Slope Stability Calculator

The factor of safety for a slope is the ratio of shear strength available along a potential failure surface to the shear stress that gravity mobilizes along it. In the infinite slope form used here, both are taken per unit area on a plane parallel to the ground surface, so slope length cancels and the result depends only on the slope angle beta, the height H, the unit weight gamma, and the Mohr-Coulomb parameters c and phi. The cohesion term, c divided by gamma times H times cos beta times sin beta, shrinks as the slide surface gets deeper, which is why cohesion controls shallow slips while friction controls deep ones. The friction term, tan phi over tan beta, is independent of depth entirely, and it shows that a purely frictional soil stays stable at any height so long as beta remains below phi. Typical inputs are moist unit weights of 100 to 135 pcf, drained friction angles of 28 to 40 degrees for sands and gravels and 20 to 30 degrees for silts, and effective cohesion ranging from 0 psf in clean sand to several hundred psf in stiff clay. Most agencies require a factor of safety of at least 1.5 for permanent slopes, 1.3 for temporary construction cuts, and 1.1 to 1.2 under pseudostatic seismic loading. The main limitation is that these closed-form expressions assume dry, homogeneous soil and a planar surface. Seepage can cut the friction term roughly in half when the water table rises to the ground surface, and layered or rotational failures require slice methods such as Bishop or Spencer.

Analytical slope stability grew out of earth pressure theory. Charles-Augustin de Coulomb presented his wedge analysis of soil failure in 1773, treating strength as cohesion plus friction acting on a plane, and that two-part form survives unchanged in the Mohr-Coulomb parameters used here. Karl Culmann, professor at the Zurich Polytechnic and founder of graphical statics, adapted the wedge idea to finite slopes in the 1860s, producing the planar-failure construction that still carries his name. Alexandre Collin, a French engineer studying failures of canal embankments, had already published observations in 1846 showing that real clay slopes tend to fail along curved rather than flat surfaces. The curved-surface insight became a method after a series of Swedish quay and railway slides, notably the 1916 Gothenburg harbor failure. The Swedish Geotechnical Commission, whose findings Wolmar Fellenius reported in 1927, formalized the circular arc and the method of slices. Karl Terzaghi supplied the effective stress principle in 1925, explaining why so many slopes fail during prolonged rain rather than under peak load. Donald Taylor of MIT published stability number charts in the late 1930s, Alan Bishop introduced his simplified method of slices in Geotechnique in 1955, and Morgenstern and Price in 1965 followed by Spencer in 1967 added rigorous formulations satisfying full equilibrium that remain the basis of modern limit-equilibrium software.