Displacements and stresses due to nonuniform circular loadings in an inhomogeneous cross-anisotropic material

t r a c t This article presents the solutions for displacement and stress components along the centerline of nonuniform circular distribution of the vertical loads in a continuously inhomogeneous cross-anisotropic material with Young's and shear moduli varying exponentially with depth. The nonuniform loading types include a conical and a parabolic circular load. Planes of cross-anisotropy are assumed to be parallel to the horizontal ground surface. The proposed solutions can be obtained by integrating the point load solutions in a cylindrical co-ordinate system for an inhomogeneous cross-anisotropic half-space, which were derived by Wang et al. [Wang, C.D., Tzeng, C.S., Pan, E., Liao, J.J., 2003. Displacements and stresses due to a vertical point load in an inhomogeneous transversely isotropic half-space. Int. J. Rock Mech. Min. Sci. 40(5), 667-685]. However, the resulting integrals of the nonuniform circular solution for displacements and stresses cannot be given in closed form; hence, numerical integrations are required. Numerical results agree very well with the analytical solutions of displacements and stresses subjected to both present loading types for a homogeneous cross-anisotropic half-space, which are also yielded in Appendix A of this work. In addition, the proposed solutions are identical with Harr and Lovell's [Harr, M.E., Lovell, C.W. Jr., 1963. Vertical stresses under certain axisymmetrical loadings. High. Res. Board Rec. 39], and Geddes's Crow