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Hydrostatic Forces on Submerged Rectangular Plate Inclined to Surface

Find the hydrostatic pressure force and its location on the submerged, inclined, rectangular

plate with dimensions b by h as shown below.

s = h s = h

F = ∫ γ x b ds = ∫ ( γ (D + s cos θ) b ds = γ [ D + (h/2) cos θ ] bh (result) --- (1)

s = 0 s = 0

Note: This result is just the pressure at the centroid of the plate times the area of the plate.

Note that the hydrostatic force acts at the centroid of the pressure prism,

not at the centroid of the plate. Let x_bar denote the location of the hydrostatic pressure

force. Apply the principle of first moments to obtain x_bar .

F x_bar= ∫ x dF = ∫ x P dA = ∫ x [ γ (D + s cos θ)] dA = ∫ γ x² b ds

s = h

F x_bar= γ ∫ b [ D + s cos θ]² ds = γb [ D²h + Dh² cos θ + (1/3) h³ cos² θ ] --- (2)

s = 0

Solve for x_bar by dividing (2) by (1). The result is

x_bar= [ (1/3) h² cos² θ + Dh cos θ + D² ] / [ D + (h/2) cos θ ] (result)

Check: For a vertical plate (θ = 0) at the surface and D = 0

x_bar= [ (1/3) h² + Dh + D² ] / [ D + h/2 ]