In many cases, when performing structural or geotechnical calculations, it is important to know the possible effects of stresses that act on materials not from just one direction, as we saw above, but from multiple directions, which happens quite a lot. The effect of stresses acting from multiple directions will cause the materials to fail differently than they would under just one dimensional stresses.

These can be shown in a graph form as below, by a circle called Mohr’s Circle, after a German engineer who used it first, at the end of 1800s. It may seem complicated at first but it really is not, as we will explain step by step below.

In the figure above,

- We have a test specimen, such as a cube or rectangular or cylindrical specimen at the bottom of the figure. We apply stresses vertically, which we call σ
_{1 }and and we apply stress horizontally, such as σ_{2 }andσ_{3}. For the sake of simplicity, in our case σ_{2 = }σ_{3}, so we just show σ_{3. }

- These applied stresses are represented on horizontal axis of the graph.

- The vertical axis is for shear stress that is produced, as a result of applying the sigma 1 and sigma 3.

- We plot the points of 500 and 100 KPa on horizontal axis.

- Now we draw a circle which passes through these two points, , which has a center on horizontal axis. So in other words, the distance between sigma 1 and 3 is the diameter of the circle.

- After the circle, now we draw a sloped line from origin, which will be tangent to the circle.
- This sloped line is called the failure envelope and that is where were were trying to reach. Any point that falls below this line, is ok, the material will not fail. Any point that falls above this line is not ok, and that material will fail under applied stresses. You go above the failure envelope for that particular material. There can not be a circle like that. Before you try to get to that point, your material will definitely fail.

So in other words, for example, if you apply a σ_{1 }andσ_{3, }and if it produces a circle that falls below this sloped line, those σ_{1 }andσ_{3 }combination is ok to apply and the material will not fail. But if the combination of _{ }σ_{1 }andσ_{3 }produces a circle that goes above this sloped line, then that sigma 1 and sigma 3 combination will cause material to fail under shear stress.

- The failure is by shear stress as we have just mentioned. The value of that shear stress, is found by looking at the intersection point of our circle and tangent line. So for instance, for a sigma 1 and 3 which has values of 100 and 500 as in our graph, the failure shear stress has a value of 149.1 as shown on the graph. Note that, our circle is capable of reaching higher shear stress value of 200 KPa. But that is not the critical here. 149.1 is more critical, and that causes failure. Why? Because the value of 200 is on a stronger plane surface (inside the material), for that given sigma 1 and 3. But 149.1 is on the most critical surface, which is called the critical plane. That is the plane where the member fails as seen in the figure. The angle of that plane happens to be half of the angle at the center, which is 132 here, so our failure plane will have 66 degrees with horizontal as you can see here.

- The two angles, shown as 42 and 132 here are related to each other as: 132 = 90 + 42. The angle of the tangent line (which is 42 here) is called angle of internal friction for that material, which we will explore more under soils, lateral earth pressure.

- Pay attention, we drew another bigger circle, which is shown by hidden lines. That is just one of many more circles that are possible. So for that bigger circle, look at σ
_{1 }andσ_{3 }values on horizontal axis. They correspond to approximately 750 and 150 KPa. So it means, that is another combination of stress we can apply, that would touch the same failure envelope. As you can see, we were able to increase sigma 1 from 500 to 750, but our material didn’t fail, just because we also increased the horizontal pressure on the material. What would happen if we only increased 500 to 750 but kept 100 the same? Then that would be a bigger circle which would cross failure envelope and our material would fail. This means, when we increase pressure from one direction, we must also increase it from perpendicular direction to keep material in place. For example retaining walls work like this. They hold the soil horizontally, so that when you build a structure nearby which applies vertical force, the earth does not fall over to the side.

- Note that we could have drawn full circles here, not half, that is the reason we left out a little portion of bigger circle under the horizontal axis. The underside of horizontal axis simply means negative shear stress, which is perfectly normal shear stress, but just in opposite direction with respect to what we accepted as positive that is all. And our calculations would not be affected if we deal with full or half circle. Only that dealing with half circle and positive values are easier.

- You can not just go on and on and draw infinitely bigger circles, even if you stay below the failure envelope. There is a limit to that, which is caused by the actual yield strength of the material, under forces from multiple directions. That three dimensional stress case is called Von Mises stresses and beyond our scope. So in other words, on this graph after you go certain distance towards right, the slope of failure envelope will start to decrease, and eventually it will become a horizontal line.

This graph is so fundamental that during my undergraduate study, our professor had told us during the strength of materials class : “it is said that when you see the Mohr’s circle, you can consider that your engineering education is now halfway complete”. This has a reason. Not only because you reach to this point after taking calculus and statics classes, but it is also because the relation of stresses on a body, versus that material’s ability to support them, is a fundamentally important in all design that we make. After all, civil engineering is mostly about designing structures that can withstand loads.

Note that we drew the tangent line right from the origin here. That may not have been the case. We can sometimes draw another circle, and simply draw the tangent connecting two circles, which may not cross origin point. But that is the subject of more advanced discussion.

To further understand why all this Mohr’s circle subject is useful, let’s give another example… Say, you have a beam, on which forces act. You have axial stresses in it, you have shear stresses in it, as we saw before, under forces (statics) section. Simple question for a non engineer: At a certain location inside that beam, can you just add these two stresses (axial and shear) together and say “this is my total stress here”?

The answer is no, and that would be meaningless. What you must do is to put that point on a Mohr’s circle, for an axial stress vs shear stress graph. Then you repeat this with many different stress combinations (different points on that graph). But these may remain inside your material’s failure limit. To know better about your material, you must also use stress combinations where material fails, in order to construct the Mohr’s circles that will define the failure condition (envelope). Then to these limiting circles, you draw a best fit tangent. Now you have valuable information. On that graph, any point that goes above this tangent line, can not exist (for that particular material). The material will fail before ever reaching to that shear vs axial stress combination. All points below this tangent line is ok. The circles touching this line is where your material fails.

In the next post of this series, we will introduce the term “Stress, Strain, Elasticity Modulus”

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