Closed-form solution for optimization of buckling column

The optimization problem for a column, loaded by compression forces is studied in this article. The direction of the applied forces coincides until buckling moment with the axis of the column. The critical values of buckling are equal among all competitive designs of the columns. The dimensional analysis eases the mathematical technique for the optimization problem. The dimensional analysis introduces two dimensionless factors, one for the total material volume and one for the total stiffness of the columns. With the method of dimensional analysis, the solution of the nonlinear algebraic equations for the Lagrange multiplier will be superfluous. The closed-form solutions for Sturm-Liouville and mixed types boundary conditions are derived. The solutions are expressed in terms of the higher transcendental functions. The principal results are the closed-form solution in terms of the hypergeometric and elliptic functions, the analysis of single- and bimodal regimes, and the exact bounds for the masses of the optimal columns. The isoperimetric inequality was formulated as the strict inequality sign, because the optimal solution could not be attained for any finite seting of the design parameter. The additional restriction on the minimal area of the cross-section regularizes the optimization problem and leads to the definite attainable shape of the optimal column.