Slope stability

The factor of safety of a simple slope against sliding on circular surfaces by Bishop's simplified method, with the ordinary method of slices for comparison (UFS 3-220-10, DM7.1 2026, 7-4). The type searches a grid of centers and radii for the critical circle, checks one entered circle, or solves an entered slice table as DM7.1 Figure 7-4 tabulates it.

Inputs of the slope #

• Analysis: search, one circle, or a slice table; Bishop simplified or the ordinary method of slices (the other is reported); the required factor of safety (the engineer's: 1.5 long-term static is common, 1.3 at the end of construction).
• Slope: height H and the face slope n horizontal to 1 vertical, with level ground at the toe (elevation 0) and above the crest; the firm stratum elevation (no circle passes below it); horizontal layers to their bottom elevations with γ, γsat, and c' and φ' (or su with φ = 0 for an undrained analysis).
• Pore water: dry, a horizontal water table (hydrostatic pore pressure at the slice base; γsat below it) or the pore pressure ratio ru (DM7.1 7-5: pore pressures belong in effective stress analyses only).
• Surcharge: a uniform load on the crest over a width from the crest.
• Search: the number of grid centers per side and radii per center. The grid spans from in front of the toe to past the crest and from just above the crest to well above it; the radii are tangent to depths from the firm stratum to the toe, plus the circle through the toe. A second pass refines around the best center.
• Circle: center (x from the toe, y elevation) and radius. Slices: b, W, α, c', φ' and u·b per foot of slope.

Checks performed #

Hard limits keep every input finite: 30 grid centers per side, 25 radii, 100 slices, 100 Bishop iterations per circle. Circles whose mα falls below 0.2 at any slice are skipped (the numerical difficulty of the simplified method at steep exit angles); a search that finds no admissible circle is NOT FEASIBLE. The drawing shows the slope, layers, firm stratum, water table and surcharge with the critical circle, its center, radius and factor of safety.

Limitations #

• Circular surfaces only; no Spencer or Morgenstern-Price (DM7.1 recommends a full equilibrium method and a thorough search for final design).
• Simple slope geometry with horizontal layers; no benches, berms, sloping layers or water table, flow nets, external water, tension cracks, reinforcement or seismic (pseudo-static) loading.

Verification examples #

Verification status: 3 values compared in 2 examples (1 published, 1 hand calculations from the cited equations): 3 within tolerance, 0 recorded deviations. Thin: only 1 published example. Code basis: the provisions it uses were read in the target edition.

The search is checked in backend/tests/test_slope_stability.py against the infinite slope (dry sand, tanφ/tanβ) and Taylor's stability number for undrained clay (about 5.5 for deep circles at 45°).