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Principal Directions, Principal Strain
The normal strains (ex' and ey') and the shear strain (ex'y') vary smoothly with respect to the rotation angle q, in accordance with the transformation equations given above. There exist a couple of particular angles where the strains take on special values.

First, there exists an angle qp where the shear strain ex'y' vanishes. That angle is given by,

This angle defines the principal directions. The associated principal strains are given by,

The transformation to the principal directions with their principal strains can be illustrated as:

Maximum Shear Strain Direction
Another important angle, qs, is where the maximum shear strain occurs and is given by,

The maximum shear strain is found to be one-half the difference between the two principal strains,

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