Stress and displacement fields for the twist-bending problem of an elbow or pipe bend

Abstract

Out of plane bending by an end couple applied to a 90° elbow or pipe bend of circular cross-section is considered. There is conversion of bending to torsion so that at 90° the elbow is held in equilibrium by a torsional couple. Using the methods of toroidal elasticity, the zero-order and first-order stress fields were determined in a previous paper. The present paper extends the analysis by determining the six second-order stresses, six second-order strains and three second-order displacements. Coupled stress fields are introduced to satisfy boundary conditions, equilibrium and compatibility equations and the requirement that displacements are single-valued. The equations in the text are derived from the general theory of toroidal elasticity by applying the method of successive approximations. All the equations of the general theory are expanded in series in I/R (where R is the toroidal radius). This leads to the working equations given in the paper. Some typical numerical values of stress are included.