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ViBest
Structural Engineering Documents
Elsa de Sá Caetano
Cable Vibrations in Cable-Stayed Bridges
International Association for Bridge and Structural Engineering
9
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Cable Vibrations in Cable-Stayed Bridges
Part 1: Assessment
Part 2: Mitigation
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Part1: Assessment of cable vibrations
Overview:
- Brief history of cable-stayed bridge construction
- Cable vibration phenomena
- Vibration phenomena due to direct action of wind and rain
- Vibration phenomena due to indirect excitation by
anchorage
- Recommendations and guidelines for dynamic design of
stay cables
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Brief history of cable-stayed bridge construction
Bluff Dale Bridge, Texas, USA, 1890Length: 61m; Span: 43 m; Width; 4 mMaterials: Wrought Iron, hand-twisted wire cables
(Images: Wikipedia, Structurae: HAER, TX,72-BLUDA,1-9)4
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Brief history of cable-stayed bridge construction
Niagara Falls Bridge, USA, 1855Span: 250 m; Deck depth; 5.5 mMaterials: Iron wire ropes
(Images: Structurae, C. Bierstadt, Publisher, Niagara Falls, NY); HAER: Brooklyn Bridge, New York County, NY)
Brooklyn Bridge, USA, 1883Span: 486.5 m, Length: 1059,9 mMaterials: Steel wire ropes
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Brief history of cable-stayed bridge construction
1912Span: 160 m; Design: Arnodin
(Images: Massinissa, Algeria; Structurae: Jacques Mossot)
Cassagne Bridge, France, 1909Span: 156 m, Length: 253 mDesign: Gisclard
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Brief history of cable-stayed bridge construction
Lézardrieux Bridge , France, 1925Span: 112 mDesign: Leinekugel le Coq
(Images: Structurae: Nicolas Janberg; K. Todeschini)7
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Brief history of cable-stayed bridge construction
Donzère Mondragon Bridge , France, 1952Span: 81 m; Length: 160 mDesign: Albert Caquot
(Images: Structurae: Jacques Mossot; Nicolas Janberg)8
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Brief history of cable-stayed bridge construction
Stromsund Bridge, 1956Span: 182 m; Length: 182 m; Materials: Steel; Design: Franz Dischinger
(Images: Andreas Stedt)9
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Brief history of cable-stayed bridge construction
(Images: Andreas Stedt; Philip Bourret in Structurae; P. )
Three generations of cable-stayed bridges (Mathivat, 1983)
(Stromsund Bridge, 1956)
(Pasco-Kennewick Bridge, 1978)(Brotonne Bridge, 1977)
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Brief history of cable-stayed bridge construction
(Images: Andreas Stedt)
0
200
400
600
800
1000
1200
1950 1960 1970 1980 1990 2000 2010 2020
Spa
n le
ngth
(m)
Year of construction
Stonecutters
Russky Island
Sutong
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Brief history of cable-stayed bridge construction
(Images: Bouygues Construction, metropol2, Nicolas Janberg for Structurae, wikipedia)
Normandy Bridge, 1994 (856 m)
(Design: M. Virlogeux)
Tatara Bridge, 1999 (890 m)
(Design: Honshu-Shikoku Bridge Auth.)
Sutong Bridge, 2008 (1088 m)
Stonecutters Bridge, 2010 (1018 m)
(Design: Ian Firth, Poul Ove Jensen)
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Brief history of cable-stayed bridge construction
(Images: Wikipedia; Grillaud)
Rion Antirion Bridge, 2004 (286 + 3 x 560 + 286m)
(Design: Ingérop)
Millau Viaduct, 2004(204 + 6 x 342 + 204 m)
(Design: M. Virlogeux)
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Cable vibration phenomena
(Image: Geurts)
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Cable vibration phenomena
(Images: Olivier Flamand; Philip Bourret in Structurae;) 15
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Cable vibration phenomena
(Videos: H. Tabatabai; E. Caetano; others, available in internet) 16
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Cable vibration phenomena
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Wind Loads on Stay Cables
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Wind Loads on Stay Cables
- Buffeting
- Vortex-shedding
- Galloping
- Wake Effects
- Rain-Wind Induced Vibrations
- Dry Galloping
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Wind Loads on Stay Cables
- Smooth flow, fixed body
- Turbulent flow, fixed body
- Turbulent flow, moving body
y
0 x
U
l
dyl
0
U
d
x
t
t
u(t)
v(t)
U
y
Ur
v(t)
Ut
u(t)
0
l
x
d
0'
(t)
l(t)d(t)
(t)(t)rV
l(t)
d(t)
20
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Wind Loads on Stay Cables
- Smooth flow, fixed body
y
0 x
U
l
d
U
x
yl
d
FL(t)(t)FD
(t)M0
BfLs(t) fDs(t)
ms (t)
)()( tfFtF DsDDs
)()( tfFtF LsLLs
)()( tmMtM ss
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Wind Loads on Stay Cables
- Smooth flow, fixed body: Mean wind forces
)(D2
D CBU21F
)(L2
L CBU21F
)(M22 CBU
21M
- Air density ;
- B body diameter;
- U mean wind velocity;
- CD, CL and CM are shape coefficients, depending on the angle of attack and on Re
22
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Wind Loads on Stay Cables
- kinematic viscosity of the air;
- B body diameter:
- U mean wind velocity
UBRe
(SETRA, 2002)B : 0.1 0.3 m
U : 5 50 m/s
CD=0.7
23
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Wind Loads on Stay Cables
- Smooth flow, fixed body: Fluctuating components
- fv Frequency of vortex shedding
- cDs, cLs and cMs are wake coefficients
- St Strouhal number (St=0.2 for circular cylinder sections)
)()( tf4cBU21tf vDs
2Ds sin
)()( tf2cBU21tf vLs
2Ls sin
)()( tf2cBU21tm vMs
22s sin
BStUfv
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(t)
l
FLtB
y
d
(t)FDt
0 x(t)Mt
Wind Loads on Stay Cables
- Turbulent flow, fixed body
FLt(t) FDt(t)
Mt (t)
)t(f)t(f)t(fF)t(F DwDvDuDtDt
)t(f)t(f)t(fF)t(F LwLvLuLtLt
)t(mw)t(m)t(mM)t(M vutt
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l
x0
d
y0'
M (t)
(t)LrF DrF (t)
r
Wind Loads on Stay Cables
- Turbulent flow, moving body
FLr(t) FDr(t)
Mr (t)
)t(f)t(f)t(f)t(f)t(fF)t(F .qD
DqDwDvDuDtDr
)t(f)t(f)t(f)t(f)t(fF)t(F .qL
LqLwLvLuLtLr
)t(m)t(m)t(m)t(m)t(mM)t(M .qqwvutr
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Wind Loads on Stay Cables
- Turbulent flow, moving body
)()()( tqKtqCtF aaa)()(
)()(
)()(
)(
.
.
.
tftm
tftf
tftf
tF
qLLq
qDDq
a
0CBCB20CCC20CCC2
BU21C
MtMt
LtDtLt
LtDtDt
a
)()()()()()()()(
'
'
'
Aerodynamic damping matrix
)(CB00)(C00)(C00
BU21K
t'M
'Lt
'Dt
2a
Aeroelastic stiffness matrix
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Buffeting
Forces due to alongwind turbulence
Forces due to acrosswind turbulence
)()()( DtDu CBtuUtf
)()()( LtLu CBtuUtf
)()()( Mt2
u CBtuUtm
))()(()()( 'LtDtDv CCBtvU
21tf
))()(()()( 'LtDtLv CCBtvU
21tf
)()()( 'Mt
2v CBtvU
21tm
28
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Buffeting
Aerodynamic damping for the kth mode of a circular cable with diameter D:
along-wind direction
across-wind direction
k
DkDaer m2
CDU,
k
DkLaer m4
CDU,
Cable D (m) m (kg/m) L (m) T (kN) fk (Hz) (%)U=15m/s U=30m/s
H01 0.160 42.9 34.7 2045 3.145 0.06 0.12
H15 0.200 74.8 147.5 4305.5 0.814 0.16 0.33
H24 0.250 100.1 226.0 6785.5 0.576 0.22 0.43
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Vortex-shedding
- Lock-in or synchronisation
Natural frequency of structure
Flow velocity U
Frequency
Lock-in region
Vortex-shedding frequency
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Vortex-shedding
(Images: Dyrbye & Hansen, 1999)
- Effect of turbulence and intrinsic damping ( )
2e
c Dm2S
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Vortex-shedding
(Images: Yamada, 1997)
- Scruton number vs cable length
Sc2= Sc/2
SC=20
32
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Vortex-shedding
- Vasco da Gama Bridge stay cables:
:160 250 mm
L: 35m 226 m U=12m/s fv=15Hz
1st cable frequency: 0.6 3Hz
St 0.2
Critical velocity:
shortest stay: Ucr=2.4m/s
Required damping to avoid lock-in:
Tabatabai & Mehrabi, 2000: 90% stay cable are stable with =0.7%
Df5U vcr
mD6
2
2
33
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Vortex-shedding
- Amplitude of oscillations
Resonant vortex-excited vibrations:
(Griffin et al., 1975) 353c
20
SSt24301291
Dy
.)(..
0
0.2
0.4
0.6
0.8
1
1.2
0 5 10 15 20 25 30
Scruton number Sc
y0/D
Vasco da Gama Bridge:
Shortest stays: 0.0085
Sc=23.7 y0 < 2% D
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Vortex-shedding
- Amplitude of oscillations
Vortex resonance model (EC1) (Ruscheweyh, 1994):
Vasco da Gama Bridge:
Shortest stays: 0.0085
1st mode:
5th mode:
(z)
LDe 6 y
D0 01.
LD
yD
e 48 12 0. 01 0 60. .yD
LDe 12
yD0 0 6.
2c
latw0
S1
S1cKK
Dy
7
65
5
0lat
10Re3010x4Re10x520
10x3Re70c
,.,.,.
,
Clat < clat,0
eL
003.0Dy0
01.0Dy0
35
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Galloping
Dynamic equilibrium in vertical direction:
U
rUy
l
B
d
0
y
x
Uy)C
dCd(BU
21ymym2ym 0D
L22
0DL C
dCdBU
21m2d )(
0CdCd
0DL )(Glauert- Den Hartog criterion
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Galloping
Susceptibility to galloping of cable with octogonal cross-section:
(Image: Simiu & Scanlan, 1996)
0CdCd
0DL )(
Circular cross sections:
0dCd L
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Galloping of inclined circular cables
- Axial flow favours instability (Matsumoto et al., 1992 2010)
Galloping occurs for yawing angles greater than 25°, when the axial flow reaches 30% U
(Image: Matsumoto et al., 2010)
Wind
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Galloping of inclined circular cables
- Typical stay cables: 30000 < Re < 100000
=0.20 m U= 20 60 m/s
Critical Re
Wind velocity+
Draf and lift force-
Cable vibration
Cable vibration
Relative wind velocity+
Cable vibration
Relative wind velocity-
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Galloping
- Prediction and mitigation measures
1
10
100
1000
1 10 100
Sc
Ucr/f
DIrwinVirlogeuxHondaCheng et al.
Red
uced
win
dve
loci
ty
32
ccr
cr S10fDUU
Honda et al. (1995):
Irwin/PTI Guide (2000):
4S40fDUU c
crcr
Virlogeux (1998):
4S35fDUU c
crcr
40
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Galloping
- Other mitigation measures: Aerodynamic
Kubo et al. (2003) Matsumoto et al. (1995)
41
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Wake Effects
(Video: Available in internet)
- Resonant buffeting
- Vortex resonance
- Galloping
- Other interference effects
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Wake Effects
Resonant buffeting
Bu
)(tu )(UBtu
B- distance between planes of
cables
U- mean wind velocity
Tt- period of the torsion mode
tcr T
B2U
Vasco da Gama Bridge : Tt = 2.3s; B=30 m Ucr=26m/s
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Wake Effects
Vortex resonance
Frequency of vortex shedding:
HStUfv
Virlogeux (1998): Evripos Bridge, Greece; Normandy Bridge, construction
x
U
H
kv ffStHfU k
cr
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Wake Effects
Interference effects
(Image: Gimsing & Nissen, 1998)
a
D
a
D
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Wake Effects
Type of associationKiv
a = D a 2D
Kiv = 1.5 Kiv = 1.5
Kiv = 4.8 Kiv = 3
Kiv = 4.8 Kiv = 3
a
D
a
D
Vortex resonance:
03Da01 ..
lativclat cKc )(
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Wake Effects
Type of associationaG
a 1.5D a 2.5D
(n=2)
aG= 1.5 aG= 3
(n=3)
aG= 6 aG= 3
(n=4)
aG= 1 aG= 2
a
D
a
D
Galloping:
DfaS2
U kG
cccG
)(
2
n
1ii
cc D
m2S )(
U25.1UcG
47
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Wake Effects
Interference galloping of free cables:
IG
c
kcIG a
SDa
Df53U . 0.3Da
Validity of formula:
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Wake Effects
Practical occurrence of interference galloping:
- Elliptical vibrations in the first mode < 3D
- Wind direction is 0 45° from transverse direction
to the bridge axis
- Depends on cable spacing:
Small spacings: D x 4D; -2D y 2D; Ucr=5 20m/s
Wide spacings: 8D y 20 D , downstream cableD
x
y
UProposed measures:
-Adopt close spacing: 1.2D 1.3 D; large spacing: 5D
-Connect parallel cables by stringers or spacers at short lengths
-Strand helically pair or ensemble of three cables
49
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Rain-Wind Induced Vibrations
(Video: Ben-Ahin Bridge, available in internet)50
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Rain-Wind Induced Vibrations
- Formation of water rivulet for specific angles of attack of wind and intensity of rainfall
- Loss of symmetry of cross section may lead to galloping instability
- Circumferential oscillation of the rivulets with the same cable frequency
- Coupling with flexural oscillation of cable may lead to intensification of vibrations
(Image: Matsumoto et al, 2010)
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Rain-Wind Induced Vibrations
Analytical and design models:
- Yamaguchi (1990)
yFykym
MI
m- cable mass per unit lengthk- generalised stiffnessFy, M- unsetady aerodynamic components of force per unit length
51
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Rain-Wind Induced Vibrations
(Image: Yamaguchi (1990))
Analytical and design models:
- Yamaguchi (1990)
00y
Ky
Cy
M
dCdUDd
21
I10
dCdUDd
21
m1
KM22
L22y
)(
)(
dCd
I2sin)Dd(
dCd
I)Dd(
dCdC
m2sin)Dd(
dCdC
m1
U)Dd(21C
M2
M
LD
LD
1001
M
52
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Rain-Wind Induced Vibrations
(Image: Peil & Nahrath (2003))
Analytical and design models:
- Peil and Nahrath (2003)
53
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Rain-Wind Induced Vibrations
Practical cases of occurrence of rain-wind vibration:
- Wind speed of 5-20 m/s. Most cases: 8-12m/s, Ucr=U/fD=20- 90
- Wind direction of 20°-60° to the bridge axis
- Cables inclined to horizontal 20°-45°
- With rainfall, mostly moderate
- Smooth cable surface
- Cable diameter of 80-200 mm
- Vibration frequencies of 0.3 3 Hz
- Amplitudes of vibration: 2D
- Structural damping low: <0.01 (0.16%)
- Cable behind pylon, declining in the direction of wind
- Low turbulence
- Vibrations orbit depends on the intensity of rainfall
Brotonne, France, 1979
Ben-Ahin and Wandre,
Belgium
Faroe Denmark
Second Severn Bridge; UK
Erasmus, Netherlands
Meiko-Nishi, Higashi Kobe,
Tempozan, Central
Meiko,Japan
Weirton-Steubenville, Fred
Hartman, Cochrane, East
Huntington, USA
Glebe Island Bridge, Australia
Allamillo, Spain
Dubrovnik, Croatia, 2002
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Rain-Wind Induced Vibrations
Mitigation measures:
Aerodynamic
(Image: H. Yamada)
Protuberated surface, Higashi-Kobe
Helical wire whirling, Vasco da Gama
Dimpled surface, Tatara
55
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Rain-Wind Induced Vibrations
Mitigation measures:
Structural
PTI Guide (2007): Sc0=m / D2>10; Sc0>5 with aerodynamic measures
Tabatabai & Mehrabi: >0.7%
(Image: Z. Savor)
56
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Indirect excitation
U sin tt
U sin sin tt
U cos sin tt
U sin sin td
U sin sin td U sin td
- External excitation
- Parametric excitation
57
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Indirect excitation
- External excitation
AB x,u
z,w
z (t)B
B'(t)w(x,t)
z(x,t) z (x,t)0
z (x,t)g
tztz BB sin)(
n
0kkkB
xksin)tsin()t()t(sinxz)t,x(z
1
B1
zmax)(
=0.1% 1=318 zB
58
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Indirect excitation
- External excitation
110,505m
AT T
B
Z sin t0
L=110.505 mm= 64.841kg/mE=210x109N/m2
T=4902.7x103N2=0.0727
F1=1.25Hz
0
0.4
0.8
1.2
1.6
0 0.05 0.1 0.15
Amplitude of oscillation at the support (m)
Ampl
itude
of o
scilla
tion
of th
e ca
ble
(m)
Non-linear model (0.5%)Non-linear model (1%)Numerical analysis (0.5%)Numerical analysis (1%)Linear model (0.5%)Linear model (1%)
59
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Indirect excitation
- External excitation
)sin()()(
)()()()()(
t1z2tX4
tX
3t2211t2t
2B
213
10
21
2
21
0
21
1242
11111
)sin()sin()( 2t2311a
X23tat 1
2
011
sinza1
B
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Indirect excitation
- External excitation
sinza1
B
0
0.4
0.8
1.2
1.6
0 0.05 0.1 0.15
Amplitude of oscillation at the support (m)
Am
plitu
de o
f osc
illatio
n of
the
cabl
e (m
)
Non-linear model (0.5%)Non-linear model (1%)Numerical analysis (0.5%)Numerical analysis (1%)Linear model (0.5%)Linear model (1%)
0
2
4
6
8
10
12
14
16
18
20
0 0.05 0.1 0.15
Amplitude of oscillation at the support (m)
Incr
emen
t of c
able
tens
ion
(%)
Non-linear model, 0.5%Non-linear model, 1%
222
B0 a42
zaTgm2AE
max
61
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Indirect excitation
- Parametric excitation
A B x (t)B
B'(t)x,u
z,w
t2xX2
tX4
t16
1X
2
tt2Xx1t2t
B0
213
10
21
221
2
0
21
10
B2
2211111
sin)()()(
)()sin()()(
Weightless taut string cable
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0
0.5
1
1.5
2
2.5
3
0 0.5 1 1.5 2 2.5 3
=2 .
(XB/2
X 0)
= 1/
1=0...5%
21222222 44121 )(
12
Indirect excitation
- Parametric excitation
0uut22u2u 32 )sin(
21
2
422
2
2
2
22 4414 )(
42
1
...),,( 21nn22
1=
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Indirect excitation
- Parametric excitation
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0 1 2 3 4 5 6
xB/(2
X0)
10
B 2X2x
Threshold amplitude for first parametric resonance:
64
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Indirect excitation
- Parametric excitation: amplitude of vibration
)sin()(21tat1
21
21
21
22
0
B420 4X2x11
3X4a )(
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
0.8 0.9 1 1.1 1.2 1.3 1.4
a
XB/2X0=0.3XB/2X0=0.05
1=1%
24
21
a122)(
tan
65
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Indirect excitation
- Parametric excitation: example
0
0.5
1
1.5
2
2.5
3
0 0.05 0.1 0.15
Amplitude of oscillation at the support (m)
Ampl
itude
of o
scilla
tion
of th
e ca
ble
(m)
Numerical, 1%
01020304050
60708090
100
0 0.05 0.1 0.15
Amplitude of oscillation at the support (m)
Incr
emen
t of c
able
tens
ion
(%)
Numerical, 1%
Analytical, 1%
110,505m
AT T
B
Z sin t0
0
5
0
Amp
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Indirect excitation
- Parametric vs external excitation
110,505m
AT T
B
Z sin t0
External, exc = c Parametric, exc = c, =1
0
0.5
1
1.5
2
2.5
3
0 0.05 0.1 0.15
Am
plitu
de o
f osc
illatio
n of
the
cabl
e (m
)
Amplitude of oscillation at the support (m)
Numerical, 0.5%Numerical, 1%Numerical, 2%Numerical, 1%
0
0.5
1
1.5
2
2.5
3
0 0.05 0.1 0.15
Am
plitu
de o
f osc
illatio
n of
the
cabl
e (m
)
Amplitude of oscillation at the support (m)
Non-linear model (0.5%)Non-linear model (1%)Numerical analysis (0.5%)Numerical analysis (1%)
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Indirect excitation
- Practical occurrence of parametric / external excitation
xB =2cm
Cable L (m) f1 (Hz) X0 (m) xB (mm) a (m)
%.201 %.501 %11 %.201 %.501 %11
V. Gama HC01 34.7 3.145 0.0783 0.6 1.6 3.1 0.43 0.43 0.43 HC24 226.0 0.576 0.7181 5.7 14.4 28.7 1.08 0.92 HC15 147.5 0.814 0.3946 3.2 7.9 15.8 0.89 0.86 0.70
Guadiana Central 1 168.5 0.763 0.5407 4.3 10.8 21.6 0.94 0.88 Central 16 49.5 3.239 0.2055 1.6 4.1 8.2 0.52 0.51 0.49
Normandy 440.9 0.257 1.0389 8.3 20.8 41.6 1.47
Ikuchi* 246.2 0.446 0.5025 4.0 10.0 20.1 1.14 1.07
xB (mm)
%.20.1 %.5%%0.1 %11
0.6 1.6 3.15.7 14.4 28.73.2 7.9 15.8
4.3 10.8 21.61.6 4.1 8.2
8.3 20.8 41.6
4.0 10.0 20.1
a (m)
%.20.1 %.5%%0.1 %11
0.43 0.43 0.431.08 0.920.89 0.86 0.70
0.94 0.880.52 0.51 0.49
1.47
1.14 1.07
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Indirect excitation
- Practical occurrence of parametric / external excitation/ cable-structure interaction
1pm-2pm: Cables 1 to 4 vibrating with large amplitude4pm: Cables 1 to 5 vibrating, cable 4 with very high amplitude (in and out-of-plane vibration in different modes)7pm: Wind velocity much lower, cable 4 persisting vibrating
9.30am: Strong vibration of cable4, dominant wind: North-NortheastNoon: Cable 4 vibrating strongly
Cable freq.:0.9Hz; Bridge freq.: 0.4 0.85 1.8 69
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Recommendations and guidelines for dynamic design of stay cables
EN 1993-1-11/ SETRA 2002
- Cable structures should be monitored after construction;
- Provisions should be made in the design to enable
implementation of suppressing measures (external
dampers, stabilizing cables);
- Rain-wind vibration must systematically be avoided (cable
texturing, added damping);
- Long cables (more than 80 m) should have a damping ratio
larger than 0.5%
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Recommendations and guidelines for dynamic design of stay cables
EN 1993-1-11/ SETRA 2002
- Parametric excitation effects should be assessed at the
design stage and overlapping of cable frequencies with
global structure frequencies should be avoided (20%
difference to s or to 2 s );
- The amplitudes of cable vibration should be limited. As
reference a maximum value equal to L/500 under a
moderate wind speed of 15 m/s.
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ViBest
Recommendations and guidelines for dynamic design of stay cables
PTI (2007)
- A minimum Scruton number of 10 (m / D2) is
recommended to prevent rain-wind induced vibration (5 if
aerodynamic measures present)
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ViBest
Cable Vibrations in Cable-Stayed Bridges
Part2: Mitigation
- Assessment of cables
- Implementation of measures for mitigation of cable vibrations
- Design of passive devices based on viscous dampers
- Examples
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