## 3 KLA in the 3D Horned Failure Mechanism
### Three-Dimensional Horned Rotational Failure Mechanism  
In the investigation of tunnel face safety through the KLA approach, it is imperative to ascertain a rational presumed failure mechanism. This mechanism should adhere to the following assumptions in the KLA: (1) the soil mass complies with pertinent regulations; (2) the soil exhibits ideal plastic behavior. Additionally, there is one need that has to be met, namely the work exerted by outside forces can be no greater than the rate of energy dissipation., as follows:  
\[\int_{V}\sigma_{ij}\dot{\epsilon}_{ij}dV\geq\int_{S}T_{i}v_{i}dS+\int_{V}X_{i}v _{i}dV \tag{5}\]  
where \(\dot{\epsilon}_{ij}\) and \(\sigma_{ij}\) indeed represent the strain and stress tensors, respectively. The failure body's volume and border are precisely denoted by the \(V\) and \(S\). And \(T_{i}\) appropriately refers to the forces acting on the boundary face, while \(X_{i}\) correctly refers to the body forces acting within the volume \(V\). \(v_{i}\) accurately signifies the velocity along the failure face. Numerous failure mechanisms have been posited within academic sphere as means to emulate tunnel collapse. When using discrete techniques, the horned mechanism differs from these others in that it is more conservative than the multi-block mechanism and easier to design than the rotating mechanism. [PERSON] initially introduced the horned failure mechanism, which subsequently evolved into a potent tool for evaluating both slope stability and tunnel face safety. Therefore, considering its efficacy in practical application, this paper opts for the adoption of the horned failure mechanism. This mechanism encompasses a pair of logarithmic spirals, denoted as AE and BE, sharing the central pivot at O and characterized by the angular velocity of \(\omega\). Schematic representations for the horizontal and longitudinal cross-sectional aspects are shown in Figure 3. In the realm of polar coordinates, the expressions for AE and BE are articulated as follows:  
\[r=r_{\rm AE}e^{-(\theta-\theta_{\rm A})\tan\varphi^{\prime}} \tag{6}\]  
\[r^{\prime}=r_{\rm BE}e^{(\theta-\theta_{\rm B})\tan\varphi^{\prime}} \tag{7}\]  
Figure 3: Three-dimensional horned failure mechanism: (**a**) horizontal cross-section; (**b**) longitudinal cross-section. In which \((r_{A},\;\theta_{A})\) and \((r_{B},\;\theta_{B})\) denote the polar coordinates of points A and B, respectively. The polar radii, \(r_{A}\) and \(r_{B}\), can be determined using the following equations:  
\[r_{A}=\frac{\sin\theta_{B}}{\sin(\theta_{B}-\theta_{A})}D \tag{8}\]  
\[r_{B}=\frac{\sin\theta_{A}}{\sin(\theta_{B}-\theta_{A})}D \tag{9}\]  
The distance between the rotational center O and the horned mechanism's center axis is as follows:  
\[r_{m}=\frac{r+r^{\prime}}{2} \tag{10}\]  
Each rotational segment's radius is as follows:  
\[R=\frac{r-r^{\prime}}{2} \tag{11}\]  
The coordinates of the intersection point of the two logarithmic spirals are as follows:  
\[\left\{ \begin{array}{l}\theta_{E}=\frac{\theta_{A}+\theta_{B}}{2}+\frac{ \ln(\sin\theta_{B}/\sin\theta_{A})}{2\tan\varphi^{\prime}}\\ r_{E}=\frac{\sqrt{\sin\theta_{A}\sin\theta_{B}}}{\sin(\theta_{B}-\theta_{A})} \frac{1}{2}(\theta_{A}-\theta_{B})\tan\varphi^{\prime}D\end{array} \right. \tag{12}\]  
When the tunnel burial depth appears to be shallow, the horned mechanism might emerge above the ground face. With such instances, the ground face CD, the tunnel face AB, and the two logarithmic curves AC and BD serve to define the bounds for this mechanism.