ch2 hydrostatic
TRANSCRIPT
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Water Pressure and Pressure Force(Revision)
The Islamic University of Gaza
Faculty of EngineeringCivil Engineering Department
Hydraulics - ECI !!""
Chapter 2Chapter 2
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2
"#$ Free %urface of &ater
• A horizontal surface upon which the pressure isconstant every where.
• Free surface of water in a vessel may be subjected to: - atmospheric pressure (open vessel or!
- any other pressure that is e"erted in the vessel (closed
vessel.
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#
"#" '(solute and Gage )ressures
• 'tmospheric pressure is appro"imately e$ual to a
%&.##-m-hi'h column of water at sea level.• Any object located (elo* the *ater surface is
subjected to a pressure greater than the atmosphericpressure ( ) atm.
Let:
dA * cross-sectional area of
the prism. the prism is at rest. +o! allforces actin' upon it must be in
e$uilibrium in all directions.
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,
+otice that,
• f the two points are on the same elevation! h = 0 P A=P .
• n other words! for water at rest! the pressure at all points
in a horizontal plane is the same.
Euili(rium in .- direction,
F x = PA dA – PB dA + γ L dA sin θ = 0
/ A * γ h The difference in pressure between anytwo points in still water is always equal to:
the product of the specific weight of water
( γ ) and the difference in elevation between
the two points (h).
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0
)ressure gages, are usually desi'ned to measure
pressures above or below the atmospheric pressure.
Gage pressure, is the pressure measured with respect to
atmospheric pressure (usin' atmospheric pressure as a
base.
'(solute pressure, abs * 'a'e 1 atm
)ressure head! h * γ
f the water body has a free surface that is e"posed to
atmospheric pressure atm. oint A is positioned on the free
surface such that A
* atm
( abs* A 1 γ h * atm 1 γ h * absolute pressure
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+otice that,
• Any chan'e in pressure at point ! would cause an e$ual
chan'e at point A! because the difference in pressure headbetween the two points must remain constant * h.
Pascal's law ,
A pressure applied at any point in a liquid at rest istransmitted equally and undiminished in all directionsto every other point in the liquid.
4his principle has been made use of in the hydraulic jac5sthat lift heavy wei'hts by applyin' relatively small forces.
4he difference in pressure heads at two points in water at
rest is always e$ual to the difference in elevation between the
two points.
( γ / ( A γ * ∆(h
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E.ample "#$
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"#! %urface of Eual )ressure
• 4he hydrostatic pressure in a body of water varies with the
vertical distance measured from the free surface of thewater body.
• All points on a horizontal surface in the water have thesame pressure.
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"#/ 0anometers
' manometer s a tube bent in the form of a 9 containin' a fluid of 5nown
specific 'ravity. 4he difference in elevations of the li$uid
surfaces under pressure indicates the difference in pressure
at the two ends.
T*o types of manometers,
$# 'n open manometer: has one end open to atmospheric
pressure and is capable of measurin' the 'a'e pressure
in a vessel.
"# ' dierential manometer: connects each end to a
different pressure vessel and is capable of measurin' the
pressure difference between the two vessels.
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%&
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%%
• 4he li$uid used in a manometer is usually heavier than thefluids to be measured. t must not mi" with the adjacent
li$uids (i.e.! immiscible li$uids.
• The most used liuids are,
- 0ercury (specific 'ravity * %#.3!
- &ater (sp. 'r. * %.&&!- 'lcohol (sp. 'r. * &.8! and
- ther commercial manometer oils of various specific'ravities.
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' simple step-(y-step procedure for pressure computation
%tep$: ;a5e a s5etch of the manometer system appro"imately
to scale.
%tep ":
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%#
' simple step-(y-step procedure for pressure computation
(1 For a differential manometersP " = P #
γ $ .h & γ w .(y h) & P ! = γ % .y & P '
∆P = P ' P ! = h ( γ $ γ w )
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%,
E.ample "#"
Determine the pressure
difference )
%olution,
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%0
%ingle-reading manometer ' differential manometer
installed in a flo* - measured system
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%3
"#2 Hydrostatic Force on a Flat %urface• 4he area '! of the bac5 face of a dam inclines at an an'le (θ ) and!
• = - a"is lies on the line at which the water free surface intersects with
the dam surface!
• > - a"is runnin' down the direction of the dam surface.
horizontal vie* pro3ection of A! on the dam surface
h
h
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%6
θsinγγ yh P ==
Aθ.sinγ d ydF =
• For a strip at depth h below the free surface:
• 4he total pressure force over the surface: yd ydF F
A A
θ.A.sinγAθ.sinγ === ∫ ∫ .A.γ h F =
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.
%here:
is the distance measured from the *a*is to the
centroid (".#.) of the plane
Aθ.sinγ d ydF =
AdA y y A
∫ =
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+otes,
• ressure forces actin' on a plane surface are distributed over
every part of the surface.
• 4hey are parallel and act in a direction normal to the surface.
• 4hey can be replaced by a sin'le resultant force F * γ h?A.actin' normal to the surface.
• 4he point on the plane surface at which this resultant force acts
is 5nown as the center o pressure (".P.)#
• The center of pressure of any submer'ed plane surface is
always below the centroid of the surface (+p , +-).
y
y A
I
y A
y A I
M
I
y A
dA y
F
dF y
Y oo
x
x A A P +=
+====
∫ ∫ 22
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%8
4he centroid! area! and moment of inertia with respect to the
centroid of some common 'eometrical plane surfaces are 'iven
below.
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2&
E.ample "#!
For the vertical trapezoidal gate4
Determine F and 5)%olution,
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2%
E.ample "#!
Determine F and 5)
%olution,
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"#6 Hydrostatic Forces on Curved %urfaces
• 4he hydrostatic force on a curved surface can be best analyzed by
resolvin' the total pressure force on the surface into its horizontal and
vertical components.
• 4hen combine these forces to obtain the resultant force and its direction.
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2#
• F $ * @esultant force on the projection of the curved surface onto a
vertical plane.
• / acts horizontally throu'h the centre of pressure of the
projection of the curved surface onto a vertical plane.
• e can use the pressure dia'ram method to calculate the positionand ma'nitude of the resultant horizontal force on a curved surface.
B A H F F '=0=∑ x F
'' ABA AAV W W F +=0=∑ y F
• FB * 4he resultant vertical force of a fluid above a curvedsurface e$ual to the wei'ht of fluid directly above the curvedsurface.
• t acts vertically downward throu'h the centre of 'ravity ofthe mass of fluid.
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2,
7esultant force
• 4he overall resultant force is found by combinin' the
vertical and horizontal components vectorialy:
• 4he an'le the resultant force ma5es to the horizontal is:
• 4he position of is the point of intersection of thehorizontal line of action of / and the vertical line of actionof .
22
V H F F F +=
= −
H
V
F
F 1tanθ
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)ressure distri(ution on a semi-cylindrical gate
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