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Principal Stresses from Mohr's Circle

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


Principal Directions from Mohr's Circle
Mohr's Circle can be used to find the directions of the principal axes. To show this, first suppose that the normal and shear stresses, sx, sy, and txy, are obtained at a given point O in the body. They are expressed relative to the coordinates XY, as shown in the stress element at right below.

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

The angle between the current axes (X and Y) and the principal axes is defined as qp, and is equal to one half the angle between the line Lxy 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.

Rotation Angle on Mohr's Circle
Note that the coordinate rotation angle qp is defined positive when starting at the XY coordinates and proceeding to the XpYp coordinates. In contrast, on the Mohr's Circle qp is defined positive starting on the principal stress line (i.e. the s-axis) and proceeding to the XY stress line (i.e. line Lxy). The angle qp has the opposite sense between the two figures, because on one it starts on the XY coordinates, and on the other it starts on the principal coordinates.

Some books avoid this dichotomy between qp on Mohr's Circle and qp on the stress element by locating (sx, -txy) instead of (sx, txy) on Mohr's Circle. This will switch the polarity of qp 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.

Stress Transform by Mohr's Circle
Mohr's Circle can be used to transform stresses from one coordinate set to another, similar that that described on the plane stress page.

Suppose that the normal and shear stresses, sx, sy, and txy, 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:

To do this we proceed as follows:

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.