## 3 Stability Analysis of an Integral Load-Bearing Structure of GER
### Interaction Mechanism of Discontinuous Arrangement Roadside Support  
#### 3.2.1 Stress Distribution between Columns  
The discontinuous arrangement of concrete-filled steel tubes forms an empty roof area between the columns, as shown in Figure 5a. According to previous studies, the stability of the roof between two columns is closely related to the span of columns; excessive span will form a tensile stress area in the middle of the two columns, resulting in an uneven deformation of the shallow roof [29; 30; 31; 32]. Meanwhile, the roadside support is located at the edge of the gob and is disturbed by multiple overburden activities. The repeated mutual extrusion of the shallow part of the roof and the roadside support will cause the roof to become broken, which is disadvantageous to gob-side entry retaining. The analysis of the mechanical characteristics of the roof between the columns focuses on the change in the roof along the direction of the roadway. When the change in stress in the roadway direction is continuous and gradual, the stress state of the roof above the adjacent roadside supports is basically the same [33; 34; 35]. Therefore, we can regard the analysis of the stress state of the roof between the columns as a mechanical problem of solving the internal stress distribution of the semi-infinite plane boundary under uniformly distributed loads. Thus, a mechanical model of the roof between two columns was established, as shown in Figure 5b, and the roof of the retained roadway was shown to be affected by support resistance and the overburden strata pressure. Before conducting the analysis of the roof between the columns, the basic assumptions on the calculation model were set as follows: (1) the model can be regarded as a plane strain problem along the direction of the roadway and (2) the supporting force of the concrete-filled steel tube to the roof is a uniform load along the direction of the roadway. Assuming that the semi-infinite plane is subjected to a distributed load concentration \(q_{b}(s)\) at the free boundary, the force element of unit length at the point distance from the coordinate origin O can be approximately regarded as a concentrated force with value \(d_{f}=q_{b}(s)dx\). Under the action of a concentrated force, the stress at any point on the semi-infinite planar body can be obtained from the theory of elastic mechanics. \[\sigma_{y}=-\frac{2}{\pi}\int_{x_{2(i-1)}}^{x_{2(i-1)}}\frac{q_{b}x^{3}d\xi}{ \left[x^{2}+(y-\xi)^{2}\right]^{2}} \tag{8}\]  
Figure 5: Mechanical model of roof between columns. (**a**) Physical model of discontinuous alley side support; (**b**) additional stress calculation model. The stress state at any point in the semi-infinite plane under the action of distributed load concentration \(q_{b}(s)\) can be obtained by integrating the mechanical response of all force elements of the distributed load concentration, as seen in Formula (8). Under the assumption that the overlying strata is isotropic elastic material, the additional stress caused by multiple adjacent concrete-filled steel tubes on the roof can be calculated by linear superposition. The additional stress caused by a discontinuous arrangement of multiple concrete-filled steel tubular columns in the roof can be calculated using Formula (9):  
\[\sigma_{yn}=-\frac{2}{\pi}\sum_{i=1}^{n}\int_{x_{2(i-1)}}^{x_{2i-1}}\frac{qy^{ 3}d\xi}{\left[y^{2}+\left(x-\xi\right)^{2}\right]^{2}}-q_{j} \tag{9}\]  
Taking the geological condition of the 91-101 working panel as the background, the relevant parameters were substituted into Formula (9) to calculate the additional stress in the roof between the columns, and a vertical stress contour map was drawn using MATLAB, as shown in Figure 6.  
For the concrete-filled steel tube roadside supports, the change in spacing had a strong influence on the distribution range and extreme value of the tensile stress zone between two columns.