Finite Element Simulation Analysis of the Influence of Pile Spacing on the Uplift Bearing Performance of Concrete Expanding-Plate Pile Groups

Abstract

Concrete expanding-plate piles (CEP piles) represent a novel type of variable cross-section concrete cast-in-place pile, wherein one or more bearing plates are added to the pile body to enhance its load-bearing capacity. Compared to traditional uniform-diameter uplift piles, the bearing plates of CEP uplift piles provide additional resistance against uplift, substantially increasing the pile’s uplift bearing capacity. CEP piles exhibit a wide range of application potential in structures such as high-rise buildings, cable-stayed bridges, and offshore platforms. However, due to changes in the load-bearing mechanism, the pile–soil interaction of CEP piles significantly differs from that of straight-shaft piles. Theories applicable to the group effect of straight-shaft piles cannot be directly applied to CEP piles, which has led to imperfections in the theoretical framework for designing CEP piles in practical engineering applications, hindering their broader adoption. Therefore, this paper employs a finite element simulation analysis to study the failure modes of three groups of symmetrically arranged CEP pile groups. The effects of pile spacing on the uplift bearing capacity of CEP pile groups are investigated, leading to a revision of the formula for calculating the uplift bearing capacity of CEP pile groups. This study enhances the theoretical understanding of the load-bearing behavior of CEP pile groups, providing a theoretical basis for their practical engineering applications.

1. Introduction

Uplift piles are increasingly employed in high-rise constructions, structures in expansive soil regions, and bridge infrastructures [1,2]. Notably, with the global shift toward renewable energy sources, the demand for uplift piles has surged in applications such as offshore wind power platforms and offshore drilling rigs [3]. Traditional bored uplift piles, which rely on shaft friction to resist uplift forces, can be effectively enhanced by increasing their diameter and length [4,5]. However, in practical engineering applications, particularly in offshore engineering, increasing the length and diameter of piles may pose challenges due to actual engineering conditions.

With the advancement of pile foundation technology, various variable-section piles have evolved, such as screw grouted piles and enlarged base piles, which enhance the uplift bearing capacity by altering the pile section [6,7,8]. The concrete expanded-plate (CEP) pile is a new type of variable-section concrete cast-in-place pile, which can be constructed in one step using a specialized drilling and expansion machine. The pile body is shown in Figure 1, and the pile drilling head is shown in Figure 2.

In studies on the bearing performance of single CEP piles, it has been observed that under vertical tensile loads, the load-bearing plate of the CEP pile provides additional end resistance, demonstrating an excellent uplift bearing performance and a significant engineering application value [9].

In the study of the bearing performance of double CEP piles, it was found that the unique load-bearing mechanism of the CEP pile causes the soil failure mode around the pile to differ significantly from that of straight bore piles, resulting in a substantial difference in the interaction between CEP piles compared to that between straight bore piles [10]. Consequently, the calculation methods for the uplift bearing capacity of straight bore pile groups cannot be directly applied to CEP piles, leading to an insufficient design theory for CEP pile groups, negatively impacting the promotion and application of CEP piles. Therefore, it is necessary to study the grouping effect of CEP piles under uplift loads to provide reliable theoretical support for engineering design.

Previous studies on single CEP piles, conducted through model tests and numerical simulations, have confirmed that finite element models established in ANSYS using the Drucker–Prager constitutive model effectively simulate soil failure around the pile in low-friction-angle cohesive soils [11]. This study considers the pile spacing—a critical factor affecting group effects—as a variable and establishes symmetrically distributed four-pile, six-pile, and nine-pile CEP pile group models in ANSYS.

By analyzing the load–displacement curves of different pile spacing groups post-loading, examining the status of pile–soil failure, and assessing the impact of pile spacing on the uplift resistance capacity of CEP pile groups, this study introduces a correction factor for the uplift group resistance of CEP piles and adjusts the formula for calculating their uplift resistance capacity [12]. This refines the uplift resistance mechanism of CEP pile groups and provides a theoretical basis for their practical application.

2. ANSYS Simulation Model Construction

2.1. Model Dimensions

The dimensions of the pile are dictated by the construction apparatus used to produce the pile body. The parameters defining the pile size include a length (L) of 9 m, a diameter (D) of 0.5 m, a distance (L₁) from the top of the pile to the top of the load-bearing plate of 6 m, an overhanging diameter of the load-bearing plate (R) of 0.7 m, an upper slope angle (α) of 35 degrees, and a lower slope angle (β) of 27 degrees. These parameters are illustrated in the schematic diagram shown in Figure 3.

This study established three sets of rectangular uprooting-resistant group pile models, each set comprising four, six, and nine piles, respectively. These models were designed with varying plate-end spacings (S₁) of 600 mm, 1100 mm, 1600 mm, 2100 mm, 3100 mm, and 4100 mm. The specific model numbers are detailed in

Table 1. Model numbers.

Distance between Plate Ends (mm)	600	1100	1600	2100	3100	4100
Four piles	MF1	MF2	MF3	MF4	MF5	MF6
Six piles	MS1	MS2	MS3	MS4	MS5	MS6
Nine piles	MN1	MN2	MN3	MN4	MN5	MN6

, and the arrangement of the group piles is depicted in Figure 4 [13].

2.2. Material Parameters and Intrinsic Modeling

2.2.1. Constitutive Model

The soil in the model exhibits a highly nonlinear behavior, making the selection of appropriate soil constitutive models pivotal for enhancing the simulation accuracy [14]. Drawing on insights from prior model test studies on concrete expanded-plate piles, the Drucker–Prager (DP) yield criterion has been identified as particularly effective. This model takes into account the effects of intermediate principal stress and hydrostatic pressure, thereby providing a precise representation of both the interactions between the pile and soil and the failure mode of the surrounding cohesive soil [15]. In the study of double CEP piles, the soil displacement under failure conditions in model tests was compared with the finite element simulation displacement cloud diagram, as shown in Figure 5.

Due to the significantly higher strength and stiffness of the concrete pile compared to the soil, the stress on the pile under vertical tensile loads remains relatively low and does not reach the plastic deformation stage. Therefore, a linear elastic model was used for the concrete pile.

After the ANSYS update, the DP model was integrated into the EDP model. Subsequently, the updated advanced solid elements, solid185 and solid186, now only support the EDP model. This update has optimized the support of solid elements for elastoplastic constitutive models, enhancing their functionality [16]. Changes were also made to the EDP model, offering a variety of yield forms and flow criteria options. Among these, the linear yield equation is similar in form to the classical DP model’s yield equation and shares the same yield surface shape. However, the input parameters have been modified, necessitating parameter conversion for the application of the EDP model. The yield equation for the DP criterion is as follows:

$$F=\frac{6\sin \phi }{\sqrt{3}\left(3-3\sin \phi \right)}{\sigma }_m+{\left[\frac{1}{2}{\left\{S\right\}}^T\left[M\right]\left\{S\right\}\right]}^{\frac{1}{2}}-\frac{6c\cos \phi }{\sqrt{3}\left(3-\sin \phi \right)}=0$$

(1)

In the equation above, φ represents the angle of internal friction and c denotes the cohesion.

The yield equation of the linear extended Drucker–Prager criterion is given by:

$$F=\alpha {\sigma }_m+{\left[\frac{3}{2}{\left\{S\right\}}^T\left[M\right]\left\{S\right\}\right]}^{\frac{1}{2}}-{\sigma }_y\left(\widehat{{\epsilon }_{pl}}\right)=0$$

(2)

It is understood from the comparison of yield functions that the stress sensitivity α is a parameter related to the angle of internal friction ϕ. The conversion formula is:

$$\alpha =\frac{6\sin \phi }{3-\sin \phi }$$

(3)

The material yield strength ${\sigma }_y\left(\widehat{{\epsilon }_{pl}}\right)$ involves parameters related to the angle of internal friction ϕ and cohesion c. The conversion formula is:

$${\sigma }_y\left(\widehat{{\epsilon }_{pl}}\right)=\frac{6c\cos \phi }{3-\sin \phi }$$

(4)

By employing these conversion formulas, it is feasible to calculate the parameters related to the EDP yield criterion using common soil parameters.

2.2.2. Material Parameters

The model includes two materials: concrete and soil [17,18]. The concrete material parameters were obtained by testing C30 concrete. The soil material parameters were determined based on geotechnical test results from a construction site in Changchun, China, as shown in

Table 2. Material parameters.

Materials	Density (t/mm3)	Young’s Modulus (MPa)	Poisson’s Ratio	Cohesion (MPa)	Angle of Friction (Angle)	Expansion Angle (Angle)
Concrete	2.25 × 10−9	3.46 × 104	0.2
Clay	1.39 × 10−9	25	0.35	0.04355	10.7	10.7

. Additionally, pull-out tests were conducted on the cast-in-place concrete piles at the construction site, resulting in a pile–soil friction coefficient of 0.3.

2.3. Construction of the Finite Element Model

2.3.1. Pile–Soil Model Establishment

As depicted in Figure 6, the geometric structure, material orientation, loads, and constraints of the pile group model in this study exhibit symmetry in two directions [19]. To enhance the computational accuracy while conserving resources, a quarter-pile group model is established. Symmetric constraints are applied on two symmetric planes for comprehensive computation.

To minimize the boundary effects on the simulation results, the dimensions of the soil for the four-pile, six-pile, and nine-pile models are set to 5 m × 5 m, 5 m × 7 m, and 7 m × 7 m, respectively, with a soil boundary distance of 3 m from the pile base [20]. Given the high-stress gradient zone around the load-bearing plate, the model employs Solid186 quadratic solid elements throughout, which significantly enhance the simulation accuracy in such zones. The piles are aligned along the Z-direction.

2.3.2. Mesh Generation

According to preliminary simulation results from earlier research, the soil mesh size is a critical factor affecting the simulation accuracy. When the soil mesh size is less than 1200 mm, the differences in simulation results become insignificant. Therefore, to balance computational accuracy and speed, the soil mesh size was set to 800 mm, with the mesh size around the load-bearing plate reduced to 500 mm due to the higher stress concentration in that region. The mesh size for the pile body was set to 300 mm. Before mesh generation, the model was segmented as much as possible along characteristic edges to form hexahedral meshes. The meshed model is illustrated in Figure 7.

2.3.3. Pile–Soil Contact Settings

Due to the significantly higher elastic modulus of the concrete pile compared to that of the soil, the contact between the model pile and the soil is defined as a rigid–flexible contact in this simulation. The pile–soil contact surface is set as “frictional”, where the normal effective stress on the contact surface is multiplied by the friction coefficient to obtain the ultimate frictional shear stress. If the shear stress along the contact surface is less than the ultimate frictional shear stress, adhesion occurs. When the shear stress along the contact surface exceeds the ultimate frictional shear stress, slippage occurs.

In the simulation, the pile is designated as the rigid target surface using the target170 element, and the soil is designated as the flexible contact surface using the contact174 element. The penalty function method is employed as the contact algorithm.

2.4. Boundary Constraints and Load Application

2.4.1. Boundary Constraint Settings

Symmetric constraints are applied to the two symmetric planes of the model to ensure stability. To prevent overall soil movement under vertical loading, fixed constraints are applied to the bottom soil boundary, while the side soil boundaries are allowed to move freely in the vertical direction.

2.4.2. Load Application

In this simulation, graded surface loads are applied incrementally at the pile top, with each load level set at 2 KN. Loading is halted once the pile top displacement exceeds 100 mm, indicating group pile failure [10,21]. A schematic diagram of boundary constraints and load application is shown in Figure 8.

3. Simulation Results Analysis

3.1. Analysis of Vertical Displacement Contour Maps of Soil

In accordance with the “Technical Specification for Building Foundation Pile Testing” (JGJ 106-2014), the condition whereby the displacement of any pile top in the pile group model under vertical tensile load surpasses 100 mm is indicative of the uplift pile group reaching a failure state.

The Z-direction displacement contour maps, delineating the failure state of the soil, were meticulously extracted and analyzed. Specifically, the displacement contour maps for soil sections around piles 1 and 2 in the four-pile group, piles 1, 2, and 3 in the six-pile group, and piles 4, 5, and 6 in the nine-pile group were obtained. For enhanced analytical clarity, these contour maps were extended along the symmetrical planes to display a complete view of the entire pile group model. These maps are depicted in Figure 8, Figure 9 and Figure 10.

An in-depth analysis of these displacement contour maps reveals that the failure modes of the soil around the piles in each group are fundamentally consistent. As illustrated in Figure 9a, the failure mode around the piles in the pile group mirrors that of single piles. However, the displacement values on the inner sides of the pile bodies are noticeably higher than those on the outer sides. Under vertical loading, the soil above the load-bearing plate experiences compression, leading to slipping phenomena, with the maximum soil displacement values observed. This compression extends outward from the top of the load-bearing plate, creating a defined area of soil slipping around the plate, where soil stress values are heightened. Conversely, the soil below the load-bearing plate detaches from the pile body, resulting in reduced soil displacement at the sides of the piles beneath the load-bearing plate.

Figure 9b illustrates that the slipping zones of soil between the piles in the group overlap, culminating in increased soil displacement. This suggests that the soil between the piles undergoes the compounded effects of multiple piles, thereby elevating the stress values on the sides of the piles compared to those of individual piles under identical loads. This phenomenon contributes to variations in the bearing capacity of each pile within the group, thereby establishing pile group effects. Moreover, in the six- and nine-pile models depicted in Figure 10 and Figure 11, respectively, the lateral and corner piles exert simultaneous forces on both sides of the central pile, increasing the soil displacement values at the sides of the central pile and precipitating earlier failure of the central pile compared to the lateral and corner piles.

Figure 9a, Figure 10a and Figure 11a show that when the spacing between the ends of the load-bearing plates is less than the overhang diameter of these plates, the soil slipping zones overlap significantly, indicating pronounced pile group effects. As the spacing between the piles increases, these slipping zones begin to separate. Figure 9b, Figure 10b and Figure 11b demonstrate that even when the net spacing between the piles reaches 1100 mm, the soil slipping zones between the piles still overlap considerably, albeit with relatively reduced soil displacement values. When this spacing is expanded to three times the overhang diameter of the load-bearing plates, as shown in Figure 9c, Figure 10c and Figure 11c, most of the soil slipping zones become distinct, and the pile group effects are diminished. As depicted in Figure 9d–f, Figure 10d–f and Figure 11d–f, with further increases in pile spacing, the soil slipping zones around the piles progressively separate more distinctly, and the pile group effects gradually decrease, aligning the failure modes of the soil around the pile group more closely with those of single piles.

The insights gleaned from the soil displacement contour maps in Figure 9, Figure 10 and Figure 11 suggest that the formation mechanism of CEP pile group effects, although distinct from that of straight-hole pile groups, still adheres to the principle that interactions decrease with increasing pile spacing, leading to a weakening of pile group effects.

3.2. Analysis of Soil Shear Stress

To delve deeper into the failure modes of the soil surrounding the piles, the XZ-directional shear stress on the left side of piles 4, 5, and 6 in model MN4 was scrutinized when the pile top displacement reached 100 mm. Nodes were strategically placed along the pile body, with node 15 located at the intersection of the upper part of the pile body and the load-bearing plate, node 16 at the end of the load-bearing plate, and node 17 at the intersection of the lower part of the load-bearing plate and the pile body. The soil XZ-directional shear stress curve alongside a schematic diagram of the nodes is illustrated in Figure 12.

The shear stress curve depicted in Figure 12 reveals a pronounced spike at node 12, where shear stress values escalate sharply, peaking at node 15—where the pile body intersects with the load-bearing plate—and subsequently declining at node 16 at the plate’s end. This specific area aligns with the soil slipping zone above the load-bearing plate, where both shear stress and displacement values are substantial, and thus this area bears the brunt of the load. Following node 17, located at the lower part of the load-bearing plate, a region of negative shear stress emerges due to the detachment of the load-bearing plate from the soil. This separation leads to negative shear stress values, stemming from overall soil deformation, resulting in minimal load bearing by the lower part of the pile body.

Additionally, since pile 4 on the left side lacks adjacent pile bodies, the soil on the upper left side of this pile normally bears the load. However, the soil on the left side of pile 5 is influenced by the negative shear stress region on the right side of pile 4, resulting in a slight reduction in shear stress values due to soil interactions. Similarly, the soil on the left side of pile 6 interacts with the soil in the negative shear stress region on the right side of pile 5. Despite this, due to the significant impact of pile group effects on pile 5, the stress values around pile 6 are elevated, leading to negative shear stress values in the upper left side soil of pile 6. Thus, under the influence of interacting pile bodies within the pile group, even the soil outside the slipping zone is affected by pile group effects, causing a decrease in stress values.

Through the detailed analysis of the soil shear stress curve, it is evident that under the action of vertical tensile loads, the load on the CEP pile group is predominantly supported by the soil within the slipping zone. This finding aligns with the conclusions drawn from the analysis of the soil displacement contour maps in Section 3.1. Moreover, it underscores that the effects of groups of CEP piles also influence the soil beyond the slipping zone, affecting the overall soil stress distribution.

3.3. Analysis of Pile Head Load–Displacement Data

Load–displacement data for the pile heads of pile 1 in the MF group, pile 2 in the MS group, and pile 5 in the MN group were extracted, and single-pile load–displacement curves were generated for a comparative analysis. These curves are presented in Figure 13.

The development trends observed in the load–displacement curves for the three groups exhibit similarities, with the slope of each curve reflecting the rate of pile head displacement. A steeper slope indicates a faster displacement rate at the pile head. Consequently, based on variations in the curve slope, the development stages of the load–displacement relationship can be categorized into linear, nonlinear, and highly nonlinear phases. For instance, in the MF group, as illustrated in Figure 13a, the load–displacement curve follows a linear trajectory when the pile head displacement remains below 20 mm, maintaining a relatively constant slope. As the displacement extends beyond 20 mm and approaches 50 mm, the slope increases alongside the load, denoting a transition into the nonlinear phase where the curves begin to diverge. When the displacement surpasses 50 mm, the slope escalates further, and with each incremental increase in load, the pile head displacement significantly intensifies, signaling an impending failure.

During the linear stage, as depicted in Figure 13a–c, when the pile head load is modest, the soil stress surrounding the pile sides is minimal, leading to negligible interactions between the piles in the group, and the curves for the three groups largely coincide, indicating a minimal pile group effect. In the nonlinear stage, as the load escalates, the stress exerted on the soil around the pile sides increases. Consequently, the slipping zone between the piles expands, and intersections start to appear, suggesting interactions between adjacent piles in the group and marking the onset of pile group effects. In models with a closer pile spacing, these slipping zones intersect at lower loads, and the curves’ slopes alter first, denoting an earlier manifestation of pile group effects.

In the highly nonlinear stage, the slipping zones between the piles significantly overlap, the rate at which the pile head displacement increases accelerates sharply, the pile group effect becomes pronounced, and the ultimate bearing capacity increases with greater pile spacings. At this juncture, the load that the uplift pile group can sustain is constrained, and the pile nears extraction in practical engineering applications.

As illustrated in Figure 13a–c, with an increase in the number of piles in the pile group, the dispersion of the curves also widens, suggesting that uplift pile groups with a larger number of piles are more influenced by the pile group effect. This observation aligns with the analysis of the soil displacement contour maps in Section 3.1, which shows that central piles in the uplift pile group are significantly affected by the collective action of multiple side piles, intensifying the pile group effect.

Through the detailed examination of the pile head load–displacement curves of the three uplift pile groups in Figure 11, it can be deduced that the effect of uplift pile groups diminishes with an increased pile spacing but intensifies with a greater number of piles. This finding corroborates the conclusions drawn from the analysis of soil displacement contour maps in Section 3.1.

3.4. Analysis of Pile Head Load–Displacement Curves for Different Pile Positions in Uplift Pile Groups

Through an in-depth analysis of the soil displacement contour maps detailed in Section 3.1, it is apparent that the pile group effect variably influences the central and side piles of the MS and MN uplift pile groups, thereby affecting the bearing capacity of CEP uplift piles at different positions within the pile group. Consequently, the pile head load–displacement curves for piles 1, 2, and 5 of the MN3 group were meticulously extracted and analyzed. The resulting pile head load–displacement curves, illustrating the different behavior of pile positions within the uplift pile group, are depicted in Figure 14.

The analysis of these curves, as shown in Figure 14, reveals a consistent trend across the three piles, albeit with notable variations in the rate of failure. During the linear stage, the slope of the curve for pile 1 is marginally less steep compared to those for piles 2 and 5, reflecting a smaller differential at lower load levels. As the load transitions into the nonlinear stage, the slope changes for piles 2 and 5 become more pronounced than that for pile 1. In the highly nonlinear stage, influenced by inter-pile interactions, the slope of the curve for pile 5 escalates dramatically, leading to its earlier failure. Specifically, when the pile head displacement of pile 5 reaches 100 mm, the corresponding displacements for piles 2 and 1 are recorded at 77.88 mm and 91.316 mm, respectively.

This analysis, coupled with insights from the soil displacement cloud diagrams in Section 3.1, indicates that the pile group’s impact on different pile positions varies significantly within the uplift pile group, resulting in observable differences in bearing capacity. Typically, in such groups, the central piles exhibit the lowest bearing capacity, followed by intermediate piles, with side piles demonstrating a relatively higher capacity.

4. Revision of the Calculation Formula for the Uplift Ultimate Bearing Capacity of CEP Pile Groups

Building upon the findings presented in this paper, pile spacing emerges as a critical factor influencing the uplift ultimate bearing capacity of CEP pile groups. Drawing from both the simulation results and prior research on single CEP pile configurations, a correction coefficient, α, for the pile spacing of CEP pile groups is proposed. This coefficient is based on the uplift ultimate bearing capacity of single CEP piles of comparable dimensions and is intended to refine the calculation formula for the uplift ultimate bearing capacity of CEP pile groups. This revised formula aims to more accurately predict the performance of CEP piles under varying conditions and configurations, thereby enhancing the reliability and applicability of CEP technology in practical engineering scenarios.

4.1. Calculation Formula for the Single-Pile Uplift Ultimate Bearing Capacity of CEP Piles

To determine the ultimate bearing capacity of concrete expanded-plate piles, three main factors are considered: the frictional resistance along the pile sides (F_Pile), the plate resistance at the pile end (F_Plate), and the self-weight of the pile body (G_Pile).

Reflecting on the simulation of single CEP piles and the analysis of soil failure states around the pile, as detailed in Section 3.1, it has been observed that under vertical tensile forces, the soil primarily fails by slipping at the plate. Consequently, a Prandtl strain field is established for the accurate calculation of the ultimate bearing capacity of the soil on the plate, as illustrated in Figure 15 [22]. This strain field is defined by two radial rays at $\theta =0$ and $\theta =\Theta$ and a logarithmic spiral given by the equation $r=R{e}^{\theta \tan \phi }$. Employing slip line theory, the ultimate bearing capacity can be calculated using the following formula:

$$F_n=\frac{2c\cot \phi R}{2R+d}(e^{2\theta \tan \phi }-1)$$

(5)

In this equation, F_n represents the ultimate bearing capacity of the soil on a unit width load-bearing plate, as calculated using the slip line theory; R denotes the overhanging diameter of load-bearing plate; d is the pile diameter; c is the cohesion of the soil around the pile; and φ is the internal friction angle of the soil around the pile.

Derive the calculation formula of disc resistance:

$$F_{Plate}=F_n\pi (\frac{R}{2}+\frac{d}{2})=\frac{\pi \left(c\cot \phi R\right)\left(e^{2\theta \tan \phi }-1\right)}{2}$$

(6)

Due to the separation between the pile body and the soil beneath the bearing plate, a tension zone of a certain length (designated as L_b) develops horizontally in the soil. Within this zone, no lateral frictional force is exerted on the pile, and thus it should be deducted when calculating the lateral frictional resistance. Concurrently, axial compression in the soil above the bearing plate leads to an increased horizontal compressive stress in the area (designated as L_a). This increases the pile side frictional resistance within this region, with an increase factor γ ranging from 1.1 to 1.2. Therefore, the formula for calculating the pile side frictional resistance is given by:

$$F_{Lateral}=f_{Lateral}\pi d\left(L-H-L_b+\gamma L_a\right)$$

(7)

where L represents the pile length, H is the height of the bearing plate, L_a is the area of increased horizontal compressive stress above the bearing plate, L_b is the tension zone beneath the bearing plate, and γ is the increase factor.

The formula for calculating the uplift bearing capacity of a single CEP pile is as follows:

$$\begin{array}{c}{F}_{single}=F_{Plate}+F_{Lateral}+G_{Pile}=\\ \frac{\pi \left(c\cot \phi R\right)\left(e^{2\theta \tan \phi }-1\right)}{2}+f_{Lateral}\pi d\left(L-H-L_b+\gamma L_a\right)+G_{Pile}\end{array}$$

(8)

4.2. Adjustment Factor for the Group Pile Uplift Bearing Capacity

The ultimate uplift bearing capacity of a CEP pile group under vertical tensile load is defined as the load borne by the pile group when the displacement at the top of any pile reaches 100 mm. The pile spacing adjustment factor, α, is computed as follows:

$$\alpha =\frac{F_{Group}}{n{F}_{single}}$$

(9)

where n denotes the number of piles in the group.

The simulation results indicate that the uplift bearing capacity of a single CEP pile of the same dimensions, F_Single, is 1719.92 kN. The adjustment factors for pile spacing, α, for various groups are presented in

Table 3. Adjustment factors α for the influence of pile spacing on the group pile bearing capacity.

Plate-End Spacing	600	1100	1600	2100	3100	4100
Four pile groups	0.739	0.814	0.843	0.861	0.884	0.931
Six pile groups	0.652	0.721	0.756	0.779	0.826	0.884
Nine pile groups	0.567	0.628	0.665	0.715	0.768	0.838

.

The formula for calculating the ultimate uplift bearing capacity of a CEP pile group is derived as follows:

$$\begin{array}{c}{F}_{Group}=\alpha n{F}_{Single}=\\ \alpha n\frac{\pi \left(c\cot \phi R\right)\left(e^{2\theta \tan \phi }-1\right)}{2}+\alpha n{f}_{Lateral}\pi d\left(L-H-L_b+\gamma L_a\right)+\alpha n{G}_{Pile}\end{array}$$

(10)

For additional pile spacing correction coefficients, values of α smaller than the specific length can be found in Table 2, which tends to provide a conservative estimate of the bearing capacity. It is crucial to note that the pile group effect on CEP piles is also influenced by other factors such as the plate overhang diameter, pile length, and pile diameter. These variables must be thoroughly considered in practical engineering calculations to ensure the accuracy and reliability of the results.

5. Conclusions

Failure Modes and Soil Interaction: Based on the finite element simulation results, it has been determined that under vertical tensile forces, the failure mode of the soil around CEP pile groups mirrors that observed in single piles. The load is predominantly borne by the soil at the bearing plate, where a slipping zone forms on the upper part of the bearing plate due to high soil stress. As the load increases, this area experiences sliding failure, while the lower part of the pile body remains mostly unloaded. The presence of overlapping sliding zones between adjacent piles leads to increased soil stress, enhancing the group effects.

Effects of Pile Spacing on the Load-Bearing Capacity: With the increase in pile spacing, the sliding zones of the soil begin to separate and the bearing capacity of the pile group members starts to approach that of single piles, thereby reducing the group effects. Nonetheless, load–displacement data indicate that even when the spacing between disk ends is 5.8 times the disk overhang diameter, differences in the bearing capacity between the pile group members and single piles still exist. Therefore, when selecting the pile spacing for practical engineering applications, comprehensive considerations are necessary to avoid displacements of CEP piles entering a highly nonlinear stage.

Influence of Pile Group Composition on Pull-out Capacity: The analysis reveals that the number of piles in the CEP pile group significantly influences the pull-out capacity. Piles located at the center and intermediate positions within the group are identified as weaker members, exhibiting lower pull-out capacities. This critical finding necessitates specific attention in engineering design.

Refinement of the Uplift Bearing Capacity Calculation Formula: From the finite element analysis, a correction coefficient α is proposed to adjust the calculation formula for the uplift bearing capacity of CEP pile groups. This modification provides a reliable theoretical basis for the engineering design of CEP piles and supports their wider application.

6. Future Work

This paper primarily analyzes the interactions between individual piles in CEP pile groups under vertical tensile loads and the soil failure modes around the piles in the group, thereby examining the mechanisms and effects of group piles. To facilitate analysis, the load conditions were simplified as much as possible, which creates discrepancies with actual engineering scenarios. In practical engineering, uplift piles typically experience horizontal loads in addition to upward loads. To make the research more applicable to engineering practice, studies on CEP pile groups under complex loading conditions should be conducted, focusing on the interaction between CEP piles, soil, and the pile cap.

Current research on CEP piles often assumes homogeneous soil layers, which facilitates the analysis of load-bearing mechanisms. However, in practical engineering, nearly all piles are embedded in heterogeneous soil. Since the load-bearing plate of the CEP pile bears most of the load, the soil properties where the load-bearing plate is located have a significant impact on the bearing capacity. Therefore, it is necessary to conduct targeted analyses on the bearing performance of CEP piles in heterogeneous soil.