## 2 Damage Model of a Karst Tunnel
### Power Exerted by the External Force  
In the upper bound theorem of limit analysis, the damage areas could be divided into a number of rigid polygonal blocks. The loading effect of the rock and soil masses around the deeply buried tunnel when the karst cave is located at the bottom of the tunnel could be represented by calculating the external power of the polygonal blocks in each damage region under the action of self-weight. According to the upper bound theorem of limit analysis for geotechnical masses, the power contributed by each computational region under the self-weight of the geotechnical mass is the product of the self-weight of the geotechnical body in that region and the vertical component of the velocity of the centre of gravity in that region. As shown in Figure 2a,b, the expression \(P_{ext1-1}\) for the power of the external work of the triangular blocks \(A_{t,0}B_{t,1}B_{t,0}\) and \(B_{t,0}B_{t,i-1}B_{t,i}\) under its own weight is given by Equation (25). \[P_{ext1-1}=\gamma_{0}S_{A_{t,0}B_{t,1}B_{t,0}}v_{t,0}+\gamma_{0}\left[\sum_{i= 1}^{n-1}\left(S_{B_{t,0}B_{t,i-1}B_{t,i}}v_{t,i}\right)\right] \tag{25}\]  
It is shown in Figure 2c,d that the expression \(P_{ext1-2}\) for the power of the external work of the polygonal block \(B_{t,j}C_{t,1}C_{t,0}O_{t,0}B_{t,n}\) and the triangle block \(C_{t,0}C_{t,i-1}C_{t,i}\) in the M\(-\)2 region under self-weight is given by Equation (26). \[P_{ext1-2}=\gamma_{0}S_{B_{t,j}C_{t,1}C_{t,0}O_{t,0}B_{t,n}}v_{r,0}+\gamma_{0} \sum_{i=1}^{n-1}\left(S_{C_{t,0}C_{t,i-1}C_{t,i}}\cdot v_{r,i}\right) \tag{26}\]  
where \(S_{B_{t,j}C_{t,j}C_{t,0}O_{t,0}B_{t,n}}\) and \(S_{C_{t,0}C_{t,j-1}C_{t,i}}\) are the area of the polygonal block \(B_{t,j}C_{t,1}C_{t,0}O_{t,0}B_{t,n}\) and the triangle block \(C_{t,0}C_{t,i-1}C_{t,i}\) respectively. The karst cave medium fills the karst cave and the karst cave medium pressure applies forces on the inner surface of the karst cave. In the PH\(-\)1 failure mode, the power of the external work performed by the pressure of the karst cave medium is expressed as:  
\[P_{ext1-3}=-KqD_{1}v_{r,n-1}\sin(\omega_{r,n-1}+\varphi)-qD_{1}v_{r,0}/2 \tag{27}\]  
where \(K\) is the lateral pressure coefficient on the geotechnical mass. It is shown in Figure 3a,b that the expression \(P_{ext2-1}\) for the power of the external work of the triangle blocks \(A_{T,0}A_{T,1}A^{\prime}{}_{T,1}\), \(A_{T,0}A_{T,j-1}A_{T,j}\) and \(A_{T,0}A^{\prime}{}_{T,i-1}A^{\prime}{}_{T,i}\) in the P\(-\)1 region under self-weight in the PH\(-\)2 failure model is given by Equation (28). \[P_{ext2-1}=\gamma_{0}S_{A_{T,0  
As shown in Figure 3e,f, the expression \(P_{ext2-3}\) for the power of the external work of the polygonal block \(B_{T,i}C_{T,1}C_{T,0}O_{T,0}B_{T,n}\) and the triangle block \(C_{T,0}C_{T,i-1}C_{T,i}\) in the P\(-3\) region under self-weight in the PH\(-2\) failure model is given by Equation (30).  
\[P_{ext2-3}=\gamma_{0}S_{B_{T,i}C_{T,1}C_{T,0}O_{T,0}B_{T,n}}v_{R,0}+\gamma_{0} \sum_{i=1}^{n-1}\left(S_{C_{T,0}C_{T,i-1}C_{T,i}}v_{R,i}\right)\,n>3 \tag{30}\]  
where \(S_{B_{T,i}C_{T,1}C_{T,0}O_{T,0}B_{T,n}}\) and \(S_{C_{T,0}C_{T,i-1}C_{T,i}}\) are the area of the polygonal block \(B_{T,i}C_{T,1}C_{T,0}O_{T,0}\) and the triangle block \(C_{T,0}C_{T,i-1}C_{T,i}\), respectively. In the PH\(-2\) failure mode, the power of the external work performed by the pressure of the karst cave medium is expressed as:  
\[P_{ext2-4}=-KqD_{1}v_{R,n-1}\sin(\omega_{R,n-1}+\varphi)-qD_{1}v_{R,0}/2 \tag{31}\]  
where \(K\) is the lateral pressure coefficient on the geotechnical mass. With a view to applying this method of calculating the instability probability in deeply buried tunnel construction when the karst cave is located at the bottom of the tunnel to practical engineering, the expressions of the analytical solutions for each damage region in these two deeply buried tunnel failure modes have to fulfil the specific geometrical constraints.