DESIGN OF FRAMES AGAINST BUCKLING USING A RAYLEIGH QUOTIENT APPROXIMATION

A Rayleigh quotient approximation is applied to the design of structures while guarding against elastic instability. It approximates buckling eigenvalues by separately estimating the modal strain energy due to the linear and geometric stiffness of the structure. Previously used during structural optimization for the fundamental natural eigenvalue, the Rayleigh quotient approximation is derived for the buckling design problem for the first time. The critical buckling load is found by solving the eigenvalue problem that arises by considering the geometric nonlinearity of the deforming structure. Rayleigh's principle is used to justify the choice of intermediate design variables for approximating terms in the Rayleigh quotient. A truss model illustrates the importance of the design space chosen for approximating the modal strain energy. A beam-column, two plane frames, and a space frame are used to verify the formulation. Special attention is paid to difficult bimodal optima.