Slope stability with the method of slices
Why we cut a sliding mass into slices, how Fellenius and Bishop differ, how a grid search finds the critical circle, and when an infinite-slope or planar-wedge check is enough.
A slope fails when the driving moment from the weight of a soil mass exceeds the resisting moment from shear strength along a slip surface. The factor of safety is the ratio of available to mobilised strength. The challenge is that the shear strength depends on the normal stress at the base, which in turn depends on how the forces between slices are shared.
Ordinary (Fellenius) slices ignore inter-slice forces, which makes the method explicit but usually conservative, sometimes by 10–20 %. Simplified Bishop assumes the inter-slice forces are horizontal and satisfies vertical equilibrium. The FS then appears on both sides of the equation and is found by iteration. It is accurate enough for circular surfaces in most practical cases. Spencer and Morgenstern–Price also satisfy force and moment equilibrium for non-circular surfaces. They are the methods commercial packages such as GEO5 offer alongside Bishop.
Finding the critical circle
No single circle is 'the' answer. Software tries a grid of centres and a range of radii, keeps the lowest FS, and contours the results so you can see that the minimum is properly enclosed. The toolkit's slope tool does the same with a grid search and draws the critical circle.
When a simpler model is enough
For a long, shallow slide parallel to the surface, typical of a residual soil mantle over rock, the infinite slope equation is exact. FS = [c′ + (γz − γw hw) cos²β tan φ′] / [γ z sin β cos β], with a pseudo-static term if a seismic coefficient applies. Planar wedges (Culmann) suit steep cuts in jointed material.