BUFFETI NG RESPONSE PREDI CTI ON BUFFETI NG RESPONSE PREDI CTI ON - - PowerPoint PPT Presentation

BUFFETI NG RESPONSE PREDI CTI ON BUFFETI NG RESPONSE PREDI CTI ON FOR CABLE FOR CABLE-STAYED BRI DGES STAYED BRI DGES

LE THAI HOA LE THAI HOA

Kyoto University Kyoto University

CONTENTS CONTENTS

• 1. I ntroduction
• 1. I ntroduction
• 2. Literature review on buffeting response
• 2. Literature review on buffeting response

analysis for bridges analysis for bridges

• 3. Basic formations of buffeting response
• 3. Basic formations of buffeting response
• 4. Analytical method for buffeting response
• 4. Analytical method for buffeting response

prediction in frequency domain prediction in frequency domain

• 5. Numerical example and discussions
• 5. Numerical example and discussions
• 6. Conclusion
• 6. Conclusion

1

I NTRODUCTI ON I NTRODUCTI ON

2

Response prediction and evaluation of long-span bridges subjected to

random 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

WI ND WI ND-I NDUCED VI BRATI ONS I NDUCED VI BRATI ONS

3

Wind- induced Vibrations And Bridge Aero- dynamics Limited-amplitude Vibrations Divergent-amplitude Vibrations Vortex-induced vibration

Buffeting vibration

Wake-induced vibration Rain/wind-induced Galloping instability

Flutter instability

Wake instability Serviceable Discomfort Dynamic Fatique Structural Catastrophe

Limited-amplitude Response Divergent-amplitude Response Response Amplitude Flutter and Galloping Instabilities Buffeting Response ‘Lock-in’ Response Karman-induced Response Resonance Peak Value

4

RESPONSE AMPLI TUDE AND VELOCI TY RESPONSE AMPLI TUDE AND VELOCI TY

Reduced Velocity Random Forces in Turbulence Wind Vortex-induced Response Forced Forces Self-excited Forces in Smooth or Turbulence Wind

nB U U re 

Self-excited Forces Low and medium velocity range High velocity range

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

I NTERACTI ON OF WI ND I NTERACTI ON OF WI ND-I NDUCED VI BRATI ONS I NDUCED VI BRATI ONS

Interaction of wind-induced vibrations and their responses is potentially happened in some certain aerodynamic phenomena. In some cases, the

interaction of them suppresses their total responses, and in

contrast, enhances total responses in the others.

5

Reduced Velocity Axis Vortex-shedding Buffeting Random Vibration Flutter Self-excited Vibration Aerodynamic Interaction Individual Phenomena

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 WI ND VELOCI TY AND FLUCTUATI ONS MEAN WI ND VELOCI TY AND FLUCTUATI ONS

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

WI ND FORCES AND RESPONSE WI ND FORCES AND RESPONSE

) ( ) ( ) ( n F t F F t F

SE B QS total

  

Total wind forces acting on structures can be computed under

superposition principle of aerodynamic forces as follows

QS

F

: Quasi-steady aerodynamic forces (Static wind forces)

) (n FSE

: Self-controlled aerodynamic forces (Flutter)

) (t FB

: Unsteady (random) aerodynamic forces (Buffeting) Aerodynamic behaviors of structures can be estimated under static

equilibrium equations and aerodynamic motion equations

QS

F KX  ) ( ) ( t F n F KX X C X M

B SE

      

: Static Equilibrium : Dynamic Equilibrium Combination of self-controlled forces (Flutter) and unsteady fluctuating forces (Buffeting) is favorable under high-velocity range

7

BUFFETI NG BUFFETI NG

The buffeting is defined as the wind-induced vibration in wind turbulence that generated by unsteady fluctuating forces as origin of the random

• nes 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

• f wind-induced vibrations for long-span bridges

8

Wind Fluctuations Fluctuating Forces Buffeting Response

Nature as Random Stationary Process Prediction of Response (Forces+ Displacement)

LI TERATURE REVI EW I N BUFFETI NG ANALYSI S (1) LI TERATURE REVI EW I N BUFFETI NG ANALYSI S (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-history turbulence simulation Time-history analysis

9

Linear analysis Linear and Non-linear

LI TERATURE REVI EW I N BUFFETI NG ANALYSI S (2) LI TERATURE REVI EW I N BUFFETI NG ANALYSI S (2)

H.W.Liepmann (1952): Early works on computational buffeting

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

Alan Davenport (1962): Aerodynamic response of suspension bridge

subjected to random buffeting loads in turbulent wind proposed by

• Davenport. Also cored in spectral analysis and statistical computation, but

associated with modal analysis. Numerical example applied for the First Severn Crossing suspension bridge (UK).

H.P.A.H I win (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

EXI STI NG ASSUMPTI ONS I N BUFFETI NG ANALYSI S EXI STI NG ASSUMPTI ONS I N BUFFETI NG ANALYSI S

(1) Gaussian stationary processes of wind fluctuations Wind 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 wind

fluctuation acting on this strip that can be representative for whole structure

(4) Correction functions and transfer function Some 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

TI ME TI ME-FREQUENCY DOMAI N TRANFORMATI ON FREQUENCY DOMAI N TRANFORMATI ON AND POWER SPECTRUM AND 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





  ) exp( ) ( ) ( dt t j t X X  





 ) exp( ) ( 1 ) (     d j X t X

Power spectrum (PSD) of physical quantities known as Fourier

Transform of correlation of such quantities





  ) exp( ) ( ) (     d j R S

X X

)] ( ) ( [ ) (     t X t X E RX

11

BASI C FORMATI ONS OF BUFFETI NG BASI C FORMATI ONS OF BUFFETI NG RESPONSE ANALYSI S RESPONSE ANALYSI S

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

) (t F KX X C X M

B

     

Fourier Transform

) ( ) ( ] [

2

   

B

F X K C j M     ) ( ) ( ) (   

B

F H X 

1 2

] [ ) (



    K C j M H   

H(): Complex frequency response matrix

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

FB(t): Buffeting forces

)] ( ) ( [ ) ( t X t X E RX  ) ( | ) ( | ) (

2

  

b X

S H S 

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

)] ( ) ( [ ) ( t F t F E R

B B F



SX(), SB(): Spectrum of response and buffeting forces

Mean square of response







2

) (    d S X

12

MULTI MODE ANALYTI CAL METHOD OF MULTI MODE ANALYTI CAL METHOD OF BRI DGES I N FREQUENCY DOMAI N BRI DGES I N FREQUENCY DOMAI N

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 i i

t B x h t x h  ; ) ( ) ( ) , (





i i i

t B x p t x p 





i i i

t x t x ) ( ) ( ) , (   

1DOF motion equation in generalized ith modal coordinate:

i b i i i i i i i

Q I

, 2

1 2            



  

L i b i b i b i b

dx x t M B x p t D B x h t L Q

,

)] ( ) ( ) ( ) ( ) ( ) ( [ 

Qb,i: Generalized force of ith mode Lb, Db, Mb: Fluctuating lift, drag and moment per unit deck length

13

RELATI ON SPECTRA OF RESPONSE AND FORCES RELATI ON SPECTRA OF RESPONSE AND FORCES AND BUFFETI NG FORCE MODEL AND BUFFETI NG FORCE MODEL

] ) ( ) ( 2 [ 2 1 ) (

' 2

U t w C U t u C B U t L

L L b

   ] ) ( ) ( 2 [ 2 1 ) (

' 2

U t w C U t u C B U t D

D D b

  

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

) ( | ) ( | ) (

, 2 ,

n S n H n S

k b k k





1 2 2 2 2 2 2 2 2

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



  

k k k k k

n n n n I n H 

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 [ 2 1 ) (

' 2 2

U t w C U t u C B U t M

M M b

  

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

Spectrum of Forces

14

Mechanical Admittance

SPECTRUM OF BUFFETI NG FORCES (1) SPECTRUM OF BUFFETI NG FORCES (1)

Spectrum of unit (point-like )buffeting forces can be computed

as such form

)] ( ) ( ) ( ) ( 4 [ ) 2 1 ( ) (

' 2

       

w Lw L u Lu L L

S C S C UBl S   )] ( ) ( ) ( ) ( 4 [ ) 2 1 ( ) (

' 2

       

w Dw D u Du D D

S C S C UBl S   )] ( ) ( ) ( ) ( 4 [ ) 2 1 ( ) (

' 2 2

       

w Mw M u Mu M M

S C S C l UB S  

Spectra of fluctuations Aerodynamic Admittance

Spectrum of spanwise buffeting forces can be computed as follows

)] ( | ) ( | | ) ( | ) ( | ) ( | | ) ( | 4 [ ) 2 1 ( ) (

2 2 2 ' 2 2 2 2 2 ,

n S n n J C n S n n J C UB n S

ww Lw Lw L uu Lu Lu L i L

     dxdx x h x h n x Coh n x x J n J n J

L B i A i L h B A L h Lw h Lu

 

   

2 2 2

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

2 2 2

| ) ( | | ) ( | | ) ( |

h L h Lw h Lu

n n n     

Joint acceptance function

Approximations

15

SPECTRUM OF BUFFETI NG FORCES (2) SPECTRUM OF BUFFETI NG FORCES (2)

Spectrum of spanwise buffeting forces can be expressed

2 2 2 2 2 2 2 1 ,

| ) ( | | ) ( | )] ( ) ( 4 [ ) (

h L h L h w h u i L

n n J n S U L n S U L n S   

2 2 2 2 2 2 2 1 ,

| ) ( | | ) ( | )] ( ) ( 4 [ ) (

p D p D p w p u i D

n n J n S U D n S U D n S   

2 2 2 2 2 2 2 1 ,

| ) ( | | ) ( | )] ( ) ( 4 [ ) (

   

 n n J n S U M n S U M n S

M M w u i M

 

2 2 1

2 1 B C U L

L

 

2 ' 2 2

2 1 B C U L

L

 

2 2 1

2 1 B C U D

D

 

2 ' 2 2

2 1 B C U D

D

 

2 2 1

2 1 B C U M

M

 

2 ' 2 2

2 1 B C U M

D

 

16

SPECTRUM OF RESPONSE SPECTRUM OF RESPONSE

Generalized response of ith mode and total generalized response

in three coordinates (response combination by SRSS principle)







, , 2 , ,

) ( dn n S

i F i F  



F= L, D or M

System response

) (

1 , , 2 2 ,







N i i F F

SQRT

 

  } )] ( ) ( [ {

1 , , 2 2 2 ,







N i k i i F k T i r F X

x r x r SQRT



        r p

• r

h r B

r

1

r= h, p or 

17

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

Background and resonance components of generalized response

• f ith mode

  

   

    

, 1 2 2 2 2 2 2 2 , 2 , 2

) ( ]} 4 ) 1 [( { ) ( | ) ( | ) ( dn n S n n n n I dn n S n H dn n S

i b i i i i i b i i i

 



2 2 2 , Ri Bi i

    







, 2 2 ,

) ( 1 dn n S I

i b i i B

 ) ( 4

, 2 2 , i i b i i i i R

n S I n    

and

Background and resonance components of total response

} 1 ) ( {

1 , 2 2 2 2

 

 



m

N i i b i k i i B

dn S I x r  

Background Resonance







m

N i i i b i i i k i i R

n S I n x r

1 , 2 2 2 2

)} ( 4 ) ( {    

18

Spectrum of Wind Fluctuations Spectrum of Point- Buffeting Forces Spectrum of Spanwise Buffeting Forces Spectrum of ith Mode Response Response Estimate

• f ith Mode

Aerodynamic Admittance Joint Acceptance Function Mechanical Admittance Power Spectral Density (PSD) Multimode Response Inverse Fourier Transform

STEPWI SE PROCEDURE OF BUFFETI NG STEPWI SE PROCEDURE OF BUFFETI NG ANALYSI S I N FREQUENCY DOMAI N ANALYSI S I N FREQUENCY DOMAI N

SRSS or CQC Combination

Background and Resonance Parts

19

Structural parameters Structural parameters:

: Pre re-stressed concrete cable stressed concrete cable-stayed bridge taken into consideration stayed bridge taken into consideration for demonstration of the flutter analytical methods for demonstration of the flutter analytical methods Layout of cable-stayed bridge for numerical example

NUMERI CAL EXAMPLE NUMERI CAL EXAMPLE

Mean wind velocity parameters:

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

20

FREE VI BRATI ON ANALYSI S (1) FREE VI BRATI ON ANALYSI S (1)

Mode 1 Mode 2 Mode 3 Mode 4 Mode 5 Mode 6 Mode 7 Mode 8 Mode 9

21

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 VI BRATI ON ANALYSI S (2) FREE VI BRATI ON ANALYSI S (2)

22

MODAL SUM COEFFI CI ENTS MODAL SUM COEFFI CI ENTS

Mode Frequency Modal Modal integral sums Grmsn shape (Hz) Character Ghihi Gpipi Gii 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 k n k s m k r k rmsn

L G

1 , ,

) ( ) (  

r, s: Modal index; m, n: Combination index r, s= h, p or : Heaving, lateral or rotational m, n= i or j : rth modal value at node k

m k r )

(

,

 23

STATI C FORCE COEFFI CI ENTS AND FI RST STATI C FORCE COEFFI CI ENTS AND FI RST- ORDER DEVI ATI VES ORDER DEVI ATI VES

CD

0.02 0.04 0.06 0.08 0.1

• 8
• 4

4 8 Attack angle (degree) Force coefficient

CL

• 0.1

0.1 0.2 0.3 0.4 0.5

• 8
• 4

4 8 Attack angle

CM

0.05 0.1 0.15 0.2 0.25 0.3 0.35

• 8
• 4

4 8 Attack angle (degree) Force coefficient

CD CL CM C’D C’L C’M 0.041 0.158 0.174 3.25 3.25 1.74 1.74

Static force coefficients above were determined by wind-tunnel

experiment [T.H.Le (2003)] 24

TURBULENCE WI ND MODEL TURBULENCE WI ND MODEL

10

• 3

10

• 2

10

• 1

10 10

1

100 200 300 400 500 600

PSD of horizontal wind fluctuation Freqency n(Hz) Su(n) m2.s/s2 Kaimal's spectrum U= 40m/s Z= 20m u*= 2.5m/s

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

 

3 / 5 2 *

50 1 200 ) ( f n fu n Su  

 

3 / 5 2 *

10 1 36 . 3 ) ( f n u f n Sw  

Horizontal fluctuation: Kaimail’s spectrum Vertical fluctuation: Panofsky’s spectrum

10

• 3

10

• 2

10

• 1

10 10

1

2 4 6 8 10 12

PSD of vertical wind fluctuation Freqency n(Hz) Sw(n) m2.s/s2

PSD

Kaimal's spectrum U= 40m/s Z= 20m u*= 2.5m/s

Kaimail’s PSD U= 40m/s;Z= 20m Panofsky’s PSD U= 40m/s;Z= 20m

25

AERODYNAMI C ADMI TTANCE AERODYNAMI C ADMI TTANCE

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

U B n n

i i 2 2

2 1 1 ) (    

ni: Modal frequency

10

• 2

10

• 1

10 10

1

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 Frequency Log(n) Aerodynamic admittance

B= 14.5m; U= 40m/s

26

COHERENCE FUNCTI ON COHERENCE FUNCTI ON

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

) exp( ) , ( U x cn x n Coh

i i u

   

C: Decay coefficient (8c16)

x: Spanwise separation

10

• 2

10

• 1

10 10

1

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Frequency Log(n) Coherence 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

JOI NT ACCEPTANCE FUNCTI ON JOI NT ACCEPTANCE FUNCTI ON







N k n k s m k r k rmsn

L G

1 , ,

) ( ) (  

Joint acceptance function can be computed by following formulas

dxdx x r x r U x cn n x x J

L B i A i h L h B A F

 

  

2

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

i ih

h h h L

G x U x cn n x J ) )( exp( | ) , ( |

2

    

Discretization

i i p

p p p D

G x U x cn n x J ) )( exp( | ) , ( |

2

    

i i

G x U x cn n x J M

   

) )( exp( | ) , ( |

2

    

i: The number of mode F= L, D or M r= h, p or  : Modal sum coefficients

m k r )

(

,



: Modal value Lk: Spanwise separation

28

MECHANI CAL ADMI TTANCE MECHANI CAL ADMI TTANCE

10

• 2

10

• 1

10 10

1

10

• 4

10

• 2

10 10

2

10

4

10

6

Frequency Log(n/ni) Amplitude Log(|H(n/ni)|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

 

1 2 2 2 2 2 2 2 2

)]} 4 ) 1 [( {



  

i i i i i

n n n n I n H 

I i: Generalized mass inertia

i a i s i , ,

    

i: Total damping ratio

(Structural s,i+ Aerodynamica,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

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 10 20 30 40 50 60 Mean wind velocity U(m/s) RMS of Rotation(Degree)

RMS of Rotation at Midspan

MODAL CONTRI BUTI ON ON RMS OF RESPONSE MODAL CONTRI BUTI ON ON RMS OF RESPONSE

0.1 0.2 0.3 0.4 0.5 0.6 0.7 10 20 30 40 50 60 Mean wind velocity RMS of Vertical disp. (m)

RMS of Vertical Displacement at Midspan

Mode 3 Mode 4 Mode 1 Mode 2 Mode 5

31

RMS OF TOTAL RESPONSE (5 MODES COMBI NED) RMS OF TOTAL RESPONSE (5 MODES COMBI NED)

RMS of total response (m)

0.1 0.2 0.3 0.4 0.5 0.6 0.7 10 20 30 40 50 60 Mean wind velocity U(m/s)

RMS of Total response (Degree)

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 BRI DGE RMS OF MODAL RESPONSE OF FULL BRI DGE

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 RMS of rotation (degree) 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 Deck nodes RMS of vertical disp. (m)

RMS of Vertical Displacement

• n Deck Nodes

RMS of Rotation

• n Deck Nodes

Mode 1 Mode 2 Mode 5 Mode 3 Mode 4

32

U= 40m/s U= 40m/s

33

CONCLUSI ON CONCLUSI ON

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 ATTENTI ON THANKS VERY MUCH FOR YOUR ATTENTI ON