Hydrostatic Force and Moment on a Gate

F = p_{c} A = γ h_{c} A, y_{c p} = y_{c} + \frac{I _{c}}{y _{c} A} sin^{2} θ

Resultant F acts at the center of pressure, below the centroid. Moment about a hinge is F times the moment arm from the hinge to the line of action. For a vertical plane, sinθ=1.

$h_{c}$ (m): Depth of centroid
$y_{c} p$: Center-of-pressure coordinate

Step-by-step solved example

Vertical 2 m × 3 m gate, top at 4 m depth, hinge at top. Fresh water. F?

Resultant F = γ hc A acts at the center of pressure on the gate.

1. Centroid depth
hc=4+1.5=5.5 m (3 m is height).
2. F
γ h A=9.81 kN/m³ ×5.5×6=324 kN.

Answer: F = 324 kN (then take moments with y_cp)

10 practice questions

0/10 correct

1.On a vertical plane I_c sin²θ / (yc A) simplifies because

2.cp is below the centroid because

3.Atmospheric pressure on both sides

4.Triangle wall (water to apex at the top): F=

5.Hinge moment = 0 at opening if

6.I_c of a rectangle b×h about centroid is

7.γ_seawater ≈

8.A curved gate’s horizontal F equals

9.If hc doubles, F

10.62.4 lb/ft³ × 10 ft × 20 ft² =