Infinite Slope Critical Depth for a Target FS
The inverse of the infinite-slope tile: the critical failure-plane depth at which the FS drops to a target, H = c' / (gamma cos beta (FS sin beta - cos beta tan phi')). A 25 degree cut in c' 200 psf, phi' 30, gamma 120 soil reaches FS 1.5 at about 16.6 ft deep. FS is high at the surface (cohesion helps) and falls with depth toward the cohesionless limit tan phi'/tan beta - a target below that is met at any depth. Needs c' > 0. Translational, no seepage; a screening aid, the geotech governs.
Formula and source
H = c' / (gamma cos(beta) (FS sin(beta) - cos(beta) tan(phi'))), the inverse of FS = (c' + gamma H cos^2(beta) tan(phi')) / (gamma H sin(beta) cos(beta)); a finite depth exists only for c' > 0 and a target FS above the deep limit tan(phi')/tan(beta).
The infinite-slope stability model (Das, Principles of Geotechnical Engineering; NAVFAC DM-7), solved for the critical failure-plane depth, by name.
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