Wind Effects

Wind effects on structures 1

Wind Effects on StructuresProf. Dr.-Ing. Udo Peil

Technische Universität Carolo-Wilhelmina Braunschweig


Wind-engineered structures


Wind-engineered structures


Wind-engineered structures ?


Wind-engineered structures ?


Wind

1. Nature of the Wind


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:


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


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



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



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



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



Auto- and cross correlation functions:

Long time measurements are needed, otherwise the result is random !



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+

−

−

= ⋅ ⋅∫



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:



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 !



A short course in Structural Dynamics


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:


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:


Wind Effects on Structures


1. Turbulence induced vibrations

2. Vortex induced vibrations

3. Self excited vibrations


Turbulence induced vibrations


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 ⋅′= + = ⋅ ⋅ ⋅ + ⋅ ⋅


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:


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


Stayed cantilever with sensors

AnemometerYoung-Monitor

Measurements of wind turbulence and system response


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


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):


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= + ⋅


.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:


Vortex induced vibrations


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= ⋅ ⋅


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:


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




If a structure tends to vibrate, counter measures are muchimportant, because the vibrationscan cause severe fatigueproblems!


• Additional dampers in resonance

Water ropes granulate

friction dash pots visko damper

• Distuning (difficult, stays) M

Vortex induced vibrations: Counter measures


• Disturbing of periodic vortexes

submarine periskope (2nd World war)

Vortex induced vibrations: Counter measures


Self Exciting Vibrations


Self excited vibrations: Galloping

Power lines under ice conditions


Stays of a guyed mast



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’




)(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


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


instableinstable stable

unstable cross sections:

Tacoma!


Self excited vibrations: Galloping, Flutter

Tacoma Suspension Bridge


Regen–Wind induzierte Schwingungen


Ermüdungsbrüche nach ca. 10 Monaten an der Elbebrücke Dömitz

Bruch an KerbstelleSituation



Der Wind und die Schwerkraft bilden Rinnsale des ablaufendenRegens.

Diese stören die Umströmung desQuerschnittes Auftriebskräfte

Trägheitskräfte verschieben dieRinnsale Selbsterregung



3 Freiheitsgradegeometrisch & physikalisch nichtlinearAdhäsion zwischen Rinnsal und Oberfläche erfaßt

Mechanisches Modell:

3 gekoppelte nichtlineare Differentialgleichungen 2. Ordnung

Mathematisches Modell:



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



Thanks for your patience !!

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Wind Effects