# Collapse Analysis of a Two-Span Reinforced Concrete Bridge Model
## 4 Analysis of Collapse Mode
### Influence of Structural Systems
#### 4.5.2 Shock Mitigation System
The peak value of longitudinal relative displacement is listed in Figure 23.  
For all the conditions except scheme 1 when \(C\) = 1 kN\(\cdot\)(s/m)\({}^{\alpha}\), schemes 30 and 37 when \(C\) = 2 kN\(\cdot\)(s/m)\({}^{\alpha}\), and scheme 38 when \(C\) = 4 kN\(\cdot\)(s/m), the setting of the damper increased the relative displacement between the side pier and the beam, but the increase was less than 0.5 mm. This shows that an excessively small damping coefficient is not conducive to the displacement control of the damper. With the increase of the damping coefficient, the relative displacement was gradually weakened by the damper. If the isolation effect of the HDR bearings in Section 4.5.1 is taken as the optimization goal, the number of damper design schemes can be reduced from 42 to 12, as listed in Table 10.  
Figure 21: The change curve of damping force peak. \begin{table}
\begin{tabular}{c} **Scheme** \\ **Number** \\ \end{tabular} & \(\alpha\) & **C/kN\(\cdot\)(s/m)\({}^{\alpha}\)** \\ \hline  
\hline \hline \end{tabular}
\end{table} 
Table 9: Damper parameter optimization scheme. The final design scheme needed to be further screened after analyzing the comprehensive stress of the bridge. Figure 24 shows that when the velocity index was constant, the bending moment of the middle pier decreased with the increase of the damping coefficient. Under the same damping coefficient, the smaller the velocity index, the better the internal force control. Then, taking the isolation effect of HDR bearings as a reference, the scheme was screened, and schemes 3, 4, 6, 7, 8, 9, 10, and 11 of Table 9 were retained. Among the eight schemes, the damping coefficient was in the range of 8-12 kN\(\cdot\)(s/m)\({}^{\alpha}\) and the velocity index was in the range of 0.2-0.4. From the design perspective, one of these could be arbitrarily selected as the final scheme. Considering the economy and nonlinear degree, scheme 8 in Table 9 was finally selected as the viscous damper design scheme in this paper: \(C=10\) kN\(\cdot\)(s/m)\({}^{0.4}\), \(\alpha=0.4\). Then, the damping material was used to simulate the dampers in LS-DYNA. \begin{table}
\begin{tabular}{c c c c c c c c c c c c} \hline \hline
**Scheme Number** & **1** & **2** & **3** & **4** & **5** & **6** & **7** & **8** & **9** & **10** & **11** & **12** \\ \hline C/kN\(\cdot\)(s/m)\({}^{\alpha}\) & 6 & 6 & 8 & 8 & 8 & 10 & 10 & 10 & 12 & 12 & 12 & 12 \\ \(\alpha\) & 0.2 & 0.3 & 0.2 & 0.3 & 0.4 & 0.2 & 0.3 & 0.4 & 0.2 & 0.3 & 0.4 & 0.6 \\ \hline \hline \end{tabular}
\end{table} 
Table 10: Damper parameter alternative scheme. Figure 23: Peak relative displacement between the side pier and the beam. Figure 22: The change curve of damper stroke peak. The El Centro wave case was taken as an example to compare the collapse modes of the damping system and the isolation system. Under the action of longitudinal unidirectional El Centro seismic waves, the bridge model of the fluid viscous damper system eventually collapsed under the condition of 1.4 g PGA, as shown in Figure 25. Compared with the original system and isolation system, the collapse mode of the shock mitigation system was obviously different. Although the seismic intensity of the resistance was less than the isolation system, thanks to the setting of the damper, the main beam did not fall even if the damage of the middle pier and the main beam was serious, which provided a guarantee for the safety of the structure and for personnel to escape.