Strength and Stiffness Properties

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    In this chapter, and the next, the soil properties are discussed under twoheadings shear strength and stiffness. This distinction is simply one ofconvenience because the behaviour of the materials forms a complete whole.

    The range of phenomena associated with the stress-strain-strength behaviourof soil is extensive, complex and potentially bewildering. The purpose of thesetwo chapters is to present an overview of these aspects.

    Note regarding notation: In this chapter and the next a considerable amountof data obtained from triaxial testing is presented. There are two sets of

    parameters in common use for stresses and strains in triaxial specimens.Unfortunately these use the same symbols! One set, which originated with thestress path method developed by the people at MIT in Massachusetts, and isused extensively in the Lambe and Whitman textbook, uses the stressescorresponding the point at the top of the Mohr stress circle where they use q

    and p, but we will denote these points as:

    1 3 1 31 1

    &2 2

    Q P

    rather than q and p. The critical state soil mechanics people also use the q and

    p symbols. In their case they want stress variables that are applicable to threedimensional stress systems. So they start with the first and second invariants of

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    Design of Earthquake Resistant Foundations


    the stress tensor. For the stress conditions associated with triaxial testing (2 =3) these become:

    1 3 1 31

    & 23

    q p

    When examining triaxial test data the Q and P stresses are very convenientbecause one does not need the hassle of drawing Mohr circles. One simplyplots the point at the top of the Mohr circle from which a best fit line can beeyeballed through the points to estimate the friction angle and cohesion asshown on the right hand sides of Figures 3.1 and 4.1.


    When we talk about the strength of a soil we mean the shear strength,

    regardless of whether we are talking of triaxial compression, extension, simpleshear, direct shear, plane strain, or unconfined compression. For the time beingwe will limit our discussion to the peak shear resistance of the soil. Performinga series of drained triaxial tests, or shear box tests or any other type of sheartest for that matter, one soon arrives at the observation that the shear strengthincreases with increasing effective confining pressure. This gives rise to theangle of shearing resistance as shown in Figure 3.1. The shearing resistance isderived from at least three different mechanisms within the soil: particlesliding, particle rolling and changes in volume in drained testing and changes inpore water pressure in undrained testing. A schematic way of considering thisis shown in Figure 3.2 (Rowe (1962)). Note that, because of these various

    mechanisms, we refer to the angle chacterising the shear resistance of the silt,sand or gravel as the angle of shearing resistance (not the friction angle because

    there is more than simple friction involved). In Figure 3.2 the angle u is thefrictional resistance of a smooth flat surface of the mineral the soil particles arecomposed of.


    ' =asin(tan( ))Mohr-Coulomb failure envelope





    Effective normal stress Effective normal stress

    Figure 3.1 Mo hr-Coulom b shea r fa i lure e nve lop for c ohe sionless soi l andwa ys of determin ing

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    Chapter 3: Strength and stiffness properties of cohesionless soils


    Figure 3.2Com po nents of shea ring resistanc e (Af ter Row e (1962))

    Figure 3.3Relat ionships be twee n an gle o f shea ring resistanc e a nd d ensity of

    san ds and grav els. (From Lam b e & Whitma n (1969))

    Investigating further this behaviour shows that the angle of shearing resistanceis a function of the void ratio or relative density of the soil. In Figure 3.3 theangle of peak shearing resistance in drained shear is seen to increase with

    decreasing void ratio (increasing relative density). The angle labeled cv in theleft hand part of Figure 3.3 is the value corresponding to constant volume

    shearing, that is the angle measured when there is no volume change while thesoil undergoes drained shear. It is also found that the angle of peak shearing

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    resistance is a function of the effective confining pressure; we will discuss thisfurther below.

    Another effect that needs consideration is the shape of the grains of

    cohesionless material. Rounded grains will produce a smaller angle of shearingresistance for a given density and effective confining pressure than angulargrains, possibly because the angularity generates larger grain to grain friction.Finally, one can ask about the effect of the particle size distribution. This islikely to be related to density effects. Well graded materials will tend to have a

    higher angle of shearingresistance than well sortedmaterials simply because anappropriate grading curve

    will produce greater density.The two effects listed in the

    above sentences willcontribute to the range offriction angles for a givenporosity shown in the righthand part of Figure 3.3.

    3.1.1 Peak shear strength

    and c onstan t vo lume shear


    Above we have talked about

    the peak shear strength ofthe soil. Figure 3.4 hasstress-strain curves for a pairof drained triaxial tests onsand consolidated at 207 kPa

    one test on dense materialand the other on loose. Thestress- strain curve for thedense specimen shows apeak but the curve for theloose specimen comes up to

    a plateau. The densespecimen has a larger peakstrength than the loosespecimen, but both tendtowards the same limiting

    value of shear stress and thesame void ratio at larger axialstrains. This behaviour isillustrated further in Figure3.5a for a suite of drainedtests on loose sand and in

    Figure 3.5b for drained testson dense sand. These testson Sacramento River sand

    Figure 3.4 Dra ined triax ial tests on lo ose a ndde nse sand, bo th spe c imens co nso l ida ted to

    207 kPa. (A fter Ta ylo r (1948))

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    (Lee (1965)) have a very wide range of consolidation pressures. Because of this

    the principal stress ratio, 1/3, is plotted instead of the shear stress. The datain Figure 3.5 illustrate the differing volume change behaviour of loose anddense sand and show that at large axial strains the volume tends to become

    constant and the stress ratio approaches a constant value regardless of theinitial state of the sand. For dense materials we thus have two angles ofshearing resistance: a peak value and another at large strains. Furthermore theangle of shearing resistance at large strains is very similar for both the initiallyloose and initially dense specimens. Figure 3.6 shows that the peak angle ofshearing resistance for these sands is affected by confining pressure; as thisincreases the angle of shearing resistance decreases. The fact that at largestrains the principal stress ratio in Figure 3.5 approaches a similar value,regardless of the initial void ratio, and at the same time the rate of volumechange of the sand approaches zero, suggests that at large strains a constant

    volume condition is approached with an associated angle of shearing resistance

    that is independent of density (the cvof Figure 3.3). We need also to observefrom Figure 3.5 that a dense sand at a sufficiently large confining pressure willexhibit similar shear characteristics to a loose specimen, in other words theeffect of increasing confining pressure is to suppress the dilatant volumechange.

    Thus we have two different ideas about the angle of shearing resistance. Theangle of peak shearing resistance is a function of effective confining pressure as

    well as density. The large strain, or constant volume, angle of shearingresistance is nearly independent of confining pressure1.

    1 With the benefit of the insight from much recent work on the effect of particle crushing onthe properties of sand we can suggest that there would have been some crushing at the higherconfining pressures used for the tests in Figure 3.5.

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    Figure 3.5 Dra ined shea ring o f Sac ram ento River san d (a) loo se spe c ime ns, (b)de nse spe c ime ns. (Af ter Lee (1965).



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    Chapter 3: Strength and stiffness properties of cohesionless soils


    Figure 3.6 Mo hr circ les and fai lure env elop es for drained triax ia l tests onSa c ram en to River sa nd illustra t ing the e ffec ts of vo id ratio on shea r stren g th.

    (A fter Le e (1965))

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    Figure 3.7 Data f rom which Bo l ton d er ived h is re la t ions betwee n pe ak and

    c onstant v olum e ang les of shea ring resistanc e. (Bol ton (1986))

    One way of representing the effect of confining pressure on angle of shearingresistance has been suggested by Bolton (1986). Having a peak angle ofshearing resistance that is a function of density and confining pressure and a

    lower limiting value for the angle of shearing resistance it is possible to relatethe various parameters. Bolton has suggested the following relationships:


    max constan t volume R

    max constan t volume R



    5I for plane strain

    3I for triaxial conditions

    d0.3I for both triaxial & plane strain




    maxR D Dmax min

    e -eI =I 10-ln p -1 and I =

    e -e


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    Chapter 3: Strength and stiffness properties of cohesionless soils


    The third of equations 3.1 gives the ratio of the volumetric to axial strainincrements. These relationships along with the data points used by Bolton areplotted in Figure 3.7 (the above three equations are Boltons equations 15, 16and 17).

    3.1.2 Crit ic a l void ratio

    We have mentioned above that it is possible for a dense material sheared at lowconfining pressure to exhibit an increase in volume, and the same materialstarting at the same relative density, but sheared at a high confining pressure, toexhibit a decease in volume at failure. This raises questions about the termsloose and dense. In Figure 3.8 the volume change data from a series of drainedtriaxial tests on sand is various initial densities and confining pressures areplotted. Looking at this data one realises that there must be some initialconditions for the sand subject to drained triaxial testing in which there is zero

    volume change at failure. Having such data it is possible to draw curve throughpoints which have zero volume change at drained shear failure. This line isknown at the critical void ratio line and was first proposed by Casagrande in1938. In Figure 3.9 the critical void ratio line estimated from the data in Figure3.8 is plotted.

    It is apparent that the critical void ratio line divides dense initial states fromloose states. Consequently the terms dense and loose are qualitative and notabsolute. No matter how dense a material is initially, shearing under a largeenough confining pressure will generate loose behaviour.

    Figure 3.9 emphasises that relative density alone is not sufficient to characterisethe condition of sand. We need something more and a state parametercombining both void ratio (or density) and hydrostatic effective stress. Onepossibility is the distance from the critical void ratio line in Figure 3.9. In other

    words we use as state variables the void ratio and current value of 3. Wedefine the state of the soil as the vertical distance between the current state

    point and the point on the critical void ratio line at the same 3. Points abovethe line have positive states and points below negative states. In qualitativeterms we regard soils with a positive state as loose and those with a negativestate as dense. This state concept is very similar to that defined by Been and

    Jefferies (1985) in a slightly different context. (A slight complication in jumping

    from discussion of the work of Lee in the 1960s and that of Been and Jefferiesin the 1980s is a change in the stress variables used. Whereas Lees results areexpressed in terms of effective confining pressure in the triaxial cell,

    subsequent work has favoured the mean principal stress, p, which for thetriaxial test conditions is defined as (1 + 23)/3.)

    Another feature of the set of axes used in Figure 3.9 is that undrained statepaths can be portrayed along with those of drained tests. In this diagram anundrained state path is a horizontal line. Paths starting from negative values ofthe state parameter (below the critical void ration line) will move to the righttowards the critical void ratio line and vice versa for undrained paths starting

    from positive values of the state parameter.

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    Figure 3.8Volum etric stra in d a ta a t fa i lure a ga inst void rat io a t end o fc on solida tion. (After Lee (1965))

    Figure 3.9Crit ic a l void ratio ve rsus co nfining p ressure. (After Lee (1965))

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    Chapter 3: Strength and stiffness properties of cohesionless soils


    3.1.3 Di lata nc y c ontr ibu t ion to shea r streng th

    We have seen that on drained shearing of a dense granular there is a peak shearresistance and then a gradual decline to a large strain value. At the peak shear

    resistance the soil is increasing in volume and at the large strain strength thereis zero volume change. Thus the question arises is there a link between thedilatant volume increase and the peak shear resistance? The answer is yes. Forthe specimen to increase in volume work is done against the confiningpressure, and this work contributes to the apparent shear resistance of the soil.In the post peak region the rate of volume increase decreases so thiscontribution to the shear strength also decreases and the mobilised shearresistance also decreases. An early analysis of this effect is given by Taylor(1948) with reference to shear box testing on sand. Once the constant volumecondition is reached there is no contribution to the shear strength generated bydilatancy. Some argue that this is a more fundamental shearing angle for the

    sand as there is no dilatancy involved. However, it has been pointed out by Leeand Seed (1967) that the constant volume angle of shearing resistance is stillnot the same as the angle of shearing resistance for the particle mineralogy.

    This is also indicated in Figure 3.3.

    3.1.4 Pha se tra nsform a tion line

    We have seen how for a dense material drained shearing results in anincrease in volume at peak shear strength and how negative pore pressures maybe produced during undrained shearing. However, if one looks at the earlystages of the shearing of these materials the onset of dilatant behaviour does

    not occur until some initial amount of shearing has taken place. Similarly inundrained stress paths on dense materials there is an initial part in which thereis a positive pore pressure response which is followed by a decrease in pore

    water pressure. This is manifested as a change in direction of the effectivestress path.

    Figure 3.10 The Ishiha ra p ha se tra nsforma tion line. (A fter Ishiha ra (1996)) (Inth is an d the fo l low ing p lots q = (1-3)/ 2 a nd p = (1+23)/ 3)

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    Figure 3.11 Und rained shea ring of sand i llustrat ing ph ase transform at ionb eh a viou r. (After Ishiha ra et a l (1975))

    However, this effect is even more marked for undrained shearing of loosematerials. Initially, as expected, there is an increase in pore water pressure andthe effective stress path moves to the left. During this phase there may be apeak in the mobilised shear resistance after which the effective stress pathcontinues to move to the left but trends downward. Then a particular stressratio there is a sudden transformation and there is a sharp change in the

    direction of the undrained stress path. Ishihara et al (1975) coined the termphase transformation for this. The line connecting phase transformation points

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    Chapter 3: Strength and stiffness properties of cohesionless soils


    is the phase transformation line. The concept is illustrated inFigure 3.10, whilstFigure 3.11 has the results of undrained tests on water sedimented sand. Theresults in Figure 3.11 indicate that the peak undrained shear strength observedearly in the tests is only a small fraction of the final strength. Although these

    specimens initially have a large pore pressure response expected of a loosematerial, the pore pressures at the end of the tests approach zero.

    3.1.5 Three d ime nsion a l fai lure c riter ia

    Above all our discussion of the shear strength of cohesionless soil has beenbased on data from triaxial testing. The final question to consider in thissection is how the angle of shearing resistance varies with more complex stresssystems. In Figure 3.12 we found that the stress conditions for the

    conventional triaxial test, 2 = 3 lie along one line in the octahedral plane.However, there are two other lines in that plane for which two principal

    stresses are equal. Thus, in the octahedral plane there are three lines ofsymmetry which divide the plane into six sectors. Thus we need to considerhow the angle of shearing resistance is affected by combinations of stress

    which fall outside these planes.

    First, though let us consider triaxial extension testing rather than triaxialcompression. Working with Mohr circles of stress the Mohr-Coulomb failureenvelope is defined in terms of the ratio of the maximum principal stress to theminor principal stress, thus:







    where: N is the principal stress ratio.

    Alternatively one can think of the octahedral shear stress and octahedralnormal stress. In this situation the failure conditions are defined by the ratio of

    oct/oct and the peak shear resistance corresponds to the peak value of thisratio. We can compare what happens in terms of triaxial extension andcompression. For both triaxial compression and extension:

    For triaxial compression and extension we have:

    1 1oct compression oct extension

    1+2/N 2+1/N

    3 3

    So that for a given angle of shearing resistance:

    1 3 1oct2 2

    = 1-1/N3 3

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    oct oct triaxial compression

    oct oct triaxial extension

    / 2+1/N

    / 1+2/N


    This shows, assuming the Mohr Coulomb failure envelope applies to a

    cohesionless soil, that for a given value ofoct the octahedral shear stress inextension is less than that in compression. Extending this into more generalstress systems reveals that in the principal stress space the Mohr-Coulombfailure envelope has the shape of a hexagonal cone, Figure 3.12.

    However, if we look at real drained shear test data it is apparent that the Mohr-Coulomb failure envelope is not followed. Figure 3.13 compares data for planestrain and triaxial tests from which it is apparent that in plane strain the angleof shearing resistance is greater than that in triaxial compression. Figure 3.14presents data for more general stress conditions which confirms that theMohr-Coulomb criterion underpredicts the angle of shearing resistance for

    non-triaxial test conditions. In that figure two other criteria, involving the thirdinvariant of the stress tensor, which bracket observed data are plotted.

    Figure 3.12Mo hr-Co ulom b fai lure env elop e in pr inc ipa l stress spa c e.

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    Chapter 3: Strength and stiffness properties of cohesionless soils


    Figure 3.13 Com pa rison o f plane stra in a nd tr iax ia l shea r streng th da ta.

    (After C ornforth (1964))

    Figure 3.14Fa ilure c riter ion for sa nd she a red u nd er ge ne ral stress c on d it ion s.

    (After Woo d (2004))


    Returning to the conceptual model of a granular material, Figure 3.1, we canask ourselves how the stiffness of a cohesionless material will be affected byincreasing confining pressure. It is not too difficult to envisage that increasedconfining pressure will hold the grains more firmly together and hence increasethe stiffness. However, it has been found that the stiffness does not increaselinearly with confining pressure but to some power of the confining pressure.

    Another approach is to consider a material composed of spherical balls invarious regular packings. Even for purely elastic behaviour this model has amodulus that depends on some power of the confining stress. We see what is

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    meant here by looking at results of oedometer tests on cohesionless materialsor even hydrostatic compression. Oedomenter data on various sands ispresented in Figure 3.15. At each stress the slope of these curves gives theconstrained modulus value corresponding to that stress. It is clear that these

    sands stiffen with increasing stress, but that the stiffness does not increaselinearly with stress. One representation of the data in Figure 3.15 is to use apower law to give the tangent stiffness:


    vtangent a


    M =App


    where: A and b are constants for the particular material and initial density andpa is atmospheric pressure.

    Figure 3.15 is drawn with a natural stress axis. Frequently in geomechanics

    such data are plotted against a logarithmic stress axis. The data from Figure3.15 are replotted this way in Figure 3.16. This comparison is important asusually oedometer data for clays is plotted with a logarithmic stress axis. Themessage from Figure 3.15 is that the sand stiffens with increasing confiningpressure. The logarithmic plot suggests a preconsolidation pressure, but thisis simply an artefact of the logarithmic stress axis and not a feature of theactual sand compressibility.

    Figure 3.15 One dim ensiona l co m pression tests on seve ral sands (note 1

    lb / i n2 6.9 kPa) . (A fter Rob e rts (1964))

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    Chapter 3: Strength and stiffness properties of cohesionless soils


    Most of the curves in Figure 3.15 are for increasing stress. If at some stageunloading takes place the curves are not retraced so we conclude that thedeformation in Figure 3.15 is not elastic and the tangent modulus given by theabove equation is not an elastic modulus since it does not apply to unloading.

    But for very small load-unload cycles the deformation is mostly recoverablewhich suggests that for small strains it is possible to have elastic behaviour.Frequently this small strain modulus is determined using dynamic testing.

    How do these moduli relate to what is observed in triaxial testing? In Figure3.15 the soil is not loaded to failure so the tangent stiffness values obtainedfrom the above power law equation relate to small strain behaviour, althoughat strains, somewhat larger than the small strain elastic range. Thus themodulus values from the oedometer tests might, perhaps, be related to theinitial linear part of a triaxial stress strain curve. If one views the whole of thetriaxial stress-strain curve it is clear that long before the failure condition is

    reached there is a decrease in the stiffness of the stress-strain curve (tangentstiffness given by the slope of the tangent to the stress-strain curve at somepoint). One suggestion that has been made is to model the stress-strain curveas an hyperbola as shown in Figure 3.17. This idea originated with Kondner(1963) and goes some way to describing the shape of the curve, although it islimited in that it cannot model stress-strain curves that have a post-peak

    Figure 3.16 One dim ensiona l co m pression da ta from Figure 3.15 rep lot ted

    on a loga ri thm ic stress ax is (note 1 lb/ in2 6.9 kPa ). (A fter Rob erts (1964))

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    axial strain -


    axial strain -

    ult =1/b


    1/a =tanb =tan


    E =1/a

    Figure 3.17 Hyp erb olic stress-strain c urve. Mea sured c urve left,t ransforme d c urve to e st im ate pa ram eters r ight .

    reduction in mobilised shear stress. This means that it may be applied to stress-strain data for loose specimens but not for dense specimens. The form of thestress-strain curve is:

    ( )1 31

    Q=2 a+b


    s s


    - = (3.6)

    where: is the axial strain and aand bare parameters for the hyperbolic curve.

    The manner in which the hyperbolic parameters are obtained from a triaxialtest curve is set-out in Figure 3.17. The hyperbolic set of parameters may fit

    well stress-strain curves for a particular confining pressure but is not

    sufficiently general to handle a whole range of stress situations. Nevertheless itwas one of the starting points in the enormous efforts that have beenexpended over the last forty or fifty years to develop mathematical models forsoil stress-strain behaviour.


    The best known aspect to the cyclic load response of saturated cohesionlesssoil is liquefaction. This is a phenomena associated with loose saturated sandhaving a particle size grading curve falling within the region shown in Figure

    3.18. However, this particle size range should not be considered as allembracing, sufficiently loose saturated materials with size distributions outsidethese limits may also to the susceptible to liquefaction. In addition there is oneother factor involved, namely the geological age of the deposit. Liquefactionresistance seems to increase with age, presumably because cementing agents aredeposited within the soil particle skeleton.

    The process of liquefaction can be viewed by plotting the rise in pore waterpressure against cycles in a cyclic undrained test. It is seen that there is agradual rise in pore pressure with each cycle and that this pore water pressureis not recovered during the unloading part of the cycle. Eventually the rise in


    axial strain - axial strain -

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    Chapter 3: Strength and stiffness properties of cohesionless soils


    Figure 3.18 Limits in grad at ion sep arat ing l iqu ef iable a nd n onl iquef iab lesoils.

    Figure 3.19Rec ords of cy c l ic und rained tests on loo se a nd d ense saturated

    sa nd . (After Ishiha ra (1985))

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    pore pressure is such that the effective confining pressure in the specimen iszero. At about this stage there is an abrupt increase in the cyclic deformation ofthe specimen. The number of cycles to liquefaction depends on the cyclicstress ratio in relation to the initial effective confining pressure and the density

    of the soil. The upper part of Figure 3.19 gives the pore pressure increase withnumber of cycles for a loose specimen of saturated sand and the lower part thesame information for a specimen of dense sand. Note the differences inbehaviour. First, the dense sand required a rather larger cyclic shear stress toproduce the changes shown. Second, the strain build up is gradual for thedense sand but occurs abruptly after about eight cycles for the loose sand.

    Third, cyclic variation in pore pressure is small for the loose sand thepressure gradually builds up till it becomes equal to the confining pressure

    when the deformation starts.

    On the other hand for the dense specimen there is a much larger variation in

    pore pressure during each cycle with only very short times when it is close tothe confining pressure.

    Figures 3.20 and 3.21 show the stress-strain loops and the effective stress pathsfor the same tests as in Figure 3.19. Although for both tests the effect of thephase transformation line is very clear, there is a big difference between theresponse of the loose and the dense specimen. In the case of the loosespecimen there is a large degradation in stiffness and the stress path loops gothrough a zero effective stress stage. For the dense case the range of strains ismuch less, despite the fact that the cyclic shear stress is larger; in other wordsthe cyclic degradation of the dense sand is not as severe as that for the loose

    sand. Part of the reason for this is that in for the dense sand the stress pathloops do not approach the origin until the last few cycles, so that the meaneffective stress in the dense sand remains higher than in the loose sand.

    After liquefaction has occurred there is some remaining undrained shearstrength in the saturated sand. This is important as it controls what happens tothe sand deposit after the earthquake. If the liquefied deposit does not have ahorizontal ground surface there is the possibility of the material flowing. Assuch flow failures have caused much damage in past earthquakes there isinterest in estimating the post-liquefaction undrained shear strength. The termthat is applied to this is residual strength.2 Figure 3.22 gives one representation

    of this information and shows how there are three possibilities, based oneffective vertical stress and Standard Penetration Test blow count noliquefaction, liquefaction but no flow deformation, and liquefaction withsubsequent flow deformation of the sand (if the ground surface is sloping).

    2 A most unfortunate term. Residual strength has long been associated with a particular aspectof the drained shearing of clays, Skempton (1964).

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    Figure 3.20Stress p a th and stress-strain c urve for c yc lic und rained she a ring

    of sa tura ted loo se sand . (After Ishiha ra (1985))


    The dynamic stiffness is usually evaluated by considering steady state cyclicloading between fixed stain amplitudes. This gives the apparent shear modulusand the equivalent viscous damping ratio. These are defined in Figure 3.23.

    The apparent shear modulus in Figure 3.23 is a secant modulus because it istaken between the end points of the loops. The damping ratio is so calledbecause the damping involved in the cyclic deformation of soil is hysteretic,that is independent of frequency, rather than being viscous in the manner of

    the usual model of damping.

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    Figure 3.21 Stress p a th and stress-strain c urve for cy c lic und rained she a ring

    of sa turated d en se sand . (A fter Ishiha ra (1985))

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    Chapter 3: Strength and stiffness properties of cohesionless soils


    Figure 3.22 Boundary curves for gravi ty- induced f low fai lure and

    lique fac tion. (After Ishiha ra (1996))

    It is found that for very small stain amplitudes the soil deforms in an elasticmanner. That is the strains are recoverable and the amount of damping is very

    small. These properties are usually measured with special testing devices suchas the resonant column or free torsional vibration. The shear modulus sodetermined is called Go. It provides an upper limit on the stiffness of the soiland is a very important property when considering the earthquake response ofa site. For round grained sands with void ratios less than 0.8, Hardin andRichart (1963) give the following expression for Go:

    ( )2

    0.5o 0

    6.9 2.17-eG = (MPa)

    1+e (3.7)

    where: e is void ratio and o is the effective confining pressure of the sand


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    Figure 3.23 Ap pa rent shea r mod u lus and eq u iva len t v isc ous dam ping ra t io.

    For cyclic testing it is observed that this elastic shear modulus is applicable forshear strain amplitudes up to about 10-4 to 10-3 %. For larger strains somedegradation takes places and the apparent shear modulus is reduced. The usual

    way of portraying this is with a modulus reduction curve such as that presentedin Figure 3.24. It is seen from this figure that the confining pressure has aneffect on the shape of the curve. At very small strain amplitudes the amountof damping is very small, it increases as the strain amplitude increases as shown

    in Figure 3.25.

    With regard to Figures 3.24 and 3.25 two important questions arise: what effectdoes the number of cycles have, and what is the effect of the density of thesand on the shape of the curves? Ishihara (1996) explains that test data indicatethat there is very little difference between the modulus values for the secondand tenth cycles after which the effect of the number of cycles disappears. Thedensity has an effect on the small strain shear modulus, Go, by virtue ofequation 3.7. However, the shapes of the normalised plots in Figs. 3.24 and3.25 are then independent of initial density. It is also known that for cyclicshearing of sand the frequency has little effect in the range 0.1 Hz to 10 Hz.

    A different approach to the modulus degradation curves is given by Ladd et al(1989). They consider the modulus degradation as a function of numbers ofcycles at varying shear strain amplitudes in cyclic triaxial testing. Figure 3.26shows how the shear modulus, normalised with respect to the shear modulusfor the first cycle, decreases with number of cycles for various strainamplitudes on sand at a relative density of 60%. This shows that for very smallstrain amplitudes, up to about 0.008%, there is negligible degradation. This isnot surprising as this strain amplitude is only just beyond the range associated

    with elastic behaviour. On the other hand when the strain amplitude is 0.3%the degradation with numbers of cycles is quite rapid. Figure 3.27 gives similar

    data for a range of relative densities. In this case the degradation is less severefor the larger relative densities. All the cyclic tests by Ladd et al were

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    undrained. They then investigated what happens to the build-up in pore waterpressure at various cyclic strain amplitudes. They found that there is athreshold shear strain for the initiation of pore pressure increase. Their data arereproduced in Figures 3.28 and 3.30. These show that for a range of relative

    densities and effective normal pressures that there is no increase in pore waterpressure until the shear strain amplitude reaches about 0.01%. What isparticularly interesting about this is the observation that this strain amplitude

    Figure 3.24 Effec ts of c onf in ing stress on stra in-d ep end ent shea r m od ulusde grad at ion c urves af ter 10 cy c les (note stra in a m pl i tude is not in %). (Af ter

    Koku sho (1980))

    Figure 3.25 Effec ts of c onf in ing stress on stra in-d ep end ent d am ping rat ioc urves after 10 cy c les (no te stra in am p litud e is no t in %). (After Kok usho


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    is well beyond the 10-4 to 10-3 % associated with truly elastic behaviour of thesand. Finally, in Figure 3.30 the data on the build-up in pore pressure forparticular numbers of cycles is plotted against strain amplitude.

    One final step is to consider strain controlled cyclic shearing of dry sand. Inthis case there is no question of pore pressure response but there are volumechanges. Youd (1972) has done cyclic simple shear testing on medium dense todense sand and found that cyclic shearing densifies the sand and that thiscontinues for large numbers of cycles. Eventually the relative density is wellbeyond 100%. Figure 3.31 shows the results for one of his tests while Figure3.32 has results for a number of tests with varying cyclic strain amplitudes. Atsmall strain amplitudes densification was still occurring after 100,000 cycles.

    This data is relevant to the settlement of sand profiles during earthquakeshaking and also to the compaction of sandy soils.

    Figure 3.26Degrada t ion o f shea r mod u lus as a funct ion o f num be r of cy c les

    for Mo nteray sand No. 0 at relat ive de nsi ty = 60%, 3= 96 kPa a nd v ar iousc yc lic she a r stra ins. (After La d d e t al (1989))

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    Figure 3.27Degrada t ion o f shea r mod u lus as a funct ion o f num be r of c yc les

    for Monteray san d No. 0 a t va rious relat ive de nsi ties, 3= 96 kPa a nd c yc l icshe a r stra in am p litude = 0.1%. (After La d d et a l (1989))

    Figure 3.28 Residu al p ore p ressure bui ld-up af ter 10 cy c les as a func t ion o fc yc l ic shea r stra in for Mo nteray No . 0 sa nd a t3 = 96 kPa an d relat ived en sit ies of 45, 60 and 80%. (After La d d et a l (1989))

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    Figure 3.29 Residua l po re p ressure bui ld-up af ter 10 c yc les as a func t ion o f

    c yc l ic shea r stra in for Mo nteray No . 0 san d a t3= 25, 96, & 192kPa a ndrelat ive d en sit ies of 45, 60 and 80%. (After La d d et a l (1989))

    Figure 3.30 Pore p ressure b ui ld- up as a funct ion of c yc l ic shea r stra in for

    Mo nteray No . 0 sand at relat ive d ensity = 60% and3= 96 kPa an d var iousnum be rs of c yc les. (After Lad d et a l (989))

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    Figure 3.31Com pa c t ion versus shea r-st ra in history for cy c l ic simp le shea r ingof san d Ottaw a sand at a ver t ic a l e f fec t ive stress of 48 kPa. (Af ter Youd



    3.5.1 Pa rtic le c rushing

    Particle crushing can exert a significant effect on the behaviour of granularsoils. This has become a recent hot topic in soil mechanics research. Oneconsequence of particle crushing is that drained triaxial tests do not reach aconstant volume condition even at very large shear strains. Materials that areknown to exhibit this are carbonate sands found offshore in many regions andpumiceous sands. Hydrostatic or Ko compression tests on materials subject tocrushing will show a kink when plotted against the logarithm of the effectivestress. Figure 3.33 has some results from Biarez and Hircher (1994) on thecrushing of petroleum coke. Figure 3.34 has some for data for sand. Thisshows that the slope of the compression curve, with a logarithmic stress scale,is steeper once particle crushing commences.

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    Figure 3.32 Effec t of shea r stra in am pl i tude on c yc l ic de nsi fica t ion o f Ottaw a

    sa nd . (A fter You d (1972))

    Figure 3.33 Oed om eter c om pression of pe t ro leum c oke at h igh ef fec t ivestresses. (Afte r Biare z & Hic he r (1996))

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    Chapter 3: Strength and stiffness properties of cohesionless soils


    Figure 3.34 Com pression o f san d with p ar t ic le c rushing. (Af ter Biarez &

    Hic he r (1994))

    3.5.2 Low c om pressib i li ty soi l and the B pa ram eter

    It is well known that for a saturated soil the B value is equal to 1.0. This comesfrom the contrast in the compressibility of water and of the soil particleskeleton. However, for very stiff soil particle skeletons this contrast is not somarked and even for a fully saturated soil B may be less than 1.0, Lee et al(1969). In this case how does one establish that the soil is saturated? Oneapproach is to consolidate the soil with a certain backpressure and thendetermine the B value. This process is repeated several times with increasingback pressures. If we arrive at a situation in which the B value reaches a steady

    value regardless of the back pressure one concludes that the material is


    Figure 3.35 compares the volume change during hydrostatic compression ofOttawa and Sacramento River sand at 100% relative density. Ottawa iscomposed of particles known to be highly resistant to crushing hence themuch smaller compressibility for this sand. Figure 3.36 gives thecompressibility of Sacramento River sand at various densities so thatcomparison can be made with Ottawa sand at 100% relative density; it alsoshows the compressibility of water and of the quartz. Finally, Figure 3.37shows how the B parameter is very close to 1.0 for the Sacramento River sandbut only 0.86 for Ottawa sand at 100% relative density. The small value for

    Ottawa sand is a consequence of the relative compressibility of this materialand water.

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    Figure 3.35 Com pressib i li ty of Ottawa sand an d Sac ram ento River sand at

    100% relat ive de nsi ty. (1kg/ c m 2 98 kPa ) (A fter Lee et a l (1969)

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    Figure 3.36 Com pressib i li ty o f Otta wa san d at 100% relat ive d ensity andSac ram ento River san d at v a rious relat ive de nsi ties and the c om pressib i li ty

    of wa ter and sand pa rt ic les. (Af ter Lee et a l (1969)

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    Figure 3.37 B Va lues for Ottaw a sand an d Sac ram ento River san d.(A fter Le e

    et a l (1969)

    3.5.3 Ca vi tat ion l imi t on ne ga t ive po re w ater p ressures

    A dense sand, on drained shearing, will increase in volume. On undrainedshearing of a dense sand negative pore water pressures will be induced. If the

    specimen has been saturated with back pressure the negative pore pressure willbe manifested with the pore pressure dropping well below the back pressurethat was used to saturate the specimen (which, for an undrained test, becomesthe pore water pressure at the start of shearing). In some cases negative pore

    water pressures of several hundred kPa can be induced. Now in the field thepore pressure before undrained shearing is controlled by the position of the

    water table or the seepage conditions taking place. These water pressures willusually be much less than the back pressures one uses in the laboratory toensure saturation. If a specimen about to be subject to undrained shear has aback pressure equivalent to what might exist in the field a definite limit isimposed on the negative pore pressure that can develop by cavitation in the

    pore water. Cavitation occurs when the water pressure reaches absolute zero (a

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    Chapter 3: Strength and stiffness properties of cohesionless soils


    Figure 3.38 In fluence of c av i ta t ion in p ore wa ter on the und ra ined shea rstren g th of sa nd . (After See d a nd Lee (1967)).

    gauge pressure of -100 kPa.). Figure 3.38 has some data from Seed and Lee(1967) which demonstrates this effect. Once the cavitation occurs the test is ineffect converted to a drained test as the volume increases. This limits theundrained shear strength as shown in Figure 3.38.


    Been, K. and Jefferies, M.G. (1985). A State Parameter for Sands.Geotechnique 35, No2, 99-112.

    Biarez J & Hicher P-Y (1994). Elementary Mechanics of Soil Behaviour. Balkema,Rotterdam.Bolton, M D (1986) The strength and dilatancy of sands, Geotechnique, Vol.

    36, No. 1, pp. 65-78.Casagrande, A (1938) Compaction tests and critical density investigations of

    cohesionless materials for Franklin Falls Dam, U. S. Engineer Corps,Boston District.

    Cornforth, D H (1964) Some experiments on the influence of strainconditions on the strength of sand, Geotechnique, Vol. 16 p. 95

    Hardin, B O & Richart, F E (1963) Elastic wave velocities in granular soils.Journal of the Soil Mechanics & Foundations Division, ASCE. Vol. 89, SM

    1, 33-65.

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    Design of Earthquake Resistant Foundations

    Ishihara, K., Tatsuoka, F. and Yasuda, S. (1975) Undrained deformation andliquefaction of sand under cyclic stresses. Soils and Foundations, Vol. 15,pp. 29-44.

    Ishihara, K (1996) Soil Behaviour in Earthquake Geotechnics. Oxford.

    Kokusho, T (1980) Cyclic triaxial test for dynamic soil properties for a widestrain range. Soils and Foundations, Vol. 22, pp. 1-18.

    Kondner, R L (1963) Hyperbolic stress-strain response: cohesive soils. ProcASCE, Journal of the Soil Mechanics and Foundations Division, Vol. 89SM1, pp. 115-143.

    Ladd, R S, Dobty, R, Dutko, P, Yokel, F Y & Ching, R M (1989) Porepressure build up in clean sands because of cyclic straining. Geotechnical

    Testing Journal, Vol. 12 No. 1, pp. 77-86.Lambe, T W & Whitman, R V (1969). Soil Mechanics. Wiley, New York.Lee, K L (1965) Triaxial compressive strength of saturated sands under

    seismic loading conditions, PhD thesis, University of California, Berkeley.

    Lee, K L and Seed, H B (1967) Drained strength characterisics of sands,Journal of Soil Mechanics and Foundations Division, Proc. ASCE, Vol. 93,No SM6, pp. 117-141.

    Lee, K L, Morrison, R A & Haley, S C (1969) A note on the pore pressureparameter B. Proc. 7th International Conference on Soil Mechanics andFoundation Engineering, Mexico City. Vol. 1, pp. 231-238.

    Roberts, J E (1964) Sand compression as a factor in oil field subsidence, ScDthesis, Massachusetts Institute of Technology.

    Rowe, P W (1962) The stress dilatancy relations for static equilibrium of anassembly of particles in contact, Proceedings of the Royal Society,London, Series A, Vol. 269, 1962, pp. 500-527.

    Seed, H B & Lee K L (1967) Undrained strength characterisistics ofgohesionless soils. Proc. ASCE, Vol. 93, No SM6, pp. 333-360.Skempton, A W (1964) Long term stability of clay slopes. Geotechnique,

    Vol. 14 No. 2, pp. 77 101.Taylor, D W (1948) Fundamentals of Soil Mechanics. Wiley, New York.Wood, D M (2004) Geotechnical Modelling. Spon, Oxfordshire.Youd, T L (1972) Compaction of sands by repeated shear straining. Journal

    of Soil Mechanics and Foundations Division, Proc. ASCE, Vol. 98, NoSM7, pp. 709-725.

    File: Chapter 3 Strength and stiffness properties of cohesionless soils 22/07/2012