# Dynamic Damage Mechanism and Seismic Fragility Analysis of an Aqueduct Structure
## 3 Seismic Fragility Analysis of the Aqueduct Structure
### IDA Method-Based Probability Demand Analysis  
Figure 8 depicts the results of a nonlinear analysis of the seismic dynamic response of an aqueduct structure under earthquake activity. When the IM remains constant, the data points in the graph reflect the maximum displacement of the pier shaft top of an aqueduct construction against different seismic waves. The quantitative indications of the four limit structural damage states are represented by the red dotted line. LS1 represents the limit state of the aqueduct structure changing from no damage to normal occupancy, LS2 represents the limit state of the aqueduct structure changing from normal occupancy to immediate occupancy, LS3 represents the limit state of the aqueduct structure changing from immediate occupancy to life safety, and LS4 represents the limit state of the aqueduct structure from life safety to collapse prevention. Since a single IDA data cannot accurately predict the limit behavior of the structure, this paper selects a series of seismic records to perform the incremental dynamic analysis and obtain multiple IDA data. The 17 seismic ground records IDA data generated during ground motion excitation were summarized in the same IM-DM [(Umax-\(S_{a}\)(\(T_{1}\), 5%))] coordinate system, as shown in Figure 8. It has been conducted using the same method as obtaining the IDA curve during ground motion excitation for the natural wave Northridge-01 (Whittier-S. Alta Dr). The response of the same structure to different seismic records input is discrete to some extent since the shape of the IDA curve is directly related to the selected seismic records. Therefore, it is necessary to summarize these different curves with the median and the 16% and 84% percentile curves to reduce the difference. The three IDA percentile curves obtained indicate that among the 17 seismic records, the 16%, 50%, and 84% seismic ground motion records exceeded the corresponding structural limit, as shown in Figure 9. The corresponding capacity values of each limit state of the structure are shown in Table 7.  
Taking the spectral acceleration \(S_{a}\)(\(T_{1}\), 5%) as the independent variable IM and the maximum displacement of aqueduct pier shaft top as the dependent variable, the functional relation between them is as follows:  
\[Ln(U_{\max})=a+bLn(S_{a}(T_{1},5\%)) \tag{22}\]  
In the formula, \(a\) and \(b\) are regression coefficients. We performed a logarithmic regression analysis with the data in Figure 10, with a correlation coefficient R = 0.998. According to the seismic design standard HAZUS99 for building fragility curve parameters [49], when  
Figure 8: Non-linear time history calculation points of 17 seismic waves. the fragility curve takes the spectral acceleration \(S_{a}(T_{1},\) 5%) as the independent variable, \(\sqrt{\beta_{c}^{2}+\beta_{d}^{2}}=0.4\). The functional relation is as follows:  
\[Ln(L_{\max})=-0.86+1.00048 Ln(S_{a}(T_{1},5\%)) \tag{23}\]  
Figure 11 shows the seismic fragility curves of the aqueduct structure calculated based on the IDA method. For the limit state LS1, the IDA method obtains that the exceedance \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{**Limit States**} & \multicolumn{3}{c}{**IM(\(S_{a}(T_{1},\) 5\%)) (g)**} & \multicolumn{3}{c}{**DM (U\({}_{\max}\)) (m)**} \\  & **16\%** & **50\%** & **84\%** & **16\%** & **50\%** & **84\%** \\ \hline (NO) & 0.189 & 0.089 & 0.026 & 0.0385 & 0.0385 & 0.0385 \\ (IO) & 0.407 & 0.189 & 0.065 & 0.0825 & 0.0825 & 0.0825 \\ (LF) & 0.657 & 0.404 & 0.130 & 0.1375 & 0.1375 & 0.1375 \\ (CP) & 1.981 & 1.442 & 0.533 & 0.2750 & 0.2750 & 0.2750 \\ \hline \hline \end{tabular}
\end{table} 
Table 7: Corresponding capacity values of each limit state of structure. Figure 10: Logistic regression analysis (IDA). Figure 9: Percentile curves (IDA). probability of the structure is 64.18% at 0.11 g and 96.77% at 0.23 g. In the limit state LS2, the exceedance probability of the structure at 0.23 g is 62.50%, and the exceedance probability at 0.51 g is 97.10%.