On the Optimal Design of Columns against Buckling

We establish existence, derive necessary conditions, 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. We prove that symmetric clamped–clamped columns possess a positive first eigenfunction and introduce a symmetric rearrangement that does not decrease the column’s buckling load. Our 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. We resolve this issue in the appendix. In our numerical treatment of the associated finite dimensional optimization problem we 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.