# Face Stability Analysis for Tunnels under Steady Unsaturated Seepage and Inhomogeneity Conditions
## 3 3D Logarithmic Spiral Failure Mechanism  
The scenario examined in this paper is illustrated in Figure 1. It involves a tunnel with a lack of face support pressure, characterized by a diameter of D and a cover depth of C. The soil surrounding the tunnel exhibits steady unsaturated seepage and inhomogeneity conditions. The horizontal plane is located at a vertical distance \(z_{0}\) from the bottom of the tunnel.  
The kinematic limit analysis method is employed in this research, with the primary focus on establishing an effective collapse mechanism. It is assumed that the soil obeys the rule of associative flow, which indicates that the velocity vector should be positioned at an angle corresponding to the effective internal friction angle of the sliding surface, hence \(v_{n}\) can be expressed as:  
\[v_{n}=v_{t}tan\varphi^{\prime} \tag{10}\]  
where \(v_{n}\) represents the normal component of the velocity on the sliding surface, while \(v_{t}\) denotes the tangential component. Equation (10) indicates that during rigid rotation, the mechanism's profile forms a logarithmic spiral curve, and the sliding surface is a  
Figure 1: Diagram of the analyzed situation in the longitudinal section of the failure mechanism.  
tangent to a cone with a summit angle of \(2\varphi^{\prime}\). Consequently, the mechanism is employed to characterize the failure of the face.  
The mechanism rotates with angular velocity \(\omega\) around the centre of rotation O. Therefore, it is necessary to set up a polar coordinate system \((\theta,\rho)\) and represent the velocity at any point within the mechanism as:  
\[\
u=\omega\rho \tag{11}\]  
The traces of this mechanism are controlled by two logarithmic spiral curves: curves AE and BE. These can be expressed as follows:  
curves AE:  
\[r=r_{A}e^{-(\theta-\theta_{A})tan\varphi^{\prime}} \tag{12}\]  
curves BE:  
\[r^{\prime}=r_{B}e^{(\theta-\theta_{B})tan\varphi^{\prime}} \tag{13}\]  
where \(r_{A}\) and \(r_{B}\) can be represented by \(\theta_{A}\) and \(\theta_{B}\), as follows:  
\[\begin{cases}r_{A}=\frac{sin\theta_{B}}{sin(\theta_{B}-\theta_{A})}D\\ r_{B}=\frac{sin\theta_{A}}{sin(\theta_{B}-\theta_{A})}D\end{cases} \tag{14}\]  
The logarithmic spiral mechanism is formed by a number of circles, which are centered on the points on the rotation axis of the two logarithmic spiral curves; the traces of the center points can be represented as:  
\[r_{m}=\frac{r+r^{\prime}}{2} \tag{15}\]  
and the radius of the circle can be determined as follows:  
\[R=\frac{r-r^{\prime}}{2} \tag{16}\]  
Two logarithmic spiral curves intersect at point E, which can be represented by the following polar coordinates:  
\[\begin{cases}\theta_{E}=\frac{\theta_{A}+\theta_{B}}{2}+\frac{ln(sin\theta_{B }/sin\theta_{A})}{2 tan\varphi^{\prime}}\\ r_{E}=\frac{\sqrt{sin\theta_{A}sin\theta_{B}}}{sin(\theta_{B}-\theta_{A})}e^{ \frac{1}{2}(\theta_{A}-\theta_{B})tan\varphi^{\prime}}D\end{cases} \tag{17}\]  
In some tunnels with shallow depths, the failure mechanism may outcrop, in which case the intersections of curves AE and BE with the ground are C and D, respectively. In this case, the mechanism is controlled by the curves AC and BD as well as the ground CD. The positions of points C and D on the ground can be represented by \(\theta_{C}\) and \(\theta_{D}\) as follows:  
\[\begin{cases}r_{B}\cos\theta_{B}-r_{A}\cos\theta_{C}e^{-(\theta_{C}-\theta_{ A})tan\varphi^{\prime}}=C\\ r_{B}\cos\theta_{B}-r_{B}\cos\theta_{D}e^{(\theta_{D}-\theta_{B})tan\varphi^{ \prime}}=C\end{cases} \tag{18}\]  
Accordingly, \(\theta_{A}\) and \(\theta_{B}\) are chosen as optimization variables describing the failure mechanism, and they need to satisfy the following conditions:  
\[\begin{cases}0<\theta_{A}<\theta_{B}<\frac{\pi}{2}\\ \theta_{B}<\theta_{E}<\pi\end{cases} \tag{19}\]