IX. The stress produced in a semi-infinite solid by pressure on part of the boundary

Abstract This paper had its origin in an attempt to throw some light on the important technical question of the safety of foundations. The question is idealised in a certain theory that has been developed as part of the mathematical theory of Elasticity. This theory sets out to give an account of the displacement and stress produced in an elastic solid body by pressure applied to part of its surface. The material of the solid is taken to be homogeneous and isotropic, and the solid is taken to be bounded by an infinite plane, and otherwise unlimited. The solid can be taken to represent the ground on which a building is raised, the pressed areas to represent the bases of walls or pillars, and the pressures to represent the weights supported on such bases. The law of distribution of pressure on the bases of walls and pillars is not known, but it would seem to be reasonable to assume that it is often not very far from being uniform, and thus the special cases of uniform pressure over rectangular and circular areas seem to be of considerable importance. A formal general solution of the problem has been known for a long time. It is applicable to any form of boundary of the pressed area, and to any law of distribution of pressure over the area. In this solution the components of displacement, and the components of stress, at any point in the solid, are expressed in terms of the space derivatives of a certain function, called by Boussinesq “le potentiel logarithmique à trois variables." This function is defined as a certain double integral taken over the pressed area. The difficulty of evaluating the integral has been a serious obstacle to the development of the formal solution in special cases. An alternative method of solution has been devised. This is applicable to a circular boundary only, and has so far been developed only in cases where the pressure is distributed symmetrically about the centre. In this form of solution the components of displacement, and the components of stress, are expressed in terms of single integrals involving Bessel’s functions. The two methods may be referred to as the “potential method” and the “Bessel’s function method.”