## 5 Conclusions  
1. The load above a THR is treated as a uniformly distributed load. A mechanical model of the bearing capacity of a THR in a UMC was built based on the bending theories for beams with single generalized displacement and the elastic foundation beam theory. We further derived mechanical equations for the rotational angle, bending moment, shear force, and deflection of the free overhang and coal-bearing zone and an equation for calculating the limiting span.
2. The foundation coefficient, roof thickness, and angle of upward mining have less of an impact on the roof bending moment. The maximum bending moment occurs at about 2 m before the coal wall. Both the foundation coefficient and the angle of upward mining have a non-linear positive correlation with the limiting span. The roof thickness has a linear positive correlation with the limiting span.
3. The roof load and overhang length significantly influence the roof bending moment. The maximum bending moment occurs 2 m in front of the coal wall. The overhang length has a non-linear positive correlation with the roof limiting span, while the roof load has a non-linear negative correlation with the limiting span.
4. The mining activities are divided into three stages based on the local geological conditions. The limiting spans of the THR are 13.18, 18.82, and 22.5 m in the three stages, respectively. The distribution characteristics of the roof bending moment differ significantly across the stages. Whatever the stage and regardless of other potential influence factors, the maximum bending moment consistently occurs at about 2 m in front of the coal wall.
5. At different extraction stages, the field-measured periodic weighting lengths were 13.33, 19.33, and 22.67 m, respectively. The periodic weighting lengths calculated from the proposed equations (the limiting spans) agree with the field measurements, proving the accuracy of the proposed mechanical model.  
Conceptualization, W.Z., L.C. and D.Z.; methodology, W.Z., L.C. and Y.H.; software, L.C., Y.H. and J.C.; validation, Y.H., J.C. and Z.Z.; formal analysis, W.Z., D.Z. and J.C.; investigation, L.C., D.Z. and Y.H.; resources, W.Z., L.C., D.Z. and Y.H.; data curation, Y.H., J.C. and Z.Z.; writing--original draft preparation, W.Z., L.C. and J.C.; writing--review and editing, D.Z., Y.H., J.C. and Z.Z.; visualization, Y.H. and Z.Z.; supervision, W.Z. and L.C.; project administration, W.Z., L.C. and D.Z.; funding acquisition, W.Z., D.Z. and Z.Z. All authors have read and agreed to the published version of the manuscript.  
This work was financially supported by the National Natural Science Foundation of China (No. 52374141), the \"Six Talent Peaks\" Project of Jiangsu Province (No. JNHB-087), the Graduate Research and Practice Innovation Program of Jiangsu Province (No. KYCX24_287), and the Graduate Innovation Program of the China University of Mining and Technology (No. 2024 WLJCRCZL011).  
Not applicable.  
The original contributions presented in the study are included in the article, further inquiries can be directed to the corresponding author.  
We wish to thank the ZCM for the support for conducting this important study, as well as [PERSON] and [PERSON] of the ZCM for their assistance with relevant geological data collection. The authors are also grateful to the three anonymous reviewers for their constructive comments and helpful suggestions.  
The authors declare no conflicts of interest.