A modified Lemke Algorithm for dynamic rigid plastic response of skeletal structures

Abstract

This paper proposes a modified Lemke algorithm to determine the non-holonomic response of rigid plastic skeletal structures subjected to extreme dynamic loading. The basic formulation for the dynamic rigid-plastic response has a mathematical form of linear complementarity problem (LCP), and, equivalently, a pair of dual quadratic programs (QP). Previous attempts using the Wolfe type LCP solver exhibited very pronounced sensitivity with semidefinite LCPs, which appear in this class of dynamics. Therefore, the current study offers a numerically robust Lemke algorithm, which is considerably adapted to deal effectively with the presence of unrestricted variables and the semidefinite characteristics of structural matrices. In addition, degeneracy in the LCP solution of rigid plastic structures is efficiently handled by using the Lexicographic Minimum Ratio test. To validate the formulation and the algorithm developed, bench tests are conducted, which involve a simply supported beam and a portal frame subjected to a uniformly distributed rectangular pulse loading.