Stiffness at small strain: research and practice · tion of ground displacements during design in...

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Clayton, C. R. I. (2011). Geotechnique 61, No. 1, 5–37 [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 l’importance 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 d’observa-tions sur le terrain. Le developpement historique de latheorie de l’elasticite est alors decrit, ainsi que les cadresconstitutifs dans lesquels il a ete propose d’appliquer lespredictions geotechniques des deformations. Les facteursaffectant la rigidite des sols et des roches tendres sontevalues, et les resultats d’une experience numerique,evaluant l’impact d’un certain nombre de parametres derigidite sur les deplacements autour d’une structure desoutenement sont decrits. On procede a l’examen, 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 d’une strategie pour la determination etl’integration des donnees de rigidite, et des developpe-ments necessaires pour l’optimisation de l’etat 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

(Young’s 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), Hooke’s 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 Hooke’s 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 Cox’s 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 ‘modulus’in his book of 1807, although his definition does not alignwith what we would now term ‘Young’s modulus’ (Todhun-ter, 1886; Timoshenko, 1953; Cooper, 1978). AlthoughYoung recognised the distinction between extension and

α�ε(1 )�

E �

10·10·010

0·2

0·4

0·6

0·8

1·2

0·0001

Axial strain: %

Nor

mal

ised

sec

ant Y

oung

’s m

odul

us,

/E

Ese

c0

Experimental data for cast ironCox’s ‘Hyperbolic law’

0·001

1·0

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 Poisson’s ratio dates back to 1814(Cooper, 1978). The Poisson–Cauchy theory of 1848 pro-duced the uni-constant theory of elasticity, a consequence ofwhich was the prediction that for homogeneous isotropicsolids Poisson’s ratio should be exactly 1/4 for all materials.In contrast Green’s (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 toGreen’s 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. Young’s modulus E and Poisson’sratio �, 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 Young’s modulus for loading in the vertical directionEh Young’s modulus for loading in the horizontal direction�vh Poisson’s ratio relating to the horizontal strain caused

by an imposed vertical strain�hv Poisson’s ratio relating to the vertical strain caused by

an imposed horizontal strain�hh Poisson’s 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 construction’case, 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 Young’s modulus andthe effective Poisson’s ratio, or the shear modulus and thedrained bulk modulus (K9 ¼ dp9/d�V, 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 Young’s modulus and Poisson’s ratio, shearmodulus and bulk modulus can be obtained from

G ¼ E9

2 1 þ �9ð Þ ¼Eu

2 1 þ �uð Þ (1)

K9 ¼ E9

3 1 � 2�9ð Þ (2)

Ku ¼ Eu

3 1 � 2�uð Þ (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 ¼ 0.5. Hence

Eu ¼ 1:5E9

1 þ �9(4)

Table 1. Isotropic drained and undrained parameter sets

Case Parameter set 1 Parameter set 2

Undrained Undrained Young’smodulus, Eu

Undrained Poisson’sratio, �u

Shearmodulus, G

Undrained bulkmodulus, Ku

Drained Effective Young’smodulus, E9

Effective Poisson’sratio, �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

�hv

Eh

¼ �vh

Ev

(5)

and

Gh ¼ Eh

2 1 þ �hhð Þ (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, Young’s modulus E9v and Poisson’s ratio �9vh

are 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 Young’s modulus E should be greater than zero, andPoisson’s 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

E9v

E9h1 � �9hhð Þ � 2 �9vhð Þ2

> 0 (8)

As in the isotropic stiffness case, Young’s moduli andPoisson’s 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 ¼ 0:5 (9)

Chowdhury & King (1971)

�uhv ¼ Eu

h

2Euv

(10)

Gibson (1974)

�uhh ¼ 1 � Eu

h

2Euv

(11)

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

Atkinson (1975), to Euv, Eu

h 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 stress–strain 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

10·10·010·001

Typical strain ranges:

Retaining walls

Foundations

Tunnels

Stif

fnes

s,G

Shear strain, : %εs

100·0001

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 G0

should 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 Bui’s survey of existingdata, a reasonable expression is

Gv0 ¼ Cp 1 þ Eð Þ�3p9=patmð Þ0:5

(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 ¼ 300 MPa and Cp ¼ 600 MPa, are shown in Fig. 4.These trends are shown without normalisation, and forCp ¼ 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

σ�: kPa

Leighton Buzzard E plus AralditeLeighton Buzzard E plus hydrate‘Typical’ rotund sand, without cement,at various stress levels, : kPaσ�

Structured Leighton Buzzard sand– no inter-grain compliance

Gv0

: MP

a

2·52·01·51·00·5

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

Gp

0a

tm0·

5/(

/)

: MP

aσ�

v

1 2 3 4

G C e pv0 p3

atm0·5(1 ) ( / ) (MPa)� � �� σ

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)



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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 0.01–0.1% (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 Young’s 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. Jardine’s linearity index (L ¼ 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 Young’s 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

10·10·01

10·10·01

0

0·2

0·4

0·6

0·8

1·0

0·001Axial strain: %

(a)

Nor

mal

ised

You

ng’s

mod

ulus

,/

EE

u vu v

0·00

1

Loading towards isotropic stress

Multiple loadings

0

200

400

600

800

0·001Local axial strain: %

(b)

Nor

mal

ised

You

ng’s

mod

ulus

,/

Eu v

0p�

t

s�

Compression

Extension

Loading in triaxialcompression

Loading intriaxial extension

Fig. 6. (a) Degradation of vertical Young’s modulus withincreasing axial strain. Triaxial compression data from intactchalk, destructured chalk, undisturbed London Clay, andundisturbed Bothkennar Clay (Heymann, 1998). (b) Degrada-tion of vertical Young’s 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 (principally‘form’; 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. Young’s 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 Young’s 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)



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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 elastic‘plateau’ 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 stress–strain 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 Young’s moduli, : MPaE�

4000 200

Dep

th: m

0

10

20

30

E�hE�v Gv

Dep

th: m

2000 100Shear moduli, : MPaG

0

10

20

30

Gh

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

εa 0·8%/� h.

εa 0·2%/� h.

εa 0·05%/� h.

4

400

Dev

iato

r st

ress

, :

kPa

q

Shear strain, : %εs

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)



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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 horizontalYoung’s 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 centre–centre 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 ¼ 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 ¼ 100 MPa.Case 2 Case 1, but with Mohr–Coulomb plastic yield

at su ¼ 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 (Eu

h . Euv, 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, Eu

0, 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 Young’s mod-ulus degradation curve computed from them, for soil at10 m depth.

The transversely isotropic cases explored in Case 5required three independent parameters (Eu

v, Euh and Gv). The

ratio Euv=Eu

h was varied from analysis to analysis, and Gv

was obtained from

8·5 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 Young’s modulus, : MPaE usec

Dep

th: m

0

5

10

15

20

Hyde Park Cavalry Barracks

0·02% � ε� 0·006%

Uniform stiffness

0·06% � ε � 0·02%0·06%�0·2% � ε

Base case (Case 4)for non-linear analyses

4 0·002%E E0 hpcb� �ε

0·006% � ε� 0·002%

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

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Gv ¼ Euh Eu

v

2Euh þ Eu

v

(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 stiffness–strain 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 A–A 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 Mohr–Coulomb 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 40.002–0.006 2.50.006–0.02 1.50.02–0.06 0.70.06–0.2 0.350.2–0.6 0.15.0.6 0.05

0

100

200

300

0·0001 0·001 0·01 0·1 1 10

Strain: %

Und

rain

ed s

ecan

t You

ng's

mod

ulus

,: M

Pa

Eu se

c

Input tangentYoung’s modulus

Resulting secantYoung’s 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

0�2 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, Mohr–Coulomb3 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 A–A (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 (B–B in Fig. 12)for four different soil models



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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, Eu

0.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 (Eu

h=Euv ) on the predicted maximum horizontal wall

movement. Maximum wall movement was normalised by thevalue from the isotropic (Case 4, base case) analysis.Horizontal Young’s 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, Mohr–Coulomb3 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

0·0001 0·001 0·01 0·1 1 10

Strain (%)(a)

Und

rain

ed s

ecan

t You

ng’s

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 0·02%

E E0 02in Case 4

� �

Case 4 base case

E0 reduced to Case4 at 0·006% strain

Base caseCase 4

E E0 02in Case 4

� �

Linear rangeextended to 0·02%

E0 reduced to Case4 at 0·006% 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

0·0001 0·001 0·01 0·1 1 10

Strain: %

Und

rain

ed s

ecan

t You

ng’s

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 andYoung’s 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

νuvh 0·49� ν u

hhuh

uv1 /2� � E E

0

0·2

0·4

0·6

0·8

1·0

1·2

1 1·5 2·0 2·5Modular ratio: /E Eh v

Ra

tio o

fm

axim

um w

all d

ispl

acem

ents

E Euh

uvGv �

E uh

Gh �2E Eu

huv�2(1 )�νu

hh

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



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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 theauthor’s 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 Dobrin’s(1960) equation for seismic velocity increasing linearly withdepth, to determine the Young’s 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 Young’s modulus values at0.01% axial strain made using local-strain instrumentationon specimens of the London Clay Formation from Canon’sPark, North London, noting that the dynamic values ofundrained Young’s 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 (20–400 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 2

s hv. Rayleigh waves travel at similar,but slightly slower, speeds. (c) Shear distortion in the horizontalplane. Gh0 rrV 2

s 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, Young’s modulus and bulkmodulus can be calculated, if Poisson’s 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 Poisson’s ratio, and, all other thingsbeing equal, for Poisson’s 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 (º), 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 stiffness–depth 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 5–7 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



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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 stiffness–depth profile.

(c) The depth of investigation is limited. Experience withlightweight vibrators suggests that it will be about 5–8 m in a stiff clay, rising to 10–20 m in weak rock.

(d ) Interpretation of CSW data relies generally on verysimplistic interpretations (for example, wavelength/3),although more sophisticated inversion (Haskell–Thomson 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 Poisson’s 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 comprising‘wad’ 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 3–4 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 5–7 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,

λ/2·6

Youn

g’s

mod

ulus

,: M

Pa

Ev

1200

0

400

800

Vertical strain: %0·01 0·1

B

A

B

AA

B

Back-analysedfield data

CSWdata

Depth0–2 m2 6 m–6 12 m–

2 6

m– 0 2

m–

6–12

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 Young’smodulus (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 Butler’s(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 Young’s 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. Young’s 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 Gv

data 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 with depth. Fig.11 has shown the horizontal shear modulus in theLondon Clay at Heathrow Terminal 5, assessed fromlaboratory tests, to be of the order of twice the verticalshear modulus.

(c) The increased scatter in Gh, compared with the scatterin Gv values, probably results from the lower energyinput available from horizontally polarised shear wavehammers. As will be seen later, noise tends to lead tolonger estimated travel times, and therefore lowerinterpreted stiffnesses. Thus some values of Gh maybe underestimates.

In the example above, down-hole and cross-hole surveysgave similar estimates of Gv, in the London Clay Formation.Although this is theoretically true for a transversely isotropicelastic medium, it not always observed in practice. A com-parison of down-hole and cross-hole determined seismicvelocities is shown in Fig. 30 (Pinches & Thompson, 1990).It can be seen that the down-hole values are almost alwayslower than those determined from cross-hole testing. This isbecause the measured down-hole travel time results fromaveraging of the velocity of layered strata, which in this casehave significantly different stiffnesses. In contrast, in across-hole survey the first arrival time results from energytravelling through the stiffest layers, provided that these are

of sufficient thickness to act as wave guides for the energy.In a cross-hole survey in layered ground, measured stiffnessis likely to represent the stiffest layers, rather than theaverage stiffness.

Laboratory testing methodsLaboratory testing plays a vital role in determining the

stiffness of geomaterials, but as already noted can sufferfrom various disadvantages.

(a) It must be possible to sample and prepare specimens ofa representative volume of soil. This will not befeasible if, for example, the stiffness of the ground iscontrolled by widely spaced discontinuities, or if itcontains very coarse material, such as cobbles andboulders.

(b) The specimens must, as far as practical, be undisturbed.Even the most ‘undisturbed’ samples will have under-gone some change in both deviatoric and meaneffective stress, which need to be compensated for insome way.

(c) Advanced laboratory testing may take many weeks ormonths, and requires sophisticated apparatus used bytechnical staff trained and experienced in its use.

While much advanced testing is carried out under quasi-static loading, the potential use of laboratory dynamic test-ing, such as the resonant column apparatus (RCA), benderelements, and cyclic triaxial testing to measure static stiff-ness, has also been recognised for some time. Some promis-ing techniques are discussed below.

50

40

30

20

10

00 100 200 300 400 500

Shear modulus, : MPaG0D

epth

bel

ow G

L: m

Gravel Down-hole Gv

Cross-hole Gv

Cross-hole Gh

Fig. 29. Cross-hole data from London Clay Formation, usingBison (Vsv) and BRE (Vsh) shear wave hammers (Butcher,personal communication)

1000 20000Shear wave velocity, : m/sVs

0

20

30

10

50

40

60

Dep

th: m

Limestone band

Mudstone

Siltstone/limestone

Fissile mudstone

Siltstone and gypsum

Cross-holemeasurements

Down-holemeasurements

Fig. 30. Comparison of down-hole and cross-hole seismicvelocities in layered ground (Pinches & Thompson, 1990)

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Bender element testing. Bender elements, which appear tohave been introduced to the UK by the Bangor marinegeophysics group (Schultheiss, 1981), were first used in theUSA by Shirley in the late 1970s (Shirley, 1978; Shirley &Hampton, 1978). The bender element test has becomeincreasingly popular over the past decade or so, thanks toits perceived simplicity. This is something of an illusion,however, as the increasing number of publications relating tonew variants, parallel testing and standardisation perhapssuggest.

The piezoelectric crystals initially used to generate shearwaves in rock specimens were very stiff compared with soil,which resulted in a large mismatch in characteristic imped-ance between the specimen and the source (Schultheiss,1982; Thomann & Hryciw, 1990). This problem was reducedthrough the use of more flexible piezo-ceramic benderelements, sometimes referred to as piezo-ceramic bimorphs.The bender element consists of two thin piezo-ceramic platesbonded rigidly together, with a conductor between them andon their outer surfaces (Fig. 31(a)). Application of a voltagecauses the plates to extend or contract, and the bimorph totry to bend, generating seismic waves in the soil in which itis embedded. Distorting a bender element generates charge,allowing a similar device (if wired appropriately) to detectincoming waves. Determination of the travel distance andtravel times allow the calculation of wave velocity and, fromdensity and orientation, the relevant shear modulus (Gv orGh).

In principle all that is required for a bender elementdetermination of stiffness is a set of (normally two) bi-

morphs, a signal generator and a (storage) oscilloscope.Because they are compact, bender elements have beeninstalled in oedometers (Schultheiss, 1981; Dyvik & Olsen,1989; Thomann & Hryciw, 1990), in direct simple shearapparatus (Dyvik & Olsen, 1989), triaxial specimens(Schultheiss, 1981; Bates, 1989), inside the resonant columnapparatus (Bennell et al., 1984; Dyvik & Madshus, 1985;Ferreira et al., 2006), and indeed on unconfined samplesimmediately after recovery from boreholes (Hight, 1998;Hight et al., 2003; Landon et al., 2007). Various configura-tions have been used (Schultheiss, 1981; Bates, 1989; Vig-giani & Atkinson, 1995; Pennington et al., 1997; Clayton etal., 2004), the more common of which are shown in Fig.31(b).

Considerable accuracy (for example �5% of G0 and �2%of Vs; e.g. Hight et al., 1997; Pennington et al., 1997) hasbeen claimed for stiffness measurements made using benderelements, but while it is true that their installation intostandard geotechnical testing apparatus is relatively straight-forward, and that with modern data logging and computerpower, high-speed acquisition and processing is not a pro-blem, it has become clear that interpretation is not easy(Ferreira & da Fonseca, 2005), or necessarily repeatable.

Figure 32 shows the input traces and received data for asingle bender element test carried out on a specimen ofnatural clay, using the conventional top cap and base pedes-tal triaxial mounting shown in Fig. 31(b), with a GDSInstruments bender element drive and data acquisition sys-tem. In order to be able to assess near-field effects theperiod of the single input pulse was varied, equivalent toinput frequencies of 10 kHz (the maximum the system couldprovide) down to 2.5 kHz. Several features can be seen inFig. 32.

(a) Although the traces are relatively free from noise, thedifficulty of picking a first break and therefore thetravel time is clear. In this exercise, first breaks werepicked at the first significant rise (i.e. movement in thesame direction as the input) in the trace, as approxi-mately indicated in the figure.

(b) Noise levels appear to increase at lower frequencies.(c) Despite the significant (fourfold) change in the period

of the input signal, the period of the received signalsdoes not alter greatly.

(d ) The periods of the input and received signals areclosest at the highest frequency (10 kHz).(a)

(b)

Flexure with applied voltage

Gv0 G Gv0 h0,

Transmitters

Receivers

Fig. 31. Bender element configurations: (a) sketch of benderelement (Shirley, 1978; Schultheiss, 1983; Dyvik & Madshus,1985); (b) layouts within a triaxial test (Schultheiss, 1981; Bates,1989; Viggiani & Atkinson, 1995; Pennington et al., 1997)

Am

plitu

de

Transmitted signals

Received signals – stiff clay

Time: ms

Firstbreaks

0 0·1 0·2 0·3 0·4 0·5

7·5 kHz

10·0 kHz

2·5 kHz

5·0 kHz

10·0 kHz

7·5 kHz5·0 kHz 2·5 kHz

Fig. 32. Example input and received waveforms at differentfrequencies in a bender element test on ‘undisturbed’ stiff clay



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The received data in Fig. 32 were examined independentlyby four engineers, with instructions to estimate the firstbreak and peak-to-peak travel times. The shear wave velo-cities calculated from these results are shown in Fig. 33.The following can be seen.

(a) For this case, perhaps as a result of noise and/or near-field effects (Clayton et al., 2004, and see below), thescatter in estimated shear wave velocities is consider-ably higher at low input frequencies than at high inputfrequencies.

(b) For the small data sample shown (a larger number ofparticipants might be expected to give a greater range),the variation in shear modulus (Gv) calculated fromfirst-break travel times is �5.4% at 10 kHz, rising to�15.4% at 2.5 kHz.

(c) The data show that it is easier to pick consistent peak-to-peak travel times than first break travel times. Thereis considerable consistency in the shear wave velocitiesestimated using the peak-to-peak method.

(d ) However, the results obtained from using peak-to-peaktravel times are not consistent with those from first-break determinations until the received period isapproximately the same as the transmitted period.

These data suggest that previous estimates of the ‘accuracy’of bender element determinations of velocity and stiffnesshave been optimistic. They also suggest that, while carryingout commercial bender element tests, there is merit in

(a) systematically using a range of input frequencies forevery test

(b) having more than one person interpret the traces, in thesame environment (e.g. in a spreadsheet)

(c) estimating shear-wave velocity and therefore stiffnesson the basis of both first-break and peak-to-peak traveltimes.

The important issue of noise is further explored in thedata shown in Fig. 34. Five noisy bender element tracestaken on the same specimen are divided into those with a(relatively) good signal-to-noise ratio (above) and those witha poor signal-to-noise ratio (below). The ratio (S/N)a shownin the figure is not the conventional value (signal powerdivided by noise power), but is the ratio of peak-to-peaknoise prior to the first break, divided by the maximum peak-to-peak amplitude in the received wave train. This value ismore easily estimated in the laboratory during testing. Theestimated position of the first breaks is shown by the arrows.The arrival of the seismic wave is detected significantly laterwhen there is more noise. A survey of about 200 traces from

tests on stiff clays in the London area, carried out by theauthor, has indicated that bender element tests with a (S/N)a

ratio of less than 10 should be considered unacceptable.Leong et al. (2005) suggest a conventional receiver S/N ratioof at least 4dB.

Reporting on the results of round robin testing carried outunder the auspices of ISSMGE Technical Committee TC29,Yamashita et al. (2009) give comparisons of shear modulus(Gv) values obtained in different laboratories across theworld on loose and dense pluviated rotund uniformly gradedToyoura sand, and compare these with the results of resonantcolumn tests by Iwasaki & Tatsuoka (1977). Fig. 35 showsan extract from their results, and indicates a large scatter inestimated shear modulus. Helpfully, however, their resultssuggest that the simplest method of bender element testing,using first-break or peak-to-peak travel times derived fromthe propagation of a single sine pulse input waveform,provides reasonably consistent results.

The wide range of issues that has been identified in themanufacture and use of bender elements is summarised inTable 3. At this stage, from a practitioner viewpoint, itwould seem sensible to use simple test and interpretationtechniques, and regard the bender element test as semi-empirical, requiring validation in each new material and testarrangement, through comparison with results from other

200

220

240

260

280

300

320

0 2 4 6 8 10 12Source frequency: kHz

She

ar w

ave

velo

city

: m/s First break - 1

First break - 2First break - 3First break - 4

Four independent peak-to-peak picks

Fig. 33. Effect of subjectivity in picking first breaks and firstpeaks, using the data presented in Fig. 32

0

Am

plitu

de: V

Am

plitu

de: V

�0·001

0·001

Time

(S/N) 12·2, 13·2, 13·7a �

0

Time

(S/N) 3·3, 4·0a �

�0·001

0·001

Firstbreaks

Firstbreaks?

Fig. 34. Effect of signal-to-noise ratio on picked arrival time:noisy bender element data

Frequency domain andcross-correlation

First breaksPeak-to-peak

50

25

100

200

400

She

ar m

odul

us,

: MP

aG

v0

0·90·6 0·7 0·8Void ratio

σ� � 200 kPaCircles: saturatedTriangles: dry

Resonant column tests(Iwasaki & Tatsuoka, 1977)

Fig. 35. TC29 international parallel bender element tests onToyoura sand (redrawn from Yamashita et al., 2009)

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Table 3. Variants in bender element testing

Issue Brief description Effect Suggested standard

Benderdimensions

Width and protrusion into thespecimen

Travel distance is generally accepted asbender tip-to-tip.Reduced travel distance causes reducedaccuracy.Installation of highly protruding benderelements is more difficult in specimens ofstiff/hard natural sediment.

Most laboratories adopt 10 mm width, 0.5–1.0 mm thickness, and a protrusion of ,5 mm(Yamashita et al., 2009). Shorter protrusion (c.1 mm) is favoured in the UK when testing stiffnatural materials, where otherwise a slot needs tobe cut and the bender filled around duringinstallation. Fixing epoxy is set back inside endcaps, and the space filled with silicone rubbercompound, to allow flexing of the benderelement.

Benderconstruction

Transmitter and receiverbender configurations,extender elements, and self-monitoring configurations.

Using different (series and parallel)bimorphs as transmitter and receiver (Dyvik& Madshus, 1985) improves shear wavegeneration and reception. Extender elements(Lings & Greening, 2001) allow P-wavegeneration. Use of a self-monitoringconfiguration, as per Schultheiss (1982),allows the movement of the bimorph (asdistinct from its driving voltage) to bedetermined.

Experience suggests that the wiring of benders asS-wave transmitters, P-wave transmitters and S-wave receivers gives advantages. Self-monitoring bender elements may be useful, asthey allow the detection of time lags and phasechanges between driving voltage and actualmovement of the transmitter, resulting fromcharacteristic impedance mismatch.

Location ofbenderelements

Bender elements may bemounted in the rigid base andtop caps of cells (e.g.Schultheiss, 1981), or throughthe triaxial membrane(Pennington et al., 1997;Clayton et al., 2004).

Mounting across or along specimens allowsshorter travel distances (giving reducedattenuation and higher S/N ratio), anddetermination of the velocity of wavespolarised in different directions. Very-small-strain anisotropy of stiffness can then beinferred from shear wave velocities.

The addition of side-mounted bender elements isrelatively easy to achieve using grommets and O-rings similar to those for mid-plane pore waterpressure measurement (Sodha, 1974).

Travel distance Small tip-to-tip travel distanceaffects travel time resolution.Large distance increasesattenuation.

Closely spaced benders display more scatterin calculated shear wave velocity (Yamashitaet al., 2009). Data from widely spacedbenders are more noisy. The estimated traveltime may be affected.

Calculated shear wave velocities may be affectedif large bender element penetrations are used inconjunction with small specimens (e.g. verticallyin oedometers or DSS specimens). Highertransmitter frequencies will be needed to keep thewavelength down (see ‘Input frequency’, below).

Input waveform

Square (Schultheiss, 1981,1982, 1983), continuous sineor pulsed sine waves (de Alba& Baldwin, 1991).

Square wave pulses contain a broad spectrumof frequencies. Low frequencies place thereceiver in the near field.

Practice suggests single sine pulses produceacceptable traces, giving repeatable first break orpeak-to-peak travel times, and permitting a morerestricted and controllable input frequency, andtherefore wavelength: see below (Thomann &Hryciw, 1990).

Input wavemode

P or S wave. In soft saturated soil P waves travel at c.1450–1550 m/s (the P-wave velocity ofwater), much faster than S waves. Inunsaturated soils P waves travel only approx.50% faster, and can obscure or be confusedwith S waves.

It is suggested that extender elements (Lings &Greening, 2001) should be routinely used to warnof misinterpretation.

Input frequency Low source frequenciesproduce long wavelengths.

Long wavelengths place the receiver in thenear field, affecting the received waveformand picked travel time.

The receiver should be at least 2–3 wavelengthsfrom the transmitter (Sanchez-Salinero et al.,1986; de Alba & Baldwin, 1991; Leong et al.,2005). It is suggested that, for routine triaxialtesting, results are returned for a range offrequencies from about 2.5 kHz to 12.5 kHz).

Dataacquisition

Low voltage and temporalresolution reduce quality ofcaptured traces.

Picked arrival times become uncertain. In order that accuracy is not degraded byresolution. the sampling time interval should beless than 1/100th of the travel time betweentransmitter and receiver. Voltage resolutionshould be better than 1/100th of the amplitude ofthe received signal (Yamashita et al., 2009).

Signal-to-noiseratio

Noise affects low-amplitudereceived signals.

Picking of first arrival or peak times issubjective. A low signal-to-noise ratioincreases scatter and tends to increase theestimated travel time.

A study of some relatively noisy UK datasuggests a minimum amplitude signal/noise ratio(S/N)a of 10. Leong et al. (2005) suggest areceiver S/N ratio of at least 4 dB.

Method ofdeterminingtravel time

First break, peak-to-peak,cross-correlation, or phase/frequency relationship.

Different processing methods lead todifferent travel times. With mixedfrequencies pulse broadening may occur,owing to attenuation of the higher-frequencycomponent, leading to increases in measuredtravel time when using peak-to-peakdetection. Receiver first breaks may behidden by noise, leading to increases inmeasured travel time when using thismethod.

First-break and first-peak-to-first-peak traveltime detection are favoured by Yamashita et al.(2009), who found significant differences insome cases where cross-correlation and phase/frequency methods were used (see also Ferreira& da Fonseca, 2005). It is suggested that resultsfrom both the first break and the peak-to-peakmethods be routinely reported.



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types of test (e.g. resonant column and field geophysicaltests, as by Dyvik & Madshus, 1985; Davis & Bennell,1986, Bennell & Taylor Smith, 1991), other laboratories’determinations on the same material, and in-laboratory com-parisons with previous tests on similar materials. The test’svalue lies in its simplicity, its relatively low cost and itspotential for determining anisotropy of shear modulus.

Resonant column testing. The resonant column apparatus,which has been successfully used for more than 40 years inthe field of soil dynamics, provides a method of determiningthe shear modulus (G) and Young’s modulus (E or Eflex) ofsoils and weak rocks at very small strain levels, and ofobtaining estimates of the rate of stiffness degradation withincreasing strain. Different apparatus configurations allowvibration of a soil specimen in torsion (Hardin & Music,1963; Stokoe et al., 1980; Menq & Stokoe, 2003), in flexure(Cascante et al., 1998) and axially (Drnevich, 1972).Torsional testing appears not to suffer from bedding effects(although compliance is an issue, as will be discussed below),and is therefore to be preferred for stiffness determinations.

Figure 36(a) shows a simple schematic diagram of aStokoe resonant column apparatus. An electromagnet drivehead, to which four magnets are attached, is bolted to thespecimen top cap. Torsion can be applied by running currentthrough the four coils in which the magnets sit, which areheld in place by a substantial support frame. Flexure can beapplied by running current through two diametrically op-posed coils. At the start of a test a relatively low sinusoidaldrive voltage is applied, and a frequency sweep is carriedout. As the frequency is increased, the amplitude of thevibrations, measured by an accelerometer mounted on thedrive head, increases up to a peak, and then decays. This isshown by the lowest curve in Fig. 37. The peak amplitude,which occurs at low levels of damping at the resonantfrequency, is recorded. Given the mass polar moment ofinertia of the drive head and top platen, the specimen massand its dimensions, and assuming linear elasticity, the shearmodulus (Gv) of the soil can be calculated.

The process is then repeated with a higher applied vol-tage. Measured amplitude, and therefore strain, increases. Atfirst, at the lowest strain levels, the peak frequency isunaffected by the increasing voltage, but as shear strainincreases shear stiffness decreases, and the peak frequencyof the system drops (Fig. 37). The shear modulus at verysmall strain (G0), and a curve of shear modulus againstshear strain (stiffness degradation), can be obtained from theresults.

Resonant column testing is the subject of ASTM standardD 4015-07 (ASTM, 2007), which provides generic guidancefor the calibration and operation of a range of resonantcolumn devices to determine both stiffness and damping, butdoes not attempt to provide engineering guidelines on theappropriate use of the apparatus, nor of problems that maybe encountered. This can be found elsewhere in the literature(e.g. Bennell et al., 1984; Bennell & Taylor Smith, 1991). Asuitable test procedure might be described as follows.

(a) Set up specimen within the apparatus. An estimate ofthe expected very-small-strain shear modulus will beuseful in judging apparatus compliance, which mayneed correction, and specimen slippage effects, whichmay be avoidable by cementing the specimen to theplatens or, for weaker materials, by using vanesprotruding into the specimen (Drnevich, 1978; Claytonet al. 2009b).

(b) Re-establish the in situ effective stress(es) on thespecimen, by applying suitable cell and (elevated) back-

pressures, and allow drainage while monitoring volumechange and specimen height.

(c) Immediately after re-establishment of in situ stresses,measure the very-small-strain shear modulus (Gv) (andnormally damping) of the specimen, and monitor it atregular intervals on a logarithmic scale (for example atapproximately 1, 2, 4, 8, 16, 32 min, 1, 2, 4, 8, 16,32 h, etc. after the start of this stage). Low-amplitude

(b)

(a)

Magnet

Cell base

Pressurevessel

Top cap

Spe

cim

en

LVDT

Pedestal

Coil Coil

Counterweight Accelerometer

Crossarm

Fig. 36. (a) Schematic drawing and (b) photograph of Stokoeresonant column apparatus

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torsional frequency sweeps (each lasting about oneminute) should be carried out under drained conditionsuntil primary consolidation is complete and the creeprate is well defined (Anderson & Stokoe, 1978),perhaps for 1000–2000 min (1–2 days). The ‘reference’shear strain level used for these measurements shouldbe as low as practical (e.g. taking into accountlimitations imposed by instrument sensitivity andbackground vibration) and should be non-destructive,with a shear strain amplitude , 0.0001%.

(d ) Determine the relationship between shear modulus Gv,damping and shear strain. With the drainage valvesopen, carry out an initial shear modulus (and damping)measurement at the reference strain level. Then closethe drainage valves.

(e) Test at a higher (say three times) strain level. Open thedrainage valves, allow a rest period of about 1 min,then repeat the stiffness (and damping) measurements atthe reference strain level, under drained conditions.

( f ) Repeat (e) with progressively higher strain levels, forexample approximately 0.00001%, 0.00003%, 0.0001%0.0003%, 0.001%, 0.003%, 0.01%, 0.03%.

(g) Cease testing when the shear modulus determined atthe reference strain level does not return to its initialvalue, since this indicates some level of specimendestructuring as a result of the torsional strains thathave been imposed.

Tests can be carried out under isotropic effective stressconditions in the Stokoe apparatus, or if required underanisotropic effective stress conditions, in the Hardin appara-tus. The advantages of the Stokoe apparatus are that it is arelatively simple piece of equipment, and it can apply highlevels of torque, thus exploring a greater torsional strainrange.

The results for Eocene sandy clay shown in Fig. 5, for anumber of specimens tested under various degrees of stressanisotropy, suggest that, apart from its impact on meaneffective stress, deviatoric stress has little effect on very-small-strain shear modulus Gv, and that for practicalpurposes this may be determined under isotropic stressconditions. This view seems to be supported by benderelement data obtained by Gasparre et al. (2007) on LondonClay, during reconsolidation to anisotropic stress states, andby the results of Yamashita & Suzuki (1999) from torsionaltests on pluviated Toyoura sand.

For the Stokoe resonant column device Clayton et al.

(2009b) report the results of extensive numerical, analyticaland experimental modelling, driven by the need to test stiffermaterials (Bennell & Taylor Smith, 1991). They providerecommendations relating to

(a) the repeated use of (appropriately designed) calibrationbars before and after testing, to check for loosening ofapparatus components

(b) the use of calibration bars with different stiffnesses toexplore and correct for apparatus compliance, whentesting stiffer materials

(c) limits on frictional fixity between stiff (e.g. cemented)specimens and the specimen platens (see also Drnevich,1978)

(d ) the base of the apparatus, which needs to have a highmass polar moment of inertia relative to that of thedrive head.

Figure 38 shows the results of some resonant columndeterminations of very-small-strain stiffness on gypsiferousmudrocks from Dubai. Note that the shear modulus valuesare plotted on a logarithmic scale; there is (roughly) a oneorder of magnitude difference between the stiffness deter-mined from the pressuremeter initial loading and that fromthe cross-hole seismic testing. Because of sample distur-bance and apparatus compliance the resonant column deter-minations of Gv fall between these values. Note the veryhigh stiffness of the ground, and that the ratio of correctedto uncorrected shear modulus for the apparatus used in thesetests is greater than 3 when the measured shear modulusrises to 2 GPa. For this particular apparatus design it wouldappear that tests on material with a shear modulus greaterthan about 500 MPa would be significantly affected byapparatus compliance. In tests on methane hydrate bearing

0·01

0·1

1

10

125 130 135 140 145Drive frequency: Hz

Out

put a

mpl

itude

: V

Increasing drive voltage

Fig. 37. Principle of operation of a resonant column apparatus;results of a Stokoe resonant column test on dense LeightonBuzzard sand

Shear modulus, : GPaGv

0

100

200

0·1 1 10

Dep

th: m

Initial loading pressuremeter shear modulus

Cross-hole shear modulus

Resonant column data, uncorrected

Resonant column data, corrected for apparatus compliance

Fig. 38. Comparison of resonant column very-small-strain stiff-ness measurements with those from pressuremeter and cross-hole seismic testing; gypsiferous mudrocks with saline porewater, Nakheel Tower, Dubai



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sediments, for which shear modulus can rise to about 5 GPa(Clayton et al., 2005), a Stokoe resonant column apparatuswith an enhanced cross-arm stiffness has been used.

Triaxial testing. A much improved appreciation of the effectsof disturbance and potential destructuring during drilling,sampling, specimen preparation and the early stages oflaboratory testing (Hight, 1986; Baligh et al., 1987; Chandleret al., 1992; Clayton et al., 1992; Hight & Jardine, 1993;Clayton et al., 1998; Clayton & Siddique, 1999), and thedevelopment and use of higher-quality drilling techniques andimproved samplers (Lefebvre & Poulin, 1979; La Rochelle etal., 1981; Scarrow & Gosling, 1986; Harrison, 1991; Tanaka& Tanaka, 1999; Tan et al., 2002), have led to a significantincrease in the availability of high-quality soil stiffness data.

The development of triaxial testing has been facilitated bythe growth in high-quality triaxial instrumentation over thepast 30–40 years. Fig. 39 shows some examples.

(a) The introduction of high-stiffness water pressure sensors(Fig. 39(a)) that could be mounted on de-airing blocksclose to the cell base allowed laboratories not only tomove away from the mercury null indicators describedby Bishop & Henkel (1962) but also to improveresponse times dramatically (Whitman et al., 1961).

(b) Pore pressure measurements were then further improvedby the introduction of ‘mid-plane’ pressure-measuringsystems. Fig. 39(b) shows the components for a mid-plane probe (Hight, 1982) using a submersible minia-ture pore pressure transducer. Experience suggests thatthis device is more applicable in the testing of softersediments, where the initial suctions are relatively low,and will not lead to cavitation behind the ceramicduring specimen set-up (Bishop & Henkel, 1962,Appendix 6). For stiffer sediments (such as, forexample, the London Clay) the flushable probe shownin Fig. 39(c) (Sodha, 1974) is preferred, since the gasbubbles that form behind the high-air-entry ceramic asa result of cavitation and dissolved-air ex-solutionduring specimen set-up can be removed to re-establishthe stiffness of the pore pressure measuring system.

(c) The introduction of submersible load cells (Fig. 39(d)),which should be mounted inside the triaxial cellbetween the specimen top cap and the ram, hasremoved the need to use rotating bushes, whileavoiding the unpredictable effects of ram friction. Thishas become more important with the introduction of Orings within the ram bushings in many cells.

(d ) The use of local strain measurement devices, such aslinear variable differential transformers (LVDTs) (Fig.39(e)) or Hall effect sensors (Fig. 39(f)). These aretypically mounted on the mid-third of the specimenheight (and in Fig. 39(e) on a radial caliper).

(e) The use of electronic sensors and computer based dataacquisition has allowed near-continuous records offorce, pressure and displacement to be made, which isparticularly important in the early stages of triaxialsmall-strain stiffness determination.

( f ) Finally, the introduction of bender elements (Fig. 39(g))has allowed independent determination of G0 (seeearlier) during stiffness testing for, for example,Young’s modulus.

Broadly, there are two classes of triaxial test currentlycarried out commercially in the UK: routine tests and ad-vanced tests.

Routine tests have been standardised in BS 1377 part 8(BSI, 1990), and in ASTM standards (for example ASTMD4767; ASTM, 2004). Standards for the more complex

cyclic triaxial test exist in the USA (ASTM D3999; ASTM,2003) and Japan (Toki et al., 1995), but not in the UK. Thetext below (see also Fig. 40) suggests how, practically, someof the disadvantages of the standard triaxial test configura-tion may be overcome in commercial testing.

Load cell and other apparatus compliance. The stiffness ofthe load cell or proving ring used to measure deviatoric load,or the loading frame itself, is not generally an issue whentesting materials with low stiffnesses, but as has been seen inthe context of the resonant column apparatus, apparatuscompliance can be very significant at the higher stiffnessesoften associated with structured soils and weak rocks whentested at small strains. For the triaxial apparatus someresearchers initially suggested reducing these effects throughseparate measurement of compliance (e.g. Atkinson & Evans,1985), but it seems now to be accepted that in practice it ismore straightforward, given the need also to remove beddingeffects (see below), to routinely use local strain measurement.

Ram misalignment. A large number of top cap arrangementsexist (e.g. Atkinson & Evans, 1985; Kuwano et al., 2001). Thearrangement that is used is known to affect the measuredstress–strain behaviour of soils (e.g. Jardine et al., 1985; Baldiet al., 1988). However well trimmed and aligned a specimenmay be at the time of set-up in the triaxial apparatus, experiencesuggests that misalignment of the specimen and top cap willoccur, for example during initial application of cell pressure, asa result of fissure closure and bedding. The use of a roundedram end engaging with a dimpled top cap, specified in theBritish Standard, then inevitably leads to sideways movementand slippage at the start of a test, which can affect even localmeasurements of strain. Fig. 40 suggests a practical method ofovercoming this, when carrying out the simplest of advancedtriaxial tests, that is, loading only in triaxial compression. Thedimple in the top cap is removed, and replaced with a flushstainless steel insert.

Air trapped around top cap and in top cap ducts. It isessential to ensure, during set-up, that air is removed not onlyfrom between the membrane, filter drains (if used) andspecimen, but also from around the specimen top cap.Experience suggests that this is made more difficult if low-air-entry porous stones and drainage ducts and leads are usedat this location. For saturated soils, failure to remove air fromblanked-off ducts, or from small voids between the edges ofthe porous stones, the specimen and the pedestal or top cap,causes the system to behave as if the specimen wereunsaturated, given a slower pore pressure response, and apotential underestimate of pore pressure change. Thearrangement in Fig. 40 removes much of the opportunity totrap air. Little is lost by removing top cap drainage providedappropriate side drains can be used, as the theoretical studiesof Bishop & Gibson (1963) showed. The amount of airtrapped at the specimen base can be significantly reducedthrough the use of a base pedestal with a flush-mounted high-air-entry ceramic (Bishop & Henkel, 1962).

Bedding between soil, porous stones and platens. The issuesof compliance, discussed above, and bedding becamegenerally recognised in the late 1970s and early 1980s (e.g.Brown & Snaith, 1974; Costa-Filho & Vaughan, 1980;Daramola, 1980). In response various devices were developedthat could be mounted on the sides of triaxial specimens tomeasure displacement locally (e.g. Jardine et al., 1984;

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Clayton & Khatrush, 1986; Goto & Tatsuoka, 1986; Ackerleyet al., 1987). In addition, standard instrumentation such assubmersible proximity sensors (Hird & Yung, 1989) andLVDTs came into use. Most research laboratories now preferthe high resolution (about 1 microstrain) that can be achievedusing LVDTs (Cuccovillo & Coop, 1997), but the stiffelectrical leads associated with their waterproofed cablingcause concern regarding conformance. The ease of use ofHall effect gauges (Clayton & Khatrush, 1986; Fig. 39(e))makes them attractive to many commercial laboratories,despite their lower resolution.

De-saturation of porous stones. Most fine-grained heavilyoverconsolidated and deep samples arrive in the laboratory

with significant suctions. When a specimen is placed ontraditional coarse low-air-entry porous stones during set-up, itimmediately starts to imbibe water. Water can be quicklyreleased by the stones because of their high permeability, andtheir low air-entry value means that, as water moves into thespecimen, it can easily be replaced by air. The use of flush-mounted high-air-entry (say 250 kPa) ceramics, fixed into thebase (and top cap, if required) with epoxy resin slows thisprocess, as well as reducing the opportunity for air to betrapped during set-up.

Pore pressure equalisation issues. In order to interprettriaxial test data it is, of course, necessary that the porepressures in the middle of the specimen are reasonably

(a) (b) (c)

(d)

(e)

(f)

(g)

Fig. 39. Examples of triaxial instrumentation: (a) external pore pressure transducer; (b) pore pressure transducer mid-plane probe;(c) flushable mid-plane pore pressure probe; (d) internal load cell; (e) LVDT local strain measurement; (f) Hall effect local strainmeasurement; (g) side-mounted bender element



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uniform, and known. Routine triaxial tests measure porepressure by connecting a pressure transducer, mounted in ade-airing block, to the cell base. The pore pressure at thebase of the specimen will not necessarily be the same as atthe mid-height of the specimen, for example duringconsolidation or shearing stages. In conventional testing,undrained loading to failure must be run sufficiently slowly toensure that sufficient equalisation of pore pressure occursbetween the mid-height of the specimen and the base by thetime that failure is reached. Drained shear stages, which forreasons of economics are generally used only for coarser,more free-draining materials, must be run sufficiently slowlyto ensure that a high degree of consolidation occurs, which inpractice means that the mid-plane and base pore pressuresshould be close to each other. To overcome these difficulties,advanced triaxial testing is normally carried out using a mid-plane probe (Sodha, 1974; Hight, 1982). A flushable probe ispreferred for stiffer materials, where the initial suction in thespecimen is likely to exceed 100 kPa, and cavitation willoccur. A miniature pore pressure transducer has theadvantage of smaller size.

Figure 41 shows the testing strategy for the simplest ofadvanced triaxial tests. The test is divided into six stages:

(a) establishment of pore pressures in the measuring system(b) consolidation or swelling to estimated in situ mean

effective stress(c) application of approximate estimated in situ deviatoric

stress, allowing for(d ) a stress path to model recent stress history, if required,

and(e) a rest period, to reduce creep to an acceptable level( f ) undrained monotonic shearing to failure, in either

triaxial compression or extension.

In the UK the standard test procedure (BSI, 1990; 1999)used for routine effective stress triaxial strength testinginvolves three stages: ‘saturation’, consolidation and mono-tonic shearing to failure in triaxial compression. The ‘satura-tion’ referred to in the standard, although not clearly worded,is that of the system (the specimen, plus voids betweenspecimen and membrane and porous stones, and in basepedestal and top cap ducts), rather than of the specimenitself. Two methods are described: back-pressure saturation,and saturation at constant water content, the latter usingapplication of cell pressure alone.

With notable exceptions, for example compacted fill andcoarse soils above the water table, many materials will have

been saturated in the ground, before sampling. While back-pressure saturation (which through the application of a verylow effective stress allows the specimen to swell) is adoptedin routine triaxial testing in the UK and elsewhere, it hasbeen found that when using advanced triaxial apparatus theapplication of large steps of cell pressure, aimed at reachingthe in situ mean effective stress as rapidly as possible beforemeasuring a B value, works well. This is shown schemat-ically in Fig. 41 (stage 0–1). Because of sampling distur-bance effects, the mean effective stress at the end of thisstage would not be expected to equal the mean effectivestress in the ground. Estimates of K0 based on these data(Skempton & Sowa, 1963) should be treated with caution.

Stages 1–2 and 2–3 (Fig. 41) aim to bring the specimenback to its in situ effective stress regime, albeit with a high(generally . 300 kPa) back-pressure, to ensure effective porepressure measurement. The most economic technique is toapply a single increment of consolidation (or swelling) underisotropic conditions (stage 1–2), and follow this with adeviatoric stress ramp (or series of small steps) (stage 2–3),during which the excess mid-plane pore pressure should bemonitored. If the excess pore pressure (the difference be-tween the measured mid-plane and base pore pressures)exceeds a specified proportion of the major principal stress(say 5%), then loading (or unloading if going into triaxialextension, as shown in Fig. 41(a)) should be slowed, toensure that the actual effective stress path does not deviateexcessively from its planned route.

The necessity of modelling recent history (stage 3–4) has

No top capdrainage or

stone

Flushable high airentry mid-plane

p.w.p. probe

Local axial straingauges Bender element

(one of a pair)

Radial straincaliper

Flush-mountedhigh air entry

flushable porousstone

Stainless steelinsert

Fig. 40. Advanced triaxial test configuration

(a)

(b)

s� � � � �( )/2σ σ1 3

t� � �( )/2σ σ1 3

21

4, 53

6

6

Triaxialextension

Triaxialcompression

Time

Cel

l pre

ssur

eP

ore

pres

sure

Dev

iato

ric s

tres

s

60 2 3 4 5

Res

t per

iod

tore

duce

cre

ep

App

lyde

via

toric

stre

ss

Shear tofailure

Mim

ic r

ecen

thi

stor

y?

Establishve pwp�

Isot

ropi

cco

nsol

ida

tion

1 month ?

1

Fig. 41. Basic testing strategy for an advanced triaxial test

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been debated for some time (Atkinson et al., 1990; Clayton& Heymann, 2001), but it now appears that small recentstrain excursions do not affect measured stiffness, whereaslarger ones do (Gasparre et al., 2007; Hight et al., 2007). Inpractice, even when they are thought to have occurred,estimating the magnitude of recent stress history (e.g. due tothe deposition of Terrace Gravels over London Clay (Bur-land et al., 1979) or fill) may be difficult.

Inaccurate measurements of stiffness will be made ifsignificant creep deformations are allowed to occur duringloading, since these will add to or subtract from the deform-ations being produced during current loading. In the testsreported by Clayton & Heymann (2001) rest periods of upto 2 weeks were used, leading to creep rates before loadingof less than 0.01%, equivalent to less than 2% of the locallymeasured shear rate at the start of each shear stage. Routinepractice for advanced testing in the UK (based on practiceat Imperial College; Jardine, 1995) has been to adopt a restperiod (stage 4–5) that extends until the creep rate becomesless than 1% of the subsequent external (machine) rate ofloading. It should be noted that this could be considerablyless conservative than the rates used by Clayton & Heymann(2001), depending upon bedding effects.

In the final stage (5–6) of this example test, deviatoricstress is applied under undrained and approximately constantrate of strain conditions, either increasing the deviatoricstress to produce failure in triaxial compression, or reducingit to produce failure in triaxial extension. In contrast withroutine testing, a more-or-less standard machine rate ofstrain of around 5% per day is typically applied. Given thatthe specimen has mid-plane pore pressure measurement,equalisation between the base and mid-plane is not a factorin deciding on the machine rate of strain.

The test described above has typically been used todetermine the very-small-strain vertical undrained Young’smodulus (Eu

v), its degradation with strain and the effectivestress failure envelopes, loading in both triaxial compressionand extension. Following Ward et al. (1959) and Atkinson(1975), it is suggested that horizontally cut specimens couldprovide estimates of the horizontal undrained Young’s mod-ulus (Eu

h). The apparatus can also be used (in slower tests)to determine the drained Young’s modulus and Poisson’sratio (E9v and �9vh), and, when loading horizontally (Lings etal., 2000), E9h=(1 � �9hh). When used with care, high-qualityinstrumentation, and excellent temperature control (Gasparre& Coop, 2006), triaxial testing can also determine the very-small-strain stiffness for other (drained) stress paths.

STRATEGIES FOR MEASUREMENT ANDINTEGRATION OF DATA

Table 4 brings together the methods that have beendiscussed above in the context of the depths for which they

can be expected to yield good data (for field seismic tests)and the parameters that can be determined.

As the table shows, there are a relatively large number ofways in which the very-small-strain shear modulus in thevertical plane (Gv0) can be determined. In addition to themethods shown in the table, if isotropy is assumed then anestimate of G0 can also be obtained from advanced un-drained triaxial testing (Poisson’s ratio being 0.5 in theundrained case). Some of these methods (e.g. cross-hole,down-hole and bender element testing) are of particularvalue in that they can be used to detect and estimatestiffness anisotropy.

Table 4 also highlights the need to carry out triaxialtesting, despite its complexity and time-consuming nature.As can be seen, this is needed for measurement of bothundrained and drained Young’s moduli, and their degradationwith strain.

Figure 42 shows a comparison of very-small-strain shearmodulus (Gv) data obtained from field geophysics, resonantcolumn and advanced triaxial testing of a site underlain byEocene sandy clay. Down to 30–40 m there is good agree-ment between Gv values determined using down-hole and

Table 4. Methods of obtaining the various stiffness parameters

Continuous surfacewave test

Down-hole shearwave survey

Cross-hole shearwave survey

Bender elementtesting

Resonantcolumn tests

Advanced triaxialtesting

Profile max. depth 8–10 m 20–40 m .100 m�Initial stiffness Gv0 Gv0 Gv0, Gh0? Gv0, Gh0 Gv, Gh

y Euv, Eu

hy

Short-term operationalstiffness

Gv0 Euv, Eu

hy

Long-term operationalstiffness

Gv0, Gh0 Gv, Gh E9v, E9h, �9vh

� Depending upon verticality of holes, change in stiffness with depth, etc.y Horizontal specimens required for Eh

u, E9h and Gh, but in situ deviatoric stress cannot be re-established.

Resonant column Gv0

CAU txl (0·01%)/3EuDown-hole Gv0

Cross-hole Gv0

0400

10

20

30

40

50

60

70

0 100 200 300Shear modulus, : MPaGv

Dep

th: m

Fig. 42. Comparison of stiffness data obtained from fieldgeophysics, resonant column and advanced triaxial testing;undisturbed samples of Eocene sandy clay



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cross-hole seismic testing. Very-small-strain stiffness reso-nant column (Gv0) results give a lower stiffness profile, eventhough the specimens were returned to their estimated in situeffective stresses – presumably because of sample distur-bance. Undrained triaxial stiffnesses, derived assuming iso-tropy (i.e. Gv ¼ Eu

v=3) from Hall effect local strainmeasurements at 0.01% axial strain, are considerably belowthe stiffnesses from resonant column testing, as might beexpected at the higher strain levels.

Estimation of the degree of stiffness anisotropy requiresgood specification and successful use of a horizontallypolarised source during cross-hole seismic testing, or ofhorizontally polarised bender element tests. The results ofbender element testing and horizontally polarised cross-holeshear wave seismic testing are not shown in Fig. 42, becauseat this site they were judged to be unreliable, and thus thedegree of stiffness anisotropy (if any) was not known. It issuggested that the chances of recognising undrained andshear stiffness anisotropy may be improved in future by theuse of horizontally cut specimens in the resonant columnand triaxial apparatuses.

Determination of the changes of stiffness parameters withstrain remains challenging. Changes in stiffness (Eu

v or Gv)can be measured under static conditions in advanced triaxialtesting, or under dynamic conditions in the cyclic triaxialtest, and in the resonant column apparatus. However, as aresult of cycling and strain rates, the different methods canbe expected to give different results, with estimates based onresonant column testing being unconservative. Determinationof stiffness degradation with increasing strain should takeplace (at least) in both compression and extension, sincedegradation rates will differ.

Very few determinations of the full set of drained (long-term) anisotropic stiffness parameters have been reported forundisturbed natural soils, to date. Given the testing time-scales, and the fact that data from several different test typesneed to be combined, this is likely to be feasible only formajor projects.

CONCLUSIONSNon-linear elasticity has proved to be an effective and

convenient basis on which, for many types of ground, todetermine geotechnical displacements. However, most soilsand weak rocks can be expected to display stiffness aniso-tropy, requiring the determination of at least three parametersfor the computation of displacements under short-term (un-drained) conditions, and five in the long term, under drainedconditions.

Despite the fact that a linear range of stress–strain behav-iour does not exist, for practical purposes it is convenient todetermine stiffness parameters at very small strains (say,0.001%) and use these as a reference. Very-small-strainstiffness can be used to establish the stiffness profile, whichhas a profound influence on displacement patterns aroundnew and existing infrastructure. And for those projects wherestrains will remain low, the operational stiffness will not begreatly different from the very-small-strain value.

To assess the sensitivity of predicted displacements andbending moments to different stiffness parameters, a proppedcantilever wall has been modelled using linear and non-linear elasticity, and it has been shown that, for this particu-lar problem, stiffness at very small strain, the change ofstiffness parameters with increasing strain, and the degree ofstiffness anisotropy had significant effects on computeddisplacements and wall bending moments.

Methods of determining very-small-strain stiffness, in thefield using seismic geophysical methods, and in the labora-tory using bender elements, the resonant column test and the

triaxial apparatus, have been described. They have beenshown to have considerable promise in determining smallstrain stiffness moduli, although they are not without prob-lems. The advantages and disadvantages of each techniquehave been reviewed, and the important potential influencesof sampling disturbance recalled.

Different methods of determining very-small-strain stiff-ness parameters cannot be expected to yield the sameresults, because of the volume affected by testing, layering,sampling disturbance, the effects of test detail, differentmethods of interpretation, and so on. The various in situseismic testing methods were shown to have different viabledepth ranges, and to be restricted to determinations of shearmodulus.

Laboratory testing, although complex, time consuming,and affected by sampling disturbance, can provide a greaterrange of stiffness data than field testing, and is unavoidableif stiffness degradation with strain is to be determined. As aresult of rate effects and cyclic loading, the rates of stiffnessdegradation with strain determined using advanced triaxialtesting, resonant column testing and cyclic triaxial testingwill not be the same.

If isotropy of stiffness is assumed, the necessary stiffnessparameters can be determined for both the undrained anddrained cases, using a limited range of loading paths in anadvanced triaxial test. If anisotropy is assumed, then in theundrained case it may be feasible to estimate the effects ofincreasing strain using a combination of resonant column,cyclic triaxial and monotonic advanced triaxial testing, withhorizontally and vertically cut specimens. Determination ofthe full suite of transversely isotropic drained stiffness para-meters, and especially Poisson’s ratio values, remains achallenge.

A great many (sometimes only partly understood) detailscontrol the precise results of stiffness tests. For futureroutine work, higher-quality sampling and more completebest practice and method specifications will be needed.Engineering judgement is currently necessary in order toweigh the different measurements in the context of a givenground model and engineering application. Comparison ofresults from different field and laboratory tests, and againstpreviously published results in similar ground conditions,should be used to check the integrity of stiffness data.

ACKNOWLEDGEMENTSThe author gratefully recognises the important contribu-

tions made by colleagues, friends and geotechnical com-panies. Individuals contributing to the work, with data,discussion, criticism and support included Jim Bennell, TonyButcher, Manolis Fleris, Chris Haberfield, Gerhard Heymann,David Hight, Qaiser Iqbal, Marcus Matthews, Rory Morti-more, Louise Otter, Justin Phillips, John Powell, Pat Power,Jeffrey Priest, Mike Rattley, Emily Rees, Chris Russell, EbenRust, Hardev Sidhu, Ken Stokoe, Jerry Sutton, RogerThompson and Xu Ming.

Data, facilities, expertise and comment have been pro-vided by AWE plc, Buro Happold, Coffey, Fugro GeoCon-sulting Ltd, Golder Associates and SGC Ltd.

Finally, this paper was reviewed and criticised by GerhardHeymann, Marcus Matthews, Jeffrey Priest, Xu Ming andAntonis Zervos. Their comments were invaluable: any re-maining errors are the author’s.

NOTATIONCp a material constantE Young’s modulus

E9 Young’s modulus in terms of effective stress

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Eh Young’s modulus for loading in horizontal directionEv Young’s modulus for loading in vertical directionEu undrained Young’s modulusE0 Young’s modulus at very small strain

e void ratioG shear modulus

Gh shear modulus for distortion in horizontal planeGv shear modulus for distortion in vertical planeG0 shear modulus at very small strainK9 bulk modulus in terms of effective stress (¼ dp9/d�V)Ku undrained bulk modulusp9 isotropic effective stress

p9atm atmospheric pressure�V volumetric strain�u undrained Poisson’s ratio�hh Poisson’s ratio relating to horizontal strain caused by imposed

horizontal strain in normal direction�hv Poisson’s ratio relating to vertical strain caused by imposed

horizontal strain�vh Poisson’s ratio relating to horizontal strain caused by imposed

vertical strain

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VOTE OF THANKSEmeritus Professor R. BUTTERFIELD, University of

Southampton.Good evening. I am very pleased to have been asked to

propose a vote of thanks to Professor Chris Clayton for hislecture on ‘Stiffness at small strain: research and practice’for many reasons – not least of which is that I have knownhim and admired his great breadth of interests for manyyears.

Above all I found his lecture refreshing, focused, clearand highly relevant – even if its embedded message was thatwe still have some way to go before we can claim to be ableto predict the service state displacements of soil structuresin general. This is in part because, in addition to their basicanisotropy and non-linearity, soil materials usually undergonon-recoverable, inelastic plastic displacements that oftenincrease over time: they creep.

Professor Clayton has, I think, provided a very significantcontribution totally in the spirit of answering Rankine’s(1858) question: ‘In practical science what are we to do?’

Historically, Roscoe in his 1970 Rankine Lecture men-tioned being similarly motivated. Apparently, in 1951 SirJohn Baker asked him to provide foundations for a steel-framed building that would collapse simultaneously with theframe, and also their predicted displacements at workingload. (Baker initiated plastic-hinge design methods for steelstructures.) Roscoe was embarrassed to find that he couldn’tdo so adequately using current soil mechanics knowledge,and proclaimed: ‘The soil mechanician should not be inter-ested only in failure; he should be concerned with beingable to predict the movements of a foundation when subjectto given working loads.’ (Apparently soil mechanics was avery masculine activity as recently as 1970!) His endeavoursled, of course, to the development of critical state soilmechanics.

Then again, Gibson, in his 1974 Rankine Lecture, pre-sented analytical solutions for elastic half-spaces in whichthe shear modulus increased linearly with depth, and hewould certainly be gratified to find that this is now a well-supported and useful model.

I must add here that I find it extremely worrying thatneither Roscoe’s nor Gibson’s work would have been sup-ported under the latest government guidelines, under which,even in universities, demonstrable short-term economic ben-efit is to be the key determinant for research funding.

I am sure we all agree that Professor Clayton hassucceeded admirably in demonstrating how the moduli re-quired in a range of non-linear elastic soil models can bedetermined from a variety of both in situ and laboratorytests, their strengths and their weaknesses and, in particular,the practical circumstances in which they are most relevant.

Please join me in expressing your appreciation of hisquite excellent lecture.


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