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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, Japan 2Department 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, are studied. This bridge is a composite cable-stayed bridge with a steel girder and concrete towers. The natural frequencies of the global vibration modes are obtained using a three-dimensional FEM model. Relations between global natural frequencies and the local natural frequencies of the stay cables are first demonstrated and the results are compared with other cable-stayed bridges. The local vibrations of the stay cables at the support during global motion are obtained and the properties of the local vibrations 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 this paper was submitted for review and possible publication on Septem- ber 12, 2008; approved on November 21, 2008

    *Corresponding author Tel: +81-95-819-2610; Fax: +81-95-819-2627 E-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