## 3 Numerical Example
### Characterization of Bridges' Seismic Demands  
Structural demand is generally defined as the damage measure of the structural components when given a certain intensity of ground motion. In this paper, both the pier columns and the LRBs are taken as vulnerable components since they may behave nonlinearly under strong ground motions. For the pier seismic demand, the curvature-based and displacement-based damage index are commonly used since bend failure usually occurs on the reinforced concrete pier columns under earthquake events. It is easier to record the displacement than the curvature. Therefore, the displacement ductility is employed as the pier seismic demand, which is defined by [34]: \(\mu_{ci}=\Delta_{ci}/\Delta_{c1}\) where \(\Delta_{ci}\) is the critical value of pier top displacement at the i-th level of damage state and \(\Delta_{c1}\) is the pier top displacement when the longitudinal reinforcement first yields. Because \(\Delta_{ci}\) is a constant value at a certain level of damage state, the influence of uncertainties to the pier seismic demand is equivalent to the max pier top displacement (MPTD). It is worth noting that the curvature ductility ratio is also a well-known pier seismic demand parameter, which can be defined as \(\mu_{\phi}=\phi/\phi_{y}\) (\(\phi\) and \(\phi_{y}\) are the sectional curvature and sectional yield curvature, respectively) [35]. [PERSON] et al. expressed the curvature ductility ratio in terms of the top displacement, the height and plastic hinge length of the pier column [36]. Therefore, the sensitivity of curvature ductility ratio is equivalent to the MPTD. In this regard, only the MPTD is analyzed in the following importance evaluation for the pier seismic demand parameter. For the LRBs' seismic demand, the shear deformation is taken as the damage index, which is defined by the ratio of bearing max displacement to the total thickness. \begin{table}
\begin{tabular}{l l l l l l} \hline \hline
**Uncertain Parameters** & **Symbol** & **Distribution** & **Mean** & **COV** & **Ref.** \\ \hline Mass per-length of superstructure (kg/m) & \(m_{g}\) & Normal & 362 & 0.10 & [26; 33] \\ Post-yield stiffness of LRB (kN/m) & \(K_{db}\) & Normal & 96 & 0.14 & [33] \\ Peak compressive strength of unconfined concrete (MPa) & \(f_{cu}\) & Lognormal & 26.1 & 0.14 & [32; 33] \\ Strain at the peak strength of unconfined concrete & \(\varepsilon_{cu}\) & Lognormal & 0.002 & 0.20 & [32; 33] \\ Strain at the crushing strength of unconfined concrete & \(\varepsilon_{uu}\) & Lognormal & 0.005 & 0.20 & [32; 33] \\ Peak compressive strength of confined concrete (MPa) & \(f_{cc}\) & Lognormal & 33.6 & 0.21 & [32; 33] \\ Strain at the peak strength of confined concrete & \(\varepsilon_{cc}\) & Lognormal & 0.004 & 0.2 & [32; 33] \\ Strain at the crushing strength of confined concrete & \(\varepsilon_{uc}\) & Lognormal & 0.015 & 0.3 & [32; 33] \\ Young’s modulus of steel rebar (MPa) & \(E_{s}\) & Lognormal & 2E5 & 0.02 & [32; 34] \\ Initial yield stress of steel rebar (MPa) & \(f_{y}\) & Lognormal & 378 & 0.07 & [32; 34] \\ Strain hardening ratio of steel rebar & \(b_{s}\) & Lognormal & 0.01 & 0.20 & [34] \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Statistical information of the uncertain parameters. In Figures 3 and 4, the 10 MC and the 100 MC represent the 10 sample MC simulation and the 100 sample MC simulation for one probability density propagation. In this paper, the 100 MC is taken as the accurate result for comparing. The MII index values among 4 PEM, 10 MC and 100 MC are different, but the index values of 4 PEM are more accurate compared to those of 10 MC. For guidance purposes, the index rankings from the three methods are shown in Table 2. The rankings from the 4 PEM and the 100 MC are match, while those from the 10 MC are not. Evidently, the 4 PEM result is acceptable. It is worth noting that the simulation samples of 4 PEM, 10 MC and 100 MC are \(11\times 11\times 4=484\),\(10\times 10\times 11+10=1110\) and \(100\times 100\times 11+100=110100\), respectively.