Optimal design of elastic rods under axial gravitational load using the maximum principle

Optimal design problems for structural elements in equilibrium often come in pairs in which the cost function for one problem becomes a constraint for the second. In particular, the problem of minimizing structural volume or weight under size and stress constraints has a dual in which potential energy is minimized for ®xed volume of material. These dual problems are solved here for a linearly elastic rod hanging from a rigid support. The design variable is the cross-sectional area of the rod. The method used is the Maximum Principle of Pontryagin and Hestenes, rather than the function space methods used by others for similar problems.