Solve the factor of safety Fs of a slope with an assumed circular slip surface using the iterative Bishop simplified method (1955). Adjust slope height, slope angle, cohesion, friction angle, pore-pressure ratio and seismic coefficient to see how rainfall and earthquakes destabilise a slope.
Parameters
Slope height H
m
Slope angle β
°
Cohesion c'
kPa
Effective-stress cohesion
Friction angle φ'
°
Unit weight γ
kN/m³
Pore-pressure ratio r_u
Rainfall / groundwater effect. r_u ≈ 0.5 means water table at the surface.
Seismic coefficient k_h
Horizontal seismic coefficient for pseudo-static analysis. Level 1 design typically 0.10–0.15.
Failure mode
This tool focuses on Bishop circular slip analysis.
Bishop simplified iterative solution. c' = effective cohesion [kPa], φ' = effective friction angle [°], W = slice weight [kN/m], u = pore pressure at slice base [kPa], b = slice width [m], α = slice-base inclination, k_h = horizontal seismic coefficient. Fs appears on both sides, so the solution is iterated; it typically converges within 3–5 iterations.
Definition of the pore-pressure ratio r_u and the critical r_u that drives Fs to 1 in the infinite-slope approximation. Rising groundwater increases u, reducing effective stress and frictional resistance.
Landslide factor of safety Fs — Bishop simplified slope stability
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You hear it in disaster news — "the factor of safety of the slope dropped after the heavy rain". But how is that factor actually calculated?
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In short, Fs is the ratio of the force that resists sliding to the force that drives it: Fs = resistance / driving. Fs > 1 means the slope is still holding on, Fs < 1 means the driving force has won and failure is imminent. In practice we assume a circular slip surface and treat the soil above it as one mass sliding along that arc. That's "circular slip", and the most widely used calculation worldwide is Bishop's simplified method (1955).
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Named after a person, then. What sets it apart from a simple force balance?
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The slope is cut into thin vertical "slices" — ten of them in this tool — and weight, pore pressure and friction are evaluated for each slice. Bishop's contribution is to consider the horizontal interslice forces and treat Fs itself as the unknown, solved iteratively. The factor m_α on the right-hand side contains Fs, so you guess Fs = 1.5, compute a new Fs, plug it back in, and after 3–5 rounds the iteration converges. This tool runs 10 iterations automatically.
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When I push the pore-pressure ratio r_u from 0.3 to 0.6, Fs drops from about 1.5 to 1.0 — that's exactly how a rainstorm triggers a landslide, isn't it?
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Exactly. Rainfall raises the groundwater table, so the pore pressure u along the slip surface increases. The (W − ub)·tan φ' term in Bishop's equation is the "effective-stress" frictional resistance, and when u grows it shrinks, dragging Fs down. That is why 70–80% of all landslides in Japan happen during the rainy season or typhoons. At about r_u = 0.5 the water table reaches the slope surface, and even a "normally Fs = 1.5" slope can fall below Fs = 1.0 there. Rainfall thresholds for evacuation orders are calibrated against this same r_u.
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And the seismic coefficient k_h — setting it to 0.15 increases the driving moment and Fs drops sharply. Earthquakes really do trigger landslides too.
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That's the "pseudo-static analysis" approach. In reality the ground motion is a time-varying wave, but design practice simplifies it to a steady horizontal inertial force of k_h times gravity. Japanese road codes use k_h = 0.10–0.15 for Level 1 design earthquakes and 0.15–0.20 for Level 2. The Vajont dam slide in Italy (1963, about 2000 people lost) and Taiwan's Typhoon Morakot landslide (2009, the village of Xiaolin destroyed) were both compound triggers — rising reservoir level or torrential rain combined with weak layers. Full dynamic analysis (FLAC, PLAXIS Dynamic) is used for detail, but pseudo-static remains the standard "first cut".
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If Fs = 1.5, is the slope safe enough? What's the target value in practice?
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Japanese building-site and road-earthworks standards typically ask Fs ≥ 1.5 for permanent conditions (no earthquake, normal water table), Fs ≥ 1.0 under design earthquake and Fs ≥ 1.2 under heavy rain. But you have to remember that the soil parameters (c', φ', γ) are very uncertain — a laboratory result of c' = 30 kPa can range from 20 to 50 kPa across the same slope in the field. Even Fs = 1.5 can drop below 1.0 if the actual strengths are 20% lower. That is why reliability-based design (with failure probability P_f) is becoming standard, and why mitigation measures aim to raise Fs from 1.2 to 1.5 with margin, rather than just past 1.0.
Frequently asked questions
The Bishop simplified method (1955) is a Limit Equilibrium technique that solves the factor of safety Fs of a circular slip surface. While the Fellenius (ordinary) method takes only the moment of each slice, Bishop accounts for the horizontal forces between slices and treats Fs itself as the unknown to be solved iteratively. Bishop typically gives Fs about 5–20% higher than Fellenius, and for circular slips it agrees closely with Spencer or Morgenstern-Price. This tool uses 10 slices with up to 10 iterations and usually converges within 3–5 iterations for typical parameters.
The pore-pressure ratio r_u is the pore pressure u at the base of a slice divided by the vertical total stress γh of the overlying soil (r_u = u / (γh)). When the groundwater table rises to the slope surface under hydrostatic conditions r_u ≈ 0.5, and with seepage it can reach 0.5–0.7. With the default conditions of this tool (H=20m, β=30°, c'=30kPa, φ'=25°), Fs ≈ 1.5 at r_u = 0.3 and drops to about Fs ≈ 1.0 at r_u = 0.6 (after a heavy storm). This is the physical reason why landslides cluster around the rainy season.
In a pseudo-static analysis, the horizontal seismic coefficient k_h = a/g (acceleration over gravity) is applied to each slice as a horizontal inertial force. Japanese road earthworks (NEXCO) typically use k_h = 0.10–0.15 for Level 1 design earthquakes and 0.15–0.20 for Level 2; the US USACE recommends 0.10–0.15. Setting k_h = 0.15 in this tool increases the driving moment by roughly 30–40% and lowers Fs to about 70% of its static value. Pseudo-static is no substitute for a full dynamic analysis (FLAC, PLAXIS Dynamic), but it remains the standard screening tool in early design.
Fs = 1.0 means the slope is on the verge of failure and is never considered safe in practice. Japanese building-site and slope-stability standards typically require Fs ≥ 1.5 for permanent (no earthquake, normal groundwater) conditions, Fs ≥ 1.0 under earthquake and Fs ≥ 1.2 under heavy rain. This tool's risk class follows the same: Fs > 1.5 stable, 1.2–1.5 caution, 1.0–1.2 warning, < 1.0 failure-imminent. Soil strength parameters carry large uncertainty, so even Fs = 1.5 can drop below 1.0 if the actual c' or φ' is 20% lower than the measured mean — a reliability-based view is recommended in practice.
Real-world applications
Steep-slope disaster zoning and hazard maps: In Japan the Sediment Disaster Prevention Act designates "warning zones (yellow)" and "special warning zones (red)" across every steep slope nationwide. Behind those designations and behind every building-permit assessment in such zones, Bishop simplified Fs calculations are the basic evaluation metric. The hazard maps published by the Geospatial Information Authority and every prefecture are backed by tens of thousands of Fs evaluations linked to digital terrain and soil-property data.
Road and railway cut/fill slope design: NEXCO (Japanese expressways) and JR companies routinely require cut slopes to satisfy Fs ≥ 1.5 (permanent), Fs ≥ 1.0 (earthquake) and Fs ≥ 1.2 (heavy rain). Counter-measures in order of cost are (1) drainage works (horizontal drains, sub-drains) to lower r_u, (2) surface protection (mortar shotcrete, vegetation), (3) ground anchors and stabilising piles, and (4) toe-buttress fills. Simulating "r_u drops from 0.5 to 0.3" in this tool quantifies the Fs recovery you can expect from drainage works.
Dam and reservoir slope stability: Rapid drawdown (sudden lowering of reservoir level) is a classic landslide trigger. The 1963 Vajont dam slide in Italy (≈ 250 Mm³ of rock plunged into the reservoir, ~2000 fatalities) was exactly this. Dam-abutment slopes must be checked for steady-state, full-reservoir, earthquake and rapid-drawdown conditions; the rapid-drawdown case is critical because pore pressures do not dissipate fast enough. Pushing r_u high while increasing β in this tool mimics that destabilisation.
Slope monitoring and early warning: IoT tiltmeters, GNSS sensors and extensometers are increasingly deployed for real-time slope monitoring. The observed displacement rate and rainfall are combined with Bishop-style Fs calculations to set thresholds — "evacuate when cumulative rainfall exceeds X mm and displacement exceeds Y mm/h". Since the 2014 Hiroshima debris flows and the 2021 Atami Izusan landslide, investment in such integrated AI-assisted warning systems has accelerated.
Common misconceptions and pitfalls
The biggest trap is that a deterministic Fs hides the uncertainty of soil parameters. Entering c' = 30 kPa gives a unique Fs, but in the field c' can vary from 10 to 60 kPa across one slope and triaxial values differ by a factor of two from in-situ tests. Fs = 1.5 can drop to 1.0 if the actual strength is just 20% lower. In practice, reliability-based design (Monte Carlo, failure probability P_f) or sensitivity analyses should be used alongside; relying on a single deterministic Fs is dangerous.
Next, do not assume that Bishop simplified works for every slope. Bishop assumes a circular slip surface, so it cannot handle planar slides along a tilted bedrock joint, wedge failures controlled by faults or weak planes, or flow-type debris movements. For weathered or colluvial slopes with bedding parallel to the slope face, planar-slide analysis or Janbu's generalised method is more appropriate. Complex geologies need FEM/FDM (PLAXIS, FLAC) with elasto-plastic models. The "planar slide" and "wedge" options in this tool still run the Bishop solver — they are an educational approximation, not a proper analysis.
Finally, do not over-trust the pseudo-static k_h. Designing for k_h = 0.15 with Fs ≥ 1.0 is convenient, but real earthquakes are tens of seconds of cyclic loading where cumulative displacement, liquefaction and post-peak strength loss matter. After the 1995 Kobe, 2011 Tohoku and 2018 Hokkaido earthquakes, many slopes that passed pseudo-static checks still suffered cumulative-displacement failures. For critical structures and large slopes, displacement-based methods (Newmark) or dynamic FEM are used in parallel. Treat the k_h in this tool as an initial screening, not a verdict.
How to Use
Enter slope height in meters (typical range: 5–50 m for natural slopes; 2–30 m for cut slopes).
Set slope angle in degrees (18–60° depending on soil type; steeper angles increase instability risk).
Input soil cohesion in kPa (clay: 15–40 kPa; silt: 5–15 kPa; sand: 0–5 kPa) and friction angle in degrees (clay: 20–30°; sand: 30–45°).
The simulator calculates Factor of safety Fs using Bishop Simplified Method, resisting moment, driving moment, and risk class (Stable if Fs > 1.3; Marginally Stable if 1.0–1.3; Unstable if Fs < 1.0).
Worked Example
A 15 m high cut slope in clayey silt with slope angle 35°, cohesion c = 22 kPa, friction angle φ = 28°, and pore pressure ratio r_u = 0.15 yields Fs ≈ 1.24 (Marginally Stable). Resisting moment ≈ 8,400 kN·m; driving moment ≈ 6,760 kN·m. Limiting slope angle at Fs = 1.0 is 31°. If groundwater rises (r_u increases to 0.35), Fs drops to 0.89 (Unstable), requiring drainage or slope flattening to 28°.
Practical Notes
Bishop Simplified Method assumes circular slip surface and is valid for slopes 20–70° with uniform soil layers; use detailed methods for complex stratigraphy.
Critical r_u identifies maximum pore pressure ratio before slope failure; monitor during monsoon or snowmelt when r_u often exceeds 0.3 in fine-grained soils.
For Fs between 1.0–1.3, implement drainage, berming, or soil nails; sheet pile walls suit urban cuts near infrastructure.
Sensitivity analysis: decreasing cohesion by 5 kPa typically reduces Fs by 0.08–0.15 depending on height and angle.