Earth Pressure Calculator

Earth Pressure Calculator: Complete Guide to Active, Passive, and At-Rest Pressures Using Rankine and Coulomb Theories

The Earth Pressure Calculator computes lateral soil pressures behind retaining structures using Rankine and Coulomb theories. Accurate earth pressure calculation is fundamental in geotechnical engineering for designing safe retaining walls, sheet piles, basement walls, and underground structures. This guide covers theory, formulas, applications, and best practices for active earth pressure, passive earth pressure, and at-rest pressure.

What is Lateral Earth Pressure?

Lateral earth pressure is the horizontal force exerted by soil on a retaining structure. It depends on soil properties (unit weight γ, friction angle φ, cohesion c), wall geometry, backfill slope, and surcharge loads. There are three primary states:

• Active Pressure (P_a): When the wall moves away from the soil (minimum pressure).
• Passive Pressure (P_p): When the wall pushes into the soil (maximum pressure).
• At-Rest Pressure (P₀): When the wall is stationary (no lateral strain).

Rankine Earth Pressure Theory

Developed by William Rankine in 1857, this theory assumes a vertical wall, frictionless wall-soil interface, and level or sloping backfill. It is widely used for simple retaining wall design.

Active Pressure Coefficient (K_a)

\[ K_a = \frac{\cos\beta - \sqrt{\cos^2\beta - \cos^2\phi}}{\cos\beta + \sqrt{\cos^2\beta - \cos^2\phi}} \]

For vertical wall (β=0): \[ K_a = \tan^2(45^\circ - \phi/2) \]

Passive Pressure Coefficient (K_p)

\[ K_p = \tan^2(45^\circ + \phi/2) \]

Pressure at Depth z

Active: \[ \sigma_h' = K_a \gamma z + K_a q \]
Passive: \[ \sigma_h' = K_p \gamma z + K_p q \]
At-Rest: \[ \sigma_h' = K_0 \gamma z + q \] where \( K_0 = 1 - \sin\phi \)

Coulomb Earth Pressure Theory

Coulomb’s wedge theory (1776) accounts for wall friction (δ) and inclined walls. It is more accurate for non-vertical walls and rough interfaces.

Active Coefficient

\[ K_{ae} = \frac{\cos^2(\phi - \beta)}{\cos^2\beta \cos(\beta + \delta) \left[1 + \sqrt{\frac{\sin(\phi + \delta)\sin(\phi - i)}{\cos(\beta + \delta)\cos(\beta - i)}}\right]^2} \]

Passive Coefficient

\[ K_{pe} = \frac{\cos^2(\phi + \beta)}{\cos^2\beta \cos(\beta - \delta) \left[1 - \sqrt{\frac{\sin(\phi + \delta)\sin(\phi + i)}{\cos(\beta - \delta)\cos(\beta - i)}}\right]^2} \]

Step-by-Step Earth Pressure Calculation

1. Determine soil parameters: γ, φ, c, δ
2. Define geometry: wall angle β, backfill slope i
3. Select theory: Rankine (simple) or Coulomb (complex)
4. Compute K_a, K_p, K₀
5. Apply surcharge and depth
6. Calculate total force: P = ½ K γ H² + K q H

Example: Rankine Active Pressure

γ = 18 kN/m³, φ = 30°, H = 5m, q = 10 kPa, vertical wall:

• K_a = tan²(15°) = 0.333
• σ_h at base = 0.333 × 18 × 5 + 0.333 × 10 = 30 + 3.33 = 33.33 kPa
• Total force P_a = ½ × 33.33 × 5 = 83.33 kN/m

Typical Earth Pressure Coefficients

Applications in Civil Engineering

• Retaining Walls: Cantilever, gravity, counterfort
• Sheet Piles: Anchored, cantilevered
• Basement Walls: Braced excavations
• Tunnels and Shafts: Lining design
• Bridge Abutments: Stability analysis

Effect of Water and Surcharge

For submerged soils, use effective stress: γ' = γ - γ_w. Hydrostatic pressure adds u = γ_w z. Surcharge q contributes uniform pressure K q across depth.

Cohesive Soils (c-φ)

Rankine active pressure: \[ \sigma_a = K_a \gamma z - 2c \sqrt{K_a} \]
Tension crack depth: z₀ = 2c √K_a / γ

Best Practices for Earth Pressure Design

• Use Coulomb for walls with friction (δ = ⅔φ)
• Apply safety factor: 1.5 on passive, 2.0 on active
• Provide drainage to reduce hydrostatic pressure
• Check global stability (sliding, overturning, bearing)
• Use geosynthetics for reinforced soil

Common Mistakes to Avoid

• Using active pressure for braced excavations (use at-rest)
• Ignoring wall friction in Coulomb analysis
• Neglecting surcharge from nearby structures
• Assuming zero cohesion in clays long-term
• Misapplying Rankine to battered walls

Advanced Topics in Earth Pressure

• Seismic Earth Pressure: Mononobe-Okabe method
• Compaction-Induced Pressure: Trapdoor effect
• 3D Effects: Corner effects in excavations
• Numerical Modeling: PLAXIS, FLAC
• Soil-Structure Interaction: Finite element analysis

Integration with Other Calculators

Pair this Earth Pressure Calculator with:

• Retaining Wall Design for stability checks
• Soil Bearing Capacity for foundation sizing
• Slope Stability for global failure

Conclusion

Accurate lateral earth pressure calculation is the cornerstone of safe retaining structure design. Whether using Rankine theory for simplicity or Coulomb method for precision, our Earth Pressure Calculator delivers instant, reliable results. Input soil properties, wall geometry, and loads to obtain active, passive, or at-rest pressures at any depth. Combine with drainage design and stability analysis for complete geotechnical solutions. Build stable, cost-effective retaining systems with confidence!

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