The coordinate directions chosen to analyze a structure are usually based on the shape of the structure. As a result, the direct and shear stress components are associated with these directions. For example, to analyze a bar one almost always directs one of the coordinate directions along the bar's axis.
Nonetheless, stresses in directions that do not line up with the original coordinate set are also important.
For example, the failure plane of a brittle shaft under torsion is often at a 45° angle with respect to the shaft's axis. Stress transformation formulas are required to analyze these stresses.
The transformation of stresses with respect to the {x,y,z} coordinates to the stresses with respect to {x',y',z'} is performed via the equations,
where q is the rotation angle between the two coordinate sets (positive in the counterclockwise direction). This angle along with the stresses for the {x',y',z'} coordinates are shown in the figure below,
