Analysis of the Upper Limit of the Stability of High and Steep Slopes Supported by a Combination of Anti-Slip Piles and Reinforced Soil Under the Seismic Effect

Abstract

The reinforcement effect of single-reinforced soil support under external loading has limitations, and it is difficult for it to meet engineering stability requirements. Therefore, the stability analysis of slopes supported by a combination of anti-slip piles and reinforced soil under the seismic loading effect needs an in-depth study. Based on the upper-bound theorem of limit analysis and the strength-reduction technique, this study establishes an upper-bound stability model for high–steep slopes that simultaneously considers seismic action and the combined reinforcement of anti-slide piles and reinforced soil. A closed-form safety factor is derived. The theoretical results are validated against published data, demonstrating satisfactory agreement. Finally, the MATLAB R2022a sequential quadratic programming method is used to optimize the objective function, and the Optum G2 2023 software is employed to analyze the factors influencing slope stability due to the interaction between anti-slide piles and geogrids. The research indicates that the horizontal seismic acceleration coefficient k_h exhibits a significant negative correlation with the safety factor F_s. Increases in the tensile strength T of the reinforcing materials, the number of layers n, and the length l all significantly improve the safety factor F_s of the reinforced-soil slope. Additionally, as l increases, the potential slip plane of the slope shifts backward. For slope support systems combining anti-slide piles and reinforced soil, when the length of the geogrid is the same, adding anti-slide piles can significantly improve the slope’s safety factor. As anti-slide piles move from the toe to the crest of the slope, the safety factor first decreases and then increases, indicating that the optimal reinforcement position for anti-slide piles should be in the middle to lower part of the slope body. The length of the anti-slip piles should exceed the lowest layer of the geogrid to more effectively utilize the blocking effect of the pile ends on the slip surface. The research findings can provide a theoretical basis and practical guidance for parameter optimization in high–steep slope support engineering.

1. Introduction

As infrastructure construction in China continues to expand, transportation corridors are extending into increasingly rugged mountainous regions. Consequently, the stability of high and steep cut slopes has become a critical concern [1,2,3]. When high-fill embankment construction is carried out in these areas, slope stability is directly related to the safety and durability of the project.

Although only a single-reinforced soil support structure can provide a certain degree of reinforcement, its limitations gradually appear under complex geological conditions and external loads. Moreover, reinforced-soil-retaining walls are subjected to traffic loading and sloping-ground conditions, which frequently lead to backfill sliding along the slope face and to wall deformation or cracking. These defects can trigger landslides and other severe consequences [4,5,6,7]. Under extreme events such as heavy rainfall or earthquakes, their stability is often insufficient for engineering demands. Zhou L. et al. [8] used a finite difference program in numerical simulations to demonstrate that earthquakes significantly reduce slope stability and amplify displacement. Excavation, anti-slip piles, and anchor cables are three treatment measures that can effectively reduce displacement and lower the internal forces in anti-slip piles, thereby improving slope stability during earthquakes and providing important reference for reinforcement strategies. Zucca M. et al. [9] conducted extensive numerical simulations to assess the seismic response of shallow, multi-supported underground structures embedded in granular soil layers with varying stiffness (shear-wave velocity 360–750 m/s). They found that the decoupled approach yields results consistent with coupled nonlinear dynamic analyses in low-stiffness soils, yet its applicability is strictly bound by the underlying assumptions of the method. Yang K.H et al. [10] investigated the failure of a 26 m high geogrid-reinforced earth slope filled with low-plasticity silty clay, which was affected by heavy rainfall. They discovered that the rapid increase in pore water pressure caused by cumulative rainfall exceeding 600 mm during a typhoon was the primary contributor to the composite failure.

To address these challenges and avoid the construction complexity of conventional high–steep slope support, a combined system of anti-slide piles and reinforced soil is adopted to improve seismic stability [11,12]. Currently, scholars in both China and abroad have carried out abundant research on reinforced-soil-reinforced slopes. Nevertheless, there are relatively few analyses and studies on combined anti-slip piles and reinforced-soil support to enhance the stability of high steep slopes and their influencing factors [13,14,15,16,17,18,19]. Cao W.Z et al. [20] proposed anti-slip piles + rigid/flexible combination, reinforced-soil-retaining wall support structure for reinforcing steep slopes in mountainous areas, and used FLAC3D 7.0 software to establish a numerical model of the support structure to compare and analyze the stability of three different anti-slip piles + bearing platforms combinations of the reinforced-soil-retaining wall support structure. Ma N. et al. [21] conducted shaking table tests at different magnitudes on soil slopes reinforced by two stabilizing structures (restrained anti-slip piles and pre-stressed anchored sheet pile walls) to investigate the dynamic soil pressure distribution and acceleration changes in response to the two models at different magnitudes, and the study showed that anchored piles can support slopes better than beam piles. Huang Y. et al. [22] explored the performance of four pile–anchor reinforcement schemes under seismic loading by means of finite element and Newmark slider analyses, and the results suggest that the pile–anchor structure can effectively reduce the slope deformation, optimize the distribution of internal forces, and improve the stability of slopes. Ren Y. et al. [23] carried out centrifugal model tests and numerical simulation calculations to analyze the factors affecting the stability of ultra-high and steep reinforced-soil slopes in mountainous airports, and the results revealed that the local stability of the middle and lower parts of the slopes should be enhanced. Zeng Y.L. et al. [24] employed the upper-bound theorem of limit analysis and the virtual-work principle to develop an analytical model for slopes reinforced with both anti-slide piles and geosynthetics. A closed-form expression for the factor of safety was derived and validated against field data and numerical simulations. The results indicate that adding anti-slide piles markedly increases the safety factor and alters the failure mode. Wang P.Y. et al. [25] constructed a three-dimensional finite difference model of a sliding-pile-reinforced gravel soil slope with different damage states under seismic action, based on the Sichuan–Qingdao Railway located in a typical work site in the strong earthquake zone, and analyzed the seismic response characteristics and damage states of sliding piles in the strong earthquake zone. To sum up, most of the existing studies carried out numerical simulation calculations and model tests for high–steep slopes supported by anti-slip piles and reinforced soil. They tended to only focus on the simple loading effect, seldom paying attention to the upper limit of the stability of slopes supported by anti-slip piles and reinforced soil under the seismic effects. Consequently, based on the upper-bound theorem of limit analysis and incorporating the strength-reduction technique, this paper establishes a computational model for the upper-bound analysis of steep slope stability. This model comprehensively accounts for seismic effects and the stabilizing action of combined anti-slide piles and reinforced-soil support. The expression for the safety factor of slopes reinforced with this combined system is derived. Finally, the objective function is optimized using the sequential quadratic programming method in MATLAB R2022a. Numerical simulations using Optum G2 2023 software are employed to analyze the influence of anti-slide piles and geogrid reinforcement on slope stability. This study aims to provide a theoretical basis and practical guidance for optimizing the design parameters of protective reinforcement for steep slopes.

2. Fundamental Principle

2.1. Limit Analysis Upper-Bound Theorem

The ultimate load is the smallest load in the permissible velocity field and strain rate field. Based on the upper-bound theorem of limit analysis, the smallest valid upper-bound solution for the objective function is obtained by equating the power of external forces to the internal energy dissipation rate under a kinematically admissible velocity field. The virtual power equation expression [26] is as follows:

$${\displaystyle \underset{A}{\int }{T}_{\mathrm{i}}}{\stackrel{·}{u}}_{\mathrm{i}}dA+{\displaystyle \underset{V}{\int }{F}_{\mathrm{i}}}{\stackrel{·}{u}}_{\mathrm{i}}dV={\displaystyle \underset{V}{\int }{\sigma }_{\mathrm{ij}}}{\stackrel{·}{\epsilon }}_{\mathrm{ij}}dV$$

(1)

where A is the area of the deformed body; V is the volume of the deformed body; T_i and F_i are the surface force and body force acting on the deformed body; u_i and ε_ij are the velocity and strain in the velocity field.

2.2. Nonlinear Mohr–Coulomb Failure Criterion

Numerous experiments and actual engineering projects have shown that geotechnical materials generally follow the nonlinear Mohr–Coulomb failure criterion. The expression for the nonlinear Mohr–Coulomb failure criterion [27] is as follows:

$$\tau =c_0\cdot {\left(1+{\sigma }_{\mathrm{n}}/{\sigma }_{\mathrm{t}}\right)}^{{\displaystyle \frac{1}{m}}}$$

(2)

where τ and σ_n are the shear stress and normal stress on the failure plane; c₀, σ_t, and m are parameters of the geotechnical material determined through experiments, where σ_t and c₀ are the intercepts of the curve with the horizontal and vertical axes. The degree of curvature of the curve is controlled by the nonlinear coefficient m. When m = 1.0, the linear Mohr–Coulomb failure criterion is considered.

Plot Equation (2) as a curve, as shown in Figure 1.

Figure 1. The schematic diagram of the nonlinear Mohr–Coulomb failure criterion.

When m ≠ 1, differentiate σ_n:

$${\displaystyle \frac{d\tau }{d{\sigma }_{\mathrm{n}}}}=c_0\cdot {\displaystyle \frac{1}{m}}\cdot {\left(1+{\displaystyle \frac{{\sigma }_{\mathrm{n}}}{{\sigma }_{\mathrm{t}}}}\right)}^{{\displaystyle \frac{1}{m}}-1}\cdot {\displaystyle \frac{1}{{\sigma }_{\mathrm{t}}}}$$

(3)

Simplified:

$${\displaystyle \frac{d\tau }{d{\sigma }_{\mathrm{n}}}}={\displaystyle \frac{c_0}{m{\sigma }_{\mathrm{t}}}}\cdot {\left(1+{\displaystyle \frac{{\sigma }_{\mathrm{n}}}{{\sigma }_{\mathrm{t}}}}\right)}^{{\displaystyle \frac{1}{m}}-1}$$

(4)

Assuming that the tangent point coordinates (σ_n0, τ₀), the tangent equation is as follows:

$$\tau -{\tau }_0={\left.{\displaystyle \frac{d\tau }{d{\sigma }_{\mathrm{n}}}}\right|}_{{\sigma }_{\mathrm{n}}={\sigma }_{\mathrm{n}0}}\left({\sigma }_{\mathrm{n}}-{\sigma }_{\mathrm{n}0}\right)$$

(5)

where τ₀ = c₀ (1 + σ_n0/σ_t)^1/m.

Substitute the derivative into the tangent equation:

$$\tau -c_0\cdot {\left(1+{\displaystyle \frac{{\sigma }_{\mathrm{n}0}}{{\sigma }_{\mathrm{t}}}}\right)}^{{\displaystyle \frac{1}{m}}}={\displaystyle \frac{c_0}{m{\sigma }_{\mathrm{t}}}}\cdot {\left(1+{\displaystyle \frac{{\sigma }_{\mathrm{n}0}}{{\sigma }_{\mathrm{t}}}}\right)}^{{\displaystyle \frac{1}{m}}-1}\cdot \left({\sigma }_{\mathrm{n}}-{\sigma }_{\mathrm{n}0}\right)$$

(6)

Simplified:

$$\tau ={\displaystyle \frac{c_0}{m{\sigma }_{\mathrm{t}}}}\cdot {\left(1+{\displaystyle \frac{{\sigma }_{\mathrm{n}0}}{{\sigma }_{\mathrm{t}}}}\right)}^{{\displaystyle \frac{1}{m}}-1}{\sigma }_{\mathrm{n}}+\left[c_0\cdot {\left(1+{\displaystyle \frac{{\sigma }_{\mathrm{n}0}}{{\sigma }_{\mathrm{t}}}}\right)}^{{\displaystyle \frac{1}{m}}}-{\displaystyle \frac{c_0}{m{\sigma }_{\mathrm{t}}}}\cdot {\left(1+{\displaystyle \frac{{\sigma }_{\mathrm{n}0}}{{\sigma }_{\mathrm{t}}}}\right)}^{{\displaystyle \frac{1}{m}}-1}\cdot {\sigma }_{\mathrm{n}0}\right]$$

(7)

Further simplification yields the tangent Equation of (7) as follows:

$$\tau =c_{\mathrm{t}}+{\sigma }_{\mathrm{n}}\cdot \tan {\phi }_{\mathrm{t}}$$

(8)

where c_t is the intercept of the tangent Equation (8); tanφ_t is the slope of the tangent Equation (8), which can be obtained using the slope formula.

$$\tan {\phi }_t={\displaystyle \frac{d\tau }{d{\sigma }_{\mathrm{n}}}}={\displaystyle \frac{c_0}{m{\sigma }_t}}\cdot {\left(1+{\displaystyle \frac{{\sigma }_{\mathrm{n}}}{{\sigma }_t}}\right)}^{{\displaystyle \frac{1-m}{m}}}$$

(9)

Combining (7), (8), and (9) yields the following:

$$c_{\mathrm{t}}={\displaystyle \frac{m-1}{m}}\cdot c_0\cdot {\left({\displaystyle \frac{m{\sigma }_t\cdot \tan {\phi }_t}{c_0}}\right)}^{{\displaystyle \frac{1}{1-m}}}+{\sigma }_{\mathrm{t}}\cdot \tan {\phi }_t$$

(10)

Using strength reduction technology [28], the slope safety factor Fs is incorporated to bring it to a critical limit equilibrium state [29]. After strength reduction, the strength index of the rock and soil material is as follows:

$$\left.\begin{array}{l}{c}_{\mathrm{f}}=c_{\mathrm{t}}/F_{\mathrm{s}}\\ {\phi }_{\mathrm{f}}=\arctan \left(\tan {\phi }_{\mathrm{t}}/F_{\mathrm{s}}\right)\end{array}\right\}$$

(11)

2.3. Dynamic Stability Pseudo-Static Analysis Method

Currently, the pseudo-static method is a commonly used simplified research method for seismic loads. This method decomposes seismic forces into two static forces in the horizontal and vertical directions, represented by k_hW and k_vW. W is the weight of the sliding soil, while the horizontal dynamic influence equivalent static force and the numerical dynamic influence equivalent static force act on a rigid body particle, represented by coefficients k_h and k_v, respectively, as shown in Equation (12):

k_v = λk_h

(12)

where k_h is the horizontal seismic acceleration coefficient, k_v is the vertical seismic acceleration coefficient, and λ is the seismic effect ratio coefficient of k_v relative to k_h.

3. Derivation of the Safety Factor for Slopes Under the Combined Support of Anti-Slip Piles and Reinforced Soil

3.1. Destruction Mode

The research findings of Chen [26], Donald [30], and Dawson et al. [31] have shown that the shape of the failure surface of a simple homogeneous slope is similar to a logarithmic spiral. Hence, this paper considers the failure surface of a slope under the combined support of anti-slip piles and reinforced soil to adopt a logarithmic spiral rotation mechanism passing through the toe of the slope as the upper limit analysis model for slope stability, as shown in Figure 2.

Figure 2. The schematic diagram of the upper limit analysis of slope stability under the action of combined anti-slip piles and reinforced soil support.

Angular parameters such as β, θ₀, θ, and θ_h, and length parameters such as L, r₀, r(θ), and r_h are all variable parameters related to the slope failure mode. ω represents the rotational angular velocity. The logarithmic spiral equation [24] is as follows:

$$r=r_0{\mathrm{e}}^{(\theta -{\theta }_0)\tan {\phi }_{\mathrm{t}}}$$

(13)

where r is the polar radius associated with θ; θ₀ is the angular parameter of the spiral rotation mechanism; r₀ is the polar radius when θ = θ₀; φ_t is the internal friction angle of the soil.

Based on the geometric relationship of the spiral line failure surface, the equations for r₀, H, and L are established as follows:

$${\displaystyle \frac{H}{r_0}}=\sin {\theta }_{\mathrm{h}}\cdot e^{\left({\theta }_{\mathrm{h}}-{\theta }_0\right)\tan {\phi }_{\mathrm{t}}}-\sin {\theta }_0$$

(14)

$${\displaystyle \frac{L}{r_0}}={\displaystyle \frac{\sin \left({\theta }_{\mathrm{h}}-{\theta }_0\right)}{\sin {\theta }_{\mathrm{h}}}}-{\displaystyle \frac{H}{r_0}}\cdot {\displaystyle \frac{\sin \left({\theta }_{\mathrm{h}}+\beta \right)}{\sin {\theta }_{\mathrm{h}}\cdot \sin \beta }}$$

(15)

To facilitate slope stability analysis, the layered reinforcement is assumed to form a uniform continuous model (as shown in Figure 2), where k₀ represents the average tensile strength of the reinforcement within the slope. Bending effects are neglected, and the reinforcement is subjected solely to tensile stress. The reinforcement density k_t is the tensile strength of the reinforcement per unit thickness within the slope, expressed by the following equation [32]:

$$k_{\mathrm{t}}=T/s$$

(16)

where T is the tensile strength of the reinforcing materials per unit width; s is the vertical spacing of the reinforcing materials. In the uniformly distributed reinforcing material model, s = H/n, where n is the number of reinforcing material layers.

For the uniform reinforcement pattern:

$$k_{\mathrm{t}}=k_0$$

(17)

where H is the slope height (m); L is the length of AB (m); β is the slope angle (°).

3.2. Energy Consumption Calculation

Energy consumption analysis consists of two parts: external work and internal energy loss. External work includes gravitational work and quasi-static seismic effect work, while internal energy loss includes energy loss at the soil interface, energy loss within the reinforcement, and work performed by the anti-sliding pile resistance.

3.2.1. Gravitational Work

Assuming that the power achieved by gravity in the OBC, OBA, and OAC zones is W₁, W₂, and W₃, the power achieved by gravity in the ABC zone is Wsoil:

$$W_{\mathrm{soil}}=\gamma r_0^3\omega \left(f_1-f_2-f_3\right)$$

(18)

where γ is the unit weight of rock and soil (kN/m³); r₀ is the length of OB; ω is the angular velocity of rotation; and the functional expressions of f₁, f₂, and f₃ related to θ₀, θ_h, and φ [29] are as follows:

$$f_1={\displaystyle \frac{\left(\sin {\theta }_{\mathrm{h}}+3\tan {\phi }_{\mathrm{t}}\cos {\theta }_{\mathrm{h}}\right){\mathrm{e}}^{3\left({\theta }_{\mathrm{h}}-{\theta }_0\right)\tan {\phi }_{\mathrm{t}}}-3\tan {\phi }_{\mathrm{t}}\cos {\theta }_0-\sin {\theta }_0}{3\left(1+9{\tan }^2{\phi }_{\mathrm{t}}\right)}}$$

(19)

$$f_2={\displaystyle \frac{L\sin {\theta }_0\left(2\cos {\theta }_0-L/r_0\right)}{6r_0}}$$

(20)

$$f_3={\displaystyle \frac{{\mathrm{e}}^{\left({\theta }_{\mathrm{h}}-{\theta }_0\right)\tan {\phi }_{\mathrm{t}}}}{6}}\cdot \left[\sin \left({\theta }_{\mathrm{h}}-{\theta }_0\right)-{\displaystyle \frac{L}{r_0}}\sin {\theta }_{\mathrm{h}}\right]\cdot \left[\cos {\theta }_0-{\displaystyle \frac{L}{r_0}}+\cos {\theta }_{\mathrm{h}}{\mathrm{e}}^{\left({\theta }_{\mathrm{h}}-{\theta }_0\right)\tan {\phi }_{\mathrm{t}}}\right]$$

(21)

3.2.2. Static Seismic Effect Work

(1) The horizontal seismic force work power is as follows:

$$W_{{\mathrm{k}}_{\mathrm{h}}-\mathrm{earthquake}}=k_{\mathrm{h}}\gamma r_0^3\omega \left(f_4-f_5-f_6\right)$$

(22)

(2) The vertical seismic force work power is the following:

$$W_{{\mathrm{k}}_{\mathrm{v}}-\mathrm{earthquake}}=k_{\mathrm{v}}\gamma r_0^3\omega \left(f_1-f_2-f_3\right)=\lambda k_{\mathrm{h}}\gamma r_0^3\omega \left(f_1-f_2-f_3\right)$$

(23)

where functions f₄, f₅, and f₆ are functions related to θ_h, θ₀, φ, and β, and their expressions [33] are as follows:

$$f_4={\displaystyle \frac{\left(3\tan {\phi }_{\mathrm{t}}\sin {\theta }_{\mathrm{h}}-\cos {\theta }_{\mathrm{h}}\right){\mathrm{e}}^{3\left({\theta }_{\mathrm{h}}-{\theta }_0\right)\tan {\phi }_{\mathrm{t}}}-3\tan {\phi }_{\mathrm{t}}\sin {\theta }_0+\cos {\theta }_0}{3\left(1+9{\tan }^2{\phi }_{\mathrm{t}}\right)}}$$

(24)

$$f_5={\displaystyle \frac{L{\sin }^2{\theta }_0}{3r_0}}$$

(25)

$$f_6={\displaystyle \frac{{\mathrm{e}}^{\left({\theta }_{\mathrm{h}}-{\theta }_0\right)\tan {\phi }_{\mathrm{t}}}\sin \left({\theta }_{\mathrm{h}}+\beta \right)}{6\sin \beta }}\cdot {\displaystyle \frac{H}{r_0}}\cdot \left(2{\mathrm{e}}^{\left({\theta }_{\mathrm{h}}-{\theta }_0\right)\tan {\phi }_{\mathrm{t}}}\sin {\theta }_{\mathrm{h}}-{\displaystyle \frac{H}{r_0}}\right)$$

(26)

3.2.3. Internal Energy Loss Power

(1) The energy dissipation power on the soil cross-section is the following:

$$W_{\mathrm{L}}={\displaystyle \frac{c{r}_0^2\omega f_7}{2\tan {\phi }_{\mathrm{t}}}}$$

(27)

(2) The energy dissipation power on the reinforcement is as follows:

$$W_{\mathrm{r}}={\displaystyle \frac{k_{\mathrm{t}}{r}_0^2\omega \left(f_8-f_9-f_{10}\right)}{\cos {\phi }_{\mathrm{t}}}}$$

(28)

where functions f₈, f₉, and f₁₀ are functions related to θ_h, θ₀, and φ, and their expressions [24] are as follows:

$$f_7={\mathrm{e}}^{2\left({\theta }_{\mathrm{h}}-{\theta }_0\right)\tan {\phi }_{\mathrm{t}}}-1$$

(29)

$$f_8={\displaystyle \frac{\cos {\phi }_{\mathrm{t}}\left(\tan {\phi }_{\mathrm{t}}\sin 2{\theta }_{\mathrm{h}}-\cos 2{\theta }_{\mathrm{h}}\right){\mathrm{e}}^{2\left({\theta }_{\mathrm{h}}-{\theta }_0\right)\tan {\phi }_{\mathrm{t}}}-\tan {\phi }_{\mathrm{t}}\sin 2{\theta }_0+\cos 2{\theta }_0}{4\left(1+{\tan }^2{\phi }_{\mathrm{t}}\right)}}$$

(30)

$$f_9={\displaystyle \frac{\cos {\phi }_{\mathrm{t}}}{4\left[{\mathrm{e}}^{2\left({\theta }_{\mathrm{h}}-{\theta }_0\right)\tan {\phi }_{\mathrm{t}}}-1\right]}}$$

(31)

$$f_{10}={\displaystyle \frac{\sin {\phi }_{\mathrm{t}}\left(\tan {\phi }_{\mathrm{t}}\cos 2{\theta }_{\mathrm{h}}+\sin 2{\theta }_{\mathrm{h}}\right){\mathrm{e}}^{2\left({\theta }_{\mathrm{h}}-{\theta }_0\right)\tan {\phi }_{\mathrm{t}}}-\tan {\phi }_{\mathrm{t}}\cos 2{\theta }_0-\sin 2{\theta }_0}{4\left(1+{\tan }^2{\phi }_{\mathrm{t}}\right)}}$$

(32)

3.2.4. Anti-Slip Piles Resistance Work

The distribution of effective soil pressure on the side of the pile plays a crucial role in the design of anti-slide piles. This paper follows the recommendations of scholars such as Liu X.Y. [34] and Han M. [35], adopting a distribution pattern where the effective soil pressure on the side of the pile is parallel to the tangent direction of the slip plane. Assume that F represents the effective anti-slide force on the side of the anti-slide piles, and a denotes the horizontal distance between the anti-slide piles and the toe of the slope. The intersection points of the anti-slide piles with the slope surface and the failure slip plane are denoted as F and E, respectively. The angle corresponding to point E in polar coordinates is θ_E, and the length between points F and E represents the pile length h of the anti-slide piles within the failure plane. The anti-slide piles are located on the slope surface, and based on geometric relationships, the following can be derived:

$$h=r_{\mathrm{E}}\sin {\theta }_{\mathrm{E}}-r_{\mathrm{h}}\sin {\theta }_{\mathrm{h}}+a\tan \beta$$

(33)

$$\left\{\begin{array}{l}{x}_{\mathrm{F}}=r_0\cos {\theta }_0-L-\left(H-a\tan \beta \right)\cot \beta \\ y_{\mathrm{F}}=r_0\sin {\theta }_0+H-a\tan \beta \end{array}\right.$$

(34)

$${\theta }_{\mathrm{F}}=\arctan \left(y_{\mathrm{F}}/x_{\mathrm{F}}\right)$$

(35)

Since x_F = x_E, in the polar coordinate system:

$$\left\{\begin{array}{l}{x}_{\mathrm{E}}=r_0{\mathrm{e}}^{\left({\theta }_{\mathrm{E}}-{\theta }_0\right)\tan {\phi }_{\mathrm{t}}}\cos {\theta }_{\mathrm{E}}\\ y_{\mathrm{E}}=r_0{\mathrm{e}}^{\left({\theta }_{\mathrm{E}}-{\theta }_0\right)\tan {\phi }_{\mathrm{t}}}\sin {\theta }_{\mathrm{E}}\end{array}\right.$$

(36)

Accordingly, from Equations (34) and (36), it can be seen that θ_E can be expressed in terms of θ₀, φ_t, β, H, a, L, and r₀:

$${\mathrm{e}}^{\left({\theta }_{\mathrm{E}}-{\theta }_0\right)\tan {\phi }_{\mathrm{t}}}\cos {\theta }_{\mathrm{E}}=\cos {\theta }_0-{\displaystyle \frac{L}{r_0}}-{\displaystyle \frac{\left(H-a\tan \beta \right)\cos \beta }{r_0\sin \beta }}$$

(37)

For the case where the effective soil pressure on the pile side is parallel to the tangent direction of the slip plane, the external work performed by the effective anti-slip force F provided by the anti-slip piles and the corresponding bending moment M is given by the following [36]:

$$W_{\mathrm{F}}=F\omega \left[\cos {\phi }_{\mathrm{t}}\cdot r_0\cdot {\mathrm{e}}^{\left({\theta }_{\mathrm{E}}-{\theta }_0\right)\tan {\phi }_{\mathrm{t}}}-m_{\mathrm{E}}h\sin \left({\theta }_{\mathrm{E}}-{\phi }_{\mathrm{t}}\right)\right]$$

(38)

where m_E is an empirical coefficient depending on the distribution pattern of the effective soil pressure on the pile side. This paper adopts a linear triangular distribution model, with m_E = 1/3 selected in reference [37].

3.3. Slope Stability Coefficient

According to the principle of conservation of internal and external energy, the power exerted by external loads is equal to the internal dissipation power. The shear strength parameters φ_t and c_t of the soil are calculated using Equation (11) to determine the strength reduction, and the critical stability height is set equal to the original slope height. The safety factor calculation formula for the reinforced-soil slope is as follows:

$$F_s={\displaystyle \frac{c_{\mathrm{f}}{f}_7}{2\tan {\phi }_{\mathrm{f}}\left\{\begin{array}{l}\gamma r_0\left[(1+\lambda k_{\mathrm{h}})\left(f_1-f_2-f_3\right)+k_{\mathrm{h}}\left(f_4-f_5-f_6\right)\right]\\ -\frac{k_{\mathrm{t}}}{\cos {\phi }_{\mathrm{f}}}\left(f_8+f_9-f_{10}\right)-\frac{F\left[\cos {\phi }_{\mathrm{f}}{r}_0{e}^{\left({\theta }_{\mathrm{E}}-{\theta }_0\right)\tan {\phi }_{\mathrm{f}}}-m_{\mathrm{E}}h\sin \left({\theta }_{\mathrm{E}}-{\phi }_{\mathrm{f}}\right)\right]}{{r_0}^2}\end{array}\right\}}}$$

(39)

From Equation (11), φ_f is a function of F_s, and the internal friction angle φ in Equation (39) is φ_f = arctan(tanφ/F_s); then, the functional expression of the safety factor F_s is actually an implicit function with respect to the unknown quantities θ_h, θ₀, φ_f, and β. As a result, F_s can be treated as the objective function, and Equation (39) can be optimized and iterated using MATLAB R2022a software to ultimately obtain the optimal upper bound solution for the safety factor F_s.

4. Comparison of Calculation Examples

4.1. Comparison of Slope Safety Factors Under Proposed Static Seismic Effects

Based on the strength reduction method, this paper compares and analyzes the slope safety factor F_s calculated using the energy dissipation method with the results obtained by Deng D.P. et al. [38] using the limit equilibrium method. The comparison results are shown in Table 1. The error between the results of this paper and those in [38] is no more than 3.20%, which verifies the accuracy of the calculation method used in this paper.

Table 1. Comparison of slope safety factors F_s under seismic loading effects.

H/m	β/(°)	γ/kN/m3	c/kPa	φ/(°)	kh	λ	Deng D.P. et al.’s [38] Fs	This Article’s Fs
10	30	17	25	20	0.25	0.5	1.234	1.206
10	40	18	30	24	0.1	0.5	1.677	1.642
14	30	18	35	32	0.25	0.5	1.564	1.514
14	40	17	40	28	0.1	0.5	1.815	1.767

4.2. Comparison of Critical Heights of Reinforced Earth Slopes

The results of this study were compared with the critical height H_cr of uniformly reinforced-soil slopes calculated by Cui X.Z. et al. [39]. The research method used in this study fully considered the energy loss rate when the reinforcement and soil are bonded together. The comparison results are shown in Table 2. The results of this study differed from the critical height obtained in [39] by only 0.43% to 8.46%, verifying the correctness of the theoretical derivation and programming calculations in this study.

Table 2. Comparison of critical heights of reinforced-soil slopes H_cr.

β/(°)	γ/kN/m3	c/kPa	φ/(°)	kt/kPa	Cui X.Z. et al.’s [39] Hcr/m	This Article’s Hcr/m
90.0	17.679	16.3	21.3	0.000	5.2	5.64
90.0	17.824	20.2	20.8	2.804	6.9	6.74
80.5	17.698	17.8	21.7	0.000	7.0	6.97
80.5	17.853	23.8	20.6	2.796	9.5	9.38

4.3. Comparison of Slope Safety Factors Under the Support of Anti-Slip Piles

The slope safety factors F_s calculated in this paper under the support of anti-slip piles were compared and analyzed with the calculation results of Tan H.H. et al. [37] and Zeng Y.L. et al. [24]. The comparison results are shown in Table 3. The results show that the slope safety factor results in this paper differ from the calculated results in the existing literature by less than 5%, proving the reliability of the calculation method in this paper.

Table 3. Comparison of slope safety factor F_s under the action of anti-slip-pile support.

H/m	β/(°)	γ/kN/m3	c/kPa	φ/(°)	a/(m)	Slope Safety Factor Fs
						Already Answered	This Article
13.7	30	19.63	23.94	10	13.7	1.63 [37]	1.58
						1.66 [24]
10	33.7	20	10	20	7.5	1.46 [37]	1.41
						1.49 [24]

4.4. Numerical Simulation Analysis of Safety Factor Comparison for Reinforced Earth Slopes

The safety factor F_s of the reinforced-soil slope obtained through numerical simulation analysis in this paper was compared with the numerical simulation results of Zeng Y.L. et al. [24]. The results are shown in Table 4. The results show that the slope safety factor results in this paper are close to the simulation results in the existing literature, which proves the reliability of the numerical simulation method.

Table 4. Comparison of the safety factor of reinforced-soil slope under different working conditions.

Working Condition	L = 6 m	L = 8 m	L = 10 m	L = 12 m
Zeng Y.L. et al.’s [24] Fs	1.124	1.140	1.159	1.208
This Article’s Fs	1.102	1.116	1.146	1.201

5. Influencing Factor Parameters and Numerical Simulation Analysis

Soil parameters selected: slope height H = 13.5 m, slope angle β = 65°, soil unit weight γ = 18.5 kN/m³, cohesion c = 20 kPa, and internal friction angle φ = 25°.

Fan F.F. et al. [40] conducted a theoretical study on the stability analysis of three-dimensional anti-slide piles and reinforced slopes under seismic loads, focusing on horizontal seismic acceleration k_h = 0.0–0.3 and proportional coefficient λ = −1.0–1.0; Li L.H. et al. [41] conducted model tests with reinforcing material tensile strength T = 1–45 kN/m and reinforcing material layers n = 0–6, verifying that reinforced soil has a significant stabilizing effect on slopes; Zhang F. et al. [42] selected reinforcement tensile strength T = 0–100 kN/m and analyzed the influence of three reinforcement lengths on the mechanical properties of reinforced-soil-retaining walls, indicating that reinforcement length has a significant impact on the maximum internal force of the reinforcement; Gu J. et al. [43] conducted numerical simulations to investigate the influence of geogrid length l = 6–9 m on slope safety factors, concluding that safety factors increase with length; Tan H.H. et al. [37] assumed an effective anti-slip force F = 500 kN/m on the side of the anti-slip piles and studied the impact of single pile positions ranging from 0 m to 26 m on slope stability. They found that as the anti-slip piles moved from the toe of the slope toward the slope crest, the safety factor first decreased and then increased; Zeng Y.L. et al. [24] set the length of the anti-slide piles to 12 m and analyzed the stability of slopes reinforced by a combination of anti-slide piles and geogrids. Based on the above research findings and combined with engineering requirements, this paper’s parameter analysis further expanded the range of key parameter values, with specific values shown in Table 5.

Table 5. Value range of parameter analysis.

kh	λ	T/kN/m	n	l/m	F/kN/m	a/m	h/m
0–0.3	−1.0–1.0	50–90	3–7	6–12	200–400	1–3	3–11

Horizontal seismic acceleration k_h is taken as 0.1, 0.2, and 0.3, and the proportional coefficient λ is taken as 0.5. The stability of the slope under different conditions is analyzed under the combined support of anti-slide piles and reinforced soil.

5.1. Analysis of the Stability of Reinforced-Soil Slopes Under Seismic Effects

When considering seismic effects, the horizontal seismic acceleration coefficient k_h = 0–0.3, and the proportional coefficient λ = 0.5. When the number of reinforcement layers n = 4 and the tensile strength of the reinforcement T = 50–90 kN/m, the safety factor of the reinforced-soil slope varies with different tensile strengths, as shown in Figure 3; when the tensile strength of the reinforcement T = 70 kN/m, and the number of reinforcement layers n = 3–7, the variation curve of the safety factor of the reinforced-soil slope under different reinforcement layer numbers is shown in Figure 4.

Figure 3. The influence of tensile strength of reinforcement on the slope safety factor.

Figure 4. The influence of reinforcement layers on the slope safety factor.

It can be seen from Figure 3 and Figure 4 that when k_h increases from 0.1 to 0.3, the maximum reductions in the slope safety factor due to the tensile strength of the reinforcing material and the number of reinforcing layers are 27.06% and 26.81%, respectively. This suggests that as the horizontal seismic acceleration increases, the safety factor of the reinforced-soil slope decreases significantly, making it highly prone to instability. When k_h = 0.1, increasing T from 50 kN/m to 90 kN/m and increasing n from three to seven layers results in respective increases in the safety factor of the reinforced-soil slope of 16.42% and 27.69%. This implies that as the tensile strength and number of layers of the reinforcing material increase, the stability of the slope is significantly improved; nevertheless, as k_h increases, the maximum improvement in the safety factor of the reinforced-soil slope decreases from 16.42% to 12.39% and from 27.69% to 24.07%, respectively, indicating that the greater the seismic effect, the more significant the adverse impact on the stability of the reinforced-soil slope. Therefore, strong seismic effects significantly weaken the reinforcement effect of geotextiles, leading to a substantial decrease in the stability of reinforced-soil slopes and an increased likelihood of instability and failure. Seismic effects are an indispensable factor in actual engineering projects, and in regions prone to strong earthquakes, it is necessary to reasonably increase reinforcement parameters to ensure slope stability.

Setting k_h = 0.1, λ = 0.5, reinforcement layer number n = 3–6, and reinforcement tensile strength T = 50–90 kN/m, the safety factor variation curves for reinforced-soil slopes under different tensile strengths and reinforcement layer numbers are shown in Figure 5.

Figure 5. The influence of tensile strength and the number of reinforcement layers on the slope safety factor.

As illustrated in Figure 5, the safety factor of reinforced-soil slopes increases significantly with higher tensile strength of the reinforcement and an increasing number of reinforcement layers. When T = 50 kN/m, increasing the n value from 3 to 6 only improves the safety factor of the reinforced-soil slope by 16.24%; nonetheless, when T = 90 kN/m, increasing n from 3 to 5 improves the safety factor by 16.66%, and increasing it to 6 further improves the safety factor by 25.10%. This demonstrates that enhancing the tensile strength of the reinforcing materials is more effective than increasing the number of reinforcing layers in improving slope stability. In consequence, in actual engineering projects, prioritizing the enhancement of the tensile strength of reinforcing materials is recommended to improve the slope stability.

5.2. Analysis of Slope Stability Under the Support of Anti-Slip Piles

For steep slopes, employing a single-reinforcement measure under seismic effects significantly affects stability, so anti-slide piles are added at the slope toe to enhance the slope safety factor.

When the horizontal distance between anti-slide piles and the slope toe a = 1.5 m, and the effective anti-slide force on the anti-slide piles F = 200–400 kN/m, and the pile length h = 3–11 m, the variation curves of the slope safety factor under different effective anti-slip forces and pile lengths are shown in Figure 6; when the pile length h = 5 m and the horizontal distance between anti-slide piles and the slope toe a = 1–3 m, the variation curves of the slope safety factor under different horizontal distances from the toe of the slope and different effective anti-slip forces of the anti-slip piles are shown in Figure 7. The effective anti-slide force on the anti-slide piles F = 250 kN/m, and the variation curves of the slope safety factor under different horizontal distances from the toe of the slope and different pile lengths are shown in Figure 8.

Figure 6. The influence of effective slip resistance and pile length of slip-resistant piles on the slope safety factor.

Figure 7. The influence of horizontal distance from the toe and effective slip resistance of anti-slip piles on the slope safety factor.

Figure 8. The influence of horizontal distance from the slope toe and pile length on the slope safety factor.

As depicted in Figure 6, as the effective anti-slip force of the anti-slip piles increases and the pile length grows, the slope safety factor significantly improves, and stability is markedly enhanced. When the effective anti-slip force F of the anti-slip piles is 200 kN/m and 400 kN/m, respectively, and the pile length h increases from 3 m to 11 m, the slope safety factor increases by 7.38% and 15.6%, respectively, indicating that increasing the pile length is beneficial for improving the slope safety factor; when the pile length h of the anti-slip piles is 3 m and 11 m, respectively, the effective anti-slip force F increases from 200 kN/m to 400 kN/m, resulting in increases in the slope safety factor of 58.30% and 70.41%, respectively. This reveals that increasing the effective anti-slip force of the anti-slip piles significantly enhances slope stability, and the longer the pile length, the greater the increase in the slope safety factor and the more pronounced the reinforcement effect. Thus, in actual construction, if the slope is generally stable but has a low safety factor, longer piles can be used to appropriately increase the slope safety factor; conversely, for slopes with poor stability, anti-slip piles with a larger effective anti-slip force can be used to reinforce the slope and significantly enhance slope stability.

As depicted in Figure 7 and Figure 8, increasing the horizontal distance between the anti-slip piles and the toe of the slope improves the slope safety factor and enhances slope stability. Increasing the horizontal distance a from the anti-slide piles to the slope toe from 1.0 m to 3.0 m resulted in safety factor increases of 0.76% and 0.99% for pile forces F of 200 kN/m and 400 kN/m, respectively. These results demonstrate that enlarging the horizontal distance a enhances the overall slope stability. Furthermore, the magnitude of the safety factor improvement associated with increasing a was greater for the higher pile force (F = 400 kN/m) than for the lower value (F = 200 kN/m). Conversely, for pile lengths h of 3 m and 11 m, the corresponding safety factor increases were 1.03% and 0.44%. This reveals that the enhancement in safety factor achieved by increasing a is more pronounced for shorter pile lengths (h = 3 m) compared to longer ones (h = 11 m). When the horizontal distance between the anti-slip piles and the slope toe is a constant value, the slope safety factor significantly increases with the increase in effective anti-slip force. For example, when a = 2.0 m, and F increases from 200 kN/m to 400 kN/m, the slope safety factor increases by 62.18%, and the increase is most pronounced when F = 350–400 kN/m, with the safety factor reaching a maximum percentage increase of 15.12%. However, the increase in the slope safety factor slightly decreases with the increase in pile length. For example, when a = 1.0 m, h = 3–5 m, and h = 9–11 m, the slope safety factor decreases from an increase of 2.59% to an increase of 2%. Hence, when using anti-slide piles to reinforce slopes, the arrangement of the anti-slide piles also has a significant impact on slope stability. The optimal reinforcement position should be arranged in the middle to lower part of the slope.

5.3. Numerical Simulation Analysis of Slope Stability Under the Combined Support of Anti-Slip Piles and Reinforced Soil

To further investigate the stability of slopes under the combined support of anti-slip piles and reinforced soil, finite element numerical simulation software Optum G2 2023 was used to conduct a simulation analysis of steep embankment slopes.

The soil parameters selected are as follows: slope height H = 13.5 m, slope angle β = 65°, soil weight γ = 18.5 kN/m³, cohesion c = 20 kPa, and internal friction angle φ = 25°. The material parameters of the model are shown in Table 6. Through references [33], the horizontal seismic acceleration k_h is taken as 0.1, and the proportional coefficient λ is taken as 0.5. The quasi-static seismic effects are simulated using fixed body loads, with units in acceleration (shown by the arrows in Figure 9). According to relevant studies [20,23,44], the slope is modeled using the Mohr–Coulomb material type, and the boundary conditions are selected as standard boundary conditions; that is, the left and right sides of the model are constrained by lateral displacement, and the bottom is constrained by both lateral and normal displacement. A uniformly distributed reinforcement grid is used within the slope, with a tensile strength of 70 kN/m, laid in layers at 1 m intervals, and a grid thickness of 5 mm.

Table 6. Model material parameters.

Materials	Gravity/kN/m3	Cohesion/kPa	Internal Friction Angle/(°)	Poisson Ratio	Elastic Modulus
Earth fill	18.5	20	25	0.3	30 MPa
Tendon	-	-	-	0.33	2.6 GPa
Slide-resistant pile	25	-	-	0.17	30 GPa

Figure 9. Anti-slide piles and geogrid combined support slope layout and grid division diagram.

During model establishment, the strength reduction limit analysis method was used to determine the safety factor of the slope. The model adopted long-term time conditions, with 3000 and 5000 units, and set adaptive meshing and three iterations. The initial number of units was 1000, and the adaptive control variable was shear dissipation. The numerical model of the combined anti-slide piles and geogrid slope support system is shown in Figure 9.

The length of geogrids is one of the key parameters affecting the stability of reinforced-soil slopes. To investigate the relationship between the length of geogrids and the location of slope failure slip surfaces, four scenarios were set up, corresponding to geogrid lengths l of 6, 8, 10, and 12 m. The shear dissipation cloud maps and safety factor comparisons of reinforced-soil slope failure modes under different working conditions are shown in Figure 10.

Figure 10. Shear dissipation cloud diagram and grid division diagram of failure mode of reinforced-soil slope under different working conditions.

As presented in Figure 10, an increase in the length of the geogrid has a significant effect on the location of the slope failure slip plane. As the geogrid length increases from 6 m to 12 m, the failure surface of the slope progressively recedes backward, indicating that the confining and restraining effect of the geogrid on the slip surface strengthens with its increasing length.

As the length of the geogrid increases, the overall displacement of the slope shows a decreasing trend. In Conditions 1 and 2, the areas with larger displacements are primarily concentrated near the toe of the slope, and the range of slope face displacement is relatively large; whereas, in Conditions 3 and 4, the range of areas with larger displacements is significantly reduced, and the peak displacement values also decrease. This signifies that increasing the length of the geogrid effectively restricts the displacement of slope soil and enhances slope stability.

The safety factor of the slope also shows a gradual increase with the increase in the length of the geogrid; when the length of the geogrid is increased from 6 m to 12 m, the safety factor of the slope increases by 28.67%.

The improvement in slope safety factor is closely related to the rearward displacement of the slip plane, changes in the morphology of the slip plane, and a reduction in displacement. An increase in the length of the geogrid causes the slip plane to shift rearward, thereby reducing the downward torque on the soil at the slip plane. Furthermore, changes in the morphology of the slip plane (deepening) increase the contact area between the soil and the geogrid, resulting in stronger anti-slip force provided by the geogrid, thereby enhancing the slope’s anti-slip stability. Moreover, the reduction in displacement signifies that the deformation of the slope soil is restricted, further improving the overall stability of the slope. Hereby, the increase in geogrid length enhances the slope safety factor and stability through multiple mechanisms, including altering the position and morphology of the slip surface and restricting soil displacement.

As can be observed from Figure 10c,d, as the length of the geogrid approaches the right boundary of the numerical model, the slope failure mode exhibits upward reversal failure. Thus, the width of the model crest was altered to further investigate the influence of the model boundary on the numerical simulation analysis results, as shown in Figure 11.

Figure 11. Shear dissipation cloud diagram and grid division diagram of reinforced-soil slope failure mode under different slope top widths.

As can be seen in Figure 11, the potential slip plane morphology of the slope changes significantly with an increase in the crest width. Figure 11a depicts the slip plane exhibits an upward curvature near the right boundary of the model; while in Figure 11b,c, as the crest width increases, the failure surface gradually normalizes, the curvature disappears, and the failure surface becomes deeper and smoother, conforming to the typical logarithmic spiral shape. This denotes that the model boundary significantly influences the development of the failure surface, and increasing the crest width helps reduce boundary constraint effects; In addition, the safety factor of the slope is significantly affected by the width of the slope top. As the anti-bending phenomenon of the sliding surface weakens, the safety factor gradually decreases, indicating that a reasonable model size is crucial for obtaining reliable numerical simulation results.

The above analysis shows that for steep embankment slopes, using a single geogrid for support results in relatively low stability. For that reason, anti-slip piles are added within the slope body to enhance the slope safety factor. Four different conditions were set up. Conditions 5 and 6 correspond to the horizontal distance a between the anti-slip piles and the toe of the slope, being 2, 4, and 6 m, respectively, with all piles having a length of 12 m. Condition 8 corresponds to a being 6 m, with a pile length of 13 m. The shear dissipation cloud diagram and safety factor comparison of the failure mode of the combined support slope of anti-slip piles and geogrids under different working conditions are shown in Figure 12.

Figure 12. Shear dissipation cloud diagram and grid division diagram of slope failure modes under different working conditions for anti-slip piles and geogrids combined support.

As indicated in Figure 12, compared to single geogrid support, the combined use of anti-slip piles and geogrids for slope support demonstrates significant advantages. Compared to Condition 2, which does not include anti-slip piles, the slope safety factors for Conditions 5–7 increased by 9.07%, 14.20%, and 4.53%, respectively. Meanwhile, when the anti-slip pile is positioned at the lower part of the slope (Condition 5) and the middle part (Condition 6), the safety factors are relatively high, especially when positioned in the middle, where the safety factor reaches its maximum value of 1.335. Nevertheless, when positioned at the upper part of the slope (Condition 7), the safety factor decreases, but by extending the pile length (Condition 8), the safety factor increases again. This establishes that the positioning and length of anti-slide piles are two interrelated factors. By reasonably positioning the piles and combining them with appropriate pile lengths, the slope safety factor can be maximized.

As indicated in Figure 12a–c, when the anti-slide pile is arranged at the lower part of the slope, the slip surface is obstructed near the piles, but there is still a large sliding area extending toward the middle of the slope; as the anti-slide pile is moved upward to the middle of the slope, the obstruction effect of the piles on the slip surface becomes more pronounced, and the range further decreases. When the pile is arranged in the upper part of the slope, the blocking effect of the sliding surface at the pile is relatively weakened, and there is a tendency to develop deeper into the slope. This establishes that the closer the anti-slide pile is positioned to the middle and lower sections of the slope, the more pronounced the blocking effect on the upper part of the slip surface, effectively limiting its upward development; when positioned at the upper section, the reinforcement effect of the anti-slide pile is relatively weakened. This also confirms that the optimal reinforcement position for anti-slide piles should be in the middle and lower sections of the slope.

Operating Conditions 7 and 8 represent whether the end of the anti-slip piles extends beyond the bottom layer of the geogrid. As shown in Figure 12c,d, increasing the pile length enhances the anti-slip pile’s ability to block the slip surface, making it difficult for the slip surface to continue developing toward the rear of the pile. The range of the slip surface further contracts near the pile. This indicates that increasing the pile length allows the anti-slip piles to penetrate deeper into the soil, enabling them to more effectively perform their anti-slip function, providing the slope with stronger anti-slip torque, and delaying the progression of slope failure.

6. Conclusions

(1)

Seismic effects have a significant adverse impact on slope stability. As the horizontal seismic acceleration k_h increases, the slope safety factor F_s gradually decreases. When conducting slope stability assessments, horizontal seismic effects are an important factor that cannot be ignored. In areas prone to strong earthquakes, appropriate reinforcement measures should be considered to ensure slope stability. Additionally, compared to single anti-slide piles or geogrid support, the combined use of anti-slide piles and geogrids demonstrates a remarkable advantage in enhancing slope stability.

(2)

As the tensile strength T of the reinforcing materials and the number of layers n increase, the safety factor F_s of the reinforced-soil slope obviously improves. Enhancing the tensile strength of the reinforcing materials is more effective than increasing the number of layers in improving slope stability. In actual engineering projects, prioritizing the enhancement of the tensile strength of the reinforcing materials is recommended to improve the slope stability. Increasing the geosynthetic length by l produces quantifiable improvements in the safety factor Fs of reinforced slopes. A progressive extension of l reduces overall slope displacement magnitude, lowers displacement peaks, and induces backward recession of the failure surface. These collective mechanisms enhance the slope stability.

(3)

As the effective anti-slip force F and pile length h of the anti-slip piles increase, the slope safety factor F_s remarkably improves. Furthermore, the longer h is, the greater the increase in the slope safety factor due to F, and the more significant the reinforcement effect. In actual construction, when the slope safety factor is low but the slope is relatively stable, longer piles can be used to moderately increase the slope safety factor. Conversely, when the slope stability is poor, anti-slide piles with higher effective anti-slide force can be employed to reinforce the slope, thereby greatly enhancing slope stability.

(4)

Increasing the horizontal distance a between the anti-slip piles and the toe of the slope improves slope stability. The larger the value of F and the smaller the value of h, the greater the increase in the slope safety factor F_s when increasing the value of a. When a is a constant value, the slope safety factor increases significantly with an increase in F, particularly during the stage where F = 350–400 kN/m. However, the rate of increase in the slope safety factor decreases slightly as h increases. In actual engineering applications, the optimal reinforcement position for anti-slide piles should be located in the middle to lower part of the slope body.

(5)

The current article’s study of the upper limit analysis of the stability of steep slopes supported by anti-slip piles and reinforced soil is limited to homogeneous, idealized slopes. The next step of the research will focus on complex soil layers or geological conditions, extending the analysis to the stability of slopes under hydraulic coupling effects and non-homogeneous soil conditions, and exploring in-depth research in combination with actual engineering cases.