## 4 Conclusions  
1. Based on the actual mining situation, assuming the main roof is fixed by the upper and lower elastic rock layers, an elastic foundation boundary model is established for this mechanical configuration. A section behind the working face is selected as the research object. A mechanical model of the action of the roof on the underlying coal and rock strata after coal seam mining is established. Through analysis, it is concluded that the fracture position of the main roof is at a location 0.5 m away from the edge of the goat.
2. During the entire process of advancing the working face, the time evolution law of vertical displacement exhibits three distinct configurations: evolving from an early-stage \"rectangle-trapezoid\" distribution, transitioning to a quasi-hyperbolic pattern, and finally forming a trapezoidal field upon reaching the surface influence threshold, which tends to stabilize when the advancement reaches 250 m.
3. Surface subsidence characteristics initiate when the working face advances to 80 m, consistent with field observations; the subsidence basin develops smoothly into a trapezoidal shape, with maximum displacement of 4.57 m and an influence radius approximately twice the mining height.
4. This study demonstrates that in shallowly buried weak rock strata scenarios, surface impacts manifest with accelerated temporal characteristics owing to enhanced strata-ground coupling. The rapid goat backfilling process effectively reduces peak subsidence magnitudes through timely stress redistribution. Roof fracturing initiates overburden delamination and collapse, establishing direct correlations between surface settlement patterns and the spatial extent of overlying strata failure along the advancement direction. Numerical simulation of progressive roof fracturing mechanisms enables predictive modeling of subsidence trough morphology, providing critical guidance for surface infrastructure protection strategies. Analysis of roof bending moment distributions under varying advancement step distances facilitates optimization of mining parameters, achieving equilibrium between operational safety and extraction efficiency. Notable limitations arise from the PFC model's simplification of material heterogeneity--actual geological sequences containing interbedded weak layers may significantly alter fracture propagation pathways. Future investigations should incorporate stochastic mechanical parameter distributions and validate nu merical predictions through field monitoring data integration, particularly regarding time-dependent deformation characteristics and fracture network evolution patterns.  
Conceptualization, Y.Y. and X.F.; software, G.H. and X.F.; validation, X.F., G.H., S.L. and K.Z.; formal analysis, Y.Y. and X.F.; data curation, Y.Y., X.F., G.H., S.L. and K.Z.; writing--original draft preparation, G.H., Y.Y. and X.F.; writing--review and editing, Y.Y. and X.F.; funding acquisition, Y.Y. All authors have read and agreed to the published version of the manuscript.  
This work was supported by the National Natural Science Foundation of China (Grant Nos. 51404167, 42302202), the General Project of the Applied Basic Research Program of Shanxi Province (202303021211057), and the Teaching Reform and Innovation Project of Higher Education Institutions in Shanxi Province (2019055).  
Not applicable.  
The data used to support the findings of this study are included within this article.  
The authors declare no conflicts of interest.