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SOILSANDFOUNDATIONSVol.48,No.5,713–725,Oct.2008

JapaneseGeotechnicalSociety

THEEFFECTOFFINESONCRITICALSTATEANDLIQUEFACTION RESISTANCECHARACTERISTICSOFNON-PLASTICSILTYSANDS

ANTHI PAPADOPOULOUi) andTHEODORA TIKAii)

ABSTRACT

Monotonicandcyclictriaxialtestswerecarriedoutonsand-siltmixturesfortheinvestigationoftheeŠectofˆnes contentontheircriticalstateandliquefactionresistancecharacteristics.Boththeundrainedandthedrainedmonotonictestsproduceauniquecriticalstatelineforeachtestedmixture,whichmovesdownwardswithincreasingˆnescontentuptoathresholdvalueof35z andthenupwards.AtagivenvoidratioandmeaneŠectivestress,theliquefaction resistanceratiodecreaseswithincreasingˆnescontentuptothesamethresholdvalueof35z,andincreasesthereafter withfurtherincreasingˆnescontent.However,atagivenintergranularvoidratio,deˆnedastheratioofthevolume of ˆnesplusvoidstothatofsandparticles,liquefactionresistanceratioincreaseswithincreasingˆnescontentuptothe thresholdvalue.Thethresholdˆnescontentvalue,whichisanimportantparameterindeterminingthetransitionfrom thesanddominatedtothesiltdominatedbehaviourofsand-siltmixtures,isrelatedtotheirparticlepacking. Anexpressionisproposedfortheestimationofthethresholdˆnescontentasafunctionofthemeandiameterratio, d50/D50, andthevoidratio.Theresults,presentedherein,alsoshowthatforeachtestedmixturetheliquefactionresistanceratio isrelatedtothestateparameterandthatthisrelationisin‰uencedbytheeŠectivestresslevelandˆnescontent.The resultsonthesand-siltmixturesaresupportedbysimilarresultsonnaturalsiltysands.

Keywords: criticalstate,ˆnes,liquefaction,non-plastic,sands,silt,stateparameter,threshold,voidratio(IGC: C3/D6/D7/E7)

INTRODUCTION

Liquefactionofsandysoilsundermonotonicandcyclicloadingconditionsisconsideredtobeoneofthe majorcausesoffailureofearthstructuresandfoundations.Fieldobservationsfromfailuresduetoliquefactionhaveshownthatbothcleansandsandsandscontainingˆnes(particleswithdiameterlessthan75 mm)aresusceptibletoliquefaction.Semi-empiricalˆeld-based proceduresforevaluatingtheliquefactionpotentialduringearthquakesarebasedoncorrelationsbetweenˆeld behaviourandin-situindextests,suchasstandard penetrationtest(SPT),conepenetrationtest(CPT), Beckerpenetrationtest(BPT)andshearwavevelocity(V s).Seedetal.(1985)proposedtheoldestandperhapsthe mostwidelyusedprocedureinwhichthecyclicstressratio,CSR＝tav/s? o,iscorrelatedwiththeSPTblowcounts, correctedforbotheŠectiveoverburdenstressandenergy, (N1)60,forcleansandsandsiltysandswithˆnescontent greaterthan5z andearthquakemagnitude, M＝7.5.Accordingtothiscorrelation,thepresenceofˆnesinsilty sandsincreasestheirliquefactionresistanceandconsequentlydecreasesthepotentialofliquefactiondevelopment.ThisbeneˆcialeŠectofˆnesontheliquefaction resistanceofsiltysandshasbeenalsoadoptedinall

moderncodes,(NCEER,1997andEurocode8)andmay beexplainedbyconsideringempiricalcorrelationsbetweentherelativedensity, Dr,and N valueoftheSPT, whichalsoaccountforthegrainsizeeŠectsonthe penetrationresistanceofsandysoils.Skempton(1986) foundthatthenormalized(N1)60/Dr 2 valueincreaseswith increasinggrainsize(or D50),whichimpliesthatfora given(N1)60 value, Dr increaseswithdecreasing D50 value (orincreasingˆnescontent).CubrinovskiandIshihara (2002)usedthevoidratiorange, emax -emin toquantifythe eŠectofgradingpropertiesontheSPTresistance.They foundthatthenormalized(N1)60/Dr 2 valuedecreaseswith increasing emax -emin.As emax -emin increaseswithincreasing ˆnescontent,theaboveˆndingsimilarlyindicatesthat foragivenvalueof(N1)60, Dr increasesastheˆnescontentincreases.

Inlaboratory,althoughnumerousstudieshavebeen performedinordertoinvestigatetheeŠectofˆnescontentontheliquefactionresistanceratioofsiltysands,the resultsappeartobecon‰icting.Ishihara(1996)andAminiandQi(2000),amongothers,foundthattheliquefactionresistanceratioincreaseswithincreasingˆnescontent(positiveeŠect),andotherslikeTroncosoandVerdugo(1985),Vaid(1994)andMiuraetal.(1995)suggestedthattheliquefactionresistanceratiodecreaseswithin-

i) ResearchStudent,DepartmentofCivilEngineering,AristotleUniversityofThessaloniki,Greece ii) Professor,ditto(tika＠civil.auth.gr).

ThemanuscriptforthispaperwasreceivedforreviewonJuly9,2007;approvedonJuly23,2008.

WrittendiscussionsonthispapershouldbesubmittedbeforeMay1,2009totheJapaneseGeotechnicalSociety,4-38-2,Sengoku,Bunkyo-ku, Tokyo112-0011,Japan.Uponrequesttheclosingdatemaybeextendedonemonth.

713

PAPADOPOULOUANDTIKA

creasingˆnescontent(negativeeŠect),whereasothers likeThevanayagametal.(2000),PolitoandMartin (2001),XenakiandAthanasopoulos(2003)andYanget al.(2004)foundthatthereisanincreaseofliquefaction resistanceratiowithincreasingˆnescontentuptoacertainvalueandadecreasethereafterwithfurtherincreasingˆnescontent.TheˆnescontentatwhichtheeŠectof ˆneschangesfrompositivetonegativehasbeentermedas threshold,orlimiting,ortransitional.

Thecriticalstate,deˆnedasthestateatwhichthesoil continuestodeformatconstantshearstressandconstant voidratio,hasincreasinglybeenusedasafundamental statetocharacterizethestrengthanddeformationpropertiesofsandsinlimitequilibrium(Casagrande,1936; Roscoeetal.,1958;SchoˆeldandWroth,1968).Atthis state,thereisauniquerelationshipamongvoidratio, ecs, meaneŠectivestress, p? cs,andshearstrength, qcs,expressedbythecriticalstateline,CSL,inthevoidratio versusmeaneŠectivestressplane.Accordingtothecriticalstateconcept,thebehaviourofasanddependsnot onlyondensity,butalsoonstresslevel.Thetruestateof asandisdescribedbythelocationofitscurrentstateof stressandvolumerelativetothecriticalstateline.When thestateofasandisabovethecriticalstateline,thesand hasthetendencytocontractuponshearing,whereas whenitsstateisbelowthecriticalstateline,ithasatendencytodilate.Variousnormalizedparametershave beenproposedtocharacterizethediŠerencebetweenthe actualstateandthecriticalstateline.BeenandJeŠeries (1985)havequantiˆedthedistanceofthecurrentstate fromthecriticalstatelinebymeansofastateparameter, c,whichisthediŠerenceinvoidratiosbetweenthecurrentstateandthecriticalstatelineatthecurrentmean eŠectivestress, p? cs,Fig.1.

Recently,inanattempttore-evaluatetheeŠectof overburdenstressconsideredinthesemi-empiricalˆeldbasedprocedures,suchasSPTandCPTtests,theliquefactionresistanceratioofcleansandsduringboth monotonicandcyclicloadinghasbeencorrelatedwith

Fig.1.Deˆnitionofstateparameter(BeenandJeŠeries,1985)

thestateparameter(PillaiandMuhunthan,2001; Boulanger,2003;IdrissandBoulanger,2004).Theextensionoftheabovestatedcorrelationfromcleansandsto sandscontainingˆneswouldbebothusefulandofgreat interest.

Thepurposeoftheworkpresentedinthispaperisˆrst toclarifytheeŠectofˆnescontent, fc,onthecriticalstate lineandtheliquefactionresistancecharacteristicsofsilty sandsandthentoinvestigatewhethertheliquefaction resistanceratioofthesesoilscanberelatedtotheircriticalstate.

TESTEDMATERIALS

Thematerialsusedinthetestingprogrammewereartiˆcialsand-siltmixturesandnaturalsiltysands. Sand-siltmixturesweremadefromacleanquartzsand (M31)withwell-roundedgrainsandanon-plasticsilt,a groundproductofnaturalquartzdepositsfromAssyros inGreece.Sampleswerepreparedbymixingthesand(S) withthesilt(F)atpercentagesof5,10,15,25,35,40and 60z ofthetotaldrymassofthemixture(notedasSF5, SF10,SF15,SF25,SF35,SF40andSF60respectively). Testswerealsoconductedonspecimensofcleansandand puresilt.

Twonaturalsiltysands(Däuzce-1,Däuzce-2)were retrievedfromliqueˆedsitesinDäuzce,Turkey,during the1999earthquakeofmagnitude Mw ＝7.1andtheother wasretrievedfromthefoundationoftheRimnioembankmentbridgeinGreece,whichfailedduringthe1995 Kozaniearthquakeofmagnitude Ms ＝6.6(Tikaand Pitilakis,1999).

Thephysicalpropertiesandgrainsizedistributionsof thetestedmaterialsarepresentedinTable1andFig.2, respectively.

AlthoughtheASTM(D4253andD4254)testmethods forthedeterminationofminimumandmaximumvoid ratiosareapplicabletosoilsthatmaycontainupto15z, bydrymass,ofsoilparticlespassingtheNo.200(75-mm) sieve,providedtheyhavecohesionlesscharacteristics, boththesemethodswereusedinthisworkinconjunction withothersinordertogetaconsistentvalue.Inparticular,forthedeterminationoftheminimumvoidratio boththevibratorytable(ASTMD4253,method2A)and thestandardproctortestmethodswereused.Themaximumvoidratiowasdeterminedinaccordancewiththe

Table1.Physicalpropertiesoftestedmaterials

(zº75 mm)

Fig.2.Grainsizedistributionsofthetestedmaterials

Fig.3.Variationofmaximum, emax,minimum, emin,andcriticalstate ( s ? o ＝100kPa), ecs,voidratioswithˆnescontent, fc,forthetested materials

ASTM(D4254,methodC)testmethodandalsobypouringdrymaterialinamouldthreetimesandconsidering theaverageasthemaximumvalue.Figure3showsthe minimumandmaximumvoidratiosofthetestedmaterials.AsshownintheaboveFigureforthesand-siltmixtures,theminimumvoidratioisobtainedataˆnescontentbetween25z and35z

TESTINGEQUIPMENTANDEXPERIMENTAL PROCEDURE

Thetestingprogrammeconsistedofmonotonicandcyclictriaxialtestsforthedeterminationofthecriticalstate lineandtheliquefactionresistanceratioofthetested materialsrespectively.Bothtypesoftestswereperformed

usingaclosed-loopautomaticcyclictriaxialapparatus (MTSSystemsCorporation,U.S.A.).Itsprinciplesof operationaregivenindetailinPapadopoulou(2008). Thespecimens(height/diameter§100mm/50mm) wereformedbymoisttampingatawatercontentvarying between4z and12z forallthetestedmaterialsand35z onlyforthesiltspecimens,usingtheundercompaction method,introducedbyLadd(1978).Moisttampingwas preferredtootherpreparationmethods,asitproduces specimensofvaryingdensities(VerdugoandIshihara, 1996).Saturationwasachievedbypercolatingthroughoutthespecimen,fromthebottomtothetopdrainage line,ˆrstcarbondioxidegas(CO2)for20minutesand thende-airedwater.Asuctionpressureof15kPawasappliedwhiledismantlingthespecimen,measuringits dimensionsandassemblingthetriaxialcell.Inorderto ensurefullsaturation,aseriesofstepsofsimultaneous increasingcellpressureandbackpressurewereperformed,whilemaintaininganeŠectiveconˆningstressof 15kPa.Inallthetests,aˆnalbackpressureof500kPa wasfoundtobesu‹cient,astheparameterofporewater pressure, B＝Du/Ds,didnotincreasebyfurtherincreasingbackpressure.Inalltheteststheparameter B had valuesfrom0.95to1.00.Nocorrectionoftheresultswas madeformembranepenetration,becauseoftheuncertainties,associatedwithsuchacorrectionandsotheraw testresultsarepresentedhere.Afterthecompletionof saturation,thespecimenswereisotropicallyconsolidated underaneŠectiveisotropicstress, s? o,rangingfrom50to 300kPa.Aperiodoftimeequaltodoubletheconsolidationtimeofthespecimenswasallowedbeforeshearing. Duringconsolidationthevolumechangeandtheaxial displacementofthespecimenswererecordedinorderto

calculatethepostconsolidationvoidratio.

Inthemonotonictests,thespecimensweresubjectedto eitherundrained(CU),ordrained(CD)compressionata constantrateofaxialdisplacementof0.05mm/min.

Inthecyclictriaxialtests,asinusoidallyvaryingaxial stress(±sd)wasappliedatafrequencyof f＝0.1Hz,underundrainedconditions.Inthiswork,cyclicstressratio, CSR＝sd/2s? o,correspondstodoubleamplitudeaxial strain, eDA§5z andliquefactionresistanceorcyclic resistanceratio,CRR15,isdeˆnedasthecyclicstressratio,CSR＝sd/2s? o,requiredtocausedoubleamplitude axialstrain, eDA§5z at15cyclesofloading.

TESTRESULTSANDDISCUSSION

Severalstudieshaveconsideredsandscontainingˆnes asconsistingoftwomatrices,thesandgrainsmatrixand theˆnesmatrix,andanalysedtheirbehaviourintermsof theinteractionwitheachother(Vaid,1994;Thevanayagametal.,2000;PolitoandMartin,2001;Xenakiand Athanasopoulos,2003;Yangetal.,2004).Thenatureof thecontributionofsandandˆnesmatricesmaybeexpressedintermsoftheintergranularandinterˆnevoid ratiosrespectively.

Theintergranularvoidratio, eg,expressestherelative contributionofsandfractiononthebehaviourofthe mixtureandisgivenbythefollowingequation(Mitchell, 1975):

eg ＝ VFINES＋VV VSAND ＝ fc＋w･(GSF/Sr) (1 fc) (G

where VFINES isthevolumeoftheˆnes, VV isthevolume ofvoids, VSAND isthevolumeofsandgrains, fc istheˆnes content, w isthewatercontentofthespecimen, GSF isthe speciˆcgravityoftheˆnesand GSG isthespeciˆcgravity ofsandgrains.Forsaturatedspecimens(Sr ＝100z)and consideringthat GSF§GSG,theintergranularvoidratio aftertheconsolidationofthespecimenisexpressedas follows:

g ＝ fc＋ec (1 fc) (2) where ec isthevoidratioofthemixtureafterconsolidation.

Formixtureswithhighˆnescontents,whenthesand grainsbecomeisolated(VSAND maybeignored),theinterˆnevoidratio, ef,maybeamoreappropriateparameter tobeused(ThevanayagamandMohan,2000):

ef＝ VV VFINES (3)

Similarly,forsaturatedspecimens(Sr ＝100z)andconsideringthat GSF§GSG,theinterˆnevoidratioafterthe consolidationofthespecimenisexpressedasfollows: ef＝ ec fc (4)

Figure4presentsthevariationofboth

eg and ef with thevoidratioofmixturesatvarious fc values.Inthefol-

Fig.4.Variationof(a)intergranularvoidratio, eg (fcÃfcth)or(b)interˆne, ef (fcÀfcth)voidratiowithvoidratio, e,atvarious fc (open andclosedsymbolsindicate emin and emax valuesrespectively)

lowingtheresultsforthecriticalstateandtheliquefactionresistancecharacteristicsofthetestedsoilsare presentedandthendiscussed.

Sand-siltMixtures

CriticalState

Inthemonotonicundrainedtestsitwasconsideredthat theonsetofcriticalstateconditionscorrespondstothe pointswheretheshearandthemeaneŠectivestresses,as wellastheporewaterpressure,remainedpracticallyconstantwithaxialstrain.Intheconductedtests,these pointscorrespondedtostrainsrangingfrom7z to42z, dependingonthetypeofbehaviour(contractive,contractive/dilative,dilative),thetypeofmaterialandthedensity.Figure5showstypicalresultsfromtheundrained testsinwhichcontractive,contractive/dilativeanddilativebehaviourofspecimenswasobserved.Inthedrained testsitwasconsideredthatcriticalstateconditionshad beenreachedataxialstrainsexceeding30z,wherethe rateofchangeofbothshearstressandvolumetricstrain decreasedconsiderablyorceased.Figure6presentsthe resultsfromthedrainedtests.Foralltestedmixtures,the CSLsinthe q-p? cs planearerepresentedbystraightlines passingthroughtheorigin:

Fig.5.Typicalresultsofmonotonicundrainedtriaxialtestswithcontractive(SF35),contractive/dilative(S)anddilativebehaviour (SF15),(a) q- e a,(b) D u- e a and(c) q/p? c -p?/p? c plots

where Mcs isanintrinsicconstantforeachmixture,Table 2.Ananalyticaldescriptionofthemonotonictestshas beenpresentedbyPapadopoulou(2008).

Figure7showstheCSLsinthe e-p? cs ofthesand-silt mixtures,aswellasthesandandthesilt.Asshowninthe aboveFigure,foreachsoilboththeundrainedandthe drainedtestsproduceauniqueCSL.AtsmallmeaneŠectivestresses,below300kPaapproximately,theCSLsare nearlyparallelandhaveasmallinclination.WithincreasingmeaneŠectivestresslevel,however,theysteepenand convergeatstressesabove1000kPa.Gradinganalyses wereperformedontheinitialmaterials,asusedinthe testsandafterthetests.Noparticlebreakagewasindicatedbothforthesandandthesand-siltmixtures.Asalso showninFig.7,theCSLsmovedownwardswithincreasing fc uptoathresholdvalue, fcth,of35z,andthenupwards.TheCSLsat fc valuesof35z and40z arevery close,andtheCSLsofthesandandthesiltpractically

Fig.6.Resultsfrommonotonicdrainedtriaxialtests,(a) q/p? c - e a and (b) e v - e a plots

Table2. Mcs valuesoftestedmaterials Soil M

Sand(S)1.35530.9970

SF51.41230.9991

SF101.44690.9997

SF151.54410.9848

SF251.40830.9999

SF351.43900.9994

SF401.43510.9999

SF601.43980.9992

Silt(F)1.51470.9998

Däuzce-11.50130.9999 Rimnio1.38640.9996

coincide.Thevariationof ecs with fc,ataneŠectivestress of s? o ＝100kPa,whichispresentedinFig.3,conˆrms the fcth valueof35z. TheCSLsofthesand-siltmixturesintermsof eg or ef, areshowninFig.8.Asshowninthisˆgure,inthiscase theCSLsmoveupwardswithincreasing fc uptothe fcth of 35z,andthendownwardsThismaybeattributedtothe increasingvaluesof eg astheincreasingpresenceofsilt loosensthestructureofthesand-siltmixture.At fc values greaterthanthe fcth＝35z,thepresenceofsiltstartsto

Fig.7.Criticalstatelinesofsand-siltmixturesintermsofvoidratio, e

Fig.8.Criticalstatelinesofsand-siltmixturesintermsofintergranular, eg ( fcÃ35 % )orinterˆne, ef ( fcÀ35 % )voidratio

Fig.9.Variationofcyclicstressratio, s d/2 s ? o,withnumberofcycles, N,atconstantvoidratio, e,andvariouslevelsofmeaneŠectivestress, s ? o, for(a)S,(b)SF15,(c)SF25,(d)SF35and(e)SF40

Fig.10.Variationofcyclicstressratio, s d/2 s ? o,withnumberofcycles, N,atconstantvoidratio, e,andvariousˆnesconents, fc,at s ? o,(a)100kPaand(b)300kPa

dictatethebehaviourofthesand-siltmixtureand ef is used.

Theabovedescribedbehaviourforthesand-siltmixtures,concerningtheinclinationoftheCSLsandthe eŠectof fc ontheirposition,isingeneralagreementwith theresultsoftheinvestigationsofZlatoviácandIshihara (1995),Thevanayagametal.(2002),NaeiniandBaziar (2001)andYangetal.(2006).However,inmostofthe aboveinvestigations,eitherthevalueof fcth wasinferred, ortheinclinationoftheCSLswasdeterminedoveranarrowstressrange.

LiquefactionResistance

Figure9showsthevariationofCSRwithnumberof cyclesatvarious fc values.Atagivennumberofcyclesa decreaseofCSRwithincreasing s? o from50kPato300 kPaisobservedfor fc valuesupto15z.Although, resultsat s? o ＝50kPaandhigher fc valuesarenotavailable,Figs.9(c),(d)and(e)showthattheeŠectof s? o onthe CSRdiminishesas fc increases.TheeŠectof fc onCSRat constantvoidratioand s? o isshowninFig.10.Atagiven numberofcycles,CSRˆrstdecreaseswithincreasing fc uptoathresholdvalueof35z andincreasesthereafter withfurtherincreasing fc

TheeŠectofboth s? o andvoidratioontheCRR15 is showninFig.11.AdecreaseofCRR15 withincreasing voidratioatagiven s? o isobservedforeachtestedmixture.ThevariationoftheCRR15 with fc isshowninFig.

Fig.11.Variationofcyclicresistanceratio,CRR15,withvoidratio, e, forvariousˆnescontents, fc,andat s ? o,(a)50kPa,(b)100kPaand (c)300kPa

12at s? o ＝100and300kPa.TheresultsinFigs.11and12 showthatonceCRR15 reachesaminimumatathreshold ˆnescontent,itremainsrelativelyconstantthereafter irrespectivelyof fc,unlessthelatterisincreasedsigniˆcantly.Thethresholdˆnescontentisabout35z and 25z at s? o ＝100and300kParespectively.Thethreshold ˆnescontentat s? o ＝100kPaisidenticaltothe fcth value, determinedfromthemonotonictests,asdescribedpreviouslyandshowninFigs.3and7.

Figure13showsthevariationofCRR15 witheither eg ( fcÃ35z),or ef ( fcÀ35z)ataconstantvalueof s? o.For eachtestedmixture,itisshownthatCRR15 decreaseswith increasing eg or ef value.However,atagiven eg value, CRR15 increaseswithincreasing fc uptothe fcth＝35z, (positiveeŠect).Athigher fc values( fcÀ35z),adecrease ofCRR15 withincreasing fc isindicatedatagiven ef value (negativeeŠect).Thismaybeattributedtothefactthatat aconstantvalueof eg,thevoidratioofthemixtures decreaseswithincreasing fc andthustheybecomestrong-

Fig.12.EŠectofˆnescontent, fc onliquefactionresistanceratio, CRR15,atconstantvoidratio, e,at s ? o,(a)100kPaand(b)300kPa

er(Fig.4(a)).Onthecontrary,ataconstantvalueof ef, thevoidratioofthemixturesincreaseswithincreasing fc abovethe fcth valueandtheybecomeweaker(Fig.4(b)).

ThepositiveeŠectofnon-plasticˆnesonCRR15 in termsof eg issimilartothatconsideredinthecorrelation betweenCSRand(N1)60,intheSPTbasedprocedure.As notedpreviously,agivenvalueof(N1)60,correspondsto greaterrelativedensitiesvalues,as fc increases,andthereforetogreatervaluesofCRR15.Similarly,foragiven valueof eg,anincreasein fc correspondstoanincreasein relativedensityandsoinincreasedvaluesofCRR15 (Fig. 4).

TheThresholdFinesContentinRelationtoParticle Packing

Asstatedabove,achangeoftheeŠectofˆnesonboth themonotonicandcyclicbehaviourofthetestedmixtureswasobservedat fcth.ThischangeoftheeŠectof ˆnesonthebehaviourofmixturescanbeexplainedby consideringthealterationoftheirstructurewithincreasing fc.As fc increasesfromzero,ˆnesenterthevoid spacesbetweenthesandgrains,whichareinfullcontact witheachother.Thisholdsuptothe fc atwhichthevoid spacesarecompletelyˆlledwithˆnes,andresultsinan increaseofthedensityofthemixture,withouthowever anyactiveparticipationofˆnesinthetransferofinterparticlecontactstresses.Withfurtherincreasing fc,the sandgrainsstarttoseparate,astheyarepushedslightly apartbytheˆnes,whichstartoccupyingalsolocations betweenthesandgrainsandparticipatinginthetransfer ofinterparticlecontactstresses.Thisisthereasonwhya decreaseof ecs andCRR15 withincreasing fc atagiven valueofvoidratio,isobserved.Thiscontinuesuntilthe

Fig.13.Variationofcyclicresistanceratio,CRR15,withintergranular voidratio, eg (fcÃ35 % )orinterˆne, ef (fcÀ35 % )voidratio

fcth,atwhichthevolumeofˆnesissogreatthatsand grainsaredisplacedfarapartfromeachotherandthey loosecontact.Thebehaviourofˆnesnowcontrolsthe behaviourofthemixture,whichbecomesunstable.As fc increasesbeyond fcth thesandgrainsbecometotallyisolatedandtheˆnesmatrixbecomesstrongerduetotheincreasingdevelopmentofcontactsbetweenˆnesparticles whichcannowtransferlargerinterparticlecontactstresses.Thispatternofbehaviourexplainswhyfor fcÀfcth an increaseof ecs andCRR15 withincreasing fc atagiven valueofvoidratio,isobserved.

Table3summarizespreviouslaboratoryinvestigations intotheeŠectof fc,eitheronthecriticalstate,oronthe cyclicbehaviourofsand-siltmixturesinwhich fcth wasdeterminedexperimentally.Accordingtothem, fcth varies from35z to50z.Sofar,therehavebeennostudieson howtoestimate fcth onatheoreticalbasis.Considering thepackingofsinglesizesphericalsandparticleswithdiameter, D,ataseparationdistance, s,arelationshipbe-

(4) Foudry0S00sand &crushed silicaˆnes Quartz—0.8000.6080.2502.1000.6270.0100.04032.5

Notation:

(1):ZlatoviácandIshihara(1995),(2a,2b):PolitoandMartin(2001),(3):XenakiandAthanasopoulos(2003),(4):Thevanayagametal.(2002), (5):NaeiniandBaziar(2004),(6):Yangetal.(2006aandb),(7):thisstudy WR:wellrounded,R:rounded,SR:subrounded,SA:subangular,A:angular * asreportedbyVerdugoandIshihara(1996)

** theexactthresholdˆnescontentwasnotdetermined

† asimpliedfromPolitoandMartin(2001)andMulilisetal.(1975)

‡ itwasassumed,asmostofthenaturalsandsinGreececontainfeldspar&micaandhavesubangularparticles

!! theexactthresholdˆnescontentwasnotdetermined.Itisestimatedtobewithintherangeof25z to40z

Fig.14.Relationshipbetweenintergranularvoidratio, eg,andtheratiooftheseparationdistancetotheparticlediameter, s/D,forsinglesizesphericalparticles

tween eg andtheratioofseparationdistancetoparticle diameter s/D canbederived,Fig.14: eg ＝b Ø s D ＋1 »3 1(6)

where b＝1.910,1.654and1.350fortheloosest(cubic), average(triangular)anddensest(tetrahedral)packingof sphericalgrains.Theseparationdistancerepresentsthe thicknessofthelayerofˆnes,whichcanbeinsertedbetweenthesphericalsandgrainsandindicatesthe probabilityofafailurezoneformingwithouttheinterferenceofsandgrains.

WhenmixingsphericalparticlesoftwodiŠerentsizes, theseparationdistanceofthelargesizeparticlesmustbe ntimesthediameterofthesmallparticles,thatis: s D ＝n･Ø d D » (7)

where D and d arethediametersofthelargeandsmall sizeparticlesrespectively.

Theformationofashearzoneatcriticalstatewouldre-

Fig.15.Relationshipbetweenˆnescontent, fc,andratioofdiameters ofsmalltolargesizeparticles, d/D forthetriangularpacking(The numbersintheˆgureindicatethenumbersofinvestigations,summarizedinTable3)

quirethedisplacementofatleasttwotothreediameters ofthesmallparticles(JeŠeriesandBeen,2006).CombiningEqs.(2),(6)and(7),thefollowingexpressionof fc as afunctionofdiameterratio, d/D,andvoidratiocanbe obtained:

Figure15showsthevariationof fc withthediameterratio, d/D,fortheaverage(triangular)packing, n＝1,2 and3andvoidratiovaluesof0.450,0.650and0.760,as wellasthe fcth values,determinedbytheinvestigations summarizedinTable3.InTable3alsothevaluesofthe parameter n,denotedas nth,forwhichthe fc valuesobtainedfromEq.(8),matchedthe fcth determinedbythe investigations,arepresented.Themeandiameterratio, d50/D50 andthesamevoidratioofthemixtures,asreportedinthestudies,wereusedtoestimate fcth.Forallthe sand-siltmixtures,summarizedinTable3,thevaluesof the nth parameterrangefrom1.15to2.30.Theabove resultsshowthatthe fcth dependsmainlyonthevoidratio ofthemixtures,onthemeandiameterratio, d50/D50,as wellasthemineralogyandtheparticleshapecharacteristics,asexpressedbysphericity,angularityandroughness. Feldsparandmicaminerals,encounteredinmanynatural siltysands(cases1–3and5–6inTable3, nth＝1.15–2.30), haveplatyshapeandtheirpresencefacilitatestheformationofashearzoneatsmallerseparationdistancesbetweenthesphericalparticles(smaller n values),ascomparedwithquartzsandswithequidimensionalparticles, (cases4and7inTable3, nth＝1.60–2.30).

Itshouldbenoted,however,thatthegradingofnaturalsiltysandsisin‰uencedbyseveralotherfactorsinadditiontothoseconsideredaboveforbinarypackingandartiˆcialsand-siltmixtures.Naturalsandshavepractically aninˆnitenumberofparticlediameterswithvarying shapecharacteristicsandmayalsocontainverysmall

Fig.16.Variationofcyclicresistanceratio,CRR15,withstate parameter, c ,forsand-siltmixturesatvariouslevelsofmeaneŠectivestress, s ? o,(a)50kPa,(b)100kPaand(c)300kPa

particleswhosebehaviourisdictatedbytheinteracting surfaceforces.

RelationshipbetweenStateParameterandLiquefactionResistanceRatio

ThevalueofCRR15 ofthetestedsand-siltmixturesis relatedtothestateparameter, c,inFig.16.Foreach testedmixture,adecreaseofCRR15 withincreasingvalue of c isobserved,duetoincreasingcontractivenessofthe soil.Thetestresultsalsoindicatethatfordilativebehaviour(cº0)andloweŠectivestresses(s? o ＝50kPa),the presenceofˆnesupto25z increasesCRR15 foragiven c.Thismaybeattributedtothefactthatfor fc upto25z besidessand,siltparticlesalsoparticipateintransferring orsustainingtheimposedstresses.However,thiscontributionofsiltparticlesdiminishesprogressivelywithincreasing s? o andat s? o ＝300kPanoapparenteŠectof fc on therelationbetweenCRR15 and c isobservedforthetestedmixtures.Therearealsoindicationsthatforcontractivebehaviour(cÀ0),at s? o valuesequalorgreaterthan 100kPa,thereisalowerboundvalueofCRR15 ofthe

Fig.17.Criticalstatelinesofnaturalsiltysandsintermsof(a)void ratio, e and(b)intergranularvoidratio, eg

Fig.18.Variationofcyclicresistanceratio,CRR15,with(a)voidratio, e and(b)intergranularvoidratio, eg,atameaneŠectivestress, s ? o ＝100kPa

Fig.19.Variationofcyclicresistanceratio,CRR15,withstate parameter, c ,atthemeaneŠectivestress, s ? o ＝100kPa

orderof0.09to0.12independentlyof fc

NaturalSiltySands

TheCSLsofthenaturalsiltysandsareshowninFig. 17bothintermsofvoidratioand eg.Thevaluesof parameter Mcs,determinedforDäuzce-1andRimniosoils aregiveninTable2.

ThevariationoftheirCRR15 withbothvoidratioand eg isshowninFig.18.ThecoincidenceofCRR15 versus voidratiorelationsbetweenDäuzce-1andthesand-silt mixturewith fc ＝15z andalsobetweenDäuzce-2and Rimniocanbeexplainedbythefactthatthesoilsinboth caseshadsimilarvaluesof emin and emax,Table1andFig. 3.CoincidenceintheCRR15 versus eg relationisobserved onlybetweenDäuzce-1andthesand-siltmixturewith fc ＝ 15z,sincebothsoilshavealsosimilar fc values.

TherelationshipbetweenCRR15 and c fornaturalsoils isshowninFig.19togetherwiththecorresponding resultsforthesand-siltmixtureswithsimilar fc values.A goodagreementisobservedbetweentheresultsfrom Däuzce-1andthecorrespondingfromthesand-siltmixturewith fc ＝15z,asthesesoilshavebothsimilar emin and emax and fc values.TheagreementoftheresultsbetweenDäuzce-2andthecorrespondingfromthesand-silt mixturewith fc ＝25z at cÀ0mayberegardedasaccidental,sincethesesoilshavediŠerentphysicalproperties(emin, emax andgradation).ForRimniosiltysand highervaluesofCRR15 atagivenvalueof c weredeterminedthanthecorrespondingforthesand-siltmixtures with fc ＝35z and40z.ThisdiŠerencemaybeattributed todiŠerencesingradation,mineralogy,andparticle shapecharacteristics.

CONCLUSIONS

Thefollowingconclusionscanbedrawnfromthetest resultsonthesand-siltmixtures:

Drainedandundrainedmonotonictestsproducea uniqueCSLforeachtestedmixture.However,thelocationoftheCSLisdiŠerentforeachmixture.Atlow stresses,theCSLsarenearlyparallelandhaveasmallinclination.WithincreasingmeaneŠectivestresslevel, however,theysteepenandconvergeatstressesabove 1000kPa.TheCSLsmovedownwardswithincreasing fc uptoathresholdvalue, fcth of35z,andthenupwards. When eg or ef isconsidered,theCSLsmoveupwardswith increasing fc upto35z andthendownwards.

Liquefactionresistanceratiodecreaseswithincreasing s? o atagiveneithervoidratio,or eg (or ef)voidratio.The eŠectof s? o onCRR15 diminisheswithincreasing fc.At constantvoidratio,CRR15 decreaseswithincreasing fc up toa fcth andincreasesthereafterwithfurtherincreasing fc Forthetestedsand-siltmixturesthe fcth is35z at s? o ＝100 kPaandisthesame,asthatdeterminedfromtheCSLs. Theresultsalsoindicatethatthe fcth reduceswithincreasing s? o.Atconstant eg,CRR15 increaseswithincreasing fc upto fcth (positiveeŠect).ThispositiveeŠectofˆneson CRR15 intermsof eg isinagreementwiththatsuggested bythesemi-empiricalproceduresincodesofpractice, whenevaluatingliquefactionpotentialduringearthquakes.

Thevalueof fcth isanimportantparameter,determiningthetransitionfromthesand-dominatedtothesiltdominatedbehaviourofthemixtures.Itisshownthatthe fcth ofsand-siltmixturesisrelatedtotheirparticlepacking.Equation(8)hasbeenproposedfortheestimationof fcth,asafunctionofthevoidratioofthemixture,the meandiameterratio, d50/D50,andtheseparationdistance ofthesandgrainswhichisequalto nth timesthediameter ofsiltparticleswith nth＝1.15–2.30.Thecomparisonof the fcth values,predictedbytheaboveequationandthe correspondingdeterminedexperimentallybythisand previousstudies,alsohighlightsthein‰uenceofother parameterson fcth suchasgradation,mineralogyandparticlesshapecharacteristicsofthemixtures,which,

however,arenotconsideredinthesemi-empiricalˆeld basedproceduresforthebehaviourofnaturalsiltysands.

TherelationbetweenCRR15 and c dependsonboth s? o and fc.ThereareindicationsthattheeŠectof fc diminisheswithincreasing s? o,i.e.,at s? o ＝300kPatheserelations coincideforallmixtures,independentlyof fc andalso thatforcontractivebehaviour(cÀ0)thereisalower boundvalueforCRR15 oftheorderof0.09to0.12forthe testedmixtures.Thebehaviourofnaturalsoilsissimilar tothatofartiˆcialmixtures,onlywhentheyhavesimilar physicalproperties,suchasgradation,mineralogyand particlecharacteristics.

Itisworthnoting,however,thatthein-siturelationshipbetweenliquefactionresistanceratioandstate parametermaybediŠerentfromthatdeterminedin laboratoryduetotheeŠectsofstresshistory,structure (fabricandbonding)andageing.

ACKNOWLEDGEMENTS

TheauthorswouldliketoexpresstheiracknowledgementstotheOnassisPublicBeneˆtFoundation,theState ScholarshipsFoundationofGreeceandtheGeneral SecretaryofResearchandTechnology,Greece,forthe ˆnancialsupportoftheˆrstauthor.Thetestsonthesoils fromDäuzcewereperformedwithintheMarmaraEarthquakeRehabilitationProject(MERP).

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'C'est entendu, Monseigneur,' replied Jacqueline softly. 'I'll come to your living-room as usual and bid you good-night after all our guests have departed.'

Far be it from me even to hint that, as she said this, Jacqueline threw more than a cursory glance on Gilles or on M. de Landas, for nothing could have looked more demure, more dignified and aloof than she did at this moment, when, having spoken, she bowed with stiff grace to the group of gentlemen before her. And even Maître Calviac would have felt that he was a mere bungler in the matter of bowings and scrapings if he could have seen these gallants responding to Madame's salute; the right leg outstretched, the left foot kept back, the hand almost touching the floor with a wide sweep of the arm, then brought back to the lips as for an imaginary kiss.

The next moment Jacqueline had turned and presently could be seen, still with that same stiff grace, receiving the adieux of her guardian's guests. She held her small head very erect and with one hand plied her fan with lazy nonchalance, whilst the other was perpetually being extended to those whose privilege it was to kiss it.

As for the group of young gallants—well! they had the immediate future to look forward to. True, that for the nonce they were forbidden to continue the quarrel for fear of incurring their host's displeasure; but it was only a matter of putting off the happy hour when one could be even with that insolent stranger. De Landas turned with a significant gesture and a knowing wink to his friends. After that, the small group dispersed and ostentatiously mingled with the rest of the departing crowd.

D'Inchy, before he left Gilles' side, managed to murmur fulsome apologies.

'I do assure Monseigneur,' he whispered earnestly in Gilles' ear, 'that these young jackanapes will not be tempted to repeat their impudence, and that I...'

'And that you, Messire,' broke in Gilles a little impatiently, 'are entirely innocent of any intention of offending me. That is, of course, understood. Believe me,' he added gaily, 'that the little incident was more than welcome

as far as I am concerned. Your lavish hospitality had made us all drowsy. M. de Landas' aggressive temper brought life and animation into the entertainment. I, for one, am grateful to him for the episode.'

Five minutes later he too had taken leave of his host. Jacqueline he did not see again. She was entirely surrounded by friends. Nevertheless, he left the banqueting hall in a state of exhilaration, and as he passed through the doors between the rows of Monseigneur's obsequious serving-men, they all remarked that Monsieur le Prince de Froidmont was humming a lively tune, the words of which appeared to be:

'Les plis de sa robe pourprée Et son teint au vostre pareil!'

CHAPTER XI

AND HOW IT ENDED

IWhen Gilles de Crohin found himself alone with Maître Jehan in the corridor which led straight to the main entrance hall, he paused for a moment, irresolute, wondering what he had best do. That there had been murder in the eyes of that gallant Marquis de Landas no one could doubt for a moment, and there lay a long stretch of dark streets and narrow lanes between the Archiepiscopal Palace and the safe shelter of 'Les Trois Rois.'

But you cannot imagine Messire Gilles de Crohin quaking even for a moment at the thought.

'Careful we must be,' he said in a whisper to his faithful alter ego; 'for my choleric friend will not, I imagine, be above lying in wait for us within the shelter of a convenient doorway, and I should ill serve the cause of the Queen of Navarre by getting spiked between the shoulders at such an early stage of the proceedings. But between that and showing that gallant Spaniard a clean pair of heels and foregoing the pleasure of threading his mask on my blade, there is a world of difference; eh, my good Jehan?'

'Above all things,' he added to himself, under his breath, so that even Jehan could not hear, 'I must find out whether a certain provoking glance, which flashed from out a pair of the most adorable blue eyes I have ever seen, were intended for me or not.'

And his thoughts flew riotously back to Jacqueline—Jacqueline, his dream, his tantalizing, exquisite dream—Jacqueline of the blue eyes and the captivating mole—Jacqueline of the roguish smile and the demure glance.

'I wonder, now!' he murmured softly. Had she perchance meant to give him a hint? Had she thrown him a warning glance? Gilles just then could have sworn that she had done both when she spoke of Monseigneur's livingroom, where she would sit prattling after the last of the guests had departed.

'Did she mean me to take refuge there against de Landas' murderous intentions?' he asked himself. But the supposition did not appear likely. Gilles was no coxcomb and had not had many dealings with women during the course of his chequered career; but he had an innate respect for them, and would not credit Jacqueline—proud, demure, stately Jacqueline—with the intention of offering a gratuitous rendezvous to an unknown gallant. Rather was her glance intended for de Landas—the assignation was for him: 'perhaps,' thought Messire Gilles with a vague stirring of hope in his heart, 'perhaps with a view to keeping that fiery lover of hers out of harm's way, till I myself was safely abed.'

Be that as it may, the most elementary dictates of prudence demanded that he should go back to his hostelry before his enemies had time to concoct any definite plans for his undoing. So, calling to Jehan to follow him, he found his way quickly out of the Palace.

It was raining heavily just then; the streets were dark and, after a while, quite deserted. Gilles and Jehan, keeping a sharp look-out around them, walked rapidly and kept to the middle of the streets. Fortunately for them both, they had had plenty of leisure in the last four days to wander through the intricate by-ways of the Flemish city. They knew the lay of the land pretty well by now, and at this moment when the thought of a possible guetapens was foremost in their minds, they were able to outwit any potential assassin who might be lurking on the direct route by going to the hostelry along devious ways usually unfrequented by strangers.

Thus it took them nearly half an hour to reach 'Les Trois Rois,' and Jehan, for one, was heartily congratulating himself that those murderous gentlemen had been comfortably thrown off the scent and were mayhap cooling their tempers somewhere in the cold and the wet, when, just as they entered the porch of the hostelry, a shadowy figure detached itself from out the gloom.

Gilles was already prepared with a quick, 'Qui va là?' but the figure proved inoffensive-looking enough: a woman, wrapped in a mantle and hood from head to foot. She had a small roll of paper in her hand, and this she held out timidly to Gilles.

'Monseigneur le Prince de Froidmont?' she inquired under her breath.

'Myself,' replied Gilles curtly. 'What is it?'

He took the paper and unrolled it. By the light of a small lanthorn which hung just inside the porch he saw that it was a letter—just a few lines— written in a small, pointed hand, and signed with the letter 'J.'

'Jacqueline!' he murmured, bewildered—so dazed that it took him some time before he was able to read. At last he deciphered the brief message.

'I do entreat you, Mesire,' it ran, 'to return to the palace within the hour. Nay! I do not entreat, I command! Go to the postern Gate: you will find it unlatched. Then cross the Courtyard till you come to a

door on the left of the main Perron—this will be unlocked. You will find yourself in one of the chief Corridors which give on the grand Staircase. Remain there concealed, and await further Orders.'

A strange enough missive, of a truth, and one, no doubt, which would have made an older and more prudent man pause ere he embarked on so dubious an adventure. But Gilles de Crohin was neither old nor prudent, and he was already up to his neck in a sea of adventure which had begun to submerge his reason. Even before he had folded up the paper again and slipped it into the inner pocket of his doublet, he had made up his mind that no power on earth, no wisdom or warning, would deter him from keeping the tryst. Did I think to remind you that he was no coxcomb? Well! he certainly was absolutely free from personal vanity, and it was not his selfconceit which was stimulated by the mysterious message; rather was it his passion for adventure, his love for the unforeseen, the unexpected, the exhilarating. The paper which he hid so tenderly inside his doublet had a delicious crisp sound about it, which seemed to promise something stimulating and exciting to come.

'Run up, Jehan,' he called to his man. 'I follow you. Let me get out of these damnable slashed and puffed rags—these velvet shoes and futile furbelows. Up, man! I follow in a trice! We have not done with adventure yet to-night.'

Then he turned, with a piece of silver in his hand ready to reward the bearer of such joyful tidings. But the messenger had disappeared into the night as quietly, as mysteriously as she had come.

II

Less than half an hour later, Gilles de Crohin once more found himself within the precincts of the Archiepiscopal Palace. He had been so quick in changing his clothes and so quick in covering the distance which separated

him from the trysting place, that he had no occasion to use the postern gate or the small door which had been indicated to him. The great entrance portals were still wide open when he arrived; some of the corridors still thronged with people—guests of Monseigneur and their servants on the point of departure—whilst others appeared entirely deserted. At one point, Gilles caught sight of M. de Landas taking elaborate leave of a group of ladies. He had his usual circle of friends around him, who—a moment or two later—followed him out of the Palace.

Gilles, with Jehan close behind him, kept well within the shadows, away from the throng. He had exchanged his elaborate and rich costume for a suit that was both plain and sombre; he had washed the perfume out of his hair and the cosmetics from off his hands. He felt unfettered in his movements now and in rare good humour. The only thing which he had borrowed from his former accoutrement was the magnificent Toledo rapier, which, after a moment's hesitation, he had buckled into his own sword-belt. It had been a parting gift from Madame la Reyne de Navarre and was a miracle of the steel-worker's art; supple as velvet, it would bend point to hilt like a gleaming arc and when it caught a ray of light upon its perfect edge, it flashed a thousand coloured rays like a streak of vivid lightning in a stormladen sky.

Jehan, on the other hand, was not altogether at his ease. Having less cause to feel exhilarated, he had a greater mistrust of the mysterious missive, had vainly tried to argue prudence where his master would only hearken to folly. But he had never succeeded in getting beyond a laboured: 'I th-th-th-think——' Upon which, he was peremptorily ordered to hold his tongue, even while Messire went merrily singing to face this questionable adventure.

At one point Gilles stopped in order to speak to a serving-man, asked him to tell him where was Monseigneur's private apartments, and when the man appeared to hesitate—for indeed he did not like to give this information to a stranger—Messire had seemingly lost his temper, and the man, trembling in his shoes, had stammered out the necessary directions. Monseigneur's private apartments and those of the household were in the right wing of the Palace. This was reached by mounting the grand staircase,

then continuing along the main corridor which connected the different portions of the vast building, until the wing containing the living-rooms was reached. No one, the man went on to explain, slept in this portion of the Palace, which held only the reception rooms and one of the chapels; but there were always night-watchmen about the place to see that no malefactors were about.

Whilst the man spoke, Jehan felt as if his eyes were searching him through and through. The worthy soul was liking this adventure less and less every moment.

Indeed, very soon after this all the corridors became deserted. Singly, in pairs, or in groups, all Monseigneur's guests and their servants had taken their departure. For awhile the varlets and wenches belonging to the household were busy clearing up the disorder and the débris attendant on so large a gathering and on so copious a supper, and one could hear them jabbering and laughing in the distant dining hall or in the offices down below. Then that noise, too, became stilled, and one felt that this portion of the vast Palace was indeed completely uninhabited.

Up at the Town Hall, the belfry of Martin et Martine had just chimed the midnight hour. Messire Gilles and his faithful Jehan found themselves in the vast hall at the foot of the grand staircase, and the main entrance with its monumental gates was then immediately behind them. A strange stillness reigned all around: the great Palace seemed here like a city of the dead.

Jehan vainly tried to protest once more. For what was Messire waiting, he wondered. Surely it was unwise and worse to linger here now, when every one had gone and all servants were abed. Presently, of course, the night-watchmen would be making their rounds. Jehan had a swift and exceedingly unpleasant foreboding that he and his master would be ignominiously turned out! and then God alone knew in what rows and quarrels they would be involved, or how hopelessly they would jeopardize their own position; not to speak of the Queen of Navarre's cherished scheme. Poor Jehan would have given five years of his life and half his savings for five minutes' glib speech with his master.

III

Even at this very moment, Jehan's vague terrors took on a definite form. Footsteps and voices raised in merry converse were heard, resounding from the distance, and the next instant two serving-men carrying torches came leisurely down the corridor in the direction of the hall. Immediately behind them walked Monseigneur the Governor, who had Madame Jacqueline on his arm. Jehan felt as if his heart had stopped its beating; his knees shook under him, whilst tiny drops of perspiration rose at the roots of his hair.

Ye gods! if they were discovered now! They would be under grave suspicion of evil intent ... burglary ... assassination.... There had been talk at the banquet of 'spy' and 'Spaniard.' Jehan's scanty hair stood up on end with horror.

Fortunately, Messire was equally aware of danger, gave a quick glance round, and perceived a door close beside him on the right. This part of the hall was, equally luckily, in shadow. There was also just sufficient time to reach the door, to open it, and to step incontinently behind it, closing it again noiselessly. Phew! it had been a narrow escape!

The footsteps and the voices came rapidly nearer; a minute or two later they passed within a foot of the door behind which Gilles and Jehan were crouching, hardly daring to breathe. The glint of the torches could be distinctly seen through a narrow chink between two panels, as well as the shimmer of Madame's white satin gown. There were but a few inches of wood and a foot of floor-space between Messire and shameful discovery, and Maître Jehan fell to wondering what particular form of torture would be applicable to a man who was found lurking at dead of night in the dark, and with obviously evil designs on the life or property of the governor of a Flemish province.

Thank Heaven and all the protecting angels, however, the footsteps passed by, and presently were heard ascending the main staircase, and whilst Maître Jehan was feeling as if his whole body would melt in a sea of cold perspiration, Madame Jacqueline's rippling laughter came only as an echo from a considerable and comparatively safe distance.

After awhile Gilles ventured to open the door very cautiously. A faint murmur of people stirring came from very far away; the shuffling footsteps of the torch-bearers died away in the distant corridors.

And once more all was still.

IV

Gilles gave vent to his feelings by a long-drawn-out 'Phew!' of obvious relief; but the next moment he said, quite coolly:

'Pardi, my good Jehan! but we did not want to be caught hiding in this place like a couple of malefactors, did we?' and made straightway to reopen the door. Jehan seized him by the arm and clung to him with all his might.

'Why shouldn't we st-st-st-stay here?' he urged almost glibly.

Gilles shrugged his shoulders. 'Why not, indeed?' he retorted. 'Something has got to happen presently,' he added carelessly. 'Somebody has got to come. If it is not Madame Jacqueline—and, honestly, my good Jehan, I have small hopes of that—If it is not Madame, then——'

He paused and frowned. For the first time a sharp suspicion had crossed his mind. Had he proved himself to be a vanity-ridden coxcomb after all? Should not the most elementary prudence have dictated....? Bah! whatever prudence had dictated, Gilles would not have listened. He was out for adventure! Whether gallant or dangerous he did not care! Once more he shrugged his broad shoulders and unconsciously his slender hand gripped the hilt of his splendid Spanish sword.

He threw a quick glance around him. Through the open door, the huge metal lamps which illumined the hall beyond threw a wide shaft of golden light into the room where he and Jehan had found such welcome refuge. It appeared to be something of a boudoir or library, for the shaft of light revealed rows of books, which lined the walls all round. There was a

window at the far end of the room, and that was closely curtained, and there was no other door save the one through which the two men had entered. The fire in the large open hearth had been allowed to die down. A massive desk stood not far from the window, and there were a few chairs about and a small, iron-bound coffer. Papers littered the desk and a finely wrought candelabra hung from the ceiling.

'The room,' said Gilles lightly, 'looks as if it had been closed for the night. There is no reason why we should not await here the future course of events.' He drew one of the chairs into a comfortable position and sat down, then added: 'I do not know, of course, how long we may have to cool our heels in this place, until the writer of the mysterious epistle chooses to explain his or her commands. I am beginning to think, as you do, my friend, that the missive should have been signed with an "L" rather than with a "J". What say you?'

'Aye! Aye!' muttered Jehan.

'Well, 'tis no matter! I'd as soon meet mine ebullient friend of the languorous eyes to-night as to-morrow, and inside this deserted Palace as out there in the rain. And a little sword-play would be very stimulating after the sentimental dalliance of the last few days.'

'H'm!' murmured Jehan equivocally.

'In the meanwhile, there is no reason why we should not have a rest. I confess to feeling rather sleepy. Just take a last look at the corridor,' concluded Gilles, as he stretched his long limbs out before him. 'And if you are satisfied that all is well, come and join me in an excursion to the land of Nod.'

Jehan went to the door as he was told and peered cautiously to right and left of him. Seeing nothing suspicious, he went as far as the great hall to listen if all was clear and still. It was whilst he was gone that something arrested Gilles' attention. Furtive footsteps this time—a number of them— moving stealthily along the corridor. With a quick gesture, he adjusted the mask over his face—instinct led him to do that first and foremost; then he jumped to his feet and went to the door, but had no time to step across the

threshold, for the next instant a compact group of moving figures emerged straight in front of him out of the gloom, intercepting him and barring the way.

'À moi, Jehan!' he called aloud.

But it was too late. From the hall beyond there came the sound of a vigorous scuffle. Jehan, caught unawares, was putting up a good fight seemingly against heavy odds; but he could no longer reach his master— whilst some half-dozen gentlemen, all wearing masks, were pushing their way into the room.

'We've run our fox to earth at last, Messeigneurs,' came with a mocking laugh from out this dense and aggressive-looking group. 'And without cooling our heels in the wet—what? I told you that this would be the better plan. His own egregious vanity hath led him straight into our trap and 'tis mighty fine sport that we'll have with this abominable spy, without fear of interruption.'

It was the voice of M. de Landas, unmistakable owing to the slight guttural pronunciation of the French language peculiar to his Spanish blood. Before Gilles could forestall him, he and his friends were all around him: six of them, fine young gallants—those who had supported de Landas in the quarrel after the banquet.

Gilles surveyed them all with a rapid glance, measured his own position, which of a truth was not an advantageous one. The light from the lamps in the hall fell, through the open doorway, full upon him, whilst his aggressors appeared only like a dense mass in the heart of the shadow. They were evidently intent on forcing him back into the room; their movements appeared like part of a concerted plan of action, to get him into a corner where they could more comfortably hold him at their mercy.

Gilles realized his position, the danger in which he stood and his best chance of defence, with the unerring rapidity of a born soldier.

'It must have taken a huge effort of intelligence, Messire,' he said ironically, 'to concoct this pretty plan. What was there in an open challenge

to frighten so many stalwart gallants?'

He gave ground, retreated into the room while he spoke. De Landas and his friends pressed in closely after him.

'I have yet to learn,' retorted the young Spaniard with a sneer, 'that you are worthy of crossing swords with one of us. You may draw, an' you have a mind; but you cannot escape the lesson which I and my friends have vowed to administer to you, and which, forsooth, you have so richly deserved.'

''Tis no use,' he added with an intaking of the breath like an angry snake, ''Tis no use calling for help. The night-watchmen are in my pay: my own men have settled with your servant, and no sound short of an earthquake could reach the distant wing of the Palace where Monseigneur and his household are abed.'

He drew his sword, and his friends immediately did likewise. Still they advanced, the solid phalanx of them, and so cunningly that Gilles was kept in the shaft of light whilst they remained under cover of the shadow.

'A murder!' said Gilles quietly.

'A lesson, first and foremost,' was de Landas' curt reply. 'After that, we shall see.'

'What shall we see, Messire?' riposted Gilles with a mocking laugh. 'A Spanish cavalier stooping to assassination——?'

'Who spoke of assassination?' queried one of the gallants.

'Why else are you here?' retorted Gilles, 'the six of you, whilst half a dozen or more of your varlets are overpowering my man outside, after ye have bribed or threatened the watchmen into silence? Methinks it looks uncommonly like projected murder.'

'Whatever it is,' broke in de Landas savagely, 'it will be a lesson which you are not like to forget.'