## 1 Introduction
### Nonlinear Finite Element Model  
A nonlinear dynamic numerical model of the extradosed cable-stayed bridge is built based on the nonlinear FEM software OpenSEES. As piers and pylons are vulnerable components under earthquakes, the main girder remains elastic. Therefore, the nonlinear beam-column element and fiber sections are used to model pylons, piers and piles, while  
Figure 4: Cross-section size and reinforcement (units: cm). Figure 5: Support arrangement (units: cm). Figure 3: Layout of the extradosed cable-stayed bridge (units: cm). the disp-beam-column element and elastic section are used to model the main girder. Due to the consolidation of the pylon and main girder, they are connected by rigid arms. The supports are modeled using a ZeroLength element which contains two nodes; the upper node is connected to the main girder by a rigid arm, and the bottom node is connected to the pier also by a rigid arm. The \"m\" method is adopted to consider the pile-soil effect, and the soil spring is also modeled using a ZeroLength element. In this bridge, the rigid arms connecting the bridge members are simulated using an elastic beam-column element. In the finite element model, the materials of the cables and main girder are Steel02 material and elastic material, respectively, while both rigid arms and soil springs use elastic material. Meanwhile, the material of the supports is elastic-perfectly plastic material. For pylons, piers and piles, the fiber sections consist of Concrete02 material and Steel02 material utilized to simulate their nonlinearity. The hysteresis model and constitutive relation of these two materials provided by the software OpenSEES are shown in Figures 6 and 7. Information about materials, simulation of components and the rigid arms is shown in Table 1, and the model parameters are summarized in Table 2. Finally, the detail of the dynamic 3D nonlinear dynamic finite element model of the employed bridge is shown in Figure 8.  
For the simulation of the pile-soil interaction, taking 13#pier as an example, the soil spring stiffness is calculated according to Equation (17) [40]  
\[k=ab_{1}mz \tag{17}\]  
where \(a\) is the equivalent soil thickness for calculating the soil spring stiffness, \(b_{1}\) represents the calculated width of pile, \(m\) denotes the proportional coefficient of foundation coefficient, and \(z\) is the calculated depth of soil below the ground or local erosion line. Along the pile length, the soil layer is divided every 2 m, and the soil spring stiffness of longitudinal direction and transverse direction is added to the corresponding pile node. The soil spring model is shown in Figure 9, and detailed information regarding the pile-soil interaction is shown in Figure 10. Finally, the soil spring stiffness is shown in Table 3. According to the geological data of the soil layer, the foundation of the bridge passes through clay, sand pebbles and glutenite. The basic bearing capacity of the clay is 180 kPa, and the shear wave velocity range of the soil layer is 250\(\geq\) vs. >140, and the shear modulus is 114.76 MPa. The basic bearing capacity of sandy pebble soil is 350 kPa, the range of shear wave velocity is 500\(\geq\) vs. >250 and the shear modulus is 147.74 MPa. The basic bearing capacity of glutenite is 500 kpa, the shear wave velocity of soil layer vs. >500 and the shear modulus is 224 MPa.