Complete Stresses and Displacements in a Cross-Anisotropic Half-Space Caused by a Surface Vertical Point Load

Abstract

The elastic solution of a loaded cross-anisotropic half-space is dependent on the type of anisotropy that is governed by whether the characteristic equation has real and distinct or equal or complex roots. Most previous solutions have been for the case of real and distinct roots and most have also been incomplete. This work presents complete expressions for all stresses and displacements in a homogeneous, linearly elastic cross-anisotropic half-space caused by a surface vertical point load for all three types of anisotropy, which are believed to be more compact and simpler than those given by the only currently existing complete solution. In the present work, only 38 intermediate parameters need to be calculated in order to define all stresses and displacements as compared with the 62 parameters required in the other existing complete solution. Also, an important discovery is made: the surface settlement of a half-space is given by the same formula irrespective of the type of anisotropy, and this parallels the previous discovery that the contact stress of a rigid punch on a half-space is independent of anisotropy. This overrides the current notion that different formulas for the settlement apply when the characteristic equation has real roots and when it has complex roots. Hence, it is concluded that all existing formulas for the surface settlement of a cross-anisotropic half-space caused by distributed surface loads—for the case of real and distinct roots—are valid for all types of anisotropy. It is discovered that a much-publicized solution for the problem of a surface vertically loaded cross-anisotropic half-space is in error. Parametric studies carried out show that all elastic constants strongly influence the horizontal normal stresses and radial displacement. It is believed that the compact formulas presented herein will be appealing to engineers in all parts of the world.