## 3 Mechanical Modeling of the Permanent Thin Wall
### Mechanical Model of the Thin Wall When Using Two Backfill Materials  
In practice, pillar slopes are mined under the condition that both sides are backfilled at the first mining step, which uses the cemented tailings backfill material, which has sufficient strength to ensure that the backfill will not fail during the mining process of pillar slopes. After the recovery of pillar slopes, to reduce costs, full-tailings backfill instead of cemented tailings materials are used for the backfill. This results in different lateral loads being applied to the thin walls at different locations. Assuming that the lateral loads applied to the wall by cemented and un-cemented backfill materials are \(q_{1}\) and \(q_{2}\), respectively, the two walls of the same barrier pillar slope will have different deflections, as shown in Figure 2(a). In this situation, the deformations of thin walls in adjacent barrier pillar slopes will affect each other, so the three-dimensional problem cannot be simplified into a two-dimensional plane problem. Since the thin wall deformation of adjacent barrier pillar slopes is different, the mutual influence is the root cause of the inability to reduce the three-dimensional problem to a two-dimensional one. Then, if it is guaranteed that the thin wall deformation of adjacent barrier pillar slopes is the same, the mutual influence between them will disappear. At  
Figure 3: Mechanical model of the thin wall when using two backfill materials. this point, they can be regarded as two mutually independent planes, each calculated. As shown in Equation (8), the wall deflection is a function of the lateral load and the wall width. The width of the thin wall can be adjusted according to the lateral loads if it is desired to keep the deflection identical to ensure no interference of wall deformations between adjacent barrier pillar slopes. Based on Equation (8), this situation occurs when the values of wall width satisfy the following relationship:  
\[\frac{q_{1}}{q_{2}}=\frac{b_{1}^{3}}{b_{2}^{3}} \tag{9}\]  
As the backfill material bears the roof loads under lateral pressure support from the thin wall, the lateral pressure \(q\) and the roof pressure \(p\) satisfy the following relationship according to the Mohr-Coulomb criterion:  
\[p=\sigma_{\text{cb}}+\frac{1+\sin\varphi_{b}}{1-\sin\varphi_{b}}q \tag{10}\]  
where \(\sigma_{\text{cb}}\) is the uniaxial compressive strength and \(\varphi_{b}\) is the internal friction angle of the backfill material. By combining Equations (9) and (10), to achieve the same wall deflection, the width of the thin wall should satisfy the following:  
\[\frac{b_{1}}{b_{2}}=\sqrt[3]{\frac{(p-\sigma_{\text{cb1}})}{(p-\sigma_{\text {cb2}})}}\cdot\sqrt[3]{\frac{(1-\sin\varphi_{\text{b1}})(1+\sin\varphi_{\text {b2}})}{(1+\sin\varphi_{\text{b1}})(1-\sin\varphi_{\text{b2}})}} \tag{11}\]  
where \(\sigma_{\text{cb1}}\) and \(\sigma_{\text{cb2}}\) are the uniaxial compressive strength values of the backfill materials in room stopes and pillar slopes, respectively; \(\varphi_{\text{b1}}\) and \(\varphi_{\text{b2}}\) are the internal friction angles of the backfill materials in room stopes and pillar slopes, respectively. From Equation (11), one can observe that the thin wall width is mainly influenced by the roof load, uniaxial compressive strength, and the internal friction angle of the backfill materials. The wall width needs to increase as the roof load increases, the uniaxial compressive strength decreases, or the internal friction angle decreases. Without considering the influence of internal friction angles, when the roof load is much larger than the uniaxial compressive strength of both backfill materials and the uniaxial compressive strength of both backfill materials is close to each other, the width ratio of the two walls would be close to 1. When the roof load is close to the uniaxial compressive strength of one backfill material, but very different from the uniaxial compressive strengths of the other, the width ratio of the two walls would be far away from 1.