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: