Solid Mechanics: Stress 
Principal Stresses from Mohr's Circle 
A chief benefit of Mohr's circle is that the principal stresses s_{1} and s_{2} and the maximum shear stress t_{max} are obtained immediately after drawing the circle,
where,

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, s_{x},
s_{y}, and t_{xy}, 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 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 saxis as shown in the schematic below,
A set of six Mohr's Circles representing most stress state possibilities are presented on the examples page. 
Rotation Angle on Mohr's Circle 
Note that the coordinate rotation angle q_{p} is defined positive when
starting at the XY coordinates and proceeding to the
X_{p}Y_{p} coordinates. In contrast, on the
Mohr's Circle q_{p} is
defined positive starting on the principal stress line (i.e. the
saxis) and proceeding to the XY
stress line (i.e. line L_{xy}). The angle q_{p} 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 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. 
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, 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:
 
To do this we proceed as follows:
 
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