A Probability Integral Prediction Model for Goaf-Induced Slope Subsidence Considering Slope-Angle Transition and an Independent Horizontal-Segment Solution

Goaf-induced slope subsidence is controlled by goaf geometry, burial conditions, slope morphology, and geotechnical parameters. Conventional probability integral models for horizontal ground cannot fully capture the effects of slope arc length, normal burial depth, and abrupt slope-angle changes. This study develops a segmented-slope probability integral model incorporating slope-angle transition correction and an independent horizontal-segment solution. A horizontal–inclined–horizontal slope profile is established using a continuous arc-length coordinate. Goaf microelements are projected onto the segmented slope, and the main influence radius is determined from their normal distance to the slope surface, with projection length correction included in the subsidence integral. Slope-angle transition and vertical projection corrections are introduced to avoid discontinuities near segment boundaries, while the left horizontal-segment solution is smoothly coupled with the segmented-slope solution. Parametric analysis shows that goaf width, equivalent mining height, and subsidence coefficient dominate maximum subsidence, contributing 76.1%. Validation using the E4600 and E5800 sections of the Fushun East Open-Pit Mine shows good agreement, with R2 values of 0.924 and 0.9237.