The tallest column — optimality revisited

摘要

The problem of the optimal design of the tallest unloaded column under selfweight is revisited with a view towards clarifying the optimality of the design proposed by Keller and Niordson (The Tallest Column, J. Math. Mech. 16 (1966), pp. 433–446). The height of the tallest column is related to the first eigenvalue of a Sturm-Liouville operator. Since the operator associated with the Keller-Niordson design does not possess a discrete spectrum, nonsmooth analysis is used to establish necessary conditions of optimality of the design. Upon solving the optimality conditions of an appropriately tapered design for the design variable, an iterative numerical scheme is developed for the optimal first eigenvalue. The scheme is then modified since it contains divergent integrals, and used to find the optimal design and height of the tallest column. The heights associated with various materials are evaluated for comparision.