hydrostatic thrust on submerged surfaces
TRANSCRIPT
Hydrostatic Thrust on Submerged Surfaces
Hydrostatic Thrust • Due to the existence of hydrostatic pressure in
a fluid mass, a normal force exerted on any part of a solid surface which is on contact with a fluid. The individual forces distributes over an area give rise to a resultant force. The determination of magnitude and the line of action of the resultant force is of practical interest to engineers.
Practical applications• In practical applications engineers is
required to determine the pressure forces on the entire surface rather than the pressure intensity at a point.– Examples are
• Forces on submerged objects such as submarines, ships and balloons.
• Forces on walls of containers such as pipes, tanks and dams.
• Forces on gates in walls of containers, submerges bodies and many other hydraulic structures.
Definition: Total Pressure • When a static mass of fluid comes in
contact with surface, either plane or curved a force is exerted by the fluid on the surface. This force is known as total pressure.
•TOTAL PRESSUREOn the walls of container
Definition: Center of Pressure • This is defined as the point of application
of the total pressure on the surface.
•TOTAL PRESSUREOn the walls of container
Cases Under Consideration• There are four cases of submerged
surfaces on which the total pressure force and center of pressure is to be determined.–The submerged surfaces may be
• Horizontal plane surface.• Vertical plane surface.• Inclined plane surface.• Curved surface.
1. Horizontal Plane Surface• Consider a plane surface immersed in a
static mass of liquid of specific weight γ, such that it is held in a horizontal position at a depth h below the free surface of the liquid as shown in Fig 1.
Click here for Fig 1
1. Horizontal Plane Surface
h
Free Surface
Pressure Intensity• Since every point on the surface is at
same depth below the free surface of the liquid the pressure intensity is constant over the entire plane surface.
Pressure Intensity P = γhA = Total Area
Total Pressure• If A is the total area of the surface then
the total pressure on the horizontal surface is F.
F = γAhA = Total Area
Direction of Force• The direction of this force is normal to
the surface as such it is acting towards the surface in the vertical direction (downwards at the centroid of the surface)
F = γAh
A = Total Area
Hydrostatic Forces on Vertical Plane Surfaces
2. Total Pressure on a Vertical Plane Surface
• Consider a plane surface of arbitrary shape and total area A, wholly submerged in a static mass of liquid of specific weight γ, such that it is held in a vertical position As shown in Fig 2.
Click here for Fig 2
1. Total Pressure on a Vertical Plane Surface
x
Free Surface
dA = Small elemental area of strip size (b.dx)
C. G.
dF
Determination of Total Pressure
= position of centroid of the surface below the free surface of the liquid
•In this case since the depth of liquid varies from point to point on the surface the. Pressure intensity is not constant over the entire surface.•Therefore determination of total pressure is done by integration method.•Consider on the plane surface a horizontal strip if the thickness dx and width b lying at a vertical depth x below the free surface of the liquid. Pressure intensity is assumed to be constant over the entire thickness (size is very small)
P = γxArea of the strip = b.dxTotal pressure on entire plane surface is F = ∫dF
Determination of Total Pressure..ContP = γ.x
Area of the strip = b.dxTotal pressure on entire plane surface is F = ∫dFWhere dF = Pressure intensity on small strip x Elemental area
OR
dF = P.dA
AxF
xdA
xA
pdAF
A
A
A
)(
F = Pressure Intensity at the center of area x Area of plane surface
xAxdAA
Area ofMoment First
Center of pressure for vertical plane surfaceFor horizontal plane surface centroid of the area and the center of pressure
coincide with each other.But for plane surface immersed vertically the center of pressure does not
coincide with the centroid of the area.Since the pressure intensity increases with the increase in the depth of the
liquid, the center of pressure for a vertically immersed plane surface lies below the centroid of the surface area.
The total pressure on the strip shown in Fig. is dF = γ.x (b.dx)Like wise, by considering a number of small strips and summing the
moments of the total pressure on these strips about free surface. The sum becomes
∫dF.x = ∫ γx. (b.dx).x = γ ∫ x2.dA
But from principle of moment--- ∫ x2.dA represents the sum of the second moment of the areas of the strips about axis passing through the free surface, which is equal to the moment of inertia.
Cont…
Ax
Ixx
AxIAxx
IxAx
dAxx
pdAx
xdFFx
cp
cp
cp
A
A
cp
2
0
)(
)(
)(
)(
Hydrostatic Forces on Inclined Plane Surfaces
Pressure on Plane Surface
sinyp
dA
x
yy
cpyCentroid
Center of pressure
F
ApF
AyF
ydA
dAy
pdAF
A
A
A
)sin(
sin
sin
Surfaces exposed to fluids experience a force due to the pressure distribution in the fluid
Line of Action of Force
• Lies below centroid, since pressure increases with depth
sinyp
dA
x
yy
cpyCentroid
Center of pressure
F
Ay
Iyy
AyIAyy
IAyy
dAyy
pdAy
ydFFy
cp
cp
cp
A
A
cp
2
0
)(
sin)sin(
)sin(
)(
Example (3.78)
m
Ay
Iyycp
4641.0
)24*464.6(
12/6*4 3
N
AyApF
000,318,1
)6*4(*)30cos33(*9810
)sin(
kNR
kN
FR
FR
M
A
A
A
05.557
1318)42265.0(6
4641.03
)4641.03(6
0
F
RA
3-0.4641
6
HW (3.87)
HW (3.92)
Example
lbf
ApFoil
370,3
)6*4(*3*4.62*75.0
Given: Gate AB is 4 ft wide, hinged at A. Gage G reads -2.17 psiFind: Horizontal force at B to hold gate.Solution:
ft
Ay
Iyycp
1)24*3(
12/6*4 3
Convert negative pressure in tank to ft of water
ftp
h 01.54.62
144*17.2
6 ft
A
B
G
OilSG=0.75
Water
Air5.01 ft
18 ft
gate
Example
lbf
ApFw
000,15
)6*4(*)01.515(*4.62
ft
Ay
Iyycp
3.024*)01.515(
12/6*4 3
6 ft
A
B
G
OilSG=0.75
Water
Air5.01 ft
18 ft
gate
4ft3.03 ft
A
B
Fw Foil
FB
lbfF
F
FFF
M
B
B
Boilw
A
6000
6*4*37003.3*15000
6*4*3.3*
0
But ∫ x.dA = represents the sum of the first
moments of areas of the strips about free surface.
Therefore ∫ x.dA = A.
• F = γ A.
Center of pressure for vertical plane surfaceFor horizontal plane surface centroid of the area and the center of pressure
coincide with each other.But for plane surface immersed vertically the center of pressure does not
coincide with the centroid of the area.Since the pressure intensity increases with the increase in the depth of the
liquid, the center of pressure for a vertically immersed plane surface lies below the centroid of the surface area.
The total pressure on the strip shown in Fig. is dF = γ.x (b.dx)Like wise, by considering a number of small strips and summing the
moments of the total pressure on these strips about free surface. The sum becomes
∫dF.x = ∫ γx. (b.dx).x = γ ∫ x2.dA
But from principle of moment--- ∫ x2.dA represents the sum of the second moment of the areas of the strips about axis passing through the free surface, which is equal to the moment of inertia.
∫ x.dA = represents the sum of the first moments of areas of the
strips about free surface.Therefore
∫ x.dA = A.
• F = γ A.
1.8 m
A
B
G
OilSG=0.75
Water
Air
5.5 m
gate