Slope Stability Calculation

Slope Stability Calculator

Preliminary slope stability analysis using the Simplified Bishop Method with automatic circular slip surface search. The calculator visualizes the slope, water table, critical failure surface, slice geometry, pore pressure, stress distribution, and calculated factor of safety.

Input Parameters

Slope Geometry
Analysis Type
Soil Parameters
Water Table
Search Settings
Higher grid values improve the search but increase calculation time.

Results and Visualization

Factor of Safety
—
Simplified Bishop
Status
—
Compared with required FS
Slope Angle
—
degrees
Valid Surfaces
—
searched candidates

Critical Slice Data

Slice x mid Height Weight Alpha u Normal Stress Effective Normal Available Shear Mobilized Shear
Run calculation to display slice data.

Formula Sheet and Engineering Basis

This calculator evaluates slope stability using the Simplified Bishop limit-equilibrium method for circular slip surfaces. The method divides the potential sliding mass into vertical slices and solves for the factor of safety using moment equilibrium with an approximate treatment of interslice forces.

1. Factor of Safety Definition

General limit-equilibrium definition
\[ FS = \frac{ \text{Available shear resistance along the slip surface} }{ \text{Mobilized shear demand along the slip surface} } \]
Mohr-Coulomb shear strength
\[ \tau_f = c’ + \sigma’ \tan \phi’ \]
Effective normal stress
\[ \sigma’ = \sigma – u \]

2. Simplified Bishop Method

For a circular slip surface divided into vertical slices, the Simplified Bishop factor of safety is solved iteratively as:

Simplified Bishop equation
\[ FS = \frac{ \displaystyle \sum_{i=1}^{n} \frac{ c’_i b_i + \left(W_i – u_i b_i\right)\tan\phi’_i }{ m_{\alpha i} } }{ \displaystyle \sum_{i=1}^{n} W_i \sin \alpha_i } \]
Bishop correction term
\[ m_{\alpha i} = \cos \alpha_i + \frac{ \sin \alpha_i \tan \phi’_i }{ FS } \]

Because \(FS\) appears on both sides of the equation, the calculator uses an iterative solution until the change in computed factor of safety is sufficiently small.

3. Slice Weight and Geometry

Slice weight
\[ W_i = \gamma_{\text{moist}} A_{\text{dry},i} + \gamma_{\text{sat}} A_{\text{sat},i} \]
Base length of slice
\[ b_i = \frac{\Delta x_i}{\cos \alpha_i} \]
Driving shear component
\[ T_i = W_i \sin \alpha_i \]

4. Water Table and Pore Pressure

The water table is modeled as either absent, horizontal, or linearly sloping between the toe and crest. Pore pressure is calculated from the vertical pressure head between the water table and the base of each slice.

Pore water pressure
\[ u_i = \gamma_w h_{w,i} \]
Effective normal force contribution
\[ N’_i = W_i \cos \alpha_i – u_i b_i \]
Available shear resistance per slice
\[ S_i = c’_i b_i + N’_i \tan \phi’_i \]

5. Undrained Clay Option

For short-term undrained clay analysis, the calculator uses total stress parameters by setting the friction angle to zero and treating the cohesion input as undrained shear strength.

Undrained shear strength
\[ \phi = 0 \] \[ \tau_f = S_u \]
Undrained resistance along slice base
\[ S_i = S_u b_i \]

6. Slope Geometry

Slope angle
\[ \beta = \tan^{-1} \left( \frac{H}{L} \right) \]

where \(H\) is the slope height and \(L\) is the horizontal slope run.

7. Automatic Critical Slip Surface Search

The calculator automatically searches trial circular slip surfaces by varying the circle center and radius. Each valid slip surface is divided into slices, analyzed using the Simplified Bishop method, and ranked by factor of safety.

Critical factor of safety
\[ FS_{\text{critical}} = \min \left( FS_1, FS_2, FS_3, \ldots, FS_k \right) \]
Primary Variables
  • \(FS\) = factor of safety
  • \(c’\) = effective cohesion
  • \(\phi’\) = effective friction angle
  • \(S_u\) = undrained shear strength
  • \(W_i\) = weight of slice \(i\)
  • \(b_i\) = base length of slice \(i\)
  • \(\alpha_i\) = inclination of slice base
Water and Stress Variables
  • \(u_i\) = pore water pressure at slice base
  • \(\gamma_w\) = unit weight of water
  • \(h_{w,i}\) = pressure head above slice base
  • \(\sigma\) = total normal stress
  • \(\sigma’\) = effective normal stress
  • \(\tau_f\) = available shear strength
  • \(\gamma_{\text{sat}}\) = saturated unit weight of soil

References

  • Bishop, A. W. (1955). The use of the slip circle in the stability analysis of slopes.
  • Duncan, J. M., Wright, S. G., and Brandon, T. L. Soil Strength and Slope Stability.
  • Das, B. M. Principles of Geotechnical Engineering.
  • U.S. Army Corps of Engineers. Engineering manuals and guidance documents for slope stability analysis.

This tool is intended for preliminary engineering screening and educational use. Final design should be verified using project-specific soil stratigraphy, groundwater conditions, seepage analysis, loading conditions, and an appropriate geotechnical standard of practice.