• A plate exposed to a liquid, such as a gate valve in a dam, the wall of a liquid storage tank, or the hull of a ship at rest, is subjected to fluid pressure distributed over its surface.
• Hydrostatic forces form a system of parallel forces,
• we need to determine the magnitude of the force;
• and its point of application,
Hydrostatic Forces On Submerged Plane Surfaces Hydrostatic Forces On Submerged Plane Surfaces
center of pressure
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When analyzing hydrostatic forces on submerged surfaces:
the atmospheric pressure can be subtracted for simplicity when it acts on both sides of the structure.
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• Hydrostatic force on an inclined plane surface completely submerged in a liquid
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• absolute pressure at any point on the plate
• resultant hydrostatic force:
• y-coord of the centroid (or center) of the surface:
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where PCis the pressure at the centroid of the surface, which is equivalent to the average pressure on the surface,
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• The pressure P0 is usually atmospheric pressure, which can be ignored in most cases since it acts on both sides of the plate.
• If not:
• Then, assume the presence of an additional liquid layer of thicknesshequiv on top of the liquid with absolute vacuum above.
Pressure magnitude: Okay Pressure point of application: ???
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• The line of action of the resultant hydrostatic force, in general, does not pass through the centroid of the surface. it lies underneath where the pressure is higher.
• Two parallel force systems are equivalent if they have the same magnitude and the same moment about any point.
• equating the moment of the resultant force to the moment of the distributed pressure force about the x-axis
line of action of the resultant force line of action of the resultant force
Center of Pressure
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• The Ixx are widely available for common shapes in engineering handbooks, but they are usually given about the axes passing through the centroid of the area.
• Fortunately, the second moments of area about two parallel axes are related to each other bythe parallel axis theorem,
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• For P0=0:
For areas that possess symmetry about the y-axis, the center of pressure lies on the y-axis directly below the centroid.
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Special Case: Submerged Rectangular Plate Special Case: Submerged Rectangular Plate
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:
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• When the upper edge of the plate is at the free surface and thus s=0
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• Vertical rectangle;
• When P0 is ignored since it acts on both sides of the plate
• and whose top edge is horizontal and at the free surface:
• Pressure center:
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pressure distribution on a submerged horizontal surface is uniform
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Hydrostatic Forces On Submerged Curved Surfaces Hydrostatic Forces On Submerged Curved Surfaces
Integration of the pressure forces
that change direction along the curved surface
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Easiest way to determine the resultant hydrostatic force:FR
Easiest way to determine the resultant hydrostatic force:FR
???Determine horizontal component:FH
???Determine horizontal component:FH
???Determine vertical component:FV
???Determine vertical component:FV
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Vertical surface of the liquid block considered is simply the projection of
the curved surface on a vertical plane,
Horizontal surface is the projection of the curved surface on a horizontal
plane
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• in static equilibrium, the force balances:
add magnitudes if both act in the same direction and subtract if they act in opposite directions
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• exact location of the line of action of the resultant force?
• The exact location of the line of action of the resultant force (e.g., its distance from one of the end points of the curved surface) can be determined by taking a moment about an appropriate point.
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The hydrostatic force acting on a circular surface always passes through the center of the circle since the pressure forces are
normal to the surface and they all pass through the center.
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• hydrostatic forces acting on a plane or curved surface submerged in a multilayered fluid of different densities can be determined by considering different parts of surfaces in different fluids as different surfaces, finding the force on each part, and then adding them using vector addition.
• For a plane surface, it can be expressed as:
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• pressure at the centroid of the portion of the surface in fluid i and Ai is the area of the plate in that fluid. The line of action of this equivalent force can be determined from the requirement that the moment of the equivalent force about any point is equal to the sum of the moments of the individual forces about the same point.
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