Elastic subsurface stress analysis for circular foundations. I

In part I of this analysis an attempt is made to determine a simple estimate of the stresses resulting from a circular foundation subjected to concentric or eccentric loading. It is assumed that the foundation loading can be modeled as combinations of uniform, linear, and quadratic tractions applied over a circular area on the surface of an elastic half space. The present analysis for quadratic and linear loading are combined with a uniform loading solution (normal or shear traction), previously derived by the authors, to provide the requisite loading conditions and resulting internal stress fields. The current analysis consists of using potential functions to derive closed form expressions for the elastic field in the half space. The half space is taken as cross-anisotropic (transversely isotropic), where the planes of isotropy are parallel to the free surface. The x- and y-axes are taken in the plane of the surface with z directed into the half space. Hence the boundary conditions within the circular loaded area are on the shear stress components τ_xz, τ_yz, and normal stress σ_zz. The solutions presented within actually comprise seven different boundary value problems for the transversely isotropic half space. The analytical solutions for a point normal or shear force are first used to write the solution for distributed loading over a circle in the form of a double integral over the loaded area. It is shown, with the aid of Hankel transform analysis, that the integrals appearing in the elastic field have been previously evaluated in terms of complete elliptic integrals. The necessary integral evaluations are provided in Appendix I. The limiting form of the expressions for an isotropic material are also included.