Displacements and stresses due to a vertical point load in an inhomogeneous transversely isotropic half-space

We present the solutions for displacements and stresses subjected to a vertical point load in a continuously inhomogeneous transversely isotropic half-space with Young's and shear moduli varying exponentially with depth. Planes of transverse isotropy are assumed to be parallel to the horizontal surface. The solutions for the half-space are obtained by superposing the solutions of two full spaces, one with a point load in its interior and the other with opposite traction of the first full space along the z=0 plane. The Hankel transform in a cylindrical co-ordinate system is employed for deriving the solutions. However, the resulting integrals for displacements and stresses involve polynomial, exponential function, and Bessel function that cannot be given in closed form; hence, numerical techniques are adopted in this work. In order to check the accuracy of numerical procedures, the comparisons are carried out with the homogeneous solutions of Liao and Wang, and the calculated results agree with those to nine decimal places. Furthermore, two illustrative examples are presented to elucidate the effect of inhomogeneity, and the type and degree of rock anisotropy on the vertical surface displacement and vertical normal stress in the inhomogeneous isotropic/transversely isotropic rocks subjected to a vertical concentrated force acting on the surface. The calculated results show that the induced displacement and stress are decisively influenced by the inhomogeneity, and the degree and type of material anisotropy. The proposed solutions can more realistically simulate the actual stratum of loading problem in many areas of engineering practice.