## 2 Mechanical Analysis Model for TSWs
### Mechanical Model  
According to previous research [9; 20; 21; 22; 23], the ratio of a plate's thickness to its minimum side length can be used to classify it as a thick plate (ratio > 1/5 to 1/8), a thin plate (ratio between 1/80 to 1/100 or 1/5 to 1/8), or a thin film (ratio < 1/80 to 1/100). For TSWs located between planned stopes and gosafs, the classical small-deflection bending theory for thin plates is appropriate. This is because the surrounding rock is brittle and exhibits limited deformation prior to failure [24]. Typically, the TSW thickness is much less than 1/10 of its minimum lateral dimension, which aligns with the thin plate assumptions in elastic mechanics [25; 26]. Therefore, thin plate theory can be used to analyze TSW stability. To simplify the analysis, the following assumptions are made:  
1. The TSW is considered a continuous, homogeneous material after discounting the in situ rock strength. 2. The TSW surface is smooth and uniform in width, which is represented by the average actual width in the model. Dynamic disturbances from blasting are neglected. Figure 1: The comprehensive 3D model of the mine, the project related to the TSW stability study and the 3D analysis model under different conditions: (**a**) comprehensive mine 3D model, (**b**) actual residual ore recovery project, (**c**) gosafs and planned stopes completely overlap, (**d**) gosafs and planned stopes do not entirely overlap. 3. Displacements induced by loading are small, and boundary displacements are minimal due to the relatively large TSW size. 4. Once plastic damage initiates, the load-bearing capacity of the TSW drops significantly, and even minor stress variations can result in large lateral deflections. This assumption is valid in hard rock mining environments. 5. There is negligible plastic deformation at the contact interfaces between the TSW and the surrounding rock, consistent with conditions in hard rock mines. 6. Stress analysis conforms to the standard sign conventions of elastic mechanics. Based on in situ geostress test results from the mine, the central section of the +330 level exhibits a maximum principal stress \(\sigma_{x}\) oriented vertically, a middle principal stress \(\sigma_{x}\) that is nearly horizontal and perpendicular to the ore body's strike direction, and a minimum principal stress \(\sigma_{y}\) that is nearly horizontal and aligned with the strike direction. Accordingly, this study adopts a lateral stress factor \(\lambda=0.8\) to represent the in situ stress condition. Following the excavation of the planned slopes, stress redistribution occurs. Stress concentration and release are observed \(\sigma_{x}\) in the z-direction, while stress release also takes place \(\sigma_{y}\) in the y-direction. In contrast, lateral stress \(\sigma_{x}\) in the x-direction remains concentrated along the inter-chamber pillars between adjacent goals. Figure 2 shows the mechanical model of the TSW, where \(\sigma_{y}\) and \(\sigma_{x}\) represent the longitudinal and transverse loads on the TSW, respectively. The lateral stresses at the retained position of the inter-chamber pillars are applied to the localized TSW surfaces with a maximum lateral load of \(q_{0}\). Since both the upper/lower and left/right boundaries of the TSW are composed of metamorphic sandstone, all edges are considered fixed and supported--i.e., no displacement or rotation is allowed. This assumption simplifies the analysis to a rectangular thin plate under combined transverse and longitudinal loading, along with localized lateral loads. The TSW is made of metamorphic sandstone, a hard-brittle rock that does not undergo significant deformation before failure. This suggests that the use of the mechanical model to compute the stress state of TSWs is consistent with the basic assumptions. Preliminary  
Figure 2: Mechanical model of the TSW under combined loading. The longitudinal and transverse loads acting on the middle of a thin plate, and the local lateral compressive stress is applied to the local surface.