Calculator · Fluid Mechanics
Hydrostatic Force Calculator
Pressure grows with depth, so a submerged plate carries a distributed load rather than a uniform one. Its resultant is the pressure at the centroid times the area — which is exact for a plane surface, whatever its inclination.
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
Resultant force F = ρg·h_c·A
Acts perpendicular to the surface, pressing against it from the fluid side.
| Area A | 6 m² |
|---|---|
| Centroid depth h_c | 2.5 m |
| Pressure at the centroid ρg·h_c | 24.5166 kPa |
| Resultant force F | 147.1 kN |
| Depth of the bottom edge | 4 m |
| Centre of pressure depth h_cp | 2.8 m |
Assumptions
- · Static fluid of constant density.
- · Gauge pressure referenced to the free surface, so atmospheric pressure acts on both faces.
Pressure on the surface
Pressure increases linearly down the plate. The resultant of 147.1 kN acts normal to the surface, below the centroid at a depth of 2.8 m.
Explore in the Lab
The plate, the pressure field it sits in, and the buoyancy view share one fluid.
Technical reference
F = ρ g h_c A
p_c = ρ g h_c
h_c = h_top + (H/2)·sinθ
- F
- resultant pressure force, normal to the surface[N]
- h_c
- depth of the area centroid[m]
- A
- area of the plane surface[m²]
- θ
- inclination of the surface from the horizontal[°]
- g
- gravity, 9.80665 m/s²[m/s²]
The pressures here are gauge pressures, measured against the ambient pressure acting on the dry side. When the same ambient acts on both faces it cancels and contributes nothing to the resultant.
The relation F = ρg·h_c·A is exact for any plane surface at any inclination — the centroid pressure is genuinely the average pressure over a plane. It is not an approximation.
The resultant acts perpendicular to the surface, and below the centroid, because the deeper half of the plate carries more pressure. Where exactly it acts is the centre of pressure.
The whole surface must be submerged in a single fluid. A partly submerged gate must be split at the waterline and handled in parts.
A curved surface does not obey this relation: its pressure forces are not parallel and must be resolved into horizontal and vertical components.