Cheatsheet: Slope Stability

1. Fundamental Concepts

1.1 Factor of Safety

• FS > 1.0: Stable slope
• FS = 1.0: Limiting equilibrium (incipient failure)
• FS < 1.0:="" unstable="" slope="" (failure="">
• Minimum FS = 1.3-1.5 for permanent slopes
• Minimum FS = 1.2-1.3 for temporary slopes

1.2 Types of Slope Failures

2. Infinite Slope Analysis

2.1 Dry or Moist Soil

2.2 Seepage Parallel to Slope

2.3 Variables

• z = depth below surface measured vertically
• β = slope angle from horizontal
• γ = unit weight of soil
• γ_sat = saturated unit weight
• γ' = submerged (buoyant) unit weight = γ_sat - γ_w
• γ_w = unit weight of water = 9.81 kN/m³ or 62.4 pcf
• c = cohesion
• φ = friction angle
• u = pore water pressure

3. Finite Slope Analysis - Circular Failure

3.1 Ordinary Method of Slices (Fellenius Method)

• W = weight of slice
• α = angle of slice base from horizontal
• l = length of slice base
• u = pore pressure at slice base
• Assumes interslice forces are zero (conservative)

3.2 Simplified Bishop Method

• b = width of slice
• Iterative solution required (FS appears on both sides)
• Assumes vertical interslice forces only
• More accurate than Fellenius method

3.3 φ = 0 Analysis (Undrained Clay)

• c_u = undrained shear strength
• L_d = developed arc length
• r = radius of failure circle
• H = slope height
• N_s = 3.8-4.0 for toe circle
• N_s = 5.5-6.0 for slope circle

3.4 Slope Circle Types

4. Effective Stress vs Total Stress Analysis

4.1 Analysis Types

4.2 Pore Pressure Parameters

• r_u = 0: Dry slope
• r_u = 0.5: Groundwater at surface with no seepage
• r_u > 0.5: Artesian conditions

5. Slope Stabilization Methods

5.1 Geometric Modifications

• Flatten slope angle: Reduces driving forces
• Reduce slope height: Reduces total weight of sliding mass
• Add berm or counterweight at toe: Increases resisting moment
• Remove material from crest: Reduces driving moment

5.2 Drainage Methods

• Horizontal drains: Reduce pore pressures within slope
• Vertical wells: Lower groundwater table
• Surface drainage: Prevent infiltration, reduce saturation
• Drainage blankets: Collect and remove seepage water
• Reducing pore pressure increases effective stress and shear strength

5.3 Structural Reinforcement

5.4 Soil Improvement

• Chemical stabilization: Lime, cement, or other additives to increase strength
• Grouting: Injection of cement or chemical grout to fill voids and increase cohesion
• Compaction: Densification to increase shear strength and reduce permeability
• Soil replacement: Remove weak soil and replace with engineered fill

6. Seismic Slope Stability

6.1 Pseudostatic Analysis

• PGA = Peak Ground Acceleration (fraction of g)
• Seismic forces treated as constant inertial forces added to static forces
• Reduces FS; FS = 1.1-1.15 acceptable for seismic conditions
• Conservative approach; assumes peak acceleration acts continuously

6.2 Newmark Sliding Block Analysis

• Displacement-based method for earthquake-induced slope movement
• Critical acceleration: a_c = (FS_static - 1)g sin α
• Permanent displacement occurs when ground acceleration exceeds a_c
• Cumulative displacement calculated from acceleration time history
• Acceptable displacement: 10-30 cm for earth dams, 5-15 cm for critical slopes

7. Special Conditions

7.1 Rapid Drawdown

• Occurs when water level drops quickly (reservoir, excavation dewatering)
• Pore pressures in soil remain high while external water support removed
• Critical for upstream slope of dams and levees
• Use undrained strength parameters for clay; effective stress for granular soils with high permeability
• Most critical when drawdown rate exceeds drainage capacity of soil

7.2 Liquefaction Susceptibility

7.3 Sensitive Clays

• Sensitivity (S_t) = undisturbed strength / remolded strength
• S_t > 4: Sensitive; S_t > 8: Extra-sensitive; S_t > 16: Quick clay
• Progressive failure possible: local failure causes remolding and strength loss
• Use residual strength for design in high-sensitivity clays

7.4 Tension Cracks

• Develop at crest of slope in cohesive soils
• Depth of tension crack: z_c = 2c/(γ√K_a) where K_a = active earth pressure coefficient
• Simplified: z_c = 2c/γ for vertical crack
• Water-filled cracks add hydrostatic pressure (destabilizing force)
• Reduce effective cohesion along failure surface

8. Chart Solutions and Correlations

8.1 Taylor's Stability Charts

• Graphical solutions for homogeneous slopes with circular failure
• Stability number: N_s = γH/(cFS) for φ = 0 analysis
• Stability factor: N_f = c/(γH×FS) for c-φ soils
• Charts provide N_s or N_f as function of slope angle and depth factor

8.2 Friction Circle Method

• Graphical method for circular failure surfaces
• Friction circle radius: r_f = r sin φ where r = failure circle radius
• Resultant of all forces must be tangent to friction circle for limiting equilibrium
• Used for preliminary analysis; less accurate than method of slices

8.3 Strength Parameters

9. Back Analysis and Monitoring

9.1 Back Analysis

• Determine strength parameters from known failure by setting FS = 1.0
• Use actual failure surface geometry if known
• Solve for unknown strength parameters (c, φ) from equilibrium equations
• Useful for residual strength determination in reactivated slides

9.2 Instrumentation

10. Design Considerations

10.1 Minimum Factor of Safety

10.2 Critical Failure Surfaces

• Perform multiple trial circles to find minimum FS
• Search routine: vary circle center location and radius systematically
• Critical circle depends on soil layering, strength distribution, and pore pressures
• Check toe, slope, and base circles
• For multilayered slopes, consider composite failure surfaces

10.3 Partial Safety Factors (LRFD Approach)

• Resistance factor for shear strength: φ = 0.65-0.85
• Load factors: Dead load = 1.25-1.5; Live load = 1.75
• Factored resistance ≥ Factored loads
• Alternative to global factor of safety approach