Hydrostatic Force & Center of Pressure

Calculate the resultant force on a submerged plane surface and locate the center of pressure — for vertical, inclined, and horizontal surfaces.

Fluid Properties & Statics

Surface Shape

A = b × H|I_G = b H³ / 12

Common fluids — click to auto-fill density

Inputs

Fluid density ρ (kg/m³)

Depth of centroid below free surface h̄ (m)

Vertical distance from the free surface to the centroid of the surface.

Surface inclination angle θ (degrees)

Measured from the horizontal. 90° = vertical surface, 0° = horizontal surface.

Width b (m)

Height H (m)

Test Case — Vertical Rectangular Gate

Fluid	Water at 20°C, ρ = 998 kg/m³
Gate shape	Rectangle, b = 1.5 m, H = 2 m
Inclination	90° (vertical)
Depth to centroid	h̄ = 3 m (top edge at 2 m depth)

A = 1.5 × 2 = 3 m²

I_G = 1.5 × 2³ / 12 = 1.0 m⁴

F = 998 × 9.81 × 3 × 3 = 88,074 N ≈ 88.1 kN

ȳ = 3 / sin(90°) = 3 m

y_cp = 3 + 1.0 / (3 × 3) = 3.111 m

h_cp = 3.111 m → e = 0.111 m below centroid

Theory

Resultant hydrostatic force:

F = ρ g h̄ A

h̄ = vertical depth to centroid · A = surface area

The magnitude equals the pressure at the centroid multiplied by the area. This is equivalent to integrating the pressure distribution over the surface.

Center of pressure (inclined surface, angle θ with horizontal):

ȳ = h̄ / sin θ (slant distance to centroid)

y_cp = ȳ + I_G / (ȳ A) (slant distance to C.P.)

h_cp = y_cp × sin θ (vertical depth to C.P.)

e = h_cp − h̄ (eccentricity, always ≥ 0)

The center of pressure is always below the centroid (h_cp ≥ h̄) because pressure increases with depth. As depth increases (h̄ → ∞), e → 0 and C.P. approaches the centroid.

Second moments of area (centroidal, I_G):

Shape	Area A	I_G
Rectangle	b × H	b H³ / 12
Circle	π D² / 4	π D⁴ / 64
Triangle	½ b H	b H³ / 36

I_G is always about the centroidal axis parallel to the free surface(horizontal axis through the centroid). The parallel-axis theorem gives the second moment about the free-surface axis: I_x = I_G + A ȳ².

Where:

ρ = fluid density [kg/m³]
g = 9.81 m/s²
h̄ = vertical depth to centroid of surface [m]
A = surface area [m²]
I_G = second moment of area about centroidal axis [m⁴]
θ = angle of surface with the horizontal [deg]
ȳ = slant distance from free-surface intersection to centroid [m]
y_cp = slant distance to center of pressure [m]
h_cp = vertical depth to center of pressure [m]
e = eccentricity = h_cp − h̄ [m]