Elastic stress analysis of partially loaded hollow discs

Abstract

This study develops and discusses solutions for the calculation of stress and displacement components in a two-dimensional elastic hollow disc. The solutions have many applications in civil, mechanical and mining engineering; such as roller disc cutter design in mechanical excavation engineering. Previously, solutions for the state of stress in circular-shaped domains have mainly considered the boundary loads as a pair of concentrated forces acting along the disc’s diameter at its circumference. In this study, the two internal and external circular boundaries of the hollow disc are under a general uniform loading, for which Lamé problem is a special case. The solution methodology is based on Michell’s expansion in polar coordinates and Fourier series representation of general boundary conditions developed for plane problems (plane strain and plane stress), encompassing all possible combinations of loading conditions at the boundaries. Displacement and stress components are constrained by equilibrium equations to ensure that they are single-valued and continuously-differentiable equations. Stresses are normalized with respect to either the applied internal pressure or the solutions from the special Lamé case, in which both boundaries at the radii r=a and r=b>a, are fully-loaded with uniform stresses p and q. Several solutions are developed in terms of design graphs. These solutions are applicable to both plane strain and plane stress problems through a conversion factor dependent on Poisson’s ratio. Results from various geometrical configurations and loading conditions show that the maximum value of the compressive normal stress is neither greater than the applied internal pressure (p) nor the external pressure (q=pa/b), while the maximum tensile stress generated in the disc reaches a value almost twice the internal pressure (p), and the maximum shear stress is not greater than one third of the internal pressure.