Coordinate Transformations

Plane State of Stress

A class of common engineering problems involving stresses in a thin plate or on the free surface of a structural element, such as the surfaces of thin-walled pressure vessels under external or internal pressure, the free surfaces of shafts in torsion and beams under transverse load, have one principal stress that is much smaller than the other two. By assuming that this small principal stress is zero, the three-dimensional stress state can be reduced to two dimensions. Since the remaining two principal stresses lie in a plane, these simplified 2D problems are called plane stressproblems.

Assume that the negligible principal stress is oriented in the z-direction. To reduce the 3D  stress matrix to the 2D plane stress matrix, remove all components with z subscripts to get,

where t_xy = t_yx for static equilibrium. The sign convention for positive stress components in plane stress is illustrated in the above figure on the 2D element.

Coordinate Transformations

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,