Effective span of continuous beams and slabs

To analyse a continuous beam or slab, we must know the 'effective span' of each of the spans of the beam or slab. Various codes gives different methods to calculate the effective span. First we will see the provisions in Eurocode-2. Then we will discuss the provisions in IS 456.

In the cl 5.3.2.2 of the Eurocode-2, the basic formula for finding the effective span is given as
Eq.7a.1: l_eff  = l_n  + a₁ + a₂  
where:
l_n is the clear distance between the faces of the supports,
a₁  and  a₂  are the quantities at the beginning and end of a member which are to be determined from the appropriate a_i  values in fig 5.4 of EC-2. This fig. is shown in Fig.7a.1 below:

Fig.7a.1
Effective span calculation

We will now do an example which will demonstrate the application of the above fig.7a.1. While doing example, we will also check the provisions given by IS 456.

Example 1:
The continuous beam ABCDE is part of a framed structure. It is a secondary beam, supported on some primary beams. The width of each of these primary beams is 300 mm. The clear spans and other details are shown in the fig.7a.2 below:

Fig.7a.2
Part plan

Calculations:
Let us first take span AB. Clear span l_n =3850mm
• Support 'A' is non-continuous. 
• So we have to use (a) of Fig.7a.1 above.
• Height of the beam is 350mm. So h = 350
• Width of the supporting beam is 300mm. So t =300
• Value of a_i is the lesser of h/2 and t/2. So a₁ =lesser of {350/2 ; 300/2} = 150

• Support 'B' is continuous. 
• So we have to use (b) of Fig.7a.1 above.
• Height of the beam is 350mm. So h = 350
• Width of the supporting beam is 300mm. So t =300
• Value of a_i is the lesser of h/2 and t/2. So a₂ =lesser of {350/2 ; 300/2} = 150

From Eq.7a.1, l_eff  = l_n  + a₁ + a₂  =3850 +150 +150 =4150mm

The above result can be summarized as in Table 7a.1 below:

Table 7a.1
Span AB, l_n =3850

	Support A	Support B
Type of support	Non-continuous support	Continuous support
Fig. to use	Fig.(a)	Fig.(b)
h	350	350
t	300	300
ai = lesser of {h/2; t/2}	150	150

l_eff = l_n + a₁ +a₂ =4150

Now let us calculate the effective span of this same span using the provisions given in IS456. The clause applicable to continuous beams and slabs is cl.22.2(b). A 'flow chart' like presentation of this clause is given below:

Fig.7a.3
Effective span using IS456

From the above 'flow chart', we can see that the first branching takes place depending on the width of the support. If the widths are less than 1/12 of the clear span, then the effective span can be calculated by the same method that we would use for a simply supported span. If the width of the support is not less than 1/12 of the clear span, we have to do more calculations:

The next branching takes place depending on whether the span that we are considering is an end span, or an intermediate span.

If it is an end span, further branching takes place depending on whether one end is 'fixed' or 'free'.

We will now apply the above chart to the span AB of our example.
Clear span l_n =3850mm.
Here we have to consider both the supports at the same time. So we will denote their widths as t₁ and t₂.
t₁ =300; t₂ =300
l_n/12 = 3850/12 =320.83. So t₁ < l_n/12 & t₂ < l_n/12
Thus we branch towards the left in the flow chart. This is shown in the fig.7a.4 below. The path taken through the chart is indicated by the yellow colored arrow line. The point of branching is indicated as 'A'.

Fig.7a.4
Application of the chart to span AB

Once we find that the widths are less than 1/12 of the clear span, the result is obvious: use cl.22.2(a). So the portion below the dashed magenta colored line is not applicable to our span AB.

Thus our next aim is to apply cl.22.2(a). We have used this clause previously for simply supported slabs and beams. The details about this clause can be seen here. We have to calculate the effective depth of the beam. For this we have to assume the diameter of the main bars and also the diameter of the links. If we assume 20mm and 8 mm respectively, we will get the effective cover as 48mm for 'moderate' exposure condition. (effective cover = 30 +8 +10 =48). So the effective depth d for our beam = 350 -48 =302mm. 

Now we calculate the following:
• c/c distance between the supports = 3850 +300 =4150
• clear span + effective depth = 3850 +302 =4152
Effective span = l_eff = Lesser of the above = 4150mm

Thus we calculated l_eff of span AB using two methods.
Now we will consider span BC. The calculations based on Eurocode-2 is shown in Table 7a.2 below:

Table 7a.2
Span BC, l_n =4200

	Support B	Support C
Type of support	Continuous support	Continuous support
Fig. to use	Fig.(b)	Fig.(b)
h	350	350
t	300	300
ai = lesser of {h/2; t/2}	150	150

l_eff = l_n + a₁ +a₂ =4200 +150 +150 =4500

The calculations based on IS 456 is given below:
Clear span l_n =4200mm.
l_n/12 = 4200/12 =350. So t₁ < l_n/12 & t₂ < l_n/12
So we branch to the left at point 'A', just as we did in the case of span AB. This is shown in the fig.7a.5 below:

Fig.7a.5
Application of chart to span BC

The portion of the chart which lies below the magenta colored dashed line is not applicable to span BC.

Now we calculate the following:
• c/c distance between the supports = 4200 +300 =4500
• clear span + effective depth = 4200 +302 =4502
Effective span = l_eff = Lesser of the above = 4500mm

Thus we calculated l_eff of span BC using two methods.
Now we will consider span CD. The calculations based on Eurocode-2 is shown in Table 7a.3 below:

Table 7a.3
Span CD, l_n =3950

	Support C	Support D
Type of support	Continuous support	Continuous support
Fig. to use	Fig.(b)	Fig.(b)
h	350	350
t	300	300
ai = lesser of {h/2; t/2}	150	150

l_eff = l_n + a₁ +a₂ =3950 +150 +150 =4250

The calculations based on IS 456 is given below:
Clear span l_n =3950mm.
l_n/12 = 3950/12 =329.17. So t₁ < l_n/12 & t₂ < l_n/12
So we branch to the left at point 'A', just as we did in the case of spans AB and BC. We can use the same fig.7a.5 above.

The portion of the chart which lies below the magenta colored dashed line is not applicable to span CD also.

Now we calculate the following:
• c/c distance between the supports = 3950 +300 =4250
• clear span + effective depth = 3950 +302 =4252
Effective span = l_eff = Lesser of the above = 4250mm

Thus we calculated l_eff of span CD using two methods.
Now we will consider span DE. The calculations based on Eurocode-2 is shown in Table 7a.4 below:

Table 7a.4
Span DE, l_n =3950

	Support D	Support E
Type of support	Continuous support	Non-continuous support
Fig. to use	Fig.(b)	Fig.(a)
h	350	350
t	300	300
ai = lesser of {h/2; t/2}	150	150

l_eff = l_n + a₁ +a₂ =3950 +150 +150 =4250

The calculations based on IS 456 is given below:
Clear span l_n =3950mm.
l_n/12 = 3950/12 =329.17. So t₁ < l_n/12 & t₂ < l_n/12
So we branch to the left at point 'A', just as we did in the case of spans AB and BC. We can use the same fig.7a.5 above.

The portion of the chart which lies below the magenta colored dashed line is not applicable to span DE also.

Now we calculate the following:
• c/c distance between the supports = 3950 +300 =4250
• clear span + effective depth = 3950 +302 =4252
Effective span = l_eff = Lesser of the above = 4250mm

Thus we calculated l_eff of span DE using two methods. All the results from the two methods are tabulated below:

Table 7a.5: Effective spans

Name of span	Based on Euro code	Based on IS456
AB	4150	4150
BC	4500	4500
CD	4250	4250
DE	4250	4250

Thus we have seen two methods for calculating the effective span.

We have branched off from the main discussion. The layout map given below will help us to navigate easily between the various sections. Links to some more examples are also given in the layout map.



                                                         

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