A nonlinear analytical model for consolidated geotextile-encased sand columns Open Access

Geosynthetics have been widely applied as construction materials for geotechnical and environmental applications (Palmeira et al. 2008). One of the applications of geosynthetics is for soil reinforcement in retaining walls (Han et al. 2018; Derksen et al. 2022; Fox 2022), slopes (Dastpak et al. 2021; Wang et al. 2022), and embankments (van Eekelen and Han 2020; Pham and Dias 2021). Geosynthetics have also been applied for the purpose of enhancing pipelines, drainage, erosion, and sediment control systems, showing great versatility and cost-effectiveness (Theisen 1992; Melo et al. 2021; Fan and Rowe 2023). Geosynthetics can be used to contain or encase coarse-grained and fine-grained materials: geotextile tubes filled with sandy materials have been used in coastal applications (Shin and Kim 2018; Kiran et al. 2022), while stone and sand columns encased by geotextiles have been employed to improve the bearing capacity of soft soils (Basack et al. 2017; Salem et al. 2017; Kadhim et al. 2022). Geotextile tubes can also be used for waste disposal (Yang et al. 2019; Karadoğan et al. 2022), presenting a means to manage wastes efficiently and economically in an environmentally responsible and sustainable manner (Bhatia 2017; Kim and Dinoy 2021).

Studies aiming to analyze the behavior of coarse-grained or fine-grained materials encapsulated by geosynthetics have been conducted, and a great deal of research exists on the tension force of geotextile tubes during filling (Leshchinsky et al. 1996; Yee 2012; Kim et al. 2020) and the modeling of their deformation during dewatering or consolidation (Yee and Lawson 2012; Ratnayesuraj and Bhatia 2018; Kim and Dinoy 2021; Zhang et al. 2022). Studies on the behavior of consolidated encased stone columns in triaxial tests conducted by Miranda and Da Costa (2016) and Xue et al. (2019) have shown that encasing stone columns results in an increase in their strength.

However, research studies on the relationship between initial stresses, relative density, geotextile properties, and the shearing behavior of soil–geotextile systems are limited. A model for encased stone columns, which may be applied for sand columns, was proposed by Pulko et al. (2011). The developed analytical closed-form solution considers the column material as a linear elasto-plastic material with constant dilatancy, while the surrounding soil is considered as an elastic material. For the geosynthetic encasement, the material is considered to be linear elastic. Pulko et al. (2011) validated the analytical model by comparing the results with PLAXIS 2D using the Mohr–Coulomb model. Kadhim et al. (2018) conducted a three-dimensional numerical analysis on dense geotextile-encased sand columns (GESACs) based on the finite-element method (FEM) using parameters calibrated from triaxial tests. In that study, a linear elastic–perfectly plastic soil model was used to simulate the behavior of the GESAC. However, soil–geotextile interaction analysis of the results of their triaxial tests was not conducted. Furthermore, the linear elastic–perfectly plastic model can show discrepancies when the observed behavior is highly nonlinear. Hence, the use of alternative models that consider nonlinear behavior can provide flexibility in modeling the behavior of GESACs.

In an effort to further analyze the soil–geotextile interaction of GESACs, a nonlinear GESAC model is proposed. The model is based on a power law, and first requires determination of the shear behavior of unconsolidated GESACs to obtain the shear behavior of consolidated GESACs at various confining pressures. Xue et al. (2019) reported that the stress–strain behavior of consolidated geotextile-encased stone columns can be obtained by superposing the uniaxial compression curve of a geotextile-encased stone column to the compressive curve of an ordinary stone column in a triaxial test; the same concept can be applied to GESACs. In this study, the terms ‘unconsolidated’ and ‘consolidated’ are used mainly because the uniaxial compression behavior of an encased column is necessary for determining the behavior of consolidated encased columns. Consolidated refers to those that have considerable effective stresses before shearing, while unconsolidated refers to those that have negligible effective stress. In triaxial testing, consolidation is performed to describe the stress state in the field, wherein the effective stress before shearing is controlled by the overburden pressure.

In the work reported in this paper, the behavior of GESACs was investigated through uniaxial compression tests and triaxial tests. Uniaxial compression testing was conducted to study the failure mechanism of the soil–geotextile system, while unconsolidated and consolidated triaxial tests on loose GESACs under low confining pressures (10, 50, and 100 kPa) were conducted to investigate the effect of initial stresses on the shear behavior of the soil–geotextile system. FEM simulations of GESACs using a nonlinear soil model and various geotextile models were performed to analyze the soil–geotextile interaction, as well as to compare the performance of the proposed GESAC model. The proposed model was then verified based on data on dense GESACs from the literature.

In a uniaxial or triaxial compression test, vertical strains (ε₁), radial strains (ε₃), and volumetric strains (ε_v) are experienced by the soil sample when compressive loading (ΔF_v) is increased. It should be noted that, in this study, compressive strains are taken as negative while tensile strains are taken as positive. Typically, the height (H) of the sample decreases while the diameter (D) and radius (r) of the sample increase, as shown in Figure 1. In Figure 1, σ₃ is the confining pressure and H₀, D₀, and r₀ are the height, diameter, and radius of the soil sample before shearing, respectively. Changes in height and volume are typically measured in a triaxial test. The height (H) during shearing is related to H₀ and ε₁, as given by Equation (1). The volume (V) is related to r₀, H₀, the volume before shearing (V₀), and ε_v, as given by Equation (2). During the test, the change in vertical pressure (Δσ_v) is obtained by dividing ΔF_v by the area (A) of the sample, as shown in Equation (3). A can be obtained by dividing V by H, as shown in Equation (4). Assuming that the specimen deforms as a cylinder during shearing, A is also related to r₀ and ε₃ by Equation (4). Rearranging Equation (4), ε₁, ε₃, and ε_v are related by Equation (5). After obtaining Δσ_v, the total vertical stress (σ_v) and the effective vertical stress (σ′_v) can be determined by Equations (6) and (7), respectively, wherein u is the pore pressure.

From the wide-width strip test (ASTM D4595), the relationship between geotextile tension force (T_g) and geotextile strain (ε_g), as well as the tension force at failure (T_g,f) and geotextile strain at failure (ε_g,f), can be obtained. Usually, the relationship between T_g and ε_g is assumed to be linear, as presented by Pulko et al. (2011). However, geotextile behavior can be nonlinear, and often varies from being nonlinearly concaving upward to nonlinearly concaving downward, as shown in Figure 2. The nonlinear behavior can be represented using a power law, as shown in Equation (8), wherein n is a curve fitting parameter. As shown in Figure 2, the exponent n controls the curvature; n < 1.0 gives a curve concaving upward, while n > 1.0 gives a curve concaving downward. For comparison, measured data reported by Wu and Hong (2009) is presented in Figure 2. It can be seen that n = 2.75 provided good agreement with the measured data, whereas using n = 1.0 (linear model) produced poor results. The main advantage of Equation (8) is its flexibility because it can consider both linear and nonlinear behavior. It is, however, important to calibrate the value of n or ε_g,f to obtain a good fit with measured data.

The geotextile strain (ε_g), which is assumed equal to ε₃, is related to the GESAC radius, as shown in Equation (9). Pulko et al. (2011) related the change in radial stress (Δσ_r) to T_g and r by Equation (10). Combining Equations (8) and (10), Δσ_r can be determined considering nonlinear geotextile behavior, as shown in Equation (11). The proposed equation takes away the limitation of the equation proposed by Pulko et al. (2011), which only considers linear elastic geotextile behavior. After obtaining Δσ_r, the total radial stress (σ_r) and effective radial stress (σ′_r) can be determined by Equations (12) and (13), respectively.

In this study, it is assumed that

• •

  shear along the soil–column interface is negligible

• •

  the top and bottom of the columns are perfectly horizontal

• •

  settlement at the base of the column does not occur

• •

  sands with unconsolidated initial states have a constant ratio between the horizontal stress and the vertical stress during shearing

• •

  sands with consolidated initial states are modeled using the hardening soil model during shearing

• •

  the geotextile material is modeled using a power law

• •

  external soil surrounding the column is not considered

• •

  lateral deformation along the column is taken as the average value.

For simplicity, uniform lateral deformation along the depth of the column was assumed. This approach has been adopted by several researchers for the analysis of traditional and encased columns, including Raithel and Kempfert (2000) and Pulko et al. (2011). In reality, bulging occurs in geotextile-encased columns, especially in the top portion of stone columns. Several researchers have studied the bulging behavior of geotextile-encased columns, including Zhang and Zhao (2015), Kong et al. (2018), and Alkhorshid et al. (2019). Zhang and Zhao (2015) incorporated the bulging characteristics of stone columns into their analytical model by introducing the ‘bulging depth’ into their equation. In that study, it was assumed that depths greater than the bulging depth will have zero lateral bulging and will have soil pressure that is at rest. It should be noted that bulging behavior was not included in the scope of the current study. Rather, it was preferred to distinguish the lateral deformation presented in this study as the average lateral deformation of the column. Nonetheless, several researchers, including Raithel and Kempfert (2000) and Almeida et al. (2013), have shown that the load–settlement behavior of encased columns can be predicted satisfactorily using the uniform lateral deformation approach.

In practice, analytical solutions can serve as a guiding tool to predict the deformation behavior of GESACs. The development of an analytical model allows for a deeper understanding of the interaction between the soil and geotextile. Hence, it is advantageous to generalize experimental results into a form that can be easily adopted in design practice for field applications. Although the FEM can be used to better represent the real-world behavior of encased columns, the analytical model can be used in the initial stages of the design process to determine the probable dimensions of the column, which will lead to cost savings. In a field application, the encapsulating geotextile, together with the in situ soil, generates continuous increasing confining pressure to the column, which is a function of the stress–strain behavior of the geotextile. The combined effect of the surrounding soil is not considered in this study. A nonlinear analytical model to predict the response of a GESAC surrounded by soil will be proposed in a future study. The explanation and derivation of equations in this study are aimed to correct commonly misinterpreted data such as the shear stress (q) and volumetric strain (ε_v), and to provide further interpretations of experimental data including the tension force (T_g), the p–q path, and the mobilized friction angle (ϕ_mob).

In designing GESACs, it is desirable that the geotextile strain during loading does not exceed the strain at failure. In addition, the use of safety factors for geotextile working-load conditions is necessary to account for installation damage, seam strength, chemical degradation, biological degradation, and creep, as discussed by Leshchinsky et al. (1996). Lastly, it is important to consider the axial strain of the GESAC during loading, such that the settlement does not exceed the allowable settlement for the given project. Hence, evaluation of the stress–strain relationship and calibration of the geotextile and soil parameters are essential in the design of GESACs.

The behavior of GESACs in uniaxial compression tests and unconsolidated triaxial tests can be similarly represented by a power law (Equation (14)), as shown in Figure 3. For unconsolidated GESACs in the triaxial test, the effective stress of the soil before shearing is negligible due to the presence of excess pore water pressure. Hence, unconsolidated GESACs in the triaxial test behave similarly to GESACs in the uniaxial compression tests. The two curves in Figure 3 are theoretical curves using exponents n = 0.6 and n = 1.2, which show possible nonlinear behavior of an unconsolidated encased column. The exponent n controls the curvature of the curve, and should be calibrated such that the behavior of the geotextile in the wide-width strip test (ASTM D4595) is well represented. In Equation (14), Δσ_v,f is the change in vertical stress at geotextile failure and ε_1,f is the axial strain at geotextile failure. In this study, Δσ_v,f is related to T_g,f, r₀, ε_g,f, and the coefficient of active lateral pressure (K_a), as shown in Equation (15), wherein K_a can be approximated using the friction angle (ϕ) of the sand (Equation (16)). In addition, ε_1,f is related to ε_g,f and Poisson's ratio of the GESAC at geotextile failure (ν_sg,f), as shown in Equation (17). Combining Equations (14), (15), and (17), ε₁ can be approximated using Equation (18).

Δσ_v is often commonly mistaken as the deviator stress or shear stress (q), such as in the works of Pulko et al. (2011), Kadhim (2016), and Kadhim et al. (2018). However, radial stresses exist due to the effect of geotextile encasement, as given by Equation (11). In this study, the shear stress of the unconsolidated GESAC (q_ucd) is obtained by subtracting Δσ_r from Δσ_v, as shown in Equation (19). Assuming that unconsolidated sands instantly mobilize ϕ during shearing, q_ucd can be approximated based on K_a, as shown in Figure 4. Combining Equations (18) and (19), the governing equation for q_ucd with respect to ε₁ is given by Equation (20). Comparing Figures 3 and 4, notice that Δσ_v, which is commonly mistaken as the shear stress, is larger than the actual shear stress q_ucd shown in Figure 4. Hence, by introducing Equation (20), the discrepancy caused by assuming Δσ_v = q is amended.

To calculate the shear stress of the consolidated GESAC (q_cd), the shear stress of the soil (q_soil) is added to the shear stress of the unconsolidated GESAC (q_ucd), as shown in Equation (21) and Figure 5. As shown in Figure 5, the shear strength of the consolidated GESAC is greater than those that are unconsolidated due to the added shear strength of the soil. The value of q_soil can be approximated based on the hardening soil model (Schanz et al. 2019) using Equations (22)–(24). The hardening soil model is an advanced model for simulating the behavior of both soft soils and stiff soils. It can simulate drained triaxial tests, wherein the observed relationship between the axial strain and the deviator stress is similar to that of a hyperbola. In Equations (22)–(24), E_i is the initial soil stiffness, q_a is the asymptotic shear stress of soil, q_soil,f is the shear strength of the soil, E^ref₅₀ is the reference secant modulus, c is cohesion, p_ref is the reference pressure, and R_f is the failure ratio. The parameter m in Equation (23) is a constant, which has a value ranging from 0.5 to 1.0, as reported by von Soos (1990). More details about the hardening soil model are provided elsewhere (Schanz et al. 2019). Substituting Equations (20) and (22) into Equation (21), the governing equation for q_cd with respect to axial strain (ε₁) is given by Equation (25).

where

The mean stress of a consolidated GESAC (p_cd), σ′_v, and σ′_r cannot be determined in a straightforward manner. To obtain p_cd, Equations (26) and (27) are proposed in this study. In Equations (26) and (27), ε_1-soil,f is the vertical strain that mobilizes q_soil,f, which can be determined using Equation (28), while M is the tangent of the critical state line, which can be obtained using Equation (29). After determining q_cd and p_cd, the linear system of equations shown in Equation (30) is used to determine both σ′_v and σ′_r. Thereafter, Δσ_v and Δσ_r can both be obtained using Equations (7) and (13), respectively. ε_g, T_g, and the coefficient of lateral pressure (K) of the GESAC can then be determined using Equations (11), (10), and (31), respectively. Given that Δσ_r is known and ε_g is unknown in Equation (11), ε_g can only be determined numerically using an approximate solution. After which, only then can Equation (10) be used to determine T_g.

When ε₁ < ε_1-soil,f

When ε₁ ≥ ε_1-soil,f

To fully evaluate the GESAC model, it is necessary to present a procedure for analyzing GESAC triaxial test data. Based on typical triaxial test data, the radial strain (ε_r), geotextile tension (T_g), the change in radial stress (Δσ_r), the mean stress (p), shear stress (q), coefficient of lateral pressure (K), and mobilized friction angle (ϕ_mob) can be determined using the flow chart presented in Figure 6.

Saemangeum silty sand was used in the experiments. The soil samples were obtained from the Saemangeum River estuary near the airport in Gunsan city, South Korea. Several laboratory tests including sieve tests, compaction tests, and basic property tests were conducted; the results are shown in Figure 7 and Table 1. The samples contained a considerable amount of fines (about 22%), with an optimum moisture content of 15% at a compaction energy of 2473 kJ/m³.

The polyester geotextiles used in this study were joined using two different methods. Geotextile 1 was joined using a flat seam with three rows of stitches while geotextile 2 was joined using a flat seam with one row of stitches. Their tensile properties were determined by conducting wide-width strip tests (ASTM D4595). The load–strain relationships of the geotextiles with joints in the wide-width strip test are shown in Figure 8. Fitted curves using Equation (8) are also shown in Figure 8; they match the measured data well using n = 0.7. The properties of the geotextiles are summarized in Table 2. For the sample in the uniaxial compression test, geotextile 1 (joined using a flat seam and three rows of stitches) was used. For the samples in the triaxial compression tests, geotextile 2 (joined using a flat seam with one row of stitches) was used.

Triaxial tests on Saemangeum silty sand with and without geotextile encasement were conducted. The samples in the triaxial test were 14 cm high and 7 cm in diameter. For the samples without geotextile encasement, the silty sand samples were covered at the top and bottom with porous stones and then covered by a rubber membrane. The relative density (D_r) of the samples was approximately 65%. The samples were consolidated under confining pressures of 100, 200, 300, and 400 kPa. For the GESAC tests, the soil samples were first encased by the geotextile and then covered with porous stones and the membrane. The relative density of these samples was about 25–30%. The samples were unconsolidated and consolidated under confining pressures of 10, 50, and 100 kPa.

The sample for the uniaxial compression test was prepared using a PVC pipe mold. A cylindrical geotextile column with an initial circumference of 64.5 cm and a height of about 40.5 cm was placed in the mold. Thereafter, Saemangeum silty sand, with a water content of 18.54%, was pluviated by air and then compacted using a rammer. A sample with a relative density of about 65% was produced. The sample was then installed in a universal testing machine and was loaded until failure was observed. During testing, measurement of the radial strain (ε₃) or area (A) of the specimen was difficult. Hence, the compressive pressure during testing could not be directly determined. To obtain A, the radial strain (ε₃) of the specimen during testing was interpolated using the measured final radial strain (ε_3,f) and the applied vertical loads during testing. After obtaining ε₃, Equation (4) was used to approximate A.

To further investigate the soil–geotextile interaction and to assess the proposed GESAC model, the uniaxial compression test and triaxial test were simulated using the FEM. The commercial program PLAXIS 2D was used for the simulations. An axisymmetric model was used (Figure 9). In the axisymmetric condition, the x-axis was used as the radius and the y-axis was used as the symmetrical axis of the model. For the left-hand boundary, groundwater flow was closed and the deformation was horizontally-fixed. The groundwater flow was also closed for the bottom boundary and the deformation of the bottom boundary was vertically fixed. For the outer boundaries, groundwater flow was open. The outer boundaries were also free to deform. The soil was set with zero unit weight since the specimen height was relatively small.

The behavior of geotextiles can be nonlinear, as shown in Figures 2 and 8. In this study, the geogrid material was used to model the geotextiles: PLAXIS 2D offers four geogrid material types – linear elastic, perfectly linear–perfectly plastic (PLPP), nonlinear elastoplastic (manual input of T_g–ε_g relationship from wide-width strip test) and the time-dependent visco-elastic. It should be noted that PLAXIS 2D does not allow geogrid behavior nonlinearly concaving upward as input data. Hence, for geotextiles with a nonlinearly concaving upward behavior, the linear elastic model was used in this study. Geotextiles can also be modeled using isotropic or anisotropic properties. In this study, EA₁ corresponds to the vertical stiffness while EA₂ corresponds to the circumferential stiffness. For the vertical stiffness of the geotextile, a value of less than 1% of the circumferential stiffness is recommended (Khabbazian et al. 2010; Kadhim et al. 2018). The vertical stiffness is a necessary parameter because it causes the calculation to fail if its value is very large. Hence, its value needed to be reduced so that it did not affect the results of the simulation.

To model the geotextile–soil interface, the strength reduction factor for interfaces (R_inter) was used. R_inter relates the strength of the soil to the strength at the interface, which results in reduced interface friction and cohesion as compared with the friction angle and cohesion of the adjacent soil. Several authors have conducted experiments related to the interface friction angle efficiencies between sands and geotextiles (e.g. Tuna and Altun 2012; Markou 2018; Rizwan et al. 2022). Tuna and Altun (2012) conducted direct shear tests on a Turgutlu sand–geotextile interface and investigated the effect of geotextile opening size and tensile strength on the value of R_inter. They concluded that the opening size and tensile strength had no pronounced effect on R_inter, for which they reported values of 0.7–0.9. Markou (2018) also conducted direct shear tests and investigated the effect of shear box size, geotextile type and properties, and grain size on a sand–geotextile interface. Based on the results of their experiments, they reported R_inter values in the range 0.6–1.0. Rizwan et al. (2022) carried out direct shear tests and vertical pullout tests, and reported R_inter = 0.67–0.97. In this study, R_inter = 0.8 was chosen, which falls in the range of values reported in the literature. However, it should be noted that vertical forces will not develop in the geotextile because geogrid elements cannot sustain compressive forces. Therefore, the effect of R_inter on the simulation of sand columns is negligible. The soil–geotextile interaction occurs in the radial direction, which results in the development of a geotextile hoop force.

Two stages were modeled. The first stage is the consolidation stage, in which confining pressures were activated while the geotextile remains deactivated. The confining pressures (σ₃) were equally applied at the top and side outer boundaries. For the uniaxial compression test, σ₃ was set to zero since there are no confining pressures in a uniaxial test. For the consolidated triaxial tests, the soil was set as drained material. For the unconsolidated triaxial tests, the soil was set as undrained material such that the effective stress of the soil was negligible before shearing due to the presence of excess pore water pressure. The second stage is the shearing stage, in which the vertical pressure was increased while σ₃ was constant. In this stage, both the geotextile and the interface are activated. For the shearing stage, the soil was set as drained material.

Consolidated triaxial tests were conducted on Saemangeum silty sand to investigate its shear strength characteristics. In the tests, drained behavior was observed during the shearing stage as the measured change in pore water pressure was negligible. Based on the results shown in Figure 10, the maximum deviator stresses were found to be about 309, 560, 870, and 1040 kPa for confining pressures (σ₃) of 100, 200, 300, and 400 kPa, respectively. Mohr circles were plotted to determine the peak friction angle (ϕ). Based on the data, ϕ was determined to be 35° for a relative density of about 65%. The parameters of the hardening soil model were also determined based on the measured data and based on general recommendations, as summarized in Table 3. Using the calibrated parameters and using Equation (22), the calculated data are also shown in Figure 10: the hardening soil model was found to be in good agreement with the measured data.

The results of the uniaxial compression test are shown in Figure 11: the sample failed at a compressive pressure of about 2250 kPa and at an axial strain at failure (ε_1,f) of about 18%. The circumference of the sample was measured at the end of the test and it was noted that the circumference was increased by about 6.2%. Hence, ν_sg,f was determined to be about 0.35. The failed sample was investigated visually, as shown in Figure 12. It was ascertained that the cause of failure was at the seams, and most of the circumferential elongation at the end of the test was experienced at the seams. In addition, lateral bulging was not well pronounced.

A simulation using the FEM was then performed to investigate the soil–geotextile interaction. The results in Figure 11 showed nonlinear behavior; hence, using a linear soil model like the Mohr–Coulomb model could result in further discrepancies since the geotextile was assumed as a linear elastic material, as discussed in Section 3.5. The hardening soil model (Schanz et al. 2019) was used for the soil. The soil parameters used in the simulation are provided in Table 3, which were obtained from analysis of the triaxial test. The same parameters were used since the relative densities were the same. For the geotextile, the parameters listed in Table 4 were used for the FEM simulation. The radial stiffness EA₂, which is the same as the geotextile stiffness (J), represents the stiffness of the geotextile along the circumference. A stiffness of 1190 kN/m was determined for geotextile 1 + joint 1 by dividing T_g,f by ε_g,f from Table 2. For the GESAC model proposed in this study, K_a was determined as 0.27 using Equation (16). The exponent n was 0.7, as determined in Figure 8. The geotextile tension force at failure (T_g,f) was 69 kN/m and the geotextile strain at failure (ε_g,f) was 0.058, as specified in Table 2. Poisson's ratio of the GESAC at failure (ν_sg,f) was 0.35, as determined after the uniaxial compression test.

A comparison of the measured and predicted axial strains with compressive pressure is shown in Figure 13. The results predicted by both PLAXIS 2D and the GESAC model were in good agreement with the measured data. Due to the confining effect of the geotextile, lateral stresses developed during compressive loading, which is a similar reaction to that of retaining walls when vertical loads are applied, as shown in Figure 14. Based on Figure 14(a), and dividing the radial stresses by the vertical stresses, K was obtained, as shown in Figure 14(b). For PLAXIS 2D, the value of K started at 1.0 and reduces to a value close to K_a = 0.27. The GESAC model differs from PLAXIS 2D because the GESAC model assumes K to be constant and equal to K_a for unconsolidated GESACs. After determining the vertical and lateral stresses, the relationship between the mean stress (p) and the shear stress (q) was obtained, as shown in Figure 15. The figure shows that both predictions (PLAXIS 2D and the GESAC model) followed the critical state line. Without geotextile encasement, the column would fail at very low compressive pressures. However, due to the confining effect of the geotextile, the p–q path continued to move along the critical state line, allowing the GESAC to carry larger compressive pressures. Figure 16 shows a comparison of the variations of geotextile tension with a change in vertical stress: PLAXIS 2D and the GESAC model showed good agreement with each other. The load at the end of the test (Δσ_v = 2250 kPa) resulted in a hoop force of about 63 kN/m, which was close to the seam strength of the geotextile.

The results of the triaxial compression tests on the GESACs are shown in Figure 17. The change in vertical stress with vertical strain was almost the same for the unconsolidated samples, despite the different confining pressures. In addition, the vertical pressures at the same axial strains were initially larger for samples that were consolidated under higher confining pressures. This phenomenon is due to the added strength of the consolidated soil. Looking at Figure 17, it may seem that the samples consolidated under lower confining pressure were stronger than those under higher confining pressures. However, a larger part of the vertical load was carried by the geotextile, as described by the geotextile tension force shown in Table 5, wherein 42 kN/m was developed in the geotextile for σ₃ = 10 kPa while only 26 kN/m was developed in the geotextile for σ₃ = 100 kPa. The radial strain and geotextile tension force in Table 5 were estimated using Equations (5) and (8), respectively. The geotextile encasement effect can be best described by the photos in Figure 18: the soil without geotextile bulged during loading while the soil encased by the geotextile was laterally constrained.

To simulate the geotextile-encased loose samples (D_r = 25–30%) in the triaxial test and to assess the GESAC model, a friction angle of 27° was used. All the simulations were performed by taking the material as drained during shearing, since the measured pore pressure rise was negligible. An axisymmetric model was used for the FEM simulation, and the boundary conditions for the FEM simulation were the same as the GESAC in the uniaxial compression test. The reference secant modulus E^ref₅₀ was determined to be 11 000 kPa. For the FEM simulation EA₁ = 6.70 kN/m and EA₂ = 670 kN/m; these value were determined from Table 2 for geotextile 2 + joint 2 by dividing T_g,f with ε_g,f to obtain EA₂. For the GESAC model, K_a was determined to be 0.38. The exponent n was 0.7, as determined in Figure 8. T_g,f was 64.31 kN/m and ε_g,f was 0.097, as specified in Table 2. ν_sg,f was assumed to be 0.35, which is the same value obtained from the uniaxial compression test.

Simulations of the geotextile-encased loose sand columns in the triaxial test are shown in Figures 19–22. The figures show that the simulation results were reasonably acceptable. The change in vertical pressure for the unconsolidated geotextile-encased loose sand column was slightly overestimated by the GESAC model, as shown in Figure 19, while Δσ_v for the geotextile-encased loose sand consolidated under a confining pressure of 10 kPa was underestimated by the GESAC model, as shown in Figure 20. The GESAC model, however, showed better predictions at higher confining pressures (Figures 21 and 22) while PLAXIS 2D showed stiffer predictions, especially for a confining pressure of 100 kPa. Nonetheless, the results show that the behavior of consolidated GESACs can be predicted by adding the strength of the soil to the strength of unconsolidated GESACs.

A series of triaxial compression tests was carried out by Wu and Hong (2009) to investigate the response of sand columns encapsulated by geotextiles. The increase in deviatoric stress, the changes in volumetric and radial strains, and the increase in confining pressure generated by the encapsulating reinforcement were measured and analyzed. Wu and Hong (2009) used three geotextiles having highly nonlinear behavior in the wide-width strip test. In the current study, a seamed geotextile with a tensile strength or maximum tension force (T_g,max) of 8.73 kN/m was analyzed; its load–strain behavior is shown in Figure 23. Three different geotextile models – linear elastic, PLPP, and nonlinear models – were used in an effort to show the effect of geotextile properties on the behavior of consolidated GESACs. In the linear elastic model, the geotextile stiffness (J) was based on the strength and strain at failure (= 13.04 kN/m), and showed poor prediction performance in comparison with the other two models. For the PLPP model, the stiffness was increased to 43.2 kN/m to better fit the experimental results. The strain at the maximum tension force (ε_g,max) was greatly underestimated as 20% using this model. For both the linear elastic and PLPP models, n = 1.0 since the stiffness (J) was assumed to be constant. For the nonlinear geotextile model, n = 2.3 was used, and ε_g,max was assumed as 48% to better fit the experimental results. As shown in Figure 23, the nonlinear model showed the best performance out of the three models.

Wu and Hong (2009) used uniformly graded angular quartz sand in their experiments. In that study, triaxial compression tests were carried out on samples having relative densities of 60% and 80%. In the current work, only samples with D_r = 60% were investigated. A summary of the shear strength of quartz sand with D_r = 60% in the triaxial test is shown in Figure 24. The parameters of the hardening soil model for quartz sand with D_r = 60% were then determined based on the measured data, as listed in Table 6. As shown by the calculated curves in Figure 24, the parameters in Table 6 were found to be in good agreement with the measured data.

FEM simulations of the triaxial tests were also performed to further assess the GESAC analytical model. For the simulations using the linear elastic and PLPP models, the stiffness obtained in Figure 23 was used as input for the geotextile material. For the nonlinear model, the T_g–ε_g relationship calculated by the nonlinear geotextile model in Figure 23 was manually input using the elastoplastic model. This was to ensure that the geotextile parameters used by PLAXIS 2D and the GESAC model were the same.

Comparisons of the measured and predicted data are shown in Figure 25. As can be seen, the predictions by PLAXIS 2D and the GESAC model were in good agreement with each other, regardless of the geotextile model used in the simulations. The geotextile model that showed poor prediction performance was obtained with the linear elastic geotextile model, corresponding to the poor fitting results shown in Figure 23. It is interesting to note that the PLPP and nonlinear geotextile models showed good prediction performance. However, upon closer inspection, the prediction trend using the PLPP model showed steeper predictions and performed slightly better than the nonlinear model. The results showed that using a linear model can still result in good ε₁–Δσ_v relationship prediction as long as the geotextile parameter is well calibrated based on experimental results. Although the results are non-favorable to promote the use of a nonlinear geotextile model, the relationship of axial strain against change in vertical stress does not depict the entire behavior of the system. Despite this, it was shown that the GESAC model performed well in predicting the various geotextile behaviors.

It should be noted from Figure 25 that Poisson's ratio of the GESAC at failure (ν_sg,f) varied for the three different geotextile models. These values were obtained by trial and error using the measured axial strain–radial strain relationship (Figure 26). The values of ν_sg,f for each geotextile model, which were quite large, applied for the various confining pressures. The large values of ν_sg,f could be due to excessive bulging of the specimens, especially because the tensile strength of the geotextile in this experiment was quite small (8.73 kN/m) and the geotextile strain could reach as much as 67% before losing strength. Nonetheless, it can be seen that ν_sg,f can be determined from the triaxial test based on measured axial and radial strains.

After determining the radial strains, the tension force (T_g) during shearing was evaluated, as shown in Figure 27. The measured tension force was obtained by means of interpolation using the measured radial strains and the measured load–strain relationship of the geotextile in the wide-width strip test. The figure shows that the PLPP model produced a larger tension force as the axial strain increased. This is because the geotextile was expected to fail at a radial strain of 20%, based on Figure 23. The nonlinear model showed the best trend of all the models, while the linear elastic model showed extremely low values of the tension force. These results show that there is merit in using a nonlinear geotextile model, as it can better predict the actual hoop force of the GESAC. The load–strain behavior of a geotextile material is innate and is not expected to change during shearing of a GESAC. However, for stone columns, the tensile strength of the geotextile may decrease due to puncture.

As a final verification, the measured p–q paths were determined (Figure 28). The p–q paths predicted by the GESAC model using a nonlinear geotextile material are also shown in Figure 28. The measured p–q path of the GESAC in the triaxial test approached the critical state line and followed the critical state line as the shear stress increased. This indicates that the procedure shown in Figure 6 for data analysis of the GESAC in the triaxial test is reliable, and that the ‘apparent cohesion’ concept may not be suitable for geotextile-encased columns. Nonetheless, the assumed p–q path used by the GESAC model was in good agreement with the measured data.

In an effort to further analyze the soil–geotextile interaction of GESACs, a nonlinear model was developed in this study. First, the behavior of GESACs was investigated in uniaxial compression and triaxial tests. Thereafter, FEM simulations were conducted to analyze the soil–geotextile interaction and assess the proposed model. The proposed model was verified based on data on dense GESACs from the literature. The following conclusions were drawn based on the results of this study.

• •

  The behavior of GESACs in uniaxial compression tests and unconsolidated triaxial tests can be represented by a power-law equation using the exponent n obtained from wide-width strip tests.

• •

  The stress–strain curve of a consolidated GESAC can be predicted by superposing the stress–strain curve of an unconsolidated GESAC with the stress–strain curve of soil alone in a consolidated triaxial test.

• •

  Based on FEM simulations, internal lateral stresses developed in the GESAC because of the confining effect of the geotextile. Due to the internal lateral stresses, the tension force on the geotextile increased while the p–q path of the GESAC followed the critical state line.

• •

  The predictions of PLAXIS 2D and the GESAC model were found to be in good agreement with each other, regardless of the geotextile model used in the simulations.

• •

  The results showed that using a linear model can still result in good prediction of the ε₁–Δσ_v relationship as long as the geotextile parameter is well calibrated based on experimental results. Although the results are non-favorable to promote the use of a nonlinear geotextile model, the relationship of axial strain against change in vertical stress did not depict the entire behavior of the system.

• •

  It was shown that Poisson's ratio of the GESAC at geotextile failure (ν_sg,f) can be determined from the triaxial test based on measured axial and radial strains.

• •

  The results showed that there is merit to using a nonlinear geotextile model as it can better predict the actual hoop force of the GESAC.

• •

  The measured p–q path of the GESAC in the triaxial test approached the critical state line and followed the critical state line as the shear stress increased. This shows that the procedure proposed in this study for data analysis of GESACs in triaxial tests is reliable and that the ‘apparent cohesion’ concept may not be suitable for geotextile-encased columns.

The authors declare that there are no conflicts of interest regarding the publication of this paper.

This research was supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (NRF-2021R1A6A1A03045185).

Basic SI units are shown in parentheses

A

area (m²)

c

cohesion (Pa)

D

diameter (m)

D₀

diameter before shearing (m)

D₁₀, D₃₀, D₆₀

grain sizes corresponding to 10%, 30%, and 60% finer, respectively (m)

D_r

relative density (dimensionless)

EA₁

vertical stiffness of geotextile in finite-element model of column (N/m)

EA₂

radial stiffness geotextile in finite-element model of column (N/m)

E^ref₅₀

reference secant modulus of soil (Pa)

E^ref_oed

reference tangent modulus of soil for primary oedometer loading (Pa)

E^ref_ur

reference unloading/reloading modulus of soil (Pa)

E_i

initial soil stiffness (Pa)

G_s

specific gravity of soil (dimensionless)

H

height (m)

H₀

height before shearing (m)

J

stiffness of the geotextile (N/m)

K

coefficient of lateral pressure (dimensionless)

K_a

coefficient of active lateral pressure (dimensionless)

M

tangent of the critical state line (dimensionless)

n

exponent that controls the curvature of the load–strain curve of a geotextile (dimensionless)

p

mean stress (Pa)

p_cd

mean stress of consolidated GESAC (Pa)

p_ref

reference pressure (Pa)

R_f

failure ratio (dimensionless)

R_inter

strength reduction factor for interfaces (dimensionless)

r

radius (m)

r₀

radius before shearing (m)

T_g

geotextile tension force (N/m)

T_g,f

geotextile tension force at failure (N/m)

q

shear stress (Pa)

q_a

asymptotic shear stress of soil (Pa)

q_cd

shear stress of consolidated GESAC (Pa)

q_soil

shear stress of soil (Pa)

q_soil,f

shear strength of soil (Pa)

q_ucd

shear stress of the unconsolidated GESAC (Pa)

u

pore pressure (Pa)

V

volume (m³)

V₀

volume before shearing (m³)

x

coordinate on x-axis (m)

y

coordinate on y-axis (m)

ΔF_v

change in compressive loading (Pa)

Δσ_r

change in radial stress (Pa)

Δσ_v

change in vertical pressure (Pa)

Δσ_v,f

change in vertical stress at geotextile failure (Pa)

ε₁

vertical strain (dimensionless)

ε_1,f

vertical strain at geotextile failure (dimensionless)

ε_1-soil,f

vertical strain that mobilizes the shear strength of the soil (dimensionless)

ε_1,ucd

axial strain of unconsolidated geotextile-encased sand (dimensionless)

ε₃

radial strain (dimensionless)

ε_3,f

measured final radial strain (dimensionless)

ε_g

geotextile strain (dimensionless)

ε_g,f

geotextile strain at failure (dimensionless)

ε_g,max

strain at maximum tension force (dimensionless)

ε_v

volumetric strain (dimensionless)

γ_dmax

maximum dry unit weight (N/m³)

γ_dmin

minimum dry unit weight (N/m³)

ν_sg

Poisson's ratio of GESAC (dimensionless)

ν_sg,f

Poisson's ratio of GESAC at geotextile failure (dimensionless)

ν_ur

Poisson's ratio of soil for unloading–reloading (dimensionless)

σ₃

confining pressure (Pa)

σ′₃

effective confining pressure (Pa)

σ_r

total radial stress (Pa)

σ′_r

effective radial stress (Pa)

σ_v

total vertical stress (Pa)

σ′_v

effective vertical stress (Pa)

ϕ

friction angle (°)

ϕ_mob

mobilized friction angle (°)

FEM

finite-element method

GESAC

geotextile-encased sand column

PLPP

perfectly linear–perfectly plastic