Maximum shear stress theory
The maximum shear stress theory is also known as the Tresca yield criterion. This failure criterion can be used to determine the failure stress of ductile materials. It states that the maximum allowable shear stress in an element which is subjected to uniaxial stress, is obtained by: Maximum shear stress
Plane stress
The maximum shear stress in a material element which is subjected to plane stress, also known as biaxial stress, depends on the direction of the principal stresses. Suppose the principal stresses have the same sign, the absolute maximum shear stress is: Absolute maximum shear stress, same sign If both principal stresses have an opposite sign, the absolute maximum shear stress is: Absolute maximum shear stress, opposite sign

When the yield stress of the material and the sign of the principal stresses is determined, the maximum allowable principal stresses according to the maximum shear stress theory can be determined by the first equation on this page. Keep in mind that this failure theory is only usable for ductile materials.