Geomechanical Factor of Safety Calculator

The factor of safety is the ratio of shear strength available on a potential failure surface to the shear stress driving movement along it. In the infinite-slope model used here the failure plane lies parallel to the ground surface at depth H, so the normal stress is gamma H cos squared beta and the driving stress is gamma H sin beta cos beta, where gamma is unit weight and beta the slope angle. Strength follows Mohr-Coulomb: cohesion c plus effective normal stress times the tangent of the friction angle phi. Pore water pressure u enters only by reducing effective normal stress, which is why saturation destabilizes a slope without contributing any driving force of its own, and why rainfall-triggered failures are so common. Typical inputs are unit weights of 18 to 22 kN/m3, friction angles of 28 to 40 degrees for granular soils and 20 to 35 degrees for clays, and cohesion ranging from near zero in clean sand to 50 kPa or more in stiff clay. Design practice generally requires a factor of safety of at least 1.5 for permanent slopes, about 1.3 for temporary excavations and 1.1 to 1.25 under seismic or short-term loading, with that margin covering parameter uncertainty rather than representing any real reserve of strength. The model's limits are genuine: it assumes a planar, shallow, translational failure in uniform material, ignores end effects and tension cracks, and is unsuitable for deep rotational failures, which require slice methods such as Bishop, Janbu or Spencer, or for jointed rock, where discontinuity orientation rather than bulk strength controls the mechanism.

Charles-Augustin de Coulomb presented an essay in 1773 that introduced the strength expression combining cohesion with a friction term proportional to normal stress, work motivated by retaining walls and fortification design. Otto Mohr supplied in 1900 the graphical failure-envelope construction, built on the stress circle he had introduced in 1882, that turned Coulomb's expression into the Mohr-Coulomb criterion still used today. The missing ingredient was water, and it was provided by Karl Terzaghi, whose 1923 work on consolidation and his 1925 Erdbaumechanik established that soil strength depends on effective stress, meaning total stress minus pore pressure. Slope analysis became quantitative in Sweden after a series of railway and quay failures around Gothenburg; the Geotechnical Commission of the Swedish State Railways reported in 1922, and Wolmar Fellenius formalized the ordinary method of slices in 1936. Donald Taylor published stability charts in 1937 that yielded critical slope heights without iteration, an approach echoed in the critical height computed here. Alan Bishop's simplified method of 1955, followed by Nils Janbu, then Morgenstern and Price in 1965 and Spencer in 1967, added interslice force assumptions and progressively fuller satisfaction of equilibrium. Evert Hoek and John Bray carried the same framework into jointed rock in Rock Slope Engineering, first published in 1974, and limit-equilibrium factors of safety remain the regulatory standard even where numerical modelling is also performed.