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Civil
Biaxial corner contact pressures of a rectangular footing
Contact pressure at the four corners of a rigid rectangular footing B by L under a vertical resultant P offset by ex along B and ey along L.
- Ref
- CIV-48
- Rev
- v1
- Field
- Civil
- Result
- q++ [Pa]
A rigid footing under a vertical load P at eccentricity ex along B and ey along L has a plane pressure distribution. The corner pressures are q = P/(BL)[1 ± 6ex/B ± 6ey/L], the four sign combinations. If every corner is in compression, the resultant lies in the kern of the rectangle, 6|ex|/B + 6|ey|/L ≤ 1. Beyond that, part of the base lifts off.
Contact pressure at the four corners of a rigid rectangular footing B by L under a vertical resultant P offset by ex along B and ey along L. The page returns the corner with both signs positive (++); the other three, in the order +−, −+, −−, are in the derived rows. A plane pressure distribution with full compression needs 6|ex|/B + 6|ey|/L ≤ 1. These are contact pressures, not bearing resistances.
Calculator
- Corner (+−) pressure100,000 Pa
- Corner (−+) pressure100,000 Pa
- Corner (−−) pressure40,000 Pa
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 largest corner pressure is 160 kPa against a mean of 100 kPa; the smallest is 40 kPa.
- The kern check 6|ex|/B + 6|ey|/L = 0.6 is within 1, so the whole base stays in compression.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing (Pa) | Local step | Change in result |
|---|---|---|---|---|---|---|
| P Vertical load | 10,000 to 1e7 N | 2,667 | 2,667,000 | 2,664,000 | +10% | +10 % |
| B Footing width | 1.2 to 10 m | 27,200 | 300,000 | 272,800 | +10% | −10.6 % |
| L Footing length | 2 to 10 m | 41,700 | 262,500 | 220,800 | +10% | −10.6 % |
| ex Eccentricity along B | −0.2 to 0.2 m | 70,000 | 190,000 | 120,000 | +0.01 m | +1.88 % |
| ey Eccentricity along L | −0.3 to 0.3 m | 70,000 | 190,000 | 120,000 | +0.012 m | +1.5 % |
Sweep of Eccentricity along B
| Eccentricity along B (m) | Pressure at the (++) corner (Pa) |
|---|---|
| −0.2 | 70,000 |
| −0.15 | 85,000 |
| −0.1 | 100,000 |
| −0.05 | 115,000 |
| 0 | 130,000 |
| 0.05 | 145,000 |
| 0.1 | 160,000 |
| 0.15 | 175,000 |
| 0.2 | 190,000 |
Sweep one variable
Variables and how each one contributes
Pressure at the (++) corner
What the equation gives you. Everything else feeds it.
Vertical load
The vertical load, in N. Scales every corner pressure in direct proportion. Raising it by 10 % (to 660,000) moves the result by +10 %, from 160,000 to 176,000 Pa.
Footing width
The footing width, in m (the direction of ex). A wider footing lowers the mean pressure and the effect of eccentricity. Raising it by 10 % (to 2.2) moves the result by −10.6 %, from 160,000 to 143,000 Pa.
Footing length
The footing length, in m (the direction of ey). A longer footing lowers the mean pressure and the effect of eccentricity. Raising it by 10 % (to 3.3) moves the result by −10.6 %, from 160,000 to 143,000 Pa.
Eccentricity along B
The load eccentricity along the width B, in m. Raises the (++) pressure and lowers the opposite one, in proportion to 6ex/B. Raising it by 10 % (to 0.11) moves the result by +1.88 %, from 160,000 to 163,000 Pa.
Eccentricity along L
The load eccentricity along the length L, in m. Raises the (++) pressure and lowers the opposite one, in proportion to 6ey/L. Raising it by 10 % (to 0.165) moves the result by +1.88 %, from 160,000 to 163,000 Pa.
Worked example with the values above
- Mean pressure: P/(BL) = 600,000 / 6 = 100,000 Pa.
- Eccentricity terms: 6ex/B = 0.3, 6ey/L = 0.3.
- q(++) = 100,000 × (1 + 0.3 + 0.3) = 160,000 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
- Using the formula when 6|ex|/B + 6|ey|/L is above 1, where a corner would be in tension and the base lifts off.
- Mixing up which eccentricity goes with which side.
- Reading the largest corner pressure as the bearing capacity.
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