Beyond 2000 in Computational Geotechnics – 10 Years of PLAXIS International ? 1999 Balkema, Rotterdam, ISBN 90 5809 040 X
A soft soil model that accounts for creep
P.A. Vermeer & H.P. Neher
Institute of Geotechnical Engineering, University of Stuttgart, Germany
Keywords: aging, creep, consolidation, soft soil ABSTRACT: The well-known logarithmic creep law for secondary compression is transformed into a differential form in order to include transient loading conditions. This 1-D creep model for oedometer-type strain conditions is then extended towards general 3-D states of stress and strain by incorporating concepts of Modified Cam-Clay and viscoplasticity. Considering lab test data it is shown that phenomena such as undrained creep, overconsolidation and aging are well captured by the model. 1 INTRODUCTION As soft soils we consider near-normally consolidated clays, clayey silts and peat. The special features of these materials is their high degree of compressibility. This is best demonstrated by oedometer test data as reported for instance by Janbu in his Rankine lecture (1985). Considering tangent stiffness moduli at a reference oedometer pressure of 100 kPa, he reports for normally consolidated clays Eoed = 1 to 4 MPa, depending on the particular type of clay considered. The differences between these values and stiffnesses for NC-sands are considerable as here we have values in the range of 10 to 50 MPa, at least for non-cemented laboratory samples. Hence, in oedometer testing normally consolidated clays behave ten times softer than normally consolidated sands. This illustrates the extreme compressibility of soft soils. Another feature of the soft soils is the linear stress-dependency of soil stiffness. According to ref the Hardening-Soil model we have Eoed = Eoed 1p ref)m , at least for c = 0, and a linear relationship is obtained for m = 1. Indeed, on using an exponent equal to one, the above stiffness law reref duces to Eoed = 1 / *, where * = pref / Eoed . For this special case of m = 1, the Hardening-Soil model yields 0 = * 1/1 , which can be integrated to obtain the well-known logarithmic compression law 0 = * ln1 for primary oedometer loading. For many practical soft-soil studies, the modified compression index * will be known and we can compute the oedometer modulus from the reref lationship Eoed = pref / *. From the above considerations it would seem that the HS-model is perfectly suitable for soft soils. Indeed, most soft soil problems can be analysed using this model, but the HS-model is not suitable when considering creep, i.e. secondary compression. All soils exhibit some creep, and primary compression is thus always followed by a certain amount of secondary compression. Assuming the secondary compression (for instance during a period of 10 or 30 years) to be a small percentage of the primary compression, it is clear that creep is important for problems involving large primary compression. This is for instance the case when constructing road or river embankments on soft soils. Indeed, large primary settlements of dams and embankments are usually followed by substantial creep settlements in later years. In some cases dams or buildings may also be founded on initially overconsolidated soil layers that yield relatively small primary settlements. Then, as a consequence of the loading, a state of normal consolidation may be reached and significant creep may follow. This is a treacherous situa-
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Beyond 2000 in Computational Geotechnics – 10 Years of PLAXIS International ? 1999 Balkema, Rotterdam, ISBN 90 5809 040 X
tion as considerable secondary compression is not preceeded by the warning sign of large primary compression. Apart from foundation-related problems, creep plays an important role in steep slopes. Many natural slopes have a relatively small factor of safety and they show continuous displacements due to creep. Gradual geometric changes and associated degradation of strength may then lead to slope slides. The various different problems that relate to creep have motivated us to develop a stressstrain relationship that takes creep into account. Buisman (1936) was probably the first to propose a creep law for clay after observing that softsoil settlements could not be fully explained by classical consolidation theory. The work on 1Dsecondary compression was continued by researchers including, for example, Bjerrum (1967), Garlanger (1972) and Mesri (1977) and Leroueil (1977). A more mathematical lines of research in the area were followed by, for example, Sekiguchi (1977), Adachi and Oka (1982) and Borja et al. (1985). This line of mathematical 3D-creep modelling was influenced by the more experimental line of 1D-creep modelling, but conflicts exist. From the authors viewpoint, 3D-creep should be a straight forward extension of 1D-creep, but this is hampered by the fact that present 1D-models have not been formulated as differential equations. For the presentation of the Soft-Soil-Creep model we will first complete the line of 1Dmodelling by conversion to a differential form, from which an extension is made to a 3D-model. In addition, attention is focused on the model parameters. Finally the model will be validated by considering both model predictions and data from triaxial tests. Here attention is focused on constant strain-rule shear tests and undrained creep tests. 2 BASICS OF ONE-DIMENSIONAL CREEP When reviewing previous literature on secondary compression in oedometer tests, one is struck by the fact that it concentrates on behaviour related to step loading even though natural loading processes tend to be continuous or transient in nature. Buisman (1936) who was probably the first to consider such a classical creep test. He proposed the following to describe creep behaviour under constant effective tress, t ε = ε c + C B log for (1) t > tc tc where 0c is the strain up to the end of consolidation, t the time measured from the beginning of loading, tc the time to the end of primary consolidation and CB is a material constant. As is often the case in soil mechanics, compression is assumed to be positive. For further consideration, it is convenient to rewrite this equation as t + t′ (2) for ε = ε c + C B log c t′ > 0 tc with t = t - tc being the effective creep time. Based on the work by Bjerrum on creep, as published for instance in 1967, Garlanger (1972), proposed a creep equation of the form τ + t′ e = ec - Cα log c with Cα = C B (1 + e0 ) for t ′ > 0 (3) τc Differences between Garlanger s and Buisman s forms are modest. The engineering strain 0 is replaced by void ratio e and the consolidation time tc is replaced by a parameter 2c. Equations (2) and (3) are entirely identical when choosing 2c = tc. For the case that 2c g tc differences between both formulations will vanish when the effective creep time t increases. For practical consulting, oedometer tests are usually interpreted by assuming tc = 24 h. Indeed, the standard oedometer test is a Multiple Stage Loading Test with loading periods of precisely one day. Due to the special assumption that this loading period coincides to the consolidation time tc, it 2
Beyond 2000 in Computational Geotechnics – 10 Years of PLAXIS International ? 1999 Balkema, Rotterdam, ISBN 90 5809 040 X
follows that such tests have no effective creep time. Hence one obtains t = 0 and the log-term drops out of equation (3). It would thus seem that there is no creep in this standard oedometer test, but this suggestion is entirely false. Even highly impermeable oedometer samples need less than one hour for primary consolidation. Then all excess pore pressures are zero and one observes pure creep for the other 23 hours of the day. Therefore we will not make any assumptions about the precise values of 2c and tc. Another slightly different possibility to describe secondary compression is the form adopted by Butterfield (1979) τc + t ′ H (4) ε H = ε c + C ln τc where 0H is the logarithmic strain defined as V 1+ e = - ln (5) ε H = - ln 1+ eo Vo with the subscript !o denoting the initial values. The superscript !H is used for denoting logarithmic strain. We use this particular symbol, as the logarithmic strain measure was originally used by Hencky. For small strains it is possible to show that Cα CB C = = (6) 1 + eo ? ln 10 ln 10
0
5
because then logarithmic strain is approximately equal to the engineering strain. Both Butterfield (1979) and Den Haan (1994) showed that for cases involving large strain, the logarithmic small strain supersedes the traditional engineering strain. 3 ON THE VARIABLES τC AND εC In this section attention will first be focussed on the variable 2c. Here a procedure is to be described for an experimental determination of this variable. In order to do so we depart from equation (4). By differentiating this equation with respect to time and dropping the superscript !H to simplify notation, one finds C 1 τ + t′ ε = or inversely (7) = c C ε τc + t ′ which allows one to make use of the construction developed by Janbu (1969) for evaluating the parameters C and 2c from experimental data. Both the traditional way, being indicated in Figure 1a, as well as the Janbu method of Figure 1b can be used to determine the parameter C from an oedometer test with constant load. The use of the Janbu method is attractive, because both 2c and C follow directly when fitting a straight line through the data. In Janbu s representation of Figure 1b,
Figure 1. Consolidation and creep behaviour in standard oedometer test.
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Beyond 2000 in Computational Geotechnics – 10 Years of PLAXIS International ? 1999 Balkema, Rotterdam, ISBN 90 5809 040 X
2c is the intercept with the (non-logarithmic) time axis of the straight creep line. The deviation from a linear relation for t < tc is due to consolidation. Considering the classical literature it is possible to describe the end-of-consolidation strain 0c, by an equation of the form εc = ε e + εc = A ln c c σ pc σ′ + B ln σ′ σ p0 0 (8)
Note that 0 is a logarithmic strain, rather than a classical small strain although we conveniently omit the superscript !H . In the above equation 10 represents the initial effective pressure before loading and 1 is the final effective loading pressure. The values 1p0 and 1pc representing the preconsolidation pressure corresponding to before-loading and end-of-consolidation states respectively. In most literature on oedometer testing, one adopt e instead of 0, and log instead of ln, and the swelling index Cr instead of A, and the compression index Cc instead of B. The above constants A and B relate for small strains to Cr and Cc as A =
0
Cr 1 + eo ? ln 10
5
B =
0C c C r 5 01+ eo5 ? ln10
(9)
Combining equations (4) and (8) it follows that σ pc σ′ τ + t′ + B ln + C ln c ε = ε e + ε c = A ln σ′ σ p0 τc 0
(10)
where 0 is the total logarithmic strain due to an increase in effective stress from 10 to 1 and a time period of tc+t . In Figure 2 the terms of equation (10) are depicted in a 0ln1 diagram. Up to this point, the more general problem of creep under transient loading conditions has not yet been addressed, as it should be recalled that restrictions have been made to creep under constant load. For generalising the model, a differential form of the creep model is needed. No doubt, such a general equation may not contain t and neither2c as the consolidation time is not clearly defined for transient loading conditions. 4 DIFFERENTIAL LAW FOR 1D-CREEP The previous equations emphasize the relation between accumulated creep and time, for a given constant effective stress. For solving transient or continuous loading problems, it is necessary to formulate a constitutive law in differential form, as will be described in this section. In a first step we will derive an equation for 2c. Indeed, despite the use of logarithmic strain and ln instead of log, the formula (10) is classical without adding new knowledge. We may also deviate a bit as we write
Figure 2. Idealised stress-strain curve from oedometer test with division of strain increments into an elastic and a creep component. For t + tc = 1 day one arrives precisely on the NC-line.
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Beyond 2000 in Computational Geotechnics – 10 Years of PLAXIS International ? 1999 Balkema, Rotterdam, ISBN 90 5809 040 X
1pc when some other authors write 1. Moreover, the question on the physical meaning of 2c is still open. In fact, we have not been able to find precise information on 2c, apart from Janbu s method of experimental determination. In order to find an analytical expression for the quantity 2c, we adopt the basic idea that all inelastic strains are time dependent. Hence total strain is the sum of an elastic part 0e and a timedependent creep part 0c. For non-failure situations as met in oedometer loading conditions, we do not assume an instantaneous plastic strain component, as used in traditional elastoplastic modelling. In addition to this basic concept, we adopt Bjerrum s idea that the preconsolidation stress depends entirely on the amount of creep strain being accumulated in the course of time. In addition to equation (10) we therefore introduce the expression ε = ε e + ε c = A ln σp σ′ + B ln σ′ σ p0 0 → σ p = σ p 0 exp
εc B
(11)
The longer a soil sample is left to creep the larger 1p grows. The time-dependency of the preconsolidation pressure 1p is now found by combining equations (10) and (11) to obtain εc - εcc = B ln τ + t′ = C ln c σ pc τc σp (12)
This equation can now be used for a better understanding of 2c, at least when adding knowledge from standard oedometer loading. In conventional oedometer testing the load is stepwise increased and each load step is maintained for a constant period of tc+t 2 , where 2 is precisely one day. In this way of stepwise loading the so-called normal consolidation line (NC-line) with 1p 1 is obtained. On entering 1p 1 and t 2Wc into equation (12) it is found that B ln σ′ τ + τ - tc = C ln c σ pc τc for OCR = 1 (13)
It is now assumed that (2c - tc) Q2. This quantity can thus be disregarded with respect to 2and it follows that τ = τc
σ′ C σ pc
B
or
τc = τ
σ pc C σ′
B
(14)
Hence 2c depends both on the effective stress 1 and the end-of-consolidation preconsolidation stress 1pc. In order to verify the assumption (2c - tc) Q 2, it should be realised that usual oedometer samples consolidate for relatively short periods of less than one hour. Considering load steps on the normal consolidation line, we have OCR = 1 both in the beginning and at the end of the load step. During such a load step 1p increases from 1p0 up to 1pc during the short period of (primary) consolidation. Hereafter 1p increases further from 1pc up to1 during a relatively long creep period. Hence, at the end of the day the sample is again in a state of normal consolidation, but directly after the short consolidation period the sample is under-consolidated with 1p1 . For the usually very high ratios of B/C 15, we thus find very small 2c-values from equation (14). Hence not only tc but also 2c tend to be small with respect to 2 It thus follows that the assumption (2c - tc) Q 2 is certainly correct. Having derived the simple expression (14) for 2c, it is now possible to formulate the differential creep equation. To this end equation (10) is differentiated to obtain C σ′ ε = εe + εc = A (15) + σ′ τc + t ′ where2c+ t' can be eliminated by means of equation (12) to obtain
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Beyond 2000 in Computational Geotechnics – 10 Years of PLAXIS International ? 1999 Balkema, Rotterdam, ISBN 90 5809 040 X
C σ′ ε = εe + εc = A + σ′ τc
σ pc C σ p σ′ C σ p
B
B
with
σ p = σ p 0 exp
εc B
(16)
Equation (14) can now be introduced to eleminate 2c and 1pc and to obtain C σ′ ε = εe + εc = A + σ′ τ where σ p = σ p 0 exp
εc B
(17)
5 THREE-DIMENSIONAL-MODEL On extending the 1D-model to general states of stress and strain, the well-known stress invariants for pressure S 1oct and deviatoric stress q = 32oct/2 are adopted, with 1oct and 2oct being the octahedral normal and shear stresses respectively. These invariants are used to define a new stress measure named peq: peq = p ′ + with p ′ = q2 M 2 p′ (18)
1 1 2 2 2 σ1′ + σ2′ + σ3′ and q = σ1 - σ2 + σ1 - σ3 + σ2 - σ3 3 2 eq In Figure 3 it is shown that the stress measure p is constant on ellipses in p-q-plane. In fact we have the ellipses from the Modified Cam-Clay-Model as introduced by Roscoe and Burland (1968). The soil parameter M represents the slope of the so-called !critical state line as also indicated in Figure 3. We use the above equation for the deviatoric stress q and 6 sin ?cv M = (19) 3 sin ?cv where Qcv is the critical-void friction angle, also referred to as critical-state friction angle. On using the above definition for q, the equivalent pressure peq is constant along ellipsoids in principal stress space. To extend the 1D-theory to a general 3D-theory, attention is now focused on normally consolidated states of stress and strain as met in oedometer testing. In such situations it yields 12 13 = K0,NC11 , and it follows from equation (18) that
0
5
0
5 0
5 0
5
Figure 3. Diagram of peq-ellipse in a p-q-plane.
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Beyond 2000 in Computational Geotechnics – 10 Years of PLAXIS International ? 1999 Balkema, Rotterdam, ISBN 90 5809 040 X
1+ 2 K 0NC 321 - K 0NC72 "# = σ′ + NC 3 M 2 21 + 2 K 0 7 # ! $ 1+ 2 K 0NC 321 - K 0NC72 "# eq = pp + σp NC 3 M 2 21 + 2 K 0 7 # ! $
peq
(20)
where 1 11 , and p eq is a generalised preconsolidation pressure, being simply proportional to the p
NC one-dimensional one. For known values of K 0 , peq can thus be computed from 1 and p eq can p thus be computed from 1p. Omitting the elastic strain in the 1D-equation (17), introducing the above expressions for peq and p eq and writing 0v instead of 0 it is found that p
v εc =
C peq C p τ peq
B
where
εc peq = peq exp v p p0 B
(21)
For one-dimensional oedometer conditions, this equation reduces to equation (17), so that one has a true extension of the 1D-creep model. It should be noted that the subscript !0 is once again used in the equations to denote initial conditions and that ε c = 0 for time t = 0. v Instead of the parameters A, B and C of the 1D-model, we will now change to the material parameters *, * and *, who fit into the framework of critical state soil mechanics. Conversion between constants follows the rules κ* ≈ 3 1 - νur A, 1 + νur
0
0
5 5
B = λ* - κ * ,
?* = C
(22)
On using these new parameters, equation (21) changes to become v εc = ?*
p τ peq
peq
λ* - κ* ?*
,
peq = peq exp p p0
εcv λ* - κ *
(23)
As yet the 3D-creep model is incomplete, as we have only considered a volumetric creep strain ε c , v whilst soft soils also exhibit deviatoric creep strains. For introducing general creep strains, we adopt the view that creep strain is simply a time-dependent plastic strain. It is thus logic to assume a flow rule for the rate of creep strain, as usually done in plasticity theory. For formulating such a flow rule, it is convenient to adopt the vector notation σ = σ1, σ2 , σ3
0
5T
and
ε = ε1, ε2 , ε3
0
5T
where T is used to denote a transpose. Similar to the 1D-model we have both elastic and creep strains in the 3D-model. Using Hook s law for the elastic part, and a flow rule for the creep part, one obtains ε = ε e + ε c = D-1 σ ′ + λ ? gc ?σ ′ (24)
where the elasticity matrix and the plastic potential function are defined as
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Beyond 2000 in Computational Geotechnics – 10 Years of PLAXIS International ? 1999 Balkema, Rotterdam, ISBN 90 5809 040 X
D-1
=
1 E ur
1 - νur - νur 1 - νur
Hence we use the so-called equivalent pressure peq as a plastic potential function for deriving the individual creep strain-rates components. The superscripts !ur are introduced to emphasize that both the elasticity modulus and Poisson s ratio will determine unloading-reloading behaviour. Now it follows from the above equations that v c 2 c εc = ε1 + εc + ε3 = λ
!- νur
- νur
"# - νur # 1# $
and
gc = peq
? peq ?σ1′
+
? peq ? peq + ?σ 2 ?σ 3 ′ ′
= λ ? peq ? ′p
? peq ?σ ′
(25)
+HQFHZHGHILQH. ?p eq / ? ′p . Together with equations (23) and (24) this leads to 1 ?* εc ? peq 1 ε = D-1 σ ′ + v = D- σ ′ + α ?σ ′ α τ peq p where peq p = peq p0
λ* - κ* * peq ?
(26)
εc exp * v * λ - κ
or inversely
εc v
= λ *
2
κ*
7 ln peq
peq p
p0
6 FORMULATION OF ELASTIC 3D-STRAINS Considering creep strains, it has been shown that the 1D-model can be extended to obtain the 3Dmodel, but as yet this has not been done for the elastic strains. To get a proper 3D-model for the elastic strains as well, the elastic modulus Eur has to been defined as a stress-dependent tangent stiffness according to Eur = 3 1- 2 νur K ur = 3 1- 2 νur
0
5
0
p 5 κ*′
(27)
Hence, Eur is not a new input parameter, but simply a variable quantity that relates to the input parameter *. On the other hand vur is an additional true material constant. Hence similar to Eur, the bulk modulus Kur is stress dependent according to the rule Kur = p*. Now it can be derived for the volumetric elastic strain that v εe = p′ K ur = κ* p′ p′ or by integration ε e = κ* ln v p′ p0 ′ (28)
Hence in the 3D-model the elastic strain is controlled by the mean stress p , rather than by principal stress 1 as in the 1D-model. However mean stress can be converted into principal stress. For oneNC dimensional compression on the normal consolidation line, we have both 3p = (1+2 K 0 )1 and
NC 3p0 = (1+2 K 0 )10 and it follows that p /p0 110. As a consequence we derive the simple rule ε e * ln(110 ), whereas the 1D-model involves ε e =A ln(110 ). It would thus seem that * coν ν incides with A. Unfortunately this line of thinking can not be extended towards overconsolidated states of stress and strain. For such situations, it can be derived that p′ 1 + νur 1 σ′ (29) = p′ 1 - νur 1 + 2 K 0 σ ′
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Beyond 2000 in Computational Geotechnics – 10 Years of PLAXIS International ? 1999 Balkema, Rotterdam, ISBN 90 5809 040 X
and it follows that v εe = κ* σ′ p′ 1 + νur κ* = p′ 1 - νur 1 + 2 K 0 σ ′ (30)
where K0 depends to a great extend on the degree of overconsolidation. For many situations, it is reasonable to assume K0 1 and together with vur 0.2 one obtains 2ε e * ln(1 10 ). Good ν agreement with the 1D-model is thus found by taking * 2A. 7 REVIEW OF THE MODEL PARAMETERS As soon as the failure yield criterion f (1, c, Q) = 0 is met, instantaneous plastic strain rates develop according to the flow rule ε p 0g/01 with g = g (1%). This gives as additional soil parameters the effective cohesion, c, the Mohr-Coloumb friction angle, Q, and the dilatancy angle %. For fine grained, cohesive soils, the dilatancy angle tends to be small, it may often be assumed that % is equal to zero. In conclusion, the Soft-Soil-Creep model requires the following material constants. Failure parameters as in the Mohr-Coulomb model: c : Q : % : * : * : * : vur : M : Cohesion Friction angle Dilatancy angle Modified swelling index Modified compression index Modified creep index Poisson's ratio for unloading-reloading Slope of the so-called !critical state line [kN/m2] [°] [°] [-] [-] [-] [-] [-]
Parameters of the Soft-Soil-Creep model:
7.1 Modified swelling index, modified compression index and modified creep index These parameters can be obtained both from an isotropic compression test and an oedometer test. When plotting the logarithm of stress as a function of strain, the plot can be approximated by two straight lines (see Figure 2). The slope of the normal consolidation line gives the modified compression index *, and the slope of the unloading (or swelling) line can be used to compute the modified swelling index *, as explained in section 6. Note that there is a difference between the modified indices * and * and the original Cam-Clay parameters and . The latter parameters are defined in terms of the void ratio e instead of the volumetric strain 0v. The parameter * can be obtained by measuring the volumetric strain on the long term and plotting it against the logarithm of time (see Figure 1). Relationship to Cam-Clay parameters: λ κ κ* = 1 + e 1 + e Relationship to internationally normalized parameters: 3 1 - νur Cc Cr κ* ≈ λ* = 2.3 ( 1 + e) 2.3 1 + νur 1 + e λ* = (31)
?* =
Cα 2.3 ( 1 + e )
(32)
As already indicated in section 6, there is no exact relation between the isotropic compression indices and * and the one-dimensional swelling index Cr, because the ratio of horizontal and vertical 9
Beyond 2000 in Computational Geotechnics – 10 Years of PLAXIS International ? 1999 Balkema, Rotterdam, ISBN 90 5809 040 X
stress changes during one-dimensional unloading. For the approximation it is assumed that the average stress state during unloading is an isotropic stress state, i.e. horizontal and vertical stress equal. For a rough estimate of the model parameters, one might use the correlation * Ip (%) / 500, the fact that */* is in the range between 15 to 25 and the general observation */* is in the range between 5 to 10. For characterising a particular layer of soft soil, it is also necessary to know the initial preconsolidation pressure 1p0. This pressure may e.g. be computed from a given value of the overconsolidation ratio (OCR). Subsequently 1p0 can be used to compute the initial value of the generalised preconsolidation pressure ppeq. 7.2 Poisson's ratio In the case of the Soft-Soil-Creep model, Poisson's ratio is purely an elasticity constant rather than a pseudo-elasticity constant as used in the Mohr-Coulomb model. Its value will usually be in the range between 0.1 and 0.2. For loading of normally consolidated materials, Poisson's ratio plays a minor role, but it becomes important in unloading problems. For example, for unloading in a onedimensional compression test (oedometer), the relatively small Poisson's ratio will result in a small decrease of the lateral stress compared with the decrease in vertical stress. As a result, the ratio of horizontal and vertical stress increases, which is a well-known phenomenon for overconsolidated materials. Hence, Poisson's ratio should be based on the ratio of difference in horizontal stress to difference in vertical stress in oedometer unloading and reloading. ? σ xx νur = (33) 1 - νur ? σ yy
8 VALIDATION OF THE 3D-MODEL This section briefly compares the simulated response of undrained triaxial creep behaviour of Haney clay with test data provided by Vaid and Campanella (1977), using the material parameters summarized in Table 1. A extensive validation is provided in Stolle et al.(1997). All triaxial tests considered were completed by initially consolidating the samples under an effective isotropic confining pressure of 525 kPa for 36 hours and then allowing them to stand for 12 hours under undrained conditions before starting the shearing part of the test. The end-of-consolidation preconsolidation pressure just before shearing, p eq , was found to be pc 373 kPa. This value was determined by simulating the consolidation part of the test. The preconsolidation pressure p eq of 373 kPa is less than 525 kPa, which would have been required for an pc OCRo of 1. It is clear that the preconsolidation pressure not only depends on the applied maximum consolidation stress, but also on time as discussed in previous section. In Figure 4 we can see the results of Vaid and Campanella s tests (1977) for different strain rates and the computed curves, that were calculated with the present creep model. It is observed that the model describe the tests very well.
Table 1. Material properties for Haney clay. * = 0.016 v = 0.25 3mc = 32o 3cs = 32.1o * = 0.105 % o * = 0.004 c = 0 kPa
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Beyond 2000 in Computational Geotechnics – 10 Years of PLAXIS International ? 1999 Balkema, Rotterdam, ISBN 90 5809 040 X
Figure 4. Results of the undrained triaxial tests (CU-tests) with different rates of strain. The faster the test the higher the undrained shear strength.
Figure 5. Strain rate dependency of the effective stress path in undrained triaxial tests.
8.1 Constant strain rate shear tests Undrained triaxial compression tests, as considered in Figure 4, are performed under constant rates of vertical strain ε1 and constant horizontal pressure 13, so the vertical stress 11 is allowed to increase. In the range of classical triaxial compression tests the shear stress q, for different strain rates increases with the increase of the strain rate, though there is a limit for very slow and very fast tests between the shear stress q can move. This behaviour is shown in Figure 5. ν ν ν ν During the undrained test the condition ε ν = ε e + ε c = 0 or ε e = - ε c is valid. Therefore the total volumetric change must be zero, but it exists a creep behaviour, though the creep compaction is compensated by an elastic swelling. The slower a test is perform, the lager the creep compaction ν is and finally the elastic swelling. The expression p ′ = Kur ε e , where Kur is the elastic bulk modulus, shows that elastic swelling implies a decrease of the effective mean stress. The slower the shear-rate is the more the stress path will curve against the origin of the p-q-plane. 11
Beyond 2000 in Computational Geotechnics – 10 Years of PLAXIS International ? 1999 Balkema, Rotterdam, ISBN 90 5809 040 X
ν For extremely fast tests there is no time for creep and it yields ε c = 0 and consequently also e ε ν = 0. Hence in this extreme case there is no elastic change and consequently neither a change of the mean stress. This implies a straight vertical path for the effective stress in p-q-plane. On inspecting all numerical results, it appears that the undrained shear strength, cu, may be approximated by equation (34), cu ≈ 1.02 + 0.09 log ε cu* (34)
* where cu is the undrained shear strength in an undrained triaxial test with a strain rate of 1% per hour. This agrees well with the experimental data summarized by Kulhawy and Mayne (1990).
8.2 Undrained triaxial creep tests In undrained triaxial creep tests the vertical and horizontal stresses are kept constant, after a initial shear stress q has been applied. The creep behaviour in these tests depends on the stress level. Under a relatively small applied shear stresses (about 30% of the shear strength as determined in con ventional tests) the creep movements are small and the creep strain rate, ε , decreases, while the excess pore-water pressure increases. After a certain time, an equilibrium condition is reached, in which the creep strain rate is zero and the excess pore-water pressure is constant. Under a shear stress on higher level (but less than 70% of the shear strength of the conventional tests), creep movements seem to continue at constant strain rates. Under shear stresses of still higher intensity, an acceleration of the creep rate takes place followed by complete failure of the specimen (!creep rupture ). As for the constant strain rate tests, all triaxial creep tests had been completed by initially consolidating the samples under an effective confining pressure of 525 kPa and for a period such that p eq = 373 kPa. Then undrained creep tests were performed at constant shear stresses of q = 278.3, pc 300.3, 323.4 kPa. Figure 6 illustrates that these tests one also matched by the model.
Figure 6. Results of triaxial creep tests. Samples were first consolidated under the same isotropic stress. Then undrained samples were loaded up to different deviatoric stresses. Creep under constant deviatoric stress is observed, being well predicted by the Soft-Soil-Creep model.
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Beyond 2000 in Computational Geotechnics – 10 Years of PLAXIS International ? 1999 Balkema, Rotterdam, ISBN 90 5809 040 X
Figure 7. Results of triaxial creep tests. All tests have different constant deviatoric stress. The creep rupture time is the creep time up to a creep rate ε = , as indicated by the assumptots in figure 6.
REFERENCES
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