Aerodynamics, stability and response Aerodynamics, stability and response of longof long--span bridges span bridges

in atmospheric turbulent flowin atmospheric turbulent flow

Le Thai HoaKyoto University

CONTENTSCONTENTS

1. Introduction1. Introduction2. Bridge aerodynamics2. Bridge aerodynamics3. Literature reviews on stability and response 3. Literature reviews on stability and response

analysesanalyses4. General formation of stability and response4. General formation of stability and response5. Analytical method for stability analysis 5. Analytical method for stability analysis 6. Analytical method for response prediction6. Analytical method for response prediction7. Numerical example and discussions7. Numerical example and discussions8. Conclusion8. Conclusion

Long-span bridges (suspension and cable-stayed bridges) are prone to

dynamic behaviors (due to traffic, earthquake and wind)

Effects of aerodynamic phenomena (due to wind):

INTRODUCTION

Computational methods for aeroelastic instability analysis and aerodynamic response prediction of long-span bridges are world-widely developed increasingly thanks to computer-aid numericalmethods and computational mechanics

Catastrophe (Instability) + Serviceability (Discomfort)

Wind-resistance Design and Analysis for Long-span BridgesPrevention and Mitigation

LONG-SPAN BRIDGES IN WORLD AND VIETNAM

Taco

ma

(USA

) 108

0

Tsin

gMa

(HK)

137

7

Gre

at B

elt (

DM

) 162

3

Seto

(Jap

an)

172

3

Akas

hi (

Japa

n) 1

991

Mes

sina

(Ita

ly)

3300

0

500

1000

1500

2000

2500

3000

3500

Span

leng

th (m

)

Suspension Bridges

Ore

sund

(DM

) 4

90

Mei

ko (J

apan

) 5

90

Yan

gpu

(Chi

na)

602

Nor

man

dy (F

ranc

e) 8

65

Tata

ra (J

apan

) 8

90

Ston

ecut

ter

101

8

Suto

ng(C

hina

)108

8

0

200

400

600

800

1000

1200

Spa

n le

ngth

(m)

Cable-stayed bridges

Binh

2

60m

Kien

27

0m

MyT

huan

35

0m

ThuT

hiem

405

m

BaiC

hay

435

m

Can

Tho

550

m0

100

200

300

400

500

600

Span

leng

th (m

)

Cable-stayed bridges in VietNam

Suspension and cable-stayed brides

RESEARCH TOOLS IN WIND ENGINEERINGRESEARCH TOOLS IN WIND ENGINEERING

Computational tools:Computational tools:Fluid and solid mechanicsFluid and solid mechanics

++

Experimental tools:Experimental tools:Wind Tunnel TestsWind Tunnel Tests

Simulation tools:Simulation tools:Computational Fluid Dynamics (CFD)Computational Fluid Dynamics (CFD)

++

WIND TUNNEL TESTSWIND TUNNEL TESTS

Some pictures of wind tunnel tests

BRIDGE AERODYNAMICS AND WINDBRIDGE AERODYNAMICS AND WIND--INDUCED INDUCED VIBRATIONSVIBRATIONS

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

Branches of bridge aerodynamics and wind-induced vibrations

Extreme vibration and failure images of Tacoma Narrow

Structural Catastrophe

Aeroelastic Instability

Flutter Instability

FAILURE OF TACOMA NARROW BRIDGE, USA 1940

Torsional modeAnsymmetric torsional modeNo heaving mode

Limited-amplitude Response Divergent-amplitude Response

ResponseAmplitude

Flutter and GallopingInstabilities

Buffeting Response

‘Lock-in’ Response

Karman-inducedResponse

ResonancePeak Value

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

FLUTTER INSTABILITYFLUTTER INSTABILITY

Flutter might be the most critical concern to bridge design at high

wind velocity causing to dynamic instability and structural catastrophe

Flutter is the divergent-amplitude self-controlled vibration

generated by the aerodynamic wind-structure interaction and

negative damping mechanism (Structural + Aerodynamic damping)

Bridge Flutter or classical Flutter are basically classified by

Type 1: Pure torsional Flutter Bluff sections: Truss, boxed…Type 2: Coupled heaving-torsional Flutter Streamlined section

The target of Flutter analysis and Flutter resistance design for long-span

bridges is to

Tracing the critical condition of Flutter occurrence

Determining the critical wind velocity of Flutter occurrence

LITERATURE REVIEW LITERATURE REVIEW IN STABILITY ANALYSIS (2)IN STABILITY ANALYSIS (2)

Empirical formulas: Bleich’s [1951], Selberg’s[1961], Kloppel’s [1967]

Modeling self-controlled aerodynamic forces:

Theodorsen’s circulation function (Potential Theory) [1935]

Scanlan’s flutter deviatives (Experiment) [1971]

2DOF Flutter problems:

Complex eigenvalue analysis: Scanlan [1976]

Step-by-step analysis: Matsumoto [1994]

nDOF Flutter problems:

Finite Differential Method (FDM) in Time Approximation:

Agar [1987]

Finite Element Method (FEM) in Modal Space:

Scanlan [1990], Pleif [1995], Jain [1996], Katsuchi [1998], Ge [2002]

BUFFETING RESPONSE BUFFETING RESPONSE

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

Wind Fluctuations Fluctuating Forces Buffeting Response

Nature as Random Stationary Process

Prediction of Response (Forces+ Displacement)

BUFFETING RESPONSE (2)BUFFETING RESPONSE (2)

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

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)].

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

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.

Reduced 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

ATMOSPHERIC TURBULENT FLOWATMOSPHERIC TURBULENT FLOW

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)

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 range

LITERATURE REVIEW LITERATURE REVIEW IN STABILITY ANALYSISIN STABILITY ANALYSIS

Analytical Methods

Empirical Formula

2DOF FlutterSolutions

nDOF FlutterSolutions

Selberg’s; Kloppel’s

ComplexEigenMethod

Step-by-Step Method

Simulation Method

Single-Mode Method

Multi-mode Method

CFD

Free Vibration Method

Flutter problems

Experiment Method

Two-Mode Method

Branches for flutter instability problems

Full-scale Bridges

Sectional modes

LITERATURE REVIEW IN BUFFETING ANALYSIS LITERATURE REVIEW IN BUFFETING ANALYSIS

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/ correctedbuffeting forces

Turbulence modeling(Power spectral density)

Spectral analysis method(Correction functions)

Quasi-steady/unsteadybuffeting model

Time-historyturbulence simulation

Time-history analysis

Linear analysis

Linear and Non-linear

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 ANALYSIS IN STEPWISE PROCEDURE OF BUFFETING ANALYSIS IN FREQUENCY DOMAINFREQUENCY DOMAIN

SRSS or CQCCombination

Background and Resonance Parts

GENERAL FORMULATION OF FLUTTER INSTABILITY

tPXKXCXM

Flutter motion equations in ordinary coordinates

XPXPPPtP sd 21

0** XKXCXM

;2* PKK 1

* PCC

Generalized coordinates and mass-matrix-based normalization X

0**

KCI

;][ ** KK T ][ **

CC T

te 0**2 KCIDet

iii i

n

iiiiiiiiiii

t tpqtqpe i

1

cos2sin2

Response in generalized coordinates

14

If any i < 0 exists then Divergence

Liapunov’s Theorem

NODE-LUMPED SELF-CONTROLLED AERODYNAMIC FORCES

XPXPPPtP sd 21

Self-controlled Forces = Elastic Aerodynamic Forces + Damping

aerodynamic Forces

Linear-lumped in bridge deck’s nodes

000000000000000000000000000000

41

*2

2*1

*2

*1

*2

*1

21 ABBA

BPPBHH

LUKBUP

000000000000000000000000000000000

41

*3

*3

*3

222 BA

PH

LBKUP

MULTIMODE FLUTTER ANALYSIS

*

0CII

A

*

00

KI

B

teY

teY

YBYA

BA

ZAZB

Z

ZZBA 1

ZZI

KC

0

**

0

**

I

KCD ZZD

0**

KCI

Generalized basic equation in the State Space

Where:

Standard form of Eigen Problem

SINGLE-MODE AND TWO-MODE FLUTTER ANALYSIS 1DOF motion equation associated with ith mode in modal space

)(2 2 tpiiiiiii

iTii

Tii PPtp 21)(

ipphhi jijijiGABGPGH

UBKUtp

][21)( *

22*

1*1

2 ijiGBABKU ][

21 *

322

nksmkrrmsnG )()(l ,,

m

1kk

02 iiiiii

jiGKAB

i

ii

)(2

1 *3

4

22

jαiαi*2

2pipji

*1hihji

*1

4

i

ii )GK(AB)GK(PG)K([H4Bρ

ωω

i

UBK i

i

Where: :Generalized force of ith mode

: Modal sums

1DOF motion equation in standard form

Critical condition: System damping ratio equal zero

TIMETIME--FREQUENCY DOMAIN TRANFORMATION AND FREQUENCY DOMAIN TRANFORMATION AND POWER SPECTRUMPOWER 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)]()([)( tXtXERX

BASIC FORMATION OF BUFFETING RESPONSE BASIC FORMATION OF BUFFETING RESPONSE

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 X

MULTIMODE ANALYTICAL METHOD OF BUFFETING MULTIMODE ANALYTICAL METHOD OF BUFFETING RESPONSERESPONSE

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

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 ForcesMechanical 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

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

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

BACKROUND AND RESONANCE COMPONENTS OF BACKROUND AND RESONANCE COMPONENTS OF SYSTEM RESPONSESYSTEM 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

)({

Structural parametersStructural parameters: : PPrere--stressed concrete cablestressed concrete cable--stayed bridge taken into considerationstayed bridge taken into considerationfor demonstration of the for demonstration of the computational procedurescomputational procedures

3/5

2*

501200)(

fnfunSu

3/5

2*

10136.3)(

fnufnSw

UBn

ni

i 22

21

1)(

NUMERICAL EXAMPLE NUMERICAL EXAMPLE

Mean wind velocity parameters:Deck elevation: z=20m

Turbulence model

Wind fluctuations modeled by the one-sided power spectral density

(PSD) functions using empirical formulas

Aerodynamic admittance approximated by Liepmann function

Coherence function proposed by Davenport (1962)

)exp(),(U

xcnxnCoh i

iu

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 9

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

H*1

H*2

H*3

-20

-15

-10

-5

0

5

10

15

20

0 1 2 3 4 5 6 7 8 9 10 11 12

Reduced Velocities

H*i

(i=

1,2

,3)

-0.5

0

0.5

1

1.5

2

2.5

3

3.5

0 1 2 3 4 5 6 7 8 9 10 11 12

Reduced Velocities

A*i

(i=

1,2,

3) A3*

A1*

A2*

H2*

H1*H3*

FFLUTTER DERIVATIVESLUTTER DERIVATIVES

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 )( ,

STATIC FORCE COEFFICIENTS AND FIRSTSTATIC FORCE COEFFICIENTS AND FIRST--ORDER ORDER DEVIATIVESDEVIATIVES

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)]

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

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)

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

Damping ratio-velocity diagram of 5 fundamental modes

10 20 30 40 50 60 70 80 90-0.2

0

0.2

0.4

0.6

0.8

1

1.2

Wind velocity (m/s)

Sys

tem

dam

ping

ratio

Mode 1 (Heaving)Mode 2 (Heaving)Mode 3 (Torsional)Mode 4 (Torsional)Mode 5 (Heaving)

Mode 1 Mode 2

Mode 5

Mode 3

Mode 4

64.5 88.5 64.5 88.5

Frequency-Velocity diagram of torsional modes

10 20 30 40 50 60 70 80 900.6

0.7

0.8

0.9

1

1.1

1.2

1.3

Wind velocity (m/s)

Freq

uenc

y (H

z)Mode 3 (Torsional)Mode 4 (Torsional)

Mode 3

Mode 4

Aerodynamic interaction

Aerodynamic interaction

U=

50m

/sU

=70m

/s

0 10 20 30 40 50 60 70 80 90 100-1

0

1 Mode 1

0 10 20 30 40 50 60 70 80 90 100-1

0

1

0 10 20 30 40 50 60 70 80 90 100-1

0

1 Mode 3

Mod

al A

mpl

itude

0 10 20 30 40 50 60 70 80 90 100-1

0

1 Mode 4

0 10 20 30 40 50 60 70 80 90 100-1

0

1 Mode 5

Time (s)

Mode 20 10 20 30 40 50 60 70 80 90 100

-1

0

1 Mode 1

0 10 20 30 40 50 60 70 80 90 100-1

0

1 Mode 2

0 10 20 30 40 50 60 70 80 90 100-2

0

2 Mode 3

Mod

al A

mpl

itude

0 10 20 30 40 50 60 70 80 90 100-1

0

1 Mode 4

0 10 20 30 40 50 60 70 80 90 100-1

0

1 Mode 5

Time (s)

(Divergence)

0 10 20 30 40 50 60 70 80 90 100-1

0

1Mode 1

0 10 20 30 40 50 60 70 80 90 100-1

0

1 Mode 2

0 10 20 30 40 50 60 70 80 90 100-5

0

5 Mode 3

Mod

al A

mpl

itude

0 10 20 30 40 50 60 70 80 90 100-1

0

1

0 10 20 30 40 50 60 70 80 90 100-1

0

1 Mode 5

Time (s)

Mode 4

0 10 20 30 40 50 60 70 80 90 100-1

0

1Mode 1

0 10 20 30 40 50 60 70 80 90 100-1

0

1 Mode 2

0 10 20 30 40 50 60 70 80 90 100-1

0

1x 10

5M

odal

Am

plitu

de

0 10 20 30 40 50 60 70 80 90 100-2

0

2 Mode 4 (Divergence)

0 10 20 30 40 50 60 70 80 90 100-1

0

1 Mode 5

Time (s)

Mode 3 (Divergence)

U= 65m

/sU

= 90m/s

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30-0.15

-0.1

-0.05

0

0.05

0.1

Deck nodes

Mod

al re

spon

se

Mode1 at 0m/sMode1 at 50m/sMode1 at 70m/sMode1 at 90m/sMode2 at 0m/sMode2 at 50m/sMode2 at 70m/sMode2 at 90m/s

Modes 1&2 - Decay

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30-0.025

-0.02

-0.015

-0.01

-0.005

0

0.005

0.01

0.015

Deck nodes

Mod

al re

spon

se

Mode3 - initialMode3 at 50m/sMode3 at 70m/sMode3 at 90m/sMode4 - initialMode4 at 50m/sMode4 at 70m/sMode4 at 90m/s

Modes 3&4 Divergence

Modal response of heaving modes and torsional modes

10-1

100

101

10-10

10-5

100

S h(n)

Node 5

10-1

100

101

10-10

10-5

100

S a(n)

Node 5

10-1

100

101

10-10

10-5

100

Frequency n(Hz)

Sh(n

)

Node 15

10-1

100

101

10-10

10-5

100

Frequency n(Hz)

Sa(n

)

Node 15

Power spectra of global responses in nodes 5 &15 at U=20m/s

0 10 20 30 40 50 600

0.05

0.1

0.15

0.2

0.25

0.3

Mean velocity (m/s)

RM

S o

f ver

tical

dis

p. (m

)

Mode 1Mode 2Mode 5Total response

0 10 20 30 40 50 600

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

Mean velocity (m/s)R

MS

of r

otat

ion

(deg

.)

Mode 3Mode 4Total response

RMS of vertical displacement (left) and rotation (right) at midpoint

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 300

0.05

0.1

0.15

0.2

0.25

Deck nodes

RM

S re

spon

se o

f ver

tical

dis

p. (m

)

Mode 1Mode 2Mode 5Total response

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 300

0.1

0.2

0.3

0.4

0.5

Deck nodes

RM

S re

spon

se o

f rot

atio

n (d

eg.)

Mode 3Mode 4Total response

RMS of vertical displacement and rotation on deck nodes

CONCLUSIONCONCLUSION

Further studies on buffeting response prediction will be focused on

1) Contribution of background and resonance components

2) Buffeting analysis method in time domain

Further studies on numerical analytical methods should be: 1) Non-linear geometry problem should be included for Flutter

time-domain analysis for ‘flexible’ long-span bridges

THANKS VERY MUCH FOR YOUR ATTENTIONTHANKS VERY MUCH FOR YOUR ATTENTION