Chapter 9 - Extension, Torsion, and Flexure of Elastic Cylinders

Abstract

This chapter will investigate particular solutions to the problem of cylindrical bars subjected to forces acting on the end planes. The general problem is that of an elastic cylindrical bar with arbitrary cross-section and lateral surface that carries general resultant end loadings of force and moment. The lateral surface is taken to be free of external loading. The cylindrical body is a prismatic bar, and the constant cross-section may be solid or contain one or more holes. Components of the general loading lead to a definition of four problem types including extension, torsion, bending, and flexure. These problems are inherently three-dimensional, and thus analytical solutions cannot be generally determined. To obtain an approximate solution in central portions of the bar, it is presumed that the character of the elastic field in this location would only depend in a secondary way on the exact distribution of tractions on the ends of the cylinder, and that the principal effects are due to the force resultants on the ends (Saint-Venant’s principle). As such, the original problem is relaxed by no longer requiring the solution to satisfy pointwise traction conditions on the ends, but rather asking for one that had the same resultant loading.