FLUTTER STABILITY ANALYSIS FOR CABLE-STAYED BRIDGES

FLUTTER STABILITY ANALYSIS
FOR CABLE-STAYED BRIDGES
LE THAI HOA
Kyoto 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

INTRODUCTION
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):
Catastrophe (Instability) + Serviceability (Discomfort)
Prevention and Mitigation
Wind-resistance Design and Analysis for Long-span Bridges
Computational methods for aerodynamic instability analysis of long-
- span bridges are world-widely developed increasingly thanks to
computer-aid numerical methods and computational mechanics
2

LONG-SPAN BRIDGES IN WORLD AND VIETNAM
Span length (m)
3500
3000
2500
2000
1500
1000
500
Tacoma (USA) 1080
TsingMa (HK) 1377
Great Belt (DM) 1623
Seto (Japan) 1723
Akashi (Japan) 1991
Messina (Italy) 3300
Span length (m)
1200
1000
800
600
400
200
Oresund(DM) 490
Meiko (Japan) 590
Yangpu (China) 602
Normandy (France) 865
Tatara (Japan) 890
Stonecutter 1018
Sutong(China)1088
0
Suspension Bridges
0
Cable-stayed bridges
Sp a n le n g th (m )
600
500
400
300
200
100
Bin h 2 6 0 m
Kie n 2 7 0 m
M yTh u a n 3 5 0 m
Th u Th ie m 4 0 5 m
Ba iC h a y 4 3 5 m
C a n Th o 5 5 0 m
0
Cable-stayed bridges in VietNam
2

BRIDGE AERODYNAMICS
Vortex-induced vibration
Bridge
Aerodynamics
Limited-amplitude
Responses
Divergent-amplitude
Responses
Buffeting vibration
Wake-induced vibration
Rain-wind-induced
Galloping instability
Flutter instability
Wake instability
Fig 1. Bridge aerodynamic branches
3

Response
Amplitude
Karman-induced
(Forced Vibration)
‘Lock-in’
Resonance
Peak
Buffeting Response
(Random Vibration)
Flutter and Galloping
(Divergence)
Reduced velocity (U re )
Limited Amplitude
Divergent Amplitude
Fig 2. Response amplitude-velocity diagram
4

FAILURE OF TACOMA NARROW
BRIDGE
Ansymmetric torsional mode
No heaving mode
Torsional mode
Structural Catastrophe
Aeroelastic Instability
Flutter Instability
Fig 3. Extreme vibration and failure images of Tacoma Narrow 5

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 occurrence 6

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).
Positive work
Negative work
Positive work
Negative work
Without initial phase
Positive work
Positive work
Lift force
Movement
Positive work
With initial phase
Positive work
Fig 4. Form of combined heaving and torsional modes
7

LITERATURE REVIEW (1)
Flutter
problems
Analytical
Methods
Empirical Formula
2DOF FlutterSolutions
Selberg’s; Kloppel’s
ComplexEigenMethod
Sectional modes
Step-by-Step Method
nDOF FlutterSolutions
Full-scale Bridges
Experiment Method
Simulation Method
Single-Mode Method
Two-Mode Method
Multi-mode Method
Free Vibration Method
CFD
Fig 5. Branches for flutter instability problems
8

LITERATURE REVIEW (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]
9

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:
L h
Zo
Zo
U
C
S
M
Zs
C
zo
S
xo
h
Kh,
K
Z
O
X
Ch,
C
Xo
h
Z
O
Xs
X
Xo
m h
C
I
C
h
h
K
K
h
h L
M
Where:L h , M : Self-controlled unit lift force and
nDOF: moment
h
Where: {P(t)}: Self-exited force vector
11

ANALYTICAL MODELS FOR SELF-EXCITED AERODYNAMIC FORCES
Theodorsen’s analytical model:
2
L h
b2FUh
2 2
( bU (1 F)
GU )
(2U
F bUG)
2UGh
2
F b b
M bbUFh
2 1
2
2
( b U U G)
( UG
bU F)
bUGh
2 2
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.
Scanlan’s experimental model:
h B
h
L
1
2 *
*
2 *
2
h
BU
KH
( K)
KH ( K)
K H ( K)
K H
*
1
2
3
4
( K)
2 U U
B
h B
h
M
1
2 2 *
*
2 *
2
B U KA
( K)
KA ( K)
K A ( K)
K H
*
1
2
3
4
( K)
2 U U
B
* *
Where: H
i
, Ai
( i 1
4) Flutter derivatives associated with self-controlled
lift force and pitching moment; K: Reduced frequency K B
/ U , K 2k
12

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
FEM
2D&3Dbridge
Modeling
Modal Space
Generalized
Coordinates
Self-controlled
aerodynamicforces
Flutter Tracing
Zero system
damping ratio
Velocity increment
Iteration
13

FLUTTER MOTION EQUATIONS IN MODAL SPACE
Flutter motion equations in ordinary coordinates
M
X
C
X
K
X
Pt
P t P
d
Ps
P X
1
P2
X
*
*
M
X
C
X
K
X
0
*
*
K
K
P
; C
C
P
2
Generalized coordinates and mass-matrix-based normalization
X
*
*
I C K
0
*
T *
* T
K [ K ] ; C [ C
* ]
t
e
Det
i
2
* *
I C K 0
Response in generalized coordinates
i
n
i
i
t
e
p
q
t q
p
i sin
2
i 1
2 cos
t
i
i
i
i
i
1
Liapunov’s Theorem
If any i < 0 exists then Divergence
i
i
i
i
i
14

NODE-LUMPED SELF-CONTROLLED AERODYNAMIC FORCES
Self-controlled Forces = Elastic Aerodynamic Forces + Damping
aerodynamic Forces
P t P
P
P X
P
X
d s 1
2
Linear-lumped in bridge deck’s nodes
1
4
0
0
0
0
0
0
0
*
1
0
2
1
2
P U
B L
1
K
U
H
BA
*
1
0
0
P
0
*
0
BH
BP
B
*
2
0
0
0
0
0 0 0 0
0 0 0 0
2
A
*
2
*
0
0
0
0
0
0
1
P2
U
4
2
BK
2
0
0
0
L
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
H
P
BA
0
0
*
3
*
3
*
3
0
0
0
0
0
0
0
0
0
0
0
0
15

A Y
BY
MULTIMODE FLUTTER ANALYSIS
Generalized basic equation in the State Space
*
*
I C K
0
Where:
A
0
I
I
*
C
B
I
0
0
*
K
Y
t
e Y
e
t
A
BZ
B
AZ
1
A BZ
Z
Z
D Z
Z
C
*
I
K
0
*
Z
Z
D
C
*
K
I 0
*
Standard form of Eigen Problem
16

SINGLE-MODE AND TWO-MODE FLUTTER ANALYSIS
1DOF motion equation associated with ith mode in modal space
i
2
Where:
rmsn
i
i
i
i
2
i
i
p i
( t)
T
T
P
P
p ( t)
1
pi
( t)
U
2
i
1 i i 2
2
BK
[ H
U
m
l k
( r,
k
)
m
( s,
k
k1
G
)
*
1
G
n
h h
i
j
P
*
1
G
p p
i
j
i
B
2
A
*
2
G
: Modal sums
:Generalized force of ith mode
i
j
]
i
1 2 2
BK
*
3
U
2
[ BA G
]
i
j
i
1DOF motion equation in standard form
i
2
i i
i
i
i
0
2
2
i
i
4
B *
1 A
3
( K i ) G
i
j
B
2
i
K i
U
4
ω
i ρ B
i *
*
2 *
[H
1
( K i ) G
hihj
P1
( K i )G
pipj
B A
2
( K i
ω
)G
i
i 4
Critical condition: System damping ratio equal zero
α iα j
17

NUMERICAL EXAMPLE
Stuctural parameters:
Pre-stressed concrete cable-stayed bridge taken into consideration
for demonstration of the flutter analytical methods.
A symmetrical span arrangement: 40.4m+97m+40.5m=178m
Fig 7. Layout of cable-stayed bridge for numerical example
18

20
3.5
15
3
H *i(i= 1 ,2 ,3 )
10
5
0
-5
-10
-15
-20
0 1 2 3 4 5 6 7 8 9 10 11 12
H*1
H 3 *
H 1 *
H*2
H*3
H 2 *
Reduced Velocities
A *i (i= 1 , 2 , 3 )
2.5
2
1.5
1
0.5
0
-0.5
A 3 *
A 1 *
A 2 *
0 1 2 3 4 5 6 7 8 9 10 11 12
Reduced Velocities
Fig 8. Flutter derivatives (By quasi-steady
formula Scanlan [1989], Pleif [1995])
19

Mode 1
f=0.60991
Mode 2
f=0.80166
Mode 1
f = 0.6099Hz
Mode 2
f= 0.801Hz
Mode 3
f= 0.8522Hz
Mode 4
f= 1.1949Hz
Fig 9. Fundamental modal shapes of 3D modeling (Mode 1 Mode 8)
20

Mode 5
f =1.2931Hz
Mode 6
f =1.4495Hz
Mode 7
f =1.5819Hz
Mode 8
f = 1.6304Hz
21

Modal Shape 1
(1st Heaving Mode)
Modal Shape 2
( 2nd Heaving Mode)
Modal Amplitude
0.06
0.04
0.02
0
-0.02
-0.04
-0.06
-0.08
-0.1
-0.12
1 3 5 7 9 11 13 15 17 19 21 23 25 27 29
Modal Amplitude
0.1
0.05
0
-0.05
-0.1
-0.15
1 3 5 7 9 11 13 15 17 19 21 23 25 27 29
Modal Amplitude
0.02
0.015
0.01
0.005
0
-0.005
-0.01
-0.015
Modal Shape 3
(1st Torsional Mode)
1 3 5 7 9 11 13 15 17 19 21 23 25 27 29
Modal Amplitude
0.015
0.01
0.005
0
-0.005
-0.01
-0.015
Modal Shape 4
(2nd Torsional Mode)
1 3 5 7 9 11 13 15 17 19 21 23 25 27 29
-0.02
-0.02
Fig 10. Modal amplitude value of fundamental modal shapes 22

Modal Shape 5
(3rd Heaving Mode)
Modal Shape 6
(4th Heaving Mode)
0.06
0.1
Modal Amplitude
0.04
0.02
0
-0.02
-0.04
-0.06
-0.08
-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
Modal Amplitude
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
-0.12
-0.15
Modal Shape 7
(3rd Torsional Mode)
Modal Shape 8
(4th Heaving Mode)
1.00E-02
0.12
Modal Amplitude
5.00E-03
0.00E+00
-5.00E-03
-1.00E-02
-1.50E-02
-2.00E-02
1 3 5 7 9 11 13 15 17 19 21 23 25 27 29
Modal A m plitude
0.1
0.08
0.06
0.04
0.02
0
-0.02
-0.04
-0.06
-0.08
1 3 5 7 9 11 13 15 17 19 21 23 25 27 29
23

Tab 1. Characteristics of free vibration
Modes Eigenvalue Frequency(Hz)
Period (s)
Modal Features
2
dao ®éng
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
Ghi chó :
S : Heaving Mode
A : Ansymmetrical
V : D¹ng dao ®éng uèn
T : Torsional Mode
P : Lateral Mode
24

Tab 2. Modal integral sums G rmsn
Modes
Freq.
Feature
Modal integral sums G rmsn
(Hz) G hihi
G pipi
G ii
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
G
N
r m s n
lk
( r,
k
)
m
( s,
k
)
n
k1
25

System damping ratio
1.2
1
0.8
0.6
0.4
0.2
Mode 1 Mode 2
Mode 1 (Heaving)
Mode 2 (Heaving)
Mode 3 (Torsional)
Mode 4 (Torsional)
Mode 5 (Heaving)
Mode 5
0
Mode 3
Mode 4
-0.2
10 20 30 40 50 6064.5 70 8088.5
90
Wind velocity (m/s)
Fig 11. Damping ratio-velocity diagram of 5 fundamental modes
39

1.3
1.2
1.1
Aerodynamic interaction
Mode 3
Mode 3 (Torsional)
Mode 4 (Torsional)
Frequency (Hz)
1
0.9
Aerodynamic interaction
0.8
Mode 4
0.7
0.6
10 20 30 40 50 60 70 80 90
Wind velocity (m/s)
Fig 12. Frequency-Velocity diagram of torsional modes
26

Critical velocity (m/s)
68
66
64
62
60
58
56
54
52
50
66
56
1
64
67
Selberg's
formula
Complex eigen
method
Mode-by-mode
method
Two-mode
method
Fig 13. Critical wind velocity resulted in some analytical methods
27

Fig 14. Modal amplitude-time diagram of 5 fundamental modes
1
0
Mode 1
1
0
Mode 1
U= 50m/s
U=70m/s
Modal Amplitude
Modal Amplitude
-1
1
0 10 20 30 40 50
Mode 2
60 70 80 90 100
0
-1
1
0 10 20 30 40 50
Mode 3
60 70 80 90 100
0
-1
1
0 10 20 30 40 50
Mode 4
60 70 80 90 100
0
-1
1
0 10 20 30 40 50
Mode 5
60 70 80 90 100
0
-1
0 10 20 30 40 50 60 70 80 90 100
Time (s)
1
0
Mode 1
-1
1
0 10 20 30 40 50 60 70 80 90 100
Mode 2
0
-1
5
0 10 20 30 40 50 60 70 80 90 100
Mode 3
0
-5
1
0 10 20 30 40 50
Mode 4
60 70 80 90 100
0
Modal Amplitude
Modal Amplitude
-1
1
0 10 20 30 40 50 60 70 80 90 100
Mode 2
0
-1
2
0 10 20 30 40 50 60 70
Mode 3 (Divergence)
80 90 100
0
-2
1
0 10 20 30 40 50 60 70 80 90 100
Mode 4
0
-1
1
0 10 20 30 40 50
Mode 5
60 70 80 90 100
0
-1
0 10 20 30 40 50 60 70 80 90 100
Time (s)
1
0
Mode 1
-1
1
0 10 20 30 40 50 60 70 80 90 100
Mode 2
0
-1
1
0 x 105 10 20 30 40 50 60 70 80 90 100
Mode 3 (Divergence)
0
-1
2
0 10 20 30 40 50 60
Mode 4 (Divergence)
70 80 90 100
0
U= 65m/s U= 90m/s
-1
1
0 10 20 30 40 50
Mode 5
60 70 80 90 100
0
-2
1
0 10 20 30 40 50
Mode 5
60 70 80 90 100
0
-1
0 10 20 30 40 50 60 70 80 90 100
Time (s)
-1
0 10 20 30 40 50 60 70 80 90 100
Time (s)
28

Fig 15. Nodes’ modal amplitude–velocity diagram
0.06
0.04
0.02
1 st Heaving mode
Modal amplitude
0
-0.02
-0.04
-0.06
-0.08
-0.1
1 3 5 7 9 11 13 15 17 19 21 23 25 27 29
Initial
50m/s
65m/s
70m/s
90m/s
-0.12
Deck nodes
0.1
0.05
2 st Heaving mode
Modal amplitude
0
-0.05
-0.1
-0.15
1 3 5 7 9 11 13 15 17 19 21 23 25 27 29
Initial
50m/s
65m/s
70m/s
90m/s
Deck nodes
29

0.01
0.005
1 st Torsional mode
Modal amplitude
0
-0.005
-0.01
-0.015
-0.02
1 3 5 7 9 11 13 15 17 19 21 23 25 27 29
Initial
50m/s
65m/s
70m/s
90m/s
-0.025
Deck nodes
0.015
0.01
3 nd Heaving mode
Modal amplitude
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
Initial
50m/s
65m/s
70m/s
90m/s
Deck nodes
30

Fig 16. Nodes’ modal amplitude–time diagram
0.06
0.04
1 st Heaving mode
Modal amplitude (at 50m/s)
0.02
0
-0.02
-0.04
-0.06
-0.08
-0.1
-0.12
1 3 5 7 9 11 13 15 17 19 21 23 25 27 29
Initial
1second
2seco nds
3seco nds
5seco nds
10seco nds
Deck nodes
0.01
1 st Torsional mode
Modal amplitude (at 70m/s)
0.005
0
-0.005
-0.01
-0.015
-0.02
-0.025
1 3 5 7 9 11 13 15 17 19 21 23 25 27 29
Deck nodes
Initial
1seco nd
2seco nds
3seco nds
5seco nds
10seconds
31

CONCLUSION
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
32

THANKS VERY MUCH FOR YOUR ATTENTION