# Plot Real-World Stress Data on a Mohr Circle

Steven Dutch, Natural and Applied Sciences, University of Wisconsin - Green Bay
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In the examples below, comments comparing one case to another are not to be interpreted as general rules! The only way to interpret stresses on a Mohr Circle diagram is to construct the diagram and read it.

In this set of diagrams, we will define compression as positive and tension as negative.

 Given the stresses shown at left, what are the normal and shear stresses on the plane shown? Recall that everything in Mohr space is done with reference to the pole to the plane, so the first thing to do is construct the pole and measure the angles between the pole and the stresses as shown.

Next (below) we construct a graph and plot the Mohr Circle. Its center is at (Smax + Smin)/2 and its radius is (Smax - Smin)/2. It passes through both Smax and Smin.

Next (below) we plot the stress. Recall that all angles on the Mohr Circle are double the values in the real world, and all angles are measured in the same sense. When we plot the angles we find we are in the lower half of the diagram. We can see that in the real world the stresses would result in right-lateral slip, so on the Mohr Circle, negative shear means right-lateral shear. Remember that you have to determine the relationship between shear sense and sign from the real-world situation. Negative may not always be right-lateral!

 Finally, we simply measure off the stresses as shown.

## Tensional Stress

 Here we have the same geometry as before, but now one of the stresses is tensional. Again we find the pole to the plane and measure angles with respect to it.

Below, we construct the Mohr Circle. If compression is positive, tension is negative.

Below, we measure angles exactly as before, with angles in Mohr space double those in the real world, and all angles measured in the same sense. In the case above, where the two stresses were pushing in opposite directions along the plane, we had to think a bit to see that the shear on the plane was right lateral. Here, both stresses are pushing in the same direction along the plane, and the sense of shear is obvious. Also, since both principal stresses are acting in the same direction along the plane, the magnitude of the shear stress will be greater.

 Read off the stresses as shown. Note that since both principal stresses are acting in the same direction along the plane, the magnitude of the shear stress is greater than in the previous example. The stress normal to the plane is slightly tensional. We have a large compressional stress at a grazing angle to the plane and a weak tension at a steep angle, so it's not intuitively obvious what the result will be. It has to be read off the diagram.

## Uniaxial Stress

 If there is stress in only one direction and zero stress in the perpendicular direction, the condition is called uniaxial stress. Uniaxial stress is handled just like any other stress situation. Measure angles with respect to the pole of the plane.

Below, we plot the Mohr Circle. The minimum stress is zero so the circle passes through the origin.

Measure angles in the Mohr Circle just like any other situation.

 Finally, read off the stresses. Since we have a stress acting along the plane and no other stress counteracting it, there is a shear stress.

## Hydrostatic Stress

If the stresses are the same in all directions, then Smax = Smin. The radius of the circle is zero and the center is at Smax (or Smin). Obviously there can be no shear stresses in this case, and the normal stress on all planes is the same.

Because this is the situation found underwater, it is termed hydrostatic. In rocks, this situation is often not far from the case, because rocks are ductile. When we refer to the burial pressure of rocks, we often use the term lithostatic.