## 4 Finite Element Analysis of Transition Zones with High Moisture
### Introduction of the Finite Element Model  
The FE model considering both the differential stiffness and settlement in transition zones is introduced in the section. The model that is used in this paper is based on the model proposed in [47] and further developed in [8; 34]. The model consists of two ballast tracks and one track on the bridge in the middle, as shown in Figure 11. It is possible to analyse both the embankment-bridge and the bridge-embankment sides of the transition zone with a single calculation. In this model, the \"bridge\" is symbolical, which is simplified to reduce the calculation costs. The length of ballast track is 48 m (one on each side) and the bridge is 24 m. Components of the ballast track are rails, fasteners, sleepers, ballast, and subgrade. The rails are modelled as beam elements with the cross-sectional and mass properties of UIC54 rails. Spring and damper elements between the rails and sleepers are used to model fasteners. The springs have bilinear properties in the vertical direction so that they have the stiffness of rail pads in compression, while the stiffness is much higher to simulate the clamping effect of fasteners in tension. Ballast, sleepers, and subgrade are modelled using fully integrated solid elements with elastic material properties. The ballast and subballast are considered as one layer, which is 0.9 m deep. There is no ballast between sleepers  
Figure 11: Finite Element (FE) model of the track transition zone: (a) full view; (b) cross-section of ballast track. since its effect on settlement is negligible. It should be noted that the ballast, subballast and subgrade are simplified in the model. Despite the fact that the responses of unbound granular materials such as ballast, subballast, and subgrade can be more accurately modelled using nonlinear constitutive models, simplifications have to be adopted in the large-scale study (e.g., transition zones) to reduce the computational expense. Many studies (e.g., [10; 48; 49; 50; 51; 52; 53]) have proved that some behaviour of ballast and hanging sleepers can be modelled accurately using simplified methods (e.g., solid elements with elastic material properties). Following [10; 48; 49; 50; 51; 52; 53], the ballast and subgrade in this model are also modelled by solid elements with elastic material properties. The element length of 75 mm is used for the sleepers and the ballast (there are eight elements within one sleeper space). The thickness of the subgrade is 2 m.  
The vehicle model is idealized as a multibody system consisting of one body, two bogies, and four wheelsets. The wheelsets are named Wheel 1 to 4 from the right to the left. Wheel 1 and 2 belong to the front bogie and the Wheel 3 and 4 belong to the rear bogie. Primary and secondary suspensions are modelled by spring-damper elements. The parameters are suggested by literature (vehicle parameters by [11; 54; 55], track parameter by [10; 49; 54; 56]), and are then tuned according to field measurements [35]. The axle load of the vehicle is 19.0 t and velocity is 144 km/h. Material properties of the track components and vehicle used in the model are collected in Table 2.  
The contact between wheels and rails is modelled using the linear Hertzian spring [56]. Silent boundaries are applied on both ends of the model in order to reduce the wave reflection effect. The nodes at the bottom of the subgrade and bridge are fixed. The model considers differential settlement between the bridge and ballast track on both sides. After construction or tamping, ballast tracks will be compacted within a short period [36; 57; 58; 59]. In this period, the large settlement appears in ballast tracks due to the volumetric compaction of particles. On the contrary, the tracks on bridges are barely settled, which generates a differential settlement in transition zones. It should be noted that the value of the differential settlement is important, but is difficult to measure precisely in reality. For instance, the void under sleepers measured in [11] had