Some anomalies of the curved bar problem in classical elasticity

Abstract

The classical formulation of the ‘homogeneous’ problem of a curved bar loaded only by and end force involves the assumption of an appropriate stress function with four arbitrary constants and the determination of these constants from the boundary conditions. Since there are five boundary conditions, four on the curved edge and one at the end, the solution is only possible because the coefficient matrix of the resulting algebraic equations is singular. This in turn means that certain inhomogeneous problems in which the curved edges are loaded by sinusoidally varying tractions cannot be solved using apparently appropriate stress functions.

Additional stress functions which resolve this difficulty are introduced and an example problem is solved, which exhibits qualitatively different behavior from that in more general cases of loading. The problem is then reconsidered as a limiting case of sinusoidal loading of arbitrary wavelength. It is shown that the solution of the latter problem appears to become unbounded as the special case is approached, but that when the end conditions have been correctly satisfied by superposing an appropriate multiple of the end-loaded solution, the limit can be approached regularly and the correct special solution is recovered. The limiting process reveals a general procedure for determining the additional stress functions required for the special case.

Similar relationships between homogeneous and inhomogeneous solutions for other common geometries are discussed.