## 4 Results and Discussion
### Parametric Studies
#### 4.2.3 Effects of Sleeper Stiffness  
In general, the permissible contact pressure on the ballast bed is theoretically below a range between 0.5 and 0.75 MPa depending on the geological type of ballast [6]. To satisfy this condition, sleepers undertake the vital role of distributing the wheel loads across the underlying ballast bed [26]. Most of the numerical track models basically rely on the theory of a beam on an elastic foundation where the rail is supported by an elastic support. For the cases where the sleeper is of interest, the theory is developed into that of a sleeper resting on an elastic foundation [6]. According to the theory, the effectiveness of the stress distribution from sleeper to ballast bed correlates with the flexibility of the sleeper, which is the function of the modulus of elasticity of the sleeper; the modulus of elasticity is a material property. The most common sleeper material is concrete; thus, a parametric study considering the changes in the moduli of elasticity of a concrete sleeper is conducted. Figure 10: Longitudinal bending moment distributions with reference to different rail pad stiffness. Concrete is mostly classified by its compressive strength. The range of the compressive strength is defined by EN 13,230 [49] that declares \"minimum compressive strength shall be class C45/55 MPa unless otherwise required by the purchaser\" for the turnout sleepers as a minimum requirement. This requirement is converted into meaningful values for the modulus of elasticity via a graph in [50] that uses the formula:  
\[\mathrm{E}=2.1\mathrm{x}10^{5}(\gamma/2.3)^{1.5}(\mathrm{f_{c}}/200)^{0.5},\]  
where \(\mathrm{E}=\mathrm{modulus}\) of elasticity (kgf/cm\({}^{2}\)), \(\gamma=\mathrm{unit}\) weight of concrete (t/m\({}^{3}\)), and \(\mathrm{f_{c}}=\mathrm{specified}\) design strength of concrete. The conversion of the minimum requirement above gives an approximate range for the modulus of elasticity of 20-50 GPa based on [50]. As can be seen from Figure 12, changing the modulus of elasticity has limited influence on the dynamic factors, ranging from 2.8 to 3.0. A similar finding could be reached for contact forces [51; 52]. (Figure 12b) By contrast, the maximum bending moment distributions along a turnout are affected by different moduli of elasticity. (Figure 13) A high modulus of elasticity elevates the stiffness of a sleeper; therefore, the effective length of the sleeper supported by ballast also changes. Figure 11: Bending moment distributions of specific sleepers with reference to different rail pad stiffness: (**a**) sleeper at the entrance; (**b**) sleeper under the switch blades; (**c**) sleeper at the closure panel; (**d**) sleeper under the crossing nose; (**e**) the longest sleeper; (**f**) the shortest sleeper. As a result, the bending moment of the sleeper is increased by higher moduli of elasticity according to the theory of sleeper on elastic foundation. The influence of the modulus of elasticity appears to be more effective on the sleepers at the closure panel and crossing panel. A high modulus of elasticity can contribute up to 7 kNm in a maximum positive bending moment whereas this contribution is around 3 kNm for maximum negative bending moments on these panels. In Figure 14, the bending moment distributions along specific sleepers for different moduli of elasticity are presented. As can be seen from the figure, the distribution characteristics vary from one sleeper to another. For instance, most of the sleepers exhibit maximum positive bending moment at the rail seat with the highest modulus of elasticity but the sleeper under the switch blade is inconsistent with this result. Another example could be given from the graph of the sleeper under the crossing nose. It is obvious that higher moduli of elasticity increase the bending moments at the section under the crossing nose and decrease the bending moments under the opposite rail.