In the current context of global climate change, geohazards such as earthquakes and extreme rainfall pose a serious threat to regional stability. We investigate a three-dimensional (3D) slope dynamic model under earthquake action, derive the calculation of seepage force and the normal stress expression of slip surface under seepage and earthquake, and propose a rigorous overall analysis method to solve the safety factor of slopes subjected to combined with rainfall and earthquake. The accuracy and reliability of the method is verified by two classical examples. Finally, the effects of soil permeability coefficient, porosity, and saturation on slope stability under rainfall in a project located in the Three Gorges Reservoir area are analyzed. The safety evolution of the slope combined with both rainfall and earthquake is also studied. The results indicate that porosity has a greater impact on the safety factor under rainfall conditions, while the influence of permeability coefficient and saturation is relatively small. With the increase of horizontal seismic coefficient, the safety factor of the slope decreases significantly. The influence of earthquake on slope stability is significantly greater than that of rainfall. The corresponding safety factor when the vertical seismic action is vertically downward is smaller than that when it is vertically upward. When considering both horizontal and vertical seismic effects, slope stability is lower.

Rainfall-induced landslides are caused by the infiltration of precipitation into the ground surface, leading to an increase in pore water pressure, hence reducing the effective stress and shear strength of the soil. Sustained rainfall or heavy rainfall events can significantly increase the risk of slope instability, especially in those areas with loose, poorly drained soils. Several landslides in the Three Gorges Reservoir area have been triggered by rainfall (Yin et al., 2012; Sun et al., 2016b). Earthquakes, as another key factor, impose additional dynamic loads on slopes through ground shaking, which may lead to instability of otherwise stable slopes. In addition, earthquake-induced landslides tend to be more destructive because they often occur without warning. Due to completely different destabilization mechanisms, studies of landslides induced by these two factors are often carried out separately. In some cases, rainfall and earthquakes may act together on slopes. And earthquake-induced landslides may occur more frequently during the rainy season, when the soil is saturated with water and its resistance to earthquakes is reduced. Further research is necessary to investigate the stability of slopes under the combined influence of rainfall and earthquake (David, 2000; Iverson, 2000; Sassa et al., 2010).

At present, the main research methods for slope stability include the limit equilibrium method (Bishop, 1955; Morgenstern and Price, 1965; Spencer, 1967), limit analysis (Farzaneh et al., 2008; Michalowski, 1995; Qin and Chian, 2018; Zhou et al., 2017), finite element method (Griffiths and Lane, 1999; Ishii et al., 2012), among others. There have been numerous studies and findings regarding the stability assessment of 3D slopes. However, most of these methods are based on extended 3D equilibrium analysis techniques (Hungr, 1987; Zhang, 1988; Chen et al., 2001; Cheng and Yip, 2007), which rarely strictly adhere to the six equilibrium conditions. Additionally, these approaches often introduce a significant number of assumptions that limit their practical engineering applications. The strict 3D limit equilibrium method proposed by Zheng (2007) is an overall analysis approach based on the natural form of slip surface stress distribution and approximation through shard interpolation. Sun et al. (2016a, 2017) combined Morgenstern–Price and Bell global analysis method to analyze the stability of reservoir bank slope, applying this method to the Three Gorges Reservoir area. Rahardjo et al. (2010) studied the effect of groundwater table position, rainfall intensities, and soil properties in affecting slope stability using the numerical analyses. Some of the defects inherent in the two-dimensional (2D) limit equilibrium method remain unresolved, and some of them are even amplified in the complex 3D analysis, which has a certain impact on the accuracy of the 3D slope stability evaluation. For the limit analysis method, it is still difficult to establish the velocity field of the motion permit in 3D space. And numerical methods often suffer from two problems: the determination criteria of the critical state of the slope and the determination of the location of the critical sliding surface. Compared with a single traditional analysis method, the mutual integration of several method theories has also been gradually developed, so as to give full play to the advantages of their respective methods and better used in slope stability analysis, such as the finite element limit analysis method (Ali et al., 2017; Lim et al., 2017; Zhou and Qin, 2022).

As a common geological hazard in seismic zones, earthquake-triggered landslides have been extensively investigated by numerous scholars (Sepúlveda et al., 2005; Chang et al., 2012; Jibson and Harp, 2016; Marc et al., 2017; Salinas-Jasso et al., 2019). At present, the stability analysis method of 3D slope is not mature, and the research on the dynamic stability of 3D slope is even more scarce. The quasi-static method (Liu et al., 2001) introduces coefficients (

Most studies only consider the role of a single factor in seepage or earthquake, neglecting the slope stability analysis under combined working conditions. Therefore, analyzing the change law of safety factors for slopes during seepage and seismic action is of great practical value in guiding slope support design and evaluating slope stability. In this paper, a 3D rigorous slice-free method considering seepage and seismic forces to solve the safety factor of bank slopes is proposed. The proposed method strictly satisfies the force balance and moment balance in three directions, without introducing other redundant assumptions.

The phreatic surface is the interface between the saturated and unsaturated zones within the slope. Physical and mechanical parameters of the sliding below the phreatic surface adopt saturated, while above the phreatic surface they adopt naturally. A differential soil slice is taken from the slip surface to the slope surface in the landslide body, shown in Fig. 1.

Relationship between rainfall and groundwater level.

The load on the soil slice is shown in Fig. 2.

Calculation sketch of forces acting on the differential soil slice.

According to the flow properties of the phreatic line perpendicular to the equipotential line, the surrounding hydrostatic pressures

Calculation sketch of hydraulic head.

The components of

The gravity of water in differential soil slice is

The permeability pressure is a pair of balancing forces with the water weight in a differential soil slice and the hydrostatic pressure around it (Zheng et al., 2004). Therefore, the weight of water in the differential soil slice and the surrounding hydrostatic pressure can be replaced by a seepage force. The force diagram in Fig. 2 can be replaced by Fig. 4.

Simplified force diagram on a differential soil slice.

The horizontal and vertical components of the seepage force

According to the geometric relation,

Therefore, the seepage force is

The direction of seepage force is consistent with groundwater flow. The direction of groundwater flow within the sliding soil mass is determined by the inclination of the phreatic surface in each differential soil slice. As shown in Fig. 4, the flow direction of groundwater is oriented at an angle

As shown in Fig. 5, taking the whole sliding body

A 2D schematic plot for force system in/on the sliding body.

Here,

According to the Mohr–Coulomb criterion,

Here,

The order is

Substituting Eqs. (

As shown with the dashed line in Fig. 5, a vertical differential cylinder is now taken from the homogeneous sliding body from the slip surface to the slope surface. The load on the differential cylinder is shown in Fig. 6.

Sketch of force acting on a vertical differential cylinder in a sliding body.

Here,

The force equilibrium condition for a differential cylinder is

Both sides of the Eq. (

Here,

The following is known:

Substituting Eq. (

The order is

Therefore,

Here,

The normal stress distribution of the slip surface can be approximated in the following (Zheng, 2009):

As shown in Fig. 7,

A triangular mesh for interpolation of normal stress on slip surface.

Substituting Eq. (

We can solve Eq. (

In Eq. (

In order to verify the accuracy of the proposed method, two examples are analyzed in this section. Different working conditions were set up for Example 2, and the results are compared with those calculated by the software.

Wedge stability in rock mechanics is a typical 3D limit equilibrium analysis problem. Examples of wedge include two cases of geometric symmetry and asymmetry. Example 1 is an asymmetric wedge. Figure 8 shows the three-dimensional model and geometric parameters of the wedge plane sliding. The sliding surface is composed of two structural planes, ABC and OAB, and the coordinates of the vertices are listed in Fig. 8. The sliding direction of the wedge sliding body is assumed to be parallel to the intersection line AB. The sliding surface of the wedge adopts the same shear strength:

Model and geometric parameters of the wedge.

In order to verify the feasibility of the proposed method for calculating the slope stability under seepage and earthquakes, a classical ellipsoid example is selected for the stability analysis, as shown in Fig. 9, which is derived from the study of Zhang (1988). Zhang's (1988) paper in 1988 provides a three-dimensional slope ellipsoid slip surface example, and the simplified three-dimensional limit equilibrium method (only three force equilibriums and one moment equilibrium are satisfied) is used for the stability analysis. Zhang's (1988) solution for the 3D limit equilibrium of a symmetric ellipsoid can be regarded as a rigorous solution since the ellipsoid has a symmetric sliding surface, and the other two moment equilibrium conditions are automatically satisfied by the symmetric bar-column method. Zhang's (1988) solution has also been used by many scholars to check the correctness of their own procedures (Hungr, 1987; Huang and Tsai, 2000; Zheng, 2009). The example is a homogeneous slope, the potential sliding surface is a part of a simple ellipsoid, the sliding surface is symmetric about the

Model of ellipsoid example.

Mechanical parameters of the slope.

Geometric parameters and middle profile with groundwater.

The ellipsoid model is shown in Fig. 9. The external load of the slope is only considered the effect of gravity, the unit gravity is 19.2

Safety factor of Example 2.

Based on the above calculation results, the comparison revealed minimal differences across all four conditions (natural, with groundwater, with seismic loading, and combined), indicating that the proposed method is also effective in assessing slope stability under seepage and seismic actions.

This section investigates slope stability evolution under the influence of rainfall and earthquake by taking an actual slope in the Three Gorges Reservoir as a case study.

Geographical location map of Woshaxi slope (©Google Maps).

Contour map of Woshaxi slope.

Geological section map of Woshaxi slope.

Average monthly rainfall from 2007 to 2009.

Figure 11 provides a depiction of the Woshaxi landslide's geographical setting. Figure 12 shows a topographic map of Woshaxi slope with contour lines, and the cross-section (I-I

According to the Seismic Ground Motion Parameter Zonation Map of China, the peak ground motion acceleration in this region is 0.05

Mechanical parameters of Woshaxi slope.

Figure 14 shows the average monthly rainfall from 2007 to 2009. Table 3 lists the physical and mechanical parameters of the landslide body. It is assumed that the reservoir water level remains unchanged. To assess the effects of different geotechnical parameters and seismic action on the safety factor, four cases are considered: (i) rainfall only, (ii) rainfall and horizontal earthquake, (iii) rainfall and vertical earthquake, and (iv) rainfall and earthquake in both horizontal and vertical directions.

The three parameters, infiltration coefficient, porosity, and saturation, have different effects on the safety factor of slopes. The safety factor varies with the monthly rainfall. The analysis indicates that an increase in rainfall does not invariably lead to a decrease in the safety factor of the slope. This phenomenon can be attributed to the fact that increased rainfall raises the phreatic surface within the slope, affecting two key aspects: firstly, it enhances the hydrodynamic forces, and secondly, it increases the pressure at the base of the slope. When the increase in pressure at the slope's base has a more pronounced impact on stability than the hydrodynamic forces, the safety factor of the slope will subsequently increase. Conversely, if the hydrodynamic forces dominate, the stability of the slope will diminish. As shown in Fig. 15a, the permeability coefficient

As shown in Fig. 15b, the porosity

Safety factors of the Woshaxi landslide under rainfall.

As shown in Fig. 15c, the saturation

Safety factors of the Woshaxi landslide under rainfall and horizontal earthquake (

Safety factors of the Woshaxi landslide under rainfall and horizontal earthquake (different horizontal seismic coefficient).

Figure 16 shows the evolution of the stability of the Woshaxi landslide under the combined effect of rainfall and horizontal earthquake with different geotechnical parameters, and the horizontal earthquake coefficient

Safety factors of the Woshaxi landslide under rainfall and vertical earthquake.

Figure 18 shows the evolution of the stability of the Woshaxi landslide with rainfall and different vertical earthquake coefficients. With other parameters unchanged, the vertical earthquake coefficient

Safety factors of the Woshaxi landslide under rainfall and earthquake (in both horizontal and vertical directions).

Figure 19 shows the evolution of the stability of the Woshaxi landslide with rainfall and different earthquake coefficients. Horizontal earthquake coefficient

In this paper, the calculation of the seepage force is studied; the normal stress expression on the sliding surface of a slope under seepage force and seismic force are also derived. Furthermore, a global analysis method that considers both seepage and seismic forces is proposed to determine the safety factor of slopes subjected to the combined effect of rainfall and earthquake. The reliability of the proposed method is also verified with two examples combining software calculations and previous results.

Taking a slope in the Three Gorges Reservoir area as an example, this study investigates the influence of soil permeability coefficient, porosity, and saturation on slope stability and analyzes the safety evolution of this slope under combined effects of rainfall and earthquakes. The results indicate that, under rainfall conditions, the porosity of the soil above the phreatic surface exerts a greater influence on safety factor than permeability coefficient and saturation. With an increase in the horizontal earthquake coefficient, the safety factor of the landslide is significantly reduced, and the impact of earthquake on slope stability surpasses that of rainfall. The safety factor corresponding to vertical downward earthquake action is smaller than that of vertical upward, and the stability of slope is lower when considering horizontal and vertical upward earthquake actions. Therefore, in order to ensure maximum safety, proper consideration should be given to vertical earthquake actions.

When considering rainfall alone, the slope safety factor is 1.04–1.09, positioning the slope in a state that between unstable and basically stable. However, upon accounting for horizontal seismic activity, the slope safety factor decreases to about 0.9 and is transformed into an unstable state. When the vertical earthquake is considered, the slope safety factor is 1.035–1.075. This represents a slight reduction but still in the unstable and basically stable state. This suggests that horizontal seismic influences exert a more pronounced impact on slope stability compared to vertical. When rainfall and earthquake act simultaneously, the safety factor calculated using the proposed method falls below 1.0, indicating an unstable condition where landslide disasters are likely to occur on the slope. The research results provide a scientific basis for slope stability analysis and prevention. Further, the proposed method can identify potential risk areas for landslide hazards, and planners in the Three Gorges Reservoir area can better consider these risks and take measures to increase the seismic and flood resilience of reservoir infrastructure.

The data used in this study are available from the first author upon request.

JW analyzed the data, conceived the paper, and wrote the paper; ZW conceived and co-wrote the paper; HL reviewed and improved the analysis and paper; and GS provided the data of the actual slope in the Three Gorges Reservoir.

The contact author has declared that none of the authors has any competing interests.

Publisher's note: Copernicus Publications remains neutral with regard to jurisdictional claims made in the text, published maps, institutional affiliations, or any other geographical representation in this paper. While Copernicus Publications makes every effort to include appropriate place names, the final responsibility lies with the authors.

This research has been supported by the National Natural Science Foundation of China (grant no. 11972043).

This paper was edited by Rachid Omira and reviewed by three anonymous referees.