## 1 Introduction  
The soil-rock binary structural slope is a common yet specialized slope in hydraulic and geotechnical engineering [1, 2, 3]. These slopes typically have an upper part composed of stockpiled soil and a lower part made of weathered rock. They are widely distributed along transportation corridors and reservoirs in mountainous and river valley areas. In these slopes, the soil body is usually exposed at the surface, while the rock body lies underground. As a result, many studies treat them as stacked layer soil slopes using traditional homogeneous slope analysis methods, which do not consider their distinct dual structural features. Consequently, research on these slopes is relatively limited compared to studies on conventional earth and rocky slopes [4]. Engineering interventions such as road construction, channel excavation, dam building, or river training, along with natural geological events like earthquakes and extreme weather, such as heavy rainfall, may gradually expose the lower rock mass. When these slopes are disturbed by construction activities, heavy rainfall, fluctuations in groundwater levels, or reservoir changes, the soil-rock structure shows a different failure mode compared to homogeneous slopes. In this failure mode, sliding occurs simultaneously within the soil, along the soil-rock contact surface, and on the rock structure surface. This complex instability can lead to larger landslides, posing significant risks to construction operations and nearby residents [5]. Addressing the complex discontinuous and nonlinear behaviors exhibited during the instability and failure process of rocky slopes, [PERSON] and [PERSON] [6] proposed a hybrid finite element method that treats the contacting bodies' nodal displacements as unknowns and uses the local nodal contact forces as iterative variables. Although this method better addresses nonlinear frictional contact problems, it remains a continuum medium numerical method and cannot simulate the complete disengagement of the rock body's structural surface. To overcome this limitation, [PERSON] et al. [7] proposed the partitioned finite elements and interface elements (PFE-IE) hybrid solution, which combines the block element method [8; 9] and the interface element method [10; 11]. This solution divides large nonlinear regions into several local regions that can be solved independently, thereby reducing computational resource consumption through parallel solving. It is also compatible with the strength reduction method for calculating stability and safety coefficients of rocky slopes [12; 13; 14]. In the PFE-IE solution process, when a rocky slope with a potential sliding surface along a weak structural plane reaches its limit state, the elastic deformation of the rock mass becomes negligible compared to the deformation at the structural plane. Thus, treating the rock mass as rigid bodies simplifies the computational model and reduces numerical integration workloads, thereby enhancing solving efficiency. Building on this rationale, [PERSON] et al. [15] proposed the partitioned rigid element and interface element (PRE-IE) method based on PFE-IE. This method decomposes the system into multiple rigid bodies and discontinuous interfaces, using interface contact forces and rigid body centroid displacements as hybrid variables. By neglecting elastic deformations in high-stiffness continuous regions and concentrating nonlinear iterative computations on discontinuous interfaces, PRE-IE maintains computational accuracy while improving the efficiency of stability analysis for rocky slopes. The instability modes of soil-rock binary structure slopes involve the coupled behavior of a continuum medium (soil body) and a discontinuous medium (structural planes in rock masses). However, in research on the stability analysis of slopes with a soil-rock dual structure, most scholars assume that the lower rock mass in soil-rock binary structure slopes is stable and consists of non-sliding bedrock.