Stress functions for a plate containing groups of circular holes

Abstract
A number of investigations, both experimental and theoretical, have been made to determine how the presence of holes in a uniform plate under given applied forces effects the distribution of the stresses in the plate (see Coker and Filon 1931, chap. iv). When there is a single hole in a plate which may be considered infinite, the problem is elementary ; but a hole near to a straight boundary or to a similar hole greatly influences the maximum stress and complicates the mathematical solution. No general method of solution has been given and we now extend methods, previously used by the present writers in particular cases, to a group of problems in which the boundaries possess a certain invariance. The boundaries we shall consider are a set of equal circles together with in some cases a pair of parallel straight lines. With each of the circles is associated a rectangular co-ordinate system, and it is essential to the method that the boundaries, boundary conditions and infinity conditions should remain invariant under a trans-formation in which each co-ordinate system and corresponding circle transforms into another system and circle of the set.

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