Yang, J. & Sze, H. Y. (2011). Geotechnique 61, No. 1, 59–73 [doi: 10.1680/geot.9.P.019]

59

Cyclic behaviour and resistance of saturated sand undernon-symmetrical loading conditions

J. YANG� and H. Y. SZE�

This paper reports findings from an experimental studythat aimed to investigate the undrained behaviour of sandin non-symmetrical cyclic loading, and to clarify the roleof initial static shear in liquefaction resistance. The test-ing programme, conducted on a standard sand undertriaxial conditions, covers a broad range of initial statesin terms of relative density, confining stress and initialshear stress ratio (Æ). Three distinct failure modes havebeen identified from the tests: flow-type failure, cyclicmobility and plastic strain accumulation. Of these, flow-type failure, characterised by abrupt runaway deforma-tions without any prior warning, is the most critical, andpertains to sand in the loose state. The tests also demon-strate that the presence of initial static shear stress isbeneficial to the liquefaction resistance of loose sand atlow Æ levels, but it becomes detrimental at high Æ levels.In this connection the concept of threshold Æ is proposed,together with the use of a no-stress-reversal line forbetter characterisation of the effect of initial static shear.Furthermore, in the conceptual framework of criticalstate soil mechanics, a fairly good linear relationship hasbeen established between the threshold Æ and the stateparameter ł that collectively accounts for the initialrelative density and mean stress level. This relationshipsuggests that the threshold Æ decreases with increasingvalues of ł, or with sand becoming looser than thecritical state. It is further proposed that the concept ofthreshold Æ also applies to sand at high relative density,as long as the confining stress becomes sufficiently high.This proposal leads to a unified and consistent interpreta-tion of the complicated static shear effect.

KEYWORDS: earthquakes; failure; laboratory tests; liquefac-tion; sands; shear strength

La presente communication fournit les conclusions d’uneetude experimentale, dont le but etait d’examiner lecomportement non draine du sable en presence decharges cycliques asymetriques, et de clarifier le role ducisaillement statique initial dans la resistance a la lique-faction. Le programme d’essais, effectue sur un sablestandard soumis a des conditions triaxiales, couvre unevaste gamme d’etats initiaux sur le plan de la densiterelative, de la contrainte de confinement et du taux decontrainte de cisaillement initial (Æ). Ces essais ont per-mis d’identifier trois modes de rupture distincts, a savoirla rupture par solifluxion, la mobilite cyclique, et l’accu-mulation d’allongement plastique. De ces differentsmodes, la rupture par solifluxion, caracterisee par desdeformations brutales sans le moindre signal de precau-tion, est la plus critique ; elle concerne le sable a l’etatmeuble. Les essais demontrent egalement que la presencede la contrainte de cisaillement statique initiale est bene-fique pour le resistance a la liquefaction du sable meublea des niveaux Æ bas, mais devient prejudiciable enpresence de niveaux Æ eleves. A cet effet, on propose unconcept, celui du Æ limite, ainsi que l’utilisation d’uneligne sans renversement de tension, afin de mieux carac-teriser l’effet du cisaillement statique initial. En outre,dans le cadre conceptuel de la mecanique des sols a l’etatcritique, on a etabli un rapport lineaire relativement bonentre le Æ limite et le parametre d’etat ł, qui exprimentcollectivement la densite relative initiale et le niveau decontrainte moyenne. Ce rapport semble indiquer que le Ælimite diminue au fur et a mesure qu’augmente ł, oulorsque le sable devient plus meuble que l’etat critique.Sur le meme plan, on propose egalement que le conceptdu Æ limite s’applique egalement au sable a densiterelative elevee, tant que la contrainte de confinementreste suffisamment elevee. Cette proposition permet d’ob-tenir une interpretation homogene de l’effet compliquedu cisaillement statique.

INTRODUCTIONAs a result of the Niigata and Alaska earthquakes of 1964,soil liquefaction has emerged as an important subject area,creating a great deal of interest among researchers andpractising engineers. A fundamental understanding of lique-faction behaviour and the mechanisms involved has beendeveloped over the past four decades, mainly through well-controlled laboratory tests (e.g. Seed & Lee, 1966; Castro,1975; Toki et al., 1986; Ishihara, 1993; Vaid et al., 2001).The liquefaction resistance of saturated sand is commonlyevaluated by undrained cyclic triaxial tests with two-way,symmetrical loading in compression and extension. In the

triaxial test a sand sample is consolidated under hydrostaticconditions and then subjected to a cyclic deviatoric stress,which alters between positive and negative values of thesame magnitude (Fig. 1(a)).

The symmetrical loading represents level ground condi-tions in the free field, where no initial static shear stressesact on the horizontal planes of the elements of soil. In manymajor projects involving earth dams, embankments or slopes,however, soil elements are subjected to static, driving shearstresses on the horizontal planes before the earthquakeloading effect is developed. The sloping ground conditionsrequire tests with initial stress conditions different fromthose shown in Fig. 1(a). Following the early work of Lee &Seed (1967), the existence of initial shear stresses has beensimulated by several researchers using triaxial tests onanisotropically consolidated samples (e.g. Vaid & Chern,1983; Hyodo et al., 1994). In so doing, anisotropic consoli-dation is conducted under the major and minor principalstresses, � 91c and � 93c, to give rise to a static shear stress �s

Manuscript received 20 February 2009; revised manuscript accepted21 December 2009. Published online ahead of print 20 July 2010.Discussion on this paper closes on 1 June 2011, for further detailssee p. ii.� Department of Civil Engineering, The University of Hong Kong,Hong Kong.

on the plane of interest in the soil sample (i.e. the 458plane); superposition of a cyclic shear stress �cyc thenproduces a cyclic loading that is non-symmetrical about thehydrostatic stress state (Figs 1(b) and 1(c)). It is evident thatshear stress reversals will occur if the magnitude of thecyclic shear stress is greater than the initial static shearstress (Fig. 1(b)); otherwise, no stress reversals will appear(Fig. 1(c)).

In such triaxial tests the level of the initial shear stressescan be measured by a parameter Æ, defined as the ratiobetween the static shear stress �s and the effective normalstress on the 458 plane, � 9nc

Æ ¼ �s

� 9nc

¼ qs

2� 9nc

¼ � 91c � � 93c

� 91c þ � 93c

(1)

where qs is the initial static deviatoric stress. Equation (1)indicates that the level of initial shear stress increases withincreasing Æ values, and the limiting case of Æ ¼ 0 repre-sents level ground conditions without initial static shearstresses.

Compared with the extensive work on level ground condi-tions, there have been only a few experimental studies of thecyclic behaviour of sands in sloping ground conditions.These studies, mainly on triaxial tests and occasionally onsimple shear tests, have produced valuable data showing thatthe presence of initial static shear stresses can have a majoreffect on the cyclic response of sands. Nevertheless, contra-dictory opinions appear to exist with respect to this effect.One is that the presence of static shear stresses tends toreduce the rate of pore water pressure generation, and thusincreases the resistance to liquefaction (e.g. Seed, 1983);

another is, however, that the effect of initial static shear isto reduce the liquefaction resistance (e.g. Yoshimi &Oh-oka, 1975).

A recent proposal by Harder & Boulanger (1997), derivedfrom published data and the earlier work of Seed & Harder(1990), appears to be more reasonable. It states that theeffect is positive for dense sand at a relative density of 55–70%, and negative for loose sand at a density of about 35%,with medium dense sand (45–50%) being in between. Notethat this proposal is limited to the conditions that the effec-tive confining pressure is less than about 300 kPa and thestatic shear stress ratio Æ is below 0.3. At the NationalCenter for Earthquake Engineering Research workshop, how-ever, it was recommended that the current proposals ‘shouldnot be used by non-specialists’ or ‘in routine engineeringpractice’ (Youd et al., 2001). This may be due to the scarcityof experimental data and the lack of convergence andconsistency in the existing data. Clearly, there is a need forcontinued research to improve understanding of the compli-cated static shear stress effects.

Before conducting new research, several observations onthe existing data are worth noting. First, most of the experi-mental work was focused on medium dense to dense sandswith a relative density above 35%, and typically 40–60%.Since sands in a looser state are more vulnerable to liquefac-tion, it is necessary and important to investigate how loosesands behave under different initial stress conditions. Second,the cyclic behaviour of sands has been studied at a relativelylow level of initial static shear, with Æ values up to 0.2 inmost cases, and occasionally to 0.33. It is necessary to haveÆ values increased to a higher level, since sands may behave

Level ground

Sloping ground

τcyc

Time

Cyc

lic s

hear

str

ess

(a)

τcyc

Time

Cyc

lic s

hear

str

ess

(b)

τs

τcyc

Time

Cyc

lic s

hear

str

ess

(c)

τs

Fig. 1. Cyclic loading conditions in the laboratory to simulate level ground and sloping ground conditions:(a) symmetrical loading; (b) non-symmetrical loading with stress reversal; (c) non-symmetrical loading without stressreversal

60 YANG AND SZE

differently when the initial shear stress becomes comparableto the cyclic stress applied. Third, it is believed that thebehaviour of sands is collectively affected by such majorfactors as relative density, confining pressure and static shearstress. Much of the early work involved investigation of onlyone or two of them, varied within a limited range, whichmay hinder development of an overall understanding. It isimportant to explore the interdependence of these factors fora wider range of states, so that a comprehensive view can bedeveloped, and the existing inconsistency can be removed.

Keeping the above observations in mind, the goal of thepresent study is, through a systematic experimental pro-gramme, to contribute to the fundamental understanding ofsand behaviour under non-symmetrical cyclic loading condi-tions, and of the initial shear effects in liquefaction resis-tance. The testing programme has several features, asfollows.

(a) It includes a series of cyclic tests on sand specimenswith post-consolidation relative density ranging from aslow as 10% to as high as 70%. Specimens with arelative density of 10% are probably the loosest evertested under non-symmetrical loading conditions. Aswill be shown later, they exhibit a significantly differentresponse from that of medium dense and dense sandsamples.

(b) It also includes a series of tests under a range ofeffective confining pressures varying from 100 to500 kPa, and a range of initial shear stresses with Ævalues up to 0.6. As the three major factors are variedin a systematic manner, and over a broad range, thedata generated are able to contribute to a fullerunderstanding of their interrelated effects and, impor-tantly, as will be shown later, to the development of aconsistent, critical state view.

Additionally, the systematic database resulting from thepresent study can be a useful reference for the calibration ofadvanced constitutive models for sands under non-symmetri-cal cyclic loading conditions.

EXPERIMENTAL PROGRAMMETest material and equipment

The test material in the study was the Japanese standardsand: Toyoura sand. It is uniform fine sand composed mainlyof quartz, with angular to sub-angular grains (Fig. 2). Thesand has a mean particle size (D50) of 0.216 mm and auniformity coefficient of 1.392, with 0% fines content. Table1 summarises the physical properties of the sand, and Fig. 2shows its particle size distribution.

All cyclic tests reported here were performed using anautomated triaxial testing system. The system is capable ofperforming a variety of functions, including isotropic andanisotropic consolidation, and various modes of shear load-ing under either stress- or strain-controlled conditions, inwhich all pressures and deformations are regulated andmonitored by computer under a closed-loop feedbackscheme.

With respect to reproducing field conditions with labora-

tory element tests, one may consider that the cyclic triaxialtest is not as good as the cyclic simple shear test, becauseof its instantaneous 908 rotations of the principal stressdirections in the loading process. Nevertheless, the triaxialtest does have advantages that make it more widely used,including its convenience and reliability in laboratory set-up,and the high repeatability of the test data it generates (Tokiet al., 1986; Ishihara, 1993). In particular, triaxial tests haveplayed an important role in the development of the conceptsof critical state soil mechanics (Schofield & Wroth, 1968;Wood, 1990), because the principal stresses in a triaxialsetting are clearly defined and measured. In simple sheartests, by contrast, there are problems with the complemen-tary shear stresses on the vertical boundaries of the sampleand the unclear lateral stress state, which can cause consid-erable difficulty in model development.

Sample preparation and saturationAll sand specimens were prepared using the moist tamp-

ing method, since this method is able to produce the widestrange in void ratio with satisfactory uniformity (Ishihara,1996). An oven-dried batch of sand was thoroughly mixedwith water and left to stand for about 1 h for the mixture toreach equilibrium. For each required relative density thepredetermined quantity of sand was placed in a mould andthen gently compacted into five layers using a flat-bottomedtamper. The concept of undercompaction (Ladd, 1974) wasemployed to ensure uniformity of specimens. Each specimenso prepared was 71.1 mm in diameter and 142.2 mm high.

Complete saturation of specimens is important in thestudy of soil liquefaction, because the resistance to liquefac-tion is sensitive to the degree of saturation (Yang, 2002a;Yang et al., 2004). In this study two stages of initialsaturation – carbon dioxide flushing and de-aired waterflushing – were carried out for each specimen, followed byback-pressure saturation. In most tests a back-pressure of190 kPa was sufficient to bring the specimen to saturation,with a B value greater than 0.99. Sample dimensions werecarefully re-measured after initial saturation to provide in-

100

80

60

40

20

0

Per

cent

age

pass

ing:

%

0·01 0·1 1 10

Particle size: mm

Fig. 2. Test material: microscopic view and particle sizedistribution

Table 1. Physical properties of the test material

Mean particle size,D50: mm

Coefficient ofuniformity

Coefficient ofcurvature

Specific gravity Maximum voidratio, emax

Minimum voidratio, emin

Fines content: %

0.216 1.392 0.961 2.64 0.977 0.605 0

CYCLIC BEHAVIOUR AND RESISTANCE OF SATURATED SAND 61

formation on specimen density prior to consolidation, andthis information was used to control the density afterconsolidation. This re-measurement is necessary, especiallyfor loose samples, as a certain amount of volume changemay occur during the process of saturation.

Isotropic and anisotropic consolidationThe consolidation process is important in controlling the

initial confining pressure and initial shear stress level in thesand specimen. It is the normal effective stress (� 9nc) andthe static shear stress (�s) on the 458 plane in the samplethat simulate the field conditions. Therefore, when no staticshear stresses are present, � 9nc is exactly equal to the minorprincipal consolidation stress � 93c. In cases where initialstatic shear stresses are present, � 9nc needs to be controlledtogether with �s according to equation (1). In so doing, theprincipal consolidation stresses � 91c and � 93c were applied insegments so that a constant initial static shear stress ratiowas maintained in each segment until the required stressconditions were reached. This method of stress control israther different from that adopted by Vaid & Chern (1983)and Hyodo et al. (1994), and is considered to be morerational in simulating the static shear effect in cyclic triaxialtests. Because of space limitations, a detailed discussion ofthis issue will be presented elsewhere.

Shearing with or without stress reversalIn each test, cyclic shearing was applied to the specimen

in a stress-controlled, sinusoidal waveform under undrainedconditions. During the shearing process, real-time data wererecorded for the deviatoric stress, effective stress, axial strainand volumetric strain. Depending on the relative magnitudesof the static shear stress �s and the applied cyclic shearstress �cyc, cyclic loading can occur with or without stressreversals. Note that the cyclic shear stress is related to thecyclic deviatoric stress by �cyc ¼ qcyc/2.

Test seriesThe programme was designed to investigate the combined

effects of initial relative density, confining pressure andstatic shear stress in a systematic manner. In this context,cyclic tests were performed at four initial relative densitiesranging from a very loose to a dense state, that isDrc ¼ 10%, 20%, 50% and 70%. The initial relative densityhere refers to the density after the consolidation stage. Thispost-consolidation relative density is a more rational indica-tor for the initial state than the relative density beforeconsolidation. In terms of the relative density prior toconsolidation, the value achieved in this study was as low as5%. For each initial relative density, the initial confiningpressure � 9nc was varied at three levels – 100, 300 and500 kPa – which went well beyond the pressure limit in thecurrent proposal of Harder & Boulanger (1997). Further-more, at each level of initial confining pressure, tests wereconducted by changing the static shear stress and the cyclicshear stress to yield a variety of values of Æ and degrees ofstress reversal. Table 2 summarises the main test seriescarried out in the study.

OBSERVED BEHAVIOUR AND FAILURE MECHANISMSThe large number of tests covering a wide range of initial

states provides a good opportunity to examine the typicalresponses and failure mechanisms of sand under both sym-metrical and non-symmetrical loading conditions.

Cyclic behaviour of medium dense to dense sandFigure 3 shows the response of a medium dense specimen

at Drc ¼ 50% and � 9nc ¼ 300 kPa under symmetrical loadingconditions (i.e. without initial static shear). This is a typicalresponse, known as cyclic mobility (Castro, 1975). As load-ing proceeded, excess pore water pressure was graduallybuilt up, and the effective stress was brought toward zero.When the effective stress reached zero momentarily at zerodeviatoric stress, initial liquefaction (Seed & Lee, 1966;Ishihara, 1996) was said to have occurred. Once the devia-toric stress became non-zero, the sand specimen substantiallyrecovered its effective stress and shear stiffness as a result ofdilation. This process was repeated in subsequent loadingcycles until unacceptably large deformations were reached.

Of particular interest here is how differently the sand willrespond if an initial shear stress is applied under otherwiseidentical conditions. Fig. 4 shows the results of a cyclic testsimilar to the above, except that an initial static shear stressof 120 kPa was added. In this test, the cyclic stress applied(155 kPa) was greater in magnitude than the initial staticstress, leading to the occurrence of stress reversal. Foranisotropically consolidated samples the mean effectivestress is determined as

p9 ¼ � 9nc �qs

6¼ 3 � Æ

3 � 3� 93c (2)

Evidently, in the absence of initial static shear stress, the meaneffective stress is exactly the confining pressure � 9nc or � 93c.

By comparing Fig. 4 with Fig. 3 it can be seen that theaxial strain development tended to be more dominant on thecompression side, and the excess pore water pressure wasunable to build up to the level of effective confiningpressure. A transient strain-softening occurred in the fifthcycle of loading, when the effective stress became close tozero; but once the cyclic stress became non-zero, the sandspecimen regained its stiffness substantially (Fig. 4(c)). It isalso worth noting that, under non-symmetrical loading con-ditions, dilation occurred at the very beginning (in the firstcycle of compressive loading), and the regain of effectivestress due to dilation was much greater in compression thanin extension.

To examine further the impact of stress reversal, a sandspecimen with the same values of Drc and � 9nc as that givenin Fig. 4 was prepared, but it was sheared at a lower cyclicstress level (qcyc ¼ 230 kPa), such that no stress reversaloccurred. The results are shown in Fig. 5. It is clear that theabsence of stress reversal brought about a response signifi-cantly different from that with stress reversal. There was notany feature of cyclic mobility in the case of no stressreversal; rather, the axial strain accumulated continuously onthe compression side (Fig. 5(b)). It is also of interest to notethat, apart from a pronounced strain in the first loadingcycle, the strain accumulation in the subsequent loadingcycles was at a more or less constant rate (Fig. 5(c)). Thetest was terminated after about 120 loading cycles, withabout 8% axial strain recorded. The excess pore waterpressure at the end of the test was less than 30% of theeffective confining pressure.

As far as denser specimens at Drc ¼ 70% are concerned,the tests have indicated that they generally share similarresponse characteristics with the medium dense specimens atDrc ¼ 50%, given otherwise identical testing conditions. Forthe sake of conciseness, the test results are not presentedhere.

Cyclic behaviour of very loose to loose sandThere is a lack of experimental data on the cyclic loading

behaviour of loose sands with relative density below 30%. It

62 YANG AND SZE

is of great interest to examine how different the behaviourof loose sand is from that of medium dense and dense sand.Fig. 6 shows a typical response of a very loose sandspecimen at Drc ¼ 10% and � 9nc ¼ 300 kPa under symmetri-cal loading conditions. The cyclic deviatoric stress wasapplied at 75 kPa. It can be seen that, although the excesspore water pressure built up gradually and the effectivestress was correspondingly reduced as cyclic loading pro-ceeded, there was no notable axial deformation (Fig. 6(b)).When loading was about to enter the eighth cycle, failurewas suddenly triggered in the specimen, manifested by run-away deformations and a complete loss of strength. Thistype of failure was difficult to predict, since the sand,immediately before failure, appeared to be sufficiently strongand stiff, and the axial deformation was negligibly small(Fig. 6(c)). If the same situation were to occur in the field,the soil deposit would fail suddenly, without any warning,and the consequences could thus be disastrous.

Figure 7 shows the results of a test similar to the oneabove but with an added initial static shear stress of 30 kPa.The cyclic deviatoric stress was applied at 95 kPa, so thatstress reversals occurred during the entire loading process. Asimilar response, characterised by abrupt runaway deforma-tions, was observed. However, the specimen failed on thecompression side (Figs 7(b) and 7(c)) rather than on theextension side as observed in the case of symmetricalloading (Figs 6(b) and (c)). This difference is consideredreasonable, because in this test cyclic deviatoric stress wasdominated by compressive loading.

Shown in Fig. 8 is the response of a loose specimen atthe same initial state (Drc ¼ 10%, � 9nc ¼ 300 kPa) as thatshown in Fig. 7, but subjected to an initial static shearstress of 75 kPa and a cyclic deviatoric stress of 60 kPa,so that no stress reversals occurred. The specimen stillshowed the abrupt failure pattern as in the symmetricalloading, but the rapid axial strain development was on the

Table 2. Undrained cyclic triaxial test series in this study

Series Initial density,Drc: %

Initial confiningpressure, � 9nc : kPa

Minor principalstress, � 93 : kPa

Initial staticdeviatoric stress,

qs: kPa

Initial static shearstress ratio, Æ

Loading type

I-a 10 100 100 0 0 Symmetrical10 100 90 20 0.1 Non-symmetrical10 100 75 50 0.25 Non-symmetrical10 100 60 80 0.4 Non-symmetrical

I-b 10 300 300 0 0 Symmetrical10 300 270 60 0.1 Non-symmetrical10 300 225 150 0.25 Non-symmetrical10 300 180 240 0.4 Non-symmetrical

I-c 10 500 500 0 0 Symmetrical10 500 450 100 0.1 Non-symmetrical10 500 375 250 0.25 Non-symmetrical

II-a 20 100 100 0 0 Symmetrical20 100 95 10 0.05 Non-symmetrical20 100 90 20 0.1 Non-symmetrical20 100 75 50 0.25 Non-symmetrical20 100 60 80 0.4 Non-symmetrical20 100 40 120 0.6 Non-symmetrical

II-b 20 300 300 0 0 Symmetrical20 300 285 30 0.05 Non-symmetrical20 300 270 60 0.1 Non-symmetrical20 300 225 150 0.25 Non-symmetrical20 300 180 240 0.4 Non-symmetrical

II-c 20 500 500 0 0 Symmetrical20 500 475 50 0.05 Non-symmetrical20 500 450 100 0.1 Non-symmetrical20 500 375 250 0.25 Non-symmetrical20 500 300 400 0.4 Non-symmetrical

III-a 50 100 100 0 0 Symmetrical50 100 90 20 0.1 Non-symmetrical50 100 60 80 0.4 Non-symmetrical

III-b 50 300 300 0 0 Symmetrical50 300 270 60 0.1 Non-symmetrical50 300 180 240 0.4 Non-symmetrical

III-c 50 500 500 0 0 Symmetrical50 500 450 100 0.1 Non-symmetrical50 500 300 400 0.4 Non-symmetrical

IV-a 70 100 100 0 0 Symmetrical70 100 90 20 0.1 Non-symmetrical70 100 60 80 0.4 Non-symmetrical

IV-b 70 300 300 0 0 Symmetrical70 300 270 60 0.1 Non-symmetrical70 300 180 240 0.4 Non-symmetrical

IV-c 70 500 500 0 0 Symmetrical70 500 450 100 0.1 Non-symmetrical70 500 300 400 0.4 Non-symmetrical

CYCLIC BEHAVIOUR AND RESISTANCE OF SATURATED SAND 63

compression side. Recalling the impact of stress reversalon the behaviour of medium dense and dense sand sam-ples, the implication here is that the stress reversal did notimpose a marked effect on the response characteristics ofloose sand.

Failure modesThe tests performed in the study have all been thoroughly

reviewed, and it has been found that the failure patterns andmechanisms can be categorised into three distinct types, asschematically illustrated in Fig. 9.

320

240

160

80

0

Exc

ess

pore

wa

ter

pres

sure

: kP

a

0 1 2 3 4 5 6 7 8

Number of loading cycles(a)

10

5

0

�10

�15

Axi

al s

trai

n: %

0 1 2 3 4 5 6 7 8

Number of loading cycles(b)

�5

200

100

0

�100

�200

Dev

iato

ric s

tres

s: k

Pa

�15 �10 �5 0 5 10

Axial strain: %(c)

200

100

0

�100

�200D

evia

toric

str

ess:

kP

a0 60 120 180 240 300 360

Mean effective stress: kPa(d)

Fig. 3. Undrained response of medium dense sand under symmetrical cyclic loading (Drc 50%, �9nc 300 kPa, qs 0, qcyc 160 kPa)

�80

240

160

80

0

Exc

ess

pore

wa

ter

pres

sure

: kP

a

0 1 2 3 4 5 6 7 8

Number of loading cycles(a)

10

5

0

�10

�15

Axi

al s

trai

n: %

0 1 2 3 4 5 6 7 8

Number of loading cycles(b)

�5

200

400

0

�200

Dev

iato

ric s

tres

s: k

Pa

�15 �10 �5 0 5 10

Axial strain: %(c)

200

400

0

�200

Dev

iato

ric s

tres

s: k

Pa

0 100 200 300 400 500

Mean effective stress: kPa(d)

15

600

15

600

Fig. 4. Undrained response of medium dense sand under non-symmetrical cyclic loading (Drc 50%, �9nc 300 kPa, qs 240 kPa,qcyc 310 kPa)

64 YANG AND SZE

The first type (Fig. 9(a)) is referred to as flow-type failureor strain-softening, which is pertinent to sand specimens in avery loose to loose state (Drc ¼ 10% and 20%), and is notaffected by stress reversal. This failure mode is featured bya sudden, runaway deformation similar to that observed inloose sand under monotonic loading conditions (e.g. Sladen

et al., 1985; Yang, 2002b; Been & Jefferies, 2004). It isbelieved that the contractive nature of loose sand facilitatesthe rapid build-up of pore water pressure and makes it moreprone to flow liquefaction.

The second failure mode (Fig. 9(b)), well known as cyclicmobility, has been observed on medium dense and dense

�40

60

40

20

0

Exc

ess

pore

wa

ter

pres

sure

: kP

a

0 30 60 90 120 150

Number of loading cycles(a)

6

4

0

Axi

al s

trai

n: %

0 30 60 90 120 150

Number of loading cycles(b)

300

450

0

Dev

iato

ric s

tres

s: k

Pa

0 6 8

Axial strain: %(c)

300

450

0D

evia

toric

str

ess:

kP

a

0 100 200 300 400

Mean effective stress: kPa(d)

8

600

10

600

�20

80

2

150

2 4

150

Fig. 5. Undrained response of medium dense sand under non-symmetrical cyclic loading (Drc 50%, �9nc 300 kPa, qs 240 kPa,qcyc 230 kPa)

300

200

100

0

Exc

ess

pore

wa

ter

pres

sure

: kP

a

0 2 4 6 8 10

Number of loading cycles(a)

�4

4

0

Axi

al s

trai

n: %

0 2 4 6 8 10

Number of loading cycles(b)

�100

50

0

Dev

iato

ric s

tres

s: k

Pa

0�6 �2

Axial strain: %(c)

50

0

Dev

iato

ric s

tres

s: k

Pa

0 100 200 300 350

Mean effective stress: kPa(d)

�8

100

2

100

400

�12

�50

�10 �4�100

�12 �8

�50

50 150 250

Fig. 6. Undrained response of loose sand under symmetrical cyclic loading (Drc 10%, �9nc 300 kPa, qs 0, qcyc 75 kPa)

CYCLIC BEHAVIOUR AND RESISTANCE OF SATURATED SAND 65

samples at Drc ¼ 50% and 70%. Apart from the sand beingsufficiently dense, another criterion for cyclic mobility tooccur is that the loading should be with stress reversal. Asdiscussed before, this condition is dependent on the relativemagnitudes of the static shear stress and the applied cyclic

shear stress. If the cyclic shear stress is less than the staticshear stress, so that no stress reversals occur, the thirdfailure mode, plastic strain accumulation, will become domi-nant (Fig. 9(c)).

There has been a general belief that stress reversal is a

240

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0

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ess

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sure

: kP

a

0 2 4 6 8

Number of loading cycles(a)

6

15

9

Axi

al s

trai

n: %

0 2 4 6 8

Number of loading cycles(b)

50

0Dev

iato

ric s

tres

s: k

Pa

6 12

Axial strain: %(c)

50

0

Dev

iato

ric s

tres

s: k

Pa

0 100 200 300 350

Mean effective stress: kPa(d)

0

100

15

100

320

�3

�500

�100�3

�50

50 150 250

1 3 5 7 7531

12

3

150

200

93

150

200

Fig. 7. Undrained response of loose sand under non-symmetrical cyclic loading (Drc 10%, �9nc 300 kPa, qs 60 kPa,qcyc 95 kPa)

200

250 250

150

100

50

0

Exc

ess

pore

wa

ter

pres

sure

: kP

a

0 2 4Number of loading cycles

(a)

6

9

Axi

al s

trai

n: %

0 2 4

Number of loading cycles(b)

50

Dev

iato

ric s

tres

s: k

Pa

6 12

Axial strain: %(c)

50

Dev

iato

ric s

tres

s: k

Pa

0 100 200 300

Mean effective stress: kPa(d)

0

100 100

250

�3

0�3 50 150 250

1 3 5 531

12

3

150

200

93

150

200

Fig. 8. Undrained response of loose sand under non-symmetrical cyclic loading (Drc 10%, �9nc 300 kPa, qs 150 kPa, qcyc 60 kPa)

66 YANG AND SZE

governing factor distinguishing the failure patterns of sandunder cyclic loading (Selig & Chang, 1981). While thisbelief is confirmed by the tests on medium dense and densesamples in this study, it does not appear to be valid for loosesand specimens at Drc ¼ 10% and 20%. The tests haveconsistently shown that stress reversal was not the key factorgoverning the failure mechanism of loose sand samples; thefailure of these samples was always dominated by the abruptrunaway pattern, regardless of stress reversal. The impact ofstress reversal for loose sand appears to lie in whether theexcessive axial deformations at failure are in compression orin extension.

To characterise the resistance to cyclic loading, the failurecriterion needs to be rationally defined for each type ofresponse. As illustrated in Fig. 9(a), the onset of failure inloose sand can be defined uniquely as the triggering ofrunaway deformations. However, there is no such unique,definite point indicating the initiation of failure or liquefac-tion in medium dense and dense sand samples. For the modeof cyclic mobility, the usual approach is to define failure asthe point that is accompanied by the occurrence of 5%double-amplitude (DA) axial strain (Toki et al., 1986;Ishihara, 1996). Although the occurrence of 10% DA strainis occasionally used as a criterion, 5% DA is considered tobe more reasonable, because it avoids unacceptably largedeformations. In the case of plastic strain accumulationresponse, since the strain development occurs only in onedirection, the occurrence of 5% peak axial strain (PS) incompression becomes as reasonable a criterion as the 5%DA is for cyclic mobility.

CYCLIC RESISTANCE AND STATE VARIABLESThe evaluation of cyclic resistance is an important step in

the analysis of liquefaction potential in engineering practice.Using the failure criteria defined previously, the cyclic stressratio CSRn can be determined for each test as a function ofthe number of loading cycles to failure

CSRn ¼qcyc

2� 9nc

(3)

Figure 10 shows the results for CSRn for loose samples(Drc ¼ 20%) at various values of � 9nc and Æ. For comparison,similar results for medium dense samples (Drc ¼ 50%) areshown in Fig. 11.

Clearly, the initial static shear stress has a significanteffect on the position of the CSRn line, and thus on theliquefaction resistance, and this effect is dependent on theinitial relative density Drc and confining pressure � 9nc. Inwhat follows, how each of the initial state variables affectsthe liquefaction resistance, and what their interdependenceis, will be investigated in detail. In so doing, the cyclicstress ratio required to cause failure at 10 loading cycles(denoted hereafter as CRRn) is introduced to characterisethe cyclic resistance of sand.

Effects of relative density and confining pressureFigure 12 shows plots of CRRn against Drc under a

variety of combinations of � 9nc and Æ. It can be seen that, nomatter how � 9nc and Æ vary, a denser sample always shows ahigher value of CRRn than a looser one under otherwiseidentical conditions. One of the features of Fig. 12 is that,under a confining pressure of 100 kPa and low static shearstress levels (Æ ¼ 0 and 0.1), the rate of increase in CRRn

Axial strain

Onset of failure

Loading cycles

Axial strain

Loading cycles

(b)

Axial strain

Loading cycles

(c)

PS

(a)

DA

Fig. 9. Deformation patterns under cyclic loading: (a) flow-typefailure; (b) cyclic mobility; (c) accumulated plastic strain

0·4

0·3

0·2

0·1

0

CS

Rn

0·1 1 10 100Number of loading cycles to failure

(a)

α 0�

α 0·25�

α 0·1�

α 0·4�

0·4

0·3

0·2

0·1

0

CS

Rn

0·1 1 10 100

Number of loading cycles to failure(b)

α 0�

α 0·25�

α 0·1�

α 0·4�

Fig. 10. Cyclic stress ratio against number of cycles to failure(Drc 20%): (a) �9nc 100 kPa; (b) �9nc 300 kPa

CYCLIC BEHAVIOUR AND RESISTANCE OF SATURATED SAND 67

resulting from an increase in the relative density from 10%to 50% was much smaller than that from 50% to 70%. Thisobservation implies that, if the sand is originally at amedium dense state, through further densification, its cyclicresistance may be substantially enhanced, but if it is origin-ally in a very loose state, the increase in resistance resultingfrom densification will be relatively small. In this respect, ifin both cases the work done in densification is the same, theeffectiveness of densification would be much higher formedium dense sand than for loose sand. However, this effectdoes not appear to be valid for sand at high static shearstress levels (Æ ¼ 0.4) (Fig. 12(c)).

With respect to the effect of confining pressure, Fig. 12indicates that increasing � 9nc always reduces the value ofCRRn, as long as Drc and Æ remain unchanged. Theimportant implication is that the already low resistance theloose sand bears would be further reduced by increasingthe confining stress level. This impact may become crucialfor high initial shear stress levels. For example, at Æ ¼ 0.4the sand specimen with Drc ¼ 20% had a CRRn of 0.2 under� 9nc ¼ 100 kPa, but it tended to lose its resistance completelywhen � 9nc was raised to 500 kPa. One more thing worthnoting in Fig. 12 is that increasing � 9nc from 100 kPa to300 kPa tends to reduce the beneficial effect of increasingDrc, particularly for medium dense to dense sand, implyingthe interdependence of the effects of relative density andconfining pressure.

The effect of high confining stress may be characterisedby introducing a so-called K� factor (Seed, 1983; Youd etal., 2001). Before doing that, the rationale behind the currentdefinition for K� needs to be carefully examined, and somemodification is required. Given the space limitations, discus-sion of this will be presented in a future paper.

Effects of initial static shearFigure 13 presents values of CRRn against Æ for a variety

of initial states in terms of Drc and � 9nc. A marked feature isthat Æ affects the cyclic resistance differently in loose sandand dense sand. In general, for the entire range of confiningpressures tested, CRRn tends to increase and then decreasewith increasing values of Æ in loose sand specimens(Drc ¼ 10% and 20%), but it continues to increase with Æ inmedium dense and dense sand samples (Drc ¼ 50% and70%).

To characterise this effect, a factor denoted by KÆ isintroduced by following the proposal of Seed (1983). It isdefined here as the ratio of CRRn at any Æ to that at Æ ¼ 0under a fixed � 9nc

KÆ ¼ CRR� ,Æ

CRR� ,0

(4)

Evidently, the variation of KÆ is the direct result of varying

0·8

0·6

0·4

0·2

0

CS

Rn

1 10 100

Number of loading cycles to failure(a)

α 0�

α 0·1�

α 0·4�

0·8

0·6

0·4

0·2

0

CS

Rn

1 10 100

Number of loading cycles to failure(b)

α 0�

α 0·1�

α 0·4�

Fig. 11. Cyclic stress ratio against number of cycles to failure(Drc 50%): (a) �9nc 100 kPa; (b) �9nc 300 kPa

1·0

0·8

0·6

0·4

0·2

0

CR

Rn

0 20 40 60 80

Drc: %(a)

100 kPa

300 kPa

500 kPa

1·0

0·8

0·6

0·4

0·2

0

CR

Rn

0 20 40 60 80

Drc: %(b)

100 kPa

300 kPa

500 kPa

1·0

0·8

0·6

0·4

0·2

0

CR

Rn

0 20 40 60 80

Drc: %(c)

100 kPa

300 kPa

500 kPa

Fig. 12. Effects of relative density and confining pressure oncyclic resistance: (a) no initial shear (Æ 0); (b) small initialshear (Æ 0.1); (c) large initial shear (Æ 0.4)

68 YANG AND SZE

CRRn with Æ, and when Æ ¼ 0, KÆ is always equal to 1. KÆ

does not indicate exactly what cyclic resistance the sandbears; rather, it indicates how great the change in cyclicresistance is due to the presence of a certain level of initialshear stress.

Typical plots of KÆ against Æ are shown in Fig. 14 forvarious values of Drc and � 9nc. For loose sand specimens thepresence of initial static shear stress can be beneficial (i.e.KÆ.1) or detrimental (KÆ,1); but for medium dense anddense specimens the effect appears to be always beneficial.However, as will be discussed later in a theoretical frame-work, the effect of initial static shear stress may alsobecome negative for sand at high relative density undersome circumstances, and a unified view can be establishedfor both loose and dense sand samples.

CRITICAL-STATE-BASED VIEWSCritical state soil mechanics (CSSM) has been shown to

be a useful conceptual framework for describing the mono-tonic behaviour of sand (e.g. Been & Jefferies, 1985; Woodet al., 1994; Yang & Li, 2004). The CSSM frameworkappears to have received less attention for cyclic loadingbehaviour of sand. Recently, Boulanger (2003) made a goodattempt to relate KÆ values to a relative state parameterindex. The focus here is on exploring whether the cyclicresistance and interrelated effects of various state variablescould be accounted for in the CSSM framework. A usefulconcept, termed threshold Æ, will be developed, along withthe use of a no-stress-reversal line, and the dependence ofthe threshold Æ on a state parameter will be established.

The central point of the CSSM framework is that there isan ultimate state of shear failure at which the sand deformscontinuously under constant stress and constant volume. Fig.15 shows the critical state line (CSL) of Toyoura sand andthe initial states of all tests performed in this study in the

1·2

1·0

0·8

0·6

0·4

0·2

00 0·1 0·2 0·3 0·4 0·5 0·6

α(a)

CR

Rn

70%

50%

20%10%

0·8

0·6

0·4

0·2

00 0·1 0·2 0·3 0·4 0·5 0·6

α(b)

CR

Rn

70%

50%

20%10%

0·8

0·6

0·4

0·2

00 0·1 0·2 0·3 0·4 0·5 0·6

α(c)

CR

Rn

70%

50%

20%10%

Fig. 13. Cyclic resistance as a function of the level of initialshear stress: (a) �9nc 100 kPa; (b) �9nc 300 kPa; (c) �9nc500 kPa

2·5

2·0

1·5

1·0

0·5

00 0·1 0·2 0·3 0·4 0·5 0·6

α(a)

Kα 70%

50%

20%10%

0 0·1 0·2 0·3 0·4 0·5 0·6α

(b)

70%

50%

20%

10%

0 0·1 0·2 0·3 0·4 0·5 0·6α(c)

70%

50%

20%10%

2·5

2·0

1·5

1·0

0·5

0

Kα

2·5

2·0

1·5

1·0

0·5

0

Kα

Fig. 14. Effects of confining pressure and relative density on therelationship between KÆ and Æ: (a) �9nc 100 kPa; (b) �9nc300 kPa; (c) �9nc 500 kPa

CYCLIC BEHAVIOUR AND RESISTANCE OF SATURATED SAND 69

e–log p9 plane, where e is the void ratio and p9 is the meaneffective stress. The CSL of Toyoura sand is reproducedfrom the undrained monotonic triaxial tests conducted byVerdugo & Ishihara (1996).

It can be seen from Fig. 15 that the initial states ofspecimens at Drc ¼ 50% and 70% all lie below the CSL; inthe CSSM concept, these samples should behave in adilative manner. All specimens at a relative density of 10%and most specimens at 20% will show contractive behaviour,because their states are above the CSL. The 20% sandspecimens at a mean effective stress of about 200 kPa maytend to exhibit a marginal response.

Cyclic resistance against state parameterThe state parameter proposed by Been & Jefferies (1985)

is adopted to account for the initial state in terms of relativedensity and stress level. It is defined as

ł ¼ e0 � ecs (5)

where e0 is the initial void ratio of the sample and ecs is thevoid ratio of the critical state at the same mean stress (Fig.15). Obviously, the parameter is a measure of how far thecurrent state is from the critical state. If ł is negative, thesand is considered to be in a dense, dilative state, whereas itis in a loose, contractive state if ł is positive.

Figure 16(a) shows the values of CRRn against the stateparameter ł for the tests performed. A striking feature ofthis plot is that there is a fairly good trend between CRRn

and the state parameter, showing that CRRn tends to de-crease as ł increases: that is, the sand will bear a highercyclic resistance when it becomes denser and/or when it issubjected to a lower confining pressure.

For the purpose of comparison, the relative state para-meter proposed by Konrad (1988) is adopted as well, todescribe the initial state. This relative state parameter isdefined as

ł1 ¼ e0 � ecs

emax � emin

(6)

where emax and emin are the maximum and minimum voidratios of a sand. Fig. 16(b) shows CRRn against ł1 for thetests performed. The trend is almost the same as that in Fig.16(a). In the following discussion, only the state parameterł will be the focus.

In Fig. 16(a), at each value of ł there is a range of datapoints scattering vertically. This scatter is the result ofvarying Æ at each set of initial states in terms of Drc and� 9nc. By re-plotting the data in terms of Æ, an improved

correlation is achieved between CRRn and ł, as shown inFig. 17.

Based on the data points in Fig. 17, and for the firstinstance, a best-fit line can be drawn to represent thevariation of CRRn with ł. If a non-linear relationship isassumed, one may arrive at the form

CRRn ¼ 8:1508ł2 � 1:2895łþ 0:1498 (7)

If a linear relationship is assumed, then

CRRn ¼ �2:3794łþ 0:1719 (8)

Threshold ÆReferring back to Fig. 13, where cyclic resistance is

plotted as a function of Æ, all loose samples show anincrease and then a decrease in CRRn with increasing valuesof Æ. This observation suggests that a threshold value of Æexists: when Æ reaches this value, the cyclic resistance tendsto reduce with further increase in Æ. Furthermore, by recal-ling the role of stress reversal, a no-stress-reversal line isintroduced herein as the line dividing between loading withstress reversals and loading without stress reversals. Byadding this line to the plot of CRRn against Æ, as shown inFig. 18 for the case of Drc ¼ 20%, it is of interest to notethat whenever the CRRn trend line touches the no-reversalline, CRRn is about to decrease with increase in Æ. In otherwords, when Æ is increased to a level such that the loadingrequired to bring the sand to failure is without stressreversals, then the cyclic resistance starts to reduce.

Apparently, the use of the no-stress-reversal line helps indetermining the threshold Æ in a rational way. One key pointto note is that this threshold Æ is a function of the relativedensity and initial confining pressure. This can be observedin Figs 13 and 18, which indicate that the plot of CRRn

1·00

0·95

0·90

0·85

0·80

0·75

0·70

e

Critical state line (CSL)

1 10 100 1000 10000

log : kPap�

ψ � �e e0 cs

10% 100 kPa

10% 300 kPa

10% 500 kPa

20% 100 kPa

20% 200 kPa

20% 300 kPa

20% 500 kPa

50% 300 kPa

50% 500 kPa

50% 100 kPa

70% 300 kPa

70% 500 kPa

70% 100 kPa

Fig. 15. Initial states of specimens with reference to the criticalstate line

1·2

1·0

0·8

0·6

0·4

0·2

0

CR

Rn

�0·25 �0·10 �0·05 0·05 0·10

State parameter,(a)

ψ

10% 100 kPa 10% 300 kPa 10% 500 kPa

20% 100 kPa 20% 200 kPa 20% 300 kPa

20% 500 kPa 50% 300 kPa

50% 500 kPa

50% 100 kPa

70% 300 kPa

70% 500 kPa

70% 100 kPa

1·4

�0·20 �0·15 0

1·2

1·0

0·8

0·6

0·4

0·2

0

CR

Rn

�0·6 �0·3 �0·2 0·1 0·2

State parameter,(b)

ψ1

10% 100 kPa 10% 300 kPa 10% 500 kPa

20% 100 kPa 20% 200 kPa 20% 300 kPa

20% 500 kPa 50% 300 kPa

50% 500 kPa

50% 100 kPa

70% 300 kPa

70% 500 kPa

70% 100 kPa

1·4

�0·5 �0·4 0�0·1 0·3

Fig. 16. Cyclic resistance as a function of: (a) state parameter;(b) relative state parameter

70 YANG AND SZE

against Æ will be lower when the sand becomes looser,because of either decreasing Drc or increasing � 9nc. Thisdependence is shown more clearly in Fig. 19, where thethreshold Æ is plotted against the confining pressure � 9nc fortwo cases of relative density. It can be seen that, at the samerelative density, the threshold Æ decreases with increasing� 9nc, while at the same confining pressure it increases withincreasing Drc.

Recognising that the state parameter can better describethe initial state, the threshold Æ (Æth) is then re-plotted as afunction of the state parameter ł in Fig. 20. Strikingly, aunique relationship exists between the two, which can berepresented by a best-fit line as

Æth ¼ �2:0854łþ 0:2205 (9)

The data points in Fig. 20 are derived from the tests onloose samples at Drc ¼ 10% and 20% only, as these samplesshow the existence of threshold Æ. An interesting questionthen arises: Does dense sand have a threshold Æ? To answerthis question, one may refer back to Fig. 13, where the testdata for Drc ¼ 50% and 70% are included along with thedata for loose samples. A first glance may suggest that theanswer is ‘No’, at least for the range of Æ values tested inthe study. However, recognising the good correlation shownin Fig. 20, and in line with the CSSM concepts, it ispostulated that the concept of threshold Æ should exist fordense samples at Drc ¼ 50% and 70% as well, if the stresslevel could become sufficiently high to lead to positivevalues of ł (i.e. a loose, contractive state).

The above inference is illustrated in Fig. 21 for the caseof dense samples at a relative density of 70%. As can beseen, the position of the CRRn lines is dependent on theconfining pressure level: with increasing � 9nc values theCRRn line tends to be lowered down and, eventually, tocross the no-stress-reversal line. From the intersection of thetwo lines the threshold Æ can subsequently be determined.

1·2

1·0

0·8

0·6

0·4

0·2

0

CR

Rn

�0·25 �0·10 �0·05 0·05 0·10ψ(a)

1·4

�0·20 �0·15 0

CRR 8·1508 1·28950·1498

n � ��

ψ ψ2

α 0�

α 0·1�

α 0·25�

α 0·4�

α 0·6�

1·2

1·0

0·8

0·6

0·4

0·2

0

CR

Rn

�0·25 �0·10 �0·05 0·05 0·10ψ(b)

1·4

�0·20 �0·15 0

CRR 2·3794 0·1719n � � �ψ

α 0�

α 0·1�

α 0·25�

α 0·4�

α 0·6�

Fig. 17. Correlation between CRRn and ł for various Æ values:(a) non-linear representation; (b) linear representation

0·6

0·5

0·4

0·3

0·2

0·1

0

CR

Rn

Drc 20%�

With stress reversal

No-reversal line

Without stress reversal

0 0·1 0·2 0·3 0·4 0·5 0·6α

Threshold α500 kPa

300 kPa100 kPa

Fig. 18. Determination of threshold Æ using test data for loosesand

0·4

0·3

0·2

0·1

0

Thr

esho

ldα

0 100 200 300 400 500 600

σ �nc: kPa

Drc 10%�

Drc 20%�

Fig. 19. Dependence of threshold Æ on confining pressure andrelative density

0·4

0·3

0·2

0·1

0

Thr

esho

ldα

�0·02 0 0·02 0·04 0·06ψ

Sand becomes looser

Threshold decreasesα

α ψ2·0854 0·2205� � �

Fig. 20. Threshold Æ as a function of state parameter

CYCLIC BEHAVIOUR AND RESISTANCE OF SATURATED SAND 71

Because of the capacity limitation of the triaxial deviceused, it was not possible to perform tests at confiningpressure levels much higher than the current 500 kPa toverify this inference. Nevertheless, by examining the existingdata for higher stress levels, it is very encouraging to seeevidence in support of the inference. Shown in Fig. 22 arethe data interpreted from the triaxial test results of Vaid &Chern (1985), who performed tests on tailings sand sampleswith a relative density of 70% under cell pressures of 200and 1600 kPa. It can be seen that, whereas the sample at200 kPa showed an increase in cyclic resistance with Æ, thesample at 1600 kPa indeed showed an increase and then adecrease of resistance with Æ. The broken line superimposedon the plot is the no-reversal line introduced previously.Interestingly, it can be seen that the cyclic resistance of thetailings sand is about to reduce when the CRRn line crossesthe no-reversal line: this observation is in good support ofthe proposed idea (see Figs 18 and 21).

As no information is available for the critical state line ofthe tailings sand, it is impossible to examine how the data fitin the relationship proposed in Fig. 20, but it is expected

that a trend similar to the proposed one should exist.Obviously, future work on different sands would be of valuein validating the idea and interpretations developed here.

SUMMARY AND CONCLUSIONSThis paper presents a systematic experimental study aimed

at investigating the cyclic behaviour of sand under non-symmetrical loading conditions, and at characterising therole of initial static shear in cyclic liquefaction resistance.An attempt has been made to develop a unified and consis-tent view of the interrelated effects of various state variablesin the framework of CSSM. The main findings resultingfrom the study can be summarised as follows.

(a) There are three distinctly different failure modes forsand under cyclic triaxial loading conditions. The mostcritical one is characterised by abrupt, runawaydeformations without any warning signal. The secondfailure mode is that in which cyclic strain in terms ofdouble amplitude accumulates to a large level ofdeformation. The third type is that in which plasticstrain accumulates in one direction to arrive at aspecified level of deformation.

(b) The runaway, flow-type failure is pertinent to suffi-ciently loose sand, and can occur under both stress-reversal and non-reversal conditions. Medium dense ordense sand at a normal confining pressure level exhibitseither the second or third mode of failure, and stressreversal is a key factor distinguishing them. When nostress reversals occur, the plastic strain accumulationpattern governs.

(c) Under otherwise identical conditions the cyclic resis-tance of sand always increases with increasing relativedensity, no matter whether initial static shear stress ispresent or not, but the effect of relative density appearsto depend on the level of static shear stress. At highstatic shear stress levels (Æ ¼ 0.4), the increase ofcyclic resistance due to the increase in relative densityis more profound than that at low static shear stresslevels (Æ ¼ 0.1 and 0).

(d ) Increasing the confining pressure always reduces cyclicresistance, as long as the relative density and the staticshear stress ratio remain unchanged. This impact can becrucial for loose sand under high static shear stresslevels (Æ ¼ 0.4), since the sand may lose its resistancetotally when the confining pressure is increased.

(e) For loose sand, the presence of initial static shear stresscan be beneficial for or detrimental to its resistance toliquefaction. In this connection, a threshold Æ existsafter which the cyclic resistance of sand tends to reducewith increasing Æ values. The magnitude of thethreshold Æ is dependent on the initial relative densityand the confining pressure.

( f ) In the CSSM framework, a fairly good linear relation-ship exists between the threshold Æ and the stateparameter ł. This relationship indicates that thethreshold Æ decreases with increasing value ł. Alongthis line, the concept of threshold Æ can be extended tosand at high relative density, as long as the confiningpressure involved is sufficiently high.

ACKNOWLEDGEMENTSThe work presented in this paper was supported by the

Research Grants Council of Hong Kong under grant number719105. This support is gratefully acknowledged. The sup-port of the University of Hong Kong through its Outstanding

1·2

1·0

0·8

0·6

0·4

0·2

0

CR

Rn

0 0·1 0·2 0·3 0·4 0·5 0·6 0·7 0·8α

Sufficiently highstress level

No-reversal line500 kPa

300 kPa

100 kPa

Fig. 21. Inference of existence of threshold Æ for dense sandsamples

0·4

0·3

0·2

0·1

0

CR

Rn

0 0·1 0·2 0·3 0·4α

Tailings sand ( 70%)Drc �

200 kPa

No-reversal line

1600 kPa

Fig. 22. Supporting evidence for existence of threshold Æ fordense sand (data interpreted from Vaid & Chern, 1985)

72 YANG AND SZE

Young Researcher Award Programme is also highly acknowl-edged.

NOTATIONCSRn cyclic stress ratio (normalised by � 9nc)CRRn cyclic resistance ratio

Drc relative density after consolidationDA double amplitude

e0 initial void ratioecs critical state void ratio

emax maximum void ratioemin minimum void ratio

p9 mean effective stressPS peak axial strain

qcyc cyclic deviator stressqs initial static deviatoric stressÆ initial static stress ratio

� 91c effective major principal consolidation stress� 93c effective minor principal consolidation stress� 9nc normal effective consolidation stress on the 45o plane�cyc cyclic shear stress�s initial static shear stressł state parameterł1 relative state parameter

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Hyodo, M., Tanimizu, H., Yasufuku, N. & Murata, H. (1994).Undrained cyclic and monotonic triaxial behavior of saturatedloose sand. Soils Found. 34, No. 1, 19–32.

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Ishihara, K. (1996). Soil behaviour in earthquake geotechnics.Oxford: Clarendon Press.

Konrad, J. M. (1988). Interpretation of flat plate dilatometer tests insands in terms of the state parameter. Geotechnique 38, No. 2,263–277, doi: 10.1680/geot.1988.38.2.263.

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