Ch2 Hydrostatic

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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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"#$ 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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)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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8

"#/ 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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%2

' 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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%7

+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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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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• 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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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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Ch2 Hydrostatic