Optimal Design of Nonlinear Structural Systems with Critical Loads

A unified continuum formulation for design sensitivity analysis of critical loads is developed for nonlinear structural systems that are subjected to conservative loading. Sizing design variables such as cross-sectional areas of beam or truss design components and thicknesses of plate or membrane design components are included in development of the continuum design sensitivity analysis method. For nonlinear structural analysis, both geometric and material nonlinear effects are considered. The total Lagrangian formulations for incremental equilibrium analysis and one-point linear eigenvalue problems for stability analysis are utilized. Numerical methods are presented to evaluate design sensitivity expressions, using structural analysis results from established finite element codes. For design sensitivity of the estimated critical load, a continuum design sensitivity analysis method is developed using the lowest eigenvalue of the linear eigenvalue problem. An adjoint variable method is used to obtain design sensitivity expressions. This method provides accurate design sensitivity of the estimated critical load. However, it is shown that the design sensitivity of the estimated critical load does not converge to the design sensitivity of the actual critical load, even though the estimated critical load converges to the actual critical load. For design sensitivity of the actual critical load, a continuum design sensitivity analysis method is developed, based on the stability equation that includes the actual critical load factor at the critical limit point. It is found that no adjoint system is necessary for design sensitivity analysis of the actual critical load. This method provides accurate design sensitivity of the actual critical load. An optimal design problem with a stability constraint is solved using the proposed continuum design sensitivity analysis method.