## 3 KLA in the 3D Horned Failure Mechanism
### Balance Equation for Work and Energy  
In accordance with the KLA method, the balance equation for work and energy arises by equating the rate of energy dissipation to the rate of external work. And the self-weight of soil is a primary contributor to the rate of outside work. Meanwhile, the internal energy dissipation consists of two constituents: effective internal cohesion \(c^{\prime}\) and matric suction-induced apparent internal cohesion \(c_{app}\). The formulation of the balance equation for work and energy is defined as follows:  
\[\dot{D}_{\rm c}+\dot{D}_{\rm s}=\dot{W}_{\gamma} \tag{15}\]  
where \(\dot{D}_{\rm c}\) and \(\dot{D}_{\rm s}\) correspondingly represent the rates of energy dissipation attributed to effective internal cohesion \(c^{\prime}\) and matric suction-induced apparent internal cohesion \(c_{app}\). \(\dot{W}_{\gamma}\) signifies the external work generated by one's own weight. And the rate of outside work equals the summation of the gravitational work rate for each component, as illustrated below:\[\dot{W}_{\gamma} =2\omega\gamma\bigg{[}\int_{\theta_{\rm A}}^{\theta_{\rm B}}\int_{d_{1} }^{R}\int_{0}^{\sqrt{R^{2}-y^{2}}}\left(r_{\rm m}+y\right)^{2}\sin\theta d{x}d{y}d \theta\ +\int_{\theta_{\rm B}}^{\theta_{\rm C}}\int_{-R}^{R}\int_{0}^{\sqrt{R^{2}-y^{ 2}}}\left(r_{\rm m}+y\right)^{2}\sin\theta d{x}d{y}d\theta \tag{16}\] \[+2\omega\gamma\int_{\theta_{\rm C}}^{\theta_{\rm D}}\int_{-R}^{d_ {2}}\int_{0}^{\sqrt{R^{2}-y^{2}}}\left(r_{\rm m}+y\right)^{2}\sin\theta d{x}d{ y}d\theta\bigg{]}\]  
It is imperative to establish the basic premise that these collapsing blocks exhibit rigidity in order to determine the rate of internal energy dissipation. This premise enables the integration for the rate of internal energy dissipation as follows:  
\[\dot{D}=\int_{S_{\rm t}}cv\cos\varphi^{\prime}{\rm d}S_{\rm t} \tag{17}\]  
Here, \(c\) signifies the cohesive strength of the soil, and \(S_{\rm t}\) signifies the failure face. Using Equation (17), it is possible to deduce the energy dissipation resulting from the effective cohesion of the soil \(c^{\prime}\).  
\[\dot{D}_{\rm c} =2\omega c^{\prime}\bigg{[}\int_{\theta_{\rm A}}^{\theta_{\rm B}} \int_{d_{1}}^{R}\frac{R\left(r_{\rm m}+y\right)^{2}}{\sqrt{R^{2}-y^{2}}}d{y}d \theta\ +\int_{\theta_{\rm B}}^{\theta_{\rm C}}\int_{-R}^{R}\frac{R\left(r_{\rm m}+y \right)^{2}}{\sqrt{R^{2}-y^{2}}}d{y}d\theta \tag{18}\] \[+\int_{\theta_{\rm C}}^{\theta_{\rm D}}\int_{-R}^{d_{2}}\frac{R \left(r_{\rm m}+y\right)^{2}}{\sqrt{R^{2}-y^{2}}}d{y}d\theta\bigg{]}\]  
where \(d_{1}\) and \(d_{2}\) can be ascertained through geometric relationships:  
\[d_{1}=r_{\rm A}\frac{\sin\theta_{\rm A}}{\sin\theta}-r_{\rm m} \tag{19}\]  
\[d_{2}=r_{\rm C}\frac{\cos\theta_{\rm C}}{\cos\theta}-r_{\rm m} \tag{20}\]  
Similarly, the energy dissipation brought about by apparent cohesion \(c_{app}\) is as follows:  
\[\dot{D}_{\rm s} =2\omega\bigg{[}\int_{\theta_{\rm A}}^{\theta_{\rm B}}\int_{d_{1} }^{R}c_{app}\frac{R\left(r_{\rm m}+y\right)^{2}}{\sqrt{R^{2}-y^{2}}}d{y}d \theta\ +\int_{\theta_{\rm B}}^{\theta_{\rm C}}\int_{-R}^{R}c_{app}\frac{R \left(r_{\rm m}+y\right)^{2}}{\sqrt{R^{2}-y^{2}}}d{y}d\theta \tag{21}\] \[+\int_{\theta_{\rm C}}^{\theta_{\rm D}}\int_{-R}^{d_{2}}c_{app} \frac{R\left(r_{\rm m}+y\right)^{2}}{\sqrt{R^{2}-y^{2}}}d{y}d\theta\bigg{]}\]  
In which the apparent cohesion \(c_{app}\) induced via matric suction can be formulated as follows:  
\[c_{app}=\left(u_{a}-u_{w}\right)\tan\varphi_{b} \tag{22}\]  
\[c_{app}=\left(u_{a}-u_{w}\right)\tan\varphi_{b}=\rho z\tan\varphi_{b} \tag{23}\]  
where \(z\) represents the vertical distance from the groundwater level to a point on the tunnel failure face, and it can be expressed as follows:  
\[z=r_{\rm A}\cos\theta_{\rm A}-\left(r_{\rm m}+y\right)\cos\theta+z_{0} \tag{24}\]