## 4 Seismic Vulnerability of Wall Pier Girder Bridges
### Time-History Analysis  
OpenSees is used for the analysis. The main girder is simulated by three-dimensional elasticBeamColumn elements. The bearings are simulated by flatSliderBearing elements and their sliding effects are considered. The foundations are simulated by EqualDOF mode, and the stiffness of each direction is calculated according to the specification. Simplified models of the abutments are established by referring to California code for seismic design of bridges [24], and their mechanical effects are calculated. The transverse and longitudinal restraint effects of pile foundations on abutment are considered by referring to Nielson's research [25]; according to the research of Xu Lueqin et al [26], simplified mechanical models are adopted for the baffle block, without considering the bounding effect between the main girder and the block. The bounding effects between the main girders, as well as between the main girder and the abutment, are simulated by linear springs, and the bounding stiffness are calculated after referring to the research of Xia Qi et al. [27]. The corresponding finite element models of 120 bridge samples were built. The entire bridge model and the mechanical models of each component are shown in Figures 7 and 8 below:  
Twenty near-field and far-field ground motions are selected from the database of Pacific Earthquake Engineering Research Center of the United States. The selection criteria are shown in Table 8. PGA is used as the intensity measure (IM) of the ground motions, and the ground motions are corrected and cut off respectively. According to the IM, 40 ground motions are modulated to 1.0, 1.5 and 2.0 times respectively, thus 120 ground motions are obtained. The ground motions and the bridge samples are matched one by one for time-history analysis, then the seismic demand of each vulnerable component is obtained.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Uncertainty Parameter** & **Distribution Type** & \multicolumn{2}{c}{**Distribution Parameter**} \\ \cline{3-4}  & & \(\alpha\) & \(\beta\) \\ \hline C35 concrete compressive strength (MPa) & normal distribution & 35 & 4.5 \\ HRB335 steel yield strength (MPa) & logarithmic normal distribution & 5.81 & 0.1 \\ abutment initial stiffness (kN/mm/m) & uniform distribution & 11.5 & 28.5 \\ Scale factor of horizontal resistance coefficient (kN/m4) & uniform distribution & 60000 & 100000 \\ damping ratio & normal distribution & 0.045 & 0.0125 \\ vertical reinforcement ratio of piers (\%) & uniform distribution & 0.55 & 0.85 \\ transverse reinforcement ratio (\%) & uniform distribution & 0.2 & 0.4 \\ pier height (m) & uniform distribution & 11 & 16 \\ expansion joint width (cm) & normal distribution & 8 & 0.5 \\ shear elastic modulus (MPa) & normal distribution & 1.18 & 0.16 \\ \hline \hline \end{tabular} Note: For the distribution parameters \(\alpha\) and \(\beta\), when the distribution type of model parameters is normal distribution, they are mean and standard deviation of the normal distribution respectively, and when the distribution type of the model parameters is uniform distribution, they are the upper and lower bounds of the uniform distribution respectively. When the distribution type of the model parameters is logarithmic normal distribution, the parameters \(\alpha\) and \(\beta\) are numerically equal to the logarithmic mean and standard deviation.  
\end{table}
Table 7: Uncertainty parameter distribution.  
Figure 7: Bridge model.