## 5 Conclusions  
The present paper aimed to predict the additional stresses in ballasted track elements caused by subgrade deformation. By predicting the effect of the settlement, planning of the maintenance process will be more accurate according to the settlement condition. This paper studied ballasted railway tracks using the FEM with the ANSYS Workbench 14.0 program. A comparison of the stresses and deformation values in the track elements conducted in three cases was carried out for stable subgrade soil without settlement. The three cases introduce the different status of ballasted track depending on the contact types between track elements. The first case is a theoretical case that has bonded contacts between all track elements. The second case introduces the status of the new tracks, for which the contacts between the superstructure elements are bonded with a 0.7 friction coefficient between sleepers and ballast stones. The third case introduces the status of the operated tracks. The third case is similar to the second case, with a 0.3 friction coefficient between the rail bottom and rail pads. Then, a case study was selected to analyze the effect of settlement, ranging from 5 mm to 10 mm, on the additional stresses in all track elements. The main findings can be summarized as follows:  
* The model has about 97.3% of agreement with the Zimmermann method. * The second case has lower stresses in most of the track elements as compared with other cases, and the contact types in the second case could also be applicable to the first case. Consequently, the second case was selected as a case study according to these characteristics for studying the effect of subgrade settlement on track elements. Moreover, it could help to decrease the maintenance costs and increase the service lifetime of the ballasted track. * The average percentages of the increased stresses due to a 1 mm increase of the settlement in the railhead, tie plate, and sleeper are 4.18%, 5.85%, and 7.21%, respectively. Moreover, the average percentages of the increased stresses at the beginning and end of the settled area due to a 1 mm increase of the settlement in ballast, subballast, and subgrade layers are 35.5%, 32.6%, and 419%, respectively. * A second-degree polynomial equation was derived to predict the possible percentage of increased stresses in each element because of subgrade deformation. This study aims to extract a simple equation that can be easily used for predicting the additional stresses (Equation (3)). \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Element** & \(\mathbf{a}\) & \(\mathbf{b}\) & \(\mathbf{e}\) \\ \hline Railhead & \(-0.03\) & \(4.82\) & \(-7.37\) \\ Rail pad & \(-0.64\) & \(16.82\) & \(-51.71\) \\ Tie plate & \(-0.73\) & \(16.99\) & \(-57.16\) \\ Sleeper & \(-0.43\) & \(13.99\) & \(-49.52\) \\ Ballast layer & \(-0.51\) & \(42.15\) & \(-78.55\) \\ Subballast layer & \(0.12\) & \(30.68\) & \(-34.55\) \\ Subgrade layer & \(-2.05\) & \(450.2\) & \(-62.56\) \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Constant parameters for all railway track elements. **Author Contributions:** Conceptualization, A.N.R. and H.N.E.-D.Z. ; methodology, A.N.R. and P.J. ; software, A.N.R. and P.J. ; validation, A.N.R., J.Z. and H.N.E.-D.Z. ; formal analysis, J.Z. and H.N.E.-D.Z. ; investigation, P.J. and A.N.R. ; writing--original draft preparation, A.N.R. and H.N.E.-D.Z. ; writing--review and editing, P.J., J.Z. and A.N.R. ; supervision, J.Z. and H.N.E.-D.Z. All authors have read and agreed to the published version of the manuscript. **Funding:** This research was funded by the National Key Research and Development Program of China (grant No. 2018 YFB1600302). **Institutional Review Board Statement:** Not applicable. **Informed Consent Statement:** Not applicable. **Data Availability Statement:** The data presented in this study are available on request from the corresponding author. **Conflicts of Interest:** The authors declare no conflict of interest. **Abbreviations**