Theories of failure

A simple tension test tells us when steel yields under one stress acting alone. In a real structure, a point may carry tension, compression and shear together. Theories of failure answer the question: what combination of stresses is as dangerous as the yield or fracture point found in the simple test? Each theory picks one quantity, such as maximum stress, strain, shear or distortion energy, and assumes failure when it reaches its value at uniaxial failure.

The maximum principal stress theory (Rankine) suits brittle materials like cast iron and, roughly, concrete in tension. The maximum shear stress theory (Tresca) says yield occurs when (σ₁ − σ₃)/2 reaches fy/2; it is conservative and simple. The distortion energy theory (von Mises) says yield occurs when the energy of shape change reaches its uniaxial value, giving the familiar equivalent stress formula. Tests on ductile steel agree best with von Mises, which is why steel codes use it for combined bending and shear in welds and plates. Maximum strain (St Venant) and total strain energy (Haigh) theories are of mainly historical interest.

For pure shear, Tresca predicts yield at τ = 0.5 fy and von Mises at τ = fy/√3 ≈ 0.577 fy. That second figure is the basis of shear yield strength used in steel design. Soils and rock use the Mohr–Coulomb criterion instead, because their strength grows with confining pressure.