On the Optimal Design of Columns Against Buckling
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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 [32]. The work of [25], [22], and [32] sought to correct the necessary conditions of Tadjbakhsh and Keller [35] 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.
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Cox, Steven J. and Overton, Michael L.. "On the Optimal Design of Columns Against Buckling." (1990) https://hdl.handle.net/1911/101678.