## 6 Conclusions  
(1) Through mechanical modeling, the force exerted by the main roof of the No. 2-1 coal seam on underlying coal and rock strata and the fracture position of the main roof were analyzed. The results revealed that the maximum force applied by the main roof on underlying strata was 11.68 MPa, gradually decreasing toward the deep coal mass and reducing to in situ stress at approximately 6.2 m. The fracture position of the main roof occurred at a distance of approximately 0.5 m from the goat edge.  
Figure 19: Roof Failure Characteristics under Different Immediate Roof Elastic Modulus Conditions.  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Immediate Roof Elastic** & & & & & \\
**Modulus/10\({}^{3}\) MPa** & 3 & 4 & 5 & 6 & 7 \\ \hline
**Fracture Position/m** & 0 & +1.4 & +1.4 & +1.5 & +1.5 \\ \hline \hline \end{tabular}
\end{table}
Table 5: Failure Locations of the Main Roof under Different Immediate Roof Elastic Modulus Conditions.  
(2) The influence of parameters on the maximum force exerted by the main roof on underlying coal and rock strata decreases in the following order: immediate roof elastic modulus, coal seam thickness, immediate roof thickness, and coal seam elastic modulus. The maximum compressive stress decreases exponentially with increasing coal seam thickness and linearly with increasing immediate roof thickness, while increasing linearly with coal seam and immediate roof elastic moduli.  
(3) We found that the main roof is most prone to primary failure near the goat edge at approximately 0.3-0.6 m. The influence of parameters on maximum bending moment decreases in the following order: immediate roof elastic modulus, coal seam thickness, immediate roof thickness, and coal seam elastic modulus. The maximum bending moment decreases linearly with increasing coal seam and immediate roof thicknesses, while increasing linearly with coal seam elastic modulus and logarithmically with immediate roof elastic modulus.  
(4) Under coal seam thicknesses of 3 m and 4 m, the surrounding rock of gob-side entry retaining can generally maintain stability. However, under thicknesses of 5 m to 7 m, maintaining stability becomes significantly more challenging. Similarly, under immediate roof thicknesses of 4 m to 8 m, stability is generally achievable, whereas under thicknesses of 10 m and 12 m, stability is difficult to maintain. Additionally, when coal seam elastic modulus ranges from 0.5 to 2.5 \(\times\) 10\({}^{3}\) MPa and immediate roof elastic modulus ranges from 3 to 7 \(\times\) 10\({}^{3}\) MPa, the surrounding rock of gob-side entry retaining can generally remain stable.  
Conceptualization, G.H. and Y.Y.; methodology, G.H. and Y.Y.; software, G.H.; validation, G.H., S.L. and K.Z.; formal analysis, G.H.; investigation, G.H.; resources, Y.Y.; data curation, G.H.; writing--original draft preparation, G.H.; writing--review and editing, G.H.; visualization, G.H.; supervision, Y.Y.; project administration, Y.Y.; funding acquisition, Y.Y. All authors have read and approved 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 (J2019055).  
Not applicable.  
The data used to support the findings of this study are included within the article.  
The authors declare no conflicts of interest.