Slope Stability Factor of Safety Calculator (Bishop)
Estimate a slope's factor of safety using Bishop's Simplified Method for a representative slice, solved iteratively since the factor of safety appears on both sides of the equation.
⛰️ What is Slope Stability Factor of Safety (Bishop)?
Slope stability factor of safety is the ratio of the total resisting (shear strength) force available along a potential failure surface to the total driving (gravity-induced shear) force trying to cause a landslide. Bishop's Simplified Method is one of the most widely used slice-based techniques for computing it, assuming a circular failure surface and satisfying vertical equilibrium slice by slice while accounting for inter-slice normal forces, at the cost of requiring an iterative solution since the factor of safety appears on both sides of its governing equation.
Geotechnical engineers use slope stability analysis to check highway cuttings and embankments, earth dams, open-pit mine walls, and natural hillsides above infrastructure, comparing the computed factor of safety against a design target, commonly 1.5 for permanent slopes under normal (static) conditions, and a lower target for temporary or seismic cases.
A common point of confusion is that a full Bishop analysis sums this same slice equation across every slice along an assumed failure circle, then repeats the whole process for many trial circles to find the most critical one. This calculator applies the method to a single representative slice, the same simplified scope used to teach the calculation by hand, letting you see exactly how the iteration converges without needing full slope geometry.
This calculator solves Bishop's equation by fixed-point iteration, reports the converged factor of safety and a plain-language stability rating, and plots how the value approaches its final answer step by step.
📐 Formula
📖 How to Use This Calculator
Steps
💡 Example Calculations
Example 1 — Marginal Highway Embankment Slice
A slice with c′ = 10 kPa, φ′ = 30°, W = 500 kN, b = 4 m, α = 25°, u = 20 kPa
Example 2 — Unstable Slice With High Pore Pressure
A slice with c′ = 15 kPa, φ′ = 28°, W = 800 kN, b = 5 m, α = 30°, u = 30 kPa
Example 3 — Stable Slice on a Gentle Slope
A slice with c′ = 20 kPa, φ′ = 32°, W = 400 kN, b = 4 m, α = 18°, u = 10 kPa
❓ Frequently Asked Questions
🔗 Related Calculators
What is Bishop's Simplified Method?
Bishop's Simplified Method is a widely used slice-based technique for slope stability analysis, assuming a circular failure surface and satisfying vertical force equilibrium for each slice while neglecting inter-slice shear forces. It requires iteration because the factor of safety appears on both sides of the governing equation.
What is the formula for Bishop's Simplified Method?
FS = [c'b + (W - ub)tan(phi')] / (m_alpha W sin(alpha)), where m_alpha = cos(alpha)[1 + tan(alpha)tan(phi')/FS]. Since FS appears inside m_alpha, the equation is solved by iteration, starting with a guess and repeating until the value stops changing.
Why is this calculator scoped to a single slice instead of a full slope?
A complete Bishop analysis sums this same equation over every slice along an assumed circular failure surface, requiring detailed slope geometry and multiple slice properties. This calculator applies the method to one representative slice, the same simplified scope commonly used to demonstrate the technique by hand, for a full multi-slice analysis, use dedicated slope stability software.
What factor of safety is considered safe for a slope?
Most geotechnical design guidelines target a factor of safety of at least 1.5 for permanent slopes under static conditions, values between 1.0 and 1.5 are considered marginal and often warrant remediation, and a factor of safety below 1.0 indicates the slope is predicted to fail.
Why does the calculation require iteration?
The factor of safety FS appears on both sides of Bishop's equation, once inside m_alpha (in the denominator) and once as the value being solved for. There is no closed-form algebraic solution, so the equation is solved by fixed-point iteration, starting from a guess (typically FS = 1.0) and repeating until the value converges.
What is pore water pressure and why does it reduce stability?
Pore water pressure u is the pressure of water within the soil pores at the slip surface. It reduces the effective normal stress available for frictional resistance (represented by the (W - ub) term), so higher pore pressure directly lowers the factor of safety, this is why drainage is such an effective slope stabilization measure.
What is the difference between total stress and effective stress parameters?
Effective stress parameters (c-prime, phi-prime) describe the soil's drained shear strength and are used with the pore pressure term as in this calculator. Total stress parameters (undrained cohesion cu, phi = 0) describe short-term undrained behavior and use a different, simpler equation without a separate pore pressure term.
How does base angle alpha affect the factor of safety?
A steeper base angle alpha increases the driving force term (W sin alpha) faster than it increases the resisting terms, so factor of safety generally decreases as the assumed failure surface gets steeper at that slice, this is part of why critical failure surfaces in real slope stability software are searched for, not assumed.
Is Bishop's Simplified Method the same as the Ordinary (Fellenius) Method?
No. The Ordinary Method of Slices ignores inter-slice forces entirely and is known to be conservative (sometimes overly so) for effective-stress analyses with high pore pressure. Bishop's Simplified Method partially accounts for inter-slice normal forces, giving generally more accurate results, at the cost of requiring iteration instead of a direct formula.
What units does this calculator use?
Cohesion and pore water pressure are entered in kilopascals, slice weight in kilonewtons, slice width in meters, and the friction angle and base angle in degrees. The factor of safety result is a dimensionless ratio.