Chapter 3 - Analysis of Beam sections by Limit State Method

In the previous section we saw the analysis of sections by the working stress method. There we saw that after the end of phase 2, the stresses are higher and 'non-linear'. We discussed it based on fig.2.13 Here we are going to discuss the analysis at ultimate state. That is., the state when the failure of a section is impending. To know the behaviour of the section at the point of impending failure, we must load the section up to that point, and then analyse the results. As the materials reach the state of impending failure, the stresses will be very high. So the stress distribution will be very much non-linear. In other words, the stress distribution graphs will be curves.

We require such a graph (graph which gives the actual non linear variation of stress with strain) for analysis and design purposes. The purpose of the graph is: To find the area enclosed by the graph and then multiply the area with the width of the section (to get the volume of the stress block), and this volume will give the magnitude of the force. To obtain the graph, compression tests are done on concrete specimens at the lab, and in these tests, the concrete is compressed to the ultimate state. Based on these test readings, we can plot the graph which gives the stress at various strains. Such a plot is given in fig.21 of the code. In this fig., the top most curve shows the results (f_ck) obtained from the tests conducted in the lab. But the strength of concrete in an actual structure may not be as good as that obtained in the lab. So only 67 percent of f_ck is taken. This is shown in the middle curve. Then we have to apply the partial safety factor for the material, which is concrete here. As seen before in chapter 1, it's value is 1.5. So we get 0.67f_ck/1.5 = 0.447f_ck. Each of the y coordinate in the topmost curve is multiplied by 0.447 to get the corresponding y coordinate of the bottom most curve.

The 'effect' of reducing the value of f_ck to 0.447f_ck can be explained as follows: When we analyse a beam section by the limit state method, we are analysing it at the ultimate state. That is., we are trying to find the magnitude of the forces in the section when it is at the state of impending failure. According to the code, that state of impending failure is reached, if the stress in concrete is 0.447f_ck. We cannot take f_ck instead of 0.447f_ck. So the stress that concrete can be subjected to, is reduced from f_ck to 0.447f_ck. In effect, the contribution that can be expected from concrete is reduced. 

The reduction from f_ck to 0.447f_ck has effect on design also. When we are designing a new section, we must expect the concrete to take a stress of only 0.447f_ck. 

The maximum stress value of 0.447f_ck corresponds to the maximum strain of 0.0035. For the fibres with lower strains, the stress will also be lower. The bottom most curve in fig.21 is the curve in which the factor 0.447 has been applied to all it's points. So from that curve, we get the required stress for the fibres with lower strains also. Thus, it is the bottom most curve that we must use in analysis and design.

So, in the Limit state method, we are considering sections at their point of impending failure, and at this point, the stress distribution in concrete is given by the graph in fig.21 of the code. And for analysis and design purposes, we use the lower most graph in that fig.

Let us now examine the various features of this curve. We can see that the curve has two portions. A parabolic portion, and a straight line portion. The parabolic portion starts from the origin (0,0) where stress = 0, and so strain is also equal to 0. From there, the stress begins to increase with strain. This increase is parabolic. The increase of stress continues upto the point where strain reaches 0.002. The maximum value of stress at this point is 0.447f_ck. When the strain exceeds 0.002, the stress remains constant at 0.447f_ck. The strain may continue to increase, but the stress will remain constant. However, the strain cannot increase indefenitely. When it reaches 0.0035, the code assumes that the concrete has reached the point of impending failure. So after this point, the concrete will fail by crushing. In a beam section, the topmost concrete fibres will be the ones that reach this strain first. The lower fibres will be having lower strains. But as the topmost fibres have reached the ultimate state, the whole section should be considered to have reached the ultimate state. This is because, the section will serve it's purpose only if strains in all the fibres in it are below the specified limits.

In a beam section which is subjected to a sagging moment, the Neutral axis NA of the beam, corresponds to the origin point (0,0) in fig.21. This is because, at the NA, both stress and strains are equal to zero. If this sagging moment is of such a magnitude that, the section is at the ultimate state, then, we can say that the stress variation in the section is given by the curve in fig.21 of the code. So, from the NA, as we move up, the stress will vary parabolically upto the level where strain equals 0.002, and then it will remain constant at 0.447f_ck. At the top most level of the beam section, the strain is equal to 0.0035. 

So now we know that the bottom most curve is our required graph. We must plot it from the NA towards the upper part of the beam section. But we already know two important points on the graph. Origin point (0,0) at the NA, and the final point (0.447f_ck, 0.0035) at the top edge. So there is no need to plot it. We can take it and directly attach it to the beam section. We attach it in such a way that the origin (0,0) is at the NA, and the strain 0.0035 is at the top most level. Then we give the graph a thickness 'b' which is equal to the width of the beam, and thus we get the 'Stress block'. This is shown in the animation below:

So the final position of the concrete stress block in the beam will be as shown in the fig.3.1 below:

Fig.3.1
Concrete stress block in compression for a beam


 














In the above stress block, the only unknowns are the total depth of the stress block (depth of NA), and the depth of the rectangular portion. In our later discussions we will see the methods to calculate these depths. 

There is also another point that we must note: We have attached the curve along the depth of the beam. The depth is a 'distance'. So it appears as if we plot the curve with 'stress' along one axis and 'distance' along the other axis. But the curve given by the code is a stress-strain curve. Not a stress-distance curve. Later in our discussions, we will see how they are related.

The graph of the stress block has a definite equation. We have seen it in chapter 1, and is given below again:

Eq.3.1

Where f_c is the stress and ε is the strain

We can make a plot of the design curve on our own, for any particular value of f_ck using Eq.3.1. For this we use the following steps:

• Choose a value for the constant f_ck, say 20 N/mm² for M20 grade concrete.

• For plotting the parabolic portion, choose values less than 0.002 for the strain ε (in convenient steps of say 0.00025) and for each of these values calculate the stress f_c, using the first part of Eq.3.1. The table 3.1 below shows the values of stress f_c when strain ε is less than 0.002

Table 3.1:

ε	fc
0	0.000
0.00025	2.095
0.00050	3.911
0.00075	5.448
0.00100	6.705
0.00125	7.683
0.00150	8.381
0.00175	8.800

• For plotting the straight line portion, we don’t have to choose values for strain and calculate the corresponding stress. This is because it is an equation of a horizontal straight line between strain (x) values ε = 0.002 and 0.0035. The y values of all the points on that straight line will be equal to 0.447 x 20 =8.94 . So we can easily draw this second part.

• plot ε along the X axis and f_c along the Y axis

Fig.3.2 below shows the plot, where f_ck = 20 N/mm² . In the plot, the parabolic portion is given blue colour and the straight line portion is given a red colour.

Fig.3.2
Design curve for M20 grade concrete

The coordinate points in the above graph are the same which are given in table 3.1. The area enclosed by the graph can be shown by shading the area between it and the X-axis. This is shown in the fig.3.3 below:

Fig.3.3
Area enclosed by the graph

In the next section we will discuss about the calculation of compressive force at a beam section by using this stress block.



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