Optimal structural design for maximum buckling load

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Virginia Polytechnic Institute and State University

Structural optimization was performed by either mathematical programming methods or optimality criteria methods. Both types of methods are based on iterative resizing of structures in the expectation that it will lead to the satisfaction of optimality conditions. Recent developments in methods for solving nonlinear equations gave a way to an alternative approach in which the optimality conditions are treated as a set of nonlinear equations and solved directly.

Two different formulations are presented; one is a conventional nested approach and the other is a simultaneous analysis and design approach.

Two procedures are explored to solve the nonlinear optimality conditions; a Newton-type iteration method and a homotopy method. Here, the homotopy method is adapted to the optimal design so that we can trace a path of optimum solutions. The solution path has several branches due to changes in the active constraint set and transitions from unimodal to bimodal solutions. The Lagrange multipliers and second-order optimality conditions are used to detect branching points and to switch to the optimum solution path.

This study specifically deals with buckling load maximization which requires highly nonlinear eigenvalue analysis and the procedure is applied to design of a column or laminated composite plate structures. A formulation to obtain multimodal solutions is given. Also, a special property in a laminate bending stiffness is found. That is, for a given stacking sequence of ply orientations, we showed an existence of a design with the same bending stiffness matrix and same total thickness even when the stacking sequence is changed.