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Rankine active earth pressure coefficient
Ratio of horizontal to vertical effective stress when a smooth vertical wall moves outward enough to mobilise the full shear strength of level, drained, cohesionless backfill.
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- Ka
Rankine active earth pressure applies when a smooth vertical wall moves outward enough for the backfill to reach limiting equilibrium. The ratio of horizontal to vertical effective stress is Ka = tan²(45° − φ′/2) = (1 − sin φ′)/(1 + sin φ′). It falls as the friction angle rises: a granular soil with φ′ = 30° has Ka = 1/3.
Ratio of horizontal to vertical effective stress when a smooth vertical wall moves outward enough to mobilise the full shear strength of level, drained, cohesionless backfill. φ′ is the effective friction angle in radians, 0 ≤ φ′ < π/2. Water pressure is computed separately.
Calculator
- Friction angle in degrees30 °
- Equivalent form, (1 − sin φ′)/(1 + sin φ′)0.3333
- Passive coefficient, 1/Ka3
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.
- With φ′ = 30° the lateral stress is 33.33 % of the vertical effective stress.
- A 3° error in φ′ would change Ka by about 11.6 %, so the friction angle is the dominant uncertainty.
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 | 0.003657 | 0.9048 | 0.9011 | +10% | −11.6 % |
Sweep of Effective friction angle
| Effective friction angle (rad) | Active coefficient |
|---|---|
| 0.05 | 0.9048 |
| 0.225 | 0.6352 |
| 0.4 | 0.4395 |
| 0.575 | 0.2955 |
| 0.75 | 0.1893 |
| 0.925 | 0.112 |
| 1.1 | 0.05753 |
| 1.275 | 0.0222 |
| 1.45 | 0.003657 |
Sweep one variable
Variables and how each one contributes
Active coefficient
What the equation gives you. Everything else feeds it.
Effective friction angle
The effective friction angle of the soil, in radians (30° = 0.5236 rad). A higher friction angle lowers the active coefficient steeply. Raising it by 10 % (to 0.576) moves the result by −11.6 %, from 0.3333 to 0.2948.
Worked example with the values above
- Angle: π/4 − φ′/2 = 0.523599 rad (30°).
- Tangent: tan = 0.57735.
- Ka = tan² = 0.333333.
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 total instead of effective stress below the water table.
- Using the active coefficient where the wall cannot move, when at-rest pressure applies.
- 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