## 1 Introduction  
Cool mine shafts serve as a critical conduit for both surface and subterranean transport within the mining industry, playing an essential role in ensuring the operational safety of mines [1, 2, 3, 4]. In regions such as Huanghuai and Northeast China, vertical shafts are predominantly utilized due to the specific geological conditions of coal seam occurrences [5, 6, 7]. Since the significant damage to nine shaft walls in the Huaibei mining area in 1987, over a hundred vertical shafts in China have experienced varying degrees of structural impairment, representing a substantial hazard to mining safety and incurring considerable economic costs [8, 9]. A primary factor contributing to numerous instances of shaft damage is the drainage induced by coal extraction activities; this process triggers secondary consolidation settlement of the soil layers adjacent to the shaft, which in turn generates vertical additional forces and leads to tensile damage in the shaft structure [10, 11, 12]. Understanding the impact of bedrock settlement on the stress distribution and deformation of the shaft is imperative for devising strategies to prevent and mitigate incidents of shaft fracturing [13; 14; 15; 16]. The phenomenon of shaft failure due to overlying rock settlement has garnered the intense focus of researchers, yielding a wealth of scholarly findings. In theoretical research, significant advancements have been made in understanding shaft behavior in complex geological settings. [PERSON] et al. [17] and [PERSON] and [PERSON] [18] employed fuzzy inversion techniques to assess negative friction forces in hydrophobic settlement formations, elucidating stress mechanics during alluvial layer settlement. Building on this foundation, [PERSON] et al. [19] and [PERSON] et al. [20] developed mechanical models to evaluate shaft stability under porous seepage conditions, considering factors such as inclination angle, azimuth, and weak plane orientation. [PERSON] and [PERSON] [21] proposed a method to determine the equivalent density of shaft collapse pressure in shale formations. Following this, [PERSON] et al. [22], based on [PERSON] and [PERSON] [21], introduced an analytical model using stability theory to calculate collapse pressure. They applied elastic-plastic mechanics to resolve stresses and displacements around the shaft, formulating an equilibrium path that defines the relationship between internal pressure and radial displacement. [PERSON] and [PERSON] [23] advanced a sensitivity analysis technique to quantify the influence of individual input parameters on shaft stability uncertainty, highlighting each parameter's singular impact amidst prevailing uncertainties. [PERSON] [24] concretized theoretical research by investigating internal force distribution and stress patterns of shaft walls under spatially axisymmetric conditions. [PERSON] et al. [25] used Midas GTX NX to construct a numerical model accurately capturing pre- and post-excavation stress and deformation in surrounding rock masses, aligning with engineering practice. In the experimental research domain, [PERSON] et al. [26] conducted model experiments on a composite shaft structure comprising double-layer steel plates and high-strength concrete. Their study elucidated the vertical stress, deformation, and strength characteristics under combined axial loads and lateral pressures. [PERSON] et al. [27] developed a novel approach to couple the stress field within shaft formations and predict stability for KT-II type unstable geologies using principal factors of shaft collapse and empirical data. [PERSON] et al. [9] employed the Mohr-Coulomb failure criterion and a multi-tiered confidence modeling framework to address three geo-stress scenarios--normal fault, strike-slip fault, and reverse fault. They crafted a visual analytical model for assessing shaft stability across various orientations and inclinations.