## 4 Bridge Model and Uncertainty  
A two-span continuous reinforced-concrete bridge was selected in order to evaluate its performance before and after strengthening. The total length of the bridge was 60 m, the height of the bridge column was 9 m, and its diameter was 1.6 m. The longitudinal steel bar area ratio of the bridge column was approximately 1.42%, and the circular hoop bar volume ratio was approximately 0.27%. The pile length and diameter were 20.0 m and 1.6 m, respectively. The details of the bridge model before and after the reinforcement are shown in Figure 1a,b. In this study, steel plates were pasted around the surface of the bridge piers, as shown in Figure 1b. In order to evaluate the performance of the bridge under different reinforcement cases, the thicknesses of the steel plates were set ranging from 1 mm to 5 mm, with an incremental value of 1 mm. OpenSees [66] was used in this study to create a finite-element model of the bridge. The beam of the superstructure was simulated using elastic elements as the beam was designed as the protective component, and the bridge columns were simulated using a displacement-based beam-column element in OpenSees [67]. The stress-strain model of the confined concrete was calculated using [PERSON]'s model [68]. The corrosion of the circular hoop bars affects the mechanical properties of the confined concrete, which were also considered in this study. For different corrosion levels, the corresponding yield strength and Young's modulus for hoop bars were calculated first and then used to calculate the properties of confined concrete using [PERSON]'s model. The strength of the confined concrete decreases with an increase in the corrosion ratio. Springs in different directions (horizontal: p-y, vertical: q-z, and t-z) were used to simulate the soil layer around the \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{**Num.**} & \multicolumn{2}{c}{**Mainshock**} & \multicolumn{2}{c}{**Aftershock**} & \multirow{2}{*}{**Earthquake**} \\ \cline{2-2} \cline{4-5}  & **Sequence** & **M** & **Sequence** & **M** & **Event** \\ \hline
54 & 985 & 6.69 & 1657 & 6.05 & Northridge \\ \hline
55 & 988 & 6.69 & 1658 & 6.05 & Northridge \\ \hline
56 & 990 & 6.69 & 1659 & 6.05 & Northridge \\ \hline
57 & 995 & 6.69 & 1660 & 6.05 & Northridge \\ \hline
58 & 996 & 6.69 & 1711 & 5.28 & Northridge \\ \hline
59 & 999 & 6.69 & 1661 & 6.05 & Northridge \\ \hline
60 & 1004 & 6.69 & 3771 & 5.28 & Northridge \\ \hline \hline \end{tabular}
\end{table} 
Table 1: _Cont._  
Figure 1: Bridge model. (**a**) Elevation view, (**b**) reinforced bridge piers. pile foundations, as presented in Figure 1b. Figure 1b shows three groups of springs. The stiffness of the soil springs was calculated using the equations from the American Petroleum Institute (1989) [69]. In this study, the uncertainty of the ground motions and materials was considered using the Latin hypercube sampling method. The sample size was set to 60, which was the same as the number of ground-motion sequences. The concrete compressive strength (which follows a normal distribution) had a mean of 31.0 MPa, and its standard deviation was 6.2 MPa; the mean strength of the steel bar was 336.2 MPa, and its standard deviation was 36 MPa, following a lognormal distribution [70]. The yield strength of the steel plate was 345 MPa, and its Young's modulus was 2.06 \(\times\) 10\({}^{5}\) MPa. Nonlinear dynamic analysis was conducted after the finite element model of the bridge was created, and the displacement at the top of the bridge pier was recorded for each case. Subsequently, the displacement was transformed into the ductility demand (\(\mu_{\Delta}\)) to determine the damaged state of the bridge. To judge the failure of the bridge, we defined four damage states according to HAZUS-MH/MR3 (FEMA 2007) [71]: slight damage \(1<\mu_{\Delta}\leq 2.5\), moderate damage \(2.5<\mu_{\Delta}\leq 4.0\), major damage \(4.0<\mu_{\Delta}\leq 7.0\), and collapse \(\mu_{\Delta}>7.0\).