## 3 Pre-Ie-Fe
### Example of Slope Stability Analysis of Soil-Rock Binary Structure  
The following example illustrates the application of the PRE-IE-FE method to a soil-rock binary structure slope using a straightforward arithmetic approach. This serves as a means of verifying the accuracy of the method in slope stability analysis. The geometric model of the example is illustrated in Figure 4a. It comprises an upper soil body and a lower rock body containing a weak structural surface. The PRE-IE-FE model of the example is illustrated in Figure 4b, where the soil body, rock body, and weak structural surface are partitioned into finite elements, rigid elements, and interface elements, respectively. As the finite elements cannot be directly coupled with the rigid elements through the nodes, the interface elements must also be divided along the soil-rock contact surfaces to ensure the correct transfer of displacements and forces. The rock mass below the slope is considered bedrock, and a fixed constraint is applied to the rigid body element at the center. The slope is only affected by self-weight, and the capacity of the soil-rock contact surface and the weak structural surface is not considered. The bulk weight of the soil is 18 \(\mathrm{kN/m^{3}}\), the cohesion is 18 \(\mathrm{kPa}\), and the friction angle is 20\({}^{\circ}\). The bulk weight of the rock is 27 \(\mathrm{kN/m^{3}}\), the cohesion at the soil-rock contact surface is 20 kPa, and the friction angle is 20\({}^{\circ}\). The example considers the two instability modes shown in Table 2. The strength parameter of the weak structural surface in the first instability mode is greater than that of the soil, the cohesion is 30 kPa, and the friction angle is 29\({}^{\circ}\). The strength parameter of the weak structural surface in the second instability mode is less than that of the soil, the cohesion is 10 kPa, and the friction angle is 20\({}^{\circ}\). In the first mode of instability, the equivalent plastic strain cloud of the soil-rock binary structure slope is illustrated in Figure 5a, and the schematic of the sliding surface is presented in Figure 5b. Given that the strength parameters of the soil body and the soil-rock contact surface are considerably lower than those of the weak structural surfaces within the rock body, the slope becomes destabilized at the upper soil body. This results in the formation of a composite sliding surface comprising curved sliding surfaces within the soil body and linear sliding surfaces on the soil-rock contact surface, as observed in the slope. Given that the lower rock body is not readily destabilized, it can be regarded as stable bedrock. The soil-rock contact surface, in contrast, is regarded as another layer of thinner soil body with different strength parameters from the upper soil body. This transformation allows the stability problem of soil-rock dichotomous structural slopes to be addressed as a stability problem of two-layer soil slopes. This approach allows for the stability of the slope to be analyzed using the limit equilibrium method, and the stability safety coefficients calculated using PRE-IE-FE can be compared. The results of the safety coefficients calculated using the different methods are presented in Table 3. As can be  
Figure 4: (**a**) Geometric modeling of slopes with soil–rock binary structures. (**b**) PRE-IE-FE modeling of slopes with soil–rock binary structures. observed from the table, the results calculated using PRE-IE-FE are in close proximity to those of the three-limit equilibrium bar-splitting methods. This indicates that PRE-IE-FE is an effective method for accurately analyzing the stability of the soil-rock binary structural slopes in the first type of destabilization mode. In the second mode of instability, the equivalent plastic strain cloud diagram of the soil-rock binary structure slope is presented in Figure 6a, and the schematic diagram of the sliding surface is shown in Figure 6b.