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A Simplified Method for Prediction of Embankment Settlement in Clays

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   JournalofRockMechanicsandGeotechnicalEngineering6(2014)61–66 JournalofRockMechanicsandGeotechnicalEngineering  Journal   of    Rock   Mechanics   and   GeotechnicalEngineering  journalhomepage:www.rockgeotech.org A   simplified   method   for   prediction   of    embankment   settlement   in   clays Chunlin   Li ∗ InstituteofCivilEngineering,TonglingUniversity,Tongling244000,China a   r   t   i   c   l   e   i   n   f   o  Articlehistory: Received22October2013Receivedinrevisedform23November2013Accepted9December2013 Keywords: SimplifiedmethodSettlementpredictionEmbankmentConsolidationtheoryClayeysoil a   b   s   t   r   a   c   t The   prediction   of    embankment   settlement   is   a   critically   important   issue   for   the   serviceability   of    subgradeprojects,   especially   the   post-construction   settlement.   Anumber   of    methods   have   been   proposed   to   predictembankment   settlement;   however,   all   of    these   methods   are   based   onaparameter,   i.e.   the   initial   timepoint.   The   difference   of    the   initial   time   point   determined   bydifferent   designers   can   definitely   induce   errorsinprediction   of    embankment   settlement.   This   paper   proposed   a   concept   named   “potential   settlement”   andasimplified   method   based   onthe   in   situ   data.   The   key   parameter   “ b ”   in   the   proposed   method   was   verifiedusing   theoretical   method   and   field   data.   Finally,   an   example   was   used   to   demonstrate   the   advantages   of the   proposed   method   by   comparing   with   other   methods   and   the   observation   data.©   2013   Institute   of    Rock   and   Soil   Mechanics,   Chinese   Academy   of    Sciences.   Production   and   hosting   byElsevier   B.V.All   rights   reserved. 1.Introduction Theone-dimensional(1D)consolidationequationsproposedbyTerzaghiarethecornerstoneofsoilmechanics.Settlementcalcu-latedusingTerzaghi’s1Dconsolidationtheory(Terzaghi,1925)has beenwidelyused,butitisnotalwayseffectiveduetotheuncer-taintyofcoefficient(Asaoka,1978).Manymethodsforsettlement predictionbasedonobservationdatahavealsobeenproposed,forexample,Asaokamethod,hyperbolicmethod(Tanetal.,1991),parabolamethod(XuandXu,2000),andinsitutests(Bergadoetal., 1991).TheAsaokamethodandhyperbolicmethodarewidelyusedduetotheirsimplicity(Andersonetal.,1994;Tan,1994,1995,1996).However,limitationsstillexistinbothmethodsthattheinitialtimepointisnecessarytobedeterminedfirst;andthedif-ferenceoftheinitialtimepointdeterminationcansignificantlyinfluencetheaccuracyofthesettlementprediction.Therefore,thispaperproposedasimplifiedmethodbasedontheTerzaghi’s1Dconsolidationequationirrelevanttotheinitialtimepointandcom-pareditwithothermethodstoverifyitseffectiveness. ∗ Tel.:+8613856250392. E-mailaddress: lichunlin111@126.comPeerreviewunderresponsibilityofInstituteofRockandSoilMechanics,ChineseAcademyofSciences. Production and hosting by Elsevier  ELSEVIER  1674-7755©2013InstituteofRockandSoilMechanics,ChineseAcademyof Sciences.ProductionandhostingbyElsevierB.V.Allrightsreserved.http://dx.doi.org/10.1016/j.jrmge.2013.12.002 2.TheoryofAsaoka’smethod In1978,Asaokaproposedanewsettlementpredictionmethod,thephilosophyofwhichisbasedon“observationalprocedure”.Thetheoryisderivedfrom1Dconsolidationequation.Hecom-binedMikasa’s(1965)equationwithTerzaghi’s(1925)equation, andobtainedtheverticalstrainas ε ( t,z  ) = T  + 12!   z  2 c  v ˙ T   + 14!   z  4 c  v ¨ T   +···+  zF  + 13!   z  3 c  v ˙ F   + 15!   z  5 c  v ¨ F   +··· (1)where ε ( t  ,  z  )istheverticalstrainof   z  attime t  ; T  and F  areunknownfunctionsoftime; c  v  isthecoefficientofconsolidation.Withthetwoboundaryconditions,i.e.drainagefrombothtopandbottomboundariesandupwarddrainage,thefollowingequa-tionscanbederived: S + 13!  H  2 c  v ˙ S  + 15!  H  4 c  v ¨ S  +···= H  2 (¯ ε + ε )(2a) S + 12!  H  2 c  v ˙ S  + 14!  H  4 c  v ¨ S  +···= H  ¯ ε (2b)where S  isthesettlement, H  isthethicknessofclaystratum,and ¯ ε istheverticalstrainatinitialtime.Thediscretetimecanbeintroducedas t   j  = t    ·  j (  j = 0 , 1 , 2 ,... )   (3)where  t    istheequaltimeinterval.  62 C.Li/JournalofRockMechanicsandGeotechnicalEngineering6(2014)61–66 Fig.1. HyperbolicplotsofTerzaghitheory(afterTan,1995). FromEqs.(2)and(3),thesettlementattime  j canbewrittenas S  j  = ˇ 0 + ˇ 1 S  j − 1  (4)where S   j  and S   j − 1  arethesettlementsattime  j and  j − 1; ˇ 0 , ˇ 1  areunknownparameters.Whenthestateisstable,thefinalsettlement S  f   canbeobtainedbythefollowingequation: S  j  = S  j − 1  = S f   (5)where S  f   isthefinalsettlement.FromEq.(5),werealizethatthefinalsettlementistheinter- sectionofrelationshiplinebetween S   j  and S   j − 1  with45 ◦ lineinthe S   j − S   j − 1  plot.If  S   j  and S   j − 1  aresubstitutedby S  f   inEq.(4),Eq.(4)canbesim- plifiedto S f   = ˇ 0 1 − ˇ 1 (6)Andthesettlement S  ( t  )attime t  canbecalculatedasfollows: S ( t  ) = ˇ 0 1 − ˇ 1 −   ˇ 0 1 − ˇ 1 − S 0  ˇ t  1  (7)where S  0  isthesettlementattheinitialtime.InEq.(7), S  0  shouldbedeterminedfirstlybeforesettlementpre-diction.Thedifferentvaluesof  S  0  canresultindifferentvaluesof  S  ( t  ),thustheprecisiondependsgreatlyontheselectionoftheini-tialtime.However,theselectionoftheinitialtimepointwillbedifferentbydifferentdesigners,whichcancausethedeviationof settlementcalculation. Fig.2. Hyperbolicplotsoffieldsettlement(Tan,1995). Fig.3. Thedeterminationofparameter b inthesectionK5+800. 3.Theoryofhyperbolicmethod ThehyperbolicmethodproposedbyTanetal.(1991)hasits originsintherectangularhyperbolicfittingmethodproposedbySridharanandRao(1981)andSridharanetal.(1987).According totheTerzaghi’stheoryofconsolidation(1925),thesettlement-timerelationshipcanbeexpressedusing U  and T  v .Therelationshipbetween T  v / U  and T  v  isshowninFig.1.FromFig.1,wecanseethat thelinearportionisbetween U  60 and U  90 ,whichcanberepresentedas T  v U   = ˛T  v + ˇ (8)where ˛ istheslopeand ˇ istheinterceptofthehyperbolicplot.Basedonthefielddata(Tan,1995),therelationshipbetween settlement ı andtime t  isshownas t  / ı vs. t  inFig.2.Theslopesof  s 60  and s 90  canbedeterminedby s 60  = s i ˛ 60 ˛ i (9) s 90  = s i ˛ 90 ˛ i (10)where s i and ˛ i aretheinitialslopeoflinearsegmentinFigs.1and2,respectively.Sothefinalsettlement ı f   canbecalculatedbythefollowingequation: ı f   = ˛ i s i = ı 60 0 . 6  = ı 90 0 . 9 (11)  C.Li/JournalofRockMechanicsandGeotechnicalEngineering6(2014)61–66 63 Fig.4. Thedeterminationofparameter b inthesectionK6+180. Thelimitationofthismethodisalsothedeterminationoftheinitialtimepoint,sincethismethodisbasedontheinitialslopeof thesettlement;thedifferenceoftheinitialtimepointcanresultinthedifferenceofsettlement.Theconstant-loadconditionwasassumedinthehyperbolicmethod,thusthesettlementbeforetheendofloadingcannotbepredicted.Duringtheloadingperiod,thesettlementratevarieswidely,andtheinitialslopeisdifficultto judge.Sunetal.(2002)proposedamethodofinitialpointdeter- minationbytheregressionanalysisofobservationdata,butitissomewhatcomplicatedtobeappliedinpractice. 4.Proposedmethod Asdiscussedabove,Asaoka’smethodandthehyperbolicmethodarenotveryadequateforthepredictionofembankmentsettlement,sincesomeparametersaredifficulttobedeterminedandtheinitialtimeisasubjectivechoice.Mostofsettlementsaretheresultsofconsolidation,soconsolidationtheoryiscommonlyusedtopredictthesettlement.Asmentionedpreviously,Terzaghi’s1Dconsolidationtheoryisnotalwayseffectiveduetotheuncer-taintyofcoefficientdetermination,butthetrendofthesettlementisconstant,thusanimprovedmethodforpredictingthetrendof thesettlementsisnecessary.Accordingtotheloadinglevels,thesettlementinducedbyloadscanbecalculatedusingTerzaghi’s1Dconsolidationequation.Thesettlementatagiventimecanbecomputedas s t   = s ∞  1 − 8  2 e bt    (12) Fig.5. Thedeterminationofparameter b inthesections(a)K6+300and(b)K6+260. where s ∞  isthefinalsettlement, s t   isthesettlementattime t  ,and b isanunknowncoefficient.Inordertosimplifythecalculation,wedefinethe“potentialsettlement”as s p  = s ∞ 8  2 e bt  = s ∞ − s t   (13)where s p  isthepotentialsettlement,whichwillhappeninthefuture,suggestingthedifferencebetweencurrentandfinalsettle-ments.InEq.(13),theparameters b and s ∞  ofTerzaghi’s1Dconsoli-dationequationshouldbedeterminedfirstly.Theparameter b canbeobtainedfrominsitudataandAsaoka’smethod,asdescribedbelow.FromEq.(13),itisclearthattherelationshipbetween ln[ s p  2 /(8 s ∞ )]and t  islinear,soparameter b canbedeterminedfromtheobservationdata.Onthescale,theparameter b representstheslopeofthestraightline.Itiswell-knownthattheparameter b representstheconditionsofdrainageinTerzaghi’s1Dconsolidationequation,whichcanbecalculatedusingtheconsolidationcoefficientanddrainagelengthundertwokindsofdrainageconditions,asshowninTable1.Itisimportanttodiscusstheconsistencyoftheparameter b acquiredbytheoreticalanalysisdataandtheobservationdatatoensuretheeffectivenessoftheproposedmethod.Fivesamp-lingpositionswerechoseninthesectionsofK5+800–K7+320of   64 C.Li/JournalofRockMechanicsandGeotechnicalEngineering6(2014)61–66 Fig.6. Thedeterminationofparameter b inthesectionsof(a)K7+106and(b)K7+110.  Table1 Thevalueof  b underdifferentdrainageconditions(afterZhangetal.,2005).NameExpressionParameter b Radialdrainage  8 c  h F  ( n ) d 2e Verticaldrainage   2 c  v 4 H  2 Vertical–radialdrainage  8 c  h F  ( n ) d 2e +   2 c  v 4 H  2 Note : c  h  and c  v  aretheradialandverticalconsolidationcoefficients,respectively; H  is   theverticaldrainagelength; n istheratioof  d e  to d w , d e  istheeffectivediameterof    well, d w  isthediameterofwell;and F  isanunknowncoefficientcorrelatedwith n . Anyang–XinxiangHighway.Theparameter b determinedbytheo-reticalmethodinTable1isshowedinTable2. Basedontheobservationdata,theparameter b canbeobtainedaccordingtoproposedmethod,andcalculationresultsoftheparameter b arepresentedinFigs.3–7.Bycomparingthevaluesof  b inFigs.3–7andTable2,theparam- eter b calculatedusingtheconsolidationtheoryisconsistentwith Fig.7. Thedeterminationofparameter b inthesectionK7+320. theproposedmethodundertwodrainageconditions(drainagefrombothtopandbottomboundariesandupwarddrainage),sotheparameter b canbederivedusingtheproposedmethod.Withtheparameter b obtainedbyEq.(13),thepotentialset- tlementcanbecalculatedfromthefinalsettlement s ∞  andtheobservationalsettlementattime t  .Thekeytopredictsettlementistoobtainthevalueof  s ∞ .AlthoughtheAsaoka’smethodhassomerestriction,thefinalsettlementpredictedbythismethodisveryprecise(Andersonetal.,1994), s ∞  may   becalculatedbyAsaoka’smethod.Basedonthetheoryandparameteranalysismentionedabove,theprocedureofthismethodissummarizedasfollows:(1)Thefinalsettlement s ∞  iscalculatedbyAsaoka’smethod.(2)Thepotentialsettlement s p  isobtainedusingtheobservationdataand s ∞ .(3)Thelinearrelationshipbetweenln[ s p  2 /(8 s ∞ )]and t  isplotted,andtheslopeofthislineis b .  Table2 Parameter b calculatedbyconsolidationtheory.SectionImprovedmethod H  (m)   c  v  (10 − 3 cm 2 /s) c  h  (10 − 3 cm 2 /s) d e  (m) nF  ( n ) b K5+800Stonepillar16.20.320.3181.5823.1640.5550.0042K6+180Compositepile14.00.27––––0.00283K6+260Compositepile16.20.324––––0.0026K7+110Compositepile15.40.258––––0.00225K7+320Stonepillar16.40.3470.3341.5823.1640.5550.0042
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