This research develops a stress-based topology optimization method (STOM) that considers various static failure criteria, including those from the maximum shear stress theory, the distortion energy theory, the ductile Coulomb–Mohr theory, the brittle Coulomb–Mohr theory, and the modified Mohr theory for ductile and brittle materials. Due to some theoretical and numerical challenges, the above static failure theories have not been implemented in topology optimization. By substituting failure formulas that are non-differentiable with respect to the stress components and design variables with differentiable approximation formulas, it is possible to utilize these failure criteria to design mechanical structures that minimize mass.