## 4 Catastrophe Analysis of Immediate Roof Instability
### Catastrophe Analysis  
Taking the compression displacement \(u\) of immediate roof rock mass as the state variable, according to the cusp catastrophe theory, obtaining of the equilibrium surface \(M\) can be expressed as  
\[V_{u}^{\prime}=-(h_{j}J_{A}\rho g+\frac{\rho_{1}g_{A}{}^{2}}{2\lambda_{z}\tan \varphi}-\frac{\tan\varphi\tan\varphi^{\prime}}{2}h_{j}J_{A}\rho g)-k_{1}(a-u )+\lambda ue^{-u/u_{0}}=0 \tag{7}\]  
Then, the singular point set equation can be expressed as  
\[V_{u}^{\prime\prime}=k_{1}+\lambda e^{-u/u_{0}}(1-\frac{u}{u_{0}})=0 \tag{8}\]  
The equilibrium surface \(M\) satisfies \(V^{\prime\prime\prime}=0\) at the cusp, and the cusp equation can be obtained as  
\[V_{u}^{\prime\prime\prime}=(2-\frac{u}{u_{0}})\frac{\lambda}{u_{0}}e^{-u/u_{0 }}=0 \tag{9}\]  
At the cusp,  
\[u=2u_{0}=u_{1} \tag{10}\]  
It was found that the cusp was the inflection point of the constitutive curve of rock mass. In order to transform the cusp catastrophe model to a standard form, we carried out a Taylor series expansion at the cusp, and the first three terms were truncated as  
\[ \begin{array}{l}-(h_{j}J_{A}\rho g+\frac{\rho_{1}g_{A}{}^{2}}{2\lambda_{z} \tan\varphi}-\frac{\tan\varphi^{\prime}}{2}h_{j}J_{A}\rho g)-k_{1}(a-u_{1})+ \lambda u_{1}e^{-u/u_{0}}\\ +[k_{1}+\lambda(1-\frac{u_{1}}{u_{0}})e^{-u/u_{0}}](u-u_{1})-\frac{\lambda}{u_ {0}}e^{-u/u_{0}}(2-\frac{u_{1}}{u_{0}})(u-u_{1})^{2}\frac{1}{2}\\ -\frac{\lambda}{u_{0}^{2}}e^{-u/u_{0}}(3-\frac{u_{1}}{u_{0}})(u-u_{1})^{3} \frac{1}{3!}\end{array}  \tag{11}\]  
Introducing dimensionless parameters \(x=\frac{u-u_{1}}{u_{1}}\), and substitute \(u_{1}=2u_{0}\), the above equation can be simplified to obtain the equation of equilibrium surface in the standard form of cusp catastrophe, which can be expressed as  
\[x^{3}+px+q=0 \tag{12}\]\[m=\frac{3}{2}(K-1) \tag{13}\]  
\[n=\frac{3}{2}[-\frac{(h_{j}l_{A}\rho g+\frac{\rho_{1}g\delta_{A}^{2}}{2\lambda_{1} \tan\varphi}-\frac{\tan\varphi\tan\varphi^{\prime}}{2}h_{j}l_{A}\rho g)}{4\lambda u _{0}e^{-2}}+K\xi-1] \tag{14}\]  
\[K=\frac{k_{1}}{\lambda e^{-2}}=\frac{k_{1}}{\lambda_{1}} \tag{15}\]  
\[\xi=\frac{a-u_{1}}{u_{1}} \tag{16}\]  
The parameter \(K\) is the ratio of the support stiffness to the slope of the constitutive relation curve of the immediate roof at the inflection point, which is called the stiffness ratio. \(\zeta\) is the total displacement parameter, which is related to the total displacement of the support and the immediate roof. According to Equations (13) and (14), the control variables \(m\) and \(n\) of the system are related to the stiffness ratio \(K\), the total displacement parameter \(\zeta\), the external load \(Q\), and the first weighting interval \(l\) of the main roof. The bifurcation set equation of cusp catastrophe with \(m\) and \(n\) as control parameters can be expressed as  
\[4m^{3}+27n^{2}=0 \tag{17}\]  
The bifurcation set Equation (17) is the critical point of the system instability. Only when \(m\leq 0\) will the system cross the bifurcate set and cause a sudden jump of deformation, and thus \(m\leq 0\) is the necessary condition of the system instability. According to Equation (13), the stiffness ratio \(K\) must be less than or equal to 1, that is  
\[K\leq 1\text{ or }\frac{k_{1}}{\lambda_{1}}=\frac{k_{1}He^{2}}{EA}\leq 1 \tag{18}\]  
According to the catastrophe theory, only when \(m\) and \(n\) satisfy the bifurcation set Equation (17) will the system have a sudden jump in instability. Therefore, the bifurcation set equation is a sufficient condition for the sudden jump in instability of the system. According to the cusp catastrophe model in Figure 10, when the bifurcation set crossed from the right branch (\(n>0\)) to the left branch (\(n<0\)), the corresponding point was in an unstable state, and the state variable \(x\) was a sudden jump. The fracture instability occurred when the deformation value of the immediate roof increased instantly.