The Limit Equilibrium Method

The Limit Equilibrium Method represents the fundamental approach to slope stability analysis, evaluating the stability of slopes by examining the equilibrium of forces acting on a potential failure mass. This method operates on the principle that a slope remains stable when the resisting forces, primarily the shear strength of the soil, exceed or equal the driving forces such as weight and other destabilizing influences.

The first step in a limit equilibrium analysis is to select a candidate failure surface which is typically circular. For slopes with narrow layers of weak soil, a non-circular failure surface may be used.

The stresses along the failure surface are then calculated assuming the soil is at the limit of static equilibrium (\(\tau\)). Next, the total available strength along the failure surface is calculated (\(s\)).Then the factor of safety \(F\) is expressed mathematically as:

\(F = \dfrac{s}{\tau}\)

The shear strength of soil is based on the Mohr-Coulomb failure criterion. For a total stress analysis:

\(s = c + \sigma \tan \phi\)

Where \(s\) represents the shear strength, \(c\) is the cohesion, \(\sigma\) is the normal stress, and \(\phi\) is the angle of internal friction. For saturated soils, typically \(\phi=0\) and \(c = S_u\) where \(S_u\) is the undrained shear strength. For an effective stress analysis:

\(s = c' + \sigma' \tan \phi'\)

where \(c'\) is the effective cohesion, \(\sigma'\) is the effective normal stress, and \(\phi'\) is the effective angle of internal friction. This expression is typically written as:

\(s = c' + (\sigma - u) \tan \phi'\)

where \(u\) is the pore pressure. In XSLOPE, pore pressures are defined by entering the geometry of a piezometric line and then for any point below the line, the pore pressure is calculated as:

\(u = \Delta y * \gamma_w\)

where \(\Delta y\) is the distance from the piezometric line to the point in question and \(\gamma_w\) is the unit weight of water.

The pore pressures can also be derived from a 2D finite element seepage analysis of the slope using the seepage tools in XSLOPE. In this case, the pore pressure at a point is interpolated from the nodes of the element containing the point using the element basis functions. This is the most accurate way to obtain pore pressures.

Developed Shear Strength

When applying the limit equilibrium method, we often utilize what is called the "developed shear stregth". The main equation:

\(F = \dfrac{s}{\tau}\)

can be rewritten as:

\(\tau = \dfrac{s}{F}\)

\(\tau = \dfrac{c + \sigma \tan \phi}{F}\)

\(\tau = \dfrac{c}{F} + \dfrac{\sigma \tan \phi}{F}\)

Then the two terms can be expressed as:

\(c_d = \dfrac{c}{F}\)

\(\tan \phi_d = \dfrac{\tan \phi}{F}\)

where \(c_d\) = the developed or mobilized cohesion and \(\tan \phi_d\) = the developed or mobilized friction. For example, if \(F = 2.0\), the mobilized cohesion and friction would be one half the total available strength values.

Method of Slices

The Method of Slices represents a numerical technique that divides the potential failure mass into a series of vertical slices for analysis. Rather than analyzing the entire mass as a single unit, each slice is examined individually, with the overall stability determined by summing the forces and moments acting on all slices. This approach allows us to handle complex geometries and varying soil conditions that would be impossible to analyze using simpler methods.

For equilibrium to be satisfied for a candidate failure surface, the following three conditions must be satisfied:

\(\Sigma F_x = 0\)

\(\Sigma F_x = 0\)

\(\Sigma M = 0\)

Typically, the number of equations is less than the number of unknowns. Therefore, simplifying assumptions must be used. Some techniques satisfy only a portion of the equilibrium conditions. If all conditions are satisfied, it is called a complete equilibrium procedure.

The basic forces on each slice and the slice geometry are as follows:

where:

\(W\) = weight of the slice
\(N\) = normal force acting on the base of the slice
\(S\) = shear force acting on the base of the slice
\(E_i\) = interslice force acting on the left side of the slice
\(X_i\) = interslice force acting on the right side of the slice
\(\Delta x\) = width of the slice
\(\Delta \ell\) = length of the slice base, calculated as \(\Delta x \sec \alpha\)
\(\alpha\) = inclination angle of the slice base
\(\beta\) = inclination angle of the slope at the top of the slice

Sometimes the side forces are represented as:

where:

\(Z_i\) = magnitude of the interslice force
\(\theta_i\) = inclination angle of the interslice force

Slopes can be either left-facing or right-facing. The diagrams above are based on a left-facing slope and the equations for each of the analysis methods are based on this convention. For right-facing slopes, the equations are the same, but the directions of the forces are reversed simply by changing the sign convention of $ \alpha$ and \(\beta\) as follows:

One exception to this is Spencer's method. Details can be found in the XSLOPE source code for Spencer's method.

Advanced Loading Conditions

In addition to the basic forces acting on each slice, modern slope stability analysis often incorporates additional loading conditions that can significantly affect the stability calculations. These advanced loading conditions include distributed loads, seismic loads, reinforcement forces, and tension cracks.

Distributed Loads: Surface loads such as traffic, construction equipment, or other surcharges can be represented as distributed forces acting on the top of slices. These loads contribute to both the driving forces and the normal forces on the slice base. Distributed loads correspond to water for submerged or partially submerged slopes or surchage loads from buildings, roads, or other structures above the slope.

In XSLOPE, these loads can be applied as a uniform load across the top of the slices or as a varying load that changes with position along the slope. Distributed loads are applied as a load per unit length, which is then converted to a force acting on each slice based on its width and the load intensity (see below).

Seismic Loads: In seismically active regions, the stability of slopes can be significantly affected by seismic forces. These forces are typically represented as horizontal accelerations acting on the mass of each slice. The pseudo-static approach is commonly used, where a horizontal acceleration coefficient \(k\) is applied to the weight of each slice to calculate the seismic force.

The seismic force is assumped to act horizontally in a direction that causes sliding and it acts through the center of gravity of each slice.

Reinforcement: Reinforcement forces are used to represent the effects of geosynthetics, soil nails, or other structural elements that provide additional stability to slopes. These forces can be modeled as tensile forces acting along the base of the slices, providing resistance to sliding and contributing to the overall stability of the slope.

For the limit equilibrium methods in XSLOPE, the reinforcement is assumed to be flexible and therefore acts in a direction parallel to the base of the slice in a direciton that resists shear. More comprehensize treatment of reinforcement is included in the finite element method.

Tension Cracks: In the upper part of the slope, the cohesion of a soil can be greater than the driving forces. Since soils can generally not withstand tension, this is unconservative. To address this problem, a tension crack can be added to a user-specified depth and the tension crack forms the upper boundary of the slices and not cohesive forces are allowed on the slice (crack) boundary. It is also possible to assume that the crack fills with water, providing a small force driving failure as an extra measure of conservative analysis.

When applied to the slices, these advanced loading conditions are treated as additional forces acting on the slices as follows:

The distributed load acting on the top of each slice is converted to a resultant force by multiplying the load intensity by the width of the slice and it acts at point \(d\). The seismic force \(kW\) acts horizontally at the center of gravity of each slice, and the reinforcement force \(P\) acts along the base of the slice in a direction that resists sliding. The load corresponding to water in the tension crack is converted to a single resultant force \(T\) acting horizontally at point \(c\) which is one third of the distance from the bottom of the slice.

Limit Equilibrium Methods Supported in XSLOPE

The following limit equilibrium methods are supported in XSLOPE. Each method has a page describing in detail the equations used, solution technique, etc.

Ordinary Method of Slices (OMS): The Ordinary Method of Slices provides the simplest approach to slope stability analysis. Since it requires no iteration, results can be obtained rapidly. However, these advantages come with significant limitations: the method only satisfies moment equilibrium, can produce unconservative or overly consrvative results, and completely neglects interslice forces. This makes OMS most suitable for preliminary analysis and simple geometries where quick results are more important than high accuracy.

Simplified Janbu Method: The Janbu Method offers a different approach by satisfying force equilibrium rather than moment equilibrium. This makes it suitable for circular or non-circular failure surfaces where moment equilibrium might be less critical. The method includes a correction factor to compensate for the missing moment equilibrium, though this factor is empirical and may not always provide accurate compensation. [Documentation]

Bishop's Simplified Procedure: Bishop's Method represents a significant improvement over OMS and Janbu by satisfying both moment and vertical force equilibrium. This approach provides more accurate results while maintaining reasonable computational efficiency. However, the requirement for circular surfaces limits its applicability, and the iterative solution process increases computational time. Bishop's method finds its niche in projects with circular failure surfaces and moderate accuracy requirements.

Force Equilibrium Methods: The General Force Equilibrium Method provides a framework for explicit treatment of interslice forces using magnitude and angle parameters. This method works on any failure surface and offers good accuracy when interslice forces are important. However, it only satisfies force equilibrium, requires an iterative solution, and depends on assumptions about interslice force angles. This method is most valuable when explicit interslice force treatment is needed and when complex geometries require more sophisticated analysis than simple methods can provide. Two variations of the force equilibrium method are available in XSLOPE:

The Corps Engineers method applies the force equilibrium framework with the assumption that interslice forces are horizontal.

The Lowe-Karafiath method assumes that the interslice force at each slice boundary is equal to the average of the slope angle at the top of the slice \(\beta\) and the failure surface angle at the bottom of the slice \(\alpha\).

Spencer's Method: Spencer's Method represents the most sophisticated approach available in XSLOPE, satisfying both force and moment equilibrium simultaneously. This comprehensive approach provides the highest accuracy and can handle both circular and non-circular failure surfaces. However, this accuracy comes at a cost: the method is the most complex to implement, requires and iterative solution, and is computationally intensive. Spencer's method is generally considered the best and most accurate of the methods supported in XSLOPE.

The primary features of the limit equilibrium methods supported in XSLOPE are summarized in the table below:

Method	Equilibrium Conditions	Failure Surface	Iterative Solution	Interslice Forces	Accuracy
Ordinary Method of Slices	Overall Moment	Circular	No	None	Low
Simplified Janbu	\(\Sigma F_x=0\)	Circular/Non-Circular	No	None	Low
Bishop's Simplified Procedure	Overall Moment, \(\Sigma F_y=0\)	Circular	Yes	None	Moderate
Corps Engineers	\(\Sigma F_x=0\), \(\Sigma F_y=0\)	Circular/Non-Circular	Yes	Horizontal	High
Lowe-Karafiath	\(\Sigma F_x=0\), \(\Sigma F_y=0\)	Circular/Non-Circular	Yes	Average of Slope and Surface Angles	High
Spencer's Method	\(\Sigma F_x=0\), \(\Sigma F_y=0\), \(\Sigma M=0\)	Circular/Non-Circular	Yes	Explicit	Very High

Automated Search for the Critical Factor of Safety

For each of the solution methods, the factor of safety can be computed either for a single failure surface or XSLOPE can perform an exhaustive search where a large number of candidate failure surfaces are considered until the surface with the minimum or critical factor of safety is found. This process is described in more detail in the search documentation.

Rapid Drawdown Analysis

Rapid drawdown analysis represents a specialized application that can use any of the other methods as its foundation. This approach is specifically designed for dam and levee analysis where water level changes create unique stability challenges. The method accounts for undrained conditions and provides multi-stage analysis that captures the complex behavior of soils during rapid water level changes. However, this specialization limits its applicability to specific scenarios, and the multi-stage approach increases computational complexity. In XSLOPE, rapid drawdown analysis can be peformed with any of the supported limit equilibrium methods. Documentation

Reliability Analysis

XSLOPE includes an option to perform a reliability analysis with any of the supported limit equilibrium methods. Rather than finding a single factor of safety, selected inputs are perturbed and the critical factor of safety is computed for each combination of inputs allowing the computation of a probability of failure. Documentation

Code Examples and Usage

To perform a limit equilibrium analsysi in XSLOPE, the user must first define the slope geometry, material properties, distributed loads, etc using the Excel Input Template as described in the Input Template page. The input template can then be loaded into Python using the 'load_slope_data' function. This function loads the input file and returns a dictionary containing the data from each sheet. The data can then be accessed using the sheet name as the key. For example:

import xslope as xslope
from xslope.fileio import load_slope_data

data = load_slope_data("input_template.xlsx")
profile_lines = data["profile_lines"]
materials = data["materials"]
piezo_line = data["piezo_line"]
gamma_w = data["gamma_water"]
circle = data["circles"][0]  # or whichever one you want
non_circ = data["non_circ"]
dloads = data["dloads"]
max_depth = data["max_depth"]
reinforce_lines = data["reinforce_lines"]

However, you don't normally need to access the data directly. In most cases you simply load the slope data and display it as follows:

import xslope as xslope
from xslope.fileio import load_slope_data
from xslope.plot import lot_inputs,

slope_data = load_slope_data(file_name)
plot_inputs(slope_data)

This loads the slope data into a dictionary and displays the inputs as follows:

The next is to select the method you want to use for the analysis. You can choose to perform a single analysis for a selected failure surface or perform an exhaustive search to find the surface with the lowest or critical factor of safety. If you are performing a single analysis, your first build a set of slices using the 'generate_slices' function. This function takes the slope data and the failure surface you want to analyze as inputs and returns a set of slices in a pandas DataFrame. These slices are then passed to the 'solve_selected' function along with a string defining the method to be used ("oms","bishop","janbu","corps_engineers","lowe_karafiath","spencer"). This function returns a dictionary containing the results of the analysis which can then be plotted using the 'plot_solution' function. For example:

import xslope as xslope
from xslope.slice import generate_slices
from xslope.plot import plot_solution
from xslope.solve import solve_selected

circle = slope_data['circles'][0] if slope_data['circular'] else None
on_circ = slope_data['non_circ'] if slope_data['non_circ'] else None
success, result = generate_slices(slope_data, circle=circle, non_circ=non_circ, num_slices=num_slices)
if success:
  slice_df, failure_surface = result
  results = solve_selected(method, slice_df, rapid=rapid_drawdown)
  plot_solution(slope_data, slice_df, failure_surface, results, save_png=save_png)
else:
  print(result)
  exit()

The 'plot_solution' function takes the slope data, the slices, the failure surface, and the results of the analysis as inputs and plots the results. The results are displayed as follows:

To perform an automated search for the critical factor of safety, you can use the 'circular_search' function for circular failure surfaces or the 'non_circular_search' function for non-circular failure surfaces. This function takes the slope data, the method string, and flag for the rapid drawdown analysis as inputs and returns a dictionary containing the results of the analysis. The results are then plotted using the 'circular_search_results' or 'non_circular_search_results' functions. You can also plot the results for the critical circle using the plot_solution function. For example:

import xslope as xslope
from xslope.plot import plot_circular_search_results, plot_noncircular_search_results
from xslope.solve import solve_all
from xslope.search import circular_search, noncircular_search

if surface_type == "circular":
  fs_cache, converged, search_path = circular_search(slope_data, method, rapid=rapid_drawdown, diagnostic=False)
  plot_circular_search_results(slope_data, fs_cache, search_path, save_png=save_png)
else:
  fs_cache, converged, search_path = noncircular_search(slope_data, method, rapid=rapid_drawdown, diagnostic=False)
  plot_noncircular_search_results(slope_data, fs_cache, search_path, save_png=save_png)

The results are displayed as follows:

You can also extract the critical factor of safety from the results dictionary as follows:

import xslope as xslope
from xslope.plot import plot_solution

critical_surface = fs_cache[0]
slice_df = critical_surface['slices']
failure_surface = critical_surface['failure_surface']
results = critical_surface['solver_result']
plot_solution(slope_data, slice_df, failure_surface, results, save_png=save_png)

The results are displayed as follows:

To perform a reliability analysis, you can use the 'reliability_analysis' function. The input template should include a set of standard deviations for each of the material properties. The results are then plotted using the 'plot_reliability_results' function. For example:

import xslope as xslope
from xslope.advanced import reliability
from xslope.plot import plot_reliability_results

circular = (surface_type == "circular")
  success, result = reliability_analysis(slope_data, method, rapid=rapid_drawdown, circular=circular, debug_level=1)
if success:
  plot_reliability_results(slope_data, result, save_png=save_png)
else:
  print(f"Reliability analysis failed: {result}")

You can find a Google Colab notebook with the code examples and usage and a set of sample problems in the Sample Problems page.