Elastic–plastic and limit-state analyses of frames with softening plastic-hinge models by mathematical programming

Frames (and more general beam systems) subjected to monotonic loading are modelled by conventional finite elements with the traditional assumption of possible plastic deformations concentrated in pre-selected "critical sections". The inelastic behaviour of these beam sections, i.e. the development of "plastic hinges", is described by piece-wise-linear constitutive models allowing for hardening and/or softening, in terms of generalized stresses and conjugate kinematic variables. The following topics are discussed: step-by-step analysis methods, both "exact" and stepwise holonomic; path bifurcations and overall stability; limit and deformation analyses combined, as an optimization problem under complementarity constraints apt to compute the safety factor (with respect to global or local failures); numerical tests of nonconventional algorithms by means of simple representative applications. The objective of the paper is to provide a unified methodology and to propose novel procedures for inelastic analyses of frames up to failure, in the light of recent results in mathematical programming, particularly on complementarity theory.