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Numerical simulation of unsaturated soil behaviour
Article in International Journal of Computer Applications in Technology · January 2009
DOI: 10.1504/IJCAT.2009.022697 · Source: DBLP
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Numerical simulation of unsaturated soil behaviour
Ayman A. Abed* and Pieter A. Vermeer
Institute of Geotechnical Engineering, University of Stuttgart,
Pfaffenwaldring 35,70569 Stuttgart, Germany E-mail: ayman.abed@igs.uni-stuttgart.de E-mail: pieter.vermeer@igs.uni-stuttgart.de
*Corresponding author
Abstract: The mechanical behaviour of unsaturated soils is one of the challenging topics in the field of geotechnical engineering. The use of finite element techniques is considered to be a promising method to solve settlement and heave problems, which are associated with unsaturated soil. Nevertheless, the success of the numerical analysis is strongly dependent on the 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 (Vermeer and Brinkgreve, 1995). This paper provides the 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.
Reference to this paper should be made as follows: Abed, A.A. and Vermeer, P.A. (2009)
‘Numerical simulation of unsaturated soil behaviour’, Int. J. Computer Applications in Technology, Vol. 34, No. 1, pp.2–12.
Biographical notes: Ayman A. Abed graduated in Civil Engineering from Al-Baath University in Syria where he also obtained his Master’s Degree in Geotechnical Engineering in the field of Unsaturated Soil. Since January 2004, he has been pursuing his PhD in Stuttgart University, Germany under Professor Pieter A. Vermeer’s supervision. He is currently working on the constitutive modelling of unsaturated soil, with special focus on the mechanical behaviour of swelling soft clays.
Pieter A. Vermeer graduated in Civil Engineering from Delft University of Technology in the Netherlands, where he obtained his PhD in 1980. He taught Geotechnical Engineering in Delft from 1980 to 1994 with research projects in computational geomechanics, dam construction and deep excavations. In 1994, he moved to Stuttgart, Germany, to become the Head of the Stuttgart Institute of Geotechnical Engineering. Since the early 1970s, he has been involved in research and consulting projects ranging from the foundation of extremely large coastal structures to the manufacturing of very small industrial powder compacts.
1 Introduction
Unsaturated soil is characterised by the existence of three different phases, namely, the solid phase, the liquid phase and the gas phase. An important consequence is the occurrence of surface tension at the air-water interface within the pores. These forces increase with continuous drying of the soil and, vice-versa, surface tension will be reduced upon wetting of the soil. Surface tension causes a difference between the air pressure ua and the pore pressure uw. The difference s = ua – uw is named the matrix suction (Fredlund and Rahardjo, 1993) In this paper, matrix suction will simply be referred to as suction.
The relation between suction and water content of a particular soil is named the soil water characteristic curve.
The water content of a soil sample is defined as its (mass) amount of pore water with respect to (mass) amount of solid material. Figure 1 shows typical characteristic curves for two different soils, namely clayey silt and fine sand.
Such curves play 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 role in the case of fine-grained soil than in the case of coarse-grained sand. Indeed, at the same water content, clay or silt exhibits much more suction than sand.
For that reason, one may expect more suction related problems for foundations on clay than on sand.
Figure 1 The soil water characteristic curves for clayey silt and fine sand
Soil shrinkage is a well recognised problem which is associated with suction increase. Similarly, soil swelling as caused by suction decrease is a main problem in foundation engineering. A decrease of suction due to wetting is especially dangerous in some very loose soils, where suction provides the stability of soil particles.
As soon as this suction is reduced the soil fabric may become unstable and cause the so-called soil fabric collapse.
Both soil shrinkage and soil swelling affect foundations if no special measures are taken during the design process.
The damage reparation cost level could reach high numbers e.g., as much as $9 billion per year in the USA only (Nelson and Miller, 1992).
Many empirical procedures have been proposed in the past to predict the volumetric changes due to suction variations, but during the last 15 years, research attention has shifted to more theoretical models. In combination with the FE method, robust constitutive models give the designer a tool to understand the mechanical behaviour of unsaturated soils and reach better design criteria.
2 Unsaturated soil modelling
While surveying the literature one can classify the modelling methods into empirical and theoretical approaches.
2.1 Empirical methods
Empirical methods are mostly based on data from one-dimensional compression tests with zero lateral strain.
These tests give clear information only about the sample’s 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. (2001).
It is believed that such empirical correlations only give 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 formulation of a constitutive stress-strain law. An early attempt was made by Bishop (1959). He extended the well-known effective stress principle for fully saturated soils to unsaturated soils.
Bishop proposed the effective stress measure
( )
a a w
u u u
Vc V F (1)
where
V: total stress ua: pore air pressure uw: pore water pressure
F: factor related to degree of saturation.
where F = 0 for dry soil and F = 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 show additional compression on wetting, which is contrary to the prediction based on Bishop’s definition of effective stress. Many critiques (Fredlund and Rahardjo, 1993;
Fredlund and Morgenstern, 1977) were expressed regarding the use of a single effective stress measure for unsaturated soils and there has been a gradual change towards the use of two independent stress state variables. Fredlund Morgenstern (1977) proposed to use the net stress V – ua
and the suction s as two independent stress state variables to describe the mechanical behaviour of the unsaturated soil.
Considering these two stress measures within the critical state soil mechanics, an elastoplastic constitutive model for unsaturated soil has been developed by Alonso et al. (1987), and later by Alonso et al. (1990). Later, other constitutive models have been proposed, but all of them remain within 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 (Alonso et al., 1990). This model uses the net stresses V – ua and the suction s as the independent stress measures. Many symbols have been used for the net stresses such as V cc and V . The latter symbol will be used * here. It is assumed that the soil has different stiffness parameters for a change of the net stress and for a change of the suction.
3.1 Isotropic loading
In soil mechanics, the soil porosity n is often measured by its void ratio e being defined as the volume of the pores
over the volume of the solid phase. For unloading-reloading, the rate of change of the void ratio is purely elastic and related to the net stress and the suction
*
*
atm e
s
p s
e e N p N s p
(2)
where N is the well known swelling index and Ns is the suction swelling index, patm is the atmospheric pressure and p* is the mean net stress
* * * *
1 2 3
1( ).
p 3 V V V (3)
In terms of volumetric strain, equation (2) yields
*
*
1 1 1 atm
e s
v v
e p s
e e p e s p
N
H H N
(4)
where compressive strains are considered positive.
For primary loading, both elastic and plastic strains develop. The plastic component of volumetric strain is given by
0 0
1 0 p p v
p
p e p
O N
H
(5)
where O0 is the compression index and pp0 is the preconsolidation pressure in the saturated state (Alonso et al., 1990). The above equation is in accordance with critical state soil mechanics. The difference from the critical state soil mechanics is the yield function
*
f p pp (6)
with
0 0 c p
p c
p p p p
O N O N
§ ·
¨ ¸
© ¹
(7)
( 0) e Es
O Of OfO (8)
where O and pp are the compression index and the suction dependent preconsolidation pressure, respectively.
Hence, for full saturation, we have s = 0, O = O0
and pp pp0. The larger the suction, the smaller the compression index O. In the limit for s = f, the above expression yields O = Of. The index ratio Of/O0 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 E 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 pp according to equation (7).
In order to study equation (5) in more detail, we consider the consistency equation f 0, as it finally leads to equation (5). In terms of partial derivatives, the consistency equation yields
*
* p pp vp 0
v
p p
f f p s
s
p H
H
w w
w
w
w w
(9)
with
* 1
f p w
w (10)
p ln p
p c
p p
s p p
O O E
O N
w f
w (11)
1 .
p p p v
p e
O N p H
w
w (12)
It follows from the above equations that 1 *
ln .
1 1
p p
v c
p
p s p
e p e p
O O O N
H f E § ·
¨ ¸
© ¹
(13)
This equation is in full agreement with equation (5), but instead of pp0 it involves the stress measures s and p*. Equation (13) shows the so-called soil collapse upon wetting. Indeed, upon wetting we have s 0 and the above equation yields an increase of volumetric strain, i.e., H !vp 0 even at constant load, i.e., for p* 0.
3.1 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 situations equation (4) can be generalised to become
* e 1 for , 1, 2, 3
i Dij j Ks j s i j
V H G (14)
where Hie is the principal elastic strain rate,Vi* is the principal net stress, Gj = 1 for j = 1, 2, 3 and
1
3 (1 ) ( atm)
s
Ks
e s p
N
(15)
1 (1 2 ) (1 ) 1
1
ij
D E
Q Q Q
Q Q Q
Q Q
Q Q Q
ª º
« »
« »
« »
¬ ¼
(16)
where Q is the elastic Poisson ratio. Young’s modulus is stress dependent
1 *
3 (1 2 ) with e .
E Q K K p
N
(17)
The term Ks1Gj in equation (14) represents the s contribution of suction loading-unloading (drying-wetting) to the elastic strain rates, whereas the other term represents the net stress loading-unloading contribution.
For formulating the plastic rate of strain, both the plastic potential and the yield function have to be considered.
For the BB-model the yield function reads
2 2( * s) ( p *)
f q M p p p p (18)
where M is the slope of the critical state line, as also indicated in Figure 2, and
* * 2 * * 2 * * 2
1 2 2 3 3 1
1 ( ) ( ) ( )
q 2 V V V V V V (19)
ps = a . s. (20)
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.
Figure 2 Yield surface of Barcelona Basic Model
The yield function (18) reduces to the Modified Cam Clay (MCC) yield function at full saturation with s = 0.
In contrast to the MCC-model, the BB-model has a non-associated flow rule, which may be written as
* ( 1, 2,3)
p i
i
g i
H V
/ w
w (21)
where Hipstands for the principal rate of plastic strain, / is a multiplier and g is the plastic potential function
2 2( * s) ( p *).
g Dq M p p p p (22)
The flow rule becomes associated for D , but Alonso et al. (1990) recommend to use
0 0
( 9)( 3) .
9(6 )
M M M
M D O
O N
(23)
In this way the crest of the plastic potential in p* – q-plane is increased. Finally it leads to realistic K0-values in one-dimensional compression, whereas the associated MCC-model has the tendency to overestimate K0-values (Roscoe and Burland, 1968).
In combination with equations (14) and (21) the consistency condition f 0 yields the following expression for the plastic multiplier
1
* *
1 T 1 T
ij j s ij j
i i
f D f K f D s
H H H s G
V V
§ ·
w w w
/ ¨ ¸
w
w © w ¹
with
* * *.
T p ij
v i j
f g f g
H D
p
H V V
w w w w
w w w w (24)
4 Numerical implementation
The numerical implementation is done using an implicit return mapping algorithm according to the well-known elastic predictor/plastic corrector strategy.
The stress correction is performed in the space of stress invariants p* and q. On Gauss point level, the question is to evaluate the values of Vi* and the hardening parameter pp
corresponding to a given strain increment 'Hi and a given suction increment 's assuming that their initial values are known at time tn.
Box 1 shows the basic steps followed during the return mapping, where [ stands for the deviatoric tensor of stresses being defined as [ij = Vij* – p*.Gij where Vij* is defined as the Cauchy stress tensor and Gij is the Kronecker delta . The symbol || . || means the Euclidean norm of a second order tensor and ˆn is the normalised deviatoric tensor.
Box 1 Return mapping for BB-model
For solving the multiplier /, one needs to construct the residual vector r and the unknown vector x defined as follows
1 1
1 1
1 1
* * * 1
1 1 *
1
/( * *)
3
,
p n v
n n
tr
n n
n n
tr n
n n
n
p
p p
q q G g
q q
g p
p p K
r p x
f p
p p eH O N
/ w ½
° w ° ½
° ° ° °
° w °
° / ° ° °
® w ¾ ®/ ¾
° ° ° °
° ° ° °
¯ ¿
° °
° °
¯ ¿
(25)
With G = 3K(1 – 2Q)/(1 + Q) is the elastic shear modulus, N* = N/(1 + e) and O* = O/(1 + e).
Using the Newton-Raphson iteration technique, one can solve the previous system of four nonlinear equations.
It is worth noting that Hvp can be expressed as / *
p
v g p
H / w w in the last residual equation. Box 2, shows solution strategy following the Newton-Raphson method.
The adopted scheme is considered as a fully implicit one in the sense that all the unknowns are updated implicitly during the iteration process. A complete convergence study is available for testing the implementation efficiency but it is beyond the scope of this paper.
By checking Step 2 in Box 1, one can see that a suction reduction under constant net stress triggers the plastic correction routine if the stress point was already on the yield surface. This feature helps in capturing the soil structure collapse phenomenon. In what follows, the implemented model is used to solve some boundary value problems to illustrate the effect of suction on both soil strength and stiffness.
Box 2 Newton-Raphson algorithm
5 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 (1991) for a particular collapsible silt.
However, as they did not specify the M-value, we assumed a critical state friction angle of 31q, 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 considered.
The distribution of saturation being shown in Figure 3, was computed using the van Genuchten model (Van Genuchten, 1980), together with additional data for the silt. For the sake of convenience however, a constant (mean) value of 17.1 kN/m3 has been used for the soil weight above the phreatic line. For the initial net stresses the K0-value of 1 has been used. The finite element mesh consists of 6-noded triangles for the soil and 3-noded beam element for the strip footing. The flexural rigidity of the beam was taken to be EI = 10 MN.m2 per metre. This value is representative for a reinforced concrete plate with a thickness of roughly 20 cm.
Figure 3 Geometry, boundary conditions and finite element mesh
Table 1 Material and model parameters
e0 1.67 E 0.0215 [kPa–1]
O0 0.22 v 0.2
N 0.006 Ns 0.008
p0
p 80 [kPa] a 0.6
M 1.24
Of 0.123
pc 18.1 [kPa]
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 BB-analysis, but we simplified the analysis by assuming no change of suction during loading. In reality, footing loading will introduce a soil compaction and thus some change of both the degree of saturation and suction. As yet this has not been taken into account.
Up to an average footing pressure of 80 kPa, both analyses yield the same load-displacement curve. This relates to the adoption of preconsolidation pressure pp0 = 80 kPa. For pressures beyond 80 kPa, Figure 4 shows a considerable difference between the results from the BB-analysis and the 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 5(a). The MCC-analysis shows
a larger plastic zone underneath the footing, as shown in Figure 5(b).
Figure 4 Footing pressure-settlement curves
Figure 5 The plastic zones from BB and MCC model for footing pressure of 150 kPa (see online version for colours)
6 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 rise of the ground water level by the MCC-model, both the footing and the adjacent soil surface are 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) the heave is relatively small, but for other (expansive) clays it may be five times as large. Similar to the MCC-analysis, the BB-analysis yields soil heave as also shown in Figure 6. In contrast to the MCC- 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 (Fredlund and Rahardjo, 1993), 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 considerable 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 nearly yield the same final settlement of about 49 mm.
Figure 6 Vertical displacement of soil surface due to wetting
7 Bearing capacity
From Figure 4 it might be 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 generalisation of the Modified Cam Clay model and a CSL-slope of M = 1.24. The applied Drucker-Prager generalisation involves circular yield surfaces in a deviatoric plane of the principal stress space, which is realistic for small friction angles 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 sinMcs/(3 sin Mcs) and we get a friction angle of Mcs = 16.4o. However, we consider the plane strain problem of a strip footing. For planar deformation it yields M 3 sinMcs (Chen and Baladi, 1985), and it follows that Mcs = 21o. Table 2 gives the soil parameters. Figure 7 shows the boundary conditions and the finite element mesh for the bearing capacity problem of a shallow footing on unsaturated soil.
Figure 7 Finite element mesh and boundary conditions for the bearing capacity problem
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
0
f c q
q c N q N J b NJ (26)
where c is the soil cohesion, q0 is the surcharge load at footing level and b is the footing width. The factors Nc, Nq
and NJ are functions of the soil friction angle 1 sin tan
, ( 1) cot . 1 sin
q c q
N M eS M N N
M M
(27)
In the present analysis, J is taken equal to zero and the corresponding NJ-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 the BB-model, the cohesion c increases with suction s linearly, according to the formula
tan .
c a s M (28)
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 8 shows the calculated load-displacement curves using the BB-model and the MCC-model. The figure shows that an increase of suction value by 20 kPa was enough to double the soil bearing capacity. Shear bands at failure, as shown in Figure 9, are typically according to the solution by Prandtl.
In Figure 10, 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 a relatively small error.
Figure 8 Loading curves for BB- and MCC-analysis
Figure 9 Incremental shear strain at failure for s = 20 kPa (see online version for colours)
Figure 10 Total displacement increments for s = 20 kPa (see online version for colours)
Table 2 Soil properties
e0 0.9 E 0.0215 [kPa–1]
O0 0.14 v 0.2
N 0.015 Ns 0.01
p0
p 60 [kPa] a 1. 24
M 0.625
Of 0.036
pc 43 [kPa]
Table 3 Bearing capacity values
Suction [kPa] 0 20
Theoretical bearing capacities [kPa] 177 327 Numerical bearing capacities [kPa] 160 315
Relative error [%] 9.6 3.7
It is believed that we can capture better bearing capacity values by adopting a more advanced failure criterion than the Drucker-Prager criterion being used in this analysis. One can use a modified version of the well-known Mohr-Coulomb failure criterion which accounts for suction effects, or Matsuoka et al. criterion (Matsuoka et al., 2002) which offers us a failure surface without singular boundaries and, as a consequence, a more suitable criterion for numerical implementation.
8 Groundwater flow
Ground water flow is governed by the ground water head h = y + uw/Jw, where y is the geodetic head and uw/Jw is the pressure head, which will be denoted as \ for the sake of
simplicity. Jw 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 may vary with time. For most footing, settlement 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
rel sat i
i
q k k h x w
w (29)
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 pore pressure head-dependent relative permeability. Gardner (1958) proposed a simple exponential relative permeability function of the form
rel rel
for 0
1 for 0
k e
k
D \ \
\
t (30)
where D is a fitting parameter. It is worth pointing out that the pressure height \ has a negative value in the unsaturated zone. Figure 11 shows a graphical representation of equation (30) for D = 2 m–1.
In order to do ground water flow calculations, one has to supplement Darcy’s equation (29) with a continuity equation of the form
0
i i
q C h
x t
w w
w w (31)
where repeated subscripts stand for summation. C is the effective storage capacity, which is often expressed as (Brinkgreve et al., 2003)
sat
d d
Sr
C C n
s (32)
where Csat is the saturated storage capacity and Sr the degree of saturation. The latter is a function of pore pressure head \. In this publication we will use the following simple equation as in Srivastava and Yeh (1991)
res ( sat res)
Sr S S S eD\ (33)
Sres is the residual degree of saturation, Ssat is the degree of saturation at full saturation which is usually taken as one.
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 (29) and (31), boundary conditions are required. For studying footing problems, one would need the water infiltration or the rate of evaporation q at the soil surface.
The PlaxFlow finite element code (Brinkgreve et al., 2003) has been used to calculate suction in unsaturated zone. In the following section, the numerical results of PlaxFlow are checked by analytical solutions in the case of one-dimensional unsaturated groundwater flow.
Figure 11 Gardner permeability model
8.1 Analytical solution for one-dimensional unsaturated stationary groundwater flow
For one-dimensional vertical steady-state flow equation (31) reduces to equation (34) in terms of pore pressure head \
( ) 1 0.
sat rel
k k
y y
\ \
ª § ·º
w w
« ¨ ¸»
w ¬ © w ¹¼
(34) Gardner (1958) gave an analytical solution for the
differential equation (34) with infiltration or evaporation boundary conditions at soil surface, as shown in Figure 12. Using equation (30) for the permeability function, the solution is given as
1 ln ( 1) y .
sat sat
q q
k e k
\ D
D
ª º
« »
¬ ¼
(35) Gardner used negative values for q to indicate infiltration
and positive values to indicate evaporation.
Figure 12 Boundary conditions used in Gardner solution
8.2 Verification of the numerical results
A silty soil with saturated permeability of ksat = 1 m/day and D = 2 m–1 has been used to compare PlaxFlow numerical results with the analytical solution of equation (34).
Figure 13 shows the finite element mesh and boundary conditions as used in the verification examples. Six noded triangular elements are used with closed vertical boundaries to recover 1-D conditions. The ground water table is at 3.0 m depth and a Neumann type boundary condition is applied at the soil surface.
Figure 13 Finite element mesh and boundary conditions used in numerical calculations
The problem has been solved for three common practical situations, namely, hydrostatic conditions with q = 0, evaporation with q = 0.002 m/day and infiltration with q = 0.1 m/day. The hydrostatic conditions represent a pore pressure distribution in unsaturated soil which has no interaction with surface water, for example a soil directly underneath a raft foundation. Evaporation represents soil moisture decrease due to temperature increase during a dry season and infiltration represents soil moisture increase due to rainfall for instance.
Figure 14 shows a very good agreement between analytical and numerical results for this particular kind of problem. At the same time, it gives a nice idea about negative pore water pressure profiles in such common cases.
Figure 14 Analytical vs. numerical results
8.3 Analytical solution for one-dimensional unsaturated transient groundwater flow
The analytical solution of equation (32) in the case of one-dimensional vertical transient flow i.e., negative pore
water pressure head \ as a function of time and vertical position is presented by Srivastava and Yeh (1991) for the particular water characteristic and permeability functions given in equations (33) and (30), respectively
1 ln( )R
\ D (36)
where
*
2 *
( ) / 2 / 4
1 2
1 4
sin( ) sin( ) 1 ( / 2) 2
i
y L y t
sat sat sat
t
i i
i i
q q q
R e e e
k k k
y L e
L L
Z Z Z
Z
f
§ ·
¨ ¸
© ¹
¦
(37)
* sat
sat res
n k t
t S S
D
(38)
Zi is the ith root of the characteristic equation (39), L is the layer thickness and t is the time.
tan(ZL) 2 Z 0. (39)
8.4 Verification of the numerical results
The same material properties mentioned in Section 8.2 and the same geometry and boundary conditions are used here considering a hydrostatic suction distribution as an initial condition. The soil is exposed to infiltration rate of q = 0.1 m/day. The analytical solution is presented by the solid lines in Figure 15, whereas the numerical results are represented by dots. This figure shows good agreement between the numerical and analytical solution. It is also interesting to notice that Gardner’s solution for the steady state is obtained after five days of continuous infiltration.
Figure 15 Negative pressure head at different time steps
Since we are confident about the numerical results of the ground water flow module, we will study the mechanical behaviour of a shallow foundation during a rainfall event in the next section.
9 Footing deformation with time
In reality, suction varies with position and time. Up to this point, we have studied only the case of shallow foundation which is exposed to a ‘sudden’ suction reduction by increasing ground water level.
In this section, a more realistic simulation is considered where the deformations of a shallow foundation are calculated during soil saturation due to rainfall. The geometry, the boundary conditions and the finite element mesh are shown in Figure 16. The mechanical properties of the soil are the same as in Table 2 with a preconsolidation pressure of pp0 = 30 kPa and pc = 10 kPa. The hydraulic properties are the same as those used to check the unsaturated ground water flow finite element code in Section 8. The finite element mesh consists of 6-noded triangles for the soil and 3-noded plate elements for the strip footing. The flexural rigidity of the plate was taken to be EI = 10 MN.m2 per metre. The initial stresses are considered to be isotropic, with K0 = 1. A hydrostatic distribution is adopted as an initial condition for the transient ground water flow calculations.
Figure 16 Finite element mesh, initial and boundary condition for transient deformation analysis (see online version for colours)
The footing is loaded upto 50 kPa. Afterwards a transient water flow calculation is done to simulate a five days rainfall event with infiltration rate of q = 0.1 m/day.
The calculated suction values at different time steps are provided to the deformation routine to evaluate the resulting displacements. Figure 17 shows suction and vertical displacements underneath the footing with time.
In this analysis, we expect a final value of suction equal to that shown in Figure 15 which is also the value given by Gardner’s solution for the steady-state situation, however it is obvious from Figure 17(a) that suction variations are negligible after two days. This has a direct consequence on the footing settlements, which take place essentially during the first two days and then reach a steady situation as shown in Figure 17(b). The rainfall causes 30 mm of additional settlement to the 60 mm due to the 50 kPa of loading. The additional deformation reproduces a well-known type of unsaturated soil behaviour upon wetting called the partial soil structure collapse, as shown in Figure 18. The word ‘partial’ means that additional wetting of the soil will lead to further settlement until the soil reaches a full saturation. A typical distribution of the net stresses underneath the footing for such type of problems is shown in Figure 19.
Figure 17 Suction and vertical displacements with time
Figure 18 Load-settlement curve at the end of infiltration
Figure 19 Net stresses distribution at the end of analysis (see online version for colours)
10 Conclusions
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 built 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 calculation of 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 (Abed, 2007). Depending on the results, the designer can pick the lowest and the highest suction values in the studied period. With this 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, but also river embankments, suffer upon wetting. High river water levels tend to occur for relatively short periods of time, so that there is partial wetting. This offers also a challenging topic of transient ground water flow and deformations in unsaturated ground.
Acknowledgements
We are grateful to Dr. Peter van den Berg of GeoDelft, the Netherlands, for providing support for this study.
Special thanks are due to Mr. John van Esch of GeoDelft, to Professor Antonio Gens from the University of Catalunia and Dr. Klaas Jan Bakker of the Plaxis company for fruitful discussions on unsaturated soil behaviour.
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