Earth Pressure Theories

Contents

**Define the following terms: i. Retaining Wall ii. Backfill iii. Wedge** 2

**Recall the Condition of Plastic Equation** 2

**Classify Lateral Earth Pressure** 3

**Describe Earth Pressure at Rest** 4

**Recall the Rankine’s Theory for Cohesionless Soil** 5

**Recall Effect of Surcharge on the Earth Pressure** 6

**Recall Effect of various Backfills on the Earth Pressure** 7

**Recall Effect of Inclined Backfill on the Earth Pressure** 8

**Recall Rankine’s Earth Pressure Theory for the Cohesive Soil** 9

**Derive an expression for the Depth of Tensile Crack** 10

**Derive an expression for Height of Unsupported Vertical cut** 11

**Recall the assumptions of Coulomb Wedge Theory** 12

**Recall the Coulomb Wedge Theory for Cohesionless Soil** 13

**Recall the Coulomb Wedge Theory for Cohesive Soil** 14

**Recall Culmann’s Theory for the Earth Pressure** 15

**Recall Rebhann’s Theory for the Earth Pressure** 16

**Define the following terms: i. Retaining Wall ii. Backfill iii. Wedge**

i. Retaining wall: A retaining wall is a structure that is built to hold back soil, rock, or other materials and prevent them from sliding or eroding. They can be used in a variety of settings, such as on slopes or hillsides, around foundations, and in landscaping. Retaining walls can be made of various materials, including stone, brick, concrete, and timber, and can be designed to be both functional and decorative.

ii. Backfill: Backfill refers to the process of filling in the area behind a retaining wall with soil, rock, or other materials. This is done to provide support for the wall and to prevent the wall from tipping or sliding. Backfill materials should be compacted to provide a stable base for the wall and to prevent settling or erosion.

iii. Wedge: In the context of retaining walls, a wedge refers to a triangular-shaped piece of soil or rock that is formed when the soil behind a retaining wall begins to slide or erode. A wedge can put pressure on the retaining wall, causing it to tip or slide. To prevent wedges from forming, proper backfill materials and compaction techniques should be used, and the retaining wall should be designed to withstand the forces of the wedge. Additionally, drainage systems can be installed to help prevent water from building up behind the wall, which can contribute to the formation of wedges.

**Recall the Condition of Plastic Equation**

The condition of plastic equilibrium, also known as the Mohr-Coulomb criterion, is a mathematical equation used in soil mechanics to determine the stability of soil and rock masses. The equation states that the shear strength of a soil or rock mass is equal to the sum of the normal stress on the material and the internal friction of the material. The equation is typically expressed as:

Shear strength = (c + σ’n)tan(φ)

Where:

- c is the cohesion of the material, which is a measure of the strength of the bonds between soil particles.
- σ’n is the normal stress on the material, which is the force per unit area acting perpendicular to the surface.
- tan(φ) is the internal friction angle of the material, which is a measure of the resistance of the material to sliding.

The condition of plastic equilibrium is used to determine the safety factor of a soil or rock mass, which is a measure of how stable the material is. A safety factor of 1 or greater indicates that the material is stable, while a safety factor less than 1 indicates that the material is unstable and may fail.

The Mohr-Coulomb criterion is widely used in geotechnical engineering and is often used to determine the stability of slopes, embankments, and retaining walls. The equation is also used to predict the behavior of soil and rock under different loading conditions and to design structures that will be stable in these conditions.

In summary, the condition of plastic equilibrium, or Mohr-Coulomb criterion, is a mathematical equation that is used to determine the stability of soil and rock masses. It takes into account the normal stress on the material, the cohesion of the material, and the internal friction angle of the material to evaluate the stability of slopes, embankments and retaining walls.

**Classify Lateral Earth Pressure**

Lateral earth pressure refers to the pressure exerted by soil or rock on a retaining wall or other structure. This pressure can cause the wall or structure to tip, slide, or collapse if it is not designed to withstand the forces.

There are several ways to classify lateral earth pressure:

- Active earth pressure: Active earth pressure occurs when the soil or rock behind a retaining wall is in a state of compression. This type of pressure is caused by the weight of the soil or rock pushing against the wall. Active earth pressure can be calculated using the Rankine or Coulomb theories.
- Passive earth pressure: Passive earth pressure occurs when the soil or rock behind a retaining wall is in a state of tension. This type of pressure is caused by the wall pushing against the soil or rock. Passive earth pressure can be calculated using the Rankine or Coulomb theories.
- At-rest earth pressure: At-rest earth pressure occurs when the soil or rock behind a retaining wall is not subject to any external forces. This type of pressure is caused by the weight of the soil or rock and is equal to the weight of the soil or rock divided by the area of the wall.

It is important to consider all types of lateral earth pressure when designing a retaining wall or other structure, as each type of pressure may have a different effect on the stability of the structure. The structural engineer will consider the soil properties and the site conditions to design the retaining wall that can withstand the lateral earth pressure.

**Describe Earth Pressure at Rest**

Earth pressure at rest, also known as “at-rest earth pressure,” refers to the pressure exerted by soil or rock on a retaining wall or other structure when the soil or rock is not subject to any external forces. This type of pressure is caused by the weight of the soil or rock and is equal to the weight of the soil or rock divided by the area of the wall.

The earth pressure at rest is calculated using the following equation:

P = γH

where:

P is the earth pressure at rest

γ is the unit weight of soil or rock

H is the height of the soil or rock behind the wall

Earth pressure at rest is a theoretical value and is usually used as a starting point for the analysis of the retaining wall stability. It assumes that the soil or rock is in a state of equilibrium and that there are no external forces acting on it.

The earth pressure at rest is important to consider when designing a retaining wall or other structure because it represents the minimum pressure that the wall or structure must be able to withstand. However, in practice, the retaining wall or structure may also be subject to other types of lateral earth pressure such as active, passive, earthquake-induced and hydrostatic pressure.

It is also important to note that the earth pressure at rest does not take into account the possibility of soil or rock movement, such as settling or sliding. Therefore, the structural engineer must also consider soil or rock movement in the design of the retaining wall or other structure.

In summary, Earth pressure at rest refers to the pressure exerted by soil or rock on a retaining wall or other structure when the soil or rock is not subject to any external forces. It is caused by the weight of the soil or rock and is equal to the weight of the soil or rock divided by the area of the wall. It is important to consider the earth pressure at rest when designing a retaining wall or other structure as it represents the minimum pressure that the wall or structure must be able to withstand, however, other types of lateral earth pressure must also be considered.

**Recall the Rankine’s Theory for Cohesionless Soil**

Rankine’s theory, also known as Rankine’s active earth pressure theory, is a method used to calculate the active earth pressure on a retaining wall for cohesionless soils. The theory was developed by William Rankine, a Scottish engineer and physicist in the 19th century.

The theory assumes that the soil behind the retaining wall is cohesionless, meaning it does not have any internal resistance to shearing. It also assumes that the soil is in a state of active pressure, meaning the soil is in compression and pushing against the retaining wall.

According to the theory, the active earth pressure on a retaining wall is equal to the weight of the soil multiplied by the horizontal component of the angle of internal friction of the soil. The formula used to calculate the active earth pressure is:

P = γH(sinϕ + cotϕcosϕ)

where:

P is the active earth pressure

γ is the unit weight of soil

H is the height of the soil behind the wall

ϕ is the angle of internal friction of the soil

The Rankine theory is widely used for cohesionless soils, particularly granular soils such as sand and gravel, but it is not suitable for cohesive soils such as clay.

It is also important to note that the Rankine theory assumes that the soil is in a state of active pressure, meaning that the soil is in compression and pushing against the retaining wall. However, in practice, the soil may also be in a state of passive pressure, meaning that the retaining wall is pushing against the soil. Therefore, it is important for the structural engineer to consider both active and passive earth pressure when designing a retaining wall.

In summary, Rankine’s theory is a method used to calculate the active earth pressure on a retaining wall for cohesionless soils. It was developed by William Rankine and assumes that the soil behind the retaining wall is cohesionless, meaning it does not have any internal resistance to shearing. The theory also assumes that the soil is in a state of active pressure, meaning the soil is in compression and pushing against the retaining wall. The Rankine theory is widely used for cohesionless soils but not for cohesive soils, and it is important for the structural engineer to consider both active and passive earth pressure when designing a retaining wall.

**Recall Effect of Surcharge on the Earth Pressure**

A surcharge is a load that is placed on the soil surface behind a retaining wall. The load can be caused by various factors such as vehicles, buildings, or other structures. The effect of a surcharge on the earth pressure is the additional pressure it exerts on the retaining wall.

When a surcharge is placed on the soil surface, the soil behind the retaining wall is compressed. This compression increases the vertical stress on the soil and, as a result, increases the active earth pressure on the retaining wall. The amount of additional pressure is dependent on the magnitude and distribution of the surcharge and the properties of the soil.

The effect of surcharge on earth pressure can be calculated using the following formula:

P = γH(sinϕ + cotϕcosϕ) + γq

where:

P is the total earth pressure

γ is the unit weight of soil

H is the height of the soil behind the wall

ϕ is the angle of internal friction of the soil

q is the intensity of the surcharge per unit area

It is important to note that the surcharge load should be applied gradually and uniformly over the soil surface to avoid any sudden changes in the earth pressure on the retaining wall. Also, the surcharge should be placed in a location that is in line with the center of the wall.

In addition, it is important to consider the duration of the surcharge, as a long-term surcharge will cause an increase in the effective stress of the soil, which will affect the strength of the soil and the earth pressure on the retaining wall.

In summary, A surcharge is a load that is placed on the soil surface behind a retaining wall. The effect of a surcharge on the earth pressure is the additional pressure it exerts on the retaining wall. When a surcharge is placed on the soil surface, the soil behind the retaining wall is compressed, increasing the vertical stress on the soil and, as a result, increases the active earth pressure on the retaining wall. The effect of surcharge on earth pressure can be calculated using a formula, it is important to apply the surcharge gradually and uniformly, and consider the duration of the surcharge as it will affect the strength of the soil and the earth pressure on the retaining wall.

** ****Recall Effect of various Backfills on the Earth Pressure**

The type of backfill behind a retaining wall can have a significant effect on the earth pressure exerted on the wall. Different types of backfill materials have different properties, such as unit weight, angle of internal friction, and permeability, which can affect the earth pressure.

- Granular backfill: Granular materials such as gravel or crushed rock have a higher unit weight than soil and are less compressible. This results in a higher active earth pressure on the retaining wall.
- Organic backfill: Organic materials such as peat or wood chips have a lower unit weight than soil and are more compressible. This results in a lower active earth pressure on the retaining wall.
- Clay backfill: Clay backfill has a higher unit weight than soil and is less compressible. This results in a higher active earth pressure on the retaining wall. However, clay backfill is also less permeable than granular or organic materials, which can lead to an increase in the pore water pressure behind the wall.
- Sand backfill: Sand backfill has a lower unit weight than soil, and it is more compressible. This results in a lower active earth pressure on the retaining wall. However, sand backfill is more permeable than clay backfill, which can lead to a decrease in the pore water pressure behind the wall.

In summary, The type of backfill behind a retaining wall can have a significant effect on the earth pressure exerted on the wall. Different types of backfill materials have different properties such as unit weight, angle of internal friction, and permeability, which can affect the earth pressure. Granular backfill has a higher unit weight and results in higher active earth pressure, Organic backfill has a lower unit weight and results in lower active earth pressure, Clay backfill has a higher unit weight and results in higher active earth pressure but also increases pore water pressure, Sand backfill has a lower unit weight and results in lower active earth pressure but also decreases pore water pressure.

**Recall Effect of Inclined Backfill on the Earth Pressure**

The effect of inclined backfill on earth pressure can be significant when designing retaining walls. An inclined backfill refers to a backfill that is placed at an angle other than vertical, such as a sloped or stepped backfill.

- Sloped backfill: When the backfill is placed at an angle, the angle of internal friction of the backfill material can affect the earth pressure on the retaining wall. A steeper slope results in a larger angle of internal friction, which can increase the active earth pressure on the wall. This can lead to increased wall design requirements and potential instability.
- Stepped backfill: When the backfill is placed in steps or terraces, the earth pressure on the wall can be reduced. This is because each step or terrace acts as a separate retaining wall, reducing the overall height of the backfill and the corresponding earth pressure on the main wall. However, the design of each step or terrace must be considered separately to ensure stability.

In summary, The effect of inclined backfill on earth pressure can be significant when designing retaining walls. An inclined backfill refers to a backfill that is placed at an angle other than vertical. A sloped backfill can affect the angle of internal friction of the backfill material and can increase the active earth pressure on the wall. A stepped backfill can reduce the earth pressure on the wall by acting as a separate retaining wall, but each step or terrace must be considered separately to ensure stability.

**Recall Rankine’s Earth Pressure Theory for the Cohesive Soil**

Rankine’s Earth Pressure Theory for cohesive soil is a method for calculating the active and passive earth pressure on a retaining wall. The theory is based on the assumption that the soil behind the wall is a cohesive soil, such as clay, and that the soil is in a state of plane strain.

Active earth pressure: The active earth pressure is the pressure exerted on the retaining wall by the soil when the wall is being pushed into the soil. It is calculated using Rankine’s formula which is:

P = c + (σ’h)tan(φ)

- where P is the active earth pressure, c is the cohesion of the soil, σ’ is the effective stress, h is the height of the backfill, and φ is the angle of internal friction of the soil.
- Passive earth pressure: The passive earth pressure is the pressure exerted on the retaining wall by the soil when the wall is being pulled out of the soil. It is calculated using the same formula as active earth pressure but with a different value for the angle of internal friction (φ’).

In summary, Rankine’s Earth Pressure Theory for cohesive soil is a method for calculating the active and passive earth pressure on a retaining wall. It is based on the assumption that the soil behind the wall is a cohesive soil and that the soil is in a state of plane strain. The active earth pressure is the pressure exerted on the retaining wall by the soil when the wall is being pushed into the soil, it is calculated using Rankine’s formula. The passive earth pressure is the pressure exerted on the retaining wall by the soil when the wall is being pulled out of the soil, it is also calculated using the same formula as active earth pressure but with a different value for the angle of internal friction.

**Derive an expression for the Depth of Tensile Crack**

The depth of a tensile crack in a retaining wall can be calculated using Coulomb’s theory of active earth pressure. Coulomb’s theory is an extension of Rankine’s earth pressure theory and takes into account the effect of a tensile crack on the active earth pressure on the wall.

Coulomb’s theory: Coulomb’s theory states that the active earth pressure on a retaining wall is equal to the sum of the normal stress and the tangential stress. The normal stress is calculated using Rankine’s formula and the tangential stress is calculated using the following formula:

σt = (Ks * P) / (1 + sinφ)

- where σt is the tangential stress, Ks is the coefficient of at-rest earth pressure, P is the active earth pressure, and φ is the angle of internal friction of the soil.

Tensile crack: A tensile crack in a retaining wall is a crack that occurs when the active earth pressure on the wall exceeds the strength of the wall. The depth of the crack is given by the formula:

a = (σt * h) / (2 * c)

- where a is the depth of the tensile crack, σt is the tangential stress, h is the height of the backfill, and c is the cohesion of the soil.

In summary, Coulomb’s theory of active earth pressure can be used to calculate the depth of a tensile crack in a retaining wall. Coulomb’s theory states that the active earth pressure on a retaining wall is equal to the sum of the normal stress and the tangential stress. A tensile crack in a retaining wall is a crack that occurs when the active earth pressure on the wall exceeds the strength of the wall. The depth of the crack is given by the formula which is (σt * h) / (2 * c) where a is the depth of the tensile crack, σt is the tangential stress, h is the height of the backfill, and c is the cohesion of the soil.

**Derive an expression for Height of Unsupported Vertical cut**

The height of an unsupported vertical cut in a retaining wall can be calculated using Coulomb’s theory of active earth pressure. Coulomb’s theory is an extension of Rankine’s earth pressure theory and takes into account the effect of an unsupported vertical cut on the active earth pressure on the wall.

Coulomb’s theory: Coulomb’s theory states that the active earth pressure on a retaining wall is equal to the sum of the normal stress and the tangential stress. The normal stress is calculated using Rankine’s formula and the tangential stress is calculated using the following formula:

σt = (Ks * P) / (1 + sinφ)

- where σt is the tangential stress, Ks is the coefficient of at-rest earth pressure, P is the active earth pressure, and φ is the angle of internal friction of the soil.

Unsupported vertical cut: An unsupported vertical cut is a cut in the soil that is not supported by a retaining wall. The height of the cut can be calculated using the following formula:

h = (σt * L) / (2 * Ks * P)

- where h is the height of the unsupported vertical cut, σt is the tangential stress, L is the length of the cut, and Ks and P are as defined above.

In summary, Coulomb’s theory of active earth pressure can be used to calculate the height of an unsupported vertical cut in a retaining wall. Coulomb’s theory states that the active earth pressure on a retaining wall is equal to the sum of the normal stress and the tangential stress. An unsupported vertical cut is a cut in the soil that is not supported by a retaining wall. The height of the cut can be calculated using the formula which is (σt * L) / (2 * Ks * P) where h is the height of the unsupported vertical cut, σt is the tangential stress, L is the length of the cut, and Ks and P are as defined above.

**Recall the assumptions of Coulomb Wedge Theory**

Coulomb Wedge Theory is a widely used method for analysing the stability of slopes and retaining walls in soil mechanics. The theory is based on the following assumptions:

- The soil is homogeneous and isotropic: This means that the soil has the same properties in all directions and at all depths.
- The soil is in a state of plane strain: This means that the soil is under equal stresses in all directions except for one (the direction of the slope or wall).
- The soil is in a state of perfect plasticity: This means that the soil can withstand any amount of stress without undergoing any permanent deformation.
- The soil is frictional: This means that the soil can resist sliding along the plane of the slope or wall.
- The soil is cohesive: This means that the soil can resist breaking apart along the plane of the slope or wall.
- The soil is dry: This means that there is no water present in the soil, which could change its properties.
- The soil is at rest: This means that the soil is not undergoing any movement or deformation.
- The slope or wall is infinitely wide: This means that the slope or wall extends infinitely in the direction perpendicular to the plane of failure.

The Coulomb Wedge Theory is based on these assumptions and uses the concept of the “wedge” to analyse the stability of slopes and retaining walls. It is also known as Coulomb’s Wedge Method, and it is a graphical method of calculating the factor of safety of a slope or retaining wall. It is one of the most widely used methods for analysing the stability of slopes and retaining walls in soil mechanics.

**Recall the Coulomb Wedge Theory for Cohesionless Soil**

The Coulomb Wedge Theory is a widely used method for analysing the stability of slopes and retaining walls in cohesionless soil. The theory is based on the following assumptions:

- The soil is homogeneous and isotropic: This means that the soil has the same properties in all directions and at all depths.
- The soil is in a state of plane strain: This means that the soil is under equal stresses in all directions except for one (the direction of the slope or wall).
- The soil is frictional: This means that the soil can resist sliding along the plane of the slope or wall.
- The soil is dry: This means that there is no water present in the soil, which could change its properties.
- The soil is at rest: This means that the soil is not undergoing any movement or deformation.
- The slope or wall is infinitely wide: This means that the slope or wall extends infinitely in the direction perpendicular to the plane of failure.

The Coulomb Wedge Theory for cohesionless soil is based on the concept of the “wedge” to analyse the stability of slopes and retaining walls. A wedge is defined as a triangular block of soil that is bounded by two planes of failure. The theory states that the stability of a slope or retaining wall is determined by the forces acting on the wedge and the strength of the soil.

In the Coulomb Wedge Theory, the factor of safety is calculated by comparing the weight of the soil in the wedge to the shear strength of the soil. The factor of safety is determined by the ratio of the weight of the soil in the wedge to the shear strength of the soil. The factor of safety is a measure of the stability of the slope or retaining wall. If the factor of safety is greater than 1, the slope or retaining wall is stable. If the factor of safety is less than 1, the slope or retaining wall is unstable.

The Coulomb Wedge Theory is widely used in the field of soil mechanics and is a valuable tool for analysing the stability of slopes and retaining walls in cohesionless soil.

**Recall the Coulomb Wedge Theory for Cohesive Soil**

The Coulomb Wedge Theory can also be applied to cohesive soil, which is soil that contains clay and has internal cohesion. The main difference between the Coulomb Wedge Theory for cohesive soil and that for cohesionless soil is the addition of an internal cohesion term.

In cohesive soil, the internal cohesion of the soil plays a role in the stability of the slope or retaining wall. The internal cohesion of the soil is the force that holds the soil particles together. This force is measured in terms of the cohesion or c value of the soil.

The Coulomb Wedge Theory for cohesive soil states that the factor of safety for a slope or retaining wall is determined by the ratio of the weight of the soil in the wedge to the shear strength of the soil. The shear strength of cohesive soil is made up of both the internal cohesion of the soil and the frictional strength of the soil.

The factor of safety for cohesive soil is calculated as:

FoS = (c + tan(phi) * (weight of soil in the wedge)) / (weight of soil in the wedge)

Where:

c = internal cohesion of the soil

phi = angle of internal friction

weight of soil in the wedge = the weight of the soil in the wedge

The factor of safety is a measure of the stability of the slope or retaining wall. If the factor of safety is greater than 1, the slope or retaining wall is stable. If the factor of safety is less than 1, the slope or retaining wall is unstable.

It is important to note that Coulomb Wedge Theory is a simplified method that assumes certain ideal conditions, it is not always accurate in real-world situations and other methods such as the limit equilibrium method should be considered as well.

**Recall Culmann’s Theory for the Earth Pressure**

Culmann’s theory of earth pressure is a method for determining the lateral earth pressure on a retaining wall. The theory is based on the concept of a “failure plane” or “wedge” of soil that is assumed to slide along the wall.

The theory assumes that the soil is homogeneous, isotropic, and that the wall is vertical. The theory also assumes that the soil is in a state of undrained conditions and that the soil is in a state of equilibrium.

According to Culmann’s theory, the lateral earth pressure on a retaining wall is equal to the weight of the soil in the wedge multiplied by the coefficient of active earth pressure. The coefficient of active earth pressure is determined by the angle of internal friction of the soil and the angle of the failure plane.

The coefficient of active earth pressure is given by :

K_{a} = sin(phi) / (1 + sin(phi))

Where:

phi = angle of internal friction

The total lateral earth pressure on the wall is given by:

P = K_{a} * H * γ

Where:

P = lateral earth pressure

K_{a} = coefficient of active earth pressure

H = height of soil in the wedge

γ = unit weight of soil

Culmann’s theory is a simplified method that assumes certain ideal conditions, it is not always accurate in real-world situations and other methods such as the limit equilibrium method should be considered as well.

**Recall Rebhann’s Theory for the Earth Pressure**

Rebhann’s theory of earth pressure is a method for determining the lateral earth pressure on a retaining wall. The theory is based on the concept of a “failure plane” or “wedge” of soil that is assumed to slide along the wall.

The theory assumes that the soil is homogeneous and isotropic, and that the wall is vertical. The theory also assumes that the soil is in a state of undrained conditions and that the soil is in a state of equilibrium.

According to Rebhann’s theory, the lateral earth pressure on a retaining wall is equal to the weight of the soil in the wedge multiplied by the coefficient of passive earth pressure. The coefficient of passive earth pressure is determined by the angle of internal friction of the soil and the angle of the failure plane.

The coefficient of passive earth pressure is given by :

K_{p} = tan(ф)

Where:

phi = angle of internal friction

The total lateral earth pressure on the wall is given by:

P = K_{p} * H * γ

Where:

P = lateral earth pressure

K_{p} = coefficient of passive earth pressure

H = height of soil in the wedge

γ = unit weight of soil

Rebhann’s theory is a simplified method that assumes certain ideal conditions, it is not always accurate in real-world situations and other methods such as the limit equilibrium method should be considered as well. It is also important to note that Rebhann’s theory gives a higher value of lateral earth pressure than Coulomb’s theory.