Understanding Stress Transformation and Mohr’s Circle

In this video we’re going to take a look at stress transformation and Mohr’s circle. Let’s start by reminding ourselves about the stress element. The stress element is a useful way of describing the stresses acting at a single point within a body. In this video we will consider a 2D stress element, corresponding to a state of plane stress. The normal stresses — and shear stresses — are shown on the four faces of the element.

 

Let’s look at a simple example of a beam under axial load. Our stress element is aligned with the direction of the applied load, and so the stress state will be very simple. We will only have a normal stress of Sigma-X. Sigma-Y and the shear stresses will be zero. But we might want to rotate our stress element to get the stress state for a specific angle. If there was a weld in our structure, for example, we might want to determine the stresses perpendicular to the weld.

 

Depending on how we choose to orient our stress element, we will get different values for the normal and shear stress components. We can calculate what the normal and shear stresses will be as we rotate our stress element by using the stress transformation equations. The inputs to these equations are the normal and shear stresses at the starting orientation of our stress element, and Theta the angle through which we are rotating the element. Theta is positive for counterclockwise rotation. Let’s look at an example of stress transformation for   the stress state described by the stress element shown here. As we rotate the stress element the normal and shear stress components will vary.

 

Once we have rotated the stress element by 180 degrees, we will return to the configuration we started with. It is important to remember that the actual stress state within the large body is not changing when we rotate our stress element – we are just rotating the axes of the coordinate system we are using to visualize the stresses at a single location. We can observe that for certain angles, the normal stress will reach maximum and minimum values, and that these maximum and minimum values are separated by an angle of 90 degrees.

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This means that when we rotate our stress element such that we have the maximum normal stress on the X face of our element, we always have the minimum normal stress on the Y face. Another interesting observation is that when the normal stresses are at their maximum or minimum values, the shear stresses are zero. These faces on which the shear stresses are zero are called the principal planes, and the corresponding maximum or minimum normal stresses acting on these planes are known as the principal stresses. Principal stresses are very important, so let’s recap.

 

They are the maximum and minimum normal stresses acting on our stress element. They always occur when the stress element is rotated such that the shear stresses are zero, as shown here. We denote the maximum and minimum principal stresses using the symbols Sigma-1 and Sigma-2 respectively. The rotation angle which gives us the principal stresses is denoted using the symbol Theta-P.

 

Because it is the maximum normal stress at the location of our stress element, being able to calculate Sigma-1 can be important for predicting failure. Mohr’s circle is a graphical method for easily determining the normal and shear stresses for different orientations of our stress element, without having to use the stress transformation equations. Let’s see how Mohr’s circle is constructed. First let’s draw the horizontal and vertical axes. Normal stress is on the horizontal axis and shear stress is on the vertical axis. We will plot positive shear stresses in the downwards direction.

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N ext we plot the stress conditions corresponding to the X face of our stress element, by plotting a point with coordinates (Sigma-X, Tau-XY). Then we plot a point corresponding to the stress conditions on the Y face by plotting a point with coordinates (Sigma-Y, Tau-XY). The sign convention we use for Mohr’s circle is that shear stresses are positive if they tend to rotate the stress element counter-clockwise, and are negative if they tend to rotate it clockwise. This is why Tau-XY is negative on the Y face of our element when we draw Mohr’s circle. Normal stresses are positive if they are tensile and negative if they are compressive. The line between these two points defines the diameter of our Mohr’s circle, which we can now draw.

 

Each point on the circle represents the normal and shear stresses for a certain orientation of our stress element. We can use Mohr’s circle to work out some interest information. We can see visually that the maximum shear stress is equal to the radius of the circle, which we can easily determine either approximately by measuring the distance on paper, or exactly by using the equation for the radius of a circle. We can also determine the principal stresses by looking at where the circle crosses the horizontal axis, as the shear stress is zero at these locations. The principal stresses can be calculated by taking the X coordinate of the center of the circle, and adding or subtracting the circle radius.

 

We can also use trigonometry to calculate angles on Mohr’s circle. For example we can calculate the angle Theta-P between our original stress element and the principal planes. An important thing to note is that angles in Mohr’s circle are doubled compared to the angle we rotate our stress element by. This is apparent by observing that in Mohr’s circle there is a 180 degree angle between the minimum and maximum principal stresses, whereas on our stress element the angle is 90 degrees.

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This is why we use the 2*Theta notation on Mohr’s circle. Theta is the angle we rotate our stress element by, and 2*Theta is the corresponding angle on Mohr’s circle. So far we have only looked at a two dimensional case, but we can extend what we have learned to three dimensions. A three dimensional stress element looks like this.

 

In three dimensions we have three principal stresses, which by convention are numbered from largest to smallest. Mohr’s circle in three dimensions is made up of three different circles, drawn as shown here. All possible combinations of normal and shear stresses for a 3D stress element lie on the boundary of or within this shaded area. I hope this video has helped you better understand stress transformation and Mohr’s circle. Please let me know what you think in the comments, and don’t forget to subscribe!

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