Formula Sheet: Slope Stability

Factor of Safety (FS)

General Definition

The Factor of Safety (FS) represents the ratio of resisting forces to driving forces along a potential failure surface.

\[FS = \frac{\text{Resisting Forces}}{\text{Driving Forces}} = \frac{\text{Shear Strength}}{\text{Shear Stress}}\]

• FS > 1.0: Slope is stable
• FS = 1.0: Slope is at limiting equilibrium (incipient failure)
• FS <> Slope is unstable
• Typical minimum acceptable FS values range from 1.3 to 1.5 for permanent slopes

Shear Strength Parameters

Mohr-Coulomb Failure Criterion:

\[\tau_f = c + \sigma_n \tan \phi\]

• τ_f: Shear strength (psf or kPa)
• c: Cohesion intercept (psf or kPa)
• σ_n: Normal stress on failure plane (psf or kPa)
• φ: Angle of internal friction (degrees)

Effective Stress Analysis:

\[\tau_f = c' + \sigma_n' \tan \phi'\]

• c': Effective cohesion (psf or kPa)
• σ_n': Effective normal stress = σ_n - u (psf or kPa)
• φ': Effective angle of internal friction (degrees)
• u: Pore water pressure (psf or kPa)

Infinite Slope Analysis

Dry or Moist Soil (No Seepage)

Factor of Safety:

\[FS = \frac{c}{\gamma z \sin \beta \cos \beta} + \frac{\tan \phi}{\tan \beta}\]

• c: Cohesion (psf or kPa)
• γ: Unit weight of soil (pcf or kN/m³)
• z: Depth to failure surface (ft or m)
• β: Slope angle (degrees)
• φ: Angle of internal friction (degrees)

For purely cohesionless soil (c = 0):

\[FS = \frac{\tan \phi}{\tan \beta}\]

Submerged Slope (Below Water Table)

Factor of Safety:

\[FS = \frac{c'}{\gamma_{sub} z \sin \beta \cos \beta} + \frac{\tan \phi'}{\tan \beta}\]

• γ_sub: Submerged unit weight = γ_sat - γ_w (pcf or kN/m³)
• γ_sat: Saturated unit weight (pcf or kN/m³)
• γ_w: Unit weight of water = 62.4 pcf or 9.81 kN/m³

Seepage Parallel to Slope

Factor of Safety:

\[FS = \frac{c'}{\gamma_{sat} z \sin \beta \cos \beta} + \frac{\gamma'}{\gamma_{sat}} \cdot \frac{\tan \phi'}{\tan \beta}\]

• γ': Effective unit weight = γ_sat - γ_w (pcf or kN/m³)
• Assumes steady-state seepage parallel to slope surface

Circular Arc (Rotational) Failure Analysis

Ordinary Method of Slices (Fellenius Method)

Factor of Safety:

\[FS = \frac{\sum [c' l_i + (W_i \cos \alpha_i - u_i l_i) \tan \phi']}{\sum W_i \sin \alpha_i}\]

• W_i: Weight of slice i (lb or kN)
• l_i: Length of arc at base of slice i (ft or m)
• α_i: Angle between base of slice and horizontal (degrees)
• u_i: Pore water pressure at base of slice i (psf or kPa)
• c': Effective cohesion (psf or kPa)
• φ': Effective friction angle (degrees)
• Note: This method neglects interslice forces and may be conservative

Simplified Bishop Method

Factor of Safety (iterative solution):

\[FS = \frac{\sum \frac{1}{m_\alpha} [c' b_i + (W_i - u_i b_i) \tan \phi']}{\sum W_i \sin \alpha_i}\]

where:

\[m_\alpha = \cos \alpha_i + \frac{\sin \alpha_i \tan \phi'}{FS}\]

• b_i: Width of slice i (ft or m)
• W_i: Weight of slice i (lb or kN)
• α_i: Angle of slice base (degrees)
• u_i: Pore water pressure at base center of slice i (psf or kPa)
• Requires iterative solution since FS appears on both sides
• Assumes vertical interslice forces only
• More accurate than Ordinary Method of Slices

Swedish (Friction Circle) Method

Factor of Safety for φ = 0 (purely cohesive soil):

\[FS = \frac{c L R}{\sum W_i x_i}\]

• c: Cohesion (undrained shear strength) (psf or kPa)
• L: Arc length of failure surface (ft or m)
• R: Radius of circular slip surface (ft or m)
• W_i: Weight of slice i (lb or kN)
• x_i: Horizontal distance from center of rotation to centroid of slice i (ft or m)
• Applicable primarily for total stress analysis (φ = 0 condition)

Translational Failure (Planar Surface)

Single Planar Failure Surface

Factor of Safety:

\[FS = \frac{c' L + (W \cos \beta - U) \tan \phi'}{W \sin \beta}\]

• W: Total weight of sliding mass (lb or kN)
• L: Length of failure plane (ft or m)
• β: Inclination of failure plane (degrees)
• U: Total uplift force from pore water pressure on failure plane (lb or kN)
• c': Effective cohesion (psf or kPa)
• φ': Effective friction angle (degrees)

Uplift Force:

\[U = u_{avg} \cdot A_{base}\]

• u_avg: Average pore water pressure on base (psf or kPa)
• A_base: Area of failure plane (ft² or m²)

Wedge Analysis

Two-Part Wedge

Force equilibrium approach: Solve by resolving forces parallel and perpendicular to each failure plane, considering:

• Weight of each wedge segment
• Normal and shear forces on each failure plane
• Interacting force between wedges
• Pore water pressures on failure surfaces

Resisting force on plane:

\[F_R = c' L + N' \tan \phi'\]

• F_R: Resisting force (lb or kN)
• N': Effective normal force = N - U (lb or kN)
• N: Total normal force (lb or kN)
• U: Pore water uplift force (lb or kN)

Pore Water Pressure and Seepage

Pore Pressure Ratio

\[r_u = \frac{u}{\gamma z}\]

• r_u: Pore pressure ratio (dimensionless)
• u: Pore water pressure (psf or kPa)
• γ: Total unit weight of soil (pcf or kN/m³)
• z: Depth below ground surface (ft or m)
• Typical values: r_u = 0 (dry), r_u = 0.5 (partially saturated), r_u approaching 1.0 (high pore pressure)

Pore Water Pressure from Water Table

\[u = \gamma_w h_w\]

• u: Pore water pressure (psf or kPa)
• γ_w: Unit weight of water = 62.4 pcf or 9.81 kN/m³
• h_w: Height of water above the point (ft or m)

Effective Stress

\[\sigma' = \sigma - u\]

• σ': Effective stress (psf or kPa)
• σ: Total stress (psf or kPa)
• u: Pore water pressure (psf or kPa)

Stability Number (Taylor's Charts)

Stability Number for Homogeneous Slopes

\[N_s = \frac{c}{\gamma H \cdot FS}\]

Rearranging for Factor of Safety:

\[FS = \frac{c}{\gamma H N_s}\]

• N_s: Stability number (dimensionless, obtained from Taylor's charts)
• c: Cohesion (psf or kPa)
• γ: Unit weight of soil (pcf or kN/m³)
• H: Height of slope (ft or m)
• FS: Factor of safety
• Taylor's charts provide N_s values based on slope angle and depth factor

Depth Factor

\[n_d = \frac{D}{H}\]

• n_d: Depth factor (dimensionless)
• D: Depth to firm stratum below toe (ft or m)
• H: Height of slope (ft or m)
• Used to determine whether failure surface extends to firm layer or is entirely within slope

Slope Geometry and Forces

Slope Angle

\[\tan \beta = \frac{V}{H} = \frac{1}{n}\]

• β: Slope angle (degrees)
• V: Vertical rise
• H: Horizontal run
• n: Slope ratio (horizontal:vertical), e.g., n = 2 for 2:1 slope

Weight of Slope Mass

For uniform cross-section:

\[W = \gamma \cdot A\]

• W: Weight per unit length (lb/ft or kN/m)
• γ: Unit weight of soil (pcf or kN/m³)
• A: Cross-sectional area of sliding mass per unit length (ft² or m²)

For slice in method of slices:

\[W_i = \gamma \cdot b_i \cdot h_i\]

• W_i: Weight of slice i per unit width (lb/ft or kN/m)
• b_i: Width of slice (ft or m)
• h_i: Average height of slice (ft or m)

Critical Height of Slope

Cohesive Soil (φ = 0)

Critical height (maximum unsupported vertical cut):

\[H_c = \frac{4c}{\gamma}\]

• H_c: Critical height (ft or m)
• c: Cohesion (undrained shear strength) (psf or kPa)
• γ: Unit weight of soil (pcf or kN/m³)
• For vertical cut (β = 90°) with FS = 1.0

For sloped cut:

\[H_c = \frac{N_s \cdot c}{\gamma}\]

• N_s: Stability number from charts (function of slope angle)
• N_s ≈ 5.5 for vertical cut (90°)
• N_s decreases as slope angle decreases

Seismic Slope Stability

Pseudostatic Analysis

Horizontal seismic force:

\[F_h = k_h \cdot W\]

• F_h: Horizontal seismic force (lb or kN)
• k_h: Horizontal seismic coefficient (dimensionless)
• W: Weight of sliding mass (lb or kN)

Vertical seismic force:

\[F_v = k_v \cdot W\]

• F_v: Vertical seismic force (lb or kN)
• k_v: Vertical seismic coefficient (dimensionless)
• Typically k_v = 0 or k_v = ±0.5k_h

Factor of Safety with seismic loading:

\[FS_{seismic} = \frac{\sum [c' l_i + N_i' \tan \phi']}{\sum [W_i \sin \alpha_i + k_h W_i \cos \alpha_i]}\]

• Seismic force acts through center of gravity of sliding mass
• Reduces factor of safety compared to static conditions

Reinforced Slopes

Factor of Safety with Reinforcement

\[FS = \frac{\text{Resisting Forces + Reinforcement Contribution}}{\text{Driving Forces}}\]

Tensile force contribution from reinforcement:

\[T = T_{allow} \cdot \cos \theta\]

• T: Component of tensile force resisting sliding (lb/ft or kN/m)
• T_allow: Allowable tensile force in reinforcement (lb/ft or kN/m)
• θ: Angle between reinforcement and failure plane (degrees)

Special Considerations

Rapid Drawdown

For rapid drawdown conditions (e.g., reservoir slopes):

• Use undrained shear strength parameters if drainage is too slow
• Pore pressures remain elevated while external water support is removed
• Most critical condition often occurs immediately after drawdown
• Analyze using total stress (φ = 0) or effective stress with appropriate pore pressures

Progressive Failure

Residual shear strength: Use for slopes in pre-sheared soils or slopes that have experienced previous movement

\[\tau_r = c_r + \sigma_n' \tan \phi_r\]

• τ_r: Residual shear strength (psf or kPa)
• c_r: Residual cohesion (often ≈ 0) (psf or kPa)
• φ_r: Residual friction angle (degrees)
• φ_r typically 20-30% lower than peak φ' for clays

Numerical Methods Concepts

Spencer Method

Satisfies both force and moment equilibrium:

• Assumes constant interslice force inclination
• Iteratively solves for FS and interslice force angle
• More rigorous than simplified Bishop
• Requires computational software

Morgenstern-Price Method

• Most general limit equilibrium method
• Satisfies complete force and moment equilibrium
• User defines interslice force function
• Requires computational software

Slope Stability Charts and Correlations

Culmann's Method (Planar Failure)

For cohesionless soil with planar failure surface:

\[FS = \frac{\tan \phi}{\tan \beta}\]

• Maximum height for given slope angle and soil properties can be determined graphically
• Applicable for simple slopes in granular soils

Friction Circle Method

Used for slopes in c-φ soils:

• Radius of friction circle: r = R sin φ
• Where R is radius of slip circle
• Graphical method requiring iterative trial