Vibration and Mode Shapes Analysis of Cable Stayed Bridges Considering Different Structural Parameters

Introduction

Construction of long-span Bridge has been very active worldwide in the past few decades. Some of the cable-stayed bridges exceeding 1000 m, such as Stonecutters, Sutong in China and Russky Bridge in Russia with central span of 1104 m were completed in the beginning of 21st century. A number of long-span bridges are now under construction in China and Korea, and the plans to build super long-span bridges in other parts of the world are also being discussed. As bridge spans get longer and pylons get taller, they become more flexible and prone to vibration. Flexible structures tend to vibrate under dynamic loading such as wind, earthquake, vehicle movement, and human motion. Vibration can have several levels of consequence, from a potentially hazardous effect such as causing immediate structural failure to a more subtle but more extended effect such as structural fatigues. In addition to that, vibration can also affect user safety and comfort and limit bridge serviceability. In the past few decades, extensive research and development has been carried out to understand the mechanisms behind bridge vibration and to reduce undesirable vibration effects through various countermeasures. Results of these research and developments have been adopted in bridge design codes and put into practices by specifying methodologies and guidelines for countermeasures, and by introducing new structural elements or devices as vibration control. Types of vibration commonly observed on the long-span bridge are discussed from structure engineering viewpoints. Some vibration mechanisms are now well understood, while some others still require further studies to achieve complete understanding. Surveys on the phenomena associated with the type of vibration reported in many long-spa n bridges are also presented as well as engineering solutions adopted as countermeasures. The progressive lengthening of the spans of cable-stayed bridges has strongly increased the importance of understanding the dynamic behavior of cable-stayed bridges. As slender, flexible and typically three-dimensional structures, cable-stayed bridges pose specific vibration problems under dynamic loading such as earthquake, wind, rain and traffic. Cables may lead to complex deck-cable and tower-cable coupling vibrations. Previous studies have revealed the importance of cable vibration in dynamics of cable-stayed bridges. It is found that the cable vibration can have a significant influence on the modal characteristics and seismic response of cable-stayed bridges when the first frequencies of cable local modes overlap with the first few frequencies of bridge deck/tower. Therefore, developing an accurate finite element model accounting for the interaction of cable vibration with bridge vibration is essential to understanding the dynamic behavior of a cable-stayed bridge.

Cable-stayed bridges usually show long fundamental periods, aspect that influences their dynamic behavior. However, their flexibility and dynamics characteristics depend on several parameters such as the main span length, stay cable layout and support conditions. First vibration modes are very long, mainly related to deck modes. They are followed by cable vibration modes or tower modes that can be coupled with the deck depending on the support conditions. An exact modeling for the deck and cables can be very important for a precise dynamic analysis, being necessary an adequate assessment of the natural frequencies and modal shapes.

Objective

The objective of this research work is to search the vibration characteristics of cable stayed bridge models so that to:

1- Discover the effect of the boundary condition on the vibration of the stay cables, deck and the pylons and to find the suitable boundary condition for each case as a result to define the regions that are much prone to early vibration and to control this mitigation in the cable stayed bridges.
2- Identify the relation between the stay cables styles and the early vibration by simulation of mode shapes as a result to get the natural frequencies of the system to increase their values by adding cross ties on the stay cables in the areas of significant detected vibrations.
3- Retrofitting of damaged cable stayed bridges that have been poorly designed through simulation of the structure vibration with negative (high vibration and deflection) of girders and pylons which will prepare a safe structure against vibration.

Natural Frequencies and Mode Shapes

In typical structural or mechanical systems, there are many multiple or nearly equal natural frequencies due to their structural symmetries or certain reasons. In this case, since eigenspace spanned by the mode shapes corresponding to the multiple natural frequencies are degenerate, any linear combination of mode shapes can be a mode shape. For the mode shape derivative to be found, the adjacent mode shapes which lie adjacent to the multiplicity of multiple natural frequency distinct mode shapes appearing when a design parameter varies must be calculated first. To do so, the approximate mode shapes could be varied continuously by varying the design parameter. For the real symmetric case, a generalization of Nelson's method was obtained by Ojalvo and amended by Mills-Curren and Dailey. Dailey's method is an exact analytical method for calculating mode shape derivatives. This method only requires knowledge of the eigenpair with multiple eigenvalues, however, the method is lengthy and complicated for finding mode shape derivatives and clumsy for programming. Dailey's method is extremely complicated for calculating the sensitivity of eigenvectors of multiple eigenvalues in the case of the damped systems. When a natural frequency has multiplicity m and a design parameter is perturbed, the corresponding mode shapes may split into as many as m distinct mode shapes. For derivatives of the mode shapes to be responsible, the mode shapes must be laid adjacent to the m distinct mode shapes that appear when a design parameter varies. The eigenvalue problem of a damped system can be expressed as

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Where M, C and K are the matrices of mass, damping and stiffness, respectively and these are order n symmetric matrices. M is positive definite and K positive definite or semi-positive definite. The first step in finding derivatives of mode shapes of multiple eigenvalues is to find corresponding adjacent mode shapes Suppose that all eigenpairs are known and multiplicity of the eigenvalue lm is m. Define the following eigenvalue problem where Fm is the matrix of eigenvectors correspond to the multiple eigenvalue, hence, its order (n*m). .

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Where the eigenvalue and eigenvectors are:

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Im is the identity matrix of order m and lm is the eigenvalue of multiplicity m for

the eigenspace.

Modal Analysis Theory

The vibration characteristics decide the dynamic response characteristics. Consequently the modal analysis of the cable-stayed bridge is essential to study the dynamic behavior of cable-stayed bridge. The structure of the cable-stayed bridge is complex, and the cables are flexible, lightweight, low damping, etc. All these features above make the mode of the cable-stayed bridge a practical engineering issue of worth attention. The structure of the cable-stayed bridge is a system with continuous distribution of mass and stiffness. It should be divided into finite elements with limited DOF. Because of the complexity of cable-stayed bridge structure, the result of three-dimensional FEM analysis is more comprehensive and more reliable than the result of the traditional empirical formula. It can be assumed that the structural of the cable-stayed bridge has N DOF. The dynamic equilibrium equation of the model can be list as:

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Where [M] = mass matrix of the architecture, [ C ] = damping matrix of the architecture, [ K ] = stiffness matrix of the architecture,

{U} = displacement vector of each node, {Ŭ} = velocity vector of each node,

{Ű} = acceleration vector of each node. Ignoring the resistance, we can get a dynamic equilibrium equation:

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take [U (t)] ={ Φ}sin t (7)

and then solve the differential equations. We can obtain:

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We can get natural frequencies of the system i (i=1, 2,…,N) and the vibration modals of the structure {Φi}{i=1, 2,…,N} from this equation.

The initial force in the cables and the gravity of the bridge has little effect on the dynamic characteristics of the structure. Therefore, the initial force in the cable and gravity of the bridge can be ignored when modal analysis carried out. By constraining the corresponding nodes degrees, the boundary conditions of the model can be defined.

Equations of Motion

The vibration of CSBs due to vehicular loading was analyzed via FEM with beam and cable elements. The substructure method proposed by Nagai was applied to consider the local vibration of cables. Each simulated cable was divided into n −1 Inter linking truss elements Fig. 1.

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Fig. 1 Finite Element Model of the Cable

To improve the computational efficiency of the dynamic analysis of CSBs, the local vibrations of cables were considered using a substructure method. According to the method of superposition, the vibrations of the cable are composed of vibrations of cable supports and several modes. The accuracy of the method depends on the quantity of modes adopted in calculation. For flexible structures such as cables, the vibration is composed primarily of the components of low orders of modes. The equation of motion of the free vibrations of each cable can be expressed as:

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where mij are the sub-matrices of the mass matrix, kij are the sub-matrices of the stiffness matrix, d 1, dn are the displacement vectors at cable supports and di represents the displacement vector of the inner nodes of the cable. According to the superposition method Fig.2, di can be separated into two parts, the movement of cable supports and modal motions. In this study, the modal motion of a cable is assumed to be a combination of several lower-order modes of the cable with fixed supports. Thus, d i is approximated as

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Where

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and q is the generalized coordinates vector. Φ is the modal matrix of a cable with two fixed ends, and is obtained by solving the eigenvalue problem from equation (7)

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