Stress Transfer Mechanism in 2D and 3D Unit Cell Models for Stone Column Improved Ground
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Abstract
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 twodimensional (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.
Keywords
Stone columns Unit cell Stress concentration ratio Arching effect Numerical analysisIntroduction
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, n_{s} is the ratio of the stress in the column, σ_{c}, to the stress in the soil, σ_{s}. 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.
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. n_{s} = 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 n_{s} as load increases, i.e. n_{s} ≈3.9 to n_{s} ≈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
Material properties of unit cell models
Parameter  Stone column  Soft soil  Embankment fill 

γ′ (kN/m^{3})  20  16  18 
\( E_{50}^{ref} \) (kN/m^{2})  40,000  3,000  30,000 
\( E_{oed}^{ref} \) (kN/m^{2})  40,000  2,500  30,000 
\( E_{ur}^{ref} \) (kN/m^{2})  120,000  10,000  90000 
c′ (kN/m^{2})  0.1  0.1  0.1 
ϕ′ (°)  45  25  35 
v_{ur} (−)  0.2  0.2  0.2 
p^{ref} (kN/m^{2})  100  100  100 
m (−)  0.5  1  1 
Results and Discussion
Settlement Performance
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 n_{2} of 3.4 assuming constant stiffness for the columns, E_{c} as 40,000 kN/m^{2} and soft soil, E_{s} as 3,000 kN/m^{2}. 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
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
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 nonlinearly 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.
Effect of Loading Stage
Results of the stress concentration ratio, n_{s} 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.
Conclusion

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, n_{s} 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 n_{s} varied from 4 to 5 throughout the column’s depth except at the column toe where the n_{s} 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.
References
 1.Baumann V, Bauer GEA (1974) The performance of foundations on various soils stabilized by the vibrocompaction method. Can Geotech J 11(4):509–530CrossRefGoogle Scholar
 2.Van Impe WF, De Beer E (1983) Improvement of settlement behaviour of soft layers by means of stone columns. In: 8th international conference on soil mechanics and foundation engineering, Helsinki, pp 309–312Google Scholar
 3.Madhav MR, Van Impe WF (1994) Load transfer through a gravel bed on stone column reinforced soil. J Geotech Eng ASCE 24(2):47–62Google Scholar
 4.Indraratna B, Basack S, Rujikiatkamjorn C (2012) Numerical solution of stone column—improved soft soil considering arching, clogging, and smear effects. J Geotech Geoenviron Eng 139(3):377–394CrossRefGoogle Scholar
 5.Han J, Ye SL (2001) Simplified method for consolidation rate of stone column reinforced foundations. J Geotech Geoenviron Eng 127(7):597–603CrossRefGoogle Scholar
 6.Ng KS, Tan SA (2014) Floating stone column design and analyses. Soil Found 54(3):478–487CrossRefGoogle Scholar
 7.Balaam NP, Booker JR (1981) Analysis of rigid rafts supported by granular piles. Int J Numer Anal Meth Geomech 5(4):379–403CrossRefGoogle Scholar
 8.Castro J, Sagaseta C (2009) Consolidation around stone columns. Influence of column deformation. Int J Numer Anal Method Geomech 33(7):851–877CrossRefGoogle Scholar
 9.Barksdale RD, Bachus RC (1983) Design and construction of stone columns. Federal Highway Administration Office of Engineering and Highway OperationsGoogle Scholar
 10.Bergado DT, Anderson LR, Miura N, & Balasubramaniam AS (1996) Granular piles. Soft ground improvement in lowland and other environments, ASCE Press, New YorkGoogle Scholar
 11.Aboshi H, Ichimoto E, Enoki M, Harada K (1979) The “Compozer”a method to improve characteristics of soft Clays by inclusion of large diameter sand columns. In: Proceeding, international conference on soil reinforcement, pp 211–216Google Scholar
 12.Goughnour RR, Bayuk AA (1979) A field study of long term settlements of loads supported by stone columns in soft ground. In: Proceeding, international conference on soil reinforcement, pp 279–286Google Scholar
 13.Mitchell JK, Huber TR (1985) Performance of a stone column foundation. J Geotech Eng 111(2):205–223CrossRefGoogle Scholar
 14.Kirsch F, Sondermann W (2003) Field measurements and numerical analysis of the stress distribution below stone column supported embankments and their Stability. Workshop on Geotechnics of Soft SoilsTheory and Practice, Essen, pp 595–600Google Scholar
 15.Ambily AP, Gandhi SR (2007) Behavior of stone columns based on experimental and FEM analysis. J Geotech Geoenviron Eng 133(4):405–415CrossRefGoogle Scholar
 16.Greenwood, DA (1991) Load tests on stone columns. In: Deep foundation improvements: design, construction and testing, pp 148–171Google Scholar
 17.Ng KS (2014) Numerical study and design criteria of floating stone columns. Dissertation, National University of Singapore, SingaporeGoogle Scholar
 18.Ichmoto E (1981) Results of design and construction of sand compaction pile method. In: 36th JSCE conference discussion, pp 51–55Google Scholar
 19.Kim TW (2001) Numerical analysis of the behavior of sand compaction pile in clay. Master thesis, Dankook University, (in Korean)Google Scholar
 20.Watts KS, Serridge CJ (2000) A trial of vibro bottomfeed stone column treatment in soft clay soil. In: Proceeding, 4th international conference ground improvement geosystems, Helsinki, pp 549–556Google Scholar
 21.Bergado DT, Panichayatum, Sampaco CL (1988) Reinforcement of soft bangkok clay using granular piles. In: Proceedings, international symposium on theory and practice of earth reinforcement, Kyushu, pp 179–184Google Scholar
 22.Schanz T, Vermeer PA, Bonnier PG (1999) The hardening soil model: formulation and verification. In: Beyond 2000 in computational geotechnics, pp 281–296Google Scholar
 23.Brinkgreve RBJ, Swolfs WM, Engin E, Waterman D, Chesaru A, Bonnier PG, Galavi V (2010) PLAXIS 2D 2010. User manual, Plaxis bvGoogle Scholar
 24.Sexton BG, McCabe BA (2013) Numerical modelling of the improvements to primary and creep settlements offered by granular columns. Acta Geotech 8(4):447–464CrossRefGoogle Scholar
 25.Killeen MM, McCabe B (2010) A numerical study of factors affecting the performance of stone columns supporting rigid footings on soft clay. In: 7th European conference on numerical methods in geotechnical engineering, Taylor and Francis, pp 833–838Google Scholar
 26.Gäb M, Schweiger HF, KamratPietraszewska D, Karstunen M (2008) Numerical analysis of a floating stone column foundation using different constitutive models. Geotechnics of soft soils—focus on ground improvement. In: Karstunen and Leoni (ed) pp 137142Google Scholar
 27.Priebe HJ (1995) The design of vibro replacement. Ground Eng 28:31–37Google Scholar
 28.Madhav MR, Miura N (1994) Soil improvement. Panel report on stone columns. In: Proceeding, 13th international conference on soil mechanics and foundation engineering, vol 5, New Delhi, pp 163–164Google Scholar
 29.Vesic AS (1972) Expansion of cavities in infinite soil mass. J Soil Mech Found Div ASCE 98:265–290Google Scholar
 30.Appendino M, Di Monaco F (1983) Use of the expanded cavity in columns group stability. Improvement of Ground. In Rathmayer HG, Saari KHO (eds) Proceeding, the 8th European conference on soil mechanic and foundantion engineering. Balkema, Rotterdam, pp 335–339Google Scholar
 31.Alamgir M, Miura N, Poorooshasb HB, Madhav MR (1996) Deformation analysis of soft ground reinforced by columnar inclusions. Comput Geotech 18(4):267–290CrossRefGoogle Scholar