Evolution Analysis of Soil-Arching Effect and Calculation of Pile–Soil Stress Ratio of Bidirectionally Reinforced Composite Foundation

Abstract

In recent years, bidirectionally reinforced composite foundations have been widely used in highway, railway, and bridge engineering with notable results. The key mechanism is the soil-arching effect, which arises from the self-adjustment of the soil and directly affects the bearing capacity of the foundation. In this study, numerical simulation was employed to analyze the vertical stress in the subgrade soil and the transfer of particle contact forces from the macro and micro perspectives. The existence of the soil-arching effect was confirmed, and its variation under loading was revealed. To quantify the degree of the soil-arching effect, the stress transfer efficiency of the soil between piles was introduced. Subsequently, a bidimensional theoretical model was established based on the coordinated deformation among the embankment, the horizontally reinforced cushion, the vertical piles, and the soil. In this model, the combined effects of the embankment soil-arching, the reinforcement of cushion net, and the stress diffusion were incorporated. A method for the calculating of the pile–soil stress ratio of bidirectionally reinforced composite foundation was proposed, and the influence of various factors on this ratio was explored. The results indicate that the soil-arching effect can be divided into three stages according to the height of the subgrade fill: no-arch stage, transition stage, and soil-arching stage. Reducing pile spacing or increasing cushion thickness can improve the stress transfer efficiency. When the pile length is appropriate, the stress in the foundation soil at 0.55 times the pile depth was contoured, enhancing stability. The pile–soil stress ratio decreases with the increase in filling weight and pile spacing, increased first and then decreased with increasing internal friction angle of filling materials, and increased with the increasing height of embankment, the number of geogrid layers, and the cohesion of filling materials.

1. Introduction

New multi-element composite foundations, such as bidirectionally reinforced (horizontally and vertically) composite foundation treatment technology (incorporating geogrid and pile), are now widely employed in highway, railway, and bridge engineering, with notable success. However, research concerning the load-transfer mechanisms and transfer paths of bidirectional composite foundations remains at a preliminary stage. Therefore, it is both timely and essential to investigate evolution process of the soil-arching effect, clarify the transfer path of the subgrade load, quantify the influence of the soil-arching effect on the stress of the soil between piles, and provide a reference for the design and optimization of bidirectionally reinforced composite foundations.

The soil-arching effect, the transfer of vertical stress from a yielding medium to the adjacent non-yielding medium, is realized through the shear strength of the soil medium in soft foundation improvement [1]. At present, Currently, some scholars have investigated the soil-arching effect using various approaches, including laboratory experiments, numerical simulations, field tests, and theoretical modeling [2,3].

Zhao et al. [4] and Wang et al. [5] studied the mechanical properties of soil arches behind anti-slide piles through theoretical analysis, examining the mechanical behavior of anti-slide piles under soil-arching action. Lu et al. [6] further revealed the mechanism of the soil-arching effect behind anti-slide piles by means of numerical simulation. Recently, Chen et al. [7] employed transparent soil technology to investigate the displacement characteristics and evolution process of soil-arching effect behind piles from a mesoscopic perspective. In addition to the above-mentioned research on the soil-arching effect behind piles, other scholars have focused on the soil-arching effect in composite foundations. For example, Zhao et al. [8] conducted a mechanical analysis of the pile-supported embankment and derived the differential settlement. The calculation formulas for the pile–soil stress ratio and arch height were verified through indoor tests and numerical simulation. However, macro- and micro-scale studies on the soil-arching effect in composite foundations remain relatively limited at present. Zhao et al. [8] conducted a mechanical analysis of the pile-supported embankment and derived the differential settlement. The calculation formulas for the pile–soil stress ratio and arch height were validated using indoor tests and numerical simulation. However, macro- and micro-scale investigations on the soil-arching effect in composite foundations.

On the other hand, the bearing behavior of a composite foundation under the soil-arching effect are manifested as the joint force of piles and the soil between piles. As an important parameter reflecting the performance of a bidirectionally reinforced composite foundation, the pile–soil stress ratio during the loading process serves as a key indicator. It describes the load sharing of pile and soil under the action of the overlying load and is used for settlement and stability analysis of the foundation. It is not only affected by the soil-arching effect in the embankment [9], but also by the net effect of the horizontal reinforcement (e.g., geogrid) [10,11], as well as the stress diffusion effect of the reinforced cushion. While many studies have examined the influence of the soil-arching effect and the net effect on the pile–soil stress ratio, the inclusion of geosynthetics in the subgrade cushion increases the cushion stiffness and the stress diffusion angle. This significantly enhances the stress diffusion effect of the cushion, thereby reducing the base pressure and substantially affecting the calculated pile–soil stress ratio. Therefore, the comprehensive influence of these effects should be fully considered in the analysis of the pile–soil stress ratio.

Many scholars have investigated calculation methods of the pile–soil stress ratio based on the net effect alone or on the coupling of net effect and the soil-arching effect. In terms of the net effect of the horizontal reinforcement, Qian et al. [12] clarified the influence of reinforcement materials on the settlement characteristics of composite foundations.

Based on the Winkler elastic foundation circular plate theory, Ma et al. [13] derived a calculation formula for the pile–soil stress ratio of bidirectionally reinforced composite foundations. Although these studies fully accounted for the effect of reinforcement deformation on load distribution, they ignored the influence of pile–soil interaction. Consequently, scholars have incorporated the influence of the soil-arching effect on load transfer mechanisms on the basis of analyzing the deformation of reinforcement. For instance, Huang et al. [14] applied an improved Terzaghi method to derive the pile–soil stress and analyzed the quantitative variation in pile-soil stress and the membrane effect with different design parameters. Although these methods better reflected the combined influence of the net effect and soil-arching effect on the pile–soil stress ratio, they neglected the stress diffusion effect of the horizontally reinforced cushion. This omission results in an overestimation of the base pressure, leading to conservative design in practical engineering. In summary, the calculation of soil-arching evolution and pile–soil stress ratio remains an open problem, with evident shortcomings in the understanding of the coupling mechanisms, global sensitivity, and statistical significance.

In this paper, the evolution process of soil-arching in a bidirectional composite foundation was first analyzed using a numerical simulation method, clarifying the influence of embankment fill height on the development of soil-arching. The concept of stress transfer efficiency was proposed, and the fitting formula was introduced to quantify the variation in filling load between piles caused by soil-arching effect. Subsequently, the influence of different pile lengths on the stress in the foundation soil within the reinforcement depth was explored, and an appropriate pile length was determined as well. On this basis, considering the soil-arching effect and the net effect, the stress diffusion effect of reinforced cushion was introduced [15], and the theoretical calculation method of pile–soil stress ratio of bidirectional reinforced composite foundation was derived. Finally, a sensitivity analysis and evaluation method were applied to evaluate the influence of multiple parameters on the pile–soil stress ratio of a bidirectional reinforced composite foundation. This approach contributes to the design and calculation theory for bidirectional reinforced composite foundations.

2. Numerical Model Establishment and Verification

In this paper, FLAC was employed to establish a numerical model of a bidirectionally reinforced composite foundation, and the bearing mechanism of the bidirectionally reinforced composite foundation was analyzed. Additionally, the discrete element software PFC2D 6.0 was used to analyze the development process of the soil-arching effect.

2.1. Establishment of FLAC Model

In this paper, a bidirectionally reinforced composite foundation was simulated according to the design for the proposed expressway in the reconstruction and expansion project of Dongying-Qingzhou Expressway. As shown in Figure 1, the subgrade was 5.3 m in height, the top width of half width was 12 m, the bottom width was 19.95 m, and the slope was 1:1.5. The geometry parameter was quoted from Shandong Province General Design Atlas SD-GS-DF-05 and had been analyzed and verified through field experiments. It represented a standard design for the widening of 120 km·h⁻¹ expressway in soft soil areas. The subgrade was filled in nine layers, with respective layer thicknesses of 0.3 m, 0.2 m, 0.3 m, 0.2 m, 0.3 m, 1 m, 1 m, 1 m, and 1 m. To avoid the influence of boundary conditions on the analysis results, the half width of the foundation model was set to 64 m, which is 3.2 times of the base width. Both practical and theoretical analyses indicate that stress changes in the foundation soils are generally below 10% at a depth of three times the width of foundation and less than 3% at a depth of five times the foundation width. In addition, Zhang [16] recommended that the vertical depth of the analysis zone was 2.5 times the pile length.

A total of three geogrid layers were installed: one on the upper surface, one on the lower surfaces of the cushion, and one in the middle. The foundation was reinforced by cement–soil mixing piles. The actual pile length was 12 m, the pile diameter was 0.5 m, the pile spacing was 2 m, the depth of foundation reinforcement was 12 m, and the length was 32 m. In the numerical model, three basic unit types were adopted: brick elements for the embankment and the non-reinforced foundation, wedge elements for the embankment slope, and cylinder elements for the pile reinforcement zone. The mesh was refined the embankment and pile-reinforced foundation area, while a coarser mesh was used in the remaining foundation region. The model comprised 615,440 elements, 619,197 nodes, organized into 21 groups, as shown in Figure 1.

2.2. Parameters of FLAC Model

Based on the field survey data and laboratory test results of the geogrid, the physical and mechanical indexes of the foundation and subgrade soil, cement mixing piles, and geogrid are shown in

Table 1. Physical and mechanical parameters of each soil layer.

Soil Layer	Elastic Modulus Es (MPa)	Density ρ (kg/m3)	Force of Cohesion c (kPa)	Angle of Friction φ (°)	Poisson Ratio μ
Filling soil	15	1850	20	28	0.3
Cushion layer	120	2580	0	30	0.3
Silty soil	6.53	1862	8.4	20.2	0.3
Silty clay	3.47	1790	23.2	24.3	0.35
Silty soil	6.86	2010	10	20	0.25
Silty clay	4.58	1880	37.7	11.8	0.3
Muddy clay	3.74	1820	23.2	10.5	0.28
Silty soil	4.5	2015	26.1	7.1	0.3

,

Table 2. Physical and mechanical parameters of pile.

Mechanical Parameter	Elastic Modulus Es (MPa)	Density ρ (kg/m3)	Force of Cohesion c (kPa)	Angle of Friction φ (°)	Poisson Ratio μ
Numerical value	200	2450	500	25	0.2

and

Table 3. Physical and mechanical parameters of geogrid.

Mechanical Parameter	Bulk Modulus K (GPa)	Poisson Ratio μ	Thickness (m)	Coupling Area Stiffness (N·m−3)	Coupling Spring Cohesion (kPa)	Angle of Internal Friction φ (°)
Numerical value	23	0.33	0.5	2.3 × 106	0	25

. According to the relevant data, the elastic modulus E of the soil was taken as approximately 3 times the compression modulus E_s [17].

The Mohr–Coulomb elastic plastic model was adopted as the constitutive model for the soil. Due to the large stiffness of the piles, the deformation under load was generally considered elastic. Therefore, an ideal elastic model was selected for the constitutive model of the pile. The pile diameter was 0.5 m and the pile spacing was set as 2 m. The piles were arranged in rectangular pattern, and a total of 32 piles were arranged in two rows. The geogrid was a two-dimensional structural geogrid element.

2.3. Establishment of PFC Model

In this paper, a particle flow model of a bidirectionally reinforced composite foundation was established with the discrete element software PFC, and the development process of the soil-arching effect was analyzed from a microscopic perspective. The formation law and microscopic mechanism of soil-arching effect for bidirectional reinforced composite foundation were further investigated by integrating the results from the FLAC macro-numerical simulation. In order to control the number of generated particles and ensure the computational efficiency, a double-pile single-arch discrete element model was adopted in Figure 2. In the PFC model, the ‘Wall’ was used to simulate the pile body, while soil particles were assigned properties corresponding to the same macroscopic parameters used in FLAC. The overlying soil above the foundation were divided into nine layers and filled sequentially. The pile diameter remained 0.5 m to maintain consistency in parameters between the macro- and micro-models.

2.4. Model Validation

The numerical simulation in this paper was conducted under the self-weight of the subgrade fill. The reliability of the model was verified by comparing the vertical effective stress distribution in the foundation soil obtained from the numerical simulation and theoretical calculation (Formula (1)).

$${\sigma }_c=\sum _{i=1}^n{\gamma }_i{h}_i$$

(1)

where ${\sigma }_c$ is the vertical effective gravity stress at any depth z under the natural ground (unit Pa); n is the total number of soil layers in the range of depth z; $h_{\mathrm{i}}$ is the thickness of the i-layer soil, m; and ${\gamma }_i$ is the natural weight of the i-layer soil, N/m³.

After the subgrade filling was completed, the vertical effective stress in the foundation soil was determined by superimposing the effective stress due to the self-weight of the foundation soil [18] (Formula (1)) and the increase in vertical stress induced by a uniform distributed rectangular load. The theoretical calculation formula for vertical stress increase [19] is as follows:

$${\sigma }_z={\alpha }_c\rho$$

(2)

$$m={\displaystyle \frac{l}{b}},n={\displaystyle \frac{z}{b}}$$

(3)

$${\alpha }_c={\displaystyle \frac{1}{2\pi }}{\displaystyle \frac{mn(m^2+2n^2+1)}{(m^2+n^2)(1+n^2)\sqrt{m^2+n^2+1}}}+{\mathit{sin}}^{-1}{\displaystyle \frac{m}{\sqrt{(m^2+n^2)(1+n^2)}}}$$

(4)

where σ_z is the vertical stress increase under the uniform load corner point, Pa; α_c is the corner stress coefficient; l, b are the width of the long and short sides of the load surface, respectively; z is any depth under the corner point, m; and ρ is the pressure on the base of substrate caused by the self-weight of the subgrade, Pa.

Figure 2 shows the distribution of vertical effective stress in the foundation soil along the depth when the original foundation was initially balanced and after the embankment filling was completed. The results show that the vertical effective stress of foundation soil obtained from the FLAC simulation is basically consistent with the theoretical calculations, thereby confirming the reliability of the numerical model.

3. Analysis of Evolution Process of Soil-Arching Effect

3.1. Macroscopic Analysis of Soil-Arching Effect

The formation of the soil arch structure results from the stiffness difference between the piles and the surrounding soil. Under the action of load, the displacement and shear stress develop at the pile–soil interface, creating a low-stress zone that reduces the stress of the soil between the piles. On this basis, FLAC 3D software was employed to simulate the layered filling of subgrade, analyzing the influence of subgrade height on the transfer and evolution of self-weight stress of soil between piles, thereby investigating the soil-arching effect and macro-stress development process of the bidirectional composite foundation.

Figure 3 shows the vertical stress variation contour of subgrade soil during the filling process. When the filling height was lower than 0.8 m (about 0.4 times the pile spacing), the vertical stress of soil between piles increased gradually without observable stress transfer, indicating that the soil arch had not yet formed. It could be defined as the non-arch stage. As the filling height increased 0.8–1.3 m (about 0.4–0.65 times the pile spacing), the vertical stress obviously deflected and concentrated toward the pile heads. When the filling reached 1.3 m, a small ‘low-stress zone’ (yellow area) appeared, which indicated that the soil-arching effect appeared. When the filling reached 2.3 m (about 1.15 times the pile spacing), the ‘low-stress zone’ evolved into a stable arch, whose shape was maintained by the overlying load. Meanwhile, the stress within the arch decreased. It can be seen that the load was primarily transmitted along the arch to the pile heads, and the soil-arching effect significantly improved the bearing capacity of the foundation.

Figure 4 illustrates the relationship between the vertical stress of subgrade soil between piles and the filling height. It can be seen that when the filling height was less than 0.8 m (0.4 times the pile spacing), the vertical stress exhibited a linear distribution, which belonged to the soil-arching stage. When it was 0.8 m–1.3 m, the stress in the embankment soil near the foundation surface decreased, forming a transition stage. With a further increase in filling height, the vertical stress in the surface of the foundation decreased significantly, forming the arch stress curve, which entered the soil-arching stage. According to the vertical stress distribution after the formation of the soil arch, the subgrade can be divided vertically into two zones: one unaffected by the soil-arching effect and the other influenced by it. And the dividing filling height was about 0.65 times the pile spacing. In the first zone, the soil vertical stress maintained a linear increase under the action of gravity. In the second one, particularly near the foundation surface, the soil vertical stress no longer increased linearly, but showed an ‘arch’ curve. The stress decreased sharply and formed a low-stress zone. These results indicate that the soil-arching effect can effectively reduce the surface stress of the soil foundation between piles, improve the bearing capacity of the foundation, and reduce the differential settlement.

3.2. Microscopic Analysis of Soil-Arching Effect

In the PFC discrete element model, particle interactions are reflected by the force chains, where a thicker chain indicates a stronger transmitted force. The soil-arching effect induces stress redistribution and affected the contact force chain. As illustrated in Figure 5, the development of the soil-arching effect can be divided into three stages: non-arch stage (after 2 layers of soil were filled, the force chains were vertically distributed, gradually sparser from bottom to top, indicating no soil-arching effect); transition stage (after 4 layers of soil were filled, the force chains at pile tops were dense, and the direction turned to the inner side of the soil between piles); and the stable stage (after 6 layers of soil were filled, the force chain at the top of the pile was the most concentrated, and an obvious ‘arch’ structure was formed, which remained stable with the increase in the overlying soil).

3.3. Stress Transfer Efficiency of Soil Arch

3.3.1. Stress Transfer Efficiency $\rho$

The soil-arching effect significantly reduces the soil stress between piles by transferring the vertical stress, which improves the bearing capacity of the foundation and reduces the settlement [20]. At present, the common characterization parameters are pile–soil stress ratio and pile load-sharing ratio, which primarily focus on stress changes in the piles. However, the soil-arching effect also changes the stress state of the soil between piles. To better improve the bearing capacity of the soft foundation, this paper aims to explore the stress changes in the soil. A new parameter, the stress transfer efficiency $\rho$ of the soil between piles, was proposed to more effectively quantify the influence of the soil-arching effect on soil between piles. The efficiency $\rho$ is defined below:

$$\rho ={\displaystyle \frac{{\sigma }_{\mathrm{c}\mathrm{z}}-{\sigma }_0}{{\sigma }_{\mathrm{c}\mathrm{z}}}}\times 100\%$$

(5)

where σ_cz is the vertical stress in the soil between piles induced by self-weight when the soil-arching effect is not considered, Pa; and σ₀ is the vertical stress in the soil between piles under the influence of soil-arching effect, Pa.

3.3.2. Analysis and Application of Stress Transfer Efficiency $\rho$

The stress transfer efficiency can not only be used as an indicator to evaluate the influence of the soil-arching effect on the stress characteristics of soil between piles, but also as a practical parameter for optimization design and safety monitoring. During the engineering design stage, the stress transfer rate can be predicted by the numerical simulation method to optimize the design parameters, thereby improving the rationality and safety. During the construction and service, the dynamic change in stress transfer efficiency can be monitored using monitoring equipment (such as earth pressure box) buried in the soil between piles. It is convenient to obtain the relevant parameters for calculating the stress transfer efficiency, and it is widely applied in practical engineering design and safety monitoring.

In this study, the stress transfer efficiency induced by the soil-arching effect in a bidirectional composite foundation was analyzed using a numerical simulation (Figure 6). The development of stress transfer efficiency was closely related to the evolution of the soil-arching effect. When the fill height of subgrade was below 0.8m, it was in the stage of no soil arch, and the stress transfer efficiency was negative, indicating that the pile-bearing capacity was not excited, and the load was mainly carried by the soil between the piles. When the height increased to 0.8–1.3 m (0.8 < H < 1.3), the soil-arching effect entered the transition stage, and the stress transfer efficiency shifted from negative to positive and rose rapidly, indicating a significant improvement in the bearing capacity of the pile. After continuing to increase to the soil-arching stage, the load was transferred along the arch to the pile heads, reducing stress in the soil stress between piles. Consequently, the transfer efficiency continued to increase steadily at a consistent rate.

3.3.3. Influence of Reinforcement Form and Pile Spacing on Stress Transfer Efficiency

Figure 7 shows the stress transfer efficiency of the soil-arching effect under four reinforcement forms. The results indicate that the inclusion of a cushion significantly improved the stress transfer efficiency in one-directional and bidirectional reinforcement. When the bidirectional reinforced composite foundation was equipped with cushion, the stress transfer efficiency was the highest, which indicated that the reinforcement method could reduce the pile quantities. The effect of soil stress was obvious.

Figure 8 shows the influence of pile spacing on the stress transfer efficiency of the soil-arching effect. The results show that an increase in pile spacing weakens the soil-arching effect, leading to a decrease in stress transfer efficiency.

4. Calculation of Pile–Soil Stress Ratio Considering Soil-Arching Effect

4.1. Analysis of Soil-Arching Effect

4.1.1. Fundamental Assumption

Fundamental assumption was made as follows:

(1)

The soil arch structure is homogeneous and isotropic.

(2)

The lateral friction resistance between the interior and exterior soil columns of the embankment is linearly distributed over the height range between the pile top and the equal settlement surface.

(3)

The soil arch foot is uniformly distributed over the pile top without superposition.

(4)

The weight of soil arch structure is negligible, and the arch thickness is less than its height.

(5)

Only the tensile properties of geogrids are considered without compressive properties.

The assumption made in the manuscript was based on relevant studies and the consideration of the actual forces acting on the earth arch [21].

4.1.2. Reasonable Arch Line Equation

The formation of the soil arch in the embankment soil of the bidirectional reinforced composite foundation is due to the uneven deformation of the soil columns interior and exterior the embankment under the vertical loading [20]. The soil surrounding the piles mobilizes its shear strength to resist the load. During this process, the soil around the pile was densified gradually, finally leading to soil arching. The soil-arching effect is formed naturally and can maximize the bearing capacity of the foundation, which is consistent with the natural equilibrium arch theory [22]. According to the analysis of structural mechanics, the arch line equation can be expressed as follows:

$$\mathrm{y}={\displaystyle \frac{4f}{l^2}}x(l-x)$$

(6)

where $l$ is the span of soil arch and it can be approximately taken as the clear distance between piles; $f$ represents the height of soil arch and can be approximately taken as 1.87 (S-a); S is the pile spacing; a is the pile cap side length.

4.1.3. Soil Arch Stress Analysis

Based on the arch line equation and the stress analysis of the soil arch shown in Figure 9, the static equilibrium equation of soil arch is obtained as follows:

$$\sum Y=0,F_{Ay}+F_{By}=ql+{\int }_0^{l}r{H}_{\left(x\right)}dx$$

(7)

$$\sum M_A=0,F_{By}l={\int }_0^{l}qxdx+{\int }_0^{l}r{H}_{\left(x\right)}xdx$$

(8)

$$\sum M_B=0,F_{Ay}l={\int }_0^{l}q\left(l-x\right)dx+{\int }_0^{l}r{H}_{\left(x\right)}\left(l-x\right)dx$$

(9)

where q is the traffic load on the top of the embankment and is simplified to uniform load; r is the weight of embankment fill; $H_{\left(x\right)}$ is the upper filling height of the arch line, $H_{\left(x\right)}=z-{\displaystyle \frac{4f}{l^2}}x\left(l-x\right)$; and z is the total height of embankment filling. From Equations (7)–(9), the following can be obtained:

$$F_{Ay}=F_{By}={\displaystyle \frac{ql}{2}}+{\displaystyle \frac{rzl}{2}}-{\displaystyle \frac{frl}{3}}$$

(10)

Taking the half arch as the isolation body (Figure 10) the static equilibrium equation of the half arch is as follows:

$$\sum M_A=0,F_{Cx}f={\int }_0^{{\displaystyle \frac{l}{2}}}qxdx+{\int }_0^{{\displaystyle \frac{l}{2}}}r{H}_{\left(x\right)}xdx$$

(11)

$$\sum X=0,F_{Cx}=F_{Ax}$$

(12)

The horizontal thrust of the arch foot can be obtained by Equations (11) and (12):

$$F_{Ax}=F_{Cx}={\displaystyle \frac{8q+8rz-5fr}{48f}}{l}^2$$

(13)

Similarly, with the static equilibrium equation, the force in the x and y directions of any point D on the arch line can be obtained as follows:

$$F_x=F_{Ax}={\displaystyle \frac{8q+8rz-5fr}{48f}}{l}^2$$

(14)

$$F_y=F_{Ay}-qx-{\int }_0^{x}r{H}_{\left(x\right)}dx$$

(15)

$$F_y=\left|\left(q+rz-{\displaystyle \frac{2}{3}}fr\right)\left({\displaystyle \frac{l}{2}}-x\right)\right|$$

(16)

The angle between the resultant force at any point on the arch line and the x-axis is

$$\alpha ={\tan }^{-1}\left({\displaystyle \frac{F_y}{F_x}}\right)={\tan }^{-1}{\displaystyle \frac{48f\left|\left(q+rz-\frac{2}{3}fr\right)\left(\frac{l}{2}-x\right)\right|}{\left(8q+8rz-5fr\right)l^2}}$$

(17)

The resultant force of any point on the arch line is

$$F=\sqrt{{F_x}^2+{F_y}^2}=\sqrt{{\left({\displaystyle \frac{8q+8rz-5fr}{48f}}\right)}^2{l}^4+{\left(q+rz-{\displaystyle \frac{2}{3}}fr\right)}^2{\left({\displaystyle \frac{l}{2}}-x\right)}^2}$$

(18)

4.1.4. Stability Analysis of Soil Arch Limit State

Analysis of Equation (18) shows that the axial force is the largest at the arch foot. That is, when the axial stress ${\sigma }_1$ at the arch foot gradually increases to the limit state, the soil arch structure reaches the limit equilibrium state [23]. According to the Mohr–Coulomb strength criterion, the major and minor principal stress should meet

$${\sigma }_1={\sigma }_3{tan}^2\left({45}^{°}+{\displaystyle \frac{\phi }{2}}\right)+2c\tan \left({45}^{°}+{\displaystyle \frac{\phi }{2}}\right)$$

(19)

Before the major and minor principal stresses ${\sigma }_1$ and ${\sigma }_3$ at the arch foot are determined, it is necessary to determine the width of the soil arch in the two-dimensional plane. According to the basic assumption (3), it is known that the contact part between the single soil-arch foot and the pile top is one quarter of the pile top section. According to the geometric relationship, the arch width corresponding to the major principal stress ${\sigma }_1$ at the arch foot is R$\sin \alpha$, then the major principal stress ${\sigma }_1$ at the arch foot is

$${\sigma }_1={\displaystyle \frac{F}{R\sin \alpha }}={\displaystyle \frac{F_{Ay}}{R{sin}^2\alpha }}$$

(20)

Similarly, the minor principal stress ${\sigma }_3$ is

$${\sigma }_3={\displaystyle \frac{{\sigma }_y}{\cos \alpha }}$$

(21)

where ${\sigma }_y$ is the vertical normal stress at the arch foot, which is obtained according to Formula (15):

$${\sigma }_y={\displaystyle \frac{F_{Ay}}{R}}$$

(22)

Substitute (22) into (21), ${\sigma }_3$ can be obtained:

$${\sigma }_3={\displaystyle \frac{F_{Ay}}{\mathrm{Rcos}\alpha }}$$

(23)

The stability equation of soil arch structure based on Mohr–Coulomb strength theory can be obtained when the major and minor principal stress Formulas (20) and (23) are put into Formula (19):

$$q={\displaystyle \frac{2R\mathrm{c}·\sin 2\alpha \cos \alpha \tan \left({45}^{°}+\frac{\phi }{2}\right)}{l\left[\cos \alpha -{sin\alpha }^2{tan}^2\left({45}^{°}+\frac{\phi }{2}\right)\right]}}-\left(z-{\displaystyle \frac{2}{3}}f\right)\gamma$$

(24)

4.2. Stress Analysis of Geogrid Reinforcement Cushion

4.2.1. Stress Diffusion

When geogrid or other reinforced materials are incorporated into the cushion between the embankment and the composite foundation, the overall stiffness of the cushion will be further improved. This increased the stress diffusion angle [24,25] and significantly improved the diffusion effect of the stress, as shown in Figure 11.

Studies indicate that the stress diffusion angle of the reinforced cushion is in the range from 25° to 55°, varying with the number of reinforced layers and materials. Owing to the influence of stress diffusion, the ultimate bearing capacity of the foundation soil increases from $p_{s1}$ to $p_{s2}$:

$$p_{s2}={\displaystyle \frac{\left(b_n+2h_c\tan {\theta }_c\right)}{b_n}}{p}_{s1}$$

(25)

where $p_{s1}$ is the ultimate bearing capacity of the original foundation soil; $b_n$ is the load width; $h_c$ is the thickness of reinforced cushion; and ${\theta }_c$ is the stress diffusion angle.

Therefore, the increment of bearing capacity caused by the stress diffusion of the reinforcement cushion $∆p_1$ is

$$∆p_1=p_{s2}-p_{s1}={\displaystyle \frac{2h_c\tan {\theta }_c}{b_n}}{p}_{s1}$$

(26)

4.2.2. Net Effect of Stiffened Body

The net effect mainly primarily attributed to the geogrid material. Under loading, the foundation produces uneven settlement, and the reinforced body sinks with the foundation soil, resulting in flexural deformation and tension [26]. The tension is decomposed into an upward jacking force along the rectangular coordinate system to bear part of the upper structure. This load-transfer mechanism improves the bearing capacity of the foundation (as shown in Figure 12). The vertical jacking force is

$$T_y=nT\sin \delta$$

(27)

where T is the tensile force of the reinforcement, which can be obtained by Formula (28).

$$T={\epsilon }_g{E}_g$$

(28)

where n is the number of geogrid layers; $\delta$ is the angle between the reinforced body at the edge of the pile and the horizontal direction; $E_g$ is the tensile modulus of geotechnical materials, which can be determined by tensile test; and ${\epsilon }_g$ is the strain at the edge of the pile.

Before the calculation of the $\delta$ angle, the flexural deformation curve of the stiffened body must be established. Given that the deformation of the stiffened body is small, it can be approximately simplified as a quadratic parabola, as shown in Figure 12.

The deflection deformation curve equation of the stiffened body is set as

$$S_x=a{x}^2+bx+c$$

(29)

The unknowns a, b, c are determined by the undetermined coefficient method. It is easy to know that when x = R or $l+R$ (that is, at the edge of the pile top), S = 0; at x = R + ${\displaystyle \frac{l}{2}}$, S = $S_{max}$ (its value can be obtained according to the test). When above variables brought into Equation (29), the parallel solution is obtained:

$$\left\{\begin{array}{c}a=-{\displaystyle \frac{4S_{max}}{l^2}}\\ b=-a\left(l+2R\right)\\ c=aR\left(l+R\right)\end{array}\right.$$

(30)

The deflection function expression of the horizontal stiffened body can be obtained when the Formula (30) is taken into (29):

$$S_x=a{x}^2-a\left(l+2R\right)x+aR\left(l+R\right)$$

(31)

where $a=-{\displaystyle \frac{4S_{max}}{l^2}}$; $S_{max}$ is the maximum deflection of embankment surface.

The tangent value of the angle at the edge is

$$\tan \delta =aR+al$$

(32)

$$\sin \delta =\sqrt{{\displaystyle \frac{{tan}^2\delta }{{1+tan}^2\delta }}}$$

(33)

Take Equations (32) and (33) into Equation (27):

$$T_y=nT\sqrt{{\displaystyle \frac{{\left(aR+al\right)}^2}{1+{\left(aR+al\right)}^2}}}$$

(34)

4.3. Pile–Soil Stress Ratio Calculation of Bidirectional Reinforced Composite Foundation

Analysis indicates that the stress arises from the combined action of the self-weight of the interior soil column [27], the surcharge load, the downward drag force exerted by the reinforcement cushion, the interface friction between the inner and outer soil columns, and the load of the arch foot. Therefore, the stress on the pile top can be expressed as

$${\sigma }_p=rz+q+\tau +{\sigma }_y+{\displaystyle \frac{T_y}{b_n}}$$

(35)

where $\tau$ is the interfacial friction between the inner and outer soil columns (as shown in Figure 13) [28], and its degree of exertion is related to the relative displacement of the inner and outer soil columns. Namely, it is 0 at the settlement surface of the embankment and it reaches the maximum at the top of the pile. Below the equal settlement surface of the embankment, the frictional resistance of the inner and outer soil columns can be calculated as follows:

$$\tau =\beta \left(c+\mu {\sigma }_p\right)$$

(36)

where $\beta$ is the degree of lateral friction; c is the cohesion of embankment soil; $\mu =f_{s}k$, $f_s$ is the friction coefficient of the interface between the inner and outer soil columns, which can be approximated by $f_s=\tan \phi$ [29]; $\phi$ is the internal friction angle of the embankment fill; $k$ is the lateral earth pressure coefficient, $k=1-\sin \phi$; and ${\sigma }_p$ is the vertical stress of the inner soil column.

When Formulas (10), (22), (34), and (36) are taken into (35), we can obtain

$${\sigma }_p={\displaystyle \frac{rz+q+\beta c+\frac{3ql+3rzl-2f\gamma l}{6R}+\frac{nT}{l}\sqrt{\frac{{\left(aR+al\right)}^2}{1+{\left(aR+al\right)}^2}}}{1-\beta f_{s}k}}$$

(37)

Similarly, the overlying load on the soil between piles is composed of the self-weight of the soil under the arch structure ${\sigma }_{\mathrm{s}\mathrm{z}}$, the vertical stress ${\sigma }_{\mathrm{s}\mathrm{g}}$ transferred from the arch structure to the soil between piles, the upward jacking force induced by the horizontal reinforcement, and the influence of stress diffusion. Therefore, the stress of the soil between piles can be expressed as

$${\sigma }_s={\displaystyle \frac{b_n}{\left(b_n+2h_c\tan {\theta }_c\right)}}\left({\sigma }_{sz}+{\sigma }_{sg}-{\displaystyle \frac{T_y}{b_n}}\right)$$

(38)

where ${\sigma }_{sz}$, ${\sigma }_{sg}$ can be obtained by

$${\sigma }_{sz}={\displaystyle \frac{F_{sz}}{l}}={\displaystyle \frac{{\int }_0^l\gamma y_{\left(x\right)}dx}{l}}={\displaystyle \frac{2}{3}}\gamma f$$

(39)

$${\sigma }_{sg}={\displaystyle \frac{2{\int }_0^{\frac{l}{2}}\left(q+\gamma z-\frac{2}{3}f\gamma \right)\left(\frac{l}{2}-x\right)dx}{l}}$$

(40)

$${\sigma }_{sg}=\left({\displaystyle \frac{q+\gamma z}{4}}-{\displaystyle \frac{1}{6}}f\gamma \right)l$$

(41)

Substituting (39), (41) into Equation (38) to obtain

$${\sigma }_s={\displaystyle \frac{b_n}{\left(b_n+2h_c\tan {\theta }_c\right)}}\left[{\displaystyle \frac{2}{3}}\gamma f+\left({\displaystyle \frac{q+\gamma z}{4}}-{\displaystyle \frac{1}{6}}f\gamma \right)l-{\displaystyle \frac{2nT}{l}}\sqrt{{\displaystyle \frac{{\left(aR+al\right)}^2}{1+{\left(aR+al\right)}^2}}}\right]$$

(42)

The pile–soil stress ratio on the horizontal surface of the pile top is

$$n_s={\displaystyle \frac{{\sigma }_p}{{\sigma }_s}}={\displaystyle \frac{rz+q+\beta c+\frac{3ql+3rzl-2f\gamma l}{6R}+\frac{nT}{l}\sqrt{\frac{{\left(aR+al\right)}^2}{1+{\left(aR+al\right)}^2}}}{\frac{b_n\left(1-\beta f_{s}k\right)}{\left(b_n+2h_c\tan {\theta }_c\right)}\left[\frac{2}{3}\gamma f+\left(\frac{q+\gamma z}{4}-\frac{1}{6}f\gamma \right)l-\frac{2nT}{l}\sqrt{\frac{{\left(aR+al\right)}^2}{1+{\left(aR+al\right)}^2}}\right]}}$$

(43)

4.4. Verification by Engineering Practice

A test section on the soft soil foundation was treated with a bidirectionally reinforced composite foundation [30]. The diameter of the mixing pile was d = 0.50 m, the pile spacing was s_d = 1.2 m, and the pile layout was plum-shaped. And the pile top was filled with 30 cm fine sand, with a layer of CAT steel-plastic composite geogrid. The geogrid exhibited a yield elongation of ≤3%, and a tensile strength of ≥54 kN/m at 2% elongation. The foundation consisted of silt and silty sand. The overlying fill had a unit weight $\gamma$ = 20 kN/m³, and the embankment height of the test section was 4 m. The measured maximum post-construction settlement value was s_max = 3.5 cm, and with a geogrid strain was s = 0.02. The cohesion of embankment fill was 30 kPa, and the internal friction angle was $\phi$ = 20$°$. The measured stress was measured by in-site earth pressure cells connected to an automatic acquisition system. The final measurement uncertainty was <5% after temperature correction, rainfall exclusion, and independent settlement verification, meeting the accuracy specified in ASTM D1143 [31] for earth-pressure monitoring. The relevant calculation and measured results are shown in

Table 4. Comparison of calculated and measured results.

Method of Calculation	Stress of Pile Top (kPa)	Stress of Soil Among Piles (kPa)	Pile–Soil Stress Ratio (ns)
Measured values	220.8	21	10.5
Ref. [30] Calculated values	341.9	26.3	13
Ref. [32] Calculated values	327.50	29.0	11.3
Calculated value in this paper	214.3	20.4	10.5

Note: Reference [30] considers the net effect, and Reference [32] comprehensively considers the soil-arching effect and net effect.

.

As shown in Table 4, when the stress diffusion effect is taken into account, the calculated values of soil stress between piles and pile–soil stress ratio are significantly lower than those obtained by considering only the net effect or the combined effect of net effect and soil-arching effect. Furthermore, these calculated results show closer agreement with the field-measured values.

5. Influencing Factors of Pile–Soil Stress Ratio

According to above derivation, the pile–soil stress ratio of the bidirectionally reinforced composite foundation depends on multiple factors. To investigate the influence of these factors on the pile–soil stress ratio, based on the above engineering example parameters, this paper focuses on the influence of filling weight $\gamma$, filling height $z$, internal friction angle of embankment filling $\phi$, pile spacing $L$, number of geogrid layers $n$, cohesion of embankment filling $c$. And the influence of each factor on the pile–soil stress ratio was analyzed, and the corresponding variation curves are presented in Figure 14, Figure 15, Figure 16, Figure 17, Figure 18 and Figure 19.

As shown in Figure 14, the pile–soil stress ratio $n_{\mathrm{s}}$ decreased with the increase in the embankment filling weight $\gamma$. The reason for this is that the soil-arching effect was active only below the equal settlement surface, but the height of the equal settlement surface was generally less than the embankment height. As the fill weight increases, the proportion of stress transferred from the outer soil column to the inner soil column via the soil-arching effect [33], leading to a gradual reduction in n_s.

The pile–soil stress ratio $n_{\mathrm{s}}$ increased with the increase in the embankment filling height $z$ (Figure 15). This is mainly because when the embankment height was low, the load transferred to the pile through the shear action of the internal. As the embankment height increased, the reinforcement provided by the pile body gradually came into play. Therefore, the pile body could bear more load, which ultimately led to an increase in the pile–soil stress ratio $n_{\mathrm{s}}$.

The pile–soil stress ratio $n_{\mathrm{s}}$ increased first and then decreased with the increase in the internal friction angle $\phi$ (Figure 16). When $\phi ={23}^{°}$, $n_{\mathrm{s}}$ reached the maximum value. The reason may be that $\beta f_{\mathrm{s}}$ in the pile–soil stress ratio Formula (43) was a function of the internal friction angle $\phi$.

The pile–soil stress ratio $n_{\mathrm{s}}$ decreased as the pile spacing L increased (Figure 17). The reason for this is that the size of the pile spacing directly affected the formation and stability of the soil-arching effect. Moreover, as analyzed from the ultimate state of the soil arch structure mentioned above, the stability of the soil arch structure decreased with the increase in the pile spacing. Therefore, it is not conducive to the transfer of the load from the outer soil column to the inner soil column by the soil arch structure, which ultimately led to the decline of the pile–soil stress ratio $n_{\mathrm{s}}$.

The pile–soil stress ratio $n_{\mathrm{s}}$ increased with the increase in both the number of geogrid layers $n$ and the cohesion $c$ of the embankment filling (as shown in Figure 18 and Figure 19). The reason for this is that the increase in the number of geogrid layers enhanced the net effect and further increased the vertical component of the pile top. Consequently, the load acting on the soil between the piles was reduced, promoting the increase in the pile–soil stress ratio. The increase in cohesion $c$ of embankment filling directly led to the improvement of soil shear strength and the stability of soil arch structure, and then the pile–soil stress ratio was increased.

6. Conclusions

In this paper, FLAC 3D 6.0 and PFC2D 6.0 software were used to simulate the evolution of the soil-arching effect from macro–micro perspectives. A calculation method for the pile–soil stress ratio of bidirectionally reinforced composite foundation was derived, and the influence of various influencing factors was investigated. The conclusions are as follows:

• The evolution process of the soil-arching effect can be divided into three stages: non-arch stage, arch transition stage, and soil-arching stage. When the filling subgrade reached 0.65 times the pile spacing, the vertical stress of the soil began to shift to the pile heads, leading to stress concentration and the forming of a stress arch. When the filling height reached 1.15 times the pile spacing, the overlying load was transmitted to the pile heads through the stress arch, a low-stress zone was formed, and the stability of the foundation was thereby enhanced.
• After the soil arch formed, the subgrade can be divided into two zones: the zone unaffected by the soil-arching effect, where the vertical stress of the subgrade was linearly distributed, and the opposite one, where the vertical stress of soil decreased with depth. It reached the minimum in the surface layer between piles, and the stress distribution was curve. The dividing filling height was about 0.65 times of the pile spacing.
• With the multi-effect coupling analysis of the stress diffusion effect, soil-arching effect, and net effect, a calculation formula of the pile–soil stress ratio of the bidirectionally reinforced composite foundation was derived. The results demonstrate that when stress diffusion effect was considered, the calculated value of soil stress between piles was effectively reduced, and the calculated value of the pile–soil stress ratio was in closer agreement with the measured value.
• The pile–soil stress ratio decreased with the increase in filling weight and pile spacing. It increased with the increase in embankment filling height, geogrid layer number, and soil filling cohesion, and increased first and then decreased with the increase in internal friction angle of soil filling.