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A NEW MODEL TO PREDICT THE SWELLING PARAMETERS

ON DEFORMABLE UNSATURATED SOIL UNDER

CONTROLLED SUCTION

H. Skandaji1 , F. Masrouri

1, M. Jamei

2and O.Touil

2

ABSTRACT

Modelling the complex behaviour of unsaturated expansive soils subjected to different stress paths(mechanical and hydraulic) constitutes a current research topic and deserves further investigation.

In this paper, the results of oedometer swelling tests, under controlled suction, are presented. These testsare conducted for measurement of swelling parameters of Ca2+- bentonite specimens which are compactedstatically.

They are carried out using a developed osmotic suction controlled oedometer. The laboratory swellingtests have a slow response. In fact, when the suction is imposed, the hydraulic equilibrium is reached withinone to three weeks. Therefore, it is interesting to develop a numerical model for quickly predicting theswelling process under different stress paths.

Recently, a new model for the simulation of swelling-pressure has been developed (Shuai andFredlund, 1998).

In this paper an extension of Fredlund’s model is proposed, in which the compressibility or the expansive

coefficients ( )v(iC and

)w( jc ) depend on the matrix suction and total net stress. The proposed model can

describe the porewater pressures as function of time and depth in a specimen as well as the volume changesduring the swelling process. Furthermore, the coefficient of permeability is a function of matrix suction andvoid-ratio.

The extended model problem has been solved by the finite element method using the Galerkin’s

technique (Zienkiewicz, 1977). The formulation of the problem leads to a non-linear algebraic system whichis solved by a Newton-Raphson Algorithm coupled with a Fixed Point Method (William H. P. and al., 1986).

INTRODUCTION

Recent research concerning the behaviour of expansive soils has been undertaken. These studies involvetwo fundamentally different states : Saturated and unsaturated expansive clays. In both cases, the expansionis produced via unloading stress paths (mechanical) or wetting paths (hydraulic). In fact, in the saturatedcase, and when the wetting path is considered, the expansion corresponds to only an increase in water

content. But when an unloading path is applied, the increase in the volume of soil corresponds to the volumechange of the structure under constant water content. If the two paths are applied together, their effects can be separated and dissociated.

On the contrary, with soil in the unsaturated state, we can’t distinguish easily the change in the volume of water from the change in the volume of soil voids because these changes depend on the interaction of phases(air, water and structure). A complete study behaviour of expansive clay should involve these interaction

phenomena. Numerous works show that the postulate of independent stress state variables ((σii-ua) ands = (ua-uw), where, σii is a component of total tensor stresses), is an adequate framework. Thus, we canapproach the expansive clay behaviour by this postulate without including microstructure interactions(Alonso et al., 1995). In fact, the specific and complete analysis leads to equations in which the parametersare very difficult to control and to determine experimentally (Lloret and Alonso, 1980).

1 Laboratoire Environnement Géomécanique & Ouvrages - ENSG - BP 40, 54501 Vandœuvre-les-Nancy cedex2 Laboratoire de modélisation et calcul des structures - Ecole Nationale d’Igénieurs de Tunis-BP37, Tunis le Belvédère

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Here, we consider a simple independent variables approach. For the purpose of the study, the Fredlund’smodel (1993 &1998) is generalised, in which the expansive or compressible parameters depend on thesuction and total net stresses.

CONSTITUTIVE RELATIONS

This model is formulated for the three dimensional case. However, given the unidimensional nature of thetests, only the one dimensional case is used in the present work. Several assumptions are made. The

behaviour of the unsaturated soil is assumed to be nonlinear elastic. The volumetric deformations are relatedto net total stress (σ-ua) and suction (ua-uw) only. The water phase is taken as incompressible. The change in

the volume of water phase " wdV " corresponds to the increase or decrease of water. When small

deformations are assumed, the change in the total volumetric strain is :

zzyyxx åååd

++=ΩΩ

(1)

where : Ω is the local elementary volume ( V⊂Ω , V is the initial total volume), (x,y,z) are the principal

axes, and εii (i =x, y, z) are the components of the strain tensor.

Let’s consider the local elementary domainsvΩ and

wΩ , which contain the point M;wwvv V;V ⊂Ω⊂Ω .

Ω

Ω+

Ω

Ω=

Ω

Ω=

ΩΩ s

vv d

d

d

d w (2)

where : vdΩ is the incremental change in the local elementary volumevΩ of soil voids, wdΩ is the

incremental change in the local elementary volumewΩ of water phase,

svdΩ is the incremental change in

the structure elementary volume of soil, vV is the volume of soil voids, wV is the volume occupied by water

( vr w VSV = ) and Sr is the degree of saturation. In the general case, strains and stresses are not

necessarily uniform. In the present formulation, the change in the volume occupied by the air phase isincluded in vdV . This consideration implies that we can’t compute the change in the volume of air phase

explicitly. Therefore, we consider only two phases : Soil structure and water phase. Moreover, this modelcannot take into account the microstructural expansion.

Soil structure :

)Is( :C:C (v)'a

(v)v21M

)( +σ=ε (3)

where :

å )(v

Mis the local strain tensor of the soil structure, '

aσ and Is are the net stress and the matrix

suction tensors at point M ,)v(

1C is the “flexibility” tensor in the given unsaturated state,)v(

2C is

the swelling (or collapse) tensor under the hydraulic path (wetting path). In the draining path,)v(

2C is the shrinking tensor ;

The components of the two latter tensors are determined through a series of suction controlled triaxial tests.

Water phase :

In the elementary local volume dΩ, the change in the volume of water is defined by :

sc)u(c d

d d (w)

2a(w)1

w(w)vol +−σ=

ΩΩ=ε (4)

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EXPERIMENTAL PROCEDURE

The soil used in the tests is a Ca2 +-bentonite reconstituted in the laboratory, with the followingcharacteristics : wl=164% ; w p=64% ; wr =8-13% ; clay fraction <2mm=89% ; montmorillonite 80% ; unitweight of solids γ s=27.4kN/m3. The specimens are compacted under static stress conditions at 1500 kPa(1kPa = 1kN/m2). The samples are normally consolidated.

The initial matrix suction determined by the "paper filter method" (ASTM D 5298-94) is nearly equal to3900 kPa. Specimen have been soaked under controlled suction using osmotic method (Modified oedometer tests- Delage et al. 1992). Samples were wetted and loaded. Swell and swelling pressure depend on theinitial conditions of the soil (height of the sample, wetting path, dry density, water content and suction).There all these initial parameters are kept constant.

In order to define the coefficients)v(

ijC and)w(

jc , different tests are carried out using different stress and

wetting paths (Guiras, 1996). Figure 1 (a and b) shows these paths.

a- Free swell oedometer tests b- Constant load –swell oedometer testsFi ure 1 : Tested stress aths

The initial and final parameters are summarised in Table 1. In the tests, one of two paths is applied(suction decrease or load paths). Changes in the volume of the sample and in the water volume aremeasured.

Table 1 : Initial state of samples “ Free swell and constant load oedometer tests”

REF Points ImposedSuction

Initial state Pathsoσ (kPa) Initial state

(kPa)

γ d

(kN/m3)

wo (%) γ d

(kN/m3)

wo (%)

Test 1 C1 1000 14,9 22,2 A 400 14,6 24,1Test 2 C2 600 14,9 22,4 B 800 14,7 23,0Test 3 C3 200 14,8 22,2 C 1200 14,7 23,4

γ d± 0,2 (kN/m3) w ± 0,2 (%)

EXPERIMENTAL RESULTS

Volume changes of soil structure in free swell tests and constant load swell tests are represented inFigure 2, versus different controlled suction (Test1, Test2 and Test3) and different load (paths A, B and C).

Experimental results using these different paths lead to the determination of the expressions of theswelling and compressibility parameters :

v'a

vv111c β+σα= ; w'

aww

111c β+σα= ; vvv222 sc β+α= and www

222 sc β+α=

0

1000

2000

3000

4000

5000

1 10 100 1000 10000

Vertical net stress ( σ -u ) kPa

C

C

C

Test 1

Test 2

Test 3

A B B'

v a

3

2

1

S u c t i o n

s = u -

u

( k P a )

a

w

0

1000

2000

3000

4000

5000

0 400 800 1200 1600

Vertical net stress ( -u ) kPa

Ao

A B C

Bo CoD

S u c t i o n s

= u

- u

( k P a )

a

w

aσv

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where wi

wi

vi

vi and,, βαβα are given in Table 2.

a – volume change of soil voids under controlledsuction

b - volume change of soil voids under constant load

Figure 2 : Swell and Compressibility of soil voids

Table 2 : Parameters for different tests

)w,v(iα 10-7(1/(kPa2))

)w,v(iβ 10-4(1/kPa)

v1

α v2

α w1α

w2α v

1β v

2β w

1βδφ

δπ=λd2

-14 -40-200

-

-400 -60 -40 -20

8 40180

-

-80100 40 40

55120351

-

582 63 42 31

-37 -81-271

-

-284-176 -97 63

The coefficient of permeability at varied hydraulic head gradient is obtained from a direct measurementof water flow using the osmotic oedometer apparatus equipped with a mariotte bottle.

For unsaturated expansive soils, the coefficient of permeability is a function of two volume mass properties (matrix suction and void ratio) (Lloret and Alonso, 1980). For tests 1, 2 and 3, under constantmatrix suction a classical method of computing the saturated coefficient of permeability at varied hydraulichead gradient is used. In fact water change and strain are varied until equilibrium is achieved. Then thecoefficient of permeability K w/a , is calculated. While for constant load oedometer tests under controlled

suction (paths A,B and C), the coefficient of permeability is determined in unsteady state (Gardner, 1956;Vicol T., 1990). Figures 3 and 4 show the variability of relative water coefficient of permeability. In thiswork, numerical divergence problems are encountered. Therefore, the relationship between the relative

permeability, suction and void ratio is not taken into account.

CONCLUSIONS

Experimental results show that along wetting paths, the compressibility coefficients of volume change)v(

ij)C( 1 ) increase. For constant suction, the swelling coefficients)v(

ij)C( 2 ) decrease in the loading path.

Along wetting and loading paths, the coefficient of water volume change)w(

jc increases.

In this model, which assumes the postulate of independent variables, we can compute the swelling parameters (change of total volume and swelling pressure), taking into account the dependence of compressibility or swelling coefficients on suction and net stress.

-50

-40

-30

-20

-10

0

101 10 100 1000 10000

σ - u (kPa)

s = 600 kPa s = 1000 kPas = 200 kPa

Computed curves

∆ V / V ( %

)

v a s = u - u (kPa)

v =10-21 kPa v = 400 kPa (Path A) v = 800 kPa (Path B) v = 1200 kPa (Path C)

Computed curves

σ

σ

σ

σ

a w

∆

V / V ( % )

-50

-40

-30

-20

-10

0

10

1 10 100 1000 10000

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This dependence which is clearly seen from the results of this study, is essential if the model is to begeneralised. Therefore, when practical applications will be studied in two or three dimensions for the

simulation of triaxial tests, a generalised form of this model can be used.

ACKNOWLEDGEMENTS

The authors thank Mr Tisot J. P. for providing the facilities of the laboratory and making it possible toconduct the experimental work in good conditions.

REFERENCES

ALONSO E. E., LLORET A., GENS A. and YANG D. Q. (1995). "Experimental behaviour of expansivedouble-structure clay". 1st Int. Conf. of Unsaturated Soils, 1995, Paris, Vol.1, pp.11-16.

ASTM D 5298-94 - (1994) - "Standard test method for measurement of soil potential (succion) using filter paper". Annual Book of ASTM Standards 1995, Vol. 04.09, pp.154-159.

Figure 3 : Water coefficient of relative permeability versus void ratio

Figure 4 : Water coefficient of relative permeability versus suction

-15

-13

-11

1 10 100 1000 10000Suction s = u - u (kPa)

Free-swell test ,σ = 11 kPa

Path AComputed curves

Path B

Path C

10

10

10

P e r m e a b i l i t y

k

( m

/ s )

W / a

a w

0,60 0,80 1,00 1,20 1,40 1,60

Void ratio

S=1000 kPaS=600 kPaS=200 kPa

Computed curves

-14

-12

-10

-08

10

10

10

P e r m e a b i l i t y

k

( m

/ s )

W / a

10

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DELAGE P.,VICOL T. and SURAJ DE SILVA G. P. R. (1992). "Suction controlled testing of non-saturatedsoils with an osmotic consolidometer. 7

th Int. Conf. On Expansive Soils. pp. 206-211.

FREDLUND, D. G. and RAHARDJO, H. (1993). "Soil mechanics for unsaturated soils". John Wiley &

Sons, Inc.

GARDNER, W. R. (1956). "Calculation of capillary conductivity from pressure plate outflow data". Soil

Science Soc. Am. Proc., Vol. 20, pp.317-320.LLORET, A. and ALONSO, E. E. (1980). "Consolidation of unsaturated soils including swelling andcollapse behavior". Geotechnique 30, N° 4, pp. 449-477.SHUAI, F. and FREDLUND, D. G. (1998). "Model for the simulation of swelling-pressure measurementson expansive soils". Canadian Geot. Journal ., Vol. 35, pp.96-114.GUIRAS- SKANDAJI, H. (1996). "Déformabilité des sols argileux non saturés : Etude expérimentale etapplication à la modélisation". Thèse de doctorat de l’I.N.PL Ecole Nationale Supérieure de Géologie-

Nancy , Juin 1996, 315p.VICOL, T. (1990). "Comportement hydraulique et mécanique d'un sol fin non saturé - Application à lamodélisation". Thèse de doctorat, Ecole Nationale des Ponts et Chaussées, CERMES, Paris, 257p.WILLIAM H. P. and al. (1986). "Numerical Recipes – the art of scientific computing – Press FlanneryTeukolsky -Vetterling, 818p.ZIENKIEWICZ O. C. (1977) "The finite element method". Third edit , Mcgraw-Hill Book UK , 783 p.

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