# Impact of Evaluation of Freeze-Thaw Cycles on Collapse Zone at Entrance and Exit of Loess Tunnel
## 2 Principle and Procedure of Network Map  
In the elastoplastic plane strain problem, there exist two intersecting shear failure surfaces for any point on the plane [36]. Two sets of slip lines can be obtained by connecting the shear failure surfaces of different points, for example, curves \(\alpha\)-\(\alpha\) and \(\beta\)-\(\beta\) in Figure 2. The tangential direction of any point on a slip line is the direction of the corresponding slip surface. The direction of each point on a slip line is related to the direction of principal stress trajectories, that is, two sets of orthogonal curves (1-1 and 2-2 in Figure 2) obtained by connecting the segments indicating the principal stress direction of each point. Taking the trajectory of the maximum principal stress \(\sigma_{1}\) as the base line, the curve with an acute angle from the base line in the clockwise direction is called line \(\alpha\), and that with an acute angle from the base line in the counterclockwise direction is called line \(\beta\). For Mohr-Coulomb (M-C) elastoplastic material, the angle between the two sets of slip lines is \(2\mu=\pi/2\,-\,\varphi\), and the angle with principal stress trajectories is \(\mu=\pi/4\,-\,\varphi/2\), where \(\varphi\) is the internal friction angle of the Mohr-Coulomb elastoplastic material. Figure 2: Slip lines and principal stress trajectories. When the stress is solved by the classic slip line network method, the two sets of slip lines form a curve coordinate (\(S_{\alpha}\), \(S_{\beta}\)). Under the coordinate system of (\(S_{\alpha}\), \(S_{\beta}\)), the following equations can be solved:  
\[\frac{dy}{dx}=\tan(\theta-\mu)\quad(line\alpha) \tag{1}\]  
\[\frac{dy}{dx}=\tan(\theta+\mu)\quad(line\beta) \tag{2}\]  
\[-\sin(2\mu)\frac{\partial P}{\partial S_{\alpha}}+2R\frac{\partial\theta}{ \partial S_{\alpha}}+\gamma\bigg{[}\sin(2\mu)\frac{\partial x}{\partial S_{ \alpha}}+\cos(2\mu)\frac{\partial y}{\partial S_{\alpha}}\bigg{]}=0 \tag{3}\]  
\[\sin(2\mu)\frac{\partial P}{\partial S_{\beta}}+2R\frac{\partial\theta}{ \partial S_{\beta}}+\gamma\bigg{[}-\sin(2\mu)\frac{\partial x}{\partial S_{ \beta}}+\cos(2\mu)\frac{\partial y}{\partial S_{\beta}}\bigg{]}=0 \tag{4}\]  
where, \(P\) is the mean stress; \(R\) is the radius of the stress circle (for M-C elastoplastic material, \(R=P\cdot\)sin\(\varphi\) + \(c\cdot\)cos\(\varphi\)); \(\gamma\) is bulk density; \(\theta\) is the angle between the direction of the first principal stress \(\sigma_{1}\) and axis \(x\). Due to the curve coordinates, the difference equation thus obtained is nonlinear. Formulas (1)-(4) can only be solved by numerical solutions through the difference method (Gong, [36]). The slip line field theory has been widely adopted to determine the slip surface of slopes. Figure 3 presents the conventional slip line field of slopes. For the collapse zone of loess arch in loess arching theory, the collapse zone of an unlined loess tunnel may be plotted according to the slip line theory (Figure 4). The collapse begins from the side walls and arch feet, and gradually expands to the vault. Since the cross-section of the tunnel is vertically symmetric, using geometric symmetry, the structure of the right half of the unlined loess tunnel was taken and rotated by 90\({}^{\circ}\) clockwise, as shown in Figure 5. It is worth noting that the slip line network of the half tunnel is very similar to the slope slip line analysis (Figure 5). Inspired by the similarity between the half tunnel slip line and slope slip line, a slip line network method for analyzing the collapse zone in a loess tunnel might be proposed, and the construction of a slip line pattern will be discussed in detail, as shown in Figure 6. With slip lines at the collapse zone, the slip line network method simplifies the curved slip lines as the straight lines, according to the collapse contours observed in loess tunnel surveys, such that the solved bearing capacity of the surrounding rock approximates the lower limit of plastic limit analysis. Figure 3: Conventional slip line field of slopes.