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