Computers and Geotechnics 9 (1990) 133-148

AXIAL RESPONSE ANALYSIS OF pF.l:.q IN VERTICALLY AND HORIZONTALLY NON-HOMOGENEOUS SOILS

C.Y. Lee Research Fellow

School of Civil and Mining Engineering University of Sydney

Australia

and

H.G. Poulos Professor

School of Civil and Mining Engineering University of Sydney

Australia

ABSTRACT

c f T h i s p p , aper presents a modified procedure for the analysis of the axial response ues embedded in multi-layered soils. The results obtained by this

procedure are compared with those computed by some previous methods and with a limited number of field test measurements.

In the determination of the group settlement interaction between piles embedded in muki-layered soils, an additional simple soil mass stiffness model is dev¢Iol~, d in order to include the horizontal non-homogeneity of the soil due to sod disturbance cause by pile installation. The predictions b y this model agree more ciosciy witll the observed field test group performance than do predictions by the conventional method which assumes lateral homogeneity of the soil.

133 Computers and Geotechnics 0266-352X/90/$03-50 © 1990 Elsevier Science Publishers Ltd, England. Printed in Great Britain

134

INTRODUCTION

Axial pile and pile group analyses using Mindlin's equations of elasticity have

provided a simple and practical means of calculating the settlement of piles and

pile groups in the past two decades (e.g. Poulos and Davis, 1980; Butterfield

and Banerjee, 1971; Banerjee and Davies, 1977). In general, these analyses lead

to adequate solutions in a soil mass with uniform or linearly increasing soil

modulus with depth (e.g. Poulos 1979a, 1979b). It has been found that they

may not give acceptably accurate solutions for piles embedded in layered soils

where the modulus of the adjacent layers differ abruptly (e.g. Poulos, 1979a, Yamashita et al, 1987). In addition, they usually overpredict group interaction

effects since they ignore the horizontal non-homogeneity in modulus in each

soil layer between piles, due to pile installation (O'Neill et al 1977). The analysis of pile groups in vertically non-homogencous soil can be modelled

more accurately by using the infinite layer method (Cheung et al 1988) or the finite element method (Chow 1987, 1989); the latter method can also be used to

model horizontally non-homogeneous soil.

In this paper, a more general approximation for piles in an arbitrary layered soil profile, involving the value of modulus in all soil layers, is developed.

The influence of pile installation on the soil modulus between piles in a group

is considered by introducing a simple empirical expression to relate the modulus

in the disturbed soil near the pile surface to the modulus in the less disturbed

soil mass further away.

These two approaches are incorporated into conventional axial pile and pile

group analyses based on the boundary element method. The modified analysis

generally leads to better agreement with field measurements than do the

conventional approaches.

Method of Analysis

(a) Single Pile

The simplified form of boundary element analysis developed by Poulos and Davis (1980) is used in which the pile is represented as an elastic cylinder and

the surrounding soil mass as an elastic continuum, as shown in Figure I.

135

The axial displacement of the pile elements may be expressed as follows:

{pp} - [AD][FE] {p) ÷ Pb {|} (1)

where

{Pp} [AD]

{P}

Pb {~}

= displacement vector

= summation matrix

= pile compression matrix

= interaction stress vector

= pile base displacement

= vector whose elements are unity

xj

Xl

- -d l - -

t ) t ( ) 1 i )~ 4 ( ) 4 ( ) t t ~pJ t I t t ( ) t i 1 b~4 t . . ) 4

t t

I t tpJ ® t t

t I t t t Pb

I r J I r r l l " ~ s r ~ F 4 1 1 r ~ J s l . . r r J i 4 1 F r l r l r i . j l . .

bteract~n shut $ ~

Soil Modulus

I I

I J v$:Constant

bE, j-..]

I I

-E,A~

I . E b . I

C k , ~ of sin] modulus ,,,~t'h dtpth I ~ t i ~ s of Sol Pie SQt ~ j t n Llytrs fn~ surface

FIG.1 ANALYSIS OF SINGLE PILE IN LAYERED SOIL

1 3 6

The displacements of the soil adjacent to each pile element may be expressed

as follows:

{ps} - [~-s]{p } (2)

where

{Ps} -"

[~1 =

soil displacement vector

matrix of soil influence factors determined from Mindlin's

equation (Mindlin 1936; Poulos and Davis, 1980); divided

by the soil Young's modulus near the pile surface.

When pile-soil interface conditions remain elastic,

(ps } - {pp}

hence

I~S - AD'FEI{pP) - Pb{l} (3)

The vertical force equilibrium condition requires:

N

_ ~ AiPi = p i - I

(4)

where

A i = surface area of element i

P = of applied load to the pile head

N = total number of pile elements.

The unknown interaction stress {p} and base displacement Pb, can be evaluated

by solving equations (1) and (2).

For vertically non-homogeneous soils, Poulos (19"/9a) proposed a simple and practical method in which the homogeneous soil modulus E s is replaced by thc mean values at the influencing and influenced elements, but this method ignores

the soil moduli of the other layers. This method does not give particularly

accurate results for a pi le embedded in layered soils in which the underlying

layers are more compressible. Yamashita et al (1987) modified this m~hod by

137

considering the soil modulus at every layer using a one -pa ramete r "a" model.

This parameter "a" depends on soil and pile properties, but no clear method is

suggested for its determination.

A similar muk i - l aye r ed soil model is developed here (termed the ML model),

which considers the effect of the soil modulus at all soil layers, but requires no

additional parameter when determining the mean soil modulus at the influencing

and influenced elements. This model postulates that, for an element i, the soil

modulus Esi j due to the influenced of element j is given as follows:

Esl j - 0 .5(Esa t + Esa j) (6)

and

N ~. 6 k Esk

k-1 Esal " N ; for I - i , j (7)

Y. 6 k k-1

where

[1 . Ixl - xkl Esk]-' 6 k

- [ ~ EslJ (8)

L = total pile length

Xl,X k = distance from ground surface of elements I

respectively

Esi,Esk ---- soil Young's modulus of layer I and k respectively

N ---- total number of elements.

and k

Basically this model assumes that the mean soil modulus depends on the relative

soil stiffness and the distance between all the influenced and the influencing

elements.

(b) Pile Groups

For a group of two identical equally loaded piles, only the calculation of the

soil displacement at each element requires modification to include the

components due to the other pile, and hence equation (2) may be re -expressed

as follows (Poulos and Davis, 1980):

138

(9)

where

F;-S 1 ,Es 2

I i , I 2

= soil Young's modulus near the surface of pile.s 1 and 2

respectively,

= matrices of displacement-influence factors for piles 1 and 2

respectively.

This conventional approach assumes that the soil moduli (i.e. Es, and Es2 ) used

to determine the matrices 11 and 12 are identical. However it has been found

that the soil closer to the pile surface is more disturbed than that further away,

due to pile installation (e.g. Cooke ¢t al 1979, Williams, 1979, Francescon, 1983)

and hence some horizontal non-homogenei ty is induced in the soil mass.

Poulos (1988a) has suggested a two-parameter soil model to modify the

calculation of group interaction effects. A simpler one parameter horizontal

non-homogeneous soil model is proposed here which includes the variation of

horizontal soil modulus used to determine the soil displacement influence

factors. The soil model is shown diagramatically in Figure 2 and Equation (9)

may be modified as follows:

{Ps} " [~'~ + -~-d]{P } (10)

and

Esd E--'~-- LQJ

where

E s = average soil Young's modulus within one pile radius from the

pile surface,

Esd = soil Young's modulus at a distance s greater than one pile radius

from pile surface,

n = soil parameter depending on pile and soil type.

S n

E,

Less

Dis

turb

ed S

oil M

ass

Dis

turb

ed S

oil d

ue

to P

ile In

stal

latio

n

I FI

G.2

HORI

ZONT

AL NO

N-HO

MOGE

NEOU

S SOIL

MOD

EL IN

PI

LE G

ROUP

INTE

RACT

ION A

NALY

SIS

0.5

L o ,Y,

u_

= 0.

25

o

0 O.S

= 0.

25

.o

n=

O

h ~

,/[,

.ml,

L

/~,S

I

_~-,.

...

,. "::

'..:.

Conv

entio

nal

- ~p

proa

ch

(a) H

om

o~

i

L~

~~

~

~

I I

~ 6

8 10

15

20

0.7

"~

{b) N

on-H

omo~

(G

ibson

) Sod

I

I I

I I

2 Z,

6 6

10

20

Pile

Spa

cing/

Diam

eter

Is/d

) FI

G3

EFFE

CT OF

n V

ALUE

S ON I

NTER

ACTIO

N FACT

OR

co

co

140

Figure 3 demonstrates the effect of the value of n in the so -called

"horizontally non-homogeneous" (HNH) soil model on the computed interaction

factor c~. For a homogeneous soil (n = 0) the values of interaction factor c~ are

equivalent to those computed by the conventional approach (Poulos and Davis,

1980). The value of interaction factor decreases as the value of n increases. It

appears to decrease more significantly with pile spacing for higher n values,

than the conventional approach (n = 0).

The values of interaction factor a in a non-homogeneous ("Gibson") soil also

vary similarly with n and pile spacing, except that the interaction factor values

from the conventional approach lie below those computed by the horizontal

non-homogeneous (HNH) soil model for n = 0.

This horizontal non-homogeneous (HNH) soil model may also be used to

analyse any general configuration of piles in a group. Using this model in

conjunction with the multi-layered soil model, it is believed that a more

realistic simulation of pile group behaviour may be made. This combined

approach will be referred to as the ML/HNH model.

Evaluation of the Modified Approaches

Single piles in layered soil

The present approach using the multi-layered (ML) soil model has been used

to analyse three idealised cases (Poulos, 1979a). The results are compared with

solutions obtained from other approaches, as shown in Figure 4. The resuks

computed by all the approaches appear to agree closely with those obtained by

the finite element approach for Case 1 and Case 3. However, for Case 2, in

which the soil modulus decreases with depth, the solutions from the present

approach and the finite element approach only differ by about 5%, whereas the

difference between the other approaches and the finite element method exceeds

20%.

Engeling and Reese (1974) performed a compression loading test on a drilled

pile shaft of length 42 ft (12.8 m) and diameter 30 inches (0.76 m), embedded

in soft to hard clay west of Bryan in Texas, USA. The soil shear strength

decreased from the ground surface to about 30 ft (9.1 m) depth and increased

beyond that depth. In the calculations performed by the authors, the soil

modulus has been assumed to be 750 times the shear strength obtained from the

triaxial tests (Aschenbrenner et al 1984). Figure 5 compares the measured

results with those predicted by the present approach, the conventional approach

(Poulos 1979a) and the Yamashita et al (1987) approach. The comparisons

~ P

1 2

Cas

e 3

l °'3

LI II

] S

oil

You

ng's

M

odul

us

Dis

trib

utio

n w

ith

Dep

th

Ep

L h

~--~

s= 1

000

~ ~

-=

25

~ ~

=

2 ~

vs

= 0.

3

",'/

,'/H

////

////

///,

','/

/,'/

////

////

H/H

/[//

~,~

////

H//

H//

///,

*//H

l/

H//

H//

////

////

/

19(f

inite

el

emen

t)

Cas

e 1

Cas

e 2

Cas

e 3

9==

PHe

settl

emen

t

~w

md

Ap

~oac

h

e et

d C

Bf/I

la=0

.SI

Ihu~

. e'

nd

Eq

u~

U

~

Sd

Y~

i et

d W

B?I

a=0.

SI

~M

nt

A~

19";9

al F.

.qub

llml U

nifc

m 5a

i e

w~

Y

amiJ

l'e e

l d

le=O

.SI

Prm

nt ~

dl

0.2

O.Z

. 0.

6 0.

8 1.

0 l

I i

i i to

12

I*

• I

• I I I

•1 I• I I"

6 E

~9

C3 "

12

15

Ap

plie

d

Load

, K

N

lJ.5

89

0

/',v

- /

} /

----

Moa

su~,

, c[

~Ii.g

a.d

-

~f

I Re

ese.

197/

,) .~

//

"-

----

[onv

entio

na[

Appr

oach

-

IPou

los

19"/9

a)

- /

!p~o

~.l.g

al

.....

- "

-Y~

ashi

ta e

t al

198

7 P

rese

nt A

ppro

ach

Con

vent

iona

l App

roac

h (P

oulo

s. 1

979a

) •

Yal

ashi

ta e

t al

(19

87J

(a=0

.51

•

Pre

sent

App

roac

h

0 0.

5

Pre

dic

fed

M

ea

sure

d

PiL

e H

ead

Se

tfle

me

nt

E 1.0

FIG

./,

CO

MPA

RIS

ON

SO

LUTI

ON

CO

MPU

TED

BY

VA

RIO

US

APP

RO

ACH

ES

FIG

S

PR

ED

ICTI

ON

S O

F C

OM

PRES

SIO

N P

ILE

IN C

LAY

SO

IL

1 4 2

indicate that the predictions of head settlement and load distribution by the

present approach agree well with the measured values, whereas the other two

approaches seem to underestimate the settlement by more than 15%.

The three approaches have also been employed to predict the behaviour of an

offshore steel tube pile driven into marine sediments at Plancoet in France (Puech, 1982). The soil profile consisted of three distinct layers, as shown in

Figure 6, and the pile was 13 m long, 0.27 m in diameter with a 6.3 mm

wall thickness. The pile was loaded in tension in three different stages. For

the theoretical calculation, the soil modulus was assumed to be 15 times the

static cone resistance (Poulos, 1988b). As shown in Figure 6, the present approach predicts the measured settlements more accurately than the other two approaches, although all three approaches underpredict the load distributions. The present approach predicts a more gradual transfer of load with depth than

the other two approaches.

"~" 0S r~

== c~ ca

Q.

1.0

Sandy Silt v-/.0% c'=0 . O*=/.2 °

Loose Sand w-/*S% #'=/.3"

Silty Clay v -~ -5% , %=57 c'=20KPa . vp=29 e'=26"30'

Conve~tkmat ,4.~oKh IPoulU I ~l?9,~J

Ymm~hita et of 11~$7) :e=O.5l

Present Approach

0 100 200

I j /a

~ s t 52

0 Leg=rid: Measured

- - - - - Eonvmltional Approach (Poulos. 19?9ai

- - * - - YamtshJto et el 1987 - - - - - - Present Approlch

Apptied Load (kN) 100 200 T'

TesP S3A I

100 200 '/I

Test S3B I

1.0 1.0 PredicPed Measured (Pite Setttement)

FIG.6 PREDICTIONS OF OFFSHORE TENSION PILES IN MARINE SOIL

1.0

143

Pile groups in layered soil

Cooke et al 0980) performed field tests on steel tube piles of 168 nun

diameter with 6.4 nun wall thickness and approximately 5 m long embedded in

London clay, and measured the settlement interaction factors. The vertical soil

modulus was assumed to be represented as a "Gibson" soil profile with

Es(z) - 35z MPa where z is the soil depth. As shown in Figure 7, the

conventional theoretical approach assuming lateral homogeneity of the soil

overestimates the interaction factor values significantly. For the horizontally

non-homogeneous soil model, various values of the parameter n have been

tried to obtain a fit with the measured values. It appears that n = 0.5 gives the

best agreement with the measured values, and this value has been used in the

predictions of another series of pile group tests.

O.SO

g o.z5

N

o 0 12

Measured . ~ " ~ - - - - Cenventic~ Approach (PouIo$, 1979a)

~ . . . . Present Approa¢h (n=03) ~ ' ; ~ ~ ' ~ - - ~ - Present Approach (n=0.51

2 I, 6 8 10 s/d

FIG.7 MEASURED ANO COMPUTED INTERACTION FACTORS

O'Neill et al (1981) have reported results of axial loading tests on full-scale

pile groups and single piles in clay. The piles were 10.75 inches (0.273 m)

diameter steel tubes with a 0.365 inches (9.3 ram) wall thickness and 43 ft

(13.1 m long). The tests were carried out at a site at the University of

Houston, Houston, Texas, USA, and the geotechnical data at the site is summarised in Figure 8.

In this case, a remoulded near-pile soil modulus of 25 times the static cone

resistance qc (Poulos, 1988a) was used for the theoretical predictions of the

behaviour of the pile groups and single piles using the following approaches:

(a) conventional approach (Poulos and Davis, 1980, with n = 0);

(b) approach using HNH model (with n = 0.5);

(c) present approach (MIdHNH model, with n = 0.5).

144

Stratigraphy

0

L~ V. stiff 9ray i tan clay Still day, sand seams

S - ~ Stiff-V. stiff r o d . . ~ i gray clay

i~XJ ----- 1Q -I~.~ Stiff-V.stiff gray &

I~XI tan sandy day. " ~ . ~ vltb sand pockets

1s-111111 o ~ , red ~ gray /11111 ~t. ~ith day. ~lt /11111 ~ sand layers

2 0 & ~ V. stiff red & gray

0

Average Cone SPT Resistance

Blows/0.3m (kN/m z) 20 ~0 0 ,000 10000

t -I Undrained Shear Strength IkN/m z)

250 500 I

i ,

t,, i ~HT

"Triaxial

FIG.8 SUMHARY OF GEOTECHNICAL DATA AT TEST SITE

Water Content OCR I'/.I

20 z,0 80 0 2 z,

' ~LL t ° , o

. : l -o "

,4-0 . ~ X 0 I ;O'JC

"FOX

~,'o~ "° ~.IOY--- .x

: ~ ; ':-°Ons°l I ~'~ j ~ iriaxialJ

~=Nat. W/C I . Consol. I

~ 9

.-~ S

z 1

i I A

x l I

I•

* I I•

x I I

Ix I I I I I

300 600 900 1000

(a) Pile Head St i f fness HN/m

- - - Measured x Conventional Approach

(n=0) (Poulos and Davis, 1980) Approach using HNH HodeI (n=0.5)

• Present Approach In=0 5)

1500

,=.9

.:- 5

cL.

z l

I I I L"

I I

&.

I

2 3

(b) Sett lement Ratio R s

FIG9 PREDICTIONS OF PILE HEAp STIFFNESS AND SETTLEMENT RATIO

Load

Av

erag

e Pi

le H

ead

Load

! ? 0.S

i@

IA

l x

I

x I !

• I

AI

x l II

0.87

5 Lo

ad

Aver

age L

oad

(a)

9 P

ile G

roup

1.2S

Lege

nd:

....

H

easu

red

x C

onve

ntio

nal

App

roac

h (n

=O)

(Pou

los

and

Dav

is,

1980

) A

App

roac

h us

ing

HN

H M

odel

In

:O.S

) •

Pre

sent

A

ppro

ach

(n=O

.S)

! O.S

.i

x I

il

0.87

5 Lo

ad

Aver

age L

oad Ix

I

1.25

(b)

S P

ile G

roup

FIG

10

PRED

ICTI

ONS

OF

PILE

HEA

D L

OAD

DIS

TRIB

UTIO

NS

Z_ 0

.S

L

10

Z [-

0S

1.0

1.1

Pred

ri[I

Dep

th

F

o,~°~J

~7

Pile

Gro

up

11

11

..

..

7

r)-

--

r

,Yr

~C

en

t r e

Pile

o} o~

,~i!~

o r n e

r Pi

le

S Pi

le G

roup

tl

11

11

Pred

rill

'?e-~tL

~'7-

/ o/

~ E

dge

Pile

!;I°

/ /'~

12"

/,Y

~/o C

entre

Pile

.I" ,J

/Cor

ner

Pile

9 P

i(e Gro

up

L_egend:

o He

asur

ed

----

-Con

vent

iona

l Ap

proa

ch In

:01

(Poulos

and

Davi

s. 1

980)

-- ~

Appr

oach

us

ing

HNH Ho

del

In:0

S)

----

-- Pr

esen

t Ap

proa

ch

In=0

S)

FIGJI PR

EOIC

TION

S OF

IO

A0 01

STRI

BUTIONS

ALON

G Ptl£ lEN

GTH

t46

The predicted and measured values of the pile head stiffness and group

settlement ratio R s are shown in Figure 9. The conventional approach (with

n = 0) appears to underestimate group stiffness and overpredict the pile

settlement, the difference increasing as the number of piles in the group

increases. However, the values predicted by the modified conventional

approach and the present approach agree much more closely with the

measurements.

Figure 10 also demonstrates that the group pile head load distributions predicted

by the HNH approach (with n = 0.5) and the present approach (ML/HNH) are

in better agreement with the measurements than those predicted by the

conventional approach.

Despite the fact that the HNH and ML/HNH approaches seem to predict the

measured pile head response similarly well, the main difference in the predicted

performance from these approaches is illustrated in Figure 11 where the load

distribution along the pile is plotted. It can be seen that the load distributions

predicted by the present ML/HNH approach agree more closely with the

measurements than do the predictions by the other two approaches.

CONCLUSIONS

The conventional approach, using Mindlin's equations for the analysis of the

settlement of a single pile in a layered soil profile, is generally adequate except

when significant differences in soil modulus exist between adjacent soil layers or

if a soil layer is underlain by a much more compressible layer. In order to

overcome this limitation, a more general soil profile approximation model (the

ML model) has been developed, involving the value of soil modulus at all

layers in the soil profile. Comparisons with some field measurements for piles

embedded in a layered soil demonstrate that this modified approach leads to

more realistic predictions of pile head response and load distribution than does

the conventional approach.

An alternative simplified pile group analysis has been developed, and involves a

computational model (the horizontal non-homogeneous or HNH model) which

relates the remoulded near-pi le soil modulus to the value for the less disturbed

soil mass further away, via the normalised pile spacing and an exponent

parameter n. The value of parameter n may depend on the pile and soil type,

but a value of n = 0.5 appears to fit limited available data. This HNH model

147

will reduce the overprediction of group interaction effects commonly experienced when using the conventional approach. Comparisons of the predictions by this modified approach with some field measurements of pile groups in layered soils have shown generally good agreement between predicted and measured group performance, although some inaccuracy remains in the predicted load distribution characteristics along the pile. This inaccuracy can be overcome by incorporating the more general soil profile approximation model into the analysis, thus leading to more realistic predictions of both the pile head performance and the load distribution characteristics within a pile group.

ACKNOWI.EDOEMENT

The work described in this paper forms part of a research project into the Mechanics of Calcareous Sediments, supported by the Australian Research Council.

REFERENCES

1. Aschenbrenner, T.B. and Oslen, R.E. (1984). Prediction of Settlement of Single Piles in Clay. Anal. and Design of Pile Foundns., ASCE, pp. 41-58.

2. Banerjee, P.K. and Davies, T.G. (1977). Analysis of Pile Groups Embedded in Gibson Soil. Prec. 9th ICSMFE, Tokyo, Vol. I, pp. 381-386.

3. Butterfield, R. and Banerjee, P.K. (1971). Elastic Analysis of Compressible Piles and Pile Groups. Geotechnique, Vol. 21, No. I, pp. 43-60.

4. Cheung, Y.K., Tham, L.G. and Guo, D.J. (1988). Analysis of Pile Group by Infinite Layer Method. Geotechnique, 38(30) pp. 41.5-431.

5. Chow, Y.K. (1987). Axial and Lateral Response of Pile Groups Embedded in Nonhomogeneous Soils. Int. Journal for Numerical and Analytical Methods in Geomechanics, 11(6), pp. 621-638.

6. Chow, Y.K. (1989). Axially Loaded Piles and Pile Groups Embedded in a Cross-Anisotropic Soil. Geotechnique, 39(2), pp. 203-211.

7. Cooke, R.W., Price, G. and Tart, K. (1979). Jacked Piles in London Clay: A Study of Load Transfer and Settlement under Working Conditions. Geotechnique 29, No. 4, pp. 461-468.

8. Cooke, R.W., Price, G. and Tart, K. (1980). Jacked Piles in London Clay: Interaction and Croup Behaviour under Working Conditions. Geotechnique 30, No. 2, pp. 97-136.

9. Engeling, D.E. and Reese, L.C. (1974). Behaviour of Three Instrumented Drilled Shafts under Short Term Axial Loading. Research Report 176-3, University of Texas, Austin, Texas. l16p.

148

10. Francescon, M. (1983). Model Pile Tests in Clay: Stresses and uDiSmPvlacements due to Installation and Axial Loading. PhD Thesis,

ersity of Cambridge.

11. Mindlin, R.D. (1936). Force at a Point in the Interior of a Semi-lnfinite Solid Physics, Vol. 7, pp. 195-202.

12. O'Neill, M.W., Ghazzaly, O.I. and Ha, H.B. (1977). Analysis of Three-Dimensional Pile Groups with Non-linear Soil Response and Pile-Soil-Pile Interaction. Proc. 9th Annual OTC, Houston, Paper OTC 2838, pp. 245-256.

13. O'Neill, M.W., Hawkins, R.A. and Mahar, L.J. (1981). Field Study of Pile Group Action. Rep. No. FHWA/Rd-81/002, Fed. Highway Admin.

14. Poulos, H,G. (1979a). Settlement of Single Piles in Non-Homogeneous Soil. Jnl. Geol. Eng. Divn., ASCE, Vol. 105, No. GTS, pp. 627-641.

15. Poulos, H.G. (1979b). Group Factors for Pile-Deflection Estimation. Jnl. Geot. Eng. Divn., ASCE, Vol. 105, No. GT12, pp. 1489-1.509.

16. Poulos, H.G. (1988a). Modified Calculation of Pile Group Settlement Interaction. Jnl. Geot. Eng., ASCE, Vol. 114, No. 6, pp. 697-106.

17. Poulos, H.G. (1988b). Marine Geotechnics. London, Unwin Hyman.

18. Poulos, H.G. and Davis, E.H. (1980). Pile Foundation Analysis and Design. John Wiley and Sons, New York.

19. Puech, A. (1982). Basic Data for the Design of Tension Piles in Silty Soils. Proc. 3rd Int. Conf. Behaviour of Offshore Structures, Vol. 1, pp. 141-157.

20 Williams, D.J. (1979). The Behaviour of Model Piles in Dense Sand Under Vertical and Horizontal Loading, PhD Thesis, University of Cambridge.

21. Yamashita, K., Tomono, M. and Kakurai, M. (1987). A Method for Estimating Immediate Settlement of Piles and Pile Groups. Soil and Foundns., Vol. 27, No. I, pp. 61-76.

Received 25 October 1989; revised version received 7 July 1990; accepted 10 July 1990