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Hydrostatic Forces on Submerged Plates (continued)

Next determine the location of the hydrostatic pressure force on the arbitrary plate by

using the principle of first moments. i.e. The moment of each elemental pressure force

about the surface must equal to the moment of the total pressure force about the surface.

Click here for a review of the principle of first moments.

Note the pressure prism shown above is just a trapezoidal pressure distribution acting

on the plate. Also note that pressure acts normal to the surface of the plate.

Let x_bar denote the location of the total hydrostatic pressure force on the plate. Then

application of the principle of first moments:

F x_bar = ∫ x dF = ∫ x P dA = ∫ x γx dA = γ ∫ x² dA ˗ ˗ ˗ ˗ ˗ ˗ (2)

Now the total hydrostatic pressure force F = γ ∫ x dA so

x_bar = γ ∫ x² dA / γ ∫ x dA = ∫ x² dA / ∫ x dA

Notice: ∫ x² dA is the second moment of area and ∫ x dA is the first moment of area.

RESULT: The location, x_bar, of the equipollent, total, hydrostatic pressure force, F,

is at the centroid of the pressure prism provided the plate is rectangular. If not,

integration is required. Also note that location of the resultant pressure force is

below the centroid of the plate.