## 4 Finite Element Analysis of Transition Zones with High Moisture
### Introduction of the Finite Element Model
\begin{table}
\begin{tabular}{l l} \hline \hline
**Parameter** & **Value** \\ \hline Sleeper Elastic Modulus (Pa) & 3.65 \(\times\) 10\({}^{10}\) \\ Sleeper Poisson’s ratio & 0.167 \\ Ballast Elastic Modulus (Pa) & 1.20 \(\times\) 10\({}^{8}\) \\ Ballast Poisson’s ratio & 0.250 \\ Subgrade Elastic Modulus (Pa) & 1.80 \(\times\) 10\({}^{8}\) \\ Subgrade Poisson’s ratio & 0.250 \\ Concrete bridge Elastic Modulus (Pa) & 3.50 \(\times\) 10\({}^{10}\) \\ Concrete bridge Poisson’s ratio & 0.167 \\ Fastening system horizontal stiffness (N/m) & 1.5 \(\times\) 10\({}^{6}\) \\ Fastening system horizontal damping (N*s/m) & 5.00 \(\times\) 10\({}^{4}\) \\ Fastening system longitudinal stiffness (N/m) & 1.5 \(\times\) 10\({}^{6}\) \\ Fastening system longitudinal damping (N*s/m) & 5.00 \(\times\) 10\({}^{4}\) \\ Fastening system vertical (compression) stiffness (N/m) & 1.20 \(\times\) 10\({}^{8}\) \\ Fastening system vertical (compression) damping (N*s/m) & 5.00 \(\times\) 10\({}^{4}\) \\ Fastening system vertical (tension) stiffness (N/m) & 1.20 \(\times\) 10\({}^{11}\) \\ Fastening system vertical (tension) damping (N*s/m) & 5.00 \(\times\) 10\({}^{4}\) \\ Distance between wheels (m) & 2.5 \\ Distance between axes (m) & 20.0 \\ Length of train body (m) & 23.0 \\ Primary suspension stiffness (N/m) & 4.25 \(\times\) 10\({}^{5}\) \\ Primary suspension damping (N*s/m) & 1.00 \(\times\) 10\({}^{6}\) \\ Secondary suspension stiffness (N/m) & 4.68 \(\times\) 10\({}^{5}\) \\ Secondary suspension damping (N*s/m) & 6.50 \(\times\) 10\({}^{4}\) \\ Secondary suspension Bending stiffness (Nm/rad) & 1.05 \(\times\) 10\({}^{4}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Material properties of the track components. an error of 3 mm. The value of the differential settlement that is used in the paper is selected from the range of values that are frequently reported in field measurements of transition zones [14; 23], which is 2-10 mm. A value of 4 mm is used, since it can represent the early period of a transition zone. Dynamic responses are expected to be larger for higher differential settlement values. The nonlinear connection between sleepers and ballast is important for modelling of the ballast degradation mechanism [48; 53]. In the case of hanging sleepers, the sleepers should generate a nonlinear interaction force to the ballast under compression and separable without loading [53]. Therefore, contact elements are applied between sleepers and ballast. According to the penalty algorithm that is employed in the contact elements, the search for penetrations between the bottom surface of sleepers and the top surface of ballast is made for every time step during the calculation. When penetration has been detected, a force that is proportional to the penetration depth is applied to resist and ultimately eliminate the penetration [60]. This method allows for simulating the impact on ballast, which is proportional to the downward acceleration of sleepers. The simulation procedure consists of two phases. First, only the gravity forces are applied to the model. When the model reaches the equilibrium state, the velocity is applied to the vehicle, so that the vehicle moves from the left end to the right end of the transition zone passing the bridge. The mass-weighted nodal damping is applied globally to the nodes and the track's random irregularity is not considered in this model. The model is solved using the commercial software LS-DYNA, version R8, developed by Livermore Software Technology Corporation, CA 94551, the US, 2015. The time-step is 1.3 \(\times\) 10\({}^{-5}\) s, and the calculation takes approximately 10 h using an 8-core (I7) workstation with High Parallel Computing (HPC). The model was validated against field measurements in [35]. The comparison between the simulated rail displacements and the calculated measurement displacements is shown in Figure 12. Rails from a close location (above the 2 nd sleeper) and a far location (above the 11 th sleeper), with respect to the bridge, are both compared.