State-Space Formulation for Structural Analysis with Coupled Degradation-Plasticity and Geometric Nonlinearity

Abstract

A nonlinear beam finite-element model is developed to account for distributed plasticity, complex degradation phenomena, axial–moment–shear interactions, and geometric nonlinearity. The formulation is derived based on consistently coupled degradation-plasticity multiaxial hysteretic laws. The strength degradations corresponding to axial, shear, and flexural capacities are treated as scalar damage functions, and the multiaxial hysteretic model is presented in the effective stress domain to satisfy the consistency criterion of the evolving yield/capacity surface. Geometric nonlinearity is incorporated through an element-level P-Δ formulation, by accounting for second-order transverse displacement effects in the beam kinematics. Constant element matrices, including elastic stiffness, geometric stiffness, and hysteretic matrices, are derived from the principle of virtual work and do not require updating throughout the analysis. Material inelasticity and degradations evolve through element-level ordinary differential equations (ODEs) based on the multiaxial hysteretic laws and are solved simultaneously with the governing equations of motion of the system. Overall, the entire system-level formulation is presented in state-space form and can be straightforwardly solved using any general first-order ODE solver without requiring gradient evaluations. Model consistency, validity, and versatility are demonstrated through several numerical illustrations and experimental verifications.