## 3 Modelling Results
### The Temporary Shutdown Condition  
In actual engineering, due to the instability of the stratum, the shielding process is often disturbed by various factors, resulting in shutdown adjustment. In the case of a breakthrough working shaft by AGF, the shield machine stays in the designated fixed position for a long time, which leads to the lower temperature of the soil around the cutter  
Figure 14: Thickness of the frozen curtain in different line. Figure 13: The bottom thickness of the cup-shaped frozen curtain. head and makes it difficult to move forward during the secondary start-up, and can even make the cutter head freeze. This study mainly analyzes the most extreme condition, that is, the relationship between the time the shield machine stays in and the freezing risk. According to the study of the shear properties of frozen clay by [PERSON] [22], the shear properties of frozen soil are mainly related to temperature and corresponding force, thus the specific relationship is shown in Equation (8). \[\tau=(-0.021T+0.637)\sigma+16.05\mathrm{e}^{-0.09T} \tag{8}\]  
In which, \(T\) is the soil temperature, \(\sigma\) is vertical stress, and \(\tau\) is shearing stress. In this model, the main resistance of the shield machine shell is from the shearing stress of the surrounding soil. Taking a point above the shield machine as an example, the plane stress state is shown in Figure 15. The vertical stress is mainly provided by the pressure of the surrounding soil, that is, the vertical stress comes from the self-weight stress and lateral earth pressure of the soil. According to Equation (8), a critical temperature value can be obtained. In the study, it is considered that when the boundary temperature of the contact surface between the shield machine and the frozen soil in the model drops to this temperature, the corresponding time is the time when the shield machine is allowed to stop. Beyond this time, the shield machine will be unable to keep tunneling due to the freezing conditions. Taking a meter of the unit strip for analysis, the three-dimensional problem is simplified to a two-dimensional problem, and the sum of the vertical stress perpendicular to the tunnel can be calculated. The calculation diagram is shown in Figure 16.  
Figure 16: Diagram of the calculation section. Figure 15: Plane stress diagram of soil element. The total pressure can be calculated as follows. \[\sigma_{\rm V}=10.65*18.424*6.75=1324.46\ \rm{kN/m} \tag{9}\]  
\[\sigma_{\rm A}=0.43*18.424*6.75*6.75=360.96\ \rm{kN/m} \tag{10}\]  
\[\sigma_{\rm H}=(107.92+176.32)*6.75/2=959.31\ \rm{kN/m} \tag{11}\]  
Figure 13:0  
where \(\sigma_{\rm V}\) represents vertical earth pressure, \(\sigma_{\rm A}\) represents soil pressure at arch back, \(\sigma_{\rm H}\) represents horizontal earth pressure, and \(\sigma\) represents the total pressure. According to [PERSON]'s research for an earth pressure shield machine with a general diameter of 6 m [23], the empirical formula of the shield total thrust is shown in Equation (13). Figure 13:1  
In this expression, \(D_{1}\) represents the excavation diameter of the shield machine. \(P_{j}\) represents the average thrust per unit of excavation face. In this model, \(D_{1}\) is taken as 6.75 m, and \(P_{j}\) is taken as 1300 kN. Based on the temperature-dependent characteristics of the freezing force at the contact surface between the frozen soil and shield machine shell, the resistance of the tunneling force and the freezing force of the shield machine can be calculated. When the shear strength of the contact surface between the frozen soil and shield machine shell is stronger than the total thrust of the shield machine, the shield machine will not be able to tunnel ahead. The critical temperature is \(-12\ ^{\circ}\)C, solved by Equation (8). The research goal of this study is to analyze the temperature change of the surrounding soil and the possible freezing risk when the shield machine is stopped due to special circumstances when the shield machine reaches the frozen soil curtain area.