The strength of materials and material systems under multiaxial loads can be predicted based on different failure criteria. Failure criteria relate the material strength allowables, defined for uniaxial tension-compression and shear, to the general stress-strain state due to multiaxial loads. Typically failure criteria are presented as mathematical expressions called failure criterion functions , which are functions of the stresses (or strains) and the material strength.

The values of failure criterion functions change with load similarly as the inverse reserve factor (values below one are non-critical and one indicates failure). However, the values are generally not equal except at the failure point. The reserve factor describes the distance from the point of the applied load to the failure point. Typically a numeric line search method is used for determining the value of based on the selected failure criterion, stresses and strains due to the applied load, and material strength allowables.