Simplified Reliability Analysis Method of Pile-Wall Combined Supporting Embankment Considering Spatial Variability of Filling Parameters

Abstract

To address the stability of high embankment slopes and investigate the influence of spatial variability of gravel soil on slope stability, this study proposes a simplified reliability analysis method of slope stability. Based on FLAC^3D, a numerical model was developed to simulate slope behavior, and a linear regression-based empirical model was formulated to quantify the relationship between soil pressure and spatial variability indicators (e.g., coefficient of variation and correlation length). By mapping the spatial variability of soil parameters to soil pressure fluctuations, the slope reliability was evaluated through the first-order second-moment method (FOSM). The results demonstrate that the mean value of internal friction angle has a significant effect on the stability of gravel soil embankment slope, whereas the coefficient of variation in this parameter is limited. Furthermore, the correlation length of soil spatial variability shows marginal influence on stability outcomes. The computational model was validated against case study data, demonstrating its applicability for practical slope stability assessments.

1. Introduction

In the context of establishing a robust transportation network, the construction of transportation infrastructure has continued to attract attention. The western region is predominantly mountainous and characterized by canyon landforms, with infrastructure that remains relatively underdeveloped. Recognized as a pivotal challenging component of national strategic implementation, innovative transportation infrastructure development strategies have been initiated in western territories [1]. With progressive implementation of highway projects, construction focus has shifted from plains to mountainous area. Mountainous routes are constrained by geomorphological features including steep gradients and erosional valleys, with alignment engineering frequently encountering challenges related to artificial fill placement and natural slope interactions. The construction of alpine highway is generally related to the stability of high-fill embankment on slope foundation [2]. When subjected to the action of natural factors such as driving load and precipitation, slope systems exhibit susceptibility to cause disasters such as rockfall, landslides, toppling failure and flow failure, which seriously threatens road driving safety. Consequently, comprehensive stability evaluation and effective control of embankment slopes represent a critical imperative in mountainous highway engineering projects.

For embankment slopes in mountainous terrain, the backfill material is usually selected caving slag, which leads to the obvious spatial variability of the embankment soil, which affects the stability of the supporting structure. The spatial variability of shear strength parameters of soil significantly affects the stability of slope structures, directly determining the location of the most critical slip surface and the scale of the sliding mass. The spatial variability of parameters also significantly influences the distribution characteristics of the critical slip surface [3]. Neglecting spatial heterogeneity in shear parameters may yield underestimated risk assessments during slope stability evaluations. When the coefficient of variation (COV) of shear strength exceeds the critical threshold, the probability of slope failure increases and the slope safety factor relatively reduces. Notably, the critical coefficient of variation in shear strength demonstrates positive correlation with safety factor magnitudes [4]. Current research on slope stability analysis of embankments considering the spatial variability of parameters is limited. Particularly for gravel-dominated embankments, explicit consideration of spatial variability constitutes an essential prerequisite for accurate stability assessment. In 2020, Dias et al. [5] considers the spatial variability impact on the behavior of a reinforced earth wall; in their study, the spatial variabilities at vertical direction of the soil properties are considered, while the horizontal spatial variability is ignored. Furthermore, the effects of soil spatial variability on the behavior of the retaining wall are reported in some other studies [6,7,8,9]. It can be seen that accounting for spatial variability leads to more realistic reliability assessments. Omission of such variability may generate biased reliability predictions, thereby increasing the risk of engineering damage [10,11,12,13,14].

Numerical simulation analyses of slope stability generally establish a random field model at first, selecting the appropriate discretization method based on different working conditions and actual engineering scenario. Current commonly used methods include the central point method, local averaging method, and Karhunen-Loève (K-L) expansion method [15,16,17,18,19,20]. When considering the spatial variability, the complexity of constructing the random field model increases significantly, resulting in a substantial increase in computational load and time. Then, the computational analysis methods of the model, such as the Monte Carlo method, response surface method, or first-order second-moment method [21,22,23]. The Monte Carlo method uses a large number of simulations to search within a set range of operating conditions. Spatial variability leads to higher computational demands, resulting in diminished efficiency. For slope reliability update problems, especially in the case of low probability, the direct application of the Monte Carlo method is excessively high [24]. Traditional methods such as the response surface method and the first-order second-order moment method are relatively simple and accurate in numerical calculations under general working conditions. When considering the spatial variability of parameters, the calculations become extremely complex, significantly increasing the computational difficulty.

This study presents a framework of replacing the random field model with a soil pressure model, leveraging an integrated anti-slide pile-retaining wall stabilization system. The proposed methodology reformulates the spatial variability of geotechnical parameters into quantifiable soil pressure effects, achieving dimensionality reduction in stochastic variables. The reliability of the CRS embankment was analyzed by the first-order second-order moment method, and the factors that may affect the reliability index were analyzed.

2. Numerical Model and Working Condition Analysis

2.1. Numerical Model

The Jiali Expressway in Guizhou comprises a structure of bridges, tunnels and embankments. Geomorphologically characterized by steeply inclined transverse slopes and high-level fill sections, this corridor exemplifies complex terrain engineering challenges. To ensure embankment stability, an integrated stabilization system combining anti-slide piles and retaining walls was implemented as the primary geotechnical intervention. This composite structure incorporates a continuous reinforced concrete footing interconnecting anti-slide piles to form a unified foundation system, with subsequent construction of platform and retaining wall.

In this manuscript, a numerical model was established using the finite difference software FLAC^3D 6.0 version, as shown in Figure 1a. The original slope of the bedrock is 1:1.5, with a height of 24.4 m. In the pile-wall combined support system, the anti-slide piles are 14 m long, with the lower 8 m embedded in the bedrock and the upper 6 m connected to the concrete platform. The retaining wall above the platform has a height of 10 m, with a top width of 4 m and a bottom width of 6.5 m. The anti-slide piles are made of the concrete C30 and reinforced steel, while the platform and retaining wall are constructed with the concrete of C15.

Due to the high hardness of the bedrock at this specific location, an elastic model is used in the numerical analysis framework to simulate the behavior of the bedrock. The supporting structures including anti-slide piles, a platform, and a gravity retaining wall are all composed of concrete members, so the elastic model is also used for simulation. The embankment filler composed of local sand is simulated by Mohr-Coulomb model. In view of the great difference between the elastic modulus of bedrock, platform, anti-slide pile and retaining wall and the elastic modulus of filling soil, an interface contact model is established at the boundary between these different materials. Figure 1b describes the schematic diagram of the contact interface model. S1, S2, … S7 represent the contact interface between fill and bedrock, fill and retaining wall, retaining wall and platform, anti-slide pile and platform, anti-slide pile and bedrock, platform and bedrock, and fill and platform. Among them, the internal friction angle (δ₁) of the contact surface between the backfill and the retaining wall is 0.7φ; the parameter (δ₂) between backfill soil and rock is 0.95φ; the parameter (δ₃) between the retaining wall and the platform is 35°; the parameter (δ₄) between the platform and the rock is 31°; the parameter (δ₅) between the pile and the rock is 39° [25], as shown in

Table 1. Friction angles of different interface elements.

Interface Friction Angle	Value (°)
δ1	28
δ2	38
δ3	35
δ4	31
δ5	39

.

Based on the previous pile-wall combined support model [25,26], this paper deduces the three-dimensional model of anti-slide pile-retaining wall combined support. Based on the three-dimensional support model verified by Bian et al. [9], considering the spatial variability of internal friction angle, the influence of the spatial variability of internal friction angle on the reliability of supporting structure is studied. The relevant parameters of retaining wall, pile, cap and retaining soil are shown in

Table 2. Material properties used in the numerical model.

Parameters	Rock (Elastic)	Pile (Elastic)	Platform (Elastic)	Wall (Elastic)	Retained Soil
Unit weight (kN/m3)	20.5	25	24	24	19.8
Cohesion (kPa)	—	—	—	—	0
Friction angle (°)	—	—	—	—	40
Modulus of elasticity (MPa)	11,000	30,000	22,000	21,000	59
Poisson’s ratio	0.26	0.2	0.2	0.2	0.3

.

2.2. Working Condition Analysis

The embankment slope in this investigation employs gravelly soil as the primary fill material. The parameters for the gravel soil slope were selected based on literature reviews, and typical parameter values were chosen to determine the working condition data. For most studies on gravel soil slopes, the internal friction angle of the gravel soil follows a logarithmic normal distribution, with the mean value μ_φ typically between 38 and 42°, and the coefficient of variation COV_φ between 0.05 and 0.5. The horizontal correlation length θ_v is between 8 and 40 m, while the vertical correlation length θ_h is between 2 and 10 m [27,28,29,30,31,32]. Given that the fill material in this study is gravel soil, the cohesion c can be assumed to be zero under the working conditions; thus, only the variability of the internal friction angle of the gravel soil needs to be considered.

The experiment employed a method of controlled variables and established 14 random operating conditions. Group A represents the base condition. Group B conditions involve altering three different vertical correlation lengths from the base condition. Group C conditions involve changing three different horizontal correlation lengths from the base condition. Group D conditions involve modifying four different coefficients of variation from the base condition. Group E conditions involve altering two different mean values of the internal friction angle from the base condition. Each random operating condition was simulated 500 times. The parameters for each group of conditions are detailed in

Table 3. Simulation conditions.

Condition	μφ	COVφ	θh/m	θv/m	Simulation Times
A0 (Baseline case)	40	0.15	40	4	500
B1	40	0.15	40	2	500
B2				8	500
B3				10	500
C1	40	0.15	8	4	500
C2			16		500
C3			32		500
D1	40	0.05	40	4	500
D2		0.3			500
D3		0.4			500
D4		0.5			500
E1	38	0.15	40	4	500
E2	42				500

Note: μ_φ is the mean value of internal friction angle; COV_φ is the coefficient of variation in internal friction angle; θ_h and θ_v are the horizontal and vertical fluctuation ranges, respectively.

.

Figure 2 presents a realization of the random field representing the friction angle of the embankment fill soil. It can be observed from the figure that the friction angle varies spatially within a range of 36° to 44°, clearly demonstrating the presence of spatial variability in the embankment fill soil parameters.

Figure 3 shows the convergence curves of the mean value and coefficient of variation in the earth pressure on the retaining wall under the baseline case (A₀) as the number of Monte Carlo simulations increases. It can be observed that both the mean and the coefficient of variation gradually stabilize after approximately 300 simulations. This indicates that performing 500 Monte Carlo simulations, as adopted in this study, is sufficient to obtain converged statistical characteristics of the earth pressure.

Based on the results derived from the working conditions, the spatial variability index has different impacts on the magnitude of lateral soil pressure on retaining walls. Specifically, the correlation length does not significantly affect the mean, 95% quantile value and coefficient of variation in the lateral soil pressure response values on retaining walls. As the COV of the internal friction angle increases, the mean and 95% quantile value of the lateral soil pressure response values on retaining walls decreases, while the COV increases. Conversely, as the mean of the internal friction angle increases, the mean, 95% quantile value, and COV of the lateral soil pressure response values on retaining walls decreases. For the lateral soil pressure on the platform, the horizontal correlation length does not significantly affect the mean, 95% quantile value, and COV of the lateral soil pressure response values. The vertical correlation length does not significantly affect the mean and 95% quantile value of the lateral soil pressure response values on the platform, but it does have a certain impact on the COV. As the COV of the internal friction angle of the embankment fill increases, the mean, 95% quantile value, and COV of the lateral soil pressure response values on the platform increase. Conversely, as the mean value of the internal friction angle increases, the mean, 95% quantile value and COV of the lateral soil pressure response values on the platform decrease. The lateral soil pressure and its application point on the retaining wall and platform under the working conditions, along with the mean and standard deviation, are presented in

Table 4. Statistics of mean and standard deviation of soil pressure on retaining wall and platform under various random conditions.

Condition	Soil Pressure of Retaining Wall Eaw/kN·m−1		Soil Pressure of Platform Eap/kN·m−1
	µEaw	COVEaw (×10−3)	µEap	COVEap (×10−3)
A0	758.87	4.20	361.83	17.00
B1	751.75	3.46	360.29	12.66
B2	754.72	3.62	361.07	14.57
B3	761.98	4.01	362.25	18.93
C1	754.83	3.33	360.42	10.77
C2	763.77	4.20	361.90	19.51
C3	770.52	4.51	361.96	21.47
D1	775.81	2.19	378.13	4.16
D2	751.91	7.38	360.44	26.06
D3	747.16	8.22	358.41	25.77
D4	743.89	8.23	358.54	25.07
E1	760.14	5.27	379.80	25.77
E2	751.99	3.26	353.69	9.58

and

Table 5. Statistics of mean and standard deviation of soil pressure application points on retaining wall and platform under various random conditions.

Condition	Soil Pressure Acting Point of Retaining Wall z1/m		Action Point of Platform Soil Pressure z2/m
	µz1	COVz1 (×10−3)	µz2	COVz2 (×10−3)
A0	9.7120	1.29	3.5137	2.36
B1	9.7124	1.02	3.5133	1.85
B2	9.7122	1.08	3.5135	2.33
B3	9.7114	1.23	3.5143	2.50
C1	9.7122	1.03	3.5133	1.76
C2	9.7116	1.26	3.5139	2.65
C3	9.7117	1.43	3.5137	2.82
D1	9.7115	0.39	3.5136	0.63
D2	9.7241	2.46	3.5062	4.45
D3	9.7285	2.57	3.5029	5.05
D4	9.7322	2.76	3.5000	4.97
E1	9.7055	1.84	3.5183	3.30
E2	9.7140	0.75	3.5123	1.65

Note: In the calculation of the reliability index, the shear strength and bending strength of the slide-resistant pile are obtained according to the actual working conditions and related specifications and based on the formula, which are 6710 kN and 39,800 kN·m⁻¹, respectively.

.

Therefore, this study establishes an empirical model using linear regression to relate four input parameters with two response parameters (soil pressure) based on simulated soil pressure data, thereby replacing the stochastic field model for reliability analysis of road embankment slopes under different conditions.

3. Soil Pressure Calculation Model

3.1. Calculation Model of Mean Soil Pressure

From the above study on the influence of spatial variability on the normalized lateral soil pressure response value (μ_Xo/X_o), it can be seen that the normalized lateral soil pressure response value has a nonlinear relationship with the spatial variability. Therefore, the polynomial equation is used to express the relationship:

$${\mu }_{X_u}/X_0=AS{v}^4+BS{v}^3+CS{v}^2+DSv+E$$

(1)

where S_v is the spatial variability; A, B, C, D and E are the coefficients of linear relationship.

By fitting the relationship between the standardized lateral soil pressure response value of the retaining wall and the spatial variability, the functional relationship is summarized in

Table 6. The functional relationship between μ_Xo/X_o and the spatial variability index.

Spatial Variability Index (Expressed by x)	μXo/Xo
	Lateral Soil Pressure Response Value of Retaining Wall	Lateral Soil Pressure Response Value of Platform
θh	fw1 = 1.01153 + 0.0000288x + 0.00000052x2	fp1 = 1.03367 + 0.000544x − 0.0000096x2
θv	fw2 = 1.0133 + 0.000175x	fp2 = 1.03401 + 0.00242x − 0.000146x2
COVφ	fw3 = 1.02202 − 0.10512x + 0.46084x2 − 0.8899x3 + 0.55365x4	fp3 = 0.99232 + 0.45797x − 0.95397x2 + 0.78901x3
μφ	fw4 = 1.9245 − 0.04228x + 0.0004875x2	fp4 = 8.2243 − 0.33955x + 0.004x2

Note: f denotes μ_Xo/X_o; x is the spatial variability index (horizontal correlation length, vertical correlation length, coefficient of variation and mean).

. It should be noted that the fitting formulas provided are based on the parameter variations defined in Table 3, and are therefore valid within specific ranges, namely: θₕ ranges from 16 m to 40 m, θᵥ ranges from 2 m to 10 m, COV_φ ranges from 0.05 to 0.5, and μ_φ ranges from 38° to 42°, as specified in Table 3.

As indicated in Table 4, the data reflect the influence of individual spatial variability index on the normalized lateral soil pressure response values. Compared with the condition of A₀, when multiple spatial variability changes at the same time, the relationship is difficult to determine. Therefore, we have found through many attempts that the influence of different indices on the response value of standardized lateral soil pressure is almost independent of each other. Therefore, this study posits that the linear superposition method can be employed to calculate the impact of multiple spatial variability indices on the normalized lateral soil pressure response values.:

$$\begin{array}{l}{\mu }_{X_u}/X_0={\left({\mu }_{X_0}/X_0\right)}^{*}+f_1\left({\theta }_v\right)-f_1\left({\theta }_v^{*}\right)-f_2\left({\theta }_h\right)-f_2\left({\theta }_h^{*}\right)\\ +f_3\left(CO{V}_{\phi }\right)-f_3\left(CO{V}_{\phi }^{*}\right)+f_4\left({\mu }_{\phi }\right)-f_4\left({\mu }_{\phi }^{*}\right)\end{array}$$

(2)

where ‘*’ denotes the reference condition (θv = 2 m; θh = 0.8 m; COV_φ = 0.15; μ_φ = 40°), f₁ (·) represents the relationship between μ_Xo/X_o and θ_v, f₂ (·) represents the relationship between μ_Xo/X_o and θ_h, f₃ (·) represents the relationship between μ_Xo/X_o and COV_φ, and f₄ (·) represents the relationship between μ_Xo/X_o and μ_φ.

3.2. Calculation Model of Standard Deviation of Soil Pressure

The coefficient of variation in the standardized response index varies with changes in the spatial coefficient of variation, and the relationship is nonlinear. Therefore, the relationship between COV_μXu/Xo and X_o can be represented by a polynomial function:

$$CO{V}_{{\mu }_{X_u}/X_0}=F\times S{v}^4+Z\times S{v}^3+H\times S{v}^2+J\times Sv+K$$

(3)

The relationship between COV_μXu/Xo and σX_u can be expressed as

$${\sigma }_{X_u}=X_0\times CO{V}_{{\mu }_{X_u}/X_0}$$

(4)

Then it will be brought into the available:

$${\sigma }_{X_u}=X_0\times CO{V}_{{\mu }_{X_u}/X_0}=\left(C\times S{v}^1+D\times S{v}^2+E\right)\times X_0$$

(5)

where S_v is the spatial variability index, and C, D and E are the coefficients of linear relationship.

By fitting formulas to the relationship between the standardized lateral soil pressure response values of retaining walls and spatial variability indices, the functional relationships are summarized in

Table 7. The functional relationship between COV_μXu/Xo and the spatial variability index.

Spatial Variability Index (Expressed by x)	COVμXu/Xo
	Lateral Soil Pressure Response Value of Retaining Wall	Lateral Soil Pressure Response Value Platform
θh	gw1 = 0.00216 + 0.00006x	p1 = 0.00804 + 0.000896x − 0.0000172x2
θv	gw2 = 0.00223 + 0.00102x − 0.000133x2 + 0.00000625x3	gp2 = 0.01753 − 0.00109x + 0.0001583x2
COVφ	gw3 = 0.00102 + 0.01991x + 0.009x2 − 0.09316x3	gp3 = −0.00351 + 0.16834x − 0.19607x2
μφ	gw4 = 0.1356 − 0.00578x + 0.0000625x2	gp4 = 2.0329 − 0.0954x + 0.000113x2

Note: g denotes COV_μXu/Xo; x is the spatial variability index (horizontal correlation length, vertical correlation length, coefficient of variation and mean).

. Subsequently, the linear superposition method is employed to determine the impact of multiple spatial variability indices on the standardized lateral soil pressure response values:

$$CO{V}_{{\mu }_{X_u}/X_0}={\left(CO{V}_{{\mu }_{X_u}/X_0}\right)}^{*}+g_1\left({\theta }_v\right)-g_1\left({\theta }_v^{*}\right)+g_2\left({\theta }_h\right)-g_2\left({\theta }_h^{*}\right)$$

(6)

Thus, substituting the formula yields:

$${\sigma }_{X_u}=\left[\begin{array}{l}{\left(CO{V}_{{\mu }_{X_u}/X_0}\right)}^{*}+g_1\left({\theta }_v\right)-g_1\left({\theta }_v^{*}\right)+g_2\left({\theta }_h\right)-g_2\left({\theta }_h^{*}\right)\\ +g_3\left(CO{V}_{\phi }\right)-g_3\left(CO{V}_{\phi }^{*}\right)+g_4\left({\mu }_{\phi }\right)-g_4\left({\mu }_{\phi }^{*}\right)\end{array}\right]X_0$$

(7)

where ‘*’ is the reference condition (θ_h = 2 m; θ_v = 0.8 m; COV_φ = 0.15; μ_φ = 40°). g₁(·) represents the relationship expression between COV_μXu/Xo and θ_v, g₂(·) represents the relationship expression between COV_μXu/Xo and θ_h, g₃(·) represents the relationship expression between COV_μXu/Xo and COV_φ, and g₄(·) represents the relationship expression between COV_μXu/Xo and μ_φ.

3.3. Verification of the Calculation Model

To fully demonstrate the accuracy of the spatial variability index and the standardized soil pressure response expression, an additional condition was introduced for validation. Statistical analysis indicates that as the correlation length increases, both the mean and the coefficient of variation in the response index escalate, trending towards a more hazardous condition. Conversely, with the increase in the coefficient of variation in the input parameters, the mean of the response index diminishes, while its coefficient of variation augments. Consequently, a condition predisposed to risk was established (horizontal correlation length θ_h = 60 m, vertical correlation length θ_v = 8 m, coefficient of variation for the internal friction angle COV_φ = 0.6). The simulation results from this new condition were analyzed and compared against the formula-derived and computational outcomes, as detailed in Table 4.

As indicated in Table 6, the actual simulation results closely align with the equivalent estimation outcomes. The error between the estimated and actual simulated values of the lateral soil pressure response index μ_Xu/X_o for the retaining wall is merely 0.797%, while the error for the lateral soil pressure response index COV_μXu/Xo is only 5.825%. This demonstrates that through formula fitting, it is possible to accurately estimate the lateral soil pressure response values μ_Xu/X_o and COV_μXu/Xo for retaining walls under specific spatial variability parameters. Furthermore, the error between the estimated and actual simulated values of the lateral soil pressure response index μ_Xu/X_o for the platform is just 1.581%, and the error for the lateral soil pressure response index COV_μXu/Xo is only 6.410%. Similarly, based on Table 6, the lateral soil pressure response values μ_Xu/X_o and COV_μXu/Xo for the platform under different spatial variability parameters can be derived, enabling the calculation of σX_u through the formula. As mentioned above, substituting the X_o obtained from the deterministic analysis into

Table 8. Verification of simulation results of new working conditions.

Soil Pressure Category		Basic Operation Condition	Actual Simulation Results	Equivalent Estimation Results	Relative Error (%)
Lateral soil pressure response value of retaining wall	μXo/Xo	1.0139	1.0293	1.0211	0.797
	COVμXu/Xo	0.0046	0.0206	0.0194	5.825
Lateral soil pressure response value of platform	μXo/Xo	1.0423	1.0121	0.9961	1.581
	COVμXu/	0.0169	0.0312	0.0292	6.410

derives the mean μ_Xu, coefficient of variation COV_μXu, and spft deviation σ_Xu of the reliability analysis response index considering spatial variability.

The comparison between the data simulation results in Table 6 and the actual operating condition data reveals that the calculated results of the established model’s empirical formula generally fall within an acceptable margin of error. This empirical model can be utilized to compute statistical data values, enabling the rapid determination of required response index through interpolation of the obtained results. This expedites the reliability analysis of embankments in the subsequent phase.

4. Simplified Reliability Analysis

4.1. Failure Mode and Limit State Equation

When a structure exceeds a certain specific condition and fails to meet the functional requirements as per design specifications, this specific condition is termed as the limit state. The functional function of the structure can be represented as

$$Z=g_X\left(X_1,X_2,\cdots ,X_n\right)$$

(8)

when Z = 0, the structure is in the limit state; when Z > 0, the structure is in a reliable state; when Z < 0, the structure is in a state of failure.

The simplified force diagram of the pile-wall combined support structure is shown in Figure 4.The design of the pile-wall-combined support structure must meet the requirements of strength and stability, and its stability generally requires anti-overturning stability and anti-slip stability. When the resistance of the pile-wall combined support structure is greater than or equal to its action effects, the structure will be in a reliable state. Consequently, in accordance with the relevant codes, an ultimate limit state equation for the stability of pile-wall combined support structures can be established.

The shear strength and bending strength of the slide-resistant pile can be obtained according to the ‘landslide prevention engineering design and construction technical specifications’:

$$F_p=0.07f_{c}b{h}_0+1.5f_{yv}{A}_{sv}/s{h}_0$$

(9)

$$M_p=f_y{A}_s/K_p\left(h_0-0.5f_y{A}_s/f_{cm}b\right)$$

(10)

where F_p is the shear strength of slide-resistant pile, M_p is the bending strength of slide-resistant pile; f_c and f_cm are the design values of axial compressive strength and flexural compressive strength of concrete, respectively, f_yv and f_y are the design values of tensile strength of stirrups and tensile steel bars, respectively. A_sv and A_s are the cross-sectional area of stirrups and steel bars, respectively, and b is the cross-sectional width of slide-resistant piles; h₀ is the effective height of the slide-resistant pile section, and S is the spacing of the slide-resistant pile stirrups; K_p is the design safety factor of flexural strength of slide-resistant pile, which is generally taken as 1.05 according to the code.

The anti-slip limit state equation of pile-wall combined support structure is

$$\begin{array}{l}Z=R_F-S_F=F_{k1}-F_f\\ ={\mu }_2\left[\left(G_1+G_2\right)\cos \alpha +E_{aw}\sin \left({\delta }_w-\alpha \right)+E_{ap}\sin \left({\delta }_p-\alpha \right)\right]+F_p/d \\ -E_{aw}\cos \left({\delta }_w-\alpha \right)-E_{ap}\cos \left({\delta }_p-\alpha \right)-\left(G_1+G_2\right)\sin \alpha \end{array}$$

(11)

The anti-overturning limit state equation of pile-wall combined support structure is

$$\begin{array}{l}Z=R_M-S_M=M_{k1}+M_{k2}-M_f\\ =G_1{x}_1+E_{aw}\sin {\delta }_w{x}_2+G_2{x}_3+E_{ap}\sin {\delta }_p{x}_4+M_p/d-E_{aw}\cos {\delta }_w{z}_1\\ -E_{ap}\cos {\delta }_p{z}_2\end{array}$$

(12)

where F_k and F_f are anti-sliding force and sliding force, respectively; M_k and M_f are anti-overturning moment and overturning moment, respectively. δ_w is the friction angle between the fill and the retaining wall, δ_p is the friction angle between the fill and the platform, E_aw is the soil pressure behind the retaining wall, α is the inclination angle of the platform base, G₁ is the self-weight of the retaining wall, E_ap is the soil pressure behind the platform, G₂ is the self-weight of the platform, and μ₂ is the friction coefficient between the platform base and the original rock. x₁ and x₂ are the horizontal length from the G₁ and E_aw action points to the wall toe, respectively; z₁ is the vertical length from the E_aw action point to the wall toe; x₃ and x₄ are the horizontal length from the G₂ and E_ap action points to the wall toe, respectively.

4.2. Reliability Analysis

Given the computational simplicity and reduced error of the improved first-order second-moment method, along with its satisfactory accuracy in most cases to meet practical engineering requirements and its widespread acceptance in the engineering community [21,33,34], this study employs the FOSM for the calculation of reliability index.

The research presented in this study indicates that there are four random variables influencing the reliability index of the anti-sliding limit state of the combined pile-wall support structure. The functional function can be expressed as

$$Z=g\left(X_{E_{aw}},X_{E_{ap}},X_{{\delta }_w},X_{{\delta }_p}\right)$$

(13)

Then the limit state equation is

$$Z=g\left(X_{E_{aw}},X_{E_{ap}},X_{{\delta }_w},X_{{\delta }_p}\right)=0$$

(14)

Expanding the limit state equation Z at the mean value using Taylor series, the first-order Taylor series expansion of the equation is

$$Z_L=g_X\left({\mu }_{X_{E_{aw}}},{\mu }_{X_{E_{ap}}},{\mu }_{X_{{\delta }_w}},{\mu }_{X_{{\delta }_p}}\right)+{\displaystyle \sum _{i=1}^4\frac{\partial g_X\left(\mu \right)}{\partial X_i}\left(X_i-{\mu }_{X_i}\right)}$$

(15)

The formula for calculating the reliability index of the anti-slip limit state of the pile-wall combined support structure can be expressed as

$$\beta ={\displaystyle \frac{{\mu }_Z}{{\sigma }_Z}}$$

(16)

The reliability index for the anti-slip limit state and the anti-overturning limit state of the combined pile-wall support structure under twelve random working conditions, calculated separately through formulas, are shown in

Table 9. Summary of support reliability index under random conditions.

Condition	Anti-Slip Limit State Reliability Index βF	Anti-Overturning Limit State Reliability Index βM
A0	2.7278	3.4855
B1	2.7602	3.5178
B2	2.7463	3.5041
B3	2.7144	3.4719
C1	2.7475	3.5043
C2	2.7082	3.4646
C3	2.6816	3.4361
D1	3.6841	4.4963
D2	1.9268	2.7576
D3	1.6184	2.3266
D4	1.3929	2.0654
E1	3.1740	4.3739
E2	2.4941	3.3224

.

4.3. Results Analysis

Based on the calculation data of embankment support reliability under different working conditions using the first-order second-order moment method, the potential impacts on the reliability index of the pile-wall combined support structure were explored by integrating experimental data, considering the spatial variability of parameters, and examining aspects such as mean values, coefficient of variation, and correlation length.

4.3.1. The Influence of Mean Value on Reliability Index of Pile-Wall Combined Supporting Structure

As illustrated in Figure 5, it is evident that the reliability index for both the anti-sliding and anti-overturning limit state of the pile-wall combined supporting structure decrease with the increase in the mean of the internal friction angle of the embankment fill. Specifically, the trend of β_M reduction accelerates sharply initially and then moderates as µ_φ increases, whereas β_F exhibits a gradual and relatively stable decrease with the rise in µ_φ. This analysis indicates that the mean of the internal friction angle has a limited impact on the reliability index of both limit states.

4.3.2. The Influence of Variation Coefficient on Reliability Index of Pile-Wall Combined Supporting Structure

As illustrated in Figure 6, as the coefficient of variation in the internal friction angle (φ) increases from 0.05 to 0.5, the reliability index for the overturning limit state (β_M) and for the sliding limit state (β_F) of the pile-wall combined support structure exhibit a declining trend. However, the rate of decline diminishes progressively, eventually leveling off. This observation indicates that the variability coefficient of the internal friction angle has a significant impact on the reliability index of both limit states for the pile-wall combined support structure. Moreover, as the uncertainty in the internal friction angle increases, the reliability of the support structure diminishes, thereby elevating the probability of failure. Nevertheless, the reliability index for both limit states progressively weakens with the escalating uncertainty in the internal friction angle.

4.3.3. The Influence of Correlation Length on Reliability Index of Pile-Wall Combined Support Structure

As illustrated in Figure 7, when the horizontal correlation length increases from 1.0 m to 4.0 m, the reliability index for the anti-overturning limit state (β_M) and the reliability index for the sliding limit state (β_F) exhibit a declining trend, though the decrease is minimal, remaining essentially constant. Similarly, within the range of vertical correlation length from 0.4 m to 1.0 m, the decline in β_M and β_F is gradual. The analysis above indicates that the correlation length of the embankment fill internal friction angle parameter has a negligible impact on the reliability index of the anti-sliding and anti-overturning limit state for the pile-wall combined supporting structure. Therefore, under the current conditions of this study, the uncertainty in the correlation length of the embankment fill internal friction angle parameter has a minor influence on the stability of the embankment slope itself, which can be disregarded, and the analytical precision meets practical engineering requirements.

5. Conclusions

This manuscript focuses on the gravel soil embankment slope considering spatial variability of the soil parameter and proposes a method that uses a soil pressure model to account for the effects of spatial variability, replacing the construction and calculation of traditional random fields. Through mathematical fitting and superposition, combined with Monte Carlo simulation and an improved first-order second-moment method, this approach enables efficient reliability analysis. Additionally, the main factors influencing the reliability index are analyzed, leading to the following conclusions:

(1)

The mean value of internal friction angle has a limited influence on the reliability index of supporting structure, but its coefficient of variation has a significant influence on reliability. When the coefficient of variation is less than 0.4, the anti-sliding and anti-overturning reliability indexes decrease rapidly, and the failure probability increases significantly, indicating that the uncertainty of filling parameters is a key factor that cannot be ignored in slope stability evaluation.

(2)

The influence of the horizontal and vertical correlation length of the filling parameters on the reliability index is negligible. Within the range of variation (horizontal 8–40 m, vertical 2–10 m), the fluctuation range of the reliability index is less than 5%, indicating that the analysis process of the correlation length can be simplified in practical engineering.

(3)

The proposed empirical formula of earth pressure and the linear superposition method can efficiently replace the traditional random field model, which can significantly reduce the computational complexity and provide theoretical support for the rapid stability evaluation of embankments with complex terrain.

(4)

It is assumed that the cohesion of filling soil is zero. In the future, the influence of cohesion and its spatial variability on stability should be further discussed to improve the applicability of the model.