Keywords: frame, frequency constraint, singular optimum, topology optimization, disjointed feasible domain, constraint relaxation..

This paper includes a study of integrated size and topology optimization of frame structures with constraints on natural fundamental frequency. Cross sectional sizes are taken as design variables. The design objective is to find feasible design or minimum weight design. By using the dynamic stiffness method, William-Wittrick algorithm and its theory for nonlinear eigenvalue problems, the frequency spectrum of the frame structure is discussed and the essential features of feasible domain in the design space are revealed quantitatively. The feasible domain consists of a set of disconnected sub-domains, some of which has degenerated dimensions. Moreover, optimum topology design of frame structures often locates at the endpoint of the degenerated feasible sub-domain and is indeed a strong singular optimum. This singular behaviour of optimum design becomes the bottleneck of solution approaches. To overcome the difficulties, an artificial relation between cross sectional area and inertia moment of beam cross section is proposed to relax the frequency-band constraints and regularize the feasible domain. Several numerical examples are given to illustrate the effectiveness of the approach.

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