Hydrostatics of fluids_forces acting on the hydraulic structures.pdf

The study of force in fluid mechanics is crucial for determining the force exerted on reservoir walls and wind-exposed surfaces like tall buildings and airplanes. In addition to being used to compute the forces exerted on dams and gates in the field of hydraulic constructions.

1. Hydraulics
Chapter 3: Hydrostatics of Fluids
Getachew Tegegne, PhD
Associate Professor of Civil and Environmental Engineering
Email: [email protected]
A Textbook of Fluid Mechanics and Hydraulic Machines
Dr. R. K. Bansal

2. ❑Introduction
✓In practical applications, the study of force in fluid mechanics is
crucial for determining the force exerted on reservoir walls and wind-
exposed surfaces like tall buildings and airplanes. In addition to being
used to compute the forces exerted on dams and gates in the field of
hydraulic constructions.
✓At the end of this chapter the student will be able to:-
•Define the hydrostatic force
•Determine the hydrostatic pressure's position, direction, and magnitude
on curved, vertical, and inclined surfaces.

3. ❑Introduction
✓Total pressure and center of pressure:
•Total pressure is the force that a static fluid applies to a planar or
curved surfaces when it makes contact with it. This force is always
perpendicular (normal) to the surface.
✓Center of pressure:
•It is the point of application of the total pressure on the surface.
✓The immersed surfaces can be:
•horizontal plane surface,
•vertical plane surface,
•inclined plane surface, and
•curved surface.

4. ❑Hydrostatic Forces on Plane Surfaces
✓On a plane surface, the hydrostatic
forces form a system of parallel forces
oFor many applications, magnitude
and location of application, which
is called center of pressure, must be
determined.
oAtmospheric pressure P
atm can be
neglected when it acts on both
sides of the surface.

5. ❑Hydrostatic Forces on Plane Surfaces
✓Take an arbitrary area AB on the back face of a dam that inclines at an angle (θ)
and then place the x-axis on the line at which the surface of the water intersects
with the dam surface, with the y-axis running down the direction of the dam
surface.
Figure (a) shows a horizontal
view of the area and
Figure (b) shows the projection
of AB on the dam surface.

6. ❑Hydrostatic Forces on Plane Surfaces
✓The total hydrostatic pressure force on any submerged plane surface is equal
to the product of the surface area and the pressure acting at the centroid of the
plane surface.
✓Pressure forces acting on a plane surface are distributed over every part of the
surface.
✓They are parallel and act in a direction normal to the surface.
✓It can be replaced by a single resultant force F of the magnitude that acts at the
centroid.
✓The resultant force also acts normal to the surface - the point on the plane
surface at which this resultant force acts is known as the center of pressure.

7. ❑Hydrostatic Forces on Plane Surfaces
✓The center of pressure of any
submerged plane surface is always
below the centroid of the surface (i.e.,
Yp > yc) – water pressure increases
with depth from the water surface.
✓The centroid, area, and moment of
inertia with respect to the centroid of
certain common geometrical plane
surfaces are given in Table.

8. ❑Hydrostatic Forces on Inclined Plane Surfaces
✓The magnitude of F
R acting on a plane surface of a completely submerged
plate in a homogenous fluid is equal to the product of the pressure P
C at the
centroid of the surface and the area A of the surface.

9. ❑Hydrostatic Forces on Inclined Plane Surfaces
✓Line of action of resultant force F
R =P
CA does not pass through the centroid of
the surface. In general, it lies underneath where the pressure is higher.
✓Vertical location of center of pressure is determined by equation - the moment
of the resultant force to the moment of the distributed pressure force.

10. ❑Hydrostatic Forces on Inclined Plane Surfaces
✓Example 1: An inclined circular gate with water on one side is shown below.
Determine the total resultant force acting on the gate and the location of the
center of pressure.

11. ❑Hydrostatic Forces on Inclined Plane Surfaces
✓Example 2: A tank contains water upto a height of 0.5 m above the base.
An immiscible liquid of specific gravity 0.8 is filled on the top of water
upto 1 m height. Calculate:
a)a) total pressure on one side of the tank,
b)b) the position of center of pressure for one side of the tank, which
is 2 m wide.

12. ❑Hydrostatic Forces on Inclined Plane Surfaces

13. ❑Hydrostatic Forces on Inclined Plane Surfaces

14. ❑Hydrostatic Forces on Inclined Plane Surfaces
✓Example 3:

15. ❑Hydrostatic Forces on Curved Surfaces
✓The hydrostatic force on a curved surface can be best analyzed by resolving
the total pressure force on the surface into its horizontal and vertical
components.
✓Because the water body in the container is stationary, every part of the water
body must be in equilibrium or each of the force components must satisfy the
equilibrium conditions

16. ❑Hydrostatic Forces on Curved Surfaces
✓F
R on a curved surface is more involved since it requires integration of the
pressure forces that change direction along the surface.
✓Easiest approach: determine horizontal and vertical components F
H and F
V
separately.

17. ❑Hydrostatic Forces on Curved Surfaces
✓Horizontal force component on curved surface: F
H = F
x.
•Line of action on vertical plane gives y coordinate of center of pressure on
curved surface.
✓Vertical force component on curved surface: F
V = F
y+ W, where W is the weight
of the liquid in the enclosed block W=γg.
•X coordinate of the center of pressure is a combination of line of action on
horizontal plane (centroid of area) and line of action through volume
(centroid of volume).
oMagnitude of force F
R = (F
H
2
+F
V
2
)
1/2
oAngle of force is α = tan
-1
(F
V/F
H)

18. ❑Hydrostatic Forces on Curved Surfaces
✓Homework: Find the magnitude and direction of the resultant
hydrostatic pressure acting on a curved face of a dam which is shaped
according to the relation ??????=
??????
2
9
as shown in the following figure. The
height of the water retained by the dam is 1 m. Consider the width of the
dam is unity.

19. ❑Hydrostatic Forces on Curved Surfaces

20. ❑Hydrostatic Forces on Curved Surfaces

21. ❑Buoyancy and Stability of Floating and Submerged Bodies
✓Buoyancy and Floatation:
•When a body is immersed wholly or partially in a fluid, it is subjected to
an upward force which tends to lift (buoy) it up.
•The tendency of immersed body to be lifted up in the fluid due to an
upward force opposite to action of gravity is known as buoyancy.
•The force tending to lift up the body under such conditions is known as
buoyant force or force of buoyancy or up-thrust.
•The magnitude of the buoyant force can be determined by Archimedes’
principle which states “When a body is immersed in a fluid either wholly
or partially, it is buoyed or lifted up by a force which is equal to the
weight of fluid displaced by the body”

22. ❑Buoyancy and Stability of Floating and Submerged Bodies
✓Buoyancy and Floatation:
•Lets consider a body submerged in water as shown in figure
•The force of buoyancy “resultant upward force or thrust exerted by fluid
on submerged body” is given
odA=Area of cross-section of element
oγ = Specific weight of liquid
??????
??????=??????�??????���?????? = Weight of volume of liquid displaced by the body (Archimedes’s Principle)

23. ❑Buoyancy and Stability of Floating and Submerged Bodies
✓Buoyancy and Floatation:
•Force of buoyancy can also be determined as difference of weight of a
body in air and in liquid.
✓Center of Buoyancy (B):
•The point of application of the force of buoyancy on the body is known as
the center of buoyancy. It is always the center of gravity of the volume of
fluid displaced.
oW
a= weight of body in air
oW
l=weight of body in liquid
oF
B=W
a-W
l
oCG or G = Center of gravity of body
oC or B= Centroid of volume of liquid
displaced by body

24. ❑Buoyancy and Stability of Floating and Submerged Bodies
✓Types of equilibrium of Floating Bodies:
•Stable Equilibrium:
oIf a body returns back to its original position due to internal forces
from small angular displacement, by some external force, then it is
said to be in stable equilibrium.
oNote: Center of gravity of the volume (centroid) of fluid displaced is
also the center of buoyancy.

25. ❑Buoyancy and Stability of Floating and Submerged Bodies
✓Types of equilibrium of Floating Bodies:
•Unstable Equilibrium:
oIf the body does not return back to its original position from the
slightly displaced angular displacement and heels farther away, then
it is said to be in unstable equilibrium
oNote: Center of gravity of the volume (centroid) of fluid displaced is
also the center of buoyancy.

26. ❑Buoyancy and Stability of Floating and Submerged Bodies
✓Metacenter and Metacentric Height:
•Center of Buoyancy (B):
oThe point of application of the force of buoyancy on the body is
known as the center of buoyancy.
•Metacenter (M):
oThe point about which a body in stable equilibrium start to oscillate
when given a small angular displacement is called metacenter.
oIt may also be defined as point of intersection of the axis of body
passing through center of gravity (CG) and original center of
buoyancy (B) and a vertical line passing through the center of
buoyancy (B’) of tilted position of body.

27. ❑Buoyancy and Stability of Floating and Submerged Bodies
✓Metacenter and Metacentric Height:
•Metacentric height (GM):
oThe distance between the center of gravity (G) of floating body and
the metacenter (M) is called metacentric height. (i.e., distance GM
shown in fig)
GM = BM - BG

28. ❑Buoyancy and Stability of Floating and Submerged Bodies
✓Metacenter and Metacentric Height:
•Condition of Stability:
oFor Stable Equilibrium
▪Position of metacenter (M) is above than center of gravity (G)
oFor Unstable Equilibrium
▪Position of metacenter (M) is below than center of gravity (G)
oFor Neutral Equilibrium
▪Position of metacenter (M) coincides center of gravity (G)

29. ❑Buoyancy and Stability of Floating and Submerged Bodies
✓Metacenter and Metacentric Height:
•Determination of Metacentric Height:
oThe metacentric height may be determined by the following two
methods
1)Analytical method
2)Experimental method

30. ❑Buoyancy and Stability of Floating and Submerged Bodies
✓Metacenter and Metacentric Height:
•Determination of Metacentric Height:
oIn Figure shown AC is the original waterline plane and B the center
of buoyancy in the equilibrium position.
oWhen the vessel is tilted through small angle θ, the center of
buoyancy will move to B’ as a result of the alteration in the shape of
displaced fluid.
oA’C’ is the waterline plane in the displaced position.

31. ❑Buoyancy and Stability of Floating and Submerged Bodies
✓Metacenter and Metacentric Height:
•Determination of Metacentric Height:
oTo find the metacentric height GM, consider a small area dA at a
distance x from O. The height of elementary area is given by xθ.
oTherefore, volume of the elementary area becomes
oThe upward force of buoyancy on this elementary area is then
oMoment of dF
B (moment due to movement of wedge) about O is
given by;

32. ❑Buoyancy and Stability of Floating and Submerged Bodies
✓Metacenter and Metacentric Height:
•Determination of Metacentric Height:
oThe change in the moment of the buoyancy Force, F
B is
oFor equilibrium, the moment due to movement of wedge = change in
moment of buoyancy force

33. ❑Buoyancy and Stability of Floating and Submerged Bodies
✓Examples:

34. ❑Buoyancy and Stability of Floating and Submerged Bodies
✓Examples:

35. ❑Buoyancy and Stability of Floating and Submerged Bodies
✓Examples:

36. ❑Buoyancy and Stability of Floating and Submerged Bodies
✓Examples: