Topology optimization of structural frames based on a nonlinear Timoshenko beam finite element considering full interaction

• Published in: International journal for numerical methods in engineering Vol. 123; no. 19; pp. 4562 - 4585
• Main Authors: Changizi, Navid, Amir, Mariyam, Warn, Gordon P., Papakonstantinou, Konstantinos G.
• Format: Journal Article
• Language: English
• Published: Hoboken, USA John Wiley & Sons, Inc 15.10.2022; Wiley Subscription Services, Inc
• Subjects: Approximation; Divergence; Equilibrium equations; Excitation; Frame design; Frame structures; Hysteresis; hysteretic beam finite element; Interpolation; material nonlinearity; multiaxial interactions; Nonlinear differential equations; Nonlinearity; Optimization; Ordinary differential equations; Stiffness matrix; time‐varying excitation; Timoshenko beam; Timoshenko beams; Topology optimization
• ISSN: 0029-5981; 1097-0207
• DOI: 10.1002/nme.7046

This article presents an approach for the topology optimization of frame structures composed of nonlinear Timoshenko beam finite elements (FEs) under time‐varying excitation. Material nonlinearity is considered with a nonlinear Timoshenko beam FE model that accounts for distributed plasticity and axial–shear–moment interactions through appropriate hysteretic interpolation functions and a yield/capacity function, respectively. Hysteretic variables for curvature, shear, and axial deformations represent the nonlinearities and evolve according to first‐order nonlinear ordinary differential equations (ODEs). Owing to the first‐order representation, the governing dynamic equilibrium equations, and hysteretic evolution equations can thus be concisely presented as a combined system of first‐order nonlinear ODEs that can be solved using a general ODE solver. This avoids divergence due to an ill‐conditioned stiffness matrix that can commonly occur with Newmark–Newton solution schemes that rely upon linearization. The approach is illustrated for a volume minimization design problem, subject to dynamic excitation where an approximation for the maximum displacement at specified nodes is constrained to a given limit, that is, a drift ratio. The maximum displacement is approximated using the p‐norm, thus facilitating the derivation of the analytical sensitivities for gradient‐based optimization. The proposed approach is demonstrated through several numerical examples for the design of structural frames subjected to sinusoidal base excitation.