Civil Slope Stability Analysis

EN 1997-1 Design Approach 1 (DA1) applies partial factors to soil strength. Combination 2 (A2+M2) governs for slope stability: γφ=1.25 and γc=1.25 are applied to tan(φ') and c' before computing FoS. The resulting factored FoS target is 1.0, equivalent to requiring FoS ≥ 1.5 in traditional analysis with unfactored strengths.

Both divide the failure mass into vertical slices. Fellenius (Swedish Slice) ignores inter-slice forces and satisfies moment equilibrium only—it's simpler but typically gives FoS 5–15% lower than Bishop. Bishop Simplified accounts for vertical inter-slice shear forces and converges iteratively, giving more accurate results. Bishop is recommended for design; Fellenius is useful for comparison.

rᵤ = u / (γ × h) is the dimensionless ratio of pore water pressure u to the overburden stress. For fully drained slopes rᵤ=0; for partially saturated slopes rᵤ=0.1–0.3; for saturated slopes with high phreatic surface rᵤ=0.3–0.5. Values above 0.5 indicate near-critical pore pressure conditions. rᵤ is an alternative to specifying the groundwater table depth.

The pseudo-static approach adds a horizontal inertia force Kₕ×W to each slice, where Kₕ is the seismic coefficient (typically 0.05–0.2) and W is the slice weight. This modifies both the driving moment (increased) and the resisting normal force (reduced). Seismic FoS ≥ 1.1 is the typical threshold per BS 6031 and CIRIA C706.

For static loading, FoS ≥ 1.5 is the standard requirement for permanent slopes (EN 1997-1 via unfactored Bishop). For temporary works, FoS ≥ 1.3 may be acceptable. For seismic (pseudo-static), FoS ≥ 1.1 is typically required. Higher factors apply to embankment dams and tailings facilities under ICOLD guidelines.

The calculator searches a grid of candidate circle centers (xₑ from -H/2 to xCrest, yₑ from H to 2H) with radii set to pass through the toe. For each candidate, it computes the Bishop FoS and retains the circle with the minimum (most critical) FoS. This grid search finds the governing failure mechanism for typical slope geometries.

Iterative moment-equilibrium method. For each slice i:
m_α = cos(α) + tan(φ_d)·sin(α)/FoS
numerator_i = (c_d·b + W·(1-rᵤ)·tan(φ_d)) / m_α
FoS = Σ[numerator_i] / Σ[W·(sin(α) + Kₕ·cos(α))]
Iterated until ΔFoS < 0.0005. EN 1997-1 DA1 C2 partial factors applied to strength before iteration.

Force-equilibrium method, no inter-slice forces:
N_i = W·(cos(α) - Kₕ·sin(α) - rᵤ)
FoS = Σ[c_d·(b/cos(α)) + N_i·tan(φ_d)] / Σ[W·(sin(α) + Kₕ·cos(α))]
More conservative than Bishop (5–15% lower FoS for typical geometries). Used for comparison.

Grid search over candidate circle centers: xc ∈ [-H/2, 0, H/2, xCrest/2, xCrest], yc ∈ [H, 1.5H, 2H]. Radius set to pass through the toe (0,0): R = √(xc²+yc²). For each candidate, slices are generated and the Bishop FoS computed. The circle yielding the minimum FoS is retained as the critical failure surface.

DA1 Combination 2 (A2+M2) governs slope stability:
γφ = 1.25: φ_d = arctan(tan(φ')/1.25)
γc = 1.25: c_d = c'/1.25
Combination 1 (A1+M1): γφ = γc = 1.0 (no reduction). Both combinations are checked; C2 typically governs for typical slopes.