Verification of the Slope Limit-Equilibrium Engine

The calculation engine is a pure, headless TypeScript module, independent of the user interface, which makes it directly scriptable for verification campaigns like this one. Slices are break-aligned to the ground polyline; impenetrable bedrock regions act as keep-outs. Slip surfaces are circular or polyline (drawn, tabulated, or found by a targeted weak-layer block search) with Mohr-Coulomb, depth-varying undrained, SHANSEP, Hoek-Brown, hyperbolic and anisotropic strength models, and a seeded Monte-Carlo / Latin-hypercube probabilistic layer (probability of failure and reliability index β) on top of the deterministic solve.

The benchmark factors of safety above use the plain factor-of-safety set (all partial factors = 1). On that baseline Slope layers seventeen built-in design-standard sets across seven code families, plus fully custom sets. Every factor is applied inside the limit-equilibrium solver (materials, soil weight, actions, reinforcement and seismic each factored at the slice, never a post-solve divisor) and each is unit-tested to its exact ratio (Eurocode 7 DA2 lowers the factor of safety by precisely 1/1.1; DA1·C2 divides tan φ′ and c′ by 1.25).

Every model carries a unit system: SI (kN, kPa, m) or US customary (lb, psf, ft). Because the engine computes in consistent units with a configurable unit weight of water, results are unit-invariant: a slope entered in US customary units reproduces its SI factor of safety exactly, a property the campaign confirms with imperial-unit benchmark problems (Appendix A).

ACADS 1(b) is the one case where the two commercial manuals themselves disagree (SLOPE/W 1.664 vs Slide2 1.596, a 4.1% spread); our value matches Slide2’s to 0.5%. The suite’s non-circular weak-layer family is also replicated on Morgenstern–Price (Bishop is gated off polylines): the specified surface of ACADS 3(b) to −0.6%, the block-searched 3(a) to −4.8% (a search-landscape divergence where our two-window search finds a more critical, conservative block than GeoStudio’s), and §2.9’s external-loading case, the same dipping weak seam carrying a piezometric surface and a new linearly-varying (trapezoidal) surcharge, to +6.7%. Here the divergence runs the other way: GeoStudio’s manual restricted its search to circular surfaces (its own published Bishop, 0.692, sits on that same circular family), while our block search finds the seam’s true non-circular mechanism, matching the identical problem’s Slide2 publication (its own block-searched GLE value) to +2.1% (§ below).

A representative 17 of the 53 replicated cases; the full set, including the reinforced micro-pile slope (§ below), the Baker Mohr-Coulomb envelopes and the six Duncan & Wright cohesive sub-cases, is in the PDF and shipped as machine-readable model files. #23’s Bishop is analyzed on the manual’s own tabulated critical circle; the published sources for that benchmark span 1.14–1.19, and our value matches Low’s original 1.14 to 0.4%.

Non-circular & composite surfaces

Slip surfaces beyond circles: the tabulated ACADS 3(b) polyline analyzed directly, the same problem block-searched (3(a) / #7, one of the campaign’s two primary comparisons beyond 5%, where our two-window search converges to a different, more critical block than the manual’s, the conservative direction for a search), the family’s fourth member (#9) adding a piezometric surface and a linearly-varying (trapezoidal) surcharge to the same dipping weak seam, matching Slide2’s published block-searched GLE to +2.1% (the companion GeoStudio §2.9 problem, on the same mechanism but a circular entry–exit search, is the campaign’s other exception beyond 5%, discussed above), and Fredlund & Krahn’s composite circular surface, the specified circle running along a thin weak seam, reconstructed as arcs joined by the weak-seam flat and analyzed dry and with rᵤ = 0.25. Bishop and Ordinary presume the common-center moment arm and are gated off polylines, so Morgenstern–Price carries the comparison.

The engine models four reinforcement element types, each resolved at the slip-surface crossing into a base-tangential force T∥ (adds to the resisting shear) and a base-normal force T⊥ (raises the base normal force, and hence friction). Published circular reinforced benchmarks are scarce (the nailed and tied-back walls use planar surfaces, the geosynthetic embankments carry figure-only tension cracks), but the one fully-circular, tabulated case is reproduced: Slide2 #54, where a single row of stabilising micro-piles (1 m spacing, 10.7 kN lateral shear) lifts the factor of safety from an unreinforced 1.102 to Bishop 1.209 vs the published 1.193 (+1.4%).

Capability demonstration: each element type applied to the #54 slope on its unreinforced critical circle (Bishop ≈ 1.100), compared on identical geometry. Illustrative of the force resolution, not a published-benchmark comparison.