## 6 Instability Predicting and Verification  
### Instability Predicting Model  
The aforementioned calculation method can judge whether the karst tunnel will eventually become unstable according to the actual state of the karst cave and surrounding rock, but it cannot predict the time of the occurrence of instability. Under specific construction conditions, the tunnel cannot keep a safe distance from the water-rich karst cave when crossing the karst area, so the prediction of the time of tunnel instability becomes particularly important. Thus, corresponding technical measures can be taken before tunnel collapse; as a result, the safety and economy of tunnel construction can be improved significantly.  
Furthermore, considering that the parameters of the surrounding rock are difficult to accurately measure in the actual construction environment of a tunnel, and the potential function \(V(x)=x^{4}/4+ux^{4}/2+vx\) is actually a quantity only related to the maximum settlement of the vault, the safety of the tunnel vault can be analyzed by combining the actual monitoring data of the tunnel vault.  
Consider the subsidence of tunnel roof as state variable \(G\); the function of state variable changing with time is expressed as follows  
\[G=g(t) \tag{18}\]  
where \(t\) is the accumulated days from tunnel excavation expanding Equation (18) as a Taylor Series; then, taking the first four terms that have a significant effect on the function, the state variable \(G\) is obtained:  
\[G=g(t)=\sum_{i=1}^{4}m_{i}t^{i} \tag{19}\]  
where \(m_{i}=\left.\frac{\partial\vec{x}f}{\partial t^{i}}\right|_{t=0}\), let \(t=x-Z\), \(Z=\frac{m_{3}}{4m_{4}}\), Equation (19) can be explained as:  
\[\overline{g}=l_{4}x^{4}+l_{2}x^{2}+l_{1}x+l_{0} \tag{20}\]  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{**Content**} & \multirow{2}{*}{**Unit**} & \multicolumn{2}{c}{**DouMo Tunnel**} & \multicolumn{2}{c}{**LuZhuBa Tunnel**} \\ \cline{3-6}  & & **Research [26]** & **This Paper** & **Research [26]** & **This Paper** \\ \hline Cavity load & MPa & 0.4 & 0.4 & 0.2 & 0.2 \\ \hline Rock gravity & KN/m\({}^{2}\) & 25 & 25 & 23 & 23 \\ \hline Modulus of Elasticity & GPa & 1 & 1 & 0.9 & 0.9 \\ \hline Cavity span & m & 13 & 13 & 8 & 8 \\ \hline Distribution Perspective of Cave & \({}^{\circ}\) & 90 & 90 & 30 & 30 \\ \hline Theoretical safe thickness & m & 7.30 & 9.54 & 5.00 & 3.56 \\ \hline Actual thickness & m & 6 & 6 & 3.6 & 3.6 \\ \hline Theoretical system state & / & collapse & collapse & collapse & stable \\ \hline Virtual tunnel condition & / & collapse & collapse & stable & stable \\ \hline \hline \end{tabular}
\end{table}
Table 1: Comparison of tunnel case calculated by different approaches and the measured data.  
In Equation (19), the matrix of the coefficients satisfies Equation (20)  
\[\left\{\begin{array}{l}l_{0}\\ l_{1}\\ l_{2}\\ l_{4}\end{array}\right\}=\left[\begin{array}{ccccc}Z^{4}&-Z^{3}&Z^{2}&-Z&1\\ -4Z^{3}&3Z^{2}&-2Z&1&0\\ 6Z^{2}&-3Z&1&0&0\\ 1&0&0&0&0\end{array}\right]\left\{\begin{array}{l}m_{4}\\ m_{3}\\ m_{2}\\ m_{1}\\ m_{0}\end{array}\right\} \tag{21}\]  
where \(l_{0}\) is a constant with no affection to system stabilization, so the state variable \(G\) can be converted into the standard form of cusp catastrophe theory:  
\[\overline{G}=\frac{1}{4}x^{4}+\frac{1}{2}\mu x^{2}+vx \tag{22}\]  
where \(\mu=l_{2}/2l_{4}\), \(v=l_{1}/4l_{4}\).  
Similarly, according to the basic theory of cusp mutation, the equilibrium equation of the system and the state standard of system instability can be obtained as Equation (23)  
\[\left\{\begin{array}{l}x^{3}+\mu x^{1}+v=0\\ 4\mu^{3}+27v^{2}=0\\ \overline{G}^{\prime\prime}=3x^{2}+\mu<0\end{array}\right. \tag{23}\]  
To solve Equation (23), the instability time of the tunnel surrounding rock can be obtained:  
\[t=\Delta x\sqrt[4]{1/4l_{4}} \tag{24}\]  
A flow chat for realizing this method is shown in Figure 5.  
Figure 5: The flow chart of prediction of instability in karst tunnels.