# An Effective Tangent Stiffness of Train-Track-Bridge Systems Based on Artificial Neural Network
## 4 Conclusions  
This paper presents a tangent stiffness calculation method based on a backpropagation neural network (i.e., BPTS) for a wheel-rail interaction (WRI) element and implements it into a general finite element software, OpenSees. The BPTS obtained using a wheelest running on a bridge at a constant speed is used for the dynamic response analysis of a realistic train-track-bridge (TTB) system with different vehicle speeds. The accuracy and efficiency of BPTS are studied by comparing the results with those using consistent tangent stiffness (CTS). In the example of a single wheelest, the lateral displacement and acceleration obtained by the two methods are compared and proven to be close. The convergence rate of BPTS is also verified by comparing the norm of the unbalance force and the total computation time. The results show that BPTS needs more convergence numbers but can significantly save the computation time, e.g., when BPTS is used, 35% of time is \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline
**Time [s]** & **Model** & **1** & **2** & **3** & **4** & **5** & **6** & **7** \\ \hline \multirow{3}{*}{0.500} & CTS & \(7.02\times 10^{4}\) & \(3.26\times 10^{3}\) & \(1.04\times 10^{1}\) & \(1.76\times 10^{-3}\) & & \\  & BPTS & \(1.05\times 10^{5}\) & \(4.76\times 10^{3}\) & \(5.30\times 10^{2}\) & \(5.55\times 10^{-1}\) & \(3.21\times 10^{-2}\) & & \\  & CTS & \(6.52\times 10^{4}\) & \(4.31\times 10^{4}\) & \(2.34\times 10^{1}\) & \(3.82\times 10^{-2}\) & \(2.97\times 10^{-3}\) & & \\  & BPTS & \(5.47\times 10^{4}\) & \(4.70\times 10^{4}\) & \(6.41\times 10^{2}\) & \(4.35\times 10^{1}\) & \(0.17\times 10^{-1}\) & \(1.01\times 10^{-2}\) & \\  & CTS & \(4.58\times 10^{5}\) & \(7.73\times 10^{4}\) & \(1.44\times 10^{3}\) & \(0.27\times 10^{1}\) & \(2.43\times 10^{-2}\) & & \\  & BPTS & \(3.31\times 10^{5}\) & \(1.21\times 10^{5}\) & \(3.05\times 10^{4}\) & \(8.61\times 10^{3}\) & \(0.45\times 10^{1}\) & \(2.41\times 10^{-1}\) & \(1.95\times 10^{-2}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 7: The norm of the unbalance force in a few representative time steps (speed is 30 m/s). Figure 16: The lateral displacement of the car body. \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline
**Time [s]** & **Model** & **1** & **2** & **3** & **4** & **5** & **6** & **7** \\ \hline \multirow{3}{*}{0.500} & CTS & \(4.19\times 10^{4}\) & \(2.66\times 10^{4}\) & \(1.84\times 10^{2}\) & \(2.81\times 10^{-1}\) & \(3.29\times 10^{-2}\) & & \\  & BPTS & \(2.17\times 10^{4}\) & \(1.37\times 10^{4}\) & \(2.97\times 10^{3}\) & \(1.14\times 10^{1}\) & \(1.16\times 10^{-1}\) & \(7.31\times 10^{-3}\) & \\ \cline{1-1}  & CTS & \(3.30\times 10^{4}\) & \(1.42\times 10^{4}\) & \(2.76\times 10^{3}\) & \(0.74\times 10^{1}\) & \(9.71\times 10^{-2}\) & & \\ \cline{1-1}  & BPTS & \(4.21\times 10^{5}\) & \(3.27\times 10^{4}\) & \(2.64\times 10^{4}\) & \(2.64\times 10^{3}\) & \(1.06\times 10^{1}\) & \(1.02\times 10^{-1}\) & \(9.39\times 10^{-3}\) \\ \cline{1-1}  & CTS & \(2.89\times 10^{5}\) & \(1.58\times 10^{5}\) & \(2.52\times 10^{4}\) & \(0.87\times 10^{3}\) & \(8.50\times 10^{-1}\) & \(2.14\times 10^{-3}\) & \\ \cline{1-1}  & BPTS & \(6.67\times 10^{5}\) & \(2.43\times 10^{4}\) & \(1.87\times 10^{4}\) & \(4.76\times 10^{3}\) & \(4.42\times 10^{2}\) & \(2.41\times 10^{1}\) & \(3.92\times 10^{-2}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 8: The norm of the unbalance force in a few representative time steps (speed is 50 m/s). saved in the example of a single wheelset. In the example of a light rail vehicle, when BPTS is used, 28% of time is saved. In summary, compared with the CTS, the BPTS can save the significant efforts of derivation and software implementation as well as improve computational efficiency and maintain good accuracy. This paper discusses a simple working condition with a TTB system, and the load on the train is relatively simple, that is, a small lateral disturbance.