Minimum-weight design of multi-purpose cylindrical bars

We solve the problem of minimizing the weight of elastic cylindrical bars under given constraints on their torsional and bending rigidities. The possible types of solutions are classed in the space of the design parameters. The necessary optimality conditions are derived and then used in the formulation of a closed-form boundary-value problem for a region with an unspecified boundary—the unknown cross-sectional shape of the optimal bar. An analytical solution of this and other related problems in terms of the given design parameters is obtained.