应用端帮采煤机开采工艺回收端帮滞留煤资源过程中，合理地设计支撑煤柱参数，是端帮开采安全、高效进行的前提。基于尖点突变理论，确立了端帮开采支撑煤柱失稳充要条件；建立了采硐顶板-煤柱力学分析模型，基于弹性地基梁理论，研究了煤柱局部失稳后顶板应力、变形空间演化规律，提出了采硐顶板垮落失稳判据，以此作为非均布载荷条件下支撑煤柱失效判据；研究了不同煤柱留设宽度下顶板极限垮落距与煤柱失稳区段长度变化规律，提出了非均布载荷下支撑煤柱参数设计方法。研究结果表明：煤柱失稳的充要条件为分叉集Δ<0；采硐顶板的失稳判据为最大拉应力σ_tmax大于抗拉强度σ_tl；煤柱的失效判据为煤柱失稳区段长度l_max大于采硐顶板极限垮落距l_2max；设计了山西某露天煤矿端帮开采煤柱宽度最终为3.0 m。

In the process of applying the mining technology of the end-wall shearer to recover the coal resources retained in the end-wall, the reasonable design of the parameters of the supporting coal pillar is the premise of the safe and efficient end-wall mining. Based on the cusp catastrophe theory, the necessary and sufficient conditions for the instability of the supporting coal pillar in the end slope mining are established. The mechanical analysis model of mining roof-coal pillar was established. Based on the elastic foundation beam theory, the spatial evolution law of roof stress and deformation after local instability of coal pillar was studied, and the instability criterion of mining roof collapse was proposed, which was used as the failure criterion of supporting coal pillar under non-uniform load. The variation law of the limit caving distance of the roof and the length of the coal pillar instability section under different coal pillar widths was studied, and the design method of the supporting coal pillar parameters under non-uniform load was proposed. The results show that the necessary and sufficient condition of coal pillar instability is the bifurcation set Δ < 0, the instability criterion of the roof is that the maximum tensile stress σ_tmax is greater than the tensile strength σ_tl , the failure criterion of the coal pillar is that the length l_max of the instability section of the coal pillar is greater than the limit caving distance l_2max of the roof of the mining cave , the width of the coal pillar in the end-wall mining of an open-pit coal mine in Shanxi is designed to be 3.0 meters.