Saint-Venant's problem for thin-walled tubes with a circular axis

Abstract

A method for analyzing a thin-walled tube acted upon by a bending moment in the plane of its curved axis has been developed by Von Karman [1]. An essentially new feature in his method was the taking into account of the flatening of the cross-section. Numerous contributions to the subject have been published later by other authors: a critical review of these later contributions is given in [2]. THe publications just referred to deal either with problems of raising the accuracy of the results derived by Von Karman for possible application to a wider interval of occurring parameters, or with some other special cases of loading. A general characteristic of all of these publications is the use of minimum principles applicable to approximate expression for displacements and stresses. A different approach was chosen by Clark and Reissner [3]; they reduce the problem of the tube acted upon by a bending moment in the plane of the curved axis to that of solving Meissner's equation. The present paper offers a uniform approach to the problem of deformation for a tube free of surface loading but carrying loads of general form along its boundary line. The problem is treated as one of the theory of thin shells. The boundary conditions of the tube ends are satisfied in accordance with Saint-Venant's principle. The notations used are fundamentally identical with those chosen in [4].