituting for pq and multiplying by 25 we get
Eq.16.3
Self wt of waist slab on a horizontal 1m²  area (in kN/m²) =

Where R, T and t are in m

Self weight of steps
Now we will determine the self wt. of steps on a horizontal area of 1m² . For this we will use the fig. below

Fig.16.19
Number of steps in a horizontal distance of 1m

In the calculations given below, all lengths are in m
• The number of steps in 1m length = 1⁄T   Eq.16.4
• Area of cross section of one step = 1⁄2 x R x T = RT⁄2 m².
• Length of one step, perpendicular to the plane of paper = 1m (since we are considering 1m strip for the design of slab). So volume of one step = RT⁄2 x 1 = RT⁄2 m³
• Multiplying this by the number of steps in 16.4, we will get the total volume of steps in 1m² of horizontal area.. Thus the volume = RT⁄2 x 1⁄T   = R⁄2 m³
• Multiplying this by the weight density of the material (denoted by γ_s ) of the steps, we will get the total weight of all the steps in 1m² of horizontal area as given below:
Eq.16.5: Self wt of steps per 1m² of horizontal area (in kN/m²)  =  (R γ_s) ⁄2   = 0.5Rγ_s. The value of γ_s for brick masonry is 20 kN/m³

In 16.4, we may get a whole number value if 1m is an exact multiple of T. For example if T = 0.25m, the number of steps will be 1/0.25 = 4. If T = 0.27 or 0.30 etc., we will not get a whole number. Instead we will get decimals. The decimal part indicates that 'a portion of one full step' is also coming in the 1m² area. So we should not discard the decimal part. However we will be using 16.5 directly to calculate the self wt. In it, T, or the number of steps is not involved. So we need not be concerned about fractions of steps. In the next section, we will discuss about the other loads coming on the stairs.

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