## 3 Analysis of the Collapse Mechanism of the Surrounding Rock of Large-Span Caverns
### Solution to the Collapse Curve of Rock Surrounding a Large-Span Cavern Vault  
According to the strain localization theory, the velocity discontinuity surface is considered a thin deformation layer with a thickness \(t\), so the normal velocity and tangential velocity at any point on the velocity discontinuity surface can be determined using the first derivative \(f(x)\) of the velocity discontinuity surface function \(f(x)\). Then, with triangular transformation, \(\dot{\varepsilon}_{n}\) and \(\dot{\gamma}_{n}\) can be expressed as  
\[\left\{ \begin{array}{l}\dot{\varepsilon}_{n}=\frac{v}{t}sin\theta=\frac{v}{t }\frac{1}{\sqrt{1+f^{\prime}(x)^{2}}}\\ \dot{\gamma}_{n}=-\frac{v}{t}cos\theta=-\frac{v}{t}\frac{f^{\prime}(x)}{\sqrt {1+f^{\prime}(x)^{2}}}\end{array} \right. \tag{5}\]  
The expression of normal stress on the speed discontinuity surface can be obtained by combining Equations (4) and (5). \[\sigma_{n}=-\sigma_{t}\ +\sigma_{c}\big{[}ABf^{\prime}(x)\big{]}^{\frac{1}{1-B}} \tag{6}\]  
According to [PERSON]'s [3] research, the internal energy dissipation rate \(D\) at any point on the velocity discontinuity surface is equal to the sum of the tangential stress dissipation power and the normal phase stress dissipation power:  
\[D=\sigma_{n}\dot{\varepsilon}_{n}\ +\tau\dot{\gamma}_{n}\ =\ \{-\sigma_{t}+\{\sigma_{c}\big{[}ABf^{ \prime}(x)\big{]}^{\frac{1}{1-B}}\Big{(}1-B^{-1}\Big{)}\}\frac{v}{t\sqrt{1\ +\ f(x)^{2}}} \tag{7}\]  
The power expression of gravity per unit length of a collapse body is  
\[W_{D}=\gamma[f(x)-c(x)]v \tag{8}\]  
According to the virtual work equation, the energy dissipation objective function composed of the independent variables of \(f(x)\) can be obtained using the upper bound analysis method. However, \(f(x)\) is itself a function, so the energy dissipation objective function is universal. In order to obtain the expression of \(f(x)\), according to the principle of variational method, the objective function \(\psi\) is introduced to represent the difference between the internal energy dissipation rate and the external force work power, and \(f(x)\) is obtained by means of the extreme value of the objective function. Since the surrounding  
Figure 1: Caving mechanism diagram of a large-span cavern and surrounding rock. rock collapse body is symmetric about the y-axis, the right half of the y-axis is taken for calculation, so the total work equation is  
\[ \begin{array}{l}\psi[f(x),f^{\prime}(x),x]=\int_{0}^{s}Dtds-\int_{0}^{L}W_{ \gamma}dx\gamma\\ =\int_{0}^{L}(Dt\sqrt{1+f^{\prime}(x)^{2}}-W_{\gamma})dx\\ =\int_{0}^{L}\Lambda[f(x),\;\;f^{\prime}(x),x]Vdx\end{array}  \tag{9}\]  
where \(ds=\sqrt{1+f^{\prime}(x)^{2}}dx\) is the length of any tiny element on the line \(f(x)\), and \(s\) is the total length of the right half of the collapse curve \(f(x)\). \(\Lambda[f(x),\,f^{\prime}(x),x]\) is a universal function of \(f(x)\), and its specific expression is  
\[\Lambda\big{[}f(x),f^{\prime}(x),x\big{]}=\sigma_{c}\big{[}ABf^{\prime}(x) \big{]}\;^{\frac{1}{1-B}}\bigg{(}1-\frac{1}{B}\bigg{)}-\sigma_{t}\;-\gamma[f( x)-c(x)] \tag{10}\]  
The extreme value problem of the universal function can be transformed for the problem of solving the Euler equation, satisfying the boundary conditions, and then the Euler equation expression of \(\Lambda\) is as follows:  
\[ \begin{array}{l}\delta\Lambda[f(x),f^{\prime}(x),x]=0\\ \Rightarrow\frac{\partial\Lambda}{\partial f(x)}-\frac{d}{dx}\Big{[}\frac{ \partial\Lambda}{\partial f^{\prime}(x)}\Big{]}=0\end{array}