Bending of an Eccentric Cicular Ring with a Cut-Out Portion by Concentrated Forces at Its Inner Edge

Abstract

To investigate stress distributions in crane hooks, we consider an eccentric circular ring with a cut-out portion undergoing equal and opposite concentrated loads at the ends of its inner diameter. Applying the theory of two-dimensional elasticity and using bipolar coordinates, we solve the problem by method of superposition of a closed ring subjected to the same loads and a ring with a cut-out portion loaded by forces and bending couples at both ends. As the result, we can obtain exact solutions for stress distributions along the inner and outer circumferences in the form of infinite series and computing residues, derive simplified formulas, from which it is verified that the outside parts of the ring bordered by the loading points are free from stresses. Further, using the solutions, we calculate stress distributions for several examples and compare them with results on "curved bars" in the elementary theory.