Owing to its geometric freedom, additive manufacturing has become the preferred technique for creating structures designed via topology optimization. However, the layer-by-layer nature of this fabrication process often results in anisotropic strength behavior that is neglected in most topology optimization formulations. To address this issue, we propose a stress-constrained topology optimization formulation tailored to designing lightweight anisotropic structures fabricated via additive manufacturing. To account for material failure, we consider two anisotropic failure criteria—the Tsai-Wu and the Liu-Huang-Stout criteria—and use them to create tailored stress constraint definitions that are imposed locally at the centroid of each finite element. We effectively and efficiently solve the optimization problem with local stress constraints via the augmented Lagrangian. In this presentation, we provide details of the formulation and present numerical examples validated through experiments to demonstrate the efficacy of our approach in effectively handling anisotropic failure.