**1. Introduction**

230 Earthquake Engineering

[13] Zatar, W. A., and Harik, I. E., "Bridge embankments: Part II - Seismic risk of I-24 in

[14] Pflazer, W. J. (1995). "Use of existing geotechnical data to supplement site investigations." *Proceedings of the Ohio River Valley Soils Seminar XXVI*, ASCE Kentucky

Kentucky." *ASCE Journal of Performance of Constructed Facilities*, June 2008.

Geotechnical Engineers Group, Clarksville, Indiana.

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

© 2012 Liu and Wang, licensee InTech. This is an open access chapter distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. © 2012 Liu and Wang, licensee InTech. This is a paper distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

by a single cable element, and the multi-element cable system (MECS), in which each stay cable is discretized into multiple cable elements.

Finite Element Analysis of Cable-Stayed Bridges with Appropriate Initial Shapes Under Seismic Excitations Focusing on Deck-Stay Interaction 233

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

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

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-

To reasonably simulate cable-stayed bridges, three types of geometric nonlinearities: the

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 sag, beam-column and large displacement effects, are considered in this study.

the appropriate initial shapes of cable-stayed bridges.

**2. Finite element formulation** 

stayed bridges are conducted in this study.

**2.1. Geometric nonlinearities** 

cable [25]

effect.

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**

232 Earthquake Engineering

cable is discretized into multiple cable elements.

source of the coupled modes [16].

research.

by a single cable element, and the multi-element cable system (MECS), in which each stay

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

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

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 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 cablestayed 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]

$$E\_{eq} = \frac{E\_c}{1 + \frac{\left(\nu l\_c\right)^2 A\_c E\_c}{12T^3}}\,\tag{1}$$

Finite Element Analysis of Cable-Stayed Bridges with Appropriate Initial Shapes Under Seismic Excitations Focusing on Deck-Stay Interaction 235

(3)

, the

0 0 ,

*tb b*

. By differentiating *<sup>j</sup> u* with respect to *q*

, (4)

. (5)

0 0

where *<sup>b</sup> E* , *Ab* , *bI* and *<sup>b</sup> L* are the modulus of elasticity, the cross-sectional area, the moment of inertia and the length of the beam-column element, respectively; *Cs* , *Ct* and *Rt* are the stability functions representing the interaction between the axial and bending stiffnesses of

*<sup>L</sup> RA I* 

*b b*

*b*

the beam-column element [26].

*jk t s*

*E I KE C C*

**Figure 2.** Beam-column element for simulating the stiffness of each deck and tower.

in both Figure 1 and Figure 2, i.e., *u uq j j*

written as

*j a*   and *<sup>j</sup>* , *a* 

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 *<sup>j</sup> u* can be expressed as a nonlinear function of the system coordinate *q*

*j*

*a*

,

*j*

 

*a*

first-order and second-order coordinate transformation coefficients can be individually

*j*

*j j*

*q qq* 

in [18], which are provided to develop the tangent system stiffness matrix in Chapter 2.2.

*a u*

2

 

for the stiffness matrices of the cable and beam-column elements can be found

*u*

*q*

*s t*

*C C*

where *<sup>c</sup> E* , *Ac* and *<sup>c</sup> l* are the effective modulus of elasticity, the cross-sectional area and the horizontal projected length of the stay cable, respectively; *w* is the weight of the stay cable per unit length; *T* is the tension in the stay cable. The stiffness matrix of a cable element in Figure 1 can be expressed as

$$KE\_{jk} = \begin{cases} \left\lfloor \frac{E\_{eq}A\_c}{L\_c} \right\rfloor, & \left. u\_1 > 0 \right.\\ \left. \left. \begin{array}{c} \\ \end{array} \right\rfloor, & \left. \begin{array}{c} \\ \end{array} \right\rfloor \\\\ \left\lceil \begin{array}{c} 0 \end{array} \right\rceil, & \left. u\_1 \leq 0 \end{array} \right. \end{cases} \tag{2}$$

where 1 *u* is the element coordinate for the relative axial deformation; *<sup>c</sup> L* is the chord length of the stay cable.

**Figure 1.** Cable element for simulating the stiffness of each stay cable.

High pretension forces in the stay cables can result in large compressive forces in the decktower 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. <sup>1</sup> *u* , 2 *u* and 3 *u* are the element coordinates for the left end rotation, the right end rotation and the relative axial deformation, respectively. The stiffness matrix of the beam-column element can be written as

Finite Element Analysis of Cable-Stayed Bridges with Appropriate Initial Shapes Under Seismic Excitations Focusing on Deck-Stay Interaction 235

$$KE\_{jk} = \frac{E\_b I\_b}{L\_b} \begin{vmatrix} \mathbf{C}\_s & & & \mathbf{C}\_t & & 0\\ & \mathbf{C}\_t & & & \mathbf{C}\_s & & 0\\ & 0 & & 0 & R\_t A\_b / I\_b \end{vmatrix} \tag{3}$$

where *<sup>b</sup> E* , *Ab* , *bI* and *<sup>b</sup> L* are the modulus of elasticity, the cross-sectional area, the moment of inertia and the length of the beam-column element, respectively; *Cs* , *Ct* and *Rt* are the stability functions representing the interaction between the axial and bending stiffnesses of the beam-column element [26].

234 Earthquake Engineering

where *<sup>c</sup> E* , *Ac* and *<sup>c</sup>*

of the stay cable.

element can be written as

Figure 1 can be expressed as

<sup>2</sup>

horizontal projected length of the stay cable, respectively; *w* is the weight of the stay cable per unit length; *T* is the tension in the stay cable. The stiffness matrix of a cable element in

> *eq c c*

where 1 *u* is the element coordinate for the relative axial deformation; *<sup>c</sup> L* is the chord length

High pretension forces in the stay cables can result in large compressive forces in the decktower 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. <sup>1</sup> *u* , 2 *u* and 3 *u* are the element coordinates for the left end rotation, the right end rotation and the relative axial deformation, respectively. The stiffness matrix of the beam-column

*E A*

*L*

12

*c*

1

*<sup>E</sup> <sup>E</sup>*

*eq*

*jk*

*KE*

**Figure 1.** Cable element for simulating the stiffness of each stay cable.

3

*c cc*

*wl A E T*

,

*l* are the effective modulus of elasticity, the cross-sectional area and the

1

,

, 0

*u*

1

*u*

0, 0

(1)

(2)

**Figure 2.** Beam-column element for simulating the stiffness of each deck and tower.

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 *<sup>j</sup> u* can be expressed as a nonlinear function of the system coordinate *q* in both Figure 1 and Figure 2, i.e., *u uq j j* . By differentiating *<sup>j</sup> u* with respect to *q* , the first-order and second-order coordinate transformation coefficients can be individually written as

$$a\_{j\alpha} = \frac{\partial u\_j}{\partial q\_{\alpha}},\tag{4}$$

$$a\_{j\alpha,\beta} = \frac{\mathfrak{d}a\_{j\alpha}}{\mathfrak{d}q\_{\beta}} = \frac{\mathfrak{d}^2 u\_j}{\mathfrak{d}q\_{\alpha}\mathfrak{d}q\_{\beta}}\,. \tag{5}$$

*j a* and *<sup>j</sup>* , *a* for the stiffness matrices of the cable and beam-column elements can be found in [18], which are provided to develop the tangent system stiffness matrix in Chapter 2.2.

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 *<sup>j</sup> u j* 1 4 in Figure 3 and that of the latter with six element coordinates *<sup>j</sup> u j* 1 6 in Figure 4 can be individually expressed as

$$ME\_{jk} = \frac{\rho\_c A\_c L\_c}{6} \begin{vmatrix} 2 & 0 & 1 & 0 \\ 0 & 2 & 0 & 1 \\ 1 & 0 & 2 & 0 \\ 0 & 1 & 0 & 2 \end{vmatrix}' \tag{6}$$

Finite Element Analysis of Cable-Stayed Bridges with Appropriate Initial Shapes Under Seismic Excitations Focusing on Deck-Stay Interaction 237

**Figure 4.** Beam-column element for simulating the mass of each deck and tower.

cable-stayed bridge can be derived from the Lagrange's virtual work principle

*M q D q Sa P*

 

The system equations in generalized coordinates of a nonlinear finite element model of a

, *j j EL*

, *jk j k EL*

 

,

1,2,3, , , *N* (8)

(9)

<sup>0</sup> , *j jk k j S KE u S* (10)

, (11)

(12)

, (13)

, (14)

 

*M ME a a*

*j j P Kb* 

*<sup>j</sup> <sup>j</sup> <sup>W</sup> <sup>b</sup> q*

*dq <sup>q</sup> dt* 

2 2 *d q <sup>q</sup> dt* 

**2.2. System equations** 

$$\begin{array}{c|ccccc} & 140 & 0 & 0 & 70 & 0 & 0 \\ & 0 & 156 & 22L\_{b} & 0 & 54 & -13L\_{b} \\ \text{ME}\_{jk} = \frac{\rho\_{b}A\_{b}L\_{b}}{420} & 0 & 22L\_{b} & 4L\_{b}^{2} & 0 & 13L\_{b} & -3L\_{b}^{2} \\ & 420 & & 70 & 0 & 0 & 140 & 0 & 0 \\ & 0 & 54 & 13L\_{b} & 0 & 156 & -22L\_{b} \\ & 0 & -13L\_{b} & -3L\_{b}^{2} & 0 & -22L\_{b} & 4L\_{b}^{2} \end{array} \tag{7}$$

where *<sup>c</sup>* and *<sup>b</sup>* are the mass densities of the cable and beam-column elements, respectively. The coordinate transformation coefficient *<sup>j</sup> a* connected between *<sup>j</sup> u* and *q* for the mass matrices of the cable and beam-column elements can be found in [20].

**Figure 3.** Cable element for simulating the mass of each stay cable.

**Figure 4.** Beam-column element for simulating the mass of each deck and tower.

#### **2.2. System equations**

236 Earthquake Engineering

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 *<sup>j</sup> u j* 1 4 in Figure 3 and that of the latter with six element coordinates *<sup>j</sup> u j* 1 6 in Figure 4 can be individually expressed as

2010

(6)

connected between *<sup>j</sup> u* and *q*

(7)

0102

140 0 0 70 0 0 0 156 22 0 54 13

0 22 4 0 13 3 , <sup>420</sup> 70 0 0 140 0 0

*b bb b b b b*

*A L LL L L*

0 54 13 0 156 22 0 13 3 0 22 4

2 2

*b b*

*L L*

2 2

*b b b b*

*L L LL*

are the mass densities of the cable and beam-column elements,

*b b*

*L L*

<sup>0201</sup> , <sup>6</sup> <sup>1020</sup>

*c cc*

for the mass matrices of the cable and beam-column elements can be found in [20].

*jk A L ME* 

*jk*

respectively. The coordinate transformation coefficient *<sup>j</sup> a*

**Figure 3.** Cable element for simulating the mass of each stay cable.

*ME*

 and *<sup>b</sup>* 

where *<sup>c</sup>* 

> 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

$$\dot{\mathbf{M}}\_{\alpha\beta}\ddot{\mathbf{q}}\_{\beta} + \mathbf{D}\_{\alpha\beta}\dot{\mathbf{q}}\_{\beta} + \sum\_{\text{EL}} \mathbf{S}\_{\dot{\boldsymbol{\beta}}} \mathbf{a}\_{\boldsymbol{j}\alpha} = \mathbf{P}\_{\alpha\prime} \quad \alpha = 1, 2, 3, \cdots, N\_{\prime} \tag{8}$$

$$M\_{\alpha\beta} = \sum\_{\text{EL}} \text{ME}\_{jk} a'\_{j\alpha} a'\_{k\beta} \,\prime \,\tag{9}$$

$$\mathcal{S}\_{\dot{j}} = \mathcal{K} E\_{\dot{j}k} \mathcal{u}\_k + \mathcal{S}\_{\dot{j}}^0 \,. \tag{10}$$

$$P\_{\alpha} = \bar{\mathbf{K}}^{\dagger} \cdot \bar{\mathbf{b}}\_{\alpha}^{\dagger} \tag{11}$$

$$
\bar{b}\_{\alpha}^{j} = \frac{\partial \bar{W}^{j}}{\partial q\_{\alpha}},
\tag{12}
$$

$$
\dot{q}\_{\alpha} = \frac{dq\_{\alpha}}{dt} \tag{13}
$$

$$
\ddot{q}\_{\alpha} = \frac{d^2 q\_{\alpha}}{dt^2} \tag{14}
$$

where *M* and *D* are the system mass and damping matrices, respectively, which both are assumed to be constant; *<sup>j</sup> S* is the element force vector; *P* is the external force vector; 0 *<sup>j</sup> <sup>S</sup>* is the initial element force vector; *<sup>j</sup> <sup>K</sup>* is the external nodal force vector; *<sup>j</sup> <sup>b</sup>* is the basis vector; *<sup>W</sup> <sup>j</sup>* is the displacement vector corresponding to *<sup>j</sup> <sup>K</sup>* ; *q* and *q* are the system velocity and acceleration vectors, respectively; *t* is the time; *N* is the number of degrees of freedom; the subscripts and denote the numbers of the system coordinates; the subscripts *j* and *k* represent the numbers of the element coordinates; the superscript *j* denotes the nodal number; *EL* represents the summation over all elements.

Under consideration of three types of geometric nonlinearities mentioned in Chapter 2.1, *KEjk* of a cable element and that of a beam-column element can be individually obtained from Eq. (2) and Eq. (3). The former and the latter are due to the cable sag effect and the beam-column effect, respectively. Similarly, *M jk E* of the cable element and that of the beamcolumn element can be individually obtained from Eq. (6) and Eq. (7). *<sup>j</sup> u* , *<sup>j</sup> a* and *<sup>j</sup> b* are nonlinear functions of *q* when the large displacement effect occurs. *<sup>j</sup> K* can be written as a function of *q* if they are displacement dependent forces. *M* and *D* are both assumed to be constant, because only nonlinearities in stiffness are considered in this system.

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)

$$M\_{\alpha\beta} \Delta \ddot{q}\_{\beta}^{\prime\prime} + D\_{\alpha\beta} \Delta \dot{q}\_{\beta}^{\prime\prime} + \,^2 K\_{\alpha\beta}^{\prime\prime} \Delta q\_{\beta}^{\prime\prime} = \,\_u P\_{\alpha}^{\prime} + \Delta P\_{\alpha}^{\prime\prime} \quad t^{\prime} \le t \le t^{\prime} + \Delta t^{\prime\prime} \tag{15}$$

$$\mathbf{k}^{2}\mathbf{K}\_{\alpha\beta}^{n} = \sum\_{\rm EL} \mathbf{K}\underline{\mathbf{E}}\_{\beta\mathbf{k}}^{n}\mathbf{a}\_{j\alpha}^{n}\mathbf{a}\_{k\beta}^{n} + \sum\_{\rm EL} \mathbf{S}\_{j}^{n}\mathbf{a}\_{j\alpha,\beta}^{n} - \,^{n}\bar{\mathbf{K}}^{j} \cdot \,^{n}\bar{\mathbf{b}}\_{\alpha,\beta}^{j} - \,^{n}\bar{\mathbf{K}}\_{\beta}^{j} \cdot \,^{n}\bar{\mathbf{b}}\_{\alpha}^{j} \tag{16}$$

$$
\bar{b}^{\dot{\jmath}}\_{\alpha,\beta} = \frac{\hat{c}\bar{b}^{\dot{\jmath}}\_{\alpha}}{\hat{c}\eta\_{\beta}},
\tag{17}
$$

Finite Element Analysis of Cable-Stayed Bridges with Appropriate Initial Shapes Under Seismic Excitations Focusing on Deck-Stay Interaction 239

is the unbalanced force vector; *<sup>n</sup> P*

are the increments of the

*uP*

 , *<sup>n</sup> q*and *<sup>n</sup> q*

 in Eq. (16) consists of four terms. The first term is the elastic stiffness matrix, while the second and third terms are the geometric stiffness matrices induced by large displacements. Furthermore, the fourth term is the geometric stiffness matrix induced by displacement

system coordinate, velocity and acceleration vectors, respectively; *<sup>n</sup> t* is the time increment; the superscript *n* and 1 *n* denote the numbers of the time (or force) steps; the superscript

Eq. (15) is a set of simultaneous second-order linear ordinary differential equations in a

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

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

> *<sup>u</sup>* <sup>1</sup> , *n n PPP*

> > , *n n nn <sup>u</sup> j j EL P P Sa*

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

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

 

(25)

(26)

small time interval, which can be solved by the direct integration method [20].

determined by incrementally solving the linearized system equations

iteration is performed using the Newton-Raphson method [18-23].

<sup>2</sup> , *nn n n Kq P P*

 

where <sup>2</sup> *<sup>n</sup> K*

<sup>2</sup> *<sup>n</sup> K* **2.3. Initial shape analysis** 

presented in this study.

is the tangent system stiffness matrix; *<sup>n</sup>*

is the increment of the external force vector; *<sup>n</sup> q*

2 represents the second-order iteration matrix.

dependent forces, which is neglected in this study.

$$
\bar{K}\_{\alpha}^{\dagger} = \frac{\partial \bar{\mathbf{K}}^{\dagger}}{\partial q\_{\alpha}} \tag{18}
$$

$$\mathbf{P}\_{\alpha} \mathbf{P}\_{\alpha}^{n} = \mathbf{P}\_{\alpha}^{n} - \mathbf{M}\_{\alpha\beta} \ddot{\mathbf{q}}\_{\beta}^{n} - \mathbf{D}\_{\alpha\beta} \dot{\mathbf{q}}\_{\beta}^{n} - \sum\_{\rm EL} \mathbf{S}\_{j}^{n} a\_{j\alpha}^{n} \tag{19}$$

$$
\Delta P\_a^n = P\_a^{n+1} - P\_a^n \tag{20}
$$

$$
\Delta \boldsymbol{q}\_{\alpha}^{n} = \boldsymbol{q}\_{\alpha}^{n+1} - \boldsymbol{q}\_{\alpha}^{n} \tag{21}
$$

$$
\Delta \dot{\boldsymbol{q}}\_{\alpha}^{n} = \dot{\boldsymbol{q}}\_{\alpha}^{n+1} - \dot{\boldsymbol{q}}\_{\alpha}^{n} \tag{22}
$$

$$
\Delta \ddot{q}\_{\alpha}^{n} = \ddot{q}\_{\alpha}^{n+1} - \ddot{q}\_{\alpha}^{n} \tag{23}
$$

$$
\Delta t^n = t^{n+1} - t^n \,\,,\tag{24}
$$

where <sup>2</sup> *<sup>n</sup> K* is the tangent system stiffness matrix; *<sup>n</sup> uP* is the unbalanced force vector; *<sup>n</sup> P* is the increment of the external force vector; *<sup>n</sup> q* , *<sup>n</sup> q* and *<sup>n</sup> q* are the increments of the system coordinate, velocity and acceleration vectors, respectively; *<sup>n</sup> t* is the time increment; the superscript *n* and 1 *n* denote the numbers of the time (or force) steps; the superscript 2 represents the second-order iteration matrix.

<sup>2</sup> *<sup>n</sup> K* in Eq. (16) consists of four terms. The first term is the elastic stiffness matrix, while the second and third terms are the geometric stiffness matrices induced by large displacements. Furthermore, the fourth term is the geometric stiffness matrix induced by displacement dependent forces, which is neglected in this study.

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**

238 Earthquake Engineering

freedom; the subscripts

denotes the nodal number;

nonlinear functions of *q*

function of *q*

and *D*

*<sup>j</sup> <sup>S</sup>* is the initial element force vector; *<sup>j</sup> <sup>K</sup>*

are assumed to be constant; *<sup>j</sup> S* is the element force vector; *P*

vector; *<sup>W</sup> <sup>j</sup>* is the displacement vector corresponding to *<sup>j</sup> <sup>K</sup>*

 and 

 

2

*EL*

are the system mass and damping matrices, respectively, which both

velocity and acceleration vectors, respectively; *t* is the time; *N* is the number of degrees of

subscripts *j* and *k* represent the numbers of the element coordinates; the superscript *j*

Under consideration of three types of geometric nonlinearities mentioned in Chapter 2.1, *KEjk* of a cable element and that of a beam-column element can be individually obtained from Eq. (2) and Eq. (3). The former and the latter are due to the cable sag effect and the beam-column effect, respectively. Similarly, *M jk E* of the cable element and that of the beam-

when the large displacement effect occurs. *<sup>j</sup> K*

 

*<sup>u</sup>* , *n nn*

, , , *n n n n nn n n n n jj j j*

, , *j*

*<sup>j</sup> <sup>j</sup> <sup>K</sup> <sup>K</sup> q*

, *n n n n nn <sup>u</sup> j j EL P P M q D q Sa*

<sup>1</sup> , *nn n PP P*

<sup>1</sup> , *nn n qq q*

<sup>1</sup> , *nn n qq q*

<sup>1</sup> , *nn n qq q*

<sup>1</sup> , *nn n*

 

  

 

 

 

*K KE a a S a K b K b*

*<sup>j</sup> <sup>b</sup> <sup>b</sup> q* 

     

> 

(16)

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)

column element can be individually obtained from Eq. (6) and Eq. (7). *<sup>j</sup> u* , *<sup>j</sup> a*

to be constant, because only nonlinearities in stiffness are considered in this system.

if they are displacement dependent forces. *M*

<sup>2</sup> , *n n nn n n MqDq Kq P P*

*jk j k j j EL EL*

 

  

represents the summation over all elements.

is the external nodal force vector; *<sup>j</sup> <sup>b</sup>*

 ; *q*and *q*

and *D*

 

(17)

(18)

(19)

(20)

(21)

(22)

(23)

*tt t* (24)

denote the numbers of the system coordinates; the

is the external force vector;

*t tt t* (15)

 and *<sup>j</sup> b*

are both assumed

can be written as a

are

is the basis

are the system

where *M*

0

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

$$\mathbf{P}^{\prime 2} K^{\prime \prime}\_{\alpha \beta} \Delta q^{\prime \prime}\_{\beta} = \,\_{u} \mathbf{P}^{\prime \prime}\_{\alpha} + \Delta \mathbf{P}^{\prime \prime}\_{\alpha \prime} \quad \mathbf{P}^{\prime \prime}\_{\alpha} \le \mathbf{P}\_{\alpha} \le \mathbf{P}^{\prime \prime + 1}\_{\alpha} \tag{25}$$

$$
\sigma\_{\mu}P\_{\alpha}^{\prime\prime} = P\_{\alpha}^{\prime\prime} - \sum\_{\rm EL} S\_{j}^{\prime} a\_{j\alpha\prime}^{\prime\prime} \tag{26}
$$

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

$$\left|\frac{q\_{\alpha}}{L\_{r}}\right| \le \varepsilon\_{r\prime} \tag{27}$$

Finite Element Analysis of Cable-Stayed Bridges with Appropriate Initial Shapes Under Seismic Excitations Focusing on Deck-Stay Interaction 241

(29)

(30)

(31)

*f* and the normalized mode shape *Yn* of the *n* th mode can

force effects. On the basis of Eq. (28) representing the free vibration of the undamped

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

*<sup>n</sup> TTT s ss d dd t tt n nn nn n*

where *<sup>j</sup> Mn <sup>j</sup> sdt* , , are the generalized mass ratios of the *<sup>n</sup>* th mode; *M<sup>j</sup> <sup>j</sup> sdt* , , are

superscripts *s*, *d* and *t* denote the quantities of the stay cable, the deck and the tower,

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

\*1 1 \* \* , *n n n n nn*

, *n n <sup>g</sup> P MI q*

 

 

*Q P M q D q Sa*

\* 2 1 2 , *n n K IM ID K*

 

*<sup>n</sup> TTT s ss d dd t tt n nn nn n*

*<sup>n</sup> TTT s ss d dd t tt n nn nn n*

*n n s*

*n n d*

*n n t*

respectively. The sum of *<sup>s</sup> Mn* , *<sup>d</sup> Mn* and *<sup>t</sup> Mn* is 1 for the corresponding *<sup>n</sup>* .

Chapter 2.3 can be conducted based on the equivalent difference equations

\* 1\* , *n nn Q Kq*

 

> 

 ,

,

,

; *<sup>j</sup> Yn <sup>j</sup> sdt* , , are the subvectors of *Yn* in the *<sup>n</sup>* th mode; the

, *n nn t tt t* (32)

*<sup>j</sup> <sup>j</sup> EL*

(33)

  (34)

(35)

*Y MY Y MY Y MY*

*<sup>T</sup> t tt*

*Y MY*

*Y MY Y MY Y MY*

*Y MY*

*<sup>T</sup> d dd*

*Y MY Y MY Y MY*

*Y MY*

*<sup>T</sup> s ss*

system, the natural frequency *<sup>n</sup>*

**2.5. Modal coupling assessment** 

presented [24] as

the submatrices of *M<sup>A</sup>*

**2.6. Seismic analysis** 

be calculated by the subspace iteration method [20].

*M*

*M*

*M*

where *q* is the displacement in a certain direction of the control point; *<sup>r</sup> L* is the reference length; *<sup>r</sup>* is the convergence tolerance. For checking the deck displacement, each control point is the node intersected by the deck and the stay cable. *q* and *<sup>r</sup> L* individually denote the vertical displacement of the control point and the main span length. Similarly, each node intersected by the tower and the stay cable, or located on the top of the tower is chosen as the control point for checking the tower displacement. *q* and *<sup>r</sup> L* represent the horizontal displacement of the control point and the tower height, respectively.

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 decktower 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

$$\mathcal{M}^A\_{\alpha\beta}\ddot{\mathfrak{q}}\_{\beta} + \,^2\mathcal{K}^A\_{\alpha\beta}\mathfrak{q}\_{\beta} = 0,\tag{28}$$

where *M<sup>A</sup>* and <sup>2</sup> *<sup>A</sup> K* are the system mass and tangent system stiffness matrices with respect to the nonlinear static state *Aq* , respectively. The initial shape obtained in Chapter 2.3 can be regarded as *Aq*. Eq. (28) is derived from Eq. (15) with negligible damping and force effects. On the basis of Eq. (28) representing the free vibration of the undamped system, the natural frequency *<sup>n</sup> f* and the normalized mode shape *Yn* of the *n* th mode can be calculated by the subspace iteration method [20].

#### **2.5. Modal coupling assessment**

240 Earthquake Engineering

where *q*

length; *<sup>r</sup>* 

initial shape of the OECS model.

initial shape of the MECS model.

**2.4. Modal analysis** 

linearized system equation

and <sup>2</sup> *<sup>A</sup> K*

respect to the nonlinear static state *Aq*

2.3 can be regarded as *Aq*

where *M<sup>A</sup>*

appropriate initial shape can therefore be obtained.

point is the node intersected by the deck and the stay cable. *q*

displacement of the control point and the tower height, respectively.

the control point for checking the tower displacement. *q*

is conducted to reduce the displacements and to smooth the bending moments, and the

A number of control points are selected for insuring that both the deck and tower

*r q L* 

the vertical displacement of the control point and the main span length. Similarly, each node intersected by the tower and the stay cable, or located on the top of the tower is chosen as

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 decktower 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

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

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

<sup>2</sup> 0, *Mq Kq A A*

 

(28)

, respectively. The initial shape obtained in Chapter

are the system mass and tangent system stiffness matrices with

. Eq. (28) is derived from Eq. (15) with negligible damping and

 

, *r*

is the displacement in a certain direction of the control point; *<sup>r</sup> L* is the reference

is the convergence tolerance. For checking the deck displacement, each control

(27)

and *<sup>r</sup> L* individually denote

and *<sup>r</sup> L* represent the horizontal

displacements satisfy the architectural design requirements in the shape iteration

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

$$\overline{\boldsymbol{M}}\_{n}^{s} = \frac{\left(\overline{\boldsymbol{Y}}\_{n}^{s}\right)^{T} \boldsymbol{M}^{s} \overline{\boldsymbol{Y}}\_{n}^{s}}{\left(\overline{\boldsymbol{Y}}\_{n}^{s}\right)^{T} \boldsymbol{M}^{s} \overline{\boldsymbol{Y}}\_{n}^{s} + \left(\overline{\boldsymbol{Y}}\_{n}^{d}\right)^{T} \boldsymbol{M}^{d} \overline{\boldsymbol{Y}}\_{n}^{d} + \left(\overline{\boldsymbol{Y}}\_{n}^{t}\right)^{T} \boldsymbol{M}^{t} \overline{\boldsymbol{Y}}\_{n}^{t}}\tag{29}$$

$$
\overline{\boldsymbol{M}}\_n^d = \frac{\left(\overline{\boldsymbol{Y}}\_n^d\right)^T \boldsymbol{M}^d \overline{\boldsymbol{Y}}\_n^d}{\left(\overline{\boldsymbol{Y}}\_n^s\right)^T \boldsymbol{M}^s \overline{\boldsymbol{Y}}\_n^s + \left(\overline{\boldsymbol{Y}}\_n^d\right)^T \boldsymbol{M}^d \overline{\boldsymbol{Y}}\_n^d + \left(\overline{\boldsymbol{Y}}\_n^t\right)^T \boldsymbol{M}^t \overline{\boldsymbol{Y}}\_n^t}\tag{30}
$$

$$\overline{\boldsymbol{M}}\_{n}^{t} = \frac{\left(\overline{\boldsymbol{Y}}\_{n}^{t}\right)^{T} \boldsymbol{M}^{t} \overline{\boldsymbol{Y}}\_{n}^{t}}{\left(\overline{\boldsymbol{Y}}\_{n}^{s}\right)^{T} \boldsymbol{M}^{s} \overline{\boldsymbol{Y}}\_{n}^{s} + \left(\overline{\boldsymbol{Y}}\_{n}^{d}\right)^{T} \boldsymbol{M}^{d} \overline{\boldsymbol{Y}}\_{n}^{d} + \left(\overline{\boldsymbol{Y}}\_{n}^{t}\right)^{T} \boldsymbol{M}^{t} \overline{\boldsymbol{Y}}\_{n}^{t}}\tag{31}$$

where *<sup>j</sup> Mn <sup>j</sup> sdt* , , are the generalized mass ratios of the *<sup>n</sup>* th mode; *M<sup>j</sup> <sup>j</sup> sdt* , , are the submatrices of *M<sup>A</sup>* ; *<sup>j</sup> Yn <sup>j</sup> sdt* , , are the subvectors of *Yn* in the *<sup>n</sup>* th mode; the superscripts *s*, *d* and *t* denote the quantities of the stay cable, the deck and the tower, respectively. The sum of *<sup>s</sup> Mn* , *<sup>d</sup> Mn* and *<sup>t</sup> Mn* is 1 for the corresponding *<sup>n</sup>* .

#### **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

$$\mathbf{t}^\* \mathbf{Q}\_{\alpha}^{n+1} = \prescript{\*}{}{\mathbf{K}}\_{\alpha\beta}^n \Delta \mathbf{q}\_{\beta}^n, \ t^n \le t \le t^n + \Delta t^n,\tag{32}$$

$$\stackrel{\*}{\prime}Q^{n+1}\_{\alpha} = P^{n+1}\_{\alpha} - M\_{\alpha\beta}\stackrel{\*}{\dot{q}}^{n}\_{\beta} - D\_{\alpha\beta}\stackrel{\*}{\dot{q}}^{n}\_{\beta} - \sum\_{\text{EL}}S^{n}\_{\dot{\beta}}a^{n}\_{\dot{\beta}\alpha^{\prime}}\tag{33}$$

$$\stackrel{\*}{}{K}\_{\alpha\beta}^{n} = I\_1 M\_{\alpha\beta} + I\_2 D\_{\alpha\beta} + \stackrel{\*}{}{K}\_{\alpha\beta}^{n} \tag{34}$$

$$P\_{\alpha}^{\pi} = -M\_{\alpha\beta} I\_{\beta}{}^{g} \ddot{q}^{\pi},\tag{35}$$

$$\stackrel{\ast}{\cdot}\dot{\boldsymbol{q}}\_{\alpha}^{n} = -I\_{4}\dot{\boldsymbol{q}}\_{\alpha}^{n} - I\_{6}\ddot{\boldsymbol{q}}\_{\alpha}^{n} \tag{36}$$

Finite Element Analysis of Cable-Stayed Bridges with Appropriate Initial Shapes Under Seismic Excitations Focusing on Deck-Stay Interaction 243

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

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.

**Figure 5.** Finite element models of the Kao Ping Hsi Bridge.

Hsi Bridge can be found in [28].

$$\stackrel{\ast}{\dot{q}}\_{\alpha}^{\ast} = -I\_3 \dot{q}\_{\alpha}^{\prime\prime} - I\_5 \ddot{q}\_{\alpha}^{\prime\prime} \tag{37}$$

$$
\eta\_{\alpha}^{n+1} = \eta\_{\alpha}^{n} + \Delta \eta\_{\alpha}^{n} \tag{38}
$$

$$
\dot{q}\_{\alpha}^{n+1} = \prescript{\*}{}{\dot{q}}\_{\alpha}^{n} + I\_2 \Delta q\_{\alpha}^{n} \tag{39}
$$

$$
\ddot{\boldsymbol{q}}\_{\alpha}^{n+1} = \prescript{\*}{}{\ddot{q}}\_{\alpha}^{n} + I\_1 \Delta \boldsymbol{q}\_{\alpha}^{n} \tag{40}
$$

$$I\_1 = \frac{1}{\beta\_1 \left(\Delta t''\right)^2},\tag{41}$$

$$I\_2 = \frac{\mathcal{V}\_1}{\mathcal{J}\_1 \Delta t^{\pi^\*}} \, \, \, \tag{42}$$

$$I\_3 = \frac{1}{\beta\_1 \Delta t^{\eta}},\tag{43}$$

$$I\_4 = \frac{1}{2\beta\_1} - 1,\tag{44}$$

$$I\_{\oplus} = \frac{\mathcal{V}\_1}{\mathcal{\beta}\_1} - 1,\tag{45}$$

$$I\_{\theta} = \left(\frac{\mathcal{V}\_1}{2\mathcal{\beta}\_1} - 1\right) \Delta t^n \,\,\,\,\,\,\tag{46}$$

where these equations are derived from Eq. (15) and Eq. (19) using the Newmark method [27]; \* 1 *<sup>n</sup> Q* is the effective force vector; \* *<sup>n</sup> K* is the effective system stiffness matrix; *<sup>g</sup> <sup>n</sup> q* is the earthquake-induced ground acceleration; *I* is the column vector in which each element is either zero or unity depending on the direction of *<sup>g</sup> <sup>n</sup> <sup>q</sup>* ; 1 and 1 are the parameters defining the variation of acceleration over a time increment and determining the stability and accuracy characteristics of the Newmark method; \* *<sup>n</sup> q* , \* *<sup>n</sup> q* and *<sup>j</sup> I j* 1 6 are the coefficients of the seismic analysis.

#### **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.

242 Earthquake Engineering

[27]; \* 1 *<sup>n</sup> Q*

\*

\*

> > *I*

1 \* <sup>2</sup> , *nn n q q Iq*

1 \* <sup>1</sup> , *nn n q q Iq*

2

3

4

5

6

is the effective force vector; \* *<sup>n</sup> K*

is either zero or unity depending on the direction of *<sup>g</sup> <sup>n</sup>*

and accuracy characteristics of the Newmark method; \* *<sup>n</sup> q*

the earthquake-induced ground acceleration; *I*

coefficients of the seismic analysis.

**2. Finite element models** 

*I*

4 6 , *n nn q Iq Iq*

3 5 , *n nn q Iq Iq*

<sup>1</sup> , *n nn q qq*

 <sup>1</sup> <sup>2</sup> 1

*t*

1 *n*

1

1 *<sup>n</sup> I t* 

1 <sup>1</sup> , *<sup>n</sup> <sup>I</sup> <sup>t</sup>*

> 1 <sup>1</sup> 1, <sup>2</sup>

1

1 *I* 1, 

 

1

2 *<sup>n</sup> I t* 

1 1

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

defining the variation of acceleration over a time increment and determining the stability

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

 

 

 

 

 

(36)

(37)

(38)

(39)

(40)

, (42)

(44)

(45)

is the effective system stiffness matrix; *<sup>g</sup> <sup>n</sup>*

 and 1 

*q* ; 1 

> , \* *<sup>n</sup> q*

is the column vector in which each element

, (46)

, (41)

(43)

*q* is

are the parameters

and *<sup>j</sup> I j* 1 6 are the

**Figure 5.** Finite element models of the Kao Ping Hsi Bridge.
