# Evaluation of Structural Stresses of Mountain-Embedded Railway Systems
## 2 Simulation Model for the Mountain-ERS Track
### Material Properties  
#### 2.3.1 Concrete  
The material strength and modulus of elasticity of concrete can be obtained by directly testing a specimen or referring to the formulas presented in previous studies and design standards. For constituting the material properties of the concrete materials, that is, the precast concrete panel, mortar filler layer (bedding layer), and subgrade layer, a formula (obtained from the literature) [11; 12] was used to calculate the values, including the modulus of elasticity according to the compressive strength, stress-strain relationship, and material properties. To calculate the elastic modulus of concrete materials, Equation (1) (presented in EN1992-1-1 [12]) was applied:  
\[E_{c}=22,000\left(\frac{f_{c}^{\
u}}{10}\right)^{0.3}, \tag{1}\]  
where \(E_{c}\) is the elastic modulus of concrete, and \(f_{c}^{\
u}\) is the maximum compressive stress at corresponding compressive strain. The compressive strengths of the concrete and bedding layers were 45 MPa and 24 MPa, respectively. The Poisson's ratio was assumed to be 0.18. The elastic modulus of each concrete component corresponding to its compressive strength is listed in Table 2.  
The concrete damaged plasticity model was applied to the plastic behavior of concrete materials in the Abaqus program. The parameters for the model were assumed as reference values [11]: 36\({}^{\circ}\) for the dilation angle (\(\upmu\)), 0.01 for eccentricity (\(\varepsilon\)), and 1.16 for the effective stress invariants value, which was calculated as the ratio \(f_{b0}/f_{c0}\). The ratio of the second stress invariant on the tensile meridian (\(K_{c}\)) was assumed to be 0.667, and the viscosity parameter (\(\upmu\)) used for the viscoplastic regularization of the concrete constitutive equations was assumed to be equal to 0.0001. For the compressive behavior of the concrete material, the stress-strain model presented by [PERSON] [13] was applied. The stress-strain relationship for compression is shown in Figure 5.  
For the ascending branch (\(0\leq\varepsilon_{c}\leq\varepsilon_{o}\)):  
\[f_{c}=f_{c}^{\prime\prime}\left[2\frac{\varepsilon}{\varepsilon_{o}}-\left( \frac{\varepsilon}{\varepsilon_{o}}\right)^{2}\right]; \tag{2}\]  
For the descending branch (\(\varepsilon_{o}\leq\varepsilon_{c}<\varepsilon_{u}\)):  
\[f_{c}=\left(\frac{-0.15f_{c}^{\prime\prime}}{\varepsilon_{c}-\varepsilon_{o}} \right)(\varepsilon-\varepsilon_{o})+f_{c}^{\prime\prime}, \tag{3}\]  
where \(f_{c}^{\prime\prime}\) is the maximum compressive stress at compressive strain (\(\varepsilon_{o}=2f_{c}^{\prime\prime}/E_{c}\)), \(f_{c}\) is the compressive stress at strain (\(\varepsilon_{c}\)), and \(\varepsilon_{u}\) is the ultimate compressive strain, which can be set to 0.0038 with 0.85 \(f_{c}^{\prime\prime}\). Concrete is a brittle material that is vulnerable to tension; therefore, its tensile behavior is determined by evaluating its tensile strength. When concrete is subjected to a tensile force, it exhibits a linear behavior up to the peak strain (\(\varepsilon_{cr}\)) at which cracks occur, and a nonlinear response thereafter. The stress tends to decrease as the strain increases. In the structural analysis model used in this research, the stress-strain model used for the tensile behavior of concrete was presented by [PERSON] and [PERSON] (2001) [14]. An equation (presented in the CEB-FIP model code) was used for the peak tensile stress of the concrete [15] (Equation (6)). A linear relationship exists until the strain reaches the peak at which the crack occurs, and \begin{table}
\begin{tabular}{c c c c} \hline
**Concrete Component** & **Compressive Strength** & **Elastic Modulus** & **Poisson’s Ratio** \\ \hline precast concrete panel & 45 MPa & 35,684 GPa & 0.18 \\ bedding layer & 45 MPa & 35,684 GPa & 0.18 \\ subgrade layer & 24 MPa & 30,160 GPa & 0.18 \\ \hline \end{tabular}
\end{table} 
Table 2: Elastic modulus of concrete components by their corresponding compressive strengths. Figure 5: Stress–strain relationship for compressive behavior of concrete material.