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

Abstract

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 [Geddes, J.D., 1975. Vertical stress components produced by axially symmetrical subsurface loadings. Can. Geotech. J. 12 (4), 482–497] solutions when the medium is isotropy. Two examples are illustrated to elucidate the effect of inhomogeneity, and the type and degree of soil anisotropy on the vertical displacement and vertical normal stress in the inhomogeneous isotropic/cross-anisotropic soils due to, respectively, a conical and a parabolic circular distribution of the vertical load acting on the ground surface. The generated solutions cannot only simulate the actual loading problem but also provide the realistic stratum in many fields of engineering practice.