A Two-dimensional Boundary Element Method for Calculating Elastic Gravitational Stresses in Slopes

Elastic stresses arising from gravitational loads in a two-dimensional slope of arbitrary shape are calculated easily using a displacement-discontinuity boundary element method (BEM). A long stress-free crack simulates the topographic surface. Gravity-induced stresses (i.e., body forces) in a laterally confined body are simulated by vertical and horizontal “far-field” stresses equal to „gy and [x/(1-x)]„gy, respectively. Here „ is material density, x is Poisson’s ratio, g is gravitational acceleration, and y is elevation, with y = 0 along the surface far from a ridge top or valley floor. BEM stress solutions compare well with analytical solutions for symmetric topography based on conformal mapping. Our analyses indicate that slope failures are likely to initiate near the bases of bedrock ridges and to be widespread along the slopes of gentle valleys cut in bedrock. The BEM method can be applied to the slopes of arbitrary shape and steepness, and it is well suited for evaluating the near-surface propagation of fractures or fracture-like structures, such as dikes and landslide failure surfaces. Our analysis also highlights the critical importance of properly accounting for the boundary conditions in a boundary value problem.