finite element modelling of rock socketed piles
TRANSCRIPT
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INTERNATIONAL
OURNAL FOR NUMERICAL AND ANALYTICAL
METHODS
IN GEOMECHANICS, VOL.
18,
25-47 1994)
FINITE ELEMENT MODELLING
OF
ROCK-SOCKETED
PILES
E. C. LEONG
School of Civil and Structural Engineering, Nanyan g Technolog ical University, Singapore 2263
M . F. RANDOLPH
Geomechanics Group , The University of Western Austra lia, Nedlands. Western Australia
6009,
Australia
SUMMARY
Rock socketed piles have a number of features which differentiate them from other types of piles. The
generally stubby geom etry leads to mo re even distribution of capacity between shaft and base. However, the
low ratio ofpile modulus to rock
modulus
leads to high compressibility and this, coupled with a tendency for
the load transfer response along the shaft to exhibit strain-softening, gives rise to an overall response where
the shaft capacity ma y be fully mobilized, an d potentially d egraded , before significant mobilization of base
load.
The paper presents results of finite elemen t analyses of the res ponse of rock-sock eted piles, with particular
attention to the shaft response with and without intimate base c ontact. The shaft interface uses a m odel,
developed from principles of tribology, that includes dilation (and strain-hardening) prior to peak shaft
friction, followed by strain-softening at larger displacements. The results of the study are shown to be
consistent with field measurements, and to capture effects of the absolute pile diameter on the peak shaft
friction. It is also shown that intimate base contact mitigates significantly the degree of strain-softening of
the shaft response.
1. INTRODUCTION
For many piles in soil, the base capacity is a small proportion of the total capacity, and under
working load conditions little load is transmitted to the base. For typical factors of safety against
failure, deflection
of
the pile will be small, and relative slip (if any) between the pile and the soil
will be confined to a small region near the ground surface. With rock-socketed piles, the situation
is rather different owing to the generally lower embedment and stiffness ratios compared with
coventional piles. The low embedment ratio leads to the base contributing a greater fraction of
the total capacity, resulting in much higher mobilization of the shaft capacity under working
conditions. The low stiffness ratio gives greater compressibility, even at moderate embedment
ratios, which can lead to significant relative movement between the shaft and the surrounding
Element tests of the response along the shaft of rock-socketed piles indicate a strain-softening
response..
t is therefore important to consider rather carefully the integrated response of the
rock-socketed pile when determining suitable working load levels. Depending on the separate
rates of strain-softening
of
the shaft response and mobilization of the base capacity, the overall
response may show a plateau, or even a decrease in load-carrying capacity, at the stage where the
shaft capacity becomes fully mobilized, even though substantial reserve capacity may be available
at large displacements.
rock.
CCC 0363-906 1/94/0 10025-23
994 by John Wiley &
Sons,
Ltd.
Received
11
January 1993
Revised
27
July 1993
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26
E. C.
LEONG AND
M. F. RANDOLPH
This paper presents results of finite element studies of the response of rock-socketed piles, All
analyses were conducted using a modified version of AFE NA .3 Details a re included of infinite
elements, tha t were introduced in order to m inimize the size of the discretized doma in, and of the
joint elements used for the interface between the pile and the surrounding rock. The interfacial
response is based on a model developed by Leong and R a n d ~ l p h , ~nd analyses are presented for
rock-socketed piles with no base, and for complete socketed piles. These analyses illustrate the
imp ortan t effect on the shaft response of the stress field resulting from load transm itted at the base
of the socket. Parametric studies and comparisons with field data are included, showing the
effects of geometry and stiffness on the stress mobilization down the rock-socket.
2.
M O D E L L I N G O F U N B O U N D E D D O M A I N S
The rock-socketed pile problem, typical of many geotechnical problems, involves an unbounded
domain. It is a common practice in finite element modelling of these problems to truncate the
finite element mesh a t a distance deem ed far enough so as not to influence the near field solutions.
These truncations are usually determined by trial and error until an acceptable solution is
obtained. Such a method places a heavy demand o n computer resources, both m emory a nd time,
as solutions for the far field which are of no interest are genera ted a s well. In the Iast decade o r so,
considerable effort has been concentrated on modelling unbounded domains. Several approaches
have been used: analytical mapping of the far-field s ~ l u t i o n ; ~apping the exterior domain
onto an interior finite one;6 boundary integral method^;^. infinite elements;-* continuous
elements; infinite boun dary elemen t;20 equivalent springs.21 (Shar an, 1992).
Of these techniques, the use of infinite elements with finite elements appears to be the most
popular. There are basically two methods in the formulation of infinite element^.'^ The first
method is the direct approach, or the displacement descent method, where the natural co-
ordinate is extended to infinity in the required direction while keeping the standard mapping
function well defined. Th e unknow n variables are expressed in terms of descent shape functions
which decay asymptotically to z ero at infinity. Th e second approa ch is the inverse method, o r the
co-ordinate ascent method, where the dom ain of the nat ural co-ordinate is maintained a s usual
while ascent mapping functions are employed to cause the physical co-ordinate to exhibit
singular behaviour at infinity. Cu rnie rI4 has shown that the tw o approach es are equivalent
provided that they are consistently applied to a linear isoparametric element. However, in the
direct approach, a special quadra ture formula is needed to accomm odate a n infinite domain of
integration. The inverse method is favoured by many researchers. 1 2 * l4 l chiefly because
(a) the same mapping is used for both geometry transformation and for transforming the
unknow n function variable from the local to the global co- ordin ate system and (b) the usual
Gauss-Legendre integration can be used for both the map ped infinite element and finite element.
In the present paper, a mapped infinite element based on the inverse method is incorporated
into AFENA. For completeness, the mapped infinite e ement is described here. Figure 1 shows
local co-ordinates, ( and
q
are related to the global co-ordinates, x and y , by
the singly and doubly infinite elements in two dime nsio
d
s, takkn from M arques a nd O wen.15 The
2 ( X j
-
i )
x - 2Xi - j )
2 ( Y j - i )
Y - 2 Y i -
Y j )
l = 1 - q = 1 -
which allows [ and
q
to approach unity as x and y , respectively, approach infinity.
The subscripts i and j in equation ( 1 ) relate to the inner nodes of the infinite element, where
node
i
is at the bo undary of the main mesh. Th e spacing between nodes and j is determined by
a pole node, o , such that x j - i
=
x i - , . The optimum position for the pole node for the
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FINITE ELEMENT MODELLING OF ROCK-SOCKETED PILES
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Shape functions
N I = $ ( l - C ) ( I - q ) ( - l - C - q )
N2 = j(l
- C 2 ) ( l -9)
N 3 = $ ( I
+C)(l -q) - l+C -q)
N4 = f (1
+ C ) ( l
- q 2 )
N s = f ( l - C ) ( 1 - q 2 )
(a) Singly infinite element
1 2
Mapping hnctions
Shape functions
(b)
Doubly infinite element
Figure 1 Two-dimensional singly and doubly infinite elements derived from eight-noded Serendipity isoparametric
element (Reference 15)
rock-socketed pile problem was found to be along the outer edge
of
the pile, the axis
of
symmetry
and the up per ground surface.22 The performance of the infinite elements was evaluated by
Leong fo r a variety of axisymmetric problems under e lastic an d elastic-perfectly plastic
conditions. Figure
2
shows different mesh arrange men ts tha t were used for the rock-socketed pile
problem. Th e resulting elastic flexibilities are sh own in Figure
3,
from which it may be seen that it
is sufficient to limit the discretized zone t o
3 4
imes the pile radius arou nd the pile, and a similar
distance below the pile, bounded by infinite elements (see Figure
2).
Although it may be argued that the mapped infinite element may be less accurate when
compared to the infinite boundary element or the equivalent springs method,2 its ease of
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28
E.
C.
LEONG AND M.
F.
RANDOLPH
28R
Figure 2. Finite element for study on
mesh
truncation
formulation and implementation into a general finite element code makes it an attractive
compromise. To date, publications of finite element solutions of pile problems have mostly been
obtained using a truncated mesh.
2.1.
Elastic
response
The elastic response of a pile socketed into rock has been investigated using the finite element
method by previous r e s e a r c h e r ~ . ~ ~ - ~ ~uch problems have been previously studied by extending
the finite element mesh to some distance.
For
example, Donald et ~ 1 . ~ ~sed a half-width of 20R
and a depth of 50R here
R
is the radius of the pile. The use of infinite elements can give superior
accuracy, with fewer degrees of freedom.
Figure 4 hows calculated flexibilities for a range of embedment and stiffness ratios of the pile,
allowing for a change in soil modulus at the level of the socket base. It may be seen that, for the
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FINITE ELEMENT MODELLING O F ROCK-SOCKETED PILES
0 . 8-
0.7-
0 . 6 -
0 . 5 -
0 . 4 -
0.3-
0.2-
29
0.1-
O
U
m
L
-
-B
- Mesh 2
. x-
-
Mesh 3
-
-- - Mesh
4
inite
+
Infinite elements
I I I 1 I I I
,
I
1.0
I
I I I
0.9 l Q
-* Mesh 1 F
E J E m
Figure 3. Comparison of finite element mesh truncation with use of infinite elements for a rock-socketed pile
stiffness ratio of E , / E ,
= 10,
where
E ,
is Youngs modu lus of the pile an d
Em
is Youngs modu lus
of the surrounding material, the flexibility coefficient becomes nearly constant for embedment
ratios of
LID
greater than about 10.
Also shown for comparison o n Figure
4
re flexibility coefficients from Do nald
et a1. 23
using
a rather coarser mesh.
As
expected, these solutions generally give lower flexibilities, the difference
being most obvious for E J E , = 10.
3.
JO IN T E L E ME N T S T O MO D E L P ILE -R O CK IN T E R F A C E
The load transfer behaviour of a rock-socketed pile through side-shear is dependent on the
behaviour
of
the pile-rock interface. The pile-rock interface usually behaves differently from eithe r
the pile o r rock material. I n finite element modelling, special elements have been used t o m odel
the pile-rock interface. Osterberg an d G ill24have used spring-loaded linkage elements. Rowe an d
Pells26 have used a series of dua l nodes based on the compatibility co ndition a t the pile-rock
interface. Donald
et
aLZ3have modelled the problem by assigning a different set of material
properties to the layer of finite elements adjacent to the pile. Anoth er element comm only used t o
model interface behaviour is the Good man-typ e joint element. Dona ld et aLz3 have also used
Good man-typ e joint elements to model the pile-rock interface but found the joint elements near
the pile base behaved erratically on reaching incipient slip.
The extent
of
the use of special elements to model th e pile-rock interface varies. Osterberg a nd
Gill24 have placed their spring-loaded linkage elements along th e edge
of
the pile (side and base),
Rowe an d Pells26 placed the dua l nodes only along the side of the pile while Do nald
et
were
not specific, but implied tha t the joint elements were just dow n the side of the pile. It is no t clear
how the finite element models hand led element connectivity for these elements, particularly at th e
base of the pile.
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30 E. C. LEONG AND M . F. RANDOLPH
1
o
0.9
0 .8
,Q 0.7
$ 0.6
w 0.5
s 0 ,4
G
0 .3
0.2
0.
L.
L 3
U
-
t
5
I
2
3
5
10
Donald et a1 (1980)
inite + Infinite elements
T l Z ' I ' I ' I
2
4
6
8
10
12 14 16
18 20
E m b e d m e n t
ratio
L/D
( a )
E,/E, =
10
1 .04 ' I . I
I
0 .9 -
0.8 -
,
0.7-
$
0.6-
0 5 -
0 . 4 -
c
Donald
e t al. (1980)
0
- 0 3 - 2
3
0 .2 -
5
0.1 - 10
5
L.
cd
1
8
-
inite + Infinite elements
I
. I ' ~ ' l ' l ' I 3 I ' ' 1
2 4
6
8 10 12 14 16 18 20
Embedment ratio LD
( b)
E,/E,
=
1000
Figure
4.
Settlement influence
factors, I,, for
complete piles
in
two layer systems
In the present study, six-noded Goodman-type joint elements were used to model the pile-rock
interface behaviour, while eight-noded isoparametric quadrilateral elements were used to model
the rock mass. Infinite elements were used to model the unbounded domain. To maintain element
connectivity, the joint elements were extended down to the bottom edge of the finite element
mesh, including the use of a one-dimensional infinite element to extend the joint into the
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FINITE ELEMENT MODELLING OF ROCK-SOCKETED PILES
31
unbounded do main. Leong has shown that the join t elements within the continuum have
negligible effect on the continuum response.
4 . PROPERTIES OF PILE, ROC K MASS AN D PILE-ROCK INTERFAC E
4.1.
Pile
The pile is assumed to be elastic with Youngs modulus, E ,
=
35 GPa and Poissons ratio
v,
= 0.2.
The diameter of the pile is assumed to be 1 m unless stated otherwise. Effects of
embedment ratio, LID, are investigated for values of LID = 2 , 5 and
10.
4.2. Rock mass
The re are several strength criteria for intact rocks, e.g. the M ohr-Co ulomb criterion, Griffith-
type criteria and empirical criteria. However, rock masses generally have naturally occurring
discontinuities such as joints, seams, faults an d bed ding planes. Therefore, the properties of a rock
mass may be very different from that of the intact rock. One way of accounting for these
discontinuities is by assigning equivalent elastic properties to the rock mass as suggested by
Go odm an an d D uncan.28 Fo r example, for a rock mass with three orthogonal discontinuity sets,
the equivalent elastic properties are
1 1
-+-+-
G, Siksi
S jk s j
(3)
for i
= x,
y, z w i t h j = y,
z ,
x and k =
z , x,
y; where
E, , G,
and
0
are the elastic properties, Youngs
modulus, shear modulus and Poissons ratio, respectively, of the intact rock; k , and k , are the
values of shea r an d n orm al stiffness of the discontinuity, respectively, and S is the joint spacing. It
is im po rta nt to estimate the stiffness of the rock mass accurately so that the stresses obtained in
any analysis involving dilatancy a t the pile-rock interface may be predicted correctly. W hen high
stresses are developed through dilatancy, the intact rock may crack and may show reduction in
~t iffn ess .~ owever, such behaviour is not considered in the present study.
The rock mass was assumed to obey a Mohr-Coulomb criterion with a shear strength,
c,
=
200 kPa, a friction angle, 4,
= 30
and zero dilation. These properties correspond t o those
reported by Williams3 for sample SM7 of a highly weathered mudstone, which had an
unconfined compressive strength, qu, of 750 kP a. Poissons ratio, v,, for the rock mass was
assumed to be 0.3.
To
investigate the effects of rock m ass stiffness on the loa d transfer beh aviour
of rock-socketed piles, two different values of Youngs modulus, Em
=
500 M P a a n d
Em
=
50 M Pa , were used. These effectively give relative stiffness ratios,
E , / E ,
=
70 and 700,
respectively, for a pile with E ,
=
35 G P a. T he effective unit w eight of the rock mass was assumed
to be 23 kN m -3. Th e initial stress state of the rock m ass was assumed to be at K O ondition w ith
K O as unity unless stated otherwise.
4.3.
Pile-rock interface
The pile-rock interface in many respects resembles a rock join t and, as such, argum ents
presented for rock joints are assum ed to be equally valid for the pile-rock interface. The simplest
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FINITE ELEMENT MODELLING
O F
ROCK-SOCK ETED PILES
33
7
a micro shear displacement
b.
increase in sliding res istance due
to dilation and plough resistance
c degradation of surface roughne ss
by wear
d steady sliding
Sliding Distance
Figure 5. Shear stress : liding distance response for irregular rock surfaces (Reference
-4)
Figure 6 shows a comparison of this model with constant normal stiffness CNS) tests from
Lam,34for the c ase of a confining s prin g stiffness,
K
= 350 kP am m -'. Parameters for the model
were: 4r=
23.5".
p
=
0.41;m/S = n/S = 0 5 mm- ' ; i / S = 1000m m - .
h
may be seen that the
model allows reasonably close fitting of the responses measured over a wide range of initial
normal stress levels. Further examples of comparisons of the model with experimental data have
b ee n g iven by L eo ng a nd R a n d ~ l p h . ~
In a recent publication by D esai and Ma ,35 a disturbed-state concept was described for the
modelling of joi nts an d interfaces. In this concept, a yield function, a critical state and a d isturbe d
state a re defined. Besides the usual elastic stiflness,k , and k , , five parameters are needed to define
the yield function, four parameters for the critical state and four parameters for the disturbed
state. The Leong and Randolph4 interfacial model uses only six parameters. However, it may be
of interest to point out that the disturbance function,
D,
proposed by Desai and Ma3' is of
a similar form to equations
(12)
and (13), and is given by
D
=
D l
-
xp(
-
~ E ]
14)
where D, is the ultimate or critical value of
D,
K and
R
are material parameters and tD s the
trajectory of plastic shear strains.
In addition to the failure criterion, it is necessary to assign values to the elastic stiffness
parameters, k , and
k,, for
the pile-rock interface. G oo dm an
et
al. ,27have suggested that these
parameters may be measured,
k,
rom a direct shear test and
k ,
from a comp ression test, with each
being the initial slope of the respective stress-displacement curve. The increm ental stress-strain
relationship is given by
where C,, and C,, are th e cross-stiffnesses of the joint. U nder elastic conditions, C,, = k , and
C,, = k , , and C,, and
C,,
are usually assumed t o be zero.27However, when slip occurs,
C, ,
and
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34
E. C. LEONG AND
M .
F. RANDOLPH
596
200
0 10 20 30
Shear
displacement
(rnm)
Figure 6 . Comparison
of
predicted values using Leong and Randolph model with experimental data for
K =
3 5 0 k P a m m -
C, , become non-zero if the joint exhibits dilatancy o r strain softening.32, 6 These values are then
dependent on the values of k, and k , .
Sun et
aL3
suggested a way of measuring the stiffness components of a rock joint through
a shear compliance test an d a no rm al compliance test. The stiffness terms of equation
(15)
are
then given by an inversion of the com pliance matrix ob tained from the tests. However, Sun et al.
cautioned that the two pairs of compliance so obtained are stress dependent and must be used
with care when predicting rock joint behaviour in a general stress path. Interestingly, Leichnitz3*
arrived at a constitutive relationship for rock joints by explicitly determining the stiffness terms
of
equation
(15)
from shear tests using curve-fitting procedures. Thus, the stiffness matrix deter-
mined by Sun et
al.
is not the elastic stiffness matrix.
The normal stiffness
of
rock joints an d discontinuities is a function of the relative stiffness,
geometry of asperities, norm al stress a nd joint fill material. T he effective modu lus, E, for elastic
contact of two joint surfaces with n o fill material m ay be given by the w ell-known Hertzs solution
where
1 ( 1
-
: (1
-
;
E E l
E2
+-
--
where
E i
and
vi
are the elastic properties of the materials m aking up the joint. Usually, the norm al
stiffness of a rock joint is measured in terms of joint closure. Goo dm an 39 suggested a hyperbolic
function linking normal stiffness to joint closure under increasing normal stress. A hyperbolic
function between normal stiffness and joint parameters of aperture strength and roughness has
also been suggested by Bandis et aL4 Swan41 showed th at the n orm al stiffness can be related t o
the normal stress
by
simple relations through assumed distribution functions for the asperity
heights.
A
review on shear stiffness for rock joints can be found in References 40 and 42. Shear
stiffness was also fo und to vary w ith the relative stiffness, geometry
of
asperities, normal stress
and joint fill material.
In sum mary, it seems likely that the values of elastic joint stiffness,
k ,
and k,, may be functions
of stress levels, in a similar manner to their co unterp arts, the shear m odulu s and Youngs modu lus
of a continuum. However, unlike a continuum , q uantification of joint stiffness is difficult as it is
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FINITE ELEMENT MODELLING O F ROC K-SOCKETED PILES 35
dependen t o n the properties of the joint fill material an d also on the physical characteristics of the
joint surfaces.
In the finite element analyses tha t follow,
k ,
and
k ,
are assumed to be
G / t
and
E,/ t
respectively,
where t is assumed to b e 1 per cent of the pile diameter,
0
and G and En re the shear and
constrained modu li of the pile-rock interface, respectively. Young's m odulu s of the pile-rock
interface is assumed t o be 1per cent of the mass mo dulus, E m .These v alues of k , and k , are used in
all subsequent analyses reported in the paper. Strictly speaking,
k,
and k , should be measured
elastic parameters as suggested by G ood ma n e t
al.
In the ir finite element modelling of the rock
socketed pile problem, Donald
e t aLZ3
used a value of 50 M P a m - ' f o r k , in their joint element,
reportedly measured from laboratory tests. Based on t
=
0.01 m(0 .010) , this would imp ly a value
of
5
M Pa of the interface shear modulus, which is only 0.1 per cent of a typical Young's m odulu s
for the rock mass of Em=
500
MPa. By contrast, Bandis et aL4' and K ~ l h a w y ~ ' , ~ ~eported
values of join t stiffness; k , and k,, which for t = 0.01 m w ould give Young's m odul us for the j6i nt
in the region of 10 per cent of the intact rock modulus, E, .
5 .
C O M P U T E D R E S PO N S E O F R O C K - SO C K E T E D P IL E S
Construction techniques for rock-sockets can play a significant role in determining the perform-
ance, both from the point of view of the roughness of the sides of the shaft (which will affect the
pile-rock interface response), and a lso the tendency for the boring ope ration t o leave a soft layer
of debris at the base of the rock-socket, necessitating large displacem ents to develop an y effective
end-bearing. The latter aspect may be addressed by specific measures to clean the base of the
socket (often involving manual removal of debris). Alternatively, it is quite common for the
load-bearing capabilities of the base to be ignored, and the design of the rock-socket is based
entirely on the shaft performance.
The first series of analyses concentrates on such 'shaft only' rock-socketed piles. In the finite
element model, a 'soft' layer of elements is placed below the pile tip. This layer, 0.150-0.250
thick, is given a Young's modulus,
Esoft,
f 0.1 per cent Young's modulus of the rock mass, Em.
Osterberg and Gillz4 used Esoft
=
Em/3000, while Rowe a nd PellsZ6used
Esoft
=
EJ15.
The performance of the rock-socketed pile will be compared for two different models of the
shaft interface: (a) a simple Mohr-Co ulomb mode l with consta nt angle of dilation, an d (b) the
model of Leong and R andolph4 outlined earlier. Th e parameters for M ohr-Coulomb interface
are c = 100kPa , 4
= 0
or 30 , =
0
or 5 , while those for the latter model are (based on
parameters deduced from tests on sample SM7 of Williams44)
4,
=
26 , $ o =
185 , p
= 0 6 2 ,
rn/S
=
0.08 m m - l , n/S
=
0.32 mm -' and
[/S =
5 mm- ' .
5 .1 .
Load transfer
through
side-shear only
The average shear stress responses of the different models using displacement loading on the
pile head are shown in Figure
7.
The model of Leong and Randolph 4 exhibits the characteristic
load transfer curves often observed in socketed piles where the load is taken in side-shear only.
Th e displacement to peak shear stress is about 0.0080, which is a typical rate of mobilization of
peak shear stress in piles. This shows that the estimated values of joint stiffness are realistic.
Th e stress profiles at the pile-rock interface for both types of models ar e shown in Figures 8
c-4, zero dilation),
9
c-4, = 5 ) , and
10.
As can be observed, the shear stress is relatively
uniform over the central part of the pile regardless of the model used. The dilatant
Mohr-Coulomb model and the Leong and Ran dolph model show the normal stress increasing
once the relative slip exceeds 2 mm . However, the dilatant M ohr-Cou lomb mod el shows the
normal and shear stresses increasing at a constant rate with slip, which is unrealistic in the
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36
E. C.
LEONG AND M .
F.
RANDOLPH
4 0 0 /
-.
m
e
2
I-
l l
L/D =
2
,eong and
Randolph
model
0 10
20
30
40
Pile head displacement m m )
Figure ?. .4verage shear stress response
of
side-shear only rock-socketed piles
0 2
0 4
0 6
0 8
-
N
1 0
1 2
1 4
1 6
2 0
( a ) Shear stress
( b ) Normal stress
Figure 8. Stress profiles at pile-rock interface of a side-shear only rock-socketed pile
using
c-c -$ Mohr-Couloinb model
physical situation. Interestingly, the Leong and Randolph model shows an increase in normal
stress past the peak shear displacem ent of 0.0080. After the peak shear displacement, the normal
stress reduces slightly a t the pile head while it increases over the lower par t of the piie and finally
reaches an equilibrium condition (see Figure 10(b)).However, the shear stress, decreases after the
peak shea r displacement and reaches a residual shear stress. The limit loads for the non-dilatant
Moh r-Coulom b models may be estimated directly from the initial stress conditions.
The effect of taking the rock mass Young's modulus, Em, as homogeneous E m
50
or
500 MPa),
or
proportional to depth (Em
=
5002 M Pa ) is shown in Figures 11 and 12. While the
average she ar stress response is only slightly affected, there is a m uch m ore significant effect on the
stresses induced along the interface. This shows clearly that the increase in normal stress, and
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FINITE ELEMENT MODELLING OF ROCK-SOCKETED PILES 37
1 (kPa) Jn (kpa)
600 500 400 300 200
LOO 0
100 200 300 400 500
600
0
0 2
0 4
0 6
+
Applied displacement
0 8
-
n + m o x X
6 m m
N o + m n x
8 m m
z 1 0
o + m o x
2
1 4
1 6
I 8
2 0
( a ) Shear
stress
(b) Normal
stress
Figure 9. Stress profiles at pile-rock interface
of
a side-shear only rock-sock eted pile using c-+$ Mohr-Coulomb model
1 ( k P a ) 5 n ( k P 4
600
500 400 300
200 100 0
100
200 300 400 500
600
0
0 2
0 4
0 6 Applied displacement
0.8
N
.E 1 0
1 2
1 4
1 6
1 8
2 0
(a ) Shea r
stress
( b )
Normal
stress
Figure 10. Stress profiles at pile-rock interface of a side-shear
only
rock-socketed pile using b o n g and Randolph model
hence the peak shaft friction, is strongly affected by the stiffness of the surrounding rock. This
point is emphasised further by the response for
Em
= 50 M P a (E,/Em
=
700) shown in Figure
11,
which gives peak shaft friction a factor of 5 lower than for Em
=
500 MPa.
The effect of embedment ratio was investigated using
LID = 10,
compared w ith
LID = 2,
as
shown in Figure
13
for the Leong and Randolph model. The average shear stress response for
LID = 10
exhibits a similar brittle response to that for
LID = 2,
even though there is a more
gradu al mobilisation of shear stress. No te th at the in itial average norm al stress at the pile-rock
interface is higher by
a
facto r of
5
for the case with
LID
=
10.
However, as may be seen from the
stress profiles in Figure 14, there is much less dilation before failure, and the peak average shea r
stress is only about
25
per cent greater than for the case of
LID
=
2.
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38
E. C. LEONG
AND M. F.
RANDOLPH
I
/D = 2
0
10
20
30 4 0
50
Pile head displacement
( m m )
Figure 11. Effect of rock mass Young s modulus
on
average shear stress response
of
side-shear only rock-socketed piles
1
(kPa)
1.1
1.6
I B
2 0
( a ) S h e a r stress
Applied displacement
X B m m
n 10mm
I
-
X
3 0 m m
a 4 0 m m
0 B Initial stress
m
+ % a
( b )
Normal
stress
Figure 12. Stress profiles at pile-rock interface of a side-shear only rock-socketed pile using Leong and Randolph model,
for Em
= 5002
MPa
Laboratory and small-scale field tests show that the pile diameter may have an effect
on
the
load transfer behaviour
of
pile^.^'.^^
Randolph4' showed that
if
the shear zone thickness a t the
interface is constant, the additional shear stress, AT (over and above that due to the in situ normal
stresses) is given
by
Aw
A z = 4 G s i n $ t a n # -
D
where G is the shear modu lus, the dilatio n angle, # the friction angle, Aw the relative
displacement between pile and surrounding m aterial an d D the pile diameter.
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F I NI TE ELEMENT MODELLl NG O F ROCK- SOCKETED P ILES
400
300
-
-
m
n
EOO
100-
39
'
I
L D =
10
L D
=
2
, I I ~ 1 ~ / .
Figure
1 3 .
Effect of LID ratio c
0
2
4
E
-
N
6
I (kPa)
I 500 400
300
200 LOO
( a )
Shear
stress
100 ZOO 300 400 500 600
700
800
Applied dsplacement
rl
5 m m
( b ) Normal
s t re ss
and
Figure
14.
Stress prof iles at pile-rock interface of a side-shear only rock-so cketed pile using Leong and Randolph model
for L / D = 10
The effect
of
pile diame ter was studied by keeping the ratio L/D at two an d increasing the scale
of the fin ite eleme nt mesh.
A
param etric study using pile diameters,
D
f
0.5,
1 ,2 ,
5
and
10
m was
performed. To ensure comparable magnitude of stress, the rock mass was assumed to have an
initially uniform isotrop ic stress
of
20 kP a. T he effect
of
pile diameter on the average peak shear
stresses, Z for the various pile diameters is shown in Figure 15. It may be seen that there is
a significant effect of the pile diam eter, in keeping with field results such a s those discussed by
Fahey et aL4* However, in contrast to the simplistic analysis of Randolph4' that indicates an
increase in shear stress that is inversely proportional to
D
(equation (17)), the computed results
indicate a variation that is closer to D-0.4 .
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40 E.
C. LEONG AND M . F. RANDOLPH
m
e
s
-
I
300 -
200 -
100-
0
1 2 3 4 5 6 7 8 9 1 0
D m)
Figure 15. Effect
of
pile diameter on average peak shear stress of side-shear only rock-socketed piles using Leong and
Randolph model
The results in Figure
15
were obtained keeping all the rock and interface parameters constant.
In reality, it is likely that some of these parameters will have a scale dependency (pa rticularly the
parameters +o and po Refere nce 22). If this sca le dependency is allow ed for, the redu ction of peak
shear stress with increasing pile size will not be as m arked as th at sh own. How ever, the effect may
still be significant, and should be borne in mind in the design of rock-socketed piles, or the
interpretation of small-scale pull out tests on grouted bars.
5 3
Load transfer through side-shear and end-bearing
Piles which carry lo ad in both side-shear and end-bearing will be termed complete piles in the
following discussion. It is a common practice in pile design to superimpose solutions from
side-shear and end-bearing to obtain the response of a complete pile. It has been pointed out that
such solutions are not valid, as the stress distributions in side-shear only piles and complete piles
are different.23.
1
Such indiscriminate superposition in cases with post-peak strain-softening
response in side-shear may result in undue conservatism.
Finite element analyses were performed for the response of
a
comp lete pile with
LI D =
2, using
an increm ental stress loading. The lo ad responses for the c-4 (non-dilatant) Mohr-Coulomb and
the Leong and Randolph models are shown in Figures 16 and 17, respectively. The base loads
were obtained by integrating the vertical stresses at the mid-level of the bottom layer of pile
elements (0.05D from the pile tip). The error between the side-shear load, obtained from the
integration of the side-shear stress, and the difference of the applied loa d a nd base load is less than
two per cent. The complete pile response is strain-hardening although the side-shear exhibits
strain-softening behaviour for the latter m odel. The average shear stress mobilized along the shaft
may be obtained from the bearing stress, q, by 5 = q/(4L/D). Thus, the peak shaft capacity
corresponds to an average shear stress
of
about 235 kPa for the complete pile, compared with
about 255 kPa for the shaft-only pile.
The stress profiles for the complete pile with the Leong and Randolph model are shown in
Figure 18. Comparison with the corresponding profiles for the side-shear only piles, Figure 10,
shows that the stress distributions are different as expected. There is an interaction of the
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FINITE ELEMENT MODELLING OF ROCK-SOCKETED PILES
41
Pile head displacement (mm)
Figure 16. Load response of a complete rock-socketed pile
using
c-&Mohr-Coulomb model
Pile head displacement
mm)
Figure 17. Load response
of
a complete rock-socketed pile using Leong and Randolph model
end-bearing and side-shear load transfer mode in a complete pile. The effect of the load
transferred to the pile base is to limit the increase in normal stress, and hence shear stress, near the
pile base. In fact, tensile failure at the pile-rock interface near the pile base is observed. The
percentage of applied load that is transferred to the pile base, up to the peak shear stress is about
50
per cent for the Mohr-Coulomb model, and 60 per cent for the Leong and Randolph model. It
is encouraging to note that the shear stress profiles are similar to those measured by Williams
et a1. 44 reproduced in Figure
19.
The above analysis was repeated for 1 m diameter socketed piles with
LID
of 5 and 10using the
Leong and Randolph interface model. The response for the latter case is shown in Figure
20.
Again, the base load was obtained by integrating the vertical stresses at the mid-level of the
bottom layer of pile elements,
0 . 2 5 0
from the pile tip. Hence, the base load is slightly over-
estimated, with the difference of the applied load and base load underestimating the side-shear
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42
E.
C. LEONG AND
M.
F. RANDOLPH
1 (kPa) O n @Pa)
0
0 2
0 4
0 6
I
MPa
0
Applied stress, q
,--.
N
1 0
1 2
1 4
1 6
I 8
2 0
( a )
Shea r
s t ress
( b )
Normal stress
Figure 18. Stress brofiles at pile-rock interface of a complete rock-socketed pile using Leong and Randolph method
I
(MPa)
Figure 19. Non-uniform development of side resistance from test pile (after Williams et
load. The difference, however, is less tha n six per cent. The load t ha t is transferred to the pile base
reduces from 60 per cent for LID of 2, o 38 per cent for LID of 5 and to 20 per ce nt for LID of 10.
Interestingly, the side-shear response in the comp lete pile has becom e m ore plastic (with less
strain-softening) comp ared with the side-shear only socketed pile (see Figure 13). This is due to
a difference in the mobilization of the side-shear stress. In the comp lete pile, the shear stress is not
fully mobilized near the pile base.
It is tempting to assume that, in the case where debris leads to a softer base response, some
strain-softening may be observed in the overall response. However, analyses conducted with
a layer, 0.15D thick, with reduced m odulu s
Esof,
= 0.1E,
do not show any such strain-softening.
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FINITE ELEMENT MODELL ING OF ROCK-SOCK ETED PILES
43
*
L
Figure
20.
Load response of
20
I8 Total
Pile
head displacement (rnm)
a complete rock-socketed pile with
LID
= 10 using Leong and Randolph
model
0
Integrated from s ide shear
0 5 10
15 20
25
Pile
head
displacement (rnrn)
Figure 21. Load response of a complete rock-socketed pile with a soft base layer of E,,, =
0.1E,
using Leong and
Randolph model
Figure 21 shows the results of an analysis with LID
=
2, from which it may be seen that the
deduced shaft response is very similar to the case without the soft layer (see Figure 16).
5.4. Comparison with field data
Donald and c o - w o r k e r ~ ~ ~ - ~ ~ave presented results from both finite element analysis an d field
loading tests of rock socketed piles. However, no direct comp arison of the two sets of results was
attempte d. At the time, the finite element analysis did no t mod el the strain-softening behaviour in
the side shear response that was observed in the field tests.
The present m odel has been used to back-analyse on e of the pile load tests (M8) presented by
Williams et ~ 1 ~ ~
s
mentioned in the original publications of Donald et al., some of the relevant
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44
E. C. LEONG AND M . F. RANDOLPH
information for the rock mass were missing, and b est estimates were made by D onald
et al.
in the
light of their experience. Th e same estimates have been adop ted here, taking Young's m odulus of
the rock mass as
610
M Pa , Poisson's ratio as 0.3, a unit weight of 23 k N m -3 , and
K O
= 1 .
The
rock mass was assumed to obey the Mo hr-Coulom b yield criterion with
c =
1.5 MPa, and
4 =
tb
= 20 . The only difference in the present analysis is tha t the dilation angle has been taken
as
I)= 0 .
The other relevant parameters for the proposed model are
(br
= 26 , I),,= 20,
n/S =
0.32 mm -'
i / S = 5
mm- ' and
pLp= 0.1.
The parameter
m
is assumed t o equal
44 ,
and the
shear and normal stiffness are taken as
G/t
and E , / t respectively with t =0.01m. These
parameters are very similar to those used to back-analyse the results of direct shear tests on
mudstone samples reported by W il li a m ~ .~ ' etails of these may be found in Reference
4.
The
concrete pile was assumed to have a Young's modulus of 35 GPa and Poisson's ratio
of
0.2.
The finite element results have been obtained using the Leong and Randolph model for the
rock socket interface. The finite element mesh com prised 286 eight-noded qu adrilat eral elements
with infinite elements at a distance of three times the radius at the side, and a distance of four
times the radius below the pile base. Two analyses are presented here, one with the base of the
rock socket in direct contact with the rock mass, and one with a soft base layer of thickness 0.140,
with Esof,
= 0.01
Em, mm ediately below the pile tip. Introduc tion of the soft layer was found to be
necessary in order to match the field response.
Figures 22 and 23 show comparisons between computed and measured results. In Figure 22,
with no softened base, the magnitude of the base response is overestimated although the
side-shear response still agrees well w ith the field data. By con trast, Figure 23 shows excellent
agreement between computed and measured responses of both shaft and base.
6. C O N C L U S I O N S
Modelling the load response of a rock-sock eted pile using the finite element method is a challeng-
ing problem. Th e problem may be divided into three modelling aspects: (a) the rock mass, (b) the
pile-rock interface and (c) the unb ound ed do mai n. In the present paper, the rock mass was
assumed to obey a Mo hr-Coulom b strength criterion, while infinite elements were used to model
the unbounded domain. The use of infinite elements may not be the best method currently
available, but the ease of formulation an d implem entation in a general finite element code renders
the approac h attractive. The rock-socket performance is dom inated by the non-linear response at
the pile-rock interface. This rem ains an active area of research, particularly as high quality field
data are scarce. However, the interfacial model of Leong and Randolph4 adopted in the paper
appe ars to have cap tured the observed lo ad response of rock-socketed piles very well, as indicated
by the comparison w ith the field load test reported by Williams
et
Fur ther verification w ith
other field data
is
desired.
Several important observations are derived in the present paper. In socketed piles with side-
shear only, the average shear stresses are depen dent o n the em bedm ent ratio,
LID,
the diam eter of
the pile, D nd the relative stiffness ratio,
EJE,,
where E , is the pile modulus.
As
L /D and
D increase, there is a decrease in the m aximu m average shea r stress mobilised along the shaft of
the rock-socket. However, no scale effects have been taken i nto acc ount for the model param eters
adopted in these analyses. Allowance for any su ch effects will result in a mo re gra dual reduction
in the average shear stress with increasing pile diameter. As the rock m odulus,
E m ,
decreases, the
pile capacity is much less as the increase in normal stress due to dilation is reduced.
In spite of the strain-softening nature of the side-shear only response, the response of a 'com-
plete' pile (including full base conta ct) gave an overall load re sponse th at was strain-harde ning,
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F lNlT E E L E M E NT M O DE L L ING O F
ROCK SOCKETED
PILES 45
14
I
Pi le se t t lement (mm)
Figure 22. Comparison of computed load response with field data of Williams et a1.,44 with n o soft layer at the pile
0 Field
t e s t
M8
-
E
results
base
10
20
30
40 50 60
Pile se t t lement (mm)
Figure 23. Comparison of computed load response with field data of Williams et al.,44 including a soft layer at the pile
base
even where a softened layer with modulus Esoft
=
OlE, was introduced below the pile base.
However, the side-shear component still has a strain-softening response, but was more plastic
compared with the side-shear only socketed piles. This is due to the interaction of the two modes
of load transfer in side-shear and end-bearing, particularly in terms
of
modification of the
stress-field around the lower part of the pile shaft.
Overall, it has been demonstrated that strain-softening models of the type proposed by Leong
and Randolph4 are capable
of
replicating a number of features commonly observed in the load
transfer behaviour of rock-socketed piles. For complete piles, the distribution
of
shear stress
down the sides of the pile
is
similar to that observed in the field (eg. Reference
44).
The common
practice of using load transfer models to estimate the complete pile response ignores interaction
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46 E. C. LEONG AND M . F. RANDOLPH
between base and shaft load transfer, and will lead to a conservative estimate of the degree of
strain-softening of the shaft co mpo nent.
REFERENCES
1. I.
W. Johnston and T.
S.
K. Lam, Shear behaviour of regular triangular concrete/rock joints
-
analysis,
J
geotech.
2.
I. W. Johnston and T. S.K. Lam, Shear behaviour of regular triangular concrete/rock join ts -evaluation, J. geotech.
3.
J. P. Carter and N. P. B alaam, AFEN A Users Manual. University
of
Sydney,
1988.
4. E. C. Leong and M.
F.
Randolph, A model
for
rock interfacial behaviour, Rock Mech. Rock Eng.,
25(3), 187-206
5.
W. L. Wood, On the finite element solution of exterior boundary value problem,
Int.
j numer. methods eng.,
10,
6. B.
Nath, The w-plane finite element method for the solution of scalar field problems in two dim ensions,Int. numer.
7. C. A. Brebbia, The Boundary Element Methodfor Engineers, Pentech Press, Plymouth. 1980.
8.
0
C.
Zienkiewicz,
D.
W.
Kelly and P. Bettess, The coupling
of
the finite element an d bound ary solution procedures,
9.
P. Bettess, Infinite elements, Int. j numer. methods eng.,
11, 53-64 (1977),
eng., diu. ASC E, 115,
711-727 (1989a).
eng.,
d i u .
ASCE,
115, 728-740 (1989b).
(1992).
885-891 (1976).
methods eng., 15,
361-379 (1980).
Int. numer. methods eng., 11,
355-375 (1977).
10. P. Bettess, More o n infinite elements,
Int.
j
numer. methods eng., 15,
1613-1626 (1980).
1 1 .
G. Beer and J. L. M eek, Infinite dom ain elements, Int . j numer. methods eng., 17,
43-52 (1981).
12.
0 C. Zienkiewicz, C. Emson and P. Bettess, A novel boundary infinite element, Int.
j
numer. methods eng.,
19,
13.
F. Medina and R. L. Taylor, Finite element techniques
for
problems of unbounded domain, Int.
j .
numer. methods
14. A. Curnier, A static infinite element,
Int.
j
numer. methods eng.,
19, 1479-1488 (1983).
15.
J. M. M. C. Marques and D. R. J. Owen , Infinite elements in quasi-static m aterially nonlinear problems, Comput.
16.
P.
Kum ar, Static infinite element form ulation,
J
Struct. Eng., 111, 2355-2372 (1985).
17. G.
R. Karpurapu and
R.
J.
Bathurst, Co mparative analysis of some geomecha nics problems using finite and infinite
18.
G. Donida,
R .
Bruschi and
R.
Bernitti, Infinite elements in pro blems
of
geomechanics, Comput. struct.
29,(1),63-67
19.
E. A. Rukos, Con tinuous elements in the finite element method, Int. j numer. methods eng.,
12, 11-33 (1978).
20. G. Beer and 1.0 Watson, Infinite boundary elements, Int . j
numer. methods eng.,
28(6), 1233-1247 (1989).
21. S. K. Sharan, Elastostatic analysis of infinite solids using finite elements,
In t .
j
numer. methods eng.,
35, 109-120
22. E. C. Leong, A study of internal an d external shaft friction ofpiles,
Ph. 0 .
Thesis, The University of Western Australia,
23.
I. B. Donald,
S. W.
Sloan an d H . K. Ch iu, Theoretical analyses of rock socketed piles, Proc. Int. Con$
on
Struct.,
24. J .0
Osterberg and S.A. Gill, Load transfer mechanism for piers socketed in ha rd soils
or
rock, Proc.
9th
Can.
Symp.
25. P. J.
N.
Pells and R. M. Turner, Elastic solutions for the design and analysis
of
rock socketed piles,
Can. Geotech.
26. R . K.
Rowe and P. J.
N.
Pells, A theore tical study of pile-rock sock et beha viou r, Proc. Int. Con$
on
Struct. Founds. on
27.
R. E. Goodman,
R.
L. Taylor and T. L. Brekke, A model
for
the m echanics
of
jointed rock,
J . soil
mech.foun d. diu.
28.
R. E. Goodman and J. M. Duncan, The role
of
structure and solid mechanics in the design
of
surface and
underground excavations in rock, Proc. Con$ S truct. Solid Me ch. and Eng. Design, Part 2, Paper 105, 1971,
1379-1404.
Wiley, London.
29.
C. M. Haberfied, The performance of the pressuremeter and socketed piles in weak rock, Ph.D. Thesis, Monash
University, Melbourne, Australia,
1987.
30.
A.
F.
Williams, The design and performance of piles socketed into weak rock, Ph.D. Thesis, Monash University,
Melbourne, Australia, 1980.
31. R . K.
Rowe and
H. H.
Armitage, Theoretical solutions
for
axial deformation of drilled shafts in rock, Can. Geotech.
32. L. H . Ooi, G. F. Boey and J.
P.
Carter, F inite element analysis of dilatant co ncrete rock interfaces, Proc. 5 th Con$
393-404 (1983).
eng.,
19, 1209-1226 (1983).
struct.,
18, 739-751 (1984).
element methods, Comput. Geotech., 5,
269-284 (1988).
(1988).
(1992).
1991.
Founds.
on Rock.
Vol.
I
Sydney,
1980,
pp.
303-316.
on Rock Mech., Montreal, Que., Canada,
1973,
pp.
235-262.
J . , 16. 481-486 (1979).
Rock, Vol.1, Sydney, 1980, pp. 253-264.
ASCE,
94, 637-659 (1968).
J . , 24, 114-125 (1987).
Australia o Finite Element Methods, Melbourne, 1987, pp. 157-162.
-
8/10/2019 Finite Element Modelling of Rock Socketed Piles
23/23
FINITE ELEMENT MODELLING OF ROCK-SOCKETED PILES
47
33. K. A.
Pease and
F.
H. Kulhaw y, Load transfer mechanisms in rock-sockets an d anchors, Report No. EL-3777, E PRI ,
34.
T.
S .
K. Lam , Shear behaviour
of
concrete-rock joints,
Ph.D.
Thesis, Monash University, Melbourne, Australia,
1983.
35.
C.
S.
Desai and
Y.
Ma, Modelling of joints and interfaces using the disturbed-state concept,
Int .
j
numer.
anal.
36.
J . Ghaboussi, E. L. Wilson and J. Isenberg, Finite element
for
rock joints and interfaces,
J
soil mech.
found.
div.
37. Z .
Sun,
C.
Gerrard and
0
Stephansson, R ock joint compliance for compression an d she ar loads, Int.
J
Rock Mech.
38. W. L eichnitz, Mechanical properties of rock-joints,
Int.
J
Rock Mech. Min . Sci. Geomech.
Abst., 22, 313-321 (1985).
39.
R.
E.
Goodman, The mechanical properties ofjoints, Proc. 3rd Congr. ISRM.
Vol. IA,
Denver,
1974,
pp.
127-140.
40. S.
C, Band is, A.
C
Lumsden an d N. R.Barton (Fundamentals of rock joint deformation,
Int.
J Rock Mech. Min. Sci.
41.
G.Swan, Determination of stiffness and other joint properties from roughn ess measurem ents,Rock Mech. Rock Eng.
42.
F. H. Kulhawy, Stress deformation properties of rock and rock discontinuities, Eng. Geol., 9
327-350
(1975).
43.
F. H. Kulhawy, Geomechanical model for rock foundation settlement, J geotech. eng., div. ASCE,
104(2), 211-227
44.
A. F. Williams,
I.
B. Donald and H. K. Chiu,
Stress
distributions in rock socketed piles, Proc. In t . Conf: on Struct.
45.
R. J. Jewell and M. F. Randolph, Cyclic rod shear tests in calcareous sediments,
Proc. Int . Conf:
on
Calcareous
46.
A.
M .
Hyden, J. M. Hulett an d A. F. Abbs, Design practice for grouted piles in Bass Strait calcareous soils, Proc. Int.
Con$
on
Calcareous Sediments,
Vol. I , Perth, Western Australia,
1988,
pp.
297-304.
47.
M . F. Randolph, The axial capacity
of
deep fo undation s in calcareous soil, Proc. Int. Con$ Calcareous Sediments,
48.
M . Fahey, R. J. Jewell, M. S. Khorshid and M. F. Randolph, Parameter selection for pile design in calcareous
Palo Alto, Ca.,
1984,
pp.
102.
methods geomech.,
16,
623-653 (1992).
A SC E ,
99, 833-848 (1973).
Min. Sci. Geomech. Abst.,
22(4), 197-213 (1985).
and
Geomech.
Abst. , 20(60), 249-268 (1983).
16,
19-38 (1983).
(1978).
Founds.
on
Rock,
Vol. I ,
Sydney,
1980,
pp.
317-325.
Sediments, Vol. I , Perth, Western Australia,
1988,
pp.
215-222.
Vol.
2,
Perth, Western Australia,
1988,
pp.
837-857.
sediments Predictive Soil Mechanics Proc
Wroth
Mem. Symp., Oxford, pp.
261-278.