Hydrostatic Force on Plane

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Hydrostatic Force on Plane Curved Surfaces
Lecture12
SCS138 Applied Physics
Dr. Bunyarit Uyyanonvara IT Department,
Sirindhorn International Institute of
Technology Thammasat University
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Sections Overview

• Lecture 11 Fluids, Density Pressure
• Lecture 12 Forces on Plane Curved surfaces
• Lecture 13 Buoyancy Stability of bodies
• Lecture 14 Fluid flow concepts
• Lecture 15 Review Tutorial

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Contacts

• Course materials can be found at
• http//www.siit.tu.ac.th/bunyarit
• Contact details
• Dr Bunyarit Uyyanonvara
• 02 5013505-20 ext 2005
• bunyarit_at_siit.tu.ac.th
• Bangkadi Campus,

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Hydrostatic force

• When a surface is submerged in a Fluid, forces
  develop on the surface due to the fluid.
• The determination of these forces is important in
  the design of storage tanks, ships, dams and
  other hydraulic structures.

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Things that we did know

• For Fluid at rest, we knew that the force must be
  perpendicular to the surface
• We knew that the pressure will vary linearly with
  depth

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Things that we did know

• For a horizontal surface, such as the bottom of
  the tank the magnitude of the resultant force is
  simply
• FR PA
• where P is a uniform pressure

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Resultant Force

• Note that if the atmospheric pressure acts on
  both sides, as illustrated,
• The resultant force on the bottom is simply due
  to the liquid in the tank

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Centroid

• Since the pressure is constant and uniformly
  distributed over the bottom, the resultant force
  acts through the centroid of the area
• as shown in the figure
• centroid is simply a center point of the area,
  in this case

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Resultant Force Calculation

• For more general case, in which a submerged plane
  surface is inclined, the determination of the
  resultant force acting on the surface is more
  involved

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Resultant Force Calculation

• Let the plane in which the surface lies intersect
  the free surface at 0 and make an angle ? with
  the plane
• The area can have arbitrary shape as shown.

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Resultant Force Calculation

• The integral is the fist moment of the area with
  respect o the x axis,
• Where yc is the y coordinate of the centroid
  measured from the x axis which passes through 0

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Resultant Force Calculation

• Where hc is the vertical distance from the fluid
  surface to the centroid area.
• Note that the magnitude of the force is
  independent of the angle ? and
• depends only on the density of fluid, the depth
  of centroid, and the total area.

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Centroids

• Centroidal coordinates and moments of inertia for
  some common areas

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Force acting on plane

• Although our intuition might suggest that the
  resultant force should pass through the centroid
  of the area, that is not actually the case.
• Resultant force is not necessarily acting on the
  centroid.
• We would like to find out where the resultant
  force acts.

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Force acting on plane

• That is the moment of the resultant force must
  equal the moment of the distributed pressure
  force, or

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Force acting on plane

• The integral is the second moment of the area
  (moment of inertia), Ix, with respect to an axis
  formed by the intersection of the plane
  containing the surface and the free surface (x
  axis), thus

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Force acting on plane

• According to the parallel axis theorem
• Where Ixc is the second moment of the area with
  respect to an axis passing through its centroid
  and parallel to the x axis, thus
• and similarly,

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Example I
A large fish-holding tank contains seawater (?
64.0 lb/ft3) to a depth of 10 ft as shown in the
figure. To repair some damage to one corner of
the tank, a triangular section is replaced with a
new section as illustrated. Determine the
magnitude and location of the force of the
seawater on this triangular area.
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Example I - solution

• Since the surface of interest lies in a vertical
  plane, and from the centroids calculation figure,
  the hc is 9 ft. The magnitude of force is

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Example I - solution
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Example II
The 4-m circular gate is located in the inclined
wall of a large reservoir containing water (?
9.8 kN/m3). For a water depth of 10m determine
(a) The magnitude and location of the resultant
force exerted on the gate by the water (b) the
moment that would have to be applied to the shaft
to open the gate.
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Example II - solution

• Since the vertical distance from the fluid
  surface to the centroid of the area is 10 m. it
  follows that

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Example II - solution

• For the coordinate system shown XR 0 since the
  area is symmetrical, and the center of pressure
  must lie along the diameter A-A. To obtain yR,

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Example II - solution

• The moment required to open the gate can be
  obtained with the aid of the free body diagram
  where W is the weight of the gate and Ox and Oy
  are the horizontal and vertical reaction fo the
  shaft on the gate. We can sum moments about the
  shaft

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Force acting on curved plane

• Many surfaces of interest (such as those
  associated with dams, pipes, and tanks) are
  non-plannar.
• Although the resultant force can be determined
  through the integration, as was done with the
  plane surface, this will be tedious work.

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Force acting on curved plane

• Alternatively, we will consider the equilibrium
  of the fluid volume enclosed by the curved
  surface of interest and horizontal and vertical
  projections of this surface.

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Force acting on curved plane

• In order for this force system to be in
  equilibrium, the horizontal component FH must be
  equal and collinear with F2.
• And vertical component Fv equal in magnitude and
  collinear with the resultant of vertical forces
  F1 and W.

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Force acting on curved plane

• In order for this force system to be in
  equilibrium,

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Force acting on curved plane

• The resultant force FR passes through the point
  O, which can be located by summing moments about
  an appropriate axis.

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Example III

• The 6-ft-diameter drainage pipe is half full of
  water at rest (specific weight ? for water is
  62.4 lb/ft3). Determine the magnitude and line of
  action of the resultant force that the water
  exerts on a 1-ft length of the curved section BC
  of the pipe wall

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Example III - Solution

• We first isolate a volume of fluid bounded by the
  curved section BC, AC and AB.
• The volume has a length of 1 ft.

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Example III - Solution
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Example III - Solution

• The force the water exerts on the pipe wall is
  equal to FR shown in the figure.
• The resultant force on act through the point O
  at the angle calculated using tan() function as
  shown in the figure.

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Exercise

• the exercise will be in the extra sheet.