The shape of the strongest column and some related extremal eigenvalue problems

Barnes, Earl R.

doi:10.1090/qam/493674

Author: Earl R. Barnes
Journal: Quart. Appl. Math. 34 (1977), 393-409
MSC: Primary 49G99; Secondary 73.49
DOI: https://doi.org/10.1090/qam/493674
MathSciNet review: 493674
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Abstract: We determine the shape of the strongest column in the class of columns of length $l$, volume $V$, and having similar cross-sectional areas $A(x)$ satisfying $a \le A\left ( x \right ) \le b$ where $a$ and $b$ are prescribed positive bounds. In the special case where there are no constraints on the areas of cross-sections the problem has been solved by Keller [1] and by Takjbakhsh and Keller [2]. These authors observed that the problem is equivalent to an extremal eigenvalue problem and developed a variational technique for solving such problems. We treat a slightly more general class of extremal eigenvalue problems and give sufficient conditions for a given function to be a solution. Our work on the strongest constrained column demonstrates a procedure for finding functions satisfying these conditions.

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