#### Vol. 3, No. 8, 2008

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

### Jyh-Jong Liao, Tin-Bin Hu and Cheng-Der Wang

Vol. 3 (2008), No. 8, 1521–1547
##### 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 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 $\left(\varphi \right)$, the type and degree of material anisotropy $\left(E∕{E}^{\prime },\nu ∕{\nu }^{\prime },G∕{G}^{\prime }\right)$, the geometric position $\left(r,\phi ,\xi \right)$, and the type of three-dimensional loading $\left({P}_{x},{P}_{y},{P}_{z}\right)$. 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 $\varphi =0$. 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.

##### Keywords
displacements, stresses, inclined transversely isotropic material, double Fourier transform, residue calculus, material anisotropy