Slope FS Calculator — Simplified Bishop Method
The Simplified Bishop method (1955) is the standard for slope stability analysis with circular failure surfaces. It divides the sliding mass into vertical slices and solves moment equilibrium about the circle centre, considering cohesion, friction, pore pressure and seismic force. This calculator processes up to 20 slices with geometry, stratum parameters and ru or pore pressure u, delivering the factor of safety FS by convergent iteration. Reference for design of dams, structural fills, deep excavations and permanent slopes in roadworks.
What is it and when to apply it?
The Simplified Bishop method assumes a circular failure surface and neglects horizontal inter-slice forces (simplification compared to the rigorous Bishop method), making it iterative but very fast. It gives practically identical results to Spencer and Morgenstern-Price in most circular cases. It is applied to embankments, mechanically stabilised walls, fills over soft ground, cut slopes on hillsides, earth dams and tailings deposits. For planar or wedge failures, the infinite slope method or specific wedge methods are used.
Applied formulas
Iterative FS by slices:
FS = Σ {[c'·b + (W·(1−ru) − u·b) · tan φ'] / mα} / Σ [W·sin α + kh·W·cos α]
where mα = cos α · (1 + tan α · tan φ' / FS)
Variables per slice:
W = γ·b·h (self-weight); α = angle of base to horizontal; b = slice width
u = pore pressure at base; ru = u / γ·h (dimensionless)
c', φ' effective Mohr-Coulomb parameters
Iteration: FS0 = 1.5 tentative → calculate mα → new FS → repeat until convergence (ΔFS < 0.01)
Minimum FS (BS EN 1997-1): Static 1.5 permanent; 1.3 temporary; Seismic 1.1-1.2
Calculation example
| Parameter | Value |
|---|---|
| Slope height H | 12.0 m |
| Cut slope | 1V:1.5H (β = 33.7°) |
| γ soil | 19 kN/m³ |
| c' | 15 kPa |
| φ' | 28° |
| ru (partial water table) | 0.20 |
| kh seismic | 0.15 |
| Critical circle | Centre (8, 16), R = 18 m, 8 slices |
Running Bishop with 8 slices of width b = 2 m each, W, α, u are tabulated, and iteration starts from FS0 = 1.5. First iteration: Σ[c·b + (W(1−ru) − u·b)·tan φ]/mα = 1,450 kN·m; Σ[W·sin α] = 820 kN·m (static). FS1 = 1,450/820 = 1.77. Second iteration with FS = 1.77: 1,462/820 = 1.78. Convergence in 2 iterations: FS_static = 1.78 > 1.5 OK. Seismic case with kh = 0.15: denominator = Σ[W·sin α + kh·W·cos α] = 820 + 310 = 1,130. FS_seismic = 1,462/1,130 = 1.29 > 1.1 OK. If the critical circle were sought by iterating over multiple centres, the final FS could drop by 5-10%; it is recommended to analyse at least 20 circles.
Result: FS static = 1.78 (OK) · FS seismic = 1.29 (OK) · Slope stable with cut 1V:1.5H.
Interpretation of results
FS static 1.78 and seismic 1.29 indicate adequate stability for a permanent road. If FS were between 1.3 and 1.5, it is advisable to review the soil parameters (is c reliable in the long term? does ru persist or increase over years?) and evaluate flattening the cut or an intermediate berm. In embankments over soft ground, Bishop with high ru and low φ is very sensitive to foundation parameters: a small variation in Su reduces FS by 20-30%.
Reference standards
- Bishop, A.W. (1955). The use of the slip circle in the stability analysis of slopes
- Duncan, J.M. & Wright, S.G. (2005). Soil Strength and Slope Stability
- BS EN 1997-1 (Eurocode 7) — Geotechnical design part global stability
- BS 8006-2
- BS EN 1997-1 (Eurocode 7) — Geotechnical design part global stability
Frequently asked questions
Simplified Bishop vs Spencer vs Morgenstern-Price?
Bishop solves only moment equilibrium and neglects inter-slice forces. Spencer and M-P satisfy forces and moments (rigorous). For circular surfaces the difference is < 5%. For non-circular surfaces rigorous methods are used. In practice, Bishop is sufficient in 90% of cases.
How do I find the critical circle?
Iterate over grids of centres and radii and return the circle with minimum FS. Software such as Slide, Geostudio or GeoSlope perform automatic search. Manually, rule of thumb: centre at ~0.3-0.5·H above the slope crest, R = 1.5-2·H. Test 10-30 circles around that point.
Should I use effective or total parameters?
Depends on the scenario. Long term (permanent analysis): c', φ' effective with u. Short term in saturated cohesive soil (during construction): UU analysis with Su = (qu)/2. The choice significantly changes results; it is recommended to evaluate both.
ru vs u per slice?
ru = u/(γ·h) is the average pore pressure coefficient. More accurate is to define the phreatic line and calculate u per slice. If the line is inclined or there are drains, the pressure u varies and it is better to model it. For preliminary analysis ru = 0.2-0.3 represents moderate water table; ru = 0.5 high water table.