# Fatigue Performance of Tunnel Invert in Newly Designed Heavy Hauilway Tunnel
## 2 Mechanical Characteristics of the Heavy Haul Railway Tunnel Base
### Numerical Model  
Based on Figure 1, a refined numerical train-track-tunnel model was established by using finite element modeling (FEM) software ABAQUS (Dassault Simulia, Providence, RI, USA). The model size was 100 m wide, 100 m tall, and 30 m long. Forming a ground arch in a shallow tunnel is difficult, and, as a result, in general, its structural stress is more unfavorable compared to a deep tunnel [16], so the distance between the tunnel vault and the ground surface was set to 15 m [17]. The system anchor bolt was ignored in this model due to its limited effect in shallow tunnels. The tunnel lining, inverted arch filling, track plate base, flexible cushion, track plate, sleeper, rail wheel, boogie, and train wagon were all elastic materials; the surrounding rock was elastoplastic material; and C3D8R solid elements were adopted to simulate these components. We used a total of 187,822 elements for this mesh. The train-track-tunnel integrated model and its mesh are presented in Figure 2.  
Fixed boundary constraints were adopted on the sides of the model. The normal stiffness of the viscoelastic boundary [18] was \(2.5\times 10^{6}\) N/m, normal damping was \(1.29\times 10^{6}\) kg:m\({}^{-2}\)-s\({}^{-1}\), shear stiffness was \(5.0\times 10^{6}\) N/m, and shear damping was \(6.35\times 10^{6}\) kg:m\({}^{-2}\)-s\({}^{-1}\).  
Figure 1: Section structure of the heavy haul railway tunnel in grade V surrounding rock (cm).  
Figure 2: Numerical model of train–track–tunnel coupling system: (**a**) Model size; (**b**) structure of the model; and (**c**) mesh of numerical model.  
The mechanical parameters of the model materials were selected from the Chinese standard [19] and are listed in Table 1.  
The contact relationships are shown in Figure 3; the constraints between the wheel, bogie, and train wagon are rigid bodies. The tie constraints were used to simulate the relationship between other components of the model, such as rocks and initial support, initial support and secondary lining, filling layer and track plate, and sleeper and track plate.  
Surface-to-surface contact was adopted to simulate the relationship between wheel and track; the wheel was the predominant surface and the normal force between the wheel and the track was determined using [PERSON] nonlinear elastic contact theory [20], as shown in Equation (1):  
\[P_{N}(t)=\left[\frac{1}{G}\delta N(t)\right]^{3/2} \tag{1}\]  
where \(P_{N}(t)\) is the wheel/rail normal force (N), \(\delta N(t)\) is the normal elastic compression of the wheel/rail contact point, \(t\) is the running time of the model, \(G\) is the constant for wheel/rail contact, LM is the wear type tread \(G=3.86\ R^{-0.115}\times 10^{-3}\) (m/N\({}^{2/3}\)), and \(R\) is the wheel rolling radius (m).  
The train axle load was 30 t, the Cartesian connector was adopted to simulate the relationship between track and sleeper, and the parameters are provided in Table 2.  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{**Material**} & **Density** & **Elastic Modulus** & **Poisson** & **Tensile Strength** & **Compressive Strength** \\  & **(\(\rho\)/kg/m\({}^{3}\))** & **E (GPa)** & **Ratio \(\mu\)** & \(f_{t}\) (MPa)** & \(f_{c}\) (MPa) \\ \hline Secondary lining (C35) & 2600 & 31.5 & 0.2 & 2.20 & 23.4 \\ Initial support (C25) & 2500 & 28.5 & 0.2 & 1.78 & 16.7 \\ Inverted arch filling (C25) & 2500 & 19.8 & 0.3 & 1.78 & 16.7 \\ Foundation plate (C30) & 2500 & 30 & 0.2 & 2.01 & 20.1 \\ Flexible cubusion (C30) & 2200 & 20 & 0.2 & 2.01 & 20.1 \\ Track plate (C45) & 2700 & 33.5 & 0.2 & 2.51 & 29.6 \\ Rail and train & 7800 & 210 & 0.3 & 210 & 210 \\ \hline \multirow{2}{*}{**Material**} & **Density** & **Elastic modulus** & **Poisson** & **Cohesion** & **Internal Friction Angle** \\  & **(\(\rho\)/kg/m\({}^{3}\))** & **E (GPa)** & **Ratio \(\mu\)** & **c (kPa)** & **\(\varphi\) (v)** \\ \hline V grade surrounding rock & 2100 & 0.50 & 0.33 & 80 & 25 \\ \hline \hline \end{tabular}
\end{table}
Table 1: Mechanical parameters.  
Figure 3: Wheel–rail contact.