Slope Stability Analysis, Modeling, and Calculation in Geotechnical Engineering

Introduction

Slope stability analysis is a critical aspect of geotechnical engineering, essential for the design and construction of infrastructure on or near slopes. It involves evaluating the potential for slope failure due to various factors such as soil properties, groundwater conditions, and external loads. This article will discuss the principles of slope stability analysis, demonstrate a modeling approach, and provide a Python-based example for calculating the Factor of Safety (FS).

Key Concepts

1. Factor of Safety (FS)

The Factor of Safety is a dimensionless number that indicates the stability of a slope. It is defined as the ratio of the resisting forces to the driving forces:

An FS greater than 1 indicates stability, while an FS less than 1 suggests a risk of failure.

2. Types of Slope Failures

• Rotational Failure: Typically occurs in cohesive soils, characterized by a curved failure surface.
• Translational Failure: Common in granular soils, involving a planar failure surface.

3. Mohr-Coulomb Failure Criterion

The Mohr-Coulomb failure criterion is used to determine the shear strength of soil, given by:

Slope Stability Analysis Method

In this article, we will focus on the Limit Equilibrium Method (LEM), which divides the slope into slices and analyzes the equilibrium of forces acting on each slice. This method is straightforward and widely used for preliminary analyses.

Example Problem

Consider a slope with the following properties:

• Height of slope (H): 10 m
• Angle of slope (β): 30 degrees
• Cohesion (c): 25 kPa
• Internal friction angle (φ): 20 degrees
• Unit weight of soil (γ): 18 kN/m3

Python Code for Factor of Safety Calculation

The following Python code calculates the Factor of Safety for the given slope properties.

import numpy as np

def calculate_fs(H, beta, c, phi, gamma):
    # Convert angles to radians
    beta_rad = np.radians(beta)
    phi_rad = np.radians(phi)
    
    # Calculate normal and shear forces
    normal_force = gamma * H * np.cos(beta_rad)
    shear_force = c + (normal_force * np.tan(phi_rad))
    
    # Calculate weight of the slope
    weight = gamma * H * np.sin(beta_rad)
    
    # Factor of Safety
    fs = shear_force / weight
    return fs

# Slope properties
H = 10  # Height in meters
beta = 30  # Angle in degrees
c = 25  # Cohesion in kPa
phi = 20  # Friction angle in degrees
gamma = 18  # Unit weight in kN/m3

# Calculate Factor of Safety
fs = calculate_fs(H, beta, c, phi, gamma)
print(f"Factor of Safety (FS): {fs:.2f}")
        

Explanation of the Code

1. Import Libraries: We use NumPy for mathematical calculations.
2. Function Definition: calculate_fs takes the slope properties as inputs and computes the Factor of Safety.
3. Angle Conversion: Angles are converted from degrees to radians for calculations.
4. Force Calculations:The normal force is determined based on the unit weight of the soil and the slope height.The shear force combines the effects of cohesion and the frictional resistance.
5. Weight Calculation: The weight of the slope is calculated based on its geometry and unit weight.
6. FS Calculation: The Factor of Safety is computed and returned.

Output

When you run the code, it will display the calculated Factor of Safety for the given slope parameters.

Modeling Slope Stability

To further understand slope stability, we can model the effects of varying soil properties on the Factor of Safety. This can be visualized by plotting the Factor of Safety against different values of cohesion and friction angle.

Extended Python Code for Modeling

import matplotlib.pyplot as plt

def model_fs_range(H, beta, gamma, c_values, phi_values):
    fs_matrix = np.zeros((len(c_values), len(phi_values)))

    for i, c in enumerate(c_values):
        for j, phi in enumerate(phi_values):
            fs_matrix[i, j] = calculate_fs(H, beta, c, phi, gamma)

    return fs_matrix

# Define ranges for cohesion and friction angle
c_values = np.linspace(10, 50, 10)  # Cohesion from 10 to 50 kPa
phi_values = np.linspace(10, 30, 10)  # Friction angle from 10 to 30 degrees

# Calculate the Factor of Safety for the ranges
fs_results = model_fs_range(H, beta, gamma, c_values, phi_values)

# Plotting
C, Phi = np.meshgrid(c_values, phi_values)
plt.figure(figsize=(10, 6))
contour = plt.contourf(C, Phi, fs_results, levels=20, cmap='viridis')
plt.colorbar(contour, label='Factor of Safety (FS)')
plt.title('Factor of Safety Contour Plot')
plt.xlabel('Cohesion (kPa)')
plt.ylabel('Friction Angle (degrees)')
plt.show()
        

Explanation of the Extended Code

1. Modeling Function: model_fs_range computes the Factor of Safety across a range of cohesion and friction angles.
2. Cohesion and Friction Angle Ranges: We define ranges for cohesion (10 to 50 kPa) and friction angle (10 to 30 degrees).
3. Contour Plotting: A contour plot visualizes how the Factor of Safety varies with different soil properties, providing insights into stability under various conditions.

Conclusion

Slope stability analysis is vital in geotechnical engineering for ensuring the safety and stability of constructions on sloped terrains. This article provided an overview of the concepts involved in slope stability analysis, demonstrated how to calculate the Factor of Safety using Python, and visualized the effects of varying soil properties through modeling.

By combining analytical and graphical methods, engineers can make informed decisions during the design process, ensuring safer and more reliable structures in geotechnical projects.