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Active force from soil weight and uniform surcharge
Resultant horizontal force per metre of wall length on a vertical wall of height H retaining homogeneous, dry, cohesionless backfill of unit weight γ with a uniform surface surcharge q.
- Ref
- CIV-45
- Rev
- v1
- Field
- Civil
- Result
- Pa [N/m]
The active pressure on a wall has two parts: the soil weight gives a triangular pressure of Ka γ z, with resultant Ka γ H²/2, and a uniform surcharge q gives a rectangular pressure Ka q, with resultant Ka q H. The total force per metre of wall length is their sum, Pa = Ka(γH²/2 + qH). Water pressure is excluded and must be added.
Resultant horizontal force per metre of wall length on a vertical wall of height H retaining homogeneous, dry, cohesionless backfill of unit weight γ with a uniform surface surcharge q. The soil gives a triangular pressure and the surcharge a rectangular one. Water is excluded.
Calculator
- Soil-weight part, KaγH²/248,000 N/m
- Surcharge part, KaqH13,330 N/m
- Overturning moment about the base90,670 N·m/m
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 thrust is 61.33 kN per metre of wall; the surcharge supplies 21.74 % of it.
- Doubling the wall height would raise the soil part fourfold and the surcharge part twofold, to 218.7 kN/m.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing (N/m) | Local step | Change in result |
|---|---|---|---|---|---|---|
| H Wall height | 0.5 to 15 m | 2,417 | 725,000 | 722,600 | +10% | +18.6 % |
| Ka Active coefficient | 0.05 to 1 | 9,200 | 184,000 | 174,800 | +10% | +10 % |
| q Surcharge | 0 to 100,000 Pa | 48,000 | 181,300 | 133,300 | +10% | +2.17 % |
| γ Unit weight of soil | 10,000 to 25,000 N/m³ | 40,000 | 80,000 | 40,000 | +10% | +7.83 % |
Sweep of Wall height
| Wall height (m) | Active force per metre (N/m) |
|---|---|
| 0.5 | 2,417 |
| 2.313 | 23,750 |
| 4.125 | 64,800 |
| 5.938 | 125,600 |
| 7.75 | 206,000 |
| 9.563 | 306,200 |
| 11.38 | 426,100 |
| 13.19 | 565,700 |
| 15 | 725,000 |
Sweep one variable
Variables and how each one contributes
Active force per metre
What the equation gives you. Everything else feeds it.
Active coefficient
The active earth pressure coefficient. Raises both parts of the force in direct proportion. Raising it by 10 % (to 0.3667) moves the result by +10 %, from 61,330 to 67,470 N/m.
Unit weight of soil
The unit weight of the backfill, in N/m³. Raises the soil-weight term in direct proportion. Raising it by 10 % (to 19,800) moves the result by +7.83 %, from 61,330 to 66,130 N/m.
Wall height
The wall height, in m. Raises the force with a mix of H² for soil weight and H for surcharge. Raising it by 10 % (to 4.4) moves the result by +18.6 %, from 61,330 to 72,750 N/m.
Surcharge
The uniform surcharge on the surface, in Pa. Raises the rectangular term in direct proportion. Raising it by 10 % (to 11,000) moves the result by +2.17 %, from 61,330 to 62,670 N/m.
Worked example with the values above
- Soil part: Ka γ H²/2 = 0.3333 × 18,000 × 16 / 2 = 48,000 N/m.
- Surcharge part: Ka q H = 0.3333 × 10,000 × 4 = 13,333.3 N/m.
- Pa = 61,333.3 N/m.
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