## 6 Conclusions  
This paper presents a nonlinear finite element model for a CWR track, with RER resting on the embankment and simple and continuous beams under the temperature change of the bridge, and then uses the method to determine the locations of RER and the fixed bearing of a continuous beam corresponding to the maximum additional forces of rail reaching minimum values and, furthermore, investigates the effect laws of any locations of RER and the fixed bearing on the maximum additional forces of rail. The obtained results can guide the design of the RER location and the fixed bearing location of a long-span continuous railway bridge to reduce the additional axial force in CWR induced by the bridge thermal effect. Several conclusions can be drawn.  
1. When RER is arranged above the two ends of the continuous beam, the minimum value of the maximum additional compression force of rail does not change with the length of the continuous beam, but that of the tension force of rail increases with the length of the continuous beam, although the increase is small. However, when RER is arranged above the middle of the continuous beam, the minimum values of the maximum additional tension and compression forces of rail increase with the length of the continuous beam; in particular, the latter increases significantly.
2. As for the temperature rise of the beam, there is an upper limit value of the additional tension force of rail but no lower limit value of additional compression force, and when the temperature falls, the result just reverses. The limit value is \(P_{\text{u}}=E\cdot A\cdot\alpha_{\text{B}}\cdot\Delta T\).
3. There is a critical length for the continuous beam. For the shorter length, RER should be arranged above the middle of the beam, while for the longer length, RER should be arranged above the two ends of the beam, which is conducive to reducing the additional forces of rail.
4. For RER arranged above the middle of the continuous beam, the fixed bearing is suggested to be located at the midpoint between the two fixed bearings of the simple beam neighbor to the continuous beam, which can reduce the maximum additional forces of rail. The further the fixed bearing is away from the midpoint, the greater is the maximum additional forces of rail.  
Figure 15: Longitudinal displacement of rail along its length, with the fixed bearing located at three different positions for the length of 180 m of continuous beam.  
When RER is arranged above one end of a continuous beam, the fixed bearing is arranged at the other end, as far as possible. This matching arrangement can ensure that the maximum additional compression force of rail reaches a minimum value.  
Conceptualization, P.L.; methodology, P.L.; investigation, P.L.; software, P.L.; supervision, P.L.; data curation, T.L.; validation, X.H. and G.H.; writing--original draft, P.L.; writing--review and editing, B.Y. All authors have contributed to the whole paper. All authors have read and agreed to the published version of the manuscript.  
This research received no external funding.  
This research was supported by the National Natural Science Foundation of China (Nos. 51578552, U1334203) and China Railway Corporation Science and Technology Research and Development Project (P2018G047).  
We declare that we have no conflicts of interest in this work. We declare that we do not have any commercial or associative interest that represents a conflict of interest in connection with the work submitted.