Elasticity solutions for constant and linearly varying loads applied to a rectangular surface patch on the elastic half-space

The solution of the point load problem in the half-space is well known in the theory of elasticity. Using direct integration, the point solution can theoretically be used to develop the solution for loading various contact areas with a variety of loading profiles. Unfortunately, anything more complicated than constant pressure loading has previously required numerical integration, and hence, no closed form solution was obtainable. Partial solutions, i.e. solutions valid only on the surface of the half-space have also been available. This paper presents the methodology to generate complete solutions to the integrals for constant and linearly varying loads applied in both the normal and tangential directions everywhere in the half-space.