Analysis of the Vertical Bearing Capacity of Pile Foundations in Backfill Soil Areas Based on Non-Stationary Random Field

Abstract

The soil parameters of artificial soil foundations exhibit significant spatial variability due to particle sorting during the filling process, which follows a trend that is distinct from that of natural soft foundations. Based on this observation, a non-stationary random field model is proposed, in which the undrained shear strength decreases along the horizontal filling direction and increases along the vertical deposition direction. The impact of the non-stationary random field model on the vertical bearing capacity of pile foundations is analyzed and discussed with Monte Carlo simulations. The results indicate that the mean and coefficient of variation of the undrained shear strength significantly affect the vertical ultimate bearing capacity of pile foundations, whereas the effects of the random field correlation distance and vertical non-stationary trend coefficient are relatively minor. Under the influence of the horizontal non-stationary trend coefficient, the mean vertical ultimate bearing capacity of pile foundations decreases with the horizontal distance between the pile and the backfill entrance. Moreover, the rate of decrease diminishes linearly with the horizontal non-stationary trend coefficient. A predictive formula for the vertical ultimate bearing capacity of pile foundations is proposed, and its applicability is verified.

1. Introduction

In the context of accelerated urbanization, China is grappling with increasingly pressing land resource constraints. Therefore, it is crucial to explore more efficient land development strategies [1]. Reclamation of soft foundations through infill (artificial soil foundations) is one approach to addressing this issue. Compared to natural foundations, artificial soil foundations offer distinct advantages, including the effective alleviation of fill resource shortages and the promotion of resource recycling, thereby contributing to the sustainable development of urbanization [2]. Natural foundations consist of layers of soil or rock that naturally exist on the foundations and serve as the primary support layer for buildings or engineering structures. Compared to natural foundations, artificial soil foundations formed by hydraulic filling of dredged silt exhibit unique spatial variability characteristics. The particle sorting phenomenon during the filling process leads to a pronounced non-stationary horizontal distribution of strength parameters. This feature fundamentally differs from the stationary random fields typically observed in natural foundations [3,4]. Current research on natural foundations is relatively comprehensive, covering aspects such as foundations bearing capacity [5], settlement, stability [6], and foundations improvement techniques [7]. Studies on the backfill soil foundations are still limited, particularly regarding the soil parameter characterization for artificial soil foundations. However, directly applying such models to artificial soil foundations can overlook the trend of strength variation with spatial location. This is particularly significant as it fails to account for the influence of pile position (the backfill entrance) on the ultimate bearing capacity. Ignoring these factors could result in predictive inaccuracies and significantly increase engineering risks.

In practical engineering, techniques such as compaction, grouting, and pile foundations are commonly employed to treat the backfill soil foundations. Among these, pile foundation engineering is widely applied to enhance the bearing capacity and stability of the soil as a common method. Numerous scholars have conducted in-depth studies on the bearing performance of pile foundations through numerical simulations, using random field theory to analyze the impact of spatial variability of the soil parameters on bearing capacity. Kasama et al. [8] investigated the effect of spatial variability of soft soil strength on the overall compressive performance of piles in cement-treated soft soil based on Monte Carlo simulations and numerical limit analysis. Wu et al. [9] developed a finite element model to analyze the bearing performance variations of piles in cement-treated soil. Their findings indicated that the spatial variability of the soil has a significant impact. Tan et al. [10] investigated the influence of spatial variability in soil properties on the bearing capacity of a single pile. Through reliability analysis, they derived an analytical formula for the characteristic length, which was ultimately validated with case studies. Dong et al. [11] optimized the reliability analysis and design methods for vertical load piles, considering the spatial variability of the soil. They also proposed a reliability sensitivity analysis method and validated its applicability.

The aforementioned studies considered the spatial variability of the soil in natural foundations when analyzing the bearing performance of pile foundations, thereby providing an important theoretical basis and practical guidance for enhancing foundations bearing capacity. However, these analytical methods are not suitable for pile structures in the backfill soil, and their direct application may lead to calculational inaccuracies. To address this issue, this study integrates random field theory [12,13], the Karhunen–Loeve expansion method [14], and a finite element–Monte Carlo hybrid algorithm to effectively tackle the numerical challenges associated with simulating non-stationary fields and the coupled analysis of pile–soil interaction. A novel random field model that incorporates a horizontal non-stationary trend component is proposed. By introducing a trend function that varies with spatial coordinates, this research quantitatively characterizes the non-stationarity of the strength of backfill soil for the first time. In contrast to traditional stationary models, the new model more accurately reflects the gradient distribution of strength caused by particle sorting. Additionally, this study establishes a relationship between pile position and bearing capacity and discusses the impact of spatial variability of the soil on the ultimate bearing capacity of pile foundations. It reveals differentiated behaviors in bearing capacity under varying pile lengths and positions. The significance of this research lies in its potential to guide the reliability design of pile foundations. For instance, adjustments to pile length or position can be made in strength degradation zones to compensate for bearing capacity loss, thereby meeting reliability performance requirements.

2. Non-Stationary Random Field of Undrained Shear Strength in Backfill Soil Area

2.1. Spatial Characterization of the Undrained Shear Strength

In the reclamation cofferdam project, the backfill soil exhibits a significant phenomenon of particle segregation. Numerous scholars have conducted extensive field experiments to investigate this phenomenon, analyzing the segregation mechanism of dredged sediment particles. Their findings indicate that the larger particles tend to settle closer to the backfill entrance, while the smaller particles are more likely to migrate to greater distances. It is influenced by their lower gravitational forces. Wang et al. reported that the permeability coefficient of dredged mud decreases with the distance from the backfill entrance, and it is based on field investigation data from Baima Lake [15]. Wu et al. conducted vane shear strength and discovered that the dredged mud at different locations within the site exhibits varying mechanical properties in the horizontal direction [16]. These related research findings indicate that the upper and distal layers of the site are predominantly composed of fine particles, while the nearer and lower layers primarily consist of coarse particles. This particle sorting phenomenon exacerbates the spatial variability of the soil parameters at the backfill soil foundations, and it leads to the strength of the soil decreasing with the distance to the backfill entrance in the horizontal direction while increasing with the depth in the vertical direction [17].

Some scholars have examined the spatial variability of the backfill and the soil parameters resulting from particle sorting. In the horizontal direction, Wang et al. conducted field tests on dredged mud, measuring the water content of the soil at two points located at the same depth (0.55 m below the surface) along the backfill direction [15]. The results indicated that the water content at the backfill entrance was 36%, whereas it increased to 107% at a distance of 100 m from the backfill entrance. This shows a gradual increase in water content within 0–100 m. Additionally, Wu et al. investigated the spatial distribution of the soil parameters in the horizontal direction by performing a cross-plate curve of field vane shear strength at the site selected from Baima Lake [16]. The site has an east–west length of approximately 850 m, with the backfill entrance situated at the eastern edge. Soil samples were taken from a depth of 5 m below the surface in the eastern, central, and western sections, yielding vane shear strength of 16.1, 12.12, and 14.26 kPa, respectively. Both field studies demonstrate that particle sorting within the backfill soil foundations leads to significant spatial variability of the soil parameters in the horizontal direction. Moreover, this variability trend correlates with the distance from the backfill entrance. Specifically, the soil strength decreases with the distance from the backfill entrance, with a decay coefficient approximately ranging from 0.02 to 0.08 kPa/m.

In the vertical direction, Cai et al. [18] conducted field tests at Nansha Port Area, focusing on the characteristics of deep silt soft soil. They employed a cross-plate curve of field vane shear strength that commenced sampling and automatic recording of test data from a depth of 5 m. The results revealed the variation of the soil strength with depth. Figure 1 indicates that the undrained shear strength (S_u) of the soil exhibits a depth-dependent trend of 0.63 kPa/m.

Additionally, Guo et al. performed extensive field tests in the Pearl River Estuary and utilized empirical geotechnical formulas to calculate the undrained shear strength S_u [19]. They established a linear correction formula for S_u of soils with depth, expressed as S_u = 0.244γz [18].

Table 1. Relationship between undrained shear strength and the depth of soil.

Soil Types	Common Interest	Literature Sources
Soft soil in Nansha Port Area	Su = 0.780z	Cai et al. [18]
Soft soil in the Pearl River Estuary (PRE)	Su = 0.405z	Guo et al. [19]

summarizes the relationship between undrained shear strength and the depth of soils from the related literature. As shown in the table, the undrained shear strength of the soil increases with depth and the growth factor approximately ranges from 0.4 to 0.8 kPa/m.

From the above analysis, it is evident that the soil parameters of artificial soil foundations are influenced by the particle sorting phenomenon. And this phenomenon leads to exhibiting spatial variability both horizontally and vertically during the blowing and filling process. Moreover, this spatial variability demonstrates a certain trend, which significantly differs from that of natural soil foundations. The conventional stationary random field models assume that the soil parameters are evenly distributed in space and do not vary with location, making them suitable for distinctly homogeneous natural soil layers. The conventional stationary random field model, which assumes a uniform distribution of the soil parameters in space without location-based variation, is applicable to clearly homogeneous natural soil layers. Therefore, it is inadequate for capturing the spatial variability trend observed in artificial soil foundations effectively, where the strength parameters diminish with increasing horizontal distance and increase with greater depth.

2.2. Non-Stationary Random Field of Undrained Shear Strength

The spatial variability of the soil parameters is characterized by the combination of a deterministic trend component and a random fluctuation component [20]. Conventional non-stationary random fields typically consider the variation of soil parameter trends in the vertical direction, with little focus on analyzing changes in the horizontal direction.

$$S_{\mathrm{u}}(x,z)=S_{\mathrm{u}0}(x,z)$$

(1)

$$S_{\mathrm{u}}(x,z)=S_{\mathrm{u}0}(x,z)-k_{\mathrm{v}}({\displaystyle \frac{h}{2}}+z)$$

(2)

where h represents the thickness of the backfill soil layer; S_u0(x, z) is a stationary random field based on the strength of the soil at point A, located at the midpoint of the soil layer with coordinates (0, −z/2); S_u(x, z) is a non-stationary random field that incorporates the strength variations in both the horizontal (x-direction) and vertical (z-direction) directions; and k_v denote the non-stationary trend components of the strength variation of the soil along the z-direction, respectively.

Equation (1) represents the strength expression of soil parameters in a two-dimensional stationary random field, while Equation (2) describes the strength expression of soil parameters in a two-dimensional non-stationary random field [20]. It can be concluded that horizontal non-stationary characterization is introduced to describe the strength variation of the soil parameters in the horizontal direction. This allows for the establishment of a non-stationary random field that aligns with the spatial distribution characteristics of the soil parameters in the backfill soil foundations.

As illustrated in Figure 2a, the foundations are blown through the backfill entrance while the backfill outlet is utilized to discharge excess water. Figure 2b shows the longitudinal section of the backfill soil layer through establishing the xoz coordinate system with the backfill entrance as the origin. In this context, the random field of the undrained shear strength of the soil in the xoz plane can be expressed as follows:

$$S_{\mathrm{u}}(x,z)=S_{\mathrm{u}0}(x,z)-k_{\mathrm{h}}x-k_{\mathrm{v}}({\displaystyle \frac{h}{2}}+z)$$

(3)

where k_h denote the non-stationary trend components of the strength variation of the soil along the x-direction, respectively.

The mean and standard deviation of this random field are given by the following:

$${\mu }_{S\mathrm{u}}(x,z)={\mu }_{S\mathrm{u}0}-k_{\mathrm{h}}x-k_{\mathrm{v}}({\displaystyle \frac{h}{2}}+z)$$

(4)

$${\sigma }_{S\mathrm{u}}(x,z)={\sigma }_{S\mathrm{u}0}$$

(5)

where μ_Su0 and σ_Su0 represent the mean and standard deviation of S_u0. Since S_u0 is a stationary random field, its mean and standard deviation are constant across different locations. Consequently, the mean value of the non-stationary random field as expressed in Equation (3) decreases with increasing x and increases with increasing z (in absolute value).

Based on the analysis in the previous subsection, k_h is assumed to range from 0.02 to 0.08 kPa/m, while k_v ranges from 0.4 to 0.8 kPa/m. The Karhunen-Loève (K-L) series discretization method is employed to derive the basic random field S_u0, which is then substituted into Equation (1) to obtain the non-stationary random field that satisfies the specified conditions.

3. Probabilistic Analyses for Vertical Bearing Capacity of Pile Foundations

3.1. Numerical Model

Figure 3 illustrates a schematic representation of the backfill soil foundations. And the backfill entrance is situated at the left (origin) of the diagram. The stratigraphy is composed of the backfill soil, gravel, and gravelly clay, arranged in descending order. The thicknesses of the backfill soil layer, the gravel layer, and the sandy gravelly clay layer are 8 m, 6 m, and 6 m, respectively. The relevant parameters are provided in

Table 2. The parameters of the soil.

Soil	High h (m)	Density γ (g/m3)	Cohesion c (kPa)	Friction φ (°)	Elastic Modulus E (MPa)	Poisson’s Ratio υ
The backfill soil	8	1.80	35	0	9.8	0.35
Gravel	6	1.90	0	30	45	0.25
Sandy gravelly clay	6	1.85	22	26	50	0.25

, where the mean undrained shear strength (S_u) at the midpoint of the backfill soil layer (designated as point A) is set as 35 kPa.

To facilitate the analysis of the vertical ultimate bearing capacity of pile foundations at various locations across the foundations, four piles were installed and analyzed in this numerical model. The piles are constructed of reinforced concrete and possess the following properties: a density of 2.6 g/cm³, a modulus of elasticity of 30 GPa, a Poisson’s ratio of 0.2, a length of 10 m, and a diameter of 800 mm. The first pile (denoted as Pile I) is positioned 12 m away from the backfill entrance, i.e., x = 12 m, while the subsequent piles (Piles II, III, and IV) are installed at intervals of 24 m.

Using the numerical data derived from the pile model, a geometric model of pile foundations was established in Flac^3D. Load–settlement curves at the top of pile foundations were generated through numerical simulations, following the methodologies outlined in the literature [21]. The vertical ultimate bearing capacity was determined based on the characteristics of the load–settlement curve according to the following criteria: (1) In the case of a steeply descending load–displacement curve, the load value at the point where the curve initiates its sharp decline is defined as the vertical ultimate bearing capacity of pile foundations; (2) if the curve exhibits a gradual descent, the load corresponding to a top settlement displacement of 40 mm is taken as the vertical ultimate bearing capacity Q in accordance with JGJ 106-2014 [22].

Considering the substantial spacing between the piles, interactions among them can be disregarded. Consequently, the vertical ultimate bearing capacity obtained from the numerical simulations is regarded as representative of the vertical ultimate bearing capacity of a single pile.

In the numerical model, an elastic model is applied to the piles while a Mohr–Coulomb model is used to represent the soil. Normal constraints are applied to the lateral boundaries of the model with three-dimensional fixed constraints imposed on the bottom surface. The foundation’s surface is designated as a free surface. Two interfaces are defined at the sides and bottom of the piles, and the fully bonded contact model is used to simulate the frictional interaction between the foundations and the pile perimeter [23,24]. The cohesion (c) and the angle of internal friction (φ) at the interface between the pile and soil are set to 0.8 times the corresponding values of the adjacent soil, thereby accounting for the spatial variability of the pile–soil interface parameters. Furthermore, the relationship between the elastic modulus of soils and undrained shear strength is expressed by the equation E/S_u = 280 [25].

Figure 4 illustrates the spatial distribution of the undrained shear strength of the backfill soil layer under two sets of random field realizations. It is evident that the undrained shear strength of the backfill soil layer exhibits significant variation across different locations. Furthermore, the variation trend in the vertical direction is more pronounced than that in the horizontal direction, which is consistent with the strength distribution pattern observed in the backfill soil foundations.

3.2. Monte Carlo Simulation Process

The basic principle of the Monte Carlo method is to treat soil characteristics as a random field, which is then discretized into multiple random elements. Through statistical analysis, a large number of sample data are generated. These samples are subsequently used as inputs for a deterministic finite element model, allowing for computations and further statistical analysis. The implementation process of Monte Carlo simulation involves the following key steps:

• Discretization of the random field

For the non-stationary random field of fill soil, a Gaussian-type correlation function is employed to generate the random field. Using the Karhunen–Loeve expansion method, the non-stationary random field of the undrained shear strength (S_u) of the soil is discretized into 5376 grid elements and 283 contact surface elements. To ensure that the expected energy ratio factor of the random field is greater than or equal to 95%, the number of truncation terms (N) in the KL series expansion is typically set to 1000 [14].

2.

Sample generation

In the probabilistic analysis of the vertical bearing capacity of pile foundations, the number of finite element simulations (n) is closely related to the accuracy of the results. To maintain the error within an acceptable range (e) at a 95% confidence level, a statistical formula is employed to determine the minimum number of simulations required [26]. When the maximum relative error is set to e = 0.1σ (resulting in an approximate average error of 10%), a minimum of 385 simulations is needed for 95% confidence. Thus, this study sets the sampling quantity to 500 groups for reasonable data collection.

3.

Finite element Solving

Each group of random field samples is input into the finite element model in FLAC^3D (version 6.0; Itasca Consulting Group, Minneapolis, MN, USA), with the Mohr–Coulomb model used to describe the soil constitutive relationship. A velocity control method is applied to compute the fixed number of steps. Throughout the first 0–5000 steps, the loading velocity gradually increases from 0 to 6 × 10⁻⁷. In the subsequent 5000–10,000 steps, the velocity remains constant at 6 × 10⁻⁷. The vertical ultimate bearing capacity of the pile foundation is then calculated based on the settlement displacement curve.

3.3. Probabilistic Simulation Cases

A stochastic analysis method is employed to investigate the impact of the spatial soil variability on the vertical ultimate bearing capacity of pile foundations. The process consists of two main parts: First, a deterministic analysis is conducted, assuming the soil to be homogeneous and isotropic in order to establish a bearing capacity assessment model for pile foundations. Second, simulating the spatial variability of soil strength and conducting stochastic condition analyses to reveal its specific impact on the vertical ultimate bearing capacity of pile foundations. In the stochastic analysis, each condition is simulated 500 times, with the specific parameter settings detailed in

Table 3. Simulated working conditions.

Working Condition Group	μSu (kPa)	COV	kv (kPa/m)	θ2 (m)	θ1 (m)	kh (kPa/m)	Pile No.
A	30, 35, 40	0.3	0.6	4	40	0.05	I
B	35	0.1, 0.3, 0.5	0.6	4	40	0.05	I
C	35	0.3	0.4, 0.6, 0.8	4	40	0.05	I
D	35	0.3	0.6	2, 4, 6, 8	40	0.05	I
E	35	0.3	0.6	4	40	0.02, 0.05, 0.08	I
F	35	0.3	0.6	4	40	0.02, 0.05, 0.08	II
G	35	0.3	0.6	4	40	0.02, 0.05, 0.08	III
H	35	0.3	0.6	4	40	0.02, 0.05, 0.08	IV

.

In the table, μ_Su represents the mean value of the undrained shear strength of the random field; COV is the coefficient of variation of the undrained shear strength; k_v denotes the vertical variability coefficient; k_h refers to the horizontal variability coefficient; θ₁ is the horizontal correlation distance; and θ₂ is the vertical correlation distance. The working condition groups A, B, C, and D focus on the effects of μ_Su, COV, k_v, and θ₂ on the vertical ultimate bearing capacity, respectively. The working condition groups E, F, G, and H explore the effects of different pile positions on the vertical ultimate bearing capacity of pile foundations.

For simplified analysis, the baseline case was defined with the following parameters: μ_Su = 35 kPa, COV = 0.3, k_v = 0.6 kPa/m, k_h = 0.05 kPa/m, θ₁ = 40 m, θ₂ = 4 m, and Pile I is selected.

4. Results and Discussions

4.1. Results of Baseline Case

Figure 5 illustrates the envelope of the pile load–settlement curves obtained from 500 Monte Carlo simulations under the baseline case. The results indicate that the load increases with the settlement at the pile top. The load–settlement curve displays distinct steep drop points, and the load at the steep drop point corresponds to the vertical ultimate bearing capacity. The maximum vertical ultimate bearing capacity is measured at 1432.57 kN, while the minimum value is recorded at 840.44 kN. The red solid line in the figure represents the result of deterministic analysis, and it lies within the range of the envelope defined under the baseline case. The vertical ultimate bearing capacity of pile foundations determined through deterministic analysis (denoted as Q_d) is calculated to be 1039.75 kN. This indicates that the deterministic analysis fails to adequately capture the variations in the bearing capacity of the pile, thereby providing an incomplete assessment of the bearing capacity of pile foundations in such conditions. This finding underscores the limitations inherent in the deterministic approach when evaluating the bearing capacity of pile backfill soil foundations.

To further analyze the distribution characteristics of the vertical ultimate bearing capacity of the pile, the frequency distribution is plotted in Figure 6. The data is fitted by a log-normal distribution, and the Kolmogorov–Smirnov (K-S) hypothesis test is conducted. At a significance level of 0.05, the p-value for the null hypothesis stating “the vertical ultimate bearing capacity of pile foundations follows a log-normal distribution” is 0.58, indicating that this hypothesis is supported. Furthermore, the histogram of the vertical ultimate bearing capacity closely aligns with the probability density function curve of the log-normal distribution, and the coefficient of determination R² is 0.96. This further validates that the log-normal distribution model can effectively describe the distribution patterns of the vertical ultimate bearing capacity of pile foundations.

Consequently, the characteristics of the overall population can be inferred when the sample size is sufficiently large. The log-normal distribution curve can be employed to calculate the cumulative distribution probabilities of the vertical ultimate bearing capacity under varying loads. From the figure, the mean vertical ultimate bearing capacity μ_Q of pile foundations is found to be 1031.21 kN under the baseline case, which is slightly lower than the deterministic value of 1039.75 kN; the standard deviation σ_Q is 109.34 kN.

4.2. Effect of the Soil Properties

This subsection analyzes the impacts of the parameters of the non-stationary random field model on the vertical ultimate bearing capacity of pile foundations. The parameters under consideration include the mean undrained shear strength (μ_Su), the coefficient of variation (COV) of the undrained shear strength (S_u), the vertical non-stationary trend coefficient (k_v), and the vertical correlation distance (θ₂). COV_Q is the coefficient of variation of the vertical ultimate bearing capacity of pile foundations. The results are summarized in Figure 7, which presents four distinct conditions (A, B, C, and D).

As shown in Figure 7a, μ_Q increases linearly with μ_Su. This is attributed to the fact that an increase in μ_Su results in higher soil unit strength, which subsequently enhances the lateral frictional resistance of the pile. Thereby elevating the overall vertical ultimate bearing capacity. Moreover, the range of variation for COV_Q is relatively small. It is indicated that changes in μ_Su have a significant impact on the mean value of the vertical ultimate bearing capacity but exert a minor influence on its variability.

As shown in Figure 7b, μ_Q increases slightly with COV, while COV_Q exhibits a pronounced linear growth trend. This behavior can be attributed to the fact that an increase in COV introduces greater variability in both pile end resistance and lateral frictional resistance, and these resistances lead to increased variability in the vertical ultimate bearing capacity.

From the observations in Figure 7c,d, it can be noted that the range of variation in μ_Q and COV_Q remains relatively small when either k_v or θ₂ changes.

In summary, θ₂ and k_v have limited influence on the vertical ultimate bearing capacity of pile foundations. In contrast, μ_Su and COV play a more significant role in the vertical ultimate bearing capacity of pile foundations. Specifically, μ_Su primarily impacts the mean vertical ultimate bearing capacity, while COV influences the variability of the vertical ultimate bearing capacity.

4.3. Effect of the Pile Length

Based on the baseline case, Figure 8 illustrates the variation curve of μ_Q as the pile length l changes, which is μ_Q increases linearly with the pile length l. However, once l reaches a certain threshold, μ_Q stabilizes and exhibits minimal change. The cause of this behavior is that the vertical ultimate bearing capacity eventually tends to stabilize or saturate as the pile length increases. Additionally, a notable relationship has been identified between l and μ_Q, written as follows:

$${\mu }_Q(l)=0.316\cdot {1.1223}^l\cdot {\mu }_{Q1}$$

(6)

where μ_Q1 represents the mean limit bearing capacity calculated for a length of l = 10 m, and so forth, and μ_Q represent the mean bearing capacities under different pile lengths.

This relationship can be utilized to model the mean vertical ultimate bearing capacity of pile foundations for varying pile lengths.

4.4. Effect of the Pile Location

Figure 9a illustrates the relationship between μ_Q at different pile locations and k_h. It can be observed that the calculated μ_Q decreases with the horizontal distance x (defined as the distance between the pile and the backfill entrance). Additionally, μ_Q decreases with k_h for the same horizontal distance. It is noteworthy that the variation in k_h at greater horizontal distances leads to larger fluctuations in μ_Q.

Figure 9b illustrates the relationship between COV_Q at different pile locations and k_h. It can be observed that the three curves for COV_Q are very close to each other. This indicates that the choice of pile location and the k_h have little influence on COV_Q.

In conclusion, the selection of pile locations and k_h significantly influence μ_Q. Pile position I demonstrates more stable and reliable load-bearing characteristics, and it is characterized by a lower coefficient of variation and a higher vertical ultimate bearing capacity. In contrast, μ_Q located further away exhibits greater fluctuations.

4.5. Discussion

Based on the analysis above, it is evident that μ_Su, COV, and x significantly affect the vertical ultimate bearing capacity of pile foundations. Among these factors, both k_h and x exhibit a strong linear relationship with μ_Q. The results obtained μ_Q for the aforementioned working condition groups with respect to x are presented in

Table 4. Linear fitting table of mean value of vertical ultimate bearing capacity.

kh (kPa/m)	μQ/kPa	R2	Range
0.02	μQ = −6.3046x + 1039.5	0.9779	0 ≤ x ≤96
0.05	μQ = −15x + 1046.7	0.9983	0 ≤ x ≤96
0.08	μQ = −23.259x + 1045.3	0.9916	0 ≤ x ≤96

. When x = 0, μ_Q of the three curves are approximately 1045.30, 1046.70, and 1039.50 kN. It indicated that they are very close to each other. In fact, these values represent the vertical ultimate bearing capacity of pile foundations at the backfill entrance, revealing that this value is independent of k_h.

Additionally, the table indicates a significant linear correlation between k_h and the gradient of bearing capacity d_μQ/dx (R² = 0.99). This statistical relationship suggests that the bearing capacity gradient d_μQ/dx can be effectively represented as a function of the horizontally non-stationary trend coefficient k_h.

To analyze the vertical ultimate bearing capacity of pile foundations at different locations, the coefficient η is defined to evaluate the vertical ultimate bearing capacity of pile foundations, written as follows:

$$\eta ={\displaystyle \frac{{\mu }_Q}{{\mu }_{Q1}}}$$

(7)

where η is the coefficient of vertical ultimate bearing capacity of pile foundations, μ_Q denotes the average value of the vertical ultimate bearing capacity of pile foundations calculated under the baseline case, and μ_Q1 represents the average value of the vertical ultimate bearing capacity of pile foundations calculated under the baseline case. Based on the above analysis, it can be assumed that μ_Q is linearly related to k_h and x. Therefore, the coefficient of vertical bearing capacity of pile foundations η can be expressed as follows:

$$\begin{array}{l}\eta =c_{1}x+c_2\\ c_1=a_1{k}_{\mathrm{h}}+b_1\\ c_2=a_2{k}_{\mathrm{h}}+b_2\end{array}$$

(8)

where a₁, b₁, a₂, and b₂ are the coefficients to be determined, k_h represents the horizontal variability coefficient, and x denotes the horizontal distance between pile foundations and the backfill entrance. Through analysis, μ_Q can be ultimately determined in the different pile positions and horizontal non-stationary trend coefficient condition:

$${\mu }_Q(k_{\mathrm{h}},x)=[(-0.01k_{\mathrm{h}}+0.0001)x+(1.005-0.0001k_{\mathrm{h}})]{\mu }_{Q1}$$

(9)

The equation presented reveals the quantitative relationship between μ_Q, x, and k_h when other parameters remain constant. When k_h = 0.04 kPa/m and x = 24, 48, and 72 m, the μ_Q calculated using this formula are 1028.94, 1021.51, and 1014.09 kN, respectively. The results with those predicted by the formula reveal no significant discrepancies, indicating that the prediction formula for the vertical ultimate bearing capacity demonstrates high accuracy and can be reliably employed in the design of pile foundation systems.

4.6. Case Study

Taking the land reclamation project in a riverside development zone of a certain city as an example, the backfill soil foundations are located on the southern side of the Yangtze River’s old dike, originally characterized by floodplain and fishpond topography. After backfilling and leveling, an area of approximately 250,000 m² was created, with ground elevations ranging from 5.110 to 7.830 m. The originally proposed design was aimed at meeting the load-bearing requirements for a rail foundation, utilizing a CFG (cement–fly ash–gravel) pile composite foundation treatment scheme. The design included a group of short CFG piles, each with a length of 10 m and a characteristic vertical load-bearing capacity of R_a = 560 kN per pile. According to empirical formula calculations, the target load-bearing capacity needs to reach 1120 kN in accordance with JGJ 106-2014 [22].

The originally proposed design adopted CFG piles, and the static load test results from three different pile locations showed ultimate bearing capacities of 1120 kN, 1680 kN, and 1540 kN. It is evident that the measured ultimate bearing capacities of the CFG piles are generally higher than expected, indicating that the design may be conservative. Additionally, the cost of CFG piles is higher than that of cast-in-place piles. The layout plan involves a uniform distribution of piles, with all piles designed to the same length.

Based on the predictive model proposed in this study, and under the condition of k_h = 0.05 kPa/m at the foundations, the average ultimate bearing capacity calculated for the 10 m long piles is μ_Q1 = 1300 kN. It should be noted that this predicted ultimate value, μ_Q1, needs to be analyzed and designed in conjunction with reliability theory. According to Equations (4) and the derived Equation (7), the predictive formula for the ultimate bearing capacity of pile foundations, as a function of pile length l and horizontal distance x, is as follows:

$${\mu }_Q(l,x)=\left(0.005x-1.004995\right)0.316\cdot {1.1223}^l\cdot {\mu }_{Q1}$$

(10)

Assuming constant variance and using the target ultimate bearing capacity Q₀ = 1120 kN as a benchmark, the probability of failure pf was calculated using a log-normal distribution model for reliability index analysis. Figure 10 illustrates the failure probability distribution curves for five different pile lengths. When the design requirement stipulates that P_f = 5.0 × 10⁻⁵, it can be observed that for a pile length of l = 11.1 m, the intersection point with the pf contour line occurs at x = 35.9 m. For a pile length of l = 11.2 m, the intersection point shifts to x = 63.7 m. These intersection points represent the optimal design locations that meet reliability requirements. Further analysis indicates that with a pile length of 11 m, the overall reliability does not meet standards; however, pile lengths of l ≥ 11.3 m fulfill the criteria across the entire site.

This research proposes a differentiated design scheme based on the spatial variability of backfill soil (shown in

Table 5. Optimized design scheme.

Region	Pile Length	Design Basis
Near zone (0–30 m)	11.1 m	Meets bearing capacity requirements and achieves optimal economy
Middle zone (30–60 m)	11.2 m	Compensates for the gradient effect on soil strength, ensuring the predefined reliability
Far zone (60–96 m)	11.3 m	Overcomes the attenuation effect of soil parameters in distant areas, ensuring stability of bearing capacity

), which quantifies the non-stationary characteristics of the soil parameters, leading to an optimized pile length design. In comparison to traditional uniform pile arrangements, this differentiated design approach effectively addresses the issue of excessively conservative pile length configurations in distant areas, significantly reducing the amount of concrete used in large-scale pile driving projects. Moreover, through precise control of failure probabilities, the entire site can achieve reliability indices that surpass regulatory requirements.

5. Conclusions

This study summarizes the spatial distribution patterns of undrained shear strength in artificial soil foundations. Based on the theory of non-stationary random fields, a characterization model for the spatial distribution of undrained shear strength in artificial soil foundations has been established. Using the vertical ultimate bearing capacity of piles as an example, numerical simulations were conducted to analyze the impact of various parameters on the vertical ultimate bearing performance of pile foundations. The following conclusions are drawn:

(1)

Based on the spatial distribution characteristics of the undrained shear strength in the backfill soil, a non-stationary random field model is proposed that incorporates the attenuation of undrained shear strength decrease with the horizontal direction and increase with the vertical direction. This model effectively characterizes the spatial distribution of strength in artificial soil foundations.

(2)

The distance between the pile and the backfill entrance, as well as the horizontal non-stationary trend coefficient, has a substantial effect on the vertical ultimate bearing capacity. The mean value of the vertical ultimate bearing capacity of pile foundations increases with the length of the pile, but there is an upper limit to this growth. When k_h (the horizontal non-stationary trend coefficient) is quantified, a linear relationship is observed between the mean vertical ultimate bearing capacity and x (the distance from the pile to the backfill entrance). It is worth mentioning that k_h has an increasingly pronounced effect on the vertical ultimate bearing capacity of pile foundations as x increases.

(3)

Through practical engineering analyses, the model can effectively guide the optimization of pile foundation design. Compared to traditional design methods, the approach adopted in this study reduces project costs while ensuring reliability. However, it is important to note that the current model’s applicability to other pile types and ultra-high water-content backfill foundations still requires further validation. Future research will focus on the following areas: improving pile body optimization by refining the design of related parameters such as pile diameter and type; expanding the applicability of the model to fill sites; and developing a three-dimensional non-stationary stochastic field model suitable for backfill soil foundations.