| The normal stresses (sx' and sy') and the shear stress
(tx'y') vary smoothly
with respect to the rotation angle q, in
accordance with the coordinate
transformation equations. There exist a couple of particular
angles where the stresses take on special values.
First, there exists an angle qp where the shear stress tx'y' becomes zero. That angle
is found by setting tx'y'
to zero in the above shear transformation equation and solving for
q (set equal to qp). The result is,
The angle qp defines
the principal directions where the only stresses are normal
stresses. These stresses are 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:
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