Vertical Drain Consolidation Using Stone Columns an Analytical Solution With an Impeded Drainage Boundary Under Multi Ramp Loading 2015 Geotextiles An

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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

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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

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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


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).

G.H. Lei et al. / Geotextiles and Geomembranes xxx (2015) 1e104



5/10

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


(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



(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


(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)




8/10

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)




9/10

To obtain the denominator term of Eq. (B1), the following

triangular orthogonal relation is needed. For m s n, itcanbe shown

that

Z h0


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


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)



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Please cite this article in press as: Lei, G.H., et al., Vertical-drain consolidation using stone columns: An analytical solution with an impeded
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