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Plenary Lecture at Fourth M.I.T. Conference on Computational Fluid and Solid Mechanics – Focus: Fluid-Structure Interactions, Boston, June 13-15, 2007. During the last decades, several studies on suspension bridges under wind actions have been developed in civil engineering and many techniques have been used to approach this structural problem both in time and frequency domain. In this paper, four types of time domain techniques to evaluate the response and the stability of a long span suspension bridge are implemented: nonaeroelastic, steady, quasi steady, modified quasi steady. These techniques are compared considering both nonturbulent and turbulent flow wind modelling. The results show consistent differences both in the amplitude of the response and in the value of critical wind velocity.
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Fourth M.I.T. Conference on Computational Fluid and Solid
Mechanics – Focus: Fluid-Structure Interactions
Boston, June 13-15, 2007
Comparison of time domain techniques
for the evaluation
of the response and the stability
of long span suspension bridges
F.Petrini, F.Giuliano, F.Bontempi*
*Professor of Structural Analysis and Design
University of Rome La Sapienza - ITALY
FB 2
PART #1
CONTEXT
FB 4
FB 5
3300183 183777 627
960 3300 m 810
+77.00 m
+383.00 +383.00
+54.00+118.00
+52.00 +63.00
3300183 183777 627
960 3300 m 810
+77.00 m
+383.00 +383.00
+54.00+118.00
+52.00 +63.00
STRUCTURAL MODEL
LOADING SYSTEM
GEOMETRY AND MATERIAL
UNCERTAINTY
3300183 183777 627
960 3300 m 810
+77.00 m
+383.00 +383.00
+54.00+118.00
+52.00 +63.00
3300183 183777 627
960 3300 m 810
+77.00 m
+383.00 +383.00
+54.00+118.00
+52.00 +63.00
CONTROL DEVICES
SOIL BEHAVIORMATERIAL NONLINEARITY
SOIL/STRUCTURE INTERFACE CONTACT
HANGERS
TOWERS
MAIN CABLES
GEOMETRIC NONLINEARITY
NONLINEARITY
3300183 183777 627
960 3300 m 810
+77.00 m
+383.00 +383.00
+54.00+118.00
+52.00 +63.00
TRAFFIC – STRUCTURE
WIND - STRUCTURE
SOIL - STRUCTURE
INTERACTION
GLOBAL/LOCAL STRUCTURAL BEHAVIOUR
FB 9
DECISIONNEGOTIATION & REFRAMING
WIND & TEMPERATURE
EARTHQUAKE
AN
TR
OP
IC A
CT
ION
S
(RA
ILW
AY
& H
IGH
WA
Y)
ST
RU
CT
UR
AL
BE
HA
VIO
R &
PE
RF
OR
MA
NC
E A
SS
ES
SM
EN
T
MODEL
FB 10
Vento = f(s,t)
Vento = f(s,t)
Vento = f(s,t)
Vento = f(s,t)
Performance level assessing in response problem
Wind Vel
(m/s)
Return
Period
(years)
Performance to be furnished Level of
performance
21 50 Complete serviceability
(roadway and railway traffic)
High
45 200 Partial serviceability (railway
traffic)
Medium
57 2000 Maintaining the structural
integrity
Low
a) COMPOUND DECK
structural complexity
FB 12
deck arrangement
FB 13
deck arrangement
FB 14
highway girder section
FB 15
railway girder section
FB 16
FB 17
transverse element section
FB 18
FB 19
FB 20
FB 21
FB 22
FB 23
FB 24
FB 25
FB 26
b) RESTRAINT DEVICES
localized nonlinearities
FB 28
FB 29
FB 30
FB 31
FB 32
FB 33
FB 34
FB 35
FB 36
FB 37
FB 38
FB 39
FB 40
FB 41
WIND
HG
TG
SICILIA’S TOWER LEG
WIND
SICILIA’S TOWER LEG CALABRIA’S TOWER LEG
CALABRIA’S TOWER LEG
TS
LS
Sicilia Calabria
RG
HG
TG
LS
TS
WIND
HG
TG
SICILIA’S TOWER LEG
WIND
SICILIA’S TOWER LEG CALABRIA’S TOWER LEG
CALABRIA’S TOWER LEG
TS
LS
Sicilia Calabria
RG
HG
TG
LS
TS
Transversal slack (TS) and longitudinal slack (LS) arrangement
along the suspension bridge.
(HG: Highway box girder; RG: Railway box girder; TG: Transverse box girder.)
FB 42
TRANSVERSAL DISPLACEMENTS
-1
0
1
2
3
4
5
6
7
-192 180 540 900 1260 1620 1980 2340 2700 3060 3420
L [m]
Uy [
m]
0 cm 30 cm 50 cm
FB 43
HORIZONTAL CURVATURE
-2.0E-05
0.0E+00
2.0E-05
4.0E-05
6.0E-05
-120 240 600 960 1320 1680 2040 2400 2760 3120
L [m]
c
[m-1
]
0 cm 30 cm 50 cm
(i)
ANALYSIS
strategies
FB 45
STRATEGY #1: SENSITIVITY
governance of priorities
FB 46
STRATEGY #2: BOUNDING
behavior governance
p
(p)
p
(p)
FB 47
Super
ControlloreProblema Risultato
Solutore #1
Solutore #2
Voting System
STRATEGY #3: REDUNDANCY
factors governance
PART #2
FLUID-STRUCTURE
INTERACTIONS
FB 49
Equation of dynamic equilibrium
By discretizing the body to a finite number of degrees of freedom (DOFs), the
equation governing the body motion is the dynamic equilibrium equation:
);;;,,;( ntVqqqshapebodyFqKqCqM (1)
where
M mass matrix of the system,
C damping matrix of the system,
K stiffness matrix of the system,
qqq ,, DOFs of the system and their first an second time derivates,
V incident wind velocity,
t time,
n oscillation frequencies of the system.
FB 50
Classification (I):
Collar
FB 51
Classification (II):
Naudascher / RockwellFLOW-INDUCED VIBRATIONS
caused by fluctuations in
flow velocity or
pressures that are
independent of any flow
instability originating
from the structure
considered and
independent of structural
movements except for
added-mass and
fluid-damping effects
brought about by a flow
instability that is intrinsic
to the flow system; in
other words, the flow
instability is inherent to
the flow created by the
structure considered
due to fluctuating forces
that arise from
movements of the
vibrating body; a
dynamic instability of the
body oscillator can gives
rise to energy transfer
from the main flow to the
oscillator
EIEExtraneously
induced excitation
MIEMovement-induced
excitation
IIEInstability-induced
excitation
es.
TURBULENCE
BUFFETING
es.
VORTEX
SHEDDING
es.
FLUTTER
FB 52
if F(t) contains negative flow-induced damping
FLOW-INDUCED FORCES
ON STATIONARY BODY
MOVEMENT-INDUCED FORCES
IN STAGNANT FLUID
Fmean
mean value
F'(t)
due to
fluctuating
fluid
F''(t)
due to
vibrating body
Extraneous
sourceFlow instability
In phase with
body velocity
In phase with
body
displacement
or acceleration
Mean
loading
systemEIE IIE MIE
Alteration of
body dynamic
characteristics
(ii)
WIND VELOCITIES
factors
FB 54
Vento = f(t)
Vento = f(s,t)
LAMINAR / TURBOLENT
FB 55
Atmosferic turbulence
Time variation
Spatial
variation
Three spatial
componentMean
component
Turbulent
component
FB 56
Componente verticale
-15
-10
-5
0
5
10
15
0 500 1000 1500 2000 2500 3000
T (secondi)
Vz (m
/s)
Velo
cit
y
time
j
kk
j
kkjk tItR sincos2(t)mc
1k
jjY
Time Histories generation by harmonic functions
superposition
Checking the spectral
compatibility
Wind velocity time histories generation (III)
FB 57
FB 58
FB 59
FB 60
Vento = f(s,t)
Vento = f(s,t)
Vento = f(s,t)
Vento = f(s,t)
Wind velocity fieldAeroelastic
theories
From
the wind
velocities
to
the sectional
forces
)()(2
1)(
2tcBtVtD Da
)(*)(2
1)(
2tcBtVtL La
)(*)(2
1)( 22
tcBtVtM Ma
a) Laminar
b) Turbulent
t1
t2
Computing of instantaneous wind forces
Velocities are stationary
Velocities are uniform at the same altitude
Velocities are non stationary and non uniform
Loading system
(iii)
AERODYNAMIC THEORIES
factors
FB 62
LES – Flow around Nude Section
FB 63
LES – Flow around a Realistic Section
FB 64
Aeroelastic theories:
qntRqntQqntPnqqqF se ),(),(),();,,(
Approximated Formulation for Aeroelastic Forces (1)
Non aeroelastic
(NO)
FB 65
(NO) AEROELASTIC THEORY
Umean U’(t)
W’(t)
α(t)
α(t)
α(t)
undeformed configuration
E )()(
2
1)(
2tcBtVtD Da
)()(2
1)( 0
2tKBtVtL La
)()(2
1)( 0
22tKBtVtM Ma
FB 66
(t)
t
0
no influence
NO
STRUCTURAL MOTION
FB 67
STEADY THEORY (ST)
Umean U’(t)
W’(t)
α(t)
α(t)α(t)
θ(t)
θ(t)
γ(t)
γ(t)
undeformed configuration
E
E
)()(2
1)(
2tcBtVtD Da
)()(2
1)(
2tcBtVtL La
)()(2
1)( 22
tcBtVtM Ma
FB 68
(t)
t
t
influence for instantaneous
effects of generalized
displacements
STRUCTURAL MOTION
FB 69
QUASI STEADY THEORY (QS) - 1
Umean
U’(t) W’(t)
β(t)
α(t)
β(t)
θ(t)
θ(t)
γ(t)
γ(t)
undeformed configuration
E
E
-p(t)
-hA(t) )()(
2
1)(
2tcBtVtD Dai
)()(2
1)(
2tcBtVtL Lai
)()(2
1)( 22
tcBtVtM Mai
FB 70
QUASI STEADY THEORY (QS) - 2
θ(t)
θ(t)
undeformed configuration
E
E
p
p(t)
hA(t)
A
A
B
biB
hA(t)=h(t)+biBθ(t)
h(t)
p(t)
FB 71
(t)
t
t
influence for instantaneous
effects of generalized
displacements and velocities(t)
STRUCTURAL MOTION
FB 72
MODIFIED QS THEORY (QSM) - 1
In respect to the QS theory, the only changes concern the aerodynamic coefficients for
the Lift and the Moment, which become dynamic as measured by wind tunnel tests.
Aeroelastic forces are expressed by the following expressions:
)()(2
1)(
2tcBtVtD DaL
)(*)(2
1)(
2tcBtVtL LaL (10)
)(*)(2
1)( 22
tcBtVtM MaM
where )(ti , 2
)(tVai ( MLi , ) and Dc , have the same meaning as the previous
expressions included in QS theory.
FB 73
MODIFIED QS THEORY (QSM) - 2In the expressions (10), aerodynamic coefficients Lc * and Mc * are dynamic and they
are computed like below:
0
0
)(*
)(*
0
0
dKcc
dKcc
MMM
LLL
(11)
where )( 0Lc e )( 0Mc are the static aerodynamic coefficients computed in the mean
equilibrium configuration ( 0 ), and LK , MK are the “dynamic derivatives”
computed like below:
M
M
L
L
caK
chK
3
3
(12)
where 3h and 3a are the Zasso’s theory coefficients [15], assessed by dynamic wind
tunnel tests. These coefficients are similar to the Scanlan’s motion derivatives (2), and
they depend both from the rotation deck angle and the “reduced wind velocity”
BVVred (depending from , which is the motion frequency).
FB 74
(t)
tt
influence of
delay/memory effects
STRUCTURAL MOTION
FB 75
Complexity
Aeroelastic theories
qntRqntQqntPnqqqF se ),(),(),();,,(
Approximated formulation for aeroelastic forces (2)
PART #3
RESULTS
(iv)
STABILITY RESULTS
for non turbulent wind
Vento = f(t)
FB 78
0,500
0,505
0,510
0,515
0,520
600 650 700 750 800 850 900 950 1000
t (sec)
stable (positive damping)
0,500
0,505
0,510
0,515
0,520
0,525
600 650 700 750 800 850 900 950 1000
t (sec)
critical (zero damping)
0,300
0,400
0,500
0,600
0,700
600 650 700 750 800 850 900 950 1000
t (sec)
unstable (negative damping)
V<Vcrit – δ>0
V~Vcrit – δ~0
V>Vcrit – δ<0
FB 79
Uz
Theta
start
final
Uz
Theta
start
final
V<Vcrit – δ>0
FB 80
V~Vcrit – δ~0
Uz
Theta
start
final
Uz
Theta
start
final
FB 81
V>Vcrit – δ<0
Uz
Theta
start
final
Uz
Theta
start
final
start
final
FB 82
Mid span oscillation envelope
to evaluate damping
0,500
0,505
0,510
0,515
0,520
0,525
600 650 700 750 800 850 900 950 1000
t (sec)
teqqq
0
Uz; Theta
q
q+q0
0,500
0,505
0,510
0,515
0,520
0,525
600 650 700 750 800 850 900 950 1000
t (sec)
teqqq
0
Uz; Theta
q
q+q0
V<Vcrit
0
FB 83
Damping and Vcrit
-1,5
-1,0
-0,5
0,0
0,5
1,0
1,5
0 10 20 30 40 50 60 70 80
Wind Velocity (m/s)
Da
mp
ing
(%
)
Total Structural Aerodynamic
-1,5
-1,0
-0,5
0,0
0,5
1,0
1,5
0 10 20 30 40 50 60 70 80
Wind Velocity (m/s)
Da
mp
ing
(%
)Total Structural Aerodynamic
FB 84
66m/s 70m/s 85m/s
0
10
20
30
40
50
60
70
80
90
NO ST QS QSM
V (
m/s
)
NO
FL
UT
TE
R
(v)
RESPONSE RESULTS
for turbulent wind
Vento = f(s,t)
FB 86
FB 87
Time history Frequencies Probability density
NO
0
2
4
6
8
10
12
14
400 900 1400 1900 2400 2900
time (sec)
Uy (
m)
0
200
400
600
800
1000
1200
2,41
3,91
5,42
6,92
8,43
9,93
11,4
3
12,9
4
Class
Fre
qu
en
cy
ST
0
2
4
6
8
10
12
14
400 900 1400 1900 2400 2900
time (sec)
Uy (
m)
0
200
400
600
800
1000
1200
2,41
3,91
5,42
6,92
8,43
9,93
11,4
3
12,9
4
Class
Fre
qu
en
cy
QS
0
2
4
6
8
10
12
14
400 900 1400 1900 2400 2900
time (sec)
Uy (
m)
0
200
400
600
800
1000
1200
2,41
3,91
5,42
6,92
8,43
9,93
11,4
3
12,9
4
Class
Fre
qu
en
cy
QS
M
0
2
4
6
8
10
12
14
400 900 1400 1900 2400 2900
time (sec)
Uy (
m)
0
200
400
600
800
1000
1200
2,41
3,91
5,42
6,92
8,43
9,93
11,4
3
12,9
4
Class
Fre
qu
en
cy
Mean wind velocity = 45 m/s
FB 88
Time history Frequencies Probability density
NO
-3,5
-2,5
-1,5
-0,5
0,5
1,5
2,5
3,5
400 900 1400 1900 2400 2900
time (sec)
Uz (
m)
0
200
400
600
800
1000
1200
-1,7
9
-1,0
2
-0,2
50,
531,
302,
072,
843,
61
Class
Fre
qu
en
cy
ST
-3,5
-2,5
-1,5
-0,5
0,5
1,5
2,5
3,5
400 900 1400 1900 2400 2900
time (sec)
Uz (
m)
0
200
400
600
800
1000
1200
-1,7
9
-1,0
2
-0,2
50,
531,
302,
072,
843,
61
Class
Fre
qu
en
cy
QS
-3,5
-2,5
-1,5
-0,5
0,5
1,5
2,5
3,5
400 900 1400 1900 2400 2900
time (sec)
Uz (
m)
0
200
400
600
800
1000
1200
1400
1600
1800
-1,7
9
-1,0
2
-0,2
50,
531,
302,
072,
843,
61
Class
Fre
qu
en
cy
QS
M
-3,5
-2,5
-1,5
-0,5
0,5
1,5
2,5
3,5
400 900 1400 1900 2400 2900
time (sec)
Uz (
m)
0
500
1000
1500
2000
2500
3000
3500
-1,7
9
-1,0
2
-0,2
50,
531,
302,
072,
843,
61
Class
Fre
qu
en
cy
Mean wind velocity = 45 m/s
FB 89
Time history Frequencies Probability density
NO
-0,055
-0,045
-0,035
-0,025
-0,015
-0,005
0,005
0,015
0,025
400 900 1400 1900 2400 2900
time (sec)
Ro
t (R
AD
)
0
200
400
600
800
1000
1200
-0,0
47
-0,0
36
-0,0
25
-0,0
14
-0,0
03
0,00
8
0,01
9
0,03
0
Class
Fre
qu
en
cy
ST
-0,055
-0,045
-0,035
-0,025
-0,015
-0,005
0,005
0,015
0,025
400 900 1400 1900 2400 2900
time (sec)
Ro
t (R
AD
)
0
200
400
600
800
1000
1200
-0,0
47
-0,0
36
-0,0
25
-0,0
14
-0,0
03
0,00
8
0,01
9
0,03
0
Class
Fre
qu
en
cy
QS
-0,055
-0,045
-0,035
-0,025
-0,015
-0,005
0,005
0,015
0,025
400 900 1400 1900 2400 2900
time (sec)
Ro
t (R
AD
)
0
200
400
600
800
1000
1200
1400
1600
1800
2000
-0,0
47
-0,0
36
-0,0
25
-0,0
14
-0,0
03
0,00
8
0,01
9
0,03
0
Class
Fre
qu
en
cy
QS
M
-0,055
-0,045
-0,035
-0,025
-0,015
-0,005
0,005
0,015
0,025
400 900 1400 1900 2400 2900
time (sec)
Ro
t (R
AD
)
0
200
400
600
800
1000
1200
1400
1600
1800
2000
-0,0
47
-0,0
36
-0,0
25
-0,0
14
-0,0
03
0,00
8
0,01
9
0,03
0
Class
Fre
qu
en
cy
Mean wind velocity = 45 m/s
FB 90
Time history Probability density Mean values
Tra
ns
ve
rsa
l
0
2
4
6
8
10
12
14
400 900 1400 1900 2400 2900
time (sec)
Uy (
m)
NO_V45 ST_V45 QS_V45 QSM_V45
0,0
1,0
2,0
3,0
4,0
5,0
6,0
7,0
NO ST QS QSM Experim
Ve
rtic
al
-3,5
-2,5
-1,5
-0,5
0,5
1,5
2,5
3,5
400 900 1400 1900 2400 2900
time (sec)
Uz (
m)
NO_V45 ST_V45 QS_V45 QSM_V45
-0,4
-0,3
-0,2
-0,1
0,0
NO ST QS QSM Experim
Ro
tati
on
-0,055
-0,045
-0,035
-0,025
-0,015
-0,005
0,005
0,015
0,025
400 900 1400 1900 2400 2900
time (sec)
Ro
t (R
AD
)
NO_V45 ST_V45 QS_V45 QSM_V45
-0,6
-0,5
-0,4
-0,3
-0,2
-0,1
0,0
NO ST QS QSM ExperimR
ota
tio
n(D
EG
)
-0,6
-0,5
-0,4
-0,3
-0,2
-0,1
0,0
NO ST QS QSM ExperimR
ota
tio
n(D
EG
)
Mean wind velocity = 45 m/s
FB 95
Envelope transv. velocity
NO_V45
ST_V45QS_V45
QSM_V45
-1,8
-0,8
0,2
1,2
0 500 1000 1500 2000 2500 3000 3500
abscissa (m)
Vy (
m/s
)
NO_V45 ST_V45 QS_V45 QSM_V45
FB 96
Envelope transv. acceleration
NO_V45ST_V45
QS_V45
QSM_V45
-0,9
-0,5
-0,1
0,3
0,7
0 500 1000 1500 2000 2500 3000 3500
abscissa (m)
ay (
m/s
^2)
NO_V45 ST_V45 QS_V45 QSM_V45
FB 97
Envelope vert. velocity
NO_V45
ST_V45
QS_V45
QSM_V45
-2,5
-1,5
-0,5
0,5
1,5
2,5
0 500 1000 1500 2000 2500 3000 3500
abscissa (m)
Vy (
m/s
)
NO_V45 ST_V45 QS_V45 QSM_V45
FB 98
Envelope vert. acceleration
NO_V45
ST_V45
QS_V45
QSM_V45
-1,5
-0,5
0,5
1,5
0 500 1000 1500 2000 2500 3000 3500
abscissa (m)
az (
m/s
^2)
NO_V45 ST_V45 QS_V45 QSM_V45
FB 99
Tiro cavi all'ancoraggio
115000
120000
125000
130000
135000
140000
600 1100 1600 2100 2600 3100
Tempo (s)
Tir
o (
To
n)
Sponda siciliana, lato nord Sponda calabrese, lato nord
Sponda siciliana, lato sud Sponda calabrese, lato sud
AXIAL FORCE IN THE MAIN CABLES (1)
Vento = f(s,t)
Vento = f(s,t)
Vento = f(s,t)
Vento = f(s,t)
FB 100
Tiro cavi all'ancoraggio
115000
120000
125000
130000
135000
140000
600 1100 1600 2100 2600 3100
Tempo (s)
Tir
o (
To
n)
Sponda siciliana, lato nord Sponda calabrese, lato nord
Sponda siciliana, lato sud Sponda calabrese, lato sud
AXIAL FORCE IN THE MAIN CABLES (2)
Vento = f(s,t)
Vento = f(s,t)
Vento = f(s,t)
Vento = f(s,t)
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CONCLUSIONS - stability
1. NO formulation can not compute the flutter phenomenon, while the other formulations can;
2. increasing the complexity of the aeroelastic forces representation, the value of the critical velocity increases;
3. the variation of aeroelastic damping with the wind incident velocity has been assessed using QS formulation, where the aerodynamic damping increases its value from zero velocity to a certain value of the wind velocity; beyond this value it starts to decrease and finally it becomes negative.
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CONCLUSIONS - response
1. with non turbulent wind, the QS and QSM formulations have a damping greater than linear; concerning the time envelopes of deck displacements, the results obtained from different formulations are very similar;
2. with turbulent incident wind, the differences between the oscillations amplitude computed by different formulations become significant.
In general, increasing the complexity of the aeroelastic forces representation (following the succession NO, ST, QS, QSM), the maximum response decrease. These differences increase with the increase of the wind mean velocity.
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ACKNOWLEDGMENTS
• The authors thank Professors R. Calzona, P.G. Malerba, and K.J. Bathe for fundamental supports related to this study.
• Thanks to the Reviewers of the present paper.
• The financial supports of University of Rome “La Sapienza”, COFIN2004 and Stretto di Messina S.p.A. are acknowledged.
• Nevertheless, the opinions and the results presented here are responsibility of the authors and cannot be assumed to reflect the ones of University of Rome “La Sapienza” or of Stretto di Messina S.p.A.
Fourth M.I.T. Conference on Computational Fluid and Solid
Mechanics – Focus: Fluid-Structure Interactions
Boston, June 13-15, 2007
Comparison of time domain techniques
for the evaluation
of the response and the stability
of long span suspension bridges
F.Petrini, F.Giuliano, F.Bontempi*
*Professor of Structural Analysis and Design
University of Rome La Sapienza - ITALY
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