A chief benefit of Mohr's circle is that the principal stresses s₁ and s₂ and the maximum shear stress t_max are obtained immediately after drawing the circle,

where,

     

The Mohr's Circle for this general stress state is shown at left above. Note that it's centered at s_avg and has a radius R, and that the two points {s_x, t_xy} and {s_y, -t_xy} lie on opposites sides of the circle. The line connecting s_x and s_y will be defined as L_xy.

The angle between the current axes (X and Y) and the principal axes is defined as q_p, and is equal to one half the angle between the line L_xy and the s-axis as shown in the schematic below,

A set of six Mohr's Circles representing most stress state possibilities are presented on the examples page.

Also, principal directions can be computed by the principal stress calculator.

Some books avoid this dichotomy between q_p on Mohr's Circle and q_p on the stress element by locating (s_x, -t_xy) instead of (s_x, t_xy) on Mohr's Circle. This will switch the polarity of q_p on the circle. Whatever method you choose, the bottom line is that an opposite sign is needed either in the interpretation or in the plotting to make Mohr's Circle physically meaningful.

Suppose that the normal and shear stresses, s_x, s_y, and t_xy, are obtained at a point O in the body, expressed with respect to the coordinates XY. We wish to find the stresses expressed in the new coordinate set X'Y', rotated an angle q from XY, as shown below:

•	Draw Mohr's circle for the given stress state (sx, sy, and txy; shown below).
•	Draw the line Lxy across the circle from (sx, txy) to (sy, -txy).
•	Rotate the line Lxy by 2*q (twice as much as the angle between XY and X'Y') and in the opposite direction of q.
•	The stresses in the new coordinates (sx', sy', and tx'y') are then read off the circle.

Stress transforms can be performed using eFunda's stress transform calculator.