Seismic Stability Analysis of Water-Saturated Composite Foundations near Slopes

Abstract

The seismic bearing capacity of water-saturated composite foundations adjacent to slopes is critical for engineering safety, yet it is significantly influenced by complex factors such as earthquakes and heavy rainfall. This paper establishes a failure mechanism model that involves both reinforced and non-reinforced zones, comprehensively considering the synergistic effects of seismic force, pore water pressure and group pile replacement rate, and thus addressing the issue that existing models struggle to account for the coupling effects of multiple factors. Based on the principle of virtual work, a general solution for ultimate bearing capacity is derived, and the optimal solution is obtained using the MATLAB R2023a exhaustive method. Findings reveal that pile group support substantially enhances bearing capacity: the improvement becomes more pronounced with higher soil strength parameters (φ, c) and replacement ratios. When the seismic acceleration coefficient increases from 0 to 0.3, the bearing capacity of the unreinforced foundation decreases by approximately 61.6% (from 134.71 kPa to 51.83 kPa), while group pile support can increase the bearing capacity by 433.2%. Notably, when soil strength is inherently high, the marginal benefit of pile group reinforcement diminishes. A case study in Fuzhou validates through numerical simulation that pile groups improve foundation stability by altering energy dissipation distribution, with the discrepancy between theoretical calculations and simulation results within 10%. The research results can directly guide the design of saturated composite foundations near slopes in earthquake-prone areas (such as Fujian and Guangdong) and enhance the seismic safety reserve by optimizing the replacement rate of group piles (recommended to be 0.2~0.3).

1. Introduction

In many landslide accidents that have occurred in China, some natural slopes or fill foundations adjacent to slopes fail due to the large-scale emergence of buildings and infrastructure construction, as well as excessive stacking loads [1,2]. In December 2015, an extremely serious landslide accident occurred at a construction waste disposal site in Guangdong Province, which was triggered by the massive accumulation of construction waste and construction debris at the site. The accident caused 73 deaths and the approved direct economic loss was 880 million yuan. In March 2019, a foundation in Shanxi Province could not bear the load of buildings, leading to a serious landslide, which caused more than 20 deaths. In May 2024, a residential building in Tongling built on a slope collapsed due to the decline of foundation bearing capacity caused by short-term heavy rainfall, resulting in 5 casualties. The state attaches great importance to this, so studying the ultimate bearing capacity of water-rich composite foundations adjacent to slopes is particularly critical [3].

The ultimate bearing capacity of slope-adjacent foundations is not only related to the safety of foundation structures but also plays a crucial role in the overall stability of slopes. Many scholars have attempted to conduct in-depth research using model test methods. Khan et al. [4] introduced indoor model tests on the load-settlement response and bearing capacity of slope-adjacent ground foundations buried on pipelines within soil slopes. Starting from the foundation failure mechanism, they analyzed the influence of the top distance from the foundation, burial depth, and pipeline diameter on the bearing capacity of slope-adjacent foundations. He et al. [5] used self-developed indoor simulated rainfall devices and dynamic load simulation devices for vehicles on the foundation to carry out seepage spatiotemporal evolution model tests. They found that the seepage expansion rate at the slope toe is high under rainfall conditions and divided the influence area of seepage into a shallow saturated zone, a seepage water filling zone, and a seepage water wetting zone. Sharma and Pain [6] analyzed the ultimate bearing capacity of shallow foundations under different influencing factors from the perspective of soil shear strength characteristics, believing that the foundation location, spatial variation of soil, and soil strength characteristics play a key role in the study of foundation bearing capacity. Hong et al. [7] took an open-pit coal mine as the research background and used the inclined strip method to calculate the bearing capacity of weak base foundations. The error between the calculation results of this method and the numerical simulation results was less than 5%.

The above scholars have conducted detailed discussions on the load response and deformation characteristics of slope-adjacent foundations using experimental methods and achieved certain results. However, some researchers have found that slope-adjacent foundations usually require the support of reinforcement technologies and have carried out a large number of studies on this. Abdi et al. [8] conducted an experimental study on the influence of geogrid-reinforced sandy slope eccentrically loaded strip foundation bearing capacity. The study shows that the position of the eccentricity position relative to the slope crest has a significant impact on the foundation performance. Yan Qing et al. [9], based on the unified strength theory, systematically analyzed the influence of the intermediate principal stress, the spacing between reinforcement layers, the number of reinforcement layers, and the tensile strength of the reinforcement on the ultimate bearing capacity of the reinforced foundation of the slope-adjacent strip foundation for the two modes of overall shear failure and composite failure of the reinforced foundation. Zheng Zheng et al. [10] established a numerical model of the granular pile composite foundation in saturated clay through the finite difference method, combined with the upper bound theorem of limit analysis, analyzed the deformation mode and mechanism of the pile body under the rigid foundation, studied the influence of pile length, burial depth, and replacement ratio on the ultimate bearing capacity of the composite foundation, and derived the equivalent strength of the group pile composite foundation and the ultimate bearing capacity coefficient under the shallow foundation failure mode. Chen Changfu et al. [11] studied the influence of granular material piles on the foundation bearing capacity for the ultimate bearing capacity problem of the horizontal reinforcement and granular material pile composite foundation under the embankment load, and derived the calculation formula of the composite foundation strength index affected by the pile replacement ratio and the degree of foundation soil consolidation degree. Hosseini et al. [12] carried out a stability analysis of the geosynthetic-reinforced shallow foundation through the finite element method, considering both the pull-out and tensile modes of geosynthetic failure, which improves the accuracy of the calculation results. Wang et al. [13] proposed a novel conical strain wedge (CSW) model for simulating the pile-cohesive soil interaction under lateral cyclic loading. The CSW model enhances the description of resistance and failure modes in the passive compressed soil zone and clarifies the performance characteristics of the pile-soil system under cyclic loading. Liang et al. [14] decomposed the pile-soil system into extended soil and fictional piles, set the compatibility conditions between the axial strain of the fictional piles and the corresponding average strain on the extended soil, and studied the interaction between piles under different consolidation conditions through a two-pile model, obtaining two pile-pile interaction factors. Qiu et al. [15] established the foundation model of the pile group and the difference equation of the p-y curve to study the horizontal force of each pile within the pile group and the horizontal bearing performance of each pile under the total thrust.

Although current studies have conducted in-depth analyses on the ultimate bearing capacity of slope-adjacent foundations under the action of support technologies and achieved a series of important and valuable results, which provide strong support for the development of this field. However, the influencing factors of the ultimate bearing capacity of slope-adjacent foundations are complex. Under extreme environmental conditions such as heavy rainfall and earthquakes, the influence of these factors on the bearing performance of the foundation is particularly significant, and the action mechanism is more complex, which may trigger a series of serious problems such as foundation instability and sudden drop in bearing capacity, bringing huge challenges to engineering construction.

In the actual construction of slope-adjacent foundation engineering, to effectively enhance the bearing capacity of the soil itself, a certain number of reinforcing bodies are usually set in the soil. These reinforcing bodies work in coordination with the original soil to form a composite foundation structure. However, at present, the research on the mechanical response, deformation characteristics, and ultimate bearing capacity of slope-adjacent composite foundations under special working conditions such as earthquakes and rainfall, when subjected to strip foundation loads and corresponding support effects, is relatively scarce. In view of this, the present study employs limit analysis to elucidate the dynamic response mechanism in the context of the coupled effects of earthquakes, rainfall, and pile group support. The study analyses the impact of earthquakes, rainfall, and replacement ratio on the stability of slope foundations, providing a reference for the seismic reinforcement of composite foundations under extreme conditions.

2. Mechanism of Equivalent Strength of Composite Foundation

Assuming that the composite formed by piles and soil in the reinforced area is regarded as a homogeneous composite material, this paper cites relevant conclusions from existing studies to derive the strength parameters of the homogeneous composite material formed by piles and soil [10].

$$\begin{array}{l}{\phi }_{\mathrm{pile}}={\displaystyle \frac{\pi }{2}}-2\arctan \left({\displaystyle \frac{1}{\sqrt{\psi K_{\mathrm{pc}}+\left(1-\psi \right)K_{\mathrm{ps}}}}}\right)\\ c_{\mathrm{pile}}={\displaystyle \frac{\psi c_{\mathrm{c}}\sqrt{K_{\mathrm{pc}}}+\left(1-\psi \right)c\sqrt{K_{\mathrm{ps}}}}{\sqrt{\psi K_{\mathrm{pc}}+\left(1-\psi \right)K_{\mathrm{ps}}}}}\end{array}$$

(1)

In the formula, c_pile and φ_pile are the cohesion and internal friction angle of the reinforced area, respectively; c_c, φ_c are the cohesion and internal friction angle of the pile body in the reinforced area, respectively; c, φ are the cohesion and internal friction angle of the soil, respectively; ψ is the group pile replacement ratio in the reinforced area; K_ps is the passive earth pressure coefficient of the soil, and Kpc is the passive earth pressure coefficient of the pile body, where

$$K_{\mathrm{ps}}={\tan }^2\left(\pi /4+\phi /2\right), K_{\mathrm{pc}}={\tan }^2\left(\pi /4+{\phi }_{\mathrm{c}}/2\right)$$

3. Calculation Model

As shown in Figure 1, the failure mechanism under the combined support of pile foundations and slope anti-slide piles under foundation load is decomposed into four dynamically changing failure bodies: I-ABF, II-1-FBF₀, II-2-FF₀C, III-CDEF. Different from the unsupported failure mode, the transition zone is divided into two parts due to the existence of the reinforcement area.

In Figure 1, the failure mechanism under the combined support of pile foundation and slope anti-slide piles under foundation load is decomposed into four real-time changing failure bodies: I-ABF, II-1-FBF₀, II-2-FF₀C, and III-CDEF. Different from the failure mode without support, the transition zone is divided into two due to the existence of the reinforcement zone. The discontinuous line BC of the damaged body II is a logarithmic helix, with the starting edge being r₀ = BF. Moreover, the θ Angle of the damaged body is not fixed and varies with the changes in relevant parameters. Due to the different internal friction Angle and cohesion within the reinforced area from those in the conventional unreinforced area, the damaged body II is divided by FF₀, and the left side is the non-reinforced area II-1-FBF₀. On the right is the reinforcement zone II-2-FF₀C, and the equation of the intermittent line BF₀ is: The equation of the intermittent line F₀C is: In the formula, r₂ is the distance from a certain calculation point on the intermittent break line F₀C to the origin, r₀ = BF, θ₀ is the Angle α formed by the intermittent break line FB and the bottom FA of the foundation, θ₂ is the Angle between the straight line of a certain calculation point of the destructive body II-1-FBF₀ and the initial pass FB, and θ₃ is the Angle between the straight line of a certain calculation point of the destructive body II-2-FF₀C and the initial pass FF₀.

As shown in the failure mechanism, the discontinuous line BC of failure body II is a logarithmic spiral, with the starting edge being r₀ = BF. The θ angle of the failure body is not fixed and changes with relevant parameters. Due to the different internal friction angles and cohesions in the reinforced area compared to the conventional unreinforced area, failure body II is divided by FF₀: the left side is the unreinforced zone II₂-FF₀C, and the right side is the reinforced zone II₁-FBF₀. The equation of discontinuous line BF₀ is:

$$r_1=r_0{\mathrm{e}}^{\left({\theta }_2-{\theta }_0\right)\tan \phi }$$

(2)

The equation of discontinuous line F₀C is:

$$r_2=F{F}_0{\mathrm{e}}^{\left({\theta }_3-{\displaystyle \frac{\pi }{2}}\right)\tan \phi }$$

(3)

In the formula, r₂ is the distance from a calculation point on the discontinuous line F₀C to the origin, r₀ = BF, θ₀ is the angle α formed by the discontinuous line FB and the foundation bottom FA, θ₂ is the angle between the straight line of a calculation point on the failure body II₁-FBF₀ and the starting edge FB, and θ₃ is the angle between the straight line of a calculation point on the failure body II₂-FF₀C and the starting edge FF₀.

4. Calculation Process

4.1. Basic Assumptions

(1)

The pile and soil reach the ultimate failure state simultaneously, and the stress of the pile body and the stress of the soil around the pile together constitute the macroscopic total stress of the pile-soil.

(2)

During the process of seismic loading, the increase in pore water pressure of the soil mass has not reached the critical state that leads to the loss of the soil’s shear strength. The soil particle framework can still maintain its structural structure and effective stress transmission capacity. The mechanical properties of the foundation soil are only affected by the seismic inertial force and the static distribution of pore water pressure, and the strength sudden change or slip effect caused by liquefaction is not considered [16].

(3)

There is no relative sliding between the pile and soil, the failure model conforms to the associated flow rule, and the slope-adjacent foundation composite reaches the ultimate failure state within the potential failure surface [17,18,19].

4.2. Velocity Field

The velocity field established through the angular relationships between velocity components is shown in Figure 2.

$$V_{01}={\displaystyle \frac{\sin \alpha }{\cos {\phi }_{\mathrm{pile}}}}\cdot V_0$$

(4)

$$V_1={\displaystyle \frac{\cos \left(\alpha -{\phi }_{\mathrm{pile}}\right)}{\cos {\phi }_{\mathrm{pile}}}}\cdot V_0$$

(5)

$$\left\{\begin{array}{l}{V}_2={\displaystyle \frac{\cos \left(\alpha -{\phi }_{\mathrm{pile}}\right){\mathrm{e}}^{\left({\theta }_2-\alpha \right)\tan {\phi }_{\mathrm{pile}}}}{\cos {\phi }_{\mathrm{pile}}}}\cdot V_0\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}{\theta }_2\in \left[0, \pi /2\right]\\ V_{20}={\displaystyle \frac{\cos \left(\alpha -{\phi }_{\mathrm{pile}}\right){\mathrm{e}}^{\left(\frac{\pi }{2}-\alpha \right)\tan {\phi }_{\mathrm{pile}}}}{\cos {\phi }_{\mathrm{pile}}}}\cdot V_0\\ V_{02}={\displaystyle \frac{\cos \left(\alpha -{\phi }_{\mathrm{pile}}\right){\mathrm{e}}^{\left(\frac{\pi }{2}-\alpha \right)\tan {\phi }_{\mathrm{pile}}}{\mathrm{e}}^{\left({\theta }_3-\frac{\pi }{2}\right)\tan \phi }}{\cos {\phi }_{\mathrm{pile}}}}\cdot V_0\hspace{1em}{\theta }_3\in \left[\pi /2, \alpha +\theta -{\displaystyle \frac{\pi }{2}}\right]\\ \theta ={\theta }_0+{\theta }_1\end{array}\right.$$

(6)

$$V_3={\displaystyle \frac{\cos \left(\alpha -{\phi }_{\mathrm{pile}}\right){\mathrm{e}}^{\left(\frac{\pi }{2}-\alpha \right)\tan {\phi }_{\mathrm{pile}}}{\mathrm{e}}^{\left(\theta +\alpha -\frac{\pi }{2}\right)\tan \phi }}{\cos {\phi }_{\mathrm{pile}}}}\cdot V_0$$

(7)

In the formula, V₀₁ is the discontinuous surface velocity of AB and FB on failure body I; V₁ is the initial sliding velocity of failure body II-1; V₂ is the discontinuous velocity at any point on discontinuous line BF₀ different from the initial velocity; V₀₂ is the discontinuous velocity at any point on discontinuous line F₀C different from the initial velocity V₂₀; V₃ is the discontinuous velocity on discontinuous line CD of failure body III in the passive zone.

4.3. External Force Power

(1)

Power of gravitational work:

As shown in the figure, the model is divided into four failure bodies: I-ABF, II₁-FBF₀ II₂-FF₀C, and III-CDEF. The total gravitational power is the sum of the self-weight power of the soil in each failure body, namely the sum of W_γI, W_γII-1, W_γII-2, and W_γIII. The specific calculations are as follows:

$$S_{\mathrm{I}}={\displaystyle \frac{b^2\tan \alpha }{4}}$$

(8)

$$W_{{\gamma }_{\mathrm{I}}}=G_{\mathrm{I}}{V}_0=S_{\mathrm{I}}\gamma V_0={\displaystyle \frac{{\mathrm{b}}^2\gamma \tan \alpha }{4}}\cdot V_0$$

(9)

$$l_{FC}={\displaystyle \frac{b{\mathrm{e}}^{\left(\frac{\pi }{2}-\alpha \right)\tan {\phi }_{\mathrm{pile}}}{\mathrm{e}}^{\left(\theta +\alpha -\frac{\pi }{2}\right)\tan \phi }}{2\cos \alpha }}$$

(10)

$$S_{\mathrm{III}}=S_{C^{\prime }EF}+S_{FC{C}^{\prime }}={\displaystyle \frac{1}{2}}\left(CC\prime \cdot CF\cos \phi +L\cdot EC\prime \sin \eta \right)$$

(11)

In the formulas:

$$W_{{\gamma }_{\mathrm{III}}}=G_{\mathrm{III}}{V}_3\cos \left(\alpha +\theta \right)={\displaystyle \frac{S_{\mathrm{III}}\gamma \cos \left(\alpha +\theta \right)\cos \left(\alpha -{\phi }_{\mathrm{pile}}\right){\mathrm{e}}^{\left(\frac{\pi }{2}-\alpha \right)\tan {\phi }_{\mathrm{pile}}}{\mathrm{e}}^{\left(\theta +\alpha -\frac{\pi }{2}\right)\tan \phi }}{\cos {\phi }_{\mathrm{pile}}}}\cdot V_0$$

(12)

where S represents the area of the failure body with the corresponding number, G represents the gravity of the corresponding failure body, and l represents the side length of the corresponding failure body.

The work achieved by failure body II can be divided into two parts: the gravitational work achieved by the reinforced zone and the gravitational work achieved by the unreinforced zone. To calculate the power, we need to combine the logarithmic spiral equations of each part with the slip angle θ. Decompose the failure body into countless tiny units, and then calculate the power through integration:

$$\begin{array}{l}{W}_{{\gamma }_{\mathrm{II}-1}}={\displaystyle {\int }_{\alpha }^{\pi /2}\left(\frac{\gamma r_0^2{V}_1{\mathrm{e}}^{3\left({\theta }_2-\alpha \right)\tan {\phi }_{\mathrm{pile}}}\cos {\theta }_2}{3}\right)}\mathrm{d}{\theta }_2={\displaystyle \frac{b^2\gamma \cos \left(\alpha -{\phi }_{\mathrm{pile}}\right)}{12\left(1+9{\tan }^2{\phi }_{\mathrm{pile}}\right){\cos }^2\alpha \cos {\phi }_{\mathrm{pile}}}}\cdot \\ \left[{\mathrm{e}}^{3\left({\displaystyle \frac{\pi }{2}}-\alpha \right)\tan {\phi }_{\mathrm{pile}}}-\sin \alpha -3\tan {\phi }_{\mathrm{pile}}\cos \alpha \right]\cdot V_0\end{array}$$

(13)

$$\begin{array}{l}{W}_{{\gamma }_{\mathrm{II}-2}}={\displaystyle {\int }_{\pi /2}^{\alpha +\theta }\left(\frac{\gamma r_1^2{V}_{20}{\mathrm{e}}^{3\left({\theta }_3-\frac{\pi }{2}\right)\tan \phi }\cos {\theta }_3}{3}\right)}\mathrm{d}{\theta }_3={\displaystyle \frac{b^2\gamma \cos \left(\alpha -{\phi }_{\mathrm{pile}}\right){\mathrm{e}}^{3\left(\frac{\pi }{2}-\alpha \right)\tan {\phi }_{\mathrm{pile}}}}{12\left(1+9{\tan }^2{\phi }_{\mathrm{pile}}\right){\cos }^2\alpha \cos {\phi }_{\mathrm{pile}}}}\cdot \\ \left\{{\mathrm{e}}^{3\left(\alpha +\theta -\pi /2\right)\tan \phi }\cdot \left[\sin \left(\theta +\alpha \right)+3\tan \phi \cos \left(\theta +\alpha \right)\right]-1\right\}\cdot V_0\end{array}$$

(14)

Therefore, the total gravitational power is:

$$W_{\gamma }=W_{{\gamma }_{\mathrm{I}}}+W_{{\gamma }_{\mathrm{II}-1}}+W_{{\gamma }_{\mathrm{II}-2}}+W_{{\gamma }_{\mathrm{III}}}$$

(15)

(2)

Power of foundation pressure:

$$W_{Q_{\mathrm{u}}}=Q_{\mathrm{u}}b{V}_0$$

(16)

(3)

Power of equivalent load:

$$W_q=qL{V}_3\cos \left(\alpha +\theta \right)={\displaystyle \frac{qL\cos \left(\alpha +\theta \right)\cos \left(\alpha -{\phi }_{\mathrm{pile}}\right){\mathrm{e}}^{\left(\frac{\pi }{2}-\alpha \right)\tan {\phi }_{\mathrm{pile}}}{\mathrm{e}}^{\left(\theta +\alpha -\frac{\pi }{2}\right)\tan \phi }}{\cos {\phi }_{\mathrm{pile}}}}\cdot V_0$$

(17)

(4)

Power of seismic force:

As shown in Figure 1, the seismic load is equivalent to a static horizontal ground load Gk_h and a vertical seismic load Gk_v acting on the damaged structure [20].

Horizontal seismic force:

$$\begin{array}{l}{W}_{k_{h\mathrm{II}-1}}=a_h{\displaystyle {\int }_{\alpha }^{\pi /2}\left(\frac{\gamma r_0^2{V}_1{\mathrm{e}}^{3\left({\theta }_2-\alpha \right)\tan {\phi }_{\mathrm{pile}}}\sin {\theta }_2}{3}\right)}\mathrm{d}{\theta }_2=a_h{\displaystyle \frac{b^2\gamma \cos \left(\alpha -{\phi }_{\mathrm{pile}}\right)}{12\left(1+9{\tan }^2{\phi }_{\mathrm{pile}}\right){\cos }^2\alpha \cos {\phi }_{\mathrm{pile}}}}\cdot \\ \left[3\tan {\phi }_{\mathrm{pile}}\cdot {\mathrm{e}}^{3\left({\displaystyle \frac{\pi }{2}}-\alpha \right)\tan {\phi }_{\mathrm{pile}}}-\cos \alpha -3\tan {\phi }_{\mathrm{pile}}\sin \alpha \right]\cdot V_0\end{array}$$

(18)

$$\begin{array}{l}{W}_{k_{h\mathrm{II}-2}}=a_h{\displaystyle {\int }_{\pi /2}^{\alpha +\theta }\left(\frac{\gamma r_1^2{V}_{20}{\mathrm{e}}^{3\left({\theta }_3-\frac{\pi }{2}\right)\tan \phi }\sin {\theta }_3}{3}\right)}\mathrm{d}{\theta }_3={\displaystyle \frac{a_h{b}^2\gamma \cos \left(\alpha -{\phi }_{\mathrm{pile}}\right){\mathrm{e}}^{3\left(\frac{\pi }{2}-\alpha \right)\tan {\phi }_{\mathrm{pile}}}}{12\left(1+9{\tan }^2\phi \right){\cos }^2\alpha \cos {\phi }_{\mathrm{pile}}}}\cdot \\ \left\{{\mathrm{e}}^{3\left(\alpha +\theta -\pi /2\right)\tan \phi }\cdot \left[\cos \left(\theta +\alpha \right)+3\tan \phi \sin \left(\theta +\alpha \right)\right]-3\tan \phi \right\}\cdot V_0\end{array}$$

(19)

$$\begin{array}{ll}{W}_{k_{h\mathrm{III}}}& =a_h{G}_{\mathrm{III}}{V}_3\sin \left(\alpha +\theta \right)\\ & =a_h{\displaystyle \frac{S_{\mathrm{III}}\gamma \sin \left(\alpha +\theta \right)\cos \left(\alpha -{\phi }_{\mathrm{pile}}\right){\mathrm{e}}^{\left(\frac{\pi }{2}-\alpha \right)\tan {\phi }_{\mathrm{pile}}}{\mathrm{e}}^{\left(\theta +\alpha -\frac{\pi }{2}\right)\tan \phi }}{\cos {\phi }_{\mathrm{pile}}}}\cdot V_0\end{array}$$

(20)

Vertical seismic force:

$$W_{k_{v\mathrm{I}}}=a_v{\displaystyle \frac{{\mathrm{b}}^2\gamma \tan \alpha }{4}}\cdot V_0$$

(21)

$$\begin{array}{l}{W}_{k_{v\mathrm{II}-1}}=a_v{\displaystyle \frac{b^2\gamma \cos \left(\alpha -{\phi }_{\mathrm{pile}}\right)}{12\left(1+9{\tan }^2{\phi }_{\mathrm{pile}}\right){\cos }^2\alpha \cos {\phi }_{\mathrm{pile}}}}\cdot \\ \left[{\mathrm{e}}^{3\left({\displaystyle \frac{\pi }{2}}-\alpha \right)\tan \phi }-\sin \alpha -3\tan {\phi }_{\mathrm{pile}}\cos \alpha \right]\cdot V_0\end{array}$$

(22)

$$\begin{array}{l}{W}_{k_{v\mathrm{II}-2}}=a_v{\displaystyle \frac{b^2\gamma \cos \left(\alpha -{\phi }_{\mathrm{pile}}\right){\mathrm{e}}^{3\left(\frac{\pi }{2}-\alpha \right)\tan {\phi }_{\mathrm{pile}}}}{12\left(1+9{\tan }^2{\phi }_{\mathrm{pile}}\right){\cos }^2\alpha \cos {\phi }_{\mathrm{pile}}}}\cdot \\ \left\{{\mathrm{e}}^{3\left(\alpha +\theta -\pi /2\right)\tan \phi }\cdot \left[\sin \left(\theta +\alpha \right)+3\tan \phi \cos \left(\theta +\alpha \right)\right]-1\right\}\cdot V_0\end{array}$$

(23)

$$W_{k_{v\mathrm{III}}}=a_v{\displaystyle \frac{S_{\mathrm{III}}\gamma \cos \left(\alpha +\theta \right)\cos \left(\alpha -{\phi }_{\mathrm{pile}}\right){\mathrm{e}}^{\left(\frac{\pi }{2}-\alpha \right)\tan {\phi }_{\mathrm{pile}}}{\mathrm{e}}^{\left(\theta +\alpha -\frac{\pi }{2}\right)\tan \phi }}{\cos {\phi }_{\mathrm{pile}}}}\cdot V_0$$

(24)

The total power of seismic forces is:

$$Wk=W_{k_{h\mathrm{II}-1}}+W_{k_{h\mathrm{II}-2}}+W_{k_{h\mathrm{III}}}+W_{k_{v\mathrm{I}}}+W_{k_{v\mathrm{II}-1}}+W_{k_{v\mathrm{II}-2}}+W_{k_{v\mathrm{III}}}$$

(25)

(5)

Total power of pore water pressure u:

The pore water pressure is vertically distributed on each discontinuous surface in the form of a uniformly distributed load. The total power is equal to the sum of the power achieved by the pore water pressure on all discontinuous surfaces AB, FB, BF₀, F₀C, and CD.

$$l_{AB}=l_{FB}={\displaystyle \frac{b}{2\cos \alpha }}$$

(26)

$$l_{F{F}_0}={\displaystyle \frac{b}{2\cos \alpha }}{\mathrm{e}}^{\left({\displaystyle \frac{\pi }{2}}-\alpha \right)\tan {\phi }_{\mathrm{pile}}}$$

(27)

$$f_2=C\prime E\sin \eta -\tan \alpha \cdot x_A$$

(28)

$$y_{AB}=\tan \alpha \cdot x-f_2$$

(29)

$$y_{CD}=-\tan \left(\alpha +\theta -\pi /2-\phi \right)\cdot x$$

(30)

$$\begin{array}{ll}{W}_{u_{AB}}& ={\displaystyle {\int }_{x_B}^{x_A}{r}_u\gamma }\left(C\prime E\sin \eta -y_{AB}\right)V_{01}\cos \left(\pi /2-{\phi }_{\mathrm{pile}}\right)\mathrm{d}x\\ & =r_u\gamma \tan {\phi }_{\mathrm{pile}}\sin \alpha {\displaystyle {\int }_{x_B}^{x_A}\left(C\prime E\sin \eta +f_1-\tan \alpha \cdot x\right)}\mathrm{d}x\cdot V_0\\ & =r_u\gamma \tan {\phi }_{\mathrm{pile}}\sin \alpha \cdot \\ & \left[\left(C\prime E\sin \eta x_A+x_A{f}_1-{\displaystyle \frac{\tan \alpha \cdot x_A^2}{2}}\right)-\left(x_{B}C\prime E\sin \eta +x_B{f}_1-{\displaystyle \frac{\tan \alpha \cdot x_B^2}{2}}\right)\right]\cdot V_0\end{array}$$

(31)

$$\begin{array}{ll}{W}_{u_{B{F}_0}}& ={\displaystyle {\int }_{\alpha }^{\frac{\pi }{2}}{r}_u\gamma l_{FB}{\mathrm{e}}^{\left({\theta }_2-\alpha \right)\tan {\phi }_{\mathrm{pile}}}\sin {\theta }_2}{V}_2\cos \left(\pi /2-{\phi }_{\mathrm{pile}}\right)\mathrm{d}{\theta }_2\\ & =r_u\gamma {\displaystyle \frac{b\tan {\phi }_{\mathrm{pile}}\cos \left(\alpha -{\phi }_{\mathrm{pile}}\right)\cdot V_0}{2\cos \alpha }}{\displaystyle {\int }_{\alpha }^{\frac{\pi }{2}}{\mathrm{e}}^{2\left({\theta }_2-\alpha \right)\tan {\phi }_{\mathrm{pile}}}\sin {\theta }_2}\mathrm{d}{\theta }_2\\ & =r_u\gamma {\displaystyle \frac{b\tan {\phi }_{\mathrm{pile}}\cos \left(\alpha -{\phi }_{\mathrm{pile}}\right)}{2\cos \alpha }}\cdot {\displaystyle \frac{2\tan {\phi }_{\mathrm{pile}}\left({\mathrm{e}}^{2\left(\frac{\pi }{2}-\alpha \right)\tan {\phi }_{\mathrm{pile}}}-\sin \alpha \right)+\cos \alpha }{1+5{\tan }^2{\phi }_{\mathrm{pile}}}}\cdot V_0\end{array}$$

(32)

$$\begin{array}{ll}{W}_{u_{F_{0}C}}& ={\displaystyle {\int }_{\frac{\pi }{2}}^{\alpha +\theta }{r}_u\gamma l_{F{F}_0}{\mathrm{e}}^{\left({\theta }_3-\frac{\pi }{2}\right)\tan \phi }\sin {\theta }_3}{V}_{02}\cos \left(\pi /2-\phi \right)\mathrm{d}{\theta }_3\\ & =r_u\gamma {\displaystyle \frac{b\sin \phi \cos \left(\alpha -{\phi }_{\mathrm{pile}}\right){\mathrm{e}}^{2\left(\frac{\pi }{2}-\alpha \right)\tan {\phi }_{\mathrm{pile}}}\cdot V_0}{2\cos \alpha \cos {\phi }_{\mathrm{pile}}}}\cdot {\displaystyle {\int }_{\frac{\pi }{2}}^{\alpha +\theta }{\mathrm{e}}^{2\left({\theta }_3-\frac{\pi }{2}\right)\tan \phi }\sin {\theta }_3}\mathrm{d}{\theta }_3\\ & =r_u\gamma {\displaystyle \frac{b\sin \phi \cos \left(\alpha -{\phi }_{\mathrm{pile}}\right){\mathrm{e}}^{2\left(\frac{\pi }{2}-\alpha \right)\tan {\phi }_{\mathrm{pile}}}}{2\cos \alpha \cos {\phi }_{\mathrm{pile}}}}\cdot \\ & {\displaystyle \frac{2\tan \phi \left[\sin \left(\alpha +\theta \right){\mathrm{e}}^{2\left(\alpha +\theta -\frac{\pi }{2}\right)\tan \phi }-1\right]-\cos \left(\alpha +\theta \right){\mathrm{e}}^{2\left(\alpha +\theta -\frac{\pi }{2}\right)\tan \phi }}{1+5{\tan }^2\phi }}\cdot V_0\end{array}$$

(33)

$$\begin{array}{ll}{W}_{u_{\mathrm{CD}}}& ={\displaystyle {\int }_0^{x_C}{r}_u\gamma }\left(C\prime E\sin \eta -y_{CD}\right)V_3\cos \left(\pi /2-\phi \right)\mathrm{d}x\\ & =r_u\gamma {\displaystyle \frac{\sin \phi \cos \left(\alpha -{\phi }_{\mathrm{pile}}\right){\mathrm{e}}^{\left(\frac{\pi }{2}-\alpha \right)\tan {\phi }_{\mathrm{pile}}}{\mathrm{e}}^{\left(\theta +\alpha -\frac{\pi }{2}\right)\tan \phi }}{\cos {\phi }_{\mathrm{pile}}}}\cdot \\ & {\displaystyle {\int }_0^{x_C}\left(C\prime E\sin \eta -\tan \left(\pi /2+\phi -\alpha -\theta \right)\cdot x\right)}\mathrm{d}x\cdot V_0\\ & =r_u\gamma {\displaystyle \frac{\sin \phi \cos \left(\alpha -{\phi }_{\mathrm{pile}}\right){\mathrm{e}}^{\left(\frac{\pi }{2}-\alpha \right)\tan {\phi }_{\mathrm{pile}}}{\mathrm{e}}^{\left(\theta +\alpha -\frac{\pi }{2}\right)\tan \phi }}{\cos {\phi }_{\mathrm{pile}}}}\cdot \\ & \left[C\prime E\sin \eta x_C-{\displaystyle \frac{\tan \left(\pi /2+\phi -\alpha -\theta \right)\cdot x_C^2}{2}}\right]\cdot V_0\end{array}$$

(34)

Therefore, the total power of pore water pressure is:

$$W_u=2W_{u_{AB}}+W_{u_{B{F}_0}}+W_{u_{F_{0}C}}+W_{u_{CD}}$$

(35)

4.4. Internal Energy Dissipation Rate

(1)

The active zone I of the failure body has internal energy dissipation above the discontinuous surfaces AB and FB, and the internal energy dissipation of the two discontinuous surfaces is completely the same. The specific calculation method is the product of the length of the discontinuous surface, the cohesive force, and the velocity component:

$$D_{AB}=D_{FB}=c_{\mathrm{pile}}{l}_{AB}{V}_{01}\cos {\phi }_{\mathrm{pile}}=c_{\mathrm{pile}}{\displaystyle \frac{b}{2}}\tan \alpha \cdot V_0$$

(36)

(2)

The energy dissipation of the transitional radiation failure zone II of the failure body occurs along the logarithmic spiral line BC. Meanwhile, internal energy consumption also arises within the failure body II due to deformation. According to relevant literature [21], it is known that:

$$\begin{array}{l}{D}_{B{F}_0}=D_{B{F}_{0}F}=c_{\mathrm{pile}}{\displaystyle {\int }_{\alpha }^{\frac{\pi }{2}}{l}_{FB}{\mathrm{e}}^{\left({\theta }_2-\alpha \right)\tan {\phi }_{\mathrm{pile}}}}{V}_2\mathrm{d}{\theta }_2\\ ={\displaystyle \frac{cb\cos \left(\alpha -{\phi }_{\mathrm{pile}}\right)}{2\cos \alpha \cos {\phi }_{\mathrm{pile}}}}{\displaystyle {\int }_{\alpha }^{\frac{\pi }{2}}{\mathrm{e}}^{2\left({\theta }_2-\alpha \right)\tan {\phi }_{\mathrm{pile}}}}\mathrm{d}\theta \cdot V_0\\ ={\displaystyle \frac{cb\cos \left(\alpha -{\phi }_{\mathrm{pile}}\right)}{4\cos \alpha \sin {\phi }_{\mathrm{pile}}}}\left[{\mathrm{e}}^{\left(\pi -2\alpha \right)\tan {\phi }_{\mathrm{pile}}}-1\right]\cdot V_0\end{array}$$

(37)

$$\begin{array}{l}{D}_{F_{0}C}=D_{F_{0}CF}=c{\displaystyle {\int }_{\frac{\pi }{2}}^{\theta +\alpha }{l}_{F{F}_0}{\mathrm{e}}^{\left({\theta }_3-\frac{\pi }{2}\right)\tan \phi }}{V}_{02}\mathrm{d}{\theta }_3\\ ={\displaystyle \frac{cb\cos \left(\alpha -{\phi }_{\mathrm{pile}}\right){\mathrm{e}}^{\left(\pi -2\alpha \right)\tan {\phi }_{\mathrm{pile}}}}{2\cos \alpha \cos {\phi }_{\mathrm{pile}}}}{\displaystyle {\int }_{\frac{\pi }{2}}^{\theta +\alpha }{\mathrm{e}}^{2\left({\theta }_3-\frac{\pi }{2}\right)\tan \phi }}\mathrm{d}\theta \cdot V_0\\ ={\displaystyle \frac{cb\cos \left(\alpha -{\phi }_{\mathrm{pile}}\right){\mathrm{e}}^{\left(\pi -2\alpha \right)\tan {\phi }_{\mathrm{pile}}}}{4\cos \alpha \cos {\phi }_{\mathrm{pile}}\tan \phi }}\left[{\mathrm{e}}^{\left(2\theta +2\alpha -\pi \right)\tan \phi }-1\right]\cdot V_0\end{array}$$

(38)

(3)

The passive failure zone III of the failure body is connected to the transitional radiation zone, and the internal energy dissipation generated by the failure body occurs on the straight line CD which is:

$$\begin{array}{ll}{D}_{CD}& =cCC\prime V_3\cos \phi \\ & =-c\left[L\sin \eta +FC\sin \left(\alpha +\theta +\eta \right)\right]\cdot \\ & {\displaystyle \frac{\cos \left(\alpha -{\phi }_{\mathrm{pile}}\right){\mathrm{e}}^{\left(\frac{\pi }{2}-\alpha \right)\tan {\phi }_{\mathrm{pile}}}{\mathrm{e}}^{\left(\theta +\alpha -\frac{\pi }{2}\right)\tan \phi }\cos \phi }{\cos \left(\alpha +\theta +\eta -\phi \right)\cos {\phi }_{\mathrm{pile}}}}\cdot V_0\end{array}$$

(39)

Therefore, the total internal energy dissipation is:

$$D=2D_{AB}+2D_{B{F}_0}+2D_{F_{0}C}+D_{CD}$$

(40)

4.5. Solution for Ultimate Bearing Capacity

According to the virtual power principle, under the ultimate failure state, the work achieved by external forces in the limit analysis theory is equal to the internal energy dissipation, and the equation can be obtained as:

$$W_{Qu}+W_{\gamma }+W_q+W_k+W_u=D$$

(41)

Therefore, the general solution for the ultimate bearing capacity can be inversely solved as:

$$Q_{\mathrm{u}}=(D-W_{\gamma }-W_q-W_k-W_u)/(b{V}_0)$$

(42)

The constraint conditions are:

$$\left\{\begin{array}{l}\phi \le \alpha \le {\displaystyle \frac{\pi }{4}}+{\displaystyle \frac{\phi }{2}}\\ 0\le \theta \le \pi \\ {\displaystyle \frac{\pi }{2}}\le \alpha +\theta \le \pi \\ 0\le \alpha +\theta +\eta -\phi -{\displaystyle \frac{\pi }{2}}\le \eta \end{array}\right.$$

(43)

Under the constraint conditions of Equation (43), the exhaustive method in MATLAB R2023a (MathWorks, Natick, MA, The United States) software can be used to solve the optimal upper limit value of the ultimate bearing capacity, and the specific program commands are no longer described in detail.

5. Parameter Analysis

According to the Technical Code for Building Pile Foundations—2008 [22], it is specified that the concrete strength used in the construction of pile foundation support should not be less than C25. Since this paper considers the stability of the horizontal foundation near slopes under complex conditions such as earthquakes and heavy rainfall, C60 high-strength concrete piles are adopted for pile group reinforcement. For C60 concrete, the corresponding cohesive force cc is generally 500–2000 kPa, and the internal friction angle φc is greater than or equal to 40° [23]. The theoretical calculation parameters used in this chapter when replacing foundation soil with pile groups are: cc = 1000 kPa and φc = 45°.

5.1. Influence of Seismic Force

Figure 3 shows the influence law of pile foundation support on the seismic ultimate bearing capacity of water-rich foundations near slopes under different internal friction angles φ. The internal friction angles of the soil are 30°, 32.5°, and 35°, and the horizontal seismic acceleration coefficients k_h are 0, 0.1, 0.2, and 0.3. Other research parameters are as follows: foundation scale coefficient a = 1, foundation width b = 2 m, soil unit weight γ = 18 kN/m³, slope angle η = 45°, soil cohesion c = 15 kPa, foundation embedment depth h = 1 m pore water pressure coefficient r_u = 0.2, and pile group replacement rate ψ = 0.1.

From an overall perspective, under different soil internal friction angles in Figure 3, by comparing the seismic ultimate bearing capacity of the water-rich foundation near the slope before and after pile group support, it can be obtained that the ultimate bearing capacity of the foundation obtained by adopting the pile group support and reinforcement method is significantly higher than that without reinforcement support. As the horizontal seismic force coefficient increases and the seismic intensity increases, the seismic ultimate bearing capacity Q_u of the water-rich foundation near the slope decreases significantly, and the two show a linear negative correlation. As the soil internal friction angle φ increases, the seismic ultimate bearing capacity Q_u of the water-rich foundation near the slope increases significantly, and the two show a linear positive correlation. In addition, under the same conditions, it is clearly seen from

Table 1. Bearing Capacity under Different Internal Friction Angles and Seismic Force Coefficients.

φ/°	c/kPa	ru	kh	kv	Ultimate Bearing Capacity of Unreinforced Foundation/kPa	Ultimate Bearing Capacity of Pile Group Reinforced Foundation/kPa	Relative Difference Δ
30	15	0.2	0	0	134.71	338.52	151.30%
	15	0.2	0.1	0.05	107.08	317.80	196.79%
	15	0.2	0.2	0.1	79.45	297.08	273.92%
	15	0.2	0.3	0.15	51.83	276.36	433.20%
32.5	15	0.2	0	0	164.46	375.88	128.55%
	15	0.2	0.1	0.05	133.34	352.78	164.57%
	15	0.2	0.2	0.1	102.22	329.69	222.53%
	15	0.2	0.3	0.15	71.10	306.60	331.22%
35	15	0.2	0	0	202.81	421.11	107.64%
	15	0.2	0.1	0.05	166.96	395.20	136.70%
	15	0.2	0.2	0.1	130.41	369.29	183.18%
	15	0.2	0.3	0.15	93.86	343.38	265.84%

that when the internal friction angle of the soil below the foundation load is large, the enhancement effect of pile group support on the bearing capacity of the foundation near the slope generally weakens.

Combining the data in Figure 3 and Table 1, taking the bearing capacity Q_u = 134.71 kPa from Figure 3a with an internal friction angle φ = 30°, no seismic effect (k_h = 0), and without pile group reinforcement as a baseline for comparison: as the horizontal seismic acceleration coefficient k_h increases to 0.1, 0.2, and 0.3, the foundation bearing capacity decreases to 107.08 kPa, 79.45 kPa, and 51.83 kPa, respectively. Using another example from Figure 3a with φ = 30°, considering seismic effects (k_h = 0.1), and with pile group reinforcement Q_u = 317.80 kPa: as the internal friction angle φ increases to 32.5° and 35°, the bearing capacity increases to 352.78 kPa and 395.20 kPa, respectively, with relative improvements of 11.2% and 24.6%. Additionally, when applying pile group support with a replacement ratio ψ = 0.1 under the same seismic conditions, the seismic ultimate bearing capacities of the water-rich slope-adjacent foundation are 338.52 kPa, 317.80 kPa, 297.08 kPa, and 276.36 kPa for k_h = 0, 0.1, 0.2, and 0.3, respectively. The increases in bearing capacity compared to the unreinforced case are 203.82 kPa, 210.72 kPa, 217.63 kPa, and 224.53 kPa, corresponding to relative enhancements of 151.30%, 196.79%, 273.92%, and 433.20%. Although seismic activity reduces the ultimate bearing capacity of slope-adjacent foundations, pile group reinforcement replaces part of the soil beneath the foundation, increasing the overall soil strength, enhancing self-stability, and thereby improving the seismic resistance of water-rich slope-adjacent foundations.

In summary, for foundation soils with a small internal friction angle, it is necessary to pay special attention and carry out targeted support and reinforcement. The ultimate bearing capacity of the slope-adjacent foundation is greatly affected by earthquakes. With the progress of construction on the water-rich slope-adjacent foundation and the superposition of foundation loads, the bearing capacity of the slope-adjacent foundation faces great challenges. In the design of the seismic bearing capacity of the water-rich slope-adjacent foundation, considering the use of pile groups directly below the foundation load for support can effectively improve the bearing capacity of the water-rich slope-adjacent foundation, enhance construction stability, and reduce the risk of foundation instability.

5.2. Influence of Pore Water

Figure 4 shows the influence law of pile foundation support on the seismic ultimate bearing capacity of water-rich foundations near slopes under different soil cohesion values c. The soil cohesion values are 10°, 12.5°, and 15°, while the pore water pressure coefficients r_u are 0, 0.1, 0.2, 0.3, and 0.4. Other research parameters are as follows: foundation scale coefficient a = 1, foundation width b = 2 m soil unit weight γ = 18 kN/m³, slope angle η = 45°, soil internal friction angle φ = 30°, foundation embedment depth h = 1 m, horizontal seismic force coefficient k_h = 0.1, and pile group replacement rate ψ = 0.1.

Visually from Figure 4, by comparing the seismic ultimate bearing capacity Q_u of the water-rich foundation near the slope before and after pile group support, it can be seen that for soil foundations with different cohesions, the ultimate bearing capacity of the foundation obtained by using the pile group support and reinforcement method is significantly higher than that without reinforcement. As the pore water pressure coefficient of the soil in the slope-adjacent foundation increases, the seismic ultimate bearing capacity Q_u of the water-rich foundation near the slope decreases significantly, and the two show an approximately linear negative correlation. As the soil cohesion c increases, the seismic ultimate bearing capacity Q_u of the water-rich foundation near the slope increases significantly, and the two show a linear positive correlation. Combined with the data in

Table 2. Bearing Capacity under Different Cohesion and Pore Water Pressure.

c/kPa	φ/°	ru	kh	kv	Ultimate Bearing Capacity of Unreinforced Foundation/kPa	Ultimate Bearing Capacity of Pile Group Reinforced Foundation/kPa	Relative Difference Δ
10	30	0	0.1	0.05	74.31	297.95	300.98%
	30	0.1	0.1	0.05	65.49	263.27	301.99%
	30	0.2	0.1	0.05	55.05	228.58	315.24%
	30	0.3	0.1	0.05	38.99	193.90	397.34%
	30	0.4	0.1	0.05	22.47	134.21	497.39%
12.5	30	0	0.1	0.05	99.51	337.80	239.47%
	30	0.1	0.1	0.05	90.69	307.88	239.47%
	30	0.2	0.1	0.05	81.88	273.19	233.66%
	30	0.3	0.1	0.05	66.57	238.51	258.27%
	30	0.4	0.1	0.05	50.51	203.82	303.53%
15	30	0	0.1	0.05	124.71	371.71	198.05%
	30	0.1	0.1	0.05	115.90	349.27	201.36%
	30	0.2	0.1	0.05	107.08	317.80	196.79%
	30	0.3	0.1	0.05	94.16	283.12	200.69%
	30	0.4	0.1	0.05	78.10	248.43	218.12%

, taking the bearing capacity Q_u = 74.31 kPa in Figure 4a with cohesion c = 10 kPa, without considering pore water pressure (r_u = 0), and without pile group reinforcement as an example for comparative analysis, with the occurrence of pore water in the soil caused by heavy rainfall, and the pore water pressure coefficient r_u gradually increasing to 0.1, 0.2, 0.3, and 0.4, the foundation bearing capacity decreases to 65.49 kPa, 55.05 kPa, 38.99 kPa, and 22.47 kPa, respectively, with relative decreases of 11.86%, 25.92%, 47.53%, and 69.77%. Similarly, in Figure 4a, under the reinforcement of pile groups, when the cohesion c = 10 kPa and the pore water pressure coefficient r_u = 0.4, the obtained bearing capacity Q_u = 134.21 kPa. As the soil cohesion gradually increases to c = 12.5 kPa and c = 15 kPa, the foundation bearing capacity increases to 203.82 kPa and 248.43 kPa, respectively.

As shown in Figure 4c and Table 2, when the pile group support with a replacement ratio ψ = 0.1 is adopted, the seismic ultimate bearing capacities of the water-rich foundation near the slope are 371.71 kPa, 349.27 kPa, 317.80 kPa, 283.12 kPa, and 248.43 kPa, respectively, under the same water-rich conditions (r_u = 0, r_u = 0.1, r_u = 0.2, r_u = 0.3, r_u = 0.4). Compared with the foundation bearing capacities of 124.71 kPa, 115.90 kPa, 107.08 kPa, 94.16 kPa, and 78.10 kPa without pile group support, the foundation bearing capacities under pile group reinforcement increase by 247.00 kPa, 233.37 kPa, 210.72 kPa, 188.96 kPa, and 170.34 kPa, respectively, with relative differences of 198.05%, 201.36%, 196.79%, 200.69%, and 218.12%. Due to the increase in soil pore water pressure, the foundation soil is saturated and softened, resulting in the decrease in soil strength and the significant reduction in the ultimate bearing capacity of the slope-adjacent foundation. However, after the adoption of pile group support, part of the soil under the foundation load is replaced by high-strength pile groups, which increases the overall self-stability of the soil and improves the seismic ultimate bearing capacity of the water-rich foundation near the slope.

In summary, under the action of different soil cohesion values c, the greater the soil cohesion, the stronger the foundation bearing capacity. The ultimate bearing capacity of the slope-adjacent foundation is significantly influenced by pore water. Meanwhile, with the progress of construction on the slope-adjacent foundation and the superposition of foundation loads, even in the presence of heavy rainfall and earthquakes, adopting pile groups for support directly below the foundation load can effectively enhance the bearing capacity of the water-rich slope-adjacent foundation.

5.3. Influence of Replacement Ratio

Figure 5 shows the influence law of foundation replacement ratio under pile group support on the seismic ultimate bearing capacity of water-rich foundations near slopes. The pore water pressure coefficients are r_u = 0.1, 0.2, 0.3, the horizontal seismic acceleration coefficients are k_h = 0, 0.1, 0.2, 0.3, and the replacement ratios are ψ = 0.1, 0.2, 0.3, 0.4, 0.5. Other research parameters are as follows: foundation scale coefficient a = 1, foundation width b = 2 m, soil unit weight γ = 18 kN/m³, slope angle η = 45°, soil cohesion c = 15 kPa, and foundation embedment depth h = 1 m.

From an overall perspective of Figure 5, the following rules can be obtained: as the foundation replacement ratio under pile group support gradually increases, more original soil of the slope-adjacent foundation is replaced by pile groups, the soil strength under the foundation load gradually increases, and the seismic ultimate bearing capacity Q_u of the water-rich foundation near the slope significantly increases, showing a linear positive correlation between the two. Combined with the data in

Table 3. Foundation Bearing Capacity under Different Pile Group Replacement Rates.

kh	kv	ξ	ru	Ultimate Bearing Capacity of Pile Group Reinforced Foundation/kPa	Relative Difference Δ2
0	0	0.1	0.1	303.84	0.00%
	0	0.2	0.1	389.98	28.35%
	0	0.3	0.1	470.61	54.89%
	0	0.4	0.1	546.43	79.84%
	0	0.5	0.1	617.96	103.38%
0.1	0.05	0.1	0.1	349.27	0.00%
	0.05	0.2	0.1	447.28	28.06%
	0.05	0.3	0.1	540.46	54.74%
	0.05	0.4	0.1	628.88	80.06%
	0.05	0.5	0.1	714.50	104.57%
0.2	0.1	0.1	0.3	331.77	0.00%
	0.1	0.2	0.3	424.91	28.08%
	0.1	0.3	0.3	513.57	54.80%
	0.1	0.4	0.3	598.53	80.41%
	0.1	0.5	0.3	680.41	105.09%
0.3	0.1	0.1	0.3	241.67	0.00%
	0.1	0.2	0.3	318.91	31.96%
	0.1	0.3	0.3	389.92	61.34%
	0.1	0.4	0.3	455.36	88.42%
	0.1	0.5	0.3	515.72	113.39%

, under the condition of keeping other research parameters unchanged, taking the bearing capacity Q_u = 317.80 kPa in Figure 5b with the horizontal seismic acceleration coefficient k_h = 0.1, pore water pressure coefficients of 0.2 and 0.3, and ψ = 0.1 as an example for comparative analysis, as the pile group replacement ratio ψ under the foundation load gradually increases to 0.2, 0.3, 0.4, and 0.5, the seismic ultimate bearing capacity of the water-rich foundation near the slope increases to 407.45 kPa, 492.09 kPa, 572.48 kPa, and 649.19 kPa, respectively, increasing by 89.64 kPa, 174.28 kPa, 254.67 kPa, and 331.39 kPa, respectively, with relative errors of 28.21%, 54.84%, 80.14%, and 104.27%.

Taking the working conditions of “k_h = 0.1, r_u = 0.1” in Table 3 as an example, when the replacement rate ψ = 0.1, the bearing capacity of the foundation is 349.27 kPa. As the replacement rate gradually increased to 0.2, 0.3, 0.4 and 0.5, the bearing capacity increased to 447.28 kPa, 540.46 kPa, 628.88 kPa and 714.50 kPa, respectively. It can be calculated that for every 0.1 increase in the replacement rate, the average increase in bearing capacity is approximately 97.6 kPa, and the increase shows a gradually expanding trend (for example, ψ increases by 98.01 kPa from 0.1 to 0.2, and by 85.62 kPa from 0.4 to 0.5). It indicates that the increase in the replacement rate has a significant effect on enhancing the bearing capacity, and the marginal benefit is more obvious in the medium and low replacement rate stage (ψ = 0.1–0.3).

It can be known from the trend in Figure 5 that this rule remains consistent under different seismic and pore water conditions. For instance, under the extreme working conditions of “k_h = 0.3, r_u = 0.3”, when the replacement rate increases from 0.1 (241.67 kPa) to 0.3 (389.92 kPa), the bearing capacity increases by 148.25 kPa, representing a relative improvement of 61.34%. The practical engineering value of optimizing the replacement rate in enhancing the seismic stability of water-rich foundations near the slope was further verified.

It can be seen that even under the conditions of earthquakes and heavy rainfall, the use of pile groups to support the foundation load can significantly reduce the occurrence of landslides and collapses in the foundation near slopes. Moreover, the more the soil below the foundation is replaced by pile groups, the higher the safety reserve and the stronger the self-stabilization ability of the water-rich foundation near slopes will be.

6. Engineering Cases

Considering the geological conditions and regional environmental characteristics of a pumped storage power station in Fuzhou, and based on the natural conditions of frequent earthquakes, typhoons, and heavy rainfall, a functional building designed on a horizontal foundation near a slope is taken as a case study. Meanwhile, the horizontal sandy foundation under water-rich conditions is reinforced by pile groups to improve foundation stability. Combined with on-site survey data, the specific parameters are shown in

Table 4. Engineering Case Parameter Table.

Parameter Type	Basic Values for Case Study
Unit Weight of Silty Sand γ	18.3 kN/m3
Cohesion c	13.7 kPa
Internal Friction Angle φ	30.6°
Slope Angle η	47.9°
Foundation Proportionality Coefficient a	1 m
Load Width of Strip Foundation b	2 m
Horizontal Seismic Acceleration Coefficient kh	0.18
Pile Group Replacement Ratio ψ	0.2

as follows:

The three-dimensional model established by FLAC^3D6.0 (Itasca Consulting Group, Minneapolis, MN, USA) is shown in Figure 6. The entire model is divided into 267,840 elements and 281,711 nodes. The overall model of the water-rich foundation near the slope is constructed based on the Mohr-Coulomb strength criterion and the associated flow criterion [24,25]. The displacement of the bottom and both sides of the model is fixed, and the uniform load Qu is gradually applied on the horizontal foundation adjacent to the slope to replace the load effect of the strip foundation. The numerical simulation is used to verify the rationality and correctness of the theoretical analysis. The material properties comply with the Mohr-Coulomb strength criterion and relevant flow rules. The boundary conditions adopt fixed displacements at the bottom and on both sides of the model. On the horizontal foundation adjacent to the slope, uniformly distributed loads are applied step by step instead of strip foundation loads to simplify the simulation of the real working conditions of strip foundation loads. The dimensional parameters are reproduced in real scale. The parameters of pile group reinforcement used in the simulation are shown in

Table 5. Example simulation parameter table.

Parameter Type	Conventional Value Range
Soil Elastic Modulus	5~50 MPa
Pile Elastic Modulus	20~40 GPa
Soil Poisson’s Ratio	0.25~0.35
Pile Poisson’s Ratio	0.15~0.25
Internal Friction Angle of Pile Foundation φc	40.8°
Cohesion of Pile Foundation cc	650 kPa
Pile Length	15 m
Pile Diameter	0.05 m
Number of Piles	8

.

For the engineering case study in the construction of the above-mentioned pumped storage power station, the limit bearing capacity analysis model of water-rich foundation near slopes under earthquake and heavy rainfall conditions, and the limit bearing capacity analysis model of water-rich composite foundation near slopes with pile group reinforcement established in this paper are used. The calculation results are shown in the following figure:

Figure 7 shows the detailed plastic deformation cloud map of the engineering example obtained through numerical calculation. As shown in the figure, when the pore water pressure coefficient ru = 0.1, the values of the energy dissipation area (i.e., the sliding surface area of the slope) in the plastic deformation cloud maps before and after the group pile reinforcement significantly decrease, and the energy dissipation concentration area in the lower left corner of the strip foundation load after reinforcement gradually dissipates. The energy dissipation that occurs at the contact surface between the top of the pile foundation and the load of the strip foundation is concentrated. This is because at this time, the pile foundation bears the main energy loss caused by the load pressure. The change in the area where energy dissipation is concentrated before and after reinforcement can verify that group pile reinforcement has a certain role in improving the seismic bearing capacity of water-rich foundations adjacent to slopes.

As shown in Figure 8, with the gradual increase in the pore water pressure coefficient r_u, the ultimate bearing capacity of the water-rich foundation near the slope under the strip foundation load gradually decreases. After pile foundation reinforcement is carried out on the horizontal foundation under the load, the ultimate bearing capacity of the horizontal silty sandy foundation is significantly enhanced. Taking the bearing capacity of 77.3 kPa obtained when the pore water pressure coefficient r_u = 0.3 in

Table 6. The relative difference in bearing capacity of pile group before and after reinforcement is calculated.

ru	Ultimate Bearing Capacity of Unreinforced Foundation/kPa	Ultimate Bearing Capacity of Pile Group Reinforced Foundation/kPa	Relative Difference Δ
0	83.8	320.7	282.70%
0.05	80.5	304.3	278.01%
0.1	77.3	288.0	272.57%
0.15	74.0	271.6	267.03%
0.2	70.7	255.2	260.96%
0.25	65.2	238.8	266.26%
0.3	58.4	222.4	280.82%
0.35	51.6	206.0	299.22%
0.4	44.8	186.1	315.40%

as an example, after pile foundation reinforcement, the bearing capacity increases to 288.0 kPa, with a relative difference of 272.57%. Through the comparison of the theoretical calculations of the ultimate bearing capacity of the water-rich foundation near the slope before and after pile foundation reinforcement, the effectiveness of pile group reinforcement can be verified to a certain extent.

The ultimate bearing capacity of the foundation is determined by analyzing the changes in the vertical displacement of the horizontal foundation. When the pore water pressure coefficient r_u = 0.1, the law of the influence of foundation load on foundation displacement is shown in Figure 9. As the strip foundation load gradually increases, the vertical displacement of the water-rich foundation near the slope increases accordingly. When displacement occurs, it indicates that the horizontal foundation adjacent to the slope gradually becomes unstable. Taking the numerical simulation foundation load of 63.8 kPa before pile group reinforcement in Figure 8 and comparing it with the horizontal foundation ultimate bearing capacity Qu = 58.4 kPa obtained from theoretical analysis in Table 6, the relative difference between the two is 8.5%. Taking the numerical simulation foundation load of 208.6 kPa after pile group reinforcement and comparing it with the horizontal foundation ultimate bearing capacity Qu = 222.4 kPa obtained from theoretical analysis, the relative difference between the two is 6.2%. In summary, the relative differences between the foundation load and the ultimate bearing capacity of the foundation under earthquake and heavy rainfall conditions are all below 10%. The theoretical model and FLAC^3D6.0 simulation error is less than 10%, enabling rapid assessment of foundation stability under extreme conditions such as earthquakes and rainfall, and providing standardized technical support for disaster-resistant design of slope foundation engineering.

The slope embankment reinforcement project of a certain expressway in Kunming, Yunnan Province is located in a hilly area, with a seismic fortification intensity of 8 degrees (horizontal seismic acceleration coefficient 0.18 g). The site is mainly composed of silty sandy soil layers. The foundation adopts a strip foundation with a width of 2 m and a proportion coefficient of 1 m. The foundation treatment adopts a group pile composite foundation, with the pile type being PHC piles. The specific parameters are shown in

Table 7. Parameter Table of Engineering Cases in Kunming City.

Parameter Type	Basic Values for Case Study
Unit Weight of Silty Sand γ	18.3 kN/m3
Cohesion c	13.7 kPa
Internal Friction Angle φ	30.6°
Slope Angle η	47.9°
Foundation Proportionality Coefficient a	1 m
Load Width of Strip Foundation b	2 m
Horizontal Seismic Acceleration Coefficient kh	0.18
Pile Group Replacement Ratio ψ	0.2
Internal Friction Angle of Pile Foundation φc	43°
Cohesion of Pile Foundation cc	700 kPa
Pile Length	15 m
Pile Diameter	0.05 m
Number of Piles	8

.

The calculation results are shown in

Table 8. The relative difference in the bearing capacity of the group piles before and after reinforcement in the engineering case of Kunming City was calculated.

ru	Ultimate Bearing Capacity of Unreinforced Foundation/kPa	Ultimate Bearing Capacity of Pile Group Reinforced Foundation/kPa	Relative Difference Δ
0	101.0543	386.7247	282.68%
0.05	98.5360	372.4662	278.01%
0.1	95.13	354.4691	272.61%
0.15	90.58	332.4583	267.03%
0.2	86.23	310.4476	260.02%
0.25	78.75	288.4368	266.26%
0.3	69.9648	266.4261	280.81%
0.35	61.25	244.4153	299.04%
0.4	53.5398	222.4046	315.39%

. It is worth noting that the relative differences between the results obtained with and without pile foundation support in this project and the theoretical results of the Fuzhou case are almost not obvious. This phenomenon indicates that under similar engineering conditions and foundation treatment methods, the effect of group pile composite foundation on enhancing the bearing capacity of the foundation has certain regularity and universality, providing a strong reference basis for related engineering design and practice.

7. Conclusions

(1)

Under rainfall and earthquake conditions, the ultimate bearing capacity of the foundation obtained by using pile group support reinforcement is significantly higher than that without reinforcement support. With the increase in the pore water pressure coefficient of the soil in the slope-adjacent foundation and the horizontal seismic force coefficient, the seismic ultimate bearing capacity Q_u of the water-rich foundation near the slope under the pile group support of the foundation load decreases significantly. With the increase in the soil internal friction angle φ and soil cohesion c, the seismic ultimate bearing capacity Q_u of the water-rich foundation near the slope under pile group support increases significantly. At k_h = 0.3, the bearing capacity of the pile group (ψ = 0.1) increased by 224.53% compared to the bearing capacity of the foundation without pile groups. Under the same water-rich conditions (r_u = 0.4), the bearing capacity of the pile group (ψ = 0.1) increased by 218.12% compared to the bearing capacity of the foundation without pile groups.

(2)

Under the same conditions, when the soil strength of the soil below the foundation load is relatively high, the enhancement effect of pile group support on the bearing capacity of the slope-adjacent foundation generally deteriorates. When designing the seismic ultimate bearing capacity of the water-rich foundation near the slope, considering the use of pile groups to support directly below the foundation load can effectively improve the bearing capacity of the water-rich foundation near the slope.

(3)

The larger the foundation replacement rate under pile group support, the more the soil below the foundation is replaced by pile groups, the higher the soil strength under the foundation load, and the significantly higher the seismic ultimate bearing capacity Q_u of the water-rich foundation near the slope. Using denser pile groups to support under the foundation load can more significantly reduce the occurrence of landslides and collapses in the foundation near the slope. Similarly, the safety reserve of the water-rich foundation near the slope will be higher, and the self-stabilization ability will be stronger. The replacement ratio of pile foundations can be optimized based on rainfall and earthquake magnitude, as demonstrated by the theory presented in this paper.