## 5 Examples
### Appropriate Location of RER  
One continuous beam with the length of 180 m is considered as an example to analyze the effect of the location of RER on the minimum values of the maximum additional tension and compression forces of rail. Using Steps (1)-(7) in Section 4, the curves of the minimum values of the maximum additional tension and compression forces of rail, changing with the locations of RER, have been obtained and plotted in Figure 6, in which the abscissa in the figure represents the distance between RER and the rail section above the left end of the continuous beam. As shown in Figure 6, the location of RER has an effect on the minimum value of the maximum additional tension force, with the difference of \(31.37\%\) for the two extreme values of \(1.8417\times 10^{5}\) and \(2.4195\times 10^{5}\) N. However, the effect on the minimum value of the maximum additional compression force is more significant, with the difference of \(125.85\%\) for the two extreme values of \(-1.2195\times 10^{5}\) and \(-2.7542\times 10^{5}\) N. RER is arranged above the end and the middle of the beam, which has the opposite effect on the minimum value of the maximum additional compression and tension forces of rail. For the former, the minimum compression force of rail is \(-1.2195\times 10^{5}\) N, but the tension force of rail is very large, with the value of \(2.4195\times 10^{5}\) N. For the latter, the minimum tension force of rail is \(1.8417\times 10^{5}\) N, but that of the compression force of rail is very large, with the value of \(-2.5877\times 10^{5}\) N. As shown in Figure 6, both cases of \(\widetilde{u}_{\rm B}\) and \(\widetilde{q}_{\rm B}\) show that RER should be arranged above the two ends and the middle of beam to reduce the maximum additional compression and tension forces of rail, respectively. The aforementioned studies show that the appropriate locations of RER are above the two ends and the middle of the continuous beam. For the case of RER located at three positions, i.e., above the left end, the middle, and the right end of the continuous beam, the effects of the lengths of the continuous beam on the minimum values of the maximum additional compression and tension forces of rail are studied. Besides 180 m, lengths of 120-300 m, with intervals of 30 m, are taken into account. Their relationship curves are drawn in Figure 7; RER is arranged above the two ends of the continuous beam, the minimum values of the maximum additional compression forces of rail do not change with the length of the continuous beam, and the minimum values of the maximum additional tension forces of rail increase with the length of the continuous beam, but the increase values are small. For example, the values in the curve of \"TF Left\" are \(2.2595\times 10^{5}\) and \(2.4395\times 10^{5}\) N, corresponding to the lengths of 120 and 300 m, respectively, with the increase of \(7.67\%\), and the values in the curve of \"TF Right\" are \(2.1534\times 10^{5}\) and \(2.4394\times 10^{5}\) N, corresponding to the lengths of 120 and 300 m, respectively, with the increase of \(13.28\%\). 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 change obviously with the length of the continuous beam; in particular, the latter increases significantly. For instance, the values in the curve of \"TF Mid\" are \(1.2242\times 10^{5}\) and \(2.3626\times 10^{5}\) N, corresponding to the lengths of 120 and 300 m, respectively, with the increase of \(92.99\%\), and the values in the curve of \"CF Mid\" are \(-1.8952\times 10^{5}\) and \(-3.8716\times 10^{5}\) N, corresponding to the lengths of 120 and 300 m, respectively, with the increase of \(104.28\%\). Figure 6: The curves of the minimum values of the maximum additional forces of rail changing with the location of RER for the length of 180 m of a continuous beam.