The optimal design of tubular structures

Given a fixed volume of material and the elastic properties of that material, the optimal design of the tallest tubular column under self-weight is sought. This problem can be formulated as an extremal eigenvalue problem because the height at which the unloaded tubular column will buckle is related to the first eigenvalue of a Sturm–Liouville operator. Considering columns of annular cross-section, the spectrum associated with physical designs is characterized. Existence of an optimal design over a particular class of designs is proven through the use of rearrangements. Necessary conditions of optimality over that class are also established.