Calculator · Fluid Mechanics
Center of Pressure Calculator
The resultant pressure force on a submerged plate does not act through the centroid. Because the deeper part of the plate carries more pressure, the line of action sits below it — at the centre of pressure.
Surface and fluid
Measured along the plate itself, not vertically — the two differ once the plate is inclined.
Depth below the free surface of the shallowest point of the surface. Zero puts the top edge at the surface.
90° is a vertical plate; 0° is a horizontal one, where every point is at the same depth.
Results
Centre of pressure depth h_cp
0.3 m below the centroid.
| Area A | 6 m² |
|---|---|
| Centroid depth h_c | 2.5 m |
| Second moment about the centroid I_G | 4.5 m⁴ |
| Centre of pressure depth h_cp | 2.8 m |
| Offset below the centroid, along the surface | 0.3 m |
| Resultant force F acting there | 147.1 kN |
Assumptions
- · Static fluid of constant density.
- · Gauge pressure referenced to the free surface, so atmospheric pressure acts on both faces.
Where the resultant acts
The centroid C sits at 2.5 m and the centre of pressure CP at 2.8 m. The resultant F acts normal to the plate at CP.
Explore in the Lab
See the resultant, its line of action and the pressure field behind it together.
Technical reference
h_cp = h_c + (I_G sin²θ) / (A h_c)
y_cp − y_c = (I_G sinθ) / (A h_c)
- h_cp
- depth at which the resultant acts[m]
- h_c
- depth of the area centroid[m]
- I_G
- second moment of area about the centroidal axis[m⁴]
- A
- area of the surface[m²]
- θ
- inclination from the horizontal[°]
The centre of pressure is always at or below the centroid, never above it: the correction term is a positive quantity divided by positive quantities.
As the surface goes deeper the correction shrinks, because the pressure difference across the plate becomes small compared with the pressure itself. Deep plates are loaded almost uniformly.
On a horizontal surface, θ = 0, the pressure is the same everywhere and the centre of pressure coincides with the centroid exactly.
This relation assumes the pressure varies linearly across the surface and that the same ambient pressure acts on both faces. A uniform extra pressure difference across the plate moves the line of action, and it is then not reported rather than guessed.
I_G is taken about the centroidal axis parallel to the free surface: bH³/12 for a rectangle of extent H along the slope, πD⁴/64 for a circle.