Effective span example 3

In this section we will calculate the effective spans of the continuous beam shown in the fig.7c.1 below:

Fig.7c.1
Plan and elevation of continuous beam

From the fig., we can see that, in this problem, all the supports have equal width. We will calculate the effective spans by the two different methods: The one based on Eurocode-2, and the other based on IS 456.

Assume dia. of bottom bars = 20 mm and dia. of links = 8 mm. So effective depth d = 400 -30 -8 -10 = 352 mm

First we will consider span AB. The calculations based on Eurocode-2 is shown in Table 7c.1 below:

Table 7c.1
Span AB, l_n =3500

	Support A	Support B
Type of support	Non-continuous support	Continuous support
Fig. to use	Fig.(a)	Fig.(b)
h	400	400
t	300	300
ai = lesser of {h/2; t/2}	150	150

l_eff = l_n + a₁ +a₂ =3500 +150 +150 =3800

The calculations based on IS 456 is given below:
Clear span l_n =3500mm.
l_n/12 = 3500/12 =291.67. So t₁ ≮ l_n/12 & t₂ ≮ l_n/12

Note the 'Not less than' symbol in the above step. So we cannot turn left. Let us see if we can turn right: For this, we have to calculate the lesser of {ln/12; 600} That is., the lesser of {291.67 ; 600}, which is 291.67. So we can write:
t₁ > lesser of {ln/12 ; 600} & t₂ > lesser of {ln/12 ; 600}

Thus we take the deviation to the right. The path taken by the calculations is shown in the fig.7c.1 below:

Fig.7c.1
Application of chart to span AB

At 'A', the deviation is towards the right. If at 'A', the deviation is to the right, we will be needing the whole portion of the chart. So proceeding downwards, at 'B', the deviation is towards the left because, the span AB that we are considering, is an End span. Next, at 'C', the deviation is towards the right because, span AB has one end free, and the other end continuous. Thus we reach a 'blue box' which gives us the steps in the final calculations that have to be made.

So we calculate the following:
• clear span + 0.5 x effective depth =3500 + 0.5 x352 =3676
• clear span + 0.5 x width of discontinuous support =3500 + 0.5 x300 =3650 
Effective span = l_eff = Lesser of the above = 3650mm

Thus we calculated l_eff of span AB using the two methods.

Now we will consider span BC. The calculations based on Eurocode-2 is shown in Table 7b.2 below:

Table 7c.2
Span BC, l_n =3200

	Support B	Support C
Type of support	Continuous support	Continuous support
Fig. to use	Fig.(b)	Fig.(b)
h	400	400
t	300	300
ai = lesser of {h/2; t/2}	115	100

l_eff = l_n + a₁ +a₂ =3200 +150 +150 =3500

The calculations based on IS 456 is given below:
Clear span l_n =3200mm.
l_n/12 = 3200/12 =266.67. So t₁ ≮ l_n/12 & t₂ ≮ l_n/12

Note the 'Not less than' symbol in the above step. So we cannot turn left. Let us see if we can turn right: For this, we have to calculate the lesser of {ln/12; 600} That is., the lesser of {266.67 ; 600}, which is 266.67. So we can write:
t₁ > lesser of {ln/12 ; 600} & t₂ > lesser of {ln/12 ; 600}

Thus we take the deviation to the right. The path taken by the calculations is shown in the fig.7c.2 below:

Fig.7c.2
Application of the chart to span BC

The path taken is different from that in the previous fig.7c.1. This is because the previous span AB is an end span. The present span BC is an intermediate span. So here, the deviation at 'B' is towards the right. After this point, there is no other deviations, and we reach a 'blue box'. Thus the effective span is the clear span between the supports. So we can write:
l_eff of span BC = 3200

Thus we calculated l_eff of span BC using the two methods.

Now we will consider span CD. The calculations based on Eurocode-2 is shown in Table 7b.3 below:

Table 7c.3
Span CD, l_n =3300

	Support C	Support D
Type of support	Continuous support	Continuous support
Fig. to use	Fig.(b)	Fig.(b)
h	400	400
t	300	300
ai = lesser of {h/2; t/2}	150	150

l_eff = l_n + a₁ +a₂ =3300 +150 +150 =3600

The calculations based on IS 456 is given below:
Clear span l_n =3300mm.
l_n/12 = 3300/12 =275. So t₁ ≮ l_n/12 & t₂ ≮ l_n/12


Note the 'Not less than' symbol in the above step. So we cannot turn left. Let us see if we can turn right: For this, we have to calculate the lesser of {ln/12; 600} That is., the lesser of {275 ; 600}, which is 275. So we can write:
t₁ > lesser of {ln/12 ; 600} & t₂ > lesser of {ln/12 ; 600}

The fig.7c.2 is applicable here also because CD is also an intermediate span just like AB. Thus the effective span is the clear span between the supports. So we can write:
l_eff of span CD = 3300

Thus we calculated l_eff of span CD using the two methods.

Now we will consider span DE. The calculations based on Eurocode-2 is shown in Table 7b.4 below:

Table 7b.4
Span DE, l_n =3650

	Support D	Support E
Type of support	Continuous support	Non-Continuous support
Fig. to use	Fig.(b)	Fig.(a)
h	400	400
t	300	300
ai = lesser of {h/2; t/2}	150	150

l_eff = l_n + a₁ +a₂ =3650 +150 +150 =3950

The calculations based on IS 456 is given below:
Clear span l_n =3650mm.
l_n/12 = 3650/12 =304.17. So t₁ < l_n/12 & t₂ < l_n/12

So for this span we need only the upper portion of the chart. This is shown in the fig.7c.3 below:

Fig.7c.3
Application of chart to span DE

So we calculate the following:
• c/c distance between the supports = 3650 +150 +150 =3950
• clear span + effective depth = 3650 +352 =4002
Effective span = l_eff = Lesser of the above = 3950mm

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

Table 7b.5: Effective spans

Name of span	Based on Euro code	Based on IS456
AB	3800	3650
BC	3500	3200
CD	3600	3300
DE	3950	3950

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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