hydrostatic force

Dams are structures whose purpose is to raise the water level on the upstream side of river, stream, or other waterway. The rising water will cause hydrostatic force which will tend the dam to slide horizontally and overturn about its downstream edge or toe. The raised water level on the upstream edge or heel will also cause the water to seep under the dam. The pressure due to this seepage is commonly called hydrostatic uplift and will reduce the stability of the dam against sliding and against overturning.
 

Situation
The figure below shows a vertical circular gate in a 3-m diameter tunnel with water on one side and air on the other side.
 

1. Find the horizontal reaction at the hinge.
   A.   412 kN
   B.   408 kN
   C.   410 kN
   D.   414 kN
2. How far from the invert of the tunnel is the hydrostatic force acting on the gate?
   A.   1.45 m
   B.   1.43 m
   C.   1.47 m
   D.   1.41 m
3. Where will the hinge support be located (measured from the invert) to hold the gate in position?
   A.   1.42 m
   B.   1.46 m
   C.   1.44 m
   D.   1.40 m

Total Hydrostatic Force on Plane Surfaces
For horizontal plane surface submerged in liquid, or plane surface inside a gas chamber, or any plane surface under the action of uniform hydrostatic pressure, the total hydrostatic force is given by
 

where p is the uniform pressure and A is the area.
 

In general, the total hydrostatic pressure on any plane surface is equal to the product of the area of the surface and the unit pressure at its center of gravity.
 

where p_cg is the pressure at the center of gravity. For homogeneous free liquid at rest, the equation can be expressed in terms of unit weight γ of the liquid.
 

where   $\bar{h}$   is the depth of liquid above the centroid of the submerged area.