A force-based formulation for the analysis of 3-dimensional inelastic structural frames

Abstract

A force-based formulation for the step-by-step non-linear (elastic-plastic) analysis of three-dimensional (3D) structural frames is presented. It uses the redundant force and moment components as primary unknowns, and approximates the non-linearity problem in an incremental pattern. Using a simple linear transformation, the equilibrium matrices are quickly formed via a partial multiplication of a subset of matrices with dimensions (3×3). The convex yield function that describes the static admissibility condition of each zero-length plastic hinge is approximated with a linear convex polyhedron (manifold), whose hyper plane equations are automatically defined with the help of De Bruijn sequences. In this way, a number of complex force/moment interaction criteria may easily be defined that take into account shear and torsion. Discontinuities (e.g. articulations) are also accounted for. Out of the partial derivatives of these yield functions with respect to the stresses, the corresponding plastic deformations are computed, with the help of Lagrange multipliers. The formulation may be solved using any non-linear optimization algorithm that solves for linear constraints. Results are compared to those of the equivalent direct stiffness method and to those of the existing literature, proving the efficiency of the proposed formulation.