## 3 Verification  
A CWR track is considered on a single continuous beam and two approach embankments. The parameters are listed in Table 1 from the Chinese code [26], except for the parameters of \(L_{\text{c}}\) and \(L_{\text{E}}\). The spring stiffness per unit length and per rail are the same with \(4.4\times 10^{6}\) N/m\({}^{2}\)/rail on the embankment and on the bridge. The fixed bearing is assumed to be located at 30 m away from the left end of the continuous beam; thus, the distance is 90 m between the fixed bearing and the right end of the continuous beam. It should be mentioned that the four parameters of \(\widetilde{u}_{\text{B}}\), \(\widetilde{q}_{\text{B}}\), \(\widetilde{u}_{\text{E}}\), and \(\widetilde{q}_{\text{E}}\) in Table 1 are not adopted in this section, but used in Section 5. In order to use the self-compiled program to calculate the additional force and displacement on the rail, the number of simple beams is set as zero. Figure 5 plots the distributions of the additional displacement and force of rail along its length, induced by the temperature change of the beam by present finite element method (FEM) solutions and analytical solutions [1], in which, the abscissa indicates the distance between a certain section and the left end of the rail. Figure 5 shows that the present FEM solutions are in good agreement with the analytical solutions. Therefore, the proposed model and the compiled program can be effective. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Notation** & **Item** & **Value** \\ \hline \(E\) & Modulus of elasticity of rail & \(2.1\times 10^{11}\) N/m\({}^{2}\) \\ \hline \(A\) & Area of the cross-section of rail & \(7.745\times 10^{-3}\) m\({}^{2}\) \\ \hline \(\alpha_{\text{R}}\) & Coefficient of thermal extension of rail & \(1.18\times 10^{-5}\)/\(c\)C \\ \hline \(L_{\text{c}}\) & Length of continuous beam & 120 m \\ \hline \(\Delta T\) & Temperature change of beam & \(15\)\({}^{\circ}\)C \\ \hline \(\alpha_{\text{B}}\) & Coefficient of thermal extension of beam & \(1.0\times 10^{-5}\)/\(c\) \\ \hline \(c\) & \begin{tabular}{c} spring stiffness per unit length and per rail of rail \\ foundation on embankment and on beam, respectively \\ \end{tabular} & \(4.4\times 10^{6}\) and \(1.6\times 10^{7}\) N/m\({}^{2}\)/rail \\ \hline \(\widetilde{u}_{\text{B}}\) & Critical value of relative displacement on beam & 0.5 mm \\ \hline \(\widetilde{q}_{\text{B}}\) & \
\begin{tabular}{c} Constant longitudinal restoring force per rail corresponds \\ to the critical value \(\widetilde{u}_{\text{B}}\) on beam \\ \end{tabular} & 8000 N/m/rail \\ \hline \(\widetilde{u}_{\text{E}}\) & Critical value of relative displacement on embankment & 2.0 mm \\ \hline \(\widetilde{q}_{\text{E}}\) &
\begin{tabular}{c} Constant longitudinal restoring force per rail corresponds \\ to the critical value \(\widetilde{u}_{\text{E}}\) on embankment \\ \end{tabular} & 8800 N/m/rail \\ \hline \(L_{\text{E}}\) & Length of CWR track on each side embankment & 100 m \\ \hline \hline \end{tabular}
\end{table} 
Table 1: The CWR track parameters on continuous beam and embankment. Figure 5: Distribution of the additional force and displacement on the rail. Procedures for Determining the Appropriate Locations of RER and the Fixed Bearing of the Continuous Beam  
The maximum additional tension and compression forces of rail vary with the locations of RER and the fixed bearing of the continuous beam. Their appropriate locations can reduce the maximum additional force. The procedures are as follows for determining their locations, corresponding to the maximum additional tension and compression forces of rail reaching their minimum values. 1. RER is located above the left end of the continuous beam. 2. The fixed bearing is also located at the left end of the continuous beam. Then, the additional force of rail along its length is calculated, and the maximum values of the tension and compression forces are obtained.