510 ON THE OPTIMAL DESIGN OF COLUMNS AGAINST BUCKING S. Cox, M. L. Overton, June 1990 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, subsume those proposed by Olhoff and Rasmussen, Masur, and Seiranian. The work of Olhoff and Rasmussen, Masur, and Seiranian sought to correct the necessary conditions of Tadjbakhsh and Keller 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 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 are exploited.