An efficient mathematical programming method for the elastoplastic analysis of frames

Abstract

The elastoplastic nonholonomic analysis of frames is a nonlinear procedure in which the magnitude of the structural loading is incrementally modified using a proportional load factor, in accordance with a certain sequence of predefined loading patterns. It is an attempt by the structural engineering profession to estimate the strength as well as the deformations of framed structures under a given loading. In this work an analysis based on the force method and mathematical programming is presented. An elastic–perfectly plastic material is assumed and conventional plastic hinges of zero length are used to model the plasticity effects. The basis of the approach is the formulation of the incremental problem as a convex parametric quadratic programming (PQP) problem between two successive plastic hinges. A novel numerical strategy is proposed that uses a fictitious load factor to convert the PQP problem to a QP one. The solution of the QP problem, by an effective standard algorithm, establishes a feasible direction on which the true solution lies. The real solution is then found, simply on the demand of the formation of a new plastic hinge that is closest to open. Possible plastic unstressing is automatically accounted for. The approach is first developed for pure bending behaviour and is then extended to cater for moment/axial force interaction. Examples of application under monotonic, variable, and cyclic loading conditions are included. The whole procedure appears to be stable, robust, and computationally efficient as it requires much less time than the alternative displacement based direct stiffness method.