Stress Systems in a Circular Disk Under Radial Forces

The transformation of inversion is employed in this paper to generalize a particular solution obtained by Michell (1), for plane stress in a circular disk. In Michell’s solution the disk is loaded in its plane with a pair of equal and opposite forces; one at the center of the disk and the other on the periphery as shown in Fig. 1. In the transformed state, the force in the interior of the disk is no longer at the center, as shown in Fig. 3 and represented by Equation [9]. By simple superposition, the transformed solution is extended to include any self-equilibrating system of radial forces applied in the plane of the disk as shown in Fig. 6b. The transformation would involve only straightforward calculation were it not for the fact that the disk is multiply connected. This condition is caused by the presence of a singularity in the interior of the disk at the point of application of the force. In the inversion of plane stress in a multiply connected plate, the displacement functions for the transformed state are not necessarily single-valued even though they may have arisen from single-valued displacements in the original plate. This difficulty is overcome by the introduction of suitable functions which remove the many valued terms in the displacements, but do not disturb the stress conditions on the boundaries of the transformed disk. The solution is shown to include, as particular cases two well-known problems: (a) A disk loaded on its periphery by forces acting along a diameter as shown in Fig. 4; and (b) a semi-infinite plate loaded near its edge by a force normal to the boundary as shown in Fig. 5.