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Vol.1 No.1 EARTHQUAKE ENGINEERING AND ENGINEERING VIBRATOIN June 2002

Article ID: 1671-3664(2002)01-0074-12

Modeling of cable vibration effects of cable-stayed bridges

S.H. Chengand David T. Lau

Department of Civil and Environmental Engineering, Carleton University, Ottawa, Ontario, Canada

Abstract: The analysis of dynamic responses of cable-stayed bridges subjected to wind and earthquake loads generallyconsiders only the motions of the bridge deck and pylons. The influence of the stay cable vibration on the responses of the bridge

is either ignored or considered by approximate procedures. The transverse vibration of the stay cables, which can be significant in

some cases, are usually neglected in previous research. In the present study, a new three-node cable element has been developed to

model the transverse motions of the cables. The interactions between the cable behavior and the other parts of the bridge

superstructure are considered by the concept of dynamic stiffness. The nonlinear effect of the cable caused by its self-weight is

included in the formulation. Numerical examples are presented to demonstrate the accuracy and efficiency of the proposed model.

The impact of cable vibration behavior on the dynamic characteristics of cable-stayed bridges is discussed.

Keywords: cable-stayed bridges; cable vibration; dynamics; finite elements; long-span structures

1 Introduction

In recent years, the innovations in the design oflong-span bridges have led to the construction of manycable-stayed bridges of increasingly long span length.The dynamic behavior of modern long span, flexiblecable-stayed bridge structures subjected to transient

dynamic loads, such as wind, earthquakes and trafficloads, are very complex. The dynamic responses of acable-stayed bridge are the combinations of andinteractions between the motions of the bridge deck andthe pylons, as well as those of the stay cables. Under theconditions of heavy traffic loads or unfavorable weatherconditions of wind accompanied by rain, the amplitudeof the stay cable vibration can be large. The observationof significant rain-wind-induced cable vibration wasfirst reported by Hikami (1986) during the constructionof the Meiko-Nishi cable-stayed bridge in Japan, with amaximum amplitude of 0.55m.

In previous analytical studies of cable-stayed bridges(Fleming and Egeseli, 1980; Parvez and Wieland, 1987;Wetyavivorn, 1987; Abdel-Ghaffar and Nazmy, 1991;Nazmy and Abdel-Ghaffar, 1992), a common approachin the modeling of the cable load carrying behavior is tomodel the individual stay cable by a truss element in the

Correspondence to: David T. Lau, Dept. of Civil andEnvironmental Engineering, Carleton University, 1125

Colonel By Drive, Ottawa, Ont., Canada, K1S 5B6.Tel: (613) 520-2600 ext 7473, Fax: (613) 520-3951

Email: [email protected] fellow, Professor

Supported by:Natural Science and Engineering Research Councilof Canada

bridge finite element model. The sagging effect due tothe self-weight of the cable is taken into account byusing an equivalent Young's modulus. The single-trusselement model can give accurate results in static orquasi-static analysis of cable-stayed bridges. Fordynamic response analysis, the truss model does notaccount for the lateral vibration motions of the cable.Thus, the interaction effects between the lateralvibration behavior of the cables and the other parts ofthe bridge superstructure, as well as the transversevibration of the stay cable itself, are ignored.

To address the interaction and vibration effects ofthe cables, research works have been carried out toimprove the modeling of the behavior of stay cables.Causevic and Screckovic (1987) have developed amodel using linear springs with lumped masses torepresent the stay cables in the analysis. Anothercommon approach is the multi-link model, whichdiscretizes a stay cable into multiple elements. This

modeling approach of the stay cables has been adoptedby several researchers to analyze the dynamic behaviorof cable-stayed bridges due to wind and seismic loads(Baron and Lien, 1973; Yiu, 1982; Tuladhar and Brotton,1987; Abdel-Ghaffar and Khalifa, 1991; Tuladharet al.,1995). However, this modeling scheme significantlyincreases the number of degree-of-freedom of theanalytical model, especially when analyzing modernmulti-stay cable-stayed bridges. Although comparisonshave been made on the accuracy between the resultsobtained from the single-truss model and the multi-linkmodel in some of the recent research (Abdel-Ghaffar,

1991; Tuladhar et al., 1995), the impact of cablevibration effect on the dynamic responses of the cable-

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No.1 S.H.Cheng et al. : Modeling of cable vibration effects of cable-stayed bridges 75

stayed bridges is still not clear.The sensitivity of aerodynamic stability behavior to

the vibration of stay cables is an important issue forconsideration in the design of modern long-span cable-supported bridges. With the trend of increasingly longerspan cable-stayed bridges, the length of the stay cables

is also getting longer and thus becomes more susceptibleto undergo vibration motions. The vibration of thecables and their interaction with the motion of thebridge superstructure become even more significant. Itis therefore essential to consider the effects and evaluatethe impact of the cable vibration on the dynamicbehavior of modern long-span cable-stayed bridges inorder to ensure safe and high performance design andoperation.

This paper presents a detailed investigation of thecable vibration effect on the dynamic behavior of cablestayed bridges. The formulation of a new and moreefficient cable model of using newly developed three-node cable elements is presented. The new cable modelcan accurately simulate the vibrational behavior of thestay cables with significantly less number of degree-of-freedom than the multi-link model. The interactionbehavior between the cables, the pylons and the bridgedeck are considered by adopting the dynamic stiffness inthe modeling of the cables. The accuracy and reliabilityof the new cable model are demonstrated by numericalexamples and a case study of a long-span cable-stayedbridge. The impact of considering cable vibrationbehavior on the dynamic characteristics of the examplebridges is discussed in the paper.

2 Modeling of stay cable vibration

2.1 Transverse motion of stay cable

A flexible cable of length l fixed at both ends isshown in Fig.1(a). The mass per unit length of the cableis . For small amplitude vibration in the transversedirection, it is assumed that the tension, S, in the cableremains constant when the cable vibrates. Theequilibrium condition of an infinitesimal small segmentof the cable shown in figure 1(b) at time t can beexpressed as follows:

P

2 2

2 2 0y yt S xP

w ww w

(1)

wherex is the longitudinal direction of the cable, andyis the transverse direction normal to the length of thecable. By separating the variables in the linear equation,and assuming motion is harmonic with fixed-fixedboundary conditions, the n transverse vibration modeshape of the cable is expressed as follows:

th

2( ) sinnn

f l n xY x

n l

S (2)

where

2n

nf S

l

P

thn

( ) is the natural

frequency of the mode.

1,2,3,n

Fig. 1 (a) Stay cable fixed at both ends; (b) equilibrium of aninfinitesimal small segment of cable

The transverse displacements at an arbitrary pointon a stay cable in three dimensions, as shown in Fig.2,can be decomposed into two parts. The first part is thelinear displacements related to the nodal displacementsat the two ends of the cable, typically connected at oneend to the girder of the bridge deck and the other end tothe pylon. The second part is the displacements due tothe transverse vibration of the cable with the fixedboundary conditions at both ends. The second part of thedisplacements can be expressed as the summation of themodal components given in Eq. (2). Thus, the totaltransverse displacements of the cable can be expressedas follows:

Fig. 2 Transverse vibration of a stay cable

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EARTHQUAKE ENGINEERING AND ENGINEERING VIBRATION Vol.176

n

nyjiL

xnav

L

x1v

L

xxv

Ssin)()( (3)

n

nzjiL

xnaw

L

x1w

L

xxw

Ssin)()( (4)

where v , w are the displacements along the Y , Zaxes

in the local coordinate system, L is the length of the

cable, iv , iw , jv and jw are the nodal displacements of

nodes and along thei j Y , Z axes in the local

coordinate system, and yna , zna are the generalized

coordinates corresponding to the n modal shape

function terms. The number of modal coordinates to be

included in Eqs. (3) and (4) is determined by the

accuracy required in the analysis.

th

2.2 Longitudinal motion of stay cable

2.2.1 Static cable stiffness

The extensional behavior of the stay cable is anotherimportant component of its dynamic motion. In practice,the nonlinear static stretching stiffness, K , has theform as follows:

c

gec

111

KKK (5)

whereKe=EA/L is the elastic stiffness due to the elastic

stretch of the cable, K is the

geometric stiffness due to the change in the cable

geometry, where E is the elastic Young's modulus, A

is the cross-sectional area of the cable, L is the chordlength of the cable, L

)/(123h

23g LWT

h is the length of the horizontal

projection of the cable, T is the mean tension in the

cable, and Wis the weight per unit length.

2.2.2 Dynamic cable stiffness

When the bridge is subjected to dynamic loads, the

response of a stay cable is influenced by the movements

of the bridge deck and the pylon through its two end

supports. Thus, the dynamic behavior of the stay cable

is affected not only by the vibration of the cable itself,

but also by the motions of the bridge deck and the pylon.

The dynamic interaction between the stay cable and the

other parts of the superstructure should therefore be

considered in the dynamic response analysis. Further,

the longitudinal and the transverse motions in the plane

of the stay cable are dynamically coupled. If the tension

of the cable changes, the sag in the cable will change

which in turn will affect the transverse acceleration and

the inertia load in the cable. The geometric nonlinearity

of the stay cable resulted from the dynamic couplingbetween the motions in the transverse and the

longitudinal direction is considered in the model by the

dynamic cable stiffness.

The dynamic extensional cable stiffness at frequencyof the structure has been derived by Davenport

(1994), which has the form as follows:

Z

2

0ge

2

0e

)/()/(])/([1)(

ZZZZZ

KK1

KK (6)

where WTL // g0 SZ is the fundamental frequency

of the taut wire.

Fig.3 Dynamic cable stretch stiffness

The variation of the dynamic cable stiffness isplotted in Fig.3. The shaded area between the frequen-

cies and represents the region of the unstable

behavior of the stay cable, where

0Z Z1

)/(1 ge01 KK ZZ .

Within this frequency range, the stiffness of the cablebecomes negative.

Studies on the massive guy cables (Davenport andSteels, 1965) have shown that the dynamic stiffnesseffect has a significant impact on the dynamic behaviorof guy cable structures. At certain frequencies when the

effective stiffness becomes negative, a dangeroussituation occurs that instead of applying a restoringforce to the mast pole, the guy cable tends to pull it over.To the best of the author's knowledge, no detailed studyof the influence of the dynamic cable stiffness on thebehavior of the cable-stayed bridges has been reportedin the literature. It is of practical importance to knowhow significant the effect of the dynamic stiffness is ascompared to the static stiffness, and the possibility ofwhen the effective stiffness of the cable becomesnegative.

In order to more accurately account for the cablebehavior in the dynamic response of the cable-stayed

bridges, the formulation of the dynamic cable stiffnessis included in the cable model employed in the current

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study. The impact of the dynamic stiffness isinvestigated in detail in the case study of the NanpuBridge.

A spring-mass-spring system, modified from the

model used in previous studies of dynamic behavior of

guy cable structures (Davenport and Steels, 1965), as

shown in Fig.4, is considered for the derivation of thedynamic cable stiffness in the present investigation. For

convenience in the derivation of the longitudinal

displacement shape function of the cable, the cable mass

which participates in the longitudinal translational

motion is assumed to be lumped at the center point m of

the cable model. Appropriate constraints are applied to

the motion of the lumped cable mass when considering

the longitudinal motion of the stay cable. The stiffness

of the springs on the two sides of the lumped mass are

the geometric stiffness,Kg, and the elastic stiffness,Ke ,

respectively. The new three-node cable model is shown

in Fig. 5. The in-plane displacement ( )u x of anarbitrary point x on the cable relative to the

displacement of the cable support point at the pylon can

be expressed as follows:

( ) u x

2 ( )/ , 0 /2

2( )( )/ (2 )( )/ , /2

d d- d d

m j

m j i j

x u u L x L

L x u u L x L u u L L x L

(7)

where iu , mu and ju are, respectively, the longitudinal

displacements at points i, m and j.

2.3 Stiffness matrix of stay cable element

During dynamic responses, the motion of the stay

cable includes components of translational movement in

the transverse direction and extension of the cable in the

longitudinal direction. For the derivation here, the first

three modal coordinates are considered in the modal

series in Eqs. (3) and (4) to express the transverse

displacements v and w along the Y and Z axes,

respectively. The displacements at an arbitrary point of

the cable element can be expressed in the matrix form as

follows:

Fig. 4 Spring-mass-spring system for modeling longitudinaldynamic behavior of stay cable

Fig. 5 Displacement parameters of three-node stay cable model

> @ e,, GG Twvu NNN (8)where

eGT

332211

zyzyzy aaaaaauwvuwvu mjjjiii

is the displacement parameter vector, T],,[ wvuGis the displacement vector,Nu, Nv andNw are the shape

function matrices for displacements u , v and w , respe-

ctively:

Nu =

-

dd

dd

LxLLxLLLx

LLL

2/],000000/)(200100/)/2[(

/2x00],00000)/x(200)/x(-200[0

Nv = )/sin(00/00)/1(0[ LxLxLx S ]0)/3sin(0)/2sin(0 LxLx SS (9)

Nw = )/sin(00/00)/1(0[ LxLxLx S ]0)/3sin(0)/2sin(0 LxLx SS

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The stiffness matrix of the cable element of size

13 13 can be derived by calculating the strain energy

of the cable due to elongation and change in geometry.The non-zero terms in the lower-triangular part of the

symmetric cable element stiffness matrix are given as

follows:

e7111 KKK

22 33 55 66 52 63 /K K K K K K T L

L

)L

g7444 KKK

ge77 KKK

2

88 99 /(2 )K K T LS 2

10,10 11,11 2 /K K TS 2

12,12 13,139 /(2K K TS (10)

2.4 Mass matrix of stay cable element

For the modeling of the longitudinal motion, thetranslational mass related to the extensional behavior ofthe cable is assumed to be lumped at the center point m.

The mass related to the transverse vibration aredetermined from the kinetic energy of the transverse

motion, i.e., T L

AL0

)(2)/(1 U xwv d)(

22

, where

is the mass density. The total mass matrix

U

cM of the

cable element can be obtained by combining thecontributions to the mass effect from all the cable

vibration motion components. The non-zero terms in the

lower-triangular part of the symmetric cable elementmass matrix are given as follows:

22 33 55 66 / 3M M M M ALU :

25 36 / 6M M ALU

77 1M

88 99 10,10 11,11 12,12 13,13

/ 2

M M M M M M

ALU

82 85 93 96 /M M M M ALU S

10,2 10,5 11,3 11,6 /(2 )M M M M ALU S

12,2 12,5 13,3 13,6 /(3 )M M M M ALU

S (11)

3 Numerical examples

To verify the validity and reliability of theproposed three-node cable model, the numericalexample of a three-span cable-stayed bridge used in aprevious study by Tuladharet al.(1995) is presented inthis section. Modal analysis of the bridge is carried out.The dead load of the bridge is calculated based on theunit weight of the bridge deck provided in the reference.In the present study, the emphasis is placed on thecomparison of the different modeling schemes of the

transverse motion of the cable, and the influence of thecable vibration effect on the dynamic modal propertiesof the bridge.

The geometry and details of the mesh of theexample bridge are shown in Fig.6. The bridge has asymmetric layout with a main span of 371m and twoside spans of 159m each. The bridge has a steel deckand concrete pylons.

Fig. 6 (a) Geometry of example bridge; (b) Mesh details of

example bridge

nalytical modeThree different a ls are employed inthe

model

Co from the twodiff

comparison study. The bridge deck and pylons in allthree models are modeled by three-dimensional beamelements. The stay cables are modeled by the proposednew three-node cable elements, the multi-link elements

and the single-truss elements in separate analyses,respectively. The tension forces in the cables areobtained from the dead load condition. The motion ofthe bridge is assumed to be restricted in the plane of thestructure, i.e., only three degrees-of-freedom (twotranslations and one rotation) are considered at eachnode of the finite element model of the bridge.

3.1 Comparison between the three-nodeand the single-truss model

mparing the results obtainederent models, numerous additional coupled cable

vibration modes are identified by the three-node model.

These cable vibration modes show up as additionalvibration modes between the frequencies of the primarystructural modes of the bridge structure. Shown in Fig. 7are the mode shapes of some typical correspondingmodes obtained from the new three-node cable modeland the single-truss cable model. The impact on themode shapes due to the vibrational behavior of thecables is clearly demonstrated. The mode shapesobtained from the three-node cable model include themotions not only of the deck and the pylons, but also ofthe stay cables. In the case of the single-truss cablemodel, only the motions of the deck and the pylons areconsidered. The cables in the mode shapes by the single-truss model remain straight as the vibrational behaviorof the cables is ignored. Fig. 8 shows the mode shapes

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of the additional vibration modes involving cablevibrations given by the three-node cable model. The

(a) (b)

Fig. 7 Vibrat tayed bridgeion mode shapes of example cable-s

(a) Single-truss cable model; (b) Proposed three-node cable model

odal frequencies obtained by the proposed three-nodemcable model and the single-truss cable model are plottedagainst the modal order in Fig.9. The relative flatregions on the curve of the three-node cable modelresult represent the additional vibration modes involvingcable vibrations.

Comparing the frequencies of the cable vibrationmodes with the frequencies obtained from the cable

vibration formula2l

n

nf S P ( 1,2,3,n ), as

shown in Table 1, it th modal

frequencies of the stay cables of the example bridge are

similar to the analytical values, which indicates that theadditional cable vibration modes are developed without

significant movement or deformation in the other partsof the bridge superstructure.

For the example bridge, w

is noted that e first three

hich has a steel deck, it isnoted that the fundamental frequencies of the stay cables

occurred between 0.484 Hz to 1.19 Hz are within thefrequency range of the first ten modes of the bridge,which is between 0.349 Hz and 1.97 Hz. This overlap inthe frequency range illustrates the importance ofconsidering the cable vibration behavior in the dynamicresponse analysis of the example cable-stayed bridge.Depending on the external load conditions, the impactof cable vibration effects on the bridge response can besignificant. For example, the total damping property ofthe bridge is the combination of structural damping andaerodynamic damping effects. The cable vibrationmodes are important for consideration of the windstability of the structure, as well as in suppressing thebridge vibration movements through contributions to theaerodynamic damping effect.

Fig.8 Additional vibration modes involving

cable vibration effects

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Fig. 9 Impact of cable vibration effect on dynamic characteristics

of example bridge

3.2 Comparison between the three-node model

e efficiency of thepro

pu Bridge

ayed bridgeloc

and the multi-link modelTo further demonstrate th

posed three-node cable model, the example bridge isalso analyzed by using multi-link elements to model thecables. Each cable is divided into two links, four linksand eight links, respectively, in three separate analyses.The fundamental cable frequencies obtained from theanalytical solution, the three-node model and the multi-link models are presented in Table 2. Results show thatfor the example here, each cable should be divided intoat least eight links in order to achieve accurate results.The cable vibration effect can be modeled accurately by

the eight-link model. The first twenty-two naturalfrequencies obtained from the three-node model and theeight-link model, including the bridge frequencies andall the cable fundamental frequencies, are presented inTable 3. There is good agreement between these twosets of results. The total number of degree-of-freedomof the three-node cable model is 171, whereas that of theeight-link model is 387. Typical of modern multi-staycable-stayed bridges, there can be more than a hundredstay cables in one bridge. If the multi-link element isemployed to model the stay cables, the number ofdegree-of-freedom of the analytical model is thereforesignificantly higher than that of the proposed cable

model. It not only requires more effort in datapreparation, but also increases significantly the cost incomputer time and storage.

4 Case study of the Nan

Nanpu Bridge is a long-span cable-stated in Shanghai, China. It has a center span of 423

m, as shown in Fig.10. The bridge girder has acomposite deck cross-section comprised of a prestressedconcrete slab supported by two I-shaped steel beams.The impact of cable vibration effect on the vibrationcharacteristics of the bridge is studied. The influence of

the dynamic cable stiffness on the extensional behavior

Fig. 10 Structural layout of Nanpu Bridge,China (symmetric half)

of the stay cables is investigated by comparing thebehavior of the longest and the shortest stay cables inthe bridge.

4.1 Modeling of the bridge

Two different numerical models are employed tostudy the cable vibration effect on the dynamiccharacteristics of the Nanpu Bridge. In these two models,the main girder of the bridge is modeled by the triple-beam finite element model (Cheng, 1999), which cantake into account the warping torsional stiffnessbehavior of the bridge deck cross-section.

The mass and stiffness properties of the bridge deckcross-section are modeled by a central beam and twoside beams in the triple-beam finite element model.Each of these three beams are then further divided into

ninety-four 3D beam elements in the longitudinaldirection separated by the cable support points, thesupports of the piers and the pylon locations. Each ofthe two H-shaped pylons is modeled by sixty-nine 3Dbeam elements. The difference between the two modelsconsidered here is due to the difference in the modelingof the stay cables. In the first model, the stay cables aremodeled by single-truss elements, whereas in the secondmodel, the cables are modeled by the proposed newthree-node cable elements.

4.2 Modal analysis

The modal analysis results obtained from these twomodels are presented in Table 4. The results show thatby considering the cable vibration in the dynamicbehavior of the Nanpu Bridge, additional cable vibrationmodes as well as new cable-deck coupled modes areidentified. The new cable-deck coupled modes involvethe transverse motions of some stay cables in the cableplane and the torsional behavior of the main girder inthe center span. Comparing the two sets of results, it isnoted that the natural frequencies of somecorresponding pairs of bridge structural modes are close,with those obtained from the three-node cable modelslightly lower due to increases in the modal masses of

the bridge system contributed from the vibrating cables.

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Table 1 First three modal frequencies of stay cables in the example bridge

/ Hz1 2 3f

0.7601 0.7526 (8) 1.5202 1.5226 (27) 2.2803 2.2833(38)2

0.7588 (9)

3 1.1332 1.1342 2.2 3 2.271 (37) 3.3995 3.4004 (48)

0.5117 0.5091 (5) 1.0234 1.0240 (16) 1.5351 1.53726

7 0.4853 0.9706 0.9707 (15) 1.4559 1.4558

1.1402 1.1410 2.2 3 2.286 (39) 3.4205 3.42179

10 1.1849 1.1847 (21) 2.3697 2.3719 3.5 6 3.555 (51)

0.5122 1.0243 1.0250 (17) 1.5365 1.537212

(19) 66 8

4 1.1913 1.1913 (22) 2.3826 2.3866 (42) 3.5739 3.5736 (52)

0.8218 0.8206 (11) 1.6435 1.6439 (32) 2.4653 2.4618 (43)5

2.4678 (44)

(29)

0.5147 (6) 1.5454 (30)

0.4888 (4) (25)

0.7622 0.7527 (3) 1.5244 1.5259 (28) 2.2865 2.2860 (39)8

0.7588 (9) 2.2915 (40)

(20) 80 0 (49)

2.2915 (40)

(41) 54 6

0.8212 0.8131 (10) 1.6423 1.6417 (31) 2.4635 2.4618 (43)11

0.8239 (12) 2.4678 (44)

0.5091 (5) (29)

0.5147 (6) 1.5454 (30)

e modal ord

f / Hz f / HzCable No.

Analytical Three-node Analytical Three-node Analytical Three-node

1 0.4838 0.4844 (3) 0.9676 0.9677 (14) 1.4515 1.4514(24)

Note: Numbers in the brackets indicate th er.

Table 2 Comparison of the fundamental frequencies of the stay cables in the example bridge

Multi-link element /HzAnalytical Three-nodeCable No.

cab z 2 links 8 links/Hz le element /H 4 links

23

4

5

6

7

8

9

10

11

12

0.76011.1332

1.1913

0.8218

0.5117

0.4853

0.7622

1.1402

1.1849

0.8212

0.5122

1.1342

1.1913

0.8206

0.4888

1.1410

1.1847

0.5091-0.5147

0.6827-0.68601.0208

0.7391-0.7415

0.4589

0.6827-0.6860

1.0208-1.0267

1.0653-1.0714

0.7391-0.7415

0.4623

0.7364

1.1623

0.8013

0.4633-0.4810

0.7408

1.1554

0.7991

0.74961.1277

1.1842

0.8163

0.4813-0.4876

0.7546

1.1347

1.1775

0.8139

1 0.4838 0.4844

0.7527-0.7588

0.5091-0.5147

0.7527-0.7588

0.8131-0.8289

0.4348-0.4364

1.0653-1.0714

0.4348-0.4364

0.4633-0.4810

1.1013-1.1083

0.4975-0.5029

1.1013-1.1176

0.4975-0.5029

0.4813-0.4876

0.5067-0.5125

0.5067-0.5125

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Table 3 Comparison of three-node cable model and multi-link cable model modal analysis results

Natural frequency /Hz

ModeProposed thre link element

Mode shape

No. e-node Multi-cable element (8 links per cable)

1

8-10

14-17

19-22

0.3481

0.4844-0.5147

0.7526-0.8131

0.9677-1.0250

1.1342-1.1913

0.3465

0.4813-0.5125

0.7496-0.8139

0.9530-0.9997

1.1277-1.1842

Heave (girder, Sym.)

H .)

Hea

H

Hea

H

2

3-6

7

11

12

13

18

0.4615

0.5892

0.8206

0.8289

0.9240

1.1017

0.4584

0.5895

0.8259

0.8306

0.9204

1.1110

eave (girder, Anti-Sym

Pure cable motion

ve (girder, Anti-Sym.)

Pure cable motion

eave (girder, Sym.)

Pure cable motion

ve (girder, Anti-Sym.)

Pure cable motion

eave (girder, Sym.)

Pure cable motion

Table 4 Impact of cable vibration behavior on the dynamic characteristics of Nanpu Bridge

Single-

c

truss Proposed three-node

able element cable element

Mode No. ency/Hz Mode No. ency/Hz

Mode shape

Frequ Frequ

1 0.1216 1 0.1216 Floating (girder, Anti-Sym.)

2 0.3433 2 0.3422 Heave (girder, Sym.)

3 0.3687 3 0.3679 Sway (girder, Sym.)

4 0.4255 4 0.4246 Heave .)

6-12 0.4758-0.4767 Pure cable motion (# C22)


5 0.4992

Be .)

23-30 0.5256-0.5262

0.5392


40-47 0.5464-0.5474 Pure cable motion (# S21)


56-63 0.6048-0.6058 Pure cable motion (# S20)

8 0.6106

(girder, Anti-Sym

5 0.4696 New mode *

13 0.4829 New mode *

21 0.4951 Torsion (girder, Sym.)

6 0.5069 22 0.5042 nding (pylon, Anti-Sym

Pure cable motion (# S22)

7 31 0.5376 Bending (pylon, Sym.)

39 0.5448 New mode *

48 0.5537 New mode *

55 0.5599 New mode *

64 0.6073 Torsion (girder, Anti-Sym.)

* w mode: Domina vibration couple th torsion of ma at the center spaNe nt cable d wi in girder n.

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Fig. 11 shows the influence of the cable vibrationeffect on the dynamic characteristics of the NanpuBridge in terms of modal order. The frequencies of thefirst 20 modes obtained from the single-truss cablemodel and the first 200 modes obtained from the three-

node cable model are plotted against the mode number.It can be observed that in the same frequency range of0-0.8 Hz, there are 14 modes extracted from the single-truss cable model, whereas more than 200 modes areidentified by the three-node cable model. Thus, byconsidering the cable vibration behavior in the modalanalysis, 186 new modes have been extracted in thefrequency range below 0.8 Hz.

Fig. 11 Impact of cable vibration effect on dynamiccharacteristics of Nanpu Bridge

The mode shapes of some typical coupled cable-deck vibration modes obtained by using the three-nodecable model are shown in Fig.12. The interactions of thecable vibration behavior with the motions of the maingirder and the pylons are clearly shown.

Fig. 12 Mode shapes of the Nanpu Bridge usingthree-node cable model

For the case of pure cable vibration modes, thefreq

sets of results is observed. The difference between the

analytical and the modal analysis results is due to the

Fundamental frequency/Hz

uencies of these modes in the center and side spansof the Nanpu Bridge are compared with the analyticalvalues, as shown in Table 5. Altogether ten stay cables,

five in the center span and another five in the side spans,are studied. Fairly good agreement between these two

difference in the boundary conditions. The analyticalsolution is based on the fixed-fixed end boundaryconditions of the cables, whereas in the modal analysis,the two ends of the cable move with the deck and the

pylons.

Table 5 Fundamental frequencies of some longeststay cables in the Nanpu Bridge

Cable

e-node

cable element

No.* Analytical Proposed thre

C22

C21

C20

0.4762

0.5418

0.4758-0.4767

C19

C18

S22

S21

S20

S19

S18

0.4894

0.5593

0.6202

0.5278

0.5495

0.6078

0.6102

0.6574

0.4890-0.4901

0.5407-0.5419

0.5571-0.5577

0.6171-0.6203

0.5256-0.5262

0.5465-0.5474

0.6048-0.6058

0.6105-0.6120

0.6587-0.6599

sest to . The sym denotes the cabl

ile 'S' den e cables in the si

* The numbering sequence of the cables starts from the one

clo the pylon bol 'C' es in

the center span, wh otes th de

span.

According to the analytical solution. the fundamentalfrequency of the longest cable in the Nanpu Bridge is0.476 Hz, whereas that of the shortest is 1.58 Hz. Thecable frequencies overlap with the frequency range ofmode 5 to 45 of the bridge structure, i.e., between 0.426Hz and 1.61 Hz. For dynamic motions of the bridge inthis frequency range, the vibration of the cables andtheir interaction with the motions of the deck and thepylons can be significant, which can affect the dynamicresponse of the bridge.

4.3 Dynamic cable stiffness

The dynamic cable stiffness accounts for the

interaction effect between the motion of the cables andthat of the other parts of the bridge superstructure. Itinfluences the extensional behavior of the cable bymodifying its longitudinal stiffness. To get an insight onthe significance of the interaction effect in the case ofthe Nanpu Bridge, the dynamic cable stiffness of thelongest and shortest cables of the Nanpu Bridge aredetermined for the bridge modes of natural frequenciesat 0.1 Hz, 0.2 Hz, 0.5 Hz and 1.0 Hz. the cable tensionsare obtained under the dead load condition, The resultsare presented in Table 6.

For the case of the longest cable, there is no

significant change in the extensional stiffness of thecable by including the interaction effect. For theextensional stiffness of the cable to reach the critical

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stat

Shortest cable

e of zero, the corresponding bridge frequency for thelongest cable is evaluated to be 7.76 Hz, and for theshortest cable is 105.6 Hz. The higher frequency islikely beyond the bridge frequency range that is of mostconcern to dynamic response analysis of bridges due towind. The results seem to indicate that the impact of

dynamic stiffness on the extensional behavior of the staycables in cable-stayed bridges is not as significant asthat pointed out by Davenport and Steels (1965) formassive guy cables. However, in reality, the stay cablesmay be subjected to deterioration due to corrosion andfatigue problems. This may affect the dynamic stiffnessbehavior of the stay cables. There are a number of casesreported that cable replacements are required due totension loss caused by corrosion and fatigue damage inthe cables. These two phenomena happen simulta-neously and reinforce each other. The tension in thecable can drop significantly due to damage to the wiresinside the stay cable. Under such conditions, thedynamic cable stiffness may have a significant impacton the longitudinal behavior of the cable.

Table 6 Effect of dynamic stiffness on the extensional stiffnessof the longest and shortest cables in the Nanpu Bridge

Item Longest cable

)m/(N 1e

aK 71.0119 10u 73.0943 10u

)1gaK m/(N

83.8815 10u 102.3517 10u

)m/(N 10

Ka70.9862 10u 73.0902 10u

0.1 2Z S u

7

0.9862 10u

7

3.0902 10u0.2 2Z S u 70.9862 10u 73.0902 10u( )dK Z

)m/(N 1 0.5 2Z S u

1.0 2Z S u

70.9861 10u70.9812 10u

73.0902 10u73.0902 10u

A 3 2/ 0 )

K

c s

/a eK Ed

, 12 [( ) ]b g hK T A LJ3 , /(c e g e g K K K K K

2 2

th e

L ,

Z

To stu ss on thedynamic behavior of the stay cables, the dynamicstiffness of e longest and the shortest cables of thNanpu Bridge are investigated for the cases of t nsion inthe

ith a f quency 2 Hz,ss

ue of 0.8

a

as fatigue related damage which results inten

vibration effect on the dynamicayed bridges is studied in this paper.

ble element has been developed.The

strength steel cables may lie withinthe

( ) ( ) /[ ( )]e g e g K K K m K mZ Z

dy the effe t of the ten ion lo

ecable taken as

DL

0.75T ,DL

0.50T andDL

0.25T res-pectively, where DLT is the initial cable tension causedby the dead load. The results are shown in table 7.

If the remaining cable tension is DL0.50T , when thebridge vibrates w re of 0 theextensional stiffne of the longest cable considering thedynamic interaction may drop by 15.2% to a val

.

nted in37107

N/m. From the data prese table 7, itcan be noticed that the value of the dynamic cablestiffness depends more on the tension in the cable thanon the external exciting frequency of the bridge motion.A negative value of the extensional stiffness of thelongest cable occurs under the condition when the

remaining tension force is DL0.25T and the oscillatingfrequency of the bridge is 1.0 Hz. Physically, this

negative stiffness indicates that instead of resisting theelongation of the cable, the deform tion tend to increase.The cable then is under an unstable condition. No suchnegative stiffness has been found in the case of theshortest cable for the same combination of the cabletensions and the bridge frequencies. The effect of the

dynamic interaction between the cables and the otherparts of the superstructure on the extensional stiffness ofthe short cables is significantly less than that on the longcables.

The results show that in the dynamic analysis ofcable-stayed bridges, if the cable is significantlycorroded or h

sion loss in the cable, the dynamic stiffness effectshould be considered, especially for the long stay cablesin cable-stayed bridges.

5 Conclusions

The impact of cablebehavior of cable-stA new three-node ca

new cable model can accurately model thetransverse motion of the cable, and the interactionbetween the motions of the cables and the other parts ofthe superstructure by adopting the dynamic stiffness inthe longitudinal direction. The new cable model iscomputationally more efficient than the multi-linkapproach, especially in the analysis of modern multi-stay cable-stayed bridges. The validity and accuracy ofthe proposed model have been verified by numericalexamples and the case study analysis of the NanpuBridge in China.

Results show that the natural frequencies of thecable vibration modes in cable-stayed bridges withconventional high

frequency range of the lower bridge structural modes.The transverse vibration motions of the cables caninfluence the dynamic response of the bridge and thusshould be considered in the dynamic response analysis.By considering the cable vibration behavior, the naturalfrequencies, mode shapes and modal order of the cable-stayed bridges are significantly altered. A large numberof additional modes involving cable vibrations as well

as new cable-deck coupled modes are identified byusing the proposed three-node cable model. Thesemodes cannot be predicted by the conventional single-truss cable model. The additional cable vibration modescan have a significant impact on the overall dampingproperties of the bridge system by contributing to theaerodynamic damping behavior of the system. Theanalysis results also show that the significance of thedynamic cable stiffness effect depends on cable tension.If there is no significant tension loss due to corrosion orfatigue problems of the cables, the influence of thedynamic cable stiffness from the longitudinal behavior

of the cables may be negligible and not as significant assuggested by previous studies of guy cables.

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Table 7 Impact of cable tension v

Tension

ariation the dynamic cable stiffness

/ (107Nm-1)

on

Dynamic cable stiffness ( )K ZZ-1/ N / ( rad s ) Longest cable Shortest cable

0.1 2Su 0.9530 3.0847

0.2 2Su 0.9529 3.0847

0.9524 3.0847

0.9507 3.0847

0.8370 3.0620

0.5 2Su

0.75DL

T

1.0 2Su

0.1 2Su

0.2 2Su 0.8365 3.0620

0.8323 3.0620

0.8156 3.0620

0.3766 2.8538

0.5 2Su0.5 DL0T

1.0 2Su

0.1 2Su

0.2 2Su 0.3689 2.8538

0.3092 2.8538

-0.3964 2.8538

0.5 2Su0.2 DL5T

1.0 2Su

43.Acknowledgements

by the National Sciences anduncil of Canada to the second

aut

, A.M. and Khalifa, M.A., (1991),Cable Vibrations in Dynamics of Cable-

of Cable-stayed Bridges,

Bridge, Computers and Structures,

Bridge Cables: Effects on Bridge

Analysis of Long-Span Cable-Stayed Bridges,

nd,

r of Massive Guy Cables,Journal of Stru-

cture Division, ASCE, 91, (ST2):

Fleming, J.F. and Egeseli, E.A., (1980), Dynamic

f Cables in Cable-

ffects

dal Dynamics, 21: 1-20.

Cable-stayed Bridges,

able-stayed Bridges,

ingdom.

The support providedEngineering Research Co

hor for this research gratefully acknowledged.

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