Rankine vs Coulomb Earth Pressure Calculator
Input Parameters
Calculated Results
Rankine Active Coefficient (Ka)
0.333
Rankine Passive Coefficient (Kp)
3.000
Coulomb Active Coefficient (Ka)
0.279
Coulomb Passive Coefficient (Kp)
5.730
Input Notes:
- α = Wall inclination from vertical
- β = Backfill slope angle
- Vertical wall → α = 0°
- Horizontal backfill → β = 0°
Rankine Theory
Rankine Active Pressure Coefficient:
Ka = (1 − sinφ) / (1 + sinφ)
Rankine Passive Pressure Coefficient:
Kp = (1 + sinφ) / (1 − sinφ)
Ka = (1 − sinφ) / (1 + sinφ)
Rankine Passive Pressure Coefficient:
Kp = (1 + sinφ) / (1 − sinφ)
Rankine theory assumes:
- Smooth vertical wall
- No wall friction
- Horizontal or simple sloping backfill
- Stress-based approach
Rankine theory is simpler and mainly used for preliminary retaining wall design.
Coulomb Theory
General Coulomb Active Pressure Equation:
Ka = [ sin²(α + φ) ] / [ sin²(α) × sin(α − δ) × ( 1 + √{ [sin(φ + δ) × sin(φ − β)] / [sin(α − δ) × sin(α + β)] } )² ]
General Coulomb Passive Pressure Equation:
Kp = [ sin²(α − φ) ] / [ sin²(α) × sin(α + δ) × ( 1 - √{ [sin(φ + δ) × sin(φ + β)] / [sin(α + δ) × sin(α − β)] } )² ]
Ka = [ sin²(α + φ) ] / [ sin²(α) × sin(α − δ) × ( 1 + √{ [sin(φ + δ) × sin(φ − β)] / [sin(α − δ) × sin(α + β)] } )² ]
General Coulomb Passive Pressure Equation:
Kp = [ sin²(α − φ) ] / [ sin²(α) × sin(α + δ) × ( 1 - √{ [sin(φ + δ) × sin(φ + β)] / [sin(α + δ) × sin(α − β)] } )² ]
Coulomb theory considers:
- Wall friction
- Inclined retaining walls
- Sloping backfill
- Failure wedge equilibrium
It provides more realistic earth pressure values and is widely used in professional geotechnical software.
Difference Between Rankine and Coulomb Theory
| Feature | Rankine Theory | Coulomb Theory |
|---|---|---|
| Wall Friction | Ignored | Considered |
| Wall Inclination | Not Properly Considered | Considered |
| Backfill Slope | Limited | Fully Considered |
| Theory Type | Stress Theory | Force Equilibrium Theory |
| Accuracy | Moderate | High |
| Complexity | Simple | Complex |
| Failure Surface | Assumed | Calculated |
| Typical Use | Preliminary Design | Real Retaining Wall Design |
Engineering Interpretation
When wall friction exists:
- Active earth pressure decreases
- Passive resistance increases
Therefore:
- Coulomb Ka is usually smaller than Rankine Ka
- Coulomb Kp is usually larger than Rankine Kp
This behavior is clearly visible in the calculator results.
Visual Understanding
RANKINE THEORY
-------------------------------
Smooth Vertical Wall
|\
| \
| \
| \ Failure Plane
| \
|_____ \________________
Features:
• Smooth wall
• No wall friction
• Simple stress theory
COULOMB THEORY
--------------------------------
Inclined Wall with Wall Friction
/
/
/ Failure Wedge
/
/____________________
Features:
• Wall friction considered
• Inclined wall possible
• Sloping backfill possible
• Wedge equilibrium theory
Applications
| Application | Recommended Theory |
|---|---|
| Simple Retaining Wall | Rankine |
| Bridge Abutment | Coulomb |
| Sheet Pile Wall | Coulomb |
| Basement Wall | Rankine |
| Dock Wall | Coulomb |
| Real Design Software | Coulomb |
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