Understanding lateral earth pressure is essential for designing safe and stable retaining walls. This calculation is particularly important when the wall movement is negligible, and the soil remains in its natural state, also known as the “at rest” condition.

Given Soil Profile and Properties

Soil Layers and Their Characteristics

• First Layer (H₁ = 2.5 m):

  • Angle of internal friction (𝜙₁) = 32°

  • Unit weight (γ) = 16 kN/m³
• Second Layer (H₂ = 3 m):

  • Angle of internal friction (𝜙₂) = 34°

  • Saturated unit weight (γ_sat) = 19 kN/m³
  • Water unit weight (γ_w) = 10 kN/m³

Submerged Unit Weight Calculation

The submerged unit weight (γ’) of the second layer:
γ’ = γ_sat – γ_w = 19 – 10 = 9 kN/m³

Earth Pressure Coefficients (At Rest Condition)

For First Layer (Ka₁):

Ka₁ = 1 – sin(𝜙₁)
Ka₁ = 1 – sin(32°)
Ka₁ ≈ 1 – 0.5299 = 0.47

For Second Layer (Ka₂):

Ka₂ = 1 – sin(𝜙₂)
Ka₂ = 1 – sin(34°)
Ka₂ ≈ 1 – 0.5591 = 0.4408

Earth Pressure Intensities

At Base of First Layer (Point B):

p₀₁ = Ka₁ × γ × H₁
p₀₁ = 0.47 × 16 × 2.5 = 18.80 kN/m²

Due to the First Layer at the Bottom of the Second Layer (Point C):

p₀₂ = Ka₂ × γ × H₁
p₀₂ = 0.4408 × 16 × 2.5 = 17.632 kN/m²

Due to Second Layer (Submerged Weight):

p₀₃ = Ka₂ × γ’ × H₂
p₀₃ = 0.4408 × 9 × 3 = 11.016 kN/m²

Water Pressure in Second Layer:

u = γ_w × H₂
u = 10 × 3 = 30 kN/m²

Note: Water pressure in the first layer is zero since the water table starts from the second layer.

Total Pressure Distribution

Pressure Components

• Triangle ABD:
  P₁ = ½ × 18.8 × 2.5 = 23.5 kN
• Rectangle BCFE:
  P₂ = 17.632 × 3 = 52.896 kN
• Triangle EFG:
  P₃ = ½ × 11.016 × 3 = 16.524 kN
• Triangle EGH (Water Pressure):
  P₄ = ½ × 30 × 3 = 45 kN

Summation of Total Active Earth Pressure

P_total = P₁ + P₂ + P₃ + P₄
P_total = 23.5 + 52.896 + 16.524 + 45 = 137.92 kN

Location of Resultant Earth Pressure

Centroid of Pressure Diagram

• P₁ (Triangle ABD):
  Acts at 3 + (⅓ × 2.5) = 3.833 m
• P₂ (Rectangle BCFE):
  Acts at ½ × 3 = 1.5 m
• P₃ (Triangle EFG):
  Acts at ⅓ × 3 = 1 m
• P₄ (Triangle EGH):
  Acts at ⅓ × 3 = 1 m

Moment Calculation

PȲ = (23.5 × 3.833) + (52.896 × 1.5) + (16.524 × 1) + (45 × 1)
PȲ = 90.076 + 79.344 + 16.524 + 45 = 230.944 kNm

Ȳ = PȲ / P_total = 230.944 / 137.92 = 1.675 m

Final Result

The total lateral earth pressure is 137.92 kN, and it acts at 1.675 meters from the base of the retaining wall.

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