AN “OPTIMAL” SOLUTION OF SAINT-VENANT’S FLEXURE PROBLEM FOR A CIRCULAR CYLINDER

摘要

In a recent paper Sternberg and Knowles established certain mini-mum strain-energy properties of Saint-Venant's solutions to the relaxed Saint-Venant problem for an elastic cylinder. They proved that Saint-Venant's solutions for the case of extension, pure bending, and torsion are uniquely distinguished, among all solutions to the appropriate relaxed problem that correspond to a fixed resultant load and to point wise vanishing shearing or normal terminal tractions, by the fact that they minimize the total strain energy. In the same paper Saint-Venant's solution for the case of bending by transverse terminal loads was shown to be no longer optimal in the foregoing sense and the optimal flexure solution was characterized implicitly as the solution to a mixed-mixed boundary-value problem for the cylinder in question. In the present investigation this optimal flexure solution is determined explicitly for a circular cylinder by means of the Papkovich-Neuber stress functions. The results obtained, which are in infinite-series form, are evaluated numerically and compared with the analogous results of Saint-Venant. The solution deduced here also supplies a quantitative illustration of Saint-Venant's principle.