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 closed-form 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
, the type and degree of material
anisotropy
, the geometric
position
, and the type of
three-dimensional loading
.
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
.
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.