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8/2/2019 Chapter 2 Hydro Static Forces
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CHAPTER 2 : FLUI D STATI CS
Hydrostati c Forces
The design of many engineering systems such as water dams andliquid storage tanks requires the determination of the forces actingon the surfaces using fluid statics. The complete description of theresultant hydrostatic force acting on a submerged surface requiresthe determination of its magnitude, its point of application or itsdirection, and the line of action of the force.
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2.7 Hydrostati c Forces On Subm erged Plane Surfaces
When a surface such as a gate valve in a dam, the wall of a liquidstorage tank, or the hull of a ship is submerged in a fluid, forces
develope on the surface due to the fluid pressure. For fluids at restwe know that the force must be perpendicular to the surface sincethere are no shearing stresses present. We also know that thepressure will vary linearly with depth if the fluid is incompressible.Fig. 2.18 below shows the hydrostatics pressure distribution on thesubmerged tilted, vertical and horizontal plates.
Fig. 2.18 : Hydrostatic force distribution on the submerged plates.
In most cases, the other side of the plate is open to the atmosphere(such as the dry side of a gate), and thus atmospheric pressure actson both sides of the plate, yielding a zero resultant. In such cases, itis convenient to subtract atmospheric pressure and work with thegauge pressureonly.
On a plane surface, we often need to determine the magnitude of theresultant force and its point of application, which is called the centerof pressure.
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2.7.1 Hydrostat ic Forces Act ing On Horizont al Subm ergedPlane Sur faces
For the horizontal surface, such as the bottom surface of a tank as
shown in Fig. 2.19, the magnitude of the resultant force is simply
FR =pA =gh.A
where ;p = uniform pressure on the bottomA = area of the bottom
Fig. 2.19 : Pressure distribution on the bottom horizontal surface oftank
Since the pressure is constant and uniformly distributed over thebottom, the resultant force acts through the centroid of the area.
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# Example 2.13 : Hydrostatic Force on the Horizontal Plane
The rigid L-shaped gate OAB is 3 m width and hinged at O and rests against a rigidsupport at B. Find the hydrostatic force acting on the plate AB.
Fig E2.13
Solution :
( )( )( )( )
ABplateofcentroidthethroughacting----kN
.
412
3278191000
=
=
== AghApFR
To be continued at Example 2.14
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2.7.2 Hydrostati c Forces Act ing On Vert ical and I nclinedSubm erged Plane Sur faces
For the more general cases in which a submerged plane surface isinclined, as is illustrated in Fig. 2.20, the determination of theresultant force acting on the surface is more involved.
Fig. 2.20 : Arbitrary shape plane submerged in liquid
For the present we will assume that the fluid surface is open tothe atmosphere. Let the plane in which the surface lies intersect thefree surface at O and make an angle with this surface. The xycoordinate system is defined so that 0 is the origin and y is directedalong the surface as shown. The area can have an arbitrary shape asshown. Now, we wish to determine the direction, location, and
magnitude of the resultant force acting on one side of this area dueto the liquid in contact with the area.
At any given depth, h, the force acting on dA is dAhdF .= andis acting perpendicular to the surface. Thus, the magnitude of the
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resultant force can be found by summing these differential forcesover the entire surface. In equation form,
with
=
AR
dAhF . sinyh= = A dAy sin
= A ydAsinThe integral of ydAis the first moment of the area with respect tothe x axis or ydA = ycA, so we can write,
AyAyF ccR .sin.sin == Or more simply as,
Agh
AhF
c
cR
= =
where ;hc= vertical distance from free surface to the centroid of the area.Note that the magnitude of the force is independent of the angle anddepends only on the specific weight of the fluid, the total area, andthe depth of the centroid of the area below the surface.
In effect, the equation indicates that the magnitude of the resultantforce is equal to the pressure at the centroid of the area multiplied bythe total area. Since all the differential forces that were summed toobtain are perpendicular to the surface, the resultant must alsobe perpendicular to the surface.
Besides the magnitude of resultant force, the location where this FR isacting on also has to be determined. This point or location is calledas center of pressure, CP. The location of CP normally is described interms of vertical distance from free surface, hRor hcp or inclineddistance from free surface, yR (or sometimes also known as ycp).
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Note : From the equation FR= ghc.A, our intuition might suggestthat the resultant force should pass through the centroid of the area,this is not actually the case.
The yRcan be determined by summing the moments aroundthe x axis. That is, the moment of the resultant force must equal themoment of the distributed pressure force, or
=
=
ARR
ARR
dAyyF
ydFyF
2sin
Thus, Ay
dAy
Ay
dAy
F
dAy
yc
A
c
A
R
AR
===
222
sin
sinsin
But, Ay2dAis the second moment of the area (moment of inertia,
Ix), with respect to an axis formed by the intersection of the planecontaining the surface and the free surface (x axis). Thus, we canwrite,
2cxc
c
xR AyI
Ay
Iy +== xIwith
where Ixc is the second moment of the area with respect to an axispassing through its centroid and parallel to the x axis. Thus,
c
c
xcR y
Ay
Iy +=
Or, in vertical distance, hcp @ hR
c
c
xcR h
Ah
Ih +=
2sinsince yR= hR/sin and yc= hc/sin
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Both result shows that the resultant force does not pass through thecentroid but is always belowthe centroid, since Ixc/ycA> 0.
Fig. 2.21 shows the Ixcproperties of some common shape.
Fig. 2.21 : Geometric properties of some common shapes.
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Procedure for comput ing t he hydrostat ic force on asubmerged plane area
1. Identify the point where the angle of inclination of the area ofinterest intersects the level of free surface of the fluid. This mayrequire the extension of the angled surface or the fluid surfaceline. Call this point as origin, 0.
2. Locate the centroid of the area from its geometry.3. Determine hc as the vertical distance from the level of the free
surface down to the centroid of the area.4. Determine yc as the inclined distance from the level of the free
surface down to the centroid of the area. This is the distance from0 to the centroid. Note that hc and yc are related by hc = yc sin .
5. Calculate the resultant force from AghAhF ccR == 6. Then calculate the Ixc, the moment of inertia of the area about itscentroidal-x axis.
7. Determine the location of cp by calculating the yR from(i) c
c
xcRcp y
Ay
Iyoy +=r
* Or hR for vertical plane cases which is using the equation of
(ii) cc
xcRcp h
Ah
Ihoh +=
2sinr
8. Sketch the FRacting on the cp, perpendicular to the area (or thewhole free body diagram if necessary).
9. Calculate other force magnitude if required using M=0 forequilibrium.
Note : For inclined plane case, follow all steps 1 to 7. For vertical plane case, follow steps 2, 3, 5, 6 and 7 (ii). Step 8 and 9 required if problem involved solving using M=0.
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# Example 2.14 : Hydrostatic Force on the Vertical Plane
The rigid L-shaped gate, OAB, of Fig. E2.14 is hinged at O and rests against a rigidsupport at B. What minimum horizontal force, P, is required to hold the gate closed ifits width is 3 m? Neglect the weight of the gate and friction in the hinge. The back ofthe gate is exposed to the atmosphere.
Fig E2.14
In this case, we should decompose the force analysis of L gate into two parts:i. determination the resultant force FR(AB) which acting on plate AB.ii. determination of the resultant force FR(OA) which acting on the plate OA.
Solution :
iii.
From Eg. E2.13, FR(OB) acting on plate AB has been calculating which isequalto 412 Kn.
iv. For plate OA,
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( )
( )( )( )(
kN6.588
34581.91000
=
=
== AghApF ccOAR
)
( ) 433 m161243
12=== bdIxc
( )
( )( )m
Ah
Ihh
c
xccOAR
27.5
345
165
.
=
+=
+=
FREE BODY DIAGRAM
Refer to free body diagram above, for equilibrium,
( )( ) ( ) ( )
( ) ( )
kNP
P
PFF
M
ABROAR
hinge
437
4141200027.2588600
)4(1327.5
0
=
=+
=+
=
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# Example 2.15 : Hydrostatic Force on the Vertical Plane
An open tank has a vertical partition and on one side contains gasoline with adensity 700 kg/m3 at a depth of 4 m, as shown in Fig. E2.15. A rectangular gate thatis 4 m high and 2 m wide and hinged at one end is located in the partition. Water isslowly added to the empty side of the tank. At what depth, h, will the gate start toopen?
Fig E2.15
Solution :
( )( )( )( )kN9.109
24281.9700
=
=
== AghApF ccgasolineR
( ) 433
m67.1012
42
12===
bdIxc
( )( )m
Ah
Ihh
c
xccgasolineR
67.2
242
67.102
.
=
+=
+=
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( )( )( )( )
/.
29810
228191000
h
hh
AghApF ccwaterR
=
=
==
( ) 4333
m167.012
2
12h
hbdIxc ===
( )( )
h
hh
hh
Ah
Ihh
c
xccwaterR
6670
22
16702
3
.
/
./
.
=
+=
+=
For equilibrium,
( ) ( )
( ) ( )
mh
hhhh
hFhhF
M
gasolineRgasolineRwaterRwaterR
hinge
563
146167733266672410990066709810
4
0
3
2
.
...
=
=
=
=
=
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# Example 2.16 : Hydrostatic Force on the Vertical Plane
A pressurized tank contains oil (SG = 0.9) and has a square, 0.6-m by 0.6-m platebolted to its side, as is illustrated in Fig. E2.16. When the pressure gauge on the topof the tank reads 50 kPa, what is the magnitude and location of the resultant force onthe attached plate? The outside of the tank is at atmospheric pressure.
Fig E2.16
Solution :
( )
( ) ( )( )( )[ ]( )kN.
....
.
325
6060328199001050 3
=
+=
+== AghpApF caircR
( ) 433
m0108.012
6.06.0
12===
bdIxx
( )( )m
Ah
I
hhc
xc
cR
31.2
6.06.03.2
0108.03.2
.
=
+=
+=
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# Example 2.17 : Hydrostatic Force on the Inclined Plane
The 4 m diameter circular gate of Fig. E2.17 is located in the inclined wall of a largereservoir containing water. The gate is mounted on a shaft along its horizontaldiameter. For a water depth of 10 m above the shaft, determine:
(a) the magnitude and location of the resultant force exerted on the gate by thewater,
(b) and the moment that would have to be applied to the shaft to open the gate.
Fig E2.17
Solution :
a. The magnitude and location of resultant force
AghF cR .=
( )( )( ) ( ) kN. 12302108191000 2 == RF ( ) 4m44
2
4
44
===R
Ixx
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Ay
Iyy
c
xxcR
.+=
( ) (( ))2260104
60
10
sin/sin+=
m611.= b. The moment that would have to be applied to the shaft to open the gate.
( ) ( ) .1007.10866.0101230 53 NmyyFM cRR ===
Pressure Prism
The pressure prism is a graphical representation of thehydrostatic force on a plain surface. The magnitude of the resultantfluid force is equal to the volume of the pressure prismand passesthrough the centroid.
Consider the pressure distribution along a vertical wall of a tankof width b, which contains a liquid having a specific weight . Sincethe pressure must vary linearly with depth, we can represent thevariation as is shown in Fig. 2.22 (a), where the pressure is equal tozero at the upper surface and equal to h at the bottom.
Fig. 2.22 : The pressure prism
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The pressure distribution shown in Fig. 2.22 (a)applies across
the vertical surface so we can draw the three-dimensionalrepresentation of the pressure distribution as shown in Fig. 2.22 (b).
The base of this volume in pressure-area space is the plane surfaceof interest, and its altitude at each point is the pressure. This volumeis called thepressure prism, and it is clear that the magnitude oftheresultant force acting on the surface is equal tothevolume of thepressure prism or,
( )( )
2
2
22
1
bh
bhh
bhh
prismofvolumeFR
=
==
=
.
pressure
JUST CHECK!....
AghAhFh
hce
Ah
bh
prismpressureofVolumeF
ccR
c
R
===
==
=
2sin
22
1 2
The resultant force must pass through the centroid of the pressureprism which is located along the vertical axis of symmetry of thesurface, and at a distance h/3 above the base (since the centroid of atriangle is located at h/3 above its base) or 2/3h from the upper end.Or it can be proved as follows,
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h
hh
bhh
bhh
Ah
Ihh
c
xccR
3
2
62
2
12
2
3
=
+=
+=
+=
/
/
Note :
The use of pressure prisms for determining the force on submerged
plane areasis convenient if the area is rectangularso the volumeand centroid can be easily determined. However, for othernonrectangular shapes, integration would generally be needed todetermine the volume and centroid. In these circumstances it is moreconvenient to use the equations developed in the previous section.
This same graphical approach can be used for plane surfaces that donot extend up to the fluid surface as illustrated in Fig. 2.23. In thisinstance, the cross section of the pressure prism is trapezoidal.
However, the resultant force is still equal in magnitude to the volumeof the pressure trapezoidal, and it passes through the centroid of thevolume.
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Fig. 2.23 : The pressure distribution on the vertical plates located farbelow from the free surface.
Specific values can be obtained by decomposing the pressure
prism/trapezoidal into two parts, ABDE and BCD. Thus,
21 FFFR += where the components can readily be determined by inspection forrectangular surfaces. The location of FRcan be determined bysumming moments about some convenient axis, such as one passingthrough A. In this instance,
2211 yFyFyF AR += Therefore the location where the FRacts measured from point A is
R
AF
yFyFy 2211
+=
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# Example 2.18 : Determination of Hydrostatic Force on the Vertical PlaneUsing Pressure Prism Technique
Solve Example 2.16 using pressure prism technique.
Solution :
Fig E2.18
( )AhpF s 11 += ( ) ( )( )[ ][ ]( )6.06.0281.9100090.050000 +=
kN4.24=
Ahh
F
=
212
2
( ) ( )6.06.026.081.9100090.0
=
kN954.0=
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Total hydrostatic force,
kNFFFR 4.25954.04.2421 =+=+=
The location of FR measured from O, yo,
( ) ( )2.03.0 21 FFyF oR +=So,
( ) ( )
R
oF
FFy
2.03.0 21 +=
( ) ( )( ) m296.0104.25 2.010954.03.0104.24 333
=
+= m296.0=
Therefore the location of FR measured from free surface, hR,mhR 3.2296.06.2 ==
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2.8 Hydrostati c Forces Act ing On Subm erged Curved
Surfaces
The equations FR=ghCA and hR=Ixc/hC.A + hC are developed for themagnitude and location of the resultant force acting on a submergedsurface only apply to plane surfaces. However,many surfaces ofinterest (such as those associated with dams, pipes, and tank) arenonplanar.
Fig. 2.24 : Examples of curved or nonplanar surfaces.
For submerged curved surface, the determination of the resultantforce (FR) typically requires the integration of the pressure force thatchange along the curve surface.
However, the easiest way to determine the FRacting on the curvedsurface by separating it into the horizontal and vertical components,FH and FV.
This is done by considering the free-body diagram of the fluid volumeenclosed by the curved surface of interest and the horizontal andvertical projections of this surface, as shown in Fig. 2.25 below.
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Fig. 2.25 : Hydrostatic force on the curved surface
The forces acting on this enclosed volume include :
1. W is the weight of the enclosed fluid volume and actsdownwardthrough the centroid of this volume which is simply given by,
W=gV
2. Fx is the hydrostatic force acting on the vertical projection surfacearea, through the centroid of this vertical surface where,Fx=gh2.Avertical
3. Fy is the hydrostatic force acting on the horizontal projectionsurface area, through the centroid of this horizontal surfacewhere,
FY=gh1.Ahorizontal
Note that :vertical surface = the projection of the curved surface on a
vertical plane,horizontal surface= the projection of the curved surface on a
horizontal plane.
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From the Fig. 2 25 (b), for the equilibrium, the force balances in thehorizontal and vertical directions give;
andxH FF = WFF yV +=
The magnitude of resultant force, FR is then given by,
22
VHR FFF +=
And its direction,
=
H
V
F
F1tan
Summary of the procedure for computing the hydrostatic
force on submerged curved sur face.
1. Isolate the volume of fluid above/under the curved surface.2. Sketch the free body diagram (FBD) of the fluid volume and
show all the forces involved with correct direction and location.3. Compute the Fx=gh2.Avertical. (identify first the h2and vertical
projection surface area,Averticalfor Fx).4. Compute FY=gh1.Ahorizontal (identify first the h1and horizontal
projection surface area,Ahorizontalfor FY).5. Compute W=gV (identify first the fluid volume,V).6. Calculate FV and FH from the FBD.7. Calculate the resultant force, FR from 22 VHR FFF += and its
direction from
=
H
V
F
F1tan .
8. Show the resultant force acting on the curved surface in such adirection that its line of action passes through the center ofcurvature of the surface.
9. Sketch the FBD and solve problem using M=0 if required*.
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# Example 2.19: Hydrostatic Force on the Curve Surface
A 5 m width curved gate is located in the side of a reservoir containing water asshown in Fig. E2.19. Determine the magnitude of the resultant force and its location.
Fig. E2.19
Solution :
Fig. E2.19 (b)From the free body diagram of the fluid on the curve surface,
xH FF =
And WFF yV +=
where,
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( )( ) ( )
kN
ghAF yx
1104
352
368191000
=
+=
=
.
( )( )( )(
kN
)
ghAF xy
883
3568191000
=
=
=
.
( )( )( )
kN
rggW
347
54
38191000
4
2
2
=
=
==
.
l
Therefore, == kN1104xH FF
=+=+=kN1230347883WFF yV
Thus,
( ) ( )
kN
FFF VHR
1653
1230110422
22
=
+=
+=
and
o
481104
123011 =
=
=
tantan H
V
F
F
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# Example 2.20: Hydrostatic Force on the Curve Surface
A long solid cylinder of radius 0.8 m hinged at point A is used as an automaticgate, as shown in Fig. E2.20. When the water level reaches 5 m, the gate opens byturning about the hinge at point A. Determine the hydrostatic force per m length ofthe cylinder and its line of action when the gate opens.
Fig. E2.20Solution :
From the free body diagram of the fluid under curve surface,
xH FF =
and WFF yV =
where,
( )( ) ( )
kN
ghAF yx
136
1802
80248191000
.
..
..
=
+=
=
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