Elastic Analysis PilesandGroups1971

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The Elastic Analysis of

Compressible Piles and Pile Groups

Article in Géotechnique · January 1971

Impact Factor: 1.87 · DOI: 10.1680/geot.1971.21.1.43

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Roy Butterfield

University of Southampton

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BUTTERFIELD, R. & BANERJEE, P.

K. (1971).

Gdotechnique1, No.

1,43-60.

THE ELASTIC ANALYSIS OF COMPRESSIBLE PILES AND

PILE GROUPS

R. BUTTERFIELD* and P. K. BANERJEEt

SYNOPSIS

The response of rigid and compressible single piles

embedded in a homogeneous isotropic linear elastic

medium has been obtained by a rigorous analysis

based on Mindlin’s solutions for a point load in the

interior of an ideal elastic medium.

The analytical method is described and is extended

to analyse axially loaded rigid and compressible pile

groups with floating caps spaced in an arbitrary

manner.

The results are presented as a series of graphs

showing the effect of variation of the ratios of pile

length to a diameter, the ratio of the modulus of

elasticity of the pile to the shear modulus of the

medium E / G and the effect of base enlargement on

the load displacement characteristics of single

axially loaded piles.

Graphs are also presented showing the effect of

length to diameter ratio, pile spacing and E / Gratio

on the response of a range of typical pile groups.

The results are compared with published data from

more approximate solutions and laboratory and full-

scale pile loading tests, The latter indicate that the

analyses may usefully calculate group settlement

ratios and allow the extrapolation of load displace-

ment data on single piles to predict group behaviour.

La reponse de pieux simples, rigides et compres-

sibles noyes dans la masse d’un milieu tlastique

lineaire isotrope a &.e obtenue par une analyse rigou-

reuse basee sur les solutions de Mindlin pour charge

ponctuelle & l’interieur d’un milieu elastique ideal.

La methode analytique est Btendue a l’analyse de

groupes de pieux rigides et compressibles, charges

axialement avec casques flottants. espaces d’une

manibre arbitraire.

Les resultats sont represent& par une gamme de

graphiques montrant l’effet de la variation des rap-

ports entre la longueur du pieu avec le diametre, le

rapport de module d’elasticite du pieu avec le module

de cisaillement du milieu

E / G

et l’effet de l’agran-

dissement de la base sur les caracteristiques de

deplacement de pieux simples, charges axialement.

Des graphiques montrent egalement I’effet du

rapport longueur a diametre et du rapport E / G et

de l’espacement des pieux, sur la reponse d’une

gamme de groupes de pieux typiques.

Les resultats sont compares avec les informations

publiees d’apres des solutions plus approximatives

et de laboratoire et des essais en vraie grandeur de

charge de pieux.

Ces derniers indiquent que l’analyse peut utile-

ment permettre le calcul du rapport de tassement de

groupe et faciliter l’extrapolation des donnees de

deplacement de charge pour des pieux simples pour

predire le comportement des groupes.

INTRODUCTION

The reliable prediction of foundation displacements at working load remains a major civil

engineering problem. However, the results of a long series of experiments at the Waterways

Experimental Station, summarized by Turnbull

et al.

(1961), showed for saturated clays close

agreement between experimentally determined values of the stresses under surface loads and

the values computed from elastic solutions based on the analysis given by Boussinesq (1885).

There is therefore some justification for attempting to obtain useful predictions of load dis-

placement characteristics of piles and pile groups based on elastic theory.

Several investigators (D’Appolonia and Romualdi, 1963; Mattes and Poulos, 1969; Nair,

1963; Poulos and Davis, 1966; Poulos, 1968; Saffery and Tate, 1961; Salas and Belzunce,

1965; Seed and Reese, 1955; Sowers et al., 1961) outlined approximate methods based on elastic

theory for analysing different aspects of the load displacement behaviour of single axially

loaded piles and piers. Poulos and Davis (1968) and Mattes and Poulos (1969) published more

* Senior Lecturer, Civil Engineering Department, University of Southampton.

t Senior Scientific Officer, Highways Engineering Computer Branch, Ministry of Transport.

43


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ELASTIC ANALYSIS OF COMPRESSIBLE PILES AND PILE GROUPS

45

short plain piles and that appreciable errors occur in the calculated values of radial stress com-

ponents in the immediate vicinity of a loaded pile due to assumption (c).

Poulos (1968) has also presented an approximate general study of axially loaded groups of

incompressible piles incorporating assumptions (u)-(c).

It is shown that pile compressibility is important in group behaviour and that the load dis-

tribution between the piles can be markedly different from that predicted by incompressible

pile analyses.

The analysis deals with arbitrarily spaced compressible piles under a rigid pile cap where the

cap is not in contact with the ground surface, although the cap group interaction problem has

been studied (Banerjee, 1970) and will be dealt with (Butterfield and Banerjee, 1971a).

Whereas the elastic analyses incorporate gross idealizations of any field situation and intro-

duce parameters which are inevitably imprecise, comparisons with published experimental

results have been made which show encouraging agreement in situations where either data

from single plain pile load tests are to be extrapolated to estimate the working load response of

single underreamed piles and groups of piles or group settlement ratios are to be estimated.

An attempt has also been made to present the theoretical analysis in an elegant and general

way since it offers a novel, powerful and versatile means of attacking many associated prob-

lems of interest to the foundation engineer which are less tractable by other methods.

METHOD OF ANALYSIS

Figure 1 (a) represents an outline of a cylindrical pile of length

L

and radius a inscribed in a

homogeneous isotropic elastic half space defined by G and II. The essence of the analysis is to

find a fictitious stress system + which, when applied to the boundaries of the figure inscribed

in the half space, will produce displacements of its boundaries which are identical to the speci-

fied boundary conditions of a real pile system of the same geometry and also satisfy identically

the stress boundary conditions on the free surface of the half space. The stresses 4 are

fictitious in that they are to be applied to the boundaries of the fictitious half space figure and

are not necessarily therefore the actual stresses acting on the real pile surfaces (Banerjee,

1970). However, once the + values have been determined it is a simple matter to calculate the

t

I

I

b 6

,

t;

I

Elemental area =

aSBSc

Fig. 1.

Integration of Mindlin’s

(1936) equations for (a) & (b)

4 and (c) 4%

(a)


5/19

46

R. BUTTERFIELD AND P. K. BANERJEE

actual stresses and displacements they produce anywhere in the half space, including those on

the real pile boundaries.

Let 4, be a vertical fictitious stress acting in the half space along the pile shaft boundary

at a depth c below the surface, the vertical and radial displacements SW,(r, z) and 6U,(r, z)

respectiveIy at a point B(r, z) due to & acting on the surface of an eIementa1 cylinder of height

6~. The radius a can be obtained by integrating Mindlin’s point load solution (Mindlin, 1936)

for (c$,&&) over the surface of the elemental cylinder.

The results can be expressed as

2n

W,(Y, z) =

s

& (KW,(c, rl, z)

6c}d6

. . .

0

2A

iw,(Y, z) =

s

a& {KU,(c, yl, z) Sc}

do

. . . . .

0

where

KW,(c, yl, z)

and

KU,(c, I ~, z)

derived from Mindlin’s soIution are given in the Appendix.

Mindlin’s solution is used here in the kernels of the integrals since it automatically satisfies the

stress boundary conditions on the unloaded surface of the half space.

The total vertical and radial displacements at B(r, z) due to all such elemental shaft inten-

sities are then obtained by integration as

L 2n

W,(y, z) =

ss

,aKW,(c, r l, z)

de dc

. . . . .

0 0

L 2n

U,(y, z) =

ss

$,aKU,(c, r l, z)

dBdc

. . . . .

0 0

Similarly taking & to be the fictitious vertical stress over the base area of the pile acting at a

point O’(E, 0, L) (Fig. l(b)) the vertical and radial displacements at

B(Y, z)

due to &, can be

expressed by analogy with equations (3) and (4) as

c$~EKW,(L , r 2, z)

de de

,eKU,(L, Y,, z)

dt’da

. . . . .

. . . . .

where

KW,(L,

y2, z) and

KU,(L , r2, z)

can be obtained by substituting

C=L

and rz=

ly2 + E2 zye cos e,]

2

for

rl

in equations (26) and (27) respectively in the Appendix.

Thus the total vertical and radial displacements at a point B(Y, z) due to both vertical

shaft and base intensities are given by

W(Y, z) =/:s; [+ ,aKW,(c, ~1, z) de dc+/;/; [& KW,(L > y2, z)] de de

- (7)

L 2n

b 21~

U(Y, z) =

ss

[ dW,(c> yl, - 41 e dc+

ss

[ , EKU, ( L,

2,Z)l de de - (8)

0 0

0 0

Equations (7) and (8) can be solved for the following boundary conditions.

At B(r, 0), o,=

~,~=0, at B(a, z), O


6/19


47

surface, Mindlin’s (1936) solution for an embedded point load acting parallel to the surface of

an elastic half space. The displacements (Fig. l(c)) are given by

L 2n

W a 2) =

ss

a ,KW,(c, Y, z) dBdc

. . . .

* (9)

0 0

U,(r, z) =ILj2’

& KU& , r, z)

dB dc

0 0

. . . .

. (10)

where KW,(c, r , z) and KU ,(c, Y, z) are given in the Appendix.

Thus from equations (7)-(10) the total vertical and radial displacements at a point B(r, z)

due to a pile loaded with an axial load are

L 2n

W(r, 2) =

ss

L 2n

q$aKW,(c, Y, z) d0 dc+ & aKW,(c, r, z) d&J c

0 0

ss

0 0

b 2n

+

ss

,eKW,(L , r2, z) d6 de

. .

* (11)

0 0

L 2n

ss

L 2R

up, 2) =

+,aKU,(c, ~1, z) de dc+

ss

& aKU,(c, Y, z) de dc

0 0

0 0

b 278

+

ss

~,EKU,(L , Y,, .z) dB de

. .

* (12)

0 0

Equations (11) and (12) can be used to calculate the displacement components at any point

within the half space if the distributions of 4,, & and b are known from prescribed displace-

ment boundary conditions at the pile soil interface.

A simple numerical treatment of integral equations similar to equations (11) and (12) has

been suggested (Butterfield and Banerjee, 1971b) in which the pile shaft is divided into n. equal

segments each of thickness G, and the base into m rings each of annular radius G,. The ver-

tical and radial displacements of any element i on the shaft can then be written in discrete

form (see the Appendix) as

(B’s)* = j I (&),(Kss)~+ , (&),(KRS)ij+ jJ

(+b),tKBs)iI - * (13)

(u.Ji = jil (A)j(KSu)i,+ j21 (+r)AKRU)i,+ j l (‘AJAKBU)t~ . . (14)

where i=l, 2, 3,.

. ., n.

Similarly the vertical base displacements (wb), are given by

twb)i = jjl (h),(KSf%j+ jgl (4r)AKW~j+ j , ( b)AKBB)i, * * (15)

wherei= 1,2,3 ,..., m.

The integrals involved in the various

K

factors (see the Appendix) can in general be

evaluated by simple quadrature formula, but a fine mesh subdivision of the field is required at

the singularities in order to obtain reliable values of the diagonal elements in the

Kij

matrices

(these singularities occur when i = j and the load points and field points coincide).

Equations (13), (14) and

{WJ

VJJ

iwb>

or

1s

SOLUTION FOR SINGLE PILE

(15) can be written in matrix notation a

H

[KSS] [KRS] [KBS]

=KSU] [KRU] [KBU]

[KSB] [KM?] [KBB]

II

I

{M

{4b)

. . .

{W} = [K](Q)

. . .

. . . . .

(17)


7/19

48


which provides a formal solution for {@I

(@} = [KJ-l(w)

. . . . . . . .

(18)

where {CD}are the required shaft (vertical and radial) and base stress intensities and {W> are

the given shaft (vertical and radial) and base displacements. [K] will always be non-singular

and diagonally dominant for the class of problems being discussed (Banerjee, 1970).

For a rigid pile the vertical displacements of all points on the shaft and the base are the

same and are equal to the displacement of the head of the pile. The radial displacement at

the shaft face is zero. Thus if unit displacement is applied to the head of the pile, from

equation (18)

{&Z} = Kl-l{ i} . . . . . . (194

If radial displacement compatibility is ignored

{ii} = [KJ-‘{i:> . . . . . .

W’b)

where

The results obtained from these equations show that

a) the solution of equations (19a) and (19b) produces values of {&} and {&,} which agree

within 3% for piles with length to diameter ratios greater than 10

(b) for pile geometries commonly encountered in practice the consideration of radial

compatibility has a negligible effect on the determination vertical displacement for

a given load

(c) if accurate evaluation of the stresses arising in the immediate vicinity of the pile is

necessary, radial displacement compatibility must be included (Butterfield and

Banerjee, 1970).

When the distributions of c$~and &, over the pile shaft and base respectively have been

obtained for a prescribed displacement of the head of the pile, the load

P

arried by the pile

at any depth below the surface is found from

P = ‘Z?ia$$dc+ b27iE bde

s

. . . . . .

L s

0

(20)

The total load P required to produce unit displacement of the head of a rigid pile is given by

substituting z= 0 into equation (20).

Also once {CD}have been obtained displacements at any

field point B(r, z) can be calculated directly from equations (11) and (12) when they are re-

stated in discrete form.

Solution fey a colrtpressible pil

The solution from equation (19a) will, if applied to a compressible pile, lead to an under-

estimation of displacement of the pile head for a given load.

If the pile is assumed to be perfectly bonded to the medium the vertical displacement of a

shaft element at a depth z will differ from that at a depth z+dz by an amount equal to the


8/19


49

Fig. 2. Compressible pile displace-

ment pattern. W is the vertic l

displacement of the pile of dia-

meter and length

L

under a

vertical load P

Fig. 3. Pile group co-ordinate

system

elastic compression of the pile length dz (Fig. 2). Since for any pile section the vertical direct

stress is much greater than other stresses, to a good approximation

aw

P

az=-A,E,

u = -/+&;a

i

. . . . . . . .

(21)

where A, is the cross-sectional area of the pile shaft, E, is Young’s modulus of the pile material

and pLps Poisson’s ratio of the pile material

Equations (21) can be written in finite difference form and used in an iterative scheme for

the solution of (16) as follows

(a) The rigid pile solution (19a) is obtained.

(b) Values of P, are found from equation (20) and substituted into equations (21) as a

first approximation, giving new {Wi} and {Wi} values.

(4 UK>, 0%) and W>

are substituted in (16) as an approximation giving new {&}, (c&}

and {c&} values.

(d) A new value

PL

is obtained for each section of pile and the cycle (b), (c), (a) is

repeated until the value of

Pg

between two successive iterations differs by an

acceptably small value.

ANALYSIS OF PILE GROUPS

The foregoing analysis can be extended directly to deal with general pile groups.

The

following approximations are introduced to reduce the order of the matrices involved.

(a)

Since the introduction of {&} produces negligible effect on the total load required for a

given settlement, radial displacement compatibility is ignored.

(b) In general, the surface intensities {&}

and {&} will be functions of (c, 0) and (E, ~9)

respectively.

Allowing for this greatly increases the number of linear equations

involved in the problem and therefore {+J and {&,}

are approximated by equivalent

rotationally symmetric distributions which are therefore independent of 0.

It is

thought that these approximations will introduce negligible errors in the cal-

culated displacements and loads for pile spacings commonly encountered in

practice.

An integral representation of the vertical displacement of a point B(y, 0, z) due to a number


9/19

50

R BUTTERFIELD AND P K BANERJEE

N

of arbitrarily spaced piles can be written, by analogy with equation (7), as (Fig. 3)

L 2n

b 2n

W(r, 8, z) = ,zl{ so/, (q ),dW’,(c, ~1, 2) de dc +~,~, (+b)aEKw2(L, “2* 2) de de

>

(22)

where rl = [yp+ a2 - 2r,a cos O,] /2

Y2 = [$+E

2 - 2TpE os e,] U2

y, = ~~2 S; - 2rs, cos

e - e,)] 12

p=1,2,3 ,...,

N, s, is the distance of the 9th pile from the origin of the global axes (Fig. 3)

and N is the number of piles in the group.

As before equation (22) can be written for discrete subdivision of the pile-medium inter-

faces as

(ws)iq = f?

ds)i~[KSSliipq+ 2

(+b)jp[KBBltjtw

p=l j=1

p=l j=l

. .

(23)

for the shaft elements and

for the base elements, where [KSS],,,,, [KBS]I,,,

and so on are analogous to similar matrices

developed for a single pile (see the Appendix).

The computational effect can often be decreased

by using the symmetry of a group, since for piles carrying identical loads the order of the matrices

can be reduced proportionately.

Equations (23) and (24) can be combined and written as

Wi, = &pq@jjp

or

{W}= [ K] { @

. . . . . . . .

(25)

which can be solved for both incompressible and compressible pile groups as before.

It is interesting that although [K] will in general be a fully populated matrix its order is

determined only by the number of surface element subdivisions adopted over the pile-

medium interfaces.

It is therefore of a much lower order than the matrices generated by

solution methods involving the use of three-dimensional elements throughout the volume of

the system.

LID

0 20 40

M 86

l o o

D =

Diamcterofrhafc

100

W = Vertical displacement of head of pile

1,”

Fig 4

Load displacement curves for compressible piles


10/19


11/19

52


k/PI x 100

[PJPJ x

IO0

0

20

40.

60

80

100

60

Fig. 7 (above and left).

Base load contribution

for compressible underreamed piles

Fig. 8 (below).

(a) Effect of

L/D on

the inter-

action between piles, (b) effect of com-

pressibility on interaction between piles

RX

I.0

1. 4 I . 8 2. 2

2.6

, - - - -

, -

, '

/ '

31

I

I

I

0

(a) (b)


12/19


53

Again, once {@I has been determined the actual displacements occurring at any field point

B(r, 19, ) can be calculated from the discrete form of equation (22).

DISCUSSION OF RESULTS

In the following Figs 4-7 refer to single compressible plain and underreamed piles and Figs

8 and 9 to groups of plain compressible piles under a rigid floating pile cap. In all cases a

range of 6000 < X 6 co is considered which covers the range of material properties of major

practical interest. (X =

E,/G

h ere G is the shear modulus of the half space material.)

Figure 4 shows the effect of the compressibility ratio X on the load displacement behaviour

of plain piles over a range of

LID

ratios. The two significant features of the curves are the

negligible effect of h for shorter piles

(L/D < 20)

and the fact that the results converge to the

rigid surface disc solution as

LID

approaches zero.

The effect of pile compressibility on the shaft surface shear stress distribution is shown in

Figs 5(a) and 5(b). For the shorter pile (L/D=20, Fig. 5(a)) the effect of h is seen to be

negligible and the stress distribution agrees closely with that obtained by more approximate

analyses (Mattes and Poulos, 1969; Poulos and Davis, 1968). Fig. 5(b) shows similar curves

for a longer pile

(L/D = 80)

in which the h = 60 000 and h = 00 results are almost identical and

similar to the short pile results. However, the shear stress distribution is radically altered in

the more compressible system (X =6000). (A

n accurate assessment of the direct radial stress

(u,) at the pile-medium interface can be made by obtaining the limiting value of the radial

stress at points in the medium as they approach the interface asymptotically, Butterfield and

Banerjee, 1970.)

Comprehensive load displacement curves are presented in Figs 6(a)-6(c) for compressible

underreamed piles over a range of base to shaft diameter ratios (16

D,/D, < 6).

Whereas these

curves enable the absolute value of the pile head displacements to be estimated, the relative

influence of

(Do/OS)

and X can be seen more clearly in Fig. 6(d) where they are related to the

ratio of the settlement of an underreamed pile to that of a plain pile under the same load (R,).

In all cases the effect of h on

R,

is seen to be very small (6000

<

h < co). The stiffening of the

system (i.e. decrease in

R,)

achieved by enlargement of the base is also very small for the longer

piles

(L/D,=SO)

and even for the shorter piles

(L/D,=20) R, .

s

only reduced to around 0.8 for

DJD, =4.

The accurate inclusion of a rigid pile base in the analysis has the effect of increasing

R,

by about 10% above more approximate predictions and also indicates a small increase in

R,

with decreasing h, for shorter piles, reversing the trend of earlier analyses (Mattes and

Poulos, 1969; Poulos and Davis, 1968).

This is also reflected in the proportion of the total

load carried by the base

PE/P

(Figs 7(a)-7(c))

w IC is considerably decreased in the more

’ h

rigorous analysis for underreamed piles with

DJD, > 24.

Figures 8(a) and 8(b) relate the settlement ratio

R,

for groups of N compressible plain piles

(N=2, 3, 4)

under a rigid cap for

LID

ratios of 20 and 40 and h values of 6000 and co, where

R, is the ratio of the vertical displacement of the pile cap under a load of N x

P

to that of a

single pile under a load

P.

Although

R,

is strongly influenced by

LID

the effect of h is

negligibly small for all the shorter piles

(L/D < 40).

It is interesting that

R,

remains at about 1.5 for the longer piles even when the pile spacing

exceeds 32 diameters, and also that the principle of superposition (Poulos, 1968) applies less

well to the results for compressible piles.

Curves showing both the load-displacement behaviour and the individual pile load sharing

of larger groups of compressible piles under a rigid floating cap are presented in Figs 9(a)-9(g).

These results have been obtained without using Poulos’s superposition approximation for the

non-symmetrical groups. A standard close spacing of s/D =23 has been adopted throughout

to indicate the likely worst case values of

R,.

Variation of h is seen to have a considerable effect on the load sharing between the piles


13/19

54


aMS d

clMi3/d

am d

I

I

I ,


14/19


55


15/19

56


Table 1 . s / D 2.5, L/D =25, y=0*5

Table 2. s/D =2*5, L/D=25, ~=0*5

Type of group 2x2 3x3 4x4 5x5

I I

__-

h=co* 0672 0.541 0.460 0.403

~~

NIR, h=coT 0.665 0.550 0.456 0.396

I

x=6OOOt j 0.620 0500 0.420 0.371

I

Type

Pile

of number

group

3x3

::

3

1

4x4

3

* The results were obtained using a uniform stress

distribution under the pile base and superposition

principle by Poulos (1968).

t

The results were obtained using the analysis

outlined in this Paper.

1

5x5

:

4

:

pIpave* plpavet

X=03 X=6000

I.520

0.74

0.050

(tension)

-

--

I

1.510

0.750

0.060

ltension)

1.380

0.765

0.120

2.020 2.020 I.840

0.960

0.965

0.965

0.05 0.044 0.180

2.580

2.520

2.300

I.180

1.190

I.190

1.160

1.160

1.141

0.010

0.048

0.145

0.010 0.106 0.119

0.190 0.095 0.095

L

within the groups but a much smaller effect on the overall group response.

A reduction in I\

from co to 6000 produces only about 10% reduction in R, (Table l), whereas the individual

pile load sharing pattern changes markedly (Table 2).

As X decreases the load carried by the

internal piles in a group increases although the contribution of these piles in 4 x 4 and 5 x 5

groups (Figs 9(f) and 9(g)) is still generally less than 10% of that of the outer piles.

COMPARISON WITH EXPERIMENTAL RESULTS

Figures 10 and 11 a)-1 1 (c) are comparisons of the previous analytical results with published

test data from model and full-scale pile tests.

The upper curve in Fig. 10 which relates LID to the percentage of load taken by the base

for plain piles (D,JDs= l), calculated for G=4000 lbf/sq. in. and ~=0*45, is a reasonable fit to

the results of Whitaker and Cooke (1966)

and a rather similar test reported by Sowers

et al.

(1961), both on full-scale piles. Measurements from full-scale underreamedpile tests (Whitaker

and Cooke, 1966) are also plotted and the theoretical curve, for Db/Ds = 2 and the elastic para-

meters given, is now an acceptable prediction of these results. The analysis has therefore

extrapolated the plain pile test results successfully to predict the response of the underreamed

piles. The curved marked slip in this Fig. 10 refers to an extension of the analysis to include

incremental slip at the pile-soil interface (Banerjee, 1970).

Figures 1I(a) and 11(b) are comparisons of calculated and measured settlement ratios for

small-scale model tests on pile groups generally at factors of safety of about 2 on ultimate load.

Unfortunately the lack of published detailed load displacement curves obtained from single

piles concurrently with the group test results, together with the variety of definitions of R, used

by different authors and the considerable scatter of the measured data, makes conclusive com-

parison of theory and experiment difficult.

However, in Fig. 1 l(a) the results of Saffery and Tate (1961) are shown, relating pile

spacing (s/D) to R, for 3 x 3 groups of 2 in. dia. piles in remoulded London Clay, and compared

with calculated curves. Agreement with the lower curve (H/L = 18) is quite good. H is the

total thickness of the compressible layer above the rigid base of the bin and H/L N 1.8 is the

experimental value. Comparison with the

H/L = co

curve emphasizes the considerable effect

of bin depth on R, in model tests. (Th

e inclusion of H/L in previous analyses is straight-

forward, Banerjee, 1970.)


16/19

L

a

b

h

b

e

F

g

1

0

C

m

p

s

o

b

w

n

c

c

u

a

e

a

m

e

s

u

e

n

o

o

p

a

n

a

u

m

e

p

e

F

g

1

r

g

C

m

p

s

o

w

h

t

e

r

u

o

3

x

3

p

e

g

a

a

b

s

e

m

e

n

r

o

c

o

s

h

n

b

w

n

p

e

.

E

E

i

=

”

4

1

1


17/19

58


Figure 11(b) is a similar comparison with model tests by Whitaker (1957, 1960) on 3 x 3

groups of & in. dia. piles also in remoulded London Clay.

Again when H/L is taken into

account reasonable agreement with the theoretical curves is obtained, although Whitaker

defines

R, as

the ratio of the settlement of the pile group to the settlement of a single pile at

half the ultimate load of each.

A more stringent test of the theoretical model is to compare the calculated and measured

load distributions between individual piles in the group and this has been done in Fig. 11 (c) for

the 3 x 3 pile group tests of Whitaker (1957) and Sowers et al. (1961).

Once more the patterns

of calculated and measured loads are similar although the general trend is for the load to be

more evenly distributed between the piles than the elastic analysis predicts with h= co.

Points are also shown indicating how a more even load distribution is produced when h is

reduced to 6000.

Poulos (1968) has made comparisons similar to those shown in Figs ll(a)-

1 I (c) using his less rigorous analysis with h = co and he arrived at essentially identical con-

clusions.

CONCLUSIONS

A rigorous elastic analysis of bonded compressible plain and underreamed piles and com-

pressible piles in general groups under a rigid floating cap has been presented in which the

truly rigid pile base and radial deformation compatibility conditions can be included.

The results of the refined analysis of this Paper have been compared with analyses by

Poulos (1968) in which at least the final two conditions were relaxed.

Overall load-displacement behaviour

For single plain piles the more rigorous results differ from those of Poulos by less than 5

whenever

LID 2 5.

For single underreamed piles with D,/D,-3, this analysis predicts a reduction in system

stiffness relative to that given by Poulos of 5-25 as

LID

is varied between 20 and 5.

For the groups of plain piles the value of

R,

is essentially the same in this Paper as that

given by Poulos and the effect of h (6000 < h < co) is negligible.

Local load distribution within and around the piles

For single piles and particularly underreamed piles the full analysis in this Paper is neces-

sary if accurate values of direct stresses are to be obtained.

If only the shaft and base load division is required then this analysis with the relaxation

of the radial compatibility requirement is adequate.

These remarks also apply to the pile group analyses where additionally the load sharing

between individual piles in the group is strongly influenced by h, the loads being more evenly

distributed between compressible piles in a group.

Comparisons with published experimental data suggest that the elastic analyses may be

applicable to the extrapolation of single pile test data to predict the response at working loads

of underreamed piles and pile groups and also to the prediction of group settlement ratios (RJ.

APPENDIX

It can be shown directly from Mindlin’s (1936) equations that

KW,(c, v, 2) =

(3-44r)+8(1-p)Z-(3-4p)

I (z-c)~ I

(3-4~)(~+~)~--2~~+6~~(~+~)~

R2

R?

RZ

G 1

P-3

(z-c)

r+

(3-4 -c)_4(1-p)(l-2~) 6c++c)

2

----+Rjl cosa . . .

R,(R,+z+c)

I

(27)

where 17, = [Y:+(z-c)~]~‘~

R, = [Y?+(z+c)~]~‘~

II = [r2+ a2-22ra cos S,]l’Z


18/19

where

Integrals for the various K matrices are as follows.

[KSS],, = 2s;;;1,0,s; aKWl(c, +,I, 2) de dc

[KRS],, = 2s;;: ,,,,-1 aKW,(c, y, 3) de dc

[KBS],, = 2/:,G-“,,,,/; cKW,(L, r2, z)

de de

[ KSUl i f= 2~~~~ , GI / ~aKUI ( c.l , de dc

[ KRUI , , =~: , o; I , GI - ; aKG( c, ,2)de&

LKB7- 3, = 2f ; t 1, 0, s; KU, ( L +, a )de de

z = (i-&)G,

here

Also

Also


59

(3-44r)(z-c) 6cz(z+c)+4(1-_c~)(1-2~)

1

- (28)

.

29)

RI = [xa+ya+(z-c)a]l’a

R, = [xa+y2+ (z+c)~]“~

x = rcos &-a

y =

--rsin&

y1 = [2a2--2a cos ep

?f=Od

Ye = [~2+~2-2~~ cos e, y

y1 =

[2a2-22a2

cos ep

Y=CZ

7, = [9+2--

2ar

cos ep

r1 = [r2+a2-2ar cos eC11 2

y1 = p2+9- 2ar cos e, y

r2 = [rr”+ 3 - 2rr cos e,y

REFERENCES

BANERJEE, P. I


19/19

60 R. BUTTERFIELD AND P. K. BANERJEE

POULOS, H. G. (1968). Analysis of the settlement of pile groups.

GCotechnique

18, No. 4, 449-471.

POULOS, H. G. & DAVIS, E. H. (1968).

The settlement behaviour of single axially-loaded incompressible

piles and piers.

GCotechnique

18, No. 3, 351-371.

SAFFERY, M. R. & TATE, A. P. K. (1961).

Model tests on pile group in a clay soil with particular reference

to the behaviour of the group when it is loaded eccentrically. Proc. 5th Int. Conf. Soil Mech. 2, 129.

SALAS, J. A. J. & BELZUNCE, J. A. (1965). Resolution theorique de la distribution des forces dans les pieux.

Proc. 6th Int. Conf. Soil Mech. 2, 309-313.

SEED, H. B. & REESE, L. C. (1955).

The action of soft clay around friction piles.

Proc. Am. Sot.

civ. Eflgrs

81,

Paper 842, December, 28 pp.

SOWERS, G. F., MARTIN, C. B. & WILSON, L. L. (1961).

The bearing capacity of friction pile groups in

homogeneous clay from model studies.

Proc. 5th Int. Conf. Soil Mech. 2, 155.

THERMAN, A. G. (1964). Computed load capacity and movement of friction and end bearing piles embedded

in uniform and stratified soils. Ph.D. thesis, Carnegie Institute of Technology.

THERMAN, A. G. & D’APPOLONIA, A. (1965).

Computed movements of friction and end bearing piles em-

bedded in uniform and stratified soils. Proc. 6th Int. Conf, Soil Mech. 2, 323-327.

TURNBULL, W. J., MAXWELL, A. & AHLVIN, R. G. (1961).

Stresses and deflections in homogeneous soil

masses. Proc. 5th Int. Conf. Soil Mech., Paris 2, 337-346.

WHITAKER, T. (1957). Experiment with model piles in groups.

Gdotechnique 7, No. 4, 147-167.

WHITAKER, T. (1960). Some experiments on model piled foundations.

Pile foundations, Proceedings of

symposium held by the International Association for Bridge and Structural Engineering,

Stockholm

p. 124.

WHITAKER, T. & COOKE, R. W. (1966). An investigation of shaft and base resistances of large bored piles

in London Clay.

Large bored piles, pp. 7-49. London: Institution of Civil Engineers.

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