## 1 Introduction  
In underground mining, the stope sidewall plays a crucial role in ensuring operational stability and safety [1]. When recovering shallow, thin residual ore bodies, mining activities are influenced by the secondary stress fields generated by nearby gos. This necessitates retaining a certain sidewall thickness to maintain stope stability during residual ore recovery [2]. To ensure safety, passive methods are typically employed to increase the sidewall thickness. However, this approach significantly reduces resource recovery rates [3, 4]. Therefore, determining the critical sidewall thickness is essential for minimizing ore loss and optimizing resource extraction [5]. In recent years, extensive research has been conducted to determine the safe thickness of mine rock structures, yielding important insights. [PERSON] et al. [6] analyzed the damage behavior of thick roofs with varying thicknesses using field measurements, physical modeling, and numerical simulations. Their findings provide theoretical guidance for maintaining roof stability. [PERSON] et al. [7] proposed a comprehensive approach for determining the blasting limit thickness of stope roofs based on mechanical modeling and numerical analysis. [PERSON] et al. [8] utilized UDEC4.0 numerical simulation software to assess the stability of slopes with varying magnesite roof characteristics. [PERSON] et al. [9] established the relationship between roof stability and slope structural parameters using elastic mechanics and proposed optimal stope designs based on actual mining conditions. [PERSON] et al. [10,11] applied elastic mechanics to evaluate the stress distribution in mine walls under combined trapezoidal lateral and vertical loads from the filling body. They developed a comprehensive stability evaluation index and a corresponding calculation formula, concluding that wall width is the primary factor influencing stability. [PERSON] et al. [12] employed a coupled modeling approach integrating 3 Dmine, Rhino 3D, and FLAC 3D to analyze stope stability around collapsed zones. Their study indicated that collapse-induced effects predominantly impact the roof. [PERSON] et al. [13] developed a mechanical model for flexural damage in thin mine pillars using elastic thin plate theory, establishing a relationship between buckling stress and the pillar's aspect ratio. [PERSON] et al. [14,15] analyzed roof stress distribution under non-symmetrical loading using fixed beam theory. They formulated a mathematical model relating safe roof thickness to the overlap ratio, width of mine pillars, and goat span. [PERSON] et al. [16] applied catastrophe theory and rheological theory to assess the stability and failure behavior of a three-dimensional roof-pillar system. They derived mathematical criteria for sudden failure and energy release under varying conditions. Further, [PERSON] et al. [17] developed an analytical model for mine wall-filling body structures based on elastic thin plate theory and analyzed the influence of filling body properties (e.g., elastic modulus, cohesion, structural parameters) on ultimate stress. These studies demonstrate that numerical simulations, physical modeling, and elastic-plastic theory are widely used to evaluate the stability and safe thickness of mine rock structures. While significant progress has been made by many scholars in the theoretical computation of sidewall stability, these theoretical methods often fail to account for the influence of boundary conditions and boundary stresses on sidewall stability. Additionally, they rarely consider the impact of uneven loading on sidewall stability. As a result, previous mechanical models are not applicable to the calculation of sidewall stability under high in situ stress and complex loading conditions. Therefore, this study proposes a method to calculate the deflection, bending moment, internal forces, and critical thickness of the thickened structural wall (TSW) under fixed support boundary conditions and various load forms.