## 3 Modal Analysis of the 3D Aqueduct Model  
A 3D FE model is established using SAP2000 for the entire aqueduct, as shown in Figure 2. The arch trusses, transverse bent frames above those trusses, moment frame supports and foundation piles are generated by frame elements, and the water-transferring U-shaped flumes and pile foundation caps are created using shell elements. A 3 cm-wide contraction joint is set between two adjacent U-shaped flume segments. The pot-type fixed rubber bearing of the arch trusses at the upstream end of each span is simulated as a rubber isolator restrained in all translational and rotational DOFs, while the pot-type sliding counterpart at the downstream end of each span is modeled with the same DOF restraints except that the shear stiffness in the longitudinal translation is set to be zero. The interaction between foundation piles and surrounding soil is simulated with node springs using \"m\" method. According to the current Chinese code \"Technical Code for Building Pile Foundations (JGJ 94-2008) [26]\", the lateral stiffness of the node spring of a pile can be calculated as follows:  
\[K=ab_{0}mZ \tag{1}\]  
where \(K\) is the lateral stiffness of the node spring (kN/m); \(a\) is the thickness of the foundation soil layer, normally taken as 1-2 m; \(b_{0}\) is the effective width of the pile (m), and when the pile diameter \(d\) is larger than 1 m, \(b_{0}\) = 0.9 \(\times\) (\(d\) + 1); Z is the depth of the foundation \begin{table}
\begin{tabular}{c c c} \hline \hline
**Nodes** & **Times** & **Frequency (Hz)** \\ \hline \multirow{3}{*}{Lateral modes} & first & 0.97 \\  & second & 1.02 \\  & third & 1.06 \\ \hline \multirow{3}{*}{Longitudinal modes} & first & 1.50 \\  & second & 1.915 \({}^{*}\) \\ \hline Vertical mode & first & 5.328 \({}^{*}\) \\ \hline \({}^{*}\)average of span 6, 8 and 10. & & \\ \end{tabular}
\end{table} 
Table 1: Dynamic characteristics of the aqueduct from field tests. soil layer (m); and \(m\) represents the proportional coefficient of horizontal resistance of foundation soil (kN/m\({}^{4}\)), and for cast-in-place concrete piles can be taken as shown in Table 2 for different foundation soils (in this case, it can be taken as 100-300 MN/m\({}^{4}\)). In the real structure, the prefabricated U-shaped concrete flume segments are laid directly on the top beams of transverse bent frames erected on the arch trusses, but the bearings only adopt two layers of thin (about 1 mm thickness) asphaltic felts instead of commonly used plate rubber supports. In the 3D model, this makeshift asphaltic bearing is simulated by an elastic link element restrained in vertical translation and rotation about the vertical axis while free in the other two rotational DOFs, but the shear stiffnesses in the transverse and longitudinal directions need to be defined, which can be assumed to be equal based on the empirical equation:  
\[K=GA/t \tag{2}\]  
where \(K\) is the bearing shear stiffness, \(G\) is the shear modulus of the bearing material, \(A\) is the plane area of the bearing, and \(t\) is the bearing thickness. The influence of the \"m\" value variation of foundation soil on the dynamic characteristics of the aqueduct was investigated. In this case, certain dynamic elastic moduli are assumed for arch trusses (40 GPa), bent frames (37.5 GPa), moment frame supports (37.5 GPa), pile caps and piles (32.5 GPa) in the 3D model based on their respective design strengths, and the shear stiffnesses of asphaltic bearings of U-shaped flumes in both transverse and longitudinal directions are set to be 1.536 \(\times\) 10\({}^{7}\) kN/m. The modal analysis results are listed in Table 3. It can be seen that the change of the \"m\" value in a code-specified range has an insignificant impact on the dynamic characteristics of the aqueduct.