Modelling the rate-sensitive characteristics of the ... · Rowe and Hinchberger (1998) have...

Modelling the rate-sensitive characteristics ofthe Gloucester foundation soil

Sean D. Hinchberger and R. Kerry Rowe

Abstract: Stages 1 (1967) and 2 (1982) of the Gloucester test embankment are studied using a fully coupled finite-element model. The rate-sensitive characteristics of the foundation soil are modelled using an elastoviscoplasticconstitutive equation based on the elliptical cap yield surface and Perzyna’s overstress theory of viscoplasticity. Theaspect ratio of the yield surface for the Gloucester foundation soil is estimated using conventional laboratory shear andconsolidation test results. Calculated and measured behaviour during consolidated isotropically undrained triaxial testsand long-term Rowe cell consolidation tests are compared and the ability of the model to describe the measuredbehaviour of stages 1 and 2 of the Gloucester test embankment is studied. This paper explores the implications ofmodelling the residual or restructured properties of the Gloucester foundation soil and demonstrates the ability of asingle elastoviscoplastic yield-surface model to describe the undrained and drained response of the Gloucesterfoundation soil during laboratory and field loading conditions.

Key words: elliptical cap, rate sensitive, elastoviscoplasticity, embankment settlements, pore pressures, field perfor-mance.

Résumé: Les étapes 1 (1967) et 2 (1982) du remblai d’essai de Gloucester sont étudiées au moyen d’un modèled’éléments finis complètement couplés. Les caractéristiques de sensibilité à la vitesse de déformation du sol defondation sont modélisées en utilisant une loi de comportement élasto-viscoplastique basée sur la surface de limiteélastique ellipsoïde et la théorie de viscoplasticité en surcontrainte de Perzyna. Le rapport de l’allure de la surfacelimite pour le sol de fondation de Gloucester est estimé au moyen de résultats d’essais conventionnels de cisaillementet de consolidation en laboratoire. L’on compare les comportements calculé et mesuré au cours des essais triaxiauxconsolidés dans des conditions isotropes et non drainées (CIU) et des essais de consolidation à long terme dans lacellule de Rowe, et l’on étudie l’habilité du modèle à décrire le comportement mesuré des étapes 1 et 2 du remblaid’essai de Gloucester. Cet article explore les implications de la modélisation des propriétés résiduelles et restructuréesdu sol de fondation de Gloucester et démontre l’habilité d’un simple modèle de surface limite élasto-viscoplastiquepour décrire la réaction non drainée et drainée du sol de fondation de Gloucester dans des conditions de chargement enlaboratoire et sur le terrain.

Mots clés: surface limite ellipsoïde, sensibilité à la vitesse de déformation, élasto-viscoplastique, tassements de rem-blais, pressions interstitielles, performance sur le terrain.

[Traduit par la Rédaction] Hinchberger and Rowe 789

The prediction of excess pore pressures, undrained defor-mations, and long-term settlements of embankments con-structed on soft cohesive soil deposits can be difficult whenembankment loads cause significant yielding within thefoundation soil (e.g., Bozozuk and Leonards 1972; Massa-chusetts Institute of Technology 1975). Nonlinearelastoplastic yield-surface models (e.g., Roscoe andSchofield 1963; Roscoe and Burland 1968; Chen 1982) haveenabled researchers to achieve considerable success in pre-dicting embankment behaviour (e.g., McCarron and Chen1987); however, as illustrated by Rowe et al. (1996), forsome soils these elastoplastic models are not sufficient to ac-

curately model embankment behaviour since, they do notconsider soil rheology and consequent rate effects. Rheologymodels such as the Gibson and Lo model (1961) enable en-gineers to predict rates of secondary compression (Lo et al.1976), and Redman and Poulos (1984) showed that aviscoelastic–plastic model could describe undrained creepduring short-term embankment loads. There is, however, anapparent need to have an integrated approach to predictingembankment performance which allows for the consider-ation of yielding, rate effects, consolidation, and creep andto test this formulation against field behaviour.

The performance of stage 1 of the Gloucester test em-bankment (constructed in 1967) has been well documentedin the literature (e.g., Bozozuk and Leonards 1972), and sev-eral researchers have investigated the behaviour of the Glou-cester foundation soil in undrained and drained laboratorytests (e.g., Law 1974; Lo et al. 1976; Leroueil et al. 1983).In June 1982, the second stage of construction for the Glou-cester test fill was commenced and the measured responsewas the subject of a Workshop on the Prediction of the Engi-neering Behaviour of the Second Stage of the Gloucester

Can. J. Geotech. J.35: 769–789 (1998) © 1998 NRC Canada

769

Received September 13, 1996. Accepted June 12, 1998.

S.D. Hinchberger. Acres International Ltd., Niagara Falls,ON L2E 6WI, Canada.R.K. Rowe. Department of Civil and EnvironmentalEngineering, The University of Western Ontario, London,ON N6A 5B9, Canada.

Test Fill in 1986. Fisher et al. (1982) successfully made aclass A prediction of the behaviour of stage 2 of the Glou-cester test embankment; however, the method of analysisused for the prediction incorporated a great deal of engineer-ing judgement.

This present paper has three objectives: (i) to study theability of an elastoviscoplastic yield-surface model to de-scribe the behaviour of the Gloucester foundation soil duringlong-term Rowe cell oedometer tests, and during consoli-dated isotropically undrained (CIU) triaxial constant rate ofstrain compression tests; (ii ) to study the ability of anelastoviscoplastic yield-surface model to describe the un-drained and drained response of stages 1 and 2 of the Glou-cester test embankment; and (iii ) to explore the implicationsof modelling the residual or restructured properties of theGloucester foundation soil. A new and important aspect ofthis paper is the demonstrated ability to describe the drainedand undrained response of the Gloucester foundation soilduring both field and laboratory loading conditions with asingle constitutive model.

A detailed subsurface profile for the Gloucester founda-tion soil has been presented in the literature by Bozozuk andLeonards (1972). In general, the Gloucester foundation soilis a very soft to firm intermediate- to high-plasticity siltyclay. In this paper, the rate-sensitive characteristics of theGloucester foundation soil were modelled using Perzyna’s(1963) theory of elastoviscoplasticity wherein the yieldfunction of classical plasticity is coupled with a time-raterule.

The theoretical formulation adopted is summarized in theAppendix and the basic concepts are illustrated in Fig. 1. An

elliptical cap yield function (Chen 1982) was used to defineyielding in the normally consolidated range. Ifσmy

(s) is theyield-surface intercept with the mean effective stress,′σm,axis ( ′σm = ( ′σ1 + ′σ2 + ′σ3 )/3), the elliptical cap is describedby the location of the centre of the ellipse at′σm = l (seeFig. 1), the yield-surface intercept,σmy

(s) , and the yield-surface aspect ratio,Rc, in (2J2)

1/2– ′σm space, whereJ2 =1/6[(σ11 – σ22)

2 + (σ11 – σ33)2 + (σ22 – σ33)2] +

(σ12)2 + (σ23)

2 + (σ31)2 is the second invariant of the stress

tensor. The expansion of the yield surface is a time-dependent process which is related to the viscoplastic volu-metric strain that occurs during a time increment (seeeq. [A7] in the Appendix). By definition, the static yield sur-face defined byσmy

(s) corresponds to the locus of yieldstresses for infinitely slow loading of the soil. For a nor-mally consolidated soil, any load that is not applied infi-nitely slow may cause the stress state in the soil to changefrom the initial stress state, such as point A in Fig. 1, to an-other stress state on the dynamic yield surface (e.g., point Bin Fig. 1). The dynamic yield surface is defined byσmy

(d) asshown in Fig. 1. This point (point B) is assumed to be over-stressed relative to the static yield surface by an amountσos

(d),which is measured in (2J)1/2– σm

′ stress space, and the dy-namic yield surface is used to define the potential resultantof the viscoplastic strain-rate vector. While a state of over-stress exists, the soil will undergo viscoplastic straining andconsequent time-dependent deformation which is propor-tional to the amount of overstress,σos

(d), relative to the staticor long-term yield curve. Given sufficient time, the full plas-tic strains associated with conventional plasticity theory willbe achieved.

The theoretical basis for elastoviscoplastic constitutivemodels has been described extensively in the literature (e.g.,Adachi and Okano 1974; Adachi and Oka 1982; Katona

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770 Can. J. Geotech. J. Vol. 35, 1998

Fig. 1. Overstress measurement and geometry of elliptical yield surface. N/C, normally consolidated; O/C, overconsolidated.

1984; Kavazanjian et al. 1985; Oka et al. 1986; Desai andZhang 1987). Only basic concepts with respect to the consti-tutive model used are provided above to allow for morecomplete comparison of calculated and measured soil behav-iour. The elastoviscoplastic elliptical cap model used in thispaper most closely resembles the constitutive models ofAdachi and Oka (1982) and Katona (1984). However, in thepresent formulation, an elliptical yield surface (Chen 1982)has been used to define the stress states at yield rather thanthe Original Cam-Clay yield surface adopted by Adachi andOka (1982). This modification was introduced to accommo-date the fact that the yield surface of clayey soils tends tovary from soil to soil (see Mitchell 1970; Tavenas andLeroueil 1977; Lew 1981; Folkes and Crooks 1985). The as-pect ratio of the elliptical yield surface can be changed toaccount for this variability. An elliptical yield surface wasalso used by Katona (1984), but the viscoplastic flow ruleadopted for the present formulation (see eq. [A6]) is closerto the flow rule proposed by Adachi and Oka (1982) thanthat adopted by Katona. For the present formulation, theflow rule of Adachi and Oka was modified to better suit theelliptical yield surface based on laboratory studies of theyielding behaviour of natural soils (Hinchberger 1996).

The elliptical cap elastoviscoplastic model described inthe Appendix was coupled with Biot consolidation theoryand incorporated into the finite-element program AFENAoriginally developed by Carter and Balaam (1990) and sub-sequently modified by the authors to allow modelling ofelastoviscoplastic behaviour. Details of the finite-elementimplementation are summarized in the Appendix and are de-scribed in more detail by Hinchberger (1996). In addition,Rowe and Hinchberger (1998) have demonstrated the abilityof the elastoviscoplastic constitutive model to describe therate-sensitive behaviour of the Sackville foundation soil.

One-dimensional consolidationFigure 2 shows a typical consolidation curve for the Glou-

cester foundation soil. The Gloucester foundation soil has anoverconsolidated range and a relatively well defined precon-solidation pressure that is dependent on strain rate duringtesting (Leroueil et al. 1983). The normally consolidatedrange may be divided into two distinct phases as suggestedby Silvestri (1984a) and shown in Fig. 2. The first phase is atransitional phase where the natural cemented structure ofthe Leda Clay is broken down. The second phase is the re-structured phase. For the purpose of modelling the deforma-tion characteristics of the Gloucester foundation soil thenormally consolidated phase was considered to be bilinear(see Fig. 2). This will be discussed further when comparingthe calculated and measured behaviour of stage 2 of theGloucester test embankment.

Behaviour during long-term Rowe cell consolidation

Selection of material propertiesLo et al. (1976) published the results of long-term Rowe

cell consolidation tests for Gloucester foundation soil fromdepths of 2.4 and 4.2 m, respectively (see Fig. 3). These

tests were modelled using the elastoviscoplastic model (seeAppendix) by discretizing the soil sample using cubic straintriangles and assuming axisymmetric conditions.

The hydraulic conductivity,k, of the Gloucester founda-tion soil was considered to be dependent on the void ratio asfollows:

[1] k ke eCk

= −

o

oexp

whereeo is the reference void ratio,ko is the hydraulic con-ductivity at the reference void ratio,e is the void ratio, andCk is the slope of thee – log (k) plot. Equation [1] was alsoused by Mesri and Choi (1979) to describe the change in hy-draulic conductivity with void ratio for the Gloucester foun-dation soil usingCk = 1.0. A value ofCk = 0.25 estimatedfrom the results of hydraulic conductivity tests at depths of2.13, 2.74, and 3.96 m (summarized in Table 1) was adoptedin this paper. The hydraulic conductivity test results listed inTable 1 were obtained at the National Research Council ofCanada (NRC). The value ofCk was found to be highly vari-able below 5.2 m; however, the hydraulic conductivity testsat 2.13, 3.74, and 3.96 m are more applicable to the long-term laboratory consolidation tests on soil from a depth of4.2 m. In addition toCk = 0.25,ko = 2.5 × 10–9 m/s was se-lected for both test simulations so that the calculated time to90% dissipation of excess pore pressures was 350 min as re-ported by Lo et al. (1976). This initial hydraulic conductiv-

© 1998 NRC Canada

Hinchberger and Rowe 771

Fig. 2. Typical oedometer consolidation behaviour of Gloucesterfoundation soil. Bilinear approximation in normally consolidatedstress range.

ity also compares favourably with the results of hydraulicconductivity tests (see Table 1) for soil from these depths.

The recompression index,κ, was estimated to be 0.025 forthe Gloucester foundation soil based on consolidation curvespublished by Leroueil et al. (1983). The compression index,λ, was selected to be 0.65 based on the results of conven-tional oedometer consolidation tests on soil from compara-ble depths(see Fig. 4). The slope of the failure envelope((2J2)

1/2– σm′ stress space) in the normally consolidatedstress range was taken to beM = 0.9, with a cohesion inter-

cept of ′cκ = 0 based on the failure envelope obtained byLaw (1974) for the Gloucester foundation soil. The aspectratio, Rc = 1.65, of the elliptical yield surface was selectedon the basis of both long-term consolidation tests and CIUtriaxial tests which will be discussed in more detail in a latersection. Table 2 contains a list of all material constants usedto calculate the long-term Rowe cell consolidation behaviourof the Gloucester foundation soil, and Table 3 summarizesthe stress increments used by Lo et al. (1976) and sample in-formation.

© 1998 NRC Canada


Initial Final

Depth(m)

Water contentw (%)

Voidratio e

Hydraulicconductivity k(×109 m/s)

Water contentw (%)

Voidratio e

Hydraulicconductivityk (×109 m/s) Ck

2.13 88.5 2.39 2.7 76.4 2.06 0.9 0.302.74 74.7 2.02 2.0 62.6 1.69 0.4 0.213.96 85.4 2.31 2.0 66.5 1.80 0.1 0.195.18 64.4 1.74 1.1 61.5 1.66 0.9 0.545.64 59.0 1.59 0.6 56.8 1.53 0.6 0.547.38 101.6 2.74 2.5 97.8 2.64 1.8 0.298.53 94.0 2.54 1.4 91.3 2.47 1.3 1.11

10.36 94.2 2.54 1.6 91.9 2.48 0.9 0.1112.19 82.8 2.23 1.2 82.1 2.21 0.9 0.0713.41 47.2 1.27 1.6 46.0 1.24 1.3 0.1515.24 56.7 1.53 1.4 55.7 1.50 0.8 0.0617.68 64.3 1.74 1.8 62.7 1.69 1.7 0.7320.73 30.6 0.82 2.7 29.6 0.80 2.7 0.0

Note: Hydraulic conductivityk was derived using eq. [1], andCk is the slope of thee – log (k) plot.

Table 1. Hydraulic conductivity test results at initial and final in situ stresses for stage 1 of the Gloucester test embankment (datasupplied by NRC).

Fig. 3. Comparison of calculated and measured compressive strains during long-term Rowe cell oedometer tests.

Comparison of calculated and experimental behaviourFigure 3 illustrates the general ability of the elastovisco-

plastic elliptical cap model (withRc = 1.65) to describe thesecondary compression behaviour of the Gloucester soil dur-ing long-term Rowe cell consolidation tests. At a depth of2.4 m, the elliptical cap constitutive model predicted moredelayed compression in the early stages of the long-termRowe cell consolidation test than that of the measured re-sponse. This difference in the early stages of the test is smalland may be the result of differences between the assumedand actual yield-surface shape for the Gloucester soil, andthe calculated and actual stress path during the test. There isbetter agreement between calculated and measured behav-

iour for soil from a depth of 4.2 m, indicating that the differ-ence between calculated and measured behaviour for soilfrom 2.4 m may also be the result of soil variability.

The yield-surface intercepts,σmy(s) , which resulted in the

best fit to the measured behaviour were 54 kPa for soil from2.4 m and 62 kPa for soil from 4.2 m. A fluidity constant,γvp, of 1.67 × 10–10 s–1 was inferred for soil from both depthsbased on the fit of Rowe cell consolidation test data and CIUtriaxial shear test results that will be discussed in a later sec-tion. The strain-rate parameter,n, was selected so that thecalculated rates of secondary compression matched the mea-sured rates (n is inversely proportional to the rate of second-ary compression). Furthermore, the yield-surface interceptsof 54 and 62 kPa are less than the preconsolidation pressuresof 62 and 69 kPa reported by Lo et al. (1976) for conven-tional oedometer consolidation on soil from 2.4 and 4.2 m,respectively. This is consistent with the concepts of overstressviscoplasticity, since the field preconsolidation pressuresgenerally correspond to infinitesimal or very slow strain rates,whereas laboratory preconsolidation pressures are typicallymeasured at higher strain rates.

Leroueil et al. (1983) established that the measured pre-consolidation pressure of the Gloucester foundation soil wasstrain-rate dependent, and the results published by both Loet al. (1976) and Leroueil et al. (1983) show that rate effectsare less significant for stress states below yield. For stressstates that exceed yield, the behaviour of the Gloucesterfoundation soil during long-term consolidation tests wasfound to be dominated by secondary compression (Lo et al.1976). Furthermore, the yield pressure of the Gloucesterfoundation soil was found to be strongly influenced by strainrate (Leroueil et al. 1983). This provides some justificationfor the use of elastoviscoplastic theory to describe the be-haviour of the Gloucester soil.

The analysis of long-term Rowe cell consolidation testsusing the elliptical cap elastoviscoplastic constitutive modelsuggests that the field yield-surface intercept of the Gloucesterfoundation soil is marginally less than the preconsolidationpressure measured during conventional oedometer consoli-dation. On average, the preconsolidation pressure obtainedfrom conventional incremental oedometer consolidation testswas found to be approximately 14% greater than the staticyield-surface intercept back-calculated using the elliptical

© 1998 NRC Canada


Poisson’s ratio,ν 0.3Viscoplastic fluidity constant,γvp (s–1) 1.67 × 10–10

Strain-rate exponent,n 30.0Reference or initial void ratio,eo 1.8Hydraulic conductivty at reference void ratio,ko (m/s) 2.5 × 10–9

Slope ofe – log (k) plot, Ck 0.25Recompression index based one – ln σ curve,κ 0.025Compression index based one – ln σ curve,λ 0.65

Table 2. Assumed material properties for long-term Rowe cell consolidation simulations.

Sampletype

Sample size(mm)

Depth(m)

Natural moisturecontent,wn (%)

Preconsolidation pressure,σp′ (kPa)

Stress increment(kPa)

Block 150 × 50 2.4 66.1 62.0 42.9–90.0Osterberg 112.5 × 50 4.2 70.6 69.0 48.2–92.3

Table 3. Sample information and stress increments for Rowe cell consolidation tests.

Fig. 4. Compression index,λ, versus depth. Data supplied bythe National Research Council of Canada (NRC).

cap elastoviscoplastic model withRc = 1.65. Figure 5 showsthe apparent preconsolidation pressure versus strain ratemeasured by Leroueil et al. (1983) for soil from a depth of4.05–4.15 m. Also included in Fig. 5 is the preconsolidationpressure reported by Lo et al. (1976) for soil from a depth of4.2 m which lies within the range of preconsolidation pres-sures obtained by Leroueil et al. (1983) for incrementaloedometer consolidation. The strain-rate dependency of theGloucester foundation soil during laboratory consolidationtests is evident in Fig. 5 and the position of the field yield-surface intercept for the elliptical cap constitutive model(Rc = 1.65) is also indicated in Fig. 5. Silvestri (1984b)showed that a simple power law (Norton 1929) could beused to correct preconsolidation pressures measured athigher strain rates during constant gradient and constant rateof strain consolidation tests back to the preconsolidationpressures obtained from conventional oedometer consolida-tion at lower strain rates. Figure 5 suggests thatpreconsolidation pressures obtained during incrementaloedometer consolidation may also require a correction forthe difference between laboratory strain rates and field strainrates.

The foregoing discussion and comparisons provide somebasis for using the elliptical cap elastoviscoplastic constitu-tive equation to model the long-term behaviour of the Glou-cester test embankment. The model can adequately describethe secondary compression behaviour of the Gloucesterfoundation soil during long-term Rowe cell consolidationtests. Recognizing that the elliptical cap elastoviscoplasticconstitutive model implies the presence of a fieldpreconsolidation pressure that is marginally lower than thepreconsolidation pressure measured during incrementaloedometer consolidation, a factor of 1/1.14 will be applied

to the laboratory oedometer preconsolidation pressure to es-timate the field yield-surface intercept. It is now necessaryto justify the use of an aspect ratio,Rc, of 1.65 for the ellip-tical cap model.

Strain-rate behaviour during undrained unconsolidatedtriaxial shear

Figure 6 shows the measured peak and postpeak strengthsof the Gloucester foundation soil during unconsolidated un-drained (UU) triaxial tests at different strain rates (Law1974). The figure illustrates two important properties of theGloucester soil, namely (i) the dependency of both the peakand postpeak strength on the axial strain rate, and (ii ) thetransition from a strain-softening to a strain-hardening be-haviour as the axial strain rate is reduced. The increasedshear strength with increased strain rate illustrates the con-cept of overstress viscoplasticity. As the strain rate in-creases, the amount of overstress above the long-term(static) shear strength increases. The shear strength of theGloucester foundation soil decreases with diminishing strainrate to a value of 13 kPa at a strain rate of 0.0001%/min.Stage 1 of the Gloucester test embankment was constructedto a height of 2.4 m above the original ground level during a10 day period (Bozozuk and Leonards 1972). Leroueil et al.(1983) estimated from field data that the in situ strain ratewithin the foundation soil at the end of construction was ap-proximately 2 × 10–5%/min, and about 2 × 10–6%/min at 1.5years after the end of construction. The low in situ strainrates combined with Fig. 6 make it reasonable to assumethat the strain-hardening range of behaviour is operative inthe field during construction of the Gloucester test embank-

© 1998 NRC Canada


Fig. 5. A comparison of the preconsolidation pressures measured by Leroueil et al. (1983) and Lo et al. (1976) with the field (static)yield-surface intercept for the Gloucester foundation soil.

ment. The strain-softening behaviour of the Gloucester foun-dation soil appears to be strain-rate induced and Gloucestersoil from a depth of 2.4 m tends towards the residual state(exhibiting no peak–postpeak response) during UU triaxialtests at axial strain rates less than approximately 0.01%/min.Lo and Morin (1972) found that Leda clay from SaintVallier and Saint Louis also tended toward the residual orrestructured state as strain rates became small during consol-idated anisotropically undrained (CAU) and consolidatedisotropically drained (CID) triaxial tests.

The intercept of the elliptical yield surface (withRc = 1.65and σmy

(s) = 54 kPa) with the failure envelope is plotted inFig. 6. The static yield-surface intercept,σmy

(s) = 54 kPa, wasinferred from the long-term Rowe cell consolidation test forsoil from a depth of 2.4 m (see Fig. 3). The measured resid-ual undrained shear strengths during UU triaxial tests tendtoward the theoretical long-term shear strength correspond-ing to the intersection of the elliptical cap (Rc = 1.65) withthe failure envelope. Based on the case records reported byBozozuk and Leonards (1972), Lo et al. (1976), and Fisheret al. (1982), the range of in situ mean effective stress at adepth of 2.4 m is approximately 20–30 kPa. The ellipticalyield surface (Rc = 1.65 andσmy

(s) = 54 kPa) intersects thefailure envelope at a mean effective stress of 21.7 kPa. Thusit is reasonable to assume that the effective stress path dur-ing UU triaxial tests at a depth of 2.4 m will tend toward astress state at failure which is similar to the intersection ofthe yield surface with the failure envelope.

Lastly, the measured residual shear strength versus strainrate data in Fig. 6 was also fit with Norton’s (1929) law,viz.,

[2] &ε γ11 =

vp u(d)

u(s)

c

c

n

where n is the strain-rate exponent (see Fig. 6),cu(d) is the

strain-rate-dependent undrained shear strength, andcu(s) is the

undrained shear strength corresponding to&ε γ11 = vp. It is im-portant to note that the strain-rate parameter,n = 30, re-quired to fit results of the long-term Rowe cell consolidationfrom a depth of 2.4 m is the same as the strain-rate parame-ter, nR = 30, obtained using Norton’s law (1929) to fit the re-sidual undrained shear strength versus strain-rate data (seeFig. 6). Hinchberger (1996) showed that eq. [2] may be usedto estimate the viscoplastic constants for the elliptical capelastoviscoplastic constitutive equation using the plot of un-drained shear strength versus strain rate.

Behaviour during CIU triaxial shearFigure 7 shows the comparison between CIU triaxial com-

pression test results for the Gloucester foundation soil froma depth of 2.4 m with the calculated behaviour using ellipti-cal cap viscoplasticity (Rc = 1.65). The consolidation phaseof the CIU tests was 24 h (Law 1974) and the consolidationpressures exceeded the measured preconsolidation pressureof the Gloucester soil. Compression occurred at an axialstrain rate of 0.017%/min. The consolidation phase of theCIU tests was modelled numerically for axisymmetric con-ditions with radial drainage. The slope of the failure enve-lope,M, in (2J)1/2– ′σm stress space was taken to be 0.9 witha cohesion of ′cκ = 0 based on the results of Law (1974). Foreach CIU triaxial compression test, the initial position of thestatic yield surface was taken to be 54 kPa based on the re-sults of fitting the long-term Rowe cell consolidation test re-sults for soil from a depth of 2.4 m. The positions of theyield surface, void ratio, and stress state at the end of the24 h consolidation phase were calculated and then used asinitial input parameters for the CIU triaxial test simulations.Modelling the consolidation phase of the CIU triaxial com-

© 1998 NRC Canada


Fig. 6. The effects of strain rate on the peak and postpeak shear strength of Gloucester foundation soil (Law 1974); constant rate ofstrain UU triaxial tests.wN, natural moisture content;wL, liquid limit; IP, index of plasticity.

pression tests using the elliptical cap elastoviscoplastic con-stitutive equation accounts for the effects of secondarycompression and aging on the position of the yield surface atthe end of the 24 h consolidation phase.

Referring to Fig. 7, it can be seen that an elliptical capwith an aspect ratio,Rc, of 1.65 adequately describes the be-haviour of the Gloucester foundation soil during CIU triaxialcompression tests. The calculated axial stress versus axialstrain response for tests corresponding to consolidation pres-sures of 83 and 103 kPa is close to the measured responseup to failure. The calculated peak shear strength is approxi-mately 3% below the measured peak shear strength at a cellpressure of 83 kPa and 8% greater than the measured peakshear strength at a cell pressure of 103 kPa.

The theoretical pore pressures are generally close to themeasured behaviour up to failure. Failure was defined byLaw (1974) to be (σ1 – σ3)max. At failure, the calculated porepressures peak while the measured pore pressures continuedto increase. The increase in pore pressure for strains in ex-

cess of the failure strains (see Fig. 7) cannot be predictedusing the elliptical cap viscoplastic model.

Finite-element discretizationThe finite-element mesh used to model stages 1 (1967)

and 2 (1982) of the Gloucester test embankment consisted of800 six-noded linear strain triangles (each with three inte-gration points) and 1702 nodes. Figure 8 shows the geome-try and finite-element mesh used to model stages 1 and 2 ofthe Gloucester test embankment. As previously noted, thedetailed subsurface profile may be found in publications byBozozuk and Leonards (1972), Lo et al. (1976), and Fisheret al. (1982). The foundation soil was modelled as an elasto-viscoplastic material coupled with Biot consolidation theoryunder plane-strain conditions. Based on field data, a rigidpermeable boundary condition was assumed at a depth of20.2 m. The foundation soil was modelled 75 m beyond theembankment centreline, and at this location a smooth rigidpermeable boundary was assumed. The interface betweenthe foundation and embankment soils was considered to bepurely cohesive and modelled using six-noded zero thick-ness joint (or slip) elements. The finite-element analysis ofthe Gloucester test embankment, however, showed that slipat this interface did not occur.

The excavation portion of stage 1 construction was simu-lated by removing excavated elements from the finite-elementmesh and incrementally removing the nodal forces associatedwith the internal stresses and unit weight of the elements.Similarly, construction of the embankment was simulated byadding rows of elements and incrementally increasing thebody forces due to gravity within these elements. The timeincrements used to march the solution forward in timeranged from 0.25 h during the short-term undrained condi-tions to 10 h near the end of stage 1 monitoring.

Material properties

Embankment fillThe embankment fill was modelled as an elastoplastic

(inviscid) material with a Mohr–Coulomb failure criterionand a flow rule of the form proposed by Davis (1968).Based on the results of Bozozuk and Leonards (1972), thefill material was assumed to be cohesionless with an effec-tive friction angleφ′ = 35°, a dilation angleψ = 0°, and aunit weight equal to 18.4 kN/m3. The Young’s modulus ofthe fill was assumed to be dependent on the minor principalstress and described using the equation of Janbu (1963), viz.,

[3]EP

KP

m

aE

a

=

σ3

whereE is the Young’s modulus of the soil,Pa is the atmo-spheric pressure,σ3 is the minor principal stress, andKE(250) andm (0.5) are based on Rowe et al. (1984).

Hydraulic conductivity profileThe hydraulic condictivity profile used to obtain the cal-

culated behaviour of the Gloucester test embankment iscompared with the measured hydraulic conductivities prior

© 1998 NRC Canada


Fig. 7. Comparison of calculated and measured behaviour duringCIU triaxial shear.

to construction of stage 1 in Fig. 9. A hydraulic conductivityvalue that was intermediate to the field and laboratory testmeasurements was assumed for much of the deposit. Higherconductivities for the upper 4 m of the foundation depositwere investigated, since laboratory tests indicated that thehydraulic conductivity near the foundation surface may besignificantly higher than those measured from field tests atgreater depths. The results of this investigation showed thatthe agreement between calculated and measured behaviourwas adequate over the range of measured conductivity val-ues. The assumed profile of initial vertical hydraulic conduc-tivity as shown in Fig. 9 was found to provide the best fit tothe measured field behaviour.

The decrease in hydraulic conductivity with decreasingvoid ratio was described using eq. [1] with a value ofCk of0.25 for the entire deposit, as previously discussed. Al-though Table 1 shows that the value ofCk is variable fordepths below 5.2 m, a constant value of 0.25 was adoptedbased on the hydraulic conductivity tests that exhibited asignificant change in void ratio between the initial and finalconsolidation pressures. The anisotropic characteristics ofthe Gloucester foundation soil were modelled and the ratioof horizontal to vertical hydraulic conductivity,kh/kv, wastaken to be 1.25 based on data supplied by NRC.

Groundwater conditionsBozozuk and Leonards (1972), among others, suggested

that the groundwater table was 1.6 m below the originalground surface. This may have been the case prior to con-struction and during the early stages of stage 1; however, theavailable case records indicate that the groundwater tablewas about 0.4 m below the original ground level during both

© 1998 NRC Canada


Fig. 9. In situ hydraulic conductivity measurements andlaboratory hydraulic conductivity results (data supplied by NRC).

Fig. 8. Geometry and finite-element mesh for stages 1 and 2 of the Gloucester test embankment.

stage 1 and stage 2 of the Gloucester test embankment con-struction (Fisher et al. 1982), and this assumption wasadopted for the present analysis.

Consolidation characteristicsThe assumed compression index,λ, profile is plotted in

Fig. 4 with the results from incremental oedometer consoli-dation tests. The assumed profile ofλ with depth is essen-tially an average of the measured values supplied by NRC.The recompression or swelling index,κ, was taken to be0.025 for the entire deposit. The value of 0.025 was selectedbased on the consolidation curves published by Leroueil et al.(1983), and a sensitivity study showed that the assumed valueof κ had only a small effect on the calculated settlements.

Figure 10 shows the measured preconsolidation pressureprofile (laboratory measurements by NRC) and the assumedyield-surface profile for the analysis of the Gloucester testembankment. The oedometer preconsolidation pressureswere corrected for strain-rate effects by applying the factor1/1.14 as discussed previously. Thus, the calculated 14% dif-ference between the measured oedometer preconsolidationpressures and the field static yield-surface intercept for theelliptical cap elastoviscoplastic model is reflected in Fig. 10.Table 4 lists all material constants used in the analysis ofstages 1 and 2 of the Gloucester test embankment.

′Ko conditionsThe selection of the coefficient of earth pressure at rest

K′o profile was based on the results of hydraulic fracturingtests by Bozozuk (1974).Based on this, the upper 5.2 mof the foundation wasassigned a ′Ko value of 1.0, and avalue of 0.8 was adopted for the remainder of the deposit. A

′Ko value of 0.8 for the entire deposit was also investigatedand found to have only a small influence on the calculatedsettlements for stage 1.

Failure envelopeBased on the research of Law (1974) for Gloucester foun-

dation soil from a depth of 2.4 m (see also Fisher et al.1982), the slope of the failure envelope in (2J)1/2– σm′ stressspace was taken to beM = 0.9 (; φ′ = 25°) with a cohesionintercept ′cκ = 0 (see Appendix) for the entire foundationsoil deposit.

Calculated behaviour

SettlementsThe calculated settlements at gauges S1, S2, S3, and S4

beneath the embankment centreline are compared with the

measured settlements in Fig. 11, and Fig. 12 shows the samecomparison for settlement gauges S6, S7, and S8 beneaththe embankment shoulder. Beneath the embankment centre-line the calculated settlements compare favourably with themeasured settlements. The settlements are slightly overesti-mated at gauges S1 and S2 and those at gauge S3 are under-estimated. This may be attributed to variations in the soilprofile. Beneath the embankment shoulder the calculatedsettlements also compare favourably with the measured set-tlements. At the location of settlement gauge S6, the calcu-lated settlement 5000 days after the end of construction is19 cm compared with a measured settlement of approxi-mately 21 cm. Although there are no long-term settlementdata available for gauge S7, the initial performance is closeto the calculated behaviour. In general, it was found that theelliptical cap elastoviscoplastic constitutive model ade-quately described the settlement behaviour of stage 1 of theGloucester test embankment. Fluctuations in the groundwa-

© 1998 NRC Canada


Fig. 10. Preconsolidation pressure versus depth (data providedby NRC).

Depth (m) κ λ eo ko (×10–9 m/s) Ck kh/kv ν γ (×10–10 s–1) n Rc M

0.0–2.0 0.025 0.65 1.8 1.2 0.25 1.25 0.3 1.67 30 1.65 0.92.0–5.2 0.025 0.65 1.8 1.2 0.25 1.25 0.3 1.67 30 1.65 0.95.2–7.2 0.025 0.32 1.8 1.0 0.25 1.25 0.3 1.67 30 1.65 0.97.2–13.4 0.025 1.35 2.4 1.0 0.25 1.25 0.3 1.67 30 1.65 0.913.4–20.2 0.025 0.72 1.8 1.2 0.25 1.25 0.3 1.67 30 1.65 0.9

Note: kh, horizontal hydraulic conductivity;kv, vertical hydraulic conductivity;Rc, yield-surface aspect ratio;M, slope of the failure envelope in(2J2)

1/2–σm′ stress space. Other symbols as defined in Table 3.

Table 4. Material properties assumed for analysis of the Gloucester test embankment.

© 1998 NRC Canada


Fig. 12. Predicted and measured displacements beneath shoulder for stage 1 of the Gloucester test embankment. Elliptical cap model.S6–S8, settlement gauges.

Fig. 11. Predicted and measured displacements at centreline for stage 1 of the Gloucester test embankment. Elliptical cap model (Rc =1.65). S1–S4, settlement gauges.

ter conditions and moisture content of the embankment fillduring the case history can account for the fluctuations inthe field measurements.

Lateral displacementsFigure 13 shows the relative horizontal displacements at

the embankment toe at the end of construction and 3.75years after the end of construction. At the end of construc-tion, the calculated horizontal displacement at the groundsurface is approximately 0.95 cm and the measured lateraldeformation is approximately 1 cm. The calculated horizon-tal displacements exceed the measured displacements be-tween 1.7 and 5.8 m below the original ground level, and theagreement between calculated and measured horizontal dis-placement at depths greater than 5.8 m is good. At 3.75years, the calculated lateral displacement at the toe of theembankment is comparable to the measured lateral displace-ments. The calculated displacements agree with measureddisplacements up to 4 m below the original ground surface.The measured displacements exceed calculated displace-ments at depths greater than 4 m. In general, the agreementbetween the calculated and measured horizontal displace-ment profile at the embankment toe is considered to be ac-ceptable.

Zones of soil at yieldFigure 14 shows the zones of normally consolidated

(yielded) soil 13 years after the end of construction. At thistime, the zones of yielded soil include much of the founda-tion layer directly beneath the test embankment between 1.2and 5.2 m below the original ground level and a portion ofthe soil at a depth greater then 5.2 m. The yielding of thelower zones at a depth greater than 5.2 m is consistent withthe findings of Lo et al. (1976), and Fig. 14 indicates thatthe calculated behaviour of the Gloucester test embankmentis dominated by yielding of the upper 5.2 m of the founda-tion soil.

Centreline settlementsThe geometry of stage 2 of the Gloucester test embank-

ment is shown in Fig. 8. Construction of stage 2 was simu-lated over the course of 5 days in accordance with theconstruction records (supplied by NRC). The prediction offield settlements for stage 2 is complicated by the bilinearconsolidation characteristic of the Gloucester foundation soilin the normally consolidated stress range and the lack of rawconsolidation data for the case history. The consolidationdata published by Leroueil et al. (1983) indicate that thetransition phase of the consolidation curve (see Fig. 2) oc-curs over a stress range of approximately 25–30 kPa. To ap-proximate the increased stiffness in the restructured range ofthe consolidation behaviour, the compression index,λ , inthe restructured range was taken to be one half of the com-pression index in the transitional phase based on the consoli-dation curves published by Leroueil et al. (1983).

Figure 15 shows the comparison of calculated settlementswith measured settlements beneath the embankmentcentreline for stage 2. The calculated settlement at the sur-face for a transitional stress range of 30 kPa is 41.5 cm com-pared with the measured settlement of approximately36.0 cm. This represents a relative error of approximately13% for the surface settlements during stage 2; however, in-cluding the deformations recorded during stage 1 of the casehistory, the relative error between measured and calculatedsettlements at centreline is approximately 7%. The calcu-lated settlements up to 1 year after the end of constructionare very close to the measured settlements. The calculatedsettlements exceed the measured settlements for timesgreater than 1 year. The rates of settlement are close to themeasured rates of settlement during the first year after con-struction; however, 4 years after the end of construction, therates of settlement and the compressive strains in the upper1.45 m of the deposit are slightly overestimated at gauge S1.

Also shown in Fig. 15 are the calculated centreline settle-ments assuming a transitional stress range of 25 kPa. Thesettlements at the surface decrease to 39 cm for a transi-tional stress range of 25 kPa compared with 41.5 cm for atransitional stress range of 30 kPa and the measuredcentreline settlement of 36 cm. At a depth of 1.45 m, thecalculated settlements are essentially the same regardless ofthe assumed transitional stress range, and the calculatedcompressive strains in the upper 1.45 m of the soil depositare closer to the measured compressive strains. The relative

© 1998 NRC Canada


Fig. 13. Lateral displacements at embankment toe for stage 1 ofthe Gloucester test embankment.

error between calculated and measured settlement 4 yearsafter the end of construction of stage 2 is approximately 8%.The calculated rates of settlement for a transitional stressrange of 25 kPa are in good agreement with the measuredrate of settlement for settlement gauges S1 and S2 beneaththe embankment centreline.

The aspect ratio,Rc, of 1.65 used above was selectedbased on two CIU compression tests at a depth of 2.4 m and

may not be representative of the entire soil deposit. To eval-uate the influence of the yield-surface aspect ratio,Rc, on thecalculated settlements, analyses were also performed assum-ing Rc = 1.55 for stages 1 and 2 and transitional stressranges of 25 and 30 kPa, respectively. Figure 15 shows thatthe centreline settlements decrease with decreasing aspectratio, Rc. For a transitional stress range of 25 kPa, the calcu-lated settlement at centreline at 4 years is 35 cm forRc =

© 1998 NRC Canada


Fig. 14. Zones of yielded soil 13 years after the end of construction for stage 1 of the Gloucester test embankment.

Fig. 15. Vertical deformations beneath centreline for stage 2 of the Gloucester test embankment. S1 and S2, settlement gauges;Rc*,

yield surface aspect ratio.

1.55 compared with 39 cm forRc = 1.65. For a transitionalstress range of 30 kPa, the calculated settlement at centrelineat 4 years was 38.5 cm forRc = 1.55 and 41 cm forRc =1.65. The measured settlement at 4 years (after the end ofconstruction) was 36 cm.

In general, the elliptical cap elastoviscoplastic constitutivemodel is able to describe the settlement behaviour of stages1 and 2 of the Gloucester test embankment for an aspect ra-tio, Rc, of 1.65 and transitional ranges of 25 and 30 kPa. Theaspect ratio of the elliptical yield surface had only a smalleffect on the calculated settlement at centreline for the casesstudied. The best agreement between calculated and mea-sured behaviour was obtained for an aspect ratio,Rc, of 1.55and a transitional stress range of 25 kPa; however, all calcu-lated centreline settlements are adequate when comparedwith the field measurements. The transitional stress rangehas only a small effect on the settlement recorded at gaugeS2 and improved the agreement between calculated andmeasured compression in the upper 1.45 m of the deposit.

Surface-settlement profileThe calculated surface-settlement profiles corresponding

to Rc = 1.65 and a transitional stress range of 25 kPa areshown in Fig. 16. The calculated surface settlements aregenerally in agreement with the measured surface settlementprofiles 1 month, 1 year, and 4 years after the end of con-struction; however, the settlements beneath the embankmentcrest are overestimated at 1 year and 4 years. Given the un-certainties associated with modelling stage 2 of the Glouces-ter test embankment, the calculated settlement behaviour ofstage 2 of the Gloucester test embankment is considered tobe good.

Excess pore pressuresFigure 17 shows the comparison of calculated and mea-

sured excess pore pressures at the end of construction and1 month after the end of construction. The backgroundgroundwater pressures at the site of the Gloucester test em-bankment were monitored during stage 2 with a series ofpiezometers installed 35 m from the location of the test em-bankment. This series of piezometers indicated that thebackground pore-pressure distribution remained relativelyconstant during the 4 year period after the end of construc-tion. At the end of construction, the calculated excess porepressures are in good agreement with the IRAD vibratingwire measurements and exceed the Geonor hydraulicstandpipe measurements in the upper 5.2 m of the founda-tion. In the lower portion of the soil deposit the calculatedpore pressures exceed both the hydraulic standpipe and vi-brating wire measurements. The hydraulic standpipepiezometers appear to suffer from lag, as the measurementsare lower than the vibrating wire measurements at all depthsimmediately after the end of construction and are in goodagreement with the vibrating wire measurements 1 month af-ter the end of construction.

One month after the end of construction the calculated ex-cess pore pressures agree with both the hydraulic andvibrating-wire measurements at all depths. Comparing theexcess pore pressures at the end of construction and 1 monthafter the end of construction shows an increase in the mea-sured excess pore pressures during this time. The field mea-surements show a maximum increase of approximately0.8 m from the end of construction to 1 month after the endof construction. The case records indicate that maximum re-corded pore pressures beneath the embankment centreline

© 1998 NRC Canada


Fig. 16. Comparison of calculated and measured surface settlement profile 1 month, 1 year, and 4 years after the end of constructionfor stage 2 of the Gloucester test embankment.

occurred in most cases approximately 1 month after the endof construction. The calculated excess pore pressures in-crease by approximately 1 m from the end of construction to1 month after the end of construction. Furthermore, the cal-culated maximum excess pore pressures occurred approxi-mately 1 month after the end of construction.

Hydrodynamic lag cannot account for the magnitude ofpore-pressure increases that occurred during this time pe-riod. Rather, the majority of the increased pore pressuresduring the initial undrained behaviour of the test embank-ment result from time-dependent plasticity. Figure 18 showscontours of overstress within the yield zone at the end ofconstruction. This overstress represents the potential forplastic strain to occur due to the construction of stage 2 ofthe Gloucester test embankment. Excess pore pressures areinduced as time-dependent plastic deformations occur andthe stress states relax toward the field (static) yield surface.

Figure 19 shows the calculated and measured excess porepressure heads 1 year and 4 years after the end of construc-tion of stage 2. The calculated excess pore pressures exceedthe measured excess pore pressures at all depths. The dissi-pation of excess pore pressures between 1 year and 4 yearsis closely predicted by the method of analysis, and the com-puted distribution of excess pore pressures agrees with thehydraulic standpipe measurements 4 years after the end ofconstruction of stage 2. The IRAD vibrating wire pressureheads are considerably lower than the more reliable hydrau-lic standpipe measurements 4 years after the end of con-struction.

Lateral displacementsThe comparison of measured and calculated horizontal

displacements at the embankment toe and shoulder 1 monthafter the end of construction are shown in Fig. 20. The cal-culated lateral displacements exceed the measured horizontaldisplacements beneath both the embankment shoulder andthe toe. This trend was also observed at the end of construc-tion, 1 year and 4 years after the end of construction. In gen-eral, 1 month after the end of construction both measuredand calculated horizontal displacements are small.

The ability of the elastoviscoplastic constitutive modelused in this paper to describe the secondary compression be-haviour of the Gloucester foundation soil during long-termRowe cell consolidation tests has been demonstrated. An el-liptical cap with an aspect ratio,Rc, of 1.65 could also ade-quately describe the behaviour of the Gloucester foundationsoil during CIU triaxial tests. Since the consolidation stageof the CIU triaxial tests exceeded the preconsolidation pres-sure of the Gloucester soil, the Rowe cell consolidation be-

© 1998 NRC Canada


Fig. 17. Comparison of calculated and measured excess porepressures (a) immediately after construction, and (b) 1 monthafter the end of construction.

Fig. 18. Contours of overstress at the end of construction ofstage 2 of the Gloucester test embankment. O.G.L., originalground level.

haviour and CIU triaxial test behaviour correspond to therestructured or residual soil fabric. The viscosity constantscalculated from the results of long-term Rowe cell consoli-dation tests were used to consider the effect of strain rateand duration of consolidation during the CIU triaxial testsusing the elliptical cap model. The aspect ratio of the ellipti-cal cap was selected so that the calculated behaviour duringCIU triaxial compression tests matched the measured behav-iour. Also, the theoretical long-term shear strength for an el-liptical cap with Rc = 1.65 was found to be comparable tothe undrained shear strength during UU triaxial tests con-ducted at very slow strain rates.

A number of researchers have investigated the in situ andlaboratory yielding of cohesive soils (see Mitchell 1970;Tavenas and Leroueil 1977; Lew 1981; Folkes and Crooks1985). In general, the existence of a yield surface for cohe-sive soils is accepted; however, the available literature indi-cates that the measured or apparent shape of the yieldsurface varies from soil to soil. The elliptical cap yield sur-

face was adopted for the analysis presented in this paper,since the aspect ratio,Rc, can be used to vary the yield-surface shape. The present study has extended the calibra-tion process described by Chen (1982) for elliptical yieldsurfaces to include the effect of time-dependent plastic de-formations. The viscoplastic constants used in this papercorrespond to the residual or restructured soil fabric and canadequately describe the strain-rate-dependent behaviour ofthe residual shear strength during UU triaxial tests as well asthe behaviour during CIU triaxial tests at consolidationstresses that exceed the preconsolidation pressure of theGloucester foundation soils.

The elliptical cap model withRc = 1.65 and other parametersbased on laboratory data were successfully used to predictthe behaviour of two stages of construction (stage 1 in 1967and stage 2 in 1982) of the Gloucester test embankment,generally giving good agreement between the calculated andmeasured settlements. During stage 1, the calculated hori-zontal displacements at the toe of the test embankment arein agreement with the measured horizontal displacements.The field measurements indicated that stage 2 of the test em-bankment performed slightly better than calculated. Forstage 2 of the Gloucester test embankment, the method ofanalysis closely described the surface-settlement profiles,centreline settlements, and the excess pore pressure responsebeneath the embankment centreline and embankment toe.Settlements beneath the embankment crest and excess porepressures beneath the embankment crest were consistentlyoverestimated.

© 1998 NRC Canada


Fig. 19. Comparison of measured and calculated excess porepressures (a) 1 year and (b) 4 years after the end ofconstruction.

Fig. 20. Comparison of calculated and measured horizontaldisplacements for stage 2, 1 month after the end of construction.

It should be noted that the impact of stage 1 settlementson the embankment geometry prior to stage 2 was not con-sidered (e.g., small strain analysis). Furthermore, the case re-cords (NRC) indicate that the embankment crest was gradedto provide drainage. It may be that the fill thickness at thecrest is less than the fill thickness at centreline as a result ofthe stage 1 settlements and grading of the embankment toprovide drainage. Differences between the assumed embank-ment geometry and actual embankment geometry may ac-count for some of the difference between calculated andmeasured behaviour. Uncertainty associated with the stressstate within the embankment fill just prior to stage 2 and theembankment loads transferred to the foundation may alsoaccount for a significant portion of the difference betweencalculated and measured behaviour. In addition to uncertain-ties associated with the embankment geometry and embank-ment loads, anisotropy of the Gloucester foundation soilmay have had an impact on the embankment performanceand may also account for some of the difference betweencalculated and measured behaviour. In particular, the agree-ment between calculated and measured horizontal displace-ments during stage 2 is limited (see Fig. 20). Given thebetter agreement between the calculated and measured hori-zontal displacements for stage 1 (see Fig. 13), some aniso-tropy may have been induced during consolidation understage 1 loads. This induced anisotropy may contribute tosome of the observed difference between calculated andmeasured horizontal displacements during stage 2. Given theuncertainties associated with predicting embankment perfor-mance, however, the agreement between calculated andmeasured behaviour is encouraging and the test embankmentperformed marginally better than calculated.

The results of this study indicate that the behaviour of theGloucester foundation soil during laboratory and field testscan be described using a single elastoviscoplastic yield-surface model. Evidence of the presence of a fieldpreconsolidation pressure (yield surface) which is less thanthe measured preconsolidation pressure has been presented,providing additional support for the concepts proposed byBjerrum (1967). Based on the available data, it would ap-pear that the Gloucester foundation soil exhibits a strain-softening response only during UU triaxial compressiontests for axial strain rates in excess of approximately0.001%/min. At axial strain rates lower than 0.001%/min,the Gloucester foundation soil behaved as a strain-hardeningmaterial (Law 1974). Leroueil et al. (1983) estimated thatthe in situ compressive strain rate within the foundation soilsfor stage 1 of the Gloucester test embankment was20 10 5. %× − /min at the end of construction and20 10 6. %× − /min at 1.5 years after the end of construction.As a result it was assumed that the strain-hardening behav-iour was operative in the field and that the Gloucester foun-dation soil tends toward the residual or restructured state inthe field. Considering the good agreement between calcu-lated and measured behaviour, this assumption appears to besupported by the present analysis.

Based on the results presented in this paper, the followingconclusions are made:

(1) Much of the behaviour of the Gloucester foundationsoil during laboratory and field loading conditions can beadequately described using yield-surface concepts and over-stress viscoplasticity.

(2) An elliptical yield surface with an aspect ratio ofabout Rc = 1.65 appears to be representative of the fieldyield surface of the Gloucester foundation soil.

(3) Both experimental and analytical evidence has beenpresented to show that the aspect ratio of the field yield sur-face can be estimated from conventional laboratory testsprovided that due consideration is given to strain-rate ef-fects. It may be possible to infer the yield-surface aspect ra-tio from a standard incremental oedometer consolidation testand triaxial compression test at similar strain rates. This ap-proach, however, warrants further study.

(4) The in situ field preconsolidation pressure of theGloucester foundation soil may be slightly overestimated us-ing conventional laboratory tests, since the strain rates arehigher than those expected in the field.

(5) The increase in excess pore pressures during the ini-tial month after the end of construction of stage 2 of theGloucester test embankment can be explained and is de-scribed well by time-dependent plasticity and yield-surfacetheory.

The research reported in this paper was funded by a re-search grant from the Natural Sciences and Engineering Re-search Council of Canada.

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The basic elastoviscoplastic formulation adopted in thispaper is summarized below. Full details are given byHinchberger (1996). The strain-rate tensor for anelastoviscoplastic material is expressed as

[A1] & & &ε ε εij ij ij= +e vp

where &εije and &εij

vp are the elastic and viscoplastic strain-ratetensors, respectively. For stress states below yield, the elasticbulk modulus,K, was considered to be stress dependent andgiven by

[A2] K e= + ′( )1

σκm

whereκ is the recompression index,e is the void ratio, andσm

′ = (σ1 + σ2 + σ3)/3 is the mean effective stress. The shearmodulus, G, is related to the bulk modulus,K, througheq. [A3]:

[A3] Gv Kv

= −+

31 221( )

( )

© 1998 NRC Canada


whereν is Poisson’s ratio and was assumed to remain con-stant. The viscoplastic strain-rate tensor is expressed as

[A4] & ( )ε γσij

ij

fvp vp F= ∂∂ ′

φ

whereγvp is the viscoplastic fluidity parameter with units ofinverse time, the scalar functionΦ(F) is called the flowfunction, and∂f /∂ ′σij is the derivative of the plastic potentialfor associated viscoplasticity. The elliptical cap yield func-tion (Chen 1982) was used to define yielding in the nor-mally consolidated stress range, and the equation of theelliptical cap in (2J2)

1/2– ′σm stress space is

[A5] ( ) ( ) ( )f (s)c my

(s)= ′ − + − − =σ σm l J R l2

22

22 0

where l is the ′σm coordinate of the centre of the ellipse, isthe yield-surface intercept with the′σm axis,J2 is the secondinvariant of the deviatoric stress tensor, and′σm is the meaneffective stress. The geometry of the elliptical yield surfaceis shown in Fig. 1. The elastoviscoplastic constitutive equa-tion associated with the elliptical cap yield surface in thenormally consolidated stress range is

[A6] &&

( )

&ε κ σσ

δijij

ijs

G e= +

+′′2 31m

m

+ − −

′

γδ

σσσ

δvp

my(d)

my(d)

m

Fφ( ) ij l3

21

2

ij

3

−′

+′

RJ s Rij ij

cm

c

m

2 2

222

3

2

σδ

σ

Φ(F) my(s)

os(d)

my(s)

=+

σ σσ

n

wheresij is the deviatoric stress tensor,G is the shear modu-lus, κ is the recompression index,e is the void ratio,δij isKronecker’s delta,γvp is the viscoplastic fluidity constant,nis the strain-rate exponent,σmy

(s) is the static yield-surface in-tercept,σmy

(d) is the dynamic yield-surface intercept,′σm is themean effective stress,Rc is the yield-surface aspect ratio in(2J2)

1/2– ′σm stress space,Φ(F) is the viscoplastic flow func-tion, andσos

(d) is the overstress (see Fig. 1).For yielding in the normally consolidated stress range,

overstress,σos(d), was calculated as the orthogonal distance

between parallel yield-surface tangents in (2J2)1/2– ′σm stress

space. At failure, the overstress,σos(d), was a function of the

second invariant of the deviatoric stress tensorJ2. The ex-pansion of the yield surface is a time-dependent process de-pending on the current viscoplastic increment of volumetricstrain,∂εvol

vp , as governed by

[A7] ∂ = +−

∂σλ κ

σ εmy(s)

my(s)

volvp( )1 e

Failure in both the normally consolidated andoverconsolidated stress ranges has been defined using aDrucker–Prager failure envelope:

[A8] f M c J(s)m= ′ + ′ − =σ κ 2 02

whereM is the slope of the failure envelope in (2J2)1/2– σm

′

stress space,′cκ is the cohesion intercept,′σm is the mean ef-fective stress, andJ2 is the second invariant of the deviatoricstress tensor.

Viscoplastic stress and strain incrementsThe viscoplastic strain-rate vector, {&εvp}, at time stepm +

1 can be obtained using Taylor series and ignoring higher or-der terms:

[A9] { & } { & }&

{ }ε ε εσ

σvp vpvp

m m

m

m+ = + ∂∂ ′

1 ∆

= +{& } [ ] { }ε σvpm m mH ∆

where {∆σ} m is the vector of stress increment, and [H]m de-notes the gradient of the viscoplastic strain-rate vector withrespect to effective stress at time m. The viscoplastic strainincrement {∆εvp} during the time interval∆tm = tm+1 – tm canbe written as follows:

[A10] [ ]{ } ( ){ & } { & }∆ ∆ε θ ε θ εvp vp vpm m m mt= − + +1 1

whereθ is a numerical constant. Forθ = 0, the simple Eulerscheme is obtained (Zienkiewicz and Cormeau 1974). Forthe analyses presented in this paper, an incremental ap-proach to the solution of elastoviscoplastic continuum prob-lems was adopted usingθ = 0.5 with small time stepincrement sizes. The time step requirements for stabilitypublished by Cormeau (1975) and Owen and Hinton (1980)were found to be helpful guidelines for stability of the solu-tions obtained.

The stress increment vector, {∆σ} m, can be written as fol-lows:

[A11] { ∆σ} m = [Ce]{ ∆εe} m = [Ce]({ ∆ε} m – {∆&εvp} m)

where {∆ε} m = [B]m{ ∆a}, [ B] is the strain-displacementtransformation matrix, and {∆a} is the incremental vector ofnodal displacements. Substitution of eq. [A10] intoeq. [A11] gives

[A12] { }∆σ m

[ ]= + −−

[ ] [ ]( [ ] ) [ ]([ ] { } { & } )I C t H C B a tm m m m m me e vpθ ε∆ ∆ ∆

1

where [I] is the identity matrix, [Ce] is the elastic constitu-tive matrix,θ is a numerical constant (0.5),∆tm is the currenttime increment, [H]m is the derivative of the viscoplasticstrain-rate vector with respect to stress, [B]m is the strain-displacement transformation matrix, {∆a} is the vector ofnodal displacements, and {&εvp} m is the current viscoplasticstrain-rate vector.

© 1998 NRC Canada


© 1998 NRC Canada


Coupled finite-element equationsApplying the principle of virtual work to the equilibrium

equation and dropping the incremental subscript, m, givesthe following relationship for the load increment:

[A13]{ } { } { } { } { } { }

{ }

VT

VT

VT

V

V V u V

d

∫ ∫ ∫∫

= ′ +

=

δε σ δε σ δε

δ

∆ ∆ ∆d d dw

Ts

TF V d s{ } { } { }∆ ∆b d d+ ∫ δ T

where {δε} is the incremental strain vector, {∆σ} is the in-cremental total stress vector, {∆σ′} is the incremental effec-tive stress vector, {∆uw} is the incremental pore-waterpressure vector, {δd} is the vector of incremental displace-ments, {∆Fb} is the incremental body force vector, and {∆T}is the applied surface load incremental vector. Within a typi-cal element, the variation of incremental displacement is

[A14] { δd} = [ N]{ δa}

where [N] is the matrix of shape functions, and {δa} is theincremental nodal displacement vector. Applying the sameshape functions, [N], the variation of excess pore-water pres-sures within a typical element are

[A15] { ∆uw} = [ N]{ ∆b}

where {δb} are the incremental nodal values of excess porepressure. The strain increment vector, {∆ε}, and the incre-mental volumetric strain vector, {∆v}, within a typical ele-ment are given by

[A16] { ∆ε} = [ B]{ ∆a}

and

[A17] { ∆v} = { m} T{ ∆ε} = { m} T [B]{ ∆a}

where [B] is the strain-nodal displacement transformationmatrix, and {m} is defined as follows:

[A18] { }m =

1

1

1

0

Substituting eqs. [A14]–[A17] into eq. [A13] results inthe following finite-element equations:

[A19] [ ]{ } [ ]{ } { }k a L b Fe ∆ ∆ ∆+ =

where

[A20] [ ] [ ] [ ][ ]K B C B VV

Te e d= ∫[A21] [ ] [ ] { }[ ]L B m N VT

V= ∫ d

[A22] { } [ ] { }∆ ∆F N F VTV

= ∫ b d

+ +∫ ∫[ ] { } { }N T s VTs V R∆ ∆d dσ

In eq. [A22], {∆σR} is the vector of viscoplastic relaxationstresses defined by

[A23] [ ]{ } [ ] { & }∆ ∆σ εRTB C t= vp

and

[A24] [ ] [ ]C I C tH C= +−

e e( ) [ ]θ∆1

Applying a similar virtual work principle to the two-dimensional equation of continuity results in the followingexpression:

[A25] { }vol

w

w

w

w d(∫ ∂∂

+ ∂∂

+ ∂∂

δ

γ γb

k ux

k ux

vt

112

112

332

332

vol) = 0

wherek11 is the hydraulic conductivity in the one-coordinatedirection, k33 is the hydraulic conductivity in the three-coordinate direction,γw is the unit weight of the pore water,v is the seepage velocity, andt is the time. The gradient ofexcess pore pressure is given by

[A26]

∂∂∂∂

=

ux

ux

E b

w

w

11

33

[ ]{ }∆

where the terms of the matrix [E] are obtained by differenti-ating the shape functions [N]. Substituting eqs. [A14]–[A17]and [A26] into eq. [A25] and solving gives

[A27] [ ]{ }

{ } [ ] { }Lat

b N v ATk

Ts n

dd

d− = ∫Φ

where

[A28] L B m N vTv

= ∫ [ ] [ ]d

[A29] Φk

T

v

E k EV= ∫ [ ] [ ][ ]

γw

d

andvn is the prescribed boundary seepage velocity,and [k] isthe hydraulic conductivity matrix

[A30] [ ]kk

k=

11

330

0

Equation [A27] is a first-order differential equation whichmay be integrated with respect to time to give

[A31] [ ] { } ( ){ } { }[ ]L a b t bT

K∆ Φ ∆− − + +1 β β( ) (t t

= [ ] ( )N v t v t t t AT

s n n∫ − + +( ){ ( )} { ( )}1 β β ∆ ∆ d

where the value ofβ is a numerical constant and defines thevariation of excess pore pressure,b, during the time interval.Booker and Small (1975) have shown that for unconditionalstability, β ≥ 0.5. A value ofβ = 1 was adopted based onBritto and Gunn (1987), reducing eq. [A31] to

[A32] [ ] { } { }L a t btk∆ Φ ∆ ∆−

= + + =∫Φ ∆ ∆ ∆ ∆KT

s n st b L v t t t A F{ } [ ] ( ) { }d

Equations [A19] and [A32] can be used to establish a so-lution at t + ∆t from the solution at timet. To summarize,the fully coupled finite-element equations can be written as

© 1998 NRC Canada


[A33][ ]

[ ]

[ ] { }

{ }

{ }

{ }

K

L

L

t

a

b

F

FTk s

e

−

=

Φ ∆∆∆

∆∆

The implementation of the coupled elastoviscoplastic con-stitutive behaviour presented above is similar to the coupledfinite-element implementation described by Britto and Gunn(1987); however, [Ke] is the elastic element stiffness matrix,and {∆F} contains additional terms related to the integrationof the viscoplastic relaxation stresses defined by eqs. [A23]and [A24].

Material propertiesThe material constants for the elastoviscoplastic constitu-

tive model can be subdivided into three categories as fol-lows: constants associated with conventional elastoplasticanalysis (κ, λ, v, Rc, M, σmy

(s) , ande), hydraulic conductivityconstants (kh, kv, eo, and Ck), and viscoplastic constants (nand γvp). The estimation of conventional elastoplastic con-

stants and hydraulic conductivity constants has been de-scribed in the literature by McCarron and Chen (1984, 1987)and Britto and Gunn (1987) and will not be described here.For the present formulation, two additional material con-stants (the strain-rate exponent,n, and the fluidity parame-ter, γvp) have been introduced to describe the time-dependentplastic (viscoplastic) deformations. These viscoplastic mate-rial constants can be estimated by fitting long-term consoli-dation test results using trial-and-error procedures. Theresults of this fitting procedure have been presented in thepresent paper for the Gloucester foundation soil. Alterna-tively, eq. [2] can be used to fit the results of conventionalundrained triaxial tests (e.g., UU or CIU) performed at dif-ferent strain rates. This latter approach can be used to pro-vide an estimate of the strain-rate exponent,n, and fluidityparameter,γvp, for use with the elastoplastic constitutivemodel described above. It is recommended, however, thatlong-term consolidation tests be used to verify and refine thematerial constants obtained using eq. [2].

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