Plastic analysis

Plastic analysis finds the load at which a ductile structure collapses, rather than the load at which its most stressed fibre first yields. As load rises, the most highly stressed section reaches its plastic moment and forms a plastic hinge. In an indeterminate structure the load can still increase, because moments redistribute to other sections, until enough hinges form to turn the structure into a mechanism.

Two theorems frame the calculation. The upper bound, or kinematic, theorem says that any assumed mechanism gives a collapse load equal to or greater than the true one, so the lowest value over all mechanisms is the answer. The lower bound, or static, theorem says that any moment distribution in equilibrium with the load and nowhere exceeding the plastic moment gives a safe load. When a mechanism's moment diagram also satisfies the yield condition everywhere, both bounds meet and the true collapse load is found.

The virtual work method is the usual hand route. Choose a mechanism, give it a small rotation, and equate the external work of the loads moving through their displacements to the internal work at the hinges, the plastic moment times the hinge rotation. Beams have beam mechanisms; frames add sway and combined mechanisms, and the combined one usually governs when vertical and horizontal loads act together.

Plastic design gives economical, robust steel frames, and the ideas carry over to the yield-line method for concrete slabs. It requires compact sections that can rotate without local buckling, lateral restraint at hinges, and checks that axial force and shear do not reduce the plastic moment. Serviceability, deflection and second-order effects are checked separately.