vertical drain consolidation using stone columns an analytical solution with an impeded drainage...
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Technical note
Vertical-drain consolidation using stone columns: An analytical
solution with an impeded drainage boundary under multi-ramp
loading
G.H. Lei a , *, C.W. Fu a, C.W.W. Ng b
a Key Laboratory of Geomechanics and Embankment Engineering of the Ministry of Education, Geotechnical Research Institute, Hohai University, 1 Xikang
Road, Nanjing 210098, Chinab Department of Civil and Environmental Engineering, The Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong, China
a r t i c l e i n f o
Article history:
Received 8 December 2014
Received in revised form
22 June 2015
Accepted 3 July 2015
Available online xxx
Keywords:
Consolidation
Pore pressures
Ground improvement
Embankments
a b s t r a c t
An analytical solution is derived to predict consolidation with vertical drains under impeded drainage
boundary conditions and multi-ramp surcharge loading. The impeded drainage is modelled by adopting
the third type boundary condition with a dimensionless characteristic factor of drainage ef ciency
developed by Gray (1945) for one-dimensional consolidation. Fully drained and undrained boundary
conditions can also be modelled by applying an innite and a zero characteristic factor, respectively. The
combined effects of drain resistance and smear are taken into account fully. An explicit, rigorous
analytical solution is derived using the method of separation of variables to calculate excess pore-water
pressure at any arbitrary point in soil and to derive the overall average degree of consolidation. The
proposed solution can also be used to analyse one-dimensional consolidation without vertical drains but
with an impeded drainage boundary. Its validity and accuracy are veried by comparing the proposed
solution with the solutions developed by Gray (1945) and Terzaghi (1943). Its practical applicability is
also evaluated by analysing a case history involving a ll embankment, which was constructed over a
crust layer of hard soil overlying soft clay improved with stone columns. The crust layer is modelled as animpeded drainage. Reasonably good agreement is obtained between the consolidation results obtained
from the proposed analytical solution and available three-dimensional nite-element predictions. With
the further consideration of smear effects, good agreement is achieved between the consolidation results
obtained from the proposed analytical solution and eld measurements.
© 2015 Published by Elsevier Ltd.
1. Introduction
Soft soil is often preloadedwith surcharge pressure as one of the
most economic and effective ways to consolidate it (Qubain et al.,
2014). Vertical prefabricated drains or sand/stone columns are
commonly utilised to accelerate the consolidation of soft soils un-der preloading (Almeida et al., 2015; Artidteang et al., 2011;
Cascone and Biondi, 2013; Chai et al., 2010; Indraratna et al.,
2010; Jang and Chung, 2014; Karunaratne, 2011; Li and Rowe,
2001; Lo et al., 2008, 2010; Rowe and Li, 2005; Rowe and
Taechakumthorn, 2008; Saowapakpiboon et al., 2009, 2010; Shen
et al., 2005; Suleiman et al., 2014; Van Helden et al., 2008;
Voottipruex et al., 2014; Xue et al., 2014). Analytical solutions
predicting the extent of consolidation in preloading play an
important role in the preliminary design of vertical drains (Abuel-
Naga et al., 2012; Bari and Shahin, 2014; Basu and Prezzi, 2009;
Chung et al., 2014; Rujikiatkamjorn and Indraratna, 2009; Sinhaet al., 2009). Since the pioneering work of Barron (1948), the
challenge of deriving an analytical solution for cylindrical unit-cell
consolidation with a vertical drain has capturedthe attention of the
ground improvement community. For consolidation of a single
layer of homogeneous soil under surcharge preloading, various
solutions have been proposed based on different assumptions and
considerations. A large number of analytical solutions were derived
for the consolidation of soil with fully drained boundary conditions
at its top and/or bottom surface (e.g., Conte and Troncone, 2009;
Deng et al., 2013a, 2013b; Indraratna et al., 2011; Kianfar et al.,
2013; Lei et al., 2015; Lu et al., 2011, 2015; Ong et al., 2012;
* Corresponding author. Tel.: þ86 13 851922201, þ86 25 83787216; fax: þ86 25
83786633.
E-mail addresses: [email protected] (G.H. Lei), [email protected]
(C.W. Fu), [email protected] (C.W.W. Ng).
Contents lists available at ScienceDirect
Geotextiles and Geomembranes
j o u r n a l h o m e p a g e : w w w . e l s e v i e r . c om / l o c a t e / g e o t e x m e m
http://dx.doi.org/10.1016/j.geotexmem.2015.07.003
0266-1144/©
2015 Published by Elsevier Ltd.
Geotextiles and Geomembranes xxx (2015) 1e10
Please cite this article in press as: Lei, G.H., et al., Vertical-drain consolidation using stone columns: An analytical solution with an impededdrainage boundary under multi-ramp loading, Geotextiles and Geomembranes (2015), http://dx.doi.org/10.1016/j.geotexmem.2015.07.003
mailto:[email protected]:[email protected]:[email protected]://www.sciencedirect.com/science/journal/02661144http://www.elsevier.com/locate/geotexmemhttp://dx.doi.org/10.1016/j.geotexmem.2015.07.003http://dx.doi.org/10.1016/j.geotexmem.2015.07.003http://dx.doi.org/10.1016/j.geotexmem.2015.07.003http://dx.doi.org/10.1016/j.geotexmem.2015.07.003http://dx.doi.org/10.1016/j.geotexmem.2015.07.003http://dx.doi.org/10.1016/j.geotexmem.2015.07.003http://www.elsevier.com/locate/geotexmemhttp://www.sciencedirect.com/science/journal/02661144mailto:[email protected]:[email protected]:[email protected]
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kh
vu
vr
¼ ksh
vusvr
; r ¼ r s (6)
The hydraulic boundary conditions can be expressed as follows:
vu
v z ¼ R
u
h and
vusv z
¼ Rush
;
z ¼ 0 for the drainage impeded top(7)
vu
v z ¼
vusv z
¼ 0; z ¼ h for the impervious bottom (8)
vu
vr ¼ 0; r ¼ r e for the impervious vertical boundary (9)
where h is the depth of the vertical drain.
The initial condition is given by
u ¼ us ¼ u ¼ us ¼ 0; t ¼ 0 (10)
Fig. 2 schematically shows the increase in total stress in soil due
to multi-ramp surcharge loading. To facilitate the derivation of the
analytical solution, a new single equation is constructed to accu-
rately describe the increase in total stress:
sðt Þ ¼XM i¼1
F iðt Þ½si si1 (11)
where
F iðt Þ ¼t t i;0
t i;1 t i;0H
t t i;0
1 H
t t i;1
þ H
t t i;1
(12)
H
t t i; j
¼
0;
t t i; j
< 0
1;
t t i; j
0; ð j ¼ 0; 1Þ (13)
where H ht t i; ji is the Heaviside step function; M is the total
number of loading ramps; t i;
0 and t i;
1 are the start time and endtime of the i-th ramp, respectively, as shown in Fig. 2; si is the in-
crease in total stress in soil at the end time of the i-th ramp, and
s0 ¼ 0.
The equations above describe the unit-cell consolidation prob-
lem to be solved.
3. The analytical solution
The governing Eqs. (2) and (3) are solved using the method of
separation of variables and the Fourier series, as presented in detail
in Appendix A. Explicit, rigorous analytical solutions are obtained
for calculating the excess pore-water pressure at any arbitrary point
in the undisturbed soil and the smeared soil:
u ¼ mvgw
kv
X∞n¼1
(½c 1nI 0ðmnr Þ þ c 2nK 0ðmnr Þ þ 1
½sinðun z Þ þ cotðunhÞcosðun z Þ
u2nX
M
i¼1
C n
;
i
ðt Þ) (14)
us ¼ msvgw
ksv
X∞n¼1
(½c 3nI 0ðmsnr Þ þ c 4nK 0ðmsnr Þ þ 1
½sinðun z Þ þ cotðunhÞcosðun z Þ
u2n
XM i¼1
C n;iðt Þ
) (15)
where
C n;iðt Þ ¼sn;i sn;i1
t i;1 t i;0
"e
8ðT hT hi;1Þvn
H hT hT hi;1i e8ðT h T hi;0Þ
vn H hT hT hi;0i
#
(16)
where un is the positive root of the transcendental Eq. (A3) in
Appendix A, which can be solved easily using commercially avail-
able packages like Matlab; I 0 and K 0 are the modied Bessel func-
tions of the rst and second kind of zero order, respectively; T h is
the time factor given by Eq. (A21) in Appendix A; the expressions
for c 1n, c 2n, c 3n, c 4n, mn, msn, T hi, j and vn are given by Eqs. (A58), (A59),
(A47), (A61), (A14), (A30), (A46) and (A22), respectively, in
Appendix A; and e is the base of the natural logarithm.
As usual, the overall average degree of consolidation is dened
as follows:
U Sðt Þ ¼ sðt Þ uo
sM (17)
where sM is the maximum increase in total stress in soil at the end
time t M ,1 of the M -th ramp of surcharge loading, as shown in Fig. 2;
and uo is the overall average excess pore-water pressure, which can
be derived based on Eqs. (14) and (15) as
uo ¼
Z h0
264Z r e
r s
2prudr þ
Z r sr d
2prusdr
375d z
p
r 2e r
2d
h
¼
1r 2e r
2d
hX∞n¼1
( 1u3n
r 2e r 2skv mvgw
Un þ
r 2s r 2d
ksv msvgwU
sn!
XM i¼1
C n;iðt Þ
)
(18)
where Un and Usn aregiven by Eqs. (A18) and (A33), respectively, in
Appendix A. Thus, by substituting Eqs. (11) and (18) into Eq. (17),
the overall average degree of consolidation can be obtained.
For ease of application of the proposed solution, a simple
Fortran program that solves the modied Bessel functions with
freeware subroutines (Press et al., 1992) has been developed. The
results are obtained through double-precision arithmetic
calculation.Fig. 2. Time-dependent increase in total stress in soil under multi-ramp loading.
G.H. Lei et al. / Geotextiles and Geomembranes xxx (2015) 1e10 3
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It is worth noting that consolidation under fully drained top
boundary conditions (i.e., R ¼∞ in Eq. (A3) in Appendix A) can also
be analysed using the proposed solution by simply letting
un ¼ ð2n 1Þp
2h (19)
Apart from this, the proposed solution can also be used to
analyse one-dimensional consolidation without vertical drains, byapplying an extremely low r d value (e.g., 0.001 m), letting kd, kh, kshand ksv be equal to kv, and letting msv be equal to mv.
4. Verication
In order to verify the validity and accuracy of the proposed
analytical solution, the results calculated from the proposed solu-
tion for one-dimensional consolidation under impeded drainage
boundary conditions are compared with those given by the
analytical solution of Gray (1945). Gray (1945) developed an
analytical solution to one-dimensional consolidation with an
impeded drainage boundary under instantaneous loading. Fig. 3
compares the degrees of consolidation calculated from the pro-
posed solution, the solution of Gray (1945) and Terzaghi's (1943)
well-known one-dimensional consolidation solution. Excellent
agreement is obtained. As expected, the characteristic factor of
drainage ef ciency has a potentially important inuence on
consolidation.
5. Case study
The proposed solution is also applied to a case history involving
a ll embankment at NewPantai Expressway in Malaysia (Tan et al.,
2008). The 1.8 m high embankment was constructed with sandy
material in 9 days. The ground consisted of a 1 m thick upper crust
layer of hard soil and a 5 m thick layer of soft clay overlying a stiff
clay layer (see Fig. 4(a)). Stone columns 800 mm in diameter, ar-
ranged in a square grid with a centre-to-centre spacing of 2.4 m,
were installed from the embankment base to a depth of 6 m above
the stiff clay layer. A settlement plate was installed to measure the
settlement at the centre of the embankment (measurement point
SP1). Using the three-dimensional nite-element method, Tan et al.
(2008) computed the settlement at SP1 and the excess pore-water
pressure at a computation point A, which was located at a depth of
3.5 m below the centre of the embankment and at the centre of the
square grid (in plan) of stone columns (see Fig. 4(a)). In the present
study, the excess pore-water pressure at point A and the overall
average degree of consolidation U S(t ) below the centre of the
embankment are calculated using Eqs. (14) and (17), respectively,
adopting the calculation parameters presented by Tan et al. (2008)
(see the rst three columns of Table 1). Although the upper crust
layer can serve as a horizontal drainage blanket for discharging the
water expelled from the stone columns installed through it, its
hydraulic conductivity is in the order of 107 m/s, which is only two
orders of magnitude higher than the hydraulic conductivity of the
underlying soft clay. For this reason, the upper crust layer is
modelled as an impeded drainage boundary for the consolidation
of soft clay. The characteristic factor R of drainage ef ciency of the
upper crust layer is derived from Eq. (1) and the adopted param-
eters, as given in Table 1. For comparison purpose, the measured
and computed settlements presented by Tan et al. (2008) are nor-
malised by their corresponding ultimate values to reect the de-
gree of consolidation. Similarly, the computed and calculated
Fig. 3. A comparison between the solution proposed in this study and those developed
by Gray (1945) and Terzaghi (1943).
Fig. 4. Comparisons of calculated results from the proposed solution and reported eld
data and computed values by Tan et al. (2008).
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excess pore-water pressures are also normalised by their corre-
sponding maximum values. The smear effects due to the installa-
tion of stone columns are not considered in the three-dimensional
nite-element analysis performed by Tan et al. (2008). Fig. 4(b)
compares the degrees of consolidation calculated from the newly
proposed solution with those measured and computed reported by
Tan et al. (2008) using the settlement data obtained at SP1. Fig. 4(c)
compares the excess pore-water pressures at point A calculated
from the newly proposed solution with those computed by Tan
et al. (2008). The dashed lines represent the calculated results for
consolidation under impeded drainage boundary conditions
without consideration of the smear effect. It can be seen that these
results are in reasonably good agreement with those computedusing the three-dimensional nite-element method, especially
during the loading period. Nevertheless, when compared with the
measured data, the calculated rates of consolidation are relatively
signicantly overestimated by both the analytical solution and the
three-dimensional nite-element method, as shown in Fig. 4(b).
This is attributed to the fact that the smear effects are not consid-
ered in both cases.
To investigate the smear effects on consolidation, back-analysis
using the newly proposed analytical solution is carried out with
back-analysed parameters for smeared soil, which are listed in the
last column of Table 1. According to Weber et al. (2010), the radius
of a smear zone is assumed to be 2.5 times the radius of the stone
column, that is, r s ¼ 2.5r d. The vertical hydraulic conductivity and
volume compressibility of smeared soil are assumed to be the sameas those of undisturbed soil, that is, ksv ¼ kv and msv ¼ mv. The
horizontal hydraulic conductivity of smeared soil is assumed to be
0.4 times that of undisturbed soil, that is, ksh ¼ 0.4kh, which is
within the range of 0.2khe0.8kh derived from experiments (Hird
and Moseley, 2000; Juneja et al., 2013; Rujikiatkamjorn et al.,
2013; Sathananthan and Indraratna, 2006; Sharma and Xiao,
2000). As shown in Fig. 4(b), it is evident that the calculated re-
sults including the smear effects (solid line) are consistent with the
measured degrees of consolidation. This indicates that the smear
effects are signicant and should not be ignored.
To investigate the effect of loading conditions on consolidation,
the degrees of consolidation and the excess pore-water pressures
under instantaneous loading are also calculated using the proposed
solution, as shown by the dotted lines in Fig. 4(b) and (c). It can beobserved that the rate of consolidation and the rate of dissipation of
excess pore-water pressure are generally overestimated if an
instantaneous loading condition is assumed. This indicates that a
realistic modelling of the loading conditions is necessary for
consolidation analysis.
Ideally, an application of the proposed analytical solution should
be compared with a case history involving consolidation with
prefabricated vertical drains. However, as far as the authors are
aware, documented case histories involving consolidation with
prefabricated vertical drains where sand blanket is generally taken
for granted as fully drained are not suitable for comparison here.
The case history reported by Tan et al. (2008) involving consoli-
dation of soft ground by stone columns is thus selected as it is
relevant to an impeded drainage boundary. The calculated results
may be considered as a rst approximation to the analysis of
consolidation of soft clay with stone columns only, as the rein-
forcement and arching effects due to the use of stone columns are
ignored (Ali et al., 2014; Castro and Sagaseta, 2011, 2013; Elsawy,
2013; Indraratna et al., 2013; Miranda et al., 2015; Shahu and
Reddy, 2014; Zhang et al., 2012). It is evident that there is a lack
of studies of in-situ permeability of sand blanket and its effect on
consolidation of soil. Further studies on this topic appear to be
warranted.
6. Conclusions
A rigorous, explicit, analytical equal-strain solution is proposed
for a unit-cell model of consolidation with a vertical drain under
impeded drainage boundary conditions and multi-ramp surcharge
loading. The solution can also be used to analyse vertical-drain
consolidation under fully drained boundary conditions and one-
dimensional consolidation under impeded drainage and fully
drained boundary conditions. Excellent agreement is obtained be-
tween the calculated results from the special cases of the proposed
solution and those from two available analytical solutions in the
literature. The practical applicability of the solution to consolida-
tion under impeded drainage boundary conditions is also explored
employing a case study involving an embankment constructed over
a crust layer of hardsoiland a layer of soft clay improved with stone
columns. The calculated results from the proposed solution are
shown to be in reasonably good agreement with measureddata and
numerically-computed results, when the crust layer is modelled as
an impeded drainage boundary and the smear effects are consid-
ered. This suggests that the proposed solution is valid for the more
general cases of drainage boundary conditions. Moreover, the case
study shows that the smear effects on consolidation with stone
columns are signicant and should not be ignored. Owing to a lack
of eld studies of in-situ permeability of sand blanket, further re-
searches on its drainage effect on consolidation are needed in order
to evaluate the practical applicability of the proposed analytical
solution to the consolidation problem with prefabricated vertical
drains.
Acknowledgements
This study was sponsored by the National Natural Science
Foundation of China (grant number 51278171), the 111 Project
(grant number B13024), the Fundamental Research Funds for the
Central Universities of China (grant number 2015B06014), and the
Chang Jiang Scholars Program of the Ministry of Education of China.
Appendix A. Derivation of Eqs. (14) and (15)
Consolidation of undisturbed soil
The excess pore-water pressure of undisturbed soil can be
expressed by introducing the Fourier sine and cosine series as
follows:
Table 1
Calculation parameters adopted from Tan et al. (2008).
Drain properties Soil properties Drainage boundary, stress and loading conditions Back-analysed parameters for smeared soil
kd ¼ 1.16 104 m/s kv ¼ ksv ¼ 1.16 10
9 m/s R ¼ 500 r s ¼ 2.5r d ¼ 1.0 m
r d ¼ 0.4 m mv ¼ msv ¼ 0.6753 103 kPa1 M ¼ 1 ksv ¼ kv ¼ 1.16 10
9 m/s
r s ¼ 0.4 m kh ¼ ksh ¼ 3.47 109 m/s s1 ¼ 32.4 kPa msv ¼ mv ¼ 0.6753 10
3 kPa1
r e ¼ 1.2 m t 1,1 ¼ 9 d ksh ¼ 0.4kh ¼ 1.388 109 m/s
h ¼ 5.0 m
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T h ¼ kht
mvgwð2r eÞ2
(A21)
vn ¼ 2Un
ðmnr eÞ2
(A22)
Based on Eqs. (A10), (A12) and (A20), Eq. (A2) can be rewritten
as
uðr ; z ; t Þ ¼X∞n¼1
½c 1nI 0ðmnr Þ þ c 2nK 0ðmnr Þ þ 1
ane
8T hvn þ mvfn f n
½sinðun z Þ þ cotðunhÞcosðun z Þ
(A23)
Consolidation of smeared soil
Again by introducing the Fourier sine and cosine series, the
excess pore-water pressure at any arbitrary point and the average
excess pore-water pressure at a given depth of smeared soil can be
expressed in accordance with Eqs. (7) and (8) of the top and bottomhydraulic boundary conditions as follows:
usðr ; z ; t Þ ¼X∞n¼1
usnðr ; t Þ½sinðun z Þ þ cotðunhÞcosðun z Þ (A24)
usð z ; t Þ ¼X∞n¼1
usnðt Þ½sinðun z Þ þ cotðunhÞcosðun z Þ (A25)
where usn and usn are their corresponding Fourier coef cients.
Using the method of separation of variables, the following
equation can be written:
usnðr ; t Þ ¼ Asnðr ÞBsnðt Þ (A26)
Following the same derivation procedures as above for the
consolidation of undisturbed soil, the following solution to Eq.(A26) for the consolidation of smeared soil can be obtained:
Asnðr Þ ¼ lsnfsn½c 3nI 0ðmsnr Þ þ c 4nK 0ðmsnr Þ þ 1 (A27)
Bsnðt Þ ¼ 1
lsnfsn
asne
8T shvsn þ msvfsn f n
(A28)
where lsn is the separation constant; c 3n, c 4n and asn are the con-
stants of integration to be determined; and
fsn ¼ gw
ksvu2n(A29)
m2sn ¼ ksvu2n
ksh(A30)
T sh ¼ ksht
msvgwð2r sÞ2
(A31)
vsn ¼ 2Usn
ðmsnr sÞ2
(A32)
Thus, Eq. (A24) can be rewritten as
usðr ; z ;t Þ ¼X∞n¼1
½c 3nI 0ðmsnr Þ þ c 4nK 0ðmsnr Þ þ 1
asne
8T shvsn þ msvfsn f n
½sinðun z Þ þ cotðunhÞcosðun z Þ
(A34)
In the following sections, the constants of integration in Eqs.(A23) and (A34) are determined according to the initial conditions
and the vertical hydraulic boundary conditions, together with the
equations of drain resistance and interface drainage.
Initial conditions
Without loss of generality, the initial average excess pore-water
pressures for undisturbed soil and smeared soil are assumed to be
uð z ; t ¼ 0Þ ¼ 1
p
r 2e r 2s
Z r er s
uðr ; z ; t ¼ 0Þ2pr dr ¼ s0 (A35)
usð z ; t ¼ 0Þ ¼ 1
p
r 2s r
2d
Z r sr d
usðr ; z ; t ¼ 0Þ2pr dr ¼ s0 (A36)
Substituting Eq. (A23) into Eq. (A35) and substituting Eq. (A34)
into Eq. (A36) yield
an ¼ sn;0
Un mvfn
sn;1
t 1;1 t 1;0(A37)
asn ¼ sn;0
Usn msvfsn
sn;1
t 1;1 t 1;0(A38)
where sn,0 is the Fourier coef cient of Fourier series expansions of
the initial increase in total vertical stress s0 as shown in Fig. 2.
In order to ensure continuity of pore-water pressure and owrate at all times, the time functions for the consolidation of un-
disturbed soil and smeared soil must be the same, i.e.,
Bnðt Þ ¼ Bsnðt Þ (A39)
Substituting Eqs. (A20) and (A28) into Eq. (A39) yields
1
lnfn
ane
8T hvn þ mvfn f n
¼
1
lsnfsn
asne
8T shvsn þ msvfsn f n
(A40)
Eq. (A40) requires that
an
asn¼ lnfn
lsnfsn(A41)
ln
lsn¼
mvmsv
(A42)
T hvn
¼ T shvsn
(A43)
Usn ¼ 1 þ2c 3n½msnr sI 1ðmsnr sÞ msnr dI 1ðmsnr dÞ 2c 4n½msnr sK 1ðmsnr sÞ msnr dK 1ðmsnr dÞ
ðmsnr sÞ2 ðmsnr dÞ
2 (A33)
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It can be readily proved that by Eqs. (A37), (A38), (A41) and
(A42), Eq. (A43) is satised.
For the initial conditions specied in Eq. (10) and Fig. 2, i.e.
s0 ¼ 0 and sn,0 ¼ 0, Eq. (A37) becomes
an ¼ mvfnsn;1
t 1;1 t 1;0(A44)
By substituting Eqs. (A9) and (A44) into Eq. (A20), the followinggeneralised time function can be derived:
Bnðt Þ ¼ mvfnlnfn
XM i¼1
(sn;i sn;i1
t i;1 t i;0
"e
8ðT h T hi;1Þvn
H hT hT hi;1i e8ðT hT hi;0Þ
vn H hT hT hi;0i
#) (A45)
where
T hi; j ¼kht i; j
mvgwð2r eÞ2; j ¼ 0; 1 (A46)
Drain resistance
Substituting Eq. (A34) into Eq. (4) yields
c 3n ¼ 1
D1I 1ðmsnr dÞ I 0ðmsnr dÞ þ D2c 4n (A47)
D1 ¼ 2
r d
kshkd
msn
u2n
(A48)
D2 ¼ D1K 1ðmsnr dÞ þ K 0ðmsnr dÞ
D1I 1ðmsnr dÞ I 0ðmsnr dÞ (A49)
For an ideal drain without drain resistance, kd ¼ ∞, and hence
D1 ¼ 0.
Interface continuity
Substituting Eqs. (A23) and (A34) into Eqs. (5) and (6) and
considering Eq. (A40) yield
lnfn½c 1nI 0ðmnr sÞ þ c 2nK 0ðmnr sÞ þ 1
¼ lsnfsn½c 3nI 0ðmsnr sÞ þ c 4nK 0ðmsnr sÞ þ 1 (A50)
khlnfnmn½c 1nI 1ðmnr sÞ c 2nK 1ðmnr sÞ
¼ kshlsnfsnmsn½c 3nI 1ðmsnr sÞ c 4nK 1ðmsnr sÞ (A51)
Substituting Eq. (A47) into Eqs. (A50) and (A51) gives
anc 1n þ bnc 2n þ D4 ¼ 0 (A52)
where
an ¼ I 0ðmnr sÞ
ffiffiffiffiffiffiffiffiffiffiffiffiffikhkv
kshksv
s I 1ðmnr sÞD3 (A53)
bn ¼ K 0ðmnr sÞ þ ffiffiffiffiffiffiffiffiffiffiffiffiffikhkv
ksh
ksv
s K 1ðmnr sÞD3 (A54)
D3 ¼ D2I 0ðmsnr sÞ þ K 0ðmsnr sÞ
D2I 1ðmsnr sÞ K 1ðmsnr sÞ (A55)
D4 ¼ msv
mv
kvksv
D3I 1ðmsnr sÞ I 0ðmsnr sÞ
D1I 1ðmsnr dÞ I 0ðmsnr dÞ 1
þ 1 (A56)
Vertical hydraulic boundary conditions
Substituting Eq. (A23) into Eq. (9) yields
c 1nI 1ðmnr eÞ c 2nK 1ðmnr eÞ ¼ 0 (A57)
The following can be derived from Eqs. (A52) and (A57):
c 1n ¼ D4K 1ðmnr eÞ
Dn(A58)
c 2n ¼ D4I 1ðmnr eÞ
Dn(A59)
Dn ¼ anK 1ðmnr eÞ bnI 1ðmnr eÞ (A60)
Substituting Eqs. (A47), (A58) and (A59) into Eq. (A50) leads to
c 4n ¼ mvmsv
ffiffiffiffiffiffiffiffiffiffiffiffiksvkhkvksh
s c 1nI 1ðmnr sÞ c 2nK 1ðmnr sÞ
D2I 1ðmsnr sÞ K 1ðmsnr sÞ
I 1ðmsnr sÞ
½D1I 1ðmsnr dÞ I 0ðmsnr dÞ½D2I 1ðmsnr sÞ K 1ðmsnr sÞ
(A61)
The nal solution
Based on Eqs. (A2), (A10), (A12) and (A45), Eq. (14) can be
formulated for calculating the excess pore-water pressure of un-
disturbed soil. Similarly, based on Eqs. (A24), (A26), (A27), (A39),
(A42) and (A45), Eq. (15) can be derived for calculating the excess
pore-water pressure of smeared soil.
Appendix B. Derivation of Eq. (A7)
According to Eq. (A5), the following equation can be derived.
The numerator term of Eq. (B1) can be easily derived as
Z h0
si½sinðun z Þ þ cotðunhÞcosðun z Þd z ¼ si
un(B2)
sn;i ¼
Z h0
si sin un z ð Þ þ cot unhð Þcos un z ð Þ½ d z
P∞
n¼1
Z h0
sin un z ð Þ þ cot unhð Þcos un z ð Þ½ sin um z ð Þ þ cot umhð Þcos um z ð Þ½ d z
( ) (B1)
G.H. Lei et al. / Geotextiles and Geomembranes xxx (2015) 1e108
Please cite this article in press as: Lei, G.H., et al., Vertical-drain consolidation using stone columns: An analytical solution with an impededdrainage boundary under multi-ramp loading, Geotextiles and Geomembranes (2015), http://dx.doi.org/10.1016/j.geotexmem.2015.07.003
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To obtain the denominator term of Eq. (B1), the following
triangular orthogonal relation is needed. For m s n, itcanbe shown
that
Z h0
½sinðun z Þ þ cotðunhÞcosðun z Þ
½sinðum z Þ þ cotðumhÞcosðum z Þd z
¼
Z h0
cos½unðh z Þ
sinðunhÞ ,
cos½umðh z Þ
sinðumhÞ d z
¼
Z h0
cos½ðum þ unÞðh z Þ þ cos½ðum unÞðh z Þ
2 sinðunhÞsinðumhÞ d z
¼
1
um þ unsin½ðum þ unÞh þ
1
um unsin½ðum unÞh
2 sinðunhÞsinðumhÞ
¼ um sinðumhÞcosðunhÞ un cosðumhÞsinðunhÞ
ðu
m þ u
nÞðu
m u
nÞsinðu
nhÞsinðu
mhÞ
¼ umh tanðumhÞ unh tanðunhÞ
ðum þ unÞðum unÞh tanðunhÞtanðumhÞ
(B3)
According to Eq. (A3), Eq. (B3) for m s n is
Z h0
½sinðun z Þ þ cotðunhÞcosðun z Þ
½sinðum z Þ þ cotðumhÞcosðum z Þd z ¼ 0
(B4)
Therefore, the denominator term of Eq. (B1) can be derived as
By substituting Eqs. (B2) and (B5) into Eq. (B1), Eq. (A7) in
Appendix A can be obtained.
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P∞n¼1
Z h0
sin un z ð Þ þ cot unhð Þcos un z ð Þ½ sin um z ð Þ þ cot umhð Þcos um z ð Þ½ d z
8<:
9=;
¼
Z h0
sin un z ð Þ þ cot unhð Þcos un z ð Þ½ 2d z
¼ unh þ sin unhð Þcos unhð Þ
2un sin2unhð Þ
(B5)
G.H. Lei et al. / Geotextiles and Geomembranes xxx (2015) 1e10 9
Please cite this article in press as: Lei, G.H., et al., Vertical-drain consolidation using stone columns: An analytical solution with an impededdrainage boundary under multi-ramp loading, Geotextiles and Geomembranes (2015), http://dx.doi.org/10.1016/j.geotexmem.2015.07.003
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