## 2 Lateral Load-Bearing Behavior of the Roadway-Side Support
### The Lateral Loading Characteristics of Gangue  
#### 2.2.1 The Strain Response Characteristics of Gangue  
Under the influence of the overlying strata load and self-weight, the immediate roof behind the gob area undergoes fracturing and collapse, forming fallen gangue, which exhibits typical swelling and compressive deformation characteristics similar to those of general rocks. The voids within the fallen gangue in the gob gradually close under the compaction effect of the overlying strata, and the gangue particles experience flow and rearrangement. Relative shearing and sliding occur between the particles [25; 26], which can be considered ideal non-cohesive particles. During the slow compaction phase, influenced by the bending and settlement of the overlying strata and the restrictions imposed via the roadway-side support structure, the bearing capacity of the gangue gradually increases, and the lateral pressure exerted on the roadway-side support transitions into active lateral pressure. The progressive compaction process of the fallen gangue in the collapse zone and its bearing mechanical properties have a critical impact on the stress state of the roof of the gob and the roadway-side support structure. Assuming that the compression strain of the gangue occurs continuously, during the phase when the roof has not yet touched the gangue, the effect of its own gravity is neglected, and the deformation of the gangue in the gob is represented as follows: \(\epsilon=\sigma=0\). During the gangue compaction phase, the stress-strain relationship is described as follows [27; 28]:  
\[\sigma=\frac{10.39\sigma_{c}^{1.042}}{K^{7.7}}\times\frac{\epsilon}{1-\frac{ \kappa}{K-1}\epsilon} \tag{11}\]  
Among them, \(\mathbf{E_{0}}\) is the initial elastic modulus of the collapsed gangue, \(\epsilon_{m}\) is the maximum strain reached after the rock mass is broken and expanded, \(\mathbf{K}\) is the expansion coefficient of the collapsed gangue in the goat, and \(\sigma_{c}\) is the initial vertical stress at the top of the coal seam. Based on the mass ratio of gangue particles of various particle sizes, the gangues with the same original gradation, porosity, and mechanical properties at different particle sizes in the gob are selected [29], as shown in Table 1. The gangue compression device specifications are chosen to be an outer diameter of 0.13 m and a cylinder depth of 0.5 m. Fine gangue particles are placed into the test container with a gangue height of approximately 0.468 m. The test process is shown in Figure 5. The results of the gangue compression test are presented in Figure 6.  
Figure 5: Ganguue compression device. As shown in Figure 6b, the stress-strain curve during the compression of gangue generally follows the trend of Equation (4). The axial strain of the gangue exhibits strong segmental behavior, namely the compaction phase and the stable consolidation phase. When the strain is less than 0.17 (the initial compression phase of the gangue), the aggregate in the gangue-bearing structure within the test device is composed of larger-sized gangue particles. The voids between the gangue blocks are relatively large, providing ample deformation space. Under load, the larger gangue blocks primarily bear the load and compress and interlock with each other. The gangue experiences rigid failure, resulting in deformation, fragmentation, and rotation, while the voids between the blocks decrease. When the strain is between 0.17 and 0.36 (the mid-compression phase), the strain rate gradually slows down. Under load, the voids within the blocks are continuously compressed, and the fine particulate gangue generated via this compression gradually fills the voids between the blocks. The load-bearing structure gradually transitions from being dominated by large-sized gangue particles to a coupled structure of large particles and fine particulates. When the strain exceeds 0.36, the voids within the block structure are almost completely compressed and filled, resulting in a significant increase in the gangue's bearing capacity.