Rotated Strains using
Strain Rotation Equations

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

Plotting these equations show that every 180 degrees rotation, the strain 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 strain 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 strains and shear strains as the rotation angle changes. To determine the actual equation for Mohr's circle, the strain transformation equations can be rearranged to give,

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

Grouping like 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 ε_ave. One advantage of Mohr's circle is that the principal strains, ε₁, ε₂ , and the maximum shear strain, (γ_max/2), are easily identified on the circle without further calculations.

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

1. On the horizontal axis, plot the circle center at ε_avg = (ε_x + ε_y)/2.
2. Plot the either the point (ε_x , γ_xy/2) or (ε_y , -γ_xy/2). Note the sign change if plotting σ_y and the vertical axis, γ_xy/2, 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 strains and maximum shear strains 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 strain state is the intersection of the new line (green in the diagram) and the circle.

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

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