Hydrostatics

K141 HYAE Hydrostatics 1

HYDRAULICS

HYDROSTATICS HYDRODYNAMICS

HYDROSTATICSLiquid in equilibrium with regard

to Earthto moving

system(reservoir)

relativeequilibrium

K141 HYAE Hydrostatics 2

PRESSURE IN LIQUID- equilibrium: only normal stress → pressure (tangential stress τij ≈ 0)

pressure force:

pressure:

SF

ΔΔ

SF[ ]Pa

dSdFp =

- at motion: both normal and tangential stress

pressure (hydrodynamic) depends on velocity

∫ ∫===S S

dS pdFF dS pdF

In gravitational field:hydrostatic pressure: ph = ρ⋅g⋅hpressure in point M : p = pe+ ph = pe + ρ⋅g⋅h

pressure force at horizontal bottom: F = p⋅S

K141 HYAE Hydrostatics 3

21

21pghp

SpghSSp=⋅ρ⋅+

⋅=⋅ρ⋅⋅+⋅

21 FgV F =⋅ρ⋅+

hSVSpFSpF

22

11

⋅=⋅=⋅=

Requirement of force equilibrium in vertical direction:

h12 pghpp =⋅ρ⋅=−

Hydrostatic pressure in liquid in vertical tube closed by pistons(considering gravity acceleration)

K141 HYAE Hydrostatics 4

equilibrium of external forces(for ρ = const.):

( ) SdppcosadsSSp ⋅+=ϕ⋅⋅⋅⋅ρ+⋅

resulting acceleration in direction schange of pressure

on path ds

pressure force pressure forcebody force

dscosadp ⋅ϕ⋅⋅ρ= Euler hydrostatic equation(one-dimensional form)

CHANGE OF PRESSURE

K141 HYAE Hydrostatics 5

Euler hydrostatic equation(component form)

component form:

( )dzadyadxadp zyx ⋅+⋅+⋅ρ=

zyx dpdpdp dp ++=dxadp xx ⋅⋅ρ=

dyadp yy ⋅⋅ρ=

dzadp zz ⋅⋅ρ=

pppp zyx ===

Gravitational force field

dhds,ga == dhgdp ⋅⋅ρ=

∫ ∫ρ=p

p

h

0e

gdhdp hgpp e ⋅⋅ρ+= hhydrostatic pressure p g h= ρ ⋅ ⋅

K141 HYAE Hydrostatics 6

SURFACE AREA

h = const.p = const. 0dhgdp =⋅⋅ρ= dh = 0

22a11e hgphgp ⋅⋅ρ+=⋅⋅ρ+hgp

pppphgp

12

21

Δ⋅⋅ρ=Δ−=Δ

=Δ⋅⋅ρ+

Perpendicular to resulting accelerationin gravitational field horizontal plane level

=⋅⋅ρ+⋅⋅ρ+ 3111e hghgp3122a hghgp ⋅⋅ρ+⋅⋅ρ+=

differential pressure gauge(U-tube)

piezometer41ae hgpp ⋅⋅ρ=−

K141 HYAE Hydrostatics 7

normal atmospheric pressure pa = 101324.72 Pa ≅ 105 Patotal static pressure ps

ps > paoverprressure

pp = (ps – pa) > 0

ps < pa underpressurepva = (ps – pa) < 0

pe = pa hgpp as ⋅⋅ρ+=

1 a 1 p1 1

2 a 2 p2 2

3 a 3 va3 3

p p g h , p g h

p p g h , p g hp p g h , p g h

= + ρ ⋅ ⋅ = ρ ⋅ ⋅

= + ρ ⋅ ⋅ = ρ ⋅ ⋅

= − ρ ⋅ ⋅ = −ρ ⋅ ⋅

OVERPRESSURE, UNDERPRESSURE

K141 HYAE Hydrostatics 8

PASCAL’S LAWGradual pressure change Δp in small closed volume of liquid expends in all directions and passes on all points of liquid without any change.

Δp = const., p = const.SFp =

force [N]

area [m2]

22

2

1

11 p

SF

SFp ===

1

212 SSFF =

21 FpF Δ⇒Δ⇒Δ

Pressure head Suction head [m, m w.c.]g

ph⋅ρ

=g

ph vava ⋅ρ

=

sfor p 0= 5va amax

p p 10 Pa= − = va maxh 10mv.sl.≈

K141 HYAE Hydrostatics 9

HYDRAULIC PRESS

PRESSURE CONVERTER

21 1

2

Fth e o re tica lly F SS

= ⋅

1

221 S

Spp ⋅⋅η=

12

21 S

SFF ⋅⋅η=

η... efficiency(0,95 –1,0)

In practice → losses

K141 HYAE Hydrostatics 10

HYDROSTATIC FORCE

Hydrostatic force = force caused by hydrostatic pressure ph.dSpdF ⋅=

S S S

F pdS gzdS, for g const. : F g zdS= = ρ ρ ⋅ = = ρ ⋅ ⋅∫ ∫ ∫

F – passes through centre of pressure body ∫S

zdS- perpendicular to loaded area

z ... vertical depth (depth bellow level)

If overpressure pp (underpressure pva) on level → to enlarge (reduce) real depth z by pressure head

gp

,g

p vap

⋅ρ⋅ρ

total pressure force

K141 HYAE Hydrostatics 11

HORIZONTAL BOTTOM

ShdShzdSSS

⋅=⋅= ∫∫ ShgF ⋅⋅⋅ρ=

h·S – volume of pressure body

hydrostaticparadoxon

K141 HYAE Hydrostatics 12

tF g z S= ρ ⋅ ⋅ ⋅INCLINED PLANE SURFACE

SzzdS tS

⋅=∫

For prismatic areas with horizontal border – possible also

ω... area of pressure diagram [m2]Ω = ω⋅b ... volume of pressure body [m3]

Ω⋅⋅ρ=⋅ω⋅⋅ρ=⋅⋅+

⋅⋅ρ=⋅⋅⋅ρ= gbgba2

zzgSzgF 21t

baS,2

zzz 21t ⋅=

+=

– volume of pressure body

tz S⋅

tz S⋅

tz S⋅T - centre of area SC - centre of force F

K141 HYAE Hydrostatics 13

DETERMINATION OF POINT OF ACTION OF HYDROSTATICFORCE

Moment condition to x-axis:

I0 ... .second moment of loaded area S about gravity centre axis o

∫=⋅S

c ydFyF

Ix ..... second moment of loaded area S about x-axis

tt

o

t

2to

c yyS

IyS

ySIy +⋅

=⋅⋅+

=t

otc yS

Iyy⋅

=Δ=−

dSsinygdSzgdSpdF

,dSysingdSzgFSS

⋅α⋅⋅⋅ρ=⋅⋅⋅ρ=⋅=

⋅α⋅⋅ρ=⋅⋅ρ= ∫∫

∫∫ ⋅α⋅⋅ρ=⋅α⋅⋅ρ⋅S

2

Sc dSysingdSysingy

t

x

S

S

2

S

S

2

c ySI

dSy

dSy

dSysing

dSysingy

⋅==

⋅α⋅⋅ρ

⋅α⋅⋅ρ=

∫

∫

∫

∫

2tox ySII ⋅+=

K141 HYAE Hydrostatics 14

Determination of point of action of hydrostatic force F on rectangular area with horizontal upper edge corresponding with water level

baS

,ab121I

,2ay

3o

t

⋅=

⋅⋅=

=

a32

2aa

61

2a

2aba

ab121

yyS

Iy3

tt

oc =+=+

⋅⋅

⋅⋅=+

⋅=

centre of loaded area S→ on gravity centre axis o

point of action of hydrostatic

force F

K141 HYAE Hydrostatics 15

pressure diagram- components

pressure diagram- complex

RESOLUTION OF HYDROSTATIC FORCE IN COMPONENTS

z

x

FtgF

α =

K141 HYAE Hydrostatics 16

nihz

n2hz

n1hz

i

2

1

=

=

=

Effective distribution of horizontal beams

K141 HYAE Hydrostatics 17

HYDROSTATIC FORCE ACTING ON CURVED AREASTwo perpendicular horizontal (Fx, Fy), and vertical component (Fz):

xix dSzgcosdSzgdF ⋅⋅⋅ρ=α⋅⋅⋅⋅ρ=

yiy dSzgcosdSzgdF ⋅⋅⋅ρ=β⋅⋅⋅⋅ρ=

ziz dSzgcosdSzgdF ⋅⋅⋅ρ=γ⋅⋅⋅⋅ρ=

,SzgF xtxx ⋅⋅⋅ρ= ytyy SzgF ⋅⋅⋅ρ=

VgSzgF ztz ⋅⋅ρ=⋅⋅⋅ρ=

ztx (zty) ... vertical depth of projection Sx (Sy)Sx (Sy) ... .projection of area S at plane YZ (XZ)

V ... ……..volume of vertical column of liquid above area S

2z

2y

2x FFFF ++=,

FFcos,

FF

cos,FFcos zyx =γ=β=α

For prismatic areas also : 0F,bgF,bgF yzzxx =ω⋅⋅⋅ρ=ω⋅⋅⋅ρ=

K141 HYAE Hydrostatics 18

Course of hydrostatic pressure:

Hydrostatic force passes through centre of curvature of cylindrical area S

Solving hydrostatic force in components:

K141 HYAE Hydrostatics 19

FLOATING BODIESApplication of Archimedes principle:buoyancy force

ρk … density of liquid [kgm-3]W … volume of displaced liquid (displacement) [m3], W = W (tn)

Vertical cylinder submerged in liquid

External surface forces:ShgF,ShgF 2211 ⋅⋅⋅ρ=⋅⋅⋅ρ=

( )vzkk

12k

1k2k12

FWgShghhSg

ShgShgFF

=⋅⋅ρ=⋅⋅⋅ρ==−⋅⋅⋅ρ=

=⋅⋅⋅ρ−⋅⋅⋅ρ=−↑

WgF kvz ⋅⋅ρ= Archimedes principle

K141 HYAE Hydrostatics 20

Fvz < G ⇒ body gravitatesFvz > G ⇒ body moves up

till Fvz = GFvz = G ⇒ body in balance

V = W – body hoversV > W – body floats(V – body volume)

Fvz goes through centre C of displacement WG goes through centre of floating body

Resolution of immersion: G = Fvz W tn

K141 HYAE Hydrostatics 21

OVERVIEW OF MAIN TERMS AND TOPICS

overpressure, underpressure, static pressure

hydrostatic pressure

pressure head, suction head

Pascal´s law

hydrostatic force acting on plane and curved surface area

(dimension, direction, point of action)

pressure body - complex

- in components

floating bodies - Archimedes principle