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:
(1)
{pp} - [AD][FE] {p) ÷ Pb {|}
where
{Pp}
=
displacement vector
[AD]
=
summation matrix
=
pile compression matrix
{P}
=
interaction stress vector
Pb
=
pile base displacement
{~}
=
vector whose elements are unity
--dl--
xj
Xl
t
(
i
(
(
t
I
(
i1
t..
) t
) 1
)~ 4
) 4
) t
~pJ t
t t
) t
b~4
) 4
Soil Modulus
I
t
t
I
I
t
J v$:Constant
tpJ ®
t
t
Ittt
Pb
I
bE, j-..]
-E,A~
I
I
t
IrJIrrll"~sr~F411r~Jsl..rrJi41Frlrlri.jl..
I.
bteract~n shut $ ~
I ~ t i ~ s of
~jtn
fn~ surface
Sol
Pie
SQt
Llytrs
Ck,~
Eb
.I
of sin] modulus ,,,~t'hdtpth
FIG.1 ANALYSIS OF SINGLE PILE IN LAYERED SOIL
136
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
(4)
i-I
where
Ai = 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 pile embedded in layered soils in which the underlying
layers are more compressible. Yamashita et al (1987) modified this m ~ h o d by
137
considering the soil modulus at every layer using a o n e - p a r a m e t e r "a" model.
This parameter "a" depends on soil and pile properties, but no clear method is
suggested for its determination.
A similar m u k i - l a y e r e d 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 Esij due to the influenced of element j is given as follows:
Esl j - 0.5(Esa t + Esa j)
(6)
and
N
~. 6k Esk
k-1
N
;
Esal "
for
I - i,j
(7)
Y. 6k
k-1
where
6k
L
[1 . Ixl - xkl Esk]-'
- [
=
Xl,Xk =
Esi,Esk
N
-------
~
(8)
EslJ
total pile length
distance from
respectively
ground
surface
of
elements
I
and
k
soil Young's modulus of layer I and k respectively
total number of elements.
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 r e - e x p r e s s e d
as follows (Poulos and Davis, 1980):
138
(9)
where
F;-S1 ,Es 2
=
Ii,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-homogeneity 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
Es
Esd
n
= average soil Young's modulus within one pile radius from the
pile surface,
= soil Young's modulus at a distance s greater than one pile radius
from pile surface,
= soil parameter depending on pile and soil type.
E,
I
FIG.2 HORIZONTALNON-HOMOGENEOUS
SOILMODELIN
PILE GROUPINTERACTIONANALYSIS
Disturbed Soil due
to Pile Installation
Less Disturbed Soil Mass
S n
O.S
0
I
I
I
6
~
0.7
{b)N o n - H o m o ~
(Gibson) Sod
I
I
(a) H o m o ~
n=O
h
L/~,SI
I
I
"~
8 10
~L i ~ ~~ ~
15 20
_~-,.... ,. "::'..:.
Conventional
~pproach
~,/[,.ml,
2
Z,
6 6 10
20
Pile Spacing/Diameter Is/d)
FIG3 EFFECTOFn VALUESONINTERACTIONFACTOR
.o= 0.25
u_
,Y,
=o 0.25 -
L
o
0.5
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
]
l
2
Ep
L
h
~--~s= 1000 ~ ~ - = 25 ~ ~ =
YamiJl'e el d
le=O.SI
Prmnt ~ d l
e w ~
19";9al
F..qubllmlUnifcm5ai
Y~i
et d WB?I
a=0.SI
~Mnt A ~
Sd
l
0.2
•
•
19(finite element)
O.Z.
0.6
0.8
I
i
i
I"
I
I•
I
I
I
•1
I
I*
1.0
ito
l/H//H//////////
2 ~ vs = 0.3
FIG./, COMPARISONSOLUTION COMPUTED BY VARIOUS APPROACHES
Case 3
Case 2
Ihu~. e'nd
Equ~ U~
la=0.SI
Ap~oach
e et d CBf/I
~wmd
9== PHe settlement
Case 1
Case 3
Soil Young's Modulus Distribution with Depth
1
",'/,'/H///////////,','//,'/////////H/H/[//~,~////H//H/////,*//H
°'3LI II
P
~
12
/
"
15
12 -
6
"
/
~f
}
//
/
I
0
0.5
Predicfed
Measured PiLe Head S e t f l e m e n t
Present Approach
•
•
----Moasu~,, c[~Ii.g a.d
Reese.197/,)
"-----[onventiona[ Approach
!p~o~.l.gal
IPoulos 19"/9a) .....
-Y~ashita et al 1987
Present Approach
Conventional Approach
(Poulos. 1979a)
Yalashita et al (1987J
(a=0.51
.~
/
/',v
1.0
E
-
-
-
890
FIGS PREDICTIONS OF COMPRESSION PILE IN CLAY SOIL
C3
~9
E
lJ.5
Applied Load, KN
142
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.
ApptiedLoad (kN)
0
100
Sandy Silt
200
T'100
200
'/I
100
200
I j
v-/.0%
/a
c'=0 . O*=/.2 °
"~"
r~
0S
==
Loose Sand
w-/*S%
#'=/.3"
c~
ca
Q.
1.0
Silty Clay
v-~-5% , %=57
c'=20KPa . vp=29
e'=26"30'
~ s t
Conve~tkmat ,4.~oKh
52
I
TesP S3A
I
Test S3B
IPoulUI ~l?9,~J
Ymm~hita et of 11~$7)
:e=O.5l
Present Approach
Leg=rid:
-----
Measured
Eonvmltional
0
Approach
(Poulos. 19?9ai
- - * - - YamtshJto et el 1987
- - - - - - Present Approlch
1.0
1.0
PredicPed
Measured (Pite Setttement)
FIG.6 PREDICTIONS OF OFFSHORE TENSION PILES IN MARINE SOIL
1.0
143
Pile groups in layered soil
Cooke et al 0980) performed field tests on steel tube piles of 168 nun
diameter with 6.4 nun wall thickness and approximately 5 m long embedded in
London clay, and measured the settlement interaction factors. The vertical soil
modulus was assumed to be represented as a "Gibson" soil profile with
Es(z) - 35z MPa where z is the soil depth.
As shown in Figure 7, the
conventional theoretical approach assuming lateral homogeneity of the soil
overestimates the interaction factor values significantly. For the horizontally
non-homogeneous soil model, various values of the parameter n have been
tried to obtain a fit with the measured values. It appears that n = 0.5 gives the
best agreement with the measured values, and this value has been used in the
predictions of another series of pile group tests.
O.SO
.~"~
~
Measured
- - - - Cenventic~ Approach(PouIo$, 1979a)
. . . .
Present Approa¢h(n=03)
~';~
~'~--~-
Present Approach(n=0.51
g o.z5
N
o
0
2
I,
6
8
10
12
s/d
FIG.7 MEASUREDANO COMPUTEDINTERACTIONFACTORS
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
Average Cone
Undrained Shear
Water Content
OCR
SPT
Resistance
Strength IkN/mz)
Blows/0.3m
(kN/m z)
I'/.I
20
~0 0
,000 10000
250
500
20 z,0
80 0 2 z,
t
I
Stratigraphy
0
-I
0
V. stiff 9ray i tan
clay
Still day, sand seams
L~
S-~
..~
Stiff-V. stiff rod
i gray clay
i~XJ
-----1Q-I~.~ Stiff-V.stiff gray &
I~XI tan sandy day.
"
~ . ~ vltb sand pockets
'
t°
~LL
,
.:l-o
i,
o
"
,4-0 . ~ X
;O'JC
"FOX
0 I
~,'o~
"°
~.IOY---.x
1s-111111o ~ , red ~ gray
t,,i~HT
/11111~t. ~ith day. ~lt
/11111~ sand layers
2 0 & ~ V. stiff red & gray
"Triaxial
FIG.8 SUMHARY OF
:~
~'~
~=Nat. W/C
;':-°Ons°l
I
j ~iriaxialJ
I . Consol. I
GEOTECHNICALDATA AT TEST SITE
i
~
I
9
l
x
I•
.-~ S
*
I•
x
z
1
A
I
- - - Measured
x Conventional Approach
(n=0) (Poulos and
Davis, 1980)
Approach using HNH
HodeI (n=0.5)
• Present Approach
In=0 5)
I
I
I
Ix
I
I
I
I
I
300
600
900
1000
1500
(a) Pile Head Stiffness HN/m
,=.9
.:- 5
cL.
L"
I
I
I
&.
I
I
zl
I
2
(b) Settlement Ratio Rs
3
FIG9 PREDICTIONSOF PILE HEAp STIFFNESS
AND SETTLEMENT RATIO
O.S
AI
x I
!
l
I
I
Ix
I
(b) S Pile Group
il
0.875
Load
AverageLoad
x
.i
(a) 9 Pile Group
II
0.875
Load
AverageLoad
x
•
i@
IA
l
I
x
1.25
1.2S
•
A
....
x
Heasured
Conventional Approach
(n=O) (Poulos and Davis,
1980)
Approach using HNH Model
In:O.S)
Present Approach
(n=O.S)
Legend:
FIG10 PREDICTIONSOF PILE HEAD LOAD DISTRIBUTIONS
!
0.S
?
!
1.0
[-Z 0S
10
Z_ 0.S
L
Edge Pile
tl
r
7 r)-
9
Pi(e Group
/
11
11
11
Pile
o~,~i!~or ner Pile
o}
/'~
12"
/,Y
~/o Centre Pile.I" ,J/Corner
!;I°
--
11
S Pile Group
~ C e n t r e Pile
,Yr
. . . .
1.1
FIGJI PREOICTIONS OF IOA0 01STRIBUTIONS ALONG Ptl£ lENGTH
o
Heasured
-----Conventional Approach In:01 (Poulos and Davis. 1980)
-- ~ Approach using HNH Hodel In:0S)
------ Present Approach In=0S)
L_egend:
o/~
/
'?e-~tL~'7-
Predrill
Pile Group
F
o,~°~J
~7
Predri[I
Depth
Load
Average Pile Head Load
t46
The predicted and measured values of the pile head stiffness and group
settlement ratio Rs 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-pile 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.
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6.
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8.
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148
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2838, pp. 245-256.
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Group Action. Rep. No. FHWA/Rd-81/002, Fed. Highway Admin.
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17. Poulos, H.G. (1988b). Marine Geotechnics. London, Unwin Hyman.
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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
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Received 25 October 1989; revisedversion received7 July 1990; accepted 10 July 1990
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