Earth pressure theories are crucial for understanding how soil pushes against structures. Rankine's theory assumes a frictionless wall and planar failure surface, while Coulomb's theory 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

• 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
  • Active earth pressure (minimum lateral pressure)
  • Passive earth pressure (maximum lateral pressure)
• Coefficient of lateral earth pressure (K) relates vertical effective stress to horizontal effective stress
• Principal stresses rotate 45° + φ/2 for active condition and 45° - φ/2 for passive condition (φ = soil's angle of internal friction)
• 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 $Ka = tan²(45° - φ/2)$
• Passive earth pressure coefficient (Kp) determined by $Kp = tan²(45° + φ/2)$
• Lateral earth pressure at depth (z) calculated using $σh = K * γ * z$ (γ = unit weight of soil)
• Total active thrust (Pa) computed by integrating active pressure distribution over wall height $Pa = 0.5 * Ka * γ H²$ (H = wall height)
• Total passive resistance (Pp) calculated similarly $Pp = 0.5 * Kp * γ H²$
• 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 cohesion (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