In
the previous section we saw the shape of the concrete stress block. In this section, we will discuss the compressive force produced by this block. The fig.3.4 below shows the elevation view of the beam with the stress block attached to it.
Fig.3.4
Elevation view showing the Stress block attached to the beam
At the first glance, we might get the impression that the stresses are plotted on the X-axis, and distances along the beam section are plotted on the Y-axis. But this is not true. The graph given by the code is a Stress-Strain curve. Not a Stress-Distance curve. So we need a method to represent strain in terms of the distance. This can be made more clear using the following fig.3.5:
Fig.3.5
Relation between strain and distance
In the above fig., we can see that the strain is plotted as equal to the vertical distance, even though they are two different quantities. This is made possible by selecting a suitable scale for plotting the strain along the vertical axis. For selecting the scale, we make use of the 'manner in which the strain variation occurs in a beam section'. So we will have a brief discussion about this variation:
Fig.3.4
Elevation view showing the Stress block attached to the beam
At the first glance, we might get the impression that the stresses are plotted on the X-axis, and distances along the beam section are plotted on the Y-axis. But this is not true. The graph given by the code is a Stress-Strain curve. Not a Stress-Distance curve. So we need a method to represent strain in terms of the distance. This can be made more clear using the following fig.3.5:
Fig.3.5
Relation between strain and distance
In the above fig., we can see that the strain is plotted as equal to the vertical distance, even though they are two different quantities. This is made possible by selecting a suitable scale for plotting the strain along the vertical axis. For selecting the scale, we make use of the 'manner in which the strain variation occurs in a beam section'. So we will have a brief discussion about this variation:
A beam will consist of numerous planes which are perpendicular to the axis of the beam. We need to know about the behaviour of these planes When the bending of the beam occurs. The X-ray views in fig.3.6 below shows some of the planes inside the beam.
Fig.3.6
View of some planes inside the beam
Fig.3.6
View of some planes inside the beam
When the beam bends, our first impression would be that, the planes will bend into curved shapes as shown in fig.3.6(b). If a plane bend into a curved shape, it can no longer be called a 'plane' because, such a curved surface will have 3 dimensions. But a real plane have only 2 dimensions: length and width. It has no thickness, height, or curvature. In our case, all the plane sections which are perpendicular to the axis of bending, remain plane even after bending. This is shown in fig.3.6(c). An elevation view of the beam after bending is shown in the fig.3.7 below:
Fig.3.7
Elevation view of the planes after bending
Fig.3.7
Elevation view of the planes after bending
This property of the beam makes our calculations a lot more easier. It may be noted that this property does not change even at very high stresses, when the beam section is at the point of impending failure. The fig.3.8 below shows the end of a section coloured in red. AB is the original edge, and PQ is the new edge. PQ is a straight line because, AB will remain plane even after bending.
Fig.3.8
End portion of a segment
Fig.3.8
End portion of a segment
We can see that the change in length Δ l of any fibre is proportional to the distance of the fibre from the NA. If Δ l is proportional to the distance, the strain (strain = Δ l / l ) will also be proportional to the distance. This proportionality would not be true if PQ is a curve. Also it is a direct proportion. That is., when distance increases, strain also increases and vice versa. So we can represent it using the equation: Strain = A constant x distance. Thus while plotting the graph, we can select a scale for the Y-axis in such a way that the strain and distance will be the same.
Thus we see that the stress block of concrete can be attached to the beam section as shown in the fig.3.4 above. In fact, we do not have to plot this stress block along the beam section. We only need to understand the 'variation of stress' (ie., the stress at various points) along the section. The above discussion was to make clear how the 'stress-strain curve' given in the code appear like a 'stress-distance' curve in the fig.3.4.
Also, from the above discussion, we see that strain is directly proportional to distance. So the strain diagram will be a straight line.
We can now proceed to calculate the area enclosed by the graph. That is., the total area of blue portion and red portion in fig.3.4. The fig.3.2 and 3.3 that we saw in the previous section are for fck =20 N/mm2. We will derive the area for the general purpose where fck can take any value.
Thus we see that the stress block of concrete can be attached to the beam section as shown in the fig.3.4 above. In fact, we do not have to plot this stress block along the beam section. We only need to understand the 'variation of stress' (ie., the stress at various points) along the section. The above discussion was to make clear how the 'stress-strain curve' given in the code appear like a 'stress-distance' curve in the fig.3.4.
Also, from the above discussion, we see that strain is directly proportional to distance. So the strain diagram will be a straight line.
We can now proceed to calculate the area enclosed by the graph. That is., the total area of blue portion and red portion in fig.3.4. The fig.3.2 and 3.3 that we saw in the previous section are for fck =20 N/mm2. We will derive the area for the general purpose where fck can take any value.
We must find the areas of the rectangular portion and parabolic portion separately. We know the length of the rectangular portion. It is equal to 0.447fck . But we do not know the height. To find the height, we make use of the same 'variation of strains' in the beam that we discussed above.
We have seen that Strain = A constant x distance. So we can represent the variation of strain with distance by a straight line. Also, if we know the distance of any fibre from the NA, we can easily calculate the strain in that fibre. We will use this property in reverse to find the height of our rectangle. That is., we know the strain at the beginning of the rectangular portion. We want to find out the distance at which this strain occurs. For this, we use the fig.3.9 below:
Fig.3.9
Distance at which the strain of 0.002 occur
Fig.3.9
Distance at which the strain of 0.002 occur
In
the fig., PQ is a straight line and it represents the strain
across the section. We have two triangles above the NA.
• Base of the larger triangle = 0.0035.
• Base of the smaller triangle = 0.002.
• Altitude of the larger triangle = xu.
• Altitude of the smaller triangle = x2.
• Base of the larger triangle = 0.0035.
• Base of the smaller triangle = 0.002.
• Altitude of the larger triangle = xu.
• Altitude of the smaller triangle = x2.
So we have
0.0035/0.002 = xu/x2.
⇒ x2 = (2/3.5) xu.
x1 = xu – x2 = xu – (2/3.5) xu
⇒
Eq.3.2
x1 = (3/7)xu
Thus we get the height of the rectangular portion. We can add this information to the fig.3.9 and get the modified fig.3.10 as shown below:
Fig.3.10
Ht. of rectangular portion in relation to the strain diagram
This force acts at the centroid of the rectangular block. The centroid will be at a distance of x1/2 from the top surface of the beam.
In the above figs.3.10 and 3.11, a quantity εst is shown in the strains diagram. This is the strain in the reinforcing steel. The yellow arrow represents the stress in the steel. We will learn about them when we take up the discussion on steel. In our present discussion, these two quantities do not come in any calculations.
In the next section we will discuss about the parabolic portion.
0.0035/0.002 = xu/x2.
⇒ x2 = (2/3.5) xu.
x1 = xu – x2 = xu – (2/3.5) xu
⇒
Eq.3.2
x1 = (3/7)xu
Thus we get the height of the rectangular portion. We can add this information to the fig.3.9 and get the modified fig.3.10 as shown below:
Fig.3.10
Ht. of rectangular portion in relation to the strain diagram
Knowing
the length and height, we can find the area of the rectangular
portion, which is equal to 0.447fck(3/7)xu. This area
multiplied by the width 'b' of the section will give the volume of
the rectangular stress block. This volume is the compressive force C1
contributed by the rectangular portion. So we get
C1 = 0.447fckb(3/7)xu
This force acts at the centroid of the rectangular block. The centroid will be at a distance of x1/2 from the top surface of the beam.
Substituting for x1 from 3.2, we can write:
The
point of application of C1
from the top most compression fibre = (3/14)xu
So
now we have all the details about the rectangular portion. These
details are shown in the fig.3.11 below:
Fig.3.11
Forces and distances related to the rectangular portion
Forces and distances related to the rectangular portion
In the next section we will discuss about the parabolic portion.
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