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Hydraulics, open channels and fluid power
Vertical depth of the centre of pressure on an inclined plane
Vertical depth below the free surface at which the resultant hydrostatic force acts on a plane of area A, centroid depth hc and centroidal second moment of area I about a horizontal in-plane axis.
- Ref
- HYD-02
- Rev
- v1
- Field
- Hydraulics, open channels and fluid power
- Result
- hp [m]
The hydrostatic force on an inclined plane acts below its centroid, because pressure is higher at depth. The vertical depth of the centre of pressure is hp = hc + I sin²θ/(A hc), where I is the second moment of area about a horizontal axis through the centroid and θ is the plane’s angle above horizontal. A vertical plane (θ = 90°) gives the largest shift, and a horizontal one gives none.
Vertical depth below the free surface at which the resultant hydrostatic force acts on a plane of area A, centroid depth hc and centroidal second moment of area I about a horizontal in-plane axis. θ is the plate angle above horizontal, in degrees.
Calculator
- Shift below the centroid, hp − hc0.3333 m
- Slope distance from the free surface to the centroid1 m
- Slope distance from the free surface to the centre of pressure1.333 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.
- The resultant acts 0.3333 m below the centroid, at a depth of 1.333 m.
- For a rectangle wetted from the surface (I = A h²/12 about its centroid with hc = h/2), the centre of pressure is at two thirds of the depth.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing (m) | Local step | Change in result |
|---|---|---|---|---|---|---|
| A Wetted area | 0.01 to 100 m² | 1.007 | 67.67 | 66.66 | +10% | −2.27 % |
| IG Centroidal second moment of area | 1e-4 to 100 m⁴ | 1 | 51 | 50 | +10% | +2.5 % |
| hc Vertical depth of the centroid | 0.05 to 50 m | 1.159 | 50.01 | 48.85 | +10% | +5.23 % |
| θ Plate angle above horizontal | 1 to 90 ° | 1 | 1.333 | 0.3332 | +10% | −0.612 % |
Sweep of Plate angle above horizontal
| Plate angle above horizontal (°) | Depth of the centre of pressure (m) |
|---|---|
| 1 | 1 |
| 12.13 | 1.015 |
| 23.25 | 1.052 |
| 34.38 | 1.106 |
| 45.5 | 1.17 |
| 56.63 | 1.232 |
| 67.75 | 1.286 |
| 78.88 | 1.321 |
| 90 | 1.333 |
Sweep one variable
Variables and how each one contributes
Depth of the centre of pressure
What the equation gives you. Everything else feeds it.
Vertical depth of the centroid
The vertical depth of the centroid, in m. A deeper centroid pushes the centre of pressure closer to the centroid, so hp rises less than in proportion. Raising it by 10 % (to 1.1) moves the result by +5.23 %, from 1.333 to 1.403 m.
Centroidal second moment of area
The second moment of area about a horizontal axis through the centroid, in m⁴. A larger I moves the centre of pressure further below the centroid. Raising it by 10 % (to 0.7333) moves the result by +2.5 %, from 1.333 to 1.367 m.
Wetted area
The wetted area, in m². A larger area moves the centre of pressure closer to the centroid. Raising it by 10 % (to 2.2) moves the result by −2.27 %, from 1.333 to 1.303 m.
Plate angle above horizontal
The plane’s angle above horizontal, in degrees. A steeper plane moves the centre of pressure further below the centroid. Raising it by 10 % (to 99) moves the result by −0.612 %, from 1.333 to 1.325 m.
Worked example with the values above
- Shift term: I sin²θ/(A hc) = 0.6667 × 1 / (2 × 1) = 0.33333 m.
- Add the centroid depth: 1 + 0.33333.
- hp = 1.3333 m.
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
- Using the plate angle from the vertical in place of the angle from the horizontal.
- Using the second moment of area about the free surface in place of the centroid.
- Measuring the depth along the slope in place of the vertical depth.
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