8.2 Rankine's and Coulomb's earth pressure theories
4 min read•august 16, 2024
Earth pressure theories are crucial for understanding how soil pushes against structures. assumes a frictionless wall and , while accounts for wall friction and complex geometries.
These theories help engineers design safe retaining walls and other structures. Rankine's theory is simpler but limited, while Coulomb's theory is more versatile for real-world applications. Both are essential tools in geotechnical engineering.
Rankine's Earth Pressure Theory
Fundamental Principles
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- Based on concept of plastic equilibrium in soils where soil mass verges on failure
- Assumes frictionless, vertical wall and planar failure surface within soil mass
- Considers two extreme states
- (minimum lateral pressure)
- (maximum lateral pressure)
- Coefficient of lateral earth pressure (K) relates vertical to horizontal effective stress
- Principal stresses rotate 45° + φ/2 for active condition and 45° - φ/2 for passive condition (φ = soil's )
- Applicable to cohesionless and cohesive soils with slight calculation differences
- Earth pressure distribution assumed linear with depth, increasing from zero at surface
Calculations for Cohesionless Soils
- Active earth pressure coefficient (Ka) calculated as
- Passive earth pressure coefficient (Kp) determined by
- Lateral earth pressure at depth (z) calculated using (γ = unit weight of soil)
- Total active thrust (Pa) computed by integrating active pressure distribution over wall height (H = wall height)
- Total passive resistance (Pp) calculated similarly
- Point of application for active thrust and passive resistance located at H/3 from wall base
- Modified equations used for sloping backfills to account for soil surface inclination angle
Active vs Passive Earth Pressures
Comparison of Active and Passive States
- Active state occurs when wall moves away from soil mass
- Soil expands horizontally
- Vertical stress remains constant while horizontal stress decreases
- Passive state occurs when wall moves towards soil mass
- Soil compresses horizontally
- Vertical stress remains constant while horizontal stress increases
- Active pressure always less than passive pressure for same soil conditions
- At-rest pressure represents intermediate state between active and passive
Practical Applications
- Active pressure used in design of retaining walls and sheet pile walls
- Determines minimum wall thickness and reinforcement requirements
- Passive pressure utilized in design of anchor blocks and sheet pile toe stability
- Provides resistance against sliding and overturning
- At-rest pressure considered for rigid structures (basement walls, bridge abutments)
- Selection of appropriate pressure state crucial for safe and economical design
- Overestimation leads to conservative but costly designs
- Underestimation results in unsafe structures
Coulomb's Earth Pressure Theory
Assumptions and Limitations
- Assumes planar failure surface and considers equilibrium of entire soil wedge behind wall
- Accounts for wall friction and non-vertical back face of retaining wall
- Assumes soil isotropic, homogeneous, and obeys Mohr-Coulomb failure criteria
- Handles complex geometries (irregular ground surfaces, non-vertical wall faces)
- Neglects curvature of failure surface
- Can lead to overestimation of passive resistance for high friction angles
- Does not consider stress distribution within soil mass
- Focuses solely on overall force equilibrium
- May not be accurate for cohesive soils or significant water pressures
- Primarily developed for dry cohesionless soils
Comparison with Rankine's Theory
- Coulomb's theory more versatile for complex wall geometries
- Rankine's theory simpler to apply but limited to vertical walls
- Coulomb's theory generally yields lower active pressures and higher passive pressures
- Both theories converge for frictionless vertical walls with horizontal backfill
- Rankine's theory provides more conservative results for active pressure design
- Coulomb's theory preferred for passive pressure calculations in most cases
Earth Pressure Problems with Coulomb's Theory
Calculation Methods
- Coulomb's active earth pressure coefficient (Ka) calculated using trigonometric functions
- Involves soil friction angle (φ), wall friction angle (δ), and wall inclination (α)
- Passive earth pressure coefficient (Kp) determined similarly to Ka
- Different angle relationships used
- Total active thrust and passive resistance computed using formulas similar to Rankine's theory
- Coulomb's coefficients substituted
- For cohesive soils, soil (c) incorporated in lateral earth pressure calculations
- Theory extended to account for surcharge loads on soil surface
- Effect added to earth pressure distribution
- Graphical methods (Culmann method) used to solve problems with complex geometries
Practical Applications and Tools
- Computer software and spreadsheets employed to facilitate calculations
- Useful for parametric studies and design optimization
- Coulomb's theory applied in design of
- Gravity retaining walls
- Cantilever retaining walls
- Mechanically stabilized earth (MSE) walls
- Used to analyze stability of slopes and excavations
- Helps determine lateral loads on bridge abutments and wing walls
- Assists in designing sheet pile walls and cellular cofferdams
Key Terms to Review (18)
Active Earth Pressure: Active earth pressure is the lateral pressure exerted by soil on a retaining structure when the soil is allowed to expand, often due to wall movement away from the soil. This condition typically occurs when the wall moves outward or when there is an increase in soil volume, leading to a reduction in stress against the wall. Understanding this concept is crucial for designing various types of retaining structures, analyzing soil behavior, and ensuring stability.
Angle of internal friction: The angle of internal friction is a measure of the shear strength of soil, which describes the resistance to sliding or deformation when stress is applied. This angle plays a critical role in understanding how soil behaves under load, influencing calculations related to stability, bearing capacity, and settlement. Higher values indicate better soil strength and stability, while lower values suggest weaker soil that may be more susceptible to failure.
At-rest condition: The at-rest condition refers to the state of soil or earth material when it is not subject to any external loads or disturbances, meaning it has not been displaced or stressed. In this state, the lateral earth pressure acting on a wall or structure is at its maximum value without any movement occurring. This condition is crucial for understanding how soil behaves in response to different forces, especially in the context of calculating earth pressures using established theories.
Cantilever Wall: A cantilever wall is a type of retaining wall that uses a cantilever design to hold back soil and other materials, relying on its own weight and structural design to resist lateral earth pressures. This type of wall consists of a vertical stem and a horizontal base slab, which work together to provide stability against forces acting on the wall, particularly in conditions where soil loads and water pressure must be countered effectively.
Cohesion: Cohesion is the property of soil that describes the attraction between soil particles, which contributes to the soil's strength and stability. This internal binding force is essential in understanding how soil behaves under different conditions, including how it interacts with moisture, external loads, and other forces acting on it.
Coulomb's Theory: Coulomb's Theory is a fundamental concept in geotechnical engineering that describes the behavior of soil under loading conditions, specifically in relation to earth pressure and stability. This theory introduces the idea of the shear strength of soil being influenced by cohesion and internal friction, which plays a crucial role in determining how soil will respond to various forces and loads applied to it.
Effective Stress: Effective stress is the stress that contributes to the strength and stability of soil, representing the difference between total stress and pore water pressure within the soil. This concept is crucial in understanding how soil behaves under various conditions, particularly in the context of fluid movement, consolidation, and strength properties of soils.
Excavation support: Excavation support refers to the methods and systems used to stabilize the sides of an excavation to prevent soil collapse and ensure safety during construction activities. This is particularly important in deep excavations where the lateral earth pressures exerted by surrounding soil can lead to instability. Proper excavation support is critical for maintaining structural integrity and protecting workers and nearby structures from potential hazards.
Foundation design: Foundation design is the process of determining the appropriate type and size of a foundation to support a structure, ensuring its stability and safety under various loads and conditions. This process involves analyzing soil properties, loads from the structure, and environmental factors to create a foundation that effectively transfers these loads to the ground.
Gravity wall: A gravity wall is a type of retaining wall that relies on its own weight to resist the lateral pressure exerted by soil or other materials behind it. These walls are typically made of heavy materials such as concrete or masonry and are designed to prevent soil movement by resisting gravitational forces. Their design involves considerations of earth pressure states and theories, ensuring stability under various conditions, including seismic events.
Horizontal backfill surface: The horizontal backfill surface is the flat area of soil or material placed behind a retaining structure that provides lateral support to the structure and prevents soil movement. This surface plays a vital role in earth pressure calculations, affecting the forces acting on retaining walls and influencing stability in the context of geotechnical engineering.
Non-cohesive soils: Non-cohesive soils are granular soils that primarily consist of particles that do not stick together, relying on friction and interlocking for stability. These types of soils include sands and gravels, which are characterized by their ability to drain water easily and exhibit high permeability. Non-cohesive soils play a significant role in various engineering applications, particularly when it comes to understanding their behavior under loading and during excavation.
Passive Earth Pressure: Passive earth pressure refers to the lateral force exerted by soil on a retaining structure when the structure moves away from the soil, such as when it is pushed or tilted. This pressure develops due to the soil’s resistance to deformation and plays a crucial role in the design and stability of various types of retaining walls. Understanding passive earth pressure is essential for calculating the forces acting on walls and ensuring they remain stable against soil movements.
Planar failure surface: A planar failure surface is a flat, distinct surface along which a mass of soil or rock can slide or fail due to gravitational forces or external loading. This concept is essential in understanding the mechanisms of slope stability and the behavior of earth materials under various stress conditions. The orientation and characteristics of a planar failure surface significantly influence the stability calculations and predictions related to earth pressure theories.
Rankine's Theory: Rankine's Theory is a method used to calculate lateral earth pressures exerted by soil on retaining structures. It assumes that the soil is cohesionless and at rest, relying on basic geometric principles to determine the horizontal and vertical stress distributions acting on a wall or structure. This theory is foundational for understanding earth pressure concepts and serves as a basis for comparison with other earth pressure theories, such as Coulomb's.
Sloped backfill: Sloped backfill refers to soil or material that is placed against a structure at an angle rather than being vertical. This configuration can affect the earth pressure acting on retaining walls and other structures. Understanding the nature of sloped backfill is essential when applying Rankine's and Coulomb's earth pressure theories, as these theories predict how soil behaves under various conditions, including different angles of backfill.
Surcharge load: A surcharge load is an additional load applied to the surface of the ground, typically resulting from structures, fill materials, or other constructions placed above the natural soil. This load can significantly influence the stress distribution within the soil, impacting the design and stability of structures such as retaining walls. Understanding how surcharge loads affect soil behavior is crucial for predicting earth pressure states and ensuring adequate support against sliding or overturning.
Total stress: Total stress is the overall stress exerted on a soil element, comprising both the weight of the soil above it and any external loads applied. Understanding total stress is crucial because it lays the foundation for assessing how soil behaves under various conditions, particularly when considering the interplay of effective stress, the distribution of stress in layered soils, and how earth pressures are calculated according to established theories.