## 4 Numerical Analysis  
### Overview  
The findings from the analysis of CWR fractures and machine learning-based parameter evaluation indicated that the primary causes of CWR weld fractures were temperature fluctuations and the distance between the fracture location and ventilation shaft. A numerical analysis was conducted based on the data from Sections 2 and 3 to assess the range of temperature variations around ventilation shafts that affect CWR fractures in tunnels. In a previous study, the influence of temperature changes around the ventilation hole was analyzed through CFD analysis [20; 21; 22; 23; 24]. To this end, finite element analysis (FEA) was performed using Ansys Workbench (Ver. 2023 R1) [25; 26], employing both CFD and structural analysis techniques. The numerical model of the ventilation shaft is illustrated in Figure 6. Additionally, the ventilation hole was modeled in the upper central part of the approximately 65 m tunnel model. Boundary conditions for the ventilation shaft were defined by setting the temperature as a load condition at the inlet section; these boundary conditions are presented in Figure 7. As shown in Figure 7, the boundary conditions for temperature and air flow were set with the blue arrow indicating the inlet and the red arrow indicating the outlet.  
\begin{table}
\begin{tabular}{c c c c c c c c c} \hline
**Output** & **Minimum** & **Maximum** & **At** & **Season** & **Curve** & **Inclination** & **Welding** & **Distance** & **(x)** \\
**Input** & **Temperature** & **Temperature** & **\(\Delta\)t** & **Season** & **Radius** & **Inclination** & **Welding** & **Distance** \\ \hline Minimum temperature & - & 1.97 & 1.36 & inf & inf & inf & inf & inf \\ Maximum temperature & 1.97 & - & 1.25 & inf & inf & inf & inf \\ \(\Delta\)t & 1.36 & 1.25 & - & 2.54 & 1.28 & 1.40 & 2.14 & 1.19 \\ Season & inf & inf & 2.54 & - & 2.38 & 1.54 & 1.20 & 2.12 \\ Curve & inf & inf & 1.28 & 2.38 & - & 1.45 & 1.48 & 1.54 \\ radius & inf & inf & 1.40 & 1.54 & 1.45 & - & 1.40 & 1.67 \\ Inclination & inf & inf & 2.14 & 1.20 & 1.48 & 1.40 & - & 2.47 \\
\begin{tabular}{c} Distance (x) \\ \end{tabular} & inf & inf & 1.19 & 2.12 & 1.54 & 1.67 & 2.47 & - \\ \hline \end{tabular}
\end{table}
Table 6: VIF analysis results.  
Figure 6: ventilation shaft numerical model.  
Figure 8a,b shows that temperature variations recorded during CWR fractures were applied as thermal load conditions in the CFD analysis. The maximum temperature variation (\(\Delta\)t) was determined using a Gaussian probability density function and set as a load condition.  
The wind speed during train operation, along with airflow from the ventilation system, was incorporated into the numerical analysis as a load condition. In addition, the airflow and temperature specifications were measured using the KA25 equipment from Kanomax. The temperature was measured inside the underground subway using the wind speed transmitted from the ventilation inlet and the train blocking time [4]. These conditions are summarized in Table 7.  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Category** & **Airflow (m/s)** & **Temperature Setting (\(\Delta\)t)** \\ \hline Case A & 0.5 m/s & \\ Case B & 1.0 m/s & \\ Case C & 2.0 m/s & 35 \({}^{\circ}\)C \\ Case D & 4.0 m/s & \\ \hline \hline \end{tabular}
\end{table}
Table 7: Numerical analysis of load conditions.  
Figure 8: Thermal load conditions based on tunnel temperature variations during CWR fractures: (**a**) temperature variation at fracture locations; (**b**) Gaussian probability density analysis results.  
Figure 7: Ventilation shaft boundary conditions.