Effective span example 5

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

Fig.7e.1
Plan and elevation of continuous slab

From the fig., we can see that, in this problem, the supports have different widths. The outer two walls on either sides have a width of 200mm and the middle wall have a width of 230mm.

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. (Fig.7a.4)

Also assume dia. of bottom bars = 10 mm and Cc = 30 mm
So effective depth d = 150 -30 -5 = 115 mm

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

Table 7e.1
Span AB, l_n =2850

	Support A	Support B
Type of support	Non-continuous support	Continuous support
Fig. to use	Fig.(a)	Fig.(b)
h	150	150
t	200	200
ai = lesser of {h/2; t/2}	75	75

l_eff = l_n + a₁ +a₂ =2850 +75 +75 =3000

The calculations based on IS 456 is given below:
Clear span l_n =2850mm.
l_n/12 = 2850/12 =237.50. 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.7e.2 below. This fig. is applicable to all the spans.

Fig.7e.2
Application of chart to span AB

Now we calculate the following:
• c/c distance between the supports = 2850 +100 +100 =3050
• clear span + effective depth = 2850 +115 =2965
Effective span = l_eff = Lesser of the above = 2965mm

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 7e.2 below:

Table 7e.2
Span BC, l_n =3300

	Support B	Support C
Type of support	Continuous support	Continuous support
Fig. to use	Fig.(b)	Fig.(b)
h	150	150
t	200	230
ai = lesser of {h/2; t/2}	75	75

l_eff = l_n + a₁ +a₂ =3300 +75 +75 =3450

The calculations based on IS 456 is given below:
Clear span l_n =3300mm.
l_n/12 = 3300/12 =275.00. So t₁ < l_n/12 & t₂ < l_n/12

Fig.7b.2 is applicable here also. Now we calculate the following:
• c/c distance between the supports = 3300 +100 +115 =3515
• clear span + effective depth = 3300 +115 =3415
Effective span = l_eff = Lesser of the above = 3415mm

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 7e.3 below:

Table 7e.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	150	150
t	230	200
ai = lesser of {h/2; t/2}	75	75

l_eff = l_n + a₁ +a₂ =3300 +75 +75 =3450

The calculations based on IS 456 is given below:
Clear span l_n =3300mm.
l_n/12 = 3300/12 =275.00. So t₁ < l_n/12 & t₂ < l_n/12

Fig.7e.2 is applicable here also. Now we calculate the following:
• c/c distance between the supports = 3300 +115 +100 =3515
• clear span + effective depth = 3300 +115 =3415
Effective span = l_eff = Lesser of the above = 3415mm

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 7e.4 below:

Table 7e.4
Span DE, l_n =2850

	Support D	Support E
Type of support	Continuous support	Non-Continuous support
Fig. to use	Fig.(b)	Fig.(a)
h	150	150
t	200	200
ai = lesser of {h/2; t/2}	75	75

l_eff = l_n + a₁ +a₂ =2850 +75 +75 =3000

The calculations based on IS 456 is given below:
Clear span l_n =2850mm.
l_n/12 = 2850/12 =237.50. So t₁ < l_n/12 & t₂ < l_n/12

Fig.7e.2 is applicable here also. Now we calculate the following:
• c/c distance between the supports = 2850 +100 +100 =3050
• clear span + effective depth = 2850 +115 =2965
Effective span = l_eff = Lesser of the above = 2965mm

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

Table 7e.5: Effective spans

Name of span	Based on Euro code	Based on IS456
AB	3000	2965
BC	3450	3415
CD	3450	3415
DE	3000	2965

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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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.


                                                         

            Copyright©2015 limitstatelessons.blogspot.in - All Rights Reserved

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.



                                                         

            Copyright ©2015 limitstatelessons.blogspot.com - All Rights Reserved