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.

1. Fourth M.I.T. Conference on Computational Fluid and Solid Mechanics –Focus: Fluid-Structure InteractionsBoston, June 13-15, 2007Comparison of time domain techniquesfor the evaluation of the response and the stabilityof long span suspension bridges
F.Petrini, F.Giuliano, F.Bontempi*
*Professor of Structural Analysis and Design
University of Rome La Sapienza -ITALY

2. FB 2

3. PART #1
CONTEXT

4. FB 4

5. FB 5

6. 777 183 3300 183 627
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

7. 777 183 3300 183 627
960 3300 m 810
+77.00 m
+383.00 +383.00
+54.00
+118.00
+52.00 +63.00
CONTROL DEVICES
SOIL BEHAVIOR
MATERIAL NONLINEARITY
SOIL/STRUCTURE INTERFACE CONTACT
HANGERS
TOWERS
MAIN CABLES
GEOMETRIC NONLINEARITY
NONLINEARITY

8. 777 183 3300 183 627
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

9. FB 9
DECISION
NEGOTIATION & REFRAMING
WIND & TEMPERATURE
EARTHQUAKE
ANTROPIC ACTIONS
(RAILWAY & HIGHWAY)
STRUCTURAL BEHAVIOR &
PERFORMANCE ASSESSMENT
MODEL

10. FB 10
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

11. a) COMPOUND DECK
structural complexity

12. FB 12
deck arrangement

13. FB 13
deck arrangement

14. FB 14
highway girder section

15. FB 15
railway girder section

16. FB 16

17. FB 17
transverse element section

18. FB 18

19. FB 19

20. FB 20

21. FB 21

22. FB 22

23. FB 23

24. FB 24

25. FB 25

26. FB 26

27. b) RESTRAINT DEVICES
localized nonlinearities

28. FB 28

29. FB 29

30. FB 30

31. FB 31

32. FB 32

33. FB 33

34. FB 34

35. FB 35

36. FB 36

37. FB 37

38. FB 38

39. FB 39

40. FB 40

41. 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
Transversal slack (TS) and longitudinal slack (LS) arrangement
along the suspension bridge.
(HG: Highway box girder; RG: Railway box girder; TG: Transverse box girder.)

42. FB 42
TRANSVERSAL DISPLACEMENTS-101234567-1921805409001260162019802340270030603420L [m] Uy [m] 0 cm30 cm50 cm

43. 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

44. (i) ANALYSIS
strategies

45. FB 45
STRATEGY #1: SENSITIVITY
governance of priorities

46. FB 46
STRATEGY #2: BOUNDING
behavior governance
p
(p)

p
 (p)


47. FB 47
Super
Controllore
Problema Risultato
Solutore #1
Solutore #2
Voting System
STRATEGY #3: REDUNDANCY
factors governance

48. PART #2
FLUID-STRUCTURE
INTERACTIONS

49. 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:
M  q  C  q  K  q  F(body shape ;q,q,q;V;t;n) (1)
where
M  mass matrix of the system,
C  damping matrix of the system,
K  stiffness matrix of the system,
q, q, q  DOFs of the system and their first an second time derivates,
V  incident wind velocity,
t  time,
n  oscillation frequencies of the system.

50. FB 50
Classification (I): Collar

51. FB 51
Classification (II):
Naudascher / Rockwell
FLOW-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
EIE
Extraneously
induced excitation
MIE
Movement-induced
excitation
IIE
Instability-induced
excitation
es.
TURBULENCE
BUFFETING
es.
VORTEX
SHEDDING
es.
FLUTTER

52. 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
source
Flow instability
In phase with
body velocity
In phase with
body
displacement
or acceleration
Mean
loading
system
EIE IIE MIE
Alteration of
body dynamic
characteristics

53. (ii) WIND VELOCITIES
factors

54. FB 54
Vento = f(t)
Vento = f(s,t)
LAMINAR / TURBOLENT

55. FB 55
Atmosferic turbulence
Time variation
Spatial
variation
Three spatial
component
Mean
component
Turbulent
component

56. FB 56
Componente verticale
-15
-10
-5
0
5
10
15
0 500 1000 1500 T (secondi)
Vz (m/s)
Velocity
time
        j  
k k
j
k j k k (t) 2     cos tR sin tI
mc
k 1
    

j j Y
Time Histories generation by harmonic functions
superposition
Checking the spectral
compatibility
Wind velocity time histories generation (III)

57. FB 57

58. FB 58

59. FB 59

60. FB 60
Vento = f(s,t)
Vento = f(s,t)
Wind velocity field
Aeroelastic
theories
From
the wind
velocities
to
the sectional
forces
( )  ( )
2
1
( )
2
D t V t B c t a D      
( ) *  ( )
2
1
( )
2
L t V t B c t a L      
( ) *  ( )
2
1
( ) 2 2
M t V t B c t a M      
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

61. (iii) AERODYNAMIC THEORIES
factors

62. FB 62
LES–Flow around Nude Section

63. FB 63
LES–Flow around a RealisticSection

64. FB 64
Aeroelastic theories:
F q q q n P t n q Q t n q R t n q se ( , , ; )  ( , )   ( , )    ( , ) 
Approximated Formulation for Aeroelastic Forces (1)
Non aeroelastic
(NO)

65. FB 65
(NO) AEROELASTIC THEORY
Umean U’(t)
W’(t)
α(t)
α(t)
α(t)
undeformed configuration
E
( )  ( )
2
1
( )
2
D t V t B c t a D      
( )  ( )
2
1
( ) 0
2
L t V t B K t a L      
( )  ( )
2
1
( ) 0
2 2
M t V t B K t a M      

66. FB 66
(t)
t
0
no influence
NO
STRUCTURAL MOTION

67. 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
( )
2
D t V t B c t a D      
( )  ( )
2
1
( )
2
L t V t B c t a L      
( )  ( )
2
1
( ) 2 2
M t V t B c t a M      

68. FB 68
(t)
t

t
influence for instantaneous
effects of generalized
displacements
STRUCTURAL MOTION

69. 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
( )
2
D t V t B c t ai D      
( )  ( )
2
1
( )
2
L t V t B c t ai L      
( )  ( )
2
1
( ) 2 2
M t V t B c t ai M      

70. 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)

71. FB 71
(t)
t

t
influence for instantaneous
effects of generalized
(t) displacements and velocities
 STRUCTURAL MOTION

72. 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
( )
2
D t V t B c t aL D      
( ) *  ( )
2
1
( )
2
L t V t B c t aL L       (10)
( ) *  ( )
2
1
( ) 2 2
M t V t B c t aM M      
where (t) i  ,
2
V (t) ai ( i  L,M ) and D c , have the same meaning as the previous
expressions included in QS theory.

73. FB 73
MODIFIED QS THEORY (QSM) -2
In the expressions (10), aerodynamic coefficients Lc* and Mc* are dynamic and they are computed like below:           00)(* )(* 00dKccdKccMMMLLL (11) where )(0Lc e )(0Mc are the static aerodynamic coefficients computed in the mean equilibrium configuration (0), and LK, MK are the “dynamic derivatives” computed like below:                     MMLLcaKchK33 (12) where3h 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).

74. FB 74
(t)
t t
influence of
delay/memory effects
STRUCTURAL MOTION

75. FB 75
Complexity
Aeroelastic theories
F q q q n P t n q Q t n q R t n q se ( , , ; )  ( , )   ( , )    ( , ) 
Approximated formulation for aeroelastic forces (2)

76. PART #3
RESULTS

77. (iv) STABILITY RESULTSfor non turbulent wind
Vento = f(t)

78. 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

79. FB 79
Uz
Theta
start
final
V<Vcrit – δ>0

80. FB 80
V~Vcrit –δ~0
Uz Theta startfinalUz startfinal

81. FB 81
V>Vcrit –δ<0
Uz Theta startfinalUz startfinalstartfinal

82. FB 82
Mid span oscillation envelope to evaluate damping
0,5000,5050,5100,5150,5200,5256006507007508008509009501000t (sec) teqqq 0Uz; Thetaqq+q00,5000,5050,5100,5150,5200,5256006507007508008509009501000t q0 V<Vcrit 0

83. 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)
Damping (%)
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)
Damping (%)
Total Structural Aerodynamic

84. FB 84
66m/s 70m/s 85m/s0102030405060708090NOSTQSQSM V (m/s) NO FLUTTER

85. (v) RESPONSE RESULTSfor turbulent wind
Vento = f(s,t)

86. FB 86

87. 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
11,43
12,94
Class
Frequency
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
11,43
12,94
Class
Frequency
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
11,43
12,94
Class
Frequency
QSM
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
11,43
12,94
Class
Frequency
Mean wind velocity = 45 m/s

88. 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
-1,79
-1,02
-0,25
0,53
1,30
2,07
2,84
3,61
Class
Frequency
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
-1,79
-1,02
-0,25
0,53
1,30
2,07
2,84
3,61
Class
Frequency
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
-1,79
-1,02
-0,25
0,53
1,30
2,07
2,84
3,61
Class
Frequency
QSM
-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
-1,79
-1,02
-0,25
0,53
1,30
2,07
2,84
3,61
Class
Frequency
Mean wind velocity = 45 m/s

89. 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)
Rot (RAD)
0
200
400
600
800
1000
1200
-0,047
-0,036
-0,025
-0,014
-0,003
0,008
0,019
0,030
Class
Frequency
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)
Rot (RAD)
0
200
400
600
800
1000
1200
-0,047
-0,036
-0,025
-0,014
-0,003
0,008
0,019
0,030
Class
Frequency
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)
Rot (RAD)
0
200
400
600
800
1000
1200
1400
1600
1800
2000
-0,047
-0,036
-0,025
-0,014
-0,003
0,008
0,019
0,030
Class
Frequency
QSM
-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)
Rot (RAD)
0
200
400
600
800
1000
1200
1400
1600
1800
2000
-0,047
-0,036
-0,025
-0,014
-0,003
0,008
0,019
0,030
Class
Frequency
Mean wind velocity = 45 m/s

90. FB 90
Time history Probability density Mean values
900 1400 1900 2400 2900
time (sec)
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
900 1400 1900 2400 2900
time (sec)
NO_V45 ST_V45 QS_V45 QSM_V45
-0,4
-0,3
-0,2
-0,1
0,0
NO ST QS QSM Experim
900 1400 1900 2400 2900
time (sec)
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 Experim
Rotation(DEG)
Mean wind velocity = 45 m/s

95. FB 95
Envelope transv. velocity
NO_V45
ST_V45
QS_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

96. FB 96
Envelope transv. accelerationNO_V45ST_V45QS_V45QSM_V45-0,9-0,5-0,10,30,70500100015002000250030003500abscissa (m) ay (m/s^2) NO_V45ST_V45QS_V45QSM_V45

97. FB 97
Envelope vert. velocity NO_V45ST_V45QS_V45QSM_V45-2,5-1,5-0,50,51,52,50500100015002000250030003500abscissa (m) Vy (m/s) NO_V45ST_V45QS_V45QSM_V45

98. FB 98
Envelope vert. acceleration NO_V45ST_V45QS_V45QSM_V45-1,5-0,50,51,50500100015002000250030003500abscissa (m) az (m/s^2) NO_V45ST_V45QS_V45QSM_V45

99. FB 99
Tiro cavi all'ancoraggio
115000
120000
125000
130000
135000
140000
600 1100 1600 2100 2600 3100
Tempo (s)
Tiro (Ton)
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)

100. FB 100
Tiro cavi all'ancoraggio
115000
120000
125000
130000
135000
140000
600 1100 1600 2100 2600 3100
Tempo (s)
Tiro (Ton)
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)

101. FB 101
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.

102. FB 102
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.

104. Fourth M.I.T. Conference on Computational Fluid and Solid Mechanics –Focus: Fluid-Structure InteractionsBoston, June 13-15, 2007Comparison of time domain techniquesfor the evaluation of the response and the stabilityof 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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