## 4 Calculation of the Progressive Collapse Process of the Surrounding Rock of a Large-Span Cavern
### Analytical Solution of the Progressive Collapse Curve of a Large-Span Cavern's Surrounding Rock  
According to the content of the previous section, the first collapse curve equation, width, and height expressions are  
\[\left\{\begin{array}{l}f_{1}(x)=k_{1}x^{\frac{1}{B}}-h_{1}\\ k_{1}=A^{-\frac{1}{B}}\left(\frac{\gamma}{\sigma_{c1}}\right)^{\frac{1-B}{B}} \end{array}\right. \tag{37}\]  
\[\left\{\begin{array}{l}L_{1}=AB^{-B}\left(1+B\right)^{\gamma-1}\sigma_{c1}^ {1-B}(\sigma_{t1}-P)^{B}\\ P_{1}=\gamma\frac{1}{2C}R^{2}\left[\alpha r\sin\frac{L}{R}-\frac{L}{R}\sqrt{1 -\left(\frac{L}{R}\right)^{2}}\right]\end{array}\right. \tag{38}\]  
\[h_{1}=(\gamma B)^{-1}(1+B)(\sigma_{t1}-P) \tag{39}\]  
In Equations (37)-(39),\(f_{1}(x)\) uses the coordinate system \(o_{1}\), where \(\sigma_{c1}\) and \(\sigma_{t1}\) are the uniaxial compressive strength and the tensile strength of the rock mass during the first fall, respectively.  
With the gradual weakening of the surrounding rock parameters, the second collapse curve equation \(f_{2}(x)\) is generated by taking the first collapse curve as the wheel line, which can be expressed as  
\[\left\{\begin{array}{l}f_{2}(x\ )=k_{2}x^{\frac{1}{B}}-h_{2}\\ k_{2}=A^{-\frac{1}{B}}\left(\frac{\gamma}{\sigma_{c2}}\right)^{\frac{1-B}{B}} \end{array}\right. \tag{40}\]  
where \(f_{2}(x)\) is the uniaxial compressive strength of the first square time rock mass using the coordinate system \(o_{2}\).  
After converting the coordinate system of the first collapse curve \(f_{1}(x)\) into \(o_{2}\), the expression of the first collapse curve is  
\[f_{12}(x)=k_{1}x^{\frac{1}{B}}-h_{1}+h_{1}-k_{1}L_{2}^{\frac{1}{B}} \tag{41}\]  
Figure 2: Solution of the progressive collapse curve of the surrounding rock of a large cavern.  
The expression of the objective function \(\psi_{2}\) in the second collapse curve is  
\[\left\{\begin{array}{l}\psi_{2}\ =vL_{2}\Big{[}(rh_{2}\ -\sigma_{t2}\ )-\gamma\ (1+B)^{-1}h_{2}+p_{1}\Big{]}\\ P_{1}=\gamma(1+B)^{-1}k_{1}L_{2}^{\frac{1}{2}}\end{array}\right. \tag{42}\]  
where \(P_{1}\) is the generalized pressure, i.e.,  
\[P_{1}=\frac{1}{L_{2}}\gamma\int_{0}^{1}f_{12}(x)dx \tag{43}\]  
Using the virtual power principle \(\psi_{2}=0\), the following can be obtained:  
\[(rh_{2}-\sigma_{t2})-\gamma(1+B)^{-1}h_{2}+p_{1}=0 \tag{44}\]  
The expressions for the height \(h_{2}\) and width \(L_{2}\) of the second collapse body are  
\[h_{2}=\frac{(1+B)\sigma_{t2}}{\gamma[B+k_{1}/k_{2}]}=\frac{(1+B)\sigma_{t2}}{ \gamma[B+\sigma_{c1}/\sigma_{c2}]^{\frac{B-1}{B}}} \tag{45}\]  
\[L_{2}=\frac{(1+B)\sigma_{t2}}{\gamma[Bk_{2}\ +k_{1}]}=\frac{A[(1+B)\sigma_{t2} ]^{B}(\sigma_{t2})^{1-B}}{\gamma\Big{[}B+(\sigma_{c1}/\sigma_{c2})^{\frac{B-1 }{B}}\Big{]}^{B}} \tag{46}\]  
It can be seen from Equations (47) and (48) that the parameters of the second collapse are only related to a/b, so the \(i\)-th collapse range is only related to the ratio of \(\sigma_{c(i-1)}/\sigma_{ci}(i\ \geq\ 2)\) of the uniaxial resistance of the rock mass. If \(n-1\) falls are carried out on the basis of the first collapse, and the uniaxial compressive strength of the rock mass is uniformly reduced, and the compressive strength of the final surrounding rock is stable at \(\sigma_{cn}\), the uniaxial compressive strength reduction value of the rock mass is  
\[\Delta\sigma_{ci}=\frac{\sigma_{c1}-\sigma_{cn}}{n-1} \tag{47}\]  
The equation for the \(i\)-th collapse curve in the coordinate system \(o\) is expressed as  
\[f_{i}(x)=k_{i}x^{\frac{1}{B}}-h_{2} \tag{48}\]  
Substituting Equation (47) into Equations (45) and (46) yields the curve equation for the nth collapse. Finally, by using coordinate system transformation, the progressive collapse process of the surrounding rock of the cave can be obtained by plotting the n-fold collapse curve onto the coordinate system \(o_{1}\).