## 1. Introduction

In the last several decades, cable-stayed bridges have become popular due to their aesthetic appeal, structural efficiency, ease of construction and economic advantage. This type of bridge, however, is light and flexible, and has a low level of inherent damping. Consequently, they are susceptible to ambient excitations from seismic, wind and traffic loads. Since the geometric and dynamic properties of the bridges as well as the characteristics of the excitations are complex, it is necessary to fully understand the mechanism of the interaction among the structural components with reasonable bridge shapes, which is used to provide the essential information to accurately calculate the dynamic responses of the bridges under the complicated excitations.

In the previous studies of bridge dynamics, the responses of a cable-stayed bridge can be categorized into global, local and coupled modes [1]. The global modes are primarily dominated by the deformations of the deck-tower system with the quasi-static motions of the stay cables; the local modes predominantly consist of the stay cable motions with negligible deformations of the deck-tower system; the coupled modes have substantial contributions from both the deck-tower system and stay cables. Since the towers are usually designed with a high rigidity to obtain an adequate efficiency of the system, the significant tower deformations do not occur in the lower modes sensitive to the ambient excitations [2]. Consequently, the coupled modes are considered to be dominated by the deck-stay interaction, while the contribution from the towers can be neglected. Numerical approaches based on the finite element method have been widely used to investigate the deck-stay interaction. The finite-element models of a cable-stayed bridge can be classified into two categories [1]: the one-element cable system (OECS), in which each stay cable is represented by a single cable element, and the multi-element cable system (MECS), in which each stay cable is discretized into multiple cable elements.

The deck-stay interaction has attracted much attention, because it not only significantly complicates both the natural frequency and mode shape characteristics of a cable-stayed bridge, but also potentially results in the large-amplitude stay cable vibrations even under the low-level deck oscillations. In the previous literature, the deck-stay interaction is due to the linear coupling (primary resonance) [3-8, 11] or the nonlinear coupling (secondary resonance), which can be further categorized into the subharmonic resonance of order 1/2 (two-to-one resonance) [3-9] and the superharmonic resonance of order 2 (one-to-two resonance) [6, 9, 10]. The primary, two-to-one and one-to-two resonances individually result in the fact that the global modes induce the direct, parametric and angle variation excitations of the local modes. Two types of simplified models: the single cable with moving anchorage [5-7] and the cable-supported cantilever beam [3, 4, 8-11], have been presented to theoretically investigate the deck-stay interaction. To extend the results of the simplified models, the OECS and MECS models of full cable-stayed bridges based on the finite element method have been widely used to explore such coupled phenomena of real structures [1, 11-16]. By focusing on the analytical and numerical study of the linear coupling, the localization factor was introduced to reveal the frequency veering phenomenon and to evaluate the mode hybridization level of a cable-stayed bridge [11]. On the basis of this research, the ambient vibration measurements were conducted to investigate the deck-stay interaction. It was suggested that the nonlinear coupling is not consistent with the measurement data. In contrast, the linear coupling is recognized as the critical excitation source of the coupled modes [16].

In parallel to the previous work [11, 16], the authors of the present paper also studied the deck-stay interaction of cable-stayed bridges based on the analytical and numerical methods as well as the long-term comprehensive full-scale measurements [17]. The measurement data indicated that the deck oscillations of small to moderate amplitudes are coupled with the large-amplitude stay cable vibrations due to the linear coupling between these two components. An analytical model of the single cable with spring-mass oscillator was presented to explain such mechanism attributed to the frequency loci veering and mode localization. Furthermore, the “pure” deck modes, “pure” cable modes and coupled modes are successfully captured by the proposed model. These phenomena are verified by the numerical simulations of the OECS and MECS models of a full cable-stayed bridge. The concepts of the indices for quantitatively assessing the degree of coupling among the structural components were also appeared in this research.

It is important to investigate the deck-stay interaction with the appropriate initial shape of a cable-stayed bridge. This is because such initial shape not only reasonably provides the geometric configuration as well as the prestress distribution of the bridge under the weight of the deck-tower system and the pretension forces in the stay cables, but also definitely ensures the satisfaction of the relations for the equilibrium conditions, boundary conditions and architectural design requirements [18-21]. The computational procedures for the initial shape analyses of the OECS and MECS models were presented for this reason [22, 23]. However, few researchers have studied the deck-stay interaction with the initial shape effect.

The objective of this study is to fully understand the mechanism of the deck-stay interaction with the appropriate initial shapes of cable-stayed bridges. Based on the smooth and convergent bridge shapes obtained by the initial shape analysis [22, 23], the OECS and MECS models of the Kao Ping Hsi Bridge in southern Taiwan are developed to verify the applicability of the analytical model and numerical formulation from the field observations [17]. For this purpose, the modal analyses of the two finite element models are conducted to calculate the natural frequency and normalized mode shape of the individual modes of the bridge. The modal coupling assessment is also performed to obtain the generalized mass ratios among the structural components for each mode of the bridge [24]. To further investigate the deck-stay interaction characteristics of cable-stayed bridges under earthquake excitations, the dynamic displacements and internal forces of the two finite element models are calculated based on the seismic analyses. These results can be used to provide a variety of viewpoints to illustrate the mechanism of the deck-stay interaction with the appropriate initial shapes of cable-stayed bridges.

## 2. Finite element formulation

On the basis of the finite element concepts, a cable-stayed bridge can be considered as an assembly of a finite number of cable elements for the stay cables and beam-column elements for both the decks and towers. Several assumptions are adopted in this study: the material is homogeneous and isotropic; the stress-strain relationship of the material remains within the linear elastic range during the whole nonlinear response; the external forces are displacement independent; large displacements and large rotations are allowed, but strains are small; each stay cable is fixed to both the deck and tower at their joints of attachment. Based on the system equations with the consideration of geometric nonlinearities, the initial shape analysis, modal analysis, modal coupling assessment and seismic analysis of cable-stayed bridges are conducted in this study.

### 2.1. Geometric nonlinearities

To reasonably simulate cable-stayed bridges, three types of geometric nonlinearities: the cable sag, beam-column and large displacement effects, are considered in this study.

A stay cable will sag into a catenary shape due to its weight and tensile force. Such cable sag effect has to be taken into consideration when the stay cable is represented by a single straight cable element. A stay cable with tensile stiffness is assumed to be perfectly elastic. The compressive, shear and bending stiffnesses of the stay cable are negligible. The cable sag nonlinearity can be simulated based on the equivalent modulus of elasticity of the stay cable [25]

where

where

High pretension forces in the stay cables can result in large compressive forces in the deck-tower system of a cable-stayed bridge. For this reason, the beam-column effect between such compressive forces and bending moments has to be considered when beam-column elements are used to simulate both the decks and towers. For a beam-column element based on the Euler-Bernoulli beam theory in Figure 2, shear strains of the element are neglected.

where

In general, large displacements occur in the deck-tower system due to the large span and less weight of a cable-stayed bridge. Such effect has to be taken into consideration when the equilibrium equations are derived from the deformed position. Under these conditions, the element coordinate

In addition to the element stiffness matrices, the element mass matrices are introduced to fully understand the essential properties of a cable-stayed bridge. Based on the consistent mass model, the mass distribution of each stay cable and that of each deck and tower can be simulated by a cable element and a beam-column element, respectively. The mass matrix of the former with four element coordinates

where

### 2.2. System equations

The system equations in generalized coordinates of a nonlinear finite element model of a cable-stayed bridge can be derived from the Lagrange’s virtual work principle

where

Under consideration of three types of geometric nonlinearities mentioned in Chapter 2.1,

Eq. (8) is a set of simultaneous second-order nonlinear ordinary differential equations. In order to incrementally solve these equations, the linearized system equations in a small time (or force) interval are derived based on the first-order Taylor series expansion of Eq. (8)

where

Eq. (15) is a set of simultaneous second-order linear ordinary differential equations in a small time interval, which can be solved by the direct integration method [20].

### 2.3. Initial shape analysis

The initial shape of a cable-stayed bridge provides the geometric configuration as well as the prestress distribution of such bridge under the weight of the deck-tower system and the pretension forces in the stay cables. The relations for the equilibrium conditions, boundary conditions and architectural design requirements should be satisfied. Under consideration of three types of geometric nonlinearities, i.e., the cable sag, beam-column and large displacement effects, the initial shape analyses of an OECS model and a MECS model are presented in this study.

For the initial shape analysis of the OECS model, the weight of the deck-tower system is considered, whereas the weight of the stay cables is neglected. The shape finding computation is performed using a two-loop iteration method: an equilibrium iteration and a shape iteration [18-23]. It can be started with an estimated initial element force (pretension force) in the stay cables. Based on the reference configuration (architectural design form) with no deflection and zero prestress in the deck-tower system, the equilibrium configuration of the whole bridge under the weight of the deck-tower system can be first determined by incrementally solving the linearized system equations

which are individually derived from Eq. (15) and Eq. (19) with negligible inertial and damping effects due to the static case. On the basis of Eq. (25) and Eq. (26), the equilibrium iteration is performed using the Newton-Raphson method [18-23].

After the above equilibrium iteration, the bridge configuration satisfies the equilibrium and boundary conditions, however, the architectural design requirements are, in general, not fulfilled. This is because large displacements and variable bending moments occur in the deck-tower system due to the large bridge span. Under these conditions, the shape iteration is conducted to reduce the displacements and to smooth the bending moments, and the appropriate initial shape can therefore be obtained.

A number of control points are selected for insuring that both the deck and tower displacements satisfy the architectural design requirements in the shape iteration

where

If Eq. (27) is not achieved, the element axial forces calculated in the previous equilibrium iteration will be taken as the initial element forces in the new equilibrium iteration, and the corresponding equilibrium configuration of the whole bridge under the weight of the deck-tower system will be determined again. The shape iteration will then be repeated until Eq. (27) is reached. Under these conditions, the convergent configuration can be regarded as the initial shape of the OECS model.

The initial shape analysis of the MECS model is also performed to reasonably simulate the bridge configuration. Based on the initial shape of the OECS model obtained previously, the both end coordinates and pretension force in each single stay cable can be used for the shape finding computation of the corresponding stay cable discretized into multiple elements using the catenary function method [22, 23]. Incorporating the interior nodal coordinates and pretension forces in each discrete stay cable into the bridge model, and then conducting the two-loop iteration method again, the convergent configuration can be regarded as the initial shape of the MECS model.

### 2.4. Modal analysis

Under the assumption that the system vibrates with a small amplitude around a certain nonlinear static state, in which the variation in such state induced by the vibration is negligible, the modal analysis of a cable-stayed bridge can be conducted based on the linearized system equation

where

### 2.5. Modal coupling assessment

According to the results of both the initial shape analysis (Chapter 2.3) and modal analysis (Chapter 2.4) with the consideration of geometric nonlinearities (Chapter 2.1) in the system equations (Chapter 2.2), three indices for quantitatively assessing the degree of coupling among the stay cables, decks and towers of a cable-stayed bridge in each mode are presented [24] as

where

### 2.6. Seismic analysis

According to the assumption that the system is under the uniform earthquake excitation, the seismic analysis of a cable-stayed bridge with respect to the initial shape obtained in Chapter 2.3 can be conducted based on the equivalent difference equations

where these equations are derived from Eq. (15) and Eq. (19) using the Newmark method [27];

## 2. Finite element models

To understand the deck-stay interaction with the appropriate initial shapes of cable-stayed bridges, an OECS model and a MECS model of the full Kao Ping Hsi Bridge are developed, as shown in Figure 5(a) and 5(b), respectively. This bridge is an unsymmetrical single-deck cable-stayed bridge with a main span of 330 m and a side span of 184 m. The deck, which consists of steel box girders in the main span and concrete box girders in the side span, is supported by a total of 28 stay cables (S1-S28), arranged in a central plane originated at the 184 m tall, inverted Y-shaped, concrete tower. A more detailed description of the Kao Ping Hsi Bridge can be found in [28].

Figure 5(a) and 5(b) illustrate the two-dimensional finite element models of the bridge. The OECS and MECS models both contain 48 beam-column elements that simulate the deck and tower. For the MECS model, each stay cable is discretized into 10 cable elements, whereas a single cable element is used to simulate each stay cable in the OECS model. This fact indicates that the OECS and MECS models individually include 28 and 280 cable elements. Figure 5(a) and 5(b) also show that 49 and 301 nodes are involved in the OECS and MECS models, respectively. A hinge, roller and fixed supports are used to model the boundary conditions of the left and right ends of the deck and the tower, respectively, and a rigid joint is employed to simulate the deck-tower connection. On the basis of the OECS and MECS models, the initial shape analysis, modal analysis, modal coupling assessment and seismic analysis of the Kao Ping Hsi Bridge are conducted in this study.

## 4. Numerical results

Based on the OECS and MECS models of the Kao Ping Hsi Bridge developed in Chapter 3, the initial shape analysis, modal analysis, modal coupling assessment and seismic analysis are conducted using the finite element formulation presented in Chapter 2. The numerical results can be used to fully understand the mechanism of the deck-stay interaction with the appropriate initial shapes of cable-stayed bridges.

### 4.1. Initial shape analysis

Based on the finite element procedures presented in Chapter 2.3, the initial shape analyses of the OECS and MECS models are conducted to reasonably provide the geometric configuration of the Kao Ping Hsi Bridge. In both Figure 5(a) and 5(b), nodes 37, 38, 40, 45 and 46 are selected as the control points for checking the deck displacement in the vertical direction, while node 19 is chosen as the control point for checking the tower displacement in the horizontal direction. The convergence tolerance ^{-4} in this study.

Figure 6(a) shows the initial shape of the OECS model of the Kao Ping Hsi Bridge (solid line), indicating that the maximum vertical and horizontal displacements measured from the reference configuration (short dashed line) are 0.038 m at node 36 in the main span of the deck and -0.021 m at node 8 in the tower, respectively. The shape of each stay cable represented by a single cable element is straight as expected. Figure 6(a) also illustrates that the overall displacement obtained by the two-loop iteration method, i.e., the equilibrium and shape iterations, is comparatively smaller than that only from the equilibrium iteration (long dashed line). Consequently, the initial shape based on the two-loop iteration method appears to be able to appropriately describe the geometric configurations of cable-stayed bridges.

Figure 6(b) shows the initial shape of the MECS model of the Kao Ping Hsi Bridge (solid line), indicating that the maximum vertical and horizontal displacements measured from the reference configuration (short dashed line) are 0.068 m at node 34 in the main span of the deck and -0.049 m at node 8 in the tower, respectively. The sagged shape occurs in the stay cables due to the fact that each stay cable is simulated by multiple cable elements.

### 4.2. Modal analysis and modal coupling assessment

According to the results of the initial shape analysis presented in Chapter 4.1, the modal analyses of the OECS and MECS models using the finite element computations developed in Chapter 2.4 are conducted to calculate the natural frequency and normalized mode shape of the individual modes of the Kao Ping Hsi Bridge. The modal coupling assessment based on the proposed formulas in Chapter 2.5 is also performed to obtain the generalized mass ratios among the structural components for each mode of such bridge. These results can be used to provide a variety of viewpoints to illustrate the mechanism of the deck-stay interaction with the appropriate initial shapes of cable-stayed bridges.

Table 1 summarizes the modal properties of the Kao Ping Hsi Bridge based on the OECS model (modes 1 to 3) and the MECS model (modes 1 to 24). In this table,

OECS | MECS | ||||||

Type | Type | ||||||

1 | 0.2877 | 1^{st} DT | G | 1 | 0.3053 | 1^{st} DT | G |

2 | 0.3382 | 1^{st} S28 | L | ||||

3 | 0.3852 | 1^{st} S27 | L | ||||

4 | 0.4274 | 1^{st} S26 | L | ||||

5 | 0.4554 | 1^{st} S1 | L | ||||

6 | 0.4653 | 1^{st} S25 | L | ||||

7 | 0.4899 | 1^{st} S24 | L | ||||

8 | 0.5067 | 1^{st} S23 | L | ||||

9 | 0.5269 | 1^{st} S22 | L | ||||

10 | 0.5378 | 1^{st} S2 | L | ||||

11 | 0.5471 | 1^{st} S21 | L | ||||

2 | 0.5455 | 2^{nd} DT | G | 12 | 0.5686 | 2^{nd} DT | G |

13 | 0.5944 | 1^{st} S3 | L | ||||

14 | 0.6040 | 1^{st} S20 | L | ||||

15 | 0.6333 | 1^{st} S4 | L | ||||

16 | 0.6346 | 2^{nd} S28 | L | ||||

17 | 0.6835 | 1^{st} S5 | L | ||||

18 | 0.6850 | 3^{rd} DT1 ^{st} S19 | C | ||||

3 | 0.6854 | 3^{rd} DT | G | 19 | 0.7171 | 3^{rd} DT1 ^{st} S19 | C |

20 | 0.7269 | 1^{st} S6 | L | ||||

21 | 0.7500 | 2^{nd} S27 | L | ||||

22 | 0.7590 | 1^{st} S7 | L | ||||

23 | 0.8008 | 1^{st} S8 | L | ||||

24 | 0.8184 | 1^{st} S18 | L |

Figure 7 shows the relationship between the natural frequency and the mode number for the first 24 modes of the MECS model of the Kao Ping Hsi Bridge. For reference, the fundamental frequency of stay S19 (0.6908 Hz) is also included. This frequency is calculated based on the assumption that stay S19 is clamped at both ends [29].

Figure 8(a) and 8(b) illustrate the normalized mode shapes of the individual modes of the OECS model (modes 1 to 3) and the MECS model (modes 1 to 24) of the Kao Ping Hsi Bridge, respectively. Each normalized mode shape (solid line) is measured from the initial shape (dashed line) obtained in Chapter 4.1.

To quantitatively assess the degree of coupling for each mode, Figure 9 depicts the variations in the generalized mass ratios with respect to the mode number for the first 24 modes of the MECS model of the Kao Ping Hsi Bridge. In this figure,

It can be seen in Table 1, Figure 7, Figure 8(a) and 8(b) that for the global modes,

It also can be seen in Figure 9 that for modes 2 to 11, modes 13 to 17 and modes 20 to 24 in the MECS model,

As shown in Table 1, Figure 7, Figure 8(a) and 8(b), the difference between

In summary, the coupled modes are attributed to the frequency loci veering and mode localization when the “pure” deck-tower frequency and the “pure” stay cable frequency approach one another, implying that the mode shapes of such coupled modes are simply different from those of the deck-tower system or stay cables alone. The distribution of the generalized mass ratios between the deck-tower system and stay cables are useful indices for quantitatively assessing the degree of coupling for each mode. These results are demonstrated to fully understand the mechanism of the deck-stay interaction with the appropriate initial shapes of cable-stayed bridges.

### 4.3. Seismic analysis

According to the results of the initial shape analysis presented in Chapter 4.1, the seismic analyses of the OECS and MECS models using the finite element computations developed in Chapter 2.6 are conducted to obtain the dynamic responses of the Kao Ping Hsi Bridge. Figure 10 shows the vertical component of the Chi-Chi earthquake accelerogram recorded in Mid-Taiwan on September 21, 1999 [30], which is selected as the earthquake-induced ground acceleration in this study. Under the excitation, the Newmark method

Figure 11 shows the horizontal and vertical displacement time histories of nodes 295, 297 and 300 in stay S28 for the MECS model. The variations in the dynamic responses among the three nodes for each direction and those between the horizontal and vertical directions for each node are observed in this figure. Consequently, the dynamic displacements of the stay cables are successfully captured by the MECS model, but not by the OECS model. Figure 12 shows the vertical displacement time histories of nodes 35, 36 and 42 in the deck, the horizontal displacement time histories of nodes 8 and 20 in the tower, and the horizontal time history of node 49 in the right end of the deck, for both the OECS and MECS models. The dynamic response of each node in the OECS model coincides with that of the corresponding node in the MECS model. Consequently, the dynamic displacements of the deck-tower system are reasonably simulated by both the OECS and MECS models.

The axial force, which is in the

In summary, the dynamic displacements of the stay cables are successfully captured by the MECS model, but not by the OECS model. Furthermore, the dynamic displacements of the deck-tower system as well as the dynamic internal forces of the stay cables and those of the deck-tower system are reasonably simulated by both the OECS and MECS models. These results are demonstrated to fully understand the deck-stay interaction characteristics of cable-stayed bridges under seismic excitations.

## 5. Conclusions

This study has provided a variety of viewpoints to illustrate the mechanism of the deck-stay interaction with the appropriate initial shapes of cable-stayed bridges. Based on the smooth and convergent bridge shapes obtained by the initial shape analysis, the OECS and MECS models of the Kao Ping Hsi Bridge are developed to verify the applicability of the analytical model and numerical formulation from the field observations in the authors’ previous work. For this purpose, the modal analyses of the two finite element models are conducted to calculate the natural frequency and normalized mode shape of the individual modes of the bridge. The modal coupling assessment is also performed to obtain the generalized mass ratios among the structural components for each mode of the bridge. To further investigate the deck-stay interaction characteristics of cable-stayed bridges under earthquake excitations, the dynamic displacements and internal forces of the two finite element models are calculated based on the seismic analyses.

The findings indicate that the coupled modes are attributed to the frequency loci veering and mode localization when the “pure” deck-tower frequency and the “pure” stay cable frequency approach one another, implying that the mode shapes of such coupled modes are simply different from those of the deck-tower system or stay cables alone. The distribution of the generalized mass ratios between the deck-tower system and stay cables are useful indices for quantitatively assessing the degree of coupling for each mode. To extend the two finite element models to be under the seismic excitation, it is evident that the dynamic displacements of the stay cables are successfully captured by the MECS model, but not by the OECS model. In addition, the dynamic displacements of the deck-tower system as well as the dynamic internal forces of the stay cables and those of the deck-tower system are reasonably simulated by both the OECS and MECS models. These results are demonstrated to fully understand the mechanism of the deck-stay interaction with the appropriate initial shapes of cable-stayed bridges.