# Dynamic Damage Mechanism and Seismic Fragility Analysis of an Aqueduct Structure
## 3 Seismic Fragility Analysis of the Aqueduct Structure
### Criteria for Amplitude Modulation of Seismic Waves  
In this paper, the hunt and fill amplitude modulation criterion proposed by [PERSON] and [PERSON] is used to balance the calculation accuracy and efficiency [48]. Using the natural wave Northridge-01 (Whititer-S. Alta Dr) as an example, we used the damping ratio of 5% to obtain the \(S_{a}(T_{1},5\%)=0.005\) g at the first amplitude modulation, and a step increment of 0.05 g. The structure's performance evolution law under ground motion excitation was determined progressively. Table 6 depicts the exact amplitude modulation process.  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Order Number** & **Calculation** & \(\mathbf{S_{a}(T_{1},5\%)}\) **(g)** & \(\mathbf{\lambda}\) & \(\mathbf{U_{max}}\) **(m)** \\ \hline
1 & - & 0.005 & 0.041 & 0.001 \\
2 & 0.005 + 0.05 & 0.055 & 0.446 & 0.009 \\
3 & 0.055 + 0.05 + 1 \(\times\) 0.05 & 0.155 & 1.256 & 0.026 \\
4 & 0.155 + 0.05 + 2 \(\times\) 0.05 & 0.305 & 2.471 & 0.050 \\
5 & 0.305 + 0.05 + 3 \(\times\) 0.05 & 0.505 & 4.092 & 0.161 \\
6 & 0.505 + 0.05 + 4 \(\times\) 0.05 & 0.755 & 6.117 & 0.193 \\
7 & 0.755 + 0.05 + 5 \(\times\) 0.05 & 1.055 & 8.548 & 0.314 \\
8 & 0.755 + (1.055 \(-\) 0.755)/3 & 0.855 & 6.928 & 0.255 \\
9 & 0.505 + (0.755 \(-\) 0.505)/3 & 0.588 & 4.764 & 0.179 \\
10 & 0.155 + (0.305 \(-\) 0.155)/3 & 0.205 & 1.661 & 0.034 \\
11 & 0.055 + (0.155 \(-\) 0.055)/3 & 0.088 & 0.713 & 0.014 \\
12 & (0.755 + 0.505)/2 & 0.63 & 5.105 & 0.182 \\
13 & (0.505 + 0.305)/2 & 0.405 & 3.281 & 0.112 \\
14 & (0.305 + 0.155)/2 & 0.23 & 1.864 & 0.038 \\
15 & (0.155 + 0.055)/2 & 0.105 & 0.851 & 0.017 \\ \hline \hline \end{tabular} Note: \(dr=\mathrm{U_{max}/H,\,}dr\) is the aqueduct pier top offset ratio, \(\mathrm{U_{max}}\) is the maximum displacement of aqueduct pier shaft top, and \(\mathrm{H}\) is the height of aqueduct pier shaft, \(\mathrm{H}=5.5\) m.  
\end{table}
Table 6: Amplitude modulation calculation table.  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Limit States** & **Damage Description** & **Discriminant Rule** & **Dr (m)** & \(\mathbf{U_{max}}\) **(m)** \\ \hline \multirow{4}{*}{(NO)} & No or few structural & \multirow{4}{*}{\(\theta_{\max}\leq\) LS1} & \multirow{4}{*}{\textless{0.007}} & \multirow{4}{*}{\textless{0.0385}} \\  & and non-structural & & & \\ \cline{1-1}  & members are damaged. & & & \\ \cline{1-1}  & Minor repair is needed & & & \\ \cline{1-1}  & for structural and & & & \\ \cline{1-1}  & non-structural & & & \\ \cline{1-1}  & members. & & & \\ \cline{1-1}  & The structure remains & & & \\ \cline{1-1}  & stable and has enough & & & \\ \cline{1-1}  & capacity. & & & \\ \cline{1-1}  & The structure does not & & & \\ \cline{1-1}  & collapse and the & & & \\ \cline{1-1}  & damage is acceptable. & & & \\ \hline \hline \end{tabular} Note: \(dr=\mathrm{U_{max}/H,\,}dr\) is the aqueduct pier top offset ratio, \(\mathrm{U_{max}}\) is the maximum displacement of aqueduct pier shaft top, and \(\mathrm{H}\) is the height of aqueduct pier shaft, \(\mathrm{H}=5.5\) m.  
\end{table}
Table 5: Values of quantitative indexes correspond to structural performance levels.