Numerical Simulation of Unsaturated Soil Behaviour
First Euro Mediterranean Conference on Advances in Geomaterials and Structures – Hammamet 3-5 May Tunisia
Numerical Simulation of Unsaturated Soil Behaviour Ayman A. Abed* Institute of Geotechnical Engineering, Universität Stuttgart, Germany E-mail: [email protected] *Corresponding author
Pieter A. Vermeer Institute of Geotechnical Engineering, Universität 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. The use of finite element technique is considered as a promising method to solve settlement and heave problems, as associated with unsaturated soil. Nevertheless, the success of the numerical analysis is strongly dependent on constitutive model being used. The well-known Barcelona Basic Model is considered to be a robust and suitable model for unsaturated soils and has thus been implemented into the PLAXIS finite element code. This paper provides results of numerical analyses of a shallow foundation resting on 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 deformation analyses.
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 numbers e.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
Figure 1 The soil water characteristic curves for clayey silt and fine sand
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UNSATURATED SOIL MODELLING
In surveying the literature one can classify the modelling methods into empirical and theoretical approaches.
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 onedimensional compression. These tests give only clear information about the sample initial conditions and final conditions 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 σsw in (kg/cm2) as a function of the liquid limit wL (%) the initial water content wn (%) and the free swelling strain ε0 (%). 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.
σ sw = 0.053 w L + 0.033 w L ⋅ ln
ε0 wn
(1)
It is believed that such empirical correlations give only satisfactory results as long as they are applied to the same soils 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. Bishop proposed the effective stress measure
σ′ = σ − u a + χ ⋅ (u a − u w )
(2)
where
σ : total stress ua : pore air pressure uw : pore water pressure χ : factor related to degree of saturation where χ = 0 for dry soil and χ = 1 for saturated soil. According to Bishop the effective stress always decreases on wetting under constant total stress. As the effective stress decreases an increase in the volume of the soil should be observed in accordance with the above definition of effective stress. However, experimental data often shows additional compression on wetting which is opposite to the prediction based on Bishop’s definition of effective stress. Many critics were expressed regarding the use of a single effective stress measure for unsaturated soil and there has been a gradual change towards the use of two independent stress state variables. It was proposed by Fredlund et al. [5] to use the net stress σ-ua and the suction s as two independent stress state variables to describe the mechanical behaviour of the unsaturated soil, where s = ua–uw. 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 σ-ua and the suction s as the independent stress measures. Many symbols have been used for the net stresses such as σ" and σ * . 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 than them 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
e& = e& e = − κ ⋅
p& * p
*
− κs ⋅
s& s + p atm
(3)
where κ is the normal swelling index and κs is the suction swelling index, patm is the atmospheric pressure and p* is the mean net stress p* =
1 (σ 1 + σ 2 + σ 3 ) − u a 3
(4)
In terms of volumetric strain equation (3) reads
ε& v = ε& ev = −
κ e& s& κ p& * ⋅ *+ s ⋅ = 1+ e 1+ e p 1 + e s + p atm
(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
ε& pv =
λ 0 − κ p& p 0 ⋅ 1 + e p p0
( 6)
where λ0 is the compression index and p p 0 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
f = p* − p p with c
pp = p ⋅ (
(7)
p p0 pc
λ0 −κ ) λ−κ
λ = λ ∞ − (λ ∞ − λ 0 ) ⋅ e − β⋅s
(8) (9 )
NUMERICAL SIMULATION OF UNSATURATED SOIL BEHAVIOUR where λ and pp are the compression index and the suction dependent preconsolidation pressure respectively. Hence, for full saturation we have s = 0, λ = λ0 and pp = p p 0 . The larger the suction the smaller the compression index λ. In the limit for s = ∞ the above expression yields λ = λ ∞ . The index ratio λ ∞ / λ0 is typically in the range between 0.2 and 0.8. The constant pc is mostly in the range from 10 to 50 [kPa]. The constant β controls the rate of decrease of the compression 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 p p according to Equation 8. In order to study Equation 6 in more detail, we consider the consistency equation f& = 0 , as it finally leads to Equation 6. In terms of partial derivatives the consistency equation reads
∂p p ∂p p ∂f f& = ⋅ p& * − ⋅ s& − p ⋅ ε& pv = 0 * ∂s ∂p ∂ε v
(10 )
ν ν 1 − ν E D ij = ⋅ ν 1− ν ν (1 − 2ν ) ⋅ (1 + ν ) ν 1 − ν ν
(17)
where ν is the elastic Poisson ratio. Young modulus is stress dependent
E = 3 ⋅ (1 − 2 ⋅ ν ) ⋅ K
with
K=
1+ e * ⋅p κ
(18 )
The term K s−1 ⋅ δ j ⋅ 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
(
)(
f = q 2 − M 2 p* + ps ⋅ p p − p*
)
(19 )
with
∂f ∂p * ∂p p ∂s
∂p p ∂ε pv
where M is the slope of the critical state line, as also indicated in Figure 2, and
= 1
(11)
pp λ − λ∞ = ⋅ p p ⋅ β ⋅ ln c λ−κ p
=
q=
(12)
(σ1 − σ 2 ) 2 + ( σ 2 − σ 3 ) 2 + ( σ 3 − σ1 ) 2
2
ps = a . s
1+ e ⋅ pp λ−κ
pp λ − λ∞ ⋅ β ⋅ ln c p 1+ e
⋅ s& + λ − κ ⋅ 1 ⋅ p& * 1 + e pp
(14 )
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 s& < 0 and the above equation yields an increase of volumetric strain, i.e. ε& pv > 0 even at constant load, i.e. for p& * = 0 .
(21)
It can be observed from Figure 2 that ps reflects the extension 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 nonassociated flow rule, which may be written as ε& ip = Λ ⋅
∂g ∂σ i
(i = 1,2,3)
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
(
σ& *i = D ij ⋅ ε& ej − K s−1 ⋅ δ j ⋅ s&
)
for i, j = 1,2,3
s=s1
s=0 (15)
* where ε& ie is a principal elastic strain rate, σ i is a principal net stress, δj = 1 for j=1,2,3 and
K s−1 =
(20)
(13)
It follows from the above equations that
ε& pv = −
1
κs 3 ⋅ (1 + e ) ⋅ (s + p atm )
(16 )
Figure 2 Yield surface of Barcelona Basic Model
( 22)
A. A. ABED AND P. A. VERMEER p
where ε& i stands for a principal rate of plastic strain, Λ is a multiplier and g is the plastic potential function
(
)(
g = α ⋅ q 2 − M 2 p* + ps ⋅ p p − p*
)
Table 1 Material and model parameters
(23)
The flow rule becomes associated for α = 1, but Gens et al. [7] recommend to use α=
M ( M − 9)( M − 3) λ 0 ⋅ 9( 6 − M ) λ0 − κ
(24)
In this way the crest of the plastic potential in p*-q-plane is increased. Finally it leads to realistic K0-values in onedimensional compression, whereas the associated MCCmodel has the tendency to overestimate K0-values [8]. In combination with Equation 15 and 22 the consistency condition f& = 0 yields the following expression for the plastic multiplier
Λ=
∂f T 1 ∂f T 1 ∂f ⋅ * D ij ε& j + ⋅ − K s−1 ⋅ * D ij δ j ⋅ s& H ∂σ i H ∂s ∂σ i
with
H=−
4
∂f ∂ε pv
⋅
∂g ∂p
*
+
∂f T ∂σ *i
⋅ D ij ⋅
∂g ∂σ *j
(25)
SETTLEMENT ANALYSIS
Figure 3 shows the geometry, the boundary conditions and the finite element mesh for the problem of a rough strip footing 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.
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 up to 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 per meter footing length. This value is MN.m2 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 Cam Clay 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 BBanalysis, 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 considerable difference between the results from the BB-analysis and the
Figure 3 Geometry, boundary conditions and finite element mesh
NUMERICAL SIMULATION OF UNSATURATED SOIL BEHAVIOUR
Figure 6 Vertical displacement of soil surface due to wetting 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 MCC150 kPa
150 kPa
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 / γw , where y is the geodetic head and γw is the unit 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 transient suction 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 q i = − k rel ⋅ k sat ⋅
(a)
(b)
Figure 5 The plastic zones from BB and MCC model for footing pressure of 150 kPa
∂h ∂x i
( 26 )
where qi is a Cartesian component of the specific discharge water, ksat is the well-known permeability of a saturated soil and krel is the suction-dependent relative permeability.
A. A. ABED AND P. A. VERMEER 7
Figure 7 Simplified van Genuchten model
A simplification of the van Genuchten model [13] leads to the equation
k rel = 10
−4⋅s sk
for
0 ≤ s < sk
(27)
where sk is a soil-dependent constant which is related to the extent hk of the unsaturated zone under hydrostatic conditions. It yields hk = sk / γw. For the saturated zone this equation 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 Darcy’s equation 26 with a continuity equation of the form ∂q i ∂h +C⋅ =0 ∂x i ∂t
( 28)
where repeated subscripts stand for summation. C is the effective storage capacity, which is often expressed as C = C sat + n ⋅
dS r ds
( 29 )
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 Cam Clay 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 = .6 ⋅ sin ϕ cs / (3 − sin ϕ cs ) and we get a friction angle of ϕcs = 16.4o . However, we consider the plane strain problem of a strip footing. For planar deformation it yields .M = 3 sin ϕ cs [14], and it follows that ϕcs = 21o. Table 2 gives the soil parameters. Figure 8, shows the boundary conditions and the finite element mesh for the bearing capacity 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 qf = c ⋅ Nc + q0 ⋅ Nq + γ ⋅ b ⋅ Nγ
(30 )
where c is the soil cohesion, q0 is the surcharge load at footing level and b is the footing width.
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.
Figure 8 Finite element mesh and boundary conditions for the bearing capacity problem
NUMERICAL SIMULATION OF UNSATURATED SOIL BEHAVIOUR Table 2 Soil properties
The factors Nc, Nq and Nγ are functions of the soil friction angle Nq
1 + sin ϕ π⋅ tan ϕ ⋅e , = 1 − sin ϕ
(
)
N c = N q − 1 ⋅ cot ϕ
Figure 10 Incremental shear strain at failure for s = 20 kPa
( 31)
In the present analysis γ is taken equal to zero and the corresponding Nγ-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 ⋅s ⋅ tan ϕ
(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 loaddisplacement curves using the BB-model and the MCCmodel. 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, the displacement 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 11
Total displacement increments for s = 20 kPa
One can use a modified version of the well-known MohrCoulomb 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 numerical implementation.
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 profile with time; typically for two or three years.
A. A. ABED AND P. A. VERMEER Table 3 Bearing capacity values
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. 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 failures are 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, AlBaath University, Syria, 2003. [4] A. W. Bishop, The principle of effective stress. Teknisk Ukeblad 39, 1959. [5] D.G. Fredlund, and N.R. Morgenstern, Stress state variables for unsaturated soils. Journal of the Geotechnical
[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 for Unsaturated 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.
Numerical Simulation of Unsaturated Soil Behaviour Ayman A. Abed* Institute of Geotechnical Engineering, Universität Stuttgart, Germany E-mail: [email protected] *Corresponding author
Pieter A. Vermeer Institute of Geotechnical Engineering, Universität 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. The use of finite element technique is considered as a promising method to solve settlement and heave problems, as associated with unsaturated soil. Nevertheless, the success of the numerical analysis is strongly dependent on constitutive model being used. The well-known Barcelona Basic Model is considered to be a robust and suitable model for unsaturated soils and has thus been implemented into the PLAXIS finite element code. This paper provides results of numerical analyses of a shallow foundation resting on 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 deformation analyses.
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 numbers e.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
Figure 1 The soil water characteristic curves for clayey silt and fine sand
Copyright © 2004 Inderscience Enterprises Ltd.
UNSATURATED SOIL MODELLING
In surveying the literature one can classify the modelling methods into empirical and theoretical approaches.
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 onedimensional compression. These tests give only clear information about the sample initial conditions and final conditions 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 σsw in (kg/cm2) as a function of the liquid limit wL (%) the initial water content wn (%) and the free swelling strain ε0 (%). 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.
σ sw = 0.053 w L + 0.033 w L ⋅ ln
ε0 wn
(1)
It is believed that such empirical correlations give only satisfactory results as long as they are applied to the same soils 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. Bishop proposed the effective stress measure
σ′ = σ − u a + χ ⋅ (u a − u w )
(2)
where
σ : total stress ua : pore air pressure uw : pore water pressure χ : factor related to degree of saturation where χ = 0 for dry soil and χ = 1 for saturated soil. According to Bishop the effective stress always decreases on wetting under constant total stress. As the effective stress decreases an increase in the volume of the soil should be observed in accordance with the above definition of effective stress. However, experimental data often shows additional compression on wetting which is opposite to the prediction based on Bishop’s definition of effective stress. Many critics were expressed regarding the use of a single effective stress measure for unsaturated soil and there has been a gradual change towards the use of two independent stress state variables. It was proposed by Fredlund et al. [5] to use the net stress σ-ua and the suction s as two independent stress state variables to describe the mechanical behaviour of the unsaturated soil, where s = ua–uw. 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 σ-ua and the suction s as the independent stress measures. Many symbols have been used for the net stresses such as σ" and σ * . 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 than them 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
e& = e& e = − κ ⋅
p& * p
*
− κs ⋅
s& s + p atm
(3)
where κ is the normal swelling index and κs is the suction swelling index, patm is the atmospheric pressure and p* is the mean net stress p* =
1 (σ 1 + σ 2 + σ 3 ) − u a 3
(4)
In terms of volumetric strain equation (3) reads
ε& v = ε& ev = −
κ e& s& κ p& * ⋅ *+ s ⋅ = 1+ e 1+ e p 1 + e s + p atm
(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
ε& pv =
λ 0 − κ p& p 0 ⋅ 1 + e p p0
( 6)
where λ0 is the compression index and p p 0 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
f = p* − p p with c
pp = p ⋅ (
(7)
p p0 pc
λ0 −κ ) λ−κ
λ = λ ∞ − (λ ∞ − λ 0 ) ⋅ e − β⋅s
(8) (9 )
NUMERICAL SIMULATION OF UNSATURATED SOIL BEHAVIOUR where λ and pp are the compression index and the suction dependent preconsolidation pressure respectively. Hence, for full saturation we have s = 0, λ = λ0 and pp = p p 0 . The larger the suction the smaller the compression index λ. In the limit for s = ∞ the above expression yields λ = λ ∞ . The index ratio λ ∞ / λ0 is typically in the range between 0.2 and 0.8. The constant pc is mostly in the range from 10 to 50 [kPa]. The constant β controls the rate of decrease of the compression 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 p p according to Equation 8. In order to study Equation 6 in more detail, we consider the consistency equation f& = 0 , as it finally leads to Equation 6. In terms of partial derivatives the consistency equation reads
∂p p ∂p p ∂f f& = ⋅ p& * − ⋅ s& − p ⋅ ε& pv = 0 * ∂s ∂p ∂ε v
(10 )
ν ν 1 − ν E D ij = ⋅ ν 1− ν ν (1 − 2ν ) ⋅ (1 + ν ) ν 1 − ν ν
(17)
where ν is the elastic Poisson ratio. Young modulus is stress dependent
E = 3 ⋅ (1 − 2 ⋅ ν ) ⋅ K
with
K=
1+ e * ⋅p κ
(18 )
The term K s−1 ⋅ δ j ⋅ 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
(
)(
f = q 2 − M 2 p* + ps ⋅ p p − p*
)
(19 )
with
∂f ∂p * ∂p p ∂s
∂p p ∂ε pv
where M is the slope of the critical state line, as also indicated in Figure 2, and
= 1
(11)
pp λ − λ∞ = ⋅ p p ⋅ β ⋅ ln c λ−κ p
=
q=
(12)
(σ1 − σ 2 ) 2 + ( σ 2 − σ 3 ) 2 + ( σ 3 − σ1 ) 2
2
ps = a . s
1+ e ⋅ pp λ−κ
pp λ − λ∞ ⋅ β ⋅ ln c p 1+ e
⋅ s& + λ − κ ⋅ 1 ⋅ p& * 1 + e pp
(14 )
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 s& < 0 and the above equation yields an increase of volumetric strain, i.e. ε& pv > 0 even at constant load, i.e. for p& * = 0 .
(21)
It can be observed from Figure 2 that ps reflects the extension 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 nonassociated flow rule, which may be written as ε& ip = Λ ⋅
∂g ∂σ i
(i = 1,2,3)
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
(
σ& *i = D ij ⋅ ε& ej − K s−1 ⋅ δ j ⋅ s&
)
for i, j = 1,2,3
s=s1
s=0 (15)
* where ε& ie is a principal elastic strain rate, σ i is a principal net stress, δj = 1 for j=1,2,3 and
K s−1 =
(20)
(13)
It follows from the above equations that
ε& pv = −
1
κs 3 ⋅ (1 + e ) ⋅ (s + p atm )
(16 )
Figure 2 Yield surface of Barcelona Basic Model
( 22)
A. A. ABED AND P. A. VERMEER p
where ε& i stands for a principal rate of plastic strain, Λ is a multiplier and g is the plastic potential function
(
)(
g = α ⋅ q 2 − M 2 p* + ps ⋅ p p − p*
)
Table 1 Material and model parameters
(23)
The flow rule becomes associated for α = 1, but Gens et al. [7] recommend to use α=
M ( M − 9)( M − 3) λ 0 ⋅ 9( 6 − M ) λ0 − κ
(24)
In this way the crest of the plastic potential in p*-q-plane is increased. Finally it leads to realistic K0-values in onedimensional compression, whereas the associated MCCmodel has the tendency to overestimate K0-values [8]. In combination with Equation 15 and 22 the consistency condition f& = 0 yields the following expression for the plastic multiplier
Λ=
∂f T 1 ∂f T 1 ∂f ⋅ * D ij ε& j + ⋅ − K s−1 ⋅ * D ij δ j ⋅ s& H ∂σ i H ∂s ∂σ i
with
H=−
4
∂f ∂ε pv
⋅
∂g ∂p
*
+
∂f T ∂σ *i
⋅ D ij ⋅
∂g ∂σ *j
(25)
SETTLEMENT ANALYSIS
Figure 3 shows the geometry, the boundary conditions and the finite element mesh for the problem of a rough strip footing 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.
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 up to 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 per meter footing length. This value is MN.m2 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 Cam Clay 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 BBanalysis, 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 considerable difference between the results from the BB-analysis and the
Figure 3 Geometry, boundary conditions and finite element mesh
NUMERICAL SIMULATION OF UNSATURATED SOIL BEHAVIOUR
Figure 6 Vertical displacement of soil surface due to wetting 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 MCC150 kPa
150 kPa
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 / γw , where y is the geodetic head and γw is the unit 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 transient suction 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 q i = − k rel ⋅ k sat ⋅
(a)
(b)
Figure 5 The plastic zones from BB and MCC model for footing pressure of 150 kPa
∂h ∂x i
( 26 )
where qi is a Cartesian component of the specific discharge water, ksat is the well-known permeability of a saturated soil and krel is the suction-dependent relative permeability.
A. A. ABED AND P. A. VERMEER 7
Figure 7 Simplified van Genuchten model
A simplification of the van Genuchten model [13] leads to the equation
k rel = 10
−4⋅s sk
for
0 ≤ s < sk
(27)
where sk is a soil-dependent constant which is related to the extent hk of the unsaturated zone under hydrostatic conditions. It yields hk = sk / γw. For the saturated zone this equation 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 Darcy’s equation 26 with a continuity equation of the form ∂q i ∂h +C⋅ =0 ∂x i ∂t
( 28)
where repeated subscripts stand for summation. C is the effective storage capacity, which is often expressed as C = C sat + n ⋅
dS r ds
( 29 )
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 Cam Clay 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 = .6 ⋅ sin ϕ cs / (3 − sin ϕ cs ) and we get a friction angle of ϕcs = 16.4o . However, we consider the plane strain problem of a strip footing. For planar deformation it yields .M = 3 sin ϕ cs [14], and it follows that ϕcs = 21o. Table 2 gives the soil parameters. Figure 8, shows the boundary conditions and the finite element mesh for the bearing capacity 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 qf = c ⋅ Nc + q0 ⋅ Nq + γ ⋅ b ⋅ Nγ
(30 )
where c is the soil cohesion, q0 is the surcharge load at footing level and b is the footing width.
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.
Figure 8 Finite element mesh and boundary conditions for the bearing capacity problem
NUMERICAL SIMULATION OF UNSATURATED SOIL BEHAVIOUR Table 2 Soil properties
The factors Nc, Nq and Nγ are functions of the soil friction angle Nq
1 + sin ϕ π⋅ tan ϕ ⋅e , = 1 − sin ϕ
(
)
N c = N q − 1 ⋅ cot ϕ
Figure 10 Incremental shear strain at failure for s = 20 kPa
( 31)
In the present analysis γ is taken equal to zero and the corresponding Nγ-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 ⋅s ⋅ tan ϕ
(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 loaddisplacement curves using the BB-model and the MCCmodel. 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, the displacement 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 11
Total displacement increments for s = 20 kPa
One can use a modified version of the well-known MohrCoulomb 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 numerical implementation.
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 profile with time; typically for two or three years.
A. A. ABED AND P. A. VERMEER Table 3 Bearing capacity values
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. 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 failures are 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, AlBaath University, Syria, 2003. [4] A. W. Bishop, The principle of effective stress. Teknisk Ukeblad 39, 1959. [5] D.G. Fredlund, and N.R. Morgenstern, Stress state variables for unsaturated soils. Journal of the Geotechnical
[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 for Unsaturated 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.
