## 3 Stability Analysis of an Integral Load-Bearing Structure of GER
### Interaction Mechanism of Discontinuous Arrangement Roadside Support
#### 3.2.2 Roof-Support Body Shear Failure Mechanism  
Assuming that the contact surface between the roadside support body and the roof is in a limit equilibrium state, the limit equilibrium zone of the roof above the roadside support support can be divided into three parts: (1) The area directly above the roadside support body is zone I, in which the contact surface is the maximum principal stress surface, the pressure in the vertical direction of the roadside support body is the maximum principal stress, and the horizontal stress perpendicular to the depth direction of the roof is the minimum principal stress. The two sets of shear surface and the roadway roof present an angle of \((45+\varphi/2)^{\circ}\), as shown in Figure 7a. (2) As the pressure of the \"large structure\" of the overlying strata diverges to the \"small structure\", zone I is squeezed to both sides, and zone III becomes a passive state zone due to the horizontal stress acting as the maximum principal stress. The shear surface is also composed of two groups of planes, and because the roof surface becomes the minimum principal stress plane, the shear surface and the roof surface present an angle of \((45-\varphi/2)^{\circ}\). (3) The middle of zone I and zone III is the transition zone II. One of the shear surfaces of zone II is the radiation line, and the other is the logarithmic spiral curve, as shown in Figure 7b.  
\[r=r_{0}e^{\theta\tan\varphi} \tag{10}\]  
In Formula (10), \(r\) represents the distance from the point \(O\) to any point \(m\), \(r_{0}\) is the distance of \(O_{m}\) along any selected axis, \(\theta\) is the angle between \(O_{n}\) and \(O_{m}\), and the radius of curvature corresponding to any point m is angle \(\varphi\) with the normal direction of the logarithmic spiral curve.  
The theoretical formula of the limit load of roadway roof is given in Formula (11):  
\[q_{u}=c^{2}\tan\varphi\Big{[}e^{\pi\cdot\tan\varphi}\cdot\tan^{2}\Big{(} \frac{\pi}{4}+\frac{\varphi}{2}\Big{)}-1\Big{]} \tag{11}\]  
Where \(c\) represents the cohesion of rock mass of the roadway roof and \(\varphi\) represents the angle of internal friction. When \(q_{b}<q_{u}\), the rock mass at the contact surface between the roadside support body and the roof will not be sheared to failure. Additionally, it can be seen from Formula (11) that the limit load of rock mass at the contact surface between the roadside support body and the roof is positively related to the \(\varphi\) and c of the roof rock mass. As the overburden pressure and the properties of rock mass are uncontrollable factors, three reinforcement measures can be considered to ensure that the contact surface between the roadside support and roof does not suffer shear failure: (1) increasing the contact area between the roadside support body and the roof of the roadway; (2) pre-grouting the roadway roof to increase the \(c\), \(\varphi\) value of the rock mass; and (3) pre-setting the additional anchor cable in advance of the working longwall. Calculations were carried out  
Figure 7: Shear slip surface of roadway roof. (**a**) Shear slip surface-radiation line; (**b**) shear slip surface-logarithmic spiral line.  
by substituting the geological parameters of the 91-101 working longwall, and the results showed that, assuming the load borne by the concrete-filled steel tube reaches 80% of its ultimate load, the diameter of the concrete-filled steel tube must be greater than 980 mm to ensure that the roof and floor of the roadway do not undergo shear failure.