The development of displacement-based toroidal elasticity.

Abstract

Toroidal Elasticity (TE) is a three-dimensional theory used for the elastostatic analysis of thick-walled curved pipes, elbows, and toroidal shells. A displacement-based approach to TE is used in the present study, and analytical solutions are set up using the method of successive approximations. Three problems are solved analytically for the first time to illustrate the approach. The first problem concerns a 90$\sp\circ$ cantilevered elbow subjected to an out-of-plane couple at the free end. A real form general solution is developed up to the third order approximation, and results obtained are compared with boundary element results. The second problem concerns an axisymmetric sectorial toroidal ring subjected to loads on its radial surfaces. A complex form solution is developed with the aid of homogeneous eigenfunctions. Numerical results obtained are compared with finite element results. The third problem concerns a toroidal shell subjected to band loads of normal pressure. The applied loading is expanded in a Fourier series, and analytical solutions are developed for each term of the series. Numerical results are computed and again are compared with finite element results. The present work demonstrates the use of displacement based TE for the analytical prediction of displacement and stress in toroidal geometries, and the agreement between the displacement based TE solution and numerical results.