## 2 Numerical Model  
A numerical model of a standard turnout with a crossing angle 1:9 and an equivalent rail profile of 54E1, which is adapted from [12], is used to investigate the dynamic effects of unsupported bearers. For clarity, the term 'bearer' is used instead of'steeper' in the following sections, where a bearer means a specific longer sleeper for turnouts. Equivalent profile is preferred to facilitate the calculations with parallel to downgrading the numerical reflection of the geometry of turnout from 3D solid model to 3D beam model. A simulation of a 48-meter turnout with a full geometry, so-called 3D solid model, is also possible but is believed to be not feasible as it consumes a lot of effort and time regarding available technology. A coupling strategy between two numerical methods, such as finite element model and multi-body simulations, is adopted here. The scope of the study is to consider the nonlinear dynamic responses of the coupling train-track and train-turnout systems, which leads to simplification of the vehicle geometry as a spring-mass system. The coupling tool between the two methods is provided by Hertzian contact theory. Figure 1 shows a descriptive model of a turnout. In the numerical model, rails and bearers are modeled with beam elements in contrast to rail pads, ballast and subgrade, where spring-dashboard is applied. It is assumed that the process leading to unsupported bearers is finished, meaning that the bearers are fully unsupported or namely hanging. In other words, there is no contact between bearers and ballast bed. To consider no-contact situation, the spring-dashboard elements representing ballast bed and substructure under the unsupported bearers are removed from numerical model. This strategy is applied separately for every scenario where the location of unsupported bearer changes. Generally, materials properties of the authors' numerical model in [12] have been re-applied in this study. The slight difference is that a turnout with FFU bearers has been found be promising in terms of performance [13; 14; 15] and is considered in this study to gain further insights into the dynamic behaviors of a turnout system with FFU bearers as a case study. The model has 77 bearers with varying lengths from 2.4 m to 4.7 m. The bearer spacing is 0.71 m on average. They are connected to rails via a fastening system, which, in reality, consists of fishplates, bolts, rail pads and clips but here, is represented by springs considering the elasticity provided by rubber rail pads. The stiffness of rail pad is in the range of practical applications, 1300 kN/mm. The damping properties of rail pad is neglected based on [16] which indicates that the effect of rail pad damping is limited when compared to an overall damping of a track having a well-compacted ballast bed. In the simulation, the concept of \"beam on an elastic foundation\" is used where ballast and subgrade is represented by series of spring-dampers. The stiffness and damping properties of ballast bed are 45 kN/mm and 32 Ns/mm per normal bearer. Additionally, those properties of ballast bed per bearer are adjusted with respect to bearer length. The earth on which the subgrade and ballast bed are laid is assumed rigid and therefore, one end of spring-dash boards are fixed as a boundary condition. On the other hand, track components are free to move or rotate on their vertical planes. By taking advantage of multibody simulation, the car body of a rolling stock is downsized to spring mass system. Furthermore, the vehicle is modeled as a single bogie due to more significant contribution of wheel mass into dynamic response of the track-train system in comparison to car body. The bogie consists of bogie frame and two wheelsets, where the mass per wheelset is 20 tons, and they are connected to each other by spring-dashboards acting as primary suspensions. The vehicle moves in facing direction on trough route of the turnout. Moreover, it travels a 45 m distance at a constant speed ranging from 10 to 50 m/s during both wheelsets on the track.