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Civil
Stress beneath centre of a uniformly loaded circular area
Vertical stress increase on the centreline at depth z below a flexible uniform circular load q of radius R, in a homogeneous, isotropic elastic half-space.
- Ref
- CIV-51
- Rev
- v1
- Field
- Civil
- Result
- Δσz [Pa]
Integrating Boussinesq’s point-load stress over a uniformly loaded disk of radius R gives, on the centreline at depth z, Δσz = q[1 − (1 + R²/z²)^(−3/2)]. At the surface (z → 0) the stress is q. Deep below, it falls as 3qR²/(2z²). The solution is for a flexible load in a homogeneous, isotropic, linearly elastic half-space.
Vertical stress increase on the centreline at depth z below a flexible uniform circular load q of radius R, in a homogeneous, isotropic elastic half-space. It follows from integrating the point-load solution over the disk. The stress is q at the surface and falls with depth.
Calculator
- Stress ratio, Δσz/q0.6464
- Depth-to-radius ratio, z/R1
- Depth at which the stress is half of q2.61 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.
- At a depth of 2 m (z/R = 1) the stress is 64.64 % of the surface pressure, 64.64 kPa.
- The stress falls to half of q at a depth of 2.61 m, about 1.3 radii.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing (Pa) | Local step | Change in result |
|---|---|---|---|---|---|---|
| q Surface pressure | 1,000 to 1,000,000 Pa | 646.4 | 646,400 | 645,800 | +10% | +10 % |
| R Radius of loaded area | 0.2 to 20 m | 1,481 | 99,900 | 98,420 | +10% | +7.61 % |
| z Depth below the centre | 0.2 to 20 m | 1,481 | 99,900 | 98,420 | +10% | −7.98 % |
Sweep of Depth below the centre
| Depth below the centre (m) | Vertical stress increase (Pa) |
|---|---|
| 0.2 | 99,900 |
| 2.675 | 48,630 |
| 5.15 | 19,000 |
| 7.625 | 9,498 |
| 10.1 | 5,606 |
| 12.57 | 3,678 |
| 15.05 | 2,592 |
| 17.52 | 1,922 |
| 20 | 1,481 |
Sweep one variable
Variables and how each one contributes
Vertical stress increase
What the equation gives you. Everything else feeds it.
Surface pressure
The uniform pressure on the circular area, in Pa. Scales the stress in direct proportion. Raising it by 10 % (to 110,000) moves the result by +10 %, from 64,640 to 71,110 Pa.
Radius of loaded area
The radius of the loaded circle, in m. A larger circle sends a larger share of q to greater depths. Raising it by 10 % (to 2.2) moves the result by +7.61 %, from 64,640 to 69,560 Pa.
Depth below the centre
The depth below the centre, in m. Greater depth lowers the stress, toward zero. Raising it by 10 % (to 2.2) moves the result by −7.98 %, from 64,640 to 59,490 Pa.
Worked example with the values above
- Ratio: (1 + R²/z²) = 1 + 1 = 2.
- Power: (…)^(−3/2) = 0.353553.
- Δσz = 100,000 × (1 − 0.353553) = 64,644.7 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