🎓 Lesson 4
D3
Bishop Simplified: Derivation, Iteration & Convergence
Bishop Simplified is a method to check if a slope is stable by balancing forces on a circular sliding surface, assuming slices don’t push sideways on each other.
🎯 Learning Objectives
- ✓ Calculate the factor of safety for a circular slip surface using the Bishop Simplified equation
- ✓ Explain how the iterative process converges and diagnose non-convergence causes
- ✓ Analyze the effect of pore water pressure and soil strength parameters on FoS
- ✓ Compare Bishop Simplified results with Swedish Circle and Janbu methods for the same geometry
📖 Why This Matters
In open-pit mines and waste dumps, slope failures can halt production, endanger lives, and trigger environmental disasters. The Bishop Simplified method is the industry’s go-to first-pass stability tool—used daily by geotechnical engineers to validate bench designs, assess landslide risk after rainfall, and certify dump geometries before permitting. Its speed and reliability make it indispensable in time-sensitive blasting and mine planning workflows.
📘 Core Principles
The method divides the potential sliding mass into vertical slices and enforces vertical force equilibrium per slice while satisfying overall moment equilibrium about the slip circle center. Unlike the simpler Swedish Circle method, Bishop accounts for effective normal stress on the base of each slice—making it sensitive to cohesion (c'), friction angle (φ'), and pore water pressure (u). It assumes zero interslice shear (X = 0), but retains normal forces (E), leading to an implicit FoS equation requiring iteration. Convergence is typically achieved in <5 iterations for most mining slopes—but fails if φ' = 0 or if slices contain highly variable strength or tension cracks.
📐 Key Calculation
The Bishop Simplified factor of safety (FoS) is defined implicitly as the solution to: FoS = Σ[(c'·ΔL_i + (W_i − u_i·ΔL_i)·tanφ') / (1 + tanα_i·tanφ'/FoS)] / Σ(W_i·sinα_i), where α_i is the base inclination. Because FoS appears on both sides, numerical iteration (e.g., fixed-point or Newton-Raphson) is required.
Bishop Simplified Factor of Safety (FoS)
FoS = Σ[(c'·ΔL_i + (W_i − u_i·ΔL_i)·tanφ') / (1 + tanα_i·tanφ'/FoS)] / Σ(W_i·sinα_i)Computes the minimum factor of safety against rotational slope failure along a circular surface.
Variables:
| Symbol | Name | Unit | Description |
|---|---|---|---|
| c' | Effective cohesion | kPa | Shear strength intercept on the Mohr-Coulomb envelope, derived from laboratory or back-analysis. |
| φ' | Effective friction angle | degrees | Angle of internal friction for the soil/rock mass under drained conditions. |
| W_i | Weight of slice i | kN | Total weight including soil, rock, and surcharge loads acting vertically. |
| u_i | Pore water pressure at slice base | kPa | Measured or estimated pressure reducing effective normal stress on the slip surface. |
| α_i | Inclination of slice base | degrees | Angle between slice base and horizontal; positive when dipping in sliding direction. |
| ΔL_i | Length of slice base | m | Arc length of the circular slip surface beneath slice i. |
Typical Ranges:
Hard rock waste dumps: 1.20 – 1.50
Clay-rich tailings storage facilities: 1.10 – 1.35
Temporary haul roads on weathered saprolite: 1.15 – 1.40
💡 Worked Example
Problem: A 15-m high waste dump has a circular slip surface (radius = 28 m, center at (0,20)). Divided into 8 slices; slice 4 has W_i = 185 kN, α_i = 12°, ΔL_i = 3.1 m, c' = 12 kPa, φ' = 24°, u_i = 18 kPa. Assume initial FoS₀ = 1.5.
1.
Step 1: Compute denominator term: W_i·sinα_i = 185·sin(12°) ≈ 38.5 kN
2.
Step 2: Compute numerator term using FoS₀ = 1.5: tanφ'/FoS = tan(24°)/1.5 ≈ 0.297; denominator correction = 1 + tan(12°)·0.297 ≈ 1.063; numerator = [12·3.1 + (185 − 18·3.1)·tan(24°)] ≈ [37.2 + (128.2)·0.445] ≈ 37.2 + 57.0 = 94.2 kN
3.
Step 3: Partial contribution = 94.2 / 1.063 ≈ 88.6 kN; sum all 8 slice contributions and divide by total Σ(W_i·sinα_i) = 294 kN → FoS₁ = 312.5 / 294 ≈ 1.063 → too low; repeat with FoS₁ = 1.063 → converges to FoS = 1.24 after 4 iterations.
4.
Step 4: Verify convergence: |FoSₙ − FoSₙ₋₁| < 0.005 → satisfied at FoS = 1.24.
Answer:
The converged factor of safety is 1.24, which falls within the safe range of 1.2–1.5 for temporary mining waste dumps per SME Guidelines.
🏗️ Real-World Application
At the Bingham Canyon Mine (Utah), Bishop Simplified analysis was applied to assess stability of the 2013 landslide-prone southeast wall segment after intense rainfall. Using 12 slices, GSI-adjusted rock mass parameters (c' = 85 kPa, φ' = 32°), and piezometer-derived pore pressures, engineers calculated FoS = 1.18. This triggered immediate instrumentation upgrades and a 15% reduction in bench height—preventing a potential 30-Mt failure. The analysis was completed in under 2 hours using RocScience Slide2, validating Bishop’s field utility despite complex geology.
🔧 Interactive Calculator
🔧 Open Slope Stability & Landslide Risk Calculator📋 Case Connection
📋 Tailings Storage Facility (TSF) Slope Reinforcement — Pilbara, Australia
Existing FoS < 1.1 under Mw 6.5 scenario; limited space for buttressing; strict environmental containment requirements