The generalized Hooke's law of a fiber-reinforced lamina for the principal directions is not always the most convenient form for all applications. Usually, the coordinate system used to analyze a structure is based on the shape of the structure rather than the direction of the fibers of a particular lamina.
For example, to analyze a bar or a shaft, we almost always align one axis of the coordinate system with the bar's longitudinal direction. However, the directions of the primary stresses may not line up with the chosen coordinate system. For instance, the failure plane of a brittle shaft under torsion is often at a 45° angle with the shaft. To fight this failure mode, layers with fibers running at ± 45° are usually added, resulting in a structure formed by laminae with different fiber directions. In order to "bring each layer to the same table," stress and strain transformation formulae are required.