Influence of Gravel Size and Geogrid Aperture on Performance of Geosynthetic-Encased Stone Column: DEM-FDM Coupled Numerical Investigation

Featured Application

The findings of this study can provide practical guidance for the design and parameter selection of geosynthetic-encased stone columns in transportation and infrastructure engineering, such as highway embankments, railway projects, and soft ground improvement in port engineering. By clarifying the influence of gravel particle size–geogrid aperture matching on lateral confinement effectiveness, the results facilitate the rational selection of gravel gradation and geogrid aperture while satisfying requirements for bearing capacity and deformation control.

Abstract

In order to investigate the effects of gravel particle size and geogrid aperture on the bearing performance of geosynthetic-encased stone columns, a discrete–continuum coupled numerical model was established based on laboratory test results, and a series of numerical simulations were conducted. The results indicate that, under the same loading level, the maximum lateral bulging of geosynthetic-encased stone columns increases with increasing geogrid aperture and decreases with increasing gravel particle size. The distance between the location of maximum lateral bulging and the pile-top decreases as the aperture increases, whereas it increases with increasing particle size. The bearing performance of geosynthetic-encased stone columns shows a positive correlation with gravel particle size and a negative correlation with geogrid aperture. The influence of particle size on bearing performance becomes insignificant when d₅₀ exceeds 40 mm. When the particle size is smaller than the geogrid aperture, contact between the gravel and the geogrid is established but remains insufficient, leading to separation as the load increases. In contrast, when the particle size is larger than the aperture, the effect of particle size on bearing performance is much more pronounced than that of aperture. Therefore, the use of gravel with a particle size slightly larger than the geogrid aperture is recommended in practical engineering applications.

1. Introduction

Geosynthetic-encased stone column (GESC) is a novel type of pile formed by wrapping a geogrid around the perimeter of a conventional stone column [1,2,3,4,5]. Building upon the advantages of traditional stone columns, such as low construction cost, rapid installation, and acceleration of soil consolidation, GESC exhibits superior bearing performance [6,7,8,9,10]. At present, the composite foundation technique using GESC has been widely applied in soft ground improvement projects for highways and railways both in China and abroad.

To gain a deeper understanding of the stress–strain behavior of GESC, a series of experimental studies have been carried out by researchers worldwide [11,12,13,14]. Rajagopal [15], Cunha [16], and Lo [17] reported that, when stone columns are encased with appropriate geosynthetic materials, both their bearing capacity and stiffness can be significantly improved. Ouyang et al. [18] found that partially encased stone columns show no obvious performance improvement compared with conventional stone columns, whereas fully encased stone columns can markedly enhance the bearing performance. Ghazavi et al. [19] investigated the influence of pile diameter on the bearing and deformation characteristics of geosynthetic-encased stone columns through laboratory model tests, and the results indicated that the ultimate bearing capacity of GESC increases with increasing pile diameter. Chen et al. [20] conducted uniaxial compression tests and observed that the strength of geosynthetic-reinforced stone columns increases with increasing geogrid strength. Hong et al. [21] reported that, when stone columns are encased with low-stiffness geosynthetics, bulging deformation mainly occurs at a depth of approximately 2.5 times the pile diameter below the ground surface. Dong et al. [22] examined the performance of GESC in marine silty clay foundations and found that, with increasing pile diameter, the lateral confinement provided by the geosynthetic decreases, resulting in a reduced contribution of the reinforcement to the bearing capacity. Li et al. [23], based on a series of laboratory model tests, found that better drainage conditions lead to smaller deformation of the stone column. Existing studies have mainly focused on the effects of reinforcement stiffness, reinforcement length, and pile diameter on the bearing performance of GESC, while relatively little attention has been paid to the influence of gravel size and geogrid aperture.

At present, most studies related to particle size and geogrid aperture mainly focus on the interface friction between particles and geogrids [24,25,26,27]. Ling et al. [28] conducted a series of direct shear tests and showed that neither excessively large nor excessively small particle sizes allow the geogrid to function effectively; optimal reinforcement can only be achieved when the particles are able to enter and interlock within the apertures of the geogrid. Hussaini [29], through large-scale monotonic direct shear tests, demonstrated that the ratio of geogrid aperture size to the mean particle size of the fill has a significant influence on the shear strength of the soil–geosynthetic interface, and pointed out that the interface shear strength reaches its peak when the aperture size is 0.95–1.2 times the mean particle size. Wang et al. [30] performed large-scale direct shear tests under different vertical stresses, shear rates, and densities, and found that both the peak shear stress and residual shear stress of the soil–geosynthetic interface increase with an increasing ratio of mean particle size to geogrid aperture. Biabani et al. [31], based on multiple direct shear tests, observed that the interface shear stress decreases with increasing geogrid aperture.

In summary, previous studies have rarely focused on the influence of particle size and geogrid aperture on the performance of geosynthetic-encased stone columns. Existing research concerning the effects of particle size and aperture has mainly emphasized interface friction at the soil–geosynthetic interface. However, in GESC, interface friction accounts for only a small part of the soil–geosynthetic interaction, and it is therefore inappropriate to explain the interaction mechanism between gravel and geogrid solely in terms of interface friction. Consequently, it is necessary to investigate the bearing behavior of GESC under different particle sizes and geogrid apertures. The main objective of this study is to investigate, at the micromechanical level, the influence of gravel particle size and geogrid aperture on the bearing performance of geosynthetic-encased stone columns using a three-dimensional DEM–FDM coupled numerical approach. Specifically, this study aims to: (i) evaluate the effects of particle size and aperture on macroscopic responses, including pile-top settlement, radial strain distribution, and bearing capacity; (ii) analyze the evolution of micromechanical indicators, such as porosity, coordination number, and contact force chain morphology, during the loading process of geosynthetic-encased stone columns; and (iii) elucidate the role of the particle–aperture interlocking mechanism in governing load transfer efficiency and lateral confinement performance from a micromechanical perspective. The findings are expected to provide a scientific basis for the rational selection of relevant design parameters in engineering practice.

2. Overview of Laboratory Tests

The single-column laboratory model test was conducted using a self-developed cubic model box, as shown in Figure 1. The experimental system mainly consisted of four components: the model box, a hydraulic loading plate, a data acquisition and control system, and earth pressure cells. The model box had dimensions of 1 m × 1 m × 1 m, while the hydraulic loading plate measured 200 mm × 200 mm × 50 mm. During the test, static loading was applied using a DJM-500 biaxial electro-hydraulic servo loading system (Sichuan Dexiang Kechuang Instrument Co., Ltd., Chengdu, China), which was capable of applying an axial load ranging from 0 to 500 kN.

Figure 1. Schematic diagram of the large-scale single column model test.

In practical engineering applications, the diameter dp of geosynthetic-encased stone columns generally ranges from 0.6 to 1.0 m, and the length-to-diameter ratio l/d typically varies between 5 and 20 [32]. To ensure that the laboratory model remains representative of field-scale behavior while satisfying construction quality control requirements and the dimensional limitations of the test apparatus, a geometric similarity ratio (prototype diameter to model diameter) of 5 was adopted based on similarity theory. This moderate scaling ratio allows key mechanical characteristics and particle–structure interactions to be reasonably preserved while minimizing scale-induced distortions commonly associated with small-scale models. Accordingly, the model tests employed stone columns with a uniform length of 800 mm and a diameter of 200 mm. The crushed stone infill consisted of naturally graded gravel with particle sizes ranging from 20 to 50 mm, ensuring material consistency with engineering practice and enhancing the reliability of extrapolating the observed mechanical behavior to field conditions [33]. A 200 mm thick rigid bearing stratum was placed at the bottom of the model box, above which the geosynthetic-encased stone columns were installed, with the surrounding soil filled with soft plastic clay. The mechanical properties of the soil were determined from triaxial tests, yielding a cohesion of 11 kPa and an internal friction angle of 20°. A single layer of biaxial geogrid was adopted as the reinforcement material, and its technical properties are summarized in Table 1.

Table 1. Geogrid specific technical indicators.

Aperture Size/mm × mm	Yield Tensile Strength/kN·m−1	Yield Elongation/%	Tensile Force at 2% Strain /kN·m−1	Tensile Force at 5% Strain/kN·m−1
20 × 20	21.5	13.4	7.5	14.8

The stone columns were constructed using a layered filling and vibration compaction method, with each layer having a thickness of 200 mm. For each layer, the crushed stone was compacted using a vibrating rod dropped freely from a height of 20 cm, with 25 compaction blows applied per layer. During the compaction process, the position of the geogrid-encased sleeve was carefully maintained to prevent displacement or wrinkling, until the column height reached the designed length.

During the tests, a stone column expansion gauge and earth pressure cells were installed to continuously monitor the bearing performance and radial deformation of the geosynthetic-encased stone columns. The earth pressure cells were positioned at the column top and within the column body. To improve load transfer uniformity and measurement accuracy, a 1 cm thick fine sand cushion was placed on both the upper and lower surfaces of each earth pressure cell, preventing direct contact between the crushed stone and the sensors and avoiding potential damage or non-uniform stress transmission. Throughout the loading process, the center of the hydraulic loading plate was aligned with the column top to ensure that the applied load was transferred axially along the column.

The tests were performed using a load-controlled method with incremental and equal load application. Each load increment was set to one-tenth of the ultimate bearing capacity of GESC obtained from the preliminary test, and the initial load was twice the magnitude of a single load increment. The preliminary test indicated that the ultimate bearing capacity of GESC was 20 KN; therefore, the first applied load in the formal test was 4 kN, followed by load increments of 2 kN at each subsequent stage. When the pile-top settlement was less than 0.1 mm within a continuous period of 1 h, the next load level was applied. GESC was considered to have reached complete failure when a sudden increase in settlement occurred. In this test, failure of GESC was observed at a pile-top load of 23 kN.

3. Numerical Modeling and Verification

3.1. Overview of the DEM–FDM Coupled Modeling Framework

To enhance methodological transparency and reproducibility, Figure 2 summarizes the numerical modeling framework and computational workflow adopted in this study. The proposed framework integrates three components: (i) a material-calibrated soft-bond representation of the geogrid, (ii) a finite-difference continuum domain for soft clay supplemented by localized discrete clay buffer regions to resolve interface kinematics and load transfer, and (iii) a discrete element representation of the granular stone column. This hybrid formulation enables efficient prediction of global stress–deformation responses while allowing refined characterization of particle-scale interactions in the vicinity of the soil–stone interface.

Figure 2. Workflow of the DEM–FDM coupled numerical modeling framework. (I) Workflow of the numerically simulated tensile test, and the stress state of the soft-bond geogrid model at the end of loading; (II) Stand-alone (single-unit) model of the geogrid-encased stone column; (III) Overall (global) model of the geogrid-encased stone column.

As shown in Figure 2, the workflow is organized into three sequential modules. Module I focuses on calibrating the micromechanical parameters of the geogrid through numerical tensile testing. The stiffness and bond-strength parameters of the soft-bond model are iteratively adjusted until the simulated tensile force–strain response reproduces the laboratory measurements, ensuring that the tensile stiffness, deformation capacity, and progressive degradation behavior of the reinforcement are represented in a physically consistent manner.

Module II constructs the coupled model of a geosynthetic-encased stone column embedded in soft clay. The far-field clay is modeled using a finite-difference continuum domain to maintain computational efficiency, whereas localized discrete clay buffer regions are introduced adjacent to the column boundary to explicitly capture particle-scale displacement compatibility and contact interactions. The stone column is generated using discrete particles with prescribed porosity and particle size distribution, and the encasement is formed by a calibrated geogrid particle-chain network arranged along the column perimeter. Mechanical interaction between the DEM and FDM domains is realized through a wall–zone coupling interface, where contact forces from the discrete particles are transferred to wall elements and subsequently mapped onto the continuum grid, while the displacement response of the continuum domain is fed back to constrain particle motion.

Module III covers model initialization, equilibrium establishment, and loading. After assembly, gravity loading and boundary constraints are applied to achieve a stable initial state. Vertical loading is then applied incrementally through an elastic loading plate following the laboratory loading protocol. At each loading stage, force equilibrium is achieved, and both macroscopic responses, including settlement and lateral deformation, and micromechanical indicators, including porosity evolution, coordination number variation, and force chain characteristics, are extracted for subsequent analysis.

By constraining model parameters at appropriate physical scales and explicitly resolving the coupling mechanisms at the soil–stone interface, the workflow supports systematic interpretation of how gravel particle size and geogrid aperture jointly influence load transfer efficiency and the overall mechanical performance of geosynthetic-encased stone columns.

3.2. Soft-Bond Model Formulation

When discrete element method is adopted for numerical simulations, most previous studies employ the parallel bond model to construct discrete element geogrids [34,35,36]. However, in this model, once the bonded contacts reach the peak load-carrying state, they immediately fail and are removed, resulting in an abrupt drop in contact forces, which makes it difficult to properly characterize the strain-softening and progressive damage behavior exhibited by geogrids during actual loading processes.

The parallel bond model is generally composed of two types of contact components, as illustrated in Figure 3. One corresponds to a point-contact component between ball–ball contacts, whose mechanical behavior is characterized by the normal and shear stiffnesses, k_n and k_s (F/L). The other corresponds to a surface-contact component between ball–facet contacts, for which the normal and shear stiffnesses are also described by k_n and k_s (F/L³). Within this modeling framework, the failure of the bonded contact occurs in the form of instantaneous breakage, and the resulting mechanical response is essentially equivalent to a fully brittle failure.

Figure 3. Schematic diagram of different types of contact elements.

Due to the failure mechanism described above, the parallel bond model has difficulty capturing the nonlinear stiffness degradation and tensile strain-softening behavior of geogrids during tensile or bending deformation. Once the bond strength reaches the prescribed threshold, the load-carrying capacity of the geogrid is rapidly lost, exhibiting an abrupt rupture behavior. This response differs significantly from the progressive damage process observed in real materials.

To address this limitation, Ma et al. [37] proposed an improved approach by introducing a softening force–displacement relationship into the conventional parallel bond model based on the Brazilian splitting test, as shown in Figure 4. In this formulation, the parallel bond is regarded as a series of elastic elements distributed over the contact area between particles, which can not only transmit normal and shear contact forces but also transfer bending moments through the contact interface. On this basis, a soft-bond model was further developed in PFC^3D to describe the progressive softening behavior of bonded contacts after reaching the peak tensile stress. Unlike the conventional parallel bond model, in which the load-carrying capacity drops instantaneously to zero once the peak strength is exceeded, the soft-bond model is capable of simulating the gradual stress degradation of bonded contacts under tensile loading, thereby providing a more realistic representation of the mechanical behavior of geogrids.

Figure 4. Comparison of the working mechanisms of the parallel bond model and the soft-bond model.

In the soft-bond contact model, the contact forces and contact moments are updated at each calculation step based on the increments of relative displacement and relative rotation. The normal force F_n and the shear force F_s are governed by the normal displacement increment ∆δ_n and the tangential displacement increment ∆δ_s, respectively, and their incremental forms are consistent with the linear elastic contact assumption. Meanwhile, the bending moment M_b and the torsional moment M_t are updated according to the corresponding increments of bending rotation ∆θ_b and torsional rotation ∆θ_t, reflecting the moment response of the bonded contact under rotational actions. The corresponding formulations for the forces and moments are given in Equations (1)–(4).

$$F_n=F_n+k_{n}A\Delta {\delta }_n$$

(1)

$$F_s=F_s-k_{s}A\Delta {\delta }_s$$

(2)

$$M_b=M_b-k_{n}I\Delta {\theta }_b$$

(3)

$$M_t=M_t-k_{s}J\Delta {\theta }_t$$

(4)

Based on the updated contact forces and moments, the internal contact actions are projected onto the bonded cross-section to evaluate the equivalent normal stress and shear stress around the bond, thereby characterizing the stress state of the bonded contact under the coupled effects of tension, bending, and shear. The corresponding expressions are given in Equations (5) and (6), respectively. Here, σ denotes the maximum normal stress, τ denotes the maximum shear stress, A is the cross-sectional area, and J is the polar moment of inertia of the bonded cross-section.

$$\sigma =-{\displaystyle \frac{F_n}{A}}+\beta {\displaystyle \frac{‖M_b‖R}{I}}$$

(5)

$$\tau ={\displaystyle \frac{‖F_s‖}{A}}+\beta {\displaystyle \frac{‖M_t‖R}{J}}$$

(6)

Based on the evaluated equivalent normal and shear stresses, the mechanical state of the bonded contact is further assessed and updated. When the bonded contact is in the elastic tensile stage and the maximum normal stress around the bonded cross-section reaches the tensile strength of the bond, the bond does not undergo instantaneous breakage but instead enters the tensile softening stage. At this stage, the maximum allowable elongation of the bonded contact is jointly determined by the critical elongation and the softening factor, as expressed in Equations (7) and (8). The critical elongation corresponds to the deformation state at which the bonded contact reaches its peak tensile strength, representing the ultimate load-carrying capacity of the bond under tensile loading.

$$l^{*}=l_c(1.0+\xi )$$

(7)

$$l_c={\displaystyle \frac{F_n}{knA}}+\beta {\displaystyle \frac{‖M_b‖R}{k_{n}I}}$$

(8)

During the tensile softening stage, the tensile load-carrying capacity of the bonded contact gradually degrades with increasing equivalent bond length, and the maximum allowable tensile stress is constrained by the softening envelope, as expressed in Equation (9). The equivalent bond length is jointly determined by the axial elongation and the accumulated bending deformation, and its formulation is given in Equation (10). When the calculated tensile stress exceeds the allowable level defined by the softening envelope, the normal internal force is adjusted to project the bond stress back onto the envelope (Equation (11)), thereby ensuring that the stress evolution of the bonded contact follows the prescribed softening rule.

Here, l_c denotes the critical elongation (peak strength), and ξ is the softening factor.

The maximum stress is given by

$${\sigma }^{*}={\displaystyle \frac{{\sigma }_c(l_c-l)}{\xi l_c}}$$

(9)

where the current bond length l is given by

$$l=l_c+{\delta }_l+R\left|\delta {\theta }_b\right|$$

(10)

where δ_l represents the bond length at the onset of softening, and $\left|{\delta }_b\right|$ denotes the accumulated bending after the initiation of softening.

If the current tensile stress is greater than σ*, it is projected back onto the softening envelope as follows:

$$F_n=F_n({\displaystyle \frac{{\sigma }^{*}}{\sigma }})$$

(11)

If the bonded contact transitions from the tensile state to the compressive state, the tensile stress level at the onset of compression and the corresponding equivalent bond length are recorded, with their formulations given in Equations (12) and (13). During subsequent loading, when the stress conditions in the compressive state again satisfy the tensile softening criterion, the bonded contact can re-enter the softening evolution process, and the tensile force and bending moment are adjusted accordingly in accordance with the softening envelope.

$${\sigma }_m=\sigma$$

(12)

$$l_m=l_c+{\delta }_l+R\left|\delta {\theta }_b\right|$$

(13)

where σ_m and l_m represent the tensile stress and the bond length at the onset of compression, respectively.

During the evolution of the bonded contact state, further failure criteria need to be evaluated. If tensile failure does not occur, shear failure is then evaluated. The shear strength is defined as $\overline{{\tau }_c}=\overline{C}-\sigma \tan \overline{\phi }$, where $\sigma =\overline{F}/\overline{A}$ represents the average normal stress acting on the parallel bond cross-section. If the shear stress exceeds the shear strength limit ($\overline{\tau }>\overline{\tau c}$), shear failure of the bond occurs.

If $\overline{\tau }>\overline{\tau c}$, then shear failure occurs.

It should be noted that if shear failure occurs, and is not reset to zero, once the bond is broken, a callback event is triggered, and the behavior reverts to that of the unbonded contact model.

3.3. Application of the Soft-Bond Model in Geogrid Modeling

A numerical tensile test of the geogrid was conducted using the soft-bond model to calibrate the micromechanical parameters and to evaluate the capability of the discrete element representation in reproducing the tensile behavior of the reinforcement. Figure 5 compares the tensile force–strain responses obtained from the discrete element simulation and the laboratory tensile test, while the corresponding micromechanical parameters are summarized in Table 2.

Figure 5. Comparison of tensile force–strain curves between discrete element geogrid and actual geogrid.

Table 2. Micromechanical properties of geogrid.

Particle Contact Modulus/kPa	Stiffness Ratio	Softening Coefficient ζ	Tensile Strength Coefficient γ	Friction Coefficient
3.5 × 107	1 × 105	1 × 103	0.1	0.5

As shown in Figure 5, the numerical and experimental curves exhibit overall consistent tensile response characteristics. In the initial loading stage (strain ≤ 0.02), the tensile force increases nearly linearly with strain, indicating an elastic-dominated response primarily controlled by the particle contact modulus and the normal-to-tangential stiffness ratio in the soft-bond model. The close agreement between the initial slopes of the numerical and experimental curves suggests that the calibrated stiffness parameters reasonably reproduce the initial tensile stiffness of the geogrid.

With further loading, the response enters a nonlinear stage where the slope of the curve gradually decreases and approaches a stable trend, reflecting progressive bond damage and stiffness degradation governed mainly by the softening coefficient. The simulated curve remains in good agreement with the experimental results in terms of peak level and overall trend, indicating that the model can reasonably capture the nonlinear deformation and load-carrying evolution of the geogrid under tensile loading.

3.4. Development and Validation of the Overall Model

The numerical model was established by coupling DEM (discrete element method) with FDM (finite difference method), namely PFC3D and FLAC3D (Version 5.0), as shown in Figure 6. In the model, spherical particles in DEM were used to simulate the crushed stone, the geogrid, and the soft plastic clay within a range of 100 mm around the pile, while the remaining soft plastic clay was simulated using FDM. Regarding the contact models between particles, a linear contact model was adopted for the crushed stone particles, the soft-bond model was used for the geogrid particles, and the soft-bond model was also employed for the clay particles. The outer clay modeled in FDM was described by the Mohr–Coulomb constitutive model. After calibration, the Young’s modulus, Poisson’s ratio, cohesion, and friction angle were determined as 7.6 × 10⁵ kPa, 0.4, 11 kPa, and 20°, respectively. All particle simulation parameters were calibrated and are listed in Table 3.

Figure 6. Numerical coupling model of DEM-FDM.

Table 3. Micromechanical properties of crushed stone and clay.

Material	Particle Contact Modulus/kPa	Stiffness Ratio	Normal Strength of Contact Bond/N·m−1	Shear Strength of Contact Bond/N·m−1	Friction Coefficient
crushed stone	9 × 107	6.0			0.7
clay	6 × 105	1.5	3 × 102	3 × 102	0.2

To ensure numerical reproducibility and consistency, a fixed random seed was adopted in all discrete element simulations. The particle assembly was generated to achieve a controlled initial packing state and then stabilized through equilibrium calculations prior to loading. Construction-induced dynamic processes, such as vibration compaction, were not explicitly modeled; therefore, the micromechanical responses, including force chain characteristics, are interpreted within this quasi-static modeling assumption.

The loading procedure was the same as that used in the laboratory tests. At each loading stage, the geosynthetic-encased stone column was considered to reach equilibrium when the unbalanced force ratio decreased to 1 × 10⁻⁴, after which the next load level was applied. It should be noted that some micromechanical indicators discussed in this study, such as porosity and particle-scale radial deformation, are extracted from numerical simulations rather than directly measured in the laboratory tests. These indicators are introduced to characterize internal structural evolution and load transfer mechanisms that are difficult to access experimentally and are therefore interpreted as complementary numerical insights rather than direct experimental validation quantities.

Figure 7 presents a comparison of the pile-top stress–settlement curves obtained from the laboratory model test and the coupled numerical simulation. The simulated response exhibits good agreement with the experimental results, with a maximum settlement deviation of less than 10 mm over the entire loading range. Both curves indicate failure at a pile-top load of approximately 702.47 kPa. These results demonstrate that the established coupled model is capable of reasonably reproducing the macroscopic mechanical response of the geosynthetic-encased stone column observed in the laboratory tests.

Figure 7. Comparison of stress–settlement curves between the experiment and the simulation.

4. Effect of Particle Size on the Bearing Performance of GESC

4.1. Analysis of Pile-Top Settlement of GESC

Most existing studies on the size relationship between granular materials and geogrids have adopted the median particle size ${\mathrm{d}}_{50}$ and the effective aperture $L_a$ as the key governing parameters. Accordingly, these two parameters were selected in this study as the primary variables to investigate the coupled effects of crushed stone particle size and geogrid aperture on the bearing performance of geosynthetic-encased stone columns. A total of 27 test cases were designed, including nine particle size groups of crushed stone and three effective apertures (20 mm, 30 mm, and 40 mm). The particle size distributions of the nine crushed stone groups are summarized in Table 4. For completeness, the corresponding gradation descriptors, including the uniformity coefficient $C_u$ and curvature coefficient $C_c$, are also reported in Table 4 to characterize the overall grading features of each particle group.

Table 4. The size data of crushed stone.

Working Condition	Gradation Parameters
	d50/mm	d30/mm	d10/mm	d60/mm	Cu	Cc
Gradation 1	15.26	13.52	11.48	16.43	1.215	1.032
Gradation 2	19.95	17.17	13.47	22.79	1.327	1.04
Gradation 3	24.49	21.85	18.49	25.77	1.179	0.998
Gradation 4	31.16	27.6	23.11	33.46	1.212	1.015
Gradation 5	40.84	35.48	27.33	53.12	1.497	1.153
Gradation 6	42.9	36.17	28.18	47.39	1.31	1.021
Gradation 7	54.43	43.68	33.18	61.95	1.418	1.077
Gradation 8	62.87	52.69	39.71	66.99	1.272	0.958
Gradation 9	70.65	52.84	39.32	83.95	1.589	1.182

The relationship between the final pile-top settlement and particle size under different geogrid apertures is illustrated in Figure 8. It can be observed that, for a constant mean particle size, the final settlement increases with increasing geogrid aperture. Conversely, for a fixed geogrid aperture, the final settlement decreases as the particle size increases; however, when d₅₀ exceeds 40 mm, the influence of further increases in mean particle size on the final pile-top settlement becomes insignificant.

Figure 8. Curve of pile-top settlement with d₅₀ under different pore sizes.

Further observation indicates that when d₅₀ is greater than 30 mm, the settlement difference caused by increasing the aperture from 20 mm to 30 mm is much smaller than that observed when d₅₀ is less than 30 mm. Similarly, when d₅₀ exceeds 40 mm, the settlement difference induced by increasing the aperture from 20 mm to 40 mm is considerably smaller than that for d₅₀ values below 40 mm. This suggests that when d₅₀ > La, the effect of aperture variation on the bearing performance of GESC is much weaker than that when d₅₀ < La. Therefore, in practical engineering applications, selecting a geogrid aperture slightly smaller than the mean particle size of the crushed stone can achieve a favorable balance between bearing performance and economic efficiency.

4.2. Analysis of Radial Strain of GESC

Failure of GESC is characterized by the occurrence of large radial strain at a certain location along the column. Under the same loading level, the magnitude of radial strain reflects the bearing performance of GESC to some extent. Figure 9 presents the maximum radial strain of the column under a load of 23 kN for different geogrid apertures and particle sizes. The results are extracted from a validated numerical framework and are used to support comparative analysis of parameter effects under consistent modeling conditions.

Figure 9. Maximum radial strain of GESC pile under different working conditions.

For convenience of interpretation, one representative value was selected in each of the four particle size intervals: d₅₀ < 20 mm, 20 mm < d₅₀ < 30 mm, 30 mm < d₅₀ < 40 mm, and d₅₀ > 40 mm—namely d₅₀ = 14.26 mm, 24.50 mm, 31.16 mm, and 42.90 mm, respectively. These representative points were kept unchanged in the subsequent analysis to facilitate consistent comparison of relative trends.

It can be observed that the maximum radial strain of the column increases with increasing particle size and decreases with increasing geogrid aperture, indicating that the bearing performance is positively correlated with particle size and negatively correlated with aperture size. The location of the maximum radial strain has also attracted considerable attention. Generally, for conventional stone columns, the maximum radial strain occurs near the pile-top; for fully encased stone columns, it typically appears within a range of approximately 2D below the pile-top (where D is the pile diameter); whereas for partially encased stone columns, it tends to occur at the interface between the encased and unencased zones.

As shown in Figure 10, the distance between the location of maximum radial strain and the pile-top increases with increasing particle size and decreases with increasing geogrid aperture. Moreover, when d₅₀ < La, the location of the maximum radial strain is very close to the pile-top, resembling the behavior of unencased stone columns, which indicates a relatively limited reinforcement effect under this condition.

Figure 10. Distance between maximum radial strain position of GESC pile and pile-top under different working conditions.

4.3. Effect of Particle Size and Aperture on Local Porosity

Porosity is commonly used to characterize the compactness of granular materials; however, its evolution is difficult to monitor directly in laboratory tests. In contrast, the porosity n can be obtained from measurement spheres in the PFC model and is defined as the ratio of the total void volume within a measurement sphere to the volume of the measurement sphere itself. Figure 11 illustrates the distribution of porosity within the column under a pile-top load of 23 kN for different particle sizes.

Figure 11. The line charts illustrate the variations in column porosity under different working conditions for three geogrid apertures with four representative gravel mean particle sizes. Subfigures (a–d) correspond to the porosity variations along the column for d₅₀ = 14.26 mm, 24.50 mm, 31.16 mm, and 42.90 mm, respectively.

It can be observed that the porosity captured by the measurement sphere at the pile-top approaches 1.0 due to downward extrusion of particles under the loading plate, resulting in an almost particle-free zone. For a constant d₅₀, increasing the geogrid aperture leads to a pronounced increase in porosity in the upper part of the column, with a maximum increment of up to 43.3%, whereas the porosity variation in the lower part remains negligible. This spatially non-uniform response indicates that aperture enlargement mainly weakens the confinement efficiency of the upper encasement zone, where radial deformation is most active.

With increasing d₅₀, the overall porosity of the GESC decreases consistently along the column. Taking La = 20 mm as an example, when d₅₀ increases from 14.26 mm to 42.90 mm, the porosity decreases from 0.900 to 0.500 at a depth of 700 mm (reduction of 44.4%), from 0.577 to 0.324 at 400 mm (reduction of 43.8%), and from 0.406 to 0.290 at 100 mm (reduction of 28.6%), corresponding to an average reduction of approximately 39.0%. These depth-dependent reductions indicate that particle size exerts a stronger densification effect in the middle and lower parts of the column, while the upper zone remains more sensitive to aperture-controlled confinement.

From a micromechanical perspective, larger particles generate fewer interparticle voids and promote stronger particle interlocking, which suppresses particle rearrangement and local dilation during compression. Similar porosity–packing density relationships and densification mechanisms have been reported in DEM simulations of geogrid-reinforced granular systems [38,39]. The enhanced interlocking stabilizes the granular skeleton and limits porosity growth under loading. At the macroscopic level, this progressive densification contributes directly to increased stiffness and reduced pile-top settlement, which is consistent with the settlement trends observed in Figure 8 and with previous DEM–FDM coupled studies on geosynthetic-encased stone columns [12].

4.4. Effect of Particle Size and Aperture on Local Coordination Number

The stress within crushed stone is transmitted through contacts between particles. As a micromechanical indicator, the coordination number can be used to evaluate the interaction and stability among particles, and it is defined as the average number of contacts per particle [40]. As shown in Figure 12, the coordination number increases monotonically with increasing particle size and decreases with increasing geogrid aperture, indicating a systematic evolution of particle contact density. For larger particle sizes, each particle participates in a greater number of load-transmitting contacts, forming a denser and more stable contact network, which enhances the continuity of force transmission paths and reduces the likelihood of localized particle instability.

Figure 12. Coordination number distribution along the column for different working conditions. Subfigures (a–d) show the variation in coordination number along the pile direction for d₅₀ = 14.26 mm, 24.50 mm, 31.16 mm, and 42.90 mm, respectively.

In contrast, increasing the geogrid aperture weakens lateral confinement and allows greater particle rearrangement and rotation, thereby reducing the coordination number. This reduction in contact density corresponds to a looser internal structure and explains the larger radial strain observed in Figure 9 as well as the higher porosity shown in Figure 11. Similar correlations between coordination number, packing density, and deformation behavior have been reported in DEM analyses of granular materials and reinforced aggregates [41,42]. These results confirm that coordination number is an effective micromechanical indicator linking internal packing structure to macroscopic deformation behavior.

4.5. Effect of Particle Size and Aperture on the Morphology of Contact Force Chains

The influence of the relationship between geogrid aperture and particle size on the bearing capacity of the GESC is clearly reflected in the force chain diagrams. Figure 13 shows the force chain distributions of the GESC under different working conditions at a pile-top load of 23 kN, where the thickness of the lines represents the magnitude of the contact forces. In Figure 13a,c, the geogrid and the crushed stone do not act effectively as an integrated system, and separation occurs at the final stage of loading. This is because when d₅₀ < La, the geogrid cannot interlock with the crushed stone and provides only weak frictional resistance during the downward movement of the stone particles.

Figure 13. Force chain distributions for cases with gravel particle size smaller than and larger than the geogrid aperture: (a) d₅₀ = 14.26 mm, La = 20 mm; (b) d₅₀ = 24.50 mm, La = 20 mm; (c) d₅₀ = 24.50 mm, La = 30 mm; (d) d₅₀ = 31.16 mm, La = 30 mm.

When d₅₀ > La, part of the crushed stone becomes interlocked within the apertures, which significantly enhances the confinement effect of the geogrid on the crushed stone. As a result, the geogrid and the crushed stone jointly bear the applied load, leading to a substantial improvement in the bearing performance of GESC.

Although qualitative variations in force chain morphology can be observed in Figure 13, a quantitative analysis is required to more clearly identify the evolution of contact force orientation. The angle $\theta$ between the contact force vector and the central axis of the column ranges from 0 to $\pi /2$ and is divided into two intervals: 0–$\pi /4$ and $\pi /4$–$\pi /2$. The interval 0–$\pi /4$ is defined as the axial-force-dominated zone, while $\pi /4$–$\pi /2$ corresponds to the radial-force-dominated zone.

Figure 14 presents stacked histograms of contact force orientation distributions within the column under different particle size and aperture conditions. It is observed that the proportion of contact forces within the axial-force-dominated zone (0–$\pi /4$) decreases with increasing geogrid aperture, whereas it increases with increasing gravel particle size. This trend indicates that a larger aperture promotes a redistribution of contact forces toward the radial direction, weakening axial force transmission along the column. In contrast, increasing particle size enhances the continuity of axial force chains, facilitating more efficient vertical load transfer from the pile head to the underlying soil. It should be noted that the geogrid stiffness-related parameters (Table 1) were calibrated at the material level and kept identical for all simulated cases; therefore, the changes in contact orientation shown in Figure 14 can be interpreted as the response of the granular contact network to particle size–aperture conditions under a consistent reinforcement stiffness baseline, rather than being influenced by varying reinforcement stiffness.

Figure 14. The bar charts present the regional distributions of contact forces within the geosynthetic-encased stone column under different working conditions. Subfigures (a–d) correspond to contact force distributions for three geogrid apertures at gravel mean particle sizes of 14.26 mm, 24.50 mm, 31.16 mm, and 42.90 mm, respectively.

It should be emphasized that the micromechanical indicators reported in this study, including porosity, coordination number, and contact force chain orientation, are extracted primarily from the post-loading (end-of-stage) states of the geosynthetic-encased stone column. These variables are therefore interpreted as resultant descriptors of the internal granular structure after loading-induced rearrangement and damage evolution, rather than as independent governing causes of bearing capacity. In particular, the purpose of introducing these indicators is to provide microstructural evidence for identifying potentially vulnerable zones along the column and for describing how the granular fabric and load transfer skeleton evolve under different particle size–aperture combinations. Since porosity reduction, coordination enhancement, and force chain alignment are intrinsically coupled outcomes of the same kinematic and contact processes, this manuscript does not attribute performance changes to any single indicator alone. Instead, the micromechanical results are interpreted through cross-comparison of multiple indicators in a consistent manner, so that the mechanistic conclusions are supported by correlated trends at both macroscopic and microscopic levels.

5. Discussion

In engineering applications of geosynthetic-encased stone columns, the selection of gravel infill and geogrid parameters is often dictated by empirical experience or material availability, whereas systematic investigations into the matching relationship between gravel particle size and geogrid aperture remain limited. Previous studies have predominantly emphasized reinforcement tensile strength, column geometry, or encasement length, and the gravel–geogrid interaction has commonly been interpreted mainly through interface friction. However, gravel is a discrete granular material; hence, geometric compatibility between particle size and aperture is expected to influence particle mobility, force transmission, and confinement efficiency. From this perspective, interface friction alone may not fully account for the diverse mechanical responses observed under different parameter combinations.

Evidence from discrete element research further supports the necessity of considering geometric characteristics beyond frictional descriptions. Gezgin [43] demonstrated that particle sphericity and roundness significantly affect shear response and contact network evolution, while Fan et al. [44] reported that particle angularity and surface roughness govern force transmission pathways and shear resistance. Chen et al. [45] additionally indicated that non-spherical particle geometry affects contact anisotropy and particle breakage mechanisms. Collectively, these findings provide an independent micromechanical basis for highlighting particle–aperture geometric compatibility as a governing factor in reinforced granular systems.

Motivated by this background, the present study performed a comparative investigation of GESC with different combinations of gravel particle size and geogrid aperture using laboratory model tests and coupled DEM–FDM numerical simulations. At the macro-scale, bearing capacity and deformation characteristics were examined; at the micro-scale, indicators such as porosity, coordination number, and contact force chains were introduced to quantify granular structural evolution. This multiscale framework enables the gravel–geogrid interaction to be interpreted not only as frictional sliding at an interface, but also as a coupled “structure–force transmission–confinement” process governed by granular rearrangement and the development of load-bearing skeletons.

From an engineering standpoint, the results underscore the practical value of a rational match between gravel particle size and geogrid aperture. Compared with approaches that simply increase reinforcement strength or indiscriminately enlarge particle size, emphasizing particle–aperture coordination may be more effective in promoting the formation of stable load-bearing structures and continuous force chain networks, thereby improving confinement efficiency and reinforcement performance. These observations are intended to provide qualitative guidance for parameter selection in GESC and may offer useful reference for other reinforced granular systems.

In practical design, the proposed particle size–aperture matching guideline should be evaluated together with regional material availability and project-specific cost constraints. Since unit prices of gravel and geogrids can vary with sourcing structure, transportation distance, and local specification requirements, a transferable approach is to adopt a screening calculation rather than a fixed cost table. Specifically, the unit-area material cost can be approximated as the sum of (1) the gravel cost converted from local volumetric quotations under a specified placement thickness and (2) the delivered unit price of the geogrid on an area basis. Under this framework, feasible scenarios (e.g., a conventional gradation versus a gradation selected to satisfy a particle–aperture matching criterion) can be compared to balance performance expectations against local cost sensitivity and supply constraints.

Several limitations should be acknowledged. For computational efficiency and numerical stability, spherical particles were adopted in the numerical simulations. Existing DEM studies indicate that particle shape can influence the absolute magnitudes of certain micromechanical indicators; therefore, the present conclusions are primarily interpreted in terms of comparative trends and mechanistic relationships associated with particle size–aperture matching, rather than as universal quantitative thresholds. Further verification using more realistic particle morphologies and, where appropriate, particle breakage representations would be valuable for refining the identified micromechanical signatures and their linkage to macroscopic performance.

To enhance practical transferability, future studies would benefit from validation against field benchmarks and full-scale tests with representative engineering dimensions, such as stone columns with diameters in the order of 0.8 m and lengths of approximately 8 m. Such tests could provide reference data for settlement, lateral deformation, and load transfer characteristics, and help assess scale effects, construction-induced variability, and material heterogeneity, which are difficult to fully capture in laboratory and numerical models. In parallel, recent studies across geotechnical and material systems have highlighted the growing importance of integrating physical understanding with advanced modeling tools and hybrid predictive frameworks, including data-driven or physics-guided learning strategies, to improve robustness, generalization, and interpretability for engineering applications [46,47,48,49]. In this context, further extensions may consider cyclic loading, long-term service behavior, multi-column interaction effects, and more advanced constitutive descriptions for reinforcement materials, thereby progressively narrowing the gap between numerical investigation and real engineering performance.

6. Conclusions

This section summarizes the main findings of the present study on the influence of gravel particle size and geogrid aperture on the performance of geosynthetic-encased stone columns based on a coupled discrete–continuous numerical framework validated against laboratory experiments. The conclusions are organized to reflect both macroscopic structural responses and micromechanical mechanisms revealed by the simulations. Specifically, the effects of particle size–aperture compatibility on settlement behavior, load transfer efficiency, and confinement performance are first highlighted, followed by insights into the associated evolution of porosity, coordination number, and contact force transmission characteristics. The conclusions also emphasize the implications of these findings for rational selection of granular materials and geogrid geometry in engineering practice, while acknowledging the modeling assumptions and scale limitations of the present study. The key conclusions are presented as follows.

(1)

Under the same loading level, a larger geogrid aperture results in a greater maximum radial strain of the column and a shorter distance between the location of maximum radial strain and the pile-top. In contrast, an increase in particle size leads to a reduction in the maximum radial strain and shifts its location farther away from the pile-top.

(2)

An increase in geogrid aperture weakens the lateral confinement provided by the geogrid to the crushed stone, thereby reducing the bearing performance of the geosynthetic-encased stone column. Conversely, increasing particle size promotes the formation of a denser and more stable internal structure of the crushed stone under loading, which enhances the bearing performance of the column. This beneficial effect reaches its maximum at d₅₀ = 40 mm, beyond which further increases in particle size have a negligible influence on bearing performance.

(3)

When d₅₀ < La, separation occurs between the crushed stone and the geogrid as the load increases, and the maximum radial strain develops near the pile-top, indicating that the reinforcement effect is poor under this condition.

(4)

When d₅₀ > La, the influence of geogrid aperture on the bearing performance of GESC is much weaker than that observed for d₅₀ < La. To achieve a balance between bearing performance and economic efficiency in practical engineering applications, crushed stone with a particle size slightly larger than the geogrid aperture is recommended.