# Stability of Slopes

### Contents

**Recall Failure of the Slope**1**Recall Assumptions of the Slope Analysis**2**List various types of Factor of Safety**3**Classify the Slope Failure**4**Recall Stability Analysis of Infinite Slope for Cohesionless Soil: i. In Dry Soil Conditions ii. In Submerged Slope iii. In Steady Seepage along the Slope**5**Recall Stability Analysis of Infinite Slope for Cohesive Soil: i. In Dry Soil Conditions ii. In Submerged Slope iii. In Steady Seepage along the Slope**6**Define Wedge Failure**7**Recall Φ<sub>u</sub> = 0 Analysis**7**Describe the Friction Circle Method for Analysis of Finite Slope**8**Recall Taylor’s Stability Number and its significance**9**Describe the Swedish Circle Method for Analysis of Finite Slope**10**Recall the Location of Most Critical Circle**11**Describe the Culmann’s Method for Stability Analysis of Homogeneous Finite Slope**12**Recall Stability of Finite Slopes under the following Conditions: i. Steady Seepage Condition ii. Sudden Drawdown Condition iii. During Construction Condition**13

Recall Failure of the Slope

Failure of a slope refers to the collapse or movement of soil or rock material down a slope due to the weight of the material and the forces acting on it. The failure of a slope can occur as a result of a variety of factors, including overloading, erosion, weathering, and changes in water levels.

There are several types of slope failure, including the following:

- Rotational failure: Rotational failure occurs when a portion of the slope rotates around a failure plane, typically caused by overloading or saturation of soil.
- Translational failure: Translational failure occurs when a portion of the slope moves down the slope as a block, typically caused by undercutting or erosion.
- Complex failure: Complex failure occurs when a combination of rotational and translational failure occurs, typically caused by multiple factors such as overloading, erosion, and changes in water levels.
- Creep failure: This type of failure occurs when soil or rock material gradually moves down a slope over time, typically caused by the weight of the material and the forces acting on it.
- Rockfall: Rockfall is the detachment and movement of a rock from a steep rock face.

The potential for slope failure can be evaluated through various methods such as stability analysis, which involves assessing the strength of the soil or rock material and the forces acting on it, as well as monitoring for signs of movement or instability.

Preventing slope failure is important in order to protect infrastructure, buildings, and human life. This can be done through various means such as reinforcing the slope with retaining walls, drainage systems or by installing anchors or cables. In some cases, slope stabilization and strengthening may be required to prevent or mitigate the potential for slope failure.

It is important to note that the failure of a slope can be triggered by natural factors, human activities, or a combination of both. It is essential to conduct regular inspections and monitoring of slopes to detect any signs of failure and take necessary actions to prevent or mitigate the potential for failure.

Recall Assumptions of the Slope Analysis

Slope analysis is the process of evaluating the stability of a slope and determining the potential for failure. In order to conduct a slope analysis, several assumptions are typically made. These assumptions include:

- The slope is made up of homogeneous and isotropic soil or rock material: This assumption assumes that the soil or rock material is the same throughout the entire slope and that its properties do not vary in different directions.
- The slope is in a state of static equilibrium: This assumption assumes that the slope is not undergoing any significant changes in loading or deformation, and that it is in a state of balance.
- The slope is not affected by external factors such as water or temperature: This assumption assumes that the slope is not being affected by external factors that could change its properties or stability.
- The slope is in a critical state: This assumption assumes that the slope is at the point of failure, and that any additional load or deformation will cause it to fail.
- The soil or rock material is linear elastic: This assumption assumes that the soil or rock material behaves in a linear elastic manner, which means that it will return to its original shape after loading and deformation.
- The slope failure is a planar failure: This assumption assumes that the failure of the slope occurs along a single plane, which makes it simpler to analyze and predict.

It is important to note that these assumptions are simplifications of the real-world scenario and may not reflect the true conditions of the slope. Therefore, it is important to conduct sensitivity analysis and consider potential variations in the assumptions to ensure the reliability of the analysis.

It is also important to note that these assumptions are subject to change depending on the type of slope and the method used for the analysis. For example, for a rock slope, it might be possible to assume that the rock is isotropic and homogeneous, but for soil slopes, it might not be the case.

List various types of Factor of Safety

The Factor of Safety (FoS) is a measure of the margin of safety against failure of a slope. It is calculated by dividing the strength of the soil or rock material by the applied load or stress. The higher the FoS, the greater the margin of safety and the less likely the slope is to fail.

There are several types of Factor of Safety that can be used in slope analysis, including:

- FoS against sliding: This type of FoS is used to determine the stability of a slope against sliding. It is calculated by dividing the shear strength of the soil or rock material by the applied shear stress.
- FoS against bearing capacity: This type of FoS is used to determine the stability of a slope against failure due to bearing capacity. It is calculated by dividing the bearing capacity of the soil or rock material by the applied load.
- FoS against liquefaction: This type of FoS is used to determine the stability of a slope against liquefaction, which is when soil loses strength and stiffness due to an increase in pore water pressure. It is calculated by dividing the effective stress at the critical state by the applied stress.
- FoS against overturning: This type of FoS is used to determine the stability of a slope against overturning. It is calculated by dividing the weight of the soil or rock material on the slope by the horizontal force acting on the slope.
- FoS against shear strength: This type of FoS is used to determine the stability of a slope against failure due to shear strength. It is calculated by dividing the shear strength of the soil or rock material by the applied shear stress.
- FoS against depth factor: This type of FoS is used to determine the stability of a slope against failure due to depth factor. It is calculated by dividing the failure depth of the soil or rock material by the applied depth factor

It is important to note that these factors of safety are approximate values and the actual factors of safety will depend on the specific conditions of the slope, such as the soil type, slope geometry, and water content. Therefore, it is important to conduct sensitivity analysis and consider potential variations in the assumptions to ensure the reliability of the analysis.

Classify the Slope Failure

Slope failure refers to the movement of soil or rock material down a slope due to a lack of stability. There are several different types of slope failure, including:

- Rotational failure: This type of failure occurs when a slope fails by rotating around a pivot point. It is characterized by a circular failure surface and typically occurs in soil or rock that is relatively homogeneous and isotropic.
- Translational failure: This type of failure occurs when a slope fails by sliding along a planar failure surface. It is characterized by a linear failure surface and typically occurs in soil or rock that is layered or has different strengths in different directions.
- Block failure: This type of failure occurs when a large block of soil or rock detaches from the slope and moves downhill. It is characterized by a rectangular failure surface and typically occurs in soil or rock that is jointed or has a high degree of weathering.
- Toppling failure: This type of failure occurs when a slope fails by rotating around a vertical axis. It is characterized by a curved failure surface and typically occurs in soil or rock that is layered or has different strengths in different directions.
- Wedging failure: This type of failure occurs when a slope fails by the wedge shaped mass of soil or rock sliding downhill. It is characterized by a triangular failure surface and typically occurs in soil or rock that is layered or has different strengths in different directions.
- Combination failure: This type of failure occurs when multiple types of failure occur simultaneously, such as a combination of rotational and translational failure.

It is important to note that in many cases, the failure of a slope is a combination of several types of failure, and that the classification of the failure can be complex, and require analysis from multiple angles, such as geotechnical, geological and hydrological analysis.

Recall Stability Analysis of Infinite Slope for Cohesionless Soil: i. In Dry Soil Conditions ii. In Submerged Slope iii. In Steady Seepage along the Slope

Stability analysis of infinite slopes for cohesionless soil is an important aspect of understanding the behavior of soil and rock slopes. There are several different methods for performing stability analysis of infinite slopes for cohesionless soil, depending on the conditions of the slope.

- Stability analysis of infinite slopes for dry cohesionless soil: In this type of analysis, the slope is assumed to be in a dry state, with no water present in the soil. Stability is determined by analyzing the internal forces and deformation of the soil, as well as the external forces acting on the slope, such as gravity and the weight of the soil itself. Factors that can affect the stability of a dry cohesionless slope include the angle of the slope, the density of the soil, and the strength of the soil.
- Stability analysis of infinite slopes for submerged cohesionless soil: In this type of analysis, the slope is assumed to be submerged in water. Stability is determined by analyzing the buoyancy of the soil and the forces acting on the soil, such as the weight of the water and the weight of the soil. Factors that can affect the stability of a submerged cohesionless slope include the density of the soil and water, the strength of the soil, and the angle of the slope.
- Stability analysis of infinite slopes for steady seepage along the slope: In this type of analysis, the slope is assumed to be experiencing steady seepage of water along the slope. Stability is determined by analyzing the forces acting on the soil, such as the weight of the soil, the weight of the water, and the seepage forces. Factors that can affect the stability of a slope experiencing steady seepage include the angle of the slope, the density of the soil and water, the strength of the soil, and the rate of seepage.

In general, stability analysis of infinite slopes for cohesionless soil can be complex and require the use of sophisticated mathematical models and computer simulations. It is important to note that there are many different methods for performing stability analysis of infinite slopes for cohesionless soil, and that the method chosen will depend on the specific conditions of the slope and the level of accuracy required.

Recall Stability Analysis of Infinite Slope for Cohesive Soil: i. In Dry Soil Conditions ii. In Submerged Slope iii. In Steady Seepage along the Slope

Stability analysis of infinite slopes for cohesive soil is an important aspect of understanding the behavior of soil and rock slopes. Cohesive soils, also known as clay soils, have a higher proportion of clay particles compared to sand and silt, which gives them unique characteristics such as high shear strength and high compressibility. This can affect the stability of slopes made of cohesive soil in different ways compared to cohesionless soil.

- Stability analysis of infinite slopes for dry cohesive soil: In this type of analysis, the slope is assumed to be in a dry state, with no water present in the soil. Stability is determined by analyzing the internal forces and deformation of the soil, as well as the external forces acting on the slope, such as gravity and the weight of the soil itself. Factors that can affect the stability of a dry cohesive slope include the angle of the slope, the density of the soil, and the shear strength of the soil. The shear strength of cohesive soil is affected by factors such as the water content and the type of clay minerals present.
- Stability analysis of infinite slopes for submerged cohesive soil: In this type of analysis, the slope is assumed to be submerged in water. Stability is determined by analyzing the buoyancy of the soil and the forces acting on the soil, such as the weight of the water and the weight of the soil. Factors that can affect the stability of a submerged cohesive slope include the density of the soil and water, the shear strength of the soil, and the angle of the slope. The shear strength of cohesive soil can be affected by factors such as the water content and the type of clay minerals present.
- Stability analysis of infinite slopes for steady seepage along the slope: In this type of analysis, the slope is assumed to be experiencing steady seepage of water along the slope. Stability is determined by analyzing the forces acting on the soil, such as the weight of the soil, the weight of the water, and the seepage forces. Factors that can affect the stability of a slope experiencing steady seepage include the angle of the slope, the density of the soil and water, the shear strength of the soil, and the rate of seepage.

In general, stability analysis of infinite slopes for cohesive soil can be complex and require the use of sophisticated mathematical models and computer simulations. It is important to note that there are many different methods for performing stability analysis of infinite slopes for cohesive soil, and that the method chosen will depend on the specific conditions of the slope and the level of accuracy required.

Define Wedge Failure

Wedge failure is a type of slope failure that occurs when a wedge-shaped mass of soil or rock becomes unstable and slides down a slope. This type of failure is characterized by a triangular-shaped mass of soil or rock that becomes detached from the slope and slides down along a pre-existing plane of weakness. This type of failure is often caused by a combination of factors such as the weight of the wedge, the angle of the slope, and the strength of the soil or rock. Wedge failure can occur in both cohesive and non-cohesive soils and can be triggered by natural factors such as erosion or heavy rainfall, or by human activities such as construction or mining. Wedge failures can cause significant damage to structures and infrastructure and can be difficult to predict and prevent.

Recall Φ<sub>u</sub> = 0 Analysis

“Φ<sub>u</sub> = 0 Analysis” refers to the analysis of the stability of a slope using the Mohr-Coulomb failure criterion, where the “phi” (Φ) represents the angle of internal friction, and “u” represents the effective normal stress. In this type of analysis, the value of “phi” is assumed to be equal to zero, meaning that the soil or rock has no internal resistance to shearing.

This type of analysis is commonly used to evaluate the stability of slopes made of cohesionless materials such as sands, gravels, and rock. In this case, the shear strength of the soil or rock is primarily provided by the weight of the soil or rock itself, and not by the internal resistance to shearing.

To perform the analysis, the following steps are typically taken:

- Determine the height and weight of the soil or rock wedge that is being evaluated.
- Estimate the angle of the slope (angle of repose)
- Determine the total normal stress acting on the base of the soil or rock wedge.
- Calculate the factor of safety against failure by dividing the total normal stress by the weight of the soil or rock wedge.
- Compare the factor of safety with a predetermined safety threshold, typically 1.3 or 1.5, to determine if the slope is stable or not.

It is important to note that this type of analysis is based on a number of assumptions, such as:

- Slope is infinite and plane
- The soil or rock is homogeneous and isotropic
- The soil or rock is in a state of undrained loading
- The soil or rock does not have any internal resistance to shearing

Therefore, it is not always applicable or accurate to real-life situations and other methods should be used as well.

Describe the Friction Circle Method for Analysis of Finite Slope

The Friction Circle Method is a technique used to analyze the stability of a finite slope in soil mechanics. The method is based on the concept of a “friction circle” which represents the maximum shear force that can be applied to a soil mass without causing failure. The radius of the friction circle is equal to the coefficient of friction of the soil multiplied by the normal force acting on the soil mass.

The method involves the following steps:

- Divide the slope into a number of slices or wedges.
- Determine the forces acting on each wedge, including the weight of the soil, the normal force, and the shear force.
- Draw a force vector diagram for each wedge, with the force vectors scaled to their relative magnitudes.
- Draw a friction circle for each wedge, with the radius equal to the coefficient of friction multiplied by the normal force.
- Check the stability of each wedge by comparing the shear force vector to the friction circle. If the shear force vector is within the friction circle, the wedge is stable. If the shear force vector is outside of the friction circle, the wedge is unstable and failure will occur.

The Friction Circle Method is widely used in the analysis of natural and man-made slopes, it is a simple, but powerful tool for understanding the stability of a slope and can be used to design stable slopes or to assess the stability of existing slopes.

Recall Taylor’s Stability Number and its significance

Taylor’s Stability Number, also known as the Factor of Safety, is a measure of the safety or stability of a slope. It is calculated by dividing the shear strength of the soil by the shear stress acting on the soil. A factor of safety greater than 1 indicates that the slope is stable, while a factor of safety less than 1 indicates that the slope is unstable.

The significance of Taylor’s Stability Number is that it allows engineers to assess the stability of a slope and determine if it is safe for construction. By calculating the factor of safety, engineers can determine if additional measures, such as reinforcing the soil or building retaining walls, need to be taken to ensure the stability of the slope. Additionally, it also helps to ensure that the slope will not fail during or after construction and will be able to withstand any external loading or forces that may be applied to it.

The Friction Circle Method for Analysis of Finite Slope is a method for analyzing the stability of a slope that takes into account the friction angle, the soil unit weight, and the height of the slope. This method involves plotting the forces acting on the slope on a special diagram called a Friction Circle Diagram. Engineers can then use this diagram to determine the factor of safety for the slope and identify any potential failure points.

Overall, Taylor’s Stability Number and Friction Circle Method are powerful tools for assessing the stability of slopes and ensuring that they are safe for construction and use. It is crucial for engineers to understand the principles and calculations behind these methods in order to make sound decisions in the design and construction of slopes.

Describe the Swedish Circle Method for Analysis of Finite Slope

The Swedish Circle Method for Analysis of Finite Slope is a method used in the field of surveying to determine the stability of slopes. The method is based on the principle of circular arc failure, which states that a slope will fail when the material on the slope is unable to hold the weight of the material above it.

The basic steps of the Swedish Circle Method include:

- Determining the center of gravity of the slope (CG)
- Drawing a circle with the CG as its center and a radius equal to the distance from the CG to the point of failure (PF)
- Dividing the circle into a number of equal segments (usually eight)
- Measuring the slope angle at each segment
- Calculating the factor of safety (FoS) for each segment by dividing the slope angle by the angle of failure
- Plotting the FoS values on a graph

The method is based on the assumption that the slope will fail in a circular pattern, with the point of failure being located at the lowest point on the circle. By determining the factor of safety at each segment, the method can identify areas of the slope that are at a higher risk of failure.

The Swedish Circle Method is a relatively simple and efficient method for analyzing the stability of slopes, and is particularly useful for natural slopes and man-made embankments. However, it does have some limitations, such as assuming a circular failure pattern, and not taking into account variations in soil properties.

It is important to note that the Swedish Circle Method should be considered as a preliminary analysis and should be used in conjunction with other techniques such as limit equilibrium method or finite element method for a more comprehensive analysis.

Recall the Location of Most Critical Circle

The location of the most critical circle in the Swedish Circle Method for Analysis of Finite Slope is determined by identifying the point of failure (PF) on the slope. The point of failure is the point at which the slope is considered to be unstable and at risk of failure.

The center of gravity (CG) of the slope is also determined, and a circle is drawn with the CG as its center and a radius equal to the distance from the CG to the point of failure (PF). This circle is known as the most critical circle, as it represents the area of the slope that is at the highest risk of failure.

To determine the location of the most critical circle, the following steps are typically followed:

- Identify the point of failure (PF) on the slope: The point of failure is the point at which the slope is considered to be unstable and at risk of failure. This point can be determined by analyzing factors such as soil properties, water content, and loading conditions.
- Determine the center of gravity (CG) of the slope: The center of gravity is the point on the slope where the slope’s weight is evenly distributed.
- Draw the circle: Draw a circle with the CG as its center and a radius equal to the distance from the CG to the point of failure (PF).

It is important to note that the location of the most critical circle can change depending on the loading conditions and soil properties. For example, if the slope is saturated with water, the point of failure may move downward, and the most critical circle will be located in a different area.

Also, the location of the most critical circle can be affected by the properties of the material in the slope, such as its strength, shear strength, and unit weight, it is important to take into account the variations in soil properties and loading conditions in order to identify the most critical circle on the slope.

Describe the Culmann’s Method for Stability Analysis of Homogeneous Finite Slope

The Culmann’s Method for Stability Analysis of Homogeneous Finite Slope is a method used in the field of surveying to determine the stability of slopes. The method is based on the principle of limit equilibrium, which states that a slope will fail when the forces acting on it reach a state of equilibrium.

The basic steps of the Culmann’s Method include:

- Determining the center of gravity of the slope (CG)
- Drawing a number of potential failure planes through the CG
- Calculating the factor of safety (FoS) for each plane by dividing the weight of the soil on the slope by the weight of the soil that would be required to cause failure.
- Plotting the FoS values on a graph

The method is based on the assumption that the slope will fail along a plane, and the point of failure is identified as the point where the factor of safety is lowest. By determining the factor of safety for a number of potential failure planes, the method can identify the plane that represents the highest risk of failure.

The Culmann’s Method is a relatively simple and efficient method for analyzing the stability of homogeneous slopes, and is particularly useful for natural slopes and man-made embankments. However, it does have some limitations such as assuming a plane failure pattern and not taking into account variations in soil properties.

It is important to note that the Culmann’s Method should be considered as a preliminary analysis and should be used in conjunction with other techniques such as limit equilibrium method or finite element method for a more comprehensive analysis. Additionally, the Culmann’s Method is suitable for homogeneous slopes only, if the slope is not homogeneous, the method may not provide accurate results.

Recall Stability of Finite Slopes under the following Conditions: i. Steady Seepage Condition ii. Sudden Drawdown Condition iii. During Construction Condition

i. Stability of Finite Slopes under Steady Seepage Condition:

When a slope is under steady seepage condition, the water is flowing through the soil at a constant rate. This can affect the stability of the slope in a number of ways, including increasing the pore water pressure and reducing the effective stress on the soil.

The stability of a slope under steady seepage condition can be determined by analyzing the seepage force acting on the soil and comparing it to the shear strength of the soil. If the seepage force exceeds the shear strength, the slope will become unstable and may fail.

To prevent failure under steady seepage conditions, measures such as installing drainage systems, or stabilizing the soil with reinforcement can be taken.

ii. Stability of Finite Slopes under Sudden Drawdown Condition:

When a slope is under sudden drawdown condition, the water level in the soil is rapidly lowered, causing a decrease in pore water pressure. This can cause an increase in the effective stress on the soil and can make the slope more unstable.

The stability of a slope under sudden drawdown condition can be determined by analyzing the change in pore water pressure and comparing it to the shear strength of the soil. If the change in pore water pressure exceeds the shear strength, the slope will become unstable and may fail.

To prevent failure under sudden drawdown conditions, measures such as installing drainage systems, or stabilizing the soil with reinforcement can be taken.

iii. Stability of Finite Slopes During Construction Condition:

During construction, the stability of a slope can be affected by the addition of weight and the excavation of soil. The additional weight can cause an increase in the stress on the soil, while the excavation of soil can cause a decrease in the stress on the soil.

The stability of a slope during construction can be determined by analyzing the changes in stress on the soil and comparing it to the shear strength of the soil. If the changes in stress exceed the shear strength, the slope will become unstable and may fail.

To prevent failure during construction, measures such as installing drainage systems, or stabilizing the soil with reinforcement can be taken. Additionally, it’s important to monitor the slope throughout the construction process and take corrective actions as necessary to ensure its stability.