Strongest Columns and Isoperimetric Inequalities for Eigenvalues

[+] Author and Article Information
I. Tadjbakhsh, J. B. Keller

Institute of Mathematical Sciences, New York University, New York, N. Y.

J. Appl. Mech 29(1), 159-164 (Mar 01, 1962) (6 pages) doi:10.1115/1.3636448 History: Received September 21, 1960; Online September 16, 2011


We consider the problem of determining what shape column has the largest critical buckling load of all columns of given length and volume. This problem was previously solved for a column hinged (pinned) at both ends. We solve it for columns clamped at one end and clamped, hinged, or free at the other end, assuming that all cross sections of the column are similar and similarly oriented. We also prove that the column previously obtained in the hinged-hinged case is actually strongest and not merely stationary. Graphs of the areas of the strongest columns as functions of distance along the columns are given for the various cases. The results are also expressed as isoperimetric inequalities for eigenvalues of second-order ordinary differential equations with various boundary conditions. Certain additional inequalities of this type are also obtained.

Copyright © 1962 by ASME
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