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Hydraulics, open channels and fluid power
Resultant hydrostatic force on a submerged plane
Normal force of a static liquid on a fully wetted plane surface of area A whose centroid lies at vertical depth hc.
- Ref
- HYD-01
- Rev
- v1
- Field
- Hydraulics, open channels and fluid power
- Result
- F [N]
The pressure of a static liquid grows linearly with depth, so the mean pressure on a plane surface equals the pressure at its centroid. The resultant force is that mean pressure times the area, F = ρ g A hc, whatever the plate’s shape or tilt. It acts normal to the surface, below the centroid (see the centre of pressure). Only the gauge pressure counts when the other side is open to the air.
Normal force of a static liquid on a fully wetted plane surface of area A whose centroid lies at vertical depth hc. It is the pressure at the centroid (gauge) times the area, with atmospheric pressure on the other side.
Calculator
- Gauge pressure at the centroid, ρ g hc14,720 Pa
- Force in kilonewtons29.43 kN
- Force if the plane is lowered to twice the depth58,860 N
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.
- A 2 m² plane with its centroid 1.5 m down carries 29,430 N (29.43 kN), the weight of 3 tonnes of liquid.
- Because only the centroid depth matters, the plane could be tilted or reshaped without changing the force, only its line of action.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing (N) | Local step | Change in result |
|---|---|---|---|---|---|---|
| A Wetted area | 0.01 to 100 m² | 147.2 | 1,472,000 | 1,471,000 | +10% | +10 % |
| hc Vertical depth of the centroid | 0.05 to 50 m | 981 | 981,000 | 980,000 | +10% | +10 % |
| ρ Liquid density | 500 to 14,000 kg/m³ | 14,720 | 412,000 | 397,300 | +10% | +10 % |
| g Gravitational acceleration | 1 to 30 m/s² | 3,000 | 90,000 | 87,000 | +10% | +10 % |
Sweep of Vertical depth of the centroid
| Vertical depth of the centroid (m) | Hydrostatic force (N) |
|---|---|
| 0.05 | 981 |
| 0.1186 | 2,326 |
| 0.2812 | 5,517 |
| 0.6668 | 13,080 |
| 1.581 | 31,020 |
| 3.749 | 73,560 |
| 8.891 | 174,400 |
| 21.08 | 413,700 |
| 50 | 981,000 |
Sweep one variable
Variables and how each one contributes
Hydrostatic force
What the equation gives you. Everything else feeds it.
Liquid density
The liquid density, in kg/m³. Raises the force in direct proportion. Raising it by 10 % (to 1,100) moves the result by +10 %, from 29,430 to 32,370 N.
Gravitational acceleration
The gravitational acceleration, in m/s². Raises the force in direct proportion. Raising it by 10 % (to 10.79) moves the result by +10 %, from 29,430 to 32,370 N.
Wetted area
The wetted area of the plane, in m². Raises the force in direct proportion. Raising it by 10 % (to 2.2) moves the result by +10 %, from 29,430 to 32,370 N.
Vertical depth of the centroid
The vertical depth of the centroid below the free surface, in m. Raises the force in direct proportion. Raising it by 10 % (to 1.65) moves the result by +10 %, from 29,430 to 32,370 N.
Worked example with the values above
- Pressure at the centroid: ρ g hc = 1,000 × 9.81 × 1.5 = 14,715 Pa.
- Multiply by the area A = 2 m².
- F = 29,430 N.
Assumptions and limits
- A liquid at rest with constant density and atmospheric pressure at the free surface.
- Gauge pressure is used, with atmospheric pressure acting on the other side of the surface.
Common mistakes
- Measuring depth from the wrong datum, instead of from the free surface to the centroid.
- Using the plate length where the vertical depth is needed.
- Using absolute pressure, which adds a force that atmospheric pressure on the other side cancels.
Statistics and uncertainty
Hydrostatic results depend on the density and on a depth that is measured from the free surface, so a 1 % error in depth gives about a 1 % error in force and a larger one in moments, which go with higher powers of depth.
Where it is used
- Sizing gates, hatches, windows and tank walls.
- Reading pressure differences from manometers and finding the surface shape of rotating liquids.
Maths you need
- Hydrostatic pressure: p = ρ g h, increasing linearly with depth
- Centroid: the geometric centre of an area
- Second moment of area: I, in m⁴, about an axis through the centroid