On the optimal design of columns against buckling

The authors establish existence, derive necessary conditions, infer regularity, and construct and test an algorithm for the maximization of a column’s Euler buckling load under a variety of boundary conditions over a general class of admissible designs. It is proven that symmetric clamped-clamped columns possess a positive first eigenfunction and a symmetric rearrangement is introduced that does not decrease the column’s buckling load. The necessary conditions, expressed in the language of Clarke’s generalized gradient [10], subsume those proposed by Olhoff and Rasmussen [25], Masur [22], and Seiranian [34]. The work of [25], [22], and [34] sought to correct the necessary conditions of Tadjbakhsh and Keller [37], who had not foreseen the presence of a multiple least eigenvalue. This remedy has been hampered by Tadjbakhsh and Keller’s miscalculation of the buckling loads of their clamped-clamped and clamped-hinged columns. This issue is resolved in the appendix.In the numerical treatment of the associated ...