## 3 Modal Analysis of the 3D Aqueduct Model
\begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline \multirow{3}{*}{\(E_{d\_arch}\) /**GPa**} & \multirow{3}{*}{\(E_{d\_frame}\) *} & \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 & 46.5 * & 32.5 & 0.885 & 0.936 & 0.946 & 1.313 & 1.615 & 4.948 \\ +0\% & +24.0\% & +0\% & +0.1\% & +0.2\% & +0.2\% & +1.2\% & +2.5\% & +0.1\% \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Dynamic characteristics of the aqueduct under different \(E_{d\_arch}\). \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\% & +Figure 11: Typical second lateral vibration mode of the aqueduct. Figure 12: Typical third lateral vibration mode of the aqueduct. Figure 10: Typical first lateral vibration mode of the aqueduct. Figure 14: Typical second longitudinal vibration mode of the aqueduct. Figure 13: Typical first longitudinal vibration mode of the aqueduct. Figure 15: Typical first (primary) vertical vibration mode of the aqueduct. The first lateral modes of the aqueduct for all analysis scenarios have modal participating mass ratios nearing 20%, which are higher than those of other lateral modes. As shown in Figure 10, the first lateral mode typically manifests itself as a lateral vibration of some interior spans (i.e., span 6 to span 11) in the same direction, whereas the motions of exterior spans including span 14 are not activated, which may justify the missing of the first lateral mode on the velocity spectrum of span 14 (see Figure 7). It can be seen from Figures 11 and 12 that the second and third lateral modes involve transverse vibration of more spans including span 6, 8, 10 and 14, which conforms to the appearances of these two modes on all velocity spectra obtained from field dynamic tests (see Figures 4-7). As shown in Figures 13 and 14, the first (normally with a modal participating mass ratio around 20-50%) and second longitudinal modes of the aqueduct for all analysis scenarios both show longitudinal vibration of most spans concurrently, which also accords with the presence of two longitudinal peak frequencies in Figures 4-7. The first vertical modes of the aqueduct for all analysis scenarios have modal participating mass ratios around 20%, typically displaying a simultaneous vertical vibration shape of arch trusses in most interior spans, as shown in Figure 15. In fact, there still exist one or two similar vertical modes which have slightly lower or higher frequencies but much less modal participation mass ratios than that primary mode, and this may explain the small differences of the first vertical mode frequencies obtained from the field dynamic tests of the four spans, as shown in Figures 4-7.