The problem of the plane elasticity theory on the diametric compression of a ring by a concentrated load

Abstract

The article considers a plane problem of the linear elasticity theory for an annular region loaded on the outer contour by two oppositely directed concentrated forces. An analytical solution to the problem is obtained using functions of a complex variable. More specifically, the solution is constructed by the Kolosov-Muskhelishvili method using the superposition principle: the known potentials of a solid disk are supplemented with Laurent series with explicitly defined coefficients. The convergence of the solution and its passage to the limiting cases of an infinitesimal hole and of a thin-walled ring are studied. Qualitative analysis demonstrates good agreement between the result of modelling and the experimental data obtained by the photoelasticity method. The reliability of the solution is confirmed by monitoring the boundary discrepancies. The solution makes it possible to determine the dependence of the tensile stress concentration coefficient on the hole contour on the ratio of the internal hole radius to the overall radius. The values of this coefficient calculated using the solution presented in this article coincide with the data obtained using the finite element method. As the radius of the central hole increases, the coefficient of tensile stress concentration on the hole contour increases nonlinearly. The differences between concentrated and contact loading, the limitations of the model, and the application of the solution to the processing of the modified Brazilian test results are discussed.