## 3 Verification of the Implemented Model
### Mohr's Circle Verification  
Stresses in the subgrade layer of the modeled track were verified with the representation of Mohr's circle. Four elements were selected in different places in the subgrade layer, which is sandy soil, as probes to get the values of the normal and shear stresses for these elements. The maximum normal stresses, the minimum normal stresses, and the maximum shear stresses are shown in Figure 9a-c, respectively. The normal and shear stresses of these four elements are introduced in Table 4. These elements were selected to cover different places that have maximum and minimum values of normal stresses and shear stresses. In the middle of the modeled track, as in Figure 9, the selected element is located in the contour plot of the minimum shear stresses (\(\tau_{max}\)) and the contour plot of the maximum values of the maximum normal stresses (\(\sigma_{1}\)). On the right side of the model, the selected elements are located in the contour plot of the maximum shear stresses (\(\tau_{max}\)) and the contour plots of different lower values of the minimum normal stresses (\(\sigma_{2}\)). That would help to ensure the numerical analysis, using ANSYS Workbench, can consider the granular nature of the subgrade soil.  
Figure 9: _Cont._  
By using Equation (2), derived from [PERSON]'s circle theory, results could be obtained as follows, and Figure 10 illustrates results of the last three elements:  
\[R_{C}=\tau_{max}=\frac{\sigma_{1}-\sigma_{2}}{2} \tag{2}\]  
where \(\sigma_{1}\) is the maximum normal stress, \(\sigma_{2}\) is the minimum normal stress, \(\tau_{max}\) is the maximum shear stress applied to the element, and \(Rc\) is the radius of Mohr's circle.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline \multirow{2}{*}{**Element No.**} & **Minimum Normal** & **Maximum Normal** & **Maximum Shear** \\  & **Stress (Mpa)** & **Stress (Mpa)** & **Stress (Mpa)** \\ \hline The 1 st element & \(-5.539\times 10^{-2}\) & 0 & 2.769 \(\times 10^{-2}\) \\ The 2 nd element & \(-5.377\times 10^{-2}\) & 4.026 \(\times\)\(10^{-3}\) & 2.281 \(\times\)\(10^{-3}\) \\ The 3 rd element & 0 & 6.264 \(\times\)\(10^{-3}\) & 3.132 \(\times\)\(10^{-3}\) \\ The 4 th element & 0 & 3.153 \(\times\)\(10^{-3}\) & 1.576 \(\times\)\(10^{-3}\) \\ \hline \hline \end{tabular}
\end{table}
Table 4: Maximum normal and shear stresses of the selected four elements in the subgrade layer.  
Figure 10: Mohr’s circles for the subgrade layer of the track model.  
Figure 9: Stresses in the subgrade layer: (**a**) minimum normal stresses; (**b**) maximum normal stresses; (**c**) maximum shear stress.  
For the first element, \(\sigma_{1}\) and \(\sigma_{2}\). equal 0 MPa and \(-\)5.539 \(\times\) 10\({}^{-2}\) MPa, respectively, so \(\tau_{max}\) is calculated to be 2.76 \(\times\) 10\({}^{-2}\) MPa. For the second element, \(\sigma_{1}\) and \(\sigma_{2}\) equal 4.026 \(\times\) 10\({}^{-3}\) MPa and \(-\)5.377 \(\times\) 10\({}^{-4}\) MPa, respectively, so \(\tau_{max}\) is calculated to be 2.2818 \(\times\) 10\({}^{-3}\) MPa. For the third element, \(\sigma_{1}\) and \(\sigma_{2}\) equal 6.264 \(\times\) 10\({}^{-3}\) MPa and 0 MPa, respectively, so \(\tau_{max}\) is calculated to be 3.132 \(\times\) 10\({}^{-3}\) MPa. For the fourth element, \(\sigma_{1}\) and \(\sigma_{2}\) equal 3.153 \(\times\) 10\({}^{-3}\) MPa and 0 MPa, respectively, so \(\tau_{max}\) is calculated to be 1.576 \(\times\) 10\({}^{-3}\) MPa. Consequently, the FE model, depending on the ANSYS Workbench V14.0 program, implements the theory of [PERSON]'s circle exactly.