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Some remarks on two and three dimensional consolidation analysis of sand-drained ground
Computers and Geotechnics 12 (1991 ) 73-87
TECHNICAL NOTE
SOME REMARKS ON TWO AND THREE DIMENSIONAL CONSOLIDATION ANALYSIS OF SAND-DRAINED GROUND
Y~ K. Cheung, P. K. K. Lee Department of Civil and Structural Engineering University of Hong Kong, Hong Kong and K. H. Xie Geotechnical Engineering Institute Zhejiang University, P. R. China
ABSTRACT Although the consolidation of sand-drained ground is three dimensional in nature, it has often been analyzed by two dimensional procedure due to the simplicity in computation. Nevertheless, whether such an approach is adequate remains questionable. In this paper, an in-depth study has been made to clarify this question by conducting a systematic two and three dimensional consolidation analysis for the same sand-drained ground using the finite element method and making comparisons of the results. It shows that the results produced by the two dimensional analysis procedure are not entirely adequate and that the simple approach should be adopted with caution.
INTRODUCTION
In order to speed up construction, it is necessary to accelerate the rate
of consolidation and to gain additional shear strength of soft ground
and the technique of vertical sand drains combined with preloading can be used.
This has been proved to be effective by a number of successful case
records for more than half a century. Usually, the consolidation behavior of sand-drained ground is predicted by analytical theories which deal with a single drain [1-3]. is
often
demonstrated
by
However,
it
case histories that there are discrepancies 73 C o m p u ~ and GeotecOnics0266-352X/91/$03.50©1991 ElsevierScience Publishers Ltd, England. Printed in Great Britain
74 between predictions edge
of
and observations,
es~ecially
in
the loading area or the improved zone.
such discrepancies
the
vicinity
of
is that while the analytical
theories are all derived with
the assumptions that preloading is uniformly distributed and vertical are
the
One of the main reasons for
drains
installed over the entire ground, only a portion of the ground is covered
in practice.
Therefore,
distribution
of
consideration.
for a realistic
both
load
analysis
and vertical
In other words,
or
drains
design,
should
the actual
be
taken
into
the entire sand drain system and not only a
single drain should be considered.
However,
this can
be realized only by
employing numerical methods such as the finite element method. Two
procedures have been developed for the consolidation
sand-drained
ground
by
the
finite
element
method.
dimensional analysis procedure in which the circular square ones of the same
cross
section
analysis procedure where the vertical a parallel sand wall system.
area. sand
One
analysis of is
a
three
drains are replaced by
The other is a two dimensional drain
system is converted
It has been shown by Zeng et ai.(1987)
into
[4] that
the former is a more reliable method because of its reality in simulating the sand-drained ground. simplicity produces
However,
in computation
good results,
the latter is more frequently used due to its
15,6].
differences
systematic
and
between the results
two
if two
dimensional
analysis
there is no need to perform a three dimensional
It is therefore of theoretical
whether
Obviously,
dimensional
practical
interest
from the two procedures
analysis
can
be
adopted
for
one.
to investigate
the
so as to ascertain practical
use.
A
study has been conducted and the results are presented herein.
COMPUTATION SCHEDULE General Description
A specific sand-drained ground used for the analysis is shown in Fig.l. It consists
of a soft
clay
layer
of 15m
Within a strip of width Bi-35m, vertical and spacing S-3.5m
are
as the improved zone. BL-28m is applied.
installed
in
On the surface
To simulate
thick with
an
impermeable
sand drains with diameter
a square
pattern
of the ground,
the practical
and this is known
a load with
the width
process of preloading,the
is time dependent with the schedule shown in Fig.l (c).
base.
dw-5OOmm
load
75
S a n d D r a i n : d~=6OOmm
S:3.6m J
111:35m
q ! alO 4) IQ el,I) ela
aid) a le • la Olil
a io alO
alo
I
!! "1" a
• a i
ee~
• a
•
e e ~o
a
a
a 1
•
Q ~
•
a
Bl~llSm
•
I Sand
H : 5r~
e[
a
!i. .'i
|
r f s
(b)
lojq (kP.) I
Bl:3Sm
o
2o
(a)
Tim*(d*~)
(c)
Figure 1 Sand-dralned ground l a y o u t (a) Plan (b) S e c t i o n (c) Loading schedule
For the soft clay, a value of 10"6nun/s is assumed to be the vertical and horizontal coefficients of permeability kv and k h.
The elastic modulus is
taken as E-3000 kPa and the Poisson's ratio, v -0.3. For sand drains, the value of E and v are assumed to be the same as that for soft clay,
thereby neglecting the rigidity of drains.
However, well
resistance, which is the resistance encountered in the seepage flow through the sand drains and thereby delaying the consolidation of sand-dralned ground, is recognized by Barron [i] as one of the most important factors affecting consolidation of sand-drained ground.
This has been included in this study
by performing two series of computations.
Well resistance is resulted from
the finite permeability of sand drains and can be indicated by the factor of well resistance, G.
Zeng et al. [3] defined G as (kb/kw)(H/dw) 2, where H is
the distance of the vertical drainage path of the soft clay layer.
The larger
the value of G, the slower will be the consolldatlon of sand-dralned ground. If the coefficient of permeability of sand drain, I%, is infinite, then G-O
76 and there is no well resistance. drains,
while
all
others
are
In this study, drains with G-0 are ideal
non-ideal
drains.
Series I
deals
with
ideal
drains in which kw is assigned a value of 104mm/s. This i s considered to be large enough to simulate ideal drain condition with G-0. Series 2 deals with non-ideal drains in which k, is taken as lO-3mm/s corresponding to G-0.9. Based on above data, carried ground.
out
with
The
an
two and three dimensional computations have been
assumption
finite
element
that
the
load
program used
is applied
directly
is a revised v e r s i o n
on
the
of the one
developed by Zeng et al. [4] based on Blot's consolidation theory [7].
It is
applicable to elastic or nonlinear elastic consolidation analysis for either two or three dimensional problems and can be run on a microcomputer advantage of the =echnique of "Frontal Solution"
Two Dimensional
The
taking
[8].
Analysis
flow of the seepage water
in a sand-dralned ground
nature due to the presence of sand drains.
is spatial
in
In a two dimensional consolidation
analysis, however, the flow is considered laminar.
It is
therefore necessary
to establish a scheme for converting sand drain systems into sand wall systems (Fig.2)
so
dimensional
as
to
change
spatial
consolidation
analysis
flow
into
can be
a
laminar
carried
out
one for
ground.
:
Soma
'I
:
I~
bw
-
-
Figure 2
de
....
4:
raU
JF
Conversion scheme from a sand d r a i n system
to a s a n d w a l l
system
before
a
two
a sand-drained
77 Shinsha approximately proper walls
al.
et
converted
analytical and
[5] proposed
into a parallel
consolidation
the horizontal
improved zone.
that a vertical
theories
coefficient
sand drain
sand wall
system
by adjusting
of permeability
system can be
(Fig.2)
based
the spacing of
soil
on
of sand
within
the
A relationship between the parameters of two systems can be
obtained and expressed as
kh'/kh - ~(Lld.) 2
(i)
where d, is the equivalent diameter of the influence zone of sand drain (for d, - 1.128S): L is half of the distance
drains arranged in a square pattern, between
two
sand
walls;
k h'
is
the
converted
horizontal
coefficient
of
permeability of soil within the improved zone: A is a conversion coefficient and is determined by considering that the time for the two systems to achieve 50% average degree of consolidation According
to Terzaghi's
is equal.
one dimensional
consolidation
theory which
is
assumed to be suitable for sand wall system and the simplified theory for sand drain system taking consideration
of well resistance
[3], A can be obtained
as follows:
I - 2.26/(F + ~G)
in which F - in(n)
(2)
- 3/4 where n - d./d w.
It can be seen in the conversion that the sand drain system, of the permeability which
the
of the drain,
permeability
of
the
computations based on Eqs.(1) sand wall,
k.',
permeability,
should
be
otherwise,
is converted wall
is
regardless
into the sand wall system in
infinite.
Accordingly,
in
and (2), the coefficient of permeability
large the
enough
sand
wall
to simulate is
treated
all of a
the wall
of
infinite
directly
as
a
fully
permeable body. In principle, range
of B! ~
between
two
horizontal
both L and b w (Fig.2) are arbitrary dimensions within the
2L+b w ~ S.
sand
walls
direction
In the finite should
be included in the analysis, the
divided
to ensure accuracy.
smaller L means more sand walls.
extreme case,
be
element into
analysis, at
Since B x is fixed,
2
mass
elements
in
selection of a
For larger value of L, lesser elements will
and the computation will become simpler.
computation will be simplest when
This is referred to as Case A.
least
the soll
As an
L - BI/2 and b, - 0.
The other extreme is to arrange the converted
78 sand wall system such that 2L+b w - S. corresponds
This
is referred
as Case B which
to
to the most complex computation in the two dimensional
analysis.
In this case , b. can be determined by considering that the volume of sand in both
systems
are
the same
and b w - ~dw2/(4S)
if the sand
drain
system
two
cases
have
is
arranged in a square pattern. In
this
study,
the
computation
of
these
extreme
performed for both ideal drains and non-ideal drains. computation drains,
referred
been
Besides, an additional
to as Case C has also been carried
out
for non-ideal
in which the sand drain system is converted into the sand wall system
with the same permeability sand volume,
(kw'-k,-10-3mm/s),
the same spacing and the same
k h' is determined by Eqs. (I) and (2) with G-0,
for cases considered
in the computation are summarized
All parameters
in Table i.
TABLE i Parameters Used in Two Dimensional Analyses Case
Ideal drains (G-0)
A B
0 56
17.5 1.72
1.71 1.71
104
3.36xi0 -5 3.24xi0 "7
A B C
0 56 56
17.5 1.72 1.72
0.55 0.55 1.71
104 10-3
1.08x10 -5 1.04xl0 -7 3.24xi0 -7
Non-ideal drains (G-0.9) Original
The
bw(mm)
k.' (totals) kh' (totals)
Series
sand drain system:
finite
element
L(m)
d~-50Omm, S-3.5m, d,-3.95m, kh-lO'Smm/s, BI-35m, H-15m
idealization
adopted
consolidation analysis of the sand-drained'ground Fig.3(a)
and Fig.3(b)
respectively. the
load
restrained fixed.
are
the
Due to symmetry,
is considered against
idealization
the
two
in Fig.l is shown in Fig.3.
for Case A a n d for Cases B,
The
in y-direction
C
two vertical
boundaries
and the bottom boundary
are is
In Case A, drainage is free at ground surface (z-0) and along the line
y-L because b,-0. In Cases B and C, drainage
is free at ground surface only.
The finite element mesh for Case A consists of 350 elements and
dimensional
only the right half from the center llne of
in computation.
displacement
in
n-7.9
for Cases B and C, 410 elements and 462 nodes.
and 396 nodes,
79
Free Boundary (Permeable)
o
__Z_Y
, , i J
illl
,Jill
IIII iilli iilli illll
(a)
,1111
IIIIIIIIIIIIII
IIIIIIIIIIIIII
IIIIIIIIIIIIII IIIIIIIIIIIIII IIIIIIIII11111 Illlllllllllll
IIII IIIIIIIIIIII IIII IIIIIIIIIIIIr
Fixed Bounda
15
lmer
m
eable
60 m
Z
Free Boundary (Permeable)
o
Y
[llllllilliiilliillil
illllllillllllillllll
(b)
IIIIIIIIIIIIIIIIIIIII IIIlillllllilllilllil Illililllllililllilll Illilillilllillllllll
15
m
IIItlllllllllllllll II IIIIIIIIIIIIIIII
Fixed Boundar ly_l!m_~ermeablel 60 m
Figure 3 Finite element idealization in two dimensional analysis (a) Case A (Mesh:36X10) (b) Cases B and C (Mesh:41xl0)
Three D~mensional Analysis
For sand-drained ground under strip load shown in Fig.l, no deformation is expected to occur in the x-direction and there will be no seepage through
8O the vertical symmetry.
planes
However,
spatial
and
normal
co x axis
at x-&mS/2
(m-O,l,2 .... ) because
between any pair of these adjacent planes,
similar.
In
the
three
dimensional
seepage
consolidation
of is
analysis,
therefore, only a cube such as that shown in Fig.4 needs to be considered. In
order
that
the
sand
drains
can
be
idealized
into
the
compatible with the 8-node cubic elements used in the program,
elements
the circular
cross section of each sand drain is converted into a square one of the same area with a side length D, - 0.8862dw. The
finite element
idealization
for the sand-dralned
ground
in three
dimensional analyses shown in Fig.4 consists of 1230 elements and 1848 nodes. The two boundaries at x-0 and x-S/2-1.75m are assumed to be impermeable and fixed
in the
x-direction
but
remaining four boundaries~
free
in the
other
two directions.
For
the
the conditions are the same as for Cases B and C
in the two dimensional analysis. Due to plane strain, all nodes are fixed in the x-direction.
(Mesh
il llll
: 3 x 41 x I0) -'W----'--------
|I|| |III |III fill
4.75 -___=--
i11111111111111111
illiilUllllllllil IIIIlIliiliUllUi lUllUliilIIIlill Iii111111111111111 1111111111119111
Y
i L5 m 5't ~r :dl d ra':n~
IIII IIII IIII IIIII
IIIIIIIIilIIIIIUl IIIIIIiilllUlIIi
i !
FIGURE 4
m
i
I
60 m
Finite element idealization in three dimensional analysis
RESULTS
AND COMPARISONS
The m a i n results of the computations described above are shown in Fig.5
81 to Fig.12
in w h i c h Figs.5,
7, 9 and Ii are for ideal drains,
and Figs.6,
i0 and 12 for n o n - l d e a l drains.
120
Pore Water Pressure (kFa) Ideal Drains •
tO0
(x=O.OOm)
3D
3D (x=O.95m) ......... 3D (x=l.75m)
80
A ;
.......Case A
6 0 ~.. A'"~ 40 20 0
.
.
.
.
0
.
5
FIGURE 5
i
A i-)~ % -. (zL
Pore
..,
•
.............
~5
,o
. z= -
2o
25
3o
3~
40
Y (m) of pore w a t e r pressure
(Ideal Drains)
(kPa)
Water Pressure
Non-Ideal Drains •
100f
_
.
Distribution
120 /
Case B z=Sm t=5Odays
!xi
3D
(x=O.OOm)
3D (x=O.95m) ......... 3D (x=t.75m)
~.. :
:.
¢'/7~I-. • ~ . " e .
11 1t
,o; V 0 o
-:
,
;,
~ase t:
~f%. /i
l;
I
5
1o
A
............
. . . . . CaseC
.=°o,=°o,.,.
I :
15
~G'"
I
20
i 25
L 30
i 35
40
Y Cm) FIGURE 6
D i s t r i b u t i o n of pore w a t e r pressure
(Non-Ideal Drains)
8,
82 Figs.5 and 6 are the distribution of pore water pressure of 5m and t-50days.
at the depth
As can be seen from the figures, both the magnitude and
the distribution of pore water pressure within the improved zone (ysl7.bm) are quite different among various cases.
In Case A, pore water pressure decreases
continuously with the increase of the distance from the center of the load and becomes zero at y-17.bm where the converted sand wall is located. cases,
In other
the distribution of pore water pressures is in a wave form declining
along the y-direction.
The crests of the waves are located at the middle of
two adjacent sand walls or drains and the troughs at the center of sand walls or drains.
In Case B, for both ideal drains and non-ldeal drains,
the pore
water pressure drops sharply to zero at the sides of sand walls because the converted sand walls are of infinite permeability.
However,
this phenomenon
does not occur in Case C for non-ideal drains, because the permeability of the converted sand wall is finite. In three dimensional x-direction
and passing
(3D) analyses, through
on
vertical planes normal to the
sand drains
such as x-Om,
the pore water
pressure at the periphery of each drain dissipates to zero for ideal drains but does not dissipate fully for non-ldeal drains due to the effect of well resistance.
On
other
planes
away
from
sand
drains
such
as
x-0.95m
and
x-l.75m, the pore water pressures are greater than those on the plane x-Om and increase with the increase of distance from the sand drain indicating clearly the characteristics of spatial seepage in sand-drained ground.
140
Pore Water Pressure (kPa) Ideal Drains .........Case B x=On~ z--bm
Case A
- -
120
•
3D
100 #
.,
_y=Om
80 60 40
2o I
I
I
I
;
i
I
20
40
80
oo
,oo
12o
[40
18o
Time (days) FIGURE
7
Variation of pore water pressure with time (Ideal Drains)
83 Pore Water Pressure (kPa)
140
Non-Ideal Drains ........ Case B .......... Case C x=Om z=Szn
Case A
120 iO0
,."
" 3D
i
_y=Om
""-..o......
W M : .......................
80
...." ' " , L
60 40 20
-"y---14m "r"
0 0
2o
40
60
80
Time FIGURE 8
loo
12o
14o
leo
(days)
Variation of pore water pressure with t i m e (Non-Ideal Drains)
Figs.7 and 8 show the variation of pore water pressure with time at two points located ac 5m below ground surface at the center (y-0) and the edge (y-14m)
of the applied load.
It can be seen from these figures
differences of pore water pressures among various cases are
significant.
Settlement (cm) -I0, ...
/~
I00
I
3D(x=Om)
z=o,,,~/l
I
...... ~'--o '
/
'
............_ c , " "
/
.
t=5Odays
•
200
300 0
I
I
5
~o
,
I
I
I
I
I
,5
20
25
30
35
Y (m) FIGURE 9
Settlement distribution
that the
curves ( I d e a l Drains)
40
84
S e t t l e m e n t (cm) -10(
0
5
tO
15
20
25
30
35
40
v (m) FIGURE 1 0
Settlement
distribution
Figs.9 and iO display
curves
(Non-Ideal
the distribution of settlement
Drains)
at t-50 days on
ground surface and at the depth z-5m. It can be seen that the differences of settlements among various cases are still distinct, but somewhat smaller than those of the pore water pressure.
The settlements in Case B for ideal drains
and Case C for non-ideal drains agree well with 3D results.The development of
S e t t l e m e n t (ram) (ram) Settlement ldeal -
Case A
Drains
....... C a s e B
'
3D
I00
200
300
400 0
I
t
i
I
I
I
20
40
60
80
100
I20
,
I
140
160
Time (days} FIGURE ii
Development of settlement with time (Ideal Drains)
85
o Settlement (mm)
200
"" "
'
.
-
~
i
300 ]
Non-ldeal Drains Case A
........Case . B
..........Case . C
"
3D
x=z=0rn
400 0
i
i
~
i
i
i
i
20
40
60
80
I00
120
140
160
Time (days) FIGURE 12
Development of settlement with time (Non-Ideal Drains)
settlement with time at the center point and at the edge point of the applied load are shown in Figs, Ii and 12. In general,
results
from two dimensional
obtained by three dimensional analysis.
analysis
differ
from those
The difference of pore water pressure
is more significant not only in magnitude but also in distribution. It can also be seen that the pore water pressure for ideal drains is smaller
than
settlements.
that
for non-ideal
drains but
the reverse
is
true
for
the
This indicates that well resistance delays the consolidation
process of sand-drained ground,
DISCUSSIONS
As shown in comparisons above, the difference between the results of two and three dimensional elastic analyses for the consolidation of sand-dralned ground is significant, in particular for the pore water pressure. The reasons may he summarized as follows: Firstly,
the
purpose
of
conducting
numerical
analysis
consolidation of sand-dralned ground, as previously suggested,
for
the
is to obtain
more accurately the loading response of the ground than those predicted by analytical
solution.
This
is realized
by
taking
into account
the actual
distributions of load and sand drains and also the interaction of each sand
86 drain.
In the two dimensional analysis, however, the actual sand drain system
is converted
into the sand wall system based on analytical
no consideration
of the distributions
Also only one drain is dealt with. dimensional conducting
analysis
in itself
a numerical
solution in which
of both load and sand drains is taken.
This implies that such a procedure for two
is in contrast
analysis.
Secondly,
to the original
intention
even for a single
drain,
of
it is
still impossible to convert this into a completely equivalent sand wall system because the boundary conditions are quite different for the two systems. fact,
the conversion
average
degree
utilizing
Eqs.(1)
of consolidation
and
(2) ensures
that only when
of the sand drain system
reaches
these two systems reach the same average degree of consolidation, average pore water pressure water pressure,
and settlement.
50%
In the can
or the same
As to other items such as pore
or to all the items at a different time, nothing can be made
to ensure that they are equal in the two systems. it is the pore water pressure, the two analyses.
Besides,
This is also the reason why
and not the settlement,
which differs more in
the choice of the conversion parameters
L and b, adopted in two dimensional
analyses
is arbitrary.
such as
Different choices
may lead to different results as can be seen from those in Case A and Case B. In
conclusion,
dimensional
the
analysis
limitations may
behavior of sand-drained In contrast
impose
on
mathematical
restrictions
simulation
in adequate
in
the
modelling
two
of the
ground.
to two dimensional
analysis,
the important characteristics
of sand-drained ground such as spatial seepage and well resistance can easily be taken
into consideration
in the three
dimensional
analysis
without
any
artificial conversion and the sand-drained ground can be analyzed as naturally as it is. makes
This forms a sound theoretical
it possible
to produce
basis for the procedure
more reliable
results,
and thus
as demonstrated
in the
study reported.
CONCLUSIONS
The following conclusions (I)
The
results
obtained
may he drawn from this study: by
the
two
dimensional
procedure
so
far
developed for the consolidation analysis of sand-drained ground are apparently affected by the choice of conversion parameters obtained
by the
available,
the
three dimensional three
dimensional
analysis. procedure
and different Until should
further be
from the ones improvement
adopted
for
is the
consolidation analysis of sand-drained ground to achieve more reliable results.
87 (2) When
only
a
two
dimensional
consolidation
analysis
program
is
available and that the consideration of load distribution is necessary, the conversion adopted in Case B may be used for ideal drains, while the one in Case C for non-ideal
drains.
However,
the results thus obtained and in
particular the pore water pressures may be doubtful and should be used with caution.
ACKNOWLEDGEMENT The
financial
support
from
the Haklng Wong
Research
Foundation
gratefully acknowledged.
REFERENCES I. Barton, R. A., Consolidation of fine grained soils by drain wells. Trans. Am. Soc.Civ. En2rs, 113 (1948) 718-742. 2. Hansbo, S., Consolidation of fine-grained soils by prefabricated drains. ~oc. 10th ICSMFE, ~ (1981) 677-682. 3. Zeng, G. X. and Xie, K. H., New development of the vertical drain ¢heories. Proc. 12th ICSMFE, ~ (1989) 1435-1438. 4. Zeng, G. X., Xie, K. H. and Shi, Z. Y., Consolidation analysis of sand-drained ground by FEM. p~oc. 8th ARCSMFE, ~ (1987) 139-142. 5. Shinsha, H., Hara, H., Abe, T. and Tanaka, A., Consolidation settlement and lateral displacement of soft ground improved by sand-dralns. Tsuchi-to-Kiso. JSSMF~, 30, No.2 (1982) 7-12 (in Japanese). 6. Kumamoto, N. Sumioka, N., Moriwaki, T. and Yoshikuni, H., Settlement behavior of improved ground with a vertical drain system. Soils and Foundations. J$$MFE, 28, NoTI (1988) 77-88. 7. Biot, M. A., General theory of three dimensional consolidation. J. A w l . Phys., 12 (1941) 155-164. 8. Cheung, Y. K. and Yeo, M. F., A Practical Introduction to Finite Element Analys~s. Pitman International Text (1979).
Received 22 July 1991; revised version received 24 September 1991; accepted 25 September 1991
is
TECHNICAL NOTE
SOME REMARKS ON TWO AND THREE DIMENSIONAL CONSOLIDATION ANALYSIS OF SAND-DRAINED GROUND
Y~ K. Cheung, P. K. K. Lee Department of Civil and Structural Engineering University of Hong Kong, Hong Kong and K. H. Xie Geotechnical Engineering Institute Zhejiang University, P. R. China
ABSTRACT Although the consolidation of sand-drained ground is three dimensional in nature, it has often been analyzed by two dimensional procedure due to the simplicity in computation. Nevertheless, whether such an approach is adequate remains questionable. In this paper, an in-depth study has been made to clarify this question by conducting a systematic two and three dimensional consolidation analysis for the same sand-drained ground using the finite element method and making comparisons of the results. It shows that the results produced by the two dimensional analysis procedure are not entirely adequate and that the simple approach should be adopted with caution.
INTRODUCTION
In order to speed up construction, it is necessary to accelerate the rate
of consolidation and to gain additional shear strength of soft ground
and the technique of vertical sand drains combined with preloading can be used.
This has been proved to be effective by a number of successful case
records for more than half a century. Usually, the consolidation behavior of sand-drained ground is predicted by analytical theories which deal with a single drain [1-3]. is
often
demonstrated
by
However,
it
case histories that there are discrepancies 73 C o m p u ~ and GeotecOnics0266-352X/91/$03.50©1991 ElsevierScience Publishers Ltd, England. Printed in Great Britain
74 between predictions edge
of
and observations,
es~ecially
in
the loading area or the improved zone.
such discrepancies
the
vicinity
of
is that while the analytical
theories are all derived with
the assumptions that preloading is uniformly distributed and vertical are
the
One of the main reasons for
drains
installed over the entire ground, only a portion of the ground is covered
in practice.
Therefore,
distribution
of
consideration.
for a realistic
both
load
analysis
and vertical
In other words,
or
drains
design,
should
the actual
be
taken
into
the entire sand drain system and not only a
single drain should be considered.
However,
this can
be realized only by
employing numerical methods such as the finite element method. Two
procedures have been developed for the consolidation
sand-drained
ground
by
the
finite
element
method.
dimensional analysis procedure in which the circular square ones of the same
cross
section
analysis procedure where the vertical a parallel sand wall system.
area. sand
One
analysis of is
a
three
drains are replaced by
The other is a two dimensional drain
system is converted
It has been shown by Zeng et ai.(1987)
into
[4] that
the former is a more reliable method because of its reality in simulating the sand-drained ground. simplicity produces
However,
in computation
good results,
the latter is more frequently used due to its
15,6].
differences
systematic
and
between the results
two
if two
dimensional
analysis
there is no need to perform a three dimensional
It is therefore of theoretical
whether
Obviously,
dimensional
practical
interest
from the two procedures
analysis
can
be
adopted
for
one.
to investigate
the
so as to ascertain practical
use.
A
study has been conducted and the results are presented herein.
COMPUTATION SCHEDULE General Description
A specific sand-drained ground used for the analysis is shown in Fig.l. It consists
of a soft
clay
layer
of 15m
Within a strip of width Bi-35m, vertical and spacing S-3.5m
are
as the improved zone. BL-28m is applied.
installed
in
On the surface
To simulate
thick with
an
impermeable
sand drains with diameter
a square
pattern
of the ground,
the practical
and this is known
a load with
the width
process of preloading,the
is time dependent with the schedule shown in Fig.l (c).
base.
dw-5OOmm
load
75
S a n d D r a i n : d~=6OOmm
S:3.6m J
111:35m
q ! alO 4) IQ el,I) ela
aid) a le • la Olil
a io alO
alo
I
!! "1" a
• a i
ee~
• a
•
e e ~o
a
a
a 1
•
Q ~
•
a
Bl~llSm
•
I Sand
H : 5r~
e[
a
!i. .'i
|
r f s
(b)
lojq (kP.) I
Bl:3Sm
o
2o
(a)
Tim*(d*~)
(c)
Figure 1 Sand-dralned ground l a y o u t (a) Plan (b) S e c t i o n (c) Loading schedule
For the soft clay, a value of 10"6nun/s is assumed to be the vertical and horizontal coefficients of permeability kv and k h.
The elastic modulus is
taken as E-3000 kPa and the Poisson's ratio, v -0.3. For sand drains, the value of E and v are assumed to be the same as that for soft clay,
thereby neglecting the rigidity of drains.
However, well
resistance, which is the resistance encountered in the seepage flow through the sand drains and thereby delaying the consolidation of sand-dralned ground, is recognized by Barron [i] as one of the most important factors affecting consolidation of sand-drained ground.
This has been included in this study
by performing two series of computations.
Well resistance is resulted from
the finite permeability of sand drains and can be indicated by the factor of well resistance, G.
Zeng et al. [3] defined G as (kb/kw)(H/dw) 2, where H is
the distance of the vertical drainage path of the soft clay layer.
The larger
the value of G, the slower will be the consolldatlon of sand-dralned ground. If the coefficient of permeability of sand drain, I%, is infinite, then G-O
76 and there is no well resistance. drains,
while
all
others
are
In this study, drains with G-0 are ideal
non-ideal
drains.
Series I
deals
with
ideal
drains in which kw is assigned a value of 104mm/s. This i s considered to be large enough to simulate ideal drain condition with G-0. Series 2 deals with non-ideal drains in which k, is taken as lO-3mm/s corresponding to G-0.9. Based on above data, carried ground.
out
with
The
an
two and three dimensional computations have been
assumption
finite
element
that
the
load
program used
is applied
directly
is a revised v e r s i o n
on
the
of the one
developed by Zeng et al. [4] based on Blot's consolidation theory [7].
It is
applicable to elastic or nonlinear elastic consolidation analysis for either two or three dimensional problems and can be run on a microcomputer advantage of the =echnique of "Frontal Solution"
Two Dimensional
The
taking
[8].
Analysis
flow of the seepage water
in a sand-dralned ground
nature due to the presence of sand drains.
is spatial
in
In a two dimensional consolidation
analysis, however, the flow is considered laminar.
It is
therefore necessary
to establish a scheme for converting sand drain systems into sand wall systems (Fig.2)
so
dimensional
as
to
change
spatial
consolidation
analysis
flow
into
can be
a
laminar
carried
out
one for
ground.
:
Soma
'I
:
I~
bw
-
-
Figure 2
de
....
4:
raU
JF
Conversion scheme from a sand d r a i n system
to a s a n d w a l l
system
before
a
two
a sand-drained
77 Shinsha approximately proper walls
al.
et
converted
analytical and
[5] proposed
into a parallel
consolidation
the horizontal
improved zone.
that a vertical
theories
coefficient
sand drain
sand wall
system
by adjusting
of permeability
system can be
(Fig.2)
based
the spacing of
soil
on
of sand
within
the
A relationship between the parameters of two systems can be
obtained and expressed as
kh'/kh - ~(Lld.) 2
(i)
where d, is the equivalent diameter of the influence zone of sand drain (for d, - 1.128S): L is half of the distance
drains arranged in a square pattern, between
two
sand
walls;
k h'
is
the
converted
horizontal
coefficient
of
permeability of soil within the improved zone: A is a conversion coefficient and is determined by considering that the time for the two systems to achieve 50% average degree of consolidation According
to Terzaghi's
is equal.
one dimensional
consolidation
theory which
is
assumed to be suitable for sand wall system and the simplified theory for sand drain system taking consideration
of well resistance
[3], A can be obtained
as follows:
I - 2.26/(F + ~G)
in which F - in(n)
(2)
- 3/4 where n - d./d w.
It can be seen in the conversion that the sand drain system, of the permeability which
the
of the drain,
permeability
of
the
computations based on Eqs.(1) sand wall,
k.',
permeability,
should
be
otherwise,
is converted wall
is
regardless
into the sand wall system in
infinite.
Accordingly,
in
and (2), the coefficient of permeability
large the
enough
sand
wall
to simulate is
treated
all of a
the wall
of
infinite
directly
as
a
fully
permeable body. In principle, range
of B! ~
between
two
horizontal
both L and b w (Fig.2) are arbitrary dimensions within the
2L+b w ~ S.
sand
walls
direction
In the finite should
be included in the analysis, the
divided
to ensure accuracy.
smaller L means more sand walls.
extreme case,
be
element into
analysis, at
Since B x is fixed,
2
mass
elements
in
selection of a
For larger value of L, lesser elements will
and the computation will become simpler.
computation will be simplest when
This is referred to as Case A.
least
the soll
As an
L - BI/2 and b, - 0.
The other extreme is to arrange the converted
78 sand wall system such that 2L+b w - S. corresponds
This
is referred
as Case B which
to
to the most complex computation in the two dimensional
analysis.
In this case , b. can be determined by considering that the volume of sand in both
systems
are
the same
and b w - ~dw2/(4S)
if the sand
drain
system
two
cases
have
is
arranged in a square pattern. In
this
study,
the
computation
of
these
extreme
performed for both ideal drains and non-ideal drains. computation drains,
referred
been
Besides, an additional
to as Case C has also been carried
out
for non-ideal
in which the sand drain system is converted into the sand wall system
with the same permeability sand volume,
(kw'-k,-10-3mm/s),
the same spacing and the same
k h' is determined by Eqs. (I) and (2) with G-0,
for cases considered
in the computation are summarized
All parameters
in Table i.
TABLE i Parameters Used in Two Dimensional Analyses Case
Ideal drains (G-0)
A B
0 56
17.5 1.72
1.71 1.71
104
3.36xi0 -5 3.24xi0 "7
A B C
0 56 56
17.5 1.72 1.72
0.55 0.55 1.71
104 10-3
1.08x10 -5 1.04xl0 -7 3.24xi0 -7
Non-ideal drains (G-0.9) Original
The
bw(mm)
k.' (totals) kh' (totals)
Series
sand drain system:
finite
element
L(m)
d~-50Omm, S-3.5m, d,-3.95m, kh-lO'Smm/s, BI-35m, H-15m
idealization
adopted
consolidation analysis of the sand-drained'ground Fig.3(a)
and Fig.3(b)
respectively. the
load
restrained fixed.
are
the
Due to symmetry,
is considered against
idealization
the
two
in Fig.l is shown in Fig.3.
for Case A a n d for Cases B,
The
in y-direction
C
two vertical
boundaries
and the bottom boundary
are is
In Case A, drainage is free at ground surface (z-0) and along the line
y-L because b,-0. In Cases B and C, drainage
is free at ground surface only.
The finite element mesh for Case A consists of 350 elements and
dimensional
only the right half from the center llne of
in computation.
displacement
in
n-7.9
for Cases B and C, 410 elements and 462 nodes.
and 396 nodes,
79
Free Boundary (Permeable)
o
__Z_Y
, , i J
illl
,Jill
IIII iilli iilli illll
(a)
,1111
IIIIIIIIIIIIII
IIIIIIIIIIIIII
IIIIIIIIIIIIII IIIIIIIIIIIIII IIIIIIIII11111 Illlllllllllll
IIII IIIIIIIIIIII IIII IIIIIIIIIIIIr
Fixed Bounda
15
lmer
m
eable
60 m
Z
Free Boundary (Permeable)
o
Y
[llllllilliiilliillil
illllllillllllillllll
(b)
IIIIIIIIIIIIIIIIIIIII IIIlillllllilllilllil Illililllllililllilll Illilillilllillllllll
15
m
IIItlllllllllllllll II IIIIIIIIIIIIIIII
Fixed Boundar ly_l!m_~ermeablel 60 m
Figure 3 Finite element idealization in two dimensional analysis (a) Case A (Mesh:36X10) (b) Cases B and C (Mesh:41xl0)
Three D~mensional Analysis
For sand-drained ground under strip load shown in Fig.l, no deformation is expected to occur in the x-direction and there will be no seepage through
8O the vertical symmetry.
planes
However,
spatial
and
normal
co x axis
at x-&mS/2
(m-O,l,2 .... ) because
between any pair of these adjacent planes,
similar.
In
the
three
dimensional
seepage
consolidation
of is
analysis,
therefore, only a cube such as that shown in Fig.4 needs to be considered. In
order
that
the
sand
drains
can
be
idealized
into
the
compatible with the 8-node cubic elements used in the program,
elements
the circular
cross section of each sand drain is converted into a square one of the same area with a side length D, - 0.8862dw. The
finite element
idealization
for the sand-dralned
ground
in three
dimensional analyses shown in Fig.4 consists of 1230 elements and 1848 nodes. The two boundaries at x-0 and x-S/2-1.75m are assumed to be impermeable and fixed
in the
x-direction
but
remaining four boundaries~
free
in the
other
two directions.
For
the
the conditions are the same as for Cases B and C
in the two dimensional analysis. Due to plane strain, all nodes are fixed in the x-direction.
(Mesh
il llll
: 3 x 41 x I0) -'W----'--------
|I|| |III |III fill
4.75 -___=--
i11111111111111111
illiilUllllllllil IIIIlIliiliUllUi lUllUliilIIIlill Iii111111111111111 1111111111119111
Y
i L5 m 5't ~r :dl d ra':n~
IIII IIII IIII IIIII
IIIIIIIIilIIIIIUl IIIIIIiilllUlIIi
i !
FIGURE 4
m
i
I
60 m
Finite element idealization in three dimensional analysis
RESULTS
AND COMPARISONS
The m a i n results of the computations described above are shown in Fig.5
81 to Fig.12
in w h i c h Figs.5,
7, 9 and Ii are for ideal drains,
and Figs.6,
i0 and 12 for n o n - l d e a l drains.
120
Pore Water Pressure (kFa) Ideal Drains •
tO0
(x=O.OOm)
3D
3D (x=O.95m) ......... 3D (x=l.75m)
80
A ;
.......Case A
6 0 ~.. A'"~ 40 20 0
.
.
.
.
0
.
5
FIGURE 5
i
A i-)~ % -. (zL
Pore
..,
•
.............
~5
,o
. z= -
2o
25
3o
3~
40
Y (m) of pore w a t e r pressure
(Ideal Drains)
(kPa)
Water Pressure
Non-Ideal Drains •
100f
_
.
Distribution
120 /
Case B z=Sm t=5Odays
!xi
3D
(x=O.OOm)
3D (x=O.95m) ......... 3D (x=t.75m)
~.. :
:.
¢'/7~I-. • ~ . " e .
11 1t
,o; V 0 o
-:
,
;,
~ase t:
~f%. /i
l;
I
5
1o
A
............
. . . . . CaseC
.=°o,=°o,.,.
I :
15
~G'"
I
20
i 25
L 30
i 35
40
Y Cm) FIGURE 6
D i s t r i b u t i o n of pore w a t e r pressure
(Non-Ideal Drains)
8,
82 Figs.5 and 6 are the distribution of pore water pressure of 5m and t-50days.
at the depth
As can be seen from the figures, both the magnitude and
the distribution of pore water pressure within the improved zone (ysl7.bm) are quite different among various cases.
In Case A, pore water pressure decreases
continuously with the increase of the distance from the center of the load and becomes zero at y-17.bm where the converted sand wall is located. cases,
In other
the distribution of pore water pressures is in a wave form declining
along the y-direction.
The crests of the waves are located at the middle of
two adjacent sand walls or drains and the troughs at the center of sand walls or drains.
In Case B, for both ideal drains and non-ldeal drains,
the pore
water pressure drops sharply to zero at the sides of sand walls because the converted sand walls are of infinite permeability.
However,
this phenomenon
does not occur in Case C for non-ideal drains, because the permeability of the converted sand wall is finite. In three dimensional x-direction
and passing
(3D) analyses, through
on
vertical planes normal to the
sand drains
such as x-Om,
the pore water
pressure at the periphery of each drain dissipates to zero for ideal drains but does not dissipate fully for non-ldeal drains due to the effect of well resistance.
On
other
planes
away
from
sand
drains
such
as
x-0.95m
and
x-l.75m, the pore water pressures are greater than those on the plane x-Om and increase with the increase of distance from the sand drain indicating clearly the characteristics of spatial seepage in sand-drained ground.
140
Pore Water Pressure (kPa) Ideal Drains .........Case B x=On~ z--bm
Case A
- -
120
•
3D
100 #
.,
_y=Om
80 60 40
2o I
I
I
I
;
i
I
20
40
80
oo
,oo
12o
[40
18o
Time (days) FIGURE
7
Variation of pore water pressure with time (Ideal Drains)
83 Pore Water Pressure (kPa)
140
Non-Ideal Drains ........ Case B .......... Case C x=Om z=Szn
Case A
120 iO0
,."
" 3D
i
_y=Om
""-..o......
W M : .......................
80
...." ' " , L
60 40 20
-"y---14m "r"
0 0
2o
40
60
80
Time FIGURE 8
loo
12o
14o
leo
(days)
Variation of pore water pressure with t i m e (Non-Ideal Drains)
Figs.7 and 8 show the variation of pore water pressure with time at two points located ac 5m below ground surface at the center (y-0) and the edge (y-14m)
of the applied load.
It can be seen from these figures
differences of pore water pressures among various cases are
significant.
Settlement (cm) -I0, ...
/~
I00
I
3D(x=Om)
z=o,,,~/l
I
...... ~'--o '
/
'
............_ c , " "
/
.
t=5Odays
•
200
300 0
I
I
5
~o
,
I
I
I
I
I
,5
20
25
30
35
Y (m) FIGURE 9
Settlement distribution
that the
curves ( I d e a l Drains)
40
84
S e t t l e m e n t (cm) -10(
0
5
tO
15
20
25
30
35
40
v (m) FIGURE 1 0
Settlement
distribution
Figs.9 and iO display
curves
(Non-Ideal
the distribution of settlement
Drains)
at t-50 days on
ground surface and at the depth z-5m. It can be seen that the differences of settlements among various cases are still distinct, but somewhat smaller than those of the pore water pressure.
The settlements in Case B for ideal drains
and Case C for non-ideal drains agree well with 3D results.The development of
S e t t l e m e n t (ram) (ram) Settlement ldeal -
Case A
Drains
....... C a s e B
'
3D
I00
200
300
400 0
I
t
i
I
I
I
20
40
60
80
100
I20
,
I
140
160
Time (days} FIGURE ii
Development of settlement with time (Ideal Drains)
85
o Settlement (mm)
200
"" "
'
.
-
~
i
300 ]
Non-ldeal Drains Case A
........Case . B
..........Case . C
"
3D
x=z=0rn
400 0
i
i
~
i
i
i
i
20
40
60
80
I00
120
140
160
Time (days) FIGURE 12
Development of settlement with time (Non-Ideal Drains)
settlement with time at the center point and at the edge point of the applied load are shown in Figs, Ii and 12. In general,
results
from two dimensional
obtained by three dimensional analysis.
analysis
differ
from those
The difference of pore water pressure
is more significant not only in magnitude but also in distribution. It can also be seen that the pore water pressure for ideal drains is smaller
than
settlements.
that
for non-ideal
drains but
the reverse
is
true
for
the
This indicates that well resistance delays the consolidation
process of sand-drained ground,
DISCUSSIONS
As shown in comparisons above, the difference between the results of two and three dimensional elastic analyses for the consolidation of sand-dralned ground is significant, in particular for the pore water pressure. The reasons may he summarized as follows: Firstly,
the
purpose
of
conducting
numerical
analysis
consolidation of sand-dralned ground, as previously suggested,
for
the
is to obtain
more accurately the loading response of the ground than those predicted by analytical
solution.
This
is realized
by
taking
into account
the actual
distributions of load and sand drains and also the interaction of each sand
86 drain.
In the two dimensional analysis, however, the actual sand drain system
is converted
into the sand wall system based on analytical
no consideration
of the distributions
Also only one drain is dealt with. dimensional conducting
analysis
in itself
a numerical
solution in which
of both load and sand drains is taken.
This implies that such a procedure for two
is in contrast
analysis.
Secondly,
to the original
intention
even for a single
drain,
of
it is
still impossible to convert this into a completely equivalent sand wall system because the boundary conditions are quite different for the two systems. fact,
the conversion
average
degree
utilizing
Eqs.(1)
of consolidation
and
(2) ensures
that only when
of the sand drain system
reaches
these two systems reach the same average degree of consolidation, average pore water pressure water pressure,
and settlement.
50%
In the can
or the same
As to other items such as pore
or to all the items at a different time, nothing can be made
to ensure that they are equal in the two systems. it is the pore water pressure, the two analyses.
Besides,
This is also the reason why
and not the settlement,
which differs more in
the choice of the conversion parameters
L and b, adopted in two dimensional
analyses
is arbitrary.
such as
Different choices
may lead to different results as can be seen from those in Case A and Case B. In
conclusion,
dimensional
the
analysis
limitations may
behavior of sand-drained In contrast
impose
on
mathematical
restrictions
simulation
in adequate
in
the
modelling
two
of the
ground.
to two dimensional
analysis,
the important characteristics
of sand-drained ground such as spatial seepage and well resistance can easily be taken
into consideration
in the three
dimensional
analysis
without
any
artificial conversion and the sand-drained ground can be analyzed as naturally as it is. makes
This forms a sound theoretical
it possible
to produce
basis for the procedure
more reliable
results,
and thus
as demonstrated
in the
study reported.
CONCLUSIONS
The following conclusions (I)
The
results
obtained
may he drawn from this study: by
the
two
dimensional
procedure
so
far
developed for the consolidation analysis of sand-drained ground are apparently affected by the choice of conversion parameters obtained
by the
available,
the
three dimensional three
dimensional
analysis. procedure
and different Until should
further be
from the ones improvement
adopted
for
is the
consolidation analysis of sand-drained ground to achieve more reliable results.
87 (2) When
only
a
two
dimensional
consolidation
analysis
program
is
available and that the consideration of load distribution is necessary, the conversion adopted in Case B may be used for ideal drains, while the one in Case C for non-ideal
drains.
However,
the results thus obtained and in
particular the pore water pressures may be doubtful and should be used with caution.
ACKNOWLEDGEMENT The
financial
support
from
the Haklng Wong
Research
Foundation
gratefully acknowledged.
REFERENCES I. Barton, R. A., Consolidation of fine grained soils by drain wells. Trans. Am. Soc.Civ. En2rs, 113 (1948) 718-742. 2. Hansbo, S., Consolidation of fine-grained soils by prefabricated drains. ~oc. 10th ICSMFE, ~ (1981) 677-682. 3. Zeng, G. X. and Xie, K. H., New development of the vertical drain ¢heories. Proc. 12th ICSMFE, ~ (1989) 1435-1438. 4. Zeng, G. X., Xie, K. H. and Shi, Z. Y., Consolidation analysis of sand-drained ground by FEM. p~oc. 8th ARCSMFE, ~ (1987) 139-142. 5. Shinsha, H., Hara, H., Abe, T. and Tanaka, A., Consolidation settlement and lateral displacement of soft ground improved by sand-dralns. Tsuchi-to-Kiso. JSSMF~, 30, No.2 (1982) 7-12 (in Japanese). 6. Kumamoto, N. Sumioka, N., Moriwaki, T. and Yoshikuni, H., Settlement behavior of improved ground with a vertical drain system. Soils and Foundations. J$$MFE, 28, NoTI (1988) 77-88. 7. Biot, M. A., General theory of three dimensional consolidation. J. A w l . Phys., 12 (1941) 155-164. 8. Cheung, Y. K. and Yeo, M. F., A Practical Introduction to Finite Element Analys~s. Pitman International Text (1979).
Received 22 July 1991; revised version received 24 September 1991; accepted 25 September 1991
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