Slope Stability Calculator
Safety Factor Against Failure · Infinite Slopes · Embankments · Cuts
Slope Stability Calculator
Internal angle of friction; typical 25–45° depending on soil type
Inclination of the slope; β < φ required for stable slopes
Formulas & Fundamentals
Safety Factor Against Failure (Infinite Slope):
η = tan(φ) / tan(β)
Ratio of available friction to required friction
η = tan(φ) / tan(β)
Ratio of available friction to required friction
Stability Criterion:
- η > 1.0 → Slope stable
- η = 1.0 → Critical case (limiting condition)
- η < 1.0 → Slope unstable (failure risk)
Friction Angle from η:
φ = arctan(η · tan(β))
Required friction angle for given safety factor
φ = arctan(η · tan(β))
Required friction angle for given safety factor
Critical Slope Angle:
β = arctan(tan(φ) / η)
Maximum allowable slope angle for specified safety factor
β = arctan(tan(φ) / η)
Maximum allowable slope angle for specified safety factor
Symbol Table
| η | Safety factor against failure [−] |
| φ | Internal friction angle of soil [°] |
| β | Slope inclination angle [°] |
| tan | Tangent function |
Note: This calculation applies to infinite, homogeneous slopes without groundwater
(dry or saturated with parallel water table). For complex geometries, layered deposits, or surface loads,
use numerical stability analyses (e.g., Bishop, Fellenius, Morgenstern-Price method).
Technical Background
Slope Stability Theory
Slope stability is a critical concern in geotechnical engineering. An infinite slope with constant inclination and homogeneous soil (frictionless) can be analyzed using the Rankine model. The safety against failure is defined as the ratio of available (resisting) friction to required friction.
Applications
- Embankments: Typical slope angle 1:2 to 1:5 (18–27°)
- Cuts/Excavations: Dependent on soil class; sand: up to ~30°, clay: up to ~20°
- Natural Hillsides: High variability; geological and hydrographic factors dominate
- Retaining Walls: Backs often inclined 1:10 to 1:3 for stability and drainage
Factors Affecting Stability
| Factor | Effect on Stability | Remark |
|---|---|---|
| Friction Angle φ | ↑ φ → ↑ Stability | Grain size, compaction, relative density |
| Slope Angle β | ↑ β → ↓ Stability | Steeper slopes are less stable |
| Groundwater | Water → ↓ Stability | Increases pore pressure, reduces effective stress |
| Cohesion c | ↑ c → ↑ Stability | Cohesive soils; time- and stress-dependent |
| Surface Load | Load → ↓ Stability | Buildings, traffic, snow at slope crest |
Typical Safety Factors
- Permanent Loads: η ≥ 1.3 to 1.5 (depends on code)
- Temporary Loads: η ≥ 1.1 to 1.2
- Seismic, Snow: Lower values, combined with load factors
Disclaimer: The formula η = tan(φ) / tan(β) presented here
is a simplified solution for friction-only (cohesionless) soils without groundwater. Real slopes
often feature cohesion, variable layers, and groundwater effects. For such cases,
use numerical methods or consult a geotechnical engineer.