## 5 Results and Discussion
### Distribution Characteristics of Dissipated Energy Density  
To visually analyze the distribution characteristics of dissipated energy density around the roadway and within the coal pillar, the results from FLAC\({}^{\text{3D}}\) were imported into Tecplot 2019 software for visualization processing, successfully generating contour maps of the dissipated energy density. Figure 14 illustrates the distribution of dissipated energy density around the roadway for coal pillar widths of 6 m, 9 m, 12 m, 15 m, 18 m, and 21 m.  
When the width is 6 m, the coal pillar has been fully penetrated by dissipated energy, posing a risk of irreversible plastic deformation throughout the entire coal pillar. The peak value is 1.89 MJ/m\({}^{3}\). The area affected by the dissipated energy in the roof extends to 7 m, with a peak value of 0.9 MJ/m\({}^{3}\), concentrated at a depth of 0.1 m in the middle of the roof. On the solid coal body side, the affected area extends to 2.1 m with a peak value of 1.14 MJ/m\({}^{3}\), concentrated at a depth of 0.2 m in the middle of the roadway. The affected area in the floor extends to 7.5 m, with a value between 0.1 MJ/m\({}^{3}\) and 0.2 MJ/m\({}^{3}\), indicating minimal influence. When the width is 9 m, an area of 4.5 m in the coal pillar remains unaffected by dissipated energy. The peak value is 1.41 MJ/m\({}^{3}\), occurring near the goat side of the coal pillar. The affected area in the roof decreases to 4.5 m, with a peak value of 0.78 MJ/m\({}^{3}\). On the solid coal body side, the affected area extends to 1.9 m with a peak value of 0.9 MJ/m\({}^{3}\), concentrated in the middle-upper part of the roadway. The affected area in the floor extends to 6.5 m, with minimal influence. When the width is 12 m, an area of 7.9 m in the coal pillar remains unaffected by dissipated energy, distributed similarly to the goat side and the opposite side of the coal pillar. The peak value is 1.1 MJ/m\({}^{3}\). The affected area in the roof is 4.5 m, with a peak value of 0.7 MJ/m\({}^{3}\). On the solid coal body side, the affected area extends to 1.9 m with a peak value of 0.81 MJ/m\({}^{3}\), concentrated in the middle-upper part of the roadway. The affected area in the floor decreases to 1.9 m, with a value of around 0.1 MJ/m\({}^{3}\). When the width exceeds 12 m, the unaffected area in the coal pillar further expands, from 11 m for a 15 m width to 17.4 m for a 21 m width. The dissipated energy in the coal pillar is concentrated near the goat side, with a peak value of 1 MJ/m\({}^{3}\), while the dissipated energy on both sides of the goat-side roadway tends to be equal, stabilizing at around 0.5 MJ/m\({}^{3}\). The affected area in the roof is 4.1 m, with a peak value of 0.5 MJ/m\({}^{3}\), while the dissipated energy in the floor decreases to below 0.1 MJ/m\({}^{3}\), indicating minimal influence. As shown in Figure 15, with the increase in coal pillar width, when the width reaches 9 m, the dissipated energy density begins to show negative values. This indicates the emergence of an elastic energy storage zone where non-plastic failure occurs, starting from the 9 m coal pillar. As the width increases, the elastic energy storage zone expands, and the peak dissipated energy values on both sides of the coal pillar decrease. The reduction in peak dissipation energy values is more pronounced on the goat-side roadway. In summary, the internal dissipated energy changes from the initial state of being continuous to a state of two independent double cores, and the instability of the coal pillar  
Figure 14: Contour map of dissipated energy density. transitions to a relatively stable state. Therefore, considering only this aspect, a coal pillar width of 9 m represents the critical stability under the geological conditions. During this period, the influence of the goaf on the dissipated energy of the roadway persists. It is not until the coal pillar width reaches 21 m that the dissipated energy density at the roof and sides of the roadway stabilizes at around 0.5 MJ/m\({}^{3}\).