View
218
Download
0
Category
Preview:
Citation preview
8/18/2019 Elastic Analysis PilesandGroups1971
1/19
See discussions, stats, and author profiles for this publication at:https://www.researchgate.net/publication/245410006
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
CITATIONS
117
READS
309
2 authors, including:
Roy Butterfield
University of Southampton
101 PUBLICATIONS 1,927 CITATIONS
SEE PROFILE
Available from: Roy Butterfield
Retrieved on: 13 April 2016
https://www.researchgate.net/profile/Roy_Butterfield?enrichId=rgreq-37b6de0e-2b3c-4236-bb7b-8437829fc90f&enrichSource=Y292ZXJQYWdlOzI0NTQxMDAwNjtBUzoxOTEzNjc3NTU5NTIxMjlAMTQyMjYzNzAzNjQyNQ%3D%3D&el=1_x_4https://www.researchgate.net/?enrichId=rgreq-37b6de0e-2b3c-4236-bb7b-8437829fc90f&enrichSource=Y292ZXJQYWdlOzI0NTQxMDAwNjtBUzoxOTEzNjc3NTU5NTIxMjlAMTQyMjYzNzAzNjQyNQ%3D%3D&el=1_x_1https://www.researchgate.net/profile/Roy_Butterfield?enrichId=rgreq-37b6de0e-2b3c-4236-bb7b-8437829fc90f&enrichSource=Y292ZXJQYWdlOzI0NTQxMDAwNjtBUzoxOTEzNjc3NTU5NTIxMjlAMTQyMjYzNzAzNjQyNQ%3D%3D&el=1_x_7https://www.researchgate.net/institution/University_of_Southampton?enrichId=rgreq-37b6de0e-2b3c-4236-bb7b-8437829fc90f&enrichSource=Y292ZXJQYWdlOzI0NTQxMDAwNjtBUzoxOTEzNjc3NTU5NTIxMjlAMTQyMjYzNzAzNjQyNQ%3D%3D&el=1_x_6https://www.researchgate.net/profile/Roy_Butterfield?enrichId=rgreq-37b6de0e-2b3c-4236-bb7b-8437829fc90f&enrichSource=Y292ZXJQYWdlOzI0NTQxMDAwNjtBUzoxOTEzNjc3NTU5NTIxMjlAMTQyMjYzNzAzNjQyNQ%3D%3D&el=1_x_5https://www.researchgate.net/profile/Roy_Butterfield?enrichId=rgreq-37b6de0e-2b3c-4236-bb7b-8437829fc90f&enrichSource=Y292ZXJQYWdlOzI0NTQxMDAwNjtBUzoxOTEzNjc3NTU5NTIxMjlAMTQyMjYzNzAzNjQyNQ%3D%3D&el=1_x_4https://www.researchgate.net/?enrichId=rgreq-37b6de0e-2b3c-4236-bb7b-8437829fc90f&enrichSource=Y292ZXJQYWdlOzI0NTQxMDAwNjtBUzoxOTEzNjc3NTU5NTIxMjlAMTQyMjYzNzAzNjQyNQ%3D%3D&el=1_x_1https://www.researchgate.net/publication/245410006_The_Elastic_Analysis_of_Compressible_Piles_and_Pile_Groups?enrichId=rgreq-37b6de0e-2b3c-4236-bb7b-8437829fc90f&enrichSource=Y292ZXJQYWdlOzI0NTQxMDAwNjtBUzoxOTEzNjc3NTU5NTIxMjlAMTQyMjYzNzAzNjQyNQ%3D%3D&el=1_x_3https://www.researchgate.net/publication/245410006_The_Elastic_Analysis_of_Compressible_Piles_and_Pile_Groups?enrichId=rgreq-37b6de0e-2b3c-4236-bb7b-8437829fc90f&enrichSource=Y292ZXJQYWdlOzI0NTQxMDAwNjtBUzoxOTEzNjc3NTU5NTIxMjlAMTQyMjYzNzAzNjQyNQ%3D%3D&el=1_x_2
8/18/2019 Elastic Analysis PilesandGroups1971
2/19
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
8/18/2019 Elastic Analysis PilesandGroups1971
3/19
8/18/2019 Elastic Analysis PilesandGroups1971
4/19
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)
8/18/2019 Elastic Analysis PilesandGroups1971
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
8/18/2019 Elastic Analysis PilesandGroups1971
6/19
ELASTIC ANALYSIS OF COMPRESSIBLE PILES AND PILE GROUPS
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)
8/18/2019 Elastic Analysis PilesandGroups1971
7/19
48
R. BUTTERFIELD AND P. K. BANERJEE
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
8/18/2019 Elastic Analysis PilesandGroups1971
8/19
ELASTIC ANALYSIS OF COMPRESSIBLE PILES AND PILE GROUPS
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
8/18/2019 Elastic Analysis PilesandGroups1971
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
8/18/2019 Elastic Analysis PilesandGroups1971
10/19
8/18/2019 Elastic Analysis PilesandGroups1971
11/19
52
R. BUTTERFIELD AND P. K. BANERJEE
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)
8/18/2019 Elastic Analysis PilesandGroups1971
12/19
ELASTIC ANALYSIS OF COMPRESSIBLE PILES AND PILE GROUPS
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
8/18/2019 Elastic Analysis PilesandGroups1971
13/19
54
R. BUTTERFIELD AND P. K. BANERJEE
aMS d
clMi3/d
am d
I
I
I ,
8/18/2019 Elastic Analysis PilesandGroups1971
14/19
ELASTIC ANALYSIS OF COMPRESSIBLE PILES AND PILE GROUPS
55
8/18/2019 Elastic Analysis PilesandGroups1971
15/19
56
R. BUTTERFIELD AND P. K. BANERJEE
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.)
8/18/2019 Elastic Analysis PilesandGroups1971
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
8/18/2019 Elastic Analysis PilesandGroups1971
17/19
58
R. BUTTERFIELD AND P. K. BANERJEE
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
8/18/2019 Elastic Analysis PilesandGroups1971
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
ELASTIC ANALYSIS OF COMPRESSIBLE PILES AND PILE GROUPS
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
8/18/2019 Elastic Analysis PilesandGroups1971
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.
Recommended