Optimal strength of a compound column

A column of fixed length and variable cross section consists of two homogeneous and isotropic components. The components are joined along their side surfaces and have different Young's moduli, but the same Poisson's ratio. One of the components encloses the other that has the smaller Young's modulus. For different values of the ratio of the moduli, the shape of the column, which has the largest critical buckling load under axial thrust, is determined, assuming that the volumes of the components are prescribed. The problem is solved for the case of pinned ends.It appears that the solution of the most general problem, in which each of the areas of the component cross sections may be varied, is a combination of the solutions of some more elementary problems. Therefore, two types of problems are discussed: the compound bar with an inner component of fixed cross section and the general compound bar.The method of solution may be extended to other boundary conditions.