## 3 Mechanical Modeling of the Permanent Thin Wall
### Optimization of the Width for the Thin Wall  
The roof of underground excavations will cave when it loses its support. Engineering practices have indicated [27; 28] that the roof will reach a stable state after an arch structure is formed by caving. The arch structure can help re-distribute the ground stresses of the roof strata to unexcavated areas on both sides of the excavation. This stable arch structure formed by the phenomenon of stress redistribution is known as a natural arch.  
For the mining operation described in this paper, backfill materials are used to replace the original rocks to support the roof strata. However, as the strength of the backfill materials is far lower than that of the original rock, they cannot completely replace the original rock to bear the stresses from the roof strata, so stress redistribution will inevitably occur. Based on the characteristics of a natural arch, the roof loads will gradually be re-distributed to rocks on both sides of the excavation if the bearing capacity of the substructure is insufficient. This is the case for the mining operation described, where the backfill will only bear the weight of caved loose rocks, as shown in Figure 4.  
Based on the theory related to natural arches and the findings of our previous studies [29; 30], the radius \(R\) of the rock mass in the disturbed region surrounding the mine with risk of falling can be determined as:  
\[R=\left[\frac{(\rho_{\mathrm{r}}gH+c_{\mathrm{r}}\cot\varphi_{\mathrm{r}})(1- \sin\varphi_{\mathrm{r}})}{c_{\mathrm{r}}\cot\varphi_{\mathrm{r}}}\right]^{ \frac{1-\sin\varphi}{2\sin\varphi}}\sqrt{\left(\frac{h}{2}\right)^{2}+\left( \frac{na}{2}\right)^{2}} \tag{12}\]  
where \(\rho_{\mathrm{r}}\) is the density of the roof rock, \(g\) is the gravity acceleration (\(9.8\) m/s\({}^{2}\)), \(H\) is the depth of the stope from the ground, \(h\) is the height of the mine wall, \(c_{\mathrm{r}}\) is the cohesion of the roof rock, \(\varphi_{\mathrm{r}}\) is the internal friction angle of the roof rock and \(n\) is the number of stopes along the direction of the stope width.  
The gravitational pressure of the caved rocks, namely, the roof load \(p\), applied to the backfill is  
\[p=\rho_{\mathrm{r}}g\left(R-\frac{h}{2}\right) \tag{13}\]  
According to the mine site test data, the mechanical parameters of different materials are shown in Table 1. Other required parameters are \(H=720\) m, \(n=19\), \(h=25\) m, and \(a=18\) m.  
To ensure the stability of the thin wall, it is required that both the maximum compressive stress and the maximum tensile stress within the mine wall are less than their corresponding strength values. From Equation (5), \(M(x)\) is a quadratic function whose maximum appears at \(x=h/2\). When \(x=h/2\), by combining Equations (2), (3) and (5), the stress inside the thin wall can be obtained:  
\[\sigma=-\frac{qh^{2}}{2 ab^{3}}y,-\frac{b}{2}\leq y\leq\frac{b}{2} \tag{14}\]  
Figure 4: Schematic diagram of load distribution.  
From this equation, the maximum compressive stress in the wall appears at \(y=b/2\), and the maximum tensile stress at \(y=-b/2\). From Equations (10), (12) and (13), when \(b_{1}<1.23\) m or \(b_{2}<3.94\) m, the tensile stress inside the wall is greater than the tensile strength, and therefore tensile damage may occur. From Equation (11), the deflections of the wall with different widths are the same when \(b_{1}\):\(b_{2}=0.63\). In this case, using \(b_{1}=2.48\) m and \(b_{2}=3.94\) m can ensure that the deflections of both walls are the same, and the wall will not be damaged as it has not exceeded its own ultimate stresses under the lateral loading of the backfill.