Complementarity framework for non‐linear dynamic analysis of skeletal structures with softening plastic hinges

Abstract

SUMMARYIn this paper, we describe an algorithm for the incremental state update of elasto-plastic systems with softening. The algorithm uses a complementary pivoting technique and is based on casting the incremental state update as a complementarity problem. In developing the algorithm, we take advantage of the special features of solid and structural mechanics problems to achieve good computational performance, and hence the ability to compute numerical solutions to practical size problems. For example, the notion of a tangent stiffness matrix arises. Numerical examples using models of skeletal structures are presented to demonstrate the practicability of the algorithm. The numerical examples also raise some interesting questions about multiplicity of the solutions.

A projected Newton algorithm for the dual convex program of elastoplasticity

“…We refer to the Fenchel–Legendre transforms of these functions as the complementary stored energy function ψ c ( σ , ξ ) and the complementary dissipation function φ c ( σ , ξ ), where σ is the stress and ξ is the generalized stress conjugate to α . With linearized kinematics, it can be shown that incremental state update can be formulated as a convex minimization problem in ( σ , ξ ) , which we term the primal problem.…”

“…This mathematical programming approach was first introduced by Maier [25]. 904 Z. LOTFIAN AND M. V. SIVASELVAN Use of mathematical programming for state update has several attractive features [26]. Models of frictional contact, plasticity, damage and phase changes are often described using different formalisms of nonsmooth mechanics, including nonsmooth energy and dissipation potentials [27][28][29], variational inequalities [30,31] and hemi-variational inequalities [32,33].…”

“…Mathematical programming formulations allow asking questions about existence, uniqueness, sensitivity and stability of solutions. For example, multiplicity of solutions to the state update problem is explored in [26,[36][37][38][39], and stability and bifurcation phenomena are studied in [40]. When incremental state update has been represented as a mathematical program, appropriate solvers or algorithms can be invoked, which typically have effective global search strategies for their specific type of problem.…”

“…With linearized kinematics, it can be shown that incremental state update can be formulated as a convex minimization problem in . , / [3,11,26], which we term the primal problem.In this paper, we further limit ourselves to quasi-static problems and, for the sake of concreteness, introduce the following additional restrictions on the energy functions.…”

“…With linearized kinematics, it can be shown that incremental state update can be formulated as a convex minimization problem in . , / [3,11,26], which we term the primal problem.…”