A robust optimality criteria procedure for cross-sectional optimization of frame structures with multiple frequency limits

An efficient optimality criteria (OC) method is presented for cross-sectional optimization of two-dimensional frame structures supporting a sizeable amount of nonstructural mass and subjected to multiple natural frequency constraints and minimum and maximum gauge restrictions. The iterative method involves alternately satisfying the constraints (scaling) and applying the Kuhn-Tucker (optimality) condition (resizing). This being the case, a criterion, which uses previous scaled designs to 'adaptively' tune the step size, is established. As the step size is tuned, the convergence rate is decreased. Hence a modified Aitken's accelerator, which extrapolates from previous scaled designs to obtain an improved one, is used. When the frequency constraints are to be satisfied, the sizing variables (cross-sectional areas) are uniformly scaled to the constraint surfaces using a closed-form formulation. This exact formulation is introduced for the first time in the open literature. Design examples are presented to demonstrate the method. The method is also used to qualitatively survey the convergence of OC recursive relations compositely used to resize and to evaluate the Lagrange multipliers. The method is robust, as it quickly dissolves the (sometimes violent) oscillations of scaled weights in the iteration history. Also, it eliminates the need for adjustments of internal parameters during the redesign phase.