Abstract

An exact solution in series form is presented for the stresses and displacements in an incomplete torus of circular center line and circular cross section under conditions of pure bending. The solution is preceded by a formal treatment of the case in which the shape of the cross section is arbitrary. This more general problem is reduced to one of axisymmetry, which is in turn attacked by a modification of the stress-function approach of Timpe to rotationally symmetric problems in elasticity theory. The modified stress-function approach, which may be of interest beyond the present application, is referred to general orthogonal axisymmetric co-ordinates, and the solution of the specific problem considered here is based on the use of toroidal co-ordinates. The normal stresses acting on the circular cross sections of the torus are evaluated numerically in an illustrative example and are compared with the results of previous approximate solutions of the same problem. The adaptation of the present solution to the determination of the initial stresses in a complete torus from which a wedge-shaped portion has been removed is indicated.

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