Cable Vibrations in Cable-Stayed Bridgesthost-iabse-elearning.org/l23new/data/downloads/... · Cable Vibrations in Cable-Stayed Bridges Part 1: Assessment Part 2: Mitigation 2 ViBest

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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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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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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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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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Lézardrieux Bridge , France, 1925Span: 112 mDesign: Leinekugel le Coq

(Images: Structurae: Nicolas Janberg; K. Todeschini)7

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Donzère Mondragon Bridge , France, 1952Span: 81 m; Length: 160 mDesign: Albert Caquot

(Images: Structurae: Jacques Mossot; Nicolas Janberg)8

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Stromsund Bridge, 1956Span: 182 m; Length: 182 m; Materials: Steel; Design: Franz Dischinger

(Images: Andreas Stedt)9

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(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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(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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(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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(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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(Images: Olivier Flamand; Philip Bourret in Structurae;) 15

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(Videos: H. Tabatabai; E. Caetano; others, available in internet) 16

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Wind Loads on Stay Cables

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- Buffeting

- Vortex-shedding

- Galloping

- Wake Effects

- Rain-Wind Induced Vibrations

- Dry Galloping

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- 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)

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- 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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- 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

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- 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

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- 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


- 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



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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)()()( tqKtqCtF aaa)()(

)()(

)()(

)(

.

.

.

tftm

tftf

tftf

tF

qq

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

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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

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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

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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

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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

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Galloping

- Other mitigation measures: Aerodynamic

Kubo et al. (2003) Matsumoto et al. (1995)

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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

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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

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Rain-Wind Induced Vibrations

(Video: Ben-Ahin Bridge, available in internet)50

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- 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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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

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(Image: Yamaguchi (1990))


- 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

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(Image: Peil & Nahrath (2003))


- Peil and Nahrath (2003)

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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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Mitigation measures:

Aerodynamic

(Image: H. Yamada)

Protuberated surface, Higashi-Kobe

Helical wire whirling, Vasco da Gama

Dimpled surface, Tatara

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Mitigation measures:

Structural

PTI Guide (2007): Sc0=m / D2>10; Sc0>5 with aerodynamic measures

Tabatabai & Mehrabi: >0.7%

(Image: Z. Savor)

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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

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Indirect 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

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Indirect 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%)

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Indirect 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


sinza1

B

0

0.4

0.8

1.2

1.6

0 0.05 0.1 0.15


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


Incr

emen

t of c

able

tens

ion

(%)

Non-linear model, 0.5%Non-linear model, 1%

222

B0 a42

zaTgm2AE

max

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Indirect 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


0uut22u2u 32 )sin(

21

2

422

2

2

2

22 4414 )(

42

1

...),,( 21nn22

1=

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Indirect 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:

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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

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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


Ampl

itude

of o

scilla

tion

of th

e ca

ble

(m)

Numerical, 1%

01020304050

60708090

100

0 0.05 0.1 0.15


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

)


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

)


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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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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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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Part2: Mitigation

- Assessment of cables

- Implementation of measures for mitigation of cable vibrations

- Design of passive devices based on viscous dampers

- Examples

73