stone columns

ELSEVIER

Computers and Geotechnics, Vol. 20, No. 1, pp. 47-70, 1997 0 1997 Elsevier Science Ltd

Printed in Great Britain. All rights reserved PII: SO266-352X(96)00013-4 0266-352X/97 $17.00+ .OO

Analysis of Behavior of Stone Columns and Lime Columns

H. B. Poorooshasb” & G. G. Meyerhofb

“Faculty of Engineering and Computer Science, Concordia University, Montreal, Quebec H3G 1 M8, Canada ‘Department of Civil Engineering, Technical University of Nova Scotia, Halifax, NS, B3J 2X4, Canada

(Received 28 May 1995; revised version received 9 September 1996; accepted 13 September 1996)

ABSTRACT

The eficiency of end bearing stone columns and end bearing lime columns in reducing the settlement of a foundation system is examined in this paper. The foundation system is assumed to consist of a large number of regularly spaced stone columns of equal length installed in a weak soil layer and supporting a rigid mat. The analysis examines the influence of such factors as the column spacing, the weak soil properties, properties of the granular medium used in constructing the column, the in situ stresses caused by the installation technique, the depth of the bedrock relative to the tip of the columns and the magnitude of the load carried by the supported raft foundation (expressed as a untformly distributed load, UDL). This paper is concluded by presenting a number of charts that may be used in the analysis and design of such a foundation scheme. 0 1997 Elsevier Science Ltd. All rights reserved.

INTRODUCTION

For low-rise buildings, lightly loaded foundations, earth structures and storage tanks that can tolerate appreciable movement, stone columns provide an economical method of support in compressible and cohesive soils [1,2]. The problem of the settlement of such a foundation system has been tackled before, notably by Poulos and Davis [3] who used the elastic theory and by Priebe [4].

A stone column is essentially a vertical cylindrical “hole” dug in the soft soil layer and filled with compacted stone fragments and gravel. The columns, which have afinished cross sectional diameter in the range of 75-150 cm, are usually extended to bedrock or a hard layer, but occasionally floating columns are also installed. This paper deals with end bearing stone columns only. A

48 H. B. Poorooshasb and G. G. Meyerhof

single column is not effective in reducing the settlement. Stone column foundations consist, usually, of a rather large number of columns installed in a uniformly and equally spaced pattern. Thus each column acts within a cylindrical cell with a radius of influence denoted by b (Fig. la). Balaam and Booker [5] relate the diameter of the cell, 26, to the actual column spacing by the relation 2b = c&, where S, is the actual spacing (from center to center of the columns) and c is a constant having values of 1.05 and 1.13 for triangular and square patterns, respectively. For most practical cases the diameter of this cell (sometimes referred to as the tributary region) may be assumed to be equal to the actual column spacing.

Each column derives its load bearing capacity (of the order of lo&500 kN) from the ambient (lateral) stresses developed at the column/soil interface, [6,7]. It is obvious, therefore, that for a systematic analysis the factors that contribute to the development of these lateral stresses must be recognized and their influence evaluated. These factors are; the radius of the column a, the radius of influence of each cell b (Fig. lc), the physical properties of the native weak soil (its modulus of deformation Es and Poisson’s ratio V) and the deformation characteristics of the compacted granular column material. This deformation of the granular material is usually represented by two curves (r,-/~t and I-‘/&i curves) where q is the ratio of the major principal stress to the minor principal stress, E~ is the axial

rigid raft

(a) (b)

oft clay

stone column

bedrock or hard layer

t 2b l

raft

UDL

P

Fig. 1. Key figure, 2a = column diameter, 2b = spacing.

Behavior of stone columns 49

strain of the column and V is the volumetric strain of the granular medium having a negative value for a dilating (expanding) material.

Other factors which influence the performance of the raft-stone column- soil system include the initial stresses caused by the installation (compaction of the material in the column) and the distance between the tip of the piles to the hard layer. Short columns installed in a deep layer of soft soil deposit will not be effective in reducing the settlement of the foundation system. It is the objective of this paper to evaluate the influence of the various factors involved and thus arrive at a practical design procedure.

Lime columns are made by mixing 7-10% of lime in situ with soft clay to make columns of up to 800 mm diameter with a maximum height of 15 meters. Lime columns are much stiffer than stone columns and are likely to behave linearly during the loading process: a fact that will facilitate the analysis considerably as will be seen later.

This paper is written in the following sequence. First the equations governing the discussions are stated without proof leaving the small amount of mathematics involved to the two Appendices provided at the end of the paper. Next the various factors which are likely to influence the performance of the system are stated and their effects evaluated. Based on these evaluations, a set of design charts are prepared and presented. Finally the results of some field tests are briefly discussed.

THE GOVERNING EQUATION

Throughout this paper geometric linearity will be assumed. Thus the small strain theory is applicable. The weak soil is assumed to possess material linearity with a Young’s modulus Es and a Poisson’s ratio v. The stress deformation characteristics of a stone column are represented by the two relations;

v = Ei + 2&3 = V(&i) (2)

where o1 is the maximum principal stress (the load intensity carried by the column), a3 is the minor principal stress, ~1 is the axial strain (the vertical strain experienced by the column and the surrounding soils), .s3 is the lateral strain and V is the volumetric strain of the stone column respectively. A formal derivation of Eqns (1) and (2) may be found in the Appendix.

50 H. B. Poorooshasb and G. G. Meyerhoj

The minor principal stress a3 appearing in Eqn (1) is the sum of two components. First, a component which is developed at the column-soil interface due to the action of the surface loads and the tendency of the stone column to expand laterally plus a residual stress component due to the self weight of the soil and those caused by the installation activities (in particular compaction of the gravel in the hole). Thus;

a, = a3 + ores (3)

Under undisturbed conditions ores = 0.5 &y’L, where K. is the at rest earth pressure coefficient, y ’ is the effective unit weight of the soil and L is the length of the column, Fig. 1 b. Note that in this analysis the value of ~7~ is assumed to remain constant with depth.

Let the load intensity carried by the weak soil be denoted by p, Fig. lc. At this stage p is assumed to be constant; that it is indeed so will be demonstrated later on. It can be shown (see the Appendix) that the following two relations hold:

(1 - v)& 2v a2

p=1_2v2_v E’-G-_E3 [ 1 (4)

o3 = &p - EsE3

(1 + v)a* + (1 - v)b2 (1 _ V2)(b2 _ a2) (5)

where c3 is defined in Eqn (3). Finally, let the intensity of the uniformly distributed load carried by the

rigid mat plus the self weight of the mat be denoted by UDL. Then, equilibrium of the mat in the vertical direction, as in Fig. lc, requires that;

uDL = a*,, + (b* - a2)p b2

The above set of six equations may be solved simultaneously to obtain the value of the six unknowns E], Ed, cl, 03, a3 and p in terms of the given value of UDL. Of these six unknowns, cl is of prime interest since the settlement of the foundation system, 6, is equal to LEE.

Since Eqns (1) and (2) are non-linear a process of trial and error must be used to solve for the unknowns. This approach is usually a lengthy process which may require many iterations and which may not converge to a solution at all. A more intelligent approach is to assume values for the unknowns appearing in the non-linear equations (thus removing them from the set) and


then solving for the other set of unknowns which are encountered in the linear equations. The process is somewhat similar to what is generally called the inverse process and is immensely easier to deal with. Thus, assuming a value of Ed, Eqn (2) may be used to evaluate c3. Equation (4) can now be solved to obtain the value of p. With the value of p just calculated cr3 from Eqn (5) can be obtained and then, with the aid of Eqn (3) and o1 of Eqn (I), substitution for the values of p and cl in Eqn (6) renders the solution. That is, the relationship between UDL, the load carried by the column/soil system and the settlement that this UDL produces is established. Obviously the above formulations apply only to the cases where the stone columns are end bearing.

In the case of lime columns the material in the columns acts as a linear material having a Young’s modulus EC and a Poisson’s ratio of v,. Thus it may be shown, as in the Appendix, that;

UDL b2 - a2 -=,4{1+B~,}~ @IL)

+ {Ec + 2v,[AC(l + Bv,) + Dvc]}; (7)

where 6 is the settlement of the system, L is the height of the column and the constants A,B,C and D have the values;

A= cl-‘) E.

1 - 29 -v S’

C=L. 1 -v’

D = (1 + v>a2 + (1 - VP2 Es

(1 - v2)(b2 - a2)

Note that if v= 0 and v, = 0 then A = Es and B = 0. Equation (7) now reduces to;

n = (UDWG) W) (8)


This last equation, in which n is the settlement ratio, A, is known as the area ratio since it represents the area of the cross section of the column to the area of the cross section of the influence cell (i.e. A, = a2/b2), is due to Priebe [4].

EVALUATIONS

Before the influence of various factors on the performance of the foundation system is evaluated, some typical results are presented in Fig. (2) which shows the behavior of a set of columns 100 cm in diameter and 10 meters long each installed in a soft clay layer with a modulus Es = 1000 kPa and a Poisson’s ratio v of 0.2. In this example the piles were at an effective distance 2b equal to 2 meters.

Figure 2a and b represents the behavior of the three types of materials used in filling the columns, that is, the stones. The most compact material

foundation settlement h. cm

foundation settlement h, cm

I I I I I 1 1 0 IO 20 3c

foundation settlement b, cm

Fig. 2. Typical results for very compact, compact and less compact stone fills, E, = 1000 kPa. v = 0.2, spacing = 2 diameters.


has an angle of friction 4 of 44” and shows a high degree of dilatation as indicated by the lowest curve in Fig. 2b. The loosest material has a friction angle of 38” and shows very little tendency to dilate (the top curve in Fig. 2b).

Figure 2c-e indicates the performance of the raft-stone column-soil system as the magnitude of the settlement 6 is increased from zero to its maximum value of 30 cm. Note that as 6 increases so does the stress ratio in the column, as the 6 values indicated in the upper left hand side of Fig. 2a show. At a settlement value of 30 cm the column has nearly exhausted all its strength. That is, for a larger magnitude settlement the stone column would, in all likelihood, develop shear bands and fail.

Figure 2c shows the relationship between the uniformly distributed load carried by the system and the resulting settlements. As expected the more compact column carries a higher load for the same settlement value.

The stiffness ratio A4 is defined as the ratio of axial stress experienced by the column, ol to the vertical stress component sustained by the soil, p, for the same settlement. This definition of the stiffness ratio is perhaps more appropriate than the parameter currently in use (called the modular ratio) which is defined as M= EC/ES where EC and ES are the deformation moduli of the column and the soil, respectively. However, EC is not a constant in the case of granular columns, as its value changes as the loading proceeds. In the linear case (e.g. lime columns) the two definitions are equivalent however, since for equal settlement of the soil and the column the relation (al)/ (EC) = (p)/(E,) must hold.

From Fig. 2d it is obvious that a well compacted gravel has a significantly higher value of M compared to that of a less compacted fill (9.5 compared to 4.7).

Finally, Fig. 2e shows the variation of the performance ratio (PR) with 6. The performance ratio is defined as the ratio of the settlement of the treated ground (ground with stone column inclusions) to that of the untreated ground under identical surcharges. In the current literature another parameter, called the settlement ratio and denoted by n is employed which is the ratio of the settlement of the untreated ground to that of treated ground. It is obvious that the relation between the performance ratio and the settlement ratio is;

performance ratio = (settlement ratio)-’ or PR = l/n (9)

For a well compacted stone fill (@= 44”) the performance ratio is 0.31. Thus if the settlement of a foundation system without stone columns is, say, 20 cm, inclusion of the columns would reduce its settlement to 0.31 x20, or about 6 cm. For a less compact stone fill ($= 38”) the settlement ratio is about 0.5. Under the circumstances just stated the settlement would be


0.5x20, or 10 cm. This clearly demonstrates the benefits of compaction of the stones used in the columns.

A point of interest is that both the stiffness ratio and the performance ratio are fairly independent of 6. This makes the preparation of certain design charts possible, which will be presented in the next section. The remainder of this section is devoted to a systematic evaluation of the different factors which are likely to influence the performance of the raft-stone column-soil foundations. The main objective is to isolate those factors which are likely to have a profound effect on the performance of the system, by evaluating their effect on the value of PR, thereby arriving at appropriate design concepts.

Physical dimensions

In this subsection the influence of the various physical dimensions of the columns and their proximity to one another will be examined.

Column spacing Column spacing is the factor which is likely to influence the behavior of the system most strongly. To demonstrate this consider the data presented in Fig. 3. The gravel in this case was assumed to be well compacted (@=44”) and all other quantities are kept the same as shown in Fig. 2 except that here, the spacing (= 2b) was increased to 6 times the diameter of a column. The following conclusions may be drawn. The system is very inefficient, note that the maximum UDL that can be sustained is only 40 kPa. At higher loads the columns develop shear bands and fail. Also note that the value of the performance ratio is very high, being equal to 0.85. Thus, in the example given

foundation settlement h, cm

z --5

B _ (b) G- z _

I I I I I I 0 IO 20 30

foundation settlement h. cm

1 30

foundation settlement b, cm

E = loo0 kPa, v=.2, +=44.2 spacing= 6 diameter

Fig. 3. Performance of the system when stone columns are installed at 6 diameter spacing.


above where the settlement of the untreated ground was assumed to be 20 cm the ground with stone columns installed at 6 diameter spacing would settle 0.85 x20, or 17 cm. This order of improvement can hardly justify the extra expense involved in installation of the stone columns.

Table 1 below shows the performance ratios for various column spacings (which it will be recalled are equal to 2b) expressed as a multiple of the column diameter (2a) and denoted by s (= b/a).

On the basis of the above finding a conclusion may be drawn, viz.: a column spacing of larger than 4 diameters makes the system inefficient even under the most favorable conditions (see the second column in Table 1). A second conclusion is that, as expected, column spacing does indeed play a major role in the performance of the system.

TABLE 1 Performance ratio (PR) for various spacings and compaction, Es = 1000 kPa, v = 0.2

Spacing, s #=&I” Angle of internal friction

#=41” #= 38”

1.5 0.15 0.2 0.27 2 0.31 0.39 0.48 3 0.56 0.64 0.72 4 0.71 0.78 0.83 6 0.85 0.89 0.92

10 0.94 0.96 0.97

6,cm I

Possibility of shear band formation

0 L=5m,2a=l m 0 L=10 m, 2a=I m + L=15 m, 2a=1 m

6 .d

3 (b) i

E < kl I I I I I I I

10 20 30 foundation settlement 6, cm

5

E 0.8 .6 0 0 L=S L=lOm, m, 2a=.S 2a=1 m m

foundation settlement 6, cm

Fig. 4. Effect of column length on the performance ratio.


Length and diameter of the stone columns In Fig. 4b the PR/6 (performance ratio vs settlement of the foundation) curves for three different column lengths (5, 10 and 15 m) are shown. The three curves almost coincide, indicating that this dimension, the length of the stone column, has negligible effect on the performance of the system. A point worthy of note is that in the case of the shortest column (length 5 m) the column is likely to “fail” at higher settlements (6 in excess of 15 cm) by forming, possibly, shear bands. This is evident from Fig. 4a.

The effect of column diameter on the behavior of the foundation is similar to that of its length. It is very mild and thus can be neglected, see Fig. 4c.

In view of the foregoing evaluations it is concluded that the only physical dimension that effects system performance is the column spacing, defined by;

b s=- a


Poisson’s ratio v Like ES, the influence of v on the performance ratio is quite small. For example, assuming a column spacing of 2 and a very compact gravel (4 = 449, the variation in this ratio (i.e. the PR value) is from 0.32 to 0.31 for a Poisson’s ratio ranging from 0 to 0.5. The corresponding values for less compact stones (4=41”) are 0.414.38 and for the uncompacted column (4= 38”) 0.524.45., as in Fig. 5. In the range of recommended values for soft clays, 0.15-0.25 [8], the variation is quite small and negligible even for an uncompact column. It is concluded, therefore, that based on the assumptions made in this paper, a soft soil’s mechanical properties do not appear to play a major role in the overall performance of the foundation system. Recall that performance is measured by the PR value only.

Mechanical properties of the stones in columns

The two relevant mechanical properties of the stone fills are the maximum angle of friction $J and a measure of the stiffness of the stone fragments after placement and compaction in the holes. It is customary to specify the stiffness of any granular material by its so called “initial tangent modulus” which is the slope of a tangent to the stress-strain curve drawn at the origin of the plot (i.e. where &1 = 0). A more objective measure is the failure strain, defined as the strain ,sl required to reach the peak stress (failure) point in the stress-strain curve. In the present paper the failure strain, which will be denoted by Er, is used. Note that in the stress-strain curves shown in Fig. 2a the failure strains are 4% for @=44”, 5% for $=41” and 7% for $= 38”. These curves (shown in Fig. 2a) will be referred to, in this paper, as the standard curves and the columns filled with these as the standard columns.

I I I 1 I .I .2 .3 .4 .5

Poisson’s ratio. v

Fig. 5. Effect of Poisson’s ratio on the performance ratio.


Failure strain ef Table 3 shows the effect of the failure strain on the computed PR values. In this table the middle column in each category represents the PR associated with the standard column. The numbers to the left of the middle columns are those associated with a stiffer column (the failure strain, Er, for the material used in their construction is chosen to be equal to one half of the standard failure strain) while those to the right belong to a “less stiff column” for which &r= 2&r (standard). The column stiffness does have an influence on the PR value (see Fig. 6) although the influence does not appear to be as profound as expected. Nevertheless its influence must be taken into account when preparing the design charts.

Before leaving this section it must be pointed out that in the preparation of Table 3 the total volumetric strain experienced by the “non-standard” samples has been assumed to remain the same as those of the standard samples. Thus, for example, for the standard sample with a $=44” (Er=4%) the total volumetric strain is, from Fig. 2b, about 10%. This value has been retained in both the case of Er= 2% and Er= 8%.

Angle of friction q!~ From Fig. 2e it is obvious that this parameter is likely to play an important role and therefore must be included in the final solution; i.e. the function PR must contain 4 as an independent variable.

Initial ambient stresses

Before the columns are installed the horizontal component of stress in the ground is given by the equation Kay ‘z where z is the depth below grade and K0 is the coefficient of the at rest earth pressure for the soft clay. Installation of the columns increases this pressure to a higher value. If the soft soil were a non-yielding material then the value of the initial ambient stress would be Key’z where K0 would now be equal to the at rest earth pressure coefficient

TABLE 3 Effect of failure strain on the performance ratio

Spacing, s Angle of internal friction $=44” f#l=41” _ (b=38”

&f=2a &f=4 Ef”5 &f= 2.5 &f=s &f= 10 &f= 3.5 &f=l &f= 14

1.5 0.1 0.15 0.21 0.15 0.20 0.27 0.24 0.27 0.34 2 0.23 0.31 0.41 0.31 0.39 0.51 0.45 0.48 0.59 3 0.46 0.56 0.67 0.56 0.64 0.75 0.70 0.72 0.81 4 0.62 0.71 0.80 0.71 0.78 0.85 0.81 0.83 0.90

“Value of .q in %.


I c

I I

I Figure 6, a- Performance Ratio vs. Spacing for Stone Columns, Standard I

= 38’, q=14% ,=41° , q=lO% = 44‘-‘, Ed =8%

Figure 6,b- Performance Ratio vs. Spacing for Stone Colnmns Less Stiff than Standard

9 = 38O, ~,=3.5%

$I=410 , q=2.5%

0 = ‘MO, Er=2%

I Figure 6, c- Performance Ratio vs. Spacing for Floating Stone Coh~mns Stiffer than Standard I

Fig. 6. Performance ratio vs spacing for (a) standard stone columns; (b) stone columns less stiff than standard; (c) floating stone columns stiffer than standard.


of the stones. However, the soft soil would yield and during the act of compaction the lateral stress would certainly rise above this level. Once the compaction is stopped relaxation causes the stresses to reduce to a lower level, the magnitude of which cannot be decided with any accuracy. In spite of this, and since the aim of the study at this stage is a sensitivity test, it was decided to run an additional test by assuming a K0 value 25% in excess of the K0 of the stone fillings. That is, Ke = 1.25( 1 -sin@). The results are shown in Table 4. A comparison of the PR values from this table and those presented in the second column of Table 1 shows that the initial stresses are not likely to be an issue. In this context it must be emphasized that the present analysis is unable to account for the initial shear stresses produced at the column-soft soil interface. The magnitudes of the inter-boundary shear stresses are prob- ably small however and may be ignored.

This completes the study of the performance of end bearing stone columns. The main conclusion to be drawn is a simple one and may be represented by the equation;

PR = PR(s, 4, q) (11)

DESIGN CHARTS

The design charts produced here are based on derived in the last section. For a proposed s, 4 and ar they provide the value of PR which is, in turn, given by the equation;

E’ HUDL (12)

In Eqn 12 6 is the settlement of the foundation system, His the height of the soft soil layer, UDL is the magnitude of the uniformly distributed load supported by the mat plus the self weight of the mat per unit area and E’ is the modified Young’s modulus for the soft soil corrected for the Poisson’s ratio effect.

Three charts (see Fig. 6) are produced here assuming a value of v= 0 as this provides the most conservative estimate of PR. The case where s = 1 represents a situation where the soft soil is completely dug out and replaced by stones! Spacings larger than 4 diameters are not considered efficient.

Spacing, s

PR value

TABLE 4 Performance ratio for & 25% in excess of K0

1.5 2 3 4

0.14 0.29 0.54 0.69 _~.

6 10

0.84 0.94

TA

BL

E

5 O

bser

ved

settl

emen

t of

fou

ndat

ions

on

st

one

colu

mns

Test

foun

dati

on

Soil

St

one

colu

mn

Obs

erve

d se

ttle

men

t ra

tio,

n

Ref

eren

ce

No.

Ty

pe

Coh

esio

n M

odul

us

Fric

tion

D

iam

eter

D

epth

Sp

acin

g A

rea

UD

L c,

(@

a)

J% (k

pa)

9 (“

) (m

m)

(m)

(m)

rati

o (k

pa)

W)

1 E

15

-

1500

-5

0 11

00

18

2.1

25

1200

2 B

25

3

ST

15

4a

S

T

20

4b

ST

20

4c

S

T

20

4d

ST

20

4e

S

T

20

4f

ST

20

- 25

00

-40

1000

10

.5

2 22

22

0 2.

5 -

1500

-4

0 10

00

6.5

2.1

20

150

2.6

- 20

00

-40

1000

6

2.1

20

150

2.8

- 20

00

-40

1000

6

1.9

33

150

3.4

- 20

00

-40

lob0

6

1.9

33

150

4.2

- 20

00

-40

1000

6

1.9

33

150

3.0

- 20

00

-40

1000

6

1.9

33

150

3.1

- 20

00

-40

1000

6

1.9

33

150

3.6

3.4

bJ E-

‘.

Cas

telli

et

al.

[9]

s

Mun

kfak

h et

al.

[13]

W

att

et a

l. [ 1

21

2

Wat

t et

al.

[ 121

R

Wat

t et

al.

[ 121

s

Wat

t et

al.

[ 121

P 3

Wat

t et

al.

[12]

1

Wat

t et

al.

[ 121

W

att

et a

l. [ 1

21

Wat

t et

al.

[ 121

Not

es:

B =

bui

ldin

g fo

unda

tion,

E

= e

mba

nkm

ent

foun

datio

n,

ST

= s

tora

ge

tank

fo

unda

tion.

TA

BL

E

6 O

bser

ved

settl

emen

t of

fou

ndat

ions

on

lim

e co

lum

ns

Test

foun

dati

on

Soil

Li

me

colu

mn

Obs

erve

d se

ttle

- R

efer

ence

m

ent

rati

o, n

N

o.

Type

C

ohes

ion

Mod

ulus

C

ohes

ion

Dia

met

er

Dep

th

Spac

ing

Are

a M

odul

us

UD

L c,

(k

pa)

E,

(kpa

) c,

(k

pa)

(mm

) (m

l (m

) ra

tio

(kpa

) (%

) (k

?a)

Sa

B

10

650

N 2

00

500

4 1.

3 11

15

000

20

4.0

Bre

denb

erg

&

Bro

ms

[15]

5b

E

10

65

0 -

200

500

5 1.

4 10

15

000

IO

3.4

Bre

denb

erg

&

Bro

ms

[15]

6

E

12

600

-150

50

0 6

1.4

9 95

00

50

1.7

Bro

ms

&

Bow

man

[l

] _

Not

es:

B =

bui

ldin

g fo

unda

tion,

E

= e

mba

nkm

ent

foun

datio

n.


FIELD CASE RECORDS

Several results of field records of stone columns of different densities in wide foundations on cohesive soils of various strengths have been published [4,9- 13]. Meyerhof [ 141 made a comparison of these results with the predictions made using the simple Eqn (8) due to Priebe. The difficulty is, of course, a correct estimation of the stiffness ratio EC/Es, since as demonstrated above EC is not a constant but a variable which is heavily dependent on the magnitude of the ambient stress as well as the physical properties of the stones in the columns (such factors as their strength and degree of compaction.)

Tables 5 and 6 present some field data from tests carried out on both stone columns and lime columns. These data are plotted in Fig. 7 and compared with the results of computations using the above analysis, curves (ak(c) of the figure. The coordinates used in Fig. 7 are the settlement ratio, n, and area ratio A,. Given the uncertainties involved in estimation of the various factors the agreement obtained between the analysis and the field observations is judged to be satisfactory.

-7

-6

Eq.

10 20 30 40 Area ratio Ar(=A,/A), %

Fig. 7. Settlement ratio vs area ratio for stone columns and lime columns. Comparison of analytical results with field data.


Also shown in the figure are certain field results from tests conducted on lime columns. Here, the Priebe equation, which as pointed out earlier, is a special case of the more general equation derived in this study, Eqn (7), agrees well with the field test results. The Priebe equation is represented by the line marked (d) for which the following data has been used; E, = 15000 kPa, E, = 650 kPa. Using the same data and upper bound values for the Poisson’s ratio of the soil (v,=O.25) and the lime column (v,=O.5), Eqn (7) is represented by the line marked (e) in the figure. Note that two of the test results fall within the zone bordered by the Priebe equation and Eqn (7) which may be viewed as the lower and the upper bounds for the settlement ratio, respectively.

CONCLUSIONS

Based on the studies presented in this paper it may be concluded that the factors that most severely effect the performance of a stone column foundation scheme are the spacing (or the area ratio) and the degree of compaction of the material in the columns which, in turn, control their strength, stiffness and dilatation properties.

REFERENCES

1. Broms, B. B. and Bowman, P., Stabilization of Soil with Lime Columns. Design handbook, Department of Soil and Rock Mechanics. Royal Institute of Tech- nology, Stockholm, 1977.

2. Mitchell, J. K., Fundamentals of Soil Behavior. J. Wiley and Sons, Inc., New York, NY, 1991.

3. Poulos, H. G. and Davis, E. H., Pile Foundation Analysis and Design. J. Wiley and Sons, Inc., New York, NY, 1980.

4. Priebe, H., Abschatzung des setzungsverhaltens eines durch stopfverdichtung ver- besserten baugrundees Die Bautechnik, 1976,54, 16&162.

5. Balaam, N. P. and Booker, J. R., Analysis of rigid rafts supported by granular piles. Int. J. for Num. and Anal. Methods in Geomech., 1981, 5, 379403.

6. Poorooshasb, H. B. and Madhav, M. R., Application of rigid plastic dilatancy model for prediction of granular pile settlement. Proceedings of the 5th ICON- MIG, Nagoya, Japan, 1985, pp. 1805-1808.

7. Schweiger, H. F. and Pande, G. N., Numerical analysis of stone column supported foundations . Computers and Geotechnics, 1986, 2(6), 347-372.

8. Das, B. M., Principles of Geotechnical Engineering. PWS-KENT Publishing Company, Boston, USA, 1990.

9. Castelli, R. J., Sarkar, S. K. and Munkfakh, G. A., Ground treatment in the design and construction of a wharf structure. Proceedings of the International Conference on Advances in Piling and Ground Treatment for Foundations. Institution of Civil Engineers, London, 1983, pp. 275-28 1.


10. Greenwood, D. A. and Kirsch, K., Special ground treatment by vibratory and dynamic methods. Proceedings of the International Conference on Advances in Piling and Ground Treatment for Foundations. Institution of Civil Engineers, London, 1983, pp. 1745.

11. Hughes, J. M. 0. and Withers, A. J., Reinforcing of soft cohesive soils with stone columns. Ground Engineering, 1974, 7, 4249.

12. Watt, A. J., deBoer, J. J. and Greenwood, D. A., Loading tests on structures founded on soft cohesive soils strengthened by compacted granular columns. Proceedings of the 3rd Asian Conference on Soil Mechanics and Foundation Engineering, Haifa, 1967, Vol. 1, pp. 248-251.

13. Munkfakh, G. A., Sarkar, S. K. and Castelli, R. J., Performance of a test embankment founded on stone columns. Proceedings of the International Con- ference on Advances in Piling and Ground Treatment for Foundations. Insti- tution of Civil Engineers, London, 1983, pp. 259-265.

14. Meyerhof, G. G., Closing address. Proceedings of the International Conference on Advances in Piling and Ground Treatment for Foundations. Institution of Civil Engineers, London, 1983, pp. 293-297.

15. Bredenberg, H. and Broms, B. B., Lime columns as foundation for buildings. Proceedings of the International Conference on Advances in Piling and Ground Treatment for Foundations. Institute of Civil Engineers, London, 1983, pp. 133-138.

APPENDIX

Derivation of Eqns (1) and (2)

The increment of the principal major strain d&i has two components; the elastic and the plastic component. The elastic component is usually small and may be ignored. The plastic component is derived from a potential function CJJ and with a magnitude depending on the amount of yielding, represented by the function5 Thus

(Al.l)

where h is a proportionality factor, and in terms of the stress parameter used in this paper, n=al/a+

The functions ~0 and f are given by co= cr3&q) and f= r]. Thus &p/&r, = d(n); df = dq, which upon substitution in Eqi Al. 1 results in;

d&l = ~(~)g-+)d~ (Al .2)


Equation (A1.2) may now be integrated to show .sl =E~(v). The inverse of this relation, viz.;

rl= +I) (Al .3)

establishes Eqn (1) Similarly it may be shown that;

d&s = WV(~) - rl&rl)dn (A1.4)

or &3 = am. And since V= &t(q) + 2~~(n) = V(q) = V [I] = V(E~) in view of Eqn (A1.3). Thus Eqn (2) is established.

Derivation of Eqns (4) and (5)

For convenience the solution will be obtained by superposition of the results from the two cases shown in Fig. A.l. The situation shown in (i) shows a thick cylinder which is constrained from movement at its base and its

(0

(ii)

Fig. A.1. Thick cylinder under internal pressure p and constrained from movement at its outer radius.


exterior wall. A rigid frictionless piston forces the upper surface of the cylinder downwards and in doing so causes the hole to reduce its diameter. This is analogous to the action of the rigid mat supported by the stone columns. The case shown in (ii) represents the column-soil interaction. It is the well known thick cylinder problem. The stone column, because of its tendency to deform laterally, exerts certain stresses on the interior wall of the cylinder and causes the hole to expand. The solution to the problem (i.e. Eqns (4) and (5) of the main text) are obtained by making the displacement of the two cases compatible.

Case (i) will be treated first. In view of the axisymmetry of the problem and with reference to the coordinate system shown in the figure the only non-zero components of the stress tensor are err, O@ and oZZ = p. It will be assumed that p is constant, a fact that will be demonstrated later on. The corresponding strain components are Ebb, &06 and E,, =E]. The use of p and &1 are in conformity with the notation employed in the main text. To ensure equilibrium and compatibility of strains the following relations must hold;

-==l(q(j-orr) da,, dr r

where

& rr = $ [Gr - +ee + P)]

and

%9 = ; [%x3 - V(Gr + P)]

Using the last two equations it may be shown that;

E CT --

[

v(1 + v) rr - 1 - y2 Err + VEee +

E ’ 1

and

(A2.1)

(A2.2)

(A2.3)

(A2.4)

(A2.5)

(A2.6a)

(A2.6b)


Substituting the values of err and 0 88-~,, from Eqns (A2.5) and (A2.6a) in Eqn (A2.1) and noting that E~Q= -u/r, F,,.= -du/dr (note the geomechanics sign convention) results in;

(A2.7)

In Eqn (A2.7) u is the component of the displacement vector along the r axis. The other non-zero component of this vector is denoted by w and is the displacement along the z axis. The solution to Eqn (A2.7) is;

B u=Ar+-

r (A2.8)

which must be solved subject to the condition that at r =b, u=O and that at r = a, B,, = 0. The first condition immediately yields;

(A2.9)

Substituting for A in Eqn (A2.8) noting that Ed@= -u/r, F,,= -du/dr, using Eqn (A2.5) and finally invoking the boundary condition at r = a results in the value of B, viz.;

B= _‘(l +‘> a2b2

E ’ (b2 + a=) - v(b2 - a=) (A2.10)

Substituting for the values of A and B in Eqn (A2.8) results in the equation;

U v(l + v) b2 - a2 - r=a = - r E ’ (b2 + a=) - v(b2 - a=)

(A2.11)

Substituting for see + E,, in Eqn (A2.6b) and using the results in conjunc- tion with the constitutive law F,, = ,sl = l/E Ip-v (CT,, + am)] yields;

‘1 = E(lp_ v) [(I - ’ - 2v2) + 2v2(1 + ‘) (b2 + a2) :=,,@2 _ a2)] (A2.12)

Note that for constant Ed, p is a constant (and does not depend on r). This justifies the claims regarding the constancy of p made earlier. Also note that if a=0 then;


Et + E’= l-v

1 -v-2$

The results expressed by Eqns (A2.11) and (A2.12) are for case (i) of Fig. A. 1. To complete the solution of the problem the results from case (ii) in Fig. A.l, must be obtained and superimposed on these results. Thus the final results obtained are;

U 1 (b2 - a2)(1 + v) _ r !.=* = -E* (b2 + a2) - v(b2 - a2) (A2.13)

2v 1 -v-2v2 & zz = El = l_v

(1 + v)u2 2v

+ (b2 + a2) - v(b2 - a2) h 1 (A2.14)

where A = [vp-(I-~)a~]. Noting that, ~3 = --u/$=~, eliminating h between the last two equations and with some rearrangement of terms, Eqn (4) of the main text is obtained. Finally, substituting for h in Eqn (A2.13) yields Eqn (5) after some rearrangement of the terms.

Derivation of Eqn (7)

For reasons of convenience Eqns (4) and (5) of the main text may be written as;

p = A(&* - B&3) (A3.1)

a3 = Cp - DEB

where A,B,C and D are as defined in the main text (following Eqn (7)). Also, the axial strain is given by the equation;

Finally the equilibrium of the raft in the vertical direction requires that;

b2 - a2 UDL=pi--to,; (A3.3)

Noting that &3 = -VIES, the first equation of the set (A3.1) reduces to;

81 = i(q - 2v,q) (A3.2) c


p = A(1 + B&)&I (A3.4)

Substituting for this value of p in the second equation, the same set provides the value of cr3;

CT3 = AC( 1 + BV,)E, + DV,EI

which when inserted in Eqn (A3.2) yields;

(T] = {E, + 2v,[AC( 1 + Bv,) + Dv,]}E, (A3.5)

Substituting for p and CT] from Eqns (A3.4) and (A3.5) in Eqn (A3.3) results in Eqn (7) of the main text.

stone columns

Documents