Soil stress-strain behaviour is highly non-linear and this hasan important in¯uence on the selection of design parametersfor simple routine geotechnical calculations. Non-linear be-haviour can be characterized by rigidity and degree of non-linearity and these can be determined from measurements ofvery small strain-stiffness, peak strength and failure strain.Very small strain-stiffness can be found from measurementsof shear wave velocity in situ or in laboratory tests. Peakstrength and failure strain can be measured in routinelaboratory tests but are strongly in¯uenced by initiation ofshear bands. The important non-linear stiffness parametersfor soil are related to its composition and to its currentstate. Back-analyses of the load settlement behaviour of full-scale and model foundations demonstrate the in¯uence ofnon-linear soil behaviour. Variations of stiffness with settle-ment calculated from full-scale and model foundations agreewell with non-linear soil stiffnesses based on rigidity anddegree of non-linearity. These results suggest a simple meth-od for routine design which takes account of soil non-linearstiffness.
KEYWORDS: Design footings/foundations; in situ testing; labora-tory tests; settlement stiffness
Le comportement contrainte-deÂformation du sol est extreÃme-ment non lineÂaire et ce fait in¯uence consideÂrablement le choixdes parameÁtres de design pour les simples calculs geÂotechni-ques de routine. Le comportement non lineÂaire peut eÃtrecaracteÂrise par la rigidite et le degre de non lineÂariteÂ, deuxparameÁtres qui peuvent eÃtre deÂtermineÂs en mesurant une treÁsfaible deÂformation-rigiditeÂ, la reÂsistance maximale et la deÂfor-mation aÁ la rupture. On trouve une treÁs faible deÂformation-rigidite en mesurant la veÂlocite de l'onde de cisaillement in situou dans les essais en laboratoire. La reÂsistance maximale et ladeÂformation aÁ la rupture peuvent eÃtre mesureÂes par des essaisde routine en laboratoire mais sont fortement in¯uenceÂes parl'initiation de bandes de cisaillement. Les parameÁtres impor-tants de rigidite non lineÂaire pour le sol sont lieÂs aÁ sa composi-tion et aÁ son eÂtat actuel. Des reÂtro-analyses du comportementde tassement pour des fondations grandeur nature et desmaquettes montrent l'in¯uence du comportement non lineÂaired'un sol. Les variations de rigidite en fonction du tassement,calculeÂes d'apreÁs des fondations reÂelles et des maquettes,correspondent bien aÁ la rigidite non lineÂaire du sol baseÂe surla raideur et le degre de non lineÂariteÂ. Ces reÂsultats suggeÁrentune meÂthode simple pour les eÂtudes de routine, meÂthode quitient compte de la rigidite non lineÂaire du sol.
INTRODUCTION
In the Autumn of 1969 when I started research at ImperialCollege on soil stiffness I had three textbooks. These were: Soilmechanics in engineering practice (Terzaghi & Peck, 1948),The measurement of soil properties in the triaxial test (Bishop& Henkel, 1957) and Critical state soil mechanics (Scho®eld &Wroth, 1968). As a young research student it was dif®cult tounderstand that these three books were all dealing with soils inground engineering. Two important themes of my work havebeen to try to clarify the principal issues covered in these threebooks and to research soil strength and stiffness.
It is now well known that the stress±strain behaviour of soilis highly non-linear and soil stiffness may decay with strain byorders of magnitude. This means that for a geotechnical struc-ture such as a foundation, retaining wall or tunnel, soil stiffnessvaries both with position and with loading.
Many aspects of non-linear soil stiffness are now well under-stood. They have been incorporated into numerical models andhave been used with success in geotechnical design. Many ofthese non-linear models and numerical analyses are relativelycomplex and require special testing and lengthy calculation.There are, however, many practical cases for which thesecomplex models and analyses are not justi®ed and familiarmethods based on load factors or simple elastic analyses aresuf®cient. These may be improved if allowance is made for soilnon-linearity.
The principal purposes of the 40th Rankine Lecture, and ofthis paper, are to consider how soil non-linearity can bequanti®ed from the results of relatively simple tests and toexamine the in¯uences of soil non-linearity on simple routinedesign methods. In characterizing non-linearity it is necessaryto consider both stiffness, strength and strain at failure and therelationships between them. Measurement of soil stiffness overthe full range of loading from very small strain to failurerequires the use of local strain gauges but stiffness at very smallstrain can be determined relatively easily from measurements ofshear wave velocity in laboratory tests or in situ. Soil has many
strengths depending mainly on drainage and strain and the peakstrength is appropriate for characterizing non-linearity. Peakstrengths are associated with slip planes or shear bands and it isnecessary to consider the in¯uence of these on strength meas-ured in laboratory tests.
Non-linear behaviour of soilOne of the major problems in ground engineering in the
1970s and earlier was the apparent difference between thestiffness of soils measured in laboratory tests and those back-calculated from observations of ground movements (e.g. Cole &Burland, 1972; St John, 1975; Wroth, 1975; Burland, 1979).These differences have now largely been reconciled through theunderstanding of the principal features of soil stiffness and, inparticular, the very important in¯uence of non-linearity. This isone of the major achievements of geotechnical engineeringresearch over the past 30 years.
Figure 1 illustrates a typical stiffness-strain curve for soil. Atsmall strains the stiffness is relatively large; at strains close tofailure the stiffness is small: this is soil being non-linear. Fig. 1includes typical ranges of strain for laboratory testing and forstructures. The ranges of strain for the different testing techni-ques in Fig. 1 are similar to those given by Atkinson & Sallfors(1991). These will be discussed later in more detail. The typicalstrain ranges for structures are those given by Mair (1993). Atypical characteristic strain in the ground is 0´1%; this repre-sents a movement of 10 mm across a gauge length of 10 m.Generally, strains in the ground will vary from zero far awayfrom the structure to relatively large values near the structureand at the edge of a rigid foundation they will be very large.The typical strain ranges proposed by Mair (1993) were basedon stiffnesses which gave reasonable designs for structures inLondon Clay.
Routine designIn geotechnical engineering there are some works which
require detailed analysis either because there are special design
487
Atkinson, J. H. (2000). GeÂotechnique 50, No. 5, 487±508
�Professor of Soil Mechanics, City University, London.
requirements or because there are substantial economies to bemade. An example would be the design of a large retaining wallin an urban environment. In this case, it would probably benecessary to calculate the distribution of horizontal and verticalground movements in front of and behind the wall, stresses inthe wall and loads in anchors or props both during constructionand in service.
Detailed analysis and design of a major geotechnical struc-ture will require special laboratory testing involving the applica-tion of complex stress paths and the measurement of smallstrains together with numerical analyses using soil modelswhich take account of the important features of soil behaviour,including current state, recent history, in-elastic deformations,anisotropy, general stress states, rotation of axes of stress andstrain, and so on (Hight & Higgins, 1995). All this is verycomplicated and demanding and requires special equipment andexpertise to obtain reliable solutions.
There are, however, very many cases where it is not soimportant to have such detailed analyses and where relativelysimple solutions are all that are needed. These routine analysescalculate only one movement in one direction; examples wouldbe the settlement of a foundation, the horizontal movement atthe top of a simple retaining wall, the surface settlement abovethe centre-line of a tunnel and so on.
Figure 2 illustrates the settlement of a loaded shallowfoundation and the two principal methods for routine design.The general principles apply also to the design and analysis ofsimple retaining walls and tunnels. For the shallow foundationillustrated the basic requirement is to determine the designbearing pressure ód which will cause a design settlement rd.
In the ®rst method the allowable bearing pressure óa iscalculated from
óa � Lfóc � 1
Fs
óc (1)
where óc is a calculated ultimate bearing capacity, Lf is a loadfactor and Fs is a factor of safety where Fs � 1=Lf . In thismethod the factor of safety or load factor is there to limitsettlements; the intention is to reduce the ultimate bearingcapacity by a factor so that the design point is in the part of theload settlement curve where settlements are relatively small.Additional partial factors may be applied to various actions andreactions.
In the second method the settlement is calculated from
ÄrB� Äó
(1ÿ í2)
Es
Ir (2)
where Är is the change of settlement due to a change ofbearing pressure, Äó , B is the width of the foundation, í isPoisson's ratio, Ir is an in¯uence factor which depends princi-pally on the geometry of the foundation (Poulos & Davis, 1974)and Es is the secant Young's modulus corresponding to theincrement of loading. Es may be related to Young's modulus forvery small strain Eo through a stiffness ratio Es=Eo. Again,partial factors may be applied to account for uncertainties.These simple routine methods may also be applied to the designof deep foundations, retaining walls and tunnels.
Load factor Lf and stiffness ratio Es=Eo are design para-meters. They will depend on, among other things, the soil, itsstate and its stress±strain behaviour, the structure and the designmovements. Since these simple routine design methods aim todetermine only one movement it must always be possible toselect load factors or stiffness ratios which give correct solu-tions. If the soil is non-linear then these will vary with loadingand movement or strain.
It must be emphasized that these simple methods are, ofcourse, limited. They can work only for calculating one move-ment in one direction for relatively simple structures and well-behaved soils. If more information is required, such as bothvertical and horizontal movements or a pro®le of settlement ordistributions of stress in the ground, then much more compli-cated analyses will be required.
The simple methods described here are applicable to drainedor to undrained loading, making use of data from drained orundrained tests but not to cyclic or repeated loadings. They areapplicable to soils which have the characteristic behaviourdescribed later. These include relatively stiff ®ne and coarsegrained soils which are not strongly bonded; they exclude verysoft soils, strongly bonded soils, and soft rocks and soils withunstable structure.
CHARACTERISTICS OF NON-LINEARITY
Figure 3 de®nes basic strength and stiffness parameters for atriaxial test. In Fig. 3(a) the cylindrical sample has axial andradial stresses óa and ó r and strains åa and år and the deviatorstress is q (� óa ÿ ó r). On loading there is a non-linear stress-strain curve as illustrated in Fig. 3(b). The sample fails at thepeak deviator stress qf at a strain åf . The stiffness is Young'smodulus E which may be de®ned as a tangent Et or as a secantEs. The stiffness at very small strains near the start of loadingis Eo.
The tangent and secant Young's moduli vary with strain asillustrated in Fig. 3(c). There are three regions de®ned byAtkinson & Sallfors (1991). In the very small strain region thestiffness is approximately constant and Et � Es � Eo and thisregion is limited by a strain åo. There is a small strain region
Stif
fnes
s: G
Typical strain ranges
Retaining walls
Foundations
Tunnels
0·0001 0·001 0·01 0·1 0 10
Shear strain, εs: %Dynamic methods
Local gauges
Conventional soil testing
Fig. 1. Characteristic stiffness±strain behaviour of soil with typicalstrain ranges for laboratory tests and structures (after Atkinson &Sallfors, 1991 and Mair 1993)
Fig. 2. Methods for routine design of simple foundations
from åo up to a strain of about 0´1% within which the stiffnessdecays rapidly. There is a large strain region beyond 0´1%within which the stiffness is relatively small. The strainå � 0:1% coincides with characteristic strains in the groundnear structures and with the smallest strain that can be meas-ured reliably in conventional soil tests. The secant modulus Es
continues to decrease gradually but remains positive evenbeyond the peak deviator stress while the tangent modulus Et iszero at failure and then becomes negative as the soil softens. Ifthe stress±strain axes are q and ås (� 2=3(åa ÿ år)) the stiffnessdq=dås � 3G where G is the shear modulus.
Figure 4(a) illustrates the stress±strain behaviour of a simplematerial which has a linear stress±strain response and whichfails at a deviator stress qf with the corresponding stiffness-strain curve. The rigidity is de®ned as the ratio of stiffness tostrength, E=qf , and this is equal to 1=åf . The ratio of stiffnessto strength is an important parameter. It appears in solutions forcavity expansion (Vesic, 1972). For linear materials it deter-mines the failure strain and characterizes brittleness or ductility.Vesic (1972) de®ned the ratio of shear modulus to undrainedstrength G=su as rigidity index.
The area beneath the stiffness-strain curve is Eåf and this isequal to the strength qf . Simpson (1992) showed that this is ageneral result and holds for non-linear materials and for drainedand undrained loading.
Figure 4(b) illustrates the stress±strain behaviour of simplenon-linear materials. The rigidity is now de®ned as Eo=qf �1=år where år is a reference strain: it is important to note thatthe reference strain år is simply de®ned from the rigidity and it
is not the strain at any characteristic point during the loading.Non-linear materials fail at strains åf which are greater than år
and the ratio åf=år � nl is a measure of the degree of non-linearity. For the particular case in which the stiffness decreaseslinearly with strain nl � 2 because the area beneath the stiff-ness-strain curve must equal the strength as shown in Fig. 4(b).
Rigidity and the degree of non-linearity together serve tocharacterize non-linear stress±strain behaviour. The parameters,very small strain stiffness Eo, strength qf , and failure strain åf ,are all easily measured.
Table 1 summarizes the stiffnesses and strengths of somecommon materials in order to put the properties of typical soilsinto context. The values are approximate and, for brittle materi-als, they are for compression. The values given for soft and stiffsoils were obtained from test results and simple correlations, asdescribed later. There are some interesting values in Table 1.The rigidity of soil is greater than that of other commonmaterials largely because soil is relatively very weak. Therigidity of stiff soil is less than that of soft soil. This is asurprising result which will be discussed later. The degree ofnon-linearity of soil is highly variable and covers almost thewhole range of all the other materials. It is this variation indegree of non-linearity which characterizes soil stiffness andwhich makes geotechnical design demanding.
MEASUREMENT OF STRAIN AND STIFFNESS IN LABORATORY
TESTS
The values of strength, stiffness and permeability parametersmeasured in laboratory tests depend on many factors including:
qfqf
qq
εr
εr εf = 2εr
εf
εf
εf
εf
ε
ε
ε
ε
(a) (b)
Eo
EoEo
E
E E
Fig. 4. Stress±strain behaviour of simple materials: (a) linearmaterial; (b) non-linear material
the quality of the sample, the procedures used to set up thesample in the apparatus prior to testing, details of the loadingpath and rate of testing, details of the design and performanceof the apparatus and instruments, procedures for analysis andinterpretation of the raw test data. Many of these factors wereconsidered by Hight (1998).
Baldi et al. (1988) found that major sources of error in themeasurement of strain and stiffness in triaxial tests, particularlyat small strains, were in bedding and seating errors and inmisalignments of the loading ram or load cell with the topplaten. These errors are avoided by the use of local gaugesattached to the sample. There is, however, still a lower limit ofstrain which can be measured reliably using local gauges butstiffness at very small strain can be measured using dynamicmethods.
Figure 5 shows the characteristic stiffness±strain curve forsoil with the three regions de®ned by Atkinson & Sallfors(1991) and shows the different laboratory equipment and testprocedures best able to measure stiffness in each region.
At strains in excess of about 0´1% secant stiffness can bemeasured with reasonable accuracy in triaxial tests using dis-placement gauges mounted in the conventional manner outsidethe cell. The accuracy of measurement can be improved byreducing the bedding, seating and misalignment errors(Atkinson & Evans, 1985).
Reliable measurement of soil stiffness throughout the smallstrain-region from strains of about 0´001% up to about 0´1%can really only be made using local gauges attached directly tothe sample (Jardine et al., 1984). If measurements of strain aremade outside the cell, the corrections to account for the errorsare often greater than the strains being measured. Local gaugesmust operate satisfactorily in water or oil under pressure andmust remain stable and accurate for long periods. (Tests on ®negrained soils to measure stiffness, taking account of recenthistory and other effects, often last several weeks.) There are anumber of recent state-of-the-art reviews on the measurement ofsoil stiffness using local gauges (e.g. Scholey et al., 1995).
Figure 6 shows the original Imperial College hydraulic stresspath apparatus (Atkinson, 1973). There are displacement gaugesinside the cell. They are not strictly local gauges because theyare attached to the platens and not directly to the sample. Thisapparatus represents the early days of research into soil stiffnessin the small strain range.
Figure 7 shows commercially available miniature displace-ment transducers, measuring axial and radial strains in a samplein an hydraulic stress path triaxial cell (Cuccovillo & Coop,1997). These instruments are relatively simple and reliable andthey are capable of very high resolution. They can resolvestiffnesses at strains of about 10ÿ6 (i.e. 0´0001%) which, formany soils, extends into the very small strain region. Thisrepresents a resolution of 0´1 micron over a gauge length of100 mm. It is hardly necessary to have more precise measure-ments and the problems are mainly in mounting the gauges onthe sample and in interpreting the data.
Figure 8 shows an example of a modern hydraulic stress pathcell capable of applying a full range of triaxial stress paths withaxis and radial strains measured using miniature displacement
Fig. 6. Imperial College stress path apparatus (Atkinson, 1973)
Fig. 5. Measurement of soil stiffness in laboratory tests Fig. 7. LVDTs used as local gauges
transducers. Tests in equipment of this kind would be requiredto measure the whole of the stiffness±strain behaviour over thefull range of strain needed for full numerical analyses of groundmovements using complex constitutive models. It is, however,still dif®cult to use local gauges routinely and there are fewengineers with the expertise required to specify and supervisetesting and who are able to interpret the results. Measurementof soil stiffness over the whole range of strain using localgauges is unlikely to be routine, at least for some time to come.
MEASUREMENT OF STIFFNESS AT VERY SMALL STRAIN USING
DYNAMIC METHODS
While it is dif®cult to measure soil stiffness in the smallstrain range using local gauges, it is much easier to measuresoil stiffness in the very small strain range using dynamicmethods. Early research in soil dynamics (e.g. Hardin &Drnevich, 1972) was associated with ground vibrations andstiffnesses measured using dynamic methods were found to beconsiderably larger than those measured using conventionaltriaxial tests with external displacement gauges. Georgiannouet al. (1991) showed that stiffnesses measured at small strainsin triaxial tests using local gauges were of the same order asthose measured in dynamic tests. Dynamic and static stiffnesseshave now been reconciled by understanding soil non-linearityand it is clear that it is the magnitude of the strain and not thestrain rate which most in¯uences soil stiffness.
The basic principles and methods for determining soil stiff-ness at very small strain from direct measurements of shearwave velocity in laboratory and in situ tests are illustrated inFig. 9. In laboratory tests shear waves are generated anddetected by bender elements (Shirley & Hampton, 1978). In situshear waves generated at the surface or below ground aredetected by instruments in boreholes or pushed in probes.
The shear modulus G of a material is related to the velocityVs of a shear wave through it by
G � rV 2s �
ã
gV 2
s (3)
where r is the density, ã is the unit weight and g is theacceleration of the Earth's gravity. The strains generated by thepassage of a shear wave will be very small, generally less than0´001% (Dyvik & Madshus, 1985), and so the shear moduluscalculated from equation (3) will be Go, the stiffness at verysmall strain. Since total and effective shear stresses are equalGo � G9o � Gu
o.For an isotropic elastic soil the effective stress elastic para-
meters are related by
E9o � 2G9o(1� í9) (4)
and, for undrained loading for which íu � 0:5
Euo � 3Gu
o (5)
There are also indirect methods for measuring shear modulusin dynamic tests such as laboratory resonant column tests(Richart et al., 1970) and in situ measurement of Rayleighsurface wave velocities (Abbiss & Ashby, 1983). Direct meas-urement of shear wave velocity using laboratory bender elementtests or in situ down-hole and cross-hole tests are relativelysimple to perform and interpret.
Measurement of shear wave velocity in laboratory tests usingbender elements
A bender element is a piece of piezo-ceramic plate whichbends if a voltage across it is changed or, if bent by an externalforce, the voltage across it changes. Bender elements are usuallyset into the top and bottom platens of a triaxial or oedometercell and penetrate about 3 mm into the sample. One element isvibrated by changing the voltage across it, shear waves propa-gate through the sample and vibrate the other element. Theinput and output voltages are continuously recorded and thetravel time determined.
Figure 10 shows bender elements set into the platens of atriaxial apparatus and Fig. 11 shows a hydraulic triaxial cellequipped with bender elements. The input and output voltagesmay be recorded on an oscilloscope or in a PC with anoscilloscope card. This equipment has been used routinely atCity University to determine shear wave velocity and Go insoils and soft rocks (Viggiani, 1992; JovicÏicÂ, 1997).
Bender elements (Shirley & Hampton, 1978; Dyvik & Mad-shus, 1985) were originally developed to measure shear wavevelocities in soft soils. The equipment was modi®ed and devel-oped for testing stiff soils and soft rocks by Viggiani (1992)and by JovicÏic (1997). Bender elements have been installed into
Fig. 8. PC-controlled stress path apparatus with local gauges
Bender elementsin triaxial oroedometersamples
Downhole
Crosshole
(b)(a)
Fig. 9. Measurement of Go from shear wave velocity: (a) laboratorytests; (b) in situ tests Fig. 10. Bender elements in the platens of a triaxial cell
separate mounts attached to the sides of the sample so the shearwaves propagate across the diameter (Pennington et al., 1997).This equipment is highly portable and has been used to measureshear wave velocities in uncon®ned samples immediately afterrecovery from the ground.
The effective distance travelled by the shear wave throughthe sample is the tip to tip distance between the benderelements (Viggiani, 1992). The quality of the measurement ofthe travel time is sensitive to the form, frequency and amplitudeof the shear wave. In early tests, the excitation was normally asingle square pulse that generated unwanted near-®eld effects.Nowadays the excitation is normally a sine wave and traveltimes can be measured reliably (JovicÏic et al., 1996).
In principle, the tests are suf®ciently simple to perform andthe results are suf®ciently reliable for routine analyses. On ascale of cost and dif®culty, simple bender element tests todetermine shear wave velocity in a sample should fall betweenconventional unconsolidated undrained (total stress) triaxial testsand consolidated drained or undrained (effective stress) tests:they are considerably less costly and easier to perform thanstress path tests using local gauges.
Direct measurement of shear wave velocity in situMeasurement of the velocities of waves propagating through
the ground is a well-established technique in geophysics and isused in ground investigations mainly for pro®ling. In commonapplications the source generates both shear waves (S-waves)and compression waves (P-waves) and it is the faster travellingP-waves which arrive ®rst and mask the arrival of the S-waves.In order to measure stiffness it is necessary to have a sourcethat generates S-waves which are essentially free from P-wavesand other fast travelling waves (JovicÏic et al., 1996).
For in situ measurement of shear wave velocity, shear wavesare normally propagated vertically downwards from the surface(down-hole tests) or horizontally from one bore hole to otherbore holes (cross-hole tests), as illustrated in Fig. 9(b). Alter-natively, a seismic cone (Campanella et al., 1986) can be usedin place of geophones in bore holes. The travel times arenormally recorded between two receivers rather than from thesource. It is important to determine the locations of the recei-vers in order to obtain accurate measurements of the wave pathlength.
ANISOTROPY IN SHEAR WAVE VELOCITY MEASUREMENTS
Many soil properties are anisotropic and shear wave velocitydepends both on the direction of propagation and on thedirection of vibration. Horizontally, propagating shear wavesmay vibrate in a horizontal or in a vertical plane, as illustratedin Fig. 9(b), and these detect anisotropy. Vvh is the velocity of a
shear wave propagating vertically with horizontal vibration, Vhv
is the velocity of a wave propagating horizontally with verticalvibration and Vhh is the velocity of a wave propagating horizon-tally with horizontal vibration.
In laboratory tests in which bender elements are installed inthe end platens, the different velocities can be measured onsamples which are installed with different orientations (Simpsonet al., 1996). Alternatively, using the equipment developed byPennington (1999), bender elements installed across a diametermeasure the velocities of shear waves which propagate horizon-tally with vibrations which may be either horizontal or vertical.
Anisotropic shear moduli are related to shear wave velocitiesby:
Govh � ã
gV 2
vh (6)
Gohh � ã
gV 2
hh (7)
and these are the same for total and effective stress. For across-anisotropic material with a vertical axis of symmetry,which is stiffer for shearing in a horizontal plane than forshearing in a vertical plane, Gohh . Govh. For a homogeneousmaterial Govh � Gohv, as both correspond to the same mode ofshearing and so Vvh � Vhv. In practice, values of Vvh and Vhv
measured in situ are often different and this is commonlyattributed to horizontal layering (Simpson et al., 1996).
The introduction of elastic anisotropy, even of the simplestkind, complicates the analyses which relate the anisotropicelastic shear moduli to the other elastic parameters. For a cross-anisotropic elastic material there are ®ve independent elasticparameters. Although both Govh and Gohh can be obtained frommeasurements of shear wave velocities in situ or in laboratorytests, determination of all ®ve effective stress elastic-parametersfor a cross-anisotropic material requires additional measure-ments (Lings et al., 2000).
For undrained loading, the condition of constant volumeimposes restrictions on the values for the undrained anisotropicPoisson's ratios (Gibson, 1974). If the degree of anisotropy forundrained loading is N (Atkinson, 1975) then
N � Euv
Euh
� íuvh
íuhv
(8)
With zero volumetric strain and equal horizontal strains
íuvh �
1
2(9)
íuhv � íu
hh � 1 (10)
and, from equation (8)
íuhv �
1
2N(11)
íuhh � 1ÿ 1
2N
� �(12)
From equation (4), with the appropriate undrained anisotropicelastic parameters, and with equation (12)
Euh � 2Gu
hh(1� íuhh) � 2Gu
hh 2ÿ 1
2N
� �(13)
and, from equation (8)
Euv � NEu
h (14)
Thus, the undrained anisotropic Young's moduli and Poisson'sratios can be found from measurements of Vhh (giving Gu
hh)together with a value for N.
The degree of anisotropy for undrained loading in terms oftotal stress N is not the same as the degree of anisotropy foreffective stresses n � E9v=E9h (Atkinson, 1975) and neither is itthe same as the ratio Govh=Gohh measured from shear wave
velocities. The measurement of N in the small and very smallstrain ranges requires precise measurement of strains using localgauges. If the degree of anisotropy for undrained loading Nremaines essentially constant with strain values for N can beobtained from measurements of Eu
h and Euv at large strain in
conventional undrained triaxial tests on oriented samples.If the soil is assumed to be isotropic, values for Young's
modulus for simple routine designs have to be calculated fromvalues of Go, obtained from measurements of shear wavevelocity using equations (4) or (5) with reasonable values ofPoisson's ratio. In this case, moduli should be determined that,as far as possible, match the directions of signi®cant loading orstraining in the structure being designed. If the soil is assumedto be cross-anisotropic, undrained total stress elastic-parameterscan be determined from equations (13) and (14) using values ofGohh, calculated from measurements of Vhh using equation (7)and values of N from conventional undrained triaxial tests. Fordrained loading, determination of the effective stress cross-anisotropic elastic-parameters requires additional measurements(Lings et al., 2000).
VALUES FOR STIFFNESS OF SOIL AT VERY SMALL STRAIN
Figure 12 shows values for Go determined from shear wavevelocities measured in situ and in laboratory bender elementtests on undisturbed samples at the in situ stress state, for avariety of soils ranging from soft silt and clay to stiff clay andtill. In all cases the in situ and laboratory values agree well.The measured stiffnesses range between Go � 10 MPa(Eu
o � 30 MPa) to Go � 120 MPa (Euo � 360 MPa).
Soil stateIn order to understand soil stiffness (and the whole of soil
mechanics) it is necessary to take account of the current stateof the soil. For isotropic stresses, the state is described by thecurrent effective stress and speci®c volume with respect to areference line; for anisotropic stress states, the stress ratioshould be included. Fig. 13 illustrates soil states with axesspeci®c volume (or water content) and effective stress. Thereference line may be the projection of the critical state line orone of the normal compression lines (Chandler, 2000). At astate at A the behaviour will be different to that at B at thesame effective stress and different to that at C at the same watercontent. However, after normalization, by dividing by the cur-rent stress, the states at B and C are equivalent. Consequently,states along any line parallel with the reference line will beequivalent and so state is measured by the distance from thereference line.
The state may be described by a stress state parameter Só ora volume state parameter Sv. If the reference line is the critical
state line, and Sv is equivalent to Ãÿ íë (Scho®eld & Wroth,1968) and to the state parameter de®ned by Been & Jefferies(1986).
If the reference line is a normal compression line, statesbelow and to the left are overconsolidated. A ®ne grained soilcan reach an overconsolidated state only by virgin compressiondown the normal compression line followed by unloading andswelling. The state of a coarse grained soil can move directlyfrom B to A by vibration or compaction at constant effectivestress and so compaction has the same in¯uence on state asloading and unloading.
Variation of very small strain-stiffness with state for soilsThe variation of very small strain-stiffness with stress, speci-
®c volume and overconsolidation has been investigated exten-sively in the past. A number of relationships have beenproposed (e.g. Hardin, 1978) and most take the general form:
Go � Af (v) p9n Rmo (15)
where f (v) is some function of speci®c volume, p9 is thecurrent effective stress, Ro is the overconsolidation ratio de®nedas p9m=p9 where p9m is the maximum past effective stress. A, nand m are material parameters. If the overconsolidation ratio Ro
is de®ned with respect to a normal compression line, the statecan be de®ned by only two of v, p9 and Ro and equation (15)can be written as:
Go
pa
� Ap9
pa
� �n
Rmo (16)
where pa is a reference pressure to make equation (16) dimen-sionally consistent. ( pa, which in¯uences the value of A, isnormally taken as 1 kPa or as atmospheric pressure.)
Figure 14 shows results obtained from a set of benderelement tests on samples of reconstituted kaolin clay carried outby Viggiani (1992). Fig. 14(a) shows results from tests onnormally consolidated samples for which Ro � 1. The data fallclose to a line given by equation (16), with n � 0:65 andA � 2000 with pa � 1 kPa. Fig. 14(b) shows the results fromtests on overconsolidated samples with Go normalized by Gonc,which is the value of Go for a normally consolidated sample atthe same effective stress. The data fall close to a line given byequation (16) with m � 0:2.
Figure 15 shows results obtained from a set of benderelement tests on samples of a carbonate sand carried out byJovicÏic (1997). The behaviour of the carbonate sand is similarto that of kaolin clay shown in Fig. 14, except the values ofthe parameter m are different for samples which reachedstates inside the normal compression line by loading andunloading (true overconsolidation) or by compaction beforeloading.
140
120
100
80
60
40
20
00 20 40 60 80 100 120
Go MPa measured in situ
Go
MP
a m
easu
red
in la
bora
tory
ben
der
elem
ent t
ests
Pentre silt
Bothkennarsoft clayChattenden;London Clay
Madingley;Gault ClayCowden Till
Fig. 12. Go measured in situ and in laboratory tests (after Butcher,2000)
The carbonate sand had relatively weak grains and reachedstates on a well-de®ned linear normal compression line atstresses in excess of about 100 kPa above which considerablechanges of grading were observed. Consequently, the gradingsof truly overconsolidated samples differed from the gradings ofcompacted samples which accounts for the different values forthe parameter m.
Material parameters for very small strain stiffnessThe parameters A, n and m in equation (16) are material
parameters and so they should depend on the nature of thegrains. Viggiani (1992) carried out bender element tests onreconstituted samples of a variety of different soils and herresults are given in Fig. 16. This shows the variations of theparameters A, n and m with plasticity for ®ne grained soils.Although there is some scatter of the data there are clear trendsshowing that A decreases and both n and m increase withincreasing plasticity index.
Coop and JovicÏic (1999) reported the results of benderelement tests on a variety of different coarse grained soils. Theyfound that the relationships between very small strain-stiffnessGo and state given by equation (16) applied equally to coarse
and ®ne grained soils. They also found that the values of thematerial parameters for coarse grained soils could be closelyapproximated by A � 4000 and n � 0:58, while the value of mdepended on the history of overconsolidation or compaction.
It should be noted that, in order to determine values ofoverconsolidation ratio Ro it is necessary to establish a truenormal compression line. For most coarse grained soils this willrequire compression to very large effective stress (Coop & Lee,1993).
STIFFNESS OF SOIL AT VERY SMALL STRAIN: SUMMARY
The stiffness of soil at very small strains can be determinedrelatively simply and reliably from measurements of shear wavevelocity in laboratory samples or in situ. The value of Go for aparticular soil varies with current state in a simple and consis-tent manner given by equation (16) in which A, n and m arematerial parameters. For soils which are not strongly bonded orhighly structured, these parameters depend principally on thenature of the grains and vary consistently with plasticity index.If the soil is assumed to be isotropic, the very small strainYoung's modulus Eo can be obtained from the shear moduluswith an assumed value for Poisson's ratio. If the soil is cross-
105
104
10310210
(a)
1
2
Go:
kP
a
Go/
Gon
c
n
m
p ′: kPa1 2 5 10
(b)Ro
Fig. 14. Variation of Go with state for reconstituted kaolin clay: (a) normally consolidated samples; (b) overconsolidatedsamples (Viggiani & Atkinson, 1995)
106
105
104102 103
1
5
Go:
kP
a
Go/
Gon
c
1
Compacted
Truly overconsolidated
10010
(b)p ′: kPa
(a)Ro
Fig. 15. Variation of Go with state for carbonate sand (Jovi i & Coop, 1997)
anisotropic and undrained, the Young's moduli and Poisson'sratios can be found from Gu
hh and the degree of undrainedanisotropy N. Very small strain-stiffness, Eo, is one of the basicparameters required to characterize rigidity and the degree ofnon-linearity.
CHOICE OF SOIL STRENGTH FOR DESIGN
The parameters that, together with the very small strain-stiffness Eo, characterize non-linearity are the strength qf andthe strain at failure åf . These are used to describe the rigidity,Eo=qf and the degree of non-linearity nl � åf=år, where år isthe reference strain which is the reciprocal of rigidity. Having
considered the measurement of Eo it is now necessary toconsider the measurement of the appropriate strength. It is wellknown that a particular soil will have a number of differentstrengths, depending on the drainage and the strain. It isimportant to consider which strength is appropriate to determinerigidity and degree of non-linearity for routine designs whichare intended to limit movements or settlements.
Figure 17 illustrates the general features of the stress±strainbehaviour of a relatively stiff soil. There is a peak strength, anultimate or critical state strength and a residual strength. Thesemay be described by undrained strengths su or by angles ofshearing resistance ö9.
For complex numerical analyses, constitutive soil modelsshould include the complete stress±strain behaviour and mightinclude coupled loading and drainage. For simple routine de-sign, it is necessary to choose either a drained or an undrainedstrength and to choose between one or other of the peak,ultimate or residual strengths. The choice between drained andundrained strength depends on the relative rates of drainage andloading but the choice between peak, ultimate and residualstrength is not so simple.
Figure 18 illustrates three typical cases. In Fig. 18(a) drivinga pile or landsliding has caused large relative displacements and
3000
2000
1000
0
0·9
0·7
0·5
0·4
0·3
0·2
0·1
Coe
ffici
ent,
mC
oeffi
cien
t, n
Coe
ffici
ent,
A
0 2010 30 50 7040 60
Plasticity index
(c)
(b)
(a )
Fig. 16. Material parameters for Go (Viggiani & Atkinson, 1995)
Shearstress
Distortion
Peak
Residual
Ultimate = critical state
Fig. 17. Strength of soil
Fig. 18. Choices of strength for design: (a) very large movements;(b) ®rst time failures; (c) small movements
the creation of well-de®ned shear bands or slip surfaces. In ®negrained soil with platy particles this might reduce the strengthto the residual which is then the strength which should be usedfor subsequent shearing in the shear bands (Skempton, 1964).For a ®rst time landslip in clay, illustrated in Fig. 18(b), themovements are initially insuf®cient to reduce the strength to theresidual and the critical state strength should be used for design.Back-analyses of shallow slips in motorway cuttings and em-bankment slopes (Crabb & Atkinson, 1991) showed that thestrength mobilized at failure was very close to the ultimate orcritical state strength: this is also very close to the fullysoftened strength (Skempton, 1970). For coarse grained soils thecritical state and residual strengths are the same.
Figure 18(c) shows a loaded foundation with a settlement r.If the bearing pressure is chosen on the basis of an ultimatebearing capacity with a load factor, the strength used tocalculate the bearing capacity should re¯ect the stiffness of thesoil. This cannot be the critical state strength. (If a foundationis designed on the basis of the critical state strength with thesame load factor then the same foundation would be designedfor dense and for loose sand which is clearly unsatisfactory.)Soil stiffness is related principally to peak strength and not toultimate or critical state strength; stiffness and peak strengthboth increase with effective stress and overconsolidation. Conse-quently, it is the peak strength and the corresponding strainwhich, together with Eo, should be used to determine rigidityand degree of non-linearity for analyses which are intended tolimit movements.
SHEAR BANDS IN SOIL SAMPLES
Figure 19 shows a sample of reconstituted overconsolidatedkaolin clay which has developed clearly de®ned discontinuitiesduring a triaxial compression test. These discontinuities areusually called slip planes but, because they have ®nite thickness(although perhaps only a few grains thick), they are knownmore properly as shear bands. It is quite obvious that oncedistinct shear bands have developed they have a profound in¯u-ence on the overall behaviour of soil in a sample in a laboratorytest and in the ground.
The occurrence of slip planes or shear bands in soils is acentral feature of geotechnical stability analyses. There isextensive literature on the initiation and development of shearbands in soils and other granular materials, including theoreticalstudies (Vardoulakis & Sulem, 1995) and experimental observa-tions (Viggiani et al., 1994; Finno et al., 1997). In his RankineLecture, Burland (1990) analysed the behaviour of soil samplescontaining shear bands. It is, however, very dif®cult to obtainmeaningful soil parameters from such distorted and discontin-uous samples. Certainly there is very little that can be learnedabout soil stiffness once a shear band has developed in asample. For the present, the question is how shear bands
in¯uence peak strengths and failure strains measured in soilsamples.
Angles of intersection of shear bands in undrained testsConventionally, directions and angles of intersection of shear
bands in soil are associated with the stress ®eld. Shear bandsare usually assumed to occur in the characteristic directionswhich are the planes in the soil on which the stresses are thosewhere the Mohr's circle touches the failure envelope: this leadsto the standard Rankine stress ®elds in which stress character-istics and shear bands intersect at angles 90ÿ ö.
For undrained loading of soils this presents a problembecause, in terms of total stress, the failure envelope is givenby öu � 0 while, at the same time in terms of effective stress,it is given by a friction angle ö9 which is non-zero. This isillustrated in Fig. 20. The two Mohr's circles, one for totalstress and one for effective stress, are for the same sample atthe same instant and the angle of intersection 2á of the stresscharacteristics and the shear bands could be either 908 or908ÿ ö9. The question is: what are the directions of shearbands in undrained tests and, if shear bands are associated withstress characteristics, how does the soil know which Mohr'scircle and failure criterion, in total or effective stress, to follow?
Formation of shear bands in triaxial testsFigure 21 shows two initially identical samples of overconso-
lidated kaolin clay after about 15% axial strain in uncon®nedcompression tests. In both cases the overall water contentsremained unchanged. The sample on the left developed a strongshear band. The shear band is inclined at about 608 to thehorizontal so complementary shear bands would intersect atabout 608. The sample on the right barrelled; there is no distinctshear band.
The difference between the two samples is that the one onthe left which had developed the shear band had been strainedvery slowly and had reached 15% axial strain in about six hourswhile the one on the right which had barrelled had beenstrained very quickly and had reached 15% axial strain in aboutone second. The shear bands which developed in the relativelyslow undrained tests were in reconstituted samples and theyappear also in coarse grained soil. They are not associated withpre-existing ®ssures.
In ®ne grained soils, rate processes are associated primarilywith drainage. In the sample that was strained very quickly itcan be assumed that there was no drainage whatsoever and thesample remained at constant volume throughout. In the rela-tively slow test, however, there is the possibility that there wassome local drainage near the developing shear band. Even if thepermeability is low, the drainage path length may be only a fewgrain diameters long.
Volumetric straining and shear bandsIn order to examine the directions of shear bands and volume
changes due to local drainage, it is necessary to considerstrains, not stresses. If the strains in material either side of ashear band are small compared with the strains inside the shearband then the material in a shear band will deform in planestrain and the directions of shear bands will coincide with thedirections of zero extension (Roscoe, 1970).
Figure 22 shows Mohr's circles of strain increment forincrements of strain äå1 and äå3. In the circle in Fig. 22(b)there is no volumetric strain (because the centre of the Mohr'scircle is at the origin of the strain axes) and shear bands, whichcoincide with directions of zero extension, intersect at 908. Inthe Mohr's circle in Fig. 22(c) there is a volumetric strain äåv
and shear bands intersect at an angle 908ÿ í, where í is theangle of dilation.
This provides an explanation for the appearance and non-appearance of shear bands in initially identical samples shearednominally undrained but at very different rates of strain. In Fig.Fig. 19. Shear bands in a soil sample
21, the sample on the left was strained relatively slowly; therewas time for local drainage, soil in the shear band dilated,softened and weakened and this allowed the shear band to grow.The direction of the shear band in the sample in Fig. 21corresponds to an angle of dilation í of about 308. The sampleon the right was strained very quickly; there was no time evenfor local drainage, there was no softening or weakening and nostrong shear band developed.
Vardoulakis (1996a, 1996b) showed theoretically that shearbands are associated with local volumetric straining and thatthis can occur in globally undrained tests. He also showed thatif there is no volumetric straining in a shear band the resultingpore pressure gradients cannot be sustained in ordinary, slow,undrained tests.
Atkinson & Richardson (1987) measured the angles of inter-section of shear bands in initially identical samples of reconsti-tuted London Clay in nominally undrained triaxial tests with
very different times to failure. They concluded that the anglebetween shear bands is governed by the angle of dilation í andnot by the angle of friction ö9 or öu. They showed that as thetime to failure increased, allowing more opportunity for localdrainage to occur and the angle of dilation í to increase, theobserved angles of intersection of shear bands decreased. Theyalso found that if the test was carried out very quickly thesample barrelled and shear bands did not develop.
Dilation in shear bandsVolume changes in shear bands have been observed directly
in laboratory tests, in model tests and in situ. Desrues et al.(1996) observed voids ratios in shear bands in sand samples intriaxial tests. They found that material within a developing shearband dilated and ultimately reached a unique critical state.
Figure 23 is an X-radiograph from the Rankine Lecture givenby Roscoe (1970). It shows passive loading of a wall in drysand as the wall moves from left to right. The tone of the imageis a negative of the voids ratio in the sand; the dark bands arelooser material. There are distinct bands of looser dilated sand.These intersect at an angle which is about 608. Fig. 24 showswater contents measured by Henkel (1956) across a shear bandin soil behind a failing retaining wall. The shear band is only afew tenths of an inch thick but the water content has increasedby about 10%. These two sets of data show clearly that shearbands have ®nite thickness in which soil has dilated andweakened.
Effects of shear bands on peak strength in undrained testsThe in¯uence of rate of strain and local drainage on the
stress±strain behaviour and peak strength observed in nominallyundrained triaxial tests was examined by Atkinson & Richard-son (1987). They showed that the directions of shear bands, theobserved stress±strain behaviour, stress paths and peak strengths
Fig. 21. Samples of kaolin clay after uncon®ned compression tests
Fig. 20. Stress characteristics for undrained loading: (a) directions of stress characteristics; (b) Mohr's circle for totalstress; (c) Mohr's circle for effective stress
observed in nominally undrained triaxial tests were all consis-tent with local drainage and dilation in the shear bands.
Figure 25 shows the interpretation proposed by Atkinson &Richardson (1987). Figs 25(a) and (c) show state paths withaxes q9, p9 and water content for overconsolidated samples withand without local drainage. Fig. 25(b) shows the correspondingstress±strain behaviour. The path ABF is fully undrained; thereis no volume change, the sample fails at F and there is no peakstrength. For the path CDE local drainage starts at D; ultimatefailure is at E, there is a peak strength at D which is the startof local dilation. (Notice that the path ABF does have a peakstress ratio and a peak ö9 but not a peak deviator stress.)
The peak state at B corresponds to initiation of a shear bandowing to local drainage in a slow undrained test. In a fastundrained test, in which there was no local drainage and noshear band, the peak strength corresponds to the ultimatestrength at F. For intermediate rates of loading, or times to
δε1
δε1
δε3δε1
δε1
δε3
δε3
ννβ
δε
δε
δε3
2β
β β
β
Pole
(b)
(a)
(c)
½δγ
½δγ
½δεv
Fig. 22. Strains in shear bands: (a) directions of zero extension; (b) Mohr's circle for constant volume straining;(c) Mohr's circle for straining with dilation
Fig. 23. Dilation in shear bands in dry sand behind a model wall(after Roscoe, 1970)
0·20 0·25 0·30 0·35Watercontent
Slipzone
Distancefrom softzone ininches
4
2
4
2
Fig. 24. Water content observed in a shear band in the groundbehind a failing wall (after Henkel, 1956)
failure, the peak strength will be between D and F. Atkinson &Richardson (1987) found that the peak strength in nominallyundrained triaxial tests decreased with the logarithm of the timeto failure, which is consistent with the interpretation based onlocal drainage in shear bands.
INITIATION OF SHEAR BANDS IN SOILS
The issue of localization and initiation of shear bands ingranular materials has been extensively researched theoretically(e.g. Vardoulakis & Sulem, 1995) and experimentally (e.g.Finno et al., 1997). At City University, Albert (1999) carriedout triaxial tests on samples of a variety of different soils usingfour local axial-strain gauges, as shown in Fig. 26(a), in the
expectation that a shear band would miss one or more of thelocal gauges. Fig. 26(b) shows strains measured by each of thefour local gauges plotted against the mean axial strain from atest on overconsolidated kaolin clay. At a mean strain of about5% one of the local gauges started to register strains signi®-cantly smaller than the others. This is taken to indicate theinitiation of a shear band.
Fig. 25. Behaviour of a sample with a dilating shear band (afterAtkinson & Richardson, 1987)
Fig. 26. Initiation of a shear band in a triaxial sample: (a)equipment; (b) Local strains in a sample of overconsolidated kaolinclay
Figure 27 shows the stress path and stress±strain curves forthe test on overconsolidated kaolin shown in Fig. 26. The pointof initiation of the shear band at an axial strain of 5%, asinterpreted from Fig. 26(b), is indicated by the arrows. Thesedata show that the start of non-uniform straining, and theinitiation of a shear band, occur a little before the peak deviatorstress and a little after the peak stress ratio. Similar results wereobtained by Albert (1999) on other soils and have also beenobtained by several others (Viggiani et al., 1994).
PEAK STRENGTH, FAILURE STRAIN AND SHEAR BANDS
Peak strength, qf , and failure strain, åf , together with stiff-ness at very small strain Eo are the parameters which areneeded for the determination of rigidity and the degree of non-linearity. There is evidence that the peak strength of a soil isassociated strongly with local drainage and dilation and withthe initiation of a shear band. It is the local deformations andvolume changes leading to the development of a shear bandwhich primarily determine peak strength and failure strain.
In most practical cases, rates of construction are suf®cientlyslow to allow shear bands to develop in the ground if thestructure comes close to failure even if the ground can be
assumed to be undrained overall. Consequently, tests to measurepeak strength and failure strain, which will be used to determinerigidity and the degree of non-linearity, should be carried outsuf®ciently slowly to allow shear bands to develop. Peakstrengths measured in samples that develop shear bands will besmaller than those measured in samples that do not developshear bands.
TYPICAL VALUES FOR SOIL NON-LINEAR PARAMETERS
Soils are essentially collections of grains with grading,mineralogy, shape and texture that are arranged in a dense orloose packing and are loaded to a current effective stress. In theabsence of strong bonding or structure, the parameters whichdescribe the mechanical behaviour of a soil should dependprimarily on its nature and its state and they should generallyfall within certain limits. Some parameters, such as critical statefriction angle ö9c, are material parameters and they depend onlyon the nature of the grains. Others, such as very small strainshear modulus Go, depend also on the state. Other parameters,such as A, n and m in equation (16), that relate mechanicalproperties to state, are themselves material parameters and arerelated to the plasticity index as shown in Fig. 16.
Typical values for stiffness and strength of soilsTatsuoka & Shibuya (1992) collected data of compressive
strength qf and very small strain Young's modulus Eo for manydifferent soils in many different states. They also included datafor other materials, including rocks, concrete and metals. Theirdata for typical soils fall within the shaded region in Fig. 28.
The range of compressive strength qf is about 80 kPa to400 kPa; the range of Eo is about 80 MPa to 600 MPa; therange of rigidity Eo=qf is about 3000 to 500. These data donot, however, distinguish between different soils and betweendifferent states of the same soil.
Variation of rigidity with plasticity and stateThe stiffness of soil at very small strain is related to its state
by equation (16) in which the parameters A, n and m dependon the plasticity index as shown in Fig. 16. For undrainedloading of isotropic elastic soil Eu
o � 3Go and equation (16)becomes
Euo
pa
� 3Ap9
pa
� �n
Rmo (17)
150
100
50
0
0
0 50 100 150
5 10 15
(b)
(c)
(a)
20
1·5
1·0
0·5
0
0
50
100
150
p ′: kPa
q′:
kPa
q: k
Pa
q′/p
′
εa: %
Fig. 27. Initiation of a shear band in a sample of overconsolidatedkaolin clay
106
105
104
104
103
103
102
102
10q f: kPa
Eo:
kP
a
Eo/q f
Fig. 28. Typical values of strength and stiffness of soils (afterTatsuoka & Shibuya, 1992)
The undrained compressive strength qf is also related to thestate by a relationship of the form (Muir Wood, 1990)
qf
pa
� 2Bp9
pa
� �Rì
o (18)
For normally consolidated soil Ro � 1 and equation (18) isequivalent to su=ó 9v � B, where su is the undrained strength, ó 9vis the vertical effective stress and B is related to the plasticityindex (Skempton, 1957). For overconsolidated soils, the param-ter ì is approximately 0´8 for a wide range of soils (MuirWood, 1990).
Dividing equation (17) by equation (18) gives
Euo
qf
� 1:5A
B
p9
pa
� �nÿ1
Rmÿìo (19)
in which the rigidity (Eo=qf ) is related to the current state(given by p9 and Ro) through material parameters whichthemselves depend on the plasticity index.
Figure 29 shows values of rigidity given by equation (19),varying with current pressure and with the overconsolidationratio for soils with different plasticity indices. To evaluateequation (19), values of A, n and m were taken from Fig. 16;values for B and ì were taken from Muir Wood (1990). Therange of values of rigidity in Fig. 29 is about 400 to 4000,which is only a little larger than that given in the data byTatsuoka & Shibuya (1992) and shown in Fig. 28.
Figure 29(a) shows that the rigidity of soil decreases withstress (for a given overconsolidation ratio). This is because thevalue of n is always smaller than 1 and so, in equation (19),nÿ 1 is always negative. Fig. 29(b) shows that the rigidity ofsoil decreases with the overconsolidation ratio (for a givenstress). This is because the value of m is always smaller thanthe value of ì and so mÿ ì is always negative. These resultsexplain why the rigidity of stiff soil is smaller than the rigidityof soft soil, as noted in Table 1.
Typical values for åo
The limiting strain within which the stiffness of soil may betaken to be constant with a value Go or Eo is åo. It may beobserved in resonant column tests (Georgiannou et al., 1991) orin triaxial tests using precise local gauges (Coop et al., 1997).
From results of resonant column tests, the limiting shearstrain ão(� 2åo) was found to increase with the plasticity indexfrom about 10ÿ3% (åo � 0:0005%) for low plasticity silts toabout 10ÿ2% (åo � 0:005%) for high plasticity clays(Georgiannou et al., 1991). From results of triaxial tests, withvery precise local gauges, Coop et al. (1997) found åo smallerthan 0´0001% for unbonded coarse grained soils. For bondedsoils and soft rocks, values of åo are relatively large; Cuccovillo& Coop (1997) found åo about 0´02% in tests on intact samplesof Greensand.
Variation of failure strain with stateFigures 30(a) and (b) show variations of strain at failure, at
the peak deviator stress, with initial state for a number ofdifferent soils for drained and undrained triaxial tests. With theexception of the data for Brasted Sand (Cornforth, 1967) thedata are from tests carried out at City University. These datashow that, in general, failure strain reduces with increasingoverconsolidation ratio and with decreasing speci®c volume.This means that, in general, failure strain, and hence degree ofnon-linearity, tends to decrease as the state moves away from areference line as indicated in Fig. 30(c) (that is the degree ofnon-linearity will tend to decrease with overconsolidation).
The data shown in Figs 29 and 30 demonstrate that, at leastfor reconstituted soils, rigidity and the degree of non-linearityvary consistently with the nature of the soil grains and with thecurrent state of the soil. Engineers like to believe that engineer-ing properties of soils are variable. Indeed they are but they dovary in a consistent and predictable way.
INFLUENCE OF SOIL NON-LINEARITY ON DESIGN
PARAMETERS
Non-linearity in soil can be described by rigidity and thedegree of non-linearity and it is interesting to examine howthese in¯uence choices of parameters for simple routine design.To do this it is helpful to make use of a simple expression fornon-linear stress±strain behaviour.
There are many expressions for non-linear stress±straincurves for soil in the literature (e.g. Kondner, 1963; Puzrin &Burland, 1998). The expression in equation (20) is about thesimplest that captures the essential features of non-linearstress±strain behaviour.
Et
Eo
�1ÿ åf
å
� �r
1ÿ åf
åo
� �r (20)
The tangent Young's modulus Et decays with strain; there is aregion of very small strain where Et � Eo up to a limitingstrain åo; there is a failure strain åf . There is also a compressivestrength qf which ®xes the value of r so that the area beneaththe stiffness±strain curve is qf . (For typical values of rigidityand degree of non-linearity for soil the value of r is generallyin the range 0´1 to 0´5.)
Equation (20) is applicable to drained or to undrained load-ing, with appropriate values for the parameters. It can beintegrated to give a simple expression q � q(å) and the secantYoung's modulus can be calculated from this. By varying theparameters Eo, qf and åf , this describes the stress±strainbehaviour of soils with different rigidities and different degreesof non-linearity.
Figure 31 shows soil behaviour given by equation (20)plotted for soil with a rigidity of 1000 and for degrees of non-
Fig. 29. Variation of rigidity with state and plasticity index
linearity in the range 10 to 100. In Fig. 31(a) the load factorLf � q=qf . The data illustrate how load factor, Lf , and secantstiffness ratio, Es=Eo, vary with the degree of non-linearity. Ata strain of 0´1%, indicated by the arrows, both load factor andstiffness ratio vary by factors of 2 to 3.
Figures 32(a) and (b) show the variations of load factor andstiffness ratio with the degree of non-linearity and rigidity for astrain of 0´1%. Over a typical range of non-linear soil parametersthe stiffness ratio needed for a design strain of 0´1% varies fromabout 0´5 to less than 0´2 and the variation of load factor isgreater. Fig. 32(c) shows how the ratio of stiffness to strength(Es=su) varies with non-linearity; this is a parameter often usedin simple routine design. Similar design curves can be easilydeveloped for other characteristic strains from a simple stress±strain equation, such as that given in equation (20).
The data given in Figs 31 and 32 were calculated for a valueof åo � 0:001%. For smaller values åo has little in¯uence on theload factor or stiffness ratio. For bonded soils and soft rocks,however, åo may be considerably larger than 0´001% and thenthe value of åo begins to have an in¯uence on the load factorand stiffness ratio. For these materials, a better basis for designmay be to avoid strains in the ground that are greater than thevalue of åo, especially if the material is brittle with a rapid dropof stiffness with strain after åo.
STRAINS IN TRIAXIAL SAMPLES AND GROUND MOVEMENTS
The data in Figs 31 and 32 relate load factor and stiffnessratio to non-linear parameters for strains in a triaxial test speci-men. But routine designs consider ground movements and so itis necessary to relate strains in triaxial specimens to groundmovements.
Figure 33(a) shows the results of ®nite element calculationsfor a triaxial sample and for shallow strip and circular founda-tions all for undrained loading (Simpson, 2000). All theanalyses were carried out using the BRICK model (Simpson,1992) with the same set of material parameters. The stress±strain curve for the triaxial sample is plotted with axes q and
Fig. 30. Variation of failure strain with state: (a) ®ne grained soils; (b) coarse grained soils; (c) general features
1·2
0·8
0·4
0
1·2
0·8
0·4
010210–110–210–3 101
(b)
(a)
ε: %
Es/
Eo
Eo /q f = 1000
Lf
n l = 100
n l = 50
n l = 20
n l = 10
Fig. 31. Variation of design parameters with strain and degree ofnon-linearity
åa. The behaviour of the foundations is plotted as bearingpressure ó and settlement to width ratio r=B. All three curvesapproach constant stress at relatively large strains or settlementsand the bearing capacity is linked to the compressive strengththrough an appropriate bearing capacity factor.
From the loadÐsettlement curves in Fig. 33(a), values ofsecant Young's modulus were calculated from
ÄrB� Äó
(1ÿ í2)
Es
Ir (21)
where Ir is an appropriate in¯uence factor (Poulos & Davis,1974) and, for undrained loading, íu � 1
2. Values for the un-
drained secant Young's modulus Eus , calculated from equation
(21) from the load±settlement curves for shallow circular andstrip foundations in Fig. 33(a), are shown in Fig. 33(b) plottedagainst the settlement to width ratio r=B. Also shown in Fig.33(b) are values for the undrained secant Young's modulus Eu
s
for a triaxial sample calculated from the triaxial stress±straincurve in Fig. 33(a).
From Fig. 33(b) the values of r=B for a shallow foundationare two to three times larger than the axial strains in a triaxialsample at the same average stiffness. These results mean thatthe stiffness at a certain strain measured in a triaxial specimenrelates to the design stiffness for a foundation at values of r=B,which are two to three times larger than the corresponding axialstrain in the triaxial sample. Bolton (1993) has obtained similarresults for shallow and deep foundations and for retaining wallsusing plasticity analyses.
NON-LINEARITY IN MODEL AND FULL-SCALE FOUNDATIONS
The in¯uence of non-linearity on foundation behaviour andon the choice of design stiffness can be illustrated by relatingthe non-linear load settlement behaviour of model and full-scalefoundations to the non-linear characteristics of the soil.
Foundations in London ClayFigure 34 shows stiffness±strain data obtained from observa-
tions of the settlements of shallow and piled raft foundations onLondon Clay made by Arup Geotechnics (1991) together withthe corresponding behaviour of London Clay in a triaxial test.For each foundation case record the equivalent undrained secantYoung's modulus Eu
s was calculated from the bearing pressureand from the observed settlement using equation (21). In Fig.34(a) the data are plotted as Eu
s=Euo and in Fig. 34(b) they are
plotted as Eus=su, which is often used to choose a value for
stiffness for routine design. Values for Euo and su were estimated
from the site investigation data for each site.Also shown in Figs 34(a) and (b) are broken lines that
represent the behaviour of London Clay in an undrained triaxialtest. The data for these were calculated using the simple modelgiven in equation (20) with parameters for London Clay whichwere a best estimate for the mean values for the many sitesconsidered. (The procedure used was to integrate equation (20),select a value of r to give the required values of qf and obtain
1·0
0·5
0
1·0
0·5
0
0
1000
2000
102 103101
(c)
(b)
(a)
Es/
s uE
s/E
o
Eo /q f = 1500
Eo/q f = 1000
Eo/q f = 500
ε = 0·1%
Lf
n l
Fig. 32. Variation of design parameters with rigidity and degree ofnon-linearity for å � 0:1%
100
80
60
40
20
00 0·5 1·0 2·01·5 2·5
q or
σ: k
Pa
ρ/B or εa: %
(a)
1
10
20
30
40
1010–110–210–3
ρ/B or εa: %
(b)
Esu :
MP
a
Strip foundation
Circular foundation
Triaxial test
Fig. 33. Finite element analyses of shallow foundations and a triaxial test for the same soil
values of the secant Young's modulus from the calculatedstress±strain curve.) The solid lines in Fig. 34 are the lines forthe triaxial test with strains increased three times to account forthe differences between axial strains in triaxial samples andvalues of r=B for foundations.
For the foundations, the stiffnesses back-calculated from the®eld observations decay with increasing values of r=B in thesame way that stiffness decays with strain in a triaxial test.The values are, however, smaller than those corresponding tothe line for the triaxial test with strains increased by threetimes. This is thought to be due to some drainage that probablyoccurred in the ground during construction and foundationloading. Drainage would have the effect of increasing settle-ments and so reducing calculated stiffnesses.
The values of r=B observed for foundations on London Clayshown in Fig. 34 are in the range of about 0´05% to 0´5%.These are comparable to the typical strain range for foundationsgiven by Mair (1993) and shown in Fig. 1.
Centrifuge model foundation on kaolin clayAlthough it is always valuable to be able to compare theor-
etical analyses with full-scale observations, it is often dif®cultto obtain all the required information about the soil, thestructure, its loads and settlements and the drainage conditions.Many of these uncertainties are avoided by observation of thebehaviour of closely monitored scale models using well docu-mented soils. Since soils are essentially frictional materials andmany of their stiffness and strength properties depend on thecurrent effective stress, geotechnical models should correctlyscale effective stress. Effective stress scaling can be achieved bytesting models in a geotechnical centrifuge (Scho®eld, 1980).
Figure 35(a) shows the London Geotechnical Centrifuge atCity University (Scho®eld & Taylor, 1988) and Fig. 35(b) showsa detail of a scale-model rigid shallow foundation on over-consolidated kaolin clay which was loaded in the centrifuge
(Stallebrass & Taylor, 1997). The foundation was 60 mm dia.and during the test the centrifuge acceleration was 100 g so theexperiment was modelling a foundation 6 m dia. The instru-ments measured the pro®le of surface settlement but only thesettlement of the rigid foundation will be considered here.
Data from a foundation loading test are shown in Fig. 36.
10
1000
500
1500
2000
0
0·2
0·4
0·6
0·8
1·0
1010–110–210–3
ρ/B or εa: %
(b)
(a)
Soil parameters:
Esu /
s uE
su /E
ou
Eou/q f = 1000
n l = 50
Shallow rafts
Piled rafts
Triaxial test
Triaxial strains × 3
Fig. 34. Settlement of foundations on London Clay (data from ArupGeotechnics, 1991)
Fig. 35(a). London Geotechnical Centrifuge. (b) Centrifuge modelfoundation
10
0·2
0·4
0·6
0·8
1·0
1010–110–210–3
ρ/B or εa: %
Soil parameters:
Esu /
Eou
Eou/q f = 5000
n l = 200
Model test
Triaxial test
Triaxial strains × 3
Fig. 36. Settlement of a centrifuge model foundation on kaolin clay(data from Stallebrass & Taylor, 1997)
The data are shown in the same form as the data for thefoundations on London Clay in Fig. 34(a). As before, the linesfor a triaxial test were calculated from equation (20) withparameters that were best estimates for the kaolin clay at thestates near the model foundation.
The stiffnesses back-calculated from the load-settlement be-haviour of the model foundation decay with increasing settle-ment. These stiffnesses are close to those given by the line fora triaxial test with strains increased by three times.
Model plate loading tests in sandsAlternatively, model tests may be carried out at elevated
effective stress in a calibration chamber. In this case stresses areassumed to be uniform with depth rather than increasing withdepth as in a centrifuge model.
Figure 37 shows data from loading tests on model plates indry carbonate sand and in dry silica sand in a calibrationchamber (Jamiolkowski, 2000). The initial mean-effective stressin these tests was generally in the range of 50 kPa to 180 kPa.The data are shown in the same form as the data for thefoundations on London Clay in Fig. 34(a), except that thesecant Young's moduli E9s are now in terms of effective stresscorresponding to drained loading of dry sand.
The lines for the triaxial tests were calculated from equation(20) with parameters that were best estimates for the effectivestress parameters for the two sands at the initial states near themodel plates in the calibration chamber. The rigidity taken forthe carbonate sand (� 1500) was the same as that taken for thesilica sand while the degree of non-linearity taken for thecarbonate sand (nl � 150) is very much larger than the degreeof non-linearity taken for the silica sand (nl � 15). This re¯ectsthe observation that the strain at failure at the peak strength incarbonate sand is often relatively large while for silica sand atthe same state it is usually relatively small.
For both sands the secant stiffnesses back-calculated from themodel plate tests from equation (21) decay with settlement andin both cases they are close to those given by the lines fortriaxial tests with strains increased by three times.
INFLUENCE OF NON-LINEARITY ON STIFFNESS RATIO FOR
DESIGN
Figure 38 shows values for the drained and undrained secantYoung's modulus, back-calculated from model tests of founda-tions and plates on silica sand, carbonate sand and kaolin clay.For the same value of r=B, the stiffness ratio is signi®cantlylarger for silica sand than for carbonate sand or kaolin clay, or,for the same stiffness ratio, the settlements for a foundation onsilica sand would be signi®cantly larger than the settlements offoundations on carbonate sand or kaolin clay. These differencesare due principally to the different degrees of non-linearity and,to a lesser extent, to the different rigidities of the threematerials.
The model plates did not fail; in each case the bearingpressure continued to increase even after very large settlements.Consequently it is dif®cult to identify a bearing capacity and toinvestigate the relationship between load factor, settlement andsoil characteristics.
SIMPLE DESIGN PROCESS
Design is always an iterative process and Fig. 39 illustrates asimple method for routine design to take account of non-linearity.
The essence is to obtain a relationship between stiffness ratioEs=Eo and r=B. This requires: measurement of very smallstrain stiffness Eo, strength qf and failure strain åf ; construction
Fig. 37. Settlements of model plates: (a) carbonate sand; (b) silicasand (data from Jamiolkowski, 2000)
Fig. 38. Settlements of model foundations on sands and clay
Es/E
o
log ε or ρ/B
ρ/B ≈ 3ε
Triaxial
Measure Eoq t and ε f Draw stiffness–strain curves
of a triaxial stress±strain curve from equation (20) or a similarrelationship; construction of a curve of stiffness ratio Es=Eo
against r=B using a relationship between triaxial strain andground movements. For a foundation, values of r=B are aboutthree times the axial strain in a triaxial test for the samestiffness.
The design process then iterates around a loop as illustrated,until the loads, dimensions, stiffnesses and settlements are allcompatible.
SUMMARY AND CONCLUSIONS
The stress±strain behaviour of soil is highly non-linear andstiffness for both drained and undrained loading decays withstrain. This has implications both for testing, to determinedesign parameters, and for calculations of ground and structuremovements. Full analyses of geotechnical structures requirespecial laboratory tests and complex numerical calculations.Alternatively, simpler routine analyses which calculate move-ment in only one direction require load factors or stiffnesseswhich, owing to soil non-linearity, depend on movements andstrains.
Simple non-linear stress±strain behaviour can be character-ized by rigidity, Eo=qf , and by degree of non-linearity, åf=år,where the reference strain, år, is the reciprocal of rigidity. Thesesoil parameters can be measured in routine and relatively simpletests.
Young's modulus at very small strain, Eo, can be obtainedfrom measurements of shear wave velocity in situ from down-hole or cross-hole tests or in laboratory samples using benderelements. For isotropic soil, relationships between shear wavevelocity and stiffness parameters are simple but for anisotropicsoil they are more complicated and for full interpretationsadditional tests are required. Eo is an important design para-meter as it contributes towards characterizing non-linear beha-viour and it is the basis for stiffness ratio Es=Eo for routinedesign.
The strength and failure strains which characterize non-linearity in soil are those at the peak state. They can bemeasured in conventional laboratory tests but they are stronglydependent on local drainage and the initiation of shear bands.
Many soil stiffness and strength parameters depend on cur-rent state, measured as the distance of the state point from areference line. For reconstituted soils and for natural soils, thatare not strongly bonded or structured, stiffness and strengthparameters depend on the nature and on the state of the soil inrelatively simple ways. Stiffness at very small strain is relatedto current state through parameters that themselves dependprincipally on the nature of the soil. The strain at failure at thepeak state also varies with state.
Soil stiffness parameters obtained from back-analyses ofobserved settlements of full-scale and model foundations arenon-linear and decay with settlement in the same way thatstiffness decays with strain in a triaxial test sample. For equiva-lent stiffnesses, values of the ratio of foundation settlement towidth (r=B) are two to three times larger than the correspond-ing axial strains in a triaxial test. By taking this difference intoaccount, stiffness±settlement relationships back-calculated fromobserved settlements of model foundations and plates are con-sistent with the non-linear stress±strain behaviour measured insimple laboratory tests. Stiffness back-calculated from observedsettlements of foundations in London Clay decayed with settle-ment in the same way but were smaller than those correspond-ing to stiffness±strain relationships measured in triaxial tests.The differences are attributable to partial drainage in the groundduring construction.
Non-linear soil behaviour can be taken into account in simpleroutine design procedures. The parameters which characterizenon-linear stiffness can be measured in simple routine tests andthe analyses are straightforward. These simple methods arehowever limited. They can determine movements in only onedirection, for relatively simple structures and for soils which arewell behaved. For complex structures, or for soils which are
strongly bonded or highly structured, or if full stress, strain anddisplacement ®elds are needed, more complex procedures willbe required.
ACKNOWLEDGEMENTS
I am indebted to many people who contributed in one way oranother to the lecture and to the written paper. I am veryfortunate to have worked for nearly 20 years in a dynamic andintellectually demanding research group at City University. Iowe much to Professor Raoul Franklin, himself originally a civilengineer, who was Vice-Chancellor for much of that time andwho had the foresight to establish and support research centresin the University. Much of the work which was the foundationfor the lecture was done by research students, research assis-tants, technicians and visitors at City University. During thepreparation of the lecture and paper, my colleagues, Neil Taylor,Matthew Coop and Sarah Stallebrass, shielded me from manydistractions and provided me with information, data and encour-agement. They and others at City University heard severalrehearsals of the lecture and commented on the paper. Collea-gues at Arup Geotechnics also heard rehearsals of the lectureand helped with calculations. I am particularly grateful toGioacchino Viggiani for help with shear bands, to Martin Lingsfor help with anisotropy, to Tony Butcher and Mike Jamiolk-owski for test data, to Vojkan JovicÏic for photographs and otherhelp and to Brian Simpson and Sarah Stallebrass for ®niteelement calculations. Finally, I am grateful to my family, Jo,Robert and Nicholas; their job was to get me to the lecture ontime and well prepared, and this they did. Nicholas made avideo which was an important part of the lecture and he alsomade many of the diagrams.
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VOTE OF THANKS
DR B. SIMPSON, Director, Arup Geotechnics, London andHonorary Editor of Geotechnique
The last time I had to speak from this rostrum, I was givensome friendly advice from Professor Peter Vaughan: `decidewhich of your slides you want the audience to understand andwhich you do not want them to understand!' Sometimes lecturesset out to bring clarity to some parts of their presentation, butto obscure in others. Contrast John Atkinson: in the Lecture wehave seen a typical example of John's workÐpart of his missionwas to take relatively complex information and to clarify andsimplify it where possible, and to make it accessible forpractical use.
The more we know about soils the more we realize their
complexity. John has referred to features such as non-linearity,anisotropy, shear bands, dependence on history, and so on. Butwe also recognize that we need to be able to carry out simplecalculations which will be approximate but nevertheless usefulas an aid to design. In fact, however sophisticated we make thecalculations, they are always only rather remote approximationsto real behaviour, which contains too many complexities foraccurate de®nition and analysis.
My own ®rst knowledge of John Atkinson, some time beforeI met him, was in studying his PhD thesis. This was on thestiffness properties of London Clay measured in triaxial testsand it provided a lot of insights into the anisotropic behaviourof the material. I noted, in particular, the very useful, andhonest, statement that he had not been able to resolve strains ofless than 0´1% (I think it was). Somehow that statement con-tained the seeds, and the suggestion, that the behaviour of theclay at very small strains might be different from what hadconventionally been measured. And John has gone on from that,over the years, to have a leading role in the business of measur-ing very small strainsÐwhich are really strains of the orderexperienced by the ground in many practical situations.
In the Lecture he has shown us the results of developmentswith small strain gauges and with bender elementsÐshear wavetestingÐto measure the smallest range of strains. He has re-minded us that interpreting the results of laboratory testing isnot always simple, and poor interpretation can make the soilbehaviour seem more complicated and less predictable than itreally is. A clear understanding of the details of test procedureand an intelligent inspection of the laboratory specimens duringand after test are vital if pitfalls are to be avoided. John and histeam at City University routinely bring these skills to theirwork, and pass them on to the students.
However, John's aim in the Lecture has been to show howthe understanding that has been developed in modern labora-tory testing can be applied to useful effect by practitioners. Hehas reduced the complexity to a small number of parameters,and it was interesting to see how soils could be compared withother common materials using these parameters. He has con-centrated on a single problemÐthat of settlement of a shallowfoundationÐand has started from the two parameters whichare easiest to measure: strength at large strains and stiffness atvery small strains. Then he has shown how settlements may beassessed for the full range of loading from zero to failure, onthe basis of the stiffness±strain relationship obtained in labora-tory tests. This is a problem that has always confronteddesigners of foundations: we are better at estimating ultimatefailure, which is generally a remote and almost irrelevantpossibility, than we are at assessing settlement in service,which affects all structures, sometimes causing signi®cantdamage. The need to assess displacements is rightly empha-sized yet again by modern codes of practice, but the means todo it are often not available.
John's principal aim, I know, was that engineers in practiceshould feel that they understand better the signi®cance of non-linear behaviour in soils, and that a simple method has beenshown to them, which they can use to make useful calculations,in line with recent developments in understanding. I think youwill agree that John has achieved what he set out to doÐandhis contribution will be tried, tested, used (and he would expectit will be challenged and improved) by engineers in practice.
I am sure you will wish to join me in thanking John for amost interesting and thought-provoking lecture, one which cantest out in everyday practice.
