FLUTTER STABILITY ANALYSIS FOR CABLE-STAYED BRIDGES

LE THAI HOAKyoto University

CONTENT

1. INTRODUCTION

2. LITERATURE REVIEW ON AERODYNAMIC

PHENOMENA AND FLUTTER INSTABILITY

3. FUNDAMENTAL EQUATIONS OF FLUTTER

4. ANALYTICAL METHODS FOR FLUTTER PROBLEMS

5. NUMERICAL EXAMPLE AND DISCUSSION

6. CONCLUSION

1

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

2

Computational methods for aerodynamic instability analysis of long-

- span bridges are world-widely developed increasingly thanks to

computer-aid numerical methods 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

2

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

m

0

100

200

300

400

500

600

Span

leng

th (m

)

Cable-stayed bridges in VietNam

BRIDGE AERODYNAMICS

3

Bridge Aerodynamics

Limited-amplitude Responses

Divergent-amplitude Responses

Vortex-induced vibration

Buffeting vibration

Wake-induced vibrationRain-wind-induced Galloping instability

Flutter instability

Wake instability

Fig 1. Bridge aerodynamic branches

4

Fig 2. Response amplitude-velocity diagram

Limited Amplitude Divergent Amplitude

Reduced velocity (Ure)

ResponseAmplitude Flutter and Galloping

(Divergence)Buffeting Response(Random Vibration)

‘Lock-in’ Resonance

Karman-induced(Forced Vibration)

Peak

Fig 3. Extreme vibration and failure images of Tacoma Narrow 5

Structural Catastrophe

Aeroelastic Instability

Flutter Instability

FAILURE OF TACOMA NARROW BRIDGE

Torsional modeAnsymmetric torsional modeNo heaving mode

FLUTTER INSTABILITY(1)

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 occurrence6

Fig 4. Form of combined heaving and torsional modes 7

FLUTTER INSTABILITY(2)

Bridge Flutter experienced dominant contribution either of one mode : fundamental torsional mode (Type 1) or of coupling between 2 modes: fundamental torsional mode and fundamental heaving mode (Type 2).

Lift forceMovement

Positive work

With initial phase

Without initial phase

Positive work

Positive workPositive work

Positive work

Positive work Negative work

Negative work

LITERATURE REVIEW (1)

8

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

Fig 5. Branches for flutter instability problems

Full-scale Bridges

Sectional modes

LITERATURE REVIEW (2)

9

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]

OBJECTIVES

Up-to-date numerical analytical methods for flutter instability

analysis of bridges will be studied, some hints of analytical methods

will be pointed out

Some investigations and discussions thanks to numerical example

of a cable-stayed bridge

Wind resistance design and analysis, especially Flutter and Buffeting

analytical methods for long-span bridges, are main interest in

research and practical application of Vietnam

10

FUNDAMENTAL EQUATIONS OF FLUTTER

2DOF:

11

nDOF:

U

Z

OX

Zo

Xo

C

h

S

Kh, K

Ch,C

Zo

Xoh

XO

Z

S

C

zo

xo

Zs

Xs

M

Lh

MKCILhKhChm hhh

Where:Lh , M: Self-controlled unit lift force and moment

Where: {P(t)}: Self-exited force vector

ANALYTICAL MODELS FOR SELF-EXCITED AERODYNAMIC FORCES

Scanlan’s experimental model:

12

Theodorsen’s analytical model:

GhUGbUFUGUFbUhFUbLh

2)2()2)1((2 22

GhbUFbUGUbGUbFUbhbUFbM

)

2()

21( 2

222

Where: F(k), G(k): Real and imaginary parts of the Theodorsen’s circulation function C(k)=F(k)+iG(k), determined by Bessel functions of first and second kind.

BhKHKKHK

UBKKH

UhKKHBULh )()()()(

21 *

42*

32*

2*1

2

BhKHKKAK

UBKKA

UhKKAUBM )()()()(

21 *

42*

32*

2*1

22

Where: Flutter derivatives associated with self-controlled lift force and pitching moment; K: Reduced frequency kKUBK 2,/

)41(, ** iAH ii

13

ANALYTICAL METHODS FOR FLUTTER PROBLEMS

Modern computational procedure for nDOF system or bridge flutter

solution consist of:

Finite Element Method (FEM)

Multimode superposition technique

‘Critical condition’ tracing technique: based on Liapunov’s

Theorem on Dynamic Stability and Instability

Full-scale bridge Modal Space

FEM

2D&3Dbridge Modeling

Flutter Tracing

Generalized Coordinates

Self-controlled aerodynamicforces

Zero system damping ratio

Velocity increment Iteration

FLUTTER MOTION EQUATIONS IN MODAL SPACE

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

15

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

16

*

0CII

A

*

00

KI

B

teY

teY

YBYA

BA

ZAZB

Z

ZZBA 1

ZZ

I

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

17

Where: :Generalized force of ith mode

: Modal sums

1DOF motion equation in standard form

Critical condition: System damping ratio equal zero

Stuctural parameters: Pre-stressed concrete cable-stayed bridge taken into considerationfor demonstration of the flutter analytical methods. A symmetrical span arrangement: 40.4m+97m+40.5m=178m

NUMERICAL EXAMPLE

18Fig 7. Layout of cable-stayed bridge for numerical example

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)

Fig 8. Flutter derivatives (By quasi-steady formula Scanlan [1989], Pleif [1995])

19

A3*

A1*

A2*

H2*

H1*H3*

Mode 1f=0.60991

Mode 2f=0.80166

Fig 9. Fundamental modal shapes of 3D modeling (Mode 1 Mode 8)

Mode 1

f = 0.6099Hz

Mode 2

f= 0.801Hz

Mode 3

f= 0.8522Hz

20

Mode 4

f= 1.1949Hz

Mode 6

f =1.4495Hz

Mode 7

f =1.5819Hz

Mode 8

f = 1.6304Hz

21

Mode 5

f =1.2931Hz

Modal Shape 1 (1st Heaving Mode)

-0.12

-0.1

-0.08

-0.06

-0.04

-0.02

0

0.02

0.04

0.06

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

Mod

al A

mpl

itude

Modal Shape 2 ( 2nd Heaving Mode)

-0.15

-0.1

-0.05

0

0.05

0.1

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

Mod

al A

mpl

itude

Modal Shape 3(1st Torsional Mode)

-0.02

-0.015

-0.01

-0.005

0

0.005

0.01

0.015

0.02

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

Mod

al A

mpl

itude

Modal Shape 4 (2nd Torsional Mode)

-0.02

-0.015

-0.01

-0.005

0

0.005

0.01

0.015

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

odal

Am

plitu

de

Fig 10. Modal amplitude value of fundamental modal shapes22

Modal Shape 5 (3rd Heaving Mode)

-0.12

-0.1

-0.08

-0.06

-0.04

-0.02

0

0.02

0.04

0.06

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

Mod

al A

mpl

itude

Modal Shape 6(4th Heaving Mode)

-0.15

-0.1

-0.05

0

0.05

0.1

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

Mod

al A

mpl

itude

Modal Shape 7(3rd Torsional Mode)

-2.00E-02

-1.50E-02

-1.00E-02

-5.00E-03

0.00E+00

5.00E-03

1.00E-02

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

Mod

al A

mpl

itude

Modal Shape 8(4th Heaving Mode)

-0.08-0.06-0.04-0.02

00.020.040.060.08

0.10.12

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

Mod

al A

mpl

itude

23

2

Ghi chó :

S : Heaving Mode

V : D¹ng dao ®éng uèn

A : Ansymmetrical

T : Torsional Mode

P : Lateral Mode

dao ®éng(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

Tab 1. Characteristics of free vibration

Modes Eigenvalue Frequency Period Modal Features

24

(Hz) Ghihi Gpipi

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

2 0.801663 A-V-2 4.95E-01 7.43-09 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 1.06-9E-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

Tab 2. Modal integral sums Grmsn

Gii

Modal integral sums GrmsnFeatureFreq.Modes

25

nksmkr

N

kksr lG

nm)()( ,,

1

Fig 11. Damping ratio-velocity diagram of 5 fundamental modes39

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

Fig 12. Frequency-Velocity diagram of torsional modes26

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

Fig 13. Critical wind velocity resulted in some analytical methods

27

66

56

64

67

50525456586062646668

Criti

cal v

eloc

ity (

m/s

)

1

Selberg'sformulaComplex eigenmethodMode-by-modemethodTwo-modemethod

Fig 14. Modal amplitude-time diagram of 5 fundamental modes U

= 50

m/s

U=7

0m/s

28

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

1st Heaving mode

Fig 15. Nodes’ modal amplitude–velocity diagram

29

-0.12

-0.1

-0.08

-0.06

-0.04

-0.02

0

0.02

0.04

0.06

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

Deck nodes

Mod

al a

mpl

itude

Initial50m/s65m/s70m/s90m/s

-0.15

-0.1

-0.05

0

0.05

0.1

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

Deck nodes

Mod

al a

mpl

itude

Initial50m/s65m/s70m/s90m/s

2st Heaving mode

1st Torsional mode

30

-0.025

-0.02

-0.015

-0.01

-0.005

0

0.005

0.01

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

Deck nodes

Mod

al a

mpl

itude

Initial50m/s65m/s70m/s90m/s

-0.02

-0.015

-0.01

-0.005

0

0.005

0.01

0.015

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

Deck nodes

Mod

al a

mpl

itude

Initial50m/s65m/s70m/s90m/s

3nd Heaving mode

31

-0.12

-0.1

-0.08

-0.06

-0.04

-0.02

0

0.02

0.04

0.06

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

Deck nodes

Mod

al a

mpl

itude

(at 5

0m/s

)Initial

1second

2seconds

3seconds

5seconds

10seconds

-0.025

-0.02

-0.015

-0.01

-0.005

0

0.005

0.01

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

Deck nodes

Mod

al a

mpl

itude

(at 7

0m/s

)

Initial1second2seconds3seconds5seconds10seconds

Fig 16. Nodes’ modal amplitude–time diagram

1st Heaving mode

1st Torsional mode

CONCLUSION

32

Flutter problem: Iteration procedure with velocity increment + Critical condition tracing technique

Bridge Flutter usually experiences to be associated with i) Pure torsional mode or ii) Coupled heaving and torsional modes. Thus single-mode and two-mode analysis methods seems to exhibit enough accuracy

Further studies on numerical analytical methods should be: 1) Aerodynamic coupling between Flutter (Self-excited forces) and Buffeting (Random forces)2) Non-linear geometry problem should be included for Flutter time-domain analysis for ‘flexible’ long-span bridges

THANKS VERY MUCH FOR YOUR ATTENTION