Earth Pressure Theories

Why it matters

Understanding Earth Pressure Theories is crucial for designing retaining walls, basement walls, and other structures that hold back soil. These theories help engineers predict how soil will behave under different conditions, ensuring the stability and safety of structures.

Key ideas

• Earth Pressure: The lateral pressure exerted by soil on a retaining structure. It varies with soil type, wall movement, and water content.
• Types of Earth Pressure:
  • Active Earth Pressure: Occurs when the wall moves away from the backfill, reducing pressure.
  • Passive Earth Pressure: Occurs when the wall moves towards the backfill, increasing pressure.
  • At-Rest Earth Pressure: Corresponds to restrained lateral strain; pressure is not generally constant with depth.
• Coulomb's Theory: Considers wall friction and slope of backfill, providing a more general solution.
• Rankine's Theory: The simplified case used here assumes a smooth vertical wall, horizontal homogeneous cohesionless backfill, no surcharge or water pressure and sufficient wall movement to mobilize the limit state.

Formulas

• Rankine's Active Earth Pressure: P_a = 0.5 * γ * H^2 * K_a where:

  • P_a = Active resultant force per unit wall length (kN/m)
  • γ = Unit weight of soil (kN/m³)
  • H = Height of the wall (m)
  • K_a = Coefficient of active earth pressure (dimensionless)

• Rankine's Passive Earth Pressure: P_p = 0.5 * γ * H^2 * K_p where:

  • P_p = Passive resultant force per unit wall length (kN/m)
  • K_p = Coefficient of passive earth pressure (dimensionless)

For this case, K_a = tan²(45° − φ/2) and K_p = tan²(45° + φ/2). The triangular resultant acts H/3 above the base; local active pressure at depth z is p_a = K_a γ z. Add pore-water pressure separately where applicable.

Worked example

Given: A retaining wall with a height of 5 m, dry cohesionless horizontal backfill of unit weight 18 kN/m³, and an angle of internal friction of 30°.

1. Calculate the coefficient of active earth pressure (K_a): K_a = tan^2(45° - φ/2) K_a = tan^2(45° - 30°/2) K_a = tan^2(30°) K_a = 0.333

2. Calculate the active earth pressure (P_a): P_a = 0.5 * γ * H^2 * K_a P_a = 0.5 * 18 kN/m³ * (5 m)^2 * 0.333 P_a = 0.5 * 18 * 25 * 0.333 P_a = 75 kN/m (using K_a = 1/3)

Final Answer: 75 kN/m (using K_a = 1/3)

Common mistakes

• Confusing active and passive earth pressures.
• Incorrectly calculating the coefficients K_a and K_p.
• Ignoring wall friction in Coulomb's theory.

For GATE CE

Questions often involve calculating earth pressures using Rankine's or Coulomb's theories. Practice problems involving different wall movements and soil conditions to strengthen understanding.

Quick check

1. What is the difference between active and passive earth pressure?
2. How does wall movement affect earth pressure?
3. What assumptions does Rankine's theory make?

Answers: 1. Active pressure occurs when the wall moves away, passive when it moves towards the soil. 2. Wall movement changes the pressure from active to passive or vice versa. 3. This simplified case uses a smooth vertical wall and horizontal cohesionless backfill.

Reference

FHWA Shallow Foundations, earth-pressure treatment.