## 4 Results and Discussion
### Bending Moment Distributions under Static and Dynamic Loadings  
In Figure 4, the maximum positive bending moment distributions of five sleepers are illustrated under both static and dynamic conditions, where the vehicle speed is 50 m/s. It is observed in current model that the speed values over 50 m/s result in so called wheel-fly, which indicates the absence of contact between wheel and rail. Here, the static condition means that an equivalent static wheel load is applied to the rail at specific locations. These are selected from the various locations of sleepers: at the entrance panel, a sleeper under the switch blade, the middle sleeper in the closure panel, a sleeper under the crossing nose, the longest sleeper and shortest sleeper. As can be seen from the Figure 4, the bending moment distribution is related to the sleeper length and the rail layout over the sleepers. Sleepers in the entrance panel, under the switch blades and the shortest sleeper show almost symmetrical distributions for both cases. Each sleeper shows small variations in bending moment distributions which are unique to their layout. For instance, the free switch blade creates a small indentation over the bending moment distribution of the sleeper under the switch blade (see Figure 4b). Likewise, there is a small difference in the distribution of the positive bending moments along the shortest sleeper, where the magnitude of the bending moment at the inside rail seat is lower than at the other side (see Figure 4f). It may result from the asymmetrical rail positioning which is carried out to place shorter sleepers or a sudden change in sleeper lengths from the longest to the shortest, which affects the bending moment distributions in the longitudinal direction. In general, the bending moment distributions of these sleepers, under dynamic loading, exhibit similar behavior to static loading. The small variations between static and dynamic cases are the consequence of track irregularities. The irregularities act as an actuator that produces force fluctuations ('dynamics') over the turnout. In the absence of irregularities, the force induced by train will be constant moving load and therefore, discrepancies cannot be observed between static and dynamic cases. For the dynamic case, the bending moment values are higher than the static values for these sleepers. Nevertheless, the difference is imperceptible and on some occasions such as in Figure 4f, dynamic bending moments at one rail seat could have similar values to the static values. Another common point for these sleepers is that the maximum positive bending moments for these sleepers occur at the rail seats and the negative bending moments at the midpoint or close to midpoint of the sleeper. The magnitudes of maximum positive and negative bending moments are respectively around 5 and 4 kNm. Figure 4c-e show that the asymmetrical distributions of bending moments are observed for the sleepers at the closure and crossing panels and at the longest sleeper due to the expanding sleeper lengths and rail layout. It seems that the layout of a turnout affects the track stiffness significantly and results in the different distributions of the bending moments; specifically, it is obvious for the static case. The magnitude of the maximum positive bending moment, under static conditions, is the highest, 11.2 kNm, for the sleeper at the middle of the closure panel and next is the longest sleeper at 9.5 kNm. The maximum positive bending moments still occur at the sleeper sections under the rail seats and they are approximately twice as high as the positive bending moments at the opposite rail seats. A similar asymmetrical behavior is observed for the maximum negative bending moments and there are also similar differences between the sleepers in closure and crossing panels in terms of maximum negative bending moment. Nevertheless, the gap, between the magnitudes of maximum negative bending moments for these sleepers, is small at about 0.5 kNm.