Yield, ultimate, and plastic collapse
Beams fail through several distinct mechanisms that require different design checks. Yield failure occurs when bending stress exceeds the material's yield strength, allowing plastic hinges to form and permanent deformation to accumulate. Ultimate failure arrives when the section moment capacity is exhausted under load. For ductile materials like steel, yielding gives warning; for brittle materials like concrete, ultimate and fracture occur nearly simultaneously without warning.
Buckling instability in compression
Slender beams loaded in compression can buckle elastically before reaching material yield, with critical buckling load depending on length, stiffness, and boundary conditions. Shorter, wider beams yield in compression; longer, slender beams buckle elastically. The Euler formula predicts elastic buckling load based on moment of inertia and length; material databases provide formulas for inelastic (plastic) buckling between the elastic and material yield limits.
Fatigue and shear-related failure
Repeated loading over many cycles (fatigue) can cause cracks at lower stress than a single static load. Shear failure in short beams (punching, web crippling) bypasses bending and fails by material shear at supports or concentrated loads. Torsional buckling complicates open-section beams (I-beams, C-channels) when lateral bracing is insufficient. Each mode triggers a dedicated check: yield under static load, buckling for slenderness, fatigue for cyclic stress, shear for high discontinuities.