Hydrostatic
Forces on Submerged Plates
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Find the hydrostatic pressure force and its
location on a rectangular plate (b by h). The top of the plate is at the surface and is
vertically oriented with respect to the surface. Use the method of integration. The elemental pressure force is dF = PdA = γ x dA =
γ x b dx where P is
the pressure acting at a depth x, b is
the width of the plate, and h is the height of
the plate. So by integration the total
pressure force, F, is x = h F =
∫ γ x b dx = ˝ γ bh2 = ( ˝ γ h)(bh) ---- (1) x = 0 Notice
that the magnitude of the total pressure force equals the pressure at the centroid of the plate times the area of the
plate. But where does this pressure force act? Let xbar
denote the location of the total
pressure force on the plate below the surface. Use the principle of first moments to
calculate the location of the pressure force as follows.
x = h F xbar
= ∫ x dF
= ∫ x P dA =
∫ x γ x dA =
γ ∫ x2 b
dx = (1/3) γ b h3 --- (2)
x = 0 So by (1) and (2) xbar = [ (1/3) γ b h3 ] / [( ˝
γ h)(bh)] =
(2/3) h Note
that the hydrostatic force acts at the centroid of
the pressure “prism” , the triangular distribution, not at the centroid of the
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