Abstract
We determine the shape of the strongest column in the class of columns of length l l , volume V V , and having similar cross-sectional areas A ( x ) A(x) satisfying a ≤ A ( x ) ≤ b a \le A\left ( x \right ) \le b where a a and b 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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