Nonlinear analysis of semirigid frames: a parametric complementarity approach

The nonlinear elastoplastic analysis of plane frames with semirigid connections is performed using a mathematical programming approach. A Lagrangian formulation suitable for any order analysis is first derived for a suitably discretized structural system through the three basic relations of statics, kinematics, and the elastoplastic constitutive law. Particular features of this formulation include the preservation of static-kinematic duality through the concept of fictitious forces and deformations, the use of a powerful class of piecewise linearized constitutive laws to model plasticity conditions in general and semirigidity in particular, and an exact governing description for the two-dimensional case which can be specialized to any order of geometrical nonlinearity. Specific consideration is then given to a simple, essentially second-order case since it can model sufficiently accurately the behaviour of most real frames. Whilst the particular mathematical programming problem takes the form of a parametric nonlinear complementarity problem involving reversible or holonomic laws, the proposed numerical algorithm which is based on an iterative adaptation of the Wolfe-Markowitz method can accommodate irreversible or nonholonomic phenomena. The scheme can trace a complete skeleton equilibrium path beyond any critical point by capturing only events involving hinge activation or unloading. Numerical examples are presented to illustrate and validate the accuracy of the approach.