X. The distribution of stress round a circular hole in a plate

Abstract

The study of stress distributions in elastic plates would seem to have many important applications in engineering practice, and from this point of view it is, at first sight, surprising that our knowledge of the subject is not more detailed than it is at present. True, the fact that the stresses are derivable from a stress-function, and the equation satisfied by this stress-function, have long been known; particular solutions, satisfying the types of boundary condition met with in practice are, however, rare. Jeffery, in his paper, “Plane Stress and Plane Strain in Bipolar Co-ordinates, says that in the problem of the equilibrium of an elastic solid“ knowledge comes by patient accumulation of special solutions rather than by the establishment of great general propositions ”and later, that“ it is of considerable importance that the two-dimensional problems should be worked out more thoroughly The present paper is an attempt to fill one gap by a fairly full examination of the stresses round a circular hole in an otherwise infinite elastic plate of uniform thickness, due to prescribed tractions in the plane of the plate, acting on the circular boundary. A general solution is obtained and particular cases are examined in detail, these cases being chosen to combine, as far as possible, mathematical simplicity with some semblance of the type of distribution of traction likely to occur in practice ; the analysis is also applied to examine some experimental results obtained in the Engineering Laboratories of University College by Prof. E. G. Coker and T. Fukuda. The attention of the author was first turned to this type of problem in 1919 by Prof. Coker , whose experimental method of solution is now well known. He suggested an attempt to calculate mathematically the stresses in the neighbourhood of a circular hole in a tension member. An exact solution was not obtained, and an approximate one is only applicable when the diameter of the hole is very small compared to the width of the member. In the course of the investigation, however, it became necessary to find the stresses due to a simple distribution of traction on the circular boundary, and difficulties were met with when the traction did not form a system in equilibrium.