Plasticity

Elastic design stops at first yield, when the extreme fibre reaches the yield stress. Ductile steel can go far beyond that point. As bending increases, yielding spreads inwards from the extreme fibres until, in the ideal elastic-perfectly plastic model, the whole section has yielded: the top half at yield in compression and the bottom half at yield in tension. The moment it then carries is the plastic moment, and the section rotates freely at that moment like a hinge, a plastic hinge.

The ratio of plastic moment to first-yield moment is the shape factor. It depends only on geometry: 1.5 for a solid rectangle, about 1.7 for a solid circle and about 1.1 to 1.2 for rolled I-sections, whose material is already concentrated in the flanges. The plastic section modulus Zp is the sum of the first moments of the two equal-area halves about the plastic neutral axis, which for an unsymmetric section is the equal-area axis rather than the centroidal axis.

For general stress states, yield criteria decide when yielding starts. Tresca uses the maximum shear stress; von Mises uses the distortion energy and is the one most codes use for steel, combining normal and shear stresses into an equivalent stress. Beyond yield, a flow rule gives the direction of plastic strain, and hardening rules describe how the yield surface grows or moves with straining. These ingredients drive the nonlinear finite element models used in research and forensic work.

In design, plasticity explains why compact steel sections can be designed for their plastic moment, why indeterminate frames can redistribute moments after the first hinge forms, and why ductile detailing in earthquake-resistant structures is about making plastic hinges form in the right places and rotate without fracture. Concrete and soil also have plasticity models, though with pressure-dependent yield surfaces.