## 5 Conclusions  
Incorporating the stratigraphic formation and structural configuration of the shaft and roadway, this study utilized FLAC\({}^{\text{3D}}\) to construct a mechanical coupling model integrating geological strata, the shaft, and the horsehead roadway. Roof subsidence displacement of the horsehead roadway was systematically set at 0.5 m, 1.0 m, and 1.5 m to assess stress response and failure dynamics. The analysis, based on Mises stress and plastic zone distribution, elucidated stress behavior and failure characteristics of the shaft structure. Examination of stress and displacement fields revealed mechanisms driving shaft instability. The findings indicate:  
1. The pulling effect on the overlying rock mass and shaft increases with greater roof subsidence displacement. Vertical displacement diminishes from the junction with the horsehead roadway to the model's apex, with Z-direction displacement at 100 m above the roadway being about one-fifth of the roof's MSD.
2. As roof subsidence displacement increases, the disturbance range of the overlying rock, shaft failure extent, and tensile failure units rise. With MSD transitions from 0.5 m to 1.5 m, the shaft failure range characterized by Mises equivalent stress expands. Tensile failure evolves from scattered occurrences to large-scale uniform distributions.
3. Shaft failure results from instability and deformation in the horsehead and connecting roadways, compounded by geological conditions. Situated in soft rock or fractured coal-bearing strata, excavation-induced disturbances lead to bending and deformation of thin bedrock, affecting the interface with loose layers and resulting in shaft rupture.
4. To prevent shaft damage from mining subsidence and water drainage, establishing protective coal pillars around the shaft and enhancing support strength at the base of the shaft parking area is recommended.  
Figure 16: Torsional deformation of rock layers along the height direction (cm).  
**Author Contributions:** J.R.: investigation, conceptualization, writing--review and editing, supervision, data curation, writing--original draft. B.W.: software, funding acquisition, formal analysis, data curation, writing--review and editing. All authors have read and agreed to the published version of the manuscript.  
**Funding:** Financial support for this work, provided by the National Natural Science Foundation of China (No. 52404108), the Engineering Research Center of the Ministry of Education for Underground Mining Engineering (No. JYBGCZX2022103), the State key Laboratory of Mining Disaster Prevention and Control (Shandong University of Science and Technology), Ministry of Education (No. JMDPC202403) and the Research Activity Funding Project for Postdoctoral Researchers in Anhui Province (No. 2023B726), is gratefully acknowledged.  
**Institutional Review Board Statement:** Not applicable.  
**Informed Consent Statement:** Not applicable.  
**Data Availability Statement:** The original contributions presented in the study are included in the article. Further inquiries can be directed to the corresponding author.  
**Conflicts of Interest:** The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.