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 principal
directions where the only stresses are normal stresses. These stre called principal
stresses and are found from the original stresses (expressed in the x,y,z
directions) via,
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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