Hydrostatic Force Calculator

When a structure retains static fluid, the fluid exerts a distributed pressure on the wetted surface. On a vertical flat gate of width b with water depth h, pressure increases linearly from P = 0 at the free surface to P = ρgh at the bottom. Integrating over height: F = ∫₀ʰ ρgy·b dy = ½ρgh²b.

The pressure distribution forms a triangle (zero at top, maximum at bottom). The total force F acts through the centroid of this triangle — at depth ȳ = 2h/3 below the surface (equivalently h/3 above the base) for a vertical rectangular gate. This is below the geometric centroid (h/2) because pressure is weighted toward greater depth.

Numerical example (water, ρ = 1000 kg/m³): h = 3 m, b = 2 m gives F = 0.5 × 1000 × 9.81 × 9 × 2 ≈ 88,300 N (about 9 tonne-force). Doubling depth quadruples force (F ∝ h²). Doubling width doubles force (F ∝ b).

For inclined flat surfaces, use depth measured normal to the surface when integrating P = ρgh. Curved dams are analyzed by resolving forces on horizontal and vertical projected areas or by integration along the arc. Horizontal force on a curved section can exceed vertical force in some geometries.

Gate hinge design requires moment M = F × (lever arm to line of action). Flood gates must include safety factors, possible hydrodynamic impact loads, and uplift on the base. Atmospheric pressure often cancels on both sides of a submerged plate — use gauge pressure in many problems.

Distinct from buoyancy (upward force on submerged objects equal to weight of displaced fluid). Hydrostatic force here acts on the container walls that hold the fluid, critical for dams, tanks, locks, and bulkheads.