## 2 The Analytical Model and Input Ground Motions
### FE Modelling of the Cable-Stayed Bridge
The YSD is designed according to the design criterion for steel dampers [29; 40; 41]. For the wind load and normal traffic loads, the YSD remains elastic and provides enough transverse stiffness. During the earthquake, the YSD starts to deform and dissipate seismic energy inelastically, thus protecting the tower columns and piers. [PERSON] et al. [42] and [PERSON] et al. [7] performed the cyclic tests of the YSD, showing that it has stable hysteretic behavior, and the hysteretic curve is assumed as the bilinear response (Figure 6), regardless of the specific form [19; 20; 21; 24; 25; 26; 30; 32]. Therefore, the YSD is simulated \begin{table}
\begin{tabular}{c c c c c c c c c c c c} \hline \hline \multirow{2}{*}{**Location**} & \multicolumn{4}{c}{**Degree of Freedom**} & \multicolumn{4}{c}{**Degree of Freedom**} \\  & \multicolumn{4}{c}{**(without YSDs)**} & \multicolumn{4}{c}{**(with YSDs)**} \\ \hline  & UX & UY & UZ & RX & RY & RZ & UX & UY & UZ & RX & RY & RZ \\ Tower & 0 & 1 & 1 & 0 & 0 & 0 & 0 & YSD (2) & 1 & 0 & 0 & 0 \\ Auxiliary pier & 0 & 0 & 1 & 0 & 0 & 0 & 0 & YSD (2) & 1 & 0 & 0 & 0 \\ Transition pier & 0 & 0 & 1 & 0 & 0 & 0 & 0 & YSD (2) & 1 & 0 & 0 & 0 \\ \hline \hline \end{tabular} Note: 1. “1” means fixed, “0” means free; yielding steel dampers (YSDs) are installed between the girder and the substructure, and numbers in () are the quantities of YSDs used on each side of the bridge. 2. X means the longitudinal direction, Y means the transverse direction, and Z means the vertical direction.  
\end{table} 
Table 2: The boundary condition of the bridge. Figure 3: Finite element (FE) model. by Plastic Wen units in SAP2000, which are bilinear constitutive models. There are four parameters in damper design, namely the yielding strength F\({}_{y}\), yielding displacement \(\Delta_{y}\), the ratio of post-yielding stiffness to pre-yielding stiffness and ultimate deformation \(\Delta_{u}\), while the ultimate strength F\({}_{u}\) is not an independent parameter. Normally, the \(\Delta_{y}\) is set to 10 mm and the ratio of post-yielding stiffness to pre-yielding stiffness is set as 0.6% (less than 5% depending on the different shapes of products), which can provide good hysteretic performance according to related studies [19; 20; 21; 24; 25; 26; 30; 32]. Therefore, F\({}_{y}\) is the key parameter which needs to be determined in this bilinear model. It is notable that the ultimate deformation \(\Delta_{u}\) reaches a certain amount in a fabricated YSD product; however, the maximum deformation of the bilinear model is assumed to be unlimited in the analysis so that the influence of F\({}_{y}\) on the relative displacement between the super- and sub-structures can be observed. In addition, it is more accurate to use complex a FE model such as shell elements or fiber elements to better consider the dynamic mechanical behavior of the girder instead of a simplified spine model. However, according to some recent studies [4; 28], acceptable dynamic behavior and corresponding seismic responses can be obtained, and we use this model to carry out our comprehensive parametric study of the proper yielding strength of the applied steel dampers for time efficiency.