## 3 Modal Analysis of the 3D Aqueduct Model
By altering one of those parameters while keeping others unchanged, the sensitivity analyses of dynamic elastic moduli of arch trusses (\(E_{d\_arch}\)), bent frames, moment frame supports (\(E_{d\_frame}\)), pile caps and piles (\(E_{d\_pile}\)) on the dynamic characteristics of the aqueduct are conducted, and the results are shown in Tables 4-6. It can be seen that under almost the same variation, \(E_{d\_pile}\) has much less influence compared with \(E_{d\_arch}\) and \(E_{d\_frame}\), which may be attributed to the large rigidity of the foundation (i.e., pile cap thickness 1.5 m and pile diameter 1.5 m). It is also found that the vertical frequency is sensitive to \(E_{d\_arch}\), while the lateral and longitudinal frequencies are sensitive to \(E_{d\_frame}\). \begin{table}
\begin{tabular}{c c} \hline \hline
**Foundation Soils** & **“\(m\)” Value/kN/m\({}^{4}\)** \\ \hline Silt and silty clay & \(2.5\times 10^{3}\)–\(6\times 10^{3}\) \\ Loose sand and earth fill & \(6\times 10^{3}\)–\(14\times 10^{3}\) \\ Medium-dense earth fill and fine sand & \(14\times 10^{3}\)–\(35\times 10^{3}\) \\ Medium-dense medium-coarse sand & \(35\times 10^{3}\)–\(100\times 10^{3}\) \\ Dense coarse sand and gravels & \(100\times 10^{3}\)–\(300\times 10^{3}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: “\(m\)” value for different nonrock foundation soils. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{**m (MN/m\({}^{4}\))**} & \multicolumn{3}{c}{**Lateral Modes**} & \multicolumn{3}{c}{**Longitudinal Modes**} & **Vertical Mode** \\ \cline{2-7}  & **First** & **Second** & **Third** & **First** & **Second** & **First** \\ \hline
300 & 0.884 & 0.934 & 0.944 & 1.297 & 1.576 & 4.942 \\
100 & 0.880 & 0.929 & 0.939 & 1.282 & 1.554 & 4.940 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Primary modal frequencies (Hz) of the aqueduct under different “m”. The plumes have the same design strength (C40) with bent frames and moment frame supports, so in the modal analysis, the dynamic elastic modulus \(E_{d\_frame}\) is assigned to be the same with \(E_{d\_frame}\). Following the same procedure, the impact of \(E_{d\_frame}\) variation on the dynamic characteristics of the aqueduct was also investigated and found to be minor, as shown in Table 7.  
Therefore, after neglecting those two parameters, the main parameters left that may affect the dynamic characteristics of the 3D aqueduct model are the shear stiffnesses of the asphaltic bearings of U-shaped flumes, the dynamic elastic modulus of arch trusses (concrete design strength C50), and the dynamic elastic modulus of bent frames and moment frame supports (C40). By arranging those 3 parameters into various combinations, a series of modal analyses of the aqueduct are performed, and the typical primary mode shapes are shown in Figures 10-15. \begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline \multirow{3}{*}{\(E_{d\_arch}\) /**GPa**} & \multirow{3}{*}{\(E_{d\_frame}\) /**GPa**} & \multirow{3}{*}{\(E_{d\_pile}\) /**GPa**} & \multicolumn{6}{c}{**Primary Modal Frequencies/Hz**} \\ \cline{4-9}  & & & & ** lateral** & & **Longitudinal** & **Vertical** \\ \cline{4-9}  & & & **1 st** & **2 nd** & **3 rd** & **1 st** & **2 nd** & **1 st** \\ \hline
40 & 37.5 & 32.5 & 0.884 & 0.934 & 0.944 & 1.297 & 1.576 & 4.942 \\
40 & 37.5 & 40 & 0.896 & 0.947 & 0.957 & 1.318 & 1.596 & 4.962 \\ +0\% & +0\% & +23.1\% & +1.4\% & +1.4\% & +1.4\% & +1.6\% & +1.3\% & +0.4\% \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Dynamic characteristics of the aqueduct under different \(E_{d\_pile}\).