Library / Civil / CIV-52
Civil
Average stress increase by the 2 to 1 method
Approximate average vertical stress increase at depth z below a B by L rectangular loaded area under net vertical load P, assuming the load spreads at two vertical to one horizontal so that the area grows to (B + z)(L + z).
- Ref
- CIV-52
- Rev
- v1
- Field
- Civil
- Result
- Δσavg [Pa]
The 2 to 1 method assumes the load spreads at a slope of two vertical to one horizontal, so at depth z the loaded area has grown from B × L to (B + z)(L + z). The average stress is the load divided by that area. It is easy to use but gives only the average over the spread area and not the peak stress under the centre of the footing.
Approximate average vertical stress increase at depth z below a B by L rectangular loaded area under net vertical load P, assuming the load spreads at two vertical to one horizontal so that the area grows to (B + z)(L + z). It gives the average over the spread area, not the stress at a point.
Calculator
- Spread area at the depth, (B + z)(L + z)20 m²
- Contact pressure at the base, P/(BL)100,000 Pa
- Stress as a fraction of the contact pressure0.3
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.
- At 2 m deep the average stress is 30 kPa, 30 % of the 100 kPa applied at the base.
- At z = B the width has doubled, and deeper layers see much less of the load, so settlement is usually small below about 2 to 3 widths.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing (Pa) | Local step | Change in result |
|---|---|---|---|---|---|---|
| P Net vertical load | 10,000 to 1e7 N | 500 | 500,000 | 499,500 | +10% | +10 % |
| z Depth below the base | 0 to 20 m | 1,186 | 100,000 | 98,810 | +10% | −8.42 % |
| L Footing length | 0.5 to 10 m | 12,500 | 60,000 | 47,500 | +10% | −5.66 % |
| B Footing width | 0.5 to 10 m | 10,000 | 48,000 | 38,000 | +10% | −4.76 % |
Sweep of Depth below the base
| Depth below the base (m) | Average stress increase (Pa) |
|---|---|
| 0 | 100,000 |
| 2.5 | 24,240 |
| 5 | 10,710 |
| 7.5 | 6,015 |
| 10 | 3,846 |
| 12.5 | 2,670 |
| 15 | 1,961 |
| 17.5 | 1,501 |
| 20 | 1,186 |
Sweep one variable
Variables and how each one contributes
Average stress increase
What the equation gives you. Everything else feeds it.
Net vertical load
The net vertical load applied at the base, in N. Scales the stress in direct proportion. Raising it by 10 % (to 660,000) moves the result by +10 %, from 30,000 to 33,000 Pa.
Footing width
The footing width, in m. A wider footing spreads the load over a larger area at every depth. Raising it by 10 % (to 2.2) moves the result by −4.76 %, from 30,000 to 28,570 Pa.
Footing length
The footing length, in m. A longer footing spreads the load over a larger area at every depth. Raising it by 10 % (to 3.3) moves the result by −5.66 %, from 30,000 to 28,300 Pa.
Depth below the base
The depth below the footing base, in m. Greater depth increases the spread area and lowers the stress. Raising it by 10 % (to 2.2) moves the result by −8.42 %, from 30,000 to 27,470 Pa.
Worked example with the values above
- Spread dimensions: B + z = 4 m, L + z = 5 m, area 20 m².
- Load P = 600,000 N.
- Δσavg = 600,000 / 20 = 30,000 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