analytical method for the analysis of stone-columns according to the rowe dilatancy theory
boStjan pulko and bojan majes
About the authors
Boštjan Pulko
University of Ljubljana,
Faculty of Civil and Geodetic Engineering
Jamova 2, 1000 Ljubljana, Slovenia
E-mail: bostjan.pulko@fgg.uni-lj.si
Bojan Majes
University of Ljubljana,
Faculty of Civil and Geodetic Engineering
Jamova 2, 1000 Ljubljana, Slovenia
E-mail: bojan.majes@fgg.uni-lj.si
Abstract
In this paper a new analytical method to analyse the behaviour of rigid foundations stabilized by end bearing stone-columns is proposed. The stone column and the surrounding soil are treated in axial symmetric conditions as a unit cell. The stone column is assumed to behave as an Mohr-Coulomb rigid-plastic material with non-associative flow rule according to the Rowe stress dilatancy theory and the soil as an elastic material. These common assumptions, combined with equilibrium and kinematic conditions, lead to the simple analytical closed-form solution for the prediction of the behaviour for rigid footings resting on stone-column reinforced ground. The parametric study is presented to show the effect of dilatancy of the granular material on the deformations and stresses in the ground and its beneficial effect on settlement reduction. The results of the new method are compared with some already known analytical methods and some published field test results and observations.
Keywords
ground improvement, stone columns, ground settlements, dilatancy theory
1 INTRODUCTION
Stone columns or granular piles are frequently used for the stabilization of soft clays and silts and loose silty sands with large amount of fines. For low-rise buildings, highway facilities, storage tanks, embankments, bridge abutments and other structures that can tolerate some settlements, stone columns are one of the most frequently used methods of support due to the cost, effectiveness and ease of the installation. The beneficial effects of stone columns are increased stiffness, reduced settlements, increased time rate of settlements, increased shear strength and reduction of the liquefaction potential of soft ground.
The available methods for the estimation of the behaviour of foundations resting on soft soil stabilised by a large number of end-bearing stone-columns can be classified as either approximate methods with important simplifying assumptions or sophisticated methods based on elasticity and/or plasticity theory such as finite element model.
The majority of the proposed approximate methods assume infinitely wide, loaded area with end-bearing stone columns having constant diameter and spacing. For such loading and geometry conditions the stone column and the surrounding soil can be treated in axial symmetric conditions. This approach is commonly known as a unit cell concept and was adopted by several researchers.
Several approximate analytical solutions are available to estimate the settlement reduction of stabilized ground and stress concentration in the stone-columns. Many of them [1-3] are based on elastic approach considering the stone-column and the surrounding soil as elastic materials. However, elastic methods give the ratio between the vertical stress in the column and in the soil (also known as stress concentration factor) approximately equal to the ratio of constrained modulus of both materials. This ratio was found to be considerably higher than measured in the field and it is believed that elastic methods may easily overestimate the effects of stone columns on settlement reduction [4].
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b. pulko & b. males: analytical method for the analysis of stone-columns according to the rome dilatancy theory
The elastic and elasto-plastic solutions presented by Balaam and Booker [2, 5] indicate that the problem can be idealized by assuming that the stone column is in a triaxial state and perhaps yielding, that there is no shear stress at the stone-soil interface, and that there is no yielding in the soil. These common assumptions have been implemented in many methods where stone column is considered to be in a state of plastic equilibrium and under a triaxial stress state [5-8]. In the majority of these methods it is assumed that when loaded, stone-column yields at constant volume. However, in the work by Van Impe and Madhav [8] the nonlinear analytical solution is presented, showing the beneficial effect of the stone-column dilation on the deformation behaviour of the stabilized ground.
The objective of this paper is to present an analytical closed-form elastic-rigid-plastic solution which takes into account confined yielding of the stone material according to the Rowe stress-dilatancy theory [9] and to show the beneficial effects of dilation on the settlement reduction.
2 METHOD OF ANALYSIS
If stone-columns are evenly distributed, a regularly shaped area around the stone-column may be consid-
ered as a "unit cell", consisting of stone-column and the surrounding soft soil in a zone of influence (Fig. 1). To simplify the analysis the zone of influence is approximated by a circle with a diameter de equal to 1.055, 1.135 and 1.295, for triangular, square or hexagonal pattern, respectively, where 5 is the column spacing.
The ratio between the area of column Ac and the area of the zone of influence Ae is represented by the area replacement ratio Ar, defined as
A = A = < = rl
r A d2 r2
(1)
Let us consider a unit cell on smooth rigid base loaded with uniform load through the smooth rigid raft. The high drainage capacity of the granular material ensures that it deforms under drained conditions. The immediate settlement of soil in undrained conditions is negligible compared to the total final settlements, and thus it will not be considered in the analysis [5].
It is assumed that the dense granular material in the column is in triaxial stress state, reaching its peak resistance and thus dilating. The self weight of the soil and the column is neglected, which is one of the main drawbacks of the proposed method.
Figure 1. Basic features of the model based on regular patterns of stone-columns
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b. pulko & b. males: analytical method for the analysis of stone-columns according to the rome dilatancy theory
Under uniform load qA applied through rigid raft the end bearing stone-column and the surrounding soil will undergo the same vertical displacement uz and radial displacement ur , thus at the soil-column interface no slippage is expected between the soil and the granular material. The vertical, radial and volumetric strains of the stone-column are defined as
^ = ^ (2) z H
u
er = - u (3)
erd = ez + 2er (4)
where H is the height and rc the radius of the column.
The relation between vertical and radial stress at the soil-column interface, azc and arc, in triaxial stress state can be simply obtained for the column material at yield:
1 +	c K
— -:—T-Kpc
arc 1-	c
(5)
where p\ represents the peak triaxial shear angle of the column material. According to the Rowe stress dilatancy theory [9], Equation (5) can also be modified to:
1--
1 + SÎn^ 'q
1- SnP
(6)
selected column material, while the ratio between the contained plastic strains of the stone-column material, svd and £z, is determined by the selected value of the dilation angle f.
The soil surrounding the stone-column can be analysed as a thick cylinder using Equations (9) and (10) relating vertical and radial displacements, uz and ur, at the soil-column interface with vertical stress in the soil and radial interaction stress, a and a [11]:
H
E 1
C2(Jzs - C1^rs

rc	- h°zs

(9)
(10)
where Eoed is the oedometric modulus of the soil and C1, C2 and C3 are constants defined as

2k0 Ar	n	1-2vs + Ar
Y-A"' 2 =~;—TT^—^> C3 = C2 -koCi (11)
(1-Ar )(1-Vs )
where vs is Poisson's ratio of the soil and k0 = vs /(1 — vs).
Applied vertical load qA must be in equilibrium with vertical stresses in the column and in the soil:
ZaA = OzcK + ^ (A - Ac ) (12)
Ur =
E
a
a
£
z
where p 'cv is a triaxial shear angle of the stone material at constant volume. The relationship between the angle of dilatancy f and the peak friction angle p\ can be obtained using Rowe's equation [9]:
sin ^ =
sin^ c - sin^ cr l-sin^ sin^
(7)
Angle of dilatancy f can also be expressed in terms of the volumetric strain due to dilation and vertical strain of the column [10]:
sin ^ = —
2 — ^L

2e„ — e.„
(8)
The stress-strain behaviour of the column is entirely defined by Equations (5) and (8), and the two material parameters, which can be arbitrary selected between p \ f and p 'cv. The ratio between vertical and horizontal stresses in the column is defined by the strength of the
Using the definition of the replacement ratio AR given by Equation (1), Equation (12) can be rewritten as
qA = azcAr + azs (1-Ar ) (13)
The stresses at the soil-column interface must be equal, thus ars = ac = ar. If kinematic relations (2), (3) and (4) are introduced in Equation (8), Equations (5), (8), (9), (10) and (13) represent a set of five equations for five unknowns: vertical stresses in the column a and in the
zc
soil a , radial stress at the soil-column interface a, verti-
zsJ	r
cal displacement uz and interface displacement ur. This set of equations can be solved to obtain simple analytical closed-form solutions for displacements and stresses:
2 HqA
Eoed C4
u = -
waKl
Eoed C4
(14)
(15)
e
e
e
z
e
z
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b. pulko & b. males: analytical method for the analysis of stone-columns according to the rome dilatancy theory
qA (C2 K, + 2fc0)
=-c- ()
C4
qAKpc (C2K, + 2k0) ^ =-~C- (17)
C4
=(18)
where constants K and CA are defined as follows:
y	4
K * =
1 + sin * 1 — sin *
C4 = (1 - Ar )(C1Kf + 2) + ArKpc (C2 Kf + 2k0)
Constant C4 depends only on material and geometrical properties of the column and the surrounding soil.
If the area of the applied load is sufficiently large, then the settlement of the untreated soil can be estimated as
Uz,0 =	(20)
Eoed
Combining Equation (20) with Equation (14), a settlement reduction factor ft, which is usually used as a measure for the improvement of the ground, can be calculated as
ß = ^ = f
Uz,0 C4
(21)
Stress concentration factor q defined as a ratio between vertical stresses in the soil a and in the column a can
zc	zs
be calculated as
Kfc (c Kf + 2kJ
n = — = —--- (22)
^ CK- +2 v ;
Similarly, stone-column stress concentration factor nc defined as a ratio between vertical stress in column azc and the applied load qA can be calculated as
Kpc (C2 K, + 2fc0)
nc = — = —--- (23)
qA

According to the above analysis the settlement reduction and the stress concentration factor depend mainly on area replacement ratio Ar , on material properties of the column material represented by the peak shear angle p 'c, the angle of dilatancy f and on Poisson's ratio vs of the soil.
3 RESULTS AND DISCUSSION
3-1 PARAMETRIC STUDY
A parametric study has been made to show the effect of area replacement ratio Ar , peak shear strength p 'c and especially the effect of dilation on settlement reduction and stress concentration. In stone column construction usually 15 to 35 percent of the soft soil is replaced [4]. However, the replacement ratios Ar from 5 up to 50 percent were considered in the present study.
When selecting basic input parameters of the method, such as peak and dilation angle of the column material, one should consider that the values for peak, critical and the dilation angle of the granular soil are not independent. The critical state shear angle p 'cv of granular material, which is sheared under constant volume, is in general a function of mineralogy and can be treated as material property. The relationship between the peak shear angle p \ , the critical state shear angle p 'cv and the angle of dilatancy f is theoretically given by Equation (7), which can be rewritten in the following form:
sinw ' + sin é sinp\ =	. é (24)
1 + sinp cv siné
In practice the difference between peak and critical state shear angle can also be correlated to the material's relative density and principal stress. According to the work of Bolton [12] the peak triaxial shear angle correlates to the relative density of the granular material and to the mean effective stress p ' as follows:
<P c
'V ' cv + 31R
(25)
where IR is a relative dilatancy index defined as
IR = Dr (10 - ln p')-1 (26)
The relative dilatancy index IR can also be used for the prediction of the angle of dilatancy as proposed by Schanz and Vermeer [10]:
sin * =
L
6.7 + IR
(27)
In the case that no laboratory data are available on the dilatancy behaviour, the strength and dilation properties of the column material can be estimated by the above relationships.
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B. PULKO & B. Mflies; ANALYTICAL METHOD FOR THE ANALYSIS OF STONE-COLUMNS ACCORDING TO THE ROUE DILflTflNCY THEORY
To investigate the effect of the dilation of the column material on the displacements and stresses, the angle of dilatancy equal to y = 0°, 5°, 10°, 15° and the representative value of the critical state shear angle p \v = 35° were adopted. Another input parameter of the method is Poisson's ratio of the soil v, which was selected to be 0.35.
The results of the parametric study are presented in Figures 2, 3, 4 and 5. The effect of area replacement ratio Ar and of angle of dilatancy y on settlement reduction factor p is shown in Figure 2. The spacing of the columns has dominant effect on settlement reduction. As the spacing of the columns increases, the replacement ratio A decreases, the unit cell becomes less stiff and the total settlement increases. The settlement reduction factors are generally low and not significant for area replacement ratios lower than 4 percent (d id > 5).
Figure 2. The effect of area replacement ratio Ar and angle of dilatancy y on settlement reduction factor ft
The angle of dilatancy has also significant effect on settlement reduction factor ft, clearly showing the importance of stone densification. High density of the column material yields high dilatancy index, hence higher dilation angle y could be achieved. The higher the value of dilation angle y, the greater is the peak shear angle p\ and more reduction of the settlement could be expected due to higher stress concentration in the stone-column (Fig. 2). However, the effect of the dilation of granular material at yield can not be clearly distinguished from the beneficial effect of peak shear strength of the stone column.
The effect of the dilation on settlement reduction is far more evident, if a constant value of peak shear strength is considered (p \ = 46,5°) and different dilation properties of the granular material are taken into account. The
effect of dilation angle y on the settlement reduction for this case is depicted in Fig. 3. The volume increase of the granular material at yield has significant effect on the settlement reduction. For example, for the area replacement ratio Ar between 0.15 and 0.35 the total settlement for dilating stone-column (y = 15°) is 16.5 to 28.0 percent lower than compared to the settlement when no dilation is taken into account.
Figure 3. The effect of area replacement ratio Ar and dilation angle y on settlement reduction factor p
Figure 4. The effect of area replacement ratio Ar and angle of dilatancy y on stress concentration factor n
The effect of area replacement ratio Ar and angle of dilatancy y on stress concentration factor n is shown in Fig. 4. The importance of the dilation of the column material on the stress concentration in the column is clearly indicated. Well densified stone-column with high dilation angle y acts stiffer and can take greater proportion of the applied load. Stress concentration factors n are generally in the range from 3 to 8 for the area replacement ratios from 0.15 to 0.35.
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1.0
0,8-
0,6-
0,4-
0,2-
0,0
1 1 1 1	1 l 1 I 1 v =0.1 s
	—O— v = 0.2 s
	^ v =0.3 s
	—v = 0.4 s
<p' = 35° ^^ T cv w =15° <p'c = 46.5°	
0,0
0,1
0,2
0,3
0,4
0,5
Figure 5. The effect of area replacement ratio Ar and Poissons ratio of the soil on settlement reduction factor ft
The effect of Poissons ratio of soil is illustrated in Fig. 5, where settlement reduction factor ft is depicted as a function of Poissons ratio against the area replacement ratio A . The value of Poisson's ratio v of soil has rather
r	s
small effect on the settlement reduction, if the soil is considered under drained conditions. The effect of Poisson's ratio on the stress concentration factor n is also small.
3.2 COMPARISON WITH OTHER METHODS
It is interesting to compare the settlement reduction factors, obtained by the proposed analytical method, with some other known analytical solutions, based on similar approaches. A comparison was firstly made with elastic methods proposed by Aboshi et al. [1] and Balaam and Booker [2] and then with elastoplastic methods, which take into account the yield of the stone-column at constant volume as proposed by Priebe [6], Van Impe and De Beer [7], and finally with the method proposed by Van Impe and Madhav [8], which takes into account the yield and the dilation of the stone-column.
A direct comparison between the results is not possible because of different input parameters. However, a comparison can be made for a regular set of parameters for stone column material and for soil. The stone and soil modulus ratio EJEs = 30, critical state shear angle P 'cv = 35°, dilation angle f = 15° and Poisson's ratio vs = 0.3 were adopted in the analysis. This combination of the critical shear angle and dilation angle of the column material leads to the peak shear angle P \ = 46.5°. This standard set of parameters was used in the prediction of settlements according to the above mentioned methods.
Figure 6. The comparison of settlement reduction factors according to different analytical methods
The comparison of settlement reduction factors is shown in Fig. 6. The differences in the settlement reduction factors obtained by different methods are quite significant, even if the general trend of the results is found to be similar.
The elastic methods [1, 2] give low settlement reduction factors due to supposed high stiffness of the stone column and consequently due to high stress concentration factors. If the column and soil are considered as elastic materials, then the ratio between the load carried by the column and by the soil is approximately the same as the stone and soil modulus ratio E /E. There is a
c s
simple approximate explanation why stress concentration factors are limited to a certain value. If vertical stress in the column vzc is a major principal stress in the column and vertical stress in soil, &zs is a minor principal stress in the soil and the radial stress = &rc = &rs at soil-column interface is a minor principal stress in the column and major principal stress in the soil, then the maximum stress concentration factor at yield can be expressed as:
= C
K

1 + sin^ \ 1 + sin^: 1 — sin^ \ 1 — sin^ '
(28)
K,
For example, peak shear angles of the granular material p' = 45° and soft soil p' = 20° give maximum stress concentration factor nmax = 11.9. If high
n
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stone and soil deformation ratio E IE is considered in
c s
the elastic analysis and if the applied load qA is high as compared to the initial (lateral) stresses in the soil, then the stress concentration factor may be too high and thus the calculated settlement underestimated. For this reason elastoplastic methods are believed to give better predictions of the settlement reduction and stress concentration factors [5].
The main difference between the proposed method and other elasto-plastic analytical methods [6-8], which also take into account column yield, is the assumption regarding the dilation of the column material. In the majority of rigid-plastic methods [6, 7] it is assumed that stone-column deforms at constant volume, thus no dilation takes place while deforming.
1.0

0,8
0,6
0,4
0,2
0,0
1 I 1 I	1 1 1 1 1 Van Impe & Madhav (1992)
	-1%, q/E =0.10 ' ^ A s
Wv --	Van Impe & Madhav (1992) ~
\N\ svd=	-1%, qA/Es = 0.01
\	Proposed new method
\	The range of solutions for
\	Van Impe & Madhav's method
\	dependig on the ratio qA /Es -
\	fore =-1% vd
\	
\	
\	
0,0
0,1
0,2 0,3 A
0,4
0,5
Figure 7. The comparison of settlement reduction factors for dilating stone-columns according to the proposed new and Van Impe and Madhav's method for a given set of parameters
The comparison of the results between the present and Van Impe and Madhav's method [8] is of special interest, since the dilatancy of the granular material is taken into account in both methods (Fig. 7). The present method is actually an extension of the Van Impe and Madhav's method [8] with three important differences. Small strains and vertical equilibrium of the undeformed geometry was assumed in the present method to simplify the problem and to get the closed form solution instead of nonlinear solution. These two differences have insignificant effect on the results as compared to the Van Impe and Mahav's method [8] as shown in Fig. 6, where the settlement reduction factors obtained with both methods are almost identical if no dilation is taken into account (evd = 0 or y = 0).
The third, and most important difference relates to the dilation. In the proposed method the dilation of the granular material at yield was considered according
to the Rowe dilantancy theory [9]. In Van Impe and Madhav's method [8] the dilation is taken into account through the final value of the volumetric strain of the column evd , which must be selected in advance as an input parameter. Inappropriate selection of the volumetric strain leads to a quite large range of settlement reduction factors for different ratios between the applied load and soil elastic modulus. If inappropriate volumetric strain is selected, it can also lead to the negative settlement reduction factor p (Fig. 7). There is a simple explanation for this phenomenon. If the load level qAIEs is low, causing only small vertical strain, it is very unlikely that large volumetric strain evd < 0 would occur. If the final volumetric strain of column material £vd < 0 is taken in advance as a constant value, then, theoretically, the dilation angle y of granular material becomes directly dependent on the vertical strain (Fig. 8). Especially for small vertical strains the dilation angle will be unrealistically high and will consequently lead to unrealistic prediction of settlements.
Figure 8. The effect of vertical strain £z on angle of dilatancy y and vertical deformation £z with the selected value of volumetric strain £vd according to Van Impe and Madhav's method
The correct value of volumetric strain at every level of axial strain can be determined from a triaxial test on the granular material. Nevertheless, Van Impe and Madhav's method [8] can be used in an iterative procedure. If the calculated vertical strain does not fit to the initially prescribed volumetric strain £vd , then the nonlinear calculation must be repeated with another value of the volumetric strain as an input parameter.
To overcome this problem, the ratio between plastic volumetric and vertical strain, defined by the dilation angle y according to Rowe's theory, was used in the proposed method, instead of assuming dilation volumetric strain £w . In this way, the ratio between the volumetric and the vertical strain of the column at
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Figure 9. Comparison of settlement reduction factors j versus area replacement ratio Ar with some field test results
yield remains constant and load independent. It should be noted, however, that the post-peak behaviour of the column material is not taken into account.
3.3 COMPARISON WITH FIELD TEST RESULTS
Figure 9 compares the predicted settlement reduction factors and corresponding area replacement ratios with some published field test results and observations. If dilation of the granular material is taken into account, then the present method is able to cover most of the field test data.
The results can not be completely conclusive due to the significant scatter of field test results and lack of well documented data in the literature. This scatter is most probably due to many factors that can affect stone-column performance, such as non-homogeneity of soil conditions, foundation shape and size, stone-column installation technique, properties of the granular material used, length of the column, different densification of the column material, different applied load, etc... Nevertheless, the predictions agree well with the field test data, thus validating the importance of the dilation in the presented method.
4 CONCLUSIONS
A simple but effective analytical method for the analysis of stone-column reinforced foundations is presented.
The stone-column and the surrounding soil are modelled as a unit cell, consisting of elastic soil and rigid plastic column material according to the MohrCoulomb failure law. The dilation of the column material according to the Rowe stress-dilatancy theory is directly incorporated into the method. An important feature of the method is a simple closed-form solution for the prediction of the effects of stone-columns on settlement reduction and stresses in the soil and column, which can be easily used in engineering practice.
Comparisons and some parametric analyses are presented to study the influence of area replacement ratio and material properties of the granular material on settlement reduction factor. The dilatancy of granular material has significant effect on the settlement reduction and stress concentration. Thus, densification of the column is not only important to achieve greater initial stiffness but also affects the column behaviour at yield and the overall performance of the stabilized ground.
The results are compared with some existing methods and with field test results and observations. The results of the present method agree well with most of the field data, showing the ability of the proposed analytical method to yield reasonable predictions of the behaviour of stone-column reinforced foundations. The present analytical model confirms the significance of the dilation of the column material and its effect on settlement reduction and on the stresses in the soil and column.
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REFERENCES
APPENDIX
[1]	Aboshi, H., Ichimoto, E., Enoki, M., and Harada, K. (1979). The compozer - a method to improve characteristics of soft clay by inclusion of large diameter sand columns. Proceedings of International Conference on Soil Reinforcement - Reinforced Earth and Other Techniques, E.N.P.C., 1, Paris, 211-216.
[2]	Balaam, N.P. and Booker J.R. (1981). Analysis of rigid rafts supported by granular piles. International Journal for Numerical and Analytical Methods in Geomechanics, 5, 379-403.
[3]	Shahu, T.F., Hayashi S., Madhav, M.R. (2000). Analysis of soft ground reinforced by non-homog-enoues granular pile-mat system. Lowland Technology International, Vol. 2, No. 2., 71-82.
[4]	Barksdale, R.D. and Bachus, R.C. (1983). Design and construction of stone columns. Report No. FHWA/RD-83/026, Federal Highway Administration, Office of Engineering and Highway Operations, Research and Development, Washington D.C.
[5]	Balaam, N.P. and Booker J.R. (1985). Effects of stone columns yield on settlement of rigid foundations in stabilized clay. International Journal for Numerical and Analytical Methods in Geomechan-ics, 9, 331-351.
[6]	Priebe, H. (1976). Abschätzung des Setzungverhaltens eines durch Stopfverdichtung verbesserten Baugrundes. Die Bautechnik, 5, 160-162.
[7]	Impe, W.F. and De Beer, E. (1983). Improvement of settlement behaviour of soft layers by means of stone columns. Proceedings of 8th European Conference of Soil Mechanics and Foundation Engineering, Helsinki, Vol. 1, 309-312.
[8]	Impe, W.F. and Madhav, M.R. (1992). Analysis and settlement of dilating stone column reinforced soil. Österreichische Ingenieur und Architekten Zeitschrift (ÖIAZ), 137(3), 114-121.
[9]	Rowe, P.W. (1962). The stress-dilatancy relation for static equilibrium of an assembly of particles in contact. Proceedings of Royal Society, 269A, 500-527.
[10]	Schanz, T. and Vermeer, P.A. (1996). "Angles of internal friction and dilatancy of sand." Géotechnique, 46(1), 145-151.
[11]	Poulos, H.G. and Davis, E.H. (1974). Elastic Solutions for Soil and Rock Mechanics. Wiley, New York.
[12]	Bolton, M.D. (1986). The strength and dilatancy of sand. Géotechnique, 36(1), 65-78.
[13]	Pulko, B. (2000), The influence of stone columns on the mechanical properties of soft ground (in Slovene), PhD Thesis, University of Ljubljana, Ljubljana.
NOTATION
The following symbols are used in the paper:
A
c
A
e
A
r
C1 ,C2 >C3 C4 dc de
D
E
oed
E
R
H
K
-pc
Kps K
f
qA p'
n
nc
v \
V 'c V cv
V
f
area of column portion; area of soil portion (influence area); replacement ratio; geometrical and material constants; diameter of stone-column; diameter of influence area; relative density; oedometric modulus of soil; elastic modulus of column material; elastic modulus of soil; relative dilatancy index; column height;
coefficient of earth pressure at rest;
passive earth pressure coefficient (column);
passive earth pressure coefficient (soil);
dilation constant;
applied vertical load;
mean effective stress;
radius of stone-column;
radius of influence area;
stone-column spacing;
radial displacement of stone-column;
vertical settlement;
vertical settlement of untreated soil;
settlement reduction factor;
radial strain;
volumetric strain of column;
vertical strain;
stress concentration factor;
column stress concentration factor;
peak shear angle of soil;
peak shear angle of column material;
shear angle of column material at critical state;
Poisson's ratio of soil;
radial stress at soil-column interface;
vertical stress in column;
vertical stress in soil;
angle of dilatancy.
E
0
r
r
u
u
z
u
ß
E
E
E
z
a, a , a r rs
a
a
zs
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