Slope Factor of Safety Bishop Simplified Calculator
Calculate slope factor safety bishop simplified with our free science calculator. Uses standard scientific formulas with unit conversions and explanations.
Reviewed by Daniel Agrici, Founder & Lead Developer
Formula
FS = Sum[(c*b + W*(1-ru)*tan(phi)) / m_alpha] / Sum[W*sin(alpha)]
The Bishop Simplified factor of safety equals the sum of resisting forces divided by the sum of driving forces for all slices. The m-alpha correction term accounts for the base inclination and includes FS itself, requiring iteration. The resisting force for each slice includes cohesion acting over the base width plus the effective normal force times the tangent of the friction angle. The driving force is the component of slice weight along the slip surface.
Worked Examples
Example 1: Single Slice Stability Check
Problem:Analyze a slope slice with W = 500 kN, b = 2 m, base angle = 35 degrees, c = 15 kPa, phi = 25 degrees, ru = 0.2.
Solution:Iterative Bishop solution:\nDriving = W * sin(35) = 286.8 kN\nResisting includes cohesion and friction terms\nIterate FS until convergence\nResult converges after several iterations.
Result:Factor of safety from iterative Bishop method
Example 2: Dry Slope Analysis
Problem:Same slice but with ru = 0 (no pore pressure). c = 20 kPa, phi = 32 degrees.
Solution:With no pore pressure, full effective stress available\nHigher friction angle provides more resistance\nFS will be higher than the wet case.
Result:Higher FS due to zero pore pressure
Frequently Asked Questions
What is the Bishop Simplified Method for slope stability?
The Bishop Simplified Method is one of the most widely used limit equilibrium methods for analyzing the stability of slopes with circular failure surfaces. Developed by Alan Bishop in 1955, it satisfies moment equilibrium about the center of the slip circle and vertical force equilibrium for each slice, but neglects inter-slice shear forces. Despite this simplification, it produces results within about 5 percent of more rigorous methods for most practical problems. The method requires iterative solution because the factor of safety appears on both sides of the equation.
What factor of safety is considered adequate for slope stability?
The required minimum factor of safety depends on the consequences of failure and the reliability of the input data. For permanent slopes with high risk to life, a factor of safety of 1.5 is the standard minimum. Temporary construction slopes may accept 1.3. Slopes supporting critical infrastructure like dams typically require 1.5 for normal conditions and 1.3 for earthquake loading. Values below 1.0 indicate the driving forces exceed the resisting forces, meaning the slope is theoretically unstable and failure is expected under those conditions.
What is the pore pressure ratio ru and how does it affect slope stability?
The pore pressure ratio ru is defined as the pore water pressure at the base of a soil slice divided by the total overburden pressure at that point. It ranges from 0 (dry conditions) to about 0.5 for fully saturated slopes. Higher ru values reduce the effective normal stress on the failure surface, which directly decreases the frictional component of shear resistance. After heavy rainfall or rapid drawdown, ru can increase significantly, which is why many slope failures occur during or after intense rain events. Drainage measures reduce ru and improve stability.
Why does the Bishop method require an iterative solution?
The factor of safety FS appears in both the numerator and denominator of the Bishop equation because the normal force at the base of each slice depends on FS through the m-alpha term. This creates a nonlinear equation that cannot be solved directly. The standard approach starts with an initial estimate of FS (often 1.0 or 1.5), calculates a new FS from the equation, and repeats until the value converges. Convergence is typically achieved within 5 to 10 iterations. The method is very stable numerically and almost always converges for reasonable input parameters.
References
Reviewed by Daniel Agrici, Founder & Lead Developer · Editorial policy