When will this hillslope fail? Forces, failure planes, and landslide prediction
2026-02-27
On the morning of 9 August 2010, a hillside above the village of Gansu in northwestern China had stood for centuries. By that evening it was gone. A combination of weeks of saturation from unusually heavy rainfall and a steep slope angle produced a landslide that buried more than 700 homes and killed over 1,700 people. The slope had failed because the pore water pressure from saturated soils had reduced the effective friction holding the soil to the bedrock below — the resisting force dropped below the driving force of gravity, and the whole mass began to move.
Slope stability analysis is the quantitative answer to a question that engineers, geologists, and land-use planners need to answer precisely: given this slope geometry, these soil properties, and these groundwater conditions, how close is this hillside to failure? The tool is the Factor of Safety — the ratio of resisting to driving forces — derived from the Mohr-Coulomb failure criterion that describes how soils and rocks resist shear. A Factor of Safety above 1 means the slope is stable; below 1 means it has already failed; exactly 1 is the critical condition. The power of the method is that it converts a qualitative question about hazard into a number that can be mapped, compared across landscapes, and updated as conditions change with rainfall, earthquake shaking, or excavation.
After heavy rain, which hillslopes will fail as landslides?
Slope stability determines whether soil/rock remains in place or slides downslope.
Driving forces: - Gravity (weight component parallel to slope) - Water pressure (reduces effective stress) - Earthquakes (dynamic loading) - External loads (buildings, traffic)
Resisting forces: - Friction (depends on normal stress) - Cohesion (soil/rock strength) - Vegetation (root reinforcement)
Critical question: Does resistance exceed driving force?
Applications: - Landslide hazard mapping - Cut/fill slope design (roads, buildings) - Dam stability - Mine pit walls - Post-earthquake assessment
By material: - Rock slide: Bedrock failure - Debris slide: Soil + rock fragments - Earth slide: Predominantly soil
By mechanism: - Rotational: Curved failure surface - Translational: Planar failure surface - Complex: Multiple failure modes
By movement: - Slide: Mass moves as coherent unit - Flow: Fluid-like behavior - Fall: Free-fall from cliff - Topple: Forward rotation
Shear strength:
\tau_f = c + \sigma' \tan\phi
Where: - \tau_f = shear strength (kPa) - c = cohesion (kPa) - \sigma' = effective normal stress (kPa) - \phi = friction angle (degrees)
Effective stress:
\sigma' = \sigma - u
Where: - \sigma = total stress - u = pore water pressure
Key insight: Water reduces effective stress → reduces shear strength!
FS = \frac{\text{Resisting Forces}}{\text{Driving Forces}} = \frac{\tau_f}{\tau}
Interpretation: - FS > 1.5: Stable (safe) - FS = 1.0-1.5: Marginal (monitor) - FS < 1.0: Failure (landslide occurs)
Design criterion: Typically require FS > 1.5 for permanent slopes.
Assumptions: - Uniform slope angle \theta - Failure parallel to surface - Infinite extent (no edge effects)
Forces on soil element:
Weight: W = \gamma z b L
Where: - \gamma = unit weight of soil (kN/m³) - z = depth to failure plane (m) - b = width (m) - L = length along slope (m)
Weight components:
Parallel to slope (driving):
W_\parallel = W \sin\theta = \gamma z b L \sin\theta
Perpendicular (normal):
W_\perp = W \cos\theta = \gamma z b L \cos\theta
Normal stress:
\sigma = \frac{W_\perp}{b L / \cos\theta} = \gamma z \cos^2\theta
Shear stress (driving):
\tau = \frac{W_\parallel}{b L / \cos\theta} = \gamma z \sin\theta \cos\theta
Shear strength (resisting):
\tau_f = c + \sigma' \tan\phi = c + \gamma z \cos^2\theta \tan\phi
(Assuming dry slope: \sigma' = \sigma)
Factor of Safety:
FS = \frac{\tau_f}{\tau} = \frac{c + \gamma z \cos^2\theta \tan\phi}{\gamma z \sin\theta \cos\theta}
Simplified:
FS = \frac{c}{\gamma z \sin\theta \cos\theta} + \frac{\tan\phi}{\tan\theta}
For cohesionless soil (c = 0):
FS = \frac{\tan\phi}{\tan\theta}
Critical slope angle: \theta_{\text{crit}} = \phi (where FS = 1)
Saturated slope with seepage parallel to surface:
Pore pressure: u = \gamma_w z_w \cos^2\theta
Where: - \gamma_w = unit weight of water (9.81 kN/m³) - z_w = depth to water table
Effective stress:
\sigma' = \gamma z \cos^2\theta - \gamma_w z_w \cos^2\theta
If fully saturated (z_w = z):
\sigma' = (\gamma - \gamma_w) z \cos^2\theta = \gamma' z \cos^2\theta
Where \gamma' = \gamma - \gamma_w = buoyant unit weight
Factor of Safety (saturated):
FS = \frac{c}{\gamma z \sin\theta \cos\theta} + \frac{\gamma'}{\gamma} \frac{\tan\phi}{\tan\theta}
Key result: \gamma'/\gamma \approx 0.5 for typical soils → FS reduced by ~50% when saturated!
Earthquake adds horizontal force:
Pseudo-static approach:
F_h = k_h W
Where k_h = horizontal seismic coefficient (~0.1-0.3 for strong shaking)
Modified driving stress:
\tau_{\text{seismic}} = \gamma z (\sin\theta + k_h \cos\theta) \cos\theta
Factor of Safety:
FS_{\text{seismic}} = \frac{c + \gamma z \cos^2\theta \tan\phi}{\gamma z (\sin\theta + k_h \cos\theta) \cos\theta}
Earthquake reduces FS by increasing driving force.
Problem: Calculate factor of safety for hillslope.
Slope conditions: - Angle: \theta = 30° - Soil depth to bedrock: z = 3 m - Unit weight: \gamma = 18 kN/m³ - Cohesion: c = 5 kPa - Friction angle: \phi = 35°
Calculate FS for: 1. Dry slope 2. Fully saturated slope 3. Saturated slope with earthquake (k_h = 0.15)
Case 1: Dry slope
\tau = \gamma z \sin\theta \cos\theta = 18 \times 3 \times \sin(30°) \times \cos(30°)
= 18 \times 3 \times 0.5 \times 0.866 = 23.4 \text{ kPa}
\sigma = \gamma z \cos^2\theta = 18 \times 3 \times (0.866)^2 = 40.5 \text{ kPa}
\tau_f = c + \sigma \tan\phi = 5 + 40.5 \times \tan(35°) = 5 + 40.5 \times 0.700 = 33.4 \text{ kPa}
FS = \frac{33.4}{23.4} = 1.43
Marginal stability (just below typical design criterion of 1.5)
Case 2: Saturated slope
Buoyant unit weight: \gamma' = 18 - 9.81 = 8.19 kN/m³
\sigma' = \gamma' z \cos^2\theta = 8.19 \times 3 \times 0.75 = 18.4 \text{ kPa}
\tau_f = 5 + 18.4 \times 0.700 = 17.9 \text{ kPa}
FS = \frac{17.9}{23.4} = 0.76
FAILURE! (FS < 1.0)
Case 3: Saturated + earthquake
\tau_{\text{eq}} = \gamma z (\sin\theta + k_h\cos\theta) \cos\theta
= 18 \times 3 \times (0.5 + 0.15 \times 0.866) \times 0.866
= 18 \times 3 \times (0.5 + 0.130) \times 0.866 = 29.4 \text{ kPa}
FS = \frac{17.9}{29.4} = 0.61
Catastrophic failure (FS well below 1.0)
Summary: - Dry: FS = 1.43 (marginal) - Saturated: FS = 0.76 (fails) - Saturated + earthquake: FS = 0.61 (catastrophic)
This explains why landslides often occur during/after heavy rain or earthquakes!
Below is an interactive slope stability calculator.
<label>
Slope angle (°):
<input type="range" id="slope-angle" min="10" max="50" step="1" value="30">
<span id="angle-val">30</span>°
</label>
<label>
Friction angle φ (°):
<input type="range" id="friction-angle" min="20" max="45" step="1" value="35">
<span id="friction-val">35</span>°
</label>
<label>
Cohesion (kPa):
<input type="range" id="cohesion" min="0" max="20" step="1" value="5">
<span id="cohesion-val">5</span>
</label>
<label>
Water table depth (m):
<input type="range" id="water-depth" min="0" max="3" step="0.5" value="3">
<span id="water-val">3.0</span> (3m = dry)
</label>
<label>
Seismic coefficient:
<input type="range" id="seismic" min="0" max="0.3" step="0.05" value="0">
<span id="seismic-val">0.00</span>
</label>
<div class="stability-info">
<p><strong>Factor of Safety:</strong> <span id="fs-value">--</span></p>
<p><strong>Status:</strong> <span id="stability-status">--</span></p>
<p><strong>Driving stress:</strong> <span id="tau-drive">--</span> kPa</p>
<p><strong>Resisting stress:</strong> <span id="tau-resist">--</span> kPa</p>
</div><canvas id="stability-canvas" width="700" height="400" style="border: 1px solid #ddd;"></canvas>Try this: - Increase slope angle: FS drops (steeper = less stable) - Lower water table (0m = saturated): FS drops dramatically! - Add seismic loading: FS drops further - Increase friction angle: Higher FS (stronger soil) - Add cohesion: Raises FS curve (especially at steep angles) - Green area: Stable (FS > 1.5) - Yellow area: Marginal (FS 1.0-1.5) - Red area: Failure (FS < 1.0) - Pink dot: Current slope condition - Watch FS cross failure threshold!
Key insight: Water + earthquakes = catastrophic combination for slope stability!
Mechanism: 1. Rain infiltrates soil 2. Water table rises 3. Pore pressure increases 4. Effective stress decreases 5. Shear strength drops 6. Failure when FS < 1.0
Threshold rainfall: - Short duration, high intensity: Shallow slides - Long duration, moderate intensity: Deep-seated slides
Example - Pacific Northwest: - Threshold: ~200mm in 3 days - Thousands of landslides triggered
Case study - 2015 Nepal: - Magnitude 7.8 earthquake - 25,000+ landslides triggered - Some on slopes previously stable (FS > 1.5 dry) - Liquefaction in saturated soils
Newmark displacement: - Permanent downslope movement during shaking - Depends on: Peak acceleration, critical acceleration, duration
Road construction:
Typical criterion: FS > 1.5 for permanent cuts
Design strategies: - Flatten slope (reduce \theta) - Improve drainage (keep dry) - Soil nails/anchors (add resistance) - Retaining walls
Cost trade-off: - Steeper → less excavation → cheaper - BUT steeper → lower FS → higher failure risk
Reality: Soil has layers with different properties.
Weak layer controls failure (lowest strength).
Example: - Strong surface soil (c=10 kPa, φ=40°) - Weak clay layer at depth (c=2 kPa, φ=25°) - Failure occurs along clay
Solution: Multilayer analysis, find critical surface.
Creep: Slow downslope movement
Residual strength: After movement starts, strength drops to residual value (< peak).
Progressive failure: Small movements → strength reduction → more movement
Solution: Use residual strength for long-term stability.
Infinite slope = 2D (no edge effects)
Real slopes: 3D geometry matters - Converging topography (gullies) → less stable - Diverging topography (ridges) → more stable
Solution: 3D limit equilibrium or FEM analysis.
Roots: Add tensile strength (resistance)
But: Trees add weight (driving force) + wind loading
Net effect: Usually stabilizing for shallow slides, destabilizing for deep slides
Deforestation: Increases landslide risk (roots decay in 2-5 years)
Combine factors spatially:
From DEM: - Slope angle - Aspect (wetness) - Curvature (convergent topography) - Upslope contributing area (water accumulation)
From soils: - Soil type (strength parameters) - Depth to bedrock
From land cover: - Vegetation (root strength) - Land use
Statistical model (logistic regression):
P(\text{landslide}) = \frac{1}{1 + e^{-(\beta_0 + \beta_1 \theta + \beta_2 c + \cdots)}}
Or physically-based (SHALSTAB, SINMAP):
Apply infinite slope model to every pixel, calculate FS.
Output: Landslide susceptibility map (high/medium/low)
Vector components:
Force F at angle \alpha:
F_x = F \cos\alpha F_y = F \sin\alpha
Translational equilibrium:
\sum F_x = 0 \sum F_y = 0
Rotational equilibrium:
\sum M = 0
(Sum of moments about any point)
Weight W on slope angle \theta:
Parallel to slope:
W_\parallel = W \sin\theta
Normal to slope:
W_\perp = W \cos\theta
These satisfy: W_\parallel^2 + W_\perp^2 = W^2