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Pseudospectral Approach to the Shape Optimization of Beams Under Buckling Constraints

In this article, a direct transcription approach to the minimization of the volume of elastic straight beams undergoing plane deformation and subject to buckling loads is presented. In particular, the so-called pseudospectral method is employed, where state variables are approximated by Lagrange interpolating polynomials and static equations are collocated at Legendre-Gauss-Radau nonuniform mesh points. The resulting shape optimization problems are thus transcribed into constrained nonlinear programming problems, which in turn are solved by developed routines. Historical benchmark and academic problems such as simply supported beams subject to a concentrated compressing force, compressed and rotating cantilever beams and simply supported beams under a non-conservative follower distributed load are revisited and numerically solved under the conditions of plane deformation theory. Numerical solutions are discussed and compared to those obtained by the shooting method, which is largely employed for these problems, emphasizing how the proposed method could forecast optimal cross sectional area distributions within a unified fashion and without resorting to accurate guesses beforehand.