BUFFETING RESPONSE PREDICTION BUFFETING RESPONSE PREDICTION FOR CABLEFOR CABLE--STAYED BRIDGESSTAYED BRIDGES

LE THAI HOALE THAI HOAKyoto UniversityKyoto University

CONTENTSCONTENTS

1. Introduction1. Introduction2. Literature review on buffeting response 2. Literature review on buffeting response

analysis for bridgesanalysis for bridges3. Basic formations of buffeting response3. Basic formations of buffeting response4. Analytical method for buffeting response 4. Analytical method for buffeting response

prediction in frequency domainprediction in frequency domain5. Numerical example and discussions5. Numerical example and discussions6. Conclusion6. Conclusion

1

INTRODUCTIONINTRODUCTION

2

Response prediction and evaluation of long-span bridges subjected torandom fluctuating loads (or buffeting forces) play very important role.

Effects of buffeting vibration and response on bridges such as:

(1) Large and unpredicted displacements affect psychologically

to passengers and drivers (Effect of serviceable discomfort)

(2) Fatique damage to structural components

Characteristics of buffeting vibration

(1) Buffeting random forces are as the nature of turbulence wind

(2) Occurrence at any velocity range (From low to high velocity).

Thus it is potential to affect to bridges

(3) Coupling with flutter forces as high sense in high velocity

range

WINDWIND--INDUCED VIBRATIONS INDUCED VIBRATIONS

3

Wind-induced

VibrationsAnd

Bridge Aero-

dynamics

Limited-amplitude Vibrations

Divergent-amplitude Vibrations

Vortex-induced vibration

Buffeting vibration

Wake-induced vibrationRain/wind-inducedGalloping instability

Flutter instability

Wake instability

Serviceable DiscomfortDynamic Fatique

Structural Catastrophe

Limited-amplitude Response Divergent-amplitude Response

ResponseAmplitude

Flutter and GallopingInstabilities

Buffeting Response

‘Lock-in’ Response

Karman-inducedResponse

ResonancePeak Value

4

RESPONSE AMPLITUDE AND VELOCITYRESPONSE AMPLITUDE AND VELOCITY

Reduced Velocity

Random Forcesin Turbulence Wind

Vortex-induced Response

Forced Forces

Self-excited Forcesin Smooth or

Turbulence Wind

nBUU re

Self-excitedForces

Low and medium velocity range High velocity range

Note: Classification of low, medium and high velocity ranges is relative together

INTERACTION OF WINDINTERACTION OF WIND--INDUCED VIBRATIONSINDUCED VIBRATIONS

Interaction of wind-induced vibrations and their responses is potentially happened in some certain aerodynamic phenomena. In some cases, theinteraction of them suppresses their total responses, and incontrast, enhances total responses in the others.

5Reduced Velocity Axis

Vortex-shedding

Buffeting Random Vibration

Flutter Self-excited Vibration

AerodynamicInteraction

IndividualPhenomena

Vortex-shedding and Buffeting (Physical Model)

Vortex and Low-speed Flutter (Physical Model)

Buffeting and Flutter (Mathematic Model)

Physical Model + Mathematic Model

Physical + Mathematic

Physical +Mathematic

Case study

Case study

MEAN WIND VELOCITY AND FLUCTUATIONSMEAN WIND VELOCITY AND FLUCTUATIONS

Mean and fluctuating velocities of turbulent wind Horizontal component: U(z,t) = U(z) + u(z,t) Vertical component: w(z,t)Longitudinal component: v(z,t)

Wind fluctuations are considered as the Normal-distributed stationary random processes (Zero mean value)

Atmospheric boundary layer (ABL)

6

Elevation (m)

ADB’s Depth d=300-500m

U(z)u(z,t)

Amplitude of Velocity

Time

U(z)

Mean

u(z,t): Fluctuation

z

Wind Fluctuations

Buffeting Forces

WIND FORCES AND RESPONSEWIND FORCES AND RESPONSE

)()()( nFtFFtF SEBQStotal

Total wind forces acting on structures can be computed under

superposition principle of aerodynamic forces as follows

QSF : Quasi-steady aerodynamic forces (Static wind forces)

)(nFSE : Self-controlled aerodynamic forces (Flutter)

)(tFB : Unsteady (random) aerodynamic forces (Buffeting)

Aerodynamic behaviors of structures can be estimated under static

equilibrium equations and aerodynamic motion equations

QSFKX

)()( tFnFKXXCXM BSE

: Static Equilibrium

: Dynamic Equilibrium

Combination of self-controlled forces (Flutter) and unsteady fluctuating

forces (Buffeting) is favorable under high-velocity range7

BUFFETING BUFFETING

The buffeting is defined as the wind-induced vibration in wind turbulence

that generated by unsteady fluctuating forces as origin of the random

ones due to wind fluctuations.

The purpose of buffeting analysis is that prediction or estimation of

total buffeting response of structures (Displacements, Sectional

forces: Shear force, bending and torsional moments)

Buffeting response prediction is major concern (Besides aeroelastic

instability known as flutter) in the wind resistance design and evaluation

of wind-induced vibrations for long-span bridges

8

Wind Fluctuations Fluctuating Forces Buffeting Response

Nature as Random Stationary Process

Prediction of Response (Forces+ Displacement)

LITERATURE REVIEW IN BUFFETING ANALYSIS (1) LITERATURE REVIEW IN BUFFETING ANALYSIS (1)

The buffeting response analysis can be treated by either:

1) Frequency-domain approach (Linear behavior) or

2) Time-domain approach (Both linear and nonlinear behaviors

Buffeting response prediction methods

Frequency Domain Methods

Time Domain Methods

Quasi-steady buffeting forces

Turbulence modeling(Power spectral density)

Spectral analysis method(Correction functions)

Quasi-steady buffeting model

Time-historyturbulence simulation

Time-history analysis 9

Linear analysis

Linear and Non-linear

LITERATURE REVIEW IN BUFFETING ANALYSIS (2)LITERATURE REVIEW IN BUFFETING ANALYSIS (2)H.W.Liepmann (1952): Early works on computational buffeting

prediction carried out for airplane wings. The spectral analysis appliedand statistical computation method introduced.

Alan Davenport (1962): Aerodynamic response of suspension bridgesubjected to random buffeting loads in turbulent wind proposed by Davenport. Also cored in spectral analysis and statistical computation, butassociated with modal analysis. Numerical example applied for the FirstSevern Crossing suspension bridge (UK).

H.P.A.H Iwin (1977): Numerical example for the Lions’ Gate suspension bridge (Canada) and comparision with 3Dphysical model inWT.

Recent developments on analytical models based on time-domain approach [Chen&Matsumoto(2000), Aas-Jakobsen et al.(2001)];aerodynamic coupled flutter and buffeting forces [Jain et al.(1995),Chen&Matsumoto(1998), Katsuchi et al.(1999)].

10

EXISTING ASSUMPTIONS IN BUFFETING ANALYSIS EXISTING ASSUMPTIONS IN BUFFETING ANALYSIS

(1) Gaussian stationary processes of wind fluctuationsWind fluctuations treated as Gaussian stationary random processes

(2) Quasi-steady assumption Unsteady buffeting loads modeled as quasi-steady forces by some simple approximations: i) Relative velocity and ii) Unsteady force coefficients

(3) Strip assumption Unsteady buffeting forces on any strip are produced by only the windfluctuation acting on this strip that can be representative for whole structure

(4) Correction functions and transfer functionSome correction functions (Aerodynamic Admittance, Coherence, Joint Acceptance Function) and transfer function (Mechanical Admittance) added for transform of statistical computation and SISO

(5) Modal uncoupling: Multimodal superposition from generalized response

is validated 10

TIMETIME--FREQUENCY DOMAIN TRANFORMATION FREQUENCY DOMAIN TRANFORMATION AND POWER SPECTRUMAND POWER SPECTRUM

Transformation processes

Time Domain Frequency Domain

Correlation Power Spectrum

Fourier Transform

Transform between time domain and frequency domain using Fourier Transform’s Weiner-Kintchine Pair

0

)exp()()( dttjtXX

0

)exp()(1)(

djXtX

Power spectrum (PSD) of physical quantities known as FourierTransform of correlation of such quantities

0

)exp()()( djRS XX)]()([)( tXtXERX11

BASIC FORMATIONS OF BUFFETING BASIC FORMATIONS OF BUFFETING RESPONSE ANALYSIS RESPONSE ANALYSIS

NDOF system motion equations subjected to sole fluctuating buffeting forces are expressed by means of Finite Element Method (FEM)

)(tFKXXCXM B

Fourier Transform )()(][ 2 BFXKCjM )()()( BFHX

12 ][)( KCjMH H(): Complex frequency response matrix

Fourier Transform of mean square of displacements and that of buffeting forces

FB(t): Buffeting forces

)]()([)0( tXtXERX

)(|)(|)( 2 bX SHS

X(), FB(): F.Ts of response and buffeting forces

)]()([)0( tFtFER BBF SX(), SB(): Spectrum of response and buffeting forces

Mean square of response

0

2 )( dS X12

MULTIMODE ANALYTICAL METHOD OF MULTIMODE ANALYTICAL METHOD OF BRIDGES IN FREQUENCY DOMAINBRIDGES IN FREQUENCY DOMAIN

Analytical method of buffeting response prediction in frequency domain for full-scale bridges based on some main computational techniques as

(1) Finite Element Method (FEM)(2) Modal analysis technique(3) Spectral analysis technique and statistical computation

For response of bridges, three displacement coordinates (vertical h, horizontal p and rotational ) can be expressed associated with modal shapes and values as follows:

;)()(),( i

ii tBxhtxh ;)()(),( i

ii tBxptxp i

ii txtx )()(),(

1DOF motion equation in generalized ith modal coordinate:

ibi

iiiiii QI ,

2 12

L

ibibibib dxxtMBxptDBxhtLQ0

, )]()()()()()([

Qb,i: Generalized force of ith mode

Lb, Db, Mb: Fluctuating lift, drag and moment per unit deck length 13

RELATION SPECTRA OF RESPONSE AND FORCESRELATION SPECTRA OF RESPONSE AND FORCESAND BUFFETING FORCE MODELAND BUFFETING FORCE MODEL

])()(2[21)( '

02

UtwC

UtuCBUtL LLb

])()(2[21)( '

02

UtwC

UtuCBUtD DDb

Transform 1DOF motion equation in generalized ith modal coordinateinto spectrum form :

)(|)(|)( ,2

, nSnHnS kbkk 1

2

222

2

222 ]}4)1[({|)(|

kk

kkk n

nnnInH

k=h; p;

Fluctuating buffeting forces (Lift, Drag and Moment) per unit deck length can be determined as follows due to the Quasi-steady Assumption

])()(2[21)( '

022

UtwC

UtuCBUtM MMb

u(t), w(t): Horizontal and vertical fluctuations

Spectrum of Forces

14

Mechanical Admittance

SPECTRUM OF BUFFETING FORCES (1)SPECTRUM OF BUFFETING FORCES (1)Spectrum of unit (point-like )buffeting forces can be computed

as such form)]()()()(4[)

21()( '

02 wLwLuLuLL SCSCUBlS

)]()()()(4[)21()( '

02 wDwDuDuDD SCSCUBlS

)]()()()(4[)21()( '

022 wMwMuMuMM SCSClUBS

Spectra of fluctuationsAerodynamic Admittance

Spectrum of spanwise buffeting forces can be computed as follows

)](|)(||)(|)(|)(||)(|4[)21()( 222'222

022

, nSnnJCnSnnJCUBnS wwLwLwLuuLuLuLiL

dxdxxhxhnxCohnxxJnJnJL

BiAi

L

hBALhLwhLu 00

222 )()(),(|),,(||)(||)(|

222 |)(||)(||)(| hLhLwhLu nnn

Joint acceptance function

Approximations

15

SPECTRUM OF BUFFETING FORCES (2)SPECTRUM OF BUFFETING FORCES (2)Spectrum of spanwise buffeting forces can be expressed

222

22

2

21

, |)(||)(|)]()(4[)( hLhLhwhuiL nnJnSULnS

ULnS

222

22

2

21

, |)(||)(|)]()(4[)( pDpDpwpuiD nnJnSUDnS

UDnS

222

22

2

21

, |)(||)(|)]()(4

[)( nnJnSUMnS

UMnS MMwuiM

20

21 2

1 BCUL L2'2

2 21 BCUL L

20

21 2

1 BCUD D 2'22 2

1 BCUD D

20

21 2

1 BCUM M2'2

2 21 BCUM D

16

SPECTRUM OF RESPONSE SPECTRUM OF RESPONSE

Generalized response of ith mode and total generalized responsein three coordinates (response combination by SRSS principle)

0

,,2

,, )( dnnS iFiF F=L, D or M

System response

)(1

,,22

,

N

iiFF SQRT

})]()([{1

,,222

,

N

ikiiFk

TirFX xrxrSQRT

r

porhrBr 1

r= h, p or

17

BACKROUND AND RESONANCE COMPONENTS BACKROUND AND RESONANCE COMPONENTS OF SYSTEM RESPONSEOF SYSTEM RESPONSE

Background and resonance components of generalized responseof ith mode

0

,1

2

222

2

22

,2

00,

2 )(]}4)1[({)(|)(|)( dnnSnn

nnIdnnSnHdnnS ib

ii

iiibiii

222, RiBii

0

,22

, )(1 dnnSI ib

iiB )(

4 ,22

, iibii

iiR nS

In

and

Background and resonance components of total response

}1)({1 0

,2222

mN

iib

ikiiB dnS

Ixr

Background Resonance

mN

iiib

ii

ikiiR nS

Inxr

1,2

222 )}(4

)({

18

Spectrum of Wind Fluctuations

Spectrum of Point-Buffeting Forces

Spectrum of Spanwise Buffeting Forces

Spectrum of ith Mode Response

Response Estimateof ith Mode

Aerodynamic Admittance

Joint Acceptance Function

Mechanical Admittance

Power Spectral Density (PSD)

Multimode Response

Inverse Fourier Transform

STEPWISE PROCEDURE OF BUFFETING STEPWISE PROCEDURE OF BUFFETING ANALYSIS IN FREQUENCY DOMAINANALYSIS IN FREQUENCY DOMAIN

SRSS or CQCCombination

Background and Resonance Parts

19

Structural parametersStructural parameters: : PPrere--stressed concrete cablestressed concrete cable--stayed bridge taken into considerationstayed bridge taken into considerationfor demonstration of the flutter analytical methodsfor demonstration of the flutter analytical methods

Layout of cable-stayed bridge for numerical example

NUMERICAL EXAMPLE NUMERICAL EXAMPLE

Mean wind velocity parameters:Mean velocity: Uz=40m/s and Deck elevation: z=20m

20

FREE VIBRATION ANALYSIS (1)FREE VIBRATION ANALYSIS (1)

Mode 1 Mode 2 Mode 3

Mode 4 Mode 5 Mode 6

Mode 7 Mode 8 Mode 921

Mode Eigenvalue Frequency Period Modal Character

2 (Hz) (s)

1 1.47E+01 0.609913 1.639579 S-V-1

2 2.54E+01 0.801663 1.247406 A-V-2

3 2.87E+01 0.852593 1.172893 S-T-1

4 5.64E+01 1.194920 0.836876 A-T-2

5 6.60E+01 1.293130 0.773318 S-V-3

6 8.30E+01 1.449593 0.689849 A-V-4

7 9.88E+01 1.581915 0.632145 S-T-P-3

8 1.05E+02 1.630459 0.613324 S-V-5

9 1.12E+02 1.683362 0.594049 A-V-6

10 1.36E+02 1.857597 0.53830 S-V-7

FREE VIBRATION ANALYSIS (2)FREE VIBRATION ANALYSIS (2)

22

MODAL SUM COEFFICIENTS MODAL SUM COEFFICIENTS

Mode Frequency Modal Modal integral sums Grmsn

shape (Hz) Character Ghihi Gpipi Gii

1 0.609913 S-V-1 5.20E-01 7.50E-11 0.00E+00

2 0.801663 A-V-2 4.95E-01 7.43E-09 1.35E-09

3 0.852593 S-T-1 3.79E-09 5.23E-05 1.14E-02

4 1.194920 A-T-2 1.78E-07 1.82E-05 1.07E-02

5 1.293130 S-V-3 5.07E-01 1.36E-07 23.62E-09

6 1.449593 A-V-4 4.99E-01 2.10E-09 9.42E-09

7 1.581915 S-T-P-3 2.67E-07 1.10E-03 1.10E-02

8 1.630459 S-V-5 5.03E-01 1.43E-07 1.27E-08

9 1.683362 A-V-6 1.64E-06 1.77E-04 1.09E-02

10 1.857597 S-V-7 4.16E-06 2.78E-03 1.11E-02

N

knksmkrkrmsn LG

1,, )()(

r, s: Modal index; m, n: Combination indexr, s=h, p or : Heaving, lateral or rotationalm, n=i or j

: rth modal value at node k mkr )( , 23

STATIC FORCE COEFFICIENTS AND FIRSTSTATIC FORCE COEFFICIENTS AND FIRST--ORDER DEVIATIVESORDER DEVIATIVES

CD

0

0.02

0.04

0.06

0.08

0.1

-8 -4 0 4 8

Attack angle (degree)

Forc

e co

effic

ient

CL

-0.1

0

0.1

0.2

0.3

0.4

0.5

-8 -4 0 4 8

Attack angle

CM

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

-8 -4 0 4 8

Attack angle (degree)

Forc

e co

effic

ient

CD CL CM C’D C’L C’M0.041 0.158 0.174 0 3.253.25 1.741.74

Static force coefficients above were determined by wind-tunnel experiment [T.H.Le (2003)]

24

TURBULENCE WIND MODELTURBULENCE WIND MODEL

10-3

10-2

10-1

100

101

0

100

200

300

400

500

600PSD of horizontal wind fluctuation

Freqency n(Hz)

Su(

n) m

2 .s/s

2

Kaimal's spectrumU= 40m/sZ= 20mu*= 2.5m/s

Wind fluctuations modeled by the one-sided power spectral density (PSD)functions using empirical formulas

3/5

2*

501200)(

fnfunSu

3/5

2*

10136.3)(

fnufnSw

Horizontal fluctuation: Kaimail’s spectrum

Vertical fluctuation: Panofsky’s spectrum

10-3

10-2

10-1

100

101

0

2

4

6

8

10

12PSD of vertical wind fluctuation

Freqency n(Hz)

Sw

(n) m

2 .s/s

2

PSD

Kaimal's spectrumU= 40m/sZ= 20mu*= 2.5m/s

Kaimail’s PSD

U=40m/s;Z=20m

Panofsky’s PSD

U=40m/s;Z=20m

25

AERODYNAMIC ADMITTANCEAERODYNAMIC ADMITTANCE

Approximated by well-known Liepmann’s function (1952)

UBn

ni

i 22

21

1)(

ni: Modal frequency

10-2 10-1 100 1010

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

Frequency Log(n)

Aer

odyn

amic

adm

ittan

ce

B=14.5m; U=40m/s

26

COHERENCE FUNCTIONCOHERENCE FUNCTIONProposed by Davenport (1962) with assumption that coherence of buffeting forces exhibits equal to that of ongoing velocity

)exp(),(U

xcnxnCoh i

iu

C: Decay coefficient (8c16)

x: Spanwise separation

10-2 10-1 100 1010

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Frequency Log(n)

Coh

eren

ce

y=0.1m

y=0.3

y=0.5

y=1

y=5

y=10

y=30

c=10; x=0.1-30m

x=6m

27

JOINT ACCEPTANCE FUNCTIONJOINT ACCEPTANCE FUNCTION

N

knksmkrkrmsn LG

1,, )()(

Joint acceptance function can be computed by following formulas

dxdxxrxrU

xcnnxxJ

L

BiAih

L

hBAF

00

2 )()()exp(|),,(|

iihhh

hL GxU

xcnnxJ ))(exp(|),(| 2

Discretization

ii ppp

pD GxU

xcnnxJ ))(exp(|),(| 2

iiGx

UxcnnxJ M

))(exp(|),(| 2

i: The number of modeF=L, D or Mr=h, p or

: Modal sum coefficients

mkr )( , : Modal value

Lk: Spanwise separation

28

MECHANICAL ADMITTANCEMECHANICAL ADMITTANCE

10-2 10-1 100 10110-4

10-2

100

102

104

106

Frequency Log(n/ni)

Am

plitu

de L

og(|H

(n/n

i)|2 )

Damping ratio 0.003

Damping ratio 0.01 Damping ratio 0.015 Damping ratio 0.02

Mechanical admittance is known as Transfer function of linear SISO

system in frequency domain in ith mode, determined as

12

222

2

222 )]}4)1[({

ii

iii n

nnnInH

Ii: Generalized mass inertia iaisi ,,

i: Total damping ratio

(Structural s,i+ Aerodynamica,i)

29

Modes s,i a,i i

Mode 1 0.005 0.00121 0.00621

Mode 2 0.005 0.000912 0.005912

Mode 3 0.005 0.0001 0.0051

Mode 4 0.005 0.0000716 0.005072

Mode 5 0.005 0.0000571 0.005057

Resonance

Background

30

00.10.20.30.40.50.60.70.8

10 20 30 40 50 60

Mean wind velocity U(m/s)

RM

S of

Rot

atio

n(D

egre

e)

RMS of Rotation

at Midspan

MODAL CONTRIBUTION ON RMS OF RESPONSE MODAL CONTRIBUTION ON RMS OF RESPONSE

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

10 20 30 40 50 60

Mean wind velocity

RM

S of

Ver

tical

dis

p. (m

)

RMS of Vertical

Displacement

at Midspan

Mode 3

Mode 4

Mode 1

Mode 2

Mode 5

31

RMS OF TOTAL RESPONSE (5 MODES COMBINED)RMS OF TOTAL RESPONSE (5 MODES COMBINED)

RMS of total response (m)

00.10.20.30.40.50.60.7

10 20 30 40 50 60

Mean wind velocity U(m/s)

RMS of Total response (Degree)

0

0.2

0.4

0.6

0.8

10 20 30 40 50 60

Mean wind velocity U(m/s)

RMS of Vertical

Displacement

at Midspan

RMS of Rotation

at Midspan

RMS OF MODAL RESPONSE OF FULL BRIDGERMS OF MODAL RESPONSE OF FULL BRIDGE

0

0.1

0.2

0.3

0.4

0.5

0.6

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29

Deck nodes

RM

S of

rota

tion

(deg

ree)

00.05

0.10.15

0.20.25

0.30.35

0.4

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29

Deck nodes

RM

S o

f ver

tical

dis

p. (m

)

RMS of Vertical Displacement

on Deck Nodes

RMS of Rotation

on Deck Nodes

Mode 1

Mode 2

Mode 5

Mode 3

Mode 4

32

U=40m/s

U=40m/s

33

CONCLUSIONCONCLUSION

Some further studies on buffeting vibration and response prediction

will be focused on

(1) Contribution of background and resonance components

to total structural response

(2) Buffeting analysis method in time domain

(Main research point)

THANKS VERY MUCH FOR YOUR ATTENTIONTHANKS VERY MUCH FOR YOUR ATTENTION