Computers and Geotechnics 36 (2009) 1072–1083

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Computers and Geotechnics

journal homepage: www.elsevier .com/ locate/compgeo

Numerical modeling of vertical drains with advanced constitutive models

Abdulazim Yildiz *

Department of Civil Engineering, The Faculty of Engineering and Architecture, University of Cukurova, 01330 Balcali/Adana, Turkey

a r t i c l e i n f o

Article history:Received 14 February 2008Received in revised form 3 April 2009Accepted 11 April 2009Available online 9 May 2009

Keywords:Soft clayEmbankmentAnisotropyDestructurationPlane strain modelingVertical drain

0266-352X/$ - see front matter � 2009 Elsevier Ltd.doi:10.1016/j.compgeo.2009.04.001

* Tel./fax: +90 322 338 67 02; mobile: + 90 506 30E-mail address: azim@cukurova.edu.tr

a b s t r a c t

The water flow into a vertical drain under an embankment loading is axisymmetric problem and there-fore, the vertical drain system must be converted into equivalent plane strain model. The matching isachieved by manipulating the drain spacing and/or soil permeability according to relatively simple rules.This paper investigates the verification of matching approaches by comparing 2D plane strain and 3D unitcell results when complex constitutive models are used in the finite element analyses. The numericalresults showed that the match was found to be satisfactory but not perfect. A full-scale plane strain anal-ysis is subsequently applied to an embankment case history taken from Haarajoki, Finland, stabilized ver-tical drains and compared with field measurements.

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1. Introduction

Natural soft clays tend to have significant anisotropy of fabric,developed during deposition and 1D consolidation. During subse-quent plastic straining, the fabric anisotropy can change. Anisot-ropy can influence both elastic and plastic behaviors of soft clays.In addition to anisotropy, natural clays exhibit the effect of inter-particle bonding. When plastic straining occurs, the bonding de-grades, referred to as destructuration. For normally consolidatedor lightly overconsolidated soft clays, plastic deformations arelikely to dominate for many problems, such as the embankmentloading considered in this paper. Advanced geotechnical designon soft clays has often been based on finite element analysis usingisotropic elasto-plastic soil models, such as Modified Cam Clay [1].However, neglecting the effects of anisotropy and/or destructur-ation may lead to highly inaccurate predictions of soft clay re-sponse in the analysis. Therefore a specific constitutive modelthat is capable of accounting for anisotropy, initial structure anddestructuration process in soft clays is desirable for the analysis.In recent years, there has been considerable development in under-standing the behavior of soft clays and a number of elasto-plasticconstitutive models incorporating such as anisotropy and/ordestructuration effects have been published in the literature [2–6]. Most of these models, however, do not take into account thecombined effect of anisotropy and destructuration. Furthermore,the application of these models to practical geotechnical designis not common due to complexity and difficulty of determination

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8 6140.

of the model input parameters which may require non-standardlaboratory tests. Two recently proposed elasto-plastic constitutivemodels for soft clays, S-CLAY1 [7] and S-CLAY1S [8] are used for thesimulations presented in this paper. The models account for plasticanisotropy and anisotropy combined with destructuration, respec-tively. S-CLAY1 is an extension of conventional critical state mod-els, with anisotropy of plastic behavior represented through aninclined yield surface and a rotational component of hardening[7]. The S-CLAY1S additionally accounts for bonding between clayparticles and destructuration [8] by using the concept of intrinsicyield surface proposed by Gens and Nova [9]. The model parame-ters can be determined from the results of standard laboratorytests by using well-defined methodologies. Furthermore, the mod-els have been successfully validated against experimental data onseveral natural and reconstituted clays [7,8,10–14].

Prefabricated vertical drains (PVDs) are often used to speed upconsolidation and to increase shear strength under embankmentson soft soil [15]. The PVD is a slender, synthetic drainage elementcomprised of a drainage core wrapped in a geotextile filter. ThePVD has a typical rectangular cross section and is installed by usinga mandrel. The installation of PVDs by means of a mandrel causessignificant disturbance in the soil surrounding the drain. The shapeof the mandrel is also rectangular and therefore, the shape of theactual disturbed zone is close to a rectangle or an ellipse. A properdesign of an embankment involving a large number of discrete ver-tical drains and their own independent influence zone should beconducted with a fully three dimensional (3D) analysis. 3D finiteelement modeling of vertical drain system is very sophisticatedand requires large computational effort when applied to a realembankment project with a large number of PVDs [16]. 2D finite

A. Yildiz / Computers and Geotechnics 36 (2009) 1072–1083 1073

element analyses of embankments have commonly been con-ducted under plane strain conditions. However, the actual fieldconditions around vertical drains are truly 3D and therefore, it isnecessary to convert the vertical drain system into equivalentplane strain condition. Several authors [17–21] have shown thatvertical drains can be effectively modeled by using appropriatematching methods to represent the typical arrangement of verticaldrains in plane strain finite element analyses. The two most usefulmatching approaches from computational point of view are thoseproposed by Chai et al. [18] and Hird et al. [19,21], as accountingfor the effects of the smear zones around the drains does not re-quire separate discretisation. In this paper the matching proce-dures proposed by Hird et al. [19,21] are used in the analyses.According to Hird et al. [19,21], the matching procedure can beachieved by manipulating either the drain spacing and/or soil per-meability. However, certain simplifying assumptions are made:each single drain is assumed to work independently, soil has a con-stant permeability and the consolidation takes place in a uniformsoil column with linear compressibility characteristics in the ab-sence of lateral movement. Such restrictive conditions are notlikely to be achieved in normally or lightly overconsolidated soilsunder embankment loading. Furthermore, the stress–strain behav-ior of natural soft soils is highly nonlinear and very complex due todifferent fundamental features of soil, such as anisotropy, creepand destructuration as mentioned above. Hird et al. [21] alsoemphasized that validation is needed when the matching methodsare applied to practical situations where the soil behavior is non-linear and varies with depth.

This paper studies the performance of the matching proceduresproposed by Hird et al. [19,21] when complex elasto-plastic mod-els are used in the plane strain analyses of vertical drains. The pro-posed methods are applied to Haarajoki test embankmentconstructed on soft soils incorporating PVDs. The finite elementanalyses are performed with S-CLAY1 models as well as the isotro-pic Modified Cam Clay model (MCC). In the paper, first, a briefdescription of the constitutive model is given, followed by infor-mation about Haarajoki test embankment, material parametersand initial conditions, which are all required as inputs to numericalanalyses. Then, the consolidation behavior of a single drain and itsinfluence zone under the embankment is analyzed for 3D case andits equivalent plane strain conditions. Finally, a full-scale planestrain analysis of PVD system under Haarajoki embankment is car-ried out with the method proposed by Hird et al. [19,21] and theresults are compared with the corresponding field observations.

2. Constitutive models

The S-CLAY1S model is a critical state model with anisotropy ofplastic behavior represented through an inclined yield surface anda hardening law to model the development or erasure of fabricanisotropy during plastic straining. Additionally, the model ac-counts for destructuration. In 3D stress space the yield surface ofthe S-CLAY1S model is a sheared ellipsoid (Fig. 1a), given by

f ¼32frd�p0adgTfrd�p0adgh i

� M2�32fadgTfadg

� �ðp0m�p0Þp0 ¼0

ð1Þ

where rd is the deviatoric stress tensor, p0 is mean effective stress,ad is a deviatoric fabric tensor (a dimensionless second order tensorthat is defined analogously to the deviatoric stress tensor, seeWheeler et al. [7] for details), M is the value of the stress ratio atcritical state and p0m defines the size of the yield surface of the nat-ural clay. Eq. (1) shows that the generalized version of the yield sur-face cannot be expressed solely in terms of stress invariants. Fig. 1aillustrates the shape of the S-CLAY1S yield surface in 3D stress

space, for the case where the principal axes of both the stress tensorand the fabric tensor coincide with the x, y and z directions.

Within the yield surface there is a notional ‘‘intrinsic yield sur-face” for the equivalent unbonded soil with the same fabric, whichis assumed to be of the same shape and orientation as the real yieldsurface, but smaller in size. The size of the intrinsic yield surface isspecified by a parameter p0mi, and this is related to the size p0m of theyield surface for the natural soil by parameter , defining the currentdegree of bonding:

p0m ¼ ð1þ vÞp0mi ð2Þ

For the simplified conditions of a triaxial test on a previously one-dimensionally consolidated sample, it can be assumed that thehorizontal plane in the triaxial sample coincides with the plane ofisotropy of the sample. In this special case, the fabric tensor canbe replaced by a scalar parameter a defined as

a2 ¼ 32fadgTfadg ð3Þ

which is a measure of the degree of plastic anisotropy of the soil.The yield surface of the S-CLAY1S model can then be visualizedby Fig. 1b. For the sake of simplicity, the S-CLAY1S model assumesisotropic elastic behavior of the same form as in the Modified CamClay model [1], and an associated flow rule:

depd

depv¼ 2ðg� aÞ

M2 � g2ð4Þ

where g = q/p0 and depd and dep

v are the increment of plastic deviator-ic and volumetric strains, respectively. Experimental evidence byGraham et al. [22] and Korhonen and Lojander [23] suggests thatthis assumption is reasonable for many soft clays.

S-CLAY1S incorporates three hardening laws. The first relatesthe change in the size of the intrinsic yield surface to the plasticvolumetric strain increment dep

v :

dp0mi ¼vp0mi

ki � jdep

v ð5Þ

where ki is the gradient of the intrinsic normal compression line (fora reconstituted soil) in the lnp0-v plane (where v is the specific vol-ume). Eq. (5) is of the same form as the equivalent hardening law inMCC and S-CLAY1, but with p0m replaced by p0mi and k replaced by ki.

The second hardening law (the rotational hardening law) de-scribes the change of orientation of the yield surface with plasticstraining:

dad ¼ x3g4� ad

� �hdep

vi þxd

g3� ad

� �dep

d

� �ð6Þ

where g = rd /p0, depd is the increment of equivalent plastic deviator-

ic strain x and xd are additional soil constants. The soil constant xd

controls the relative effectiveness of plastic shear strains in settingthe overall instantaneous target values for the components of ad,whereas the soil constant x controls the absolute rate of rotationof the yield surface towards the current target values of the compo-nents of ad (see Wheeler et al. [7] for details).

The third hardening law in S-CLAY1S (the destructuration law)describes the degradation of bonding with plastic straining. It issimilar in form to the rotational hardening law (Eq. (6)), exceptthat both plastic volumetric strains and plastic shear strains(whether positive or negative) tend to decrease the value of thebonding parameter v towards a target value of zero:

dv ¼ n ½0� v�jdepv j þ nd½0� v�dep

d

� �¼ �nv jdep

v j þ nddepd

� �ð7Þ

where n and nd are additional soil constants. Parameter n controlsthe absolute rate of destructuration, and parameter nd controlsthe relative effectiveness of plastic deviatoric strains in destroying

Fig. 1. S-CLAY1S yield surface in (a) 3D stress space; (b) triaxial stress space.

1074 A. Yildiz / Computers and Geotechnics 36 (2009) 1072–1083

the bonding (see Koskinen et al. [8] for details). Theoretically, aspointed out in Zentar et al. [14], a monotonic reduction in v dueto plastic straining which modifies the anisotropy and graduallybreaks the bonding due to slippage at interparticle contacts andsubsequent rearrangement and realignment of particles impliedby the hardening law (Eq. (7)) can sometimes result in reductionof p0m during hardening for certain combinations of parameters nnd and v. As a result, the peak undrained shear strength of the nat-ural soil may actually be predicted to decrease during consolidation.There is field evidence for such behavior on a moderately sensitiveFinnish clay, as discussed in Koskinen et al. [13].

2.1. Model parameters and initial state

The use of S-CLAY1S model requires values of eight soil con-stants (ki, j, m0, M, x, xd, n and nd) and information on initial state(initial values for void ratio e0, a0, v0 and p0m). Three of these areconventional parameters (j, m0, M) and can be directly measuredfrom standard laboratory tests on undisturbed natural samples asin the Modified Cam Clay model while the intrinsic value ki shouldbe determined from an oedometer test on a reconstituted sample.The initial size and inclination of the yield curve (p’m and a) can bedetermined using a simple procedure that the previous history ofthe soil deposit is restricted to simple 1D (K0) loading, and possibleunloading, to a normally consolidated or lightly overconsolidatedstate (see Wheeler et al. [7] for details). If the normally consoli-dated value of K0 can be estimated by using Jaky’s simplified for-mula (K0 = 1�sin /0), this can be used to calculate acorresponding value of stress ratio gK0. The model predicts thatonly one value of yield curve inclination a would produce 1Dstraining for continuous loading at this stress ratio gK0, and thistherefore provides an estimate for the initial value of a. Assumingthat elastic strains are much smaller than plastic strains, which isoften true for very soft soils, 1D straining (zero lateral strain) cor-responds to:

depd

depv¼ 2

3ð8Þ

Combining the above equation with the flow rule of Eq. (4), theyield curve inclination corresponding to 1D consolidation aK0 is gi-ven by:

aK0 ¼g2

K0 þ 3gK0 �M2

3ð9Þ

There are two possible ways to determine the value of initialamount of bonding v0 for a natural soil [11]. A conservative esti-mate can be obtained based on measured sensitivity, which is alower bound estimate of 1 + v0, because some proportion of thebonding can be lost during sampling and testing. The second way

of determining a value for v0 is based on the initial void ratioand the size of the initial yield curve for the natural soil (e0 andp0m0) and the position of the intrinsic normal compression line forreconstituted soil in lnp’-e space. Since different samples of thesame clay have different void ratios due to disturbance and/or vari-ations in the sampling depth and location, the values of v0 for dif-ferent samples can vary significantly and a geometrical average ofv0 should be used. In this study, the first method is used to deter-mine the value of v0.

The parameters x and xd are related to the rotational harden-ing in the S-CLAY1S model. The model parameter xd defines therelative effectiveness of plastic shear strains in rotating the yieldcurve. For plastic loading along any constant xd stress path, a willultimately tend to a final equilibrium value, which can be found bysetting da = 0 in Eq. (6) and combining with Eq. (4).

xd ¼3 M2 � g2

K0 �3gK0

4

� 2ðg2

K0 �M2 þ 2gKOÞð10Þ

Knowledge of the value of K0 (and hence gK0) can therefore beused to determine an appropriate value for xd, using Eq. (10).The model parameter xd controls the rate at which a tends to-wards its current target value. It is difficult to devise a simpleand direct method for experimentally determining the value of xfor a given soil (see Wheeler et al. [7] for details). The values ofx for soft soils in this paper were estimated based on the apparentvalues of k, as suggested by Zentar et al. [14].

Determination of the destructuration parameters n and nd re-quires an optimization procedure by comparing the experimentalresults with model simulations, as explained by Koskinen et al.[8] and Karstunen and Koskinen [11]. The effect of parameter nd,which controls the relative effectiveness of plastic deviatoricstrains in destructuration, is negligible in a triaxial test with astress ratio g close to zero, where the deviatoric strains are verysmall. Therefore, the most appropriate value for parameter n canbe chosen first by simulating a triaxial test with a small stress ratio,such as an isotropic loading test. Using that value of parameter n,the effect of parameter nd can then be studied by simulating a tri-axial test with a large stress ratio g, where the deviatoric strains,and hence the effect of parameter nd, are significant.

By setting the initial value of the state parameter v to zero andusing an apparent value of k (determined from an oedometer teston a natural clay sample), instead of the intrinsic value ki of areconstituted clay, S-CLAY1S reduces to the S-CLAY1 model thataccounts for plastic anisotropy only. Furthermore, if the initial va-lue of the state parameter a (used for calculating the initial valuesof the components of the fabric tensor) and the value of the soilconstant x are set to zero, the model ultimately reduces to the iso-tropic Modified Cam Clay (MCC) model.

Table 2Values for MCC soil constants.

Layer c /0 m0 j k M

1 17.0 36.9 0.35 0.006 0.12 1.502 17.0 36.9 0.35 0.009 0.21 1.503a 14.0 28.8 0.18 0.033 1.33 1.153b 14.0 28.8 0.18 0.033 1.33 1.153c 14.0 28.8 0.18 0.033 1.33 1.154 14.0 27.7 0.10 0.037 0.96 1.105 15.0 27.0 0.10 0.026 0.65 1.076 15.0 27.0 0.28 0.031 1.16 1.077 15.0 28.8 0.28 0.033 1.06 1.158 16.0 36.9 0.28 0.026 0.45 1.509 17.0 36.9 0.28 0.009 0.10 1.50

Table 1The initial values for the state parameters.

Layer Depth (m) e0 K0 POP (kN/m2) a0 v0

1 0.0–1.0 1.4 1.0 76.5 0.58 42 1.0–2.0 1.4 1.0 60.0 0.58 43a 2.0–3.0 2.9 0.70 38.0 0.44 223b 3.0–4.0 2.9 0.70 34.0 0.44 223c 4.0–5.0 2.9 0.70 30.0 0.44 224 5.0–7.0 2.8 0.70 24.0 0.42 305 7.0–10.0 2.3 0.70 21.0 0.41 456 10.0–12.0 2.2 0.70 28.5 0.41 457 12.0–15.0 2.2 0.70 33.5 0.44 458 15.0–18.0 2.0 0.55 17.0 0.58 459 18.0–22.0 1.4 0.45 1.0 0.58 45

A. Yildiz / Computers and Geotechnics 36 (2009) 1072–1083 1075

3. Haarajoki test embankment

In 1997, the Finnish National Road Administration organized aninternational competition to calculate and predict the behavior of aroad embankment at Haarajoki. Haarajoki Test Embankment wasbuilt as a noise barrier in Finland. The geometry of the embank-ment is shown in Fig. 2. The embankment is 2.9 m high and100 m long. The crest of the embankment is 8 m wide and theslopes have a gradient of 1:2. The data concerning site investiga-tion and the results of the associated laboratory tests and fieldmonitoring data provided by FinnRA [24] are very useful for thevalidation of different constitutive models and methods of analy-ses. The results of some finite element analyses also have beenpublished in the literature [25–28]. Based on the ground investiga-tion data [24] and these numerical studies [25–28], Haarajoki testembankment is founded on a 2 m thick dry crust layer overlying a20.2 m thick soft clay deposit and the subsoil is divided into ninesub-layers with different compressibility parameters and overcon-solidation ratios. The water content of the soft clay layer varies be-tween 75% and 112% depending on the depth, and is almost thesame as, or greater than, the liquid limit. The bulk density variesfrom 14 to 17 kN/m3 and specific gravity varies from 2.73 to2.79. The undrained (undisturbed) shear strength determined byfall cone tests and field vane tests was between 15 and 42 kN/m2

[24]. The Haarajoki deposits can be characterized as a sensitiveanisotropic soft soil with sensitivity values (determined with fallcones tests) between 20 and 55. The organic content is between1.4% and 2.2% at the depth of 3–13 m. Half of the embankment isconstructed on area improved with vertical drains and the otherhalf on natural deposits without any additional ground improve-ment. In the improved area, the vertical drains were installed ina regular pattern with 1 m spacing before embankmentconstruction.

4. Input data

The embankment, which was made of granular fill, was mod-eled with a simple Mohr Coulomb model assuming the followingmaterial parameters: E0 = 40,000 kN/m2, m0 = 0.35, u0 = 40�, w0 = 0�,c0 = 2 kN/m2, and c = 21 kN/m3 (where E0 is the Young’s modulus,m0 is the Poisson’s ratio, u0 is the friction angle, w0 is the dilatancyangle and c is the unit weight of the embankment material). The

8m

2.9m1:2

2m

20.2m

2~3m

2~3m

Dry crust

Soft soil deposit

Silt

Till

Fig. 2. Cress-section of Haar

problem is dominated by the soft clay response and is hence ratherinsensitive to the embankment parameters.

The values of the input parameters and state variables for themodels listed in Tables 1–3 were estimated for each layer basedon the laboratory results provided by FinnRA [24] using the bestpractice for the determination of model and state parameters forthe S-CLAY1S model and its simplifications (i.e. S-CLAY1 andMCC). The initial values for the state parameters are shown inTable 1. Pre-overburden pressure (POP) defines the differencebetween the vertical preconsolidation stress ðr0mpÞ and the in situvertical effective stress ðr0moÞ. The in situ stresses were calculatedby assuming a horizontal effective stress distribution, using K0

1:2

GWT

VerticalDrains 15m

ajoki test embankment.

Table 3Values for additional soil constants for S-CLAY1 and S-CLAY1S.

Layer xd x ki n nd

1 1.00 50 0.04 8 0.22 1.00 50 0.06 8 0.23a 0.70 20 0.38 8 0.23b 0.70 20 0.38 8 0.23c 0.70 20 0.38 8 0.24 0.64 20 0.27 8 0.25 0.60 20 0.19 8 0.26 0.60 20 0.33 8 0.27 0.70 20 0.30 8 0.28 1.00 20 0.13 8 0.29 1.00 20 0.03 8 0.2

De dw

S

wt

S

Fig. 3. Square installation pattern and influence zone of PVDs.

Table 4Haarajoki test embankment. Drain properties.

Drain pattern Square netModel SOLPACK C634Spacing (S) 1 mAv. width of the drain (w) 98.7 mmThickness at 20 kPa (t) 6.83 mmDischarge capacity, qw 157 m3/year

1076 A. Yildiz / Computers and Geotechnics 36 (2009) 1072–1083

values estimated with equation K0 = (1 � sin /0) OCRsin/0 [29],where u0 is the critical state friction angle in triaxial compressionand OCR is the vertical overconsolidation ratio ðOCR ¼ r0p=r0moÞ.Due to natural variability, there was some scatter in the valuesand for each layer average values have been chosen. The valuesof permeability used for calculations were reported by Näätänenet al. [27] based on vertical and horizontal CRS oedometer tests.The values of j and k were determined from oedometer test re-sults. The values for the initial inclination a0 of the yield surfaceand parameter xd were determined following the procedure de-scribed in Section 2, using KNC

0 values corresponding to Jaky’s sim-plified formula. The values of x were estimated based on apparentvalues of k, as suggested by Zentar et al. [14]. The use of S-CLAY1Smodel requires, additionally, values for the two destructurationparameters (n and nd) and information on the initial amount ofbonding v0. The values of ki need to be measured from oedometertests on reconstituted samples. In the absence of these tests, ki –values were estimated throughout the deposit assuming similark/ki ratios to other Finnish clays, and similarly, past experienceon Finnish clays was used in fixing the values for parametersn and nd that control the rate of degradation of bonding [13]. Modelsimulations of different Finnish natural clays have indicated simi-lar values for soil constants n and nd [13] and typically n is about 8–12 and nd about 0.2–0.3. The initial values of v0 were estimatedbased on sensitivity (v0 = St � 1), as suggested in Koskinen et al.[8]. The sensitivity values were determined with fall cones testsand reported at FinnRA [24].

5. Numerical modeling

5.1. 3D modeling

The 3D behavior of a single drain and its influence zone (a unitcell) under the centerline of Haarajoki test embankment was ini-tially investigated by a 3D finite element code PLAXIS 3D Founda-tion V2 [30]. Three different constitutive models (MCC, S-CLAY1and S-CLAY1S) are used to represent the soft clay behavior in thefinite element modeling. The models have been implemented intoPLAXIS program as user-defined soil models by Wiltafsky [31]. Thesoil parameters and layering given in Tables 1–3 are used in the 3Dunit cell simulations.

The length of PVDs is 15 m and they were installed in a squaregrid with 1 m spacing underneath the embankment (Fig. 3). Thedrain parameters relevant for the analysis are summarized inTable 4. In 3D unit cell analyses, a single drain is represented byan equivalent cylindrical drain. Different equations for equivalentdrain diameter have been derived and reported in the literature[32–35]. The equivalent drain diameter (dw) of a rectangularshaped drain can be calculated based on ‘‘perimeter equivalence”proposed by Hansbo [32]:

dw ¼2ðwþ tÞ

pð11Þ

where w and t are the width and thickness of the drain, respectively.Eq. (11) has been widely used in practice and was also verified by3D finite element analysis in this study. The equivalent drain diam-eter (dw) is calculated as 67 mm based on Eq. (11). The equivalentdiameter of the mandrel (dm) is assumed to be 100 mm. These val-ues of dm and dw correspond to a value of dm/dw ffi 1.5. The finite ele-ment mesh used for the numerical simulations is illustrated inFig. 4. The 15-node wedge elements were used to model foundationsoils in the analysis. The 15-node wedge element is composed ofsix-node triangles in horizontal direction and eight-node quadrilat-erals in vertical direction. A total of 2160 3D elements were used inthe finite element analysis.

5.2. 2D modeling

Fig. 5 shows an axisymmetric unit cell with the total radius, Rand its equivalent plane strain unit cell with half width, B. Theeffective diameter of a drain influence was taken to be De = 1.13Sfor a square configuration [34] where S is the drain spacing(Fig. 3). The equivalent drain radius (rw) and unit cell radius (R)were calculated as 0.034 m and 0.565 m, respectively.

Analytical solutions already developed for consolidation ofground improved with vertical drains invariably employ the ‘‘unitcell” model (Fig. 5). The theory for radial drainage consolidationhas been developed by many researchers e.g., [36–39]. Based onHansbo’s [37] solution, Hird et al. [19] showed that the average de-grees of consolidation U, at any depth and time in the two unit cellswere theoretically identical and the axisymmetric unit cell can beconverted into equivalent plane strain unit cell (Fig. 5). Hird et al.[19] proposed that the matching can be achieved by any one of thefollowing three methods:

Fig. 4. Finite element mesh for 3D unit cell analyses.

(a) Axisymmetric unit cell (b) Plane strain unit cell

rs

rw

R

Smea

r zon

e

Und

istu

rbed

zon

e

B

bs

Und

istu

rbed

zon

e

Smea

r zon

e bw

L L

Fig. 5. Unit cell model for 2D finite element analyses.

A. Yildiz / Computers and Geotechnics 36 (2009) 1072–1083 1077

(i) Geometric matching: the drain spacing is matched whilemaintaining the same permeability coefficient. The geomet-ric matching was done according to the following equationin the absence of well resistance:

BR¼ 3

2

� �ln

Rrs

� �þ kh

ks

� �ln

rs

rw

� �� 3

4

� �� � �12

ð12Þ

where B is the half-width of the plane strain unit cell; R, rw and rs

are the radius of the axisymmetric unit cell, the drain and the smearzone, respectively; kh and ks are the horizontal permeability of theundisturbed and smeared soil, respectively. Well resistance was ne-glected in the analyses.

(ii) Permeability matching: the coefficient of permeabilityis matched while keeping the same drain spacing. The

technics 36 (2009) 1072–1083

permeability matching can be achieved by using the follow-ing equation:

1078 A. Yildiz / Computers and Geo

kpl

kax¼ 2

3 ln Rrs

� þ kax

ks

� ln rs

rw

� � 3

4

� �h i ð13Þ

where kax and kpl are the values of horizontal permeability ofaxisymmetric and plain strain conditions, respectively.

(iii) Combined matching: the combination of (i) and (ii), with theplane strain permeability calculated for a convenient drainspacing. A value of B is preselected and then kpl is calculatedby using the following equation:

kpl

kax¼ 2B2

3R2 ln Rrs

� þ kax

ks

� ln rs

rw

� � 3

4

� �h i ð14Þ

-2.0

-1.5

-1.0

-0.5

0.00 100 200 300 400 500 600

Time (days)

Settl

emen

t (m

)

MCCS-CLAY1S-CLAY1S

Fig. 6. Time settlement curves predicted by 3D finite element analyses.

Table 5The results of matching procedures.

Geometry matching Permeability matching Combined matching

R (m) 0.57 0.57 0.57 0.57B (m) 0.99 0.57 0.50 2.00kpl/kax 1.00 0.32 0.26 4.07

These matching procedures assume that each drain works inde-pendently around a circular zone of influence and soil has linearcompressibility characteristics and constant permeability in theabsence of lateral movements.

The 3D unit cell model described in 5.1 is converted into equiv-alent plane strain unit cell according to Eqs. (12)–(14) and ana-lyzed with finite element code PLAXIS V8.2 [40] using threeelasto-plastic models (MCC, S-CLAY1 and S-CLAY1S). The problemwas discretized by using a finite element mesh with about 277six-noded triangular elements. The soil parameters and layering gi-ven in Tables 1–3 are used in the 2D plane strain simulations.

In both 3D and 2D finite element analyses, the lateral bound-aries are restrained horizontally and the bottom boundary is re-strained in both directions. Drainage boundaries are assumed tobe at the ground surface and at the bottom of the mesh whereasthe lateral boundaries are closed. The embankment constructionconsists of two phases: first, the embankment loading is appliedunder undrained conditions, assuming the embankment to bedrained material and next, a consolidation phase is simulated viafully coupled consolidation analysis. The construction of Haarajokiembankment was done in 0.5 m layers, each taking 2 days, whilethe foundation layer was constructed in 5 days, and the real con-struction schedule has been simulated in the calculation. Afterthe construction of each layer, a consolidation phase is introducedto allow the excess pore pressures to dissipate. Hence, a total oftwelve calculation phases were defined in the analyses. The con-struction of embankment was completed in 35 days. After the con-struction of last layer, the calculations have been performed untilthe excess pore pressures had dissipated to a residual value of1 kPa to find out the final consolidation settlement. Mesh sensitiv-ity studies were done to confirm that the mesh was dense enoughto give accurate results for all of the constitutive models con-cerned. In PLAXIS it is possible to decrease the permeability asthe void ratio decreases, using the formula by Taylor [41]:

logkk0

� �¼ De

ckð15Þ

where De is the change in void ratio, k is the soil permeability in thecalculation step, k0 is the initial value of the permeability and ck isthe permeability change index. It was assumed in the calculationsthat ck = 0.5e0 [42].

6. Numerical results and comparisons

6.1. Unit cell analyses

The first set of finite element analyses was conducted with 3Dmodeling of a single drain and its influence zone underneath thecenter line of the embankment. The predicted vertical displace-ments versus time for all models (MCC, S-CLAY1 and S-CLAY1S)

are presented in Fig. 6. Smear effect is neglected in the analysis.Differences between the three models are relatively minor imme-diately after construction of the embankment, but become signifi-cant during consolidation. The isotropic MCC model predictssmaller settlements than the two anisotropic models. The S-CLAY1S model predicts larger vertical settlements than the S-CLAY1 model.

Although 2D equivalent plane strain analyses are subsequentlycarried out for each constitutive model, the results have been illus-trated for the MCC and S-CLAY1S models in the following analyses.The values B and kpl for three matching procedures are calculatedunder perfect drain conditions by using Eqs. (12)–(14). The match-ing results are presented in Table 5.

The results of equivalent plane strain analyses show that allthree matching procedures produce identical settlement responsefor each constitutive model. Therefore, only the combined match-ing procedure was adopted in the settlement analyses as it is com-putationally most convenient and enables controlling meshgeometry in the finite element analyses.

To verify the matching methods proposed by Hird et al. [19,21],the predicted vertical displacements by 3D and equivalent 2Dplane strain analyses are compared for both MCC and S-CLAY1Smodels in Fig. 7. It can be seen that the rate of consolidation inthe equivalent plane strain analyses is faster than that in the cor-responding 3D analysis. Fig. 8 illustrates the differences betweenthe results of 3D and 2D equivalent plane strain analyses for twoconstitutive models. During consolidation, the maximum differ-ence in the predicted settlements between the 3D and equivalentplane strain analyses is about 6.5% with MCC, while difference ofabout 6% result with S-CLAY1S. The difference changes from modelto model, and varied during consolidation from 0.06% to 6.5%,being the largest after a few months of consolidation.

Excess pore pressures (EPP) are also calculated by 3D and 2Dplane strain analyses at a depth of 5 m below the ground surface.In the 3D numerical analyses, the values of EPP predicted by theMCC and S-CLAY1S models are almost identical at a depth of5 m. Hence, the values of EPP predicted by equivalent 2D planestrain and 3D simulations are compared for only S-CLAY1S modelin Fig. 9. As expected, the values of EPP increase during the

-1.5

-1.0

-0.5

0.00 200 400 600

Time (days)Se

ttlem

ent (

m)

3-D analysis2-D analysis

(a) MCC model results

-2.0

-1.5

-1.0

-0.5

0.00 200 400 600

Time (days)

Settl

emen

t (m

)

3-D analysis2-D analysis

(b) S-CLAY1S model results

Fig. 7. Comparison of time settlement curves predicted by 3D and 2D equivalentplane strain analyses.

Fig. 8. Differences between the results of 3D and 2D equivalent plane strainanalyses.

0

5

10

15

20

25

30

35

400 100 200 300 400 500 600

Time (days)

EPP

(kN/

m²)

Geometry mapping (B=0.99m)Permeability mapping (B=0.57m)Combined mapping (B=2.00m)3-D

x = 0.25my = -5.00m

Fig. 9. Comparison of EPP predicted by 2D and 3D finite element analyses.

A. Yildiz / Computers and Geotechnics 36 (2009) 1072–1083 1079

embankment construction and then gradually dissipate with time.The maximum value of EPP is predicted at the end of construction.It can be seen from Fig. 9 that the agreement between the results of3D and 2D equivalent plane strain analyses is not satisfactory. Inaddition, three matching procedures predicted different maximumvalue of EPP whereas all three matching procedures producedidentical settlement response. The maximum values of EPP in thegeometry, permeability and combined matching are predicted asabout 29, 37 and 19 kN/m2, respectively, while the 3D model pre-dicts a value of about 31 kN/m2. The excess pore pressure dissipa-tion rate after the construction is also different from one matchingto another. This is because either geometry or soil permeability ischanged for matching taking the average degree of consolidation

into account while compressibility characteristics of soil remainconstant. Similar results were also reported by Hird et al. [19,21]expressing that excess pore pressures in a plane strain analysisdo not matched at corresponding points in axisymmetric unit cell.However, it can be seen that the agreement between the perme-ability matching and 3D model is reasonably good.

Other uncertainties in the FE analysis of PVDs are related to thewell resistance and the smear zone around vertical drains. Wellresistance refers to the finite permeability of the vertical drain withrespect to the soil. The limited discharge capacity of drains cancause a serious delay in the consolidation process. In general, lab-oratory and field data indicate that the discharge capacities of mostcommercial PVDs have little influence on the consolidation rate ofclay, especially for drains that are not too long [43]. For values ofqw > 100–150 m3/year (in the field) and where drains are shorterthan 30 m, there should be no significant increase in the consolida-tion time. According to Hansbo [44], the discharge capacity ofmodern prefabricated vertical drains is considered to be high en-ough (qw > 150 m3/year) and the effect of well resistance can be ig-nored in the design. Hence, the effect of well resistance is neglectedin the numerical analyses presented in this study.

In the field, installation of vertical drains by means of a mandrelcauses significant remolding of the subsoil, especially in the imme-diate vicinity of the mandrel. Haarajoki deposits can be character-ized as very sensitive anisotropic soft clay. The water content isoften higher than the liquid limit. Hence, considerable disturbanceis expected in the subsoil during the installation of vertical drains.In the numerical analyses, the permeability of soil in the disturbedzone is reduced, because the structure of the soil is destroyed bymechanical disturbance [45] and other properties may also beinfluenced. The extent and permeability of the smear zone are dif-ficult to determine from laboratory tests, and so far, there is nocomprehensive or standard methods to measure them. There isno test data available about the key parameters relating to thesmear zone (ds/dm and kh/ks) for this particular soil. The extent ofthe smear zone and its permeability depend on the installationprocedure, size and shape of the mandrel as well as the type andsensitivity of soil. Several investigations have been made on thesefactors [46–52]. Indraratna and Redana [49] estimated smear zoneto be about (4–5) dm in the laboratory, using a large-scale consol-idometer. Chai and Miura [50] suggested that the value of ds = 3dm

can be used in practice when there are no test data for evaluatingthe smear zone size. The studies of Bo et al. [51] and Xiao [52] indi-cate that the smear zone can be as large as four times of the size ofthe mandrel (dm), or 5–8 times of the equivalent diameter of drain(dw). According to Jamiokowski et al. [46] the value of kh/ks can varyfrom 1 to 15. Bergado et al. [48] proposed the ratio of kh/ks between5 and 20 based on full-scale field test. Laboratory tests may be acorrect way to determine the value of kh/ks, but they generallyunderestimate the hydraulic conductivity of field deposits because

1080 A. Yildiz / Computers and Geotechnics 36 (2009) 1072–1083

of sample disturbance and sample size effect. It is suggested that kh

/ks can be expressed as (Chai and Miura [50])

kh

ks¼ kh

ks

� �lCf ð16Þ

where subscript l represents the value determined in the labora-tory; and Cf = hydraulic conductivity ratio between field and labora-tory values. The value of Cf needs to be determined by back-analysisor field and laboratory hydraulic conductivity tests. Chai and Miura[50] reported that the values of Cf were ranged from 1 to 25 for dif-ferent clays. It is seen that there is no general agreement in the lit-erature, and the size of smear zone and its permeability are still notexactly known. In the following analyses, the size of smear zone andits permeability were identified from the back-calculations assum-ing Cf = 1. Therefore, the ratios of ds/dm or kh/ks were iteratively var-ied to show how well the analysis simulated the field performance.Only S-CLAY1S model was used in the back analyses.

Figs. 10 and 11 show the finite element predictions obtainedfrom 3D modeling, which considers the effect of smear. As seenin Fig. 10, an increase in the diameter of smear zone causes a de-crease in the settlement rate of the soft subsoil. The results inFig. 10 demonstrate that when the ratio of ds/dw is greater than 7(or ds/dm > 5), there is not much effect on the settlement behavior.The time–vertical displacement relationships for different kh/ks

values predicted by 3D analyses are shown in Fig. 11. The effectof reduced horizontal permeability in the smear zone on settle-ment behavior of the embankment is clearly significant. The rateof settlement decreases with an increase in the kh/ks ratio. It isobvious that the smear effect has major influence on the settle-

-2.0

-1.5

-1.0

-0.5

0.00 1000 2000 3000 4000

Time (days)

Settl

emen

t (m

) 1

23

57

10

ds/dw ratios

Fig. 10. The effect of ds/dw on the settlement behavior (with S-CLAY1S model).

-2.0

-1.5

-1.0

-0.5

0.00 1000 2000 3000 4000 5000

Time (days)

Settl

emen

t (m

) 1

2

5

10

20

kh/ks ratios

Fig. 11. The effect of kh/ks on the settlement behavior (with S-CLAY1S model).

ment behavior. In addition, the kh/ks ratio was found to affect thesettlement rate more than the ds/dw ratio.

Vertical displacement versus time as predicted by 3D and 2Dequivalent plane strain analyses is compared for various ds/dw

and kh/ks ratios and only the results of ds/dw = 7 and kh/ks = 20 arepresented in Fig. 12. Again, all matching procedures produce effec-tively identical settlement responses for each constitutive modeland combined matching results are only presented in Fig. 12. Themaximum differences between 3D and 2D equivalent plane strainpredictions for various kh/ks ratios are also shown in Table 6,assuming the ratio of ds/dw = 7. It can be seen that the rate of con-solidation in the plane strain analysis is faster than that in 3Danalysis.

Fig. 13 illustrates the calculated differences in the plane strainanalyses with smear effect compared to the 3D unit cell analysis.The results of combined matching procedure are taken as referencefor the difference calculations. The maximum differences are about6.7% and 5.9% for MCC and S-CLAY1S, respectively. At the end ofconsolidation, the differences are about 0.1% and 2.1% for MCCand S-CLAY1S, respectively. It is seen that the match is reasonable,but not perfect. Given differences vary from model to model. In or-der to improve the match, the permeabilities for the equivalentplane strain model need to be adjusted further. As the predictedcoefficients of consolidation differ from model to model, it mightbe necessary to consider each model separately to improve thematch.

6.2. Full-scale plane strain analyses

The boundaries of the geometry used in the full-scale finite ele-ment analyses have an extent of 40 m in the horizontal directionfrom the symmetry axis and 22.2 m in the vertical direction. Thetest embankment was assumed symmetric and only half of the

-1.5

-1.0

-0.5

0.00 1000 2000 3000 4000

Time (days)

Settl

emen

t (m

) 3-D analysis2-D analysis

(a) MCC model results

-2.0

-1.5

-1.0

-0.5

0.00 1000 2000 3000 4000 5000

Time (days)

Settl

emen

t (m

) 3-D analysis2-D analysis

(b) S-CLAY1S model results

Fig. 12. Comparison of time-settlement curves predicted by 3D and 2D equivalentplane strain analyses.

Fig. 13. Differences between the results of 3D and equivalent 2D equivalent planestrain analyses.

Table 6Predicted maximum differences for various kh/ks ratios (ds/dw = 7).

Constitutive model kh/ks = 1 (%) kh/ks = 2 (%) kh/ks = 5 (%) kh/ks = 10 (%) kh/ks = 20 (%)

MCC 6.5 6.2 5.9 5.8 6.0S-CLAY1S 5.8 5.6 5.7 5.8 6.1

0

200

400

600

800

0 500 1000Time (days)

Settl

emen

t (m

m)

MCCS-CLAY1S-CLAY1Sobserved

(a)

0

200

400

600

800

0 500 1000Time (days)

Settl

emen

t (m

m)

MCCS-CLAY1S-CLAY1Sobserved (right)observed (left)

(b)

Fig. 15. Time settlement curves for 2D full-scale plane strain analysis: (a) under thecenterline; (b) 4 m from centerline (under the crest of the embankment).

A. Yildiz / Computers and Geotechnics 36 (2009) 1072–1083 1081

embankment is considered in the finite element analyses. Theplane strain condition and 15-noded triangular elements wereused and all simulations were done as large strain analyses. A finiteelement mesh with 953 elements is used to model the subsoil andthe embankment (Fig. 14). Mesh sensitivity studies were done toconfirm that the mesh was dense enough to give accurate resultsfor all of the constitutive models concerned. Updated mesh analy-sis taking into account the effects of large deformations was usedin the numerical analyses.

Based on the results given above, a combined matching proce-dure was adopted to simulate full-scale plane strain analysis ofHaarajoki embankment. The equivalent plane strain width of thedrain (B) was preselected as 0.5 m in the analyses. Based on theback calculations, the smear effect is taken into consideration byusing ds/dw = 7 (or ds/dm = 5) and kh/ks = 20, within the ranges pro-posed by several investigators [48–52]. For ds/dw = 7 and kh/ks = 20,the kpl was calculated as 0.013kax for the full plane strain analysisof the embankment, by using Eq. (14). The observed settlementsand predicted vertical displacements by three constitutive modelsversus time at the surface of the ground underneath the center line(Fig. 15a) and the crest (Fig. 15b) of Haarajoki test embankment on

Fig. 14. Finite element mesh used for 2D plane strain analyses.

1082 A. Yildiz / Computers and Geotechnics 36 (2009) 1072–1083

PVD-improved soft soil are compared in Fig. 15. The settlementspredicted by the two anisotropic models (S-CLAY1 and S-CLAY1S)after 3 years of consolidation are in good agreement with the fieldmeasurements. However, the rate of consolidation in the calcula-tions is faster than that in measured values. This is due to thematching effect since the agreement between 3D and equivalentplane strain analyses is not perfect as mentioned before. Fig. 15again highlights the role of anisotropy in the predicted soil re-sponse. However, the calculated time settlement curve predictsthe settlements to slow down, whereas the observed settlementssuggest that the embankment keeps on settling with a constantrate. This could be due to creep effects, which are not accountedfor in the analysis.

7. Conclusions

This paper investigates the performance of matching proce-dures developed by Hird et al. [19,21] for 2D plane strain analysisof vertical drains compared to the results of 3D numerical analyses.The actual 3D behavior of a vertical drain was converted intoequivalent plane strain model, considering both perfect drain andthe smear effect due to the installation of vertical drains. A full-scale plane strain finite element analysis of Haarajoki test embank-ment built on PVD-improved soft soil was carried out and theresults were compared to field observations. The soft clay wasmodeled with three different constitutive models, the isotropicModified Cam Clay (MCC) model, the S-CLAY1 model which ac-counts for plastic anisotropy and its extension, the S-CLAY1S mod-el that additionally accounts for bonding and destructuration.

The numerical simulations demonstrate that for this particularboundary value problem, ignoring the effect of anisotropy and mi-cro-structure leads to notable underprediction of vertical displace-ments. The isotropic MCC model predicts notably smaller verticalsettlements than the two anisotropic models.

Two dimensional equivalent plane strain simulations show thatthe three alternative matching procedures proposed by Hird et al.[19,21] produce identical results for each constitutive model. Com-bined matching procedure is the most convenient as it enablescontrolling mesh geometry in the finite element analysis. The rateof consolidation in the equivalent plane strain analyses tended tobe faster than that in the 3D analyses. The difference changes frommodel to model, and varied during consolidation from 0.06% to6.5%, being the largest after a few months of consolidation. How-ever, the match is found to be satisfactory, but not perfect.

The smear effect must be taken into consideration in the finiteelement analyses. The full-scale plane strain analyses showed thatthe settlements calculated with two anisotropic models (S-CLAYand S-CLAY1S) agreed with the field measurements whenrs/rw = 7 (ratio of the radius of the smear zone to the radius ofthe equivalent drain) and kh/ks = 20 (ratio of the intact permeabilityto the permeability in the smear zone) which agree with the valuesproposed in literature. The models seem to be a significantimprovement compared with the MCC model.

In the future analyses, it would be advisable to consider the 3Dnature of the whole embankment. In addition, the effects of creepbecome significant because the vertical drains speed up the consol-idation in the vertically drained area. Further investigations shouldconsider the creep effect, using the time dependent extensions ofthe S-CLAY1 model proposed by Leoni et al. [53] and Yin andKarstunen [54].

Acknowledgements

The work presented was carried out as part of a Marie CurieResearch Training Network ‘‘Advanced Modelling of Ground

Improvement on Soft Soils (AMGISS)” (MRTN-CT-2004-512120)supported by the European Community through the program ‘‘Hu-man Resources and Mobility”. In addition, the author was sup-ported by the Scientific and Technological Research Council ofTurkey (TUBITAK). The author wishes to thank Dr. Minna Karstun-en, University of Strathclyde, Glasgow, UK, for her valuable com-ments and suggestions.

References

[1] Roscoe KH, Burland JB. On the generalized stress–strain behaviour of ‘wet’ clay.Engineering plasticity. Cambridge University Press; 1968. p. 553–609.

[2] Rouainia M, Muir Wood D. A kinematic hardening model for natural clays withloss of structure. Géotechnique 2000;50(2):153–64.

[3] Kavvadas M, Amorosi A. A constitutive model for structured soils.Géotechnique 2000;50(3):263–74.

[4] Liu MD, Carter JP. A structured Cam Clay model. Can Geotech J2002;41:1313–32.

[5] Nova R, Castellanza R, Tamagnini C. A constitutive model for bondedgeomaterials subject to mechanical and/or chemical degradation. Int J NumAnal Meth Geomech 2003;27(9):705–32.

[6] Dafalias YF, Manzari MT, Papadimitriou AG. SANICLAY: simple anisotropic clayplasticity model. Int J Numer Anal Meth Geomech 2006;30:1231–57.

[7] Wheeler SJ, Näätänen A, Karstunen M, Lojander M. An anisotropic elasto plasticmodel for soft clays. Can Geotech J 2003;40:403–18.

[8] Koskinen M, Karstunen M, Wheeler SJ. Modeling destructuration andanisotropy of a natural soft clay. In: Proceedings of the 5th Europeanconference on numerical methods in geotechnical engineering (NUMGE),Paris; 2002. p. 11–20.

[9] Gens A, Nova R. Conceptual bases for a constitutive model for bonded soils andweak rocks. In: Proceedings of the international symposium on hard soils –soft rocks, Athens; 1993. p. 485–94.

[10] Koskinen M, Zentar R, Karstunen M. Anisotropy of reconstituted POKO clay. In:Proceedings of the 8th international symposium on numerical models ingeomechanics (NUMOG), Rome; 2002. p. 99–105.

[11] Karstunen M, Koskinen M. Anisotropy and destructuration of Murro clay. In:Jardine RJ et al., editors. Advances in geotechnical engineering – the Skemptonconference, vol. 1. London: Thomas Telford; 2004. p. 476–87.

[12] Näätänen A, Wheeler S, Karstunen M, Lojander M. Experimental investigationof an anisotropic hardening model. In: Proceedings of the 2nd internationalsymposium on prefailure deformation characteristics of geomaterials, Turin,Balkema; 1999. p. 541–8.

[13] Koskinen M, Karstunen M. The effect of structure on the compressibility ofFinnish clays. Proc XIV Nordic Geotech Meet 2004;1:11–22.

[14] Zentar R, Karstunen M, Wheeler SJ. Influence of anisotropy and destructurationon undrained shearing of natural clays. In: Proceedings of the 5th Europeanconference on numerical methods in geotechnical engineering (NUMGE),Paris; 2002. p. 21–6.

[15] Li AL, Rowe RK. Combined effects of reinforcement and prefabricated verticaldrains on embankment performance. Can Geotech J 2001;38:1266–82.

[16] Borges JL. Three-dimensional analysis of embankments on soft soilsincorporating vertical drains by finite element method. Comp Geotech2004;31:665–76.

[17] Chai JC, Miura N, Sakajo S, Bergado DT. Behavior of vertical drain improvedsubsoil under embankment loading. Soils Found 1995;35(4):49–61.

[18] Chai JC, Shen SL, Miura N, Bergado DT. Simple method of modeling pvd-improved subsoil. ASCE J Geotech Geoenv Eng 2001;127(11):965–72.

[19] Hird CC, Pyrah IC, Russell D. Finite element modeling of vertical drains beneathembankments on soft ground. Géotechnique 1992:499–511.

[20] Indraratna B, Redana IW. Plane strain modeling of smear effects associatedwith vertical drains. J Geotech Geoenv Eng 1997;123(5):474–8.

[21] Hird CC, Pyrah IC, Russell D, Cinicioglu F. Modelling the effect of vertical drainsin two-dimensional finite element analyses of embankments on soft ground.Can Geotech J 1995;32:795–807.

[22] Graham J, Noonan ML, Lew KV. Yield states and stress–strain relations innatural plastic clay. Can Geotech J 1983;20:502–16.

[23] Korhonen KH, Lojander M. Yielding of Perno clay. In: Proceedings of the 2ndinternational conference on constitutive laws for engineering materials,Arizona, vol. 2; 1987. p. 1249–55.

[24] Finnish National Road Administration. Competition to calculate settlements atthe Haarajoki test embankment. Competition programme, Competitionmaterials, FinnRA, Finland: 1997.

[25] Näätänen A, Puumalainen N, Saarelainen K, Aalto A, Lojander M, Vepsäläinen P.Estimation of settlement of Haarajoki test embankment. In: Proceedings of 6thfinnish mechanical days, Oulu, Finland; September 5-6, 1997. p. 295–310.

[26] Aalto A. The calculations on Haarajoki test embankment with the finiteelement program Plaxis 6.1. In: Proceedings of the 4th European conference onnumerical methods in geotechnical engineering (NUMGE), Udine; 14–16October, 1998.

[27] Näätänen A, Vepsäläinen P, Lojander, M. Finite element calculations onHaarajoki test embankment. In: Proceedings of the 4th European conferenceon numerical metchods in geotechnical engineering NUMGE98, Udine; 1998.p. 151–60.

A. Yildiz / Computers and Geotechnics 36 (2009) 1072–1083 1083

[28] Cundy M, Neher H. Numerical analysis of a test embankment on soft groundusing an anisotropic model with destructuration. In: International workshopon geotechnics of soft soils-theory and practice, Noordwijkerhout,Netherlands; 17–19 September, 2003. p. 265–70.

[29] Mayne PW, Kulhawy FH. K0 – OCR relationship in soil. J Geotech Eng1982;18(6):851–72.

[30] Brinkgreve RBJ, Swolfs WM. PLAXIS, finite element code for soil and rockanalyses, 3D foundation – V2. Rotterdam: Balkema; 2007.

[31] Wiltafsky C. S-CLAY1S. User defined soil model for plaxisdocumentation. University of Glasgow; 2003.

[32] Hansbo S. Consolidation of clay by band-shaped vertical drains. Ground Eng1979;12(5):16–25.

[33] Atkinson MS, Eldred PJL. Consolidation of soil using vertical drains.Geotechnique 1981;31(1):,33–43.

[34] Rixner JJ, Kramer SR, Smith AD. Prefabricated vertical drains, vol. II, Summaryof research effort. Federal highway administration research, report no. FHWA/RD-86/169, Washington, DC; 1986.

[35] Long RP, Covo A. Equivalent diameter of vertical drains with an oblong crosssection. J Geotech Eng Div ASCE 1994;120(9):1625–30.

[36] Barron RA. Consolidation of fine-grained soils by drain wells. Trans Am SocCivil Eng 1948;113:718–43.

[37] Hansbo S. Consolidation of fine-grained soils by prefabricated drains. In:Proceedings of the 10th international conference on soil mechanics andfoundation engineering, Stockholm, vol. 3; 1981. p. 677–82.

[38] Onoue A. Consolidation by vertical drains taking well resistance and smearinto consideration. Soils Found 1988;28(4):165–74.

[39] Zeng GX, Xie KH. New development of vertical drain theories. In: Proceedingsof the 12th ICSMFE, Rio de Janeiro, vol. 2; 1989. p. 1435–8.

[40] Brinkgreve RBJ. PLAXIS, finite element code for soil and rock analyses, 2D –version 8. Rotterdam: Balkema; 2002.

[41] Taylor DW. Fundamentals of soil mechanics. New York: Wiley; 1948.[42] Tavenas F, Jean P, Leblond P, Leroueil S. The permeability of natural clays, Part

II: permeability characteristics. Can Geotech J 1983;20(4):645–60.

[43] Indraratna B, Balasubramaniam AS, Ratnayake P. Performance of embankmentstabilized with vertical drains on soft clay. ASCE J Geotech Eng1994;120(2):257–73.

[44] Hansbo S. Aspects of vertical drain design: Darcian or non-Darcian flow.Géotechnique 1997;47:983–92.

[45] Zhou W, Hong HP, Shang JQ. Probabilistic design method of prefabricatedvertical drains for soil improvement. J Geotech Geoenviron Eng1999;125(8):659–64.

[46] Jamiolkowski M, Lancellotta R. Consolidation by vertical drains: uncertaintiesinvolved in prediction of settlement rates. In: Proceedings of the 10thinternational conference on soil mechanics and foundation engineering, vol.4; 1981. p. 593–5.

[47] Hansbo S. Design aspects of vertical drains and lime column installation. In:Proceedings of 9th Southeast Asian geotechnical conference, vol. 2(8); 1987. p.1–12.

[48] Bergado DT, Mukherjee K, Alfaro MC, Balasubramaniam AS. Prediction ofvertical-band-drain performance by the finite-element method. GeotextilesGeomembranes 1993;12:567–86.

[49] Indraratna B, Redana IW. Laboratory determination of smear zone due tovertical drain installation. J Geotech Geoenviron Eng 1998;124:180–4.

[50] Chai JC, Miura N. Investigation of factors affecting vertical drain behavior. ASCEJ Geotech Geoenv Eng 1999;125(3):216–26.

[51] Bo MW, Bawajee R, Chu J, Choa V. Investigation of smear zone around verticaldrain. In: Proceedings of the 3rd international conference on groundimprovement techniques, Singapore; 2000. p. 109–14.

[52] Xiao DP. Consolidation of soft clay using vertical drains. Ph.D. Thesis, NanyangTechnological University, Singapore; 2001.

[53] Leoni M, Karstunen M, Vermeer PA. Anisotropic creep model for soft soil.Géotechnique 2008;58(3):215–26.

[54] Yin ZY, Karstunen M. Influence of anisotropy, destructuration and viscosity onthe behaviour of an embankment on soft clay. In: 12th internationalconference of international association for computer methods and advancesin geomechanics (IACMAG), India; 2008.