## 2 The Analytical Model and Input Ground Motions  
### FE Modelling of the Cable-Stayed Bridge  
The bridge has two H-shaped reinforced concrete towers with a height of 93 m, and the total span is 640 m with a main span of 380 m and two side spans of 70 m and 60 m, respectively. The elevation of the bridge is shown in Figure 1. The dimension and section of the tower/pier are shown in Figure 2. The major materials and their characteristics are listed in Table 1.  
A three-dimensional FE model had been developed in a Structural Analysis Program 2000 version 19.0.0 (SAP2000 v 19.0.0) [36; 37]. The simulation of a cable-stayed bridge consists of four parts, namely the girder, the tower/pier, the cable and the boundary conditions. Generally, the girder is equipped with a closed box section with large free torsional stiffness, and a single girder model could be adopted considering the axial stiffness, bending stiffness, torsional stiffness, distributed mass and mass moment of inertia concentrated on the central axis. In addition, the girder would be elastic even under occasionally occurring earthquakes. Simultaneously, the tower/pier is also elastic due to the energy dissipation system. Therefore, the girder and tower/pier could be modeled by an elastic frame element which includes the effects of biaxial bending, torsion, axial deformation, and biaxial shear deformations [38]. Obviously, cables should be simulated by a truss element, and the modulus of elasticity modified by the Ernst formula [39], considering the mechanical properties and the sag effect. For boundary conditions, since only the overall response of the structure is considered rather than \begin{table}
\begin{tabular}{c c c} \hline \hline
**Material and Component** & **Standard Strength (MPa)** & **Young’s Modulus (MPa)** \\ \hline Concrete (tower) & 32.4 & \(3.45\times 10^{4}\) \\ Concrete (transition pier) & 26.8 & \(3.25\times 10^{4}\) \\ Concrete (auxiliary pier) & 26.8 & \(3.25\times 10^{4}\) \\ Cable & 1770 & \(2.05\times 10^{5}\) \\ Rebar & 400 & \(2.06\times 10^{5}\) \\ \hline \hline \end{tabular} Note: 1. The standard strength of concrete is the axial compressive strength of concrete measured with a 150 mm \(\times\) 150 mm \(\times\) 300 mm prism as the standard specimen, and the standard strength means the strength characteristic value with a certain guarantee rate (95%). 2. Statistical analysis shows that the test value of the rebar and cable strength conforms to normal distribution, and the standard strength of is the strength characteristic value with a certain guarantee rate (97.73%).  
\end{table} 
Table 1: Main materials of the bridge. Figure 1: Elevation of the cable-stayed bridge (unit m). Figure 2: Dimensions of the major structure and their sections (unit: cm). the pile-soil effect, the node constraint of SAP2000 can be directly used to simulate the boundary conditions; specifically, both the tower bottom and pier bottom are fixed. In order to simulate a double-cable-stayed bridge, a rigid arm can be used to connect the girder and the cable; thus, a spine model should be adopted. In the longitudinal and transverse directions, the connections between the girder and the tower/pier are shown in Table 2. The FE model of SAP 2000 in this paper is shown in Figure 3. The girder, tower and pier are divided into around 4-m sections as a unit, and the model has a total of 607 nodes and 447 units. It is necessary to note that ordinary constraints and steel dampers are simulated in different ways. An elastic link element with large stiffness would be employed when the girder is fixed in the transverse direction at the tower locations. By contrast, the simulation of steel damper is relatively complex. As shown in Figure 4, the YSD is connected at the bottom of the deck through the sliding groove to accommodate the deck movement longitudinally. The YSDs are arranged symmetrically on the two sides of the bridge, and their connections at longitudinal locations on the bridge are shown in Figures 4 and 5.