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Clayton, C. R. I. (2011). Geotechnique 61, No. 1, 537 [doi: 10.1680/geot.2011.61.1.5]

5

Stiffness at small strain: research and practice

C. R. I . CLAYTON

This paper provides the background to the 50th RankineLecture. It considers the growth in emphasis of the predic-tion of ground displacements during design in the pasttwo decades of the 20th century, as a result of the lessonslearnt from field observations. The historical developmentof the theory of elasticity is then described, as are theconstitutive frameworks within which it has been pro-posed that geotechnical predictions of deformation shouldbe carried out. Factors affecting the stiffness of soils andweak rocks are reviewed, and the results of a numericalexperiment, assessing the impact of a number of stiffnessparameters on the displacements around a retaining struc-ture, are described. Some field and laboratory methods ofobtaining stiffness parameters are considered and criti-cally discussed, and the paper concludes with a suggestedstrategy for the measurement and integration of stiffnessdata, and the developments necessary to improve theexisting state of the art.

KEYWORDS: anisotropy; deformation; elasticity; geophysics;ground movements; in situ testing; laboratory equipment;laboratory tests; stiffness

Cet article presente le contexte de la 50eme conference deRankine. Il se penche sur limportance croissante accor-dee, au cours des vingt dernieres annees du 20eme siecle,a la prediction des deplacements du sol en phase dedimensionnement, a la suite des lecons tirees dobserva-tions sur le terrain. Le developpement historique de latheorie de lelasticite est alors decrit, ainsi que les cadresconstitutifs dans lesquels il a ete propose dappliquer lespredictions geotechniques des deformations. Les facteursaffectant la rigidite des sols et des roches tendres sontevalues, et les resultats dune experience numerique,evaluant limpact dun certain nombre de parametres derigidite sur les deplacements autour dune structure desoutenement sont decrits. On procede a lexamen, et aune discussion critique, de certaines methodes adopteesin situ et en laboratoire pour la determination de para-metres de rigidite. La communication se termine avec laproposition dune strategie pour la determination etlintegration des donnees de rigidite, et des developpe-ments necessaires pour loptimisation de letat actuel desconnaissances.

INTRODUCTIONThe rapid development of computing power and of numer-ical modelling software over the past 40 years has madesophisticated analysis of geotechnical problems accessible tomost engineering practices. Typically, computer packagesnow offer a wide range of constitutive models, which thedesign engineer needs to choose among, and then obtainparameters for. For structures designed to be far from fail-ure, for example supporting urban excavations, strains in theground are small. A sound knowledge of stiffness parametersat small strain is essential, if realistic predictions of theground movements that may affect adjacent buildings orunderlying infrastructure are to be made.

This paper discusses the geotechnical background to themeasurement of stiffness parameters, briefly reviewing thelessons learnt from field observation and back-analysis offoundation and deep excavation behaviour. It describes thehistorical development of elastic theory, and the constitutiveframeworks within which it can be applied to soil and weakrock behaviour. It reviews what is now known about thecomplex stiffness behaviour of soil and weak rocks in thecontext of what is, arguably, the simplest of constitutivemodels. A numerical experiment, to assess the importance ofdifferent parameters for the displacement of a particular struc-ture, a singly propped retaining wall, is described. It is shownthat for this particular problem most parameters significantlyaffect predicted displacements. Methods of determining therequired stiffness parameters are then explored, and the useful-ness of seismic field testing, dynamic laboratory testing andadvanced triaxial testing is examined. Finally, strategies forintegrating the data are discussed, and conclusions are drawn.

GEOTECHNICAL BACKGROUNDJames Bell (1989) has described the 19th century as the

Age of Design by Disaster. According to him, surprisinglyfew engineers working in this period carried out analyses oftheir design concepts before beginning construction. Giventhe significant construction problems faced by civil engineersat the beginning of the 20th century, early soil mechanicsunderstandably focused on preventing failure.

But by the late 1970s the emphasis had changed. Formany practising engineers soil mechanics was becoming amature science, because most failure mechanisms wereunderstood, and with good practice could be identified andavoided with some certainty. The start of global urbanisationchanged all that, as the pressure to redevelop inner cityinfrastructure produced more and more challenges, many ofwhich now related to ground movements and their effects onadjacent structures and buried infrastructure. At the sametime the need to build nuclear and other key facilitiesincreased, requiring analysis for the effects of large, albeitsometimes infrequent, seismic events. The rise of numericalmodelling in the 1960s, and the huge increase in computingpower since then, has given us increasingly sophisticatedanalytical tools for use in practice (e.g. Zienkiewicz et al.,1968; Simpson, 1981; Britto & Gunn, 1987; Potts, 2003).The determination of the parameters needed for such ana-lyses has, perhaps understandably, lagged behind thedevelopment of numerical modelling.

Burland (1989) gives a good account of how the inter-action of field observations and numerical modelling of thedeformations associated with foundations and excavations inthe London area led, in the UK, to the development of moreappropriate stiffness models for the ground. Back-analysis ofconstruction in London showed that field stiffnesses weremuch greater than those obtained from routine laboratorytests, for example in the oedometer or triaxial apparatus(Cole & Burland, 1972; St John, 1975; Clayton et al., 1991),

School of Civil Engineering and the Environment, University ofSouthampton, UK

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that stiffness increased significantly with depth (Marsland &Eason, 1973; Burland & Kalra, 1986), that stiffness wasanisotropic (Cole & Burland, 1972), and greater at smallstrains than at large (Burland & Hancock, 1977; Simpson etal., 1979). These concepts seem to be true for weak rockssuch as the chalk, as well as for stiff clays (Ward et al.,1968; Burland & Lord, 1970; Kee, 1970; Burland et al.,1973; Hobbs, 1975; Matthews & Clayton, 2004).

Atkinson & Sallfors (1991) noted that, as far as theycould determine, no previous state-of-the-art or generalreports to major international conferences in the previousdecade had considered specifically the determination of soildeformation parameters. The 1991 International Society forSoil Mechanics and Foundation Engineering conference inFlorence was a turning point, at which the importance ofsoil stiffness to both theoreticians and practising engineersbecame accepted. Now, almost 20 years on, this paperrevisits the issue of stiffness determination, and in particularis concerned with small-strain stiffness, which was for manythe defining feature of the research in the period leading upto the Florence conference.

CONSTITUTIVE FRAMEWORKS FOR STIFFNESSThe stiffness of a body (or structure) is defined as the

resistance of that body to deformation under applied force.It is derived from:

(a) the shape of the body(b) boundary conditions, such as fixities and load positions(c) the stiffness properties of the constituent materials

(Youngs moduli, etc.).

Thus deformation depends upon stiffness, which in turndepends on the stiffness properties that are the subject ofthis paper. In geotechnical engineering practice stiffness isnormally defined within the context of the mathematicaltheory of elasticity, although this is not strictly necessary.The development of the theory of elasticity is describedbelow.

Historical developmentThe recognition of linear load/deformation behaviour is

widely attributed to Hooke (1676), who wrote at the end ofhis A description of helioscopes that

To fill the vacancy of the ensuing page, I have here addeda decimate of the centesme of the Inventions I intend topublish, though possibly not in the same order, but as I canget opportunity and leasure; most of which, I hope, will beas useful to Mankind, as they are yet unknown and new.

The third of these Inventions was on

The true Theory of Elasticity or Springiness, and aparticular Explication thereof in several Subjects in whichit is to be found: And the way of computing the velocity ofBodies moved by them. ceiiinosssttuu.

In his treatise De Potentia Restitutiva, or of Spring, Hooke(1678) explained his anagram as Ut tensio sic vis, which isroughly translated as extension is proportional to force. Aswe would see it today, this is a description of linearity.Hooke also recognised elastic behaviour, that is, the behav-iour of a material that returns to its original shape afterloading is removed. In the same work he states that

. . . it is very evident that the Rule or Law of Nature inevery springing body is, that the force or power thereof torestore it self to its natural position is always proportionateto the Distance or space it is removed therefrom.

In reality, according to Bell (1989), Hookes measurementson long iron wires were too insensitive to show linearity. Asearly as 1687 James Bernoulli produced data for the gutstring of a lute that suggested a parabolic relationship be-tween load and deformation at small strains (although Leib-nitz assumed his data were hyperbolic). Over 100 years later,in about 1810, two independent sets of experiments, byDuleau and by Dupin, led to conflicting conclusions. Duleau(1820), testing forged iron for a bridge over the Dordogneriver, found linear behaviour at small strain. Dupin (1815),testing wooden beams for ships, found a non-linear response.

By the end of the 19th century, and following the findingsof the Royal Commission on Application of Iron to RailwayStructures (1849), which recommended that Hookes lawshould be replaced by experimentally based, well-documen-ted non-linearity, several non-linear laws (parabolic, byEaton Hodgkinson, a member of the Royal Commission;hyperbolic, by Homersham Cox; and non-linear exponential,by Carl Bach) had been proposed. Nineteenth-century datafor cast iron, showing Coxs hyperbolic law, are given inFig. 1. The results are similar in form to those obtainedfrom triaxial testing on intact chalk (Heymann et al., 2005).Writing in his classic work on The experimental foundationsof solid mechanics, Bell (1989) has noted that

The dilemma of Leibniz in the 17th century over theapparently conflicting experiments of Hooke and JamesBernoulli has been resolved in favor of the latter. Theexperiments of 280 years have demonstrated amply forevery solid substance examined with sufficient care, thatthe strain resulting from small applied stress is not a linearfunction thereof.

However, the impact of this realisation has generally beensmall, for as Viggiani (2000) notes, the achievements oflinear elasticity theory are well known to all of us; modernengineering is still largely based on it.

ParametersAlthough Hooke recognised the concept of the stiffness of

a body, the idea of an elastic property was not developeduntil 1727, by Leonhard Euler, and not measured until 1782,by Giordano Riccati. The concept of stresses in solids hadbeen introduced by Coulomb in 1773, in his classic paper,which also dealt with the pressure of soil on retainingstructures. Young later published the idea of his modulusin his book of 1807, although his definition does not alignwith what we would now term Youngs modulus (Todhun-ter, 1886; Timoshenko, 1953; Cooper, 1978). AlthoughYoung recognised the distinction between extension and

(1 )

E

1010010

02

04

06

08

12

00001

Axial strain: %

Nor

mal

ised

sec

ant Y

oung

s m

odul

us,

/E

Ese

c0

Experimental data for cast ironCoxs Hyperbolic law

0001

10

Reference modulus, E0

Stiffness continues to rise

Fig. 1. Normalised stiffness data for cast iron (Royal Commis-sion on Application of Iron to Railway Structures, 1849; Cox,1856)

6 CLAYTON

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shear stiffnesses, he did not suggest a different measure forthese modes of deformation.

The concept of a Poissons ratio dates back to 1814(Cooper, 1978). The PoissonCauchy theory of 1848 pro-duced the uni-constant theory of elasticity, a consequence ofwhich was the prediction that for homogeneous isotropicsolids Poissons ratio should be exactly 1/4 for all materials.In contrast Greens (1828) work, although largely unknownat the time, had identified an elastic system with 21 elasticconstants, reduced to two in the case of isotropy. Laterexperimental work by Wertheim, Kupffer, Neumann andKirchhoff (Timoshenko, 1953) subsequently lent support toGreens findings. By the end of the 19th century the frame-work of elasticity was fully developed.

Application in geotechnical engineeringProbably the most commonly assumed behaviour in prac-

tical geomechanics is that of isotropic linear elasticity.Characterisation of an isotropic elastic solid requires thedetermination of only two material parameters (from fourpossible measurements, i.e. Youngs modulus E and Poissonsratio , or shear modulus G and bulk modulus K) forcalculations of strain or deformation, and therefore anassumption of isotropic elasticity has the merit of simplicity.However, as noted by Bishop & Hight (1977), there aremany reasons to believe that the ground will generally beanisotropic, or at least transversely isotropic.

As found by Green, the characterisation of an anisotropicelastic solid requires the determination of 21 independentelastic constants. Given the complexity of subsurface geom-etry, and the spatial variability of soil and rock, this isbeyond the reach of practical soil mechanics. But in manycases it may be sufficient to assume transverse isotropy, orcross-anisotropy as it is also known, for which there areseven measurable parameters, and a further two (for exam-ple, dip and dip direction) necessary to define the orientationof the plane of isotropy in the general case where it is nothorizontal. For a transversely isotropic material where theplane of isotropy is horizontal, the seven elastic parametersare

Ev Youngs modulus for loading in the vertical directionEh Youngs modulus for loading in the horizontal directionvh Poissons ratio relating to the horizontal strain caused

by an imposed vertical strainhv Poissons ratio relating to the vertical strain caused by

an imposed horizontal strainhh Poissons ratio relating to the horizontal strain caused

by an imposed horizontal strain in the normal directionGv shear modulus in the vertical planeGh shear modulus in the horizontal plane

where the subscripts v and h refer to the vertical andhorizontal directions.

Skeleton and pore fluid interaction. The discussion so far hasignored the fact that most geomaterials are not solid, but are

particulate or voided, and have at least two and often threephases:

(a) a skeleton, or frame, for example an assembly ofparticles in contact with, and often cemented to, eachother

(b) pore fluid, which will normally be water for a saturatedmaterial, and water and air for an unsaturated material.

The skeletal stiffness of uncemented soil is a function ofeffective stress, and is often low in comparison with thestiffness of water, which may then be considered incompres-sible (Bishop & Hight, 1977). For a saturated soil, therefore,there are two cases and two sets of stiffness parameters thatmay be required:

(a) the undrained, short-term, or end of constructioncase, where shear strains have occurred but excess porepressures remain, and volumetric strain is assumed tohave been prevented because of the low permeability ofthe soil relative to the rate of loading/unloading, andthe incompressibility of the pore fluid relative to thesoil skeleton

(b) the drained, long term or effective stress case,where both volumetric and shear strains have occurred,and any excess pore pressures set up during loadinghave fully dissipated (Bishop & Bjerrum, 1960).

The stiffness of an isotropic soil material can be definedin terms of a number of different sets of parameters, themost commonly used in soil mechanics being shown inTable 1. For the isotropic drained case the engineer canchoose to measure either the effective Youngs modulus andthe effective Poissons ratio, or the shear modulus and thedrained bulk modulus (K9 dp9/dV, where p9 is meaneffective stress and V is the volumetric strain). Parameterset 1 (Table 1) can be readily measured (assuming isotropy)in the triaxial test. The computational convenience of par-ameter set 2 lies in the fact that G remains the same in theundrained and drained cases, since it involves change inshape without change in volume, and the contribution tostiffness of the shear modulus of water can be assumed tobe negligible at low rates of strain.

In the isotropic case, the relationships between drainedand undrained Youngs modulus and Poissons ratio, shearmodulus and bulk modulus can be obtained from

G E92 1 9

Eu

2 1 u (1)

K9 E93 1 29

(2)

Ku Eu

3 1 2u (3)

If the pore fluid is assumed incompressible (but see Bishop& Hight, 1977), then the undrained bulk modulus is infinite,and from equation (3) u 0.5. Hence

Eu 1:5E9

1 9 (4)

Table 1. Isotropic drained and undrained parameter sets

Case Parameter set 1 Parameter set 2

Undrained Undrained Youngsmodulus, Eu

Undrained Poissonsratio, u

Shearmodulus, G

Undrained bulkmodulus, Ku

Drained Effective Youngsmodulus, E9

Effective Poissonsratio, 9

Shearmodulus, G

Drained bulkmodulus, K9

STIFFNESS AT SMALL STRAIN: RESEARCH AND PRACTICE 7

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Options for anisotropic parameter determination. In the caseof a transversely isotropic elastic solid it is only necessaryto measure five of the seven parameters identified above,since Love (1892) (see Pickering, 1970) proved that

hvEh

vhEv

(5)

and

Gh Eh

2 1 hh (6)

In the case of a drained material the engineer may choose towork with one of a number of parameter sets (Pickering,1970; Lings et al., 2000; Lings, 2001): for example,

E9v, E9h, 9vh, 9hh and Gv, or

E9v, E9h, 9vh, Gh and Gv, or

E9v, E9h, 9hv, 9hh and Gv, or

E9v, 9hv, 9hh, Gh and Gv:

The choice is arbitrary from the point of view of computa-tion, since conversion between different parameter sets canreadily be achieved using equations (5) and (6). However, aswill be seen later, some parameters are more readily meas-ured than others, and a combination of field and/or laboratorytechniques will be required to obtain a full five-parameterset. For example, Youngs modulus E9v and Poissons ratio 9vhare readily obtained from a drained triaxial compression testwith local axial and radial strain measurement. The deter-mination of Gh and E9h is more challenging.

Limits. Thermodynamic considerations require that the strainenergy of an elastic material should always be positive(Pickering, 1970). It follows that for an isotropic elasticmaterial Youngs modulus E should be greater than zero, andPoissons ratio should fall between 1.0 and +0.5 (Pickering,1970; Gibson, 1974). For a drained transversely isotropicelastic material, E9v, E9h, Gv and Gh must all be greater thanzero,

1 < 9hh < 1 (7)and

E9vE9h

1 9hh 2 9vh 2 > 0 (8)

As in the isotropic stiffness case, Youngs moduli andPoissons ratios are different in the drained (long-term) andundrained (short-term, or end of construction) conditions,while shear moduli remain the same (Table 1). In theundrained case fewer parameters are required, because

uvh 0:5 (9)Chowdhury & King (1971)

uhv Euh

2Euv(10)

Gibson (1974)

uhh 1 Euh

2Euv(11)

Gibson (1974).Thus the parameter set can be reduced, as noted by

Atkinson (1975), to Euv, Euh and Gv.

FACTORS AFFECTING MEASURED STIFFNESSAny measurement of stiffness, whether made in the field

or in the laboratory, needs to be critically reviewed in thecontext of those factors that will control the stiffness of theground around a structure. If conditions are not the same,then the measured stiffness will be different, and may be oflimited value or require modification when making predic-tions of displacements. Therefore, in the following para-graphs, key factors affecting stiffness are reviewed.

The effect of strain level has already been noted in thesection on Historical background above. Experimentalphysicists have established beyond doubt that (even formaterials much more competent than soil and weak rock),there is no linear stressstrain behaviour. Superficially atleast, soils and weak rocks appear to behave in a similarway to other materials, and it has been observed that for awide range of stiffness (e.g. Clayton & Heymann, 2001)behaviour is sufficiently constant below a strain level ofabout 0.001% for this to be taken, for practical purposes, asthe strain range within which to measure the very-small-strain reference modulus values (E0 or G0) (Fig. 1).

Stiffness parameters may therefore, for practical purposes,be considered constant at very small strains, but can beexpected to reduce as strains increase above this level. Thiswas the approach of Atkinson & Sallfors (1991). Becausethe strain levels around well-designed geotechnical structuressuch as retaining walls, foundations and tunnels are gener-ally small (Fig. 2), measurements are required in order todetermine two sets of parameters:

(a) Parameters at very small (ideally reference) strain levels(e.g. E0, 0 and G0). These depend upon, for example,(i) void ratio(ii) grain characteristics such as particle size and

shape(iii) current effective stresses(iv) structure (here used in the sense of Kavvadas &

Anagnostopoulos, 1998)(v) stress history(vi) fabric (in the sense of Rowe, 1972) and particle

arrangement(vii) discontinuities(viii) rate of loading

(b) Stiffness parameters are altered by increasing strain andchanging stress levels, during loading or unloading.Factors controlling stiffness under operational condi-tions include(i) strain level(ii) loading path and changes in effective stress(iii) changes in loading path(iv) recent stress history

1010010001

Typical strain ranges:

Retaining walls

Foundations

Tunnels

Stif

fnes

s,G

Shear strain, : %s

1000001

Fig. 2. Typical stiffness variation and strain ranges for differentstructures (redrawn from Mair, 1993)

8 CLAYTON

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(v) destructuring(vi) changes in loading rate.

(See also Hight & Higgins, 1995; Jardine, 1995.)

Stiffness at very small strainTests on reconstituted materials have shown the important

influence that state (void ratio, current effective stresses)and stress history can have on stiffness. The shear modulusof a granular material at very small strain levels is affectedfundamentally by three factors:

(a) the void ratio of the specimen(b) interparticle contact stiffness, which will depend upon

particle mineralogy, angularity and roughness, andeffective stress

(c) deformation and flexing within individual particles,which will depend on particle mineralogy and shape.

If interparticle stiffness is removed, for example by cement-ing, then for a given particle shape and mineralogy thecombined effect of void ratio and particle flexing can beseen. Fig. 3 shows shear modulus (Gv0) measured in theresonant column apparatus, for cemented Leighton Buzzardfraction E sand (rotund, uniform, D50 0.1 mm). Two typesof cement have been used: methane hydrate (Clayton et al.,2010) and epoxy resin (Fleris, personal communication). Thelocation of the cement in both cases is primarily at the graincontacts. In the case of methane hydrate this was achievedusing the excess gas method, where damp sand wascompacted into a mould to form an unsaturated specimen,and the water then combined (under suitable thermobaricconditions) with methane gas to form disseminated hydrate.The epoxy-bonded specimens were formed by tumbling thesand grains in a pre-measured quantity of epoxy resin, andthen either compacting them, or rubbing them through agrillage into a mould. This allowed very high void ratios tobe obtained, which for both types of specimen were calcu-lated taking into account the volume of the sand grains andof the cement.

The results show, for this sand, a unique relationship forupper-bound shear modulus against void ratio, independentof the effective stress applied during testing. Lower valuesof stiffness were obtained for hydrate volumes of less thanabout 5% of the void space, below which stiffness issensitive to effective stress, suggesting that cementing of theinterparticle contacts is incomplete (Clayton et al., 2005).The epoxy resin data are for 2%, 4% and 6% Araldite by

weight of sand, equivalent at 4%, for example, to between4.5% and 13% of the void space between the sand grains.For fully cemented sand, over the range tested, void ratiohas an approximately 20-fold effect on stiffness.

Hardin (1961) suggested that the shear wave velocity (andtherefore the shear modulus) of sands was influenced notonly by void ratio but also by mean effective stress. Experi-mental work by Hardin & Richart (1963), Hardin & Drne-vich (1972) and Iwasaki & Tatsuoka (1977), carried out inthe resonant column apparatus, subsequently, as might beexpected, supported this view. For this reason, the results ofresonant column testing on pluviated or compacted sandshave classically been shown normalised by a function ofeffective stress, in addition to being plotted against voidratio.

Figure 4 shows the results of a recent survey of reporteddata from resonant column testing (Bui, 2009) for bothsands and clays. For dimensional consistency the effectivestress applied during testing needs to be normalised, in thiscase by atmospheric pressure, patm. The introduction ofHertzian contact theory into expressions for the shear mod-ulus of a pack of identical elastic spheres suggests that G0should approximately be a function of effective stress to thepower of 1/3 (Duffy & Mindlin, 1957; Goddard, 1990) Inexperiments, the exponents of individual pluviated sandshave been found to vary between approximately 0.4 and 0.6,and (as in Fig. 4) a value of 0.5 has been observed by manyresearchers and used to normalise their stiffness data (Hardin& Black, 1966, 1968; McDowell and Bolton, 2001).

A number of equations have been derived to capture thetrends of such data. Based upon Buis survey of existingdata, a reasonable expression is

Gv0 Cp 1 E 3 p9=patm 0:5

(MPa) (12)

where Gv0 is the very-small-strain shear modulus in thevertical plane (MPa), Cp is a constant (in MPa), e is thevoid ratio of the specimen under test, p9 is the (isotropic)effective stress applied to the specimen, and patm is atmos-pheric pressure (in the same units as p9).

Trend lines for the data normalised by effective stress, forCp 300 MPa and Cp 600 MPa, are shown in Fig. 4.These trends are shown without normalisation, and forCp 450 MPa, in Fig. 3. From equation (12) and Fig. 3 itcan be seen that a tenfold increase in effective stressproduces only a threefold increase in stiffness, which is

0

1000

2000

3000

4000

5000

0Void ratio, e

200

1600800

100

400

: kPa

Leighton Buzzard E plus AralditeLeighton Buzzard E plus hydrateTypical rotund sand, without cement,at various stress levels, : kPa

Structured Leighton Buzzard sand no inter-grain compliance

Gv0

: MP

a

2520151005

Fig. 3. Effect of void ratio on cemented Leighton Buzzardfraction E sand

Gp

0a

tm0

5/(

/)

: MP

a

v

1 2 3 4

G C e pv0 p3

atm05(1 ) ( / ) (MPa)

0

100

200

0Void ratio

Cp 300

Cp 600

Fig. 4. Stiffness Gv0 normalised by effective stress, as a functionof void ratio. From resonant column tests on pluviated sandsand reconstituted clays (Bui, 2009)

STIFFNESS AT SMALL STRAIN: RESEARCH AND PRACTICE 9

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small in comparison with the increase that can be producedby quite modest amounts of cement.

For clays, overconsolidation produces significant decreasesin void ratio, even at quite modest stress levels. Perhaps forthis reason, very-small-strain stiffness data for clays (in thiscase from bender element testing) have been normalised bya void ratio function before being plotted as a function ofstress (e.g. Pennington et al., 1997). For sands, relativelylarge stresses are needed in the short term to induce thegrain crushing necessary to produce major changes in voidratio, although in the longer term and over the geologicaltimescale this is not the case, as shown by the evidencefrom aged natural materials, such as locked sands (Dus-seault & Morgenstern, 1979). Fig. 5 shows resonant columntest results for two undisturbed specimens of natural mater-ials: a sandy facies of the Eocene London clay, from a siteto the west of London, and a Lower Cretaceous (FolkestoneBeds) locked sand from a site to the south of London(Cresswell & Powrie, 2004). Superimposed on these data arepredictions made using equation (12), with a value of Cp of300 adopted for the sandy clay, and 1200 for the sand. Thedifference in value presumably reflects that fact that forsands the development of flats between particles will havetaken much of the compressibility out of the grain contacts,whereas for the clay, particle flexure is an important mech-anism. These data show how preliminary estimates of shearmodulus can be made on the basis of mean effective stress,void ratio, stress history, particle grading and particle shape.

As noted above, there are other factors affecting stiffness,so it is not surprising that equation (12) cannot completelynormalise the data. For example, data presented by Claytonet al. (2006) and by Xu et al. (2007) show the influence ofparticle arrangement on stiffness under horizontal cyclictriaxial loading.

Change of stiffness with increasing strainJardine et al. (1986) and Mair (1993) have shown that the

typical strain levels around geotechnical structures such asretaining walls, spread foundations, piles and tunnels fall inthe range where soil stiffness changes most dramaticallywith strain, and that for many structures they are in therange 0.010.1% (Fig. 2). Thus both stiffness at very small

strain, and stiffness degradation data, are required for predic-tions of ground movements.

Figure 6(a) shows, as an example, the degradation ofnormalised vertical Youngs modulus with increasing strain,for triaxial compression data taken from Heymann (1998).The test results given in the figure show a remarkableconsistency, which is increased once the higher values, fortests involving reversed and repeated loadings, are excluded.Given that E0 for these materials varied from approximately24 MPa for the Bothkennar clay to 240 MPa for the LondonClay, and to 4800 MPa for the intact chalk, it is notable thatthere is so little scatter around E0:01/E0:001 0.8, and E0:1/E0:001 0.4. Jardines linearity index (L E0:1/E0:01; Jardineet al., 1984) is approximately 0.5.

If identical specimens are tested, or the same specimen istested several times without significant destructuring, thenundrained triaxial tests will produce the same very-small-strain Youngs modulus, E0, regardless of the approach path,and whether tested in triaxial compression or extension.Loading path direction does, however, have some effect atslightly higher strains. Fig. 6(b) shows, as might be ex-pected, that when soil is loaded towards the nearest failure

8007006005004003002001000

100

200

300

400

500

600

700

800

900

1000

0Mean effective stress, : kPap

Locked sand (measured)Locked sand (predicted 1200)Cp Eocene sandy clay (measured)Eocene sandy clay (predicted 300)Cp

She

ar m

odul

us,

: MP

aG

v0

Fig. 5. Shear modulus G0v from resonant column tests on twonatural undisturbed materials. Eocene sandy clay results on anumber of specimens reconsolidated to their approximate in situstress levels. Lower Cretaceous locked sand results for a singleblock sample tested at a range of isotropic effective stress levels

101001

101001

0

02

04

06

08

10

0001Axial strain: %

(a)

Nor

mal

ised

You

ngs

mod

ulus

,/

EE

u vu v

000

1

Loading towards isotropic stress

Multiple loadings

0

200

400

600

800

0001Local axial strain: %

(b)

Nor

mal

ised

You

ngs

mod

ulus

,/

Eu v

0p

t

s

Compression

Extension

Loading in triaxialcompression

Loading intriaxial extension

Fig. 6. (a) Degradation of vertical Youngs modulus withincreasing axial strain. Triaxial compression data from intactchalk, destructured chalk, undisturbed London Clay, andundisturbed Bothkennar Clay (Heymann, 1998). (b) Degrada-tion of vertical Youngs modulus with strain, for the samespecimen tested with the same initial effective stress, undertriaxial compression and extension (Clayton & Heymann, 2001)

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envelope, the initial rate of stiffness degradation with strainis higher than when loading takes place away from thenearest failure envelope. The influence of approach loadingpath (the recent stress history of Atkinson et al., 1990),while apparently unimportant at very small strain levels (lessthan about 0.05% in tests by Clayton & Heymann, 2001),can be significant at higher strains (Gasparre et al., 2007).

AnisotropyAnisotropy can be recognised at a number of scales. At

the very small (laboratory) scale, anisotropic effects havebeen variously described as inherent and induced. Inher-ent anisotropy results from grain characteristics (principallyform; Abbireddy et al., 2009; Clayton et al., 2009a) andthe depositional process. Casagrande & Carillo (1944) de-scribed this type of anisotropy as a physical characteristicinherent in the material and entirely independent of theapplied stresses and strains. Fig. 7 shows a computedtomography (CT) scan of pluviated platy sand-sized materi-al. The orientation of the particles normal to the direction ofgravity is clear, and suggests that stiffness will be higher inthe horizontal than in the vertical direction.

Induced anisotropy is caused by stress or strain changesfollowing deposition, particularly those resulting from thepost-depositional application (as is normal) of different ef-fective stresses in the horizontal and vertical directions (forexamples in relation to the London Clay, the reader isreferred to Burland et al., 1979). Changes in principal stressdirections can cause disruption of strong force chains withingranular materials (Thornton & Zhang, 2010), and changesin memory as a result of particle rotation.

As a result of the in situ stress regime, most materials arelikely to exhibit anisotropic stiffness. Youngs modulus meas-ured in a laboratory specimen is controlled primarily by theeffective stress in the direction of loading (Hardin & Bland-

ford, 1989; Yamashita & Suzuki, 1999), although because ofthe Poisson effect it will also be somewhat influenced by theeffective stresses in the normal directions. Shear modulus iscontrolled by the effective stresses acting in the plane ofdistortion (Roesler, 1979; Yu & Richart, 1984; Stokoe et al.,1995; Bellotti et al., 1996). This means that in a transverselyisotropic material, horizontal shear modulus Gh is a functionof horizontal effective stress alone (Butcher & Powell,1997), whereas vertical shear modulus Gv is a function ofboth vertical and horizontal effective stress.

Anisotropy also needs to be assessed at larger scales. Thestiffness of softer materials may be increased by the inclu-sion, for example, of more sandy or cemented layers (e.g.claystones within the London Clay, and hydrate sheetswithin deep ocean sediments; Fig. 8). The stiffness of weakrocks is significantly reduced by fracturing, jointing and (inthe case of the Chalk) dissolution associated with stressrelief, and weathering (Lord et al., 2002, Matthews &Clayton, 2004). In the unusual example shown in Fig. 9 thedominant joint set, probably associated with a plane ofstiffness isotropy, is sub-vertical. More normally in the chalkit is sub-horizontal, associated with the shallow dip ofbedding.

At the largest scale, relevant for example to the volume ofsoil loaded by a large foundation, or unloaded during deepbasement or tunnel excavation, many soils and rocks showevidence of heterogeneity in the form of bedding and oflayering of different materials within that bedding. Fig. 10shows piezocone results in gold tailings; the rhythmicdeposition of finer and coarser materials results from varia-tions in the position of the central pool, which (as a resultof the management of the dam) moves around the deposi-tional area with time. Similar rhythmic deposition can beseen in many natural deposits, for example varved claysdeposited in glacial lakes, and in the Cenomanian Chalk,where the layering is driven by global climate changesresulting from Milankovitch cycles (Hart, 1987).

As might be expected from the discussion above, aniso-tropy of stiffness has been widely recognised in naturalmaterials, both in seismic geophysical testing (Butcher &Powell, 1997) and in laboratory measurements (Ward et al.,1959; Atkinson, 1975; Graham & Houlsby, 1983). However,few studies have been carried out in sufficient depth todetermine the full set of anisotropic stiffness parameters,two notable exceptions being reported by Lings et al. (2000)for the Gault Clay, and by Gasparre et al. (2007) for theLondon Clay. Values of effective Youngs moduli and ofshear moduli for the London Clay at Heathrow Terminal 5are shown in Fig. 11. Bearing in mind the likely variation ofthe ratio of effective horizontal to vertical stress (K0) to beexpected over the 30 m profile shown in Fig. 11 (Burland etal., 1979), these observations suggest that anisotropy ofvery-small-strain stiffness seen here is likely to be domi-nated by factors other than effective stress ratio.

The effect of loading, and ultimately of destructuring, onthe anisotropy of stiffness remains a matter of some debate.Jovicic & Coop (1998) suggest that very large plastic strainsare necessary to affect the inherent anisotropy of weak rocksand stiff clays, while test data for isotropic effective stressloading of undisturbed Bothkennar Clay (Clayton et al., 1992)and chalk (Clayton & Heymann, 2001) suggest that even smallstrains may be sufficient to change the degree of stiffnessanisotropy as a result of comprehensive destructuring.

Cyclic loading and rate effectsIt has long been held that the observed stiffness of soil is

strongly dependent on the rate at which it is tested. As aresult, the stiffness values obtained from field seismic or

(b)

(a)

Fig. 7. CT scan showing preferred particle orientation of 1 mmpluviated glass glitter (Abbireddy, 2008): (a) horizontal section(view from top); (b) vertical section (view from side)

STIFFNESS AT SMALL STRAIN: RESEARCH AND PRACTICE 11

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(b)

(a)

Hydrate veins

Fig. 8. Sub-vertical orientation of methane hydrate veins in very soft deep oceansediment: (a) CT scan of methane hydrate veins in a very soft deep oceansediment core; (b) lower hemisphere projections from three core sections,showing preferred orientations of hydrate veins

Fig. 9. Structured chalk, showing preferred orientations ofdiscontinuities (image courtesy of Professor R. N. Mortimore,University of Brighton)

u

10864

1000800600400

20

21

22

23

24

25

Dep

th: m

qc

qc: MPa20

2000u: kPa

Slimes

Sands

Fig. 10. CPT profiles of pore pressure and cone resistance fromgold tailing (Obuasi, Ghana) showing interlayered slimes (fines)and sands (data courtesy of Professor E. Rust, University ofPretoria)

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laboratory dynamic tests were thought for many years to belarge overestimates of the static stiffnesses required for mostengineering predictions of ground deformations. Two factorsare now known to have been behind the development of thisview.

(a) The effects of sampling disturbance in many cases ledto reductions in the stiffness measured in laboratorytests, through destructuring during sampling and as aresult of associated decreases in effective stress.

(b) Tests carried out on reconstituted and destructuredmaterials did indeed demonstrate significant rate effects,but the material tested was not representative of naturalmaterial.

Rate effects are now considered to be relatively unimportantat very small strain levels. For example, for the tests on stiffclays and mudstones reported by Tatsuoka & Shibuya(1992), stiffness was found to be almost independent ofstrain rate for strains ,0.001%. At higher strains, threesignificant effects have been observed: see for exampleIsenhower & Stokoe (1981), Tatsuoka & Shibuya (1992) andLo Presti et al. (1997). First, the extent of the elasticplateau increases with strain rate, so that the results ofresonant column tests, cyclic and monotonic loading testscannot be expected to be the same at small (as distinct fromvery small) strains. Second, shear stiffness becomes moresensitive to rate of loading at intermediate strains, saybetween 0.01% and 0.1% strain (see also Sorensen et al.,2007; Fig. 12), and finally the stressstrain response undercyclic loading can be expected to be stiffer than undermonotonic loading. Because of this, Lo Presti et al. (1997)conclude that the very high cyclic strain rates imposed byresonant column testing make it not very suitable for themeasurement of static monotonic stiffness degradation. The

results are likely to provide a lower limit when comparedwith other measurements.

SIGNIFICANCE OF PARAMETERS FOR PREDICTEDPERFORMANCE: A NUMERICAL EXPERIMENT

As noted above under Geotechnical background, back-analysis of monitored construction, and particularly of deepexcavations in the London Clay, has indicated the complex-ity of soil behaviour, and has suggested that, particularly ifdisplacement patterns are to be predicted, soil needs to be

Bender element tests

Static torsional test

Resonant column apparatus

Triaxial tests

Hollow cylinder apparatus

Effective Youngs moduli, : MPaE

4000 200

Dep

th: m

0

10

20

30

EhEv Gv

Dep

th: m

2000 100Shear moduli, : MPaG

0

10

20

30

Gh

Fig. 11. Profiles of small-strain effective Youngs moduli and shear moduli for the London Clay atHeathrow Terminal 5 (modified from Gasparre et al., 2007)

a 08%/ h.

a 02%/ h.

a 005%/ h.

4

400

Dev

iato

r st

ress

, :

kPa

q

Shear strain, : %s

200

0

600

30 521

Intact London Clay

Fig. 12. Effects of changes of strain rate during shear ondeviatoric stress. Intact London Clay (Sorensen et al., 2007)

STIFFNESS AT SMALL STRAIN: RESEARCH AND PRACTICE 13

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treated as a non-linear transversely isotropic material. Asignificant number of stiffness parameters are required forsuch a material, as described in the section on Constitutiveframeworks for stiffness, and the question then arises as towhether all of these need be determined with the sameaccuracy. For example, given that they relate to strain levelsmuch smaller than are expected in the zone of influence, arethe values of E0 and G0 really significant when attemptingto predict deformations close to such structures?

The answer to such a question must be that the impor-tance of different parameters depends on the ground, on thestructure, and on the aspect of performance to be predicted.For example, it might intuitively be expected that horizontalYoungs modulus would be particularly significant in control-ling the horizontal displacement of retaining structures,while vertical stiffness may be more significant when pre-dicting the settlement of spread foundations. Therefore inpractice some kind of sensitivity analysis will be required inmany cases, in order to identify which parameters dominatethe particular problem under consideration.

As a demonstration of the significance of different stiff-ness parameters, a numerical experiment has been carriedout, to estimate the ground deformations around a singlypropped retaining wall. The underpinning methodology isdescribed below. The analyses were carried out using two-dimensional FLAC version 5.0 (Itasca, 2005). Stiffnessdegradation was implemented using a FISH function.

Problem geometryFigure 13 shows the geometry of the selected problem.

Dimensions are similar to those of a dual-carriageway high-way underpass. The excavated depth is 8 m, and the fullwidth of the excavation is 30 m (i.e. the distance from thewall face to the centreline of the excavation is 15 m). The0.6 m thick retaining wall is 16.5 m long, and is supportedby props at 1 m below ground level, with loads equivalent toan 8.75 m centrecentre spacing. A preliminary parametricstudy was undertaken to explore the effects of mesh sizeand boundary locations (Iqbal, personal communication). Forthe analyses reported here, computational time was reducedby placing the vertical boundary 80 m back from the face ofthe retaining wall, with the basal boundary 40 m belowground level. A single soil type was used in each analysis.The wall was wished in place (Gunn et al., 1992), and auniform value of K0 1 was therefore used to calculatestarting in situ stress levels (Gunn & Clayton, 1992). Ex-cavation was modelled in 1 m stages, with the prop beinginstalled after the first excavation step.

Soil modelsAnalyses were run with two sets of variables:

(a) Uniform stiffness, or stiffness increasing with depth.(b) A range of constitutive models:

Case 1 Linear elastic soil, with Eu 100 MPa.Case 2 Case 1, but with MohrCoulomb plastic yield

at su 100 kPa.Case 3 Linear elastic soil, with stiffness increasing

with depth.Case 4 As in Case 3, but with stiffness decreasing

with strain. A base case was used to explorethe effects of some variables (e.g. very-small-strain stiffness, and rate of stiffness degrada-tion) on the displacements predicted by thismodel.

Case 5 As in Case 4 base case, but with variousdegrees of transverse isotropy (Euh . E

uv, etc.).

Analysis Case 3 was based on the short-term parametersdeduced by Hooper (1973) from movement around the HydePark Cavalry Barracks excavation.

As the predictions were for the undrained (short-term)case, only one parameter (Eu; recall that u 0.5, and Gand K are dependent on Eu) was required for the isotropicCases 1, 2, 3 and 4. In Case 3 Eu varied with depth, asshown by the dashed line in Fig. 14.

In Case 4 stiffness increased with depth but reduced withincreasing strain. The base case adopted a reference stiff-ness, Eu0, arbitrarily taken as four times the values back-analysed from Hyde Park Cavalry Barracks (hpcb) (Hooper,1973). Stiffness degradation was modelled by assumingconstant values of tangent stiffness above, below and be-tween fixed octahedral strain limits shown in Table 2.

Figure 14 shows the variations of stiffness with depth atdifferent strain levels, and Fig. 15 compares the steppedinput tangent stiffness values with the secant Youngs mod-ulus degradation curve computed from them, for soil at10 m depth.

The transversely isotropic cases explored in Case 5required three independent parameters (Euv, E

uh and Gv). The

ratio Euv=Euh was varied from analysis to analysis, and Gv

was obtained from

85 m

15 m

AB B

A

Props at 1 m depth

8 m

CL

Fig. 13. Selected retaining wall geometry for numerical analysis

5004003002001000

Undrained secant Youngs modulus, : MPaE usec

Dep

th: m

0

5

10

15

20

Hyde Park Cavalry Barracks

002% 0006%

Uniform stiffness

006% 002%006%02%

Base case (Case 4)for non-linear analyses

4 0002%E E0 hpcb

0006% 0002%

Fig. 14. Comparison of undrained secant Youngs moduli valuesat different strain levels, as a function of depth, for non-linearbase case, E0 4Ehpcb

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Gv Euh E

uv

2Euh Euv(13)

Lekhnitskii (1981)As noted by both Simpson (1992) and Atkinson (2000),

the mobilised strength at any strain level is equal to the areaunder the tangent stiffnessstrain curve, up to that strainlevel. Therefore, even though the soil is modelled as elastic,there are restrictions on the values of stiffness that can beused as input. For example, increasing the rate of stiffnessdegradation with strain will reduce the available strength ata given strain level, and in an undrained retaining wall orspread foundation, analysis may prevent stability withinreasonable deformation limits. For the base case describedabove, the mobilised undrained shear strength at 1% strain isof the order of 60 kPa at 10 m depth, which is a relativelylow value for the London Clay (Marsland, 1972; Hight,1986).

Impact of model and parametric variations on predicteddisplacements

In order to simplify the discussion, the following keyoutputs are compared below for different soil models andparametric values:

(a) horizontal wall displacements(b) vertical displacements at original ground level(c) vertical displacements at excavation level(d ) bending moments and prop loads.

Figure 16 shows horizontal displacements on the plane ofthe back of the wall (shown as AA in Fig. 13). The use ofa constant stiffness profile with depth (Case 1) leads tounrealistic predictions of the pattern of displacement of thewall, when compared with field observations in the London

Clay. Introduction of MohrCoulomb yielding (Case 2) haslittle effect. Increasing stiffness with depth (Case 3) has amajor impact on the shape of wall deflections, predictingmaximum horizontal movements at about excavation level(as observed in practice, e.g. by Burland & Hancock, 1977).The predicted shape is further enhanced by the introductionof higher stiffnesses at small strains (Case 4). It is clear that,for a problem of this type, determination of the stiffnessprofile must be a priority.

Figure 17 shows the vertical displacements at originalground level, behind the wall. Again, there is little differencebetween Case 1 and Case 2. Cases 1, 2 and 3 show heave,and tilt away from the wall, at between approximately 10 mand 20 m. In contrast, Case 4 shows settlement between 5 mand 25 m behind the wall, associated with tilt towards theexcavation. This mirrors the case record at New Palace Yard,where on the basis of a linear-elastic analysis Big Ben waspredicted to tilt away from the excavation for the new House

Table 2. Base case reduction in tangent stiffness values

Octahedral strain level: % Stiffness ratio, Eu=Euhpcb

,0.002 40.0020.006 2.50.0060.02 1.50.020.06 0.70.060.2 0.350.20.6 0.15.0.6 0.05

0

100

200

300

00001 0001 001 01 1 10

Strain: %

Und

rain

ed s

ecan

t You

ng's

mod

ulus

,: M

Pa

Eu se

c

Input tangentYoungs modulus

Resulting secantYoungs modulus

At 10 m below ground level

E0 uhpcb4. E

Fig. 15. Secant moduli resulting from input tangent moduli, forbase case at 10 m below ground level

30

20

10

02 0 2 4 6 8 10 12 14

Horizontal displacement: mm

Dep

th b

elow

gro

und

leve

l: m

1 Uniform linear elastic, 100 MPa2 Uniform linear elastic, MohrCoulomb3 Linear, stiffness increasing with depth4 Non-linear elastic, stiffness

increasing with depth

Eu

Base of excavation

Base of wall

Case 4 Case 3 Cases 1 and 2

Case 1

Case 2

Fig. 16. Horizontal displacements on plane AA (back of wall)for four soil models

5

0

5

10

15

20

60 40 20 0 20

Distance behind wall: m

Hea

ve: m

m

Back of wall

Excavation

Cases 1 and 2

Case 4

Case 3

Fig. 17. Heave of soil at original ground level (BB in Fig. 12)for four different soil models

STIFFNESS AT SMALL STRAIN: RESEARCH AND PRACTICE 15

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of Commons car park, but was observed in reality to tilttowards it (Burland & Hancock, 1977; Simpson et al., 1979)

Figure 18 shows predicted bending moments in the wall,for Cases 1 to 4. Again, there is little difference between thepredictions for Case 1 and Case 2. As would be expectedfrom the deformed shapes in Fig. 16, the introduction of astiffness increase with depth leads to a significant increase(more than 50%) in maximum bending moment, and this isfurther enhanced by the increase of stiffness at small strains.The use of the simple models leads to lower predictions ofbending moment. These changes do not affect prop load,however, which for this example were found to remain fairlyconstant (10%), being largely a product of initial horizon-tal effective stress and wall geometry.

In addition to the variations in constitutive models de-scribed above, a number of parametric variations have beenused in conjunction with Case 4 (isotropic stiffness increas-ing with depth and decreasing with strain) in order toexplore the sensitivity of key outputs to uncertainties in

(a) reference stiffness moduli (E0 and G0)(b) rate of stiffness degradation.

Figure 19(a) shows four variations of stiffness at very smallstrains. The effects on surface settlement can be seen in Fig.19(b). These variants were produced by changing the magni-tude and range of the very-small-strain tangent modulus, Eu0.Both the shape and the magnitudes of settlements behind thewall are affected. Fig. 20 shows different rates of stiffnessdegradation. The shaded area is taken from the experimentalresults previously shown in Fig. 6(a). The left-hand curveshows Case 4 base case values, and the other two curvesshow additional lower rates of stiffness degradation assumedfor additional analyses. At any given intermediate strainlevel the expected stiffness varies very significantly, depend-ing upon the line adopted. For example, at 0.02% strain thestiffness increases by 50%, and then doubles, as one movesfrom the base case through to the reduced rates of stiffnessdegradation shown by the other two lines in Fig. 20. For thisproblem, there are very significant associated reductions inthe predicted deformations of the wall, the ground surface,

and (Fig. 21) the vertical movements at base of excavationlevel.

Finally, Fig. 22 shows the effect of undrained modularratio (Euh=E

uv ) on the predicted maximum horizontal wall

movement. Maximum wall movement was normalised by thevalue from the isotropic (Case 4, base case) analysis.Horizontal Youngs modulus has a large effect, and a mod-ular ratio of 2.5, approximately the value expected in theLondon Clay Formation (Fig. 11), halves the predictedmagnitude of wall movement.

0

10

20 0 20 40 60 80 100 120 140 160 180

Bending moment: kNm

Dep

th: m

Case 4Case 3

Case 2Case 1

1 Uniform linear elastic, 100 MPa2 Uniform linear elastic, MohrCoulomb3 Linear, stiffness increasing with depth4 Non-linear elastic, stiffness

increasing with depth

Eu

Fig. 18. Predicted bending moments in the wall, for fourdifferent soil models

0

100

200

300

400

500

00001 0001 001 01 1 10

Strain (%)(a)

Und

rain

ed s

ecan

t You

ngs

mod

ulus

,: M

Pa

Eu se

c

60 40 20 0 20

Distance in front of wall: m(b)

Hea

ve: m

m

Back of wall

4

2

0

2

4

6

8

10

Excavation

Linear rangeextended to 002%

E E0 02in Case 4

Case 4 base case

E0 reduced to Case4 at 0006% strain

Base caseCase 4

E E0 02in Case 4

Linear rangeextended to 002%

E0 reduced to Case4 at 0006% strain

Fig. 19. Effect of changes in very-small-strain stiffness onvertical movement behind the wall: (a) variations in Eu at10 m depth; (b) predictions of surface settlement behind thewall

0

100

200

300

00001 0001 001 01 1 10

Strain: %

Und

rain

ed s

ecan

t You

ngs

mod

ulus

,: M

Pa

Eu se

c

At 10 m below ground levelShaded area from Fig. 6(a)

Reduced ratesof stiffnessdegradation

E E0 uhpcb4

Base caseCase 4

Fig. 20. Undrained secant moduli against strain, showing basecase 4 and two reduced rates of stiffness degradation, comparedwith observed values in the triaxial test (Fig. 6(a))

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In summary, for the particular problem that has beenanalysed here, an assumption of uniform stiffness with depthleads to unrealistic wall deflections and low predictions ofbending moments. Increasing stiffness with depth givesbetter estimates, when compared with field observations inthe London Clay. It is clear that the determination of areliable stiffness profile must be a priority in any investiga-tion. Although a number of different combinations of soilstiffness model may seem from the figures to give broadlysimilar estimates of ground movements, high initial stiffness,coupled with stiffness degradation with increasing strain, isneeded to mimic the pattern of observed ground surfacemovements for structures that take the soil to intermediatestrain levels, for example at the House of Commons carpark. Predicted displacement patterns are sensitive to mostparameters, including very-small-strain stiffness, rate of stiff-ness degradation, and anisotropy.

These observations may not be generally true, however. Instiffer materials, strain levels may be much smaller, andcloser to the elastic plateau. In softer materials, significantdestructuring may take place, and an elastic approach todeformation modelling may not be appropriate.

MEASURING STIFFNESS PARAMETERSThe selection of methods for measuring stiffness at any

given site needs to be made in the context of a number offactors:

(a) the variability of the ground(b) the relative merits of field and laboratory measurement

techniques(c) prior experience of the use of the technique in the

given ground conditions(d ) the availability of equipment and personnel in the

country or region where the work is to be carried out(e) the need for redundancy of data.

This section first discusses these issues, before passing on togive examples of a range of techniques that have been usedby the author.

The heterogeneity of the ground is important, becauseeven in the most intensely investigated site it is unlikely thatmore than one part in one million of the volume of groundaffected by construction will be sampled, seen (for examplein trial pits or as core), or mechanically explored (e.g. usingpenetrometers) (Broms, 1980). If, as is frequently the case,there is a high degree of vertical variability but relativelylittle lateral variability (e.g. as a result of stratification orweathering), then, having established lateral correlations be-tween different layers (for example by profiling, by indextesting, or by classification testing) it may be practical todetermine the stiffness of the different layers. But if lateralcontinuity cannot be established, then the priority must be tocarry out profiling, perhaps deducing stiffness from simpleand approximate correlations (e.g. between CPT or SPT andYoungs modulus). In such situations the advanced andgenerally more reliable methods of stiffness measurementdescribed in the paper are unlikely to be of practical use.

The relative merits of field and laboratory testing havebeen well rehearsed over the years (e.g. Dyer et al., 1986;Clayton et al., 1995b). In terms of stiffness determinations(as will be discussed further below), field seismic testingtechniques can be significantly affected by background noise.But because they can be very effective in determiningsubsoil geometry and heterogeneity, are carried out at the insitu stress level, and can test large volumes of soil (soincluding the effects of smaller-scale heterogeneities, such asfractures, and large particle sizes), they remain attractive formajor projects, such as deep excavations, tall structures andseismically sensitive projects (e.g. nuclear power plants). Insitu test methods can, in most cases, avoid the worst effectsof borehole and sampling disturbance, although installationand bedding effects can still be significant when relativelysmall volumes of soil are tested close to the wall of anexploration hole (for example during pressuremeter testing).

Laboratory tests can also suffer from background noise(of various types), and can be impractical, because longtesting times can delay the design process. In addition, alllaboratory test specimens will have been disturbed to someextent by drilling and sampling. Sample disturbance canmake the results of laboratory tests unrepresentative, throughthree mechanisms:

(a) removal of total stresses (so called perfect sampling;Skempton & Sowa, 1963), which includes the removalof any shear stresses that exist in the ground

(b) changes in effective stress, as a result of tube samplingstrains (Clayton et al., 1998), or air entry and swelling

(c) destructuring (Clayton et al., 1992; Hight & Jardine,1993).

On the positive side, as will be seen, laboratory testsgenerally have controlled boundary conditions, and for thisreason can be used to obtain a wider range of parameters

0

10

20

30

80 60 40 20 0 20Distance in front of wall: m

Hea

ve: m

m

Linear elastic, stiffnessincreasing with depth

Base case Case 4Non-linear elastic,

stiffness increasing withdepth

Reduced rates ofstiffness degradation

(see Fig. 20)

Fig. 21. Effects of rate of stiffness degradation on predictedvertical movements at excavation level

uvh 049 uhh

uh

uv1 /2 E E

0

02

04

06

08

10

12

1 15 20 25Modular ratio: /E Eh v

Ra

tio o

fm

axim

um w

all d

ispl

acem

ents

E EuhuvGv

E uhGh 2E Euh

uv2(1 )

uhh

Fig. 22. Case 5: effect of stiffness anisotropy on maximum walldisplacement

STIFFNESS AT SMALL STRAIN: RESEARCH AND PRACTICE 17

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than in situ tests. In addition they are (hopefully) carried outin a better-regulated (laboratory) environment than are fieldtests.

The availability of test equipment, experienced test per-sonnel and written standards and method statements isclearly very important in all testing, but is critical for manystiffness tests, which can be complex. Experience of testingin similar soil conditions is essential. In this respect labora-tory testing has the advantage that samples can be flown tokey laboratories, while field seismic testing must sometimesrely on (reasonably) local skills.

Finally, when good determinations of stiffness are essen-tial, for example because the range of measured valuesproduce significantly different designs, there is a need fordata redundancy. Poorly conducted tests, or tests affected bybackground noise (for example) can then be identified andignored. Combinations of field and laboratory tests, tests fordifferent stiffness parameters (Eu and G, for example) andtests at different strain levels are helpful in this respect.

Marsland (1986) has stated that

the choice of test methods and procedures is one of themost important decisions to be made during the planningand progress of a site investigation. . . . In assessing thesuitability of a particular test it is necessary to balance thedesign requirements, the combined accuracy of a test andassociated correlations, and possible differences betweentest and full-scale behaviour.

A great many techniques exist from which stiffness param-eters can be derived, ranging from the simple SPT to thesophisticated self-boring pressuremeter. This paper considersa limited selection of more unusual techniques, based on theauthors experience and belief that they will have value inmany situations. In particular, two classes of test arereviewed:

(a) field geophysics(i) continuous surface wave testing(ii) down-hole geophysics(iii) cross-hole geophysics

(b) laboratory methods(i) bender element testing(ii) resonant column testing(iii) advanced triaxial testing.

Field geophysicsUp until the 1980s it seems to have been widely assumed

that stiffnesses measured in dynamic (laboratory and fieldseismic) tests might be about one order of magnitude higherthan those needed for analysis of ground movements, andwere therefore only of practical significance for dynamicproblems, such as the effects of machinery vibration, orearthquake loading on construction (Ballard & MacLean,1975; ASCE, 1976). During the late 1970s and the 1980s,and partly as a result of the realisation by geotechnicalresearchers that statically measured small-strain stiffness wasmuch higher than previously thought, it became apparentthat field seismic testing might be used to determine stiff-ness values for more routine, static, geotechnical design.

Abbiss (1979) used first arrival times in a seismic refrac-tion survey, coupled with an interpretation based on Dobrins(1960) equation for seismic velocity increasing linearly withdepth, to determine the Youngs modulus values of thefractured Chalk Mundford, and found encouraging agreementwith stiffness values obtained from both down-hole(865 mm) plate tests, and values back-figured from observedground movements beneath an 18.6 m diameter tank loadingtest. He later reported (Abbiss, 1981) stiffness values derivedfrom continuous surface wave and seismic refraction shear

wave testing on the London Clay at Brent, which Burland(1989) compared with undrained Youngs modulus values at0.01% axial strain made using local-strain instrumentationon specimens of the London Clay Formation from CanonsPark, North London, noting that the dynamic values ofundrained Youngs modulus were only about 30% greaterthan the values of Eu(0:01).

Over a period, Hoar & Stokoe (1978), Abbiss (1981), Chuet al. (1984), Sully & Campanella (1995), Bellotti et al.(1996), Hight et al. (1997) and others have demonstrated thepotential for measuring stiffness anisotropy. But despite thepractical potential for seismic field tests to provide valuablestiffness data, seismic techniques remain relatively unknownin general geotechnical engineering practice. The followingsections describe the technical background, and some testmethodologies, and give examples of their application.

Background. The seismic field geophysical techniques usedin geotechnical engineering make use of two types of seismicwave:

(a) body waves, which travel through the body of a solid,unaffected by its surface, with a velocity and ray pathcontrolled only by the density and stiffness, and theirvariation

(b) surface waves, which in general propagate along theinterfaces between materials with different densitiesand/or stiffnesses, or along the ground surface.

There are two types of body waves: primary (P), firstarriving, compressional waves; and secondary (S), or shearwaves. P waves induce volumetric strain (Fig. 23(a)), andtherefore travel at a speed related to the undrained volu-metric stiffness of the ground, since the dominant frequen-cies (20400 Hz, according to Woods, 1994) do not allowdrainage. In saturated near-surface soils, values of compres-sional wave velocity are typically found to be of the orderof 1500 m/s, the calculated undrained bulk modulus beingsimilar to that of water rather than that of the volumetric

(a)

(b)

(c)

Direction of wave travel

Shear wave

Compression wave

Shear wave

Fig. 23. Compressional and shear wave travel: (a) volumetricdistortion. Vp depends upon the volumetric compressibility ofboth soil skeleton and pore water. (b) Shear distortion in thevertical plane; Gv0 rrV 2s hv. Rayleigh waves travel at similar,but slightly slower, speeds. (c) Shear distortion in the horizontalplane. Gh0 rrV 2s hh

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skeletal stiffness of the soil. This makes the measurement ofP-wave velocity unattractive in most geotechnical surveys.

Shear waves (Figs 23(b) and 23(c)) induce change inshape without change in volume, and (provided the correctbulk density can be estimated or measured) the stiffnessesdetermined from them are then independent of whether ornot the ground is saturated. Shear waves travel at a velocitythat is a function of soil density and shear stiffness in theplane of distortion, arriving after the compressional waves.Thus, if a seismic source rich in both P and S waves is used,the S-wave first arrivals may be obscured by the P waves,and the travel time overestimated. Provided that they can bedetected, shear-wave arrivals can be used to determine theshear modulus, G. From this, Youngs modulus and bulkmodulus can be calculated, if Poissons ratio is known or canbe estimated, and the ground stiffness is assumed to beisotropic. In anisotropic ground, shear wave velocities (fromdifferent modes of distortion; compare Figs 23(b) and 23(c))can in principle be used to determine both Gv and Gh.

Most of the energy input by a source at the groundsurface will travel away from the point of input as aRayleigh wave. The Rayleigh wave is a species of surfacewave (the other being the Love wave) that results from theinteraction of compressional and shear waves at the groundsurface, propagating away from a surface energy source withan elliptical motion in the vertical plane. In given groundconditions the Rayleigh wave will travel a little slower thanthat of a vertically polarised shear wave. It is a function ofbulk density Gv and Poissons ratio, and, all other thingsbeing equal, for Poissons ratios of 0.25 and 0.5 the shearwave velocities will be greater than the Rayleigh wavevelocities by 9% and 5% respectively. Rayleigh waves aredispersive; when (as is usual) stiffness varies with depth,their velocity (VR) varies with wavelength (), because long-er wavelength energy engages with deeper, stiffer ground.

Figure 24 illustrates the layouts and principles of threeestablished field geophysics techniques that will be discussedbelow. Fig. 24(a) shows continuous surface wave (CSW)testing. A vibrator, which may be mechanical, servo-hydraul-ic or electro-magnetic, applies a single-frequency sinusoidalforce at the ground surface. Rayleigh waves travel awayfrom the vibrator, and are detected by co-linear geophonesat a range of distances from the source. By varying the inputfrequency a profile of phase velocity against wavelength isobtained, from which a stiffnessdepth profile can be com-puted.

Figure 24(b) shows down-hole seismic testing. This uses asurface source (a sledgehammer striking a weighted metalbeam, for example) to input shear wave energy to theground. For practical reasons, and to avoid significant energytravelling down the borehole and its casing, the energysource is offset from the top of the hole, such that the traveldistance (typically calculated on the basis of a straight ray)is greater than the depth. In a noisy environment, data froma number of blows can be stacked (i.e. added to eachother) to improve the signal-to-noise ratio of the receivedsignal. The arrival of seismic energy is detected at deptheither by geophones clamped within a plastic-cased borehole(to avoid borehole collapse while allowing transfer of energyfrom the ground to the geophones), or by geophones withina seismic CPT. In either case, it is desirable to have two setsof three orthogonally orientated geophones in each detectorarray, separated vertically by about 1 m. This allows thetravel time to be determined from waveforms detected atboth sensors from the same hammer blow.

Figure 24(c) shows the principle of cross-hole seismictesting. Three co-linear boreholes, lined with grouted plastic(ABS) casing and at a 57 m separation, are generally used.A borehole verticality survey is required in order to calcu-

late the actual distance between the boreholes at each testdepth (typically 1 m intervals), since some deviation fromvertical will have occurred during drilling and casing instal-lation. A down-hole shear wave energy source and two setsof three-component geophones are lowered to the bottom ofthe holes and progressively raised, and clamped to the bore-hole walls to generate shear waves and take data, typicallyat 1 m intervals. The use of two sets of receivers avoids theissue of trigger accuracy, but increases the cost of this typeof test. The inter-borehole distance is divided by the traveltime at each depth, determined either on a first break orpeak-to-peak basis, to calculate the shear wave velocity.Most commonly, the energy source is clamped in the bore-hole and struck vertically, to produce a vertically polarisedhorizontally travelling shear wave, from which Gv can becalculated. Horizontally polarised, horizontally travellingshear wave sources have also been used (Hoar & Stokoe,1978; Woods & Henke, 1979; Sully & Campanella, 1995;Butcher & Powell, 1997), from which Gh can, in favourableconditions, be determined.

(a)

(b)

Seismic CPT

Three-component geophones

(c)

Three-component geophones

Down-hole hammer

G Gv h,

Gv

Gv

Geophones

Fig. 24. Three established field seismic testing techniques:(a) continuous surface wave; (b) down-hole; (c) cross-hole

STIFFNESS AT SMALL STRAIN: RESEARCH AND PRACTICE 19

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Continuous surface wave testing. An introductory text on thepracticalities of geotechnical surface wave testing is providedby Matthews et al. (1996). Although spectral analysis ofsurface waves (SASW) (Nazarian & Stokoe, 1984; Stokoe &Nazarian, 1985) is the more economical in terms ofequipment and test time, and can allow greater depths tobe explored, the CSW method, which uses a mono-frequencysource, has been found to provide better data in ageotechnical setting, because unwanted background noise ismore easily recognised, avoided or filtered.

CSW has several disadvantages.

(a) The test data, once processed, produces estimates ofonly a single parameter, Gv, at very small strain.

(b) On a noisy site it may be very difficult to recordsignals with the necessary coherence to avoid thegeneration of a scattered stiffnessdepth profile.

(c) The depth of investigation is limited. Experience withlightweight vibrators suggests that it will be about 58 m in a stiff clay, rising to 1020 m in weak rock.

(d ) Interpretation of CSW data relies generally on verysimplistic interpretations (for example, wavelength/3),although more sophisticated inversion (HaskellThomson method referred to by Lai & Rix, 1998) ordynamic finite-element modelling methods (Clayton etal., 1995a) can be used.

(e) The ground may vibrate in a number of modes, whichin routine testing may not be recognised; and near-fieldeffects may be significant.

( f ) The Poissons ratio uncertainty leads to a possible errorin predicted stiffness of about 10%.

(g) In complex ground, interpretation may be madeuncertain by aliasing.

(h) A complex (irregular) ground surface can significantlyaffect Rayleigh wave propagation, leading to difficultyin interpreting data.

Despite these limitations there are many situations where theadvantages of CSW make it an invaluable tool.

(a) It is a relatively low-cost technique.(b) It is non-intrusive, which contributes to its low cost, but

is also an advantage when working on contaminatedland.

(c) The test requires relatively little space, for shallowdepths.

(d ) It can be used to determine the stiffness profiles ofnear-surface materials, which are important (for exam-ple) in linear and low-rise projects (highways, pipelines,housing).

(e) It can provide stiffness profiles in highly weathered andfractured ground, and where coarse particles (e.g.boulders) prevent most other methods being used.

Figure 25 shows a recent example (Heymann et al.,2008), where CSW testing was carried out for the SouthAfrican Gautrain project, and the results compared withstiffnesses back-analysed from the ground movements be-neath a 20 m 3 20 m 3 10 m high load provided by concretekentledge. The material tested was composed of chert grav-els and boulders in a matrix of hillwash sand to approxi-mately 2 m, underlain by dolomite residuum comprisingwad and chert in highly variable proportions down tobedrock. It is almost impossible to obtain values of stiffnessin such materials, except through expensive and time-consuming area load tests. Stiffnesses back-figured from thedata from two extensometers (A and B) located under thekentledge are shown, for three depth ranges, on the right-hand side of Fig. 25. The reduction of stiffness with increas-ing strain can clearly be seen, and the stiffnesses of deepermaterials tend to be greater. Stiffnesses derived from CSW

testing, interpreted in two ways (Gazetas, 1982; Butcher &Powell, 1996; Lai & Rix, 1998), are shown on the left-handside of the graph. As might be expected in such difficultground conditions, there is considerable variation in meas-ured stiffness, but the values obtained from CSW testingappear to give a reasonably conservative estimate whencompared with those from full-scale measurements at smal-ler strain levels. Matthews et al. (2000) have similarly showngood agreement between CSW stiffness measurements andthose obtained from 1.8m diameter plate loading tests onweathered and fractured chalk.

Down-hole geophysics. The potential problems of measuringstiffnesses in an urban setting and on a live construction siteare illustrated by a case history given by Hope et al. (1998).At the time of the seismic surveys reported in this paper ahighway was under construction in an old railway cutting,which created complex ground surface geometry in the areaof testing. The available space within which to carry out thesurveys was very limited, and lay immediately alongside thesite. Ground conditions consisted of 34 m of made groundand glacial till, overlying weathered mudstones and sand-stones. Six seismic methods were applied in an attempt to getstiffness data, in order to enhance a dataset previouslydeveloped using dilatometers and pressuremeters. Problemsof ground-borne vibration were created by traffic on nearbyroads, by construction plant, and by a nearby electricitysubstation.

Of the six methods initially proposed (parallel cross-holewith vertically polarised shear waves, CSW, downhole seis-mic profiling, SASW, shear wave refraction, and upholeseismic profiling), only two (CSW and downhole seismicprofiling) could achieve a good enough signal-to-noise ratioto produce credible results. The data from these are shownin Fig. 26, along with the dataset from dilatometer andpressuremeter testing along the length of the road. A numberof lessons can be learnt.

(a) The limited effective depth of CSW testing (about 57 m) can be seen.

(b) The difficulties of down-hole testing near to groundsurface are obvious, as the scatter of this dataset within3 m of ground level shows.

(c) Despite all the problems, data were produced, but thiswas only because flexible and varied arrangementscould be used to obtain them. A rigid contract,

/26

Youn

gs

mod

ulus

,: M

Pa

Ev

1200

0

400

800

Vertical strain: %001 01

B

A

B

AA

B

Back-analysedfield data

CSWdata

Depth02 m2 6 m6 12 m

2 6

m 0 2

m

612

m

Lai & Rix (1998)

Fig. 25. Comparison of stiffnesses derived from CSW with thoseback-analysed from ground movements beneath a loaded area.Test site 55: extensometers A and B (redrawn from Heymann etal., 2008)

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preventing repeat visits to site and restricting themethod to be used, would probably have failed.

(d ) The value of the seismic dataset is illustrated by thenumber of data points obtained, the tight grouping ofthe seismic dataset (as compared with the pressuremeterand dilatometer datasets), and the agreement betweentwo different methods of obtaining stiffness fromseismic data.

Cross-hole geophysics. During the 1990s the author andcolleagues from the University of Surrey carried out anumber of seismic surveys in the south-east of England. Thedata from three cross-hole surveys in the London Clay aresupplemented by those from a later survey reported by Hightet al. (2007) in Fig. 27. Also shown in Fig. 27 is an insetmap, showing the locations of the four sites. Some sites (e.g.the Surrey Research Park at Guildford, to the south-west ofLondon) would normally be considered relatively quiet, butthe presence of several railway lines and a trunk road, allwithin a couple of kilometres from the site, meant that carehad to be taken when recording data, and much had to berejected on the basis of observed background noise. The A1North Circular Road site was urban, located on a trunk road,and data were therefore taken at the quietest time, in the earlyhours of the morning.

Despite these difficulties, and the fact that the varioussites are located tens of kilometres from each other, there isa remarkable consistency between the datasets for verticalshear modulus (Gv), with only a few measurements fallingoutside the shaded area (at the Heathrow site, the shallowestare probably due to a layer of gravel at ground surface). Inthe early 1990s we observed that the stiffness parameters wewere obtaining from geophysics were similar to those thatwe had been using in numerical modelling, based on back-analysis of excavations in the London Clay. Fig. 28 thereforetakes the data from these surveys and compares them withdata from back-analysis, and with estimates of Youngsmodulus (E) based on the results from routine laboratorytests. As might be expected from the early work of Ward etal. (1959), the stiffnesses back-analysed from measurementsof foundation and retaining wall movements are muchgreater (by about an order of magnitude) than the stiffnesses

obtained from routine laboratory testing (in this case theoedometer testing). Even the enhanced values, using Butlers(1975) proposed correlation with undrained shear strength,are four or five times too low. Thus, even though the very-small-strain stiffness values obtained from seismic geophy-sics overpredict the back-figured results, they are relativelyclose to them, and at the very least provide a benchmarkagainst which to assess the stiffnesses provided by othermethods of measurement.

Two difficulties potentially arise with the interpretation of

0 100 200

20

15

10

5

0

Gv0: MPaD

epth

: m

Down-hole profiling

Continuous surface wave

Dilatometer

Weak rock pressuremeter

Self-boring pressuremeter

300

Fig. 26. Comparison of CSW and down-hole stiffness measure-ments for a noisy weak-rock site with complex surface geometry(from Hope et al., 1998)

00

10

200Shear modulus, : MPaGv0

Dep

th: m

30

20

100

A1 North Circular (Gordon, 1997)

Chattenden (Hope, 1993)

Surrey Research Park (Gordon, 1997)

Heathrow T5 (Hight ., 2007)et al

London

Site locations

Fig. 27. Vertical shear moduli (Gv) against depth, from fourcross-hole seismic surveys in the London Clay around London

600

Undrained Youngs modulus, or : MPaE Euuv

Dep

th b

elow

gro

und

leve

l: m

30

20

10

02000 100 300 400 500

Back-analysis of case recordsCross-hole geophysics

Constrained modulus from oedometer

E su u220. (triaxial) (Butler, 1975)

Fig. 28. Youngs moduli against depth for the London Clay,from cross-hole geophysics (assuming isotropy and v 0.5; seeshaded area in Fig. 27), back-analysis of case records, androutine laboratory testing at Grand Buildings, Trafalgar Square(modified from Clayton et al., 1991)

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cross-hole seismic data in terms of stiffness. First, there isthe issue of noise, which even though a reasonable datasetmay be obtained can still affect results. Fig. 29 shows theresults of down-hole and cross-hole testing carried out in theLondon Clay using three sources. A surface source and aBison vertically polarised down-hole shear wave hammer(see Clayton et al., 1995b) were used to determine values ofVsv and hence Gv from down-hole and cross-hole surveys.The BRE horizontally polarised shear wave hammer wasused to determine Vsh (and hence Gh). Two sets of data werecollected: that is, there were two down-hole surveys and fourcross-hole surveys, each in different boreholes. Three pointscan be made from Fig. 29.

(a) Below the gravel and above about 30 m depth the Gvdata are tightly grouped, and the down-hole andvertically polarised cross-hole surveys yield similarvalues of shear modulus.

(b) In general, values of horizontal shear modulus (Gh) arehigher than those of Gv, indicating significant stiffnessanisotropy. The dashed line in Fig. 29 is not intended torepresent the trend of Gh with depth; it has been drawnat a stiffness of twice the solid line, which has beenused to represent the trend of Gv data wit