Chapter 7 (cont..2)

In the previous section we completed the analysis of a continuous slab. In this section we will do the analysis of a continuous beam. A view of a four span continuous beam ABCDE is given in fig 7.17 below:

Fig.7.17
View of a continuous beam ABCDE

Total depth of the beam = 400 mm
Width of the beam = 230 mm
Width of all outer supports A,B, D & E = 230 mm
Width of inner support C = 300 mm

In the previous example of a continuous slab, we were required to calculate the loads acting per unit length of the slab (unit length of a 1000mm wide strip). But here, the loads acting per meter length of the beam is given with the problem: 

Span	DL	LL	Factored DL	Factored LL
AB	16.63	6.82	24.94	10.23
BC	16.48	6.75	24.72	10.13
CD	16.25	6.64	24.37	9.96
DE	16.63	6.82	24.94	10.23

Effective spans: But unlike in the previous example, here, only the clear spans and the widths of supports are given. We have to calculate the effective span of the various spans.

We will calculate the effective spans by the two different methods: The one based on Eurocode-2, and the other based on IS 456. For this problem, it is convenient to mention before hand that, each of the spans have their support widths less than it's ln/12. So while using the cl.22.2(b) of IS 456, we will not have to look to the portion below the magenta colored dashed line of the chart. (see fig.7a.4)

Also assume dia. of bottom bars = 16 mm, dia. of links =8mm and Cc = 30 mm
So effective depth d = 400 -30 -8 -8 = 354 mm

First we will consider span AB. The calculations based on Eurocode-2 is shown below: (fig.7a.1)

Span AB, l_n =4000

	Support A	Support B
Type of support	Non-continuous support	Continuous support
Fig. to use	Fig.(a)	Fig.(b)
h	400	400
t	230	230
ai = lesser of {h/2; t/2}	115	115

l_eff = l_n + a₁ +a₂ =4000 +115+115=4230

The calculations based on IS 456 is given below:
Clear span l_n =2850mm.
l_n/12 = 4000/12 =333.33. So t₁ < l_n/12 & t₂ < l_n/12

As mentioned above, we only need the portion above the magenta colored dashed line for all spans of the beam. This is shown in the fig.7.18 below. This fig. is applicable to all the spans.

Fig.7.18
Application of chart to span AB

Now we calculate the following:
• c/c distance between the supports = 4000 +115 +115 =4230
• clear span + effective depth = 4000 +354 =4354
Effective span = l_eff = Lesser of the above = 4230mm

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 below:

Span BC, l_n =3900

	Support B	Support C
Type of support	Continuous support	Continuous support
Fig. to use	Fig.(b)	Fig.(b)
h	400	400
t	230	300
ai = lesser of {h/2; t/2}	115	150

l_eff = l_n + a₁ +a₂ =3900 +115+150=4165

The calculations based on IS 456 is given below:
Clear span l_n =3300mm.
l_n/12 = 3900/12 =325.00. So t₁ < l_n/12 & t₂ < l_n/12

Fig.7.18 is applicable here also. Now we calculate the following:
• c/c distance between the supports = 3900 +115 +150 =4165
• clear span + effective depth = 3900 +354 =4254
Effective span = l_eff = Lesser of the above = 4165mm

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 below:

Span CD, l_n =3800

	Support C	Support D
Type of support	Continuous support	Continuous support
Fig. to use	Fig.(b)	Fig.(b)
h	400	400
t	300	230
ai = lesser of {h/2; t/2}	150	115

l_eff = l_n + a₁ +a₂ =3800 +150+115=4065

The calculations based on IS 456 is given below:
Clear span l_n =3800mm.
l_n/12 = 3800/12 =316.67. So t₁ < l_n/12 & t₂ < l_n/12

Fig.7.18 is applicable here also. Now we calculate the following:
• c/c distance between the supports = 3800 +150 +115 =4065
• clear span + effective depth = 3800 +354 =4154
Effective span = l_eff = Lesser of the above = 4065mm

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 below:

Span DE, l_n =4000

	Support D	Support E
Type of support	Continuous support	Non-Continuous support
Fig. to use	Fig.(b)	Fig.(a)
h	400	400
t	230	230
ai = lesser of {h/2; t/2}	115	115

l_eff = l_n + a₁ +a₂ =4000 +115+115=4230

The calculations based on IS 456 is given below:
Clear span l_n =4000mm.
l_n/12 = 4000/12 =333.33. So t₁ < l_n/12 & t₂ < l_n/12

Fig.7.18 is applicable here also. Now we calculate the following:
• c/c distance between the supports = 4000 +115 +115 =4230
• clear span + effective depth = 4000 +354 =4354
Effective span = l_eff = Lesser of the above = 4230mm

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

Effective spans:

Name of span	Based on Euro code	Based on IS456
AB	4230	4230
BC	4165	4165
CD	4065	4065
DE	4230	4230

Now the beam is analysed for the ten possible combinations of the loads, using Kani's method. The first combination (where LL is applied on span AB only) is denoted as case1, and it's line diagram is shown in fig.7.19 below:

Fig.7.19
Load arrangement for case 1

In this way, a total of ten combinations are analysed. The results are shown in the fig.7.20 below:

Fig.7.20
Results of analysis

Extreme values are picked from the above fig., and are tabulated below in fig.7.21:

Fig.7.21
Extreme values of BM and SF

In the above figs., '1' denotes the maximum BM at supports; '2' denotes the maximum BM at midspans, and '3' denotes the maximum SF at supports.

*2.a denotes the minimum BM at midspans. We have to take special care about minimum BM in BC and CD because, for some load combinations (cases 8 and 9), these spans are bending upwards. 

So now we have the BM and SF at the various points of the continuous beam. Just as in the previous example, let us calculate them again using the coefficients given in the code. We will do this in the next section

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