Elastic solutions for an inclined transversely isotropic material due to three-dimensional point loads

Abstract

We present the elastic solutions for displacements and stresses due to three-dimensional point loads in a transversely isotropic material (rock), for which the transversely isotropic full planes are inclined with respect to the horizontal loading surface. The closed-form solutions are derived by applying an efficient method, the double Fourier transform, to obtain the integral expressions for displacements and stresses. Subsequently, the double inverse Fourier transform and residue calculus are utilized to integrate the contours. Utilizing the double Fourier transform in a Cartesian coordinate system is a new approach to solving the displacement and stress components that result from three-dimensional point loads applied to an inclined transversely isotropic medium. In addition, it is the first presentation of the exact closedform characteristic roots for this special material anisotropy. The proposed solutions demonstrate that the displacements and stresses are profoundly influenced by the rotation of the transversely isotropic planes (φ), the type and degree of material anisotropy (E/E , ν/ν , G/G ), the geometric position (r, ϕ, ξ ), and the type of three-dimensional loading (P x , P y , P z ). The present solutions are identical to previously published solutions if the planes of transverse isotropy are parallel to the horizontal loading surface. A parametric study is conducted to elucidate the influence of the aforementioned factors on the displacements and stresses. The computed results reveal that the induced displacements and stresses in the inclined isotropic/transversely isotropic rocks by a vertical point load are quite different from the displacements that result from previous solutions in which φ = 0. The numerical results presented here are interesting for their ability to describe the physical features of inclined transversely isotropic rocks. Hence, the dip at an angle of inclination should be considered in computing the displacements and stresses in a transversely isotropic material due to applied loads.

Analytical Solution of Stress in a Transversely Isotropic Floor Rock Mass under Distributed Loading in an Arbitrary Direction

“…As can be seen from Figure 10a, the curve displayed a positive and negative stage. Obvious stress concentration is formed in the loading area, in which the value reached the peak stress, then it attenuates with the stress propagation along the X [46]. It is observed that with the increasing values of E1/E2, the magnitude of stress became bigger.…”

Boundary element analysis of transversely isotropic bi‐material halfspaces with inclined planes of isotropy and interfaces

“…Effects of such anisotropy on displacements and stresses can be essential to the design of structures. Hu et al and Liao et al presented the elastic solutions induced by the point load in a transversely isotopic fullspace whose isotopic plane is inclined to the horizontal loading plane. These solutions show that the displacements and stresses in the inclined transversely isotropic rocks by a point load are quite different from those induced in the horizontal transversely isotropic rocks.…”