Study of Load Calculation Models for Anti-Sliding Short Piles Using Finite Difference Method

Abstract

Anti-sliding short piles, a novel technique for slope stabilization, have been applied in engineering practices. Nonetheless, a mature structural calculation theory for these piles is still lacking. In this paper, the study presents an internal force solution model for anti-sliding short piles using the finite difference method. By extending the Euler–Bernoulli beam theory and defining boundary conditions, this study develops a set of finite difference equations for computing the structural forces of anti-sliding short piles. Furthermore, this study conducted laboratory model tests on soil landslide cases reinforced with anti-sliding short piles. By comparing the internal forces and deformations of these piles, the test validates the proposed calculation model for anti-sliding short piles. The results suggest that treating the load-bearing and embedded sections as a unified entity during the calculation process, instead of applying continuity conditions separately at the sliding surface as performed in traditional methods, simplifies the complex solving procedure. Moreover, under identical loading conditions, the displacement, bending moment, and shear force data obtained through the finite difference method closely coincide with the measurements from the model tests, confirming the reliability of the anti-sliding short pile calculation model. Additionally, this study demonstrates that reducing the spacing between nodes along the entire anti-sliding short pile, i.e., decreasing the value of the differential segment length ‘h’, results in more precise computational outcomes. This research offers valuable insights and references for sustainable solutions in the realm of geological disaster prevention and control.

1. Introduction

Currently, landslides have emerged as among the most prevalent and severe geological disasters globally [1]. In landslide management practices worldwide, the utilization of anti-sliding piles for reinforcement has become the favored option. Anti-sliding piles, serving as slope support structures, find extensive application in landslide control due to their flexible placement and robust anti-sliding capabilities. In engineering practice, the majority of employed anti-sliding piles are cantilevered ones, relying primarily on the anchoring force exerted at the pile base, embedded in stable strata, to counteract the landslide thrust. Nonetheless, traditional cantilevered anti-sliding piles fail to fully realize their slope protection potential, leading to material wastage and elevated costs. Recently, in certain extensive and intricate landslide control projects, buried anti-sliding short piles [2,3], also referred to as embedded anti-sliding piles, have been employed to reduce pile length and economize on construction expenses. Anti-sliding short piles present several advantages, such as shorter pile lengths, a more logical distribution of internal forces within the pile body, reduced cross-sectional dimensions, and diminished reinforcement needs. These factors, combined with their capacity to mitigate environmental and ecological impacts, especially when slope reduction and load reduction strategies lead to substantial disturbances, render the use of anti-sliding short piles exceedingly promising in the context of sustainable geological disaster control project development.

Numerous scholars have performed qualitative analyses of the stress mechanisms and characteristics of short anti-sliding piles through model experiments. Fu [4] conducted indoor model tests on fully buried anti-sliding piles, analyzing the internal forces, deformations, and displacement curves of these piles. They determined the distribution patterns of internal forces within anti-sliding short piles under the influence of landslide thrust and identified their failure modes. Zeng [5] explored the question of an appropriate pile length for anti-sliding piles through model tests and numerical analyses. The focus was on efficiently increasing pile length to ensure they effectively contribute to slope stabilization without extending beyond the slope’s crest. Li [6] carried out model tests on anti-sliding short piles and delineated three stages in the loading process: the zero displacement stage, effective deformation stage, and failure stage. They defined the boundaries of these stages as the critical starting load and critical anti-sliding load. With the rapid advancement of computer technology, numerical analysis has gained widespread use in geotechnical engineering analysis and calculations due to its cost-effectiveness and swift results [7,8,9,10,11,12]. The study of anti-sliding short piles follows suit. Lei, Y. [13] used the distribution of horizontal thrust behind anti-sliding short piles, and the magnitude of the resistance force in front of these piles was examined using ANSYS finite element software (https://www.ansys.com/, accessed on 8 November 2023). Liang and Chen [14,15,16,17] investigated the mechanisms, conditions, and optimal pile spacing related to the soil arching effect created by anti-sliding piles through computer numerical simulation. Hu [18], utilizing FLAC3D software (https://www.itasca.fr/en/software/flac3d, accessed on 8 November 2023), performed numerical simulations of anti-sliding short piles. The analysis included an examination of the impact of parameters such as pile length, pile diameter, pile wall thickness, and soil properties on the stress performance of the piles. These experiments and finite element analyses primarily concentrate on the stress characteristics, interactions between the piles and the soil, optimal pile spacing, and the determination of pile length for anti-sliding short piles. Nonetheless, the design and calculation theory for anti-sliding short piles remain insufficiently developed. Current calculation theories do not comprehensively capture the genuine stress characteristics of anti-sliding short piles. Furthermore, unified regulations or guidelines for their design calculations are lacking. Engineers frequently depend on their individual calculation methodologies and engineering expertise for analysis. To gain an accurate understanding of the stress characteristics and stress distribution patterns of anti-sliding short piles, it is imperative to establish a scientifically effective structural calculation method for these piles from a theoretical perspective.

Extensive engineering practice has demonstrated the safety and reliability of the finite difference method for structural calculations [19,20,21,22,23,24]. The finite difference method’s principle entails approximating the differential equations of the elastic foundation beam using finite difference equations. This is achieved by substituting the function values at finite nodes in the difference equations for the derivatives of various orders in the original differential equations. This approach converts the differential solving process into algebraic solving, rendering it an approximate numerical solution method. Compared to differential equations, the finite difference method offers advantages in terms of its intuitive mathematical concepts, ease of expression, and its status as a relatively mature numerical solution method with a lengthy history of development. Nonetheless, existing research is typically focused on cantilever anti-sliding piles, and there is limited research on finite difference calculation methods for anti-sliding short piles. Fan QY et al. [25] employed the m-k method to derive finite difference equations for the displacement and internal forces of deeply buried anti-slide piles, enabling the calculation of the entire pile’s internal forces and displacements through a programmed approach. Their research confirmed the suitability of the finite difference method for calculating the internal forces of anti-sliding short piles. However, this study segregated the load-bearing section and the embedded section, resulting in a relatively intricate internal force calculation process using the continuity condition at the sliding surface. Additionally, the m-k method is primarily tailored for landslide control scenarios where the upper sliding surface consists of soil and the lower part is rock [26], a relatively specific circumstance. In practical engineering, the embedded section of anti-sliding piles often traverses multiple geological layers. Conventional “m methods” and “K methods” are designed for single-layer soil or rock foundations. Consequently, industry regulations recommend employing weighted averaging techniques to convert layered foundations into equivalent single-layer homogeneous foundations. For instance, the railway industry standard [27] employs the area-equivalent method for foundation coefficient calculation, while the highway industry standard [28] utilizes the flexural curve weighting method. Therefore, Dai [29] utilized a weighted conversion based on the pile’s deflection curve to compute the proportionate average coefficient of multi-layer foundations, providing precise results for internal forces in anti-slide piles. This paper builds upon previous research on anti-sliding short piles and finite difference methods for cantilever anti-sliding piles. It analyzes the landslide thrust acting on anti-sliding short piles and the distribution of soil resistance on the pile body [30]. The paper commences with a stress analysis of anti-sliding short piles for reinforcing cohesive soil landslides. Subsequently, it applies the Euler–Bernoulli elastic foundation beam theory [31,32]. The embedded section is treated as a linear elastic foundation, the sliding surface is simplified as a fixed support, and the load-bearing section is considered as a cantilever section. Simultaneously, with an understanding of the applicable conditions and stress characteristics of anti-sliding short piles [33,34,35,36,37], the paper addresses the boundary conditions at the pile’s top and bottom. It employs an enhanced finite difference method for structural stress calculations in anti-sliding short piles. The paper proposes a comprehensive model for calculating the internal forces of anti-sliding short piles and validates this model through large-scale indoor physical model experiments. This validation ensures the model’s reliability and provides design theory references for future landslide control engineering.

2. Structural Calculation Model for Anti-Sliding Short Piles

Traditional differential methods frequently partition anti-sliding piles into two segments: the load-bearing section and the embedded section. They rely on iterative techniques to address the problem, incorporating the continuity condition at the sliding surface, which can be a cumbersome process. To circumvent this computational methodology and account for the load characteristics of anti-sliding short piles, as well as principles from differential theory, this study embraces the Euler–Bernoulli beam theory. It conceptualizes anti-sliding short piles as vertically oriented elastic foundation beams and incorporates soil springs to simulate the soil in front of the pile, thus modeling its interaction with the anti-sliding short pile. Additionally, this study uniformly discretizes the support structure from top to bottom, bypassing the conventional differential algorithms that partition the anti-sliding short pile into two sections at the sliding surface. This streamlines the solving process, consequently augmenting both efficiency and precision in the computation of internal forces within anti-sliding short piles.

2.1. Basic Assumptions

The model’s derivation process is grounded in the following assumptions: (1) The deformation of the anti-sliding pile conforms to the Euler–Bernoulli beam theory and fulfills the assumption of a planar cross-section. (2) A uniformly distributed load q(z) is applied along the entire length of the loaded section, and the pile material is assumed to be homogeneous and linearly elastic. (3) The model does not incorporate axial pressure from the soil above the loaded section.

2.2. Analysis of the Forces on Anti-Sliding Short Piles

The primary thrust of the landslide is concentrated in the upper region of the slip surface, causing significant bending deformation in the anti-sliding short piles, particularly near the slip surface. Near the top of the pile, deformation is relatively minimal due to the soil’s restraining effect, resulting in nearly parallel movement. The anchoring section below the slip surface is located within the sliding bed and experiences minimal deformation. Considering the bending deformation characteristics of anti-sliding short piles and stress patterns obtained from prior experiments, the distribution of landslide thrust can assume various shapes, such as a rectangle, triangle, or trapezoid. To facilitate calculations, this study simplifies the landslide thrust as a triangular shape. The JANBU method is utilized to calculate the landslide thrust q(z) [38]. Figure 1 illustrates the forces acting on the blocks. In this figure, E_i and E_i+1 denote the inter-block normal forces, T_i and T_i+1 represent the inter-block shear forces, W_i is the block’s self-weight, and Q_i is the horizontal action force. Additionally, N_i and S_i correspond to the total normal (comprising effective normal force and pore pressure) and shear forces at the base of the block. The remaining symbols are displayed in the diagram. The equilibrium conditions for forces and moments are provided below.

Transverse,

$$E_{i+1}=E_i+U_{i,i-1}+Q_i+N_i\sin {\alpha }_i+U_i\sin {\alpha }_i-U_{i,i+1}-S_i\cos {\alpha }_i$$

(1)

Longitudinal,

$$T_{i+1}=T_i+S_i\sin {\alpha }_i+N_i\cos {\alpha }_i+U_i\cos {\alpha }_i-W_i$$

(2)

The final derived formula for calculating the landslide thrust is as follows:

$$P_i=\sqrt{E_{i+1}{}^2+T_{i+1}{}^2}$$

(3)

In this formula, P_i signifies the thrust of the landslide, while E_i+1 and T_i+1 represent the resulting forces acting in the transverse and longitudinal directions of the block.

In the load-bearing section, the anti-slip short pile experiences bending moments and shear forces at the sliding surface, effectively acting as a fixed support, while the pile head remains unconstrained, serving as a free end. Therefore, the loaded section is treated as a cantilever beam, and calculations for cantilever beams are applied. Below the sliding surface, there is the anchorage section, treated as an elastic foundation beam. Depending on the type of soil strata and following elastic foundation beam theory, various methods such as the ‘m’ method, ‘K’ method, ‘C’ method, and others are used for calculations. To derive formulas, this paper utilizes the ‘m’ method to determine the resistance of the sliding mass and the foundation coefficient, which is calculated as $K_{\mathrm{j}}(z)=mz$ Here, ‘m’ represents the calculation coefficient, and ‘z’ denotes the depth, taking into account soil foundation conditions. Specific stress conditions are illustrated in Figure 2.

Utilizing the Euler–Bernoulli beam theory, we can express the unified differential governing equation for both the loaded and embedded sections of the anti-sliding short pile as follows:

$$EI\frac{d^{4}x}{d{y}^4}x+B_0{K}_{\mathrm{j}}x=q(z)$$

(4)

In this equation, EI represents the stiffness of the anti-sliding short pile structure. B₀ denotes the calculated width of the pile. K_j represents the foundation coefficient. x signifies the horizontal displacement of the pile. y denotes the distance from the pile head. q(z) represents the landslide thrust in kN/m.

2.3. Finite Difference Solution for the Anti-Sliding Short Pile Model

As illustrated in Figure 3, in the application of the finite difference method, it becomes essential to discretize the entire anti-sliding pile into n segments, where each segment possesses a length of h. Nodes are sequentially numbered from the pile head to the pile end as 0, 2, …, n. The node corresponding to the pile body at the sliding surface is designated as m. Furthermore, two virtual nodes are introduced above the pile head and beneath the pile base, marked as −2, −1, n + 1, and n + 2, resulting in a total of n + 5 nodes.

The governing difference equation at any arbitrary node can be formulated as follows:

$$\frac{EI}{h^4}\left(x_{i-2}-4x_{i-1}+6x_i-4x_{i+1}+x_{i+2}\right)+B_0{K}_{j}ih{x}_i=q(z)$$

(5)

For the anti-sliding short pile, the boundary conditions are determined by the bending moment M₀ and shear force Q₀ applied at the pile head due to the overlying soil pressure. These conditions are specified as follows:

$$\begin{array}{l}\left\{\begin{array}{l}{x}_{-1}-2x_0+x_1=\frac{M_0{h}^2}{EI}\\ x_{-2}-2x_{-1}+2x_1-x_2=\frac{-2Q_0{h}^3}{EI}\end{array}\right.\\ \end{array}$$

(6)

In the case of an elastic foundation, it is assumed that the pile base of the embedded section of the anti-sliding short pile acts as a free end, with both the bending moment and shear force being zero. Consequently, the boundary conditions at the pile base can be defined as follows:

$$\left\{\begin{array}{l}{x}_{n-1}-2x_n+x_{n+1}=0\\ x_{n-2}-2x_{n-1}+2x_{n+1}-x_{n+2}=0\end{array}\right.$$

(7)

When expressing the pile body nodes using a central difference format, Equation (4) can be restated as follows:

$$\begin{array}{l}\left\{\begin{array}{l}{x}_{i-2}^{}-4x_{i-1}^{}+(6+\frac{B_0{K}_{\mathrm{j}}h{}^4}{EI})x_i^{}-4x_{i+1}^{}+x_{i+2}^{}=\frac{q(z)h^4}{EI}\\ (0\le i\le m)\\ x_{i-2}^{}-4x_{i-1}^{}+(6+\frac{B_0{K}_{\mathrm{j}}h{}^4}{EI})x_i^{}-4x_{i+1}^{}+x_{i+2}^{}=0\\ (m<i\le n+2)\end{array}\right.\\ \end{array}$$

(8)

By substituting the n + 1 nodes along the pile body into Equation (4) and incorporating the four boundary conditions at the pile head and pile base, it becomes feasible to determine the horizontal displacements (x_i) at each node along the pile body. Let:

$$a_i=6+\frac{B_0{K}_{\mathrm{j}}h{}^4}{EI}$$

(9)

As a result, the linear equation system for the entire pile matrix of the rear pile can be represented as follows:

$$\left[k_i\right]\left[X_i\right]=\left[Q_i\right]$$

(10)

In the equation, [k_i] signifies the coefficient matrix, [x_i] denotes the pile body displacement matrix, and [Q_i] represents the load matrix. More specifically:

$$\left[k_i\right]={\left[\begin{array}{ccccccccccccc}0& 1& -2& 1& 0& \cdots & \cdots & \cdots & 0& 0& 0& 0& 0\\ 1& -2& 0& 2& 1& \cdots & \cdots & \cdots & 0& 0& 0& 0& 0\\ 1& -4& a_0& -4& 1& \cdots & \cdots & \cdots & 0& 0& 0& 0& 0\\ 0& 1& -4& a_1& -4& 1& \cdots & \cdots & 0& 0& 0& 0& 0\\ ⋮& ⋮& ⋮& ⋮& ⋮& \ddots & & & ⋮& ⋮& ⋮& ⋮& ⋮\\ ⋮& ⋮& ⋮& ⋮& ⋮& & \ddots & & ⋮& ⋮& ⋮& ⋮& ⋮\\ ⋮& ⋮& ⋮& ⋮& ⋮& & & \ddots & ⋮& ⋮& ⋮& ⋮& ⋮\\ 0& 0& 0& 0& 0& \cdots & \cdots & 1& -4& a_{n-1}& -4& 1& 0\\ 0& 0& 0& 0& 0& \cdots & \cdots & \cdots & 1& -4& a_n& -4& 1\\ 0& 0& 0& 0& 0& \cdots & \cdots & \cdots & 0& 1& -2& 1& 0\\ 0& 0& 0& 0& 0& \cdots & \cdots & \cdots & -1& 2& 0& -2& 1\end{array}\right]}^{}$$

(11)

$$\left[X_i\right]={\left[x_{-2},x_{-1},\cdots x_m⋮x_{m+1},\cdots x_{n+1},x_{n+2}\right]}^T$$

(12)

$$\left[Q_i\right]=\frac{h^4}{EI}{\left[\begin{array}{cccccccccc}\frac{M_0}{h^{{}_2}},& -\frac{2Q_0}{h},& q_0,& \cdots ,& q_m,& \cdots & 0,& 0,& 0,& 0\end{array}\right]}^T$$

(13)

By incorporating Equations (11)–(13) into Equation (10), a consolidated system of differential equations is established for both the loaded section and the embedded section of the anti-sliding short pile. This unified system, referred to as Equation (14), can be employed to determine the internal forces throughout the entire pile of the anti-sliding short pile.

$$\left[\begin{array}{l}{x}_{-2}\\ x_{-1}\\ x_0\\ x_1\\ \cdots \\ x_m\\ \cdots \\ x_{n-1}\\ x_n\\ x_{n+1}\\ x_{n+2}\end{array}\right]={\left[\begin{array}{ccccccccccccc}0& 1& -2& 1& 0& \cdots & \cdots & \cdots & 0& 0& 0& 0& 0\\ 1& -2& 0& 2& 1& \cdots & \cdots & \cdots & 0& 0& 0& 0& 0\\ 1& -4& a_0& -4& 1& \cdots & \cdots & \cdots & 0& 0& 0& 0& 0\\ 0& 1& -4& a_1& -4& 1& \cdots & \cdots & 0& 0& 0& 0& 0\\ ⋮& ⋮& ⋮& ⋮& ⋮& \ddots & & & ⋮& ⋮& ⋮& ⋮& ⋮\\ ⋮& ⋮& ⋮& ⋮& ⋮& & \ddots & & ⋮& ⋮& ⋮& ⋮& ⋮\\ ⋮& ⋮& ⋮& ⋮& ⋮& & & \ddots & ⋮& ⋮& ⋮& ⋮& ⋮\\ 0& 0& 0& 0& 0& \cdots & \cdots & 1& -4& a_{n-1}& -4& 1& 0\\ 0& 0& 0& 0& 0& \cdots & \cdots & \cdots & 1& -4& a_n& -4& 1\\ 0& 0& 0& 0& 0& \cdots & \cdots & \cdots & 0& 1& -2& 1& 0\\ 0& 0& 0& 0& 0& \cdots & \cdots & \cdots & -1& 2& 0& -2& 1\end{array}\right]}^{-1}\left[\begin{array}{c}{h}^2{M}_0/EI\\ -2h^3{Q}_0/EI\\ q_0{h}^4/EI\\ q_1{h}^4/EI\\ ⋮\\ q_m{h}^4/EI\\ ⋮\\ 0\\ 0\\ 0\\ 0\end{array}\right]$$

(14)

These formulas can be employed with MATLAB software (https://ww2.mathworks.cn/en/products/matlab.html, accessed on 8 November 2023) to calculate displacements at different nodes along the pile, enabling the determination of shear forces and bending moments along the pile.

$$M_i=\frac{EI}{h^2}\left(x_{i+1}-2x_i+x_{i-1}\right)$$

(15)

$$Q_i=-\frac{EI}{2h^2}\left(x_{i+2}-2x_{i+1}+2x_{i-1}-x_{i-2}\right)$$

(16)

3. Model Experiment

To validate the anti-slide short pile calculation method proposed in this paper, we conducted model tests on anti-slide short piles. These tests involved analyzing and organizing experimental data and then comparing the obtained results with calculations based on the method developed in this paper. This process ensures the effectiveness of the calculation method in practical engineering applications.

3.1. Experimental Design

This paper conducts a similarity design for the model test, building on design insights from previous anti-slide pile model tests [39]. The cross-sectional dimensions of the full-scale anti-slide pile in the practical project are 1.0 × 1.6 m², constructed of concrete. Using similarity theory and the deflection equation for anti-slide piles, the corresponding conditions for the model test are determined as follows:

$$f={\frac{qL}{8EI}}^4$$

(17)

In actual engineering:

$$fp=\frac{qpL{p}^4}{8EpIp}$$

(18)

In the model experiment:

$$fm=\frac{qmL{m}^{{}^4}}{8Em\mathrm{Im}}$$

(19)

To establish a comparison between the model pile and the pile used in the practical engineering, the following equation must be met:

$$Cf=\frac{CqCL}{CECI}$$

(20)

$$Cq=\frac{qp}{qm},CE=\frac{Ep}{Em},CI=\frac{Ip}{\mathrm{Im}},CL=\frac{Lp}{Lm},Cf=\frac{fp}{fm}$$

(21)

Following similarity theory, the calculation for reinforcing the short pile used in the test is conducted. The bending moment similarity constant is determined as follows:

$$C_m=\frac{M_m}{M_p}=\frac{f_{my}{A}_{ms}{h}_{m0}^{}}{f_{py}{A}_{ps}{h}_{p0}}=\frac{A_{ms}}{A_{ps}}{C}_{fy}{C}_L$$

(22)

Reinforcement configuration with equivalent cross-sectional area:

$$A_{ms}=\frac{A_{ps}{f}_{py}}{A_{pc}{f}_{pc}}·\frac{A_{mc}{f}_{mc}}{f_{my}}=\frac{1}{C_L^{ 2}}·\frac{f_{mc}{f}_{py}}{f_{pc}{f}_{my}}·A_{ps}$$

(23)

Shear force similarity coefficient:

$$C_V=\frac{V_m}{V_p}=\frac{f_{my}{A}_{ms}}{f_{py}{A}_{ps}}=\frac{A_{ms}}{A_{ps}}·C_{fy}$$

(24)

Equivalent cross-sectional area for the tie reinforcement configuration:

$$A_{svm}=\frac{A_{psv}{f}_{pyv}{h}_{p0}}{f_{myv}{h}_{m0}}{C}_v$$

(25)

In the equation, the variables represent the following: A_ps: cross-sectional area of the prototype reinforcement; A_pc: design compressive strength of concrete in the prototype structure; f_py: design tensile strength of steel reinforcement in the prototype structure; A_ms: cross-sectional area of the reinforcement in the model structure; A_mc: cross-sectional area of gypsum in the model structure; f_mc: design compressive strength of gypsum in the model structure; f_my: design tensile strength of aluminum bars in the model structure.

In this model test, when f_y = f_y’ = f_my = 100 N/mm², M = 607.61 N·m, b = 50 mm, h₀ = 69 mm, α₁ = 1.0, ξ_b = 0.6, f_c = f_mc = 7.3 N/mm² by means of Equation (26):

$${\alpha }_s=\frac{M}{{\alpha }_1{f}_{c}b{h}_0{}^2}$$

(26)

α_s = 0.348, by means of Equation (27):

$$\xi =1-\sqrt{1-2{\alpha }_{\mathrm{s}}}$$

(27)

ξ = 0.4488, by means of Equation (28):

$${\gamma }_s=0.5·(1+\sqrt{1-2{\alpha }_s})$$

(28)

γ_s = 0.776, plug into Equation (29):

$$A_s=\frac{M}{f_y{\gamma }_s{h}_0}$$

(29)

determined that A_s = 109.43 mm². Following the principles of similarity theory and the deflection equation for anti-slip short piles, we selected five Φ6 aluminum bars for longitudinal reinforcement in the model test, while maintaining a protective layer thickness of 10 mm. The tie reinforcement comprised two Φ1 helical wires with a 125 mm spacing. Based on the calculated reinforcement for the anti-slip short pile, we completed the arrangement of the aluminum bar cage. Figure 4 depicts the reinforcement configuration for the anti-slip pile, ensuring compliance with the similarity conditions for the model test.

Following the principles of similarity theory and the deflection equation for anti-slip piles, the model test attained similarity conditions with a fundamental geometric similarity ratio C₁ = 20 and a fundamental stress similarity ratio C₀ = 20. The precise similarity relationships are detailed in

Table 1. Model test similarity relationships.

Type	Simulation Quantity	Dimension	General Model	Practical Model	Experiment Model
Material property	Stress σ	FL−2	Kσ	1	20
	Strain ε	-	1	1	1
	Elastic modulus E	FL−2	Kσ	1	20
	Shear modulus Gm	FL−2	Kσ	1	20
	Compressive strength R	FL−2	Kσ	1	20
	Cohesive strength C	FL−2	Kσ	1	20
	Friction angle φ	-	1	1	1
	Bulk density γ	FL−3	Kγ	1	1
Geometric property	Length L	L	KL	KL	20
	Linear displacement δ	L	KL	KL	20
	Angular displacement β	-	1	1	1
	Area A	L2	KL2	KL2	202
Load	Point load P	F	KσKL2	KL2	203
	Distributed load W	FL−1	KσKL	KL	202
	Surface load q	FL−2	Kσ	1	20
	Torque M	FL	KσKL3	KL3	204

.

After establishing the fundamental physical properties of the landslide soil material utilized in the model test, the material for the anti-slip pile model was selected based on the provided table. Ultimately, high-strength gypsum was chosen to mimic actual concrete material, and aluminum bars were employed to replicate actual steel reinforcement. Given a geometric similarity constant of 20, the model pile had cross-sectional dimensions of 50 × 80 mm². In line with the geometric similarity ratio, the anti-slip pile was constructed as a cast-in-situ concrete pile measuring 1300 mm in length, with an embedded section of 550 mm, as depicted in Figure 5 and Figure 6.

For the experiment, the sliding bed and sliding mass materials were sourced from the homogenous loess found in the northern suburbs of Xi’an, representing the prototype soil. The loess was saturated with water to attain its optimal moisture content, layered, and compacted to establish a specific bearing capacity [40]. The fundamental physical properties of the soil, as detailed in

Table 2. Experimental soil physical parameters.

	Bulk Density γ (kN/m3)	Moisture Content ρ	Cohesive Strength C (kPa)	Internal Friction Angle φ (°)
Sliding mass	18.6	15.2	10.2	20.4
Sliding bed	18.8	14.7	10.2	20.6
Sliding strip	--	--	7	13

, were determined through geotechnical testing.

The material for the anti-slip pile was composed of high-strength gypsum, cement, and water in a ratio of 10:7:2. Test specimens were prepared and cured under standard conditions, resulting in a uniaxial compressive strength of 15 GPa. To closely replicate the prototype, reinforcement was integrated into the anti-slip pile. The steel cage was constructed using aluminum bars to represent longitudinal reinforcement, with a 6 mm diameter. Three longitudinal bars were placed on the backside of the pile and two on the front side. Fine iron wires were used for tie reinforcement, with a 1 mm diameter, spaced at 10 cm intervals. The anti-slip pile was cast in pre-prepared molds made of hard wood panels. Following casting, it was cured for 30 days in appropriate environmental conditions.

In the model experiments, a variety of measuring instruments and tools were utilized, such as resistance strain gauges, soil pressure cells, displacement meters, and more. The arrangement of these instruments is illustrated in the schematic diagrams presented in Figure 7, Figure 8 and Figure 9.

The obtained strain data are based on the principles of moment calculation in materials mechanics. Utilizing Equation (30), it is possible to calculate the bending moment at different measurement points along the pile:

$$M(x)=\frac{2EI({\epsilon }_{+}-{\epsilon }_{-})}{x}$$

(30)

In the equation: EI—pile’s flexural stiffness, N·m²; ε₊, ε₋—tensile and compressive strains at various measuring points; x—distance between two measuring points on the same section, m; M—bending moment, N·m.

Calculating the derivative of the bending moment will result in the shear force along the pile:

$$Q(x)=-\frac{2EI({\epsilon }_{+}-{\epsilon }_{-})}{x^2}$$

(31)

The displacement measurement instrument employs rebound-type displacement sensors, offering a measurement accuracy of 0.01 mm.

The distribution of soil pressure values obtained from the experiments is shown in the following Figure 10:

Figure 11 illustrates the selection of measurement instruments and the preparatory steps for the experimental setup. This includes casting and installing anti-sliding piles, creating the landslide model, and arranging loading devices, strain monitoring equipment, soil pressure boxes, and displacement gauges, as outlined in the preliminary phase of the experimental plan.

In the indoor model experiments designed to address landslides in loess regions, measurements were conducted for pile body soil pressure, pile stress, and slope displacement. The experimental data were compiled to discern the distribution patterns of pile body soil pressure and pile bending moment at different loading levels. Subsequently, using the experimental data as a basis, the same conditions were employed to calculate the overall internal forces of the pile, enabling a more scientifically grounded design for anti-sliding short piles.

3.2. Analysis of Experiment Results

Following the load test on the anti-slip piles, a layered excavation of the slope was carried out to remove the anti-slip piles from the landslide mass. Data regarding pile cracks, deformation, and other variables were gathered to analyze how the piles behaved under load and their impact on soil reinforcement. Figure 12 illustrates that under the influence of landslide thrust, the anti-slip piles consistently displayed outward movement, with a stable angle of 4–5° at failure sites. Failures in the piles consistently occurred in proximity to the area just above the sliding surface. Furthermore, each anti-slip pile experienced bending failures immediately above the sliding surface, resulting in the formation of plastic hinge supports both above and below the failure site, which were located close to peak moment values. Regarding crack development, cracks above the sliding surface extended from the front to the back of the pile, suggesting that the anti-slip piles were in tension in front of the pile with respect to the sliding surface. Below the sliding surface, cracks extended from behind the pile to the front, indicating tension in the anti-slip piles behind the pile beneath the sliding surface. Due to the impact of landslide thrust, the concrete protection layer consistently suffered damage at the failure sites. This was a result of the maximum soil compression experienced by the anti-slip piles near the sliding surface, resulting in the fracture of the concrete protection layer.

4. Data Comparison

The stress calculation model for anti-sliding short piles under a load ranging from 0 to 60 kPa was employed to collect displacement, bending moment, and shear force data. These data were subsequently compared and analyzed alongside the experimental measurements to assess the validity of the developed anti-slip short pile force calculation model.

4.1. Comparison of Pile Head Displacement

Figure 13 displays the comparison between model test values and theoretical calculations for pile head displacement as the load increases. It can be observed that under the same loading conditions, the model test values for pile head displacement closely align with the differential method theoretical calculations, showing a close resemblance in the curves. Furthermore, the displacement of the anti-sliding pile exhibits three distinct stages as the load is incrementally applied: the soil compaction stage (0–20 kPa), pile elastic deformation stage (20–40 kPa), and pile failure stage (>40 kPa). Pile head displacement experiences a notable increase at approximately 45–60 kPa of load, signifying substantial deformation of the pile body within this phase.

Figure 14 depicts the comparison between the calculated model values and the experimental values of pile body displacement at various depths under loads of 20 kPa, 40 kPa, and 60 kPa. It is evident that displacement consistently rises as the load increases, and the rate of displacement increase is more pronounced in the vicinity of the sliding surface. The pile head displacement at various depths under different loads, calculated using the finite difference-based anti-slide pile force model, closely matches the results obtained from model tests, with a minor margin of error.

4.2. Comparison of Bending Moments

Figure 15 illustrates the comparison between model test and calculated model bending moments under varying loads, recorded at a depth of 20 cm both above and below the sliding surface (at the location of maximum bending moment). It is evident that, when subjected to identical loading conditions, the bending moments obtained from model testing and the theoretical calculations employing the finite difference method demonstrate relatively minor disparities. Furthermore, the fluctuation in bending moment corresponds to the three stages of pile deformation, namely the soil compaction stage, pile elastic deformation stage, and pile failure stage. As the load approaches 45–60 kPa, there is a notable increase in bending moment; this is consistent with the underlying elastic-plastic theory upon which the force calculation model relies, signifying substantial deformation of the pile body. This phenomenon aligns with the findings derived from the model experiments.

Figure 16 presents a comparison between the calculated model values and the model test values of pile bending moment at varying depths, under loads of 20 kPa, 40 kPa, and 60 kPa. It can be inferred that the overall bending moment consistently rises with the increment in load, and the rate of bending moment increase is more pronounced in the vicinity of the sliding surface. The maximum bending moment is attained at a depth of roughly 20 cm above and below the sliding surface, signifying pile failure at this specific depth. The fluctuation of bending moment with depth under different loads demonstrates a resemblance between the finite difference anti-sliding short pile calculation model and the model test results. Both exhibit an “S” shape, with relatively minor discrepancies.

Table 3. Maximum bending moment values under different loads.

Load (kpa)			Numerical Value (N·m)	Distance from Sliding Surface (cm)	Error
20	Calculated Value	Positive Bending Moment	572	20	5.0%
		Negative Bending Moment	−395	20
	Experimental Value	Positive Bending Moment	513	19.6
		Negative Bending Moment	−408	21.5
40	Calculated Value	Positive Bending Moment	748	20	12.4%
		Negative Bending Moment	−634	20
	Experimental Value	Positive Bending Moment	659	21.4
		Negative Bending Moment	−570	22.3
60	Calculated Value	Positive Bending Moment	882	20	3.3%
		Negative Bending Moment	−642	20
	Experimental Value	Positive Bending Moment	819	18.7
		Negative Bending Moment	−747	21.6

allows us to observe that the comparison between the bending moment values obtained from model tests and those calculated in this article reveals the following findings: (1) The maximum bending moment value in the pile above the sliding surface is considerably higher than that in the pile below the sliding surface. (2) The numerical values of the maximum positive and negative bending moments in the pile closely match the experimental values, with a relative error of approximately 6.9%. The positions of the maximum bending moments are both situated roughly 20 cm from the sliding surface. In summary, the finite difference calculation results for the bending moments of anti-sliding short piles closely align with the experimental results.

4.3. Shear Force Comparison

Figure 17 illustrates a comparison between the calculated model and the model test results for pile shear forces under loads of 20 kPa, 40 kPa, and 60 kPa. The shear force in the anti-sliding pile exhibits an upward trend with increasing load. The shear force at the maximum bending point, both above and below the sliding surface, is zero. In proximity to the sliding surface, the shear force experiences a rapid increase, and the maximum shear force of the anti-sliding pile is observed near the sliding surface, with a point of inflection in shear force occurring at the sliding surface. The variation of shear force with depth exhibits a close resemblance between the finite difference calculation model for anti-sliding short piles and the model test results, with a minimal margin of error.

Table 4. Maximum shearing force values under different loads.

Load (kpa)			Numerical Value (kN)	Error
20	Calculated Value	Positive Shear	5.973	8.5%
		Negative Shear	−2.085
	Experimental Value	Positive Shear	6.545
		Negative Shear	−2.258
40	Calculated Value	Positive Shear	7.366	9.8%
		Negative Shear	−2.535
	Experimental Value	Positive Shear	8.024
		Negative Shear	−2.947
60	Calculated Value	Positive Shear	8.188	8.0%
		Negative Shear	−2.986
	Experimental Value	Positive Shear	8.792
		Negative Shear	−3.358

reveals that when comparing the moment values obtained from model tests with the shear force values calculated in this article, (1) shear force values measured in model tests under various loading conditions consistently exceed those obtained from the calculation model, showing an overall tendency to be higher, and (2) minimum shear force values universally occur in the section of the anti-slide short pile under load, and the shear force variation trend in this section exceeds that in the embedded section. The numerical values of the maximum positive and negative shear forces within the pile body closely match the calculated values, with a relative error of approximately 8.8%. In summary, the finite difference calculations for shear forces in anti-slide short piles exhibit strong alignment with experimental results.

A comparison between the calculated model and the model test results for displacement, moment, and shear force reveals that the error values fall within a narrow range, and the trends are consistent. Thus, the calculated model for anti-sliding short piles developed in this paper is suitable for engineering design calculations. An analysis of the disparities between the model test values and the values obtained from the finite difference calculation model reveals several contributing factors:

The choice of the finite difference segment length (h) impacts the accuracy of the calculation results. For enhanced precision, the segment length should be minimized in future analyses. The finite difference calculation model approximates the distribution of landslide thrust as triangular, which also contributes to the presence of errors. The coefficient (m) for the ground soil utilized in the calculations is determined empirically, resulting in disparities between the calculated results and the model test data. In practical applications, it might be essential to compute the bottom boundary conditions individually for each ground layer with its specific coefficient.

The arrangement of strain gauges in the model test imposes constraints, as the positioning of the strain gauges affects the location of the maximum bending moment, making it difficult to obtain a precise value for the maximum bending moment. Furthermore, the model test’s restricted number of strain gauges may lead to discrete data points in the bending moment curve.

The finite difference calculation model assumes no axial deformation in the pile and at the pile top, treating the pile bottom as a free end and disregarding the pile bottom reaction force. Conversely, in the model test, the pile bottom encounters non-zero shear forces and moments, along with soil reactions. These variations in boundary conditions can result in discrepancies in the results.

5. Discussion

5.1. The Calculation Method for ‘m’ in Multilayered Foundations

The quality of the soil within a specific depth range h_m, close to the ground or the lowest erosion line, significantly influences the lateral soil resistance of the pile. Therefore, an average value ‘m’ for the foundation coefficients of various soil layers within the h_m depth can be determined. This can be achieved either as a weighted average based on layer thickness or by ensuring equal areas before and after conversion to represent the ‘m’ value throughout the entire buried depth of the pile. As illustrated in Figure 18, in cases where there are three soil layers surrounding the pile, the areas under the original and converted foundation coefficient curves are equal within the h_m depth, allowing ‘m’ to be determined as follows:

$$m=\frac{m_1{h}_1{}^2+m_2(2h_1+h_2)h_2+m_3(2h_1+h_2+h_3)h_3}{h_m{}^2}$$

(32)

If there are only two different foundation layers, it can be expressed as:

$$m=\frac{m_1{h}_1+m_2(2h_1+h_2)h_2}{h_m{}^2}$$

(33)

In the equation:

$$h_m=\left\{\begin{array}{l}2(d+1)(H>2.5/\alpha ),\\ H(H\le 2.5/\alpha ),\end{array}\right.$$

(34)

‘α’ is the pile’s deformation coefficient (flexibility factor).

5.2. Numerical Simulation of Anti-Slip Short Piles

In the realm of numerical simulation for anti-sliding short piles, it is imperative to acknowledge the constraints and obstacles linked to indoor model testing. These tests are vulnerable to uncontrollable variables that may introduce inaccuracies into the outcomes. For instance, displacement sensors, strain gauges, and soil pressure sensors may not offer real-time measurements of the internal forces and displacements of anti-sliding short piles, resulting in discrete outcomes. Therefore, in an effort to enhance the precision of the findings, this paper will utilize ABAQUS numerical simulation software to replicate the observed loading conditions in the experiments. This endeavor seeks to attain precise values for internal forces and displacements, thus providing additional validation for the stress and force models calculated for anti-sliding short piles.

In this numerical model, the anti-sliding piles, derived from gypsum casting in the experiments, are treated as elastic materials, whereas the loess slide bed and sliding body are regarded as elastoplastic materials. The constitutive model utilized in this calculation is based on the elastoplastic model, specifically the Mohr–Coulomb model. The numerical simulation model replicates the dimensions of the model test at a 1:1 scale, as illustrated in Figure 19. Particular parameters for the experimental design are delineated in

Table 5. ABAQUS model calculation parameters.

	Bulk Density γ (N·m−3)	Elastic Modulus E (MPa)	Internal Friction Angle Φ(°)	Poisson’s Ratio μ	Cohesion c (kPa)
Sliding body	18,600	80	20.4	0.3	10.2
Sliding bed	18,800	80	20.6	0.3	10.2
Anti-sliding piles	21,000	18,000	--	0.2	--
Sliding strip	--	--	13	--	7

. Vertical self-weight is applied to the model, and stress equilibrium is upheld through the inclusion of normal constraints on the model’s sides, above, below, and in front of the sliding bed.

The computational model for numerical simulation comprises three constituents: the sliding bed, sliding body, and anti-sliding piles, as depicted in Figure 20. The model encompasses a total of 15,352 nodes and 13,384 elements, with all elements meshed utilizing the C3D8R formulation. The simulation is segregated into six loading steps, with each step progressively augmenting the load. To ensure the stability of the anti-sliding piles and avert convergence problems, a load of 10 kPa is administered in each step, culminating in a cumulative load of 60 kPa before conclusion. The specifics of the loading steps are outlined in

Table 6. Loading steps.

Loading Times	Earth Stress	1	2	3	4	5	6
1	found	transmit	transmit	transmit	transmit	transmit	transmit
2		10 kPa	10 kPa	10 kPa	10 kPa	10 kPa	10 kPa
3			10 kPa	10 kPa	10 kPa	10 kPa	10 kPa
4				10 kPa	10 kPa	10 kPa	10 kPa
5					10 kPa	10 kPa	10 kPa
6						10 kPa	10 kPa
7							10 kPa

.

Based on the parameters outlined in Table 5, the ABAQUS software (https://www.3ds.com/products-services/simulia/products/abaqus/, accessed on 8 November 2023) parameters have been set up to create the Mohr–Coulomb computational model. Utilizing this model, simulations have been performed to generate displacement contour plots for both the sliding body and anti-sliding piles, as illustrated in Figure 21. Additionally, an analysis of the internal forces and displacement data for both the anti-sliding piles and the sliding body has been carried out.

Figure 22 presents a summary of the displacement, bending moment, and shear force data obtained through numerical simulations.

It is evident that the force and displacement curves closely align with the stress calculation model, showing a high degree of consistency. Furthermore, the numerical values at loading stages of 20 kPa, 40 kPa, and 60 kPa are nearly identical. This provides further validation for the rationality and applicability of the stress calculation model for anti-sliding short piles.

6. Conclusions

This paper performs a stress analysis of short anti-slide piles, formulates fundamental assumptions, and outlines boundary conditions. It derives the governing equations for deflection curves in both the load-bearing and embedded sections of short anti-slide piles, subsequently solving for the internal forces of the entire pile through a finite difference calculation model. Large-scale indoor model tests are carried out to confirm the reliability of the proposed method. The primary findings can be summarized as follows:

(1)

Leveraging the Euler–Bernoulli beam theory, this study formulates a finite difference calculation model tailored for short anti-slide piles. Equation (10) enables the unified calculation of internal forces across the entire pile. This method obviates the requirement for intricate iterative computations between the load-bearing and embedded sections while relying on continuous conditions, thereby markedly enhancing computational efficiency.

(2)

Both experimental and calculated data demonstrate that short anti-slide piles undergo three discernible stages of displacement variation when subjected to identical loading conditions. The distribution of bending moments along the pile follows an “S” shape, with the maximum bending moments occurring in proximity to the sliding surface. Simultaneously, shear values peak at the sliding surface, while they attain zero values at positions corresponding to maximum positive and negative bending moments. These observations suggest that short anti-slide piles effectively strengthen the soil in the vicinity of the sliding surface.

(3)

The finite difference calculation model is utilized to independently compute displacements, bending moments, and shear values, subsequently subjecting them to comparison with experimental data. The observed discrepancies fall within an acceptable range, affirming the reliability and precision of the calculation model introduced in this paper. This novel approach offers a rapid method for determining the internal forces across short anti-slide piles. The study underscores the significance of the stress analysis model for short anti-slide piles in advancing sustainable engineering practices. It furnishes insights into the stress state and distribution patterns of short anti-slide piles, thereby providing valuable references for geological disaster prevention and control, as well as the promotion of sustainable engineering practices.