Proceedings of Indian Geotechnical Conference December 15-17,2011, Kochi (Invited Talk-6.)

CONSOLIDATION OF SOFT GROUND WITH DRAINS WITH NO (PVDs) AND FINITE

STIFFNESS (GRANULAR PILES)

M.R. Madhav, Professor Emeritus, JNTUH CE and Visiting Professor, IIT, Hyderabad, madhavmr@gmail.com

P. Ayub Khan, Associate Professor, ACE Engineering College, Hyderabad, akp1468@gmail.com

ABSTRACT: PVDs and granular piles are the most popular techniques for improving soft soil deposits. While the former

improves the ground by accelerating consolidation the latter improves the ground by not only accelerating consolidation but by

reinforcing the soft ground as well. This paper presents review of the different analyses especially of non-linear theory of

radial consolidation due to PVDs in thick clay deposits and the consolidation of granular pile-reinforced soils considering the

stiffness of the granular pile.

INTRODUCTION

India has a long coast line of nearly 5,000 km length. Soft,

weak, highly compressible soft soils are prevalent along this

stretch. Considerable infrastructure development is taking

place along the coast for obvious reasons. Construction of

facilities on these deposits is very challenging because of

their low strength, high compressibility, etc. Several

alternatives, e.g. preloading without or with PVDs, granular

piles, heavy tamping, etc., are available to engineer the

ground. Amongst these, preloading with PVDs and

reinforcing with granular piles/stone columns are the most

preferred alternatives.

Prefabricated Vertical Drains (PVDs)

These are band or strip shaped plastic drains about 100 mm

wide and 4 mm thick usually installed in square or triangular

arrays (Fig. 1a). Consequently, the zone of influence of each

drain is either square or hexagonal area. The strip drain and

the zone of influence of each drain are replaced by a unit cell

of equivalent circular shapes. The flow pattern around the

drain is considered to be axi-symmetric (Fig. 1b). The

equivalent diameter of the drain, dw =2(a+b)/ , where ‘a’ and

‘b’ are the width and thickness of the PVD respectively and

the equivalent diameter of the zone of influence, de = 1.13S

& 1.05S for square and triangular patterns respectively,

where S is the spacing of drains.

Fig. 1(a) Triangular Arrangement of PVDs and (b) Flow in Unit

Cell

Barron [1] presented analytical solutions for the radial

consolidation due to sand drains for both free and equal strain

conditions. This classical theory is based on the assumptions

of small strains, linear void ratio-effective stress relationship

and constant coefficients of volume compressibility, mv, and

horizontal permeability kh. Hansbo [2] presented a simple

solution for radial consolidation with band shaped vertical

drains. Lekha et al. [3] presented a non-linear theory of

consolidation with sand drains under time dependent loading

for equal strain case. Full-scale test conducted by Bergado et

al. [4] on soft Bangkok clay with PVDs revealed that the

degree of consolidation obtained from pore pressure

measurements is lower than the corresponding values

obtained from settlement measurements. The various

modeling aspects of PVDs are comprehensively discussed by

Indraratna et al. [5] along with the evaluation of their

effectiveness in practice. Indraratna et al. [6] developed a

theory for consolidation with radial flow using log - e (Cc

and Cr) & e – log kh (Ck) relationships and for different

loading increment ratios )/( i . Indraratna et al. [7]

developed a modified consolidation theory for vertical drains

incorporating vacuum preloading for both axi-symmetric and

plane strain conditions. Two- and three-dimensional

multidrain finite-element analyses of a case study of

combined vacuum and surcharge preloading with vertical

drains is presented by Rujikiatkamjorn et al. [8]. The

numerical predictions compared well with observed data.

Indraratna et al. {9] developed a new technique to model

consolidation by vertical drains beneath a circular loaded area

by transforming system of vertical drains into a series of

concentric cylindrical drain wall. Walker et al. [10] presented

the spectral method for analysis of vertical and radial

consolidation in multilayered soil with PVDs by assuming

constant soil properties within each layer.

For relatively large applied stress range the void ratio is not

proportional to effective stress and the coefficients of

compressibility and permeability decrease during

consolidation. A non-linear theory of consolidation was

developed by Davis & Raymond [11] considering e – log

relationship but it is valid only for vertical flow and thin layer

of clay. A theory of non-linear consolidation for radial flow

de 1 1

S

S

(a) (b)

dw

HDrain

de

33

M.R. Madhav & P. Ayub Khan

around a vertical drain is developed by Ayub Khan et al. [12]

based on linear void ratio-log effective stress relationship but

assuming constant coefficient of consolidation for thin clay

layers. This theory is further extended by Ayub Khan et al.

[13] for thick clay deposits as well.

The general equation of non-linear consolidation with radial

flow [13] in terms of a parameter, w, is

r

w

rr

wc

t

wr

1

2

2

(1)

with

f10logw or

f

f10

ulogw

)(, where and u are

the effective vertical stress and the excess pore pressures

respectively at time, t and radial distance, r, cr - coefficient of

consolidation for radial flow and f the final effective

vertical stress. The parameter ‘w’ varies with the depth in

thick deposits of clay as the initial effective in-situ stress, o

and the final effective stress, f (= o +qo) vary with depth

due to overburden stresses, and qo is the applied load

intensity. The thick clay layer of thickness, H is divided into

m layers of thickness, H=H/m (Fig. 2). The equation

governing the consolidation process of each layer is

r

w

rr

wc

t

w jjr

j 1

2

2

(2)

where the subscript j refers to the layer number. Eqs. (1 and

2) are of the same form as that given by Barron [1]. Even

though the initial and final effective stresses are different in

each layer, but the flow in each layer is assumed to be purely

radial and independent of the flows in the adjacent layers.

Degree of Settlement of the layer j is,

e

w

e

w

e

w

e

w

r

rjo

r

rj

r

rfo

r

ro

jS

drrw

drrw

drree

drree

U

2

2

1

2 )(

2 )(

,

, (3)

where jf

j10j logw

, ,

jf

jo10jo logw

,

,, , the initial

effective in-situ stress in the jth layer, o,j = .zj.

Fig. 2 Radial Flow in Clay Layers in a Thick Deposit

Average degree of consolidation for the entire thickness,

H

HU

U

m

jjjs

s1

, ).(

(4)

Normalized average excess pore pressure for the layer j is:

e

w

e

w

r

ro

r

rj

javg

drru

drru

U

2

2

* , (5)

Degree of dissipation of average excess pore pressure at the

layer, j is:

)*1( ,, javgjP UU (6)

Average degree of dissipation of excess pore pressure for the

entire thickness is:

H

HU

U

m

jjjP

P1

, ).(

(7)

Initial and Boundary Conditions

jf

jo10joj

jojfj

logrwrw

ru

,

,,

,,ew

)0,()0,(or

)()0,( ; rr r and 0For t

(8)

where f,j= ( o,j+ ) and =qo

0),(or 0),( ; rr and 0For t w trrwtrru wjwj

(9)

0or 0 ; and 0For t

ee rr

j

rr

j e

r

w

r

urr

(j=1, 2, 3…m) (10)

The radius of influence zone, re = 0.5de and radius of

equivalent drain, rw =0.5dw. Eq. (2) is re-written in non-

dimensional form as

R

w

RR

w

T

w jjj 1

2

2

( j=1, 2, 3……m) (11)

where R = r/de and time factor, Th = cr.t./de2.

RESULTS AND DISCUSSION

A thick clay layer of thickness, H, is divided into ‘m’ (=20)

layers of equal thickness, H= H/m, as shown in Figure 2.

The numerical analysis is carried out for each layer

independently using the corresponding ratio f/ o and the

results obtained in terms of degree of settlement, Us, and

excess pore pressures, u. Variations of these results along the

Centre Line of Drain

Layer- m

H/m

Surcharge, qo Drain

Layer-1

rw

re

o & f

Impervious

Pervious

z

Flow Path

34

Consolidation of soft ground with drains with no (PVDs) and finite stiffness (granular piles)

0

20

40

60

80

100

0.001 0.01 0.1 1 10

Time Factor, ThD

egre

e o

f S

ett

lem

ent,

Us

n=40

15

10

5

25

For all q*o

&

All Depths

0

0.2

0.4

0.6

0.8

1

0 20 40 60 80 100 120

f / o

No

rmali

zed

Dep

th,

Z=

z/H

.

q*o=3

1

0.5

2

0

20

40

60

80

100

0.001 0.01 0.1 1 10

Time Factor, Th

Us a

nd

U

p (%

)

Us For all q*o

Up For

q*o =2

1

0.5

n =15

Fig. 3 Variation of f / o with Depth

depth and radial distance are studied. As non-linear

consolidation is mainly influenced by the stress ratio, f / o,

the variation of f/ o with depth is shown in Fig. 3 for

different values of normalized applied load intensity,

q*o=qo/( .H). The decrease of f / o with depth is very

sharp as the initial effective stress is very small near the top.

The decrease of f / o with depth is significant for depths in

the range 0.1H to 0.4H and relatively very small for z>0.4H

for all q*o. Hence, the effect of non-linearity in the void ratio

- effective stress on consolidation in a thick clay layer can be

pronounced at shallow depths compared to that at greater

depths. Increases in q*o can be either due to increase of load

intensity, qo for a given thickness, H, or due to decrease of

thickness of clay deposit for a given load intensity. In either

case, only the non-dimensional stress ratio, f/ o, influences

the rate of consolidation and not the thickness of the deposit

or the magnitude of loading individually. The degree of settlement, Us, is determined for different

layers along the depth for various qo* values for different

values of n ranging from 5 to 40 and shown in Fig.4. The

degree of settlement is identical at all the layers in given thick

clay for all values of n and q*o. The degree of settlement

obtained from the present non-linear theory is identical to that

from the linear theory [1] for free strain case. The degree of

settlement, Us, is independent of f / o and varies only with

n as is the case for vertical flow [11]. The degree of

settlement decreases with increase of n since it takes

relatively long time for dissipation of pore pressure for larger

radial distances. Us decreases from 58% to 24.8% for n

increasing from 5 to 40 at a time factor of 0.10. The degree of dissipation of average excess pore pressure for

the entire thickness, Up is presented in Fig. 5 for n=15 along

with the degree of settlement, Us. While the degree of

settlement is independent of q*o, the degree of dissipation of

pore pressure, Up is sensitive to q*o values. Thus for a given

thickness of clay deposit, the increase of load intensity, qo

results in decrease of degree of dissipation of average excess

pore pressure or for a given load intensity the decrease of

thickness of clay deposit results in decrease of degree of

dissipation of average excess pore pressure.

Fig. 4 Us versus Th for both Linear and Non-Linear Theories

– Effect of ‘n’

Fig. 5 Variation of Us and Up with Th

The normalized average excess pore pressure, U*avg(z) =

(uavg(z)/uo).100 at different depths determined and its

variation with time shown in Fig. 6 for n=15 and q*o=1 along

with the degree of settlement. The excess pore pressure is

relatively high at shallow depths where the f / o ratio is

relatively high compared to the values at greater depths. The

importance of the above finding is that when large load is

applied on soft ground, the possibility of shallow seated

rotational bearing failure is to be examined in view of the

large residual pore pressures at these depths over longer

periods of time.

35

M.R. Madhav & P. Ayub Khan

0

20

40

60

80

100

0 5 10 15

r/rw

P

ore

Pre

ssu

re, U

*(r/

rw, Z

) (%

)

z/H=0.025

0.10

0.5

0.25

0.975

n=15

q*o=1

Th=0.20

0

20

40

60

80

100

0.001 0.01 0.1 1 10

Time Factor, Th

De

gre

e o

f S

ettle

me

nt, U

s

0

20

40

60

80

100

Avg

. P

ore

Pre

ssu

re, U

* avg( Z

) (

%)

Us at all

Depths

U*av g (Z) at

z =0.025H

0.25H

0.975H

0.5H

n=15

q*o=1

Fig. 6 Variation of Us & U*avg(z) with Th –- Effect of Depth

The normalized average excess pore pressures, U*avg (z) at

different depths of the thick clay are presented in Fig. 7 for

n=15 at time factor, Th=0.20 along with the results of linear

theory. The remarkable phenomenon observed is that average

pore pressure values from the non-linear radial consolidation

theory vary with depth in contrast to the depth-independent

U*avg values of linear theory. The differences between the

pore pressures of non-linear and linear theories are relatively

large at shallow depths due to large values of f / o

compared to those at greater depths. This difference increases

with increase of q*o. Moreover, at shallow depths the

variation of pore pressure with depth is relatively very large

compared to that at greater depths due to sharp variation of

f / o at shallow depths. While the excess pore pressures in

the linear theory are independent of, q*o, the pore pressures

according to non-linear theory are dependent on q*o as the

variation of q*o influences the ratio f / o..

0

0.2

0.4

0.6

0.8

1

0 20 40 60 80 100

Avg. Pore Pressure, U*av g(Z) (%)

De

pth

, z/H

n=15

Th=0.20

Lin

ea

r T

he

ory

0.5

0

21

q* o

=4

Fig. 7 Variation of U*avg(z) with Depth—Effect of Load

Intensity

The residual average excess pore pressures thus are

underestimated in the conventional linear theory. In view of

the above, instead of applying the entire preload

instantaneously on the soft ground, it may be applied in

increments with proper time lag to allow quicker dissipation

of pore pressures and gain of shear strength.

Fig. 8 shows the pore pressure variation with radial distance

at different depths for q*o=1, n=15 and Th=0.20. The excess

pore pressures are relatively large at shallow depths wherein

the stress ratio is extremely large but decrease with depth

since the ratio of final to initial stress decreases with increase

of depth. The pore pressure variation with radial distance, r,

is relatively more significant in the upper half of the deposit

compared to that in the lower half.

Fig. 8 Variation of Pore Pressures with Radial Distance-

Effect of Depth

GRANULAR PILES (GP)

Granular piles (GP)/stone columns increase bearing capacity

of foundations laid on the ground, reduce settlement and

accelerate the rate of consolidation of soft soils. The increase

of bearing capacity and reduction of settlement are attributed

to the higher stiffness of GP compared to that of the

surrounding soil. The acceleration of consolidation is credited

to higher or larger permeability of stone columns and shorter

drainage path. In addition, GPs relieve excess pore pressures

by transferring loads from soft soil to GP due to the latter’s

higher stiffness. Typical drained deformation modular ratio

of GP to soft clay ranges from 10 to 20, Han and Ye [14].

The increase of modular ratio of GP to soil accelerates the

consolidation rate under a rigid raft but not under a flexible

one [15]. Though the PVDs and GPs involve radial flow,

basic differences between them are (i) GPs have higher

deformation modulus than the surrounding soil while the

PVDs have no stiffness and cannot sustain any loads.

36

Consolidation of soft ground with drains with no (PVDs) and finite stiffness (granular piles)

Fig. 9 Definition Sketch

The classical solutions for radial consolidation developed by

Barron [1] and Hansbo [2] do not consider the effect of

stiffness of the drains as they do not have any. (ii). Granular

piles have smaller diameter ratio (= unit cell/GP diameters)

compared to that for PVDs. Typical diameter ratios for stone

columns range from 1.5 to 5 while the range varies from 10

to 30 in practice in the case of PVDs. Han and Ye [16] and

Xie et al. [17] observed that the effectiveness of GPs as

drains to accelerate consolidation is reduced due to

contamination of stones with fine grained soil, the

disturbance of surrounding soil and the intrusion of stones

into the surrounding soil.

GP reinforced soil is analyzed by treating it as an

axisymmetric problem (Figs. 1 & 9), similar to that of

consolidation due to radial flow towards sand drain in fine

grained soil. The classical solution of Barron [1] without well

resistance and smear effects under equal strain condition can

be used for approximate results. However, Barron’s solution

with well resistance and smear effects is relatively complex

to use. Hansbo [2] developed a simplified solution for radial

consolidation accounting for well resistance and smear

effects. The above two theories ignore the influence of the

stiffness of the sand drain/GP on the consolidation of the

treated ground.

Han and Ye [14] developed a closed form solution for equal

strain case without and with smear and well resistance but

considering the stiffness of the GP. The equation presented

without smear and well resistance is

2

2

2

21

z

uc

r

u

r

u

rc

t

uvr (12)

where cr and cv are the modified coefficients of

consolidation in the radial and vertical directions

respectively, u the excess pore water pressure at location (r,

z) in the soil, u the average excess pore water pressure at a

depth z in soil, r, z the cylindrical coordinates as defined in

Fig. 1 and t the time.

1

11

)1(

)1(

2,,

,,

Nnc

amm

amamkc sr

scvsv

ssvscv

w

rr (13)

)1(

)1(

,,

,,

scvsv

ssvscv

w

vv

amm

amamkc (14)

where kr and kv are the coefficients of permeabilities in the

radial and vertical directions respectively, mv,c and mv,s are

the coefficients of volume compressibility for the GPs and

the soil respectively and as the replacement ratio of GPs over

the total influence area (=Ac/A, Area of GP/area unit cell), cr

the coefficient of consolidation for radial flow in

unreinforced soil, ns the stress concentration ratio and N the

diameter ratio (=de/dc). Eqn. (12) can be divided into two

equations, one for radial flow and the other for vertical flow.

These equations are in the same form as those of the classical

theories developed by Barron [1] for radial flow and Terzaghi

for 1D consolidation except for the modified coefficients of

consolidations used herein. The overall rate of consolidation

due to the combined effect of radial and vertical

consolidation is expressed as

Ur,v = 1-(1-Ur)(1-Uv) (15)

The stress concentration ratio, ns is defined as [14]

s

c

cv

sv

ss

css

E

E

m

mn

,

, (16)

where cs and ss are the vertical stresses within the GP and

the surrounding soil respectively, Ec and Es the deformation

moduli of GP and the surrounding soil respectively, the

Poisson’s ratio factor which depends upon the Poisson’s

ratios of the GP and the surrounding soil. The variation of

stress concentration ratio with the modular ratio (=Ec/Es) is

depicted in Fig. 10 along with the results of experimental

study by Barksdale and Bachus [18]. The stress concentration

ratio increases with the increase in the modular ratio.

However, many field studies have shown that the stress

concentration ratios for GP-reinforced foundations to be in

the range of 2 to 6, Mitchell [19].

The increase in effective stress in the soft soil over a certain

time period is

)1(1

1

ss

sros

na

aUu (17)

where, uo is the initial excess pore water pressure, Ur the

degree of consolidation for radial flow. The variations of

average total stress on the soil and the GP with time are

depicted in Fig. 11 for ns of 5 and diameter ratio, N of 5.

Stone column

Drainage surface

Drainage surface

37

M.R. Madhav & P. Ayub Khan

Fig. 10 Stress Concentration Ratio vs. Modular Ratio (Han

and Ye, 2001)

Fig. 11 Variation of Vertical Stress on Soil and GP with

Time (Han and Ye, 2001)

Since it is assumed that the entire load applied is initially

taken by water in the surrounding soil, the stress on the

column (GP) increases from zero and the average stress on

the soil is slightly more than the applied pressure, p, over the

influence area. The stress on GP increases while the stress on

the soil decreases with time and consolidation of soft soil.

This phenomenon of stress transfer from the soil to GP is

referred to “stress concentration/transfer”. The

commencement of consolidation initiates the process of

vertical stress transfer from the soft soil onto the GP. Thus

the stress concentration ratio is zero at the beginning and

increases with time and approaches the steady stress

concentration ratio, ns at the end of consolidation.

Fig. 12 shows that excess pore pressure in the soft ground

reduces because of vertical stress transfer. The magnitude of

this reduction can be about 40% of the total applied stress for

the parameters considered. The phenomenon of vertical stress

transfer from soft soil to the drain doesn’t exist in the case of

PVDs as the latter have no stiffness. Higher stress

concentrations induce rapid dissipation of pore pressure

because of larger vertical stress reduction in the soil.

Fig. 12 Dissipation of Excess Pore Pressure with Time (Han

and Ye, 2001)

If the smear and well resistance effects are considered, the

degree of consolidation for radial flow towards the stone

column can be expressed as [16]

m

rm

F

T

eU

8

1 (18)

where the modified time factor for radial flow, 2

e

rmrm

d

tcT ,

crm-the modified coefficient of consolidation for radial flow,

the same as cr in Eq. (13).

2

222

2

2

2

2

2

2

32

4

11

1

1

411

14

3lnln

1

cc

r

s

r

s

r

s

rm

d

H

k

k

NNk

k

N

S

k

k

N

SS

k

k

S

N

N

NF

(19)

where S- the ratio of diameter of the smeared zone (ds) to the

diameter of GPs (dc), kr and ks – the coefficients of radial

permeability of the undisturbed soil and smeared soil

respectively, H the longest drainage distance in the vertical

direction. Eq. (19) is similar in form to that by Hansbo [2].

However the present equation takes into account the

characteristics of GP-reinforced soil. The effect of some of

the important parameters incorporated in Eq. (19) is

discussed below. The influence of the diameter ratio on the rate of

consolidation is presented in Fig. 13 for typical diameter

ratios of 1.5 to 5. The rate of consolidation is accelerated with

the decrease in the diameter ratio due reduction in the

drainage path. The diameter ratio can be decreased either by

decreasing the spacing between the GPs or/and by enlarging

the diameter of the GPs. The effect of the stress concentration ratio on the rate of

consolidation of GP treated ground is depicted in Fig. 14.

Consolidation is accelerated with increase in stress

concentration ratio.

38

Consolidation of soft ground with drains with no (PVDs) and finite stiffness (granular piles)

Han & Ye (2002) Han & Ye (2001)

Fig. 13 Influence of Diameter Ratio on the Rate of

Consolidation (Han and Ye, 2002)

Fig. 14 Influence of Stress Concentration Ratio on the Rate

of Consolidation (Han and Ye, 2002)

The theory proposed by Han and Ye [16] considers the

excess pore pressures developed even inside GPs unlike in

the classical theories of Barron [1] and Hansbo [2]. The rate

of dissipation of excess pore pressures within the GP depends

on the length of the column. This aspect is embedded in the

function F’m. As expected, the increase in thickness of soft

soil retards the rate of consolidation due to an increase of the

drainage path for water draining out after it enters the GP.

Fig. 15 depicts the reduction of rate of consolidation due to

increase in the thickness of soft soil for a given set of

parameters involved.

During consolidation the average vertical effective stress in

the GP increases while the average vertical total stress in the

soil decreases with time due to stress transfer from soil to GP.

This change of stress is illustrated in Fig. 16. In the case of

PVD treated ground, the entire stress applied is borne by the

soft soil alone and the total stress remains constant during the

consolidation process.

Fig. 15 Influence of the Thickness of Soft Soil on the Rate of

Consolidation (Han and Ye, 2002)

Fig. 16 Variation of Vertical Stress in Soil with Time (Han

and Ye, 2002)

Fig. 17 Comparison of Rate of Consolidation by Different

Models (Han and Ye, 2002)

The comparison of different methods to evaluate the rate of

consolidation of GP-reinforced soil is shown in Fig. 17. The

39

M.R. Madhav & P. Ayub Khan

solutions used herein, Han and Ye [14] and Barron [1]

ignore the smear and well resistance effects while the other

two solutions, Han and Ye [16] and Hansbo [2] consider the

smear and well resistance effects. Han and Ye [14] solution

overpredicts the rate of consolidation compared to that by

Barron due to the reason that the former theory considers the

stiffness of GP or the stress concentration ratio which

accelerates the rate of consolidation. The same reason

accounts for the faster the rate of consolidation predicted by

Han and Ye [16] compared to that by Hansbo [2] even though

both the theories consider smear and well resistance effects.

CONCLUSIONS Consolidation of soft ground treated by PVDs and granular

piles are compared and contrasted. The former are soft and

cannot carry any of the applied stress while the latter

accelerate consolidation further by transfer of load from soft

ground to stiff granular piles during the consolidation phase.

Thus consolidation by GPs is extremely fast compared to that

by PVDs. A simple non-linear theory of radial consolidation

developed for thick deposit of clay treated with PVDs

considering linear void ratio-log effective stress relationship

predicts that while the degree of settlement is independent of

the final to initial stress ratio, the degree of dissipation of

pore pressure is very much dependent on the stress ratio. The

residual excess pore pressures are under-estimated in the

conventional linear theory. The proposed nonlinear theory

substantiates the actual in-situ slower rate of degree of

dissipation of excess pore pressures compared to that of the

degree of settlement. The non-linear consolidation effect is

pronounced at shallow depths compared to the effect at

greater depths. The excess pore pressures due to radial

drainage vary not only with time and radial distance but also

with depth in contrast to depth-independent pore pressures

from the conventional theory for radial flow. The significance

of the proposed theory is that it can explain failure of high

embankments constructed rapidly on thick deposits of fine

grained soils.

The rate of consolidation due to GPs can reasonably be

predicted considering the modular ratio of GPs to that of the

soil, the smear and well resistance effects. The smear and

well resistance effect is significant in the case of GPs. The

rate of consolidation is accelerated because of increasing

stress concentration ratio.

REFERENCES

1. Barron, R.A. (1947). Consolidation of fine grained soils

by drain wells. Trans. ASCE, 113(2346), 718-754.

2. Hansbo, S. (1981). Consolidation of fine grained soils by

prefabricated drains. Proc. of the 10th ICSMFE,,

Stockholm, Sweden, June, 3, 667-682.

3. Lekha, K.R., Krishnaswamy, N.R. and Basak, P. (1998).

Consolidation of clay by sand drain under time–

dependent loading. J of Geotech. and Geoenv., 124(1),

91-94.

4. Bergado, D.T., Balasubramaniam, A.S., Fannin, R.J. and

Holtz, R.D. (2002). Prefabricated vertical drains (PVDs)

in soft Bangkok clay: a case study of the new Bangkok

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