We compare two- and three-dimensional approaches to assessing the risk of limit-equilibrium failure of long earth slopes, embankments, dams or levees. The writer's earlier simplified three-dimensional stochastic slope stability model is extended by considering different types of spatial variation of the sliding resistance along the axis of the slope, including multi-scale and self-similar variation. Evaluating the probability Pf(b) of a sliding failure involving failure-zone segments of different width b, the model predicts that slope failures extending over a very long or a very short segment are highly improbable, and quantifies the most likely width. This also enables comparison, highly relevant in practical applications, of slope reliability measures obtained based on 2-D and 3-D stability analyses. Using stochastic failure-threshold-crossing analysis, it is further shown that the probability P_F(B) of a sliding failure anywhere along a long slope or embankment of given total length B grows linearly with the length, when P_F(B) 1, and the mean rate of increase in this "system failure" risk per unit length is quantified.
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