## 2 Analysis of Roof Instability Effect
### Coupling Mechanical Model of Roof-Top Coal-Hydraulic Support  
In the collapsed region, the goaf is generated as the LTCC face advances. This process is accompanied by the breaking of the overlying strata. Hydraulic support in a short period will have a huge impact pressure, leading to the bearing components' severe deformation and pin shaft fracture, as shown in Figure 1.  
As shown in Figure 2, there is a layer of top coal above the hydraulic support in the LTCC. The existence of top coal increases the thickness of overlying strata, making it simple to build a voussoir beam on the main roof. According to Table 1, regarding sandy mudstone, its lithology is weak and has a good energy absorption effect. Therefore, the gangue and the rock with low lithology absorb most of the impact energy generated by the \begin{table}
\begin{tabular}{c c c c c c} \hline \multicolumn{5}{c}{**Stratigraphic System**} & \multicolumn{1}{c}{**Lithology**} & \multicolumn{1}{c}{**Thickness (m)**} & \multicolumn{1}{c}{**No.**} \\ \hline
**Period** & **Series** & **Formation** & & & \\ \hline Quaternary system & Holocene series & Aeolian layer & Aeolian sand & 33.5 & 1 \\  & Pleistocene series & Diluvial layer & Conglomeratic coarse sand & 24.2 & 2 \\  & & & Fine sandstone & 68.1 & 3 \\  & & & Silstone & 43.1 & 4 \\  & & & Medium sandstone & 85.7 & 5 \\ Cretaceous system & Lower series & Zhidan group & Fine sandstone & 54.3 & 6 \\  & & & Silstone & 24.2 & 7 \\  & & & Medium sandstone & 22.2 & 8 \\  & & & Fine sandstone & 28.6 & 9 \\  & & & Coarse sandstone & 34.2 & 10 \\  & & & Sandy mudstone & 31.2 & 11 \\  & & & Mudstone & 26.7 & 12 \\  & & & Fine sandstone & 37.9 & 13 \\  & & & Sandy mudstone & 29.1 & 14 \\  & & & Medium sandstone & 60.7 & 15 \\  & & & Silstone & 22.0 & 16 \\  & & & Fine sandstone & 8.5 & 17 \\  & & & Silstone & 5.5 & 18 \\ Jurassic system & & & Coarse sandstone & 14.5 & 19 \\  & & & Medium sandstone & 10.2 & 20 \\  & & & Fine sandstone & 8.3 & 21 \\  & & & Sandy mudstone & 12.1 & 22 \\  & Middle–lower series & Yan’an group & 2-2 upper coal & 5.2 & 23 \\  & & & Mudstone & 0.4 & 24 \\  & & & 2-2 middle coal & 4.8 & 25 \\  & & & Fine sandstone & 14.8 & 26 \\  & & & Coarse sandstone & 12.7 & 27 \\ Triassic system & Upper series & Yanchang group & Fine sandstone & 40.2 & 28 \\ \hline \end{tabular}
\end{table} 
Table 1: Overlying strata structure of the 08 mining area in the Shilawusu coal mine. movement of overlying strata. Additionally, there is a delay in gangue filling in the goat, which leads to the coal strata near the surface not forming a stable structure, showing a cantilever beam state [39; 40]. The rock strata that cannot be self-stabilized and collapse in the caving zone are collectively referred to as the immediate roof [14]. Due to the rock strata's different fracture locations and lithology, there are two types of immediate roofs: the lower immediate roof and the upper immediate roof. In order to quantitatively analyze the impact load caused by the instability of the upper immediate roof on the hydraulic support [41], the thickness \(H_{Z}\) of the immediate roof can be defined as Equation (1):  
\[H_{Z}=\frac{H+T-S_{A}-C}{K_{A}-1} \tag{1}\]  
where \(H_{Z}\) is the thickness of the immediate roof, m; \(H\) is the cutting height, m; \(T\) is the thickness of the top coal; \(S_{A}\) is the subsidence of the main roof, m; \(C\) is the thickness of residual coal, m; \(K_{A}\) is a constant under certain roof conditions [17], which is set to 1.4. The subsidence of the main roof \(S_{A}\) can be defined as Equation (2):  
\[S_{A}=K_{S}H \tag{2}\]  
where \(K_{S}\) is the main roof subsidence coefficient [15], which is set to 0.2. The thickness of residual coal \(C\) can be defined as Equation (3):  
\[C=(1-\eta)TK_{T} \tag{3}\]  
where \(\eta\) is the recovery rate of the top coal, which is set to 80%; \(K_{T}\) is the bulking coefficient of the top coal [15], which is set to 1.2. Figure 1: Failure modes of hydraulic support: (a) Failure modes of the canopy, (b) Failure modes of the pin shaft.