In
the previous section we learned the methods to calculate the effective flange width bf. So in this section we can start the discussion on the actual analysis of flanged sections.
When we analysed a beam with a simple rectangular cross section, the first step that we did was to find xu, the depth of NA. If it is an under reinforced section, the calculation of xu is easy. But if it is an over reinforced section, then we will have to use the strain compatibility method. Here, in the analysis of flanged sections also, the first step is to calculate xu. But there is an important difference: There are two possible positions for the NA. (1) In the web or (2) In the flange. That is., at the ultimate state, the NA of a flanged beam may be situated in the flange, or in the web. These two possibilities are shown in the fig. below:
Fig.9.5
Two possible positions of NA
So we will first learn 'how the beam section will behave under the different possible positions of the NA'. In this discussion, we will come across a number of cases. So it is convenient to give particular names to the different cases. We will call the first case that we are about to see as 'Case 1: Df ≥ xu'
Case 1: Df ≥ xu
In this case, the NA lies within the flange. So the depth of flange is greater than or equal to the depth of the NA. That is., Df ≥ xu. This condition is shown in the fig.9.5(a) that we saw above. We know that in a beam section, the portion below the NA will be in tension, and the portion above will be in compression. So in our present case, all the concrete in the web will be in tension. Not only the concrete in the web, some portion of concrete in the flange, which comes below the NA will also be in tension. We know that the concrete which is in tension below the NA, does not have any role to play in the analysis. Thus the flanged section will, in effect, become a rectangular section of width bf and effective depth d. This is shown in the fig. below:
Fig.9.6
All the concrete below the NA is in Tension
Fig.9.7 below shows the stress and strain distribution across the section.
Fig.9.7
Stress and strain distribution
We can see that the concrete stress block is completely within the flange. The fig. below shows a 3D view of the concrete stress block in compression when the NA lies within the flange.
Fig.9.8
3D view of the stress block
However, we do not have to go into greater details of this case. Because from fig.9.6, it is clear that the section will act as a rectangular section. When we get a flanged section that belongs to this case, we can use all the equations that we derived for a rectangular section. The only change that has to be made is this: where ever 'b' appears in those equations, change it into bf.
Thus we complete the discussion on 'Case 1: Df ≥ xu'. In the next section we will see the other case.
When we analysed a beam with a simple rectangular cross section, the first step that we did was to find xu, the depth of NA. If it is an under reinforced section, the calculation of xu is easy. But if it is an over reinforced section, then we will have to use the strain compatibility method. Here, in the analysis of flanged sections also, the first step is to calculate xu. But there is an important difference: There are two possible positions for the NA. (1) In the web or (2) In the flange. That is., at the ultimate state, the NA of a flanged beam may be situated in the flange, or in the web. These two possibilities are shown in the fig. below:
Fig.9.5
Two possible positions of NA
So we will first learn 'how the beam section will behave under the different possible positions of the NA'. In this discussion, we will come across a number of cases. So it is convenient to give particular names to the different cases. We will call the first case that we are about to see as 'Case 1: Df ≥ xu'
Case 1: Df ≥ xu
In this case, the NA lies within the flange. So the depth of flange is greater than or equal to the depth of the NA. That is., Df ≥ xu. This condition is shown in the fig.9.5(a) that we saw above. We know that in a beam section, the portion below the NA will be in tension, and the portion above will be in compression. So in our present case, all the concrete in the web will be in tension. Not only the concrete in the web, some portion of concrete in the flange, which comes below the NA will also be in tension. We know that the concrete which is in tension below the NA, does not have any role to play in the analysis. Thus the flanged section will, in effect, become a rectangular section of width bf and effective depth d. This is shown in the fig. below:
Fig.9.6
All the concrete below the NA is in Tension
Fig.9.7 below shows the stress and strain distribution across the section.
Fig.9.7
Stress and strain distribution
We can see that the concrete stress block is completely within the flange. The fig. below shows a 3D view of the concrete stress block in compression when the NA lies within the flange.
Fig.9.8
3D view of the stress block
However, we do not have to go into greater details of this case. Because from fig.9.6, it is clear that the section will act as a rectangular section. When we get a flanged section that belongs to this case, we can use all the equations that we derived for a rectangular section. The only change that has to be made is this: where ever 'b' appears in those equations, change it into bf.
Thus we complete the discussion on 'Case 1: Df ≥ xu'. In the next section we will see the other case.
i need to read this example
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