## 2 Model  
Figure 2 shows a schematic planar model of a CWR track with RER resting on multiple span simple beams, one continuous beam, multiple span simple beams, and two approach embankments, in which \(L_{\mathrm{s}}\) represents the length of the simple beam, \(L_{\mathrm{c}}\) represents the length of the continuous beam, \(L_{0}\) represents the distance between the two fixed bearings of the simple beam neighbor to the continuous beam, \(l_{\mathrm{B}}\) represents the distance between the fixed bearing of the continuous beam and that of the simple beam neighbor to the left side of the continuous beam, \(\mathrm{o_{c}}\) represents the midpoint of the continuous beam, \(\mathrm{o_{0}}\) represents the midpoint of \(L_{0}\), and SB and CB represent the simple and continuous beams, respectively. The continuous beam has only one fixed bearing, and the fixed and movable bearings of simple beams on two sides of the continuous beam are arranged alternately from left to right. It is assumed that the rightward longitudinal displacement, additional tension force in rail, and the temperature rise of the beams are taken as positive. In order to find the appropriate locations of RER and the fixed bearing, RER and fixed support can be anywhere in the range of the continuous beam in the numerical simulation. With CWRs, simple and continuous beams are modeled as beam elements. The rail at RER is simulated as disconnected without constraining its longitudinal displacement, and its additional force is zero. The track components between rails and bridge, including fasteners, sleepers, and bed, are modeled as coupling shear elements with nonlinear stiffness, as shown in Figure 3. The rail-beam thermal interaction descriptions can be found in [6; 7; 19]. To facilitate the reading, the main contents are listed here. The nonlinear characteristic of the difference \(u_{\mathrm{D}}\) of the coupling element depends on the critical value \(\overline{u}\)[6; 7]. \[u_{\mathrm{D}}= \begin{cases}\begin{aligned} u_{\mathrm{R}}-u_{ \mathrm{B}}&\mathrm{for}|u_{\mathrm{R}}-u_{\mathrm{B}}|<\overline{u}\\ \overline{u}\cdot\mathrm{sign}(u_{\mathrm{R}}-u_{\mathrm{B}})& \mathrm{for}|u_{\mathrm{R}}-u_{\mathrm{B}}|\geq\overline{u}\end{aligned} \end{cases}  \tag{1}\]  
where, \(u_{\mathrm{R}}\) and \(u_{\mathrm{B}}\) represent the longitudinal displacements of the rail and the upper surface of the beam, respectively, \(u_{\mathrm{D}}\) represents the displacement difference between \(u_{\mathrm{R}}\) and \(u_{\mathrm{B}}\), and \(\overline{u}\) represents the critical value of displacement difference in elastic and plastic phases. It is assumed that the simple and continuous beams are not influenced by the rails during the temperature change of the beams and that they can move freely. The value of \(u_{\mathrm{B}}\) of the simple and continuous beams can be written as (Reference [1]). \[u_{\mathrm{B}}=\alpha_{\mathrm{B}}\cdot\Delta T\cdot x_{\mathrm{B}} \tag{2}\]  
where \(\alpha_{\mathrm{B}}\) represents the coefficient of thermal extension of the beam, \(\Delta T\) represents the temperature change with reference to the initial temperature of the beam, and \(x_{\mathrm{B}}\) represents the distance between the calculated section and the fixed bearing of the beam. If the absolute value of displacement difference \(u_{\mathrm{D}}\) is less than this critical value \(\overline{u}\), there is a linear relationship between the difference \(u_{\mathrm{D}}\) and the longitudinal restoring force \(q\), as shown in Figure 4, they can be expressed as (Refernces [6; 7])  
Figure 2: Schematic planar model of a continuously welded rail (CWR) track with the rail expansion regulator (RER) resting on bridges and embankments.