## 1 Introduction  
Stability analysis for slopes has invariably been one of the most classic and hottest subjects in geotechnical engineering. New and numerous innovative approaches have been employed, and feasible engineering conditions are considered in the investigation of slope stability. Most of these research works are directed to slopes in the two-dimensional (2D) plane-strain situation. Nevertheless, slope instability and failure frequently exhibit distinct three-dimensional (3D) properties. Therefore, it is necessary to analyze the 3D stability for slopes in order to obtain more reasonable results. The stability analysis of 3D slopes can normally be carried out by the following three measures: (1) limit equilibrium method (LEM), (2) finite element method (FEM), and (3) limit analysis method (LAM). The existing 3D LEM is generally promoted on the basis of the 2D method; that is, the 2D slices are enlarged to the columns under the 3D space [1, 2]. The assumptions of internal forces must be introduced in the analysis so as to ascertain the positive stress distribution on the sliding surface, yet the results obtained from this calculation and the actual stress state often have a large difference [3]. In numerical simulations based on the FEM, the strength reduction method is commonly utilized for calculations [4, 5, 6]. However, after discounting the slope strength, the stress state inside the slope will be changed so that the location of the most critical sliding surface is somewhat different from the actual situation. Recently, the LAM has been widely implemented in the stability assessment of slopes, which avoids the unreasonable assumptions on internal forces in the LEM and the tedious calculations in the FEM [7, 8, 9, 10]. The primary challenge of applying the LAM to stability evaluation for 3D slopes is to construct a 3D kinematical admissible failure mechanism, and there has been a large amount of literature written on this field of research. The failure mechanisms adopted in the investigation of 3D limit analysis for slope stability can be summarized into two categories: (1) the block sliding mechanism, and (2) the integral sliding mechanism. The multi-block 3D sliding mechanism and the improved multi-block 3D sliding mechanism [11] belong to the first category, and the 3D horn-like rotational mechanism [12] and spherical mechanism belong to the second category. The application of the above two categories of sliding mechanisms in the stability assessment for 3D slopes can obtain more satisfactory results. However, the integral sliding mechanism can ensure calculation accuracy while greatly simplifying the calculation process. Relevant studies have shown that slope geometry has a significant effect on slope stability. Therefore, in practical excavation engineering, the single-step is often modified into the double-step by adding a step platform to improve slope stability. In addition, earthquakes are also a factor that cannot be ignored in the stability assessment for 3D slopes [13]. The seismic stability of the 3D steep slope was investigated on the basis of the pseudo-static method. Nevertheless, the pseudo-static method evaluates the seismic effect in a highly simplified way, and it does not account for the influence of seismic excitation features on slope stability. A simplified pseudo-dynamic approach was proposed by [PERSON] and [PERSON] [14; 15], which considers the temporal effects of seismicity but overlooks critical factors such as boundary conditions and material damping. On the basis of the existing methods, [PERSON] [16; 17] presented a modified pseudo-dynamic method, which has been frequently utilized in seismic stability analysis of geotechnical structures. Furthermore, the linear Mohr-Coulomb (M-C) criterion, which has previously commonly been used in the stability assessment of soil slopes, cannot reflect the mechanical properties of soil material.