**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.

Finite Element Analysis of Cable-Stayed Bridges with Appropriate Initial Shapes Under Seismic Excitations Focusing on Deck-Stay Interaction 245

**Figure 6.** Initial shapes of the Kao Ping Hsi Bridge.

Table 1 summarizes the modal properties of the Kao Ping Hsi Bridge based on the OECS

represent the natural frequency and the normalized mode shape of the *n* th mode, respectively. As expected, the MECS model reveals the global, local and coupled modes, whereas the OECS model only yields the global modes. The modal properties of modes 1 and 2 in the OECS model are individually similar to those of modes 1 and 12 in the MECS model, because these modes represent the global modes. While mode 3 in the OECS model is identified as the global mode, mode 19 in the MECS model is the coupled mode. The other coupled mode can also be observed in mode 18 in the MECS model. These results suggest that the interaction between the deck-tower system and stay cables can be captured by the MECS model, but not by the OECS model. Also due to the limitations of the OECS model, modes 2 to 11, modes 13 to 17 and modes 20 to 24, which represent the local modes of the stay cables, are successfully captured by the MECS model, but not by the OECS model.

*f* and *Yn*

model (modes 1 to 3) and the MECS model (modes 1 to 24). In this table, *<sup>n</sup>*

## **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 *<sup>r</sup>* is set to 10-4 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.

Finite Element Analysis of Cable-Stayed Bridges with Appropriate Initial Shapes Under Seismic Excitations Focusing on Deck-Stay Interaction 245

244 Earthquake Engineering

bridges.

**4. Numerical results** 

**4.1. Initial shape analysis** 

appropriate initial shapes of cable-stayed bridges.

in the horizontal direction. The convergence tolerance *<sup>r</sup>*

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

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

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

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

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

cables due to the fact that each stay cable is simulated by multiple cable elements.

**4.2. Modal analysis and modal coupling assessment** 

the appropriate initial shapes of cable-stayed bridges.

is set to 10-4 in this study.

**Figure 6.** Initial shapes of the Kao Ping Hsi Bridge.

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, *<sup>n</sup> f* and *Yn* represent the natural frequency and the normalized mode shape of the *n* th mode, respectively. As expected, the MECS model reveals the global, local and coupled modes, whereas the OECS model only yields the global modes. The modal properties of modes 1 and 2 in the OECS model are individually similar to those of modes 1 and 12 in the MECS model, because these modes represent the global modes. While mode 3 in the OECS model is identified as the global mode, mode 19 in the MECS model is the coupled mode. The other coupled mode can also be observed in mode 18 in the MECS model. These results suggest that the interaction between the deck-tower system and stay cables can be captured by the MECS model, but not by the OECS model. Also due to the limitations of the OECS model, modes 2 to 11, modes 13 to 17 and modes 20 to 24, which represent the local modes of the stay cables, are successfully captured by the MECS model, but not by the OECS model.


Finite Element Analysis of Cable-Stayed Bridges with Appropriate Initial Shapes Under Seismic Excitations Focusing on Deck-Stay Interaction 247

> **OECS MECS**

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

*n*

**Figure 7.** Relationships between natural frequencies and mode numbers of the MECS model of the Kao

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

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, *<sup>s</sup> Mn* , *<sup>d</sup> Mn* and *<sup>t</sup> Mn* represent the generalized mass ratios of the stay cable, the deck and the tower of the *n* th mode, 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> <sup>n</sup>* 1 24 . It is evident that *<sup>t</sup> Mn <sup>n</sup>* 1 24 approaches 0 for the first 24 modes due to the high rigidity of the concrete tower, resulting in the insignificant tower deformations in the lower modes sensitive to the ambient excitations, as can also be seen in Figure 8(b). These results are in

**0 5 10 15 20 25**

based on the assumption that stay S19 is clamped at both ends [29].

*nf*

**0.0**

shape (dashed line) obtained in Chapter 4.1.

agreement with the literature [2].

Ping Hsi Bridge.

**0.1**

**0.2**

**0.3**

**0.4**

**0.5**

**0.6**

**0.7**

**0.8**

**0.9**

**1.0**

DT: Deck-tower system

S: Stay cable

G: Global mode

L: Local mode

C: Coupled mode

**Table 1.** Comparisons between corresponding modal properties of the OECS and MECS models of the Kao Ping Hsi Bridge.

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].

246 Earthquake Engineering

*n <sup>n</sup>*

DT: Deck-tower system

Kao Ping Hsi Bridge.

S: Stay cable G: Global mode L: Local mode C: Coupled mode OECS MECS

1 0.2877 1st DT G 1 0.3053 1st DT G 2 0.3382 1st S28 L 3 0.3852 1st S27 L 4 0.4274 1st S26 L 5 0.4554 1st S1 L 6 0.4653 1st S25 L 7 0.4899 1st S24 L 8 0.5067 1st S23 L 9 0.5269 1st S22 L 10 0.5378 1st S2 L 11 0.5471 1st S21 L 2 0.5455 2nd DT G 12 0.5686 2nd DT G 13 0.5944 1st S3 L 14 0.6040 1st S20 L 15 0.6333 1st S4 L 16 0.6346 2nd S28 L 17 0.6835 1st S5 L

18 0.6850 3rd DT

3 0.6854 3rd DT G 19 0.7171 3rd DT

 20 0.7269 1st S6 L 21 0.7500 2nd S27 L 22 0.7590 1st S7 L 23 0.8008 1st S8 L 24 0.8184 1st S18 L

**Table 1.** Comparisons between corresponding modal properties of the OECS and MECS models of the

*f* (Hz) *Yn* Type

1st S19 <sup>C</sup>

1st S19 <sup>C</sup>

*f* (Hz) *Yn* Type *n <sup>n</sup>*

**Figure 7.** Relationships between natural frequencies and mode numbers of the MECS model of the Kao Ping Hsi Bridge.

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, *<sup>s</sup> Mn* , *<sup>d</sup> Mn* and *<sup>t</sup> Mn* represent the generalized mass ratios of the stay cable, the deck and the tower of the *n* th mode, 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> <sup>n</sup>* 1 24 . It is evident that *<sup>t</sup> Mn <sup>n</sup>* 1 24 approaches 0 for the first 24 modes due to the high rigidity of the concrete tower, resulting in the insignificant tower deformations in the lower modes sensitive to the ambient excitations, as can also be seen in Figure 8(b). These results are in agreement with the literature [2].

Finite Element Analysis of Cable-Stayed Bridges with Appropriate Initial Shapes Under Seismic Excitations Focusing on Deck-Stay Interaction 249

> **Stay Cable Deck Tower**

> > *f* and *Yn*

*f* and *Yn n* 1,12 in the MECS

*n* **0 5 10 15 20 25**

**Figure 9.** Variations in generalized mass ratios with respect to mode numbers of the MECS model of

model. It is consistent with the results in Figure 9 that for modes 1 and 12 in the MECS model, the sum of *<sup>d</sup> Mn* and *<sup>t</sup> Mn <sup>n</sup>* 1,12 is close to 0.9, whereas *<sup>s</sup> Mn <sup>n</sup>* 1,12 approaches 0.1. Consequently, these modes are primarily dominated by the deformations of the deck-tower system with the quasi-static motions of the stay cables. This type of response

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, *<sup>s</sup> Mn <sup>n</sup>* 2 11,13 17,20 24 is close to 1, whereas the sum of *<sup>d</sup> Mn* and *<sup>t</sup> Mn <sup>n</sup>* 2 11,13 17,20 24 approaches 0. It is consistent with the results in Table 1,

Figure 7 and Figure 8(b) that *Yn n* 2 11,13 17,20 24 in the MECS model is the local mode predominantly consisting of the stay cable motions with negligible deformations of the deck-tower system. This type of response can be recognized as the "pure" cable mode in

As shown in Table 1, Figure 7, Figure 8(a) and 8(b), the difference between 19 *f* in the MECS

in the MECS model is the coupled mode, but *Y*3 in the OECS model is the global mode, i.e., the "pure" deck-tower mode. Similarly, 18 *f* in the MECS model (0.6850 Hz) branches from the fundamental frequency of stay S19 clamped at both ends (0.6908 Hz). This is because

*f* in the OECS model (0.6854 Hz) is evident due to the fact that *Y*<sup>19</sup>

It can be seen in Table 1, Figure 7, Figure 8(a) and 8(b) that for the global modes, *<sup>n</sup>*

*n* 1,2 in the OECS model are individually similar to *<sup>n</sup>*

can be identified as the "pure" deck mode in the analytical model [17].

**0.0**

the Kao Ping Hsi Bridge.

the analytical model [17].

model (0.7171 Hz) and 3

**0.1**

**0.2**

**0.3**

**0.4**

**0.5**

**0.6**

*s Mn d Mn t Mn*

**0.7**

**0.8**

**0.9**

**1.0**

**Figure 8.** Normalized mode shapes of the Kao Ping Hsi Bridge.

**Figure 8.** Normalized mode shapes of the Kao Ping Hsi Bridge.

**Figure 9.** Variations in generalized mass ratios with respect to mode numbers of the MECS model of the Kao Ping Hsi Bridge.

It can be seen in Table 1, Figure 7, Figure 8(a) and 8(b) that for the global modes, *<sup>n</sup> f* and *Yn n* 1,2 in the OECS model are individually similar to *<sup>n</sup> f* and *Yn n* 1,12 in the MECS model. It is consistent with the results in Figure 9 that for modes 1 and 12 in the MECS model, the sum of *<sup>d</sup> Mn* and *<sup>t</sup> Mn <sup>n</sup>* 1,12 is close to 0.9, whereas *<sup>s</sup> Mn <sup>n</sup>* 1,12 approaches 0.1. Consequently, these modes are primarily dominated by the deformations of the deck-tower system with the quasi-static motions of the stay cables. This type of response can be identified as the "pure" deck mode in the analytical model [17].

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, *<sup>s</sup> Mn <sup>n</sup>* 2 11,13 17,20 24 is close to 1, whereas the sum of *<sup>d</sup> Mn* and *<sup>t</sup> Mn <sup>n</sup>* 2 11,13 17,20 24 approaches 0. It is consistent with the results in Table 1, Figure 7 and Figure 8(b) that *Yn n* 2 11,13 17,20 24 in the MECS model is the local mode predominantly consisting of the stay cable motions with negligible deformations of the deck-tower system. This type of response can be recognized as the "pure" cable mode in the analytical model [17].

As shown in Table 1, Figure 7, Figure 8(a) and 8(b), the difference between 19 *f* in the MECS model (0.7171 Hz) and 3 *f* in the OECS model (0.6854 Hz) is evident due to the fact that *Y*<sup>19</sup> in the MECS model is the coupled mode, but *Y*3 in the OECS model is the global mode, i.e., the "pure" deck-tower mode. Similarly, 18 *f* in the MECS model (0.6850 Hz) branches from the fundamental frequency of stay S19 clamped at both ends (0.6908 Hz). This is because

*Y*18 in the MECS model is the coupled mode, while the fundamental mode shape of stay S19 can be regarded as the "pure" stay cable mode. These observations are attributed to the frequency loci veering when the natural frequency of the "pure" deck-tower mode (0.6854 Hz) approaches that of the "pure" stay cable mode (0.6908 Hz). As illustrated in Figure 9, the sum of 19 *<sup>M</sup><sup>d</sup>* and 19 *<sup>M</sup><sup>t</sup>* is relatively higher than 19 *<sup>M</sup><sup>s</sup>* , whereas the sum of 18 *<sup>M</sup><sup>d</sup>* and 18 *<sup>M</sup><sup>t</sup>* is comparatively lower than 18 *<sup>M</sup><sup>s</sup>* . Consequently, *Y*18 and *Y*19 in the MECS model are the pair of coupled modes with the similar configurations, which have substantial contributions from both the deck-tower system and stay cables. These phenomena correspond to the mode localization. This type of response coincides with the coupled mode in the analytical model [17].

Finite Element Analysis of Cable-Stayed Bridges with Appropriate Initial Shapes Under Seismic Excitations Focusing on Deck-Stay Interaction 251

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.

**Figure 11.** Displacement time histories of the stay cable of the Kao Ping Hsi Bridge.

The axial force, which is in the 1 *u* coordinate of the cable element in Figure 1, is the unique internal force of the stay cable. Figure 13 shows the internal force time history of element 28 in stay S28 for the OECS model and those of the corresponding elements 271, 275 and 280 in stay S28 for the MECS model. The variations in the dynamic responses among the three elements of the MECS model are negligible. In addition, the dynamic response of each element in the MECS model is in agreement with that of the corresponding element in the OECS model, which can be considered as the "nominal" dynamic axial force of the stay cable. Consequently, the dynamic internal forces of the stay cables are successfully captured by both the OECS and MECS models. The internal forces of the deck-tower system include the left moment, right moment and axial force, which are individually in the 1 *u* , 2 *u* and 3 *u* coordinates of the beam-column element in Figure 2. Figure 14 shows the internal force time histories of element 69 (321) in the deck and those of element 40 (292) in the tower for the

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 1 1 1 4, 1 2 is used to calculate the displacement and internal force time histories of the system. The duration of the simulation is set to 30.0 s.

**Figure 10.** The Chi-Chi earthquake accelerogram.

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.

250 Earthquake Engineering

[17].

*Y*18 in the MECS model is the coupled mode, while the fundamental mode shape of stay S19 can be regarded as the "pure" stay cable mode. These observations are attributed to the frequency loci veering when the natural frequency of the "pure" deck-tower mode (0.6854 Hz) approaches that of the "pure" stay cable mode (0.6908 Hz). As illustrated in Figure 9, the sum of 19 *<sup>M</sup><sup>d</sup>* and 19 *<sup>M</sup><sup>t</sup>* is relatively higher than 19 *<sup>M</sup><sup>s</sup>* , whereas the sum of 18 *<sup>M</sup><sup>d</sup>* and 18 *<sup>M</sup><sup>t</sup>* is comparatively lower than 18 *<sup>M</sup><sup>s</sup>* . Consequently, *Y*18 and *Y*19 in the MECS model are the pair of coupled modes with the similar configurations, which have substantial contributions from both the deck-tower system and stay cables. These phenomena correspond to the mode localization. This type of response coincides with the coupled mode in the analytical model

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

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

1 1 1 4, 1 2 is used to calculate the displacement and internal force time histories of

appropriate initial shapes of cable-stayed bridges.

the system. The duration of the simulation is set to 30.0 s.

**Figure 10.** The Chi-Chi earthquake accelerogram.

**4.3. Seismic analysis** 

 

**Figure 11.** Displacement time histories of the stay cable of the Kao Ping Hsi Bridge.

The axial force, which is in the 1 *u* coordinate of the cable element in Figure 1, is the unique internal force of the stay cable. Figure 13 shows the internal force time history of element 28 in stay S28 for the OECS model and those of the corresponding elements 271, 275 and 280 in stay S28 for the MECS model. The variations in the dynamic responses among the three elements of the MECS model are negligible. In addition, the dynamic response of each element in the MECS model is in agreement with that of the corresponding element in the OECS model, which can be considered as the "nominal" dynamic axial force of the stay cable. Consequently, the dynamic internal forces of the stay cables are successfully captured by both the OECS and MECS models. The internal forces of the deck-tower system include the left moment, right moment and axial force, which are individually in the 1 *u* , 2 *u* and 3 *u* coordinates of the beam-column element in Figure 2. Figure 14 shows the internal force time histories of element 69 (321) in the deck and those of element 40 (292) in the tower for the OECS (MECS) model. The dynamic responses of each element in the OECS model coincide with those of the corresponding element in the MECS model. Consequently, the dynamic internal forces of the deck-tower system are reasonably simulated by both the OECS and MECS models.

Finite Element Analysis of Cable-Stayed Bridges with Appropriate Initial Shapes Under Seismic Excitations Focusing on Deck-Stay Interaction 253

**Figure 14.** Internal force time histories of the deck-tower system of the Kao Ping Hsi Bridge.

cable-stayed bridges under seismic excitations.

are calculated based on the seismic analyses.

**5. Conclusions** 

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

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

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

**Figure 12.** Displacement time histories of the deck-tower system of the Kao Ping Hsi Bridge.

**Figure 13.** Internal force time histories of the stay cable of the Kao Ping Hsi Bridge.

**Figure 14.** Internal force time histories of the deck-tower system of the Kao Ping Hsi Bridge.

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**

252 Earthquake Engineering

MECS models.

VerticalDisp lacement (m)

Vertical Displa cement (m)

Vertical Disp lacement (m)

OECS (MECS) model. The dynamic responses of each element in the OECS model coincide with those of the corresponding element in the MECS model. Consequently, the dynamic internal forces of the deck-tower system are reasonably simulated by both the OECS and

> Horizontal Displa cement (m)

HorizontalDisplacement

HorizontalD isplacem ent (m)

**Figure 12.** Displacement time histories of the deck-tower system of the Kao Ping Hsi Bridge.

**Figure 13.** Internal force time histories of the stay cable of the Kao Ping Hsi Bridge.

 (m)

> 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.

Finite Element Analysis of Cable-Stayed Bridges with Appropriate Initial Shapes Under Seismic Excitations Focusing on Deck-Stay Interaction 255

[10] Gattulli, V., Lepidi, M., Macdonald, J.H.G., and Taylor, C.A. (2005). "One-to-two globallocal interaction in a cable-stayed beam observed through analytical, finite element and experimental models." *International Journal of Non-Linear Mechanics*, 40(4), 571-588. [11] Gattulli, V., and Lepidi, M. (2007). "Localization and veering in the dynamics of cable-

[12] Tuladhar, R., Dilger, W.H., and Elbadry, M.M. (1995). "Influence of cable vibration on seismic response of cable-stayed bridges." *Canadian Journal of Civil Engineering*, 22(5),

[13] Caetano, E., Cunha, A., and Taylor, C.A. (2000a). "Investigation of dynamic cable-deck interaction in a physical model of a cable-stayed bridge. Part I: modal analysis."

[14] Caetano, E., Cunha, A., and Taylor, C.A. (2000b). "Investigation of dynamic cable-deck interaction in a physical model of a cable-stayed bridge. Part II: seismic response."

[15] Au, F.T.K., Cheng, Y.S., Cheung, Y.K., and Zheng, D.Y. (2001). "On the determination of natural frequencies and mode shapes of cable-stayed bridges." *Applied Mathematical* 

[16] Caetano, E., Cunha, A., Gattulli, V., and Lepidi, M. (2008). "Cable-deck dynamic interactions at the International Guadiana Bridge: On-site measurements and finite

[17] Liu, M.Y., Zuo, D., and Jones, N.P. (2005). "Deck-induced stay cable vibrations: Field observations and analytical model." *Proceedings of the Sixth International Symposium on* 

[18] Wang, P.H., Tseng, T.C., and Yang, C.G. (1993). "Initial shape of cable-stayed bridges."

[19] Wang, P.H., and Yang, C.G. (1996). "Parametric studies on cable-stayed bridges."

[20] Wang, P.H., Lin, H.T., and Tang, T.Y. (2002). "Study on nonlinear analysis of a highly

[21] Wang, P.H., Tang, T.Y., and Zheng, H.N. (2004). "Analysis of cable-stayed bridges during construction by cantilever methods." *Computers and Structures*, 82(4-5), 329-346. [22] Wang, P.H., Liu, M.Y., Huang, Y.T., and Lin, L.C. (2010). "Influence of lateral motion of cable stays on cable-stayed bridges." *Structural Engineering and Mechanics*, 34(6), 719-

[23] Liu, M.Y., Lin, L.C., and Wang, P.H. (2011). "Dynamic characteristics of the Kao Ping Hsi Bridge under seismic loading with focus on cable simulation." *International Journal* 

[24] Liu, M.Y., Zuo, D., and Jones, N.P. "Analytical and numerical study of deck-stay interaction in a cable-stayed bridge in the context of field observations." *Journal of* 

[25] Ernst, H.J. (1965). "Der E-modul von Seilen unter Berücksichtigung des Durchhanges."

element modelling." *Structural Control and Health Monitoring*, 15(3), 237-264.

*Cable Dynamics*, 175-182, Charleston, South Carolina, USA, September 19-22.

redundant cable-stayed bridge." *Computers and Structures*, 80(2), 165-182.

stayed bridges." *Computers and Structures*, 85(21-22), 1661-1678.

*Earthquake Engineering and Structural Dynamics*, 29(4), 481-498.

*Earthquake Engineering and Structural Dynamics*, 29(4), 499-521.

1001-1020.

738.

*Modelling*, 25(12), 1099-1115.

*Computers and Structures*, 46(6), 1095-1106.

*Computers and Structures*, 60(2), 243-260.

*of Structural Stability and Dynamics*, 11(6), 1179-1199.

*Engineering Mechanics*, *ASCE*. (under review).

*Der Bauingenieur*, 40(2), 52-55. (in German).
