Factor of Safety Definition
The factor of safety against slope failure is defined as the ratio of resisting forces (or moments) to driving forces (or moments) along a potential failure surface:
FS = Resisting shear strength / Driving shear stress
This definition applies to any geometry and failure mechanism. A FS of 1.0 means the slope is at the point of failure. Regulatory agencies and codes establish minimum acceptable FS values — not as absolute guarantees of stability, but as consistent measures of safety margin relative to our uncertainty in soil properties and loading.
Infinite Slope Analysis
For shallow, planar failure surfaces parallel to a slope (common in residual soils, colluvium, and fill slopes where failure depth << slope length), the infinite slope model gives:
For dry sand: FS = tanφ' / tanβ (independent of depth)
For saturated slope with seepage parallel to slope: FS = (γ' / γsat) · (tanφ' / tanβ)
For c-φ soil: FS = [c' + (γH cos²β − u) tanφ'] / (γH sinβ cosβ)
where β = slope angle, H = depth to failure plane, u = pore water pressure = γw·hw·cos²β for seepage parallel to slope (hw = depth of water table above slip surface). The ratio γ'/γsat ≈ 0.5 for typical soils, showing that submerged seepage through a slope roughly halves the FS in sand.
Swedish Circle / Fellenius Method
For circular failure surfaces in homogeneous soils, the Swedish Circle (Fellenius, 1927) method divides the soil mass into vertical slices and sums moments about the center of the trial circle:
FS = Σ[c'·ΔL + (W cosα − u·ΔL) tanφ'] / Σ(W sinα)
where W = slice weight, α = base angle of slice, ΔL = arc length of slice base, u = pore pressure at base. The method neglects interslice forces and therefore underestimates FS by 5–20% in steep slopes. It is still widely used for hand calculations and as a check method.
Bishop's Simplified Method
Bishop (1955) accounts for interslice normal forces (but neglects interslice shear) and significantly improves accuracy. The equation is solved iteratively:
FS = Σ{[c'·b + (W − u·b) tanφ'] / mα} / Σ(W sinα)
where b = slice width, mα = cosα + (sinα · tanφ')/FS (hence the iteration), and all other variables are as in Fellenius. Bishop's simplified method matches more rigorous methods (Spencer, Morgenstern-Price) within 2–5% for circular slip surfaces and is the standard for hand or spreadsheet analysis.
Computer programs such as SLOPE/W, SLIDE, and STABL automatically search for the critical slip surface by minimizing FS over many trial circles (or non-circular surfaces using Spencer or Morgenstern-Price methods for complex geometries).
Undrained vs Drained Analysis
The choice of strength parameters is critical and one of the most common errors on the PE exam:
- Undrained (φ=0) analysis: Use Su (undrained shear strength) directly. Applicable during and immediately after rapid loading on saturated clays where drainage cannot occur. Also called the "end-of-construction" case. FS = Σ(Su·ΔL) / Σ(W sinα).
- Drained (c'-φ') analysis: Use effective strength parameters with actual pore pressures (steady seepage, measured piezometers). Applicable for long-term stability, for sands and gravels at any rate of loading, and for overconsolidated clays subject to long-term swelling.
The critical case depends on soil type and loading rate. For embankments on soft clay, the undrained (φ=0) end-of-construction case is typically critical. For cuts in stiff OC clay, the drained long-term case with softened residual strength is often more critical (progressive failure).
Tension Cracks
At the crest of a slope in cohesive soils, tension cracks form to a depth z_c = 2c / (γ√Ka) ≈ 2Su/γ (for φ=0). Tension cracks reduce the effective arc length available for resistance and, when filled with water, add a significant destabilizing hydrostatic force. FHWA and SLOPE/W both recommend including tension cracks in stability analysis of cohesive slopes. In practice, a conservatively positioned tension crack filled with water can reduce FS by 15–25%.
Seismic (Pseudostatic) Analysis
For the pseudostatic method (used in AASHTO LRFD Section 11 for retaining structures and embankments), a horizontal seismic force kh·W is added to each slice, where kh is the seismic coefficient:
- AASHTO and FHWA suggest kh = 0.5·PGA/g for screening; kh = PGA/g for design of critical facilities.
- The pseudostatic FS = Σ[(c'·ΔL + N'tanφ')] / Σ(W sinα + kh·W cosα)
- Minimum FS ≥ 1.1 under seismic loading (AASHTO); FS ≥ 1.5 under static loading for permanent fill slopes (FHWA Geotechnical Engineering Circular 7).
For critical facilities or large embankments, Newmark sliding block analysis should supplement or replace pseudostatic methods to estimate permanent displacement.
Minimum FS Requirements
| Condition | Minimum FS | Reference |
|---|---|---|
| Static, permanent slope | 1.5 | FHWA GEC 7, AASHTO |
| Static, temporary cut | 1.25–1.3 | OSHA, FHWA |
| Seismic (pseudostatic) | 1.1 | AASHTO LRFD |
| Retaining wall global stability | 1.5 | AASHTO LRFD 11.6 |
| Dam / levee (steady seepage) | 1.5–2.0 | USACE EM 1110-2-1902 |
Common Failure Modes and Mitigations
- Translational slide on weak layer (e.g., bentonite seam): address with shear keys or deep drainage to reduce pore pressure.
- Rotational slide in soft clay: stage construction, allow strength gain, install wick drains to accelerate consolidation.
- Planar slide in colluvium: surface and subsurface drainage (French drains, horizontal drains), geogrid reinforcement.
- Seepage-induced failure: install interceptor drains, reduce phreatic surface with pump wells.
See Retaining Wall Design for the application of these methods to wall global stability.