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Civil
Boussinesq vertical stress from a surface point load
Increase in vertical stress at depth z and radial offset r from a downward point load P on the surface of a homogeneous, isotropic, linearly elastic half-space.
- Ref
- CIV-50
- Rev
- v1
- Field
- Civil
- Result
- Δσz [Pa]
Boussinesq’s solution gives the vertical stress increase in an elastic half-space from a point load: Δσz = 3P/(2πz²) × 1/[1 + (r/z)²]^(5/2). Directly under the load (r = 0) it is 3P/(2πz²) = 0.477P/z², falling with the square of depth. The stress is independent of the soil stiffness, which cancels out in the solution.
Increase in vertical stress at depth z and radial offset r from a downward point load P on the surface of a homogeneous, isotropic, linearly elastic half-space. Use it for the stress increment away from the load, since the stress becomes infinite at the surface point itself.
Calculator
- Stress directly below the load, 3P/(2πz²)11,940 Pa
- Ratio to the centreline value0.5724
- Influence factor, Δσz·z²/P0.2733
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.
- At 2 m deep and 1 m to the side the stress rises by 6.833 kPa, 57.24 % of the value directly below the load.
- Doubling the depth would reduce the axis stress by a factor of four, so loads spread out quickly with depth.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing (Pa) | Local step | Change in result |
|---|---|---|---|---|---|---|
| P Point load | 1,000 to 1e7 N | 68.33 | 683,300 | 683,200 | +10% | +10 % |
| r Radial offset | 0 to 20 m | 0.1164 | 11,940 | 11,940 | +10% | −9.77 % |
| z Depth | 0.2 to 20 m | 118.6 | 8,410 | 8,291 | +10% | −9.72 % |
Sweep of Radial offset
| Radial offset (m) | Vertical stress increase (Pa) |
|---|---|
| 0 | 11,940 |
| 2.5 | 1,136 |
| 5 | 84.34 |
| 7.5 | 13.56 |
| 10 | 3.463 |
| 12.5 | 1.175 |
| 15 | 0.4813 |
| 17.5 | 0.2253 |
| 20 | 0.1164 |
Sweep one variable
Variables and how each one contributes
Vertical stress increase
What the equation gives you. Everything else feeds it.
Point load
The point load on the surface, in N. Scales the stress in direct proportion. Raising it by 10 % (to 110,000) moves the result by +10 %, from 6,833 to 7,516 Pa.
Depth
The depth below the surface, in m. Lowers the stress with the square of the depth, and spreads it laterally. Raising it by 10 % (to 2.2) moves the result by −9.72 %, from 6,833 to 6,168 Pa.
Radial offset
The horizontal distance from the load axis, in m. Lowers the stress sharply as the point moves off the load axis. Raising it by 10 % (to 1.1) moves the result by −9.77 %, from 6,833 to 6,165 Pa.
Worked example with the values above
- Axis stress: 3P/(2πz²) = 3 × 100,000 / (2π × 4) = 11,936.6 Pa.
- Offset factor: [1 + (r/z)²]^(−5/2) = (1 + 0.25)^(−2.5) = 0.572433.
- Δσz = 6,832.92 Pa.
Assumptions and limits
- A rigid or flexible footing on a homogeneous elastic half-space with a level surface.
- A full-contact plane pressure distribution unless the page says that one edge lifts off.
Common mistakes
- Reading contact pressure as bearing capacity.
- Using the full-contact formula when the eccentricity is outside the middle third.
- Using the point-load formula close to the load, where the stress becomes unrealistically large.
Statistics and uncertainty
Elastic stress solutions idealise the soil as homogeneous, isotropic and linear. Measured stresses under real footings differ from these by 20 % to 30 % because of soil layering and non-linearity. Use them for settlement estimates, not for precise stress values.
Where it is used
- Checking footing pressures and stress increments for settlement calculations.
- Estimating stresses under tanks and silos.
Maths you need
- Middle-third rule: a footing stays in full compression if the load acts within the middle third
- Boussinesq solution: stress in an elastic half-space from a point load
- Net load: the load applied less the weight of soil removed