Transformation of Stress

For a stress state expressed in term of a coordinate system x-y, we want to determine the stress state relative to the system 1-2, corresponding to a rotation of θ, as shown below.

The transformation can be conveniently expressed as a matrix operation as follows

where the T is a transformation matrix given by:

Expanded, the transformed stresses are as follows:

Sometimes it is convenient to express the powers and products of trigonometric functions using multi-angle identities. The following is one possible form: