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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
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