Principal Stress for the Case of Plane Stress
engineering fundamentals Principal Stress (2D)
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Principal Directions, Principal Stress
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:

Maximum Shear Stress Direction
Another important angle, qs, is where the maximum shear stress occurs. This is found by finding the maximum of the shear stress transformation equation, and solving for q. 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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