Equivalent Forces Due to Hydrostatic Pressure on a Dam

A dam is filled with water at height on the left and on the right; arrows represent the hydrostatic pressure on the walls. The graphic shows a cross section of the dam; assume the dam has unit horizontal length (perpendicular to the cross section). Hydrostatic pressure increases linearly with depth. On the right, the pressure is perpendicular to the surface, with an angle of inclination . The pressure distributions can be represented as triangles, with the effective forces (shown in red and green) acting through their centroids. Atmospheric pressure can be neglected, since it balances out on the opposite sides of the dam.

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The force on the left face is given by , where is the hydrostatic pressure at the bottom and is the length of the dam. The net force acts at the centroid height .
The force on the right face is , where is the hydrostatic pressure at the bottom; and are the usual unit vectors in the and directions.
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