On the Optimal Design of Columns Against Buckling

Abstract

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 finite-dimensional optimization problem the authors build on the work of Overton [26] in devising an efficient means of extracting an ascent direction from the column’s least eigenvalue. Owing to its possible multiplicity, this is indeed a nonsmooth problem and again the ideas of Clarke [10] are exploited.