The maximum potential energy of elastic rods under axial gravitational load

An optimal design problem solved previously for an elastic rod hanging under its own weight found the distribution of the crosssectional area that minimized the total potential energy stored in an equilibrium state, with the admissible designs bounded above and below and also subject to the constraint of prescribed total volume. This work solves the companion problem of the design that stores the maximum potential energy under the same constraint conditions. The method used is based on a comparison theorem for sandwich structures.