## 6 Conclusions  
This study used GeoStudio software to establish a 2D model of the slope, and based on this model, a 10 h short-duration heavy rainfall simulation was conducted for single-step slopes with heights ranging from 5 to 30 m. The M-P method was applied to analyze the stability of slopes with slope ratios of 1:1.5, 1:1.75, 1:2, and 1:2.25, leading to the following conclusions:As rainfall infiltrates the slope soil, the soil moisture content gradually increases. When the soil pores approach saturation, the capillary tension and suction in the soil decrease, leading to a reduction in matric suction and weakening the soil strength. During rainfall, if the infiltration rate exceeds the drainage rate, water accumulates within the soil, increasing pore water pressure. As pore water pressure rises, the effective stress in the soil decreases, causing the slope to lose its original shear strength. 2 Steeper slopes are more prone to accumulating large amounts of surface runoff under heavy rainfall or prolonged precipitation, which increases the soil moisture content and pore water pressure, thereby weakening the soil's shear strength and increasing the risk of instability. In contrast, gentler slope ratios allow for a more uniform distribution of stresses, reducing localized stress concentrations and enhancing the overall stability of the slope. Additionally, gentler slopes aid in drainage, reducing pore water pressure buildup and mitigating the negative impacts of rainfall infiltration on slope stability. 3 According to the stability analysis results of single-step slopes under sustained rainfall, when the slope height is less than or equal to 5 m, a slope ratio of 1:1.5 is appropriate. For step heights between 5 and 15 m, a slope ratio of 1:1.75 meets the stability requirements. For step heights ranging from 15 to 25 m, a slope ratio of 1:2.0 can be selected. A slope ratio of 1:2.25 is suitable for single-step slopes with heights up to 30 m. 4 In slope design and remediation, the slope ratio should be chosen based on the specific conditions. For high and steep slopes, measures such as slope reduction, drainage systems, and counterpressure at the slope foot should be implemented to improve stability. In areas with dense cracks, it is recommended to enhance the drainage design to prevent excessive water accumulation within the soil, which may lead to instability. 5 To address local instability within the internal slopes of the dump site, several stabilization measures can be implemented. One approach is to construct a placement platform at the toe of the dump slope to exert a certain amount of counteracting horizontal pressure, which is particularly suitable for areas with sufficient space at the slope toe and where additional support for the toe embankment is required. Another method involves slope trimming to reduce the overall slope gradient and increase platform areas, thereby decreasing the slope height. If the earthwork volume required for slope trimming is excessive, a combined approach incorporating counterweight berms at the slope toe can be considered. In addition to these two remediation methods, further measures should be taken to ensure the long-term stability of the dump site. These include the installation of retaining structures, drainage interception and discharge systems, monitoring facilities, and land reclamation initiatives. These measures help ensure that the dump site meets the requirements of relevant standards and regulations, thereby enhancing overall stability and safety. F.Z. : Methodology, Investigation, Writing--review and editing and Data curation and Funding acquisition. L.T. : Methodology, Investigation, Conceptualization, Writing--original draft and Visualization. Y.C. : Writing--review and editing. Y.X. : Supervision and Funding acquisition. All authors have read and agreed to the published version of the manuscript. This research was funded by the National Natural Science Foundation of China (Grant No.