## 4 Modeling and Results with Discussion
### States without Settlement of the Three Cases
In the state without settlement, the maximum stress in the subballast layer is around 1.2 MPa, and the maximum stress in the subgrade layer is less than 0.07 MPa. [PERSON] et al. [35] concluded that the maximum stresses reached 0.08 MPa and 0.04 MPa at the surfaces of the first and second parts of the subgrade layer, respectively. By comparing the results of the maximum stress in the  
Figure 17: Wheel/rail pressure in states with and without settlement. subgrade layer of the 2D model with the results obtained by [PERSON] et al. [35], we can conclude that the subgrade layer stresses are logical. [PERSON] et al. [35] introduced the 3D track model conditions as follows: 80 km/h of wheel speed, 25 tons of axle load, 70 kg/m of rail, 50 cm of ballast thickness, a subgrade first layer of 0.6 cm, and a subgrade second layer of 1.9 m. It could be concluded that the stress values of the substructure layers are considerable; however, the recorded maximum stresses are located around the beginning and the end of the settled part. Figure 18: _Cont._  
Figure 18: Stresses of the track elements in states with and without settlement: (**a**) railhead; (**b**) the 11 th sleeper; (**c**) ballast layer; (**d**) subballast layer; (**e**) subgrade layer. Table 5 introduces the average values of the maximum stresses in the states with and without settlement. Stresses of the track elements were analyzed to conclude the rates of the additional stresses in each part. For substructure layers, these rates express the increase of stresses in each layer as the maximum stresses that exist at the beginning and the end of the settled zone. Figure 19 explains the rate of additional stresses for superstructure elements with polynomial equations and the coefficient of determination values (R\({}^{2}\)). However, [PERSON] [24] studied the effect of increasing pier settlement, from 1 mm to 5 mm, on the tensile stresses in the concrete slab and the base of a ballastless track. He concluded that the rate of increasing stresses has a semi-regular increase, expressed in linear relationships. Figure 12:0 Rate of additional stresses in different elements of the track model: (**a**) railhead; (**b**) the 11 th rail pad; (**c**) the 11 th tie plate; (**d**) the 11 th sleeper. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Element** & **without** & **Settlement** & **5 mm** & **6 mm** & **7 mm** & **8 mm** & **9 mm** & **10 mm** \\ \hline Wheel/rail & 781.76 & 782.38 & 782.47 & 780.75 & 778.03 & 776.46 & 775.72 \\   pressure & 157.19 & 182.51 & 188.25 & 195.83 & 202.90 & 209.48 & 215.37 \\   ReLU & 32912 & 3.804 & 4.1735 & 4.4873 & 4.6842 & 4.7404 & 5.0802 \\   The 11 th tie plate & 27.216 & 30.184 & 31.112 & 34.816 & 36.603 & 36.138 & 38.158 \\   The 11 th sieeper & 0.3772 & 0.4202 & 0.4332 & 0.484 & 0.5210 & 0.5238 & 0.5564 \\   Ballast layer & 0.3553 & 0.6781 & 0.8173 & 0.9182 & 1.0124 & 1.1172 & 1.2353 \\   Subballast layer & 0.0963 & 0.2140 & 0.2446 & 0.2757 & 0.3085 & 0.3379 & 0.3712 \\   Subgrade layer & 0.0497 & 1.1113 & 1.3228 & 1.5338 & 1.7423 & 1.9481 & 2.1523 \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Stress values of railway track elements in states with and without settlement. Equation (3) is a second-degree equation that could be used to predict the percentages of additional stresses in track elements, depending on the settlement amount in the subgrade soil:  
\[y=az^{2}+bz+e \tag{3}\]  
where \(y\) is the increasing percentage of stresses in the track element (%), \(z\) is the settlement amount (mm), and \(a\), \(b\), and \(e\) are the constant parameters. Values of the constant parameters (\(a\), \(b\), and \(e\)) are introduced in Table 6 to predict the percentage of additional stresses of the second case for all railway track elements.