Vermeer+Abed Tunisia2006

8/8/2019 Vermeer+Abed Tunisia2006

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First Euro Mediterranean Conference on Advances in Geomaterials and Structures Hammamet 3-5 May Tunisia

Copyright 2004 Inderscience Enterprises Ltd.

Numerical Simulation of Unsaturated Soil

Behaviour

Ayman A. Abed*Institute of Geotechnical Engineering,

Universitt Stuttgart, Germany

E-mail: [email protected]

*Corresponding author

Pieter A. VermeerInstitute of Geotechnical Engineering,

Universitt Stuttgart, Germany

E-mail: [email protected]

Abstract: The mechanical behaviour of unsaturated soils is one of the challenging topics in the field of geotechnical engineering. Theuse of finite element technique is considered as a promising method to solve settlement and heave problems, as associated withunsaturated soil. Nevertheless, the success of the numerical analysis is strongly dependent on constitutive model being used. Thewell-known Barcelona Basic Model is considered to be a robust and suitable model for unsaturated soils and has thus beenimplemented into the PLAXIS finite element code. This paper provides results of numerical analyses of a shallow foundation restingon an unsaturated soil using the implemented model. Special attention is given to the effect of suction variation on soil behaviour.

Keywords: unsaturated soil, constitutive modelling, finite element method, shallow foundation.

1 INTRODUCTION

Unsaturated soil is characterized by the existence of three

different phases, namely the solid phase, the liquid phase

and the gas phase. The important consequence is the

development of suction force at the solid-water-air interface.

This force increases with continuous drying of the soil and

vice versa suction forces will be reduced upon wetting of

the soil. This relation between the suction in the gas phase

and the soil water content is named the Soil Water

Characteristic Curve (SWCC). Figure 1 gives a graphical

representation for two different soils, namely clayey silt and

fine sand. This curve plays a key role in unsaturated ground

water flow calculations and unsaturated soil deformationanalyses.

Figure 1 The soil water characteristic curves for

clayey silt and fine sand

It can be seen from Figure 1 that suction plays a more

important rule in the case of fine-grained soil than in the

case of a coarse-grained sand. Indeed at the same water

content, clay exhibits much more suction than sand. For that

reason, one can expect more suction related problems

during construction on clay than on sand.

Soil shrinkage is a well recognized problem which is

associated with suction increase, i.e. soil drying. On the

other hand, soil swelling and soil structure collapse is

considered as a main engineering problem during suction

decrease under constant load, i.e. soil wetting. These

phenomena would affect the foundations if no special

measures would have been taken during the design process.

The damage reparation costs level could reach high numberse.g. as much as $9 billion per year in the USA only [1].

Many empirical procedures have been proposed during the

past to predict the volumetric changes due to suction

variations, but during the last fifteen years research attention

has shifted to more theoretical models. In combination with

robust constitutive models the FE method gives the designer

a nice tool to understand the mechanical behaviour of

unsaturated soils and reach better design criteria.

2 UNSATURATED SOIL MODELLING

In surveying the literature one can classify the modelling

methods into empirical and theoretical approaches.


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A. A. ABED AND P. A. VERMEER

2.1 Empirical methods

Empirical methods are based on direct fitting of test data

for clays or silts. Especially poorly graded silts (loess) are

renown as collapsible soil. These empirical methods are

mostly based on data from oedometer apparatus for one-

dimensional compression. These tests give only clear

information about the sample initial conditions and finalconditions but no information about the suction variation

during the saturation process. A nice review and evaluation

of these methods can be found for example in the paper by

Djedid et al. [2]. As an example, Equation 1 is proposed by

Kusa and Abed [3] to predict the swelling pressure sw in(kg/cm2) as a function of the liquid limit wL (%) the initial

water content wn (%) and the free swelling strain 0 (%).This strain is defined as the ratio of the soil sample height

after saturation (without any external load) and the initial

sample height before saturation.

n

0LLsw

w

lnw033.0w053.0

+= (1)

It is believed that such empirical correlations give onlysatisfactory results as long as they are applied to the samesoils which are used to derive them. This reduces their use

to a very narrow group of soils.

2.2 Theoretical methods

This category uses the principles of soil mechanics together

with sophisticated experimental data for the formation of a

constitutive stress-strain law. An early attempt was made by

Bishop [4]. He extended the well-know effective stress

principle for fully saturated soil to unsaturated soil. Bishopproposed the effective stress measure

)uu(u waa += (2)

where

: total stress

ua : pore air pressure

uw : pore water pressure

: factor related to degree of saturation

where = 0 for dry soil and = 1 for saturated soil.According to Bishop the effective stress always decreaseson wetting under constant total stress. As the effective stressdecreases an increase in the volume of the soil should beobserved in accordance with the above definition ofeffective stress. However, experimental data often showsadditional compression on wetting which is opposite to the

prediction based on Bishops definition of effective stress.Many critics were expressed regarding the use of a singleeffective stress measure for unsaturated soil and there has

been a gradual change towards the use of two independentstress state variables.

It was proposed by Fredlund et al. [5] to use the net stress

-ua and the suction s as two independent stress statevariables to describe the mechanical behaviour of theunsaturated soil, where s = uauw. Considering the two

stress measures together with the critical state soil

mechanics, an elastoplastic constitutive model for

unsaturated soil has been developed by Alonso et al. [6],

and later by Gens et al. [7]. Later other constitutive models

have been proposed, but all of them remain in the

framework of the Alonso and Gens model, which became

known as Barcelona Basic Model (BB-model).

3 BARCELONA BASIC MODEL

The BB-model is based on the Modified Cam Clay model

for saturated soil with extensions to include suction effects

in unsaturated soil [7]. This model uses the net stresses -uaand the suction s as the independent stress measures. Many

symbols have been used for the net stresses such as " and* . The latter symbol will be used here. It is assumed that

the soil has different stiffness parameters and different

mechanical response for the changes in net stresses thanthem for the changes in suction.

3.1 Isotropic loading

For unloading-reloading the rate of change of the void ratio

is purely elastic and related to the net stress and the suction

where is the normal swelling index and s is the suctionswelling index, patm is the atmospheric pressure and p

* is the

mean net stress

In terms of volumetric strain equation (3) reads

(5)

where compressive strains are considered positive.

For primary loading both elastic and plastic strains develop.

The plastic component of volumetric strain is given by

where 0 is the compression index and 0pp is the preconsolidation pressure in saturated state. The above

equation is in accordance with critical state soil mechanics.

The difference with the critical state soil mechanics is the

yield function

(7)

with

(8)

)3(ps

s

p

pee

atms*

*e

+==

&&&&

)4(u)(3

1p a321

* ++=

atm

s

*

*evv

ps

s

e1p

p

e1e1

e

+

+

+

+

=+

==&&&

&&

)6(p

p

e1 0p

0p0pv

&&

+

=

=0

)p

p(pp

c

0pc

p

( ) )9(e s0

=

p* ppf =


3/8

NUMERICAL SIMULATION OF UNSATURATED SOIL BEHAVIOUR

where and pp are the compression index and the suctiondependent preconsolidation pressure respectively. Hence,

for full saturation we have s = 0, = 0 and pp = . Thelarger the suction the smaller the compression index . Inthe limit for =s the above expression yields .= The index ratio / 0 is typically in the range between 0.2and 0.8. The constant pc is mostly in the range from 10 to 50

[kPa]. The constant controls the rate of decrease of thecompression index with suction; it is typically in the range

between 0.01 and 0.03 [kPa-1]. The monotonic increase of

soil stiffness with suction is associated with an increase of

the preconsolidation pressure according to Equation 8.

In order to study Equation 6 in more detail, we consider

the consistency equation 0f =& , as it finally leads toEquation 6. In terms of partial derivatives the consistency

equation reads

with

(11)

(12)

(13)

It follows from the above equations that

This equation is in full agreement with Equation 6, but

instead of pp0 it involves the stress measures s and*p .

Equation 14 shows the so-called soil collapse upon wetting.

In deed, upon wetting we have 0s & even at constant load, i.e. for 0p* =& .

3.2 More general states of stress

For the sake of convenience, the elastic strains will not be

formulated for rotating principal axes of stress and strain.

Instead, restriction is made to non- rotating principal

stresses. For such situation Equation 5 can be generalized to

become

whereei& is a principal elastic strain rate,

*i is a principal

net stress, j = 1 for j=1,2,3 and

(17)

where is the elastic Poisson ratio. Young modulus isstress dependent

The term sK j1

s &

in Equation 15 represents the

contribution of suction loading-unloading (drying-wetting)

to the elastic strain rates, whereas the other term represents

the net stresses loading-unloading contribution.

For formulating the plastic rate of strain, both the plastic

potential and the yield function have to be consider. For the

BB-model the yield function reads

where M is the slope of the critical state line, as also

indicated in Figure 2, and

(20)

ps = a. s (21)

It can be observed from Figure 2 that ps reflects theextension of the yield surface in the direction of tension part

due to apparent cohesion. The constant a determines the rate

of ps increase with suction.

The yield function (19) reduces to the Modified Cam Clay

(MCC) yield function at full saturation with s = 0. In

contrast to the MCC-model, the BB-model has a non-

associated flow rule, which may be written as

Figure 2 Yield surface of Barcelona Basic Model

)10(0p

ss

pp

p

ff pvp

v

pp*

*=

= &&&&

1p

f*

=

c

pp

p

p

plnp

s

p

=

ppv

pp

e1p

+

=

)14(pp

1

e1s

p

pln

e1

*

pc

ppv &&& +

+

+

=

( ) )15(3,2,1j,iforsKD j1sejij*i == &&&

( ) ( ))16(

pse13K

atm

s1s ++

=

( ) ( )

+

=

1

1

1

121

ED ij

( ) ( ) )19(ppppMqf*

ps*22

+=

213

232

221 )()()(

2

1q ++=

s=0

s=s1

0pp

pp

( ) )18(pe1

KwithK213E *+

==

)22()3,2,1i(g

i

pi =

=&


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wherepi& stands for a principal rate of plastic strain, is a

multiplier and g is the plastic potential function

The flow rule becomes associated for = 1, but Gens et al.

[7] recommend to use

In this way the crest of the plastic potential in p *-q-plane is

increased. Finally it leads to realistic K0-values in one-

dimensional compression, whereas the associated MCC-

model has the tendency to overestimate K0-values [8].

In combination with Equation 15 and 22 the consistency

condition 0f =& yields the following expression for theplastic multiplier

with

(25)

4 SETTLEMENT ANALYSIS

Figure 3 shows the geometry, the boundary conditions and

the finite element mesh for the problem of a rough stripfooting resting on partially saturated soil. The material

properties shown in Table 1 are the same as those given by

Compas and Vargas [9] for a particular collapsible silt.

However, as they did not specify the M-value, we assumed

a critical state friction angle of 31o, which implies M = 1.24.

The ground water table is at a depth of 2 m below the

footing. The initial pore water pressures are assumed to be

hydrostatic, with tension above the phreatic line. For the

suction, this also implies a linear increase with height above

the phreatic line, as in this zone the pore air pressure ua is

assumed to be atmospheric, i.e. s = ua-uw = -uw. Below the

phreatic line pore pressures are positive and we set ua = uw,as also indicated in Figure 3.

Table 1 Material and model parameters

For uw < 0 the linear increase of uw implies a decreasing

degree of saturation, as also indicated in Figure 3. In fact,

the degree of saturation is not of direct impact to the present

settlement analysis, as transient suction due to deformation

and changing degrees of saturation are not be considered.

The distribution of saturation being shown in Figure 3, was

computed using the van Genuchten model [10] together

with additional data for the silt. Using the empirical van

Genuchten relationship the soil is found to be saturated upto some 50 cm above the phreatic line. For the sake of

convenience, however, a constant (mean) value of 17.1

kN/m3 has been used for the soil weight above the phreatic

line. For the initial net stresses the K0-value of 1 has been

used. The finite element mesh consists of 6-noded triangles

for the soil and 3-noded beam element for the strip footing.

The flexural rigidity of the beam was taken to be EI = 10

MN.m2 per meter footing length. This value is

representative for a reinforced concrete plate with a

thickness of roughly 20 cm.

Computed load-settlement curves, are shown in Figure 4

both for the Barcelona Basic model and the Modified CamClay model. For the latter MCC-analysis, suction was fully

neglected. In fact it was set equal to zero above the phreatic

line. On the other hand suction is accounted for in the BB-

analysis, but we simplified the analysis by assuming no

change of suction during loading. In reality, footing loading

will introduce a soil compaction and thus some change of

both the degree of saturation and suction. As yet this has not

been taken into account.

Up to an average footing pressure of 80 kPa both analyses

yield the same load-displacement curve. This relates to the

adoption of preconsolidation pressure pp0 = 80 kPa. For

pressures beyond 80 kPa, Figure 4 shows a considerabledifference between the results from the BB-analysis and the

)24()M6(9

)3M)(9M(M

0

0

=

sDf

Ks

f

H

1D

f

H

1jij*

i

T1

sjij*i

T

&&

+

=

*j

ij*i

T

*pv

gD

f

p

gfH

+

=

Figure 3 Geometry, boundary conditions and finite element mesh

( ) ( ) )23(ppppMqg *ps*22 +=


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Figure 4 Footing pressure-settlement curves

MCC-analysis. Indeed, the BB-analysis yields much smaller

settlements than the MCC-model. Hence settlements are

tremendously overestimated when suction is not taken into

account. The impact of suction is also reflected in the

development of the plastified zone below the footing. For

the BB-analysis the plastic zone with f = 0 is indicated in

Figure 5a. The MCC-analysis shows a larger plastic zone

underneath the footing, as shown in Figure 5b.

5 INCREASE OF GROUND WATER LEVEL

Having loaded the footing up to an average pressure of

150 kPa, we will now consider the effect of soil wetting by

increasing the ground water table up to ground surface. This

implies an increase of pore water pressures and thus a

decrease of effective stresses, being associated with soil

heave. On simulating this raise of the ground water level by

the MCC-model, both the footing and the adjacent soil

surface is heaving, as plotted in Figure 6. Due to the fact

that we adopted an extremely low swelling index of only

0.006 (see Table 1) heave is relatively small, but for other

(expansive) clays it may be five times as large.

Similar to the MCC-analysis, the BB-analysis yields soil

heave as also shown in Figure 6. In contrast to the MCC-

Figure 5 The plastic zones from BB and MCC model for footingpressure of 150 kPa

Figure 6 Vertical displacement of soil surface due to wetting

analysis, however, the footing shows additional settlements.

Here it should be realised that Figure 6 shows vertical

displacements due to wetting only, i.e. an extra footing

settlement of about 25 mm. The BB-analysis yields this

considerable settlement of the footing, as it accounts for the

loss of so-called capillary cohesion as soon as the suction

reduces to zero. In text books [11] this phenomenon is

referred to as soil (structure) collapse.

The different performance of both models is nicely

observed in Figure 4. Here the BB-analysis yields a

relatively stiff soil behaviour when loading the footing up to

150 kPa, followed by considered additional settlement upon

wetting. In contrast, the MCC-model yields a relatively soft

response upon loading and footing heave due to wetting.

Finally both models yield nearly the same final settlement

of about 49 mm.

6 GROUND WATER FLOW

Ground water flow is governed by the ground water head

h = y + uw / w , where y is the geodetic head and w is theunit pore water weight. In most practical cases there will not

be a constant ground water head, but a variation with depth

and consequently ground water flow. Indeed, in reality there

will be a transient ground water flow due to varying rainfall

and evaporation at the soil surface. This implies transientsuction fields and footing settlements that vary with time.

For most footing, settlements variations will be extremely

small, but they will be significant for expansive clays as

well as collapsible subsoil. In order to analyse such

problems, we will have to incorporate ground water flow.

Flow in an isotropic soil is described by the Darcy equation

where qi is a Cartesian component of the specific discharge

water, ksat is the well-known permeability of a saturated soiland krel is the suction-dependent relative permeability.

150 kPa 150 kPa

(a) (b)

)26(x

hkkq

isatreli

=


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Figure 7 Simplified van Genuchten model

A simplification of the van Genuchten model [13] leads to

the equation

(27)

where sk is a soil-dependent constant which is related to theextent hk of the unsaturated zone under hydrostatic

conditions. It yields hk = sk / w. For the saturated zone thisequation yields krel = 1 and for s = sk it gives krel = 10

-4. In

numerical analyses 10-4 is a suitable threshold value that

may be used for s > sk. Figure 7 shows a graphical

representation of Equation 27 for sk = 50 kPa , i.e. a

capillary height of hk= 5 m.

In order to do ground water flow calculations, one has to

supplement Darcys equation 26 with a continuity equation

of the form

where repeated subscripts stand for summation. C is the

effective storage capacity, which is often expressed as

where Csat is the saturated storage capacity, n the porosity

and Sr the degree of saturation. The latter is a function of

saturation and one often adopts the van Genuchten

relationship [10].

Strictly speaking soil deformation implies changing soil

porosity n and pore fluid flow cannot be separated from soil

deformation. For many practical problems, however, the soil

porosity remains approximately constant and flow problems

may be simulated without consideration of coupling terms.

In order to solve the differential equations 26 and 28,

boundary conditions are required. For studying footing

problems, one would need the water infiltration or the rate

of evaporation at the soil surface, qsurface, as a function of

time. For the footing in section 4, the surface discharge was

taken to be zero, being accounted for by a constant ground

water head.

7 BEARING CAPACITY

From Figure 4 it might seen that the bearing capacity of

the footing is nearly reached, at least for the MCC-analysis

without suction. However, the collapse load is far beyond

the applied footing pressure of 150 kPa, at least for a

Drucker-Prager type generalization of the Modified CamClay model and a CSL-slope of M=1.24. The applied

Drucker-Prager generalization involves circular yield

surfaces in a deviatoric plane of the principle stress space,

which is realistic for small friction angels rather than large

ones. For this reason we will analyse the bearing capacity of

a strip footing for a relatively low CSL-slope of M = 0.62.

Under triaxial compression conditions we have M =

. and we get a friction angle of cs =16.4o . However, we consider the plane strain problem of a

strip footing. For planar deformation it yields

. [14], and it follows that cs = 21o. Table

2 gives the soil parameters. Figure 8, shows the boundary

conditions and the finite element mesh for the bearingcapacity problem of shallow footing on unsaturated soil.

In this analysis, the soil has been loaded up to failure

using again both the BB-model and the MCC-model. In

order to be able to compare the numerical results with

theoretical values, we used a uniform distribution for

suction in the unsaturated part of s = 20 kPa. The soil is

considered to be weightless and the surcharge soil load is

replaced by a distributed load of 25 kN/m2 per unit length

which is equal to a foundation depth of about 1.5 m. A value

of K0 = 1 is used to generate the initial net stresses. The

same finite element types as in the previous problem are

used here for the soil and the footing.

According to Prandtl, the bearing capacity is given by

where c is the soil cohesion, q0 is the surcharge load atfooting level and b is the footing width.

Figure 8 Finite element mesh and boundary conditions for the

bearing capacity problem

ks

s4

rel ss0for10kk


7/8


Table 2 Soil properties

The factors Nc, Nq and N are functions of the soil frictionangle

In the present analysis is taken equal to zero and thecorresponding N-factor is not needed. For the zero-suction

case we have c = 0 and the bearing capacity qf is found to

be 177 kPa.

According to BB-model, the cohesion c increases with

suction s linearly, according to the formula

c = a tans (32)

On using a = 1.24 and s = 20 kPa we find c = 9.5 kPa. For

this capillary cohesion of 9.5 kPa the Prandtl equation yields

qf =327 kPa. Figure 9 shows the calculated load-

displacement curves using the BB-model and the MCC-

model. The figure shows that an increase of suction value

by 20 kPa was enough to double the soil bearing capacity.

Shear bands at failure as shown in Figure 10 are typically

according to the solution by Prandtl. In Figure 11, thedisplacement increments show the failure mechanism

represented by footing sinking which is associated with soil

heave at the edges. By comparing the theoretical bearing

capacity values with the computed ones (Table 3), it is clear

that the results are quite satisfactory with relatively small

error.

It is believed that we can capture better bearing capacity

values by adopting more advanced failure criterion than the

Drucker-Prager criterion being used in this analyses.

Figure 9 Loading curves for BB- and MCC-analysis

Figure 10 Incremental shear strain at failure for

s = 20 kPa

Figure 11 Total displacement increments for

s = 20 kPa

One can use a modified version of the well-known Mohr-

Coulomb failure criterion which accounts for suction

effects, or Matsuoka et al. criterion [15] which offers us a

failure surface without singular boundaries and as a

consequence a more suitable criterion for numericalimplementation.

8 CONCLUSION

The present study illustrates the possibility of simulating

the mechanical behaviour of unsaturated soil using the finite

element method with a suitable constitutive model. On

incorporating suction, soil behaviour was shown to be much

stiffer than without suction. Moreover, it has been shown

that soil collapse was well simulated. This phenomenon is

well-known from laboratory tests, but it also applies to

footings as shown in this study.

In general shallow foundations will not be build on

collapsible soils, but many footings have been constructed

on swelling clays and this will also be done in the future.

From an engineering point of view, pile foundations may be

preferred, but they are often too costly for low-rise

buildings. Therefore heave and settlement of shallow

foundations on expansive clays will have to be studied in

full detail. At this point, a one dimensional transient flow

calculations for an infiltration and evaporation processes

can be very helpful. By applying transient boundary

conditions one can simulate the variation of a suction profilewith time; typically for two or three years.

( ) )31(cot1NN,esin1

sin1N qc

tanq =

+=


8/8


Table 3 Bearing capacity values

Depending on the results, the designer can pick the lowest

and the highest suction values in the studied period. With

these information in hand, deformation analyses for these

cases can be done to determine the absolute foundation

deformation variations as well as the differential settlements

with respect to neighbouring footings. Such movements due

to suction variations can introduce quite high bending

moments in the beams, columns and walls of

superstructures if they have not been considered in design.

Another important application of unsaturated soil

mechanics is seen in the field of slope stability. Many

natural slopes have low factors of safety and slope failuresare especially imminent after wetting by rainfall. Hence, soil

collapse computations would seem to be of greater interest

to slopes than to footings, as considered in this study. Not

only natural slopes suffer upon wetting, but also river

embankments. High river water levels tend to occur for

relative short period of time, so that there is partial wetting.

This offers also a challenging topic of transient ground

water flow and deformations in unsaturated ground.

ACKNOWLEDGEMENT

We are grateful for GeoDelft, the Netherlands, for

providing support for this study. Special thanks are due to

Mr. John van Esch of GeoDelft and to Prof. Antonio Gens

from the University of Catalunia and Dr. Klaas Jan Bakker

of the Plaxis company for fruitful discussions on

unsaturated soil behaviour.

REFERENCES

[1] J.D. Nelson, D.J. Miller, Expansive Soils, John Wiley &

Sons, 1992.

[2] A. Djedid, A. Bekkouche, S.M. Mamoune, Identification

and prediction of the swelling behaviour of some soils from

the Telmcen region of Algeria, Bulletin des Laboratories

des Ponts et Chaussees , (233), July August, 2001.

[3] Issa.D. Kusa, Ayman.A.Abed, The effect of swelling

pressure on the soil bearing capacity, Master thesis, Al-

Baath University, Syria, 2003.

[4] A. W. Bishop, The principle of effective stress. Teknisk

Ukeblad39, 1959.

[5] D.G. Fredlund, and N.R. Morgenstern, Stress state

variables for unsaturated soils. Journal of the Geotechnical

Engineering Division, Proceedings, American Society of

Civil Engineering (GT5), 1977

[6] E. E. Alonso, A. Gens, and D. W. Hight, Special

Problem Soils, General report, Proc. 9th Eur. Conf. Soil

Mech., Dublin, 1987.

[7] E. E. Alonso, A. Gens, and A. Josa, A constitutive

model for partially saturated soils. Geotechnique (40), 1990

[8] K.H. Roscoe, and J.B. Burland, On the generalized

stress-strain behaviour of wet clay.Engineering Plasticity,

Cambridge University Press, 1968.

[9] T. M. de Compas, E. A. Vargas, Discussion : A

constitutive model for partially saturated soils.

Geotechnique (41), 1991.

[10] M. Th. Van Genuchten, A closed-form equation for

predicting the hydraulic conductivity of unsaturated soils.

Soil Sci. Soc. Am. J(44), 1980.

[11] D.G. Fredlund, and H. Rahardjo, Soil Mechanics forUnsaturated Soils, John Wiley & Sons, 1993.

[12] P. A. Vermeer, R. Brinkgreve, PLAXIS Finite

element code for soil and rock analysis, Balkema,

Rotterdam, 1995.

[13] R. Brinkgreve, R. Al-Khoury and J. van Esch,

PLAXFLOW User Manual,Balkema, Rotterdam, 2003.

[14] W.F. Chen, G.Y. Baladi, Soil Plasticity,Elsevier, 1985

[15] H. Matsuoka, D. Sun, A. Kogane, N. Fukuzawa and W.

Ichihara, Stress-Strain behaviour of unsaturated soil in true

triaxial tests, Can. Geotech. J. (39), 2002.

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