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Steel Structures 8 (2008) 277-284 www.ijoss.org

Analysis of Local Vibrations in the Stay Cables of the

Hizen-Takashima Bridge

Kensuke Tanaka1, Kazuo Takahashi2,*, and Shozo Nakamura2

1Graduate School of Science and Technology, Nagasaki University, 1-14, Bunkyo-machi, Nagasaki 852-8521, Japan2Department of Civil Engineering, Nagasaki University, 1-14, Bunkyo-machi, Nagasaki 852-8521, Japan

Abstract

Local parametric vibrations of the stay cables of the Hizen-Takashima Bridge, which is currently under construction, arestudied. This bridge is a composite cable-stayed bridge with a steel girder and concrete towers. The natural frequencies of theglobal vibration modes are obtained using a three-dimensional FEM model. Relations between global natural frequencies andthe local natural frequencies of the stay cables are first demonstrated and the results are compared with other cable-stayedbridges. The local vibrations of the stay cables at the support during global motion are obtained and the properties of the localvibrations are discussed.

Keywords: cable-stated bridge, stay cable, local vibration, cable vibration

1. Introduction

Previous studies have shown that local vibrations of

large amplitude are induced in the stay cables of cable-

stayed bridges under wind and traffic loading (Fujino et

al., 1997). This phenomenon has been confirmed in

vibration tests on the Hitsuishijima Bridge (Okauchi et

al., 1992), the Yokura Bridge (Fujino et al., 1993) and the

Tatara Bridge (Manabe et al., 1999) in Japan. These

vibrations are considered to be local parametric vibrations

(i.e., dynamic instabilities) in the stay cable due to excitation

at the support by girder and/or tower oscillations. Since

multi-cable systems are widely used in cable-stayed

bridges, the natural frequencies of global vibration modes

can be close to the natural frequencies of the cables, so

bridges are prone to large-amplitude cables vibration. It is

clear that further research is necessary on the parametric

vibrations of cables in cable-stayed bridges.

Kovacs (Kovacs, 1982) was the first to point out the

possibility of such parametric vibrations. The problem

has been analyzed recently from a number of perspectives

(Fujino et al., 1993; Nagai et al., 1992; Pinto da Costa et

al., 1995; Takahashi et al., 2000). Generally, large-amplitude

vibrations of the cables in a cable-stayed bridge can be

induced when the natural frequency of global vibration

modes is either close to that of the cables (the second

unstable region) or twice that of the cables (the principal

unstable region). In their study, the cables were given

periodic time-varying displacement at the supports.

However, the study did not explain local vibrations in the

cables under environmental and service loadings (such as

wind and earthquake) in terms of the vibration characteristics

of the whole bridge system.

In this paper, the authors analyze the Hizen-Takashima

Bridge (tentative name) using finite cable elements

(Kimura et al., 2005) to investigate parametric vibrations

of the stay cables. In the finite element model of a cable-

stayed bridge, each stay cable is represented by a certain

number of finite cable elements so that parametric

vibrations can be included in the stay cable vibrations.

The overall response of the girder, towers and stay cables

can be simultaneously obtained by using a whole-bridge

approach.

The dynamic characteristics of cable parametric vibrations

in a steel cable-stayed bridge are discussed using this

whole-bridge approach. Because parametric vibrations of

the stay cables may occur under periodic excitation,

dynamic analysis is first performed for the case where the

cable-stayed bridge is subjected to periodic sinusoidal

excitation. Comparing the responses obtained by this whole-

bridge method with those obtained using the approximate

approach, the effect of cable vibrations, including parametric

vibrations, on the overall dynamic characteristics of a

composite cable-stayed bridge with a steel girder and

reinforced concrete (RC) towers is discussed in detail.

Furthermore, the possibility of using chord elements to

simulate parametric of stay cable is noted.

Note.-Discussion open until May 1, 2009. This manuscript for thispaper was submitted for review and possible publication on Septem-ber 12, 2008; approved on November 21, 2008

*Corresponding authorTel: +81-95-819-2610; Fax: +81-95-819-2627E-mail: [email protected]

278 Kensuke Tanaka et al.

2. The Bridge

The Hizen-Takashima Bridge (tentative name) is an

isolated island bridge. It crosses the Hibi strait, connecting

Hizencho, Karatsushi, Saga and Takashimacho, Matsurashi,

Nagasaki. As shown in Fig. 1(a), the main span is 400 m

and the two side spans are 220 m. The bridge is 9.75 m

wide, as shown in Fig. 1(c). It is scheduled for

completion in 2007. The purpose of the bridge is to

promote sightseeing and industrial development in the

area.

A composite cable-stayed bridge with a steel girder and

RC towers was chosen to minimize construction costs, as

shown in Figure 1(c). The towers are of the inverted-Y

design and the cables consist of a two-plane multiple-

cable system arranged in a fan pattern. The bridge data

are summarized in Table 1. The cable numbers are shown

in Fig. 1(a). The cables are numbered sequentially from

the side span to the main span.

3. Analysis Model and Dynamic Analysis Procedure

3.1 Analysis model

Three-dimensional finite element models are used to

model the girder and towers. The girder is treated as a

single central spine with offset links to the cable anchor

points. Two approaches are employed for the cables in

this analysis: a divided model and a non-divided model.

The stay cables in the divided model are split into eight

cable elements (Kimura et al., 2005) using a method

proposed by the authors. The non-divided model uses a

single string element for each cable. The equivalent

modulus concept proposed by Ernst (Ernst 1965) is used

to describe the behavior of the stay cables. The natural

vibration of each stay cable can be obtained by the

equation of motion of a single cable (Pinto da Costa et

al., 1995).

Figure 1. General view of the Hizen-Takashima Bridge unit (mm).

Table 1. Data of the bridge

Length of bridge (m) 840

Center span (m) 400

Side span (m)Left 220

Right 220

Width of bridge　(m) 9.75

Height of girder (m) 2

Shape of main tower Reversed Y

Height of main tower (m) 105

Cable Fan shape (two plane)

Analysis of Local Vibrations in the Stay Cables of the Hizen-Takashima Bridge 279

Table 2 shows the number of nodes and the number of

elements used in this study.

The effect of pier foundations on their displacement

and rotation in the longitudinal and out-of-plane directions

is modeled by a linear spring. Rubber bearings for

distributing earthquake shear force are installed at the

joints between girder and piers and these are modeled by

linear springs.

The mass of each member is concentrated at the nodal

points as a lumped mass. In addition, the beam elements,

which consist of geometric non-linear beam elements,

and the material characteristics take account only of the

elastic domain so as to make linear analysis applicable.

Table 3 shows the material characteristics of the beam

elements used in the study.

3.2. Analytical technique

3.2.1. Modeling of the cable

Because the natural frequency of a cable is not obtained

by FEM analysis when it is modeled by a string element,

the natural frequency of stay cables must be solved using

another method. The equation of motion of a flat cable

can be obtained using a partial differential equation (Pinto

da Costa et al., 1995).

The first natural frequency f1 and second natural

frequency f2, taking into account the sag of a fixed cable,

are given by following equations:

(1)

(2)

Here, , , , L is the

L is the span of the cable, F0 is the initial tension of the

cable, s is the cross-sectional area of the cable, γ is the

angle of inclination of the cable to the vertical, E is the

Young’s modulus of the cable and m is the mass per the

unit length of the cable.

3.2.2. Selection of cable prone to parametric vibration

Large amplitude parametric vibrations may be induced

in a stay cable when the natural frequency of the global

vibration mode of a cable-stayed bridge is either close to

that of the cables (the second unstable region) or twice

that of the cables (the principal unstable region). The

relationship between the natural frequencies of the global

vibration mode and of the cables must be checked.

3.2.3 Periodic excitation

In this paper, the discussion focuses on the characteristics

of local parametric vibrations in the cables under

sinusoidal excitation. Such excitation may be induced by

an exciter during a vibration test. The amplitude of the

excitation force is assumed to be 50 kN.

4. Analytical Results

4.1. Natural frequency of the whole bridge system

Table 4 shows the computed natural frequencies of the

whole bridge system. The tower mode has a low natural

frequency of 0.353 Hz, thought to be attributable to the

high mass of the main tower, which is made of RC,

compared with a steel tower.

Furthermore, the 1st torsional symmetric mode appears

with a relatively higher frequency of 1.251 Hz since the

bridge is narrow. Furthermore, Table 5 illustrates some of

the modes of global girder-tower vibration. As this

demonstrates, the difference between the two modeling

approaches is no more than 2%. The natural frequency

obtained using the string element is slightly larger than

that obtained using the cable element.

f1

1

2L------

F0

m----- 1

1

2---

2

π---⎝ ⎠⎛ ⎞

4

+ λ2

⎩ ⎭⎨ ⎬⎧ ⎫

1

2---

=

f2

1

L---

F0

m-----=

λκ

F0Es⁄

-----------------= κ χL= χ mg γ F0

⁄sin=

Table 2. Number of point and element

Number of nodal point

Main girder 94

Rigid body 178

Main tower 136×2=272

CableChord element 0

Cable element 7×18×4=504

TotalChord element 544

Cable element 1048

Number of element

Main girder Beam element 93

Rigid body Beam element 144

Main tower Beam element 78×2=156

Bridge pier Beam element 142

Support Spring element 76

CableChord element 72

Cable element 576

TotalChord element 683

Cable element 1187

Table 3. Characteristic of using material

ContentMaterial

Steel Concrete

Young’s modulus (kN/m2) 2.0×108 3.1×107

Elastic shear modulus (kN/m2) 7.7×107 1.4×107

Poisson’s ratio 0.3 0.2

Table 4. Frequency of whole bridge system (Division model)

Modenumber

Frequency(Hz)

Mode

1 0.214 1st out-of-place symmetric mode

2 0.353 Tower mode

3 0.443 1st vertical asymmetric mode

43 0.665 2nd vertical symmetric mode

162 1.251 1st torsional symmetric mode


4.2. Natural frequency of the cable

Table 6 shows the in-plane natural frequencies of the

stay cables using the cable element method and using

analytical approach using equation (1). The difference

between the two methods is again no more than 2%. The

previous our approach which calculates natural frequency

of stay cable by using differential equation of flat cable is

useful to calculate the local natural frequency of the stay

cable.

4.3. Influence of material employed

By analyzing cable-stayed bridges constructed in Japan,

the relationship between overall natural frequency and

that of the stay cables is investigated to examine the

effects of construction material. The steel bridges used in

this study are the Ohshima Bridge (350 m) and the

Megami Bridge (480 m) while the PC bridges are the

Yobuko Bridge (250 m) and the Aomori Bay Bridge

(240 m). Table 7 shows the materials employed in each

bridge and the natural frequencies of the global modes

and of local cable vibration for the longest and shortest

cables. Since a steel bridge has a lower total weight than

a PC bridge, the cable tension is relatively smaller. The

relationship between natural frequencies for the Hizen-

Takashima bridge are given as those of a steel bridge

since the girder material is steel.

4.4. Relationship between global and cable natural

frequencies

Figure 2 describes the relationship between the natural

frequencies of the global modes and those of the cables.

The figure shows the natural frequencies of the cables

(corresponding to the second unstable region) and the

doubled natural frequencies (corresponding to the principal

unstable region). Parametric vibration may occur when

Table 5. Natural girder-tower modes

Table 6. Natural frequency of cable

Cablenumber

1st natural frequency (Hz)

Divisonmodel

Non-divided model

Difference(%)

C1 0.576 0.586 1.7

C2 0.614 0.626 2.0

C3 0.685 0.695 1.5

C4 0.784 0.792 1.0

C5 0.922 0.929 0.8

C6 1.042 1.049 0.7

C7 1.182 1.188 0.5

C8 1.396 1.400 0.3

C9 1.656 1.659 0.2


the natural frequency or twice the natural frequency of a

stay cable is close to that of the global natural vibration

mode of the whole bridge. Table 8 summarizes the cable

numbers in which local parametric vibration might occur.

There is a possibility of parametric vibrations resulting

from the second unstable region in stay cable C3, C4, C5,

C13, C15 and others at lower modes of global vibration.

There is also a possibility that parametric vibrations

arising from the primary region of C2 and C15 will be

generated in response to the 1st torsional mode of vibration.

Table 7. Relation between natural frequency of whole system and cable for different materials

Bridge nameLength of

Span(m)

Main girdermaterial

Main towermaterial

Natural frequency of whole system

(Hz)

1st frequency ofThe longest cable

(Hz)

1st frequency ofThe shortest cable

(Hz)

Hizen-TakashimaBridge

400 Steel R C

0.3260.3590.4430.6770.781

0.576 1.539

Ohshima Bridge 350 Steel Steel

0.2330.306 0.4200.6971.133

0.618 1.468

Megami Bridge 480 Steel Steel

0.2540.3590.5190.6280.665

0.427 1.488

Yobuko Oohashi 250 PC PC

0.3150.3620.7240.8463.230

0.854 8.017

Aomori BayBridge

240 PC PC

0.3510.5180.7840.9330.791

1.142 2.940

Figure 2. Relationship between natural frequencies the global modes and those of the cables.


4.5. Dynamic characteristics under periodic excitation

4.5.1. Vertical periodic excitation

This section discusses the dynamic properties of the

cable-stayed bridge under vertical sinusoidal loading

corresponding to P=Asinωt. The 2nd vertical symmetric

mode of 0.665 Hz is adopted as the excitation frequency.

Since this mode is symmetric over the bridge length, the

excitation point is made the center of the main span. The

excitation direction is vertical and the amplitude 50 kN.

Figures 3 and 4 show the maximum displacement response

of the girder in the vertical and axial directions. The

origin of the abscissa represents the center point of the

center span. The response of the bridge in the axial

direction is large in the vicinity of the center of the side

spans. On the other hand, the response in the vertical

direction is greatest at the excitation point in the center of

the main span. Figure 5 shows the girder response in the

axial direction and the response at the mid-point of the

cable. The ordinate expresses the response in the axial

direction and the abscissa the coordinate of the axial

direction. Also △ shows the displacement response of the

girder at the cable anchorage point and ▲ is the

displacement response of the cable mid-point. The

response in the axial direction is no more than 0.05 m in

many of the cables, as shown in Fig. 5. However, it is

greater than the girder response in cables C2, C3, C16

and C17, which are those where the possibility of

parametric vibration was noted. The response exceeds

0.35 m in C16. From Figure 6, the shorter cables exhibit

little response in the vertical direction. However, a large

vertical response occurs in cables C2, C3, C16 and C17

as compared with the girder response. Overall, large

amplitude vibrations occur in stay cables C2 and C17 and

the adjacent C3 and C16.

Table 8. Cables in which local parametric vibration may be occur

Vibration mode Natural frequency (Hz) Cable number

2nd vertical symmetric mode 0.677 C3 (Sub), C16 (Sub)

2nd vertical asymmetric mode 0.781 C4 (Sub)

3rd vertical asymmetric mode 0.905 C5 (Sub)

4th vertical symmetric mode 0.959 C13 (Sub)


1st torsional symmetric mode 1.252 C2 (Primary,Sub of 2nd)C17 (Primary,Sub of 2nd)

6th vertical symmetric mode 1.280 C11 (Vice)

5th vertical asymmetric mode 1.331 C16 (Primary,Sub of 2nd)

2nd torsional symmetric mode 1.341 C3 (Primary,Sub of 2nd)

6th vertical asymmetric mode 1.376 C16 (Primary,Sub of 2nd)

C8 (Sub)

3rd torsional symmetric mode 1.435 C15 (Primary,Sub of 2nd)


Figure 5. Maximum axial response of the girder and staycables under vertical excitation (0.665 Hz).

Figure 6. Maximum vertical response of the girder andstay cables under vertical excitation (0.665 Hz).

Figure 3. Maximum axial response of the girder undervertical excitation (0.665 Hz).

Figure 4. Maximum vertical response of the girder undervertical excitation (0.665 Hz).


4.5.2. Torsional periodic excitation

This section discusses the dynamic properties of the

cable-stayed bridge under torsional loading. Since this

mode is also symmetric over the bridge length, the

excitation point is set to the center of the main span. The

freedom of the excitation point is rotation along the

longitudinal direction and the amplitude of the sinusoidal

excitation is 500 kNm. Figures 7 and 8 show the maximum

displacement responses in the axial direction and in the

out-of-plane direction of the girder. The origin of the

abscissa represents the center of the main span. The

maximum response in the axial direction is large at the

center of main span and is small in the side spans. From

Fig. 9, the response in the axial direction appears only in

the main span, and does not appear in the side span.

Under torsional excitation, the greatest axial response

occurs in stay cables C11, C16, and C17. From Fig. 10,

displacement response in the out-of-plane direction

occurs in cables C2, C11, C16 and C17.

It is understood through this study that no large response

in either the axial direction or the out-of-plane direction

occurs in the side span.

Conclusions

Local parametric vibrations in the stay cables of an

actual cable-stayed bridge are examined when the bridge

is subjected to sinusoidal excitation. The focus is on

evaluating the dynamic characteristics of a steel cable-

stayed bridge using a whole-bridge approach. Through

analysis and resulting discussions, the characteristics of

local parametric cable vibrations are determined. The

results of the study can be summarized as follows:

(1) Comparing the use of cable elements and chord

elements in modeling the cables, it is confirmed that the

cable finite elements proposed by the authors for

modeling the stay cables are preferable in the whole-

bridge approach since they lead to correct evaluation of

non-linear cable vibrations including parametric vibrations.

(2) The dynamic characteristics of the stay cables can

be evaluated using either a whole-bridge approach or an

approximate approach. However, the response values of

the cables are different by the two methods. This

difference may be responsible for the different axial force

fluctuant of stay cables at both ends of one girder. In the

method used here, the cable is divided into elements and

the natural frequency is calculated for the cable as one

continuum.

(3) The parametric vibration characteristics of the

present bridge, which is a composite bridge, are similar to

those of a steel bridge.

Acknowledgment

The authors wish to express their gratitude to Nagasaki

Prefecture for providing the materials used in this study.

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Figure 7. Maximum axial response of the girder undertorsional excitation (1.251 Hz).

Figure 8. Maximum out-of-plane response of the girderunder torsional excitation (1.251 Hz).

Figure 9. Maximum axial response of the girder andcables under torsional excitation (1.251 Hz).

Figure 10. Maximum out-of-plane response of the girderand cables under torsional excitation (1.251 Hz).


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