Download - Wind Effects
Wind effects on structures 1
Wind Effects on StructuresProf. Dr.-Ing. Udo Peil
Technische Universität Carolo-Wilhelmina Braunschweig
Wind effects on structures 2
Wind-engineered structures
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Wind-engineered structures
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Wind-engineered structures ?
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Wind-engineered structures ?
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Wind
1. Nature of the Wind
Wind effects on structures 10
Near the gound the mean wind speed is decreasing much. At the ground level the wind speed is zero!
The mean wind is superimposed by transient gusts (turbulence).
The profile depends on the roughness of the surface, the level of the gradient wind speed is higher the rougher the surface is.(α, z0 factors are given in the codes)
( ) (10)10zu z u
α⎛ ⎞= ⋅ ⎜ ⎟⎝ ⎠
α =0,28
300
200
100
α =0,40600
500
400
u G
Gu
α =0,16
Gu
z[m]
u u u
Description: or:0
( ) (10) ln zu z uz
⎛ ⎞= ⋅ ⎜ ⎟
⎝ ⎠Potential law logarithmic law
Wind Speed Profile:
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Description of the turbulence:
Due to the stochastic character of the turbulence only a statisticaldescription is possible.The turbulent part of the wind is Gauss distributed:
For description of a Gauss-process we need:
mean value
standard deviation
auto correlation
cross correlation0 10 20 30 40 50 60 70 80 90
t[s]
10
20
30
40u [m/s]
0
u
2
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2 2 2 2 21 2 3
2
1 1( ) .....1
u t dt u u uT N
σ
σ σ
⎡ ⎤= ≈ + + +⎣ ⎦−
=
∫
[ ]1 2 31 1( ) .....u u t dt u u uT N
= ≈ + + +∫Mean:
Variance:
Standard dev. (root mean square: rms)
v
v v v
v1 2
3
i
t
dt
Description of the turbulence:
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The rms of the turbulence is decreasing with the height. Theinfluence of the rough surface becomes less important.
The turbulence intensity is a measure for the turbulence. It is definedto be:
( )I zuσ
=
I(z) is decreasing with the height.
It reaches values of about 20%.
z [m]
I = u
34032030028026024022020018016014012010080604020
0.05 0.1 0.15 0.2
I [-]
04
03
σ
Description of the turbulence:
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The turbulent wind process is a correlated processes, because a passing gust increases the wind speed for a certain time as well as for a certain 3D-area:
v
t
t
v(t+ )
v(t)
v(t) v(t+ )
v(t+ )v(t)
auto correlation-functioncross correlations-function
Rxx
=R ( )xx
Description of the correlation:
0
1( ) ( ) ( )xxR u t u t dtT
τ τ∞
= ⋅ +∫Both processes are shifted and themean of the product of bothfunctions is determined. Thus itmust be:
2( 0)xxR τ σ= =
If a process is periodic, the auto-correlation function must beperiodic as well
Description of the turbulence:
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For long time intervals τ the autocorrelation functions tends to zero (the gust „ball“ has a finite lenght)
If 2 different processes are analysed a so called cross-correlationfunction is determined. If wind speeds on 2 levels are measured, the wind speed hits oneanemometer earlier than the second one (ball shape of the gust):
The maximum of the cross correlationfunction is shifted to τ=3s.
0 20 40 60 80
t[s]
10
20
30
40
50
60
0100
u[m/s]
u(66m)+20m/su(48m)+10m/su(30m) 3 s
66m
30 m3 s
Description of the turbulence:
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Auto- and cross correlation functions:
Long time measurements are needed, otherwise the result is random !
Description of the turbulence:
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If the Auto- or Cross correlation functions are Fourier-transformed, theso called Power Spectral Density functions (PSD) are determined:
( ) ( ) i txx xxS R e dωω τ τ
+∞−
−∞
= ⋅ ⋅∫
( ) ( ) i txy xyS R e dωω τ τ
+∞−
−∞
= ⋅ ⋅∫
Auto-Spectrum
Cross-Spectrum
The PSD can be determined approximately (as an estimation) fromthe square of the amount of the complex amplitude spectra of themeasured function:
21( ) ( )2xx TS XT
ω ω≈ With: ( ) ( )T
i tT
T
X x t e dtωω+
−
−
= ⋅ ⋅∫
Description of the turbulence:
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0.003 0.01 0.1 1.0 2.0
f [Hz]
S (f)1000
100
10
1
0.1
0.01
h =138mv= 27.3 m/so= 3.54 m/sL =2500mx
modif. Davenport-Spektrum
Example of a PSD of wind speed (double logarithmic):
The PSD shows the energy of theanalysed process as function of thefrequency.
The wind energy is very high at frequencies of 0.01 Hz (T=100s)
The energy in a frequency range of the eigenfrequencies of buildings(>0.1 Hz) is (happily) less in an order of 2 to 3 magnitudes.
2 1 ( )2 xxS dσ ω ωπ
+∞
−∞
= ⋅∫
The area under the PSD equals thevariance:
Description of the turbulence:
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0.003 0.01 0.1 1.0 2.0 5.0
1
5
10f*S(f)
f[Hz]
DavenportDavenportmodif.
Kaimal
Simiu
Proposals for the PSD of wind speed:
Davenport Spectrum:
For practical use the measured (rough) spectra are fitted by a function:
2 2
2 4 / 3
10
2( )3 (1 )
: x
xS ffx
L fwith xu
σ= ⋅ ⋅
+⋅
=
Lx : charact. Length =1200 m
This spectrum is heightindependent !
Description of the turbulence:
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A short course in Structural Dynamics
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Simple Model with 1 degree of freedom:
If a structure is moving, forces must act!
Equilibrium conditions:
F uK= ⋅K(Hookes law)
FFtF DM ++=)( FK
(Differential Equation))(tFuKu‘Du“M =⋅+⋅⋅ +
F D= ⋅D v D= ⋅ u‘F M= ⋅M a (Newton Axiome)M= ⋅ u‘‘
u, u‘, u‘‘K
FM
FK
FDD
MF(t)
A short course in dynamics:
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A short course in dynamics: Forced vibrations
Resonance: Small forces can causelarge vibrations!
0.0 0.2 0.4 0.6 0.8 1.00.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
4.5
5.0
5.5
6.0
0.8 0.6 0.4 0.2 0.0
0.1
12
48
d=0
A
Log. damping d
0.50.4
0.2
w/WW/w
Counter measures:
• Increase of Damping
• Distune Eigenfrequency = Κ /Μw
Damping δ = 0: ∞amplitudesDamping δ > 0: finite amplitudes
v πδ
=Amplification factor:
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Wind Effects on Structures
Wind effects on structures 33
1. Turbulence induced vibrations
2. Vortex induced vibrations
3. Self excited vibrations
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Turbulence induced vibrations
Wind effects on structures 35
Determination of the wind force from wind speed:
The wind pressure results from the wind speed as follows:
22 21 1( ) ( ( ))
2 1600 2total
totaluq t u u u tρ ρ= ⋅ ⋅ = = ⋅ ⋅ + ρ : density of the air 1,25 kg/m³
2 2 2 21 1( ) ( ) ( ( )) ( ) ( 2 ( ) ( ))2 2d dW t c f A u u t c f A u u u t u tρ ρ= ⋅ ⋅ ⋅ ⋅ + = ⋅ ⋅ ⋅ ⋅ + ⋅ ⋅ +
The wind force results in:
Products of small values are neglected. Splitting up the aerodynamicforce coefficient into a stationary and non stationary part, it follows:
20
0
( ) ( )( ) (1 2 )2
dd
d
u t c fW t u c Au c
ρ= ⋅ ⋅ ⋅ ⋅ + ⋅ ⋅
20
0
2 ( )( ) ( ) ( ))2
dd
d
W c fW t W W t u c A u tu c
ρ ⋅′= + = ⋅ ⋅ ⋅ + ⋅ ⋅
Wind effects on structures 36
0
( )dl
d
c fRc
= is called aerodynamic admittance function
The function is determined via measurements. In Eurocode 1, 2.4 (Wind):
( )1 1 2122lR e η
η η−= − −
1lR =
( )4,6 1f h
L zi effη
⋅ ⋅=
It describes the effect, that small gust „balls“ belong to higher frequencies, they can only cover a small area of the structure, the overall wind force isreduced.
0.01 0.1 1 10 1000.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0Rl
η
0η >
0η =
with:
Determination of the wind force from wind speed:
Wind effects on structures 37
Measurements of wind turbulence and system response
Response Wind Action
Acceleration Wind Speed Direction
TemperatureLeg Strains
Transverse
Rope Force
Control PC (Modem)
On-Site Computer
Deflection
CCD-Camera
Rope Force
US-A
60 m
132 m
216 m
312 m
344 m
Biggest wind measurement and system responseequipment of the world
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Stayed cantilever with sensors
AnemometerYoung-Monitor
Measurements of wind turbulence and system response
Wind effects on structures 39
Covering
Pressure Sensor
Stayed cantilever
Young-Wind Monitor
Shaft cross section
Elevator
Edge Leg
Stay
Cable Way
Covering
Inner Part Supported
on Load-Cells
Pressure sensor
60 m
4,0 m
100 m
132 m
Covering
Pressure Sensor
Stayed cantilever
Young-Wind Monitor
Shaft cross section
Elevator
Edge Leg
Stay
Cable Way
Covering
Inner Part Supported
on Load-Cells
Pressure sensor
60 m
4,0 m
100 m
132 m
Turbulence induced vibrations (Measurements):
Actual enlargement
Measurements of aerodynamic admittance
Wind effects on structures 40
Exponential law exponent
W(48m) / (m/s)
0.3
0.4
0.5
0.2
0.1
10 11 12 13 14 15 16 17 18
(1990 - 1996, Sektor 4)1247 evaluated measurements
0.5
0.1
0.2
0.3
0.4
0 2 4 6 8 10 12 0 2 4 6 8 10 12 0 2 4 6 8 10 12
"
"
Probab.dens.Class width
@ (m/s)
a db
858values
295values
66Values
19 20 21 22
21Values
0.5
0.1
0.2
0.3
0.4
0.5
0.1
0.2
0.3
0.4
0.5
0.1
0.2
0.3
0.4
0 2 4 6 8 10 12
c
302520151050
350
300
250
200
150
100
50
0
Height [m]
W [m/s]
measured values1993, Sektor 5
exponential law
α⎟⎠
⎞⎜⎝
⎛⋅=
BB zzzWzW )()(
18.0119:24
09.1221:17
" = 0,34 " = 0,19
0.35
0.30
0.25
0.20
0.1510 15 20 25
"
W(48m) [m/s]
Sektor 4: "(W48) = 0,40 ! 0,01 @ W48 [m/s] for W48 # 19 m/s
0,21 for W48 > 19 m/s
Sektor 5: "(W48) = 0,303 ! 0,007 @ W48 [m/s] for W48 # 19 m/s
0,17 for W48 > 19 m/s
Sector 4 (SW)
Sector 5 (W)
Mean values of "
Turbulence induced vibrations (Measurements):
Wind effects on structures 41
Turbulence induced vibrations:
Treated as Random Vibrations:
f[Hz]
S w
0.1 1.0
1.00.1 f[Hz]
f[Hz]0.1 1.0
H 2
S A
0.01
0.01
0.01
LeistungsspektrumBelastung
SystemUbertragungsfunktion
2
AntwortLeistungsspektrum
1- 22
+2
21
k
2
H =
PSD and the mechanical admittancefunction are multiplied. Result is the PSD of the structureresponse:
2 1 ( )2resp AS dσ ω ωπ
+∞
−∞
= ⋅∫
*
2
( ) ( ) ( ) ( )
( ) ( )
A w
w
S H H S
H S
ω ω ω ω
ω ω
⎡ ⎤= ⋅ ⋅⎣ ⎦
= ⋅
SDOF-Structure:
The variance results from the integral over the response PSD:
rms: 2.resp respσ σ=
Total response: .resp resp respA A g σ= + ⋅
Wind effects on structures 42
.resp resp respA A g σ= + ⋅
g: peak factor, belongs to the choosen fractile of the normal distributedrandom response:
1 2 3A123
f(A)
12
AA+ A+ A+A-A-A-
probability of excedence:
( ) ( )u
F A f A dA∞
= ∫2A A σ= + ⋅
g* σ F(A)1∗σ 0,1586552∗σ 0,0227503∗σ 0,0013504∗σ 0,000032
In table the probabilities of excedence are given for different peak factors g:
A usual peak factors is 3,5
With this the responsecan be determined!
.resp resp respA A g σ= + ⋅
Turbulence induced vibrations:
Wind effects on structures 43
Vortex induced vibrations
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Vortex induced vibrations:
Circular (and angular) cross sections produce vortexes, which leaves orseparates the cross section periodically:
][HzduSf ⋅=vortex frequency:
Strouhal No. : 0 2S ,≈
critical velocity: 5 [ / ]cr iu f d m s= ⋅ ⋅
Wind effects on structures 46
1.0
0.5
010 10 10 104 5 6 7
clat
Re
sub critical
trans criticalsuper critical
tfcqp latlat π2d ⋅ sin⋅⋅=harmonic lateral force:
with:
clat according to the diagram:
Vortex induced vibrations of a chimney: a resonance problem!
Maximum value: sin2πft = 1: cqp latlat ⋅d⋅=
with: νcritud⋅=Re
ucr: critical windspeed: 5cr iu f d= ⋅ ⋅[ / ]1600
cruq m s=
v πδ
=
Resonance amplificationfactor:
reson latp q c dπδ
= ⋅ ⋅ ⋅approxim. resonance response:
Wind effects on structures 47
Simple approximation: (chimney from the movie)
f = 0,6 Hz, d = 6,0 m: δ = 0,01
vcrit = 5 * f * d = 5 * 0,6 * 6,0 = 18m/s
qcrit= 18² /1600 = 0,20 kN/m²
Flow state ? Re = v * d / ν = 18*6 / 15*10-6 = 7,2*106 (trans critical)
From diagram: clat = 0,2
qlat = 0,20* 0,2* 6,0=0,24 kN/m
qres= π / δ*0,24 = π / 0,01*0,24 = 75,4 kN/m
Vortex induced vibrations of a chimney: a resonance problem!
Wind effects on structures 48
Vortex induced vibrations of a chimney: a resonance problem!
If a structure tends to vibrate, counter measures are muchimportant, because the vibrationscan cause severe fatigueproblems!
Wind effects on structures 49
• Additional dampers in resonance
Water ropes granulate
friction dash pots visko damper
• Distuning (difficult, stays) M
Vortex induced vibrations: Counter measures
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• Disturbing of periodic vortexes
submarine periskope (2nd World war)
Vortex induced vibrations: Counter measures
Wind effects on structures 51
Self Exciting Vibrations
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Self excited vibrations: Galloping
Power lines under ice conditions
Wind effects on structures 53
Stays of a guyed mast
Self excited vibrations: Galloping
Wind effects on structures 54
wA Ice
A
u,u’
Ice vanes on ropes:
Symmetrical flow: no lift forces!
)(u‘FLuKu‘Du“M =⋅+⋅⋅ +
uKu‘(Du“M = 0⋅+−⋅ + FL )
System damping is reduced by the flow forces, can become negative!
Increase of vibrations
• u’ increases a increases Lift A increases
• The profile “feels” the relative wind
• Relative wind produces lift forces in same direction as u.
• lateral movement: u, u’
Self excited vibrations: Galloping
Wind effects on structures 55
Self excited vibrations: Galloping
)(tFuKu‘Du“M =⋅+⋅⋅ +
Exciting force in the transverse direction:
2( ) ( )2 A yF t u d cρ α= ⋅ ⋅ ⋅
cy(α): aerodynamic coefficient from wind tunnel tests. W, A are measured:
50 10 15 20 25
0
-0.2
-0.4
-0.6
-0.8
0.2
W
cy
0 0 0 0 00
dL
L/d=1,0
( ) cos sinF t A Wα α= ⋅ + ⋅
u A IceW
F
A
α
αα
,y y
arctanA
yu
α =transverse speed
Wind effects on structures 58
50 10 15 20 25
0
-0.2
-0.4
-0.6
-0.8
0.2
W
cy
0 0 0 0 00
dL
L/d=1,0
Self excited vibrations: Galloping
instableinstable stable
unstable cross sections:
Tacoma!
Wind effects on structures 59
Self excited vibrations: Galloping, Flutter
Tacoma Suspension Bridge
Wind effects on structures 60
Regen–Wind induzierte Schwingungen
Wind effects on structures 61
Ermüdungsbrüche nach ca. 10 Monaten an der Elbebrücke Dömitz
Bruch an KerbstelleSituation
Regen–Wind induzierte Schwingungen
Wind effects on structures 62
Der Wind und die Schwerkraft bilden Rinnsale des ablaufendenRegens.
Diese stören die Umströmung desQuerschnittes Auftriebskräfte
Trägheitskräfte verschieben dieRinnsale Selbsterregung
Regen–Wind induzierte Schwingungen
Wind effects on structures 63
3 Freiheitsgradegeometrisch & physikalisch nichtlinearAdhäsion zwischen Rinnsal und Oberfläche erfaßt
Mechanisches Modell:
3 gekoppelte nichtlineare Differentialgleichungen 2. Ordnung
Mathematisches Modell:
Regen–Wind induzierte Schwingungen
Wind effects on structures 64
Rechenbeispiel:Tenpozan Brücke, Japan, 1988
Seilparameter:- Länge: 50m- Neigung: 60°- Anströmwinkel: 45°- Windgeschwindigkeit: 14m/s
Regen-Wind ind. Schwingungen:Messung:
Frequenz: 0,82 Hz (f1)
z-Amplitude: 0,55m
Rechnung:0,85 Hz (f1) 0,6m
-0 ,8 0
-0 ,6 0
-0 ,4 0
-0 ,2 0
0 ,0 0
0 ,2 0
0 ,4 0
0 ,6 0
0 ,8 0
2 0 0 2 2 0 2 4 0 2 6 0 2 8 0 3 0 0
Ze i t [s ]
Am
plitu
de [m
]
yz
Regen–Wind induzierte Schwingungen
Wind effects on structures 65
Thanks for your patience !!