In
the previous section we saw the details of the design curves. We also saw the tables. Now
we will see in detail, how the design curves are used in the
analysis.
Before we proceed with our discussion, let us try to understand the variation of stresses across a beam section. The fig.3.17 below gives the details:
Fig.3.17
Variation of stresses across the section
We know that the stress varies only in the vertical direction. So the stress in any fibre will be different from the stress in the fibres above and below it. (However, In the regions near the top of the section, where the strain has exceeded 0.002, the stress will have a constant value of 0.447fck). There will be no variation in the horizontal direction. So all the fibres in any one horizontal layer will experience the same stress. We have already seen the details of the stress-strain curve which give variation in concrete (fig.21 of the code). We saw (fig.3.4) how it is attached to a beam section. In a similar way, the stress-strain curve of steel can also be attached to the beam section as shown in fig.3.18 below:
Fig.3.18
Stress-strain curve of steel attached to the beam section.
In the above fig., the origin (0,0) of the curve for steel is placed at the NA. From there the tensile stress and strain increase as we move downwards. But below the NA, the major portion consists of cracked concrete. This cracked concrete does not take any tension. Even if it does take any tension, the curve is not applicable because the curve is prepared for tension in steel only. But we have steel further down, near the bottom edge of the section. The curve is applicable there. We can find the strain in the steel, and from that, calculate the stress from the curve.
We have seen that the strain variation can be represented by a straight line because plane sections remain plane after bending. (We discussed this based on figs.3.6,3.7 and 3.8.) So we can easily calculate the strain in the steel. This is shown in the fig.3.19 below:
Fig.3.19
Strain values in steel
In
the above fig., PQ is the straight line which represents the strain. As PQ is straight, using similar triangles, we can calculate the strain in any fibre, if we know the distance of that fibre from the NA. The strain at the topmost fibre of steel will be less
than that in the bottom most fibre. So the stresses at these points
will also be different. But as the area of steel is small, we can
take the strain at the centroid of steel. This strain at the centroid is denoted as εst . The stress corresponding to
this strain value at the centroid is calculated from the
stress-strain curve. This stress value is denoted as fst . And this stress is assumed to be acting
uniformly on the entire area of steel. If there are more than one
layer of bars, the centroid of the 'bar group' should be considered.
Note that the strain in steel εst , that we obtain from the line PQ in the above fig. is the strain at the ultimate state. This is clear from the fact that the strain at the top most compression fibre is marked as 0.0035.
So now we know how to calculate fst, the stress in steel at the ultimate state. Let us examine the different values of fst that can possibly occur in the steel at the ultimate state. In the above fig., the horizontal dashed line from the centroid of the bar meets the stress-strain curve at the magenta line. This is the final horizontal region of the stress-strain curve. In this case, we can easily calculate fst because, there is only one value of stress in this region, and this value is the design yield strength (fy/1.15 = 0.87fy) of steel. We get this value when the strain εst in steel is equal to or greater than the yield strain: 0.00380 if it is Fe415 (Table for Fe415) and 0.00417 if it is Fe500 (Table for Fe500) . That is., when the steel has yielded. Now consider the fig.3.20 below:
Fig.3.20
Strain value falls in the curved region
The horizontal dashed line meets the design stress-strain curve at the blue curve. This means that the steel has not yielded. This happens when either the steel used is of very high strength, and/or, the steel has a larger area so that it is more strong and does not yield even when the concrete in the portion above NA has reached the ultimate state.
In the next section, we will see the explanation for the different values of stresses.
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Before we proceed with our discussion, let us try to understand the variation of stresses across a beam section. The fig.3.17 below gives the details:
Fig.3.17
Variation of stresses across the section
We know that the stress varies only in the vertical direction. So the stress in any fibre will be different from the stress in the fibres above and below it. (However, In the regions near the top of the section, where the strain has exceeded 0.002, the stress will have a constant value of 0.447fck). There will be no variation in the horizontal direction. So all the fibres in any one horizontal layer will experience the same stress. We have already seen the details of the stress-strain curve which give variation in concrete (fig.21 of the code). We saw (fig.3.4) how it is attached to a beam section. In a similar way, the stress-strain curve of steel can also be attached to the beam section as shown in fig.3.18 below:
Fig.3.18
Stress-strain curve of steel attached to the beam section.
In the above fig., the origin (0,0) of the curve for steel is placed at the NA. From there the tensile stress and strain increase as we move downwards. But below the NA, the major portion consists of cracked concrete. This cracked concrete does not take any tension. Even if it does take any tension, the curve is not applicable because the curve is prepared for tension in steel only. But we have steel further down, near the bottom edge of the section. The curve is applicable there. We can find the strain in the steel, and from that, calculate the stress from the curve.
We have seen that the strain variation can be represented by a straight line because plane sections remain plane after bending. (We discussed this based on figs.3.6,3.7 and 3.8.) So we can easily calculate the strain in the steel. This is shown in the fig.3.19 below:
Fig.3.19
Strain values in steel
Note that the strain in steel εst , that we obtain from the line PQ in the above fig. is the strain at the ultimate state. This is clear from the fact that the strain at the top most compression fibre is marked as 0.0035.
So now we know how to calculate fst, the stress in steel at the ultimate state. Let us examine the different values of fst that can possibly occur in the steel at the ultimate state. In the above fig., the horizontal dashed line from the centroid of the bar meets the stress-strain curve at the magenta line. This is the final horizontal region of the stress-strain curve. In this case, we can easily calculate fst because, there is only one value of stress in this region, and this value is the design yield strength (fy/1.15 = 0.87fy) of steel. We get this value when the strain εst in steel is equal to or greater than the yield strain: 0.00380 if it is Fe415 (Table for Fe415) and 0.00417 if it is Fe500 (Table for Fe500) . That is., when the steel has yielded. Now consider the fig.3.20 below:
Fig.3.20
Strain value falls in the curved region
The horizontal dashed line meets the design stress-strain curve at the blue curve. This means that the steel has not yielded. This happens when either the steel used is of very high strength, and/or, the steel has a larger area so that it is more strong and does not yield even when the concrete in the portion above NA has reached the ultimate state.
In the next section, we will see the explanation for the different values of stresses.
PREVIOUS
NEXT
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