On an optimization problem for elastic rods

Optimal shape of an elastic rod loaded by extensional force is determined. It is assumed that the rod is described by a classical Bernoulli–Euler rod theory. The optimality conditions are obtained by using Pontriyagin's maximum principle. It is shown that the optimal shape (cross-sectional area as a function of an arc length) is determined from the solution of a nonlinear second-order differential equation. The solution of this equation is given in the closed form. It is shown that for the same buckling force, the savings of the material are of the order of 30%. An interesting feature of the problem is that for certain values of parameters, there is no optimal solution.