## 4 Double-Layer Flexible Support Technique and its Key Parameters  
### Anti-Disturbance Principle of Thick-Layer Anchorage  
The front abutment pressure is induced by coal seam mining. When the distance between the working face and the withdrawal roadway is sufficiently small, the front abutment pressure overlaps with the abutment pressure of the withdrawal roadway, resulting in a further increase in peak stress [19; 20; 21]. When any point in the rock mass evolves into a new stress equilibrium state, it will produce frequent stress fluctuations, leading to cumulative damage to the rock mass. It can be seen that the reasons for the large deformation and failure of the withdrawal roadway caused by mining disturbance are complex. At this point, it is necessary to build a stable anti-disturbance support structure.  
The coal seam roof exhibits typical layered characteristics, and, even with the same lithology, it is often subdivided into multiple rock layers due to the presence of weak interlayers within it. The individual layers are comparatively thinner, exhibit low stability, and are susceptible to deformation. By using an anchor bolt support, multiple thinner rock layers can transform into thicker and more integrated anchorage layers, and the stability and disturbance resistance of the roof can be greatly improved.  
The roof rock layer can be assumed to be a simply supported beam to reflect the actual stress situation of the layered roof. Supposing that the thickness and the width of the anchored rock beam on the roof are \(h\) (m) and \(b\) (m), respectivley, \(I\) is the moment of inertia of the cross-section to the neutral axis (for a rectangular section \(I=bl^{3}/12\), m\({}^{4}\)), the load borne is \(q\) (MPa), the elastic modulus is \(E\) (MPa), and the roadway width is \(w\) (m), then the maximum roof subsidence \(s_{\max}\) can be obtained as follows:  
\[s_{\max}=5 qw^{4}/384 EI=5 qw^{4}/32 Ebl^{3} \tag{2}\]  
During the roof subsidence, the maximum bending moment is \(M_{\max}=qw^{2}/8\) and the bending interface coefficient is \(W=bl^{2}/6\). Therefore, the maximum tensile stress borne by the anchored rock beam of the roof \(\sigma_{\max}\) is as follows:  
\[\sigma_{\max}=M_{\max}/W=3 qw^{2}/4 bl^{2} \tag{3}\]  
As shown in Equations (2) and (3), the thickness \(h\) of the anchored rock beam is a sensitive factor affecting the maximum subsidence and maximum tensile stress of the roof. If typical values are taken for the other parameters in the equation (\(q=1.2\) MPa, \(w=4.8\) m, \(E=220\) MPa, \(b=1\) m), then \(h\) has a significant impact on \(s_{\max}\) and \(\sigma_{\max}\), as presented in Figure 8. It is obvious that the larger the \(h\), the smaller the maximum settlement and maximum tensile stress of the roof. Therefore, increasing the thickness \(h\) of the anchored rock beam is crucial for ensuring the safety of the roadway roof. The thicker anchorage not only mitigates roof deformation, but also alleviates the tensile stress by enhancing the overall bearing capacity of the roof.  
Front abutment pressure can lead to rock failure in some cases. The important attribute of thick layer anchoring at this stage is that it can maintain the stability of the rock structure surrounding the roadway and provide support for further strengthening measures, such as grouti