Semi-analytical method for solving stresses in slope under general loading conditions

Abstract

Assessing the stress distribution within the slope in geotechnical engineering is critical. Despite the widely available numerical methods, no analytical solutions are available for determining the stress distribution within a slope under general loading conditions. This study presents a method of analytically approximating elastic stresses within a slope of arbitrary inclination subject to general surcharges and supporting forces. The prototype model of this problem is equivalent to a superposition of two sub-models: a half-plane body subjected to an initial earth stress field as well as surcharges on the crest (Model I) and a slope loaded by the release stresses caused by excavation, together with supporting forces on its inclined surface and bottom (Model II). The former stresses can be calculated analytically using Flamant’s solution, and the latter stresses can be further thought of as being composed of two additional components: one in an infinite plane with a half-infinite hole loaded by virtual tractions upon hole’s boundary (Model II1), which can be analytically approximated, and the other in a half-plane subjected to virtual tractions along the ground surface (Model II2), which can be calculated analytically as well. The two sets of virtual tractions that lead to stresses in Model II are calculated using an iterative process. The current approach provides analytical approximations of elastic stress solutions for slopes that are sufficiently close to the exact ones as accurate as much. A case study demonstrates that such solutions are in good agreement with those of the finite-element method’s over the entire region, the stresses within the region up to 10−11 times the slope’s height away from the slope toe can also be accurately determined using the current method. With this method, contour plots of stresses within a slope inclined at various angles are presented, which can be applied directly in practical engineering.

Keywords

• semi-analytical stress solution
• half-infinite hole
• slope
• half-plane