Fluid Mechanics_hydro pprddddvdvfvffvfv.pdf

hydro pressure

1. By Samar Singhal

2. Hydrostatic Force on Inclined Surface

3. First Moment of Area
•The first moment of area of a shape, about a certain axis, equals the
sum over all the infinitesimal parts of the shape of the area of that
part times its distance from the axis .
•First moment of area is used to determine the centroid of an area.

4. Second and Product of moment of Area
•The second moment of area I
xx (often denoted as I
x) can be
computed as


•The product moment of area is defined as

5. Product of moment of Area



•Green area need to balance the red
area.

•Automatically satisfied when x and y
are symmetry axes.

6. Hydrostatic Force on Inclined Surface

The pressure on any submerged surface varies linearly
with depth.
If h is the depth to any element area dA of the plate,
the pressure there is
From the figure
First Moment of Area
Therefore
θ is constant along the plate
Force on the Plate

7. Hydrostatic Pressure on Inclined
Surface

But,
Therefore Force on the plate can be written as


whereas h
cg the depth straight down from the surface to the
plate centroid p
cg is the pressure at centroid of the plate.

The force on one side of any plane submerged surface in a
uniform fluid equals the pressure at the plate centroid times the
plate area, independent of the shape of the plate or the angle θ at
which it is slanted.

8. Pressure Distribution

The hydrostatic pressure force on a plane surface is equal, regardless
of its shape, to the resultant of the three-dimensional linear
pressure distribution on that surface.

9. Special Cases

10. Center of Pressure
•To find the coordinates ( x
CP , y
CP ), we sum moments of the
elemental force p dA about the centroid and equate to the moment
of the resultant F .



•vanishes due to on centroid of centroidal axes

•Let

•Then

•I
xx is the area moment.

11. Center of Pressure
•The negative sign in Eq. shows that y
CP is below the centroid at a
deeper level and, unlike F , depends on angle θ . If we move the
plate deeper, y
CP approaches the centroid because every term
remains constant except p
CG , which increases.

•determination of x
CP






•For positive I
xy, x
CP is negative because the dominant pressure force
acts in the third, or lower left, quadrant of the panel. If I
xy = 0,
usually implying symmetry, x
CP = 0 and the center of pressure lies
directly below the centroid on the y axis.

12. Hydrostatic Force on Inclined Surface


•In most cases the ambient pressure pa is neglected because it acts
on both sides of the plate; for example, the other side of the plate is
inside a ship or on the dry side of a gate or dam. In this case p
CG =
γh
CG , and the center of pressure becomes independent of specific
weight:

13. Pressure Prism


•Length of prism is the linearly varying pressure.
Its volume is equal to the magnitude of the resultant hydrostatic
force acting on the plate since FR = ʃ P dA, and the line of action of
this force passes through the centroid of this homogeneous prism.
The projection of the centroid on the plate is the pressure center.

14. Centroidal moments of inertia for various
cross sections:

15. Hydrostatic thrust on a submerged
curved surface

16. Hydrostatic thrust on a submerged
curved surface
•The hydrostatic force on the elemental area dA is

•The force acts in a direction normal to the area dA.

•The components of the force dF in x, y and z directions are




•The components of the surface element dA projected on yz, xz and
xy planes are, respectively

17. Hydrostatic thrust on a submerged
curved surface







Forces can be written as:



Integrating the forces:



z
c is the z cordinate of centroid of area A
x and A
y the projected
areas on yz and and xz planes respectively.

18. Center of Pressure







Point of action of F
x




Point of action of F
y






For a curved surface, the component of hydrostatic force in a
horizontal direction is equal to the hydrostatic force on the
projected plane surface perpendicular to that direction and acts
through the centre of pressure of the projected area.

19. Vertical component of force






















Vertical component of the hydrostatic force:

where V is the volume of the body of liquid within the region
extending vertically above the submerged surface to the free
surface of the liquid.
Therefore, the vertical component of hydrostatic force on a
submerged curved surface is equal to the weight of the liquid
volume vertically above the solid surface to the free surface of
the liquid and acts through the centre of gravity of the liquid
in that volume.

20. Surface submerged in a multilayered
fluid






















The hydrostatic force on a surface submerged in a
multilayered fluid can be determined by considering parts of
the surface in different fluids as different surfaces.

21. Hydrostatic thrust on a submerged
curved surface






















•We could sum the separate three components of these elemental
pressure forces, but it turns out that we need not perform a
laborious three-way integration.

•The horizontal component of force on a curved surface equals the
force on the plane area formed by the projection of the curved
surface onto a vertical plane normal to the component.

•The vertical component of pressure force on a curved surface
equals in magnitude and direction the weight of the entire column
of fluid, both liquid and atmosphere, above the curved surface.

22. Hydrostatic thrust on a submerged
curved surface

23. Hydrostatic thrust on a submerged
curved surface

24. Hydrostatic thrust on a submerged
curved surface






















The horizontal component of the hydrostatic force acting on a curved surface is
equal (in both magnitude and the line of action) to thehydrostatic force acting on
the vertical projection of the curved surface.
The vertical component of the hydrostatic force acting on a curved surface is
equal to the hydrostatic force acting on the horizontal projection of the curved
surface, plus (minus, if acting in the opposite direction) the weight of the fluid
block.

25. Buoyancy







When a body is either wholly or partially immersed in a fluid,
the hydrostatic lift due to the net vertical component of
hydrostatic pressure forces experienced by the body is called
the buoyant force and the phenomenon is called buoyancy.

26. Buoyancy







•The resultant horizontal force in any direction for such a closed
surface is always zero.
•The vertical forces acting on the two ends of such a prism



•Therefore, the buoyant force (the net vertically upward force)
acting on the elemental prism is

•Integrating will give net buoyant force:

27. Line of action of the force







•The line of action of the force can be found by taking moment of
the force with respect to z-axis.


•Substituting the values:

•Which is the centroid of the displaced volume.

•Buoyant force F
B equals to the weight of liquid displaced by the
submerged body of volume V and known as the Archimedes
principle.
•The principle states that the buoyant force on a submerged body is
equal to the weight of liquid displaced by the body, and acts
vertically upward through the centroid of the displaced volume.

28. Float, Suspended and Sinking Body

29. Numericals
•A tank of oil has a right-triangular panel near the bottom, as in
Fig. Omitting p
a , find the (a) hydrostatic force and (b) C
P on
the panel.

30. Numericals
A long solid cylinder of radius 0.8 m hinged at point A is used as an
automatic gate, as shown in Fig. When the water level reaches 5 m,
the gate opens by turning about the hinge at point A. Determine (a)
the hydrostatic force acting on the cylinder and its line of action when
the gate opens and (b) the weight of the cylinder per m length of the
cylinder.