## 5 Conclusions  
For the aquduct under investigation in this study, \(K_{flume}\), \(E_{d\text{-}arch}\) and \(E_{d\text{-}frame}\) are among the main parameters to define its dynamic behavior, but there is no easy approach to get them tested directly on site. Since these parameters have direct correlation with the vibrational characteristics of the aquduct, it is practicable to apply a BP neural network to evaluate them using the primary modal frequencies of the structure as inputs. However, it is impossible to obtain a large amount of experimental data to train the network from the real structure. To solve this data deficiency problem, a full-scale 3D FE model which can simulate the real structure to its best is established, and various modal analyses under different combinations of \(K_{flume}\), \(E_{d\text{-}arch}\) and \(E_{d\text{-}frame}\) are conducted to create the analytical dataset for the network. Through cross-validation procedure, the network architecture with two hidden layers is determined, and it shows that with appropriated transfer functions for the hidden and output layers, the trained BP neural network can provide very good approximations for \(K_{flume}\), \(E_{d\text{-}arch}\) and \(E_{d\text{-}frame}\) using the primary modal frequencies obtained from the FE model. The actual primary modal frequencies acquired from in situ dynamic tests are then put into the BP neural network to estimate \(K_{flume}\), \(E_{d\text{-}arch}\) and \(E_{d\text{-}frame}\) of the aquduct. By substituting the \(K_{flume}\), \(E_{d\text{-}arch}\) and \(E_{d\text{-}frame}\) obtained from two BP neural networks with different sizes (6 and 3) of input frequency vectors into the 3D FE model, it is found that more inputs of modal frequencies can improve the approximation accuracy of the BP neural network. Finally the \(E_{d\text{-}arch}\) and \(E_{d\text{-}frame}\) of the aquduct are approximated to be 47.8 GPa and 46.1 GPa, respectively, and \(K_{flume}\) is estimated to be the maximum value of the corresponding training data, implying that the makeshift asphalt bearings of U-shaped plumes basically can be treated as a three-directional hinge (i.e., fixed in vertical, lateral and longitudinal translations) in the FE model. Conceptualization, X.L. ([PERSON]) and X.L. ([PERSON]); methodology, J.X. ; software, M.L. ; validation, X.L. ([PERSON]) and M.L. ; formal analysis, X.L. ([PERSON]); investigation, J.X. ; resources, X.L. ([PERSON]); data curation, X.L. ([PERSON]); writing--original draft preparation, X.L. ([PERSON]); writing--review and editing, X.L. ([PERSON]); visualization, J.X. ; supervision, M.L. ; project administration, [PERSON] ([PERSON]); funding acquisition, [PERSON] ([PERSON]). All authors have read and agreed to the published version of the manuscript. This research was funded by the major science and technology project plan of the Ministry of Water Resources (SKS-2022111), IWHR Research&Development Support Program (SM0145B022021) and the science and technology project of the Xinjiang Erjsi River Investment Development (Group) Co., Ltd. (QBT087/FY36). Not applicable. The data that support the findings of this study are available on request from the corresponding author. \"Research on static and dynamic safety monitoring and health evaluation for large aquducts\" sponsored by the Water Diversion Operation and Maintenance Center of Shandong Province. The authors declare no conflict of interest. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Modal Frequencies** & \(f_{\text{TI}}\)**/Hz** & \(f_{\text{T2}}\)**/Hz** & \(f_{\text{T3}}\)**/Hz** & \(f_{\text{L1}}\)**/Hz** & \(f_{\text{L2}}\)**/Hz** & \(f_{\text{V1}}\)**/Hz** \\ \hline Field test results & 0.970 & 1.020 & 1.060 & 1.500 & 1.915 & 5.328 \\ FE modal analysis & 0.964 & 1.019 & 1.030 & 1.424 & 1.745 & 5.396 \\ Relative error & \(-\)0.62\% & \(-\)0.10\% & \(-\)2.83\% & \(-\)5.07\% & \(-\)8.88\% & 1.28\% \\ \hline \hline \end{tabular}
\end{table}