Stress Transfer Mechanism in 2D and 3D Unit Cell Models for Stone Column Improved Ground

  • K. S. NgEmail author
  • S. A. Tan
Original Paper


Stone column has been used widely to improve the foundation for many structures. Many designs of stone column are based on the unit cell concept. However, the intrinsic mechanism of stress transfer between the column and the surrounding soil has not been investigated thoroughly. This paper presents the important features of stress sharing mechanism in unit cell concept under an embankment loading. The arching effect, the deformation mode, the stress concentration ratio and the plastic straining in the unit cell are the main focus of this paper. Finite element software PLAXIS was used to examine these features. Unit cell was simulated as a two-dimensional (2D) axisymmetrical model and a representative three dimensional model in the numerical analysis. Drained loading condition was analyzed in this study in which the embankment is assumed to be built slowly with no excess pore pressure buildup. The change of the stress concentration ratio as the embankment height increases was also studied. From this study, it was found that the bulging happened near the column head accompanied by multiple shear bands progressing along the column. Generally, stone column in the unit cell shared about 4–5 times more the loads than the surrounding soils throughout the column depth. In most cases, 2D and 3D models give results that are similar to each other especially on the settlement performance and the failure mechanism.


Stone columns Unit cell Stress concentration ratio Arching effect Numerical analysis 


Stone column is a very popular ground improvement method to improve the subsoil condition. Its ability to increase the bearing capacity, reduce settlement and speed up the consolidation process has made it the top choices among the geotechnical experts when considering alternative solutions for soft soil problems. The method requires replacing of the weak subsoil with stiffer granular materials of typically 10–40 %. The application of stone column includes the road embankment, building foundations, bridge abutment, chemical or oil storage tanks and reservoir.

Most of the stone column designs have adopted the unit cell concept [1, 2, 3, 4, 5, 6]. The unit cell model comprises a single stone column and its equivalent circular influence zone. It is used to represent a column located on the interior of an infinitely large group of stone columns. The idealization is made to simulate the case of rigid raft or large uniform loaded area as in the case of embankment supported on soft soils with uniformly spaced stone column group. The equivalent diameter for triangular, square and hexagonal arrangement of stone columns are 1.05 s, 1.13 s, and 1.29 s respectively, where s is the spacing of the columns [7].

Since the load and geometry are symmetrical in the unit cell, the boundary conditions at the outer wall are: zero shear stress, zero radial displacement, and no water flow [8]. Following these assumptions, total stress applied on the top of the unit cell must remain within the unit cell although the stress distribution between the column and the soil can be varied with depth [9]. Uniform loading applied over the unit cell is analogous to one dimensional (1D) consolidation test [10].

Since the stone column is stiffer than the native soil, concentration of stress occurs in the stone column with accompanying reduction of stress in the surrounding soil [11]. The stress concentration ratio, ns is the ratio of the stress in the column, σc, to the stress in the soil, σs. The stress distribution occurs when the settlement of the column and the surrounding soil is roughly equal. The stress concentration ratio is the most important factor in the unit cell concept. However, there is no rigorous solution available to give a rational estimate of this ratio, so that it has to be chosen either by empirical estimation on the basis of field measurements by means of load tests using earth pressure cells or from an engineer’s experience. This ratio is important in predicting the beneficial effects of stone column reinforced ground especially in the settlement and stability analysis.

Aboshi et al. [11] proposed the average stress, σ over the unit cell area corresponding to a given area replacement ratio as:
$$ \sigma = \sigma_{s} \alpha + \sigma_{c} \left( { 1 - \alpha } \right) $$
Area replacement ratio α = Ac/(Ac + As), where Ac and As are cross-section areas of the column and the surrounding soil i.e. clay respectively. The stresses in the clay, σc and stone column, σs are given as:
$$ \sigma_{c} = \frac{\sigma }{{\left[ {1 + (n_{s} - 1)\alpha } \right]}} = \mu_{c} \sigma $$
$$ \sigma_{s} = \frac{n\sigma }{{\left[ {1 + (n_{s} - 1)\alpha } \right]}} = \mu_{s} \sigma $$
where μc and μs are the ratio of stresses in the column and clay, respectively, to the average stress, σ over the tributary area.

Numerous publications have shown that the steady stress concentration ratio for stone column reinforced foundations is typically in the range of 2–6, with the usual values of 3–4 [12, 13, 14, 15]. On the other hand, Greenwood [16] reported a much higher ratio, i.e. ns = 25 being measured in very soft clay at low stress level. The effect of applied loading on the stress concentration ratio has been examined by some researchers. Ng [17] reported a very small increase in the ns as load increases, i.e. ns ≈3.9 to ns ≈4 for q = 50 kPa to q = 400 kPa. Ichmoto [18] and Kim [19] drew the same conclusions while other researchers like Watts et al. [20] reported the increase of the stress concentration ratio due to the increase of loads based on a field load test but Bergado et al. [21] suggested the opposite trend.

This paper focuses on the load transfer mechanism between the column and the soil in the unit cell model. The change of stress concentration ratio with load is also examined. This paper also highlights the differences in the results for two dimensional (2D) and three dimensional (3D) numerical models.

Numerical Model and Analysis

In this study, a problem where stone column is used to support a large embankment was investigated. The columns were installed in a square grid with spacing, s = 1.5 m supporting an embankment of 5.0 m high. The 10.0 m long column was fully penetrating and resting on a rigid stratum. The column diameter is fixed at 0.85 m and the area replacement ratio, α was calculated to be 0.25. Numerical analysis was performed using the finite element program PLAXIS 2D 2012 and PLAXIS 3D 2011. The 2D axisymmetric model using 1164 15-node triangular elements was built with equivalent diameter, De of 1.70 m. While in 3D analysis, a representative unit cell consists of 11432 10-node tetrahedral elements. Mesh sensitivity studies were performed to ascertain that the mesh was fine enough to give reliable results for the models concerned. Roller boundaries were used for the sides and fixed both horizontally and vertically for the bottom boundary. The phreatic level was set at the top surface. Figure 1 shows the 2D and 3D models used in the analysis.
Fig. 1

Numerical models used in the analysis a 2D, and b 3D

The hardening soil (HS) model was chosen in this study. The HS model is an advanced elasto-plastic hyperbolic type model which is capable of capturing both the compression and shear hardening [22]. The HS model adopts Mohr–Coulomb failure criterion and the yield surface can expand due to plastic strains. The material properties of the unit cell model are shown in Table 1. The friction angle of stone column, \( \phi_{\text{c}}^{\prime } \) was taken as 45° with non-associated flow rule (ψ = 0). The value of dilation angle, ψ is less certain since it is much affected by the confining stress of the surrounding soil and it is also very difficult to separate out the effect of high friction angle and dilation angle for an actual constructed stone column. To avoid numerical instability, the cohesion value of 0.1 kN/m2 was used. Typical soft soil stiffness, E50 was adopted in this study. The columns stiffness parameters are more than ten times stiffer than the surrounding soil (reference pressure taken as 100 kN/m2) which is within the typical range. Barksdale and Bachus [9] stated that the typical elastic modulus ratio, Ec/Es of stone columns range from 10 to 20, where Ec is the Young’s Modulus of the column, and Es is the Young’s modulus of the soft soil. The HS model possesses the ability to capture the stress dependency of soil stiffness, and it is controlled by the parameter ‘m’. The value of m = 1 is appropriate for the soft soil [23] but for the stone column, the m value is an assumed value since it has not been measured before but probably between 0.3 and 0.5. The surrounding soft soil was assumed to be normal consolidated with OCR = 1.0. Initial stresses were generated by Ko procedure with the proposed value of lateral earth pressure, K = 1.0 for both column and soil reflecting wish-in-place approach adopted in the model. This approach has also been adopted by many researchers [24, 25, 26]. Drained analysis was performed in this study where the embankment fill was assumed to be constructed slowly and there was no excess pore pressure generation in the model.
Table 1

Material properties of unit cell models


Stone column

Soft soil

Embankment fill

γ′ (kN/m3)




\( E_{50}^{ref} \) (kN/m2)




\( E_{oed}^{ref} \) (kN/m2)




\( E_{ur}^{ref} \) (kN/m2)




c′ (kN/m2)




ϕ′ (°)




vur (−)




pref (kN/m2)




m (−)




Results and Discussion

Settlement Performance

The deformation behavior is first investigated. Figure 2 shows the settlement induced by the embankment loading. Excellent agreement is obtained for 2D and 3D models. The final settlement under 90 kPa embankment loads is about 260 mm (measured at the centre of the column). Due to the nature of embankment loading, it is not a pure rigid condition but an intermediate between the ‘equal strain’ and ‘free strain’ state. This is observed when the settlement of the soft soil is more than for the column (about 20–30 mm) as shown in Figs. 3 and 4. Line A, B and C in Fig. 4 are the outer boundaries for the column, 2D model and 3D model respectively, in which the x-distance is measured from the centre of the column. Distance to C boundary is from the column’s centre to the corner of the unit cell in the 3D model. The difference of results for 2D and 3D is very small (<2 mm) and in the practical point of view, the difference can be ignored.
Fig. 2

Load-settlement curve for 2D and 3D models at centre of column

Fig. 3

Settlement shading for 3D model

Fig. 4

Settlement profile at ground surface

Separate numerical analysis on the soft soil performance without stone column was carried out and the final settlement obtained was 594 mm. This indicates that with stone column inclusion, the reduction of settlement is about 56 % or equivalent to settlement improvement factor, n of 2.3. Settlement improvement factor is the ratio of the settlement without improvement over the settlement with improvement. For comparison, calculation with Priebe [27] method gives final settlement improvement factor n2 of 3.4 assuming constant stiffness for the columns, Ec as 40,000 kN/m2 and soft soil, Es as 3,000 kN/m2. This is much higher than the value obtained in the present study. On the other hand, a simplified design method proposed by Ng and Tan [6] predicts n equal to 2.3, a result similar to this study. The method was developed from two dimensional numerical analysis using Mohr–Coulomb model for the soil materials in a unit cell. This method takes into accounts the influential parameters such as the column’s length, column’s friction angle, area replacement ratio, loading intensity and post installation lateral earth.

Deformation Modes

The deformation mode for stone column can be seen in Fig. 5a. Bulging is noticed in both the 2D and 3D models near the upper part of the column. The bulging of the stone column takes place when the applied load is higher than the confining stress. The surrounding soil provides some lateral support to prevent further expansion of the column. The confining stress increases with depth, so the bulging occurs in the upper part of the stone column [28]. Other than the bulging mode, multiple shear plane is found in the column with the initial shear band develops at the edge of the column and inclined toward the centre forming a shear cone (Fig. 5b). The shear band developed at \( 45^\circ \; + \;\phi_{c}^{\prime } /2 \) (where \( \phi_{c}^{\prime } \) is the friction angle of stone column) and extend downward with zip-zap pattern.
Fig. 5

a Deformation mode, b incremental strain, and c horizontal displacement profile

Horizontal displacement profile along the column edge depicted in Fig. 5c shows that the maximum displacement in 2D model is slightly larger and located slightly higher compared to 3D model. The magnitude of maximum bulging is less than 15 mm or about 3 % radial strain in the column and it happens at 0.5 m below ground surface. Below that, the radial strains are small and may not be able to fully develop the passive resistance along the column. Few methods have been proposed to estimate the ultimate bearing capacity of the improved ground assuming full bulging along the column length by adopting cylindrical cavity expansion theory [9, 29, 30]. However, it is doubtful to use these methods for columns under wide spread loading since the failure has occurred in the upper part of the column before full passive resistance are mobilized in the lower part of the column. On the other hand, Priebe [27] also assume constant bulging along the column using cavity expansion theory to predict the settlement improvement factor. Again, observation in this study shows the bulging is not constant along the column length and this may be the reason for the overestimation in the settlement improvement factor obtained using Priebe’s method.

Load Transfer Mechanism

In stone column supported embankment, arching in the embankment is an important mechanism for load transfer, results in less differential settlement on the surface. Figure 6a shows the principal stress due to an arching effect where the stresses are redistributed. Arching initiates a reduction in the total stress on the soft soil but an increment on the column. The orientation of the principal stress shows inverse catenary-like shape arches. The arching height is roughly at 0.8 m for both 3D and 2D models. The arching height is much dependent on the flexibility of the embankment. Moreover, the differential vertical displacements in the embankment only occur up to a certain height, beyond that the uniform settlement can be seen as shown in Fig. 6b. Similarly, uniform settlement is also noticeable in the column and the surrounding soil at some distance below the ground surface.
Fig. 6

a Orientation of principal stress, b total displacement shading

Through arching action, the embankment loads are largely transferred to the column due to high column’s stiffness. Figure 7 shows the stress distribution in the unit cells for the 2D and 3D models at the ground surface. The stress concentration ratio, ns was calculated to be about 6.5 and 4.1 for 3D and 2D models respectively, which falls within normal range. The changes of the stress concentration ratio over depth are shown in Fig. 7a. There is a slight increment (ns = 4–5) in this value for 2D model throughout the column depth (except at the column toe where ns = 6.0 was recorded) but a reduction in the value is noticed in 3D model near the column head to a point where 2D and 3D results coincide with each other and it increases slightly afterwards. It can be suggested that the stress concentration ratio of 4–5 may be a good value to be adopted in the design to predict the settlement of the embankment foundation.
Fig. 7

Stress distribution in a 3D model, and b 2D model

At the depth of 5.0 m (referring to Fig. 8b), the 3D results gives the column stress, σc of 217 kN/m2 and the soil stress, σs of 48 kN/m2, while Eqs. 2 and 3 give σc = 218 kN/m2 and σs = 47 kN/m2 using ns = 4.62 (from 3D results). The good match of results has further validated the simple analytical method proposed by Aboshi et al. [11]. in calculating the average stress in the stone column and its surrounding soil. Figure 8 also shows higher stress concentration ratio at the column toe due to the remaining load that cannot shed along the side of the column.
Fig. 8

a Stress concentration ratio profiles. b Stress increment profiles

Alamgir et al. [31] proposed an elastic method to predict the load sharing for the improved ground assuming ‘free strain’ condition in the unit cell. In their study, the stress concentration ratio was found to vary from unity at the surface and after that it increased non-linearly with depth up to the bottom. This finding is unacceptable since the stress concentration ratio is never a unity value measured on the ground surface based on the result obtained in the current study and also from the actual measurement at field and laboratory tests.

The yielding in the column with substantial Mohr–Coulomb points near the upper part was observed as shown in Fig. 9. Deeper along the column, the stress state in the column has not reached Mohr–Coulomb failure line but significant shear hardening (expanding shear locus) is occurred in the 2D and 3D models where 2D model indicates additional compressive hardening, a cap yield surface. On the other hand, no yielding is occurring in the soft soil as the stress state is still within the elastic region. The assumption made by Priebe [27] considering only plastic deformation for column and elastic deformation for surrounding soil could be true at least for the range of loading applied in the present study.
Fig. 9

Plastic points in the numerical models

Effect of Loading Stage

In this study, the effect of loading stage was examined. The 5.0 m embankment was constructed in ten stages where each stage consists of 0.5 m embankment height, simulated in drained condition. The stress concentration ratio was measured at the ground surface and 5.0 m below the surface. Figure 10a shows that the 3D model gives overall higher stress concentration ratio than 2D models. The rate of increment in the stress concentration ratio is significantly higher in the early stage and stabilizes thereafter. This is due to the bulging and plastic straining that slowly developed until a state where a full lateral restraint and substantial yielding have achieved near the upper column. The kink in the 3D curve happened at stage 5 was not readily understood, possibly due to the shearing plane that cut through the column material that reduces the sustained load carried by the column.
Fig. 10

Change of the stress concentration ratio with the increasing of embankment height at ad = 0 m, and bd = 5.0 m

Results of the stress concentration ratio, ns at depth 5.0 m are the opposite compared to that of ground surface. Higher values are obtained for the 2D model than the 3D model as indicated in Fig. 10b. The 2D axisymetrical model takes advantage of the symmetry but sometimes the intrinsic mechanism such as the continuous shearing plane that occurred in the column as seen in the 3D model cannot be reproduced correctly by the axisymmetrical model. This roughly explains why there are differences in the 2D and 3D results. On the other hand, the increment of embankment height produces a reduced trend in the stress concentration ratios, more steeply in the 2D model than that in the 3D model. This phenomenon is also due to the increase of plastic straining in column, as explained in the above paragraph which reduces the proportion load sharing on the column. In the early stages of construction, the lower stress concentration ratio obtained at ground surface compared to the value at depth of 5 m may be attributed to the low surcharge at the beginning that has not caused the full arching effect to develop.


Numerical analysis was performed to investigate the load sharing mechanism between column and surrounding soil for unit cell in 2D and 3D models under drained condition. In general, 2D model results agree very well with 3D model despite some discrepancy in the details. A few conclusions can be drawn from this study:
  • Bulging is observed in the numerical models near the column head. Shear bands developed from the top of the column and progressing downwards.

  • Soil arching reduces the vertical stress acting on the relatively soft soil while increasing the vertical stress acting on the stone column. Arching height is about 0.8 m.

  • The stress concentration ratios, ns for 3D and 2D models are 6.5 and 4.1 respectively at the ground surface. Generally, the stress concentration ratio increases with depth and the ns varied from 4 to 5 throughout the column’s depth except at the column toe where the ns is about 6.0.

  • Stone column experiences substantial plastic straining compared to the surrounding soil where the stress state is still within the elastic region.

  • Increment of embankment height results in the increase in the stress concentration ratio at the ground surface especially in the early loading stage. On the other hand, the stress concentration ratio at 5.0 m below the ground surface reduces when the embankment height increases.


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Copyright information

© Springer International Publishing AG 2014

Authors and Affiliations

  1. 1.Faculty of Civil EngineeringUniversity Teknologi MARAPermatang PauhMalaysia
  2. 2.Department of Civil and Environmental EngineeringUniversity of SingaporeSingaporeSingapore

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