## 1 Introduction
They proposed a stability calculation method based on the limit equilibrium principle and preliminarily assessed the toppling risk of the slope using the geometric area of the S-zone. [PERSON] et al. [12] treated the failure surface of reinforced soil-retaining walls as composed of multiple line segments. Based on the mechanical equilibrium conditions of horizontal soil strips under seismic action, they derived a calculation formula for reinforcement tension related to failure surface parameters and determined the critical failure surface corresponding to the maximum tension value. [PERSON] et al. [13] conducted theoretical research on the stability of soil slopes by combining the variational method and the limit equilibrium method. They employed the shooting method for numerical solutions, obtaining precise results for the slip surface location, normal stress distribution, and stability coefficient. The effectiveness of the proposed method was validated through comparisons with the slice method and the finite element strength reduction method. The potential fracture surface of a soil nailing wall can be determined by calculating the maximum axial force of layered soil nailing in the sliding zone. There have also been analyses derived from full-scale tests and model tests. For example, [PERSON] et al. [14] analyzed the deformation of a soil nailing wall and the internal force characteristics of soil nailing through a full-scale test. [PERSON] et al. [15] and [PERSON] [16] tested the horizontal displacement of the top of the soil nailing wall, the internal force of soil nailing, and the soil pressure during the whole test process through similar model tests. Numerical analysis has also been conducted. [PERSON] et al. [17] used the finite element method to study the evolution of the failure surface and earth pressure distribution during the active limit state of backfill soil under the translation of counterweight-retaining walls. By comparisons with centrifuge tests, they proposed a modified earth pressure calculation model based on the shear state of the failure surface. [PERSON] et al. [18] applied strength reduction and limit equilibrium theories to analyze the failure mechanisms and stability of homogeneous soil slopes under different geometries using finite difference software. [PERSON] and [PERSON] [19] simulated the toppling failure of anti-dip layered rock slopes under seismic action, finding that increasing the internal friction angle reduces the time taken to reach a new equilibrium state and decreases the maximum horizontal displacement of sliding blocks. [PERSON] et al. [20] and [PERSON] et al. [21] investigated the seismic failure modes of rock slopes with intermittent bedding joints and the deformation mechanisms of fault-controlled creeping bedding landslides, respectively. They revealed the mechanism of soil nailing support. Studies integrating engineering practice have also been conducted. [PERSON] et al. [22] discussed and optimized the calculation formula of the overall stability safety factor of a soil nailing wall in the technical regulations for building foundation pit support according to different types of sliding surfaces of the soil nailing wall. [PERSON] et al. [23] established the corresponding function of reliability analysis by studying the stability of a soil nailing wall against overall slip and overturning. The corresponding reliability index was calculated using the JC method combined with engineering examples. [PERSON] and [PERSON] [24] monitored the force of a soil nailing support structure in combination with a soft-rock high-slope project. It was found that the working principle of the unearthed nailing support was different from that of the anchor structure and soil nailing wall. Based on the change in soil nailing axial force, the failure mode of the soft-rock high slope was proposed. Scholars from other countries have also conducted extensive research on this issue. [PERSON] et al.