BUFFETING RESPONSE PREDICTION
FOR CABLECABLE-STAYED BRIDGES
LE THAI HOA
Kyoto University
CONTENTS
1. Introduction
2. Literature review on buffeting response
analysis for bridges
3. Basic formations of buffeting response
4. Analytical method for buffeting response
prediction in frequency domain
5. Numerical example and discussions
6. Conclusion
1
INTRODUCTION
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
2
WIND--INDUCED VIBRATIONS
WIND
Serviceable Discomfort
Dynamic Fatique
Windinduced
Vibrations
And
Bridge
Aerodynamics
Limited-amplitude
Vibrations
Vortex-induced vibration
Buffeting vibration
Wake-induced vibration
Rain/wind-induced
Galloping instability
Divergent-amplitude
Vibrations
Flutter instability
Wake instability
Structural Catastrophe
3
RESPONSE AMPLITUDE AND VELOCITY
Response
Amplitude
Vortex-induced
Response
Karman-induced ‘Lock-in’
Response
Response
Forced Forces
Self-excited
Forces
Buffeting
Response
Random Forces
in Turbulence Wind
Flutter and Galloping
Instabilities
Self-excited Forces
in Smooth or
Turbulence Wind
Resonance
Peak Value
Reduced Velocity U re 
Limited-amplitude Response
Low and medium velocity range
U
nB
Divergent-amplitude Response
High velocity range
Note: Classification of low, medium and high velocity ranges is relative together
4
INTERACTION OF WINDWIND-INDUCED VIBRATIONS
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.
Physical Model + Mathematic Model
Vortex-shedding
Individual
Phenomena
Physical + Mathematic
Buffeting Random Vibration
Case study
Flutter Self-excited Vibration
Physical +
Mathematic
Vortex-shedding and Buffeting (Physical Model)
Aerodynamic
Interaction
Vortex and Low-speed Flutter (Physical Model)
Case study
Buffeting and Flutter
(Mathematic Model)
Reduced Velocity Axis
5
MEAN WIND VELOCITY AND FLUCTUATIONS
Atmospheric boundary layer (ABL)
Elevation (m)
Amplitude of
Velocity
ADB’s Depth
d=300-500m
z
U(z)
u(z,t)
u(z,t): Fluctuation
U(z)
Mean
Time
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)
Buffeting Forces
Wind Fluctuations
Wind fluctuations are considered as the Normal-distributed
stationary random processes (Zero mean value)
6
WIND FORCES AND RESPONSE
Total wind forces acting on structures can be computed under
superposition principle of aerodynamic forces as follows
Ftotal (t )  FQS  FB (t )  FSE (n)
: Quasi-steady aerodynamic forces (Static wind forces)
FQS
FSE (n) : Self-controlled aerodynamic forces (Flutter)
FB (t ) : Unsteady (random) aerodynamic forces (Buffeting)
Aerodynamic behaviors of structures can be estimated under static
equilibrium equations and aerodynamic motion equations
: Static Equilibrium
KX  F
QS
MX  CX  KX  FSE ( n)  FB (t )
: Dynamic Equilibrium
Combination of self-controlled forces (Flutter) and unsteady fluctuating
forces (Buffeting) is favorable under high-velocity range
7
BUFFETING
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
Nature as Random
Stationary Process
Fluctuating Forces
Buffeting Response
Prediction of Response
(Forces+ Displacement)
8
LITERATURE REVIEW IN BUFFETING ANALYSIS (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
Linear analysis
Frequency Domain Methods
Quasi-steady buffeting
forces
Turbulence modeling
(Power spectral density)
Spectral analysis method
(Correction functions)
Buffeting response
prediction methods
Quasi-steady buffeting
model
Time Domain Methods
Time-history
turbulence simulation
Linear and Non-linear
Time-history analysis
9
LITERATURE REVIEW IN BUFFETING ANALYSIS (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 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)].
10
EXISTING ASSUMPTIONS IN BUFFETING ANALYSIS
(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
TIME--FREQUENCY DOMAIN TRANFORMATION
TIME
AND POWER SPECTRUM
Transformation processes
Time Domain
Frequency Domain
Fourier
Transform
Correlation
Power Spectrum
Transform between time domain and frequency domain using Fourier
Transform’s Weiner-Kintchine Pair

X ( )   X (t ) exp( jt ) dt
0

1
X (t )   X ( ) exp( j ) d
 0
Power spectrum (PSD) of physical quantities known as Fourier
Transform of correlation of such quantities

R X ( )  E[ X (t ) X (t   )]
S X ( )   R X ( ) exp(  j )d
0
11
BASIC FORMATIONS OF BUFFETING
RESPONSE ANALYSIS
NDOF system motion equations subjected to sole fluctuating buffeting
forces are expressed by means of Finite Element Method (FEM)
MX  CX  KX  FB (t )
FB(t): Buffeting forces
[  2 M  jC  K ] X ( )  FB ( )
Fourier Transform
X ( )  H ( ) FB ( )
H(): Complex frequency response matrix
H ( )  [  2 M  jC  K ] 1
X(), FB(): F.Ts of response and
buffeting forces
Fourier Transform of mean square of displacements and that of
buffeting forces
RF (0)  E[ FB (t ) FB (t )]
R X (0)  E[ X (t ) X (t )]
SX(), SB(): Spectrum of response
and buffeting forces
S X ( ) | H ( ) | 2 S b ( )
Mean square of response


2
  S X ( ) d
0
12
MULTIMODE ANALYTICAL METHOD OF
BRIDGES IN FREQUENCY DOMAIN
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:
h( x, t )   hi ( x) B i (t );p ( x, t )   p i ( x) B i (t ); ( x, t )   i ( x)i (t )
i
i
i
1DOF motion equation in generalized ith modal coordinate:
1
2



Qb,i: Generalized force of ith mode
 i  2 ii i  i  i  Qb ,i
Ii
L
Qb ,i   [ Lb (t )hi ( x ) B  Db (t ) pi ( x) B  M b (t ) i ( x )]dx
0
Lb, Db, Mb: Fluctuating lift, drag and moment per unit deck length
13
RELATION SPECTRA OF RESPONSE AND FORCES
AND BUFFETING FORCE MODEL
Transform 1DOF motion equation in generalized ith modal coordinate
into spectrum form :
2
2
2
S , k (n) | H ( nk ) | Sb , k ( n)
Mechanical Admittance
| H (n k ) | 2  {I k2 [(1 
Spectrum of Forces
n 2
2 n
1
)

4

]}
k
n k2
n k2
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
1
2u (t )
2
' w(t )
Lb (t )  U B[C L 0
 CL
]
2
U
U
1
2u (t )
w(t )
Db (t )  U 2 B[C D 0
 C D'
]
2
U
U
1
2u (t )
2 2
' w(t )
M b (t )  U B [C M 0
 CM
]
2
U
U
u(t), w(t): Horizontal and vertical fluctuations
14
SPECTRUM OF BUFFETING FORCES (1)
Spectrum of unit (point-like )buffeting forces can be computed
as such form
1
S L ( )  ( UBl ) 2 [4C L 0  Lu ( ) S u ( )  C L'  Lw ( ) S w ( )]
2
1
S D ( )  ( UBl ) 2 [4C D 0  Du ( ) Su ( )  CD'  Dw ( ) S w ( )]
12
S M ( )  ( UB 2 l ) 2 [4C M 0  Mu ( ) S u ( )  C M'  Mw ( ) S w ( )]
2
Spectra of fluctuations
Aerodynamic Admittance
Spectrum of spanwise buffeting forces can be computed as follows
1
S L ,i (n)  ( UB 2 ) 2 [ 4C L20 | J Lu (n) |2 |  Lu (n) |2 S uu ( n)  C L'2 | J Lw ( n) |2 |  Lw (n) |2 S ww (n)]
2
Joint acceptance function
L
Approximations | J Lu (n h ) | | J Lw (n h ) | | J L ( x A , x B , n h ) |  
2
2
2
0
L
 Coh(x, n)h
i
( x A )hi ( x B )dxdx
0
|  Lu (n h ) | 2 |  Lw (n h ) | 2 |  L (n h ) | 2
15
SPECTRUM OF BUFFETING FORCES (2)
Spectrum of spanwise buffeting forces can be expressed
4 L12
L22
S L ,i (n)  [ 2 S u (nh )  2 S w ( nh )]| J L (n h ) | 2 |  L (n h ) | 2
U
U
2
4 D1
D22
S D ,i (n)  [ 2 S u (n p )  2 S w (n p )]| J D ( n p ) | 2 |  D (n p ) | 2
U
U
2
4M 1
M 22
S M ,i (n)  [ 2 S u (n )  2 S w (n )] | J M (n ) | 2 |  M (n ) | 2
U
U
1
1
2 '
2
2
2
L


U
C
B
L1  U C L 0 B
2
L
2
2
1
1
2
2
D2  U 2 C D' B 2
D1  U C D 0 B
2
2
1
2 '
2
1
2
2
M


U
C
B
2
D
M 1  U C M 0 B
2
2
16
SPECTRUM OF RESPONSE
Generalized response of ith mode and total generalized response
in three coordinates (response combination by SRSS principle)

 2, F ,i   S  , F ,i (n)dn
F=L, D or M
0
N
 2, F  SQRT (  2  , F ,i )
i 1
System response
N
2
 X2 , F  SQRT {  r [riT ( x k )  2  , F ,i ri ( x k )]}
i 1
r= h, p or 
r 
B r  h or p
1 r 
17
BACKROUND AND RESONANCE COMPONENTS
OF SYSTEM RESPONSE
Background and resonance components of generalized response
of ith mode


2
i
   S  ,i (n) dn  
0
2
0
2
Bi
  ,i    
2
Ri

2
n2 2
2 n
| H (ni ) | S b ,i (n)dn   {I [(1  2 )  4 i 2 ]}1 S b ,i (n)dn
ni
ni
0
2
2
i

 B,i
2
1
 2  Sb ,i (n)dn
I i0
Background
and
 R ,i
2
ni

Sb, i (ni )
2
4 i I i
Resonance
Background and resonance components of total response

Nm
1
2
2 2
 B   { i ri ( x k ) 2  S b,i dn }
Ii 0
i 1
Nm
ni
  { r ( xk )
S (ni )}
2 b,i
4 i I i
i 1
2
R
2 2
i i
18
STEPWISE PROCEDURE OF BUFFETING
ANALYSIS IN FREQUENCY DOMAIN
Power Spectral
Density (PSD)
Aerodynamic
Admittance
Spectrum of Wind
Fluctuations
Joint Acceptance
Function
Spectrum of PointBuffeting Forces
Spectrum of Spanwise
Buffeting Forces
Mechanical
Admittance
Background and
Resonance Parts
Multimode Response
SRSS or CQC
Combination
Response Estimate
of ith Mode
Spectrum of ith Mode
Response
Inverse Fourier
Transform
19
NUMERICAL EXAMPLE
Structural parameters:
parameters:
Pre
re--stressed concrete cablecable-stayed bridge taken into consideration
for demonstration of the flutter analytical methods
Mean wind velocity parameters:
Mean velocity: Uz=40m/s and Deck elevation: z=20m
Layout of cable-stayed bridge for numerical example
20
FREE VIBRATION ANALYSIS (1)
Mode 1
Mode 2
Mode 3
Mode 4
Mode 5
Mode 6
Mode 7
Mode 8
Mode 9
21
FREE VIBRATION ANALYSIS (2)
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
22
MODAL SUM COEFFICIENTS
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
Grmsn   Lk (r , k ) m (s , k ) n
k 1
r, s: Modal index; m, n: Combination index
r, s=h, p or : Heaving, lateral or rotational
m, n=i or j
( r , k ) m : rth modal value at node k
23
STATIC FORCE COEFFICIENTS AND FIRSTFIRSTORDER DEVIATIVES
Force coefficient
CD
CL
0.1
0.5
0.08
0.4
0.06
0.3
0.04
0.2
0.1
0.02
0
0
-8
-4
0
4
8
-0.1 -8
-4
0
4
8
Attack angle
Attack angle (degree)
CM
0.35
CD
CL
CM
C’D
C’L
C’M
0.041
0.158
0.174
0
3.25
1.74
Force coefficient
0.3
0.25
0.2
0.15
0.1
0.05
0
-8
-4
0
Attack angle (degree)
4
8
Static force coefficients above
were determined by wind-tunnel
experiment [T.H.Le (2003)]
24
TURBULENCE WIND MODEL
Wind fluctuations modeled by the one-sided power spectral density (PSD)
functions using empirical formulas
200 fu*2
S u ( n) 
5/3
n1  50 f 
3.36 f u*2
S w ( n) 
n1  10 f 5 / 3 
Horizontal fluctuation: Kaimail’s spectrum
Vertical fluctuation: Panofsky’s spectrum
PSD of horizontal wind fluctuation
PSD of vertical wind fluctuation
600
12
500
10
400
8
Sw(n) m 2 .s/s 2
Su(n) m 2.s/s 2
PSD
300
200
6
4
100 Kaimal's spectrum
2 Kaimal's spectrum
Kaimail’s
PSD
U=
40m/s
Z= 20m
U=40m/s;Z=20m
u*=
2.5m/s
0
-3
10
-2
10
Panofsky’s
PSD
U=
40m/s
Z= 20m
U=40m/s;Z=20m
u*=
2.5m/s
-1
10
Freqency n(Hz)
0
10
1
10
0
-3
10
-2
10
-1
10
0
10
1
10
Freqency n(Hz)
25
AERODYNAMIC ADMITTANCE
Approximated by well-known Liepmann’s function (1952)
1
 (ni ) 
2 2ni B
1
U
2
ni: Modal frequency
0.9
0.8
Aerodynamic admittance
0.7
0.6
0.5
0.4
0.3
0.2
0.1
B=14.5m; U=40m/s
0
-2
10
-1
0
10
10
Frequency Log(n)
1
10
26
COHERENCE FUNCTION
Proposed by Davenport (1962) with assumption that coherence of
buffeting forces exhibits equal to that of ongoing velocity
 cni x
Cohu (ni , x)  exp(
)
U
C: Decay coefficient (8c16)
x: Spanwise separation
1
y=0.1m
0.9
y=0.3
0.8
y=0.5
0.7
Coherence
y=1
0.6
y=5
x=6m
0.5
y=10
0.4
0.3
y=30
0.2
0.1
c=10; x=0.1-30m
0
-2
10
-1
0
10
10
1
10
Frequency Log(n)
27
JOINT ACCEPTANCE FUNCTION
Joint acceptance function can be computed by following formulas
L
2
| J F ( x A , x B , nh ) |  
0
L
 cn h x
0 exp( U )ri ( x A )ri ( x B )dxdx
Discretization
i: The number of mode
F=L, D or M
r=h, p or 
 cnh x
| J L ( x, n h ) |  exp(
)(x)G hi hi
U
 cn x
2
| J M (x, n ) |  exp(
)(x)G i i
U
 cn p x
2
| J D (x, n p ) |  exp(
)(x)G pi pi
U
N
Grmsn   Lk (r , k )m (s , k ) n : Modal sum coefficients
k 1
( r ,k ) m : Modal value
2
Lk: Spanwise separation
28
MECHANICAL ADMITTANCE
Mechanical admittance is known as Transfer function of linear SISO
system in frequency domain
in ith mode,
determined as
2
2
H ni 
2
n 2
2 n
 {I [(1  2 )  4 i 2 )]}1
ni
ni
2
i
Ii: Generalized mass inertia
i: Total damping ratio
(Structural s,i+ Aerodynamica,i)
6
10
Resonance
Damping ratio 0.003
4
10
Amplitude Log(|H(n/ni)|2)
 i   s ,i   a ,i
Damping ratio 0.01
Damping ratio 0.015
Damping ratio 0.02
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
2
10
0
10
Background
-2
10
-4
10
-2
10
-1
0
10
10
Frequency Log(n/ni)
1
10
29
MODAL CONTRIBUTION ON RMS OF RESPONSE
RMS of Vertical disp. (m)
0.7
RMS of
Vertical
Displacement
at Midspan
0.6
0.5
Mode 1
0.4
0.3
0.2
Mode 2
0.1
0
10
20
30
40
Mode 5
50
60
50
60
RMS of
Rotation
at Midspan
RMS of Rotation(Degree)
Mean wind velocity
0.8
0.7
0.6
Mode 3
0.5
0.4
0.3
Mode 4
0.2
0.1
0
10
20
30
40
Mean wind velocity U(m/s)
30
RMS OF TOTAL RESPONSE (5 MODES COMBINED)
RMS of total response (m)
RMS of
Vertical
Displacement
at Midspan
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
10
20
30
40
50
60
Mean wind velocity U(m/s)
RMS of Total response (Degree)
0.8
RMS of
Rotation
0.6
0.4
0.2
at Midspan
0
10
20
30
40
50
60
Mean wind velocity U(m/s)
31
RMS of Vertical
Displacement
on Deck Nodes
RMS of vertical disp. (m)
RMS OF MODAL RESPONSE OF FULL BRIDGE
0.4
0.35
Mode 1
0.3
0.25
Mode 2
0.2
0.15
U=40m/s
0.1
0.05
0
Mode 5
1
3
5
7
9 11 13 15 17 19 21 23 25 27 29
RMS of
Rotation
on Deck Nodes
RMS of rotation (degree)
Deck nodes
0.6
0.5
Mode 3
0.4
0.3
U=40m/s
0.2
0.1
Mode 4
0
1
3
5
7
9 11 13 15 17 19 21 23 25 27 29
Deck nodes
32
CONCLUSION
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)
33
THANKS VERY MUCH FOR YOUR ATTENTION