modelling glacial till under triaxial conditions using a brick soil model

Modelling glacial till under triaxial conditionsusing a BRICK soil model

Barry M. Lehane and Brian Simpson

Abstract: The paper presents some findings from recent laboratory research aimed at improving ground-movement pre-dictions in a hard, heavily overconsolidated lodgement till. These findings are subsequently applied to a new three-dimensional version of the BRICK soil model to investigate the suitability of this model for the till. It is shown thatBRICK captures the essential features of the material’s behaviour under triaxial conditions and is capable, when incor-porated in a finite element code, of providing realistic predictions for the settlement of shallow foundations.

Key words: glacial till, stiffness, strength, BRICK model, footing settlement.

Résumé: Cet article présente des résultats d’une recherche récente en laboratoire visant à améliorer les prédictions dumouvement de terrain dans un till hôte raide et fortement surconsolidé. Ces résultats sont subséquemment mis en appli-cation dans une nouvelle version 3-D du modèle de sol BRICK pour étudier si ce modèle convenait au till. On montreque BRICK reproduit les caractéristiques essentielles du comportement du matériau dans des conditions triaxiales et estcapable, lorsqu’incorporé dans un code d’éléments finis, de fournir des prédictions réalistes du tassement de fondationssuperficielles.

Mots clés: till glaciaire, rigidité, résistance, modèle BRICK, tassement de semelle.

[Traduit par la Rédaction] Lehane and Simpson 1088

Introduction

The glacial till discussed in this paper underlies much ofthe Greater Dublin area in Ireland and is referred to locallyas Dublin black boulder clay (DBC). The material’s rela-tively high strength and stiffness, which is typical of manylodgement tills worldwide, has enabled most structures inDublin to be founded on shallow footings or short drivenpiles. However, as more challenging geotechnical designsbecome commonplace, there has been a need to gain an im-proved understanding of the mechanical characteristics ofthe till. Proposed new deep basements and tunnels have re-quired finite element analyses, and consequently there hasbeen a need to select and use an appropriate constitutivemodel for the till.

A laboratory testing programme was put in place with oneof its primary aims being to establish the stiffness character-istics of the till at low and intermediate strain levels. Thefirst part of this paper presents some typical data andgeotechnical parameters obtained during this programme.The nature of these data prompted the authors to investigatethe suitability of the BRICK soil model for the till.

The paper then presents a new generalized three-dimensional (3D) formulation for BRICK; this model is anextension of the original plane strain model summarised bySimpson (1992). The stiffness data measured in triaxial tests

on the till are compared with this formulation (in itsaxisymmetric form) to assess the potential of the model formovement predictions. As an illustrative example, predic-tions made in finite element analyses using BRICK 3D arecompared with settlement measurements obtained in a loadtest on a shallow footing founded on the till.

Geology

Dublin boulder clay (DBC) was deposited beneath an icesheet that covered much of Ireland during the Pleistoceneperiod. The grinding action of this sheet as it eroded the un-derlying carboniferous limestone coupled with itspreconsolidation effect resulted in the formation of a verydense or hard low-permeability deposit which contains occa-sional pockets and lenses of coarse gravel, particularly atdepth. DBC is often found at or close to ground level and istypically 10–15 m in thickness and underlain by relativelyintact, homogeneous limestone.

No significant chemical weathering has taken place otherthan that in the top 2–3 m of the stratum where oxidation ofthe iron content has resulted in a change in colour fromblack to brown. The properties of this (lower strength)brown boulder clay are not discussed here.

Geotechnical properties of DBC

CompositionThe low permeability (which is often less than 1 × 10–10 m/s)

and high gravel content of DBC have resulted in the till be-ing referred to as a boulder clay. However, the clay fractionof the material is only between about 12 and 20% and thefines content (<63µm) is typically between 40 and 50%.

Can. Geotech. J.37: 1078–1088 (2000) © 2000 NRC Canada

1078

Received May 17, 1999. Accepted December 20, 1999.Published on the NRC Research Press website on October 10,2000.

B.M. Lehane. Department of Civil Engineering, TrinityCollege Dublin, Dublin 2, Ireland.B. Simpson.Arup Geotechnics, London W1P 6BQ, U.K.

I:\cgj\Cgj37\Cgj05\T00-032.vpWednesday, October 04, 2000 12:22:38 PM

Color profile: Generic CMYK printer profileComposite Default screen

Significantly, Hartford (1987) showed that the clay fractioncontains a negligible proportion of clay minerals and con-sists predominantly of angular calcite and quartz “rock flour.”

Typical classification index properties for the black(unweathered) till are provided in Table 1. The Atterberglimits plot as a low-plasticity clay on the Casagrandeplasticity chart. It is noteworthy that these limits, althoughindicative of the engineering behaviour of the till, are deter-mined on the fraction passing the 0.425 mm sieve, whichrepresents only about 60% of the material.

Sampling and in situ testing in DBC are made difficult bythe presence of cobbles and boulders. Cone penetration test(CPT) values of end resistance (qc) and standard penetrationtest (SPT)N values are typically 15 ± 5 MPa and 50 ± 30,respectively. These very highqc and N values coupled withthe low liquidity indices are indicative of a very hard ordense overconsolidated deposit.

In situ stressNo reliable measurements of the in situ horizontal stresses

have been made due to the unsuitability of a self-boringpressuremeter in DBC. However, overconsolidation ratios(OCRs) inferred from in situ and laboratory test data are inexcess of 20 in the top 5 m of thestratum, suggesting thatthe coefficient of earth pressure at rest (Ko) is typically at orclose to the passive pressure coefficient (Kp). Fissures, indi-cating passive failure of DBC, have been observed byHanrahan (1977), but these usually take the form of micro-cracks and are not anticipated to have an important influenceon the mass behaviour of the material.

CompressibiltyTypical examples of the behaviour of DBC in one-

dimensional (1D) compression are provided in Fig. 1, whichplots the variations of void ratio (e) with vertical effectivestress (σv′) recorded in four standard oedometer tests: (i) arotary-cored sample of DBC from 5 m depth, (ii ) a reconsti-tuted mix of the same sample from 5 m excluding the sandand gravel fraction and mixed at an initial water contentequal to the liquid limit, (iii ) a rotary-cored sample from7.5 m, and (iv) a driven 100 mm diamter tube sample from12 m. These and other compression curves were correctedcarefully for system compliance in the oedometer because ofthe very stiff nature of DBC.

Although there is a shortage of test results at the very highstress levels, the vertical yield stresses (σvy′ ) can be identi-fied approximately by plotting compression data such asthose in Fig. 1 using a log(1 +e) versus logσv′ representa-tion (Butterfield 1979); theseσvy′ values are typically 1.3 ±0.3 MPa and do not display a systematic variation withdepth. The mean compressibility parameters,λ and κ , de-rived from Fig. 1 and other 1D compression curves for DBCare listed in Table 1. Isotropic compression and swellingtests on reconstituted DBC indicatedλ andκ values whichwere the same as the equivalent reconstituted 1D values.

It is evident from Fig. 1 and the parameters in Table 1 thatthe 1D compressibility of reconstituted DBC is higher thanthat of the intact material, particularly during recompression.Lower recompression indices are often exhibited by naturalsoil samples possessing interparticle bonding (e.g., seeBurland 1990). Such bonding is not, however, considered tobe significant for DBC, as the measuredκ values were rela-tively insensitive to the sampling technique and maximumapplied stress. In addition, direct shear tests on high-quality,300 mm square block samples of DBC did not reveal evi-dence of any significant effective cohesion (c ′) componentof strength (Treacy 1996). The difference between the com-pressibility of the intact and reconstituted material is thoughtto be a result of the high gravel content of the (small)oedometer samples of intact DBC and the complex cyclicdepositional history of lodgement tills. The gravel presenthas a much lower saturation water content than the remain-der of the “matrix” material and consequently led to the in-ference of a lowe value (which varies with gravel content)from the overall water content determination. Gens andHight (1979) observed similar features in another glacial tilland concluded that the water content of the matrix materialwas a better indicator of the mass strength and stiffness.

Triaxial shear stiffnessLocal strain instrumentation was used in triaxial tests on

rotary-cored samples of DBC to provide information on thesmall strain stiffness. The axial strain (εa) instrumentation

© 2000 NRC Canada

Lehane and Simpson 1079

Bulk density (kN/m3) 21.5±0.5Global water content (%) 11±3Liquid limit (%) 25±4Plastic limit (%) 14±2Plasticity index (%) 11±2Liquidity index –0.2±0.03Clay fraction (%) 15±5Gravel fraction (%) 30±5Permeability (m/s) 1×10–11 to 1×10–8

Coefficient of consolidation,cv (m2/year) 40±20Preconsolidation stress,σvy′ (MPa) 1.3±0.3In situ OCR 8–30λ (reconstituted, intact) 0.04, 0.030±0.005κ (reconstituted, intact) 0.008, 0.004±0.001

Table 1. Typical properties of Dublin boulder clay. Fig. 1. Behaviour of DBC in 1D compression.



system employed was similar to that described byCuccovillo and Coop (1997) and comprised submersible dis-placement transducers mounted on miniature pedestalspinned to the membrane. Radial strain belts with displace-ment transducers were used to monitor radial strain (ε r ).The resolution of both axial and radial strains was assessedto be at least 0.01%.

As expected, these triaxial tests revealed a strong depend-ence of the secant shear stiffness (Gsec) on the levels oftriaxial shear strain (εs = 2/3(εa – ε r )) and mean effective

stress (p ′) but also exhibited a significant dependence onthe direction of the previous stress path. A sample from theavailable database of small-strain triaxialGsecmeasurementson intact DBC is summarised in Fig. 2. All of these datapoints are representative of measurements made in drainedprobing tests on rotary-cored samples after they had beensubjected to initial isotropic or anisotropic consolidation andswelling. The initial stress state of all samples was at, orclose to, an isotropic condition and maximum appliedconsolidation stresses were less than 400 kPa. Figure 2 plots

© 2000 NRC Canada

1080 Can. Geotech. J. Vol. 37, 2000

Fig. 2. Dependence of normalized shear stiffness onθ for rotary-cored DBC samples.



the variation ofGsec normalized by the current values ofp ′at variousεs values with the rotation (θ) of the stress pathdirection from the previous direction. The value ofθ isdefined in the inset diagrams in Fig. 2 and represents the ro-tation of the stress path in thep ′–q plane (whereq is the de-viator stress). AsGsecdoes not vary in proportion top ′ at allstrains (and typically for DBC in proportion to (p ′)1/2 at lowstrains; see Faulkner 1998), Figs. 2a and 2b display datameasured over the relatively narrowp ′ ranges of 50–150and 150–250 kPa, respectively.

The data in Fig. 2 are confirmation of the very stiff natureof DBC. Lehane and Farrell (1997), for example, show thattheGsec/p ′ ratios are over three times higher than equivalentratios for (high plasticity) London Clay. The strongnonlinearity of the shear stiffness of DBC is also evident,with Gsec/p ′ values typically reducing by a factor of five asεs increases from 0.01 to 0.1%. In addition,Gsec/p ′ ratios re-duce significantly as the rotation from the previous directionof loading (θ) reduces. The dependence onθ is most pro-nounced at lower shear strain levels, e.g.,Gsec at εs = 0.01%reduces almost six-fold with a reduction inθ from 180° to30°. The dependence of the nonlinearity of shear stiffnesswith strain level depends on the value ofθ, with greatestnonlinearity occurring after a complete reversal in loadingdirection (i.e., whenθ is 180°).

Strength of intact DBCIntact DBC in shear exhibits a response typical of a

heavily overconsolidated low-plasticity material. For exam-ple, samples tested in isotropically consolidated undrainedtriaxial compression (CIUC) experience an initial increase inpore pressure until peak obliquity (φp′) is developed; porepressures subsequently fall until the peak deviator stress (qf)and maximum mean effective stress (p f′ ) are attained at crit-ical state conditions when the friction angleφ′ = φcv′ .

A Hvorslev-type normalization of strength data usingequivalent effective stresses on the critical state line of DBCdid not reveal a unique bounding surface. This is most prob-ably because the unknown saturation water content of thegravel within samples (and the unknown gravel content, in

many cases) meant that the void ratio of the (controlling)matrix material could not be inferred to an acceptable accu-racy. Effective stress strength data are therefore representedin Fig. 3 using secant peak (φp′) and constant-volume (φcv′ )friction angles, assumingc′ of zero. The angles plotted weremeasured in CIUC tests on 100 mm diameter rotary-coredsamples of DBC (from between 5 and 20 m depth) and areplotted against the initial consolidation stress (p0′ ) in Fig. 3along with φp′ values measured in a number of drainedtriaxial compression tests.

It is evident thatφcv′ is relatively constant for all samplesand its mean value of 32° compares well with the criticalstate angle of 34 ± 1° shown by reconstituted samples(Lehane and Faulkner 1998). This relatively high frictionangle is typical of a well-graded granular deposit and is con-sistent with the deficiency of clay minerals in the clayfraction. φp′ exhibits a relationship with logp0′ typical oflow-plasticity soil (e.g., see Been and Jefferies 1985) andapproachesφcv′ when p0′ is in excess of 1 MPa.

The undrained strength ratios (=cu /p0′ ) and pore-pressureparameters at failureaf (defined as (p0′ – p f′ ) /qf ) are plot-ted againstp0′ in Fig. 4 for the same series of CIUC tests.These variations are typical of a critical state material andmay be expressed in the form of the following standard rela-tionships (see Muir Wood 1990) with the isotropic OCR,np =pmax′ /p0′ :

[1]c

p

M n

ru p

02′

=⋅ Λ

Λ

[2] a

n

r

Mf

p

=

−

−Λ

1

whereM = qf /pf′, Λ = 1 –κ /λ, andr is the ratio ofp ′ on thenormal consolidation line (pmax′ ) to p ′ at the intersection ofthe critical state line and the unload–reload line passingthrough pmax′ . The mean trend lines shown in Fig. 4 werederived takingM = 1.27 (; φcv′ = 32°) andr = 1.6 (assessedfor reconstituted DBC by Lehane and Faulkner 1998) andestimating a best fitΛ value of 0.7 andpmax′ (in kPa) from

[3] pmax′ = 1000 + 25z

wherez is the sample depth in metres.Equation [1] implies an undrained strength ratio of 0.45

for normally consolidated DBC. This ratio is 50% higherthan the average value of 0.3 quoted by many workers fornormally consolidated clays tested in triaxial compression.Lehane and Faulkner (1998) show that the ratio of 0.45 atnp =1 also arises in reconstituted DBC due to the tendency of thenormally consolidated material to dilate as shear failure isapproached.

Equation [3] implies a meanpmax′ value of 1300 kPa forthe database included in Fig. 4; this value, which may be an-ticipated from Fig. 3, is compatible with the preconsolida-tion inferred from oedometer tests (e.g., Fig. 1). The spreadof results about the mean trend lines in Fig. 4 is also inkeeping with the variable dilatant tendencies observed intests on reconstituted DBC.

© 2000 NRC Canada


Fig. 3. Friction angles of rotary-cored DBC samples.



Modelling DBC using BRICK

The insensitive nature of DBC and the strong dependenceof stiffness on the previous direction of loading, as seen inFig. 2, prompted the authors to investigate the suitability ofthe BRICK constitutive model (Simpson 1992) for DBC.

The BRICK 3D modelBRICK was originally formulated to model the depend-

ence of stiffness on the previous stress path direction.Simpson (1992) published a two-dimensional (2D) versionof the concept, which has since been extended to 3D. Themodel is essentially elastic–plastic with multiple kinematicyield surfaces. These are expressed in strain space, which isconsidered to provide a more convenient and possibly morefundamental framework than the more usual use of stressspace. Simpson (1992) describes the model by analogy to a“man” walking around in strain space and pulling behindhimself a set of bricks on strings; some possible paths forthe man and strings are shown in Fig. 5a. The man repre-sents the current strain state of an element of soil, and thebricks represent theplastic current strains in proportions of

the soil within the element. When the man moves withoutmovement of the bricks, the strain is entirely elastic; whenall the bricks are moving by the same amount as the man,the behaviour is entirely plastic. As strains increase, it is as-sumed that an increasing proportion of them is plastic; thisleads to the “S-shaped” or “backbone” curve of normalizedtangent stiffness against strain shown in Fig. 5b. Stiffness isassumed to vary in direct proportion top ′, and the deriva-tion of plastic volumetric strains follows the ideas of theCam clay model. BRICK uses a parameterβ to account fordifferences between the stiffness and strength of normallyconsolidated and overconsolidated soils; details on howβmay be assessed are described by Simpson (1992).

The 3D version of the model incorporates the followingmodifications:

(1) Axes of volumetric strain (εv ) and shear strain (γ) inthe 2D model are replaced with anεv axis and five shear

© 2000 NRC Canada


Fig. 4. Variation of cu /p 0′ and af with p 0′ for rotary-cored sam-ples.

Fig. 5. (a) Some paths travelled by man and bricks. (b) BRICKrepresentation of stiffness reduction with strain.



strain axes. Inx, y, z orthogonal axes, these shear strains areεz – εx, (2εy – εx – εz)/(3)1/2, γxy, γyz, γzx. A compatible set ofstress axes is adopted.

(2) The “string lengths,” which govern the strain to failureand derived angle of shearing resistance, are varied as afunction of the relative proportions of the developing shearstrains, giving a failure envelope relative to the five shearcomponents of the type proposed by Drucker and Prager(1952). The Drucker–Prager criterion is usually expressed interms of stresses. However, by expressing it in terms ofstrains the form of the stiffness–strain curve can be main-tained without any discontinuities as failure is approached.

(3) In the BRICK model presented by Simpson (1992), allstiffness terms increased with OCR. The swelling index (κ)consequently reduced as overconsolidation increased and themodel predicted very low swelling potential as stresses be-came smaller. This behaviour is not typical, and a straightline in εv – ln p ′ space for isotropic swelling is now adoptedby the model.

(4) Behaviour is assumed to be elastic at very smallstrains. The elastic shear modulus (Gmax) may be determinedfrom shear wave velocity measurements or from the initialstages of laboratory tests with high-resolution local strain in-strumentation; such laboratory tests can also assist in esti-mating the elastic strain range, which is typically less than0.005% for many soils.

(5) The elastic bulk modulusKmax is related toGmax usingthe following function of the small-strain Poissons ratio (ν):

[4]GK

vv

max

max

( )( )

= −+

3 1 22 1

(6) It is assumed thatKmax is directly proportional to themean effective stress (p ′ ), whereKmax is given asp ′/ ι , andι is the very small strain equivalent ofλ andκ of the Camclay models. In BRICK, all similar equations are expressedin terms of volumetric strain, rather than void ratio or spe-cific volume.

BRICK provides a model which covers a wide range ofsoil behaviour, including both normally consolidated andheavily overconsolidated states. For a given soil, it is un-likely that it will fit the whole range of behaviour equallywell, and so it is appropriate to select parameters by fittingdata from the range which is of interest to the user, and for

which data are available. For example, as BRICK assumesstiffness varies directly withp ′ at all strain levels, separateBRICK parameters need to be derived for specificp ′ valuesif a varying dependence of stiffness onp ′ is to be modelled.For the same reason, values ofι andβ may need to be ad-justed to yield an area under the S-shaped stiffness–straincurve which is compatible with the peak friction angle of thematerial.

The BRICK model has been implemented into the OasysSAFE finite element program (Ove Arup & Partners 1996), a2D program which uses the 3D model for axisymmetricwork. At the start of each run, the materials are takenthrough their stress history to set up the current stresses andrelative positions of the bricks.

Testing BRICK against triaxial stiffness dataThe BRICK parameters estimated for DBC atp ′ = 100

and 200 kPa are summarised in Table 2. The tangent shearstiffness (Gt) decay curves, designated in Table 2 as sets 1and 2, are those assessed to correspond to reload pathsfollowing a complete reversal of the stress path direction(i.e.,θ = 180°) and theι values were selected to yield repre-sentative peak friction angles at these stress levels. TheBRICK β factor was assumed to be zero and therefore theparameters quoted are only relevant to the state ofoverconsolidation of DBC at these stress levels.

The BRICK computer code (Ove Arup & Partners 1996)was used in its axisymmetric mode, in conjunction with theparameters provided in Table 2, to predictGsec/p ′ ratios fortests on three rotary-core samples, designated DBCS1,DBCS2, and DBCS3. The predictions for these samples,when subjected to the stress paths shown in Fig. 6, are com-pared in Table 3 with measuredGsec/p ′ ratios at varioustriaxial shear strains (εs). To take approximate account of thestress-level dependence of the shear stiffness, predictions forsamples DBCS1 and DBCS3 were made using the BRICKparameters provided forp ′ = 100 kPa, and predictions forsample DBCS2 were made using those forp ′ = 200 kPa.The stress state prior to tests DBCS2 and DBCS3 was at Aand that for DBCS1 had been at A but was taken to B priorto swelling to C (see Fig. 6).

Predictions forGsec/p ′ are seen in Table 3 to compare wellwith measured ratios over the full range of imposedθ andεs

© 2000 NRC Canada


Gt /Gmax

String γ

Set 1 (p ′ = 100kPa, ι = 0.00022,β = 0, φ p′ = 45°)

Set 2 (p ′ = 200kPa, ι = 0.000315,β = 0, φ p′ = 41°)

Set 3 (allp ′,ι = 0.00024,β = 12,φ ′ = 31°)

1 0.00001 0.80 0.80 0.82 0.00003 0.38 0.45 0.383 0.00009 0.12 0.25 0.124 0.0005 0.05 0.055 0.0525 0.0015 0.02 0.025 0.0166 0.0035 0.01 0.014 0.0067 0.006 0.002 0.0023 0.0018 0.02 0.0008 0.001 0.00029 0.1 0 0 0

Note: λ* = 0.03,κ* = 0.006, ν = 0.2, andpmax′ = 1300 kPa, whereλ* = λ /(1 + e) andκ* = κ/(1 + e).

Table 2. BRICK parameters for Dublin boulder clay atp ′ = 100 and 200 kPa (sets 1 and 2) and for largestrain applications (set 3).



values. This agreement is particularly encouraging given thatmeasuredGsec/p ′ values vary by a factor of 10 over theεsinterval 0.01–0.1%. It is clear that BRICK captures the de-pendence of shear stiffness on both the previous loading di-rection and the shear strain extremely well. This arisesbecause of the apparent ability of the model to predict a re-alistic dependence onθ of the rate of degradation of shearstiffness with shear strain.

Modelling the strength of DBC with BRICKThe BRICK parameter sets 1 and 2 in Table 2, while pro-

viding a good representation of stiffness at small and inter-mediate strain levels, are not suitable if an accurateprediction of the strength of DBC is required over a largerange of p′ such as 100–1300 kPa. The BRICK parameterset 3, given in Table 2, was derived to achieve this and pre-dict a dependence of the peak friction angle onp0′ similar tothe measured variation in Fig. 3. This parameter set, how-ever, overpredicts stiffness at low strain levels.

BRICK predictions for the variations withp0′ of cu /p0′andaf are compared in Fig. 7 with the mean measured varia-tions as given by eqs. [1] and [2]. The analyses were per-

formed using the set 3 BRICK parameters and assuming iso-tropic consolidation to pmax′ = 1300 kPa followed byisotropic swelling top0′ before undrained shearing in triaxialcompression. It is evident that the predicted trends withp0′are comparable to those measured, and computed undrainedstrength ratios are close to those measured at typical in situstress levels (i.e.,p ′ < 200 kPa). However, the BRICKparameters selected lead to (i) underpredictions of theundrained strength ratio at low OCRs (or at highp0′ values),and (ii ) overestimates ofaf at typical in situ stresses. More-over, BRICK predicts peak obliquity (φp′) at peak deviatorstress (qf) while test samples mobiliseφp′ well beforeqf. It isclear, therefore, that BRICK requires further development tomatch the strength characteristics of DBC more precisely.

© 2000 NRC Canada


Fig. 6. Stress paths for the BRICK predictions in Table 3.

Gsec/p ′Sample No. Path Previous path θ (°) εs (%) Measured Predicted

DBCS1 C–D B–C 123 0.01 1000 1070DBCS1 C–E D–C 15 0.01 480 520

0.1 170 198DBCS2 F–G A–F 50 0.01 415 580DBCS2 G–F F–G 180 0.005 1620 1700DBCS2 F–H G–F 159 0.01 1080 1190

0.1 230 2690.5 89 81

DBCS3 C–I A–C 50 0.01 565 785DBCS3 I–C C–I 180 0.005 3200 2600DBCS3 C–J I–C 159 0.01 1250 1440

0.1 245 2900.5 80 84

Note: The shear strain origin is taken at the beginning of each path.

Table 3. Comparison of measured and predicted secant shear stiffness ratiosGsec/p ′.

Fig. 7. BRICK predictions at failure in undrained triaxial com-pression.



Predictions of the settlement of a shallowfoundation

The ability of the BRICK 3D model when incorporated inthe SAFE finite element (FE) code (Ove Arup & Partners1996) to provide realistic estimates of ground movements isnow investigated by comparing its predictions for the settle-ments of a shallow footing with those measured in a drainedload test. This test did not bring the underlying DBC closeto failure and its response was controlled by the stiffness ofDBC at low and intermediate strain levels.

Footing experimentLehane and Farrell (1997) report results from a load test

on a 1.5 m square reinforced concrete footing that wasfounded on DBC at a site in Dublin. The SPTN values re-corded during the site investigation are plotted in Fig. 8,which reveals a highly variable, although typical,N profilein DBC. Much of the apparent variability is due to drillingdisturbance and the influence of coarse gravel and cobbles(Faulkner et al. 1998).

The complete site had been lowered by 3 m one year priorto the test, leaving unweathered DBC at the surface.Oedometer data indicated that the coefficient of consolida-tion (cv) for DBC at the site was in the range 20–60 m2/year,and a simple calculation indicated that over 90% of swellingin the vicinity of the footing was completed within 2 monthsof the site excavation. Despite the relatively highcv value, toaccelerate consolidation during the footing test, four 1.5 mdeep 50 mm diameter holes were drilled at 0.7 m centresprior to casting the footing at 0.2 m depth. The water tablelevel was also at 0.2 m depth.

The measured bearing pressure (qb) versus settlement (s)response of the footing is shown in Fig. 9. Loading to amaximum bearing pressure of 1020 kPa was achieved withnine load increments over a period of 3 months. Creep set-

tlement rates reduced to less than 0.02 mm/day prior to ap-plication of the next load increment, illustrating the essen-tially drained nature of the test. The curves in Fig. 9 startfrom a point at whichqb = 90 kPa whens is zero because nosettlements could be measured during application of the ini-tial seating pressure of 90 kPa.

Despite the variability of the SPTN value, practitioners inDublin continue to employ empirical correlations with theNvalue and, for shallow foundations, assume an equivalentelastic shear modulus (Geq) in kilopascals of 600N (Farrell1989). Such an approach is clearly difficult to apply giventhe range ofN values measured (see Fig. 8) and leads to set-tlement predictions at the maximum appliedqb stress rang-ing from about 10 mm forN = 80 to almost 30 mm forN =30.

FE–BRICK predictionsThe shear stiffness of DBC varies with (p ′) x, wherex in-

creases from a value of-0.5 at very small strains to a valueof unity at shear strains in excess of-0.1%. Because BRICKassumesx is unity at all strains, parameters specific to agiven stress level (e.g., parameter sets 1 and 2 in Table 2)have been used to achieve good estimates of stiffness in thecomparative exercise summarised in Table 3. It is clearly notexpedient to specify separate sets of BRICK parameters forall expectedp ′ values in a FE analysis. BRICK parametersover the stress and strain levels of interest may, however, bederived from Gt /p ′ values that are measured directly instress path tests which approximate the rate of increase ofshear stress withp ′ in a given loading case. The triaxialcompression stress path is considered here to be a reasonableapproximation to the stress paths imposed on soil beneath ashallow foundation.

The drained triaxial compression tests performed on ro-tary-cored samples DBCS2 and DBCS3 (along paths F–Hand C–J, as shown in Fig. 6) are used as the basis for thesettlement prediction presented here. The data from thesetests are plotted in Fig. 10 as variations ofGt /p ′ with εs overthe strain levels of interest. Figure 10 also presents twoGt /p ′ variations withεs selected for analysis; these are des-ignated runs A and B. Run A adopted the best-fit BRICK

© 2000 NRC Canada


Fig. 8. SPT N values recorded at the site of the footing test. Fig. 9. Bearing pressure versus settlement for trial footing.



approximation to the triaxial data, and, rather arbitrarily andto provide an indication of sensitivity, run B employed thesame rate of degradation of stiffness with strain but usedGt /p ′ ratios which were 33% higher than those in run A.The BRICK input parameters used in the analyses are sum-marised in Table 4.

The footing test was modelled in a FE axisymmetric anal-ysis and adopted an equivalent foundation radius of 0.85 m(see Fig. 11). Full friction between the footing base and thesoil was allowed and the elements used were eight-nodedquadrilaterals with four Gauss points. The lower and edgeboundaries of the mesh were fully fixed and located at a dis-tance of at least 6 m from the footing. Given the dependenceof stiffness on stress level, initialKo values in the soil arecritical to the settlement predictions. These values werecomputed automatically by BRICK in the first stage of eachanalysis, both of which assumed a vertical preconsolidationstress (σ vy′ ) of 1300 kPa was applied at the footing level.The predictedKo values, which are plotted in Fig. 12, areconsistent with expectations in a very heavilyoverconsolidated deposit and indicate slightly higher initialstresses immediately beneath the footing in run B. Subse-quent stages of the analyses involved applying drained incre-

ments of load. The computedqb versuss relationships de-rived in these stages are compared with the experimentaldata in Fig. 9.

It is evident that, despite specifying a strongly nonlinearvariation of stiffness with strain, both run A and run Bdisplay a near-linear variation of bearing pressure withsettlement. This linearity is compatible with the observedvariation of drained settlement and arises because the stiff-ness degradation with strain is approximately compensatedfor by the increase in mean stress in the ground. It is alsoapparent from Fig. 9 that theqb /s ratio predicted in run A isnearly half of that in run B, illustrating the importance of pa-rameter selection in nonlinear analyses of this type.

The drained footing settlements are typically 30% lessthan those predicted by run A. Reasons for thisoverprediction can only be speculative because of the

© 2000 NRC Canada


γ Gt /Gmax

0.000015 0.8000.000075 0.1600.0015 0.0530.0023 0.0320.0055 0.0170.0135 0.00530.03 0.0020.045 0.0

Note: σ vy′ = 1300 kPa,λ* = 0.03,κ* =0.006, i (= ′p /κ max) assigned 0.0004 for runA and 0.0003 for run B, andβ = 0.

Table 4. BRICK parameters used in theOasys SAFE analyses of footing.

Fig. 10. BRICK parameter match with triaxial compression stiff-ness data.

Fig. 11. Finite element mesh for trial footing analyses.

Fig. 12. Ko values predicted by BRICK.



absence of additional site records such as extensometer dataand surface settlement distributions. The assumption that allparameters can be derived from triaxial compression data isclearly, at best, an approximation in view of the variety of3D stress paths imposed on the ground beneath a footing. Inaddition, factors such as the anisotropy of DBC in situ andthe inevitable disturbance caused to samples can contributeto prediction errors. Finally, part of the discrepancy may bebecause the experimental stiffness measurements in Fig. 10correspond to aθ value of 159°, whereas the specified stiff-ness variation with strain in the analysis should correspondto that preceded by vertical unloading withθ = 180°.

The difference between the measured settlement and therun A predictions, although relatively significant, compareswell with the reliability of the currently used linear elasticmethod for shallow footings on DBC. Clearly, however, themain advantage of the BRICK approach described is that itis has the capacity to provide realistic predictions of grounddeformation patterns. The inclusion of well-designed instru-mentation systems in upcoming projects in Dublin will assistin the testing and refining of the BRICK parameter selection.

Conclusions

(1) Dublin boulder clay (DBC) is a dense and heavilyoverconsolidated well-graded lodgement till which is classi-fied as a clay, despite the virtual absence of clay minerals,because of its low permeability. In many respects, it displaysthe characteristics of a very low permeability dense sand.

(2) It has been possible to derive parameters which enablethe BRICK model to provide a good representation of thestiffness characteristics of DBC. Best predictions are ob-tained when parameters are fitted to data obtained within therange of interest to a particular problem.

(3) The model has been used successfully to reproduce thestiffness–strain curves measured in drained triaxial probingtests on rotary-cored samples, and the same model, when tai-lored to suit specific triaxial compression stiffness data, gaverelatively good agreement between the computed and mea-sured settlement of an incrementally loaded pad footing onDBC.

(4) BRICK is evidently a most useful analytical tool formodelling the behaviour of DBC at small and intermediatestrain levels. However, further development is required to re-produce the strength and dilatant characteristics of DBCmore accurately.

Acknowledgements

The authors would like to acknowledge the contributionsof Andrea Faulkner, Patrick Murphy, Brian O’Shea, andPatrick Treacy, each of whom assisted in the laboratory testprogramme. Dublin Corporation is also acknowledged formaking some of the test data presented here available to thepublic.

References

Been, K., and Jeffries, M.G. 1985. A state parameter for sands.Géotechnique,35(2): 99–112.

Burland, J.B. 1990. On the compressibility and shear strength ofnatural clays. Géotechnique,40(3): 329–378.

Butterfield, R. 1979. A natural compression law for soils (an ad-vance one – log p). Géotechnique,29(4): 469–480.

Cuccovillo, T., and Coop, M.R. 1997. Measurement of local axialstrains in triaxial tests using LVDTs. Géotechnique,47(1): 167–171.

Drucker, D.C., and Prager, W. 1952. Soil mechanics and plasticanalysis in limit design. Quarterly of Applied Mathematics,10:157–165.

Farrell, E.R. 1989. Settlement parameters of Dublin Black BoulderClay. Ground Engineering,22(5): 33–35.

Faulkner, A. 1998. Application to Irish conditions of new field andlaboratory testing techniques. M.Sc. thesis, University of Dublin(Trinity College), Dublin, Ireland.

Faulkner, A., Lehane, B.M., and Farrell, E.R. 1998. Cone penetra-tion testing in Irish soils.In Proceedings of the 1st InternationalConference on Geotechnical Site Characterisation, Atlanta, Ga.,pp. 1033–1038.

Gens, A., and Hight, D.W. 1979. The laboratory measurement ofdesign parameters for a glacial till.In Proceedings of the 7thEuropean Conference on Soil Mechanics and Foundation Engi-neering, Brighton, U.K., Vol. 2, pp. 57–65.

Hanrahan, E.T. 1977. Irish glacial till: origin and characteristics.Special Publication No. 1, FORBAIRT, Irish GovernmentAgency, Dublin, Ireland.

Hartford, D. 1987. Some aspects of the engineering behaviour ofwell graded soils. Ph.D. thesis, University of Dublin (TrinityCollege), Dublin, Ireland.

Lehane, B.M., and Farrell, E.R. 1997. A serviceability limit statedesign approach for footings.In Proceedings of the 14th Inter-national Conference on Soil Mechanics and Foundation Engi-neering, Hamburg, Vol. 1, pp. 359–362.

Lehane, B.M., and Faulkner, A. 1998. Stiffness and strength char-acteristics of a hard lodgement till.In Proceedings of the 2nd In-ternational Conference on Hard Soils and Soft Rocks, Naples,Vol. 2, pp. 637–646.

Muir Wood, D. 1990. Soil behaviour and critical state soil mechan-ics. Cambridge University Press, Cambridge, U.K.

Ove Arup & Partners. 1996. Oasys: geotechnical programs manual.Ove Arup & Partners, London, U.K.

Simpson, B. 1992. Retaining structures: displacement and design:32nd Rankine Lecture. Géotechnique,42(4): 541–576.

Treacy, P. 1996. Movements of vertical cuts in Dublin BoulderClay. M.Sc. dissertation, University of Dublin (Trinity College),Dublin, Ireland.

List of symbols

af (p 0′ – p f′ ) /qfc′ effective cohesioncu undrained shear strength

cu /p 0′ undrained strength ratiocv coefficient of consolidatione void ratio

Geq equivalent linear elastic shear modulusGmax small strain (elastic) shear modulusGsec secant shear modulus

Gt tangent shear modulusqc CPT end resistance

Kmax small strain (elastic) bulk modulusKo coefficient of earth pressure at restKp passive pressure coefficientnp isotropic OCR =p max′ /p 0′M qf /p f′

© 2000 NRC Canada




© 2000 NRC Canada


N SPT blowcount/300 mmp ′ mean effective stress

p f′ p ′ when q = qfp 0′ p′ prior to shear

p max′ maximum previously appliedp ′q deviator stress

qb average footing bearing pressureqf maximumqr Cam clay parameter as defined in the texts average footing settlementz sample depthβ empirical BRICK parameter to account for stiffness

and strength dependence on OCRε n direct strain inn direction, e.g.,εa = axial strain and

ε r = radial strainεs triaxial shear strain (= 2γ/3)εv volumetric strain

φ cv′ constant-volume critical state friction angle

φ p′ peak friction angleγ string length in BRICK =ε a – ε r for triaxial conditions

γnm shear strain inn–m planeι p ′/Kmaxκ slope of unload–reload line ine versus lnp ′ or e versus

ln σ v′ paceκ slope of unload–reload line inεv versus lnp ′ or ε v

versus lnσ v′ spaceλ slope of normal consolidation line ine versus lnp ′ or

e versus lnσ v′ spaceλ* slope of normal consolidation line inεv versus lnp ′ or

ε v versus lnσ v′ spaceν Poisson’s ratio

σ v′ vertical effective stressσ vy′ vertical preconsolidation stress

θ rotation of stress path direction in (p ′, q) space, asdefined in Fig. 2

Λ 1 – κ/λ



modelling glacial till under triaxial conditions using a brick soil model

Documents