Finite Element Modelling of Rock Socketed Piles

8/10/2019 Finite Element Modelling of Rock Socketed Piles

1/23

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


2/23

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


3/23

FINITE ELEMENT MODELLING OF ROCK-SOCKETED PILES

27

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


4/23

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


5/23

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.


6/23

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


7/23


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


8/23


9/23

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


10/23

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


11/23

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


12/23

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


13/23

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.


14/23

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.


15/23

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 .


16/23

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


17/23


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


18/23

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.


19/23

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


20/23

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,


21/23

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


22/23

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.

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Finite Element Modelling of Rock Socketed Piles

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