Axis transformation

If the material and local axes are not aligned the constitutive relationship needs to be transformed into the local axes of the element. If are the direction cosines relating the material axes, to the local axes such that

In the local axes the stress and strain relationship becomes

where is

As an example if the material and axes are rotated 90° from the local axes the transformation matrix becomes

Multiplying the constitutive relationship results in an updated constitutive matrix