Tsai-Hill failure criterion

The Tsai-Hill criterion is based on an energy theory with interactions between stresses. Ply rupture appears when:[1][2]

${\displaystyle {\begin{aligned}\left({\cfrac {\sigma _{11}}{X_{11}}}\right)^{2}-\left({\cfrac {\sigma _{11}\sigma _{22}}{X_{11}^{2}}}\right)+\left({\cfrac {\sigma _{22}}{X_{22}}}\right)^{2}+\left({\cfrac {\tau _{12}}{S_{12}}}\right)^{2}\geq 1\end{aligned}}}$

Where:

${\displaystyle {\begin{aligned}X_{11}\end{aligned}}}$ is the allowable strength of the ply in the longitudinal direction (0° direction)

${\displaystyle {\begin{aligned}X_{22}\end{aligned}}}$ is the allowable strength of the ply in the transversal direction (90° direction)

${\displaystyle {\begin{aligned}S_{12}\end{aligned}}}$ is the allowable in-plane shear strength of the ply between the longitudinal and the transversal directions${\displaystyle }$

The Tsai hill criterion is interactive, i.e. the stresses in different directions are not decoupled and do affect the failure simultaneously.[2] Furthermore, it is a failure mode independent criterion, as it does not predict the way in which the material will fail, as opposed to mode-dependent criteria such as the Hashin criterion, or the Puck failure criterion. This can be important as some types of failure can be more critical than others.