Performance analysis of geotextile-encased stone columns using a simplified analytical approach

Abstract

Geosynthetic-encased stone columns (GESCs) effectively improve soft soils where traditional stone columns lack sufficient lateral confinement. The complex interaction among the column, encasement, and surrounding soil necessitates an analytical model for efficient prediction and design. This study presents a simplified semi-analytical iterative solution to evaluate ground reinforced with GESCs. The system is modeled as a unit cell with regularly arranged end-bearing stone columns in soft soil. The stone column is idealized as a rigid-plastic material, yielding at active stress and deforming plastically without volume change, while the geosynthetic encasement is represented as linear-elastic, and the surrounding soil is modeled using semi-empirical assumptions in which horizontal stresses are linearly proportional to vertical stresses. The soil profile is divided into horizontal slices to account for depth-dependent behavior, applicable to both homogeneous and layered soils, and to un-encased, partially encased, or fully encased columns. Validation with field tests and finite element analysis shows that the proposed solution provides comparable results for settlements, stress distribution in both soil and stone columns, and encasement radial expansion. The model is effective across a wide range of area replacement ratios (i.e., 0 to 0.35) and encasement stiffnesses (0 to 5000 kN/m), with optimal stiffness effects calculated between 2000 and 3000 kN/m. Parametric analyses reveal that settlement improvement and stress concentration are most sensitive to the area replacement ratio, encasement stiffness, and soil stiffness, while increasing column diameter beyond an optimal value reduces effectiveness. An optimum partial encasement length ratio of 0.45 was identified, beyond which the Settlement improvement factor (SIF) increases only marginally. The proposed model also provides an analytical design chart for evaluating the settlement of GESC-reinforced soft soils, along with a Python-based code for calculation.

Supplementary Information

The online version contains supplementary material available at 10.1038/s41598-025-34737-2.

Introduction

Structures on soft soils often face excessive settlement, lateral spreading, and stability problems. Stone columns, arranged in square, triangular, or hexagonal patterns, are commonly used in cohesive soils with undrained shear strength above 15 kPa¹. However, their effectiveness relies on lateral confinement, which is limited in very soft clays. Field tests by Rahman et al. on ordinary stone columns (OSCs) showed low stress concentration ratios (2.1–2.5) for area replacement ratios of 0.14 and 0.28, indicating poor load transfer². McKenna et al. reported high pore pressures and settlement beneath an embankment, attributed to construction disturbance and insufficient confinement³. To overcome such limitations, additional confinement is often needed. Encasement of stone columns with geosynthetics provides lateral support, improving performance. Several studies have been conducted to understand the behavior of GESCs using numerical analysis^4,5 experimental investigations^6,7, and field tests^5,8–11, numerically studied steel slag as a sustainable column infill in soft clay, comparing ordinary and encased columns with respect to settlement, excess pore pressure, stress concentration, and lateral deformation, and found that encased columns achieved higher stress concentration ratios, indicating greater structural performance. Benefits of encasement include increased stiffness¹⁰, reduced lateral deformation¹², prevention of stone loss, improved drainage¹³, and enhanced liquefaction resistance¹⁴. Field, experimental and numerical studies validate these advantages^4,11,15. Yoo and Lee showed reduced bulging and recommended encasement lengths at least four times the column diameter¹². Ouyang et al.¹⁰ reported up to 16% settlement reduction with partial encasement. Studies by Wang et al.¹³ and Chen et al.¹⁶ observed faster pore pressure dissipation, promoting long-term stability under embankments supported by geosynthetic-encased columns. Li et al.¹⁷ conducted centrifuge tests and found that, under undrained conditions, encasing stone columns with high and low-stiffness geogrids reduced column settlement by 50% and 34%, respectively, compared to ordinary stone columns. Zhao et al.¹⁸ investigated bio-inspired geocells (pocket diameter 187.6 mm) and found that all types increased soil bearing capacity by 1.2 to 1.7 times, reaching maximum pressures of 350 to 400 kPa compared to unreinforced beds.

Despite their advantages, the performance of GESCs depends strongly on key design parameters, requiring careful analytical evaluation. Several models have been developed to assess GESCs behavior, including those by Raithel and Kempfert¹⁹, Zhang and Zhao²⁰, Castro and Sagaseta²¹, Pulko et al.²², Castro et al.²³, and recently Yu et al.²⁴.

Castro and Sagaseta²¹ proposed an axisymmetric unit-cell model incorporating elastic soil, an elasto-plastic column, and an elastic geosynthetic encasement, applying the Mohr-Coulomb (MC) yield criterion under undrained loading and consolidation. Their predictions agreed well with experimental settlement reduction ratios (β) for various encasement stiffnesses. Pulko et al.²² developed an analytical model including initial stresses, treating the soil as elastic, the encasement as linear-elastic, and the column as elasto-plastic with constant dilatancy, and provided design charts for column spacing and encasement stiffness to improve settlement performance. Both studies assumed elastic soft soil across the stress range, MC behavior for the column with constant dilation angle, and no shear transfer at the soil-column interface. While Castro and Sagaseta considered consolidation, these models mainly addressed settlement prediction, omitting stress concentration, lateral GESC deformation, and broad field validation. Raithel and Kempfert¹⁹ modeled the column as rigid-plastic and the surrounding soil and encasement as linear-elastic, but their approach was limited to homogeneous soil profiles and did not account for depth-dependent behavior. Ignoring depth-dependent behavior oversimplifies soil conditions, leading to inaccurate stress distribution, unreliable settlement prediction, misrepresentation of bulging and encasement tension, and limited applicability to layered soils.

This paper extends the analytical approaches of Balaam and Booker²⁵ and Raithel and Kempfert¹⁹ by introducing an iterative solution that incorporates the depth-dependent behavior of GESC-improved ground. The developed model accounts for loading from footings and widespread loads, differential settlement beneath embankments, depth-dependent settlement behavior, stress concentration ratios, and lateral deformation patterns of geosynthetic-encased stone columns (GESCs), using a simplified axisymmetric unit cell framework. It applies to both homogeneous and layered soil conditions and accommodates un-encased, partially encased, and fully encased stone columns. The analysis focuses on drained conditions, which typically result in the largest settlements and highest geotextile tensions. The predictions have been validated against various field test data and numerical simulations, showing reasonable agreement with the observed behavior of GESC-improved ground.

Assumptions of the analytical model

Stone columns are usually arranged in regular grid patterns, commonly in triangular, square, or hexagonal layouts. In this study, a 2D axisymmetric model based on the unit cell concept is used, with boundary conditions shown in Fig. 1. The radius of the unit cell varies depending on the layout pattern and is taken as 0.525 C, 0.565 C, and 0.645 C for triangular, square, and hexagonal arrangements, respectively, where C is the center-to-center spacing between columns. In this approach, the stone column is idealized as a rigid-plastic material with a very high modulus of elasticity. It behaves elastically at low stress levels and yields once the active stress state is reached, after which it undergoes plastic deformation at constant volume, implying zero dilatancy. The geosynthetic encasement is modeled as a linear-elastic membrane, where the tensile response is directly proportional to the radial strain, thereby providing confining pressure to the column. The surrounding soil is represented through semi-empirical assumptions: the horizontal stresses are taken to be linearly proportional to the vertical stresses, expressed through lateral earth pressure coefficients that account for installation effects and soil-structure interaction. To simplify the problem and enable an analytical solution, the following assumptions are made:

Fig. 1.

Analytical calculation model of a geotextile-encased sand column: (a) Basic features of the model; (b) Equilibrium and compatibility conditions of the encasement along Section A–A.

1. The geotextile encasement, assumed to behave as a linear elastic material, provides radial confinement by balancing the pressure difference between the active expansion of the column material and the opposing resistance of the soil pressure;

2. The geosynthetic encasement is perfectly bonded to the column material. Therefore, after deformation, the outer radius of the expanded column must equal the inner radius of the expanded encasement.

3. Only drained conditions are considered, as they lead to the largest settlements and geotextile tension. Undrained and consolidation behaviors of GESCs are not analyzed in this study;

4. Lateral support for the column is mainly provided by the lateral earth pressure of the surrounding soil and the lateral confinement by the encasement;

5. The active earth pressure coefficient is valid within the stone column¹⁹.

6. The at-rest earth pressure coefficient can be used for the calculation of the lateral pressure from the surrounding soil. However, a higher value is adopted to account for the effects of column installation;

7. Settlement at the base of the column is assumed negligible for GESC with a length equal to or greater than the critical length. For floating columns shorter than the critical length, potential punching failure may occur, and this current analytical model is not applicable;

8. Under rigid footings, equal settlement is assumed for both the column and the surrounding soil. For widespread loadings, a simplified method based on the area replacement ratio and composite stiffness is used to estimate differential settlement between the column and the surrounding soil;

9. The volume of the column material is assumed to remain constant, whereas the volume of the surrounding soil changes due to its high compressibility as the column compresses it;

10. The analytical model accounts for depth by dividing the column into vertical slices. Depth-dependent parameters include stresses, oedometric modulus, lateral expansion, and geotextile tension. While designed for homogeneous soils, it can be adapted to layered soils by assigning each slice properties matching the average soil conditions at that depth.

Limitations of the analytical model

The analytical model is based on several simplifying assumptions, which impose the following limitations on its applicability:

1. Shear stresses at the interfaces between the column, geotextile, and surrounding soil are neglected. The contribution of shear stresses at the column-geotextile-soil interfaces is generally secondary compared with the dominant radial confinement effect provided by the encasement. Neglecting this mechanism allows the governing equations to remain simple while still capturing the essential mechanics of encased column behavior. Similar assumptions have been widely adopted in other analytical formulations^19,21.

2. Consolidation effects are ignored, considering only drained conditions; therefore, pore water pressure dissipation during loading is not considered. The assumption of drained loading is appropriate for long-term settlement analysis and ultimate load-carrying capacity, which are the primary design concerns in practice. While it leads to overestimation of settlements in the early stages of loading (when consolidation is incomplete), the model is not intended for predicting time-dependent consolidation responses. Instead, it provides an estimate of the final settlement and stress distribution under steady-state conditions.

3. The nonlinear elastoplastic behavior of the soil is not considered;

4. Dilatancy of the granular column during shearing is neglected.

5. The analytical model assumes the column material to be incompressible, which simplifies the computation of radial expansion and settlement. This assumption may slightly overestimate load transfer to the column and underestimate column compression compared with fully compressible Numerical models.

Range of validity

The proposed analytical model is developed under specific assumptions and is applicable within the following ranges: It is suitable for soft cohesive soils with an undrained shear strength greater than 10 to 15 kPa, under drained, static loading conditions, and is not intended for undrained, rapid, cyclic, or dynamic loading. The model applies to end-bearing columns or floating columns with lengths equal to or exceeding the critical length, but is not applicable to short columns susceptible to punching failure. The validation covers area replacement ratios from 0.05 to 0.35 and geosynthetic encasement stiffness values up to 5000 kN/m. However, the observed trends in settlement improvement factor (SIF) and stress concentration ratio (SCR) were found to continue even when area ratio exceeded 0.35. It should be noted, however, that for dense column configurations (area replacement ratios greater than 0.3–0.35), group effects due to overlapping stress bulbs become significant. Since the current model does not explicitly account for these interactions, its predictions in this range should be applied with caution.

Type and length of SC

Columns are assumed to rest on a rigid base (end-bearing) or have lengths equal to or exceeding the critical length for floating columns. The critical length ensures full load transfer and sufficient lateral confinement, based on the pressure bulb depth and plastic deformation extent. This analytical model does not consider punching failure. Settlement at the column base is assumed negligible, as it remains minimal for end-bearing columns and floating columns meeting or exceeding the critical length²⁶.

Simplified axisymmetric model

Analytical modeling of GESC groups under a rigid footing in full three-dimensional analysis is complex; therefore, this study adopts a simplified 2D axisymmetric model. According to Castro²⁶, if the area replacement ratio and encasement stiffness-to-diameter ratio remain the same, the difference in load and settlement between 3D and 2D models is minimal. Therefore, the group of columns can be represented by a single central column with the same area replacement ratio and encasement stiffness-to-diameter ratio. The square footing can also be replaced with a circular one of equal area.

Lateral earth pressure coefficient

In GESCs, the coefficient of active earth pressure () governs the lateral behavior of the granular column as it expands laterally under vertical loading, mobilizing active earth pressure conditions. This coefficient reflects the mobilization of active earth pressure conditions and is defined as: , where, is the effective frictional angle of the granular column material. When GESC’s are installed using the Vibro-replacement method, the surrounding soil remains in an at-rest state, defined by (, where is the effective friction angle of the surrounding soil), since no lateral displacement occurs initially. The GESC installed using the Vibro-displacement method induces lateral compaction, leading to an enlarged lateral coefficient in the surrounding soil ​, though stress relaxation may occur over time. The installation effect of the granular column may increase the lateral earth pressure coefficient in the surrounding soil. According to Mitchell and Huber²⁷, the installation effects of stone columns can be considered by increasing the at-rest lateral earth pressure coefficient () to a higher post-installation value (). Elshazly et al.²⁸ examined the impact of stone column behavior on the post-installation values, finding a range between 0.7 and 2.0. Kirsch simulated the effects of stone column installation by applying cylindrical expansion and increasing stiffness around the footing²⁹. The FEM analysis showed good alignment with field data and numerical methods, with an average post-installation earth pressure coefficient of 1.30. Rahman et al.¹¹ found that a post-installation earth pressure coefficient, = 1.25, provided results that most closely matched the field data. The current study uses a modified lateral earth pressure coefficient for the surrounding soil, denoted as, , to account for the effects of installation, as follows:

1

This adjustment is intended to represent the installation effects for both vibro-replacement and vibro-displacement techniques. A single ​ value was adopted as a simplifying assumption to keep the model formulation general and broadly applicable. If installation-specific values were considered, the model would predict greater radial expansion and higher stress concentration ratios (SCR) for vibro-displacement, and comparatively lower values for vibro-replacement.

Stress-dependent stiffness

In the present analytical model, the stress-dependent stiffness formulation is adopted to realistically capture the variation of soil stiffness with depth and effective stress, following the formula:

2

Where, is the oedometric modulus at stress; is the existing effective stress at depth (including an additional cohesion term , as suggested by Schad (1979)); is the reference oedometric modulus of soil at reference stress ; is the stress-dependency exponent (Vermeer and Neher, 1999 suggested that the stress-dependency exponent typically ranges from 0.5 to 1.0 for normally consolidated (NC) clays and 0.3 to 0.5 for over-consolidated (OC) clays, indicating how rapidly the soil stiffness increases with applied effective stress). Incorporating this relationship improves the accuracy of deformation predictions under varying loads. The approach is supported by the PLAXIS Hardening Soil model³⁰. The existing effective stress can be calculated using the following equation:

3

If the oedometer modulus is not available, the following empirical equation can be used as an alternative:

4

Where, is the initial void ratio of the soil; is the compression index of the soil.

Composite stiffness modulus

The composite stiffness modulus () characterizes the vertical stiffness of a granular column-soil system under axisymmetric loading. It accounts for strain compatibility, stress equilibrium, radial confinement, and incorporates both the area replacement ratio () and Poisson’s ratio () effects, as formulated by Raithel & Kempfert¹⁹.

5

Derivation of this equation is given the supplementary file: APPENDIX-A section.

Analytical solution

This section outlines the analytical approach used to evaluate vertical and lateral stresses, radial expansion, and settlements in Geotextile-Encased Stone Columns (GESCs) under vertical loading. The column-soil system is analyzed slice by slice along the depth, assuming axisymmetric conditions.

Vertical stress distribution

The distribution of vertical stress between the column () and the surrounding soft soil , in equilibrium with the applied stress (), depends on the area replacement ratio () and can be expressed as follows:

6

Where, , with and representing the radius of the column and the unit cell, respectively.

Lateral stress distribution

For a given vertical applied stress (), the resulting lateral stress in the column (), soil (), and encasement () can be calculated as:

7

8

9

where indicates the bottom depth of each slice (from the ground surface), where indicates the slice number from to ; is the tension force in the encasement due to the applied stress at each slice; is the outer radius of the encasement within the unit cell; is the radial expansion increment of the geotextile at a given depth slice due to ; is the encasement stiffness; and the subscript “”, “”, “”, and “” refers to the column, soil, geotextile encasement, and unit cell, respectively.

The subsoil is divided into equal-thickness layers (i.e., ) below the ground surface, as illustrated in Fig. 2. Here, represents the bottom depth of the first layer, and denotes the bottom depth of the second layer, where .

Fig. 2.

Analytical model for calculating GESC behavior using equal-thickness depth slices.

The net lateral stress mobilized from the vertical stress in the column, transferred through the geotextile to the surrounding soil, can be calculated as:

10

Radial expansion of column and encasement

The radial expansion of the column’s outer surface is directly transferred to the encasement, with the encasement behaving as a thin membrane that develops tension through hoop strain. The initial gap between the column and the encasement is accounted for in the deformation relationship. The radial expansion of encasement at any depth () can be computed from the relation of the radial expansion of the column materials (), and the thickness of the encasement (), as follows:

11

Where and are the initial outer radius of the encasement and the column, respectively. Equation (11) shows that the encasement expands only in response to column bulging, with its tensile stiffness mobilizing hoop stresses that confine the column, thereby limiting bulging and enhancing load capacity.

GESC-improved composite ground exhibits a stress concentration ratio higher than OSC. As the column carries a greater portion of the applied vertical stress compared to the surrounding soft soil, resulting in a higher lateral stress in the column than in the surrounding soil .

Assuming the column material is incompressible, its volume remains constant before and after deformation, which yields the following Eq. 

12

Equation (12) can be rearranged as follows:

13

Where is the total settlement of the column at the bottom depth of the first slice ; is the initial radius of the column; is the radial expansion of the column at depth ; is the bottom depth of the first slice. For the settlement of the column at the bottom of the next slice depth, Eq. (13) can be modified as follows:

14

The total settlement of the surrounding soil at the bottom of each slice () are calculated based on the formulation proposed by Ghionna and Jamiolkowski³¹, originally developed for a hollow cylinder subjected to combined radial and axial loading, which was later adopted by Raithel and Kempfert¹⁹ to estimate the settlement of soft soil improved with GESCs. The Ghionna & Jamiolkowski model³¹ was specifically developed for a hollow cylinder subjected to combined radial and axial loading, which directly represents the unit cell of an improved soil mass. The formulation explicitly incorporates both vertical stress increments and the influence of lateral confinement through the soil’s Poisson’s ratio, making it well-suited for application in the present analytical model. The expression incorporates the Poisson’s ratio of the surrounding soft soil () and is given as:

15

The term in the Eq. (15) indicates additional vertical strain in the surrounding soil induced by net lateral stress transfer from the column to the soil through the encasement. The factor highlights the role of Poisson’s ratio, which couples radial stresses to axial deformation: soils with higher are more sensitive to lateral stress changes. For the settlement of the soil at the bottom of the second slice depth, Eq. (15) can be modified as follows:

16

For rigid loading, the settlement of the column and soil is equal, resulting .

From Eqs. (13) and (15),

17

Equation (17) can be rearranged as follows:

18

The detailed derivation of this equation is also given in the supplementary file: APPENDIX-A section.

Differential settlement

The combined stiffness of the stone column improved ground () based on the area replacement ratio can be computed from Eq. (6) as follows:

19

Considering both the column and the GESC composite system are assumed to behave as linear elastic materials. Therefore, settlement is related to stress and stiffness as, , which yields . The settlements of the stone column and GESC composite ground are both considered over the same loaded depth, typically the full length of the column. By equating the normalized stiffness responses (settlement × modulus / stress) of the column and the composite ground, the equation becomes:

20

Where denotes the settlement of the composite ground, obtained by equating Eqs. (13) and (15). The combined stress of the composite ground () can be determined from Eq. (6) as follows:

21

Where is the stress concentration ratio,

From Eqs. (20) and (21), the settlement of the individual stone column () can be calculated as follows:

22

The differential settlement () can be calculated as follows:

23

This formulation explicitly shows how the area replacement ratio, stress concentration, and material stiffness of the column and surrounding soil control the differential settlement.

Calculation procedure

The proposed analytical method adopts a rigorous approach by assuming an initial vertical stress in the soil that ensures equal settlement between the column and the surrounding soil at the bottom of each slice. To simplify the calculation process and facilitate the derivation of final results, the column can be divided into number of segments, each with a thickness of , corresponding to depths z₁, z₂,…, z_n at the bottom of each segment. The overall calculation flow chart is presented in Fig. 3, and the detailed stepwise procedure adopted to compute the settlement and stress distribution in GESCs using a depth-slicing approach is described as follows:

Fig. 3.

Calculation flow chart.

For the calculation, all relevant geometric, loading, and material parameters must be defined based on laboratory test results or design specifications. These include the diameter of the column (); diameter of the unit cell/spacing of the column (); outer radius of the encasement (); column length (); unit weight of stone aggregates () and surrounding soil (); reference oedometric modulus of soil (); geotextile stiffness (); effective frictional angle of stone aggregates () and soil (); effective cohesion of the surrounding soil (); applied vertical load on GESC-improved ground (). A step-by-step calculation procedure up to the segment is as follows:

1. The first step involves calculating the area replacement ratio, ; the active earth pressure coefficient in the column, ; and the modified lateral earth pressure coefficient for the surrounding soil, , as given in Eq. (1).

2. Divide the total length of the column into equal slices, each with a thickness of . The calculation steps for the first slice are outlined as follows.

3. Assume an initial vertical stress in the soil (), which may be taken equal to the applied vertical stress for the initial assumption.

4. Determine the vertical stress in the column () using Eq. (6), and calculate the lateral stresses in the soil () and the column () using Eqs. (8) and (9), respectively.

5. Calculate oedometric modulus () using Eq. (2), where the existing effective stress at depth () is determined using Eq. (2); stress-dependency exponent () can be considered as 1 for soft soil.

6. Reference stress () be assumed as 100 kPa, following Raithel and Kempfert¹⁹. The composite stiffness modulus () can be determined using Eq. (5).

7. Determine the radial expansion of the column () at depth from Eq. (18). Radial expansion of encasement () at can be calculated from Eq. (11).

8. Calculate the radial stress in the encasement () using Eq. (9); The net lateral stress () mobilized, can be calculated from Eq. (10).

9. Determine the settlement of the column () and the surrounding soil () from Eqs. (13) and (15), respectively.

10. Repeat steps (4) through (7) iteratively until the settlements of the column and the surrounding soil become equal.

For the calculation of the next slice, repeat steps (3) through (9), substituting, for in all equations used from steps (3) to (8), where .

To calculate the differential settlement () between the column and the soil, Eq. (23) is applied at each depth. The calculation involves an iterative procedure, which can be time-consuming when performed manually. To simplify the process, using an Excel spreadsheet or Python code is recommended. In this study, the analytical model was implemented using Python code (provided in the supplementary file: APPENDIX-B section).

Validation of the model

Field test case by Raithel³²

The proposed model was validated against field test results by Raithel³², with input parameters summarized in Table 1. Both soil and column were divided into ten horizontal slices of equal thickness ( = 0.125 m). This number was sufficient, as the predicted load-settlement response closely matched field results (see Fig. 4a). Figure 4b compares lateral encasement deformation at 150 kPa with the analytical model of Raithel and Kempfert¹⁹. The earlier-described calculation procedure was applied, and results are given in Table 2. Although the model reproduced Raithel’s results well, limited documentation in that field study made it unsuitable as a benchmark for the parametric analysis.

Table 1.

Parameters used for analytical calculation (Raithel and Kempfert, 2000).

[-]	h [m]	[m]	[m]	[kN/m]	[kN/m2]	[kN/m2]	[	[	[kN/m3]	[kN/m3]	[-]	[kN/m2]	[-]	[-]
Large-scale model test (scale 1:1)
0.125	1.25	0.3375	0.0125	800	725	9	38	24	8	18	0.7	100	1	0.4

Fig. 4.

Comparison of the current analytical model with field test results from Raithel³²: (a) Load-settlement response; (b) Radial deformation pattern.

Table 2.

Calculation example of the current model.

Depth, (m)	(kPa)	(kPa)	(kPa)	(kPa)	(kPa)	(kPa)	(m)	(m)	(kPa)	(kPa)	(m)	(m)
0.125	34.2	24.6	960.6	229.0	154.0	867.2	0.0415	0.0290	189.2	15.2	0.0259	0.0259
0.250	32.9	24.5	969.5	231.7	161.4	909.1	0.0418	0.0293	191.1	16.1	0.0423	0.0423
0.375	34.0	25.9	961.6	230.4	168.8	950.9	0.0413	0.0288	187.8	16.6	0.0621	0.0621
0.500	35.2	27.5	953.6	229.0	176.3	992.8	0.0407	0.0282	184.4	17.1	0.0809	0.0809
0.625	36.3	29.0	945.7	227.6	183.7	1034.6	0.0402	0.0277	181.0	17.6	0.0996	0.0996
0.750	37.4	30.5	938.0	226.4	191.1	1076.5	0.0397	0.0272	177.8	18.1	0.1179	0.1179
0.875	38.5	31.9	930.5	225.1	198.6	1118.3	0.0392	0.0267	174.6	18.6	0.1359	0.1359
1.000	39.5	33.4	923.1	223.9	206.0	1160.2	0.0388	0.0263	171.4	19.0	0.1537	0.1537
1.125	40.6	34.8	915.9	222.7	213.4	1202.0	0.0383	0.0258	168.4	19.5	0.1711	0.1711
1.250	41.6	36.3	908.8	221.5	220.9	1243.9	0.0378	0.0253	165.4	19.9	0.1883	0.1883

Note: Each slice thickness, = 0.125 m; indicates each slice number from 1 to 10, for the first slice, =1; All symbols used are defined in the list of notations.

Field test case by Lee et al.⁸

A validation case was developed for a foundation reinforced with GESCs embedded in a 5.0-m clayey sand layer underlain by a rigid base. The columns, 0.8 m in diameter and 5.0 m in length, were arranged in a square pattern with a spacing of three times the column diameter and topped with a 0.25-m crushed-stone cushion. Soil and column parameters were adopted from reported values for soft Bangkok clay^8,33, while the geotextile stiffness was taken as = 2,500 kN/m⁸ with an encasement thickness of 1.0 mm²⁰. The parameters used in the analysis are summarized in Table 3.

Table 3.

Parameters used for analytical calculation.

[m]	h [m]	[m]	[m]	[kN/m]	[kN/m2]	[kN/m2]	[	[	[kN/m3]	[kN/m3]	[-]	[kN/m2]	[-]	[-]
1.2	5.0	0.4	0.001	2500	15,000	5	28	45	9	23	0.7	100	1	0.3

Note: All parameters were adopted from Lee et al.⁸, except for , , and .

Settlements were evaluated for applied loads of 50, 100, 150, and 200 kPa, and the results were compared with predictions from Raithel and Kempfert¹⁹, Pulko et al.²², Castro et al.²³, and Zhang and Zhao²⁰. As illustrated in Fig. 5, the proposed AM shows excellent agreement with the field data (), outperforming previous models whose values range from 3.58 to 5.39. Settlements were assessed under applied loads of 50–200 kPa, with the current AM predicting 48.7 mm at 200 kPa, closely matching field observations and improving accuracy compared with earlier approaches. Due to limited data on radial expansion and stress concentration ratios, that field study was deemed unsuitable as a benchmark for the parametric analysis conducted in this study.

Fig. 5.

Comparison of the current analytical model with field test results by Lee et al.⁸ and other published analytical models.

Field test case by Almeida et al.⁹

To support further model validation, a well-documented field case reported by Almeida et al.⁹ was selected, involving a 5.3 m high embankment supported by GESCs, constructed in 2012 at the ThyssenKrupp stockyard in Itaguaí, Brazil. The embankment was built over a 9 to 10 m thick soft clay layer with an undrained shear strength of approximately 15 kPa. The foundation was reinforced with 36 GESCs, each 0.8 m in diameter and 11 m in length, arranged in a 6 × 6 grid with a 2 m center-to-center spacing. The columns were installed in 2008 using Ringtrac 100/250 geotextile and 10 to 35 mm crushed stone, allowing sufficient time for pore pressure dissipation before loading.

Figure 6 illustrates the embankment cross-section and instrumentation layout, which includes devices for monitoring settlement (S), lateral deformation (IN), pore pressure (PZ), vertical stress (CP), and encasement strain (EX). While the proposed model can incorporate layered soil conditions by applying the actual parameters of each layer, this significantly increases the calculation complexity. Therefore, for the validation of the field case reported by Almeida et al.⁹, average properties of the layered soil, such as the Young’s modulus and effective friction angle of the surrounding soil, were used to represent the entire soft deposit in the analytical model. The input parameters adopted for the validation are summarized in Table 4, based on the data from Almeida et al.⁹. A 3D numerical analysis was also conducted for this field case, with material properties, construction stages, and boundary conditions selected following the numerical studies conducted by Hosseinpour et al.^34,35.

Fig. 6.

Embankment geometry and instrumentation from Almeida et al.⁹ reference case.

Table 4.

Parameters used for analytical calculation by Almeida et al.⁹.

[-]	h [m]	[m]	[m]	[kN/m]	[kN/m2]	[kN/m2]	[	[	[kN/m3]	[kN/m3]	[-]	[kN/m2]	[-]	[-]
0.125	11.0	0.4	0.408	1750	3000	5	40	27	14.4	20	0.7	100	1	0.3

Note: Definitions of all symbols can be found in the notations section.

Figure 7a shows the developed 3D numerical model used in the analysis, while Fig. 7b illustrates the time-settlement response, comparing the current analytical model (AM), the models proposed by Raithel and Kempfert¹⁹, Pulko et al.²², and Castro et al.²³, along with numerical simulation results and field measurements reported by Almeida et al.⁹. The comparison indicates that the current AM, with an of 0.816, predicts the field results more accurately than the model of Raithel and Kempfert¹⁹ ( = 0.636) and performs comparably to Pulko et al.²² ( = 0.835) and Castro et al.²³ ( = 0.827). As part of the model comparison, it is important to note that consolidation effects were not considered in the current analytical model; however, to illustrate the settlement corresponding to the end of each consolidation stage in the time-settlement response, the analytical results were included for comparison with the field test data, as shown in Fig. 7b. The figure also presents the differential settlement between the encased column and the surrounding soil, as calculated using the proposed analytical method. Since the current analytical model is capable of accounting for layered soil conditions, the corresponding soil settlement results are also presented considering layered soil condition, yielding an of 0.977 and an of 27.43. Figure 7c shows a comparison of the vertical stress carried by the column and the surrounding soil between the current AM and the field tests by Almeida et al.⁹, while Fig. 7d compares the radial expansion of the column under applied loading with the field test results. The comparison demonstrates good agreement between the current AM results and the findings reported by Almeida et al.⁹. Table 5 presents the differential settlement between the encased column and soft soil, comparing field test data with analytical results.

Fig. 7.

(a) 3D Numerical model used for validation; (b) Comparison of time-settlement response between the analytical model (AM) and the field test reported by Almeida et al.⁹; (c) Comparison of time-vertical stress response between the AM and the field test⁹; (d) Comparison of the response of radial expansion of the column with time between the AM and the field test⁹.

Table 5.

Differential settlement calculation by the current model for the field case study by Almeida et al.⁹.

Time (days)	Embankment height (m)	Current Analytical Model			Almeida et al. [9]
13	1.5	235	191.3	43.7	111.5	93.0	18.5
47	3.0	333	271	62.0	237.0	189.0	48.0
63	4.2	395	322.5	72.5	322.0	236.0	86.0
245	5.3	512	417	95.0	524.0	436.0	88.0

Note: kPa; = 512 mm, obtained by equating Eqs. (13) and (15) for column depth of 11 m; indicates the bottom depth of 11 m.

Figure 8 compares the load-settlement behavior of the GESC-improved ground as obtained from the field test, the current analytical model, the numerical model, and the Raithel and Kempfert¹⁹ model. The proposed analytical model shows good agreement with both the benchmark model and the numerical simulation, as well as with the general trend of the field data. At a low applied stress of 38 kPa, the analytical model predicted a settlement of 235 mm, compared to 111.5 mm observed in the field. This discrepancy is likely due to the short consolidation period in the field (approximately 10 days), which resulted in partial drainage. In contrast, the analytical model assumes fully drained conditions, leading to higher predicted settlements. Despite this discrepancy, the overall agreement among the analytical, numerical, and field results demonstrates the reliability and practical applicability of the proposed method for preliminary design and performance evaluation of GESC-reinforced ground.

Fig. 8.

Load-settlement comparison between analytical model and Almeida’s field test⁹.

Sensitivity analysis of the AM

An Artificial Neural Network (ANN) based sensitivity analysis has been conducted to understand the influence of stone friction angle , stress-dependent exponent (), , and Poisson’s ratio of soil on SCR and SIF. Four One at time (OAT) predictor responses were concentrated into single block sets. For each fitted model, relative importance was obtained from the weightage of each hidden layer. Matrices based on absolute weightage were multiplied along the input-to-output range for an effective normalized influence on variables. The entire analysis has been adopted based on a modified Garson’s algorithm that adapts deep neural networks by combining all algorithms along the path^36,37. The algorithm provides the following equation:

24

Here, indicates the relative importance, and denote the weightage connection between the input and hidden neurons , and between hidden and output neurons respectively. The parameters considered in the sensitivity analysis are summarized in Tables 6 and 7 for SIF and SCR calculation, respectively. The corresponding results for each parameter were then used to evaluate their relative importance on SIF and SCR using Eq. (24), as illustrated in Fig. 9.

Table 6.

Parameters used for the sensitivity analysis on SIF.

36	2.594945	0.5	3.089047	20	6.215374	0.3	2.8576
38	2.720319	0.6	3.037753	30	4.797679	0.35	2.846155
40	2.8576	0.7	2.989093	40	4.095514	0.4	2.831378
42	3.008256	0.8	2.942933	60	3.401851	0.45	2.811934
44	3.173966	0.9	2.899142	80	3.06018
46	3.356641	1	2.8576	100	2.8576

Table 7.

Parameters used for the sensitivity analysis on SCR.

36	16.4153	0.5	15.08187	20	4.99557	0.3	19.06521
38	17.68031	0.6	15.79844	30	6.69801	0.35	18.5216
40	19.06521	0.7	16.55304	40	8.4274	0.4	17.89515
42	20.58481	0.8	17.34764	60	11.93754	0.45	17.16482
44	22.25598	0.9	18.18431	80	15.48921
46	24.09801	1	19.06521	100	19.06521

Fig. 9.

Sensitivity analysis of each parameter influencing (a) SCR and (b) SIF.

Figure 9 illustrates the influence of key parameters on the SCR and SIF for an area replacement ratio of 0.125 and an encasement stiffness of 1750 kN/m, chosen based on the field test case reported by Almeida et al.⁹. Detailed discussion of these two parameters (i.e., area replacement ratio and encasement stiffness) on the effect of SCR, SIF, radial expansion is provided in the later section of the manuscript. The results highlight that the stone friction angle has the greatest impact on both SIF and SCR, as a higher friction angle enhances intergranular resistance and shear strength, thereby improving load transfer and reducing settlement^38,39. The reference pressure () also shows notable influence, primarily due to its effect on reducing normalized stiffness at a given confining stress and improving the compressibility behavior of the soil⁴⁰. In contrast, the soil Poisson’s ratio has only a minor influence, as it primarily governs lateral deformation under vertical loading, while the stress-dependent exponent shows a moderately low effect by controlling stiffness variation with confining pressure.

Parametric study

To better investigate the influence of key parameters, a parametric study was conducted using a simplified homogeneous soil profile. Using the field test by Almeida et al.⁹ as a reference, only the soft clay I layer surrounding the 11 m long GESC was considered, replacing the layered soil profile observed in the field for simplicity. The analytical results were validated by comparison with the numerical analysis performed using the finite element method (FEM). The numerical models used in the parametric study are shown in Fig. 10. These analyses accounted for a range of column spacings, diameters, friction angles, and encasement stiffnesses to validate the analytical model applied in the current study.

Fig. 10.

Numerical model used for the parametric study.

Overview of the numerical model

In this study, 3D finite element analysis was conducted using PLAXIS 3D with coupled consolidation to capture time-dependent behavior, validated against measured data. The Mohr-Coulomb (MC) model was applied to the encased column and embankment, while the Soft Soil (SS) model was used for soft clay based on parameters from Hosseinpour et al.³⁴. The geotextile encasement was modeled as a linear elastic material with tensile modulus = 1750 kN/m. A 15-node element mesh was used for soils, and 5-node line elements for the geogrid. The boundary conditions, mesh configuration, and material models were kept consistent with the numerical study by Hosseinpour et al.³⁴ on the field test conducted by Almeida et al.⁹.

The embankment was modeled by sequentially activating each layer, with consolidation intervals established to achieve at least 95% degree of consolidation. Since the analytical model assumes drained conditions, each stage in the numerical analysis was allotted sufficient time to allow for at least 95% dissipation of excess pore pressures. The embankment construction stages used in the numerical model are summarized in Table 8.

Table 8.

Details of calculation steps in numerical modeling.

Phase	Steps	Type of analysis	Loading type	Embankment height (m)	Interval (days)	Event
0	Initial Step	procedure	Staged construction	--	--	Initial Phase
1	Step 1	Consolidation	Staged construction	1.5	3	Construction
	Step 2	Consolidation	95% degree of consolidation	1.5	--	Consolidation
2	Step 3	Consolidation	Staged construction	3	2	Construction
	Step 4	Consolidation	95% degree of consolidation	3	--	Consolidation
3	Step 5	Consolidation	Staged construction	4.2	2	Construction
	Step 6	Consolidation	95% degree of consolidation	4.2	--	Consolidation
4	Step 7	Consolidation	Staged construction	5.3	2	Construction
	Step 8	Consolidation	95% degree of consolidation	5.3	--	Consolidation

Effect of applied load

Figure 11 compares the load-settlement behavior of GESC-improved ground from analytical calculations and numerical simulations for different area replacement ratios, with a constant encasement stiffness of 1750 kN/m. Using the composite stiffness approach (Eq. 5), the result indicates that increasing the area replacement ratio raises the proportion of stiff column material, thereby enhancing the overall stiffness and reducing settlement. For an area ratio of 0.09, the settlement is calculated as 1280 mm. Increasing the area ratio from 0.09 to 0.12, 0.20, and 0.35 reduces the settlement by 15.2%, 32.2%, and 49.2%, respectively. Settlements are calculated slice by slice by distributing vertical stress based on the area ratio (Eq. 6) and computing deformations in both column and soil (Eqs. 13 and 15), while ensuring compatibility through iterative adjustment. The strong agreement with numerical simulations confirms its reliability across varying area ratios.

Fig. 11.

Load-settlement behavior of GESC-improved ground for different area replacement ratios.

A comparison between the analytical results and the numerical outcomes of vertical stress distribution under various loading conditions, for an area replacement ratio of 0.125 and an encasement stiffness of 1750 kN/m, is presented in Fig. 12a-d. The figure indicates that the vertical effective stress in the column reaches 175, 446, 683, and 877 kPa under applied embankment stresses of 42, 84, 120.4, and 150 kPa, respectively. The vertical stress distribution described by Eq. (6) is based on a simplified theory of stress sharing between the soil and column, influenced solely by the area replacement ratio. In the analytical model, vertical stress is shared between the column and surrounding soil based on the area replacement ratio, treating them as two separate materials. This creates a sharp stress transition in Fig. 12, as the model does not account for gradual interaction or plastic behavior like in the numerical results. In contrast, the numerical simulations reveal a more continuous stress transition, without a distinct boundary between loaded and unloaded soil zones, since the vertical stresses in the columns are also affected by shear band formation. Despite these differences, the analytical solution effectively captures the overall stress distribution between the soil and the column, and a good agreement between the two approaches is observed.

Fig. 12.

Comparison of vertical stresses under different embankment loadings: (a) 42 kPa; (b) 84 kPa; (c) 120.4 kPa; (d) 150 kPa.

Figure 13 shows a comparison between the analytically calculated radial expansion of the geotextile encasement and finite element results for an area replacement ratio of 0.125 under various load levels. The predicted radial deformations, which are directly related to the hoop tension in the encasement, closely align with the FEM results. The analytical model computes radial expansion based on the lateral deformation of the column material and the encasement stiffness, as described by Eq. (11). This expansion serves as a proxy for the hoop force in the geotextile, which is proportional to the radial strain.

Fig. 13.

Comparison of radial expansion of the encasement under various embankment loadings.

The radial stress in the encasement, calculated using Eq. (9), increases with applied load and is resisted by the tensile stiffness of the geotextile, thereby providing lateral confinement that limits column bulging. Although the analytical model assumes elastic soil behavior and neglects shear stresses at the column-soil interface, it effectively predicts the settlement, stress distribution, and radial deformation across all load levels. Slight deviations observed at higher load levels in the FEM results are attributed to nonlinear behaviors such as column yielding, strain softening, and localized bulging-effects that are not captured in the simplified, linear-elastic analytical approach. Nonetheless, the proposed model demonstrates reliable performance within practical load ranges and offers a rapid yet accurate means of estimating encasement deformation.

Influence of area replacement ratio

The comparison between analytical calculations and numerical analyses was extended across a range of area replacement ratios () by varying the column diameter. To simplify the study, the encasement stiffness was kept constant at =1750kN/m. Figure 14a illustrates that the settlement of the improved ground decreases from 1474 mm to 613 mm as the area replacement ratio increases from 0.05 to 0.35 under 150 kPa loading, emphasizing its significance as a key design parameter. A strong agreement was observed between the analytical model and numerical results. To evaluate the settlement behavior of unimproved ground, the analytical model was also applied by setting the column diameter, encasement stiffness, and encasement thickness to zero. Based on this approach, the settlement improvement factor (), defined as the ratio of the settlement of the unimproved ground to the GESC-improved ground, was calculated for each area ratio and is presented in Fig. 14b. The area replacement ratio was limited to a maximum of 0.35. In practice, values beyond 0.3 are rarely adopted, as they tend to increase construction costs and may cause overlapping of pressure bulbs between adjacent columns, which is typically undesirable. Therefore, an upper limit of 0.35 was chosen for illustration purposes in this study.

Fig. 14.

Influence of area replacement ratios on: (a) Settlement; (b) Settlement Improvement Factor.

The variation of maximum radial expansion of the encasement () with different area replacement ratios, as obtained from both analytical calculations and numerical analyses, is shown in Fig. 15. The radial expansion decreases as the area ratio increases. A radial expansion of 52 mm is calculated for an area ratio of 0.05 under an embankment loading of 150 kPa, which decreases to 9 mm when the area ratio is increased to 0.35 under the same loading condition. However, beyond an area ratio of 0.25, the rate of decrease becomes more gradual. This trend was also observed in the present numerical modeling as well as in the study by Yoo et al.³³.

Fig. 15.

Influence of area replacement ratios on the maximum radial deformation of the encasement.

Figure 16 illustrates the effect of area replacement ratios on the stress concentration ratio (SCR), defined as the ratio of vertical stress in the stone column () to that in the surrounding soil (), in GESC improved composite ground. The analytical and numerical results indicate that the SCR decreases from 25 to approximately 11.4 as the area replacement ratio increases from 0.05 to 0.35 under an applied stress of 150 kPa. Although a higher area ratio results in a greater portion of the load being carried by the columns due to their increased cross-sectional area, the reduced spacing between columns enhances interaction among them. Simultaneously, the surrounding soft soil has less area to deform laterally. This increased interaction reduces the differential stress transfer between the column and the soil, leading to a more uniform stress distribution. Consequently, the SCR tends to decrease with increasing area ratio. This trend aligns with experimental findings reported by Almikati et al.⁴¹, where consolidated drained (CD) triaxial tests demonstrated that higher area replacement ratios lead to lower SCR due to enhanced soil–column interaction and confinement effects.

Fig. 16.

Influence of area replacement ratios on the stress concentration ratio of GESC-improved ground.

Influence of encasement stiffness

Under vertical loading, the tension in the encasement () induces additional lateral confining stress () due to the encasement stiffness effect, as expressed in Eq. (9). This increased lateral confinement further enhances the vertical load-bearing capacity of the system. Murugesan and Rajagopal⁴² proposed that this additional confinement improves the vertical stress capacity () of the GESC-reinforced composite ground, expressed as:

25

where represents initial radial stress, is the undrained shear strength, and is the passive earth pressure coefficient of the column material. The effect of various encasement stiffness values was evaluated in the present analytical model for a fixed area replacement ratio of 0.125.

Figure 17 shows that the SCR increases from 5 to 32 as the encasement stiffness rises from 0 to 5000 kN/m under a 150 kPa loading, indicating improved load transfer to the stone column with enhanced lateral confinement. The SCR value of ordinary stone columns ( = 0 kN/m) is relatively low compared to GESCs because OSCs lack lateral confinement. Without geosynthetic encasement, the column can expand radially under axial load, transferring more stress to the surrounding soil and reducing the proportion of load carried by the column itself. Ouyang et al.¹⁰ also reported field test results showing an SCR of 10 for pESCs, while the SCR for OSCs was below 5. Figure 17 shows that at lower stiffness levels (e.g., below 2000 kN/m), the SCR increases sharply from around 6.0 to over 19, as greater restriction of column bulging allows the column to carry more vertical stress. However, as encasement stiffness increases beyond 2000 kN/m and up to 5000 kN/m, the rate of SCR growth slows, and the curve gradually levels off. This suggests that once a certain level of stiffness is reached, further increases have only a limited effect on enhancing stress concentration. A similar trend is observed for the SIF, which increases with encasement stiffness up to a value of about 3 at encasement stiffness of 2000 kN/m (see Fig. 18), after which the rate of increase also becomes less pronounced. As the encasement stiffness increases, the radial expansion of both the column material () and the encasement () in the analytical model decreases. This reduction in radial expansion, in turn, decreases the settlement of the column (), as described by Eq. (13).

Fig. 17.

Influence of encasement stiffness on the Stress Concentration Ratio of GESC-improved ground.

Fig. 18.

Effect of encasement stiffness on the Settlement Improvement Factor of GESC-improved ground.

The additional confining stress provided by the geotextile encasement increases confinement in encased columns compared to non-encased ones, significantly reducing column bulging, as shown in Fig. 19. Under an applied stress of 150 kPa, the maximum lateral bulging decreases sharply from 65 to 36, 21, 15, 10, and 8 for encasement stiffness values of 0, 1000, 1750, 2500, 4000, and 5000 kN/m, respectively. The optimum encasement stiffness was identified between 2000 and 3000 kN/m, beyond which further reduction in bulging becomes minimal.

Fig. 19.

Effect of encasement stiffness on the maximum radial deformation of the encasement.

Influence of column diameter

The influence of column diameter was analyzed for a constant area replacement ratio of 0.125 and an encasement stiffness of 1750 kN/m. At this constant replacement ratio, increasing the column diameter () requires a proportional increase in the unit cell area or spacing, meaning each column supports a larger surrounding zone of soft soil. Initially, as increases, the stiffer column more effectively reduces settlement, increasing the SIF. However, beyond a certain diameter (e.g., 0.6 m under 150 kPa loading), the larger unit cell introduces more soft soil into the system, reducing load transfer efficiency due to weaker lateral confinement. As the surrounding soil becomes more dominant in stress sharing (as described in Eq. 6), the overall composite stiffness decreases, and further increases in column diameter yield diminishing returns. Consequently, the SIF begins to decline as the benefit of added column stiffness is outweighed by the adverse effects of increased soft soil influence, as illustrated in Fig. 20. In this analysis, keeping the same area replacement ratio indicates that the spacing-to-diameter ratio remains constant, so larger pile diameters correspond to proportionally larger spacing. As the column diameter increases, each column must support a larger surrounding zone of soft soil. While the lateral stresses in the column and surrounding soil remain unchanged at constant area ratio, the net lateral confinement on the column is reduced because the softer soil is more compressible and less able to transfer stress to the encasement. This reduction in lateral confinement weakens the column’s effectiveness in limiting settlement, leading to lower settlement improvement factor and stress concentration ratio values for larger columns. Therefore, a similar trend was also observed in the variation of the SCR with column diameter, as shown in Fig. 21.

Fig. 20.

Effect of column diameter on the Settlement Improvement Factor for an area ratio of 0.125, = 1750 kN/m.

Fig. 21.

Effect of column diameter on the Stress Concentration Ratio of GESC-improved ground for an area ratio of 0.125, = 1750 kN/m.

Effect of oedometric modulus of soil

The reference oedometric modulus of the surrounding soil was varied to evaluate its influence on the SIF and SCR. As expected, the SIF increased with higher oedometric modulus values, as shown in Fig. 22. However, while the numerical analysis exhibited a nonlinear response, the analytical model assumes purely elastic behavior. It does not account for plasticity and considers the soil to behave as a linearly elastic material throughout the loading process. The variation of SCR with the reference oedometric modulus, as calculated by the analytical model, showed good agreement with the numerical results, as illustrated in Fig. 23.

Fig. 22.

Effect of oedometric modulus of the soil on the SIF of GESC-improved ground.

Fig. 23.

Effect of oedometric modulus of the soil on the SCR of GESC-improved ground.

Effect of partial encasement

Partial encasement is modeled by assigning encasement stiffness () and setting zero thickness in the unencased column portion. Figure 24 illustrates the effect of the partial encasement ratio of pESC on the SIF under an applied stress of 150 kPa. The results indicate an optimum encasement length ratio of 0.45, beyond which the SIF increases only marginally with further increases in encasement length. Lateral deformation profiles were computed for various partial encasement lengths. Figure 25 shows deformation for an encasement length of 3 m, 5 m, and 9 m. The analytical model agrees well with numerical analysis but predicts slightly higher lateral deformation in the unencased zone. This difference arises because the numerical model includes nonlinear soil behavior, confinement effects, and shear resistance at the soil-column interface, which reduce deformation. The analytical model is more conservative, assuming elastic soil, uniform loading, and no shear transfer.

Fig. 24.

Effect of partial encasement ratio on SIF for an applied stress of 150 kPa.

Fig. 25.

Radial deformation profile of a partially encased stone column with an encasement length () of 9, 5, 3 m for an applied stress of 150 kPa.

Design chart

Figure 26 presents a design chart developed for stone columns with an internal friction angle of 40°, showing the influence of area replacement ratio () and normalized encasement stiffness () on Settlement Improvement Factor (SIF, ) under normalized loading () of 0.5 to 1.0. The encasement stiffness is normalized by dividing by the product of column radius () and soil elastic modulus () to account for column size and soil stiffness. The chart indicates that higher SIF values require much stiffer encasement, particularly for closely spaced columns with higher area replacement ratios, while lower area replacement ratios (widely spaced columns) lead to greater reductions in SIF, demanding even stiffer encasement to achieve similar improvement. It also shows that increasing normalized loading amplifies the need for higher encasement stiffness, especially when substantial settlement improvement is targeted. Comparison with previously published test results demonstrates close agreement with the trends, confirming the reliability of the current AM-based design charts.

Fig. 26.

Design chart for the settlement improvement factor ( indicating ).

Conclusions

This paper presents a simplified semi-analytical iterative solution to evaluate ground reinforced with GESCs. The system is modeled as a unit cell with regularly arranged end-bearing stone columns in soft soil. The stone column is idealized as a rigid–plastic material, yielding at active stress and deforming plastically without volume change. Key findings from the parametric study include:

• Area replacement ratio is the most influential parameter: increasing it from 0.05 to 0.35 reduces settlement by up to 49% and decreases the SCR from 25 to 11.4, highlighting its critical role in design.

• Encasement stiffness significantly improves load transfer and limits radial expansion of the column, with optimal effects observed between 2000 and 3000 kN/m; higher stiffness provides minimal additional benefit.

• Column diameter beyond the optimal range, at constant area replacement ratio, reduces settlement improvement due to the increased influence of surrounding soft soil, resulting in lower SIF and SCR.

• The analytical model predictions show good agreement with FEM and field test results, validating the underlying assumptions and demonstrating its effectiveness across a wide range of area ratios and encasement stiffness.

• Although the model is primarily intended for homogeneous conditions, the solution can be adapted to layered soil profiles and partially encased stone columns. An optimum partial encasement length ratio of 0.45 was identified, beyond which further increases provide only marginal improvement in SIF.

The model neglects shear transfer at column-geotextile-soil interfaces, consolidation, pore pressure dissipation, soil nonlinear elasto-plasticity, and column dilatancy. Future work should address these aspects and group effects to improve accuracy. Despite these simplifications, the analytical approach captures key trends and provides essential guidance for practical design of GESCs.

Supplementary Information

Below is the link to the electronic supplementary material.

Author contributions

Nibir Rahman - Conceptualization, Methodology, Software, Validation, Investigation, Writing – Original Draft, VisualizationMd. Rokonuzzaman - Supervision, Project Administration, Formal Analysis, Writing – Review & EditingAshiqur Rahman - Software, Validation, Formal Analysis, Visualization, Investigation.

Data availability

Data generated or analyzed during this study are provided in full within the published article.

Declarations

Competing interests

The authors declare no competing interests.