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Civil
Rankine passive earth pressure coefficient
Ratio of horizontal to vertical effective stress when a smooth vertical wall moves into level, drained, cohesionless soil far enough to mobilise passive resistance.
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- CIV-44
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- v1
- Field
- Civil
- Result
- Kp
Rankine passive pressure is the limiting lateral stress when a wall moves into the soil. The coefficient is Kp = tan²(45° + φ′/2) = 1/Ka, so it rises quickly with friction angle: φ′ = 30° gives Kp = 3. Passive resistance needs a large wall movement to mobilise, often several per cent of the wall height, so designs apply a reduction.
Ratio of horizontal to vertical effective stress when a smooth vertical wall moves into level, drained, cohesionless soil far enough to mobilise passive resistance. Kp = 1/Ka. No wall friction, cohesion or strength reduction is included, and the movement needed to reach it depends on the soil.
Calculator
- Friction angle in degrees30 °
- Active coefficient, 1/Kp0.3333
- Equivalent form, (1 + sin φ′)/(1 − sin φ′)3
What drives the result
Each variable is moved across its typical range while the others stay at the values above. The longest bar is the key driver.
- The limiting lateral stress is 3 times the vertical effective stress, 9 times the active pressure for the same soil (Kp/Ka = Kp²), so passive resistance is far larger than active thrust.
- Because large movements are needed to mobilise it, it is common to apply a factor of 2 or more to the passive resistance in design.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing () | Local step | Change in result |
|---|---|---|---|---|---|---|
| φ′ Effective friction angle | 0.05 to 1.45 rad | 1.105 | 273.5 | 272.4 | +10% | +13.1 % |
Sweep of Effective friction angle
| Effective friction angle (rad) | Passive coefficient |
|---|---|
| 0.05 | 1.105 |
| 0.225 | 1.574 |
| 0.4 | 2.276 |
| 0.575 | 3.384 |
| 0.75 | 5.282 |
| 0.925 | 8.932 |
| 1.1 | 17.38 |
| 1.275 | 45.05 |
| 1.45 | 273.5 |
Sweep one variable
Variables and how each one contributes
Passive coefficient
What the equation gives you. Everything else feeds it.
Effective friction angle
The effective friction angle, in radians. A higher friction angle raises the passive coefficient steeply. Raising it by 10 % (to 0.576) moves the result by +13.1 %, from 3 to 3.392.
Worked example with the values above
- Angle: π/4 + φ′/2 = 1.0472 rad (60°).
- Tangent: tan = 1.73205.
- Kp = tan² = 3.
Assumptions and limits
- A vertical smooth wall, level drained cohesionless backfill, and enough wall movement to mobilise the active or passive state.
- Effective stresses for soil strength, with water pressure computed separately.
Common mistakes
- Using full passive resistance in design without allowing for the large wall movement needed to develop it.
- Applying it where the soil in front of the wall may be removed, by erosion or excavation.
- Entering the friction angle in degrees where the page needs radians.
Statistics and uncertainty
Earth pressure is among the most uncertain quantities in structural design. The effective friction angle is rarely known to better than ±2° to 3°, which alone changes Ka by 10 % to 15 %, and the wall movement needed to reach the active state is rarely checked. Treat the numbers as estimates with generous margins.
Where it is used
- Designing retaining walls, basement walls and sheet piles.
- Finding overturning moments and sliding forces on walls.
Maths you need
- Effective stress: total stress minus pore water pressure
- Friction angle: the angle whose tangent is the shear to normal stress ratio at failure
- Active and passive states: minimum and maximum lateral stress at limiting equilibrium