vertical drain 2.pdf

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T

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10e 3

tricmer onear

2007 Elsevier Ltd. All rights reserved.

faster consolidation in soft clay subsoil can be achieved;

surcharge) can be represented by an axisymmetric unit cell[2,6] as shown in Fig. 2. To simulate the performance of

unit cell (see Fig. 2), then an equivalent full-scale plane

and plane strain cells. Based on these models, theyproposed two solutions for development of the averageexcess pore water pressure with time within these cells.

Afterwards, by equating these two solutions, theyobtained a conversion expression of permeability fromthe axisymmetric cell to the equivalent plane strain cell.

* Corresponding author. Tel.: +81 11 706 6194; fax: +81 11 706 7204.E-mail addresses: [email protected] (T.A. Tran), mitachi@

eng.hokudai.ac.jp (T. Mitachi).

Available online at www.sciencedirect.com

Computers and Geotechnicsin short, we call this preloading technique vacuum-sur-charge preloading.

In vacuum-surcharge preloading, the vacuum pressurepropagates along the vertical drains, and a nearly isotropiccompression zone is created within the subsoil zonebeneath the embankment; therefore, this technique canenhance the stability of the subsoil during consolidationunder the embankment.

It is widely agreed that the performance of a verticaldrain under conventional embankment (or conventional

strain simulation can be made. Consequently, the timeneeded for computing the full-scale plane strain simulationis much shorter than that needed in a full-scale 3D simula-tion. In fact, Chai et al. [3], Hird et al. [8], Indraratna andRedana [9,10], and Indraratna et al. [11] conrmed the fea-sibility of this idea.

In 2005, Indraratna et al. [12] proposed a conversionmethod for cases of single layer subsoil under vacuum-surcharge preloading condition. In this method, theydeveloped two analytical models for the axisymmetricKeywords: Analytical solution; Consolidation; Finite element analysis; Plane strain modeling; Vacuum-surcharge preloading; Vertical drain

1. Introduction

It is widely recognized that vertical drains are able tospeed up consolidation in soft clay subsoil that usuallyhave extremely slow natural drainage. In recent years, thevertical drains have been combined with vacuum preload-ing under embankment (see Fig. 1), and therefore a much

multiple vertical drains under an embankment by FEM,we have to carry out a 3D full-scale simulation, in whicha lot of cubic elements have to be employed [3,10]. As aresult, the time needed for computation becomes very long,and a powerful computer is needed.

However, if we assume that the performance of a verti-cal drain can be equivalently represented by a plane strainEquivalent plane strain modelingunder embankment combin

Tuan Anh Tran *,

Hokkaido University, Graduate School of Engineering, Soil Mecha

Received 18 July 2007; received in revised formAvailable onlin

Abstract

A conversion method is proposed to convert from an axisymmeloading combined with vacuum preloading. To verify the proposedis a subsoil having only one homogeneous clay layer, and the othethat the eects of both well resistance (in the vertical drain) and smproposed plane strain unit cell in both cases.0266-352X/$ - see front matter 2007 Elsevier Ltd. All rights reserved.doi:10.1016/j.compgeo.2007.11.006f vertical drains in soft groundd with vacuum preloading

oshiyuki Mitachi

Laboratory, Room A6-53, Building A, Sapporo 060-8628, Japan

November 2007; accepted 14 November 2007January 2008

unit cell to an equivalent plane strain unit cell under embankmentthod, we have conducted FE analyses for two cases of subsoil, onee is a subsoil having two clay layers. The analyzed results showedzone (around the vertical drain) are satisfactorily modeled by the

www.elsevier.com/locate/compgeo

35 (2008) 655672

Surcharge or embankment

Vacuum pumpVertical drain Geomembrane

Fig. 1. Cross-section of subsoil improved by vertical drains undervacuum-surcharge preloading.

Nomenclature

a width of the prefabricated vertical drainb thickness of the prefabricated vertical drainB half-width of the plane strain unit cellbs half-width of the smear zone of the plane strain

unit cellbw half-width of the drain-wallCha coecient of consolidation for horizontal drain-

age in axisymmetric caseChp coecient of consolidation for horizontal drain-

age in plane strain caseds smear zone diameterdw equivalent drain diameterk1 maintaining factor of vacuum pressure

(0 6 k1 6 1)kha horizontal permeability coecient in undis-

turbed zone of the axisymmetric unit cellkhp horizontal permeability coecient in undis-

turbed zone of the plane strain unit cellksa horizontal permeability coecient in smear zone

of the axisymmetric unit cellksp horizontal permeability coecient in smear zone

of the plane strain unit celll The length of drainage path of the drain inside

the axisymmetric unit cellna ratio R/rw of the axisymmetric unit cellp0 vacuum pressure applied to the top of the drain

in axisymmetric case, and to the top of thedrain-wall in plane strain case

qwa discharge capacity of the drain of the axisym-metric unit cell

qwp discharge capacity of the drain-wall of the planestrain unit cell

R radius of the axisymmetric unit cellrs radius of the smear zone of the axisymmetric

unit cellrw radius of the drain of the axisymmetric unit cellsa ratio rs/rw of the axisymmetric unit cellTha dimensionless time factor for horizontal drain-

age in the axisymmetric unit cellThp dimensionless time factor for horizontal drain-

age in the plane strain unit cellu average excess pore water pressure of the unit cellr1 initial overburden pressure due to surcharge

656 T.A. Tran, T. Mitachi / Computers and Geotechnics 35 (2008) 655672preloadinglza parameter depending on depth z, including well

resistance, smear eect, and the geometry of theaxisymmetric unit cellAs a result, by using this plane strain cell, a full-scale planestrain modeling of soft ground under vacuum-surchargepreloading can be conducted.

2D cell with smear zone 2D cell without smear zone

2B 2B

Axisymmetric cell

De

Fig. 2. Axisymmetric unit cell and its equivalent plane strain unit cells.

lzp parameter depending on depth z, including wellresistance, smear eect, and the geometry of theplane strain unit cell

Subscripts

a axisymmetricp plane strains smear zoneh horizontalz depth z

l (

presd dis

- (1

axis

s anWe see that, in their method, the inclusion of theplane strain smear zone in the plane strain cell is deemedto be needed (see Figs. 2 and 4a). However, in our expe-rience, the inclusion of plane strain smear zones in planestrain nite element simulation increases the number ofelements and material parameters. In particular, for afull-scale simulation of a multi-layer subsoil incorporat-

(a) Analytical model of the axisymmetric unit cel

R

- k1 p

0

C

undi

sturb

ed zo

ne

un

distu

rbed

zo

ne

smea

r zo

ne

smea

r zo

ne

0- p

- p [1 - (1 - k )z /l]0 1

vacuum pressureAssumed distribution of

impe

rmea

ble

1

vacuumAssume

- p [1 0

rw

sr

impermeable

The

len

gth

of t

he dr

ainag

e pa

th of t

he dr

ain (l

)

impermeable

L

Fig. 3. Analytical models of the

T.A. Tran, T. Mitachi / Computering a large number of small vertical drain elements, thenumber of elements and the material parameters forthe plane strain smear zones become very large. In addi-tion, for determining the equivalent permeability by theirmethod, the determination of the length of the drainagepath of the drain (l) in each soil layer of that multi-layersubsoil (where the drain is driven through) is required(see Fig. 3a).

In 2006, Chai et al. [5] presented a plane strain numeri-cal modeling of a soft ground improved by vertical drainsunder vacuum-surcharge preloading. In their numericalmodeling, they used a conversion method from the axisym-metric cell to the plane strain cell proposed by Chai et al.[4].

We have observed that, this conversion method is devel-oped under the condition of conventional surcharge load-ing, and the boundary condition of excess pore pressureof the vertical drain dened in this method is that the excesspore pressure at the top of the drain is equal to zero, only.Therefore, this method might not apply well in the case ofvacuum-surcharging.

In reality, the subsoil usually has many layers, and, insome cases, there is a sandy silt layer having high perme-ability just below an upper clay layer having low perme-ability. In this case, the vacuum pressure propagates fromthe vertical drain to the surrounding soil in the sandy siltlayer will be much faster than the propagation of the vac-uum pressure in the surrounding soil of the upper claylayer. This means that the vacuum pressure in the sandy siltwill reach the maximum value much sooner than that in theupper clay layer.

However, in Chai et al. [4] method, their plane strainunit cell has only an equivalent vertical permeability and

b) Analytical model of the plane strain unit cell

suretribution of

equi

val

ent z

on

e

- k1 p

0

0- p

- k )z /l]1

equi

val

ent z

on

e

B

bw

1

impermeable

impe

rmea

ble

impermeable

The

len

gth

of t

he dr

ainag

e pa

th of t

he dr

ain (l

)

CL

ymmetric and plane strain cells.

d Geotechnics 35 (2008) 655672 657has no drain-wall. Therefore, even though the lower soillayer has a much higher permeability than that of the upperclay layer, and if the vacuum pressure is applied to the topof the cell, then the vacuum pressure would only able topropagate gradually from the upper layer to the lower layerin their plane strain cell. This means that, in their planestrain cell, there is no way to let the vacuum pressure inthe lower clay layer reach the maximum value sooner thanthat in the upper clay layer.

It is observed that there are various ways to develop aconversion method. In this paper, by modifying Indraratnaet al. [12] method (developed for single layer cases undervacuum-surcharge preloading), a new conversion methodhas been proposed, which is dierent to those of Hirdet al. [8] and Indraratna and Redana [9,10].

2. The proposed conversion method under vacuum-surcharge

preloading condition

To nd the conversion expression of permeability fromthe axisymmetric unit cell to the equivalent plane strainunit cell under vacuum-surcharge preloading condition,we conducted the mathematical formulation as follows.

Firstly, based on the analytical model of Indraratnaet al. [12] for the axisymmetric unit cell, we used analyticalmathematics to nd a function of development of excess

5. The displacement at outer boundaries of the verticaldrain and the cell are xed in horizontal direction, i.e.

anonly vertical displacement is allowed at theseboundaries.

6. Darcys law is considered to be valid, and the solutionsare based on the Darcys law.

7. The change in volume corresponds to the change in voidratio, and coecient of volume compressibility, mv, isconstant during consolidation process.

8. Indraratna et al. [12] assumption about the loss of vac-uum pressure along the vertical drain is employed, i.e.the vacuum loss is a linear increase with depth (seeFig. 3), in which p0 is the vacuum pressure at thepore water pressure with time at any given depth z withinthis cell.

Subsequently, we built another analytical model for theplane strain unit cell excluding plane strain smear zone,and then the function of development of excess pore waterpressure for this cell was developed.

Finally, by equating two obtained functions of excesspore water pressure, we found the conversion expressionof permeability from the axisymmetric unit cell to theequivalent plane strain unit cell under vacuum-surchargepreloading condition.

2.1. Analytical models of the axisymmetric and plane strainunit cells under vacuum-surcharge preloading

In general, these two models are depicted in Fig. 3. Inthese gures, r1 denotes the surcharge; p0 is the vacuumpressure applied to the top of the drain as well as thedrain-wall; k1 is the maintaining factor of vacuum pressure(0 6 k1 6 1); z the depth; l the length of the drainagepath of the drain under the condition that the bottom ofthe drain is undrained, and the excess PWP at the top ofthe drain is equal to p0; therefore, l is also equal tothe length of the unit cell in this ideal case (one homoge-neous soil layer); rw the equivalent drain radius; rs thesmear zone radius; R the equivalent radius of the inuencezone; bw the half-width of the drain-wall; B is the half-width of the plane strain unit cell.

Main assumptions of these models are:

1. The soil within the cell is fully saturated andhomogeneous.

2. The permeability of the soil is assumed to be constantduring consolidation.

3. The vertical ow within the soil of the relatively longunit cell is insignicant, i.e. it is assumed that only radialow occurs within the soil.

4. Equal strain hypothesis of Kjellman [14] is followed, i.e.the horizontal sections of the axisymmetric and planestrain unit cells remain horizontal during the consolida-tion process.

658 T.A. Tran, T. Mitachi / Computerstop of the drain, and k1p0 is the corresponding valueat the bottom of the drain.2.1.1. Analytical solution for the axisymmetric unit cell

(modied by the present authors after Indraratna et al.

[12])

It is observed that Indraratna et al.s [12] function of theexcess pore water pressure (PWP) is an average functionfor the whole axisymmetric cell at a given time. Therefore,we modied it to become the following function, which canshow the average value of the excess PWP throughout ahorizontal cross-section of the cell at any given depth zfor a given time t

u r1 p0 1 1 k1zl

h i exp 8T ha

lza

p0 1 1 k1zl

h i1

The detailed analytical formulation of Eq. (1) is given inAppendix A.

In Eq. (1)

lza lnnasa

khaksa

ln sa 34 pz2l z kha

qwa2

in which

na Rrw ; sa rsrw

where kha and ksa are horizontal permeability coecients ofthe axisymmetric unit cell in undisturbed zone and in smearzone, respectively; qwa the discharge capacity of the drain;Tha is the dimensionless time factor for horizontal drainageand

T ha Chat4R2

khamvcw

t

4R23

where Cha is the coecient of consolidation for horizontaldrainage; mv the coecient of volume compressibility forone-dimensional compression; t the time; cw is the unitweight of water.

2.1.2. The proposed analytical solution for the plane strain

unit cell excluding the plane strain smear zone

In 1992, Hird et al. [8] introduced a plane strain unit cellexcluding plane strain smear zone under conventionalembankment loading condition. By adapting Hird et al.s[8] unit cell for the case subjected to embankment com-bined with vacuum preloading, we developed the followingsolution for the average excess PWP within the adaptedunit cell

u r1 p0 1 1 k1zl

h i exp 8T hp

lzp

!

p0 1 1 k1zl

h i4

d Geotechnics 35 (2008) 655672The detailed analytical formulation of Eq. (4) is given inAppendix B.

In Eq. (4)

lzp 2

3 2khpBqwp

2lz z2 5

T hp Chpt4B2

t4B2

khpmvcw

6

in which khp is horizontal permeability coecient in theequivalent zone of the plane strain cell; qwp is dischargecapacity of the drain-wall; Thp and Chp are dimensionless

qwp 2B

pR2qwa 11

And, by using Eq. (10) as a condition for Eq. (9), the con-version expression of permeability for the equivalent planestrain unit cell can be obtained

khp 2B2

3R2kha

ln nasa 34 khaksa

ln sa12

In conclusion, our proposed method to convert from the

b n2n 1 npnp sp 1 3 sp sp 1

ratn

T.A. Tran, T. Mitachi / Computers and Geotechnics 35 (2008) 655672 659time factor and coecient of consolidation for horizontaldrainage, respectively.

2.2. The proposed conversion expression of permeability for

the proposed plane strain unit cell

By adapting the matching procedure of Hird et al. [8]under conventional surcharge preloading condition forthe proposed plane strain cell under vacuum-surcharge pre-loading condition, the following steps are conducted.

Equating Eqs. (1) and (4), the following equation can beobtained

T halza

T hplzp

7

Then, substituting Eqs. (3) and (6) into Eq. (7), and aftersome rearrangements, we can obtain

khaR2lza

khpB2lzp

8

Subsequently, substituting the expressions for lza and lzpinto Eq. (8), and the terms rearranged to give

2

3khaB2 ln nasa

3

4 kha

ksaln sa

khpR2

khpkhapR2

qwa khpkha2B

qwp

!2lz z2 9

If setting

khpkhapR2

qwa khpkha2B

qwp 0 10

then, the eect of well resistance is matched independently,as follows

Table 1Comparison of the conversion expressions of permeability between Indrapreloading condition

The conversion expression of permeabilityproposed by Indraratna et al. [12]

khpkha

a khpksp b hh i

ln nasa

khaksa

lnsa 34 p 2kha3qwa l2

h i p p

h 4khp3Bqwp

1 1np

l2; np Bbw ; sp

bsbw

a et al. [12] method and the proposed method under vacuum-surcharge

The conversion expression of permeabilityproposed by the present authors

khp 2B2

3R2kha

ln nasa 34 khaksa

ln sa12

andaxisymmetric unit cell to the plane strain unit cell undervacuum-surcharge preloading condition includes the com-bined use of both Eqs. (11) and (12).

2.3. The way to use our proposed conversion method

By inputting R, B, na, sa, kha, and the ratio kha/ksa intoEq. (12), we can obtain the horizontal permeability coe-cient for the equivalent zone between two vertical drainsin the full-scale plane strain simulation.

In the next step, usually, the information on dischargecapacity, qwa, of the vertical drain is available; therefore,by inputting parameters qwa, B, and R into Eq. (11), thevalue of qwp for the drain-wall of the plane strain cell canbe obtained.

Finally, inputting this qwp into the following equation,we can obtain kwp for the drain-walls in the full-scale planestrain simulation

kwp qwp2bw

13

In order to make clear the dierence between our conver-sion expression of permeability and that of Indraratnaet al. [12], we made the Table 1 as follows.

In Table 1, the meaning of symbols is

na Rrw ; sa rsrw

; a 23

np sp3n2pnp 1

;

2sp 1 1 2 qwp 2B

pR2qwa 11

khp and ksp are horizontal permeability coecients in theundisturbed and smear zones of Indraratna et al. [12] planestrain unit cell, respectively; in our proposed method, khp ishorizontal permeability coecient in the equivalent zone of

condition: Case 1 is a homogeneous clay layer that has thethickness being 10 m, and Case 2 is the case having two claylayers, in which the upper clay layer and the lower clay layerhave the thickness being 4.95 and 5.05 m, respectively.

a b

dy = 1

Equivalent zone

dz

B

dz

wb

qwp

Plane-strain undisturbed zone

dy = 1

wpq

sb

B

wb

Plane-strain smear zone

Drain wall Drain wall

Fig. 4. (a) A horizontal cross-sectional slice of Indraratna et al. [12] plane strain unit cell, and (b) that of our proposed plane strain unit cell.

660 T.A. Tran, T. Mitachi / Computers and Geotechnics 35 (2008) 655672our plane strain unit cell; bs is half-width of the plane strainsmear zone in the plane strain unit cell of Indraratna et al.[12] method (see Fig. 4). The meaning of other symbols waspreviously mentioned and is shown in Figs. 4 and A1 inAppendix A.

3. Verication of the proposed method via nite element

method

To verify the proposed method, we have conducted FEanalyses for two cases under vacuum-surcharge preloading

CEmbankment LThe case of one clay layer

Thic

knes

s H =

10

m

One homogeneousclay layer

k 1ha= k1va 1vsa= k = 4.30 E-4 (m/day)

1sak 1ha= (1/5)k

a

Fig. 5. Schematic description of (a) the case of one homogeneous clay layer andof each soil layer.In general, the illustration of these two cases is shownschematically in Fig. 5. In Fig. 5a, k1ha and k1sa are the hor-izontal permeability in the undisturbed and smear zones ofthe axisymmetric cell, respectively. k1va and k1vsa are thevertical permeability in the undisturbed and smear zones,respectively.

In Fig. 5b, the horizontal and vertical permeability ofthe upper clay layer is the same as those of the homoge-neous clay layer in Fig. 5a, but the horizontal and verticalpermeability of the lower clay layer is ve times higher thanthose of the upper clay layer.

Embankment CLH =

4.9

5 m

Lower clay layer

The case of two clay layers

H =

5.0

5 m

k1sa= (1/5)k1ha

= k= kk 2vsa2va 2ha2ha= (1/5)k2sak

= 5 (k )1ha

Upper clay layer

b

k = k = k = 4.30 E-4 (m/day) 1ha1vsa1va

of (b) the case of two clay layers, together with the permeability coecient

s an3.1. Detailed description of the case of one clay layer

Regarding the equivalent radius of the band-shapeddrain, rw, FE analyses performed by Rixner et al. [17]and supported by Hansbo [7] indicated that the equivalentdiameter of the band-shaped drain for use in practice canbe determined by

dw a b2

14

where a and b are the width and the thickness of the rect-angular cross-section of the band-shaped drain,respectively.

Commonly, the band-shaped drain or PVD has dimen-sions being 10 cm 0.4 cm. Using Eq. (14), we can obtaindw 5 cm and therefore rw 2.5 cm or 0.025 m.

Concerning the smear zone radius, rs, according to Jam-iolkowski and Lancellotta [13], the diameter of the smearzone, ds, can be in the range of 2.5dm to 3dm, where dm isthe equivalent diameter of the mandrel used for drivingPVD into the soft ground. Referring to the practice of vac-uum-surcharging in Japan, the mandrel usually has anequivalent diameter dm = 12 cm, and, in this study, wechose ds = 3dm. This leads to the smear zone diameter usedin this study is 36 cm or 0.36 m; therefore, rs = 0.18 m.

PVDs are commonly driven on a square grid; therefore,the equivalent radius of the inuence zone, R, needs to becalculated. Logically, it can be dened to be the radius of acircle having the same area as that of a square. Therefore,the following expression is obtained

R 1:128 S2

15

where S is the spacing between two vertical drains.We observed that, in Japan, S is usually 0.8 m. Hence,

by using Eq. (15), the equivalent radius R was determinedto be 0.45 m for FE analyses in this paper.

Regarding B for FE analyses in this paper, we made twoplane strain unit cells, one has B = 0.45 m and the otherone has B = 0.75 m, in which their permeability was con-verted by our conversion method from the axisymmetricunit cell having R = 0.45 m. For convenience, bw of thesetwo plane strain cells was chosen equal to rw (i.e.0.025 m) of the axisymmetric unit cell.

Concerning well resistance of vertical drains, we havetested our proposed conversion method with various valuesof well resistance that are selected on the basis of Mesri andLo [16] discharge capacity factor as follows

F d qwakhal

2 p kwa

kha

rwl

216

where kha is the horizontal permeability coecient inundisturbed zone of the axisymmetric unit cell; qwa the dis-charge capacity of the drain; l the drainage length of the

T.A. Tran, T. Mitachi / Computercell; rw the drain radius; kwa is the equivalent vertical per-meability coecient of the drain.Mesri and Lo [16] reported that well resistance is consid-ered to be insignicant if Fd is larger than 5. Therefore, wehave tested our conversion method for two values of Fd,which are 0.1 (i.e. very high well resistance) and 20 (i.e.no well resistance), respectively.

3.1.1. Boundary conditions and computed cases

Because one of the basic assumptions of our analyticalsolutions is based on equal strain hypothesis, therefore,we have conducted FE analyses as follows:

(1) All the drain, the smear and undisturbed zones of theaxisymmetric cells, and both the drain-wall and theequivalent zone of the plane strain cells convertedby our method were simulated by linear elastic mod-els having the same elastic modulus (E = 1000 kN/m2) and zero Poissons ratio.

(2) In the case of the axisymmetric cell, to avoid horizon-tal displacement from the smear zone to the drain orvice versa, the nodes on the boundary between thedrain and the smear zone are allowed to move inthe vertical direction only. Similarly, the nodes onthe boundary between the drain-wall and the equiva-lent zone of the plane strain cell are also allowed tomove in the vertical direction only.

(3) An undrained rigid plate was put on the surface of eachof these axisymmetric and plane strain cells to ensurethe uniform settlement at the surface of the cells (seeFig. 7). In addition, in accordance with the boundarycondition of the analytical solutions, the vacuum pres-sure is applied to the top of the drain, not to the surfaceof the cell; besides, all the vertical permeability of boththe axisymmetric and plane strain cells are set equal tozero. This case, we named it VTD-ES (vacuum at thetop of the drain with equal strain).

In addition, we carried out other FE analyses to checkthe applicability of our method in the case of free-strainis allowed at the surface of the cell; in this case, theundrained rigid plate in Fig. 7 is removed from the surfaceof the cells, and the vacuum pressure is applied to both thetop of the drain and the surface of the cell. We named thiscase VS-FS (vacuum pressure applied to the surface withfree strain).

In 1986, Rixner et al. [17] reported that, of clay soil withno or slightly developed macrofabric, essentially homoge-neous deposits, the ratio of the horizontal permeability tothe vertical permeability, kh/kv, can be in the range of 11.5. Based on this report, for the case VS-FS, the verticalpermeability of the axisymmetric and plane strain cells ischosen as follows:

(1) We assume the ratio, kha/kva, equal to 1 for the undis-turbed zone of the axisymmetric unit cell.

(2) For the smear zone of the axisymmetric unit cell, we

d Geotechnics 35 (2008) 655672 661assume its vertical permeability, kvsa, is equal to thevertical permeability of the undisturbed zone, kva.

(3) For the plane strain cells converted by our method,the vertical permeability of the equivalent zone, kvp,is assumed to be equal to kva.

According to Kobayashi et al. [15], the horizontal per-meability of the smear zone of the clay ksa can be decreasedto 1/5 of that of the undisturbed zone kha. Therefore, forthe axisymmetric cell, we assumed the ratio ksa/kha, to beequal to 1/5.

In summary, all computed cases are illustrated in Fig. 6,and all input parameters used for the case of one clay layerare tabulated in Table 2. And, the boundary conditions andmeshes of the cells are shown in Fig. 7.

Note that, in engineering practice, the value of qwa isavailable and then qwp can be calculated by using Eq.(11). In this paper, for the purpose of investigating theeect of well resistance on each of cases computed byIndraratna et al. [12] method and by our method, thedischarge capacity factor Fd was rstly assumed asshown in Table 2, and then qwa was calculated basedon Eq. (16).

Regarding the FEM program used, the Sage Crisp pro-gram developed by the CRISP Consortium Ltd. and SAGEEngineering Ltd. [18] on the basis of consolidation theoryof Biot [1] was employed; we used linear strain quadrilat-eral elements that incorporate quadratic displacementnodes together with linearly interpolated pore pressure

662 T.A. Tran, T. Mitachi / Computers annodes in this program (see Fig. 8).

The case of two clay layers

The case of one homogeneous clay layer

k = 11

k = 11

k = 0.51

1k = 0.5

o0.5p

op

opp

op

o

op

op

VTD-ES VS-FS

0.5po

oo

o0.5p 0.5p

o

ppoo

op

op

pp

VS-FSVTD-ES

VS-FSVTD-ESVS-FSVTD-ESFig. 6. Illustration of computed cases.3.2. Detailed description of the case of two clay layers

We observed that the conversion method proposed byIndraratna et al. [12] showed very good matching resultsfor the case of one clay layer, in their paper. Therefore,a check on the applicability of both our proposedmethod and Indraratna et al. [12] method for the caseof two clay layers would be worthwhile; for this reason,FE analyses for the case of two clay layers were con-ducted in this study.

In this two-clay-layer case, the geometric parameters ofthe axisymmetric unit cell, R, rs, rw, were chosen the sameas those of the axisymmetric unit cell in the case of one claylayer listed in Table 2. In general, the boundary conditionsand the mesh of the cells are shown in Fig. 9.

With regard to the soil properties and material param-eters of the axisymmetric unit cell in this two-clay-layercase.

For the vertical drain, all material parameters wereassumed to be the same as those in the case of one homo-geneous clay layer.

For the upper and lower clay layers, all soil propertiesexcept permeability were assumed to be the same as thoseof the case of one homogeneous clay layer. For the perme-ability of the smear and undisturbed zones in each layer,the assumption is illustrated in Fig. 5b.

Note that, for the plane strain cell converted by Ind-raratna et al. [12] method, we assumed the ratio of thesmear zone permeability to the undisturbed zone perme-ability, ksp/khp, equal to 1/5. On the other hand, the planestrain unit cell of our proposed method has no plane strainsmear zone; therefore, such a kind of ratio is not requiredin our plane strain unit cell.

Regarding the assumption of vertical permeability, ofboth the two clay layers, the vertical permeability in thesmear and undisturbed zones of both the axisymmetric celland Indraratna et al. [12] plane strain cell is assumed to bezero in the case of VTD-ES. Similarly, the vertical perme-ability in the equivalent zone of our proposed plane straincell is also assumed to be zero.

In the case of VS-FS, the vertical permeability of theplane strain cells was assumed as follows:

(1) For the plane strain cells converted by Indraratnaet al. [12] method, for the upper and lower clay layers,both the vertical permeability of the smear zone, kvsp,and of the undisturbed zone, kvp, are equal to the ver-tical permeability of the undisturbed zone, kva, of thecorresponding layer in the axisymmetric cell. Thismeans that kvsp = kvp = kva for each of the upperand lower clay layers.

(2) For the plane strain cells converted by our proposedmethod, in each of the two clay layers, the verticalpermeability of the equivalent zone, kvp, is the sameas that of the undisturbed zone in the axisymmetric

d Geotechnics 35 (2008) 655672cell. It should be noted here that kvsp is not neededin our proposed plane strain cell.

Table 2Input parameters of the axisymmetric and plane strain unit cells for the case of one clay layer

Unit cell parameter Symbol Computed cases

VTD-ES VTD-ES VS-FS VS-FS

The axisymmetric unit cell

Radius of the axisymmetric cell (m) R 0.45 0.45 0.45 0.45Smear zone radius (m) rs 0.180 0.180 0.180 0.180Drain radius (m) rw 0.025 0.025 0.025 0.025Discharge factor Fd 20 0.1 20 0.1

Elastic modulus (kN/m2) and Poissonsratio of the drain

E and m E = 1000 and m = 0

Horizontal permeability of the undisturbedzone (m/day)

kha 4.30E04 4.30E04 4.30E04 4.30E04

Horizontal permeability of the smear zone(m/day)

kha (1/5)kha (1/5)kha (1/5)kha (1/5)kha

Vertical permeability of the undisturbed andsmear zones (m/day)

kva andkvsa

0.0 0.0 4.30E04 4.30E04

Elastic modulus (kN/m2) and Poissonsratio of the undisturbed and smear zones

E and m E = 1000 and m = 0

Computed cases

VTD-ES VTD-ES VTD-ES VTD-ES VS-FS VS-FS VS-FS VS-FS

The plane strain unit cell converted by our proposed method

Half-width of the plane strain cell (m) B 0.45 0.75 0.45 0.75 0.45 0.75 0.45 0.75Half-width of the drain-wall (m) bw 0.025 0.025 0.025 0.025 0.025 0.025 0.025 0.025Elastic modulus (kN/m2) and Poissons

ratio of the drain-wallE and m E = 1000 and m = 0

Horizontal permeability of the equivalentzone (m/day)

k#1hp 2.86E05 7.93E05 2.86E05 7.93E05 2.86E05 7.93E05 2.86E05 7.93E05

Vertical permeability of the equivalent zone(m/day)

kvp 0.0 0.0 0.0 0.0 4.30E04 4.30E04 4.30E04 4.30E04

Elastic modulus (kN/m2) and Poissonsratio of the equivalent zone

E and m E = 1000 and m = 0

Note: The superscript #1 in the table means that this value was calculated based on Eq. (12).

(a) Axisymmetric unit cell, R = 0.45 m

w

Impermeable boundary

Vertical roller boundaryVertical roller boundary

Centreline of the drainImpermeable boundary

Fixed, impermeable boundary

Horizontal roller boundaryHalf drain, r = 2.5 cm

Smear zone, 15.5 cmFixed, impermeable boundary

Undisturbed zone, 27 cm

Impermeable boundaryPeriphery of the cell

50 kPaC

Undrained rigid plateTop of the cell

Periphery of the drainVertical roller boundary

(b) Plane strain unit cell, B = 0.45 m

Equivalent zone, 42.5 cm

Half drain wall, b = 2.5 cmHorizontal roller boundary


Impermeable boundaryw

Negative 50 kPa excess pwpTop of the drain

Top of the cellUndrained rigid plate

50 kPa

Periphery of the cellImpermeable boundaryVertical roller boundary

Half drain wall, b = 2.5 cmHorizontal roller boundary

Impermeable boundaryw

Fixed, impermeable boundaryEquivalent zone, 72.5 cm

Top of the cellUndrained rigid plate

CL50 kPa


Vertical roller boundaryPeriphery of the drain

Impermeable boundaryCentreline of the drain

Vertical roller boundary

H=

10

m

Vertical roller boundary

Centreline of the drainImpermeable boundary

Vertical roller boundaryPeriphery of the drain

Top of the drainNegative 50 kPa excess pwp


(c) Plane strain unit cell, B = 0.75 m

L CL

Fig. 7. Boundary conditions and meshes of the axisymmetric and plane strain cells used in VTD-ES cases of one homogeneous clay layer.

T.A. Tran, T. Mitachi / Computers and Geotechnics 35 (2008) 655672 663

In summary, all computed cases for this two-clay-layercase are illustrated in Fig. 6, and all input parameters usedfor these plane strain cells are shown in Tables 3 and 4.

In this two-clay-layer case, when Indraratna et al. [12]conversion expression was used for the lower clay layer,then a question arose as to whether we should choose thelength of the drainage path of the drain (l) of the axisym-metric unit cell in the lower clay layer equal to the totalthickness of both two clay layers (l = 10 m) or equal to

In Table 4, in the case that the plane strain cell hasB = 0.75 m together with qwp = 0.0043 m

3/day (i.e. corre-sponding to Fd = 0.1 of the axisymmetric cell), the con-verted permeability of the lower clay layer based onIndraratna et al. [12] method becomes negative, eventhough using either l = 10 or 5.05 m. Therefore, we couldnot model this case by Indraratna et al. [12] method. Forthis reason, the comparison between our method andIndraratna et al. [12] method was not conducted for thiscase.

Also in Table 4, when the plane strain cell hasB = 0.45 m together with qwp = 0.0043 m

3/day, the con-verted permeability of the lower clay layer based on Ind-raratna et al. [12] method also becomes negative if usingl = 10 m (i.e. equal to the total thickness of both two claylayers). Hence, for the lower clay layer, we did not choosel = 10 m, but chose l = 5.05 m (i.e. equal to the thickness ofthis layer) to input into the conversion expression of Ind-raratna et al. [12].

4. Results and discussion

4.1. In the case of one homogeneous clay layer

Pa exop of

l rolhery

Undrained rigid plateTop of the cell

late

ndaryndar

Integration pointPore pressure unknownDisplacement unknown

Fig. 8. The element type used for FE analyses.

664 T.A. Tran, T. Mitachi / Computers and Geotechnics 35 (2008) 655672the thickness of the lower clay layer only (l = 5.05 m). Allpermeability for the lower clay layer of plane strain cellsconverted by Indraratna et al. [12] conversion expression,in accordance with both l = 10 and 5.05 m, are listed inTable 4.

Vertical roller boundaryImpermeable boundary

Periphery of the drainVertical roller boundary


Negative 50 kT

VerticaPerip

Top of the cellUndrained rigid p

C50 kPa

Impermeable bouVertical roller bou

LCentreline of the drain

wHalf drain, r = 2.5 cmImpermeable boundary

Horizontal roller boundary


Fixed, impermeable boundarySmear zone, 15.5 cm

Undisturbed zone, 27 cm

(a) Axisymmetric unit cell, R = 0.45 m

Centreline

Vertical rolImpermeab

Periphery of the cell

Fig. 9. Boundary conditions and meshes of the axisymmetric andw

Impermeable boundaryHalf drain wall, b = 2.5 cmHorizontal roller boundary

Fixed, impermeable boundaryEquivalent zone, 42.5 cm

(b) Plane strain unit cell, B = 0.45 m

H=

4.

95 mler boundary

of the drain

of the drain

ler boundaryle boundary


H=

5.

05 m

Upp

er c

lay

laye

rLo

wer

cla

y la

yer

yFE results of the degree of consolidation of the axisym-metric cell, and of our proposed plane strain cell are shownin Figs. 10 and 11, in which all curves of degree of consol-idation were calculated based on the surface settlement ofthe soil layer.

cess pwp the drain

50 kPaCLplane strain cells used in VTD-ES cases of two clay layers.

or th

ES

1

05

04

q. (1s va

s anThe proposed conversion method was validated in thecase of VTD-ES with k1 = 1. As shown in Fig. 10a, itcan be seen that a good agreement in the degree of

Table 3Input parameters of the plane strain unit cells converted by our method f

Parameters of the proposed plane strain unitcell

Symbol Computed cases

VTD-ES VTD-

Half-width of the plane strain cell (m) B 0.45 0.75Half-width of the drain-wall (m) bwDischarge capacity of the drain-wall (m3/

day)qwp 1.222

1 1.222

Elastic modulus (kN/m2) and Poissonsratio of the drain-wall

E and m

The upper clay layer


k#1hp 2.86E05 7.93E


kvp 0.0 0.0


E and m

The lower clay layer


k#1hp 1.43E04 3.97E


kvp 0.0 0.0


E and m

Note: The superscript #1means that this value was calculated based on E(11) and is corresponding to Fd = 20. The superscript 2 means that thi

T.A. Tran, T. Mitachi / Computerconsolidation, between the proposed plane strain cell andthe axisymmetric cell, was obtained not only under no-well-resistance condition, but also under high wellresistance condition. After that, the proposed methodwas also examined in the case of VS-FS with k1 = 1; asshown in Fig. 10b, the same good agreement as that inFig. 10a was obtained under both no-well resistance andhigh well resistance conditions.

Further, we tested the proposed method in the case ofVTD-ES with k1 = 0.5. The results in Fig. 11a indicatedthat, under both no well resistance and high well resistanceconditions, the proposed method produced good matchingresults. Finally, we tested the proposed method in the caseof VS-FS with k1 = 0.5. As can be seen in Fig. 11b, thesame good matching results as those in the case of VTD-ES (k1 = 0.5) were also obtained.

4.2. In the case of two clay layers

The FE results of the degree of consolidation of theaxisymmetric cell, of the Indraratna et al. [12] planestrain cell, and of our proposed plane strain cell are pre-sented in Fig. 12, in which all curves of degree of consol-idation were calculated based on the surface settlementof soil layers.

Besides, the dierence in the degree of consolidation(Ua Up) between the axisymmetric unit cell (Ua) andthe plane train unit cell (Up) of Indraratna et al. [12]method, and the plane strain cell (Up) of our proposedmethod is illustrated in Fig. 13.

As shown in Fig. 12a and b, the results of degree of con-

e case of two clay layers

VTD-ES VTD-ES VS-FS VS-FS VS-FS VS-FS

0.45 0.75 0.45 0.75 0.45 0.750.025

0.00612 0.00612 1.2221 1.2221 0.00612 0.00612

E = 1000 and m = 0

2.86E05 7.93E05 2.86E05 7.93E05 2.86E05 7.93E05

0.0 0.0 4.30E04 4.30E04 4.30E04 4.30E04

E = 1000 and m = 0

1.43E04 3.97E04 1.43E04 3.97E04 1.43E04 3.97E04

0.0 0.0 2.15E03 2.15E03 2.15E03 2.15E03

E = 1000 and m = 0

2). The superscript 1means that this value was calculated based on Eq.lue was calculated based on Eq. (11) and is corresponding to Fd = 0.1.

d Geotechnics 35 (2008) 655672 665solidation revealed that both methods are very good underno well resistance condition (Fd = 20).

Fig. 13a shows that, under the condition of Fd = 20, themaximum dierence in the degree of consolidation(Ua Up), of the whole two clay layers is 2% for Ind-raratna et al. [12] method, and 4% for the proposedmethod. This means that our method produced matchingresults which are almost as good as those of Indraratnaet al. [12] method (under condition of Fd = 20). Besides,also under condition of Fd = 20, Fig. 13b showed thatthe maximum dierence in the degree of consolidation ofthe lower clay layer is 1% for Indraratna et al. [12] method,and 2.5% for the proposed method.

Under high well resistance condition, Figs. 12a and13a revealed that, of the whole two clay layers, the max-imum dierence (Ua Up) of Indraratna et al. [12]method is 9%, which is considered to be fairly high,whereas the maximum dierence of the proposed methodis 2%. Figs. 12b and 13b show that, of the lower claylayer, the maximum dierence (Ua Up) under high wellresistance condition of Indraratna et al. [12] method evenreach 14%, whereas the dierence of the proposedmethod is less than 1%.

Further, we examined both methods in the case of VS-FS with k1 = 0.5 as shown in Fig. 12c, d and 13c, d. Ascan be seen in these gures, almost the same results as thosein the case of VTD-ES (k1 = 0.5) are obtained. However, inthis VS-FS case, the maximum dierence in the degree of

Table 4Input parameters of the plane strain unit cells converted by Indraratna et al. [12] method for the case of two clay layers

Parameters of the plane strain unitcell of Indraratna et al. [12]

Symbol Computed cases

VTD-ES VTD-ES VTD-ES VTD-ES VS-FS VS-FS VS-FS VS-FS

Half-width of plane strain cell (m) B 0.45 0.75 0.45 0.75 0.45 0.75 0.45 0.75Half-width of drain-wall (m) bw 0.025Discharge capacity of the drain-wall

(m3/day)qwp 0.864

1 0.8641 0.00432 0.00432 0.8641 0.8641 0.00432 0.00432

Elastic modulus (kN/m2) andPoissons ratio of the drain-wall

E and m E = 1000 and m = 0

The upper clay layer; we chose l = 4.95 m (i.e. equal to the thickness of this layer)Horizontal permeability of the

undisturbed zone (m/day)k#2hp 1.01E04 2.27E04 2.27E04 6.55E04 1.01E04 2.27E04 2.27E04 6.55E04

Horizontal permeability of the smearzone (m/day)

ksp (1/5)khp

Vertical permeability of theundisturbed and smear zones(m/day)

kvp andkvsp

0.0 0.0 0.0 0.0 4.30E04 4.30E04 4.30E04 4.30E04

Elastic modulus (kN/m2) andPoissons ratio of the undisturbedand smear zones

E and m E = 1000 and m = 0

The lower clay layer; if choosing l = 5.05 m (i.e. equal to the thickness of this layer)Horizontal permeability of the



ksp (1/5)khp

Vertical permeability of theundisturbed and smear zones(m/day)

kvp andkvsp

0.0 0.0 0.0 0.0 2.15E03 2.15E03 2.15E03 2.15E03

Elastic modulus (kN/m2) andPoissons ratio of the undisturbedand smear zones

E and m E = 1000 and m = 0

The lower clay layer; if choosing l = 10 m (i.e. equal to the total thickness of the two clay layers)Horizontal permeability of the



ksp (1/5)khp

Note: The superscript #2means that this value was calculated based on Eq. (*) in Table 1. The superscript 1 means that this value was assumed to beequal to qwa, which is corresponding to Fd = 20. The superscript 2 means that this value was assumed to be equal to qwa, but, in this case, qwa iscorresponding to Fd = 0.1.

(b) VS-FS with k1=1

0

20

40

60

80

100

0.1 1 10 100Time factor T h

Deg

ree

of c

on

so

lidat

ion

Uh (%

)

(a) VTD-ES with k1=1

0

20

40

60

80

100

0.1 1 10 100Time factor T h

Deg

ree

of c

on

so

lidat

ion

Uh (%

)

B0.45: ProposedR0.45: Axisymmetric

B0.75: Proposed


B0.75: Proposed

R0.45: Axisymmetric

B0.45: ProposedB0.75: Proposed

R0.45: Axisymmetric


o

op

po

op

p

i

h

i

: =

:

i

i

:

:

Fig. 10. Comparison of FEM results of the axisymmetric unit cell having R = 0.45 m (R0.45: Axisymmetric) with that of the plane strain unit cell havingB = 0.45 m (B0.45: Proposed) and with that of the plane strain unit cell having B = 0.75 m (B0.75: Proposed); these graphs corresponding to the case ofone clay layer and k1 = 1.

666 T.A. Tran, T. Mitachi / Computers and Geotechnics 35 (2008) 655672

0.1 1 10 100

(a) VTD-ES with k1=0.5

0.1 1 10 100

Time factor Th

= 0.1>


B0.75: Proposed


B0.75: Proposed


R0.45: AxisymmetricB0.45:

B0.75:

o

o0.5p

po

0.5po

p

1

f

= 0.1>

.7

:

. 5:

(b) VS-FS with k1=0.5 1

0

20

40

60

80

100

Time factor Th f

Deg

ree

ofc

on

so

lida

tion

Uh (%

)0

20

40

60

80

100

Deg

ree

ofc

on

so

lida

tion

Uh (%

)

R0.45: Axisymmetric

Proposed

:

5: Proposed

o

Fig. 11. Comparison of FEM results of the axisymmetric unit cell having R = 0.45 m (R0.45: Axisymmetric) with that of the plane strain unit cell havingB = 0.45 m (B0.45: Proposed) and with that of the plane strain unit cell having B = 0.75 m (B0.75: Proposed); these graphs corresponding to the case ofone clay layer and k1 = 0.5.

0 0

60

80

100

0.1 1 10Time factor T

Deg

ree

of c

on

so

lida

tion

Uh (%

)

o0.5p

op

o

0.5po

of lower clay layerDegree of consolidation

o0.5p

op p

o

0.5po

Degree of consolidationof lower clay layer

B0.45:R0.45:

B0.45: In


B0.45: Indraratna


B0.45: Indraratna


B0.45: Indraratna


B0.45: Indraratna


B0.45: Indraratna

VTD-ES with k1=0.5; Uh for two clay layers

20

40

100

0.1 1 10 100Time factor T

0.1 1 10Time factor T

100


0

60

80

100

Deg

ree

of c

on

so

lida

tion

Uh (%

)

20

40

0

60

80

100

Deg

ree

of c

on

so

lida

tion

Uh (%

)

20

40

60

80

100

Deg

ree

of c

on

so

lida

tion

Uh (%

)

20

40

(a) VTD-ES with k1=0.5; Uh for lower clay layer (b)

VS- FS with k1=0.5; Uh for lower clay layer (d) VS- FS with k1=0.5; Uh for two clay layers (c)

p

p

of lower clay layerDegree of consolidation

o0.5p

op p

o

0.5po

Degree of consolidationof lower clay layer

ProposedAxisymmetric

draratna

B0.45:R0.45:

B0.45: In ProposedAxisymmetric

draratna

B0.45:R0.45:

B0.45: In ProposedAxisymmetric

draratna


B0.45: Indraratna


B0.45: Indraratna


B0.45: Indraratna


B0.45: Indraratna


B0.45: Indraratna

Fig. 12. Comparison of FEM results of the axisymmetric unit cell having R = 0.45 m (R0.45: axisymmetric) with that of the plane strain unit cellconverted by Indraratna et al. [12] method (B0.45: Indraratna) and with that of the plane strain unit cell converted by the proposed method (B0.45:Proposed); these graphs corresponding to the case of the two clay layers and k1 = 0.5.

T.A. Tran, T. Mitachi / Computers and Geotechnics 35 (2008) 655672 667

(b

U a

U p

(%)

(b

U a

U p

(%)

an(a) VTD-ES with k1=0.5; (U a - U p) for two clay layers


U a

U p (%

)

14

Indraratna

Proposed

Indraratna

Proposed

(a) VTD-ES with k1=0.5; (U a - U p) for two clay layers-14-12-10-8-6-4-202468

101214 10 100

Th


10 100 Th

14

Indraratna

Proposed

Indraratna

Proposed

Indraratna

Proposed

668 T.A. Tran, T. Mitachi / Computersconsolidation in the lower clay layer of Indraratna et al.[12] method is 12%, i.e. little smaller than the value 14%in the case of VTD-ES, but this value 12% is still be consid-ered to be signicant.

5. Conclusions

By modifying Indraratna et al. [12] method, which wasdeveloped for cases of single layer subsoil, a new conver-sion method has been proposed in this study to convertan axisymmetric unit cell to an equivalent plane strain cellunder vacuum-surcharge preloading condition.

In the proposed method, the widths and the permeabil-ity of the smear and undisturbed zones in the axisymmetriccell are converted theoretically into an equivalent perme-ability of the equivalent zone in the proposed plane straincell. And, as the proposed cell has no plane strain smearzone, a full-scale plane strain simulation of soft groundimproved by vertical drains under vacuum-surcharge pre-loading can be conveniently made.

(c) VS-FS with k1=0.5; (U a - U p) for two clay layers-14

-12

-10-8-6-4

-2

02

4

68

10

12

U a

U p (%

)

(-

-

-

U a

U p

(%)

Indraratna

Proposed

Indraratna

Proposed

(c) VS-FS with k1=0.5; (U a - U p) for two clay layers-14

-12

-10-8-6-4

-2

02

4

68

10

12

U a

U p (%

)

(-

-

-

U a

U p

(%)

Indraratna

Proposed

Indraratna

Proposed

Indraratna

Proposed

Indraratna

Proposed

Fig. 13. The dierence in the degree of consolidation between the axisymmetrimethod, and the plane strain unit cell converted by the proposed method; thesshown in Fig. 12.) VTD-ES with k1=0.5; (U a - U p) for lower clay layer

14

Indraratna

Proposed

Indraratna

Proposed

-14-12-10-8-6-4-202468

1012


10 100 Th


10 100 Th

) VTD-ES with k1=0.5; (U a - U p) for lower clay layer

14

Indraratna

Proposed

Indraratna

Proposed

Indraratna

Proposed

d Geotechnics 35 (2008) 655672Besides, in the proposed method, the determination ofthe length of the drainage path of the drain (l) in each soillayer (of a multi-layer subsoil) is not required in determin-ing the equivalent permeability for each soil layer.

The proposed method was validated via analyzing con-solidation of the axisymmetric and plane strain unit cells intwo cases, one is a homogeneous clay layer, and the otherone is a two-clay-layer case. The vertical drains analyzed inthese cases are under two conditions, no well resistance(Fd = 20), and high well resistance (Fd = 0.1). The analyzedresults showed that the proposed method produced verygood agreements in all the cases. In addition, the resultsalso indicated that the proposed method can be used well,not only under equal strain condition, but also under freestrain condition.

Acknowledgements

The nancial support from JICA (Japan InternationalCooperation Agency) for this study through AUN/

d) VS-FS with k1=0.5; (U a - U p) for lower clay layer14

12

10-8

-6-4

-2

02

4

6

810

12

Indraratna

Proposed

Indraratna

Proposed

d) VS-FS with k1=0.5; (U a - U p) for lower clay layer14

12

10-8

-6-4

-2

02

4

6

810

12

Indraratna

Proposed

Indraratna

Proposed

Indraratna

Proposed

Indraratna

Proposed

c unit cell and the plane strain unit cell converted by Indraratna et al. [12]e graphs corresponding to the case of two clay layers with k1 = 0.5 that is

SEED-Net Project (ASEAN University Network/South-east Asia Engineering Education Development NetworkProject) is greatly appreciated. Besides, the authors wouldlike to thank Associate Professor, Dr. Hiroyuki Tanakaof Hokkaido University very much for his various com-ments and suggestions on this study.

Appendix A. Formulation of the analytical solution for the

average excess pore pressure of the axisymmetric unit cell

Consider a horizontal cross-sectional slice with a thick-ness dz from the radius r to the outer radius R of the axi-symmetric unit cell (see Fig. A1a).

The centripetal ow rate (into the drain) in the slice atradius r can be dened by

oq kh ou 2prdz A1

For smear zone, rw 6 r 6 rs we obtain:ousor

cw2

oeot

1

ksa

R2 r2r

A3b

where us and ksa are excess pore water pressure and hori-zontal permeability coecient of soil in the smear zone,respectively.

Considering a horizontal cross-sectional slice with athickness dz of the circular cylindrical drain with radiusrw (Fig. A1b), the change of vertical ow of water withinthe drain, in the z direction, from the entrance face to theexit face of the slice can be calculated by

dqz qwacw

o2uoz2

dzdt A4

where qwa is the discharge capacity of the drain.The horizontal inow of water, in the radial direction,

T.A. Tran, T. Mitachi / Computers and Geotechnics 35 (2008) 655672 669ot cw or

where q, u, cw, and kh are horizontal ow of water in thesoil mass, excess pore water pressure, unit weight of water,and horizontal permeability coecient of soil, respectively.

The rate of the soil volume change of the slice, from theradius r to the outer radius R, in the vertical direction canbe determined by

oVot

oeot

pR2 r2dz A2

where V and e are volume of the soil mass and verticalstrain, respectively.

Assuming that water is incompressible, therefore Eq.(A1) = Eq. (A2).

For undisturbed zone, rs 6 r 6 R we can obtainouor

cw2

oeot

1

kha

R2 r2r

A3a

where u and kha are excess pore water pressure and hori-zontal permeability coecient of the soil in the undisturbedzone, respectively.

Undisturbed zoneUndisturbed zone

Smear zone

q q

dz

r

R

rs

wrFig. A1. A horizontal cross-sectionalfrom the outer face of the slice of the circular cylindricaldrain is determined by

dqr 2prwksa

cw

ouor

dzdt for r rw A5

where rw is the equivalent radius of the drain.For continuity of ow, the following equation must be

satised

dqz dqr A6therefore

ouor

rw

qwa2prwksa

o2uoz2

rw

0 for r rw A7

Substituting Eq. (A3b) into Eq. (A7), we obtain

o2uwoz2

cwpr2w

qwan2a 1 oe

otA8

where na is the ratio R/rw; uw is the excess pore water pres-sure at rw.

rw

sr

R

zq

zzq + dq

dz

rdqdqr

Smear zone

Undisturbed zoneslice of the axisymmetric unit cell.

By using the following boundary conditions:

At z 0 : uw p0.At z l : ouwoz p0 1k1l

.

The solution of Eq. (A8) can be given by

uw p0 1 1 k1zl

h i

cwpr2w

qwan2a 1 oe

otlz z

2

2

A9

After that, integrating Eq. (A3b) in the r direction with the

Let u be the average excess pore water pressure throughouta horizontal cross-section at depth z and for a given time, t

u R rsrw2pusrdr

R Rrs2purdr

p R2 r2w A13

After substituting Eqs. (A11) and (A12) into Eq. (A13),and integrating it, nally Eq. (A13) becomes

u p0 1 1 k1zl

h i cw

2

oeot

R2

khaln

nasa

khaksa

ln sa 34 pz2l z kha

qwa

A14

at t = 0, u r1, then the following expression can be

670 T.A. Tran, T. Mitachi / Computers and Geotechnics 35 (2008) 655672boundary condition that at r = rw, us = uw (i.e. Eq. (A9)).Therefore, for rw 6 r 6 rs, the following equation is

obtained

us p0 1 1 k1zl

h i cw

2

oeot

1ksa

R2 lnrrw

r2 r2w2

ksa pr2w

qwan2a 12lz z2

A10

or

us pz cw2

oeot

1ksa

R2 lnrrw

r2 r2w2

ksaqz

A11

where

pz p0 1 1 k1zl

h iand qz

pr2w

qwan2a 1 2lz z2

Then, integrating Eq. (A3a) along the r direction with theboundary condition that at r = rs, u = us (i.e. Eq. (A11)).

Hence, for rs 6 r 6 R, we can obtain the followingequation

u pz cw2

oeot

1

ksaR2 ln

rrs r

2 r2s2

khaksa

R2 lnrsrw

r2s r2w2

khaqz

A12

q q

x

B

wb

Equivalent zone

dy =1Fig. B1. A horizontal cross-sectionobtained

u r1 p0 1 1 k1zl

h i exp 8T ha

lza

p0 1 1 k1zl

h iA16

where

lza lnnasa

khaksa

ln sa 34 pz2l z kha

qwaA17

T ha Chat4R2

khamvcw

t

4R2A18

Appendix B. Formulation of the analytical solution for theaverage excess pore pressure of the plane strain unit cell

Let us consider a horizontal cross-sectional slice with athickness dz from the vertical cross-section x to the widthB of the plane strain unit cell (see Fig. B1a).

In the slice, the horizontal ow rate into the drain, atcross-section x, can be dened by

B

dz

qz dqxxdq

wb

q + dqz zdz

Equivalent zone

dy= 1We assume that

oeot

mv or0

ot mv ouot A15

where mv is the coecient of volume compressibility forone dimensional compression.

Substituting Eq. (A15) into Eq. (A14) then integrating itwith time, t, and introducing the boundary condition thatal slice of the plane strain cell.

2cw B bw oe lz z B9

s anoqot

khpcw

ouox

dzdy khpcw

ouox

dz 1 khpcw

ouox

dz B1

where q is the horizontal ow of water in the soil mass; khpis the equivalent horizontal permeability coecient of thesoil in the plane strain cell.

The rate of the soil volume change of the slice in the ver-tical direction, within the space from width x to the widthB, can be determined by

oVot

oeotB xdz B2

where V, e, and B are volume of the soil mass, verticalstrain, and half-width of the equivalent plane strain cell,respectively.

Assume that water is incompressible, therefore Eq.(B1) = Eq. (B2), then the following equation can beobtained

ouox

cwkhp

oeotB x B3

where u is excess pore water pressure in the equivalent zoneof the plane strain cell; khp is the equivalent horizontal per-meability coecient of the equivalent zone of the planestrain cell.

The change of vertical water ow within the drain-wall,in the z direction, from the entrance face to the exit face ofthe slice (see Fig. B1b) can be calculated by

dqz qwpcw

o2uoz2

dzdt B4

where qwp is the discharge capacity of the drain-wall.The horizontal inow of water, from the outer face of

the drain-wall slice, ows into the drain-wall, can be deter-mined by

dqx khpcw

ouox

dzdt for x bw B5

where bw is the half-width of the drain-wall.For continuity of ow, the following equation must be

satised

dqz 2dqx B6

therefore, it leads to

o2uoz2

2khp

qwp

ouox

bw

for x bw B7

Substituting Eq. (B3) into Eq. (B7), we obtain

o2uoz2

2cwqwp

oeotB bw at x bw B8

By using the following boundary conditions:

At z = 0 and x = bw: u = p0.

T.A. Tran, T. Mitachi / ComputerAt z = l and x bw: ouoz p0 1k1l

.qwp ot 2

Integrating Eq. (B3) with the following boundarycondition.

At x = bw, we have u = uw being Eq. (B9).Therefore, for bw 6 x 6 B, the following equation is

obtained

u pz cwoeot

1

2kp2Bx x2 B bw

qwp2lz z2

"

12kp

2Bbw b2w # B10

where

pz p0 1 1 k1zl

h iLet u be the average excess pore water pressure through-

out a horizontal cross-section at a given depth z and for agiven time, t

u R Bbwudx

B bw B11

After substituting Eq. (B10) into Eq. (B11), and integratingit, nally Eq. (B11) becomes

u p0 1 1 k1zl

h i B bw

2

khp

cw2

oeot

2

3 2khpB bwqwp

2lz z2" #

B12

Substituting Eq. (A15) (in Appendix A) into Eq. (B12),then integrating it with boundary condition that at t = 0,u r1, nally the following expression can be obtained

u r1 p0 1 1 k1zl

h i exp 8T hp

lzp

!

p0 1 1 k1zl

h iB13

where

lzp 2

3 2khp

Bqwp2lz z2 B14

T hp Chpt4B2

t4B2

khpmvcw

B15

References

[1] Biot MA. General theory of three-dimensional consolidation. J ApplPhys 1941;12:15564.The solution of Eq. (B8) for uw at x = bw can be given by

uw p0 1 1 k1zl

h i2

d Geotechnics 35 (2008) 655672 671[2] Barron RA. Consolidation of ne-grained soils by drain wells. ASCETrans 1948;113:71854.

[3] Chai JC, Miura N, Sakajo S, Bergado DT. Behavior of vertical drainimproved subsoil under embankment loading. Soil Found1995;35(4):4961.

[4] Chai JC, Shen SL, Miura N, Bergado DT. Simple method ofmodeling PVD improved subsoil. J Geotech Geoenviron Eng, ASCE2001;127(11):96572.

[5] Chai JC, Carter JP, Hayashi S. Vacuum consolidation and itscombination with embankment loading. Can Geotech J 2006;43:98596.

[6] Hansbo S. Consolidation of ne-grained soils by prefabricateddrains. In: Proceedings of 10th international conference onsoil mechanics and foundation engineering, vol. 3; 1981. p.67782.

[7] Hansbo S. Design aspects of vertical drains and lime columninstallations. In: Proceedings of the ninth southeast Asian geotech-nical conference, vol. 2; 1987. p. 812.

[8] Hird CC, Pyrah IC, Russel D. Finite element modeling of verticaldrains beneath embankments on soft ground. Geotechnique1992;42(3):499511.

[9] Indraratna B, Redana IW. Plane strain modeling of smear eectsassociated with vertical drains. J Geotech Geoenviron Eng, ASCE1997;123(5):4748.

[10] Indraratna B, Redana IW. Numerical modeling of vertical drainswith smear and well resistance installed in soft clay. Can Geotech J2000;37:13245.

[11] Indraratna B, Bamunawita C, Khabbaz H. Numerical modeling ofvacuum preloading and eld applications. Can Geotech J 2004;41:1098110.

[12] Indraratna B, Rujikiatkamjorn C, Sathananthan I. Analytical andnumerical solutions for a single vertical drain including the eects ofvacuum preloading. Can Geotech J 2005;42:9941014.

[13] Jamiolkowski M, Lancellotta R. Consolidation by vertical drains-uncertainties involved in prediction of settlement rates, Panel discus-sion. In: Proceedings of the 10th international conference on soilmechanics and foundation engineering; 1981.

[14] Kjellman W. Consolidation of ne-grained soils by drain wells. TransASCE 1948;113:74851 [Contribution to the discussion].

[15] Kobayashi M, Minami J, Tsuchida T. Determination method ofhorizontal consolidation coecient of clay. Tech Rep Res CenterHarbor Eng Transport Ministry 1990;29(2):6383.

[16] Mesri G, Lo DOK. Field performance of prefabricated verticaldrains. Proceedings of the international conference on geotechnicalengineering for coastal development theory and practice on softground Yokohama, vol. 1. Japan: Coastal Development Institute ofTechnology; 1991. p. 2316.

[17] Rixner JJ, Kraemer SR, Smith AD. Prefabricated vertical drains.Engineering Guidelines vol. 1, Report No. FHWA-RD-86/168.Federal Highway Administration, Washington, DC; 1986.

[18] SAGE Engineering Ltd. SAGE CRISP users manual. Bath, UnitedKingdom: SAGE Engineering Ltd.; 1999.

672 T.A. Tran, T. Mitachi / Computers and Geotechnics 35 (2008) 655672

Equivalent plane strain modeling of vertical drains in soft ground under embankment combined with vacuum preloadingIntroductionThe proposed conversion method under vacuum-surcharge preloading conditionAnalytical models of the axisymmetric and plane strain unit cells under vacuum-surcharge preloadingAnalytical solution for the axisymmetric unit cell (modified by the present authors after Indraratna et blank al. [12])The proposed analytical solution for the plane strain unit cell excluding the plane strain smear zone

The proposed conversion expression of permeability for the proposed plane strain unit cellThe way to use our proposed conversion method

Verification of the proposed method via finite element methodDetailed description of the case of one clay layerBoundary conditions and computed cases

Detailed description of the case of two clay layers

Results and discussionIn the case of one homogeneous clay layerIn the case of two clay layers

ConclusionsAcknowledgementsFormulation of the analytical solution for the average excess pore pressure of the axisymmetric unit cellFormulation of the analytical solution for the average excess pore pressure of the plane strain unit cellReferences

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