The normal stresses ( x' and y') and the shear stress ( x'y') vary smoothly with respect to the rotation an accordance with the coordinate transformation equations.

**Principal planes
and stresses**

Principal stresses and planes

**Principal Directions, Principal Stress**

The normal stresses ( x' and y') and the
shear stress ( x'y') vary smoothly with respect to the rotation an accordance
with the coordinate transformation equations. There exist a couple of
particular angles where the take on special values.

First, there exists an angle p where the
shear stress x'y' becomes zero. That angle is found by setting x'y' t the above
shear transformation equation and solving for (set equal to p). The result is,

The angle * _{p}* defines the

The transformation to the principal
directions can be illustrated as:

**Maximum Shear
Stress Direction**

Another important angle, * _{s}*, is where the maximum
shear stress occurs. This is found by finding the maximu shear stress
transformation equation, and solving for
. The result is,

The maximum shear stress is
equal to one-half the difference between the two principal stresses,

The transformation to the
maximum shear stress direction can be illustrated as:

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