Hydrostatic force on curved surface

On a curved surface the pressure acts normal to each element, so the elemental forces have different directions. The total force is found by resolving it into horizontal and vertical components.

   free surface ==================
        :  water above   |
        :  (volume ABCD) |  F_V = weight of
        :                |  this water
        D________________A
                         \    <- curved surface AB
                          \
                           B
     F_H = force on vertical projection of AB

Horizontal component

$$F_H = \rho g A_v \bar h_v$$

• $A_v$ = area of the vertical projection of the curved surface, $\bar h_v$ = depth of its centroid.
• $F_H$ acts through the centre of pressure of this projected area, at depth $\bar h_v + \dfrac{I_G}{A_v \bar h_v}$.

Vertical component

$$F_V = \rho g V$$

• $V$ = volume of liquid (real or imaginary) lying vertically above the curved surface up to the free surface.
• $F_V$ acts through the centroid of that volume.
• If the liquid is above the surface, $F_V$ acts downward. If the liquid is below the surface, $F_V$ acts upward and equals the weight of the imaginary liquid above it.

Resultant

$$F = \sqrt{F_H^2 + F_V^2}, \qquad \tan\theta = \frac{F_V}{F_H}$$

For a circular (cylindrical) surface, every elemental force is normal to the surface and passes through the centre, so the resultant also passes through the centre of curvature.

Applications: radial (Tainter) gates, sector gates and drum gates on spillways, curved dam faces and pipe bends.