8/8/2019 Graz 2006

http://slidepdf.com/reader/full/graz-2006 1/7

1 INTRODUCTION

Unsaturated soil is characterized by the existence of three different phases, namely the solid phase, theliquid phase and the gas phase. The important con-

sequence is the development of suction at the solid-water-air interface. Suction increases with continu-ous drying of the soil and vice versa suction will bereduced upon wetting of the soil. The relation be-tween the suction and the water content is named theSoil-Water Characteristic Curve (SWCC). Figure 1gives a graphical representation for two differentsoils, namely clayey silt and fine sand. This curve

 plays a key role in unsaturated groundwater flow ca-

Figure 1. Soil-water characteristic curves for clayey silt and fine sand

lculations and unsaturated soil deformation analyses.It can be seen from Figure 1 that suction plays amore important rule in the case of fine-grained soilthan in the case of coarse-grained sand. Indeed at thesame water content, clay exhibits much more suction

than sand. For that reason, one can expect more suc-tion related problems for foundations on clay thanon sand.Soil shrinkage is a well recognized problem which isassociated with suction increase, i.e. soil drying. Onthe other hand, soil swelling and soil structure col-lapse is considered as a main engineering problemduring suction decrease under constant load, i.e. soilwetting. These phenomena would affect the founda-tions if no special measures would have been takenduring the design process. The damage reparationcosts level could reach high numbers e.g. as much as$9 billion per year in the USA alone, as reported by

 Nelson & Miller (1992).In the past many empirical procedures have been

 proposed to predict soil volume changes due to suc-tion variations, but during the last fifteen years re-search attention has shifted to more theoretical mod-els. In combination with robust constitutive modelsthe FE method gives the designer a nice tool to un-derstand the mechanical behaviour of unsaturatedsoils and it offers the opportunity to reach better de-sign criteria.

Foundation analyses with unsaturated soil modelfor different suction profiles

Ayman A. Abed & Pieter A. Vermeer  Institute of Geotechnical Engineering, Universität Stuttgart, Germany

ABSTRACT: The well-known Barcelona Basic Model is considered to be a robust and suitable model for un-saturated soils and has thus been implemented into the Plaxis finite element code. After verifying its results

 by analytical solutions the unsaturated-saturated groundwater flow code PlaxFlow is used to calculate suction profiles required for the deformation analyses. This paper provides results of numerical analyses of a shallow

foundation resting on an unsaturated soil using the implemented model. The unsaturated soil is considered tohave common suction profiles. Special attention is given to the effect of suction variation on soil behaviour.

8/8/2019 Graz 2006

http://slidepdf.com/reader/full/graz-2006 2/7

2 UNSATURATED SOIL MODELLING In surveying the literature one can classify the mod-elling methods into empirical and theoretical ap-

 proaches.

2.1  Empirical methods  Empirical methods are based on direct fitting of testdata for clays or silts. Especially poorly graded silts(loess) are renown as collapsible soil. These empiri-cal methods are mostly based on data from one-dimensional compression in the oedometer appara-tus. These tests give only clear information about thesample initial conditions and final conditions but noinformation about the suction variation during thesaturation process. A nice review and evaluation of these methods can be found for example in the paper 

 by Djedid et al. (2001).

As an example, Equation 1 is proposed by Kusa& Abed (2003) to predict the swelling pressure σsw in (kPa) as a function of the liquid limit wL (%) theinitial water content wn (%) and the free swellingstrain ε0 (%). This strain is defined as the ratio of thesoil sample height after saturation (without any ex-ternal load) and the initial sample height before satu-ration.

n

0LLsw

wlnw3.3w3.5

ε⋅+=σ (1)

It is believed that such empirical correlations giveonly satisfactory results as long as they are appliedto the same soils as used to establish the particular considered correlation. This reduces their use to avery narrow group of soils.

2.2 Theoretical methods This category uses the principles of soil mechanicstogether with sophisticated experimental data for the

formulation of a constitutive stress-strain law. Anearly attempt was made by Bishop (1959). He ex-tended the well-know effective stress principle for fully saturated soil to unsaturated soil. Bishop pro-

 posed the effective stress measure

( )waa uuu −⋅χ+−σ=σ′ (2)

where σ is the total stress with compression reck-oned positive, ua is the pore air pressure, uw is the

  pore water pressure and χ is a factor related to de-

gree of saturation. It yields χ = 0 for dry soil and χ =1 for saturated soil. According to Bishop the effec-tive stress always decreases on wetting under con-stant total stress. As the effective stress decreases anincrease in the volume of the soil should be ob-

served in accordance with the above definition of ef-fective stress. However, experimental data oftenshows additional compression on wetting which isopposite to the prediction based on Bishop’s defini-tion of effective stress. Many critics were expressedregarding the use of a single effective stress measurefor unsaturated soil and there has been a gradualchange towards the use of two independent stress

state variables.It was proposed by Fredlund et al. (1977) to use

the net stress σ-ua and the suction s as two inde-  pendent stress state variables to describe the me-chanical behaviour of the unsaturated soil, where s =ua –uw. In this study pore air pressure is considered to

 be atmospheric and soil suction is equal to negative  pore water pressure. In the rest of this paper theterms suction and negative pore water pressure havethe same meaning. On integrating the two stressmeasures into critical state soil mechanics, an elas-

toplastic constitutive model for unsaturated soil has been developed by Alonso et al. (1987) and later byGens et al. (1990). Later other constitutive modelshave been proposed, but many of them remain in theframework of the Alonso and Gens model, which

 became known as Barcelona Basic Model (BBM).

3 BARCELONA BASIC MODEL

The BB-model is based on the Modified Cam Claymodel for saturated soil with extensions to includesuction effects in unsaturated soil (Gens et al. 1990).This model uses the net stresses σ-ua and the suc-tion s as the independent stress measures. Manysymbols have been used for the net stresses such asσ" and σ*. The latter symbol will be used here. It isassumed that the soil has different stiffness parame-ters for changes of net stress and changes of suction.

3.1  Isotropic loading 

For isotropic unloading-reloading the rate of changeof the void ratio is purely elastic and related to thenet stress and the suction:

atm

s*

*e

 ps

s

 p

 pee

+⋅κ−⋅κ−==

&&&& (3)

where κ is the usual (modified) swelling index andκs is the suction swelling index, patm is the atmos-

 pheric pressure and p* is the mean net stress

a321* u)(

31 p −σ+σ+σ= (4)

In terms of volumetric strain equation (3) reads

8/8/2019 Graz 2006

http://slidepdf.com/reader/full/graz-2006 3/7

atm

s

*

*e

vv ps

s

e1 p

 p

e1e1

e

+⋅

+

κ+⋅

+κ

=+

−=ε=ε&&&

&& (5)

where compressive strains are considered positive.For primary loading both elastic and plastic strainsdevelop. The plastic component of volumetric strainis given by the equation

0 p

0 p0 pv

 p

 p

e1

&

& ⋅+ κ−λ=ε (6)

where λ0 is the usual (modified) compression indexand p p0 is the preconsolidation pressure in saturatedstate. The above equation is in accordance with criti-cal state soil mechanics, but we write p p0 instead of 

 p p and λ0 instead of λ to emphasize that they relateto the saturated state. The difference with criticalstate soil mechanics is the yield function

 p

*  p pf  −= (7)

where p p is the suction dependent preconsolidation pressure

κ−λ

κ−λ

⋅=0

) p

 p( p p

c

0 pc

 p (8)

where pc is a reference pressure and

( ) s

0 e ⋅β−∞∞ ⋅λ−λ−λ=λ (9)

λ is the suction dependent compression index.Hence, for full saturation we have s = 0, λ = λ0 and p p = p p0. The larger the suction the smaller the com- pression index λ. In the limit for s = ∞ the aboveexpression yields λ = λ∞. The index ratio λ∞/ λ0 istypically in the range between 0.2 and 0.8.The constant pc is mostly in the range from 10 to 50kPa. The constant β controls the rate of decrease of the compression index with suction; it is typically inthe range between 0.01 and 0.03 kPa-1. The mono-tonic increase of soil stiffness with suction is associ-ated with an increase of the preconsolidation pres-

sure p p according to Equation 8. In order to studyEquation 6 in more detail, we consider the consis-tency equation 0f =& , as it finally leads to Equation6. In terms of partial derivatives the consistencyequation reads

0 p

ss

 p p

 p

f f   p

v p

v

 p p*

*=ε⋅

ε∂

∂−⋅

∂

∂−⋅

∂∂

= &&&& (10)

with

1 p

f 

*

=∂

∂(11)

c

 p

 p

 p

 p

 pln p

s

 p⋅β⋅⋅

κ−λ

λ−λ=

∂

∂∞ (12)

 p p

v

 p p

e1 p⋅

κ−λ+

=ε∂

∂(13)

It follows from the above equations that

*

 pc

 p p

v  p p

1

e1s

 p

 pln

e1&&& ⋅⋅

+κ−λ

+⋅ 

  

 ⋅β⋅

+λ−λ

−=ε ∞ (14)

This equation is in full agreement with Equation6, but instead of p p0 it involves the stress measuress and p*. Equation 14 shows the so-called soil col-lapse upon wetting. Indeed, upon wetting we have

0s <& and the above equation yields an increase of volumetric strain, i.e. 0 p

v >ε& even at constant load,i.e. for  0 p* =& .

3.2 More general states of stress

For the sake of convenience, the elastic strains willnot be formulated for rotating principal axes of stress and strain. Instead, restriction is made to non-rotating principal stresses. For such situation Equa-tion 5 can be generalized to become

3,2,1 j,ifor sK D  j

1

s

e

 jij

*

i =⋅δ⋅−ε⋅=σ −&&& (15)

wheree

iε& is a principal elastic strain rate, σi* is a

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

( ) ( )atm

s1

s pse13

K +⋅+⋅

κ=− (16)

( ) ( )

ν−νν

νν−ν

ννν−

⋅ν+⋅ν−

=

1

1

1

121

ED ij (17)

where ν is the elastic Poisson ratio. Youngmodulus is stress dependent:

( ) K 213E ⋅ν−⋅= with * pe1

K  ⋅κ+

= (18)

The term sK   j

1

s&⋅δ⋅− 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-unloadingcontribution.

For formulating the plastic rate of strain, both the  plastic potential and the yield function have to beconsider. For the BB-model the yield function reads

( ) ( )* ps

*22  p p p pMqf  −⋅+−= (19)

where M is the slope of the critical state line, as alsoindicated in Figure 2, and

2

13

2

32

2

21 )()()(2

1q σ−σ+σ−σ+σ−σ= (20)

 ps = a · s (21)

8/8/2019 Graz 2006

http://slidepdf.com/reader/full/graz-2006 4/7

 

Figure 2. Yield surface of Barcelona Basic Model

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 constanta determines the rate of ps increase with suction.At full saturation with s = 0 the yield function inEquation 19 reduces to the Modified Cam Clay

(MCC) yield function. In contrast to the MCC-model, the BB-model has a non-associated flow rule,which may be written as

i

 p

i

g

σ∂∂

⋅Λ=ε& (i = 1,2,3) (22)

where p

iε& stands for a principal rate of plastic strain,Λ is a multiplier and g is the plastic potential func-tion

( ) ( )* ps

*22  p p p pMqg −⋅+−⋅α= (23)

The flow rule becomes associated for  α = 1, butGens et al. (1990) recommend to use

κ−λ

λ⋅

−−−

=α0

0

)M6(9

)3M)(9M(M(24)

In this way the crest of the plastic potential in p*-q-  plane is increased. Finally it leads to realistic K 0-values in one-dimensional compression, whereas theassociated MCC-model has the tendency to overes-

timate K 0-values (Roscoe & Burland 1968)

In combination with Equation 15 and 22 the con-sistency condition 0f =& yields the following ex-

 pression for the plastic multiplier 

sDf 

K s

f 

H

1D

f 

H

1 jij*

i

T1

s jij*

i

T

&& ⋅ 

  

 δ

σ∂

∂⋅−

∂∂

⋅+εσ∂

∂⋅=Λ −  

with

* j

ij*i

T

* pv

g

D

f 

 p

gf 

H σ∂

∂

⋅⋅σ∂

∂

+∂

∂

⋅ε∂

∂

−= (25)

Figure 3. Gardner permeability model

4 GROUNDWATER FLOW

Ground water flow is governed by the ground water head h = y + uw / γw , where y is the geodetic headand uw / γw is the pressure head which will be de-noted as ψ for the sake of simplicity. γw is the unit

 pore water weight. In most practical cases there will

not be a constant ground water head, but a variationwith depth and consequently ground water flow.

Indeed, in reality there will be a transient groundwater flow due to varying rainfall and evaporation atthe soil surface. This implies transient suction fieldsand footing settlements that may vary with time. For most footing, settlements variations will be ex-tremely small, but they will be significant for expan-sive clays as well as collapsible subsoil. In order toanalyse such problems, we will have to incorporateground water flow. Flow in an isotropic soil is de-scribed by the Darcy equation

i

satrelix

hk k q

∂∂

⋅⋅−= (26)

where qi is a Cartesian component of the specificdischarge water, k sat is the well-known permeabilityof a saturated soil and k rel is the pore pressure head-dependent relative permeability. Gardner (1958)

  proposed a simple but suitable exponential relative permeability function of the form

0for 1k 

0for ek 

rel

rel

≥ψ=

<ψ= ψ⋅α

(27)

where α is a fitting parameter. It is worthy to pointout that the pressure height ψ has a negative value inthe unsaturated  zone. Figure 3 shows a graphicalrepresentation of Equation 27 for α = 2 m-1.

In order to do ground water flow calculations, onehas to supplement Darcy’s equation 26 with a conti-nuity equation of the form

0=

∂

∂⋅+

∂

∂

t

hC

x

q

i

i (28)

where repeated subscripts stand for summation. C isthe effective storage capacity, which is often ex- pressed as (Brinkgreve et al. 2003)

8/8/2019 Graz 2006

http://slidepdf.com/reader/full/graz-2006 5/7

ds

dSnCC r 

sat ⋅+= (29)

where Csat is the saturated storage capacity, n the po-rosity and Sr  the degree of saturation. The latter is afunction of pore pressure head ψ,  Russo (1988)  proposed the following equation which is compati- ble with Gardner model (1958) 

( )ressateresr  SSSSS −⋅+= (30)

Sres is the residual degree of saturation, Ssat is the de-gree of saturation at full saturation which is usuallytaken as 1 and Se is the effective degree of saturationgiven as

[ ] m2

25.0

e e)5.01(S +ψ⋅α⋅ψ⋅α⋅−= (31)

where m is a fitting parameter.Strictly speaking soil deformation implies changingsoil porosity n and pore fluid flow cannot be sepa-rated from soil deformation. For many practical  problems, however, the soil porosity remains ap-  proximately constant and flow problems may besimulated without consideration of coupling terms.

In order to solve the differential equations 26 and28, boundary conditions are required. For studyingfooting problems, one would need the water infiltra-tion or the rate of evaporation at the soil surface, q.PlaxFlow finite element code has been used tocalculate suction in unsaturated zone. In the follow-ing section, the numerical results of PlaxFlow are

checked by analytical solutions in the case of onedimensional unsaturated groundwater flow.

4.1  Analytical solution for one dimensional unsaturated stationary groundwater flow

For one dimensional vertical steady flow Equation28 reduces to Equation 32 in terms of pore pressurehead ψ 

( ) 01

y

k k 

y

relsat =

 

 

 

 +

∂

ψ∂⋅ψ⋅

∂

∂(32)

Figure 4. boundary conditions used in Gardner solution 

Figure 5. Finite element mesh and boundary conditions used innumerical calculations

Gardner (1958) gave an analytical solution for thedifferential Equation 32 with infiltration or evapora-tion boundary conditions at soil surface, as shown in

Figure 4. Using Equation 27 for the permeabilityfunction, the solution is given as

−⋅+⋅

α=ψ ⋅α−

sat

y

sat k 

qe)1

k 

q(ln

1(33)

Please note that Gardner used negative values for qto indicate infiltration and positive values to indicateevaporation.

4.2 Verification of the numerical results

A silty soil with saturated permeability of k sat = 1m/day and α = 2 m-1 has been used to comparePlaxFlow numerical results with the analytical solu-tion of Equation 32. Figure 5 shows the finite ele-ment mesh and boundary conditions used in the veri-fication examples. Six noded triangular elements areused with closed vertical boundaries to recover 1-Dconditions. The ground water table is at 3.0 depthand a Neumann type boundary condition is appliedat the soil surface.

The problem has been solved for three common  practical situations, namely hydrostatic conditions

with q = 0, evaporation with q = 0.002 m/day and in-filtration with q = -0.1 m/day. The hydrostatic condi-tions represent pore pressure distribution in unsatu-rated soil which has no interaction with surfacewater, for example a soil directly underneath a raftfoundation. Evaporation represents soil moisture de-crease due to temperature increase during a dry sea-son and infiltration represents soil moisture increasedue to rainfall for instance.

Figure 6 shows very good agreement betweenanalytical and numerical results for this particular 

kind of problems. At the same time it gives a niceidea about negative pore water pressure profiles insuch common cases. These profiles are used as inputfor deformation analyses using the Barcelona BasicModel.

8/8/2019 Graz 2006

http://slidepdf.com/reader/full/graz-2006 6/7

 

Figure 6. Analytical versus numerical results 

5 DEFORMATION ANALYSES

Figure 7 shows the geometry, the boundary condi-tions and the finite element mesh for the problem of 

a rough strip footing resting on partially saturatedclay of 3.0 meters thickness followed by a very stiff sand layer. The material properties shown in Table 1are the same as those given by Gens (1990), exceptfor the preconsolidation pressure p p0 and the refer-ence pressure pc. For the sake of convenience, a con-stant (mean) value of 17.1 kN/m3 has been used for the soil weight above the phreatic line. For the initialnet stresses K 0-value of 1 has been used.The finite element mesh consists of 6-noded trian-gles for the soil and 3-noded plate element for thestrip footing. The flexural rigidity of the plate wastaken to be EI = 10 MN.m2 per meter footinglength. This value is representative for a reinforcedconcrete plate with a thickness of roughly 20 cm.The footing is loaded by a uniform pressure of 50kPa and the corresponding settlements are deter-mined for unsaturated soil with suction profiles sat-isfy those calculated in section 4.2. in addition ananalysis without suction is to be considered. Com- puted load-settlement curves are shown in Figure 8.

Loading the soil during dry season results in a stiff  behaviour due to high suction values caused by con-tinuous evaporation.

Figure 7. Finite element mesh, initial and boundary conditionfor deformation analysis

Table 1. Material and model parameters

e0 0.9 [-] β 0.0164 [kPa-1]λ0 0.14 [-] ν 0.15 [-]κ 0.015 [-] κs 0.01 [-]

 p p0 30 [kPa] a 1.24 [-]M 0.82 [-] pc 10 [kPa]λ∞

0.036 [-]

 Nevertheless, it is only slightly different from load-ing the soil with hydrostatic suction distribution, both analyses yield about 70 mm footing settlement.In contrast, rainfall will reduce soil stiffness dra-matically leading to final settlement of 126 mmwhich is two times higher than the previous cases.When soil reaches full saturation it loses all suctioncontribution to stiffness which weaken the soil moreand more. Loading the soil in such condition will re-sult in settlement of 210 mm which is about four times higher than the settlement in the case of drysoil.

Figure 9 shows the principal stresses distribution atthe end of loading in the case of hydrostatic suction profile. The distribution suggests that for a continu-ous clay layer with depth, a deeper finite elementmesh will be preferred.A typical plastified zone below the footing is indi-cated in Figure 10.

The previous calculations consider that we areloading the soil with a specific suction profile andthis profile stays constant during and after construc-tion. In reality, after construction on unsaturated

soil, the decisive factor for footing movement will  be suction variation with the variation of environ-mental factors such as temperature and rainfall.These variations lead to different phenomena such assoil shrinkage associated with soil drying and soilswelling or even soil structure collapse upon wet-ting. Soil potential for swelling or collapse should begiven enough attention during design to avoid anyunexpected consequences. A detailed numericalstudy on unsaturated soil deformation and stabilityin the case of shallow foundations can be found in arecent paper by Vermeer & Abed (in press).

Figure 8. Load-settlement curves for footing on unsaturatedsoil with different suction profiles

8/8/2019 Graz 2006

http://slidepdf.com/reader/full/graz-2006 7/7

 

Figure 9. Principal stresses for hydrostatic pore pressure distri- bution at the end of loading.

6 CONCLUSIONS

The present study addresses the effect of evapora-tion and infiltration on negative pore water pressure

values in the unsaturated zone. This has been clari-fied using a simple 1-D vertical unsaturated flowmodel which has been solved analytically and nu-merically. Furthermore, the effect of changing suc-tion on soil mechanical behaviour has been illus-trated by studying the settlement of shallow footingon unsaturated soil with different suction profiles.The study gives clear idea about the strong influenceof suction on soil stiffness. Further studies about soildeformation especially during wetting under con-stant load will improve too much our understanding

for the mechanics of unsaturated soils and pave theway to incorporate such relatively new science in thereal geotechnical engineering practise.

7 ACKNOWLEDGEMENT

We are grateful to Dr. Peter van den Berg of GeoDelft, the Netherlands for providing support tothis study. We are also indebted to Dr. Klaas JanBakker of the Plaxis company for fruitful discus-sions on unsaturated soil behaviour.

Figure 10. Hardening points distribution for hydrostatic pore pressure conditions at the end of loading.

8 REFERENCES

Abed, Ayman. A. & Kusa, Issa. D. 2003.   Effect of swelling  pressure on the soil bearing capacity. Master thesis. Syria:Al-Baath University.

Alonso, E.E. & Gens, A. & Hight, D.W. 1987. Special prob-lem soils : General report.   In proc. 9th Euro. Conf. on Soil Mech. and Found. Engin.Vol.3:1087-1146. Rotter-dam:Balkema

Bishop, A.W. 1959. The principle of effective stress. Teknisk Ukeblad 39:859-863.

Brinkgreve, R. & Al-Khoury, R. & van Esch, J. 2003.  PLAX- FLOW User Manual . Rotterdam:Balkema.

Djedid, A. & Bekkouche, A. & Mamoune, S.M. 2001. Identifi-cation and prediction of the swelling behaviour of somesoils from the Telmcen region of Algeria.  Bulletin des La-boratories des Ponts et Chausses 233(July-August).

Fredlund, D.G. & Morgenstern, N.R. 1977. Stress state vari-ables for unsaturated soils. Journal of the Geotechnical En- gineering Division 103: 447-466.

Fredlund, D.G. & Rahardjo, H. 1993. Soil Mechanics for Un- saturated Soils. John Wiley & Sons.

Gardner, W.R. 1958. Some steady-state solutions of the unsatu-

rated moisture flow equation with applications to evapora-tion from a water table. Soil Science 85 : 228 - 232.

Gens, A. & Alonso, E.E. & Josa, A. 1990. A constitutivemodel for partially saturated soils. Géotechnique 40:405-430.

 Nelson, J.D. & Miller, D.J. 1992. Expansive Soils. John Wiley& Sons.

Roscoe, K.H. & Burland, J.B. 1968. On the generalized stress-strain behaviour of ‘wet’ clay.   Engineering Plasticity.Cambridge University Press.

Russo, D. 1988. Determining soil hydraulic properties by pa-rameter estimation: On the selection of a model for the hy-draulic properties. Water Resources Research 24:453–459.

Vermeer, P.A & Brinkgreve, R. 1998.  PLAXIS User Manual .Rotterdam:Balkema.

Vermeer, P.A & Abed, A. A. (in press). Numerical simulationof unsaturated soil behaviour. In proc. First Euro Mediter-ranean Conference on Advances in Geomaterials and Structures, Hammamet, 3-5 May 2006. Tunisia.