## 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
 \tag{11}\]  
According to formula (10), it can be concluded that  
\[\frac{\partial\Lambda}{\partial f(x)}=-\gamma,\;\;\frac{\partial\Lambda}{ \partial f^{\prime}(x)}=-\sigma_{c}(AB)^{\frac{1}{1-B}}B^{-1}f(x)^{\frac{B}{1-B}} \tag{12}\]  
Substituting Equation (12) into Equation (11) yields the body expression of the Euler equation as  
\[\sigma_{c}(1-B)^{-1}(AB)^{\frac{1}{1-B}}f^{\prime}(x)^{\frac{2B-1}{1-B}}f^{ \prime\prime}(x)-\gamma=0 \tag{13}\]  
Equation (13) is a nonlinear second-order differential equation with respect to the first integral of the equation, producing  
\[\sigma_{c}B^{-1}(AB)^{\frac{1}{1-B}}f^{\prime}(x)^{\frac{B}{1-B}}-\gamma x-C_{ 0}=0 \tag{14}\]  
where \(C\) is the integration constant. From Equation (14), the specific expression of the first derivative \(f(x)\) is  
\[f^{\prime}(x)=\Big{[}B\sigma_{c}^{-1},(AB)^{\frac{1}{1-B}}\Big{]}\;^{\frac{1- B}{B}}(\gamma x+C_{0})\;^{\frac{1-B}{B}} \tag{15}\]  
After integrating the above equation again, the final analytical expression of the collapse curve \(f(x)\) can be obtained:  
\[f(x)=k\Big{(}x+\gamma^{-1}C_{0}\Big{)}^{\frac{1}{B}}-h \tag{16}\]  
where \(k=A^{-\frac{1}{B}}\Big{(}\frac{\gamma}{c_{c}}\Big{)}^{\frac{1-B}{B}}\), \(h\) is also an integral constant, and in a geometric sense, \(h\) is the maximum height of the collapsed body since there are integral constants, \(C_{0}\) and \(h\), in \(f(x)\). For convenience of calculation, \(C_{0}\) should be found first. If \(\overrightarrow{n}\) and \(\overrightarrow{s}\) are the normal and tangent directions of the collapse curve f(x), respectively, \(\overrightarrow{n}=[cos\theta,\ -sin\theta]^{T}\) and \(\overrightarrow{s}=[-sin\theta,cos\theta]^{T}\), which can be expressed in the partial coordinate system \(\{n,s\}\) as  
\[\left\{ \begin{array}{l}\frac{\partial\tau_{n}}{\partial n}+\frac{\partial \tau_{n}}{\partial s}=\gamma sin\theta\\ \frac{\partial\tau_{n}}{\partial s}=\gamma cos\theta\end{array} \right. \tag{17}\]  
By substituting Formulas (6) and (16) into Equation (2), it can be concluded that  
\[\tau_{n}=\gamma x+C_{0} \tag{18}\]  
Then, the following can be introduced:  
\[\frac{\partial\tau_{n}}{\partial\overrightarrow{n}}=\
abla\tau_{n}\cdot \overrightarrow{n}=\gamma cos\theta \tag{19}\]  
where \(\
abla\tau_{n}=\left[\frac{\partial\tau_{n}}{\partial x},\frac{\partial\tau_ {n}}{\partial y}\right]^{T}\). Since the collapse body curve \(f(x)\) is symmetrical with respect to the \(y\)-axis, the shear stress at \(x=0\) and \(y=-h\) is zero, and then  
\[\tau_{xy}(x=0,y=-h)=0 \tag{20}\]  
Using the stress field coordinate transformation formula,  
\[\tau_{xy}=\tau_{n}cos2\theta-\frac{1}{2}\sigma_{n}sin\theta \tag{21}\]  
Since \(cot\theta=f^{\prime}(x)\),  
\[\left. \begin{array}{l}cos2\theta=\frac{f^{\prime}(x)^{2}-1}{f^{\prime}(x)^{ 2}+1},\ \ \frac{1}{2}sin2\theta=\frac{f^{\prime}(x)}{f^{\prime}(x)^{2}+1}\\ cot\theta|_{x=0}=f^{\prime}(x=0)=Qc_{0}\frac{1-h}{\theta}\end{array} \right. \tag{22}\]  
where \(Q=\left[B\sigma_{c}{}^{-1}(AB)^{\frac{-1}{B}}\right]^{\frac{1-B}{B}}\). Substitute Equations (15) and (22) into Equation (21), and then  
\[\left(Q^{2}c_{0}\frac{2(1-B)}{B}+1\right)^{-1}\left\{c_{0}\left[Q^{2}c_{0} \frac{2(1-B)}{B}\ -1\right]+\left[\sigma_{t}(ABQ)^{\frac{1}{1-B}}c_{0} \frac{1}{B}\right]Qc_{0}\frac{(1-B)}{B}\right\}=0\]  
Equations (26) and (27) can be substituted into Equation (20), and the principle of virtual work can be combined to obtain  
\[ \begin{array}{c}\psi[f(x),f^{\prime}(x),x]=\int_{0}^{L}\Lambda[f(x),f^{\prime}( x),x]vdx=0\\ \Rightarrow vL\left\{\alpha-\beta B(1+B)^{-1}L^{\frac{1}{B}}+P\right\}=0\\ \Rightarrow\alpha-\beta B(1+B)^{-1}L^{\frac{1}{B}}+P=0\end{array}