## 3 Computation of Stand-Up Time of Bearing Structure
### Protective Roof Failure Process
#### 3.2.4 Third Stage  
At this time, both the long side and the short side of the protective roof have been damaged, and the constraint condition of the contact surface between the protective roof and the surrounding rock has been changed into four hinged sides. The overall damage of the protective roof will further occur and its boundary condition can be expressed as:  
\[\left\{\begin{array}{l}\omega|_{x=\pm m}=0,\;\omega|_{y=\pm n}=0\\ \frac{\partial^{2}\omega}{\partial x^{2}}|_{x=\pm m}=0,\;\frac{\partial^{2} \omega}{\partial y^{2}}|_{y=\pm n}=0\end{array}\right. \tag{18}\]  
At this stage, the deflection distribution function of the protective roof can be approximately expressed as:  
\[f(x,y)=\cos\frac{\pi x}{2m}\cos\frac{\pi y}{2n} \tag{19}\]  
At this time, the boundary conditions of the four sides of the fixed support are all broken, the stress is further transferred to the center of the protective roof and the failure of the protective roof continues to extend which is manifested as the rock collapse of the protective roof. Finally, the stress state of the protective roof reaches the limit state.  
The bending moment of any section of the protective roof can be obtained by differentiating the Equation (19) of the deflection distribution function of the protective roof, andthe maximum stress value \(\sigma_{\text{max}}\) on the protective roof can be calculated after taking the extreme value. Then, the maximum stress \(\sigma_{\text{max}}\) on the protective roof can be expressed as  
\[\left\{\begin{array}{l}\sigma_{\text{xmax}}=\frac{6|M_{\text{x}}|_{\text{max} }}{h^{2}}=\frac{3\pi^{2}D}{2h^{2}}(\frac{1}{m^{2}}+\frac{\mu}{n^{2}})\omega_{2 }(t_{3})\leq[\sigma_{\text{t}}]\\ \sigma_{\text{ymax}}=\frac{6|M_{\text{y}}|_{\text{max}}}{h^{2}}=\frac{3\pi^{2} D}{2h^{2}}(\frac{h}{m^{2}}+\frac{1}{n^{2}})\omega_{2}(t_{3})\leq[\sigma_{ \text{t}}]\end{array}\right. \tag{20}\]  
In combination with (8) and (20), the stand-up time \(t_{3}\) of the bearing structure in the third stage is:  
\[t_{3}=\omega^{-1}(\frac{[\sigma_{5}]2m^{2}n^{2}h^{2}}{3\pi^{2}D(\mu n^{2}+m^{ 2})}) \tag{21}\]  
Similarly, according to the Formula (24) for calculating the stand-up time of the third stage, it can be seen that the influencing factors of the stand-up time of the bearing structure are the same as \(t_{1}\) and \(t_{2}\) (Figure 6).  
In conclusion, when the bearing structure in the third stage reaches its limit state, a large area of fracture will occur on the protective roof, resulting in multiple lumpiness bodies, overall dismemberment and destruction. Large area collapse may be formed at any time. At this time, it can be considered that the bearing structure has lost its bearing capacity and the goaf will be in a dangerous state.  
Based on the above Equations (13), (17) and (21), the stand-up time \(T\) of bearing structure can be expressed as:  
\[T=t_{1}+t_{2}+t_{3} \tag{22}\]  
For the goaf bearing structure of a specific bauxite mine, the composition of the overlying rock layer is determined by the physical properties of each layer of rock and soil. Therefore, when analyzing the stand-up time factors of the goaf bearing structure, the influencing factors such as the thickness of the protective roof, the coefficient of pillar uniform surface force, the span of the goaf and the thickness of the overlying rock layer are mainly selected for analysis.  
Figure 6: Algorithm flow chart of stand-up time \(T\).