Rotated Stresses using
Stress Rotation Equations

Previously, the stress transformation equations were developed to calculate the stress state at different orientations. These equations were

Plotting these equations show that every 180 degrees rotation, the stress state repeats. In 1882, Otto Mohr noticed that these relationships could be graphically represented with a circle. This was a tremendous help in the days of slide rulers when using complex equations, like the stress transformation equations, was time consuming.

Mohr's circle is not actually a new derived formula, but just a new way to visualize the relationships between normal stresses and shear stresses as the rotation angle changes. To determine the actual equation for Mohr's circle, the stress transformation equations can be rearranged to give,

Each side of these equations can be squared and then added together to give

Grouping similar terms and canceling other terms gives

Using the trigonometry identity, cos²2θ + sin²2θ = 1, gives

This is basically an equation of a circle. The circle equation can be better visualized if it is simplified to

where

This circle equation is plotted at the left using r and σ_ave. One advantage of Mohr's circle is that the principal stresses, σ₁, σ₂ , and the maximum shear stress, τ_max, are easily identified on the circle without further calculations.

In addition to identifying principal stress and maximum shear stress, Mohr's circle can be used to graphically rotate the stress state. This involves a number of steps.

1. On the horizontal axis, plot the circle center at σ_avg = (σ_x + σ_y)/2.
2. Plot either the point (σ_x , τ_xy) or (σ_y , -τ_xy). Note the sign change if plotting σ_y and the vertical axis, τ_xy, is positive downward.
3. Draw a line from the center to the point plotted in step two (blue line in the diagram). This line should extend from one side of the circle to the other. Radius, r, can now be measured from the graph.
4. The circle itself can be drawn since the center and one point on the circle is known (compass works well for this).
5. The principal stresses and maximum shear stress can be identified on the graph.
6. The line drawn in step 3 can be rotated twice the rotation angle, 2θ, in the counter-clockwise direction. It is important that the angle is twice the desired rotated angle.
7. The new stress state is the intersection of the new line (green in the diagram) and the circle.

The angle, 2θ_p, for the principal stresses is simply the half the angle from the blue line to the horizontal axis.

Remember, Mohr's circle is just another way to visualize the stress state. It does not give additional information. Both the stress transformation equations and Mohr's circle will give the exactly same values.