Numerical analysis of Paris-Erdogan law parameters of self-compacting concrete...

Thomas Thienpont

self-compacting concreteNumerical analysis of Paris-Erdogan law parameters of

Academic year 2015-2016Faculty of Engineering and ArchitectureChair: Prof. dr. ir. Luc TaerweDepartment of Structural Engineering

Master of Science in de industriële wetenschappen: bouwkundeMaster's dissertation submitted in order to obtain the academic degree of

of Materials of the Academy of Sciences of the Czech Republic, Brno, Tsjechië)Supervisors: Prof. dr. ir. Wouter De Corte, Prof. Stanislav Seitl (Institute of Physics

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Preface

ThisthesisishandedinasafulfilmentoftherequirementstoobtainthedegreeofMasterof Science inCivil Engineering Technology at the Faculty of Engineering andArchitecture,GhentUniversity.TheauthoracknowledgesthesupportofCzechSciencesfoundationprojectNo.15-07210SandBrnoUniversityofTechnologyProjectNo.FAST-S-16-3475.TheresearchwasconductedintheframeofIPMinfrasupportedthroughprojectNo.LM2015069ofMEYS.Thisthesishasbeen worked out under the “National Sustainability Programme I” project “AdMaS UP –AdvancedMaterials,StructuresandTechnologies”(No.LO1408)supportedbytheMinistryofEducation,YouthandSportsoftheCzechRepublic.TheauthoracknowledgesthesupportoftheERASMUS+program,grantedbytheEuropeanCommission.The author gratefully acknowledges the support of his main supervisor, Assoc. Prof.Stanislav Seitl, BrnoUniversity of Technology and Institute of Physics ofMaterials of theAcademy of Sciences of the Czech Republic. The author acknowledges the support of hissupervisorinGhent,Prof.dr. ir.WouterDeCorte,GhentUniversity.Moreover,theauthorwishestothanking.OndřejKrepl,whohelpedhimusingtheANSYSsoftware.A special acknowledgement to dr. Sara Korte, who provided the test data on which theresearchinthisthesisisbased.SpecialthankstoMarijkeDevos,MathiasWillemsandmyparents,whohelpedmewiththeEnglishtranslationofthistext.

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tocopypartsofthismasterdissertationforpersonaluse.�Inthecaseofanyotheruse,the

copyright terms have to be respected, in particularwith regard to the obligation to state

expresslythesourcewhenquotingresultsfromthismasterdissertation."01/06/2016

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Abstract

ThisworkcontainsanumericalanalysisofParis-Erdoganlawparametersofself-compactingconcrete,writtenbyThomasThienpont,undersupervisionofassoc.prof.StanislavSeitlandprof.dr. ir.WouterDeCorte.Thismaster'sdissertation ishanded inasafulfilmentoftherequirementstoobtainthedegreeofMasterofScienceinCivilEngineeringTechnologyattheFacultyofEngineeringandArchitecture,GhentUniversity.Duringthe last twodecades,concretetechnologyhasmadeanenormousadvanceduetothe introduction of self-compacting concrete. Its unique properties of high workabilitywithout loss of stability have allowed complex construction and rigorous constructionschedules.Lately,fatiguebehaviourofconcretehasbecomemoreimportantforthedesignof structures due to slimmer structures, which are more sensitive to fatigue loading.Concrete is a complex heterogeneous material. In order to predict its behaviour inapplications which involve millions of load cycles (e.g. bridges, beam cranes, offshoreconstructions),thoroughresearchisessential.Thisworkaimstoevaluateandcomparethefatiguecrackpropagationrateofself-compactingconcrete(SCC)tovibratedconcrete(VC)underdifferentstressratios.Therefore,testdataobtainedinthree-pointbendtests(3PBT)andwedgesplittingtests(WST) iscorrelatedtotheParis-Erdoganlaw.The3PBTandWSTdatawasobtainedfromstatictestsandcyclictestsonsingleedgenotchedspecimens,whilemeasuringthecrackmouthopeningdisplacement(CMOD)foreachloadcycle.Withtheuseoffiniteelementanalysis,datafromthesetestswasthencorrelatedwiththeappliedstressintensityrange,correspondingtotheParis-Erdoganlaw.Theresultinggraphs,depictingtherelationshipbetweenthecrackpropagationrateandthestress intensityrange,wereusedtoevaluatethefatiguecrackpropagationbehaviourofSCCcomparedtoVC.Keywords:Fatiguecrackpropagation,Paris-Erdoganlaw,self-compactingconcrete,stressratio.

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Extendedabstract

ThisworkcontainsanumericalanalysisofParis-Erdoganlawparametersofself-compactingconcrete,writtenbyThomasThienpont,undersupervisionofassoc.prof.StanislavSeitlandprof.dr. ir.WouterDeCorte.Thismaster'sdissertation ishanded inasafulfilmentoftherequirementstoobtainthedegreeofMasterofScienceinCivilEngineeringTechnologyattheFacultyofEngineeringandArchitecture,GhentUniversity.Duringthe last twodecades,concretetechnologyhasmadeanenormousadvanceduetothe introductionof self-compactingconcrete (notedasSCC). Itsuniquepropertiesofhighworkability without loss of stability have allowed complex construction and rigorousconstruction schedules. For example, SCCwas used in the construction of the anchorageblocksoftheAkashi-KaiyobridgeinJapan,asuspensionbridgewiththelargestmainspanintheworld. The use of this newmaterial shortened the anchorage construction period by20%,from2.5to2years(Ouchi,2001).Moreover,designersnowhavetheopportunitytocreatecomplicatedshapesandintricatestructuresmoreeasilythankstoconcrete'sabilitytoflow intodenselyreinforcedareas,constrictedspaces,orover longdistances(Szecsy&Mohler,2009).Itcanthereforebeusedinthinnerconcretesectionsandindirectaccessibleconcretesections(AmrutGroup,2011).Lately, fatiguebehaviourofconcretehasbecomeamore important issue inthedesignofconstructions due to the desire to build slimmer structures, which aremore sensitive tofatigue loading. Ingeneral, fatiguecanbedefinedasaprocessofprogressive,permanentinternal structural changes in a material subjected to repeated loading. Concrete is acomplex heterogeneous material in which these changes are mainly associated with theprogressive growth of internal micro cracks, which results in a significant increase ofirrecoverabledamage (Lee&Barr,2004). Inorder topredict thebehaviourofconcrete inapplications which involve millions of load cycles (e.g. bridges, beam cranes, offshoreconstructions),thoroughresearchisessential.Thisstudyaimsforabetterunderstandingofthe fatigue performance of SCC, in comparison to vibrated concrete, in order to attain acorrectandreliableapplicationofthematerial.Inthiswork,thefatiguecrackpropagationrateinSCCandvibratedconcrete(notedasVC)isevaluated and compared under different stress ratios. Therefore, test data obtained inthree-point bend tests (3PBT) and wedge splitting tests (WST) is correlated to the well-established Paris-Erdogan law. To investigate the fatigue crack propagation rate in SCC,threetypesofconcreteweretested:atraditionalVC,aSCCwithcomparablestrengthandaSCC with a comparable water/cement ratio. Herein, VC is used for comparison, since itsfracturepropertiesarewellknown.

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The3PBTandWSTdatawasobtainedfromtheresearchofS.Korte(Korte,2014).Herein,static tests and cyclic tests were performed on notched specimens, whilemeasuring thecrack mouth opening displacement (CMOD) for each load cycle, using a clip gauge. The3PBT’swereperformedonbeamshapedspecimens,witha standardvalueofS/W=3, inwhichS is the spanbetween the supports, andW thedepthof the specimen. The singleedgenotchedbeamgeomertyisausefulconfigurationforfracturetoughnesstestingsinceitcanbeeasilyshapedandtested.Itsgeometryisincludedinallinternationalstandardsforfracture toughness testing (Guinea,Pastor,Planas,&Elices,1998). Just like the3PBT, theWSTisaninterestingtestset-up;itcanbeperformedusinganordinaryelectromechanicaltestingmachinewithaconstantactuatordisplacementrate(Seitl,Veselý,&Řoutil,2011).TheWST’swereperformedonconcretecubespecimens,whichwerealsoprovidedwithanotch.UsingtheCMODmeasurementfromthe3PBTandWST,thefatiguecrackpropagationratesof the three concrete types can be compared using the Paris-Erdogan law parameters.However,neithercrackpropagationrateda/dNnorstress intensity range∆K,used in thislaw,canbedirectlymeasuredduringa3PBTorWST.TheymustthereforebeobtainedusingacombinationoffiniteelementanalysisinANSYSandseveralcalculationprocedures.First,amathematicalrelationshipbetweentherelativecracklengthα(=a/W)andtheCMODwascalculated trough inverse analysis. This was achieved by calculating the CMOD for fixedvalues of α. From the inverse analysis, the crack propagation da/dN rate can be easilyderived.Inanextstep,thestressintensityratio∆K,whichisusedtodescribethestressfieldintheregionofthecracktip,wascomputedusingabuilt inANSYScommand.Afterwards,thecrackpropagationrateda/dNisplottedagainstthestressintensityratio∆K.Fromtheselog-log plots, the Paris-Erdogan law parameters can easily be derived and used forcomparisonofthethreeconcretetypes.Asageneralconclusionfromthe3PBT’sandtheWST’s,itcanbestatedthatbothtestscanbeused toobtain valuable informationabout the fatigue crackpropagationpropertiesofboth vibrated concrete and self-compacting concrete. For small stress ratios, the 3PBTdeliversvaluableresultswhile theWST ismoreuseful for thesehigherstressratios.Fromthe data correlation it can be concluded that VC performs better under cyclic loadingsituations compared to SCC with comparable strength. In nearly all tests, VC performsbetter.FromthetestsonSCCwithacomparablew/c-ratiothedifferenceswithVCarelesspronounced. Overall, it can be stated that SCC is more brittle than VC. This conclusioncorresponds with the conclusions stated in the work of Sara Korte. It can therefore beconcluded that the Paris-Erdogan law is a valuable tool for the evaluation of the fatiguecrackpropagationincementbasedcomposites.Duetoitsalternatebehaviour,precautionisrequiredwhenusingSCCinsteadofVCincyclicloadingconditions.

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TableofContents

Preface..............................................................................................................................ii

Abstract...........................................................................................................................iv

Extendedabstract.............................................................................................................v

TableofContents............................................................................................................vii

ListofFigures....................................................................................................................x

ListofTables...................................................................................................................xii

ListofSymbolsandAbbreviations...................................................................................xiii

Introductionandproblemstatement.........................................................................11.1 Introduction...................................................................................................................11.2 Problemstatement........................................................................................................31.3 Goals..............................................................................................................................3

Literatureoverview...................................................................................................42.1 Self-compactingconcrete...............................................................................................4

2.1.1 Generaldefinition.............................................................................................................42.1.2 Materialproperties...........................................................................................................4

2.2 Durabilityandfatiguecracking.......................................................................................52.2.1 Durability..........................................................................................................................52.2.2 Fatigueinconcrete...........................................................................................................52.2.3 Crackgrowth.....................................................................................................................6

2.3 Paris-Erdoganlaw..........................................................................................................72.3.1 Earlycrackpropagationlaws............................................................................................72.3.2 Paris-Erdoganlaw.............................................................................................................72.3.3 Paris-Erdoganlawapplication..........................................................................................92.3.4 Stressintensityfactor.....................................................................................................10

Materialsandmethods............................................................................................113.1 Concretemixtures........................................................................................................11

3.1.1 Mixdesign.......................................................................................................................113.1.2 Mechanicalproperties....................................................................................................12

3.2 Three-pointbendtest..................................................................................................133.2.1 Testspecimen.................................................................................................................133.2.2 Statictests......................................................................................................................133.2.3 Cyclictests......................................................................................................................14

3.3 Wedgesplittingtest.....................................................................................................153.3.1 Testspecimen.................................................................................................................153.3.2 Statictests......................................................................................................................163.3.3 Cyclictests......................................................................................................................17

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Conclusionsofpreviousresearch.............................................................................184.1 Experimentalresearch.................................................................................................18

4.1.1 Wedgesplittingtest........................................................................................................184.1.2 Three-pointbendtest.....................................................................................................184.1.3 3PBTversusWST............................................................................................................18

4.2 Generalconclusions.....................................................................................................19

Dataevaluation3PBT...............................................................................................205.1 Introduction.................................................................................................................205.2 MeasuredCMODandcracklengthrelationship............................................................21

5.2.1 FEM-analysis:ANSYSmodel...........................................................................................215.2.2 Cracklengthcalculation..................................................................................................235.2.3 Fittingcurves..................................................................................................................255.2.4 Meshsize........................................................................................................................26

5.3 Crackpropagationrate.................................................................................................285.3.1 Crackpropagationcurve.................................................................................................285.3.2 Crackpropagationratethroughcurvefitting.................................................................295.3.3 Conclusions.....................................................................................................................30

5.4 Stressintensityfactor..................................................................................................315.4.1 ModellinginANSYS........................................................................................................315.4.2 Fittingcurves..................................................................................................................32

5.5 Paris-Erdoganlawparameters.....................................................................................335.5.1 Plottingdatapoints........................................................................................................335.5.2 Curvefitting....................................................................................................................355.5.3 Overviewofresults.........................................................................................................36

DataevaluationWST...............................................................................................376.1 Introduction.................................................................................................................376.2 MeasuredCMODandcracklengthrelationship............................................................38

6.2.1 FEM-analysis:ANSYSmodel...........................................................................................386.2.2 Modelcomparison..........................................................................................................396.2.3 Cracklengthcalculation..................................................................................................416.2.4 Meshcomparison...........................................................................................................43

6.3 Crackpropagationrate.................................................................................................456.3.1 Crackpropagationratethroughcurvefitting.................................................................456.3.2 3PBT:Discussionofresults.............................................................................................46

6.4 Stressintensityfactor..................................................................................................476.4.1 ModellinginANSYS........................................................................................................476.4.2 Fittingcurves..................................................................................................................47

6.5 Paris-Erdoganlawparameters.....................................................................................496.5.1 Plottingdatapoints........................................................................................................496.5.2 WST:Discussionofresults..............................................................................................49

Discussionofresults................................................................................................517.1 3PBT:stressratiocomparison......................................................................................51

7.1.1 Overviewtablecyclic3PBT.............................................................................................52

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7.1.2 Vibratedconcrete...........................................................................................................537.1.3 Self-compactingconcrete1............................................................................................547.1.4 Self-compactingconcrete2............................................................................................55

7.2 3PBT:concretetypecomparison..................................................................................567.2.1 Stressratio10-70%.........................................................................................................567.2.2 Stressratio10-75%.........................................................................................................57

7.3 WST:stressratiocomparison.......................................................................................587.3.1 OverviewtablecyclicWST..............................................................................................597.3.2 Vibratedconcrete...........................................................................................................607.3.3 Self-compactingconcrete1............................................................................................617.3.4 Self-compactingconcrete2............................................................................................62

7.4 WST:concretetypecomparison...................................................................................637.4.1 Stressratio10-70%.........................................................................................................637.4.2 Stressratio10-75%.........................................................................................................647.4.3 Stressratio10-80%.........................................................................................................657.4.4 Stressratio10-90%.........................................................................................................66

Conclusions..............................................................................................................678.1 Three-pointbendtest..................................................................................................678.2 Wedgesplittingtest.....................................................................................................688.3 3PBTandWSTcomparison...........................................................................................698.4 Generalconclusion.......................................................................................................70

References...............................................................................................................719.1.1 Authorreferences...........................................................................................................749.1.2 AuthorCV.......................................................................................................................75

Appendices..........................................................................................................7710.1 AppendixA:3PBTmodel-ANSYSAPDLcode...............................................................7710.2 AppendixB:WSTmodel-ANSYSAPDLcode................................................................80

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ListofFigures

Figure 2-1: Comparison between regular vibrated concrete (Regular mix) and self-compactingconcrete(SCC)mixture(NationalReadyMixedConcreteAssociation,2011)..........................................................................................................................................4

Figure2-2:Thestagesoffatiguelife(Schijve,1977)................................................................6Figure2-3:Crackmodes(Mach,Nelson,&Denny,2007)........................................................7Figure2-4:Paris-Erdoganlaw:alog-logrelationshipbetweentherateofcrackpropagation

rateda/dNandtheSIFrange∆".....................................................................................9Figure2-5:Paris-Erdoganlawregions(Charles,Crane,&Furness,1997)...............................9Figure3-1:Three-pointbendtestspecimen(dimensionsinmm).........................................13Figure3-2:Three-pointbendtestset-up(Korte,2014).........................................................14Figure3-3:Wedgesplittingtestspecimen(dimensionsinmm)............................................15Figure 3-4: Schematic of WST geometry: Specimen is placed on line support, two roller

bearingloadingdevicesaremounted,andwedgeappliessplittingload......................16Figure5-1:CMODdatafromthree-pointbendtests(Korte,2014).......................................20Figure5-2:Lefthalfofthe3PBT-modelinANSYS..................................................................21Figure5-3:Detailedviewof(a)thestressfieldnearthecracktipand(b)(c)themeshdensity

nearthecracktip...........................................................................................................22Figure5-4:Horizontaldeformationfrom3PBTcalculationsfor⍺=0.3,0.5and0.7.............23Figure5-5:CMODvsa/WgraphforVCunder70%load,(a)linearverticalaxis,(b)logaritmic

verticalaxisandexponentialfittingcurve......................................................................24Figure5-6:ExponentialfittingcurvecomparisonforVCunder70%load.............................24Figure5-7:(a)comparisonofstressratiosonVC,(b)comparisonofconcretetypesundera

90%load.........................................................................................................................26Figure5-8:3PBTmeshcomparison:relativedifferenceinCMODcomparedto1mmmesh.27Figure5-9:CrackpropagationinSCC2TPBsample,under70%cyclicload...........................28Figure5-10:LinearcurvefittingoncrackpropagationcurvefromaSCC2sampleunder10-

70%stressratio..............................................................................................................29Figure5-11:Increasedmeshdensitynearthecracktip,using‘KSCON’command...............31Figure5-12:3PBT:CalculatedSIFKforVCunder70%load...................................................32Figure5-13:Paris-Erdoganlawdataplotsobtainedfrom(a)dN=40(15datapoints),(b)dN

=20(32datapoints),(c)dN=10(65datapoints),(d)dN=5(134datapoints)..........34Figure5-14:(a)originaland(b)manipulatedParis-Erdogangraphdatapoints....................35Figure6-1:WST:CMODdatafrom10-80%stressratio(Korte,2014)...................................37Figure6-2:(a)WSTlaboratoryset-up(Korte,2014),(b)simplifiedFEAmodelinANSYS......38Figure6-3:ANSYSWSTmodels1,2and3.............................................................................39Figure6-4:RelativedifferenceinCMODresultscomparedmodel3.....................................40Figure6-5:HorizontaldeformationfromWSTcalculationsfor⍺=0.2,0.4,0.6and0.8.......41

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Figure6-6:CMODvs⍺graphforSCC2under70%load,(a)linearverticalaxis,(b)logaritmicverticalaxisandexponentialfittingcurve......................................................................42

Figure6-7: (a)comparisonoftested loadsonSCC1, (b)comparisonofthetestedconcretetypesundera90%load..................................................................................................43

Figure6-8:WST:model3meshcomparison..........................................................................44Figure 6-9: (a) crack propagation curve fromWST on VC under 10-90% stress ratio, (b)

detailedviewbetweenN=275andN=375.................................................................46Figure6-10:CalculatedSIFKforVCin70%load....................................................................47Figure6-11:Crackpropagationcurvesfor(a)VC10-70%and(b)VC10-75%.......................50Figure7-1:Paris-Erdoganlawfittingcurvesfor3PBTVC.......................................................53Figure7-2:Paris-Erdoganlawfittingcurvesfor3PBTSCC1...................................................54Figure7-3:Paris-Erdoganlawfittingcurvesfor3PBTSCC2...................................................55Figure7-4:Paris-Erdoganlawfittingcurvesfor3PBT10-70%stressratio............................56Figure7-5:Paris-Erdoganlawfittingcurvesfor3PBT10-75%stressratio............................57Figure7-6:Paris-ErdoganlawfittingcurvesforWSTVC........................................................60Figure7-7:Paris-ErdoganlawfittingcurvesforWSTSCC1....................................................61Figure7-8:Paris-ErdoganlawfittingcurvesforWSTSCC2....................................................62Figure7-9:Paris-ErdoganlawfittingcurvesforWST10-70%stressratio.............................63Figure7-10:Paris-ErdoganlawfittingcurvesforWST10-75%stressratio...........................64Figure7-11:Paris-ErdoganlawfittingcurvesforWST10-80%stressratio...........................65Figure7-12:Paris-ErdoganlawfittingcurvesforWST10-90%stressratio...........................66Figure8-1:WST:crackpatternsinVCspecimens(Korte,2014)............................................68

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ListofTables

Table3-1:Concretemixtures,inkg/m3..................................................................................11Table3-2:Mechanicalpropertiesoftheconcretemixtures..................................................12Table3-3:loadvaluesforthetestedconcretetypes.............................................................14Table3-4:loadvaluesforthetestedconcretetypes.............................................................17Table5-1:MathematicalrelationshipbetweenCMODandα,obtainedthroughexponential

curvefittinginMAPLE....................................................................................................25Table5-2:3PBTmeshcomparison:CMODvaluesfordifferentmeshsizes(inmm).............27Table5-3:3PBT:Crackpropagationratesobtainedthroughlinealcurvefitting...................30Table5-4:3PBT:MathematicalrelationshipbetweenKandα,obtainedthroughexponential

curvefittinginMAPLE....................................................................................................32Table5-5:Mathematicalrelationshipbetweenda/dNand∆K,obtainedthroughlinearcurve

fittingfrommanipulatedParis-Erdogangraphdatapoints...........................................36Table6-1:CMODvaluesandrelativedifferencecomparedtomodel3................................40Table6-2:MathematicalrelationshipbetweenCMODandα,obtainedthroughexponential

curvefittinginMAPLE....................................................................................................43Table6-3:WST:meshcomparisonformodel3.....................................................................44Table6-4:WST:crackpropagationrates...............................................................................45Table6-5:WST:MathematicalrelationshipbetweenKandα,obtainedthroughexponential

curvefittinginMAPLE....................................................................................................48Table6-6:Mathematicalrelationshipbetweenda/dNand∆K,obtainedthroughlinearcurve

fittingfrommanipulatedParis-Erdogangraphdatapoints...........................................49Table7-1:3PBT:Paris-Erdoganlawfittingcurves..................................................................52Table7-2:Paris-Erdoganlawfittingcurvesfor3PBTVC........................................................53Table7-3:Paris-Erdoganlawfittingcurvesfor3PBTSCC1....................................................54Table7-4:Paris-Erdoganlawfittingcurvesfor3PBTSCC2....................................................55Table7-5:Paris-Erdoganlawfittingcurvesfor3PBT10-70%stressratio..............................56Table7-6:Paris-Erdoganlawfittingcurvesfor3PBT10-75%stressratio..............................57Table7-7:WST:Paris-Erdoganlawfittingcurves...................................................................59Table7-8:Paris-ErdoganlawfittingcurvesforWSTVC.........................................................60Table7-9:Paris-ErdoganlawfittingcurvesforWSTSCC1.....................................................61Table7-10:Paris-ErdoganlawfittingcurvesforWSTSCC2...................................................62Table7-11:Paris-ErdoganlawfittingcurvesforWST10-70%stressratio............................63Table7-12:Paris-ErdoganlawfittingcurvesforWST10-75%stressratio............................64Table7-13:Paris-ErdoganlawfittingcurvesforWST10-80%stressratio............................65Table7-14:Paris-ErdoganlawfittingcurvesforWST10-90%stressratio............................66Table8-1:3PBTandWSTcomparison....................................................................................69

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ListofSymbolsandAbbreviations�

AbbreviationsVC VibratedconcreteSCC Self-compactingconcrete3PBT Three-pointbendtestWST WedgesplittingtestCMODCrackmouthopeningdisplacementSIF Stressintensityfactorw/c Water/cement

3PBTandWSTrelatedsymbolsW Heightofthetestspecimen[mm]S Distancebetweensupports[mm]θ Wedgeangle(wedgesplittingtest)[°]

Fatiguetestrelatedsymbols

a Cracklength[mm]⍺ Relativecracklength[-]da/dNCrackpropagationrate[mm]�Ntot Totalnumberofloadcyclesforacertainfatiguetest[-]��Ni Loadcyclei[-]�K Stressintensityfactor[Pa.√m]KI StressintensityfactorinmodeI[Pa.√m]KII StressintensityfactorinmodeII[Pa.√m]�KIII StressintensityfactorinmodeIII[Pa.√m]∆K Stressintensityrange[Pa.√m]∆Kx-y Stressintensityrangebetweenx%andy%[Pa.√m]�

�

MaterialpropertiesrelatedsymbolsE Young’smodulus[MPa]��ν Poissonratio[-]fcm meanvalueofthecompressivestrengthfromoncylinders[MPa]fc,cub,m meanvalueofthecompressivestrengthfromoncubes[MPa]

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Introductionandproblemstatement

1.1 Introduction

Sincemanyyears,concretehasbeenusedforconstructinglargecivilengineeringstructureslike bridges, tunnels andutility buildings.Due to years of extensive research, itsmaterialproperties and behaviour are well known, in both static and dynamic loading situations(Karihaloo,1995).Concretehas interestingstructuralpropertiesandarelatively lowprice.However, the amount of work required when casting concrete is a major disadvantage,sincevibrationofthefreshlycastconcrete isnecessarytoremovetheentrappedair fromthemixture. Therefore, a new type of concrete was developed in 1988: self-compactingconcrete (hereafternotedasSCC) (Szecsy&Mohler,2009).Thishighlyworkableconcretetype can flow throughdensely reinforcedand complex structural elementsunder itsownweight, without the need for vibration or othermechanical consolidation (Amrut Group,2011). Since the self-consolidating property eliminates the need for vibration, in somecases, the labour required for conventional concrete can be halved (Baumgartner, 2003).This inherent reduction in labournecessaryhasenormouspotential to reduce theoverallprojectcost.Duringthe last twodecades,concretetechnologyhasmadeanenormousadvanceduetotheintroductionofSCC(DeSchutter,Gibbs,Domone,&Bartos,2008).Thisnewkindofhefthigh performance concrete has had a tremendous impact on the concrete constructionindustry. The use of SCC in civil engineering is growing year after year, especially in theprecast concrete industry. Its market share is rapidly growing because of the economicopportunities and improvements of the quality of the concrete and the workingenvironment(Bonen&Shah,2005).Theuniquepropertiesofhighworkabilitywithoutlossofstabilityhaveallowedforcomplexconstructionandrigorousconstructionschedules.Forexample, SCC was used in the construction of the anchorage blocks of the Akashi-KaiyobridgeinJapan,asuspensionbridgewiththelargestmainspanintheworld.Theuseofthisnewmaterial shortened the anchorage construction period by 20%, from 2.5 to 2 years(Ouchi,2001).Moreover,designersnowhavetheopportunitytocreatecomplicatedshapesandintricatestructuresmore easily thanks to concrete's ability to flow into densely reinforced areas,constrictedspaces,oroverlongdistances(Szecsy&Mohler,2009).Itcanthereforebeusedinthinnerconcretesectionsandindirectaccessibleconcretesections(AmrutGroup,2011).However,themostimportantbenefitofSCCisthedurabilityincrease.Theuniformityofthemixturereducesthepermeabilityandenhancestheoveralldurabilityoftheconcrete.This,

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inturn,enhancesthelifespanoftheSCCbeyondthatofconventionalconcrete(Corinaldesi&Moriconi,2005).Overtheyears,thefatiguebehaviourofconcretehasbecomemoreimportantinthedesignof all kinds of structures, since designers aim to build slimmer and slenderer structures,which aremore sensitive to fatigue loading. In order to use SCC in applications ofwhichinvolve millions of load cycles (e.g. bridges, beam cranes, offshore constructions), it isnecessary that its fatigue properties are fully understood. This study aims for a betterunderstanding of the fatigue performance of SCC, in comparison to vibrated concrete, inorder to attain a correct and reliable application of the material. Numerical research isperformedbasedonexperimental researchperformedbySaraKorte,obtainedduringherdoctoral thesis: Experimental and Numerical Investigation of the Fracture Behaviour andFatigueResistanceofSelf-CompactingConcrete(Korte,2014).

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1.2 Problemstatement

Since its firstdevelopment in1988,various investigationshavebeencarriedout,and thistypeofconcretehasbeenusedinpracticalstructuresinJapan,mainlybylargeconstructioncompanies. In order to make SCC a standard concrete, investigations for establishing arationalmix-designmethodandself-compactabilitytestingmethodshavebeencarriedout(Okamura, 2003).Moreover, extensive research has been carried out on fresh, hardenedand transport properties, as well as on durability aspects of SCC, showing that thesubstantially different composition, opposed to vibrated concrete, sometimes causes analteredmechanicalbehaviour(DeSchutter&Audenaert,2008).Aftertwodecades,theshorttermpropertiesandbehaviourofthisnewconcretetypearethoroughlyresearched(Khayat&DeSchutter,2014),whileitslongtermfatiguebehaviourisnotyetfullyunderstood.Theavailabilityofdatafromlongtermfieldperformanceisscarcebecauseoflimiteduseandrelativelyrecentintroduction(DeSchutter&Audenaert,2008).However,adistinct fracturebehaviourcanbeexpected,since thestrengthof thecementpaste and the location and size of the aggregates play an important role in the crackpropagationphenomenon(deOliveiraeSousa&Bittencourt,2001).Inordertopredictthematerial behaviour of SCC in applications which involve millions of load cycles, moreresearchisrequired.It’simportanttohaveconsistentmodelsthatdescribeandpredictthelong term behaviour and the durability of this concrete type. Accuratemodels for crackpropagationcanbeavaluabletoolintryingtoevaluatetheremaininglifetimeofastructureinwhichcracksarealreadypresent.

1.3 Goals

Themaingoalofthisworkistoevaluatewhetherthree-pointbendtestsorwedgesplittingtestscanbeusefulindeterminingthecrackfatiguebehaviour,andmorespecificthecrackpropagation rate of cement based composites like SCC. Therefore, crack mouth openingdisplacement (noted as CMOD) measurements from fatigue tests on vibrated concrete(notedasVC)andtwotypesofSCCareanalysedandcorrelatedtotheParis-Erdogan law.TheCMODdatawillbemodifiedandusedtocompareandevaluatethefatiguebehaviourofVC and SCC under different stress ratios. Afterwards the three-point bend tests and thewedgesplittingtestsresultswillbecomparedandevaluated.

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Literatureoverview

2.1 Self-compactingconcrete

2.1.1 Generaldefinition

Self-compacting concrete (SCC), also known as self-consolidating concrete, is a highlyworkable high performance concrete type that can flow through densely reinforced andcomplexstructuralelementsunderitsownweight–withouttheneedforvibrationorothermechanical consolidation – and adequately fill all voids without segregation, excessivebleeding,airmigrationorotherseparationofmaterial(AmrutGroup,2011).Toobtainthesetypical pre-casting properties, smaller aggregates, filler material and superplasticizer areused, which change the overall behaviour of the material. Figure 2-1 depicts the maindifferencesbetweenatraditional,vibratedconcrete(VC)andaSCCmixdesign.

Figure2-1:Comparisonbetweenregularvibratedconcrete(Regularmix)andself-compactingconcrete(SCC)mixture(NationalReadyMixedConcreteAssociation,2011)

2.1.2 Materialproperties

In the three decades following its first introduction, SCC and its mechanical properties(compressivestrength,shrinkage,creep,durability…),aswellasitsbehaviourinfreshstateandduring transportationhavebeen intensively researched. In2001,PerssonpublishedacomparativestudyregardingthedifferenceinmechanicalpropertiesofSCCandofVC.Thestudy showed that the Young’s modulus, creep and shrinkage of SCC did not differsignificantlyfromthecorrespondingpropertiesofVC(Persson,2001).Therefore,onemightconcludethatthehardenedconcretecharacteristicsremainfundamentallythesameasVC;the raw materials are relatively similar and the hardened concrete behaviour is notappreciablydifferent.Therefore,inawiderangeofapplications,SCCcanbeaninteresting

5

substitute for regular concrete, thus saving time and money due to its economicaladvantages(Szecsy&Mohler,2009).However, insomecases,thehardenedconcretebehaviourwillactuallyexceedthatoftheconventionalconcrete.Moreover,in2008,DeSchutteretal.publishedabookinwhichitisshownthatthereareindeedsignificantdifferencesinthematerialpropertiesofVCandSCC.ThespecificmixdesignofSCC,whichguarantees its self-compactingability in freshstate,inevitably influences the performance of the hardened concrete. For instance, the highercontent of fine particles (for example by adding fillers) affects thewholemicrostructure,making the interfacial transition zone of SCC stronger and consequently increasing thecompressiveandtensilestrength,comparedtoVCwiththesamewater-cementratio(notedasw/c-ratio).Moreover,incomparingthepropertiesofSCCtoconventionalconcrete,somecautionisadvisable.TheRILEMTechnicalCommittee205-DSCcautionsaboutthedifficultyin making this comparison due to the lack of an accepted basis for comparison. Until astandard basis for comparison is developed, comparisons between SCC and conventionalconcretemustbeconsideredonacase-by-casebasis.EvencomparisonsbetweendifferentSCC typesmust be carefully considered given thewide range of proportions available tosatisfytheself-consolidatingqualifier(DeSchutter,Gibbs,Domone,&Bartos,2008)

2.2 Durabilityandfatiguecracking

2.2.1 Durability

One of the original drivers in the development of SCC was increased durability of thefinishedproduct.Thedurabilityofconcreteisgovernedbythepermeabilityanduniformityof consolidation. SCC typically has a lowwater-cement ratio and high paste content. Thehighquality transitionzonebetweenthepasteandaggregatealsodecreasespermeability(Szecsy&Mohler,2009).Moreover,theuseofsupplementarycementingmaterialslikeflyash or silica fume in SCC, as in conventional concrete, has also been shown to reducepermeability (Suksawang, Nassif, & Najm, 2005) (Nehdi, Pardhan, & Koshowski, 2004).When it comes to the design of long lifetime structure, durability and especiallypermeability play an important role.A lowerpermeability decreases the chanceofwaterpenetration, which might otherwise lead to corrosion of the reinforcement. Lowpermeability thus increases the lifetime of the structure. Furthermore, research on thedurabilityaspectsofSCCshowsthatthesubstantiallydifferentcomposition,opposedtoVC,sometimescausesanalteredmechanicalbehaviour(DeSchutter&Audenaert,2008).

2.2.2 Fatigueinconcrete

Fatiguemaybedefinedasaprocessofprogressive,permanentinternalstructuralchangesin a material subjected to repeated loading. In concrete, these changes are mainlyassociated with the progressive growth of internal micro cracks, which results in a

6

significant increase of irrecoverable damage (Lee & Barr, 2004). Each load cycle inducesmicroscopic cracks in the cement matrix, which gradually propagate during the loadingprocess until an extended crack pattern is formed, leading to a significant change of thematerial properties (Seitl, Keršner, Bílek, & Knésl, 2009). The more cycles a concretestructure has to sustain, the more the material irreversibly gets damaged and the lessstrength and stiffness. Fatigue is the most common cause of crack initiation and crackgrowthtocriticalsize(Wells,1963),atwhichsuddenfracturetakesplace.(Vikram&Kumar,2013)Unlikemetalsandfine-grainedceramics,themicrostructureofconcreteisdisorderedonallscales, fromthenanoscaletothemacroscaleofarepresentativevolumeelement,whosesizeistypically0.1m(assumingnormalsizeaggregates).Onallscales,thematerialisfullofflawsandpre-existingcracks(Bazant&Hubler,2014).Forthisreason,thestudyofconcretefatigueisrathercomplex.

2.2.3 Crackgrowth

In literature, fatigue lifehasbeen considered tobe composedof threephasesorperiods(Nirpesh & Raghuvir, 2013): crack nucleation, crack propagation and final failure. Someresearchers furtherdividethecrack initiationperiod into anucleationand amicro-crackgrowthperiod,asshowninFigure2-2.Thebeginningandtheendofeachperiodcan’tbeeasilydefined,exceptforthelastone.Finalfailureoccursatthelastcycleandusuallythispartoffailureissupposedtobequasi-staticratherthanfatigue(Schijve,1977).

Figure2-2:Thestagesoffatiguelife(Schijve,1977)

Fatiguecrackpropagationhasbeenshowntoresult incrackextensionineveryloadcycle.Details of the fracturemechanismonanatomic level areunknown,butona larger scaleseveral observations have been made. The most important one being the fact that inconcrete, cracks usually grow perpendicularly to the direction of themaximum principalstress(Alfaiate,Pires,&Martins,1997).ThisprocessiscalledmodeIcrackgrowth,themostcommoncrackgrowthmechanism(Figure2-3),whichistheresultofatensilestressnormaltotheplaneofthecrack.

Nucleation Microcrackgrowth

Macrocrackgrowth Finalfailure

Completefatiguelife(N)

Nucleationperiod Crackgrowthperiod

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Figure2-3:Crackmodes(Mach,Nelson,&Denny,2007).

Oncecrackgrowthhas initiated,therateof itsgrowthcanbedenotedasda/dN (slopeofthe crack growth curve). In otherwords, da/dN is the crack extension ∆a after one loadcycle. Research has shown that ∆a depends on a number of factors, themost importantones being: the cyclic stress on the crack tip area and the elasto-plastic response of thematerialinthecracktiparea.Theplasticdeformationsaroundthecracktipdependonthestrainhardeningbehaviourofthematerial(Schijve,1977).Inconcrete,however,theplasticflowisnonexistentsinceconcrete intension isnotcapableofplasticdeformation(Korte,2014). Herein lays a big difference between the fatigue fracture process in concrete andsteel.

2.3 Paris-Erdoganlaw

2.3.1 Earlycrackpropagationlaws

Inordertopredictthecrackpropagationrateda/dN inconstructionmaterialsundercyclicloadingconditions,crackpropagationlawsareused.Sincetheearlyfifties,researchershaveproposed numerous crack-propagation laws. Amongst them: (Head, 1953), (Frost &Dugdale, 1958), (McEvily& Illg, 1958) and (Paris,Gomez,&Anderson, A rational analytictheory of fatigue, 1961). In 1963, Paris and Erdogan published a well-established paper,comparingthemostimportantcrackpropagationlawsknownatthattime(Paris&Erdogan,ACriticalAnalysisofCrackPropagation Laws, 1963). Their research showed thatmanyofthe thus far known crack propagation laws were only valid for specific stress ratios orgeometries. In some cases, extrapolation of these laws leads to highly incorrect or evencontradictingresults.

2.3.2 Paris-Erdoganlaw

Awarenessarosethatcrackextensiontakesplaceduetostressconcentrationatthecracktip and due to failure ofmaterial during cyclic loading. An effortwasmade to relate thecrackgrowthwithstressintensityfactor(SIF)(seeparagraph2.3.4)atthecracktip(Nirpesh

8

& Raghuvir, 2013). Paris and Erdogan proposed a very simple, yet highly useablerelationshipbetweentherateofcrackpropagationda/dNandtheSIFrange∆K(Figure2-4),expressedas:

eq.2-1:#$#%= '(∆")+.

Herein, C and m depend on the material, the specimen geometry and the loadingconditions. They are therefore different for each material and must be obtainedexperimentally. Furthermore ∆K is the difference between the minimal and maximal SIF∆Kmaxand∆Kminforacertainstressratioappliedintheloadset-up.Forexample:ina10-70%cyclic load test, the minimal and maximal value K10 and K70 are calculated. Then, theaccordingSIFratioiscalculatedusingthefollowingequation:

eq.2-2: ∆K=∆K70-10=K70-K10.

TheParis-Erdoganlawcanalsobewritteninalog-logrelationship:

eq.2-3: ,-. /0 /1 = 2 ∗ ,-. ∆" + ,-.(').

Whenplottingthecrackpropagationrateda/dNandtheSIFrange∆Kinalog-loggraph,theparametersmandlog(C)ofeq.2-3canbeimmediatelyderivedfromalinearfittingcurve.Then,inafinalstep,theparametersCandmcanbeobtainedbytransformingequation2-2back into equation 2-1. This method was used in the correlation of the CMOD data inChapter5and6.

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Figure2-4:Paris-Erdoganlaw:alog-logrelationshipbetweentherateofcrackpropagationrateda/dNandtheSIFrange∆"

2.3.3 Paris-Erdoganlawapplication

Theformula inequation2-1 isapplicabletoawiderangeofmaterialsanddescribestheircrackpropagationbehaviourinarelativelycorrectwayoverawiderangeofstressratios.Ifthe crack propagation law for a certain material is known, it is possible to calculate byintegrationthenumberofcyclesrequiredforthecracktogrowfromonelengthtoanother.However, theapplicationsof this lawarenotendless. TheParis-Erdogan lawappliesonlyover themiddle range of crack growth rates. A plot of log(da/dN) against log(∆K) showsthree regimes of behaviour, only in the central region (zone B), the Paris-Erdogan lawappliesasshowninFigure2-5(Charles,Crane,&Furness,1997).Avalueof∆Kexistsbelowwhichthecrackisnotpropagating;itopensandcloseswithoutpropagating. This value is called the threshold for fatigue crack growth∆KTH, as shown inFigure2-5.Onecouldtrytodesignastructureinwhich∆Kremainssmallerthen∆KTH,soitcouldwithstandaninfinitenumberofloadcycles.Howevermoreresearchisneededsincecurrent understanding of threshold effects is rather limited (Charles, Crane, & Furness,1997).

Figure2-5:Paris-Erdoganlawregions(Charles,Crane,&Furness,1997)

On the other hand, designing a structure that can withstand an infinite number of loadcyclesisinmanycasesveryuneconomical.Adesignerwhoknowsthenumberofloadcyclesa certain structure has to withstand during its lifetime, can try to create a slimmerconstruction. Consequently, this structure cannot withstand an infinite number of loadcycles,but isstrongenoughnottocollapseduring itsproposed lifetime.Adesign likethiscansavealotofmaterialwhichalsoreducesitscost.Inordertocreatedesignslikethis,thefatiguebehaviourofthebuildingmaterialmustbewellunderstood.

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2.3.4 Stressintensityfactor

IntheParis-Erdoganlaw,thecrackpropagationrateisrelatedtoarangeofstressintensityfactors, for a certain geometry and load set-up. The stress intensity factor (SIF) K is ameasureforthesingularstresstermoccurringnearthetipofacrackanddefinedby:

eq.2-4: 567(8, :) =;<=>

∗ ?67(:),

whererand:arepolarcoordinateswiththeoriginatthecracktip.TheangularfunctionsformodeIaregivenbelow.ForcrackmodesIIandIII,similarfunctionscanbefound:

eq.2-5: ?@@ = A-B C<

1 − BFG C<BFG HC

<,

eq.2-6: ?II = A-B C<

1 + BFG C<BFG HC

<,

eq.2-7: ?@I = A-B C<BFG C

<BFG HC

<.

FormodeItheSIF,thenotationKIisused,whichcanbeexpressedas:

eq.2-8: "J = 5K L0MJ(0/O),

whereaisthecracklength,Wisthewidthofthecomponentandσ’ischaracteristicstressinthecomponent,forexampletheouterfibrestressinabendingbar.FI(a/W)isafunctionoftheratioofthecracklengthtothespecimenswidthaswellasofthetypeofloadapplied(Fett,1998).ThereareseveralwaystoobtainavaluefortheSIF.Overtheyears,numerousresearchershavepublishedempiricalformulasforalltypesofspecimenandcrackgeometries:(Gross&Srawley,1965),(Newman&Raju,1983),(AlLaham,Branch,&Ainsworth,1998),(Fett,1998)and many others. In recent years three-dimensional finite element analyses have beenperformedbyanumberofanalystsasanalternative toempirical formulas. In this thesis,finiteelementanalysis softwareANSYS (ANSYS Inc.,2011)wasused for thecalculationofstressintensityfactors.

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Materialsandmethods

Thischaptergivesabriefoverviewofthematerialsandmethodsusedintheexperimentalthree-pointbend tests (3PBT)andwedge splitting test (WST). First, themixdesignof thetested concrete types is given, alongwith the accordingmechanical properties. Then thegeometry of the concrete specimens and test set-ups used in the 3PBT and WST arediscussed.

3.1 Concretemixtures

3.1.1 Mixdesign

Inordertocompareself-compactingconcretetoclassicvibratedconcrete,severalmixtureswere prepared for performing a number of static and fatigue tests. The static and cyclic3PBTandWSTwereperformedonthreedifferenttypesofconcrete:

• Vibratedconcrete(VC)• Self-compactingconcretewithsimilarstrength(SCC1)• Self-compactingconcretewithsimilarw/c-ratio(SCC2)

Vibrated concretewasused for comparison since itsmaterialpropertiesandbehaviour iswell know fromextensive research. The composition of the concretemixtures is given inTable3-1(Korte,2014).Foreachbatch, consistingofaVCandaSCC type, identical aggregate typesand sizes, aswellasthesamecementtypewereusedinordertoavoidapossibleinfluenceontheresultsinthecomparativestudy(VCversusSCC).Theonlydifference istheadditionof limestonefiller and a larger amount of superplasticizer in case of SCC (Korte, 2014). Due to theadditionoflimestonefiller,therelativeamountofcourseaggregatesdecreases,asshowninFigure2-1andTable3-1.

Composition VC SCC1 SCC2

CEMIII/A42.5LA 360 293 360Water 161 161 161Sand 759 651 651

Crushedlimestone2/6.3 433 523 523Crushedlimestone6.3/14 610 321 321

Limestonefiller 0 377 377Superplasticizer(PCE) 2.9 9.0 9.5

Retardingagent 1.2 0.0 0.0

Table3-1:Concretemixtures,inkg/m3

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3.1.2 Mechanicalproperties

For each batch of 3PBT beams andWST cubes, at least six control specimenswere cast;cubeswithside150mmandcylinderswithradius150mmandheight300mm.Thesewereused toobtain the compressive strengthof thedifferent concrete types, aswell as someotherproperties(Table3-2).Inthistable,fcmrepresentsthemeanvalueofthecompressivestrength,obtained through testsoncylinders.On theotherhand, fc,cub,m isobtained fromtestsoncubes(Korte,Boel,DeCorte,&DeSchutter,2014).

VC SCC1 SCC2

fcm[MPa] 53.4±2.3 53.9±7.9 65.0±8.3fc,cub,m[MPa] 54.3±4.7 54.6±12.1 63.8±4.8

Young’smodulusEcm[MPa] 38400±300 38100±500 35300±4200Poissonratiov[-] 0.2 0.2 0.2

Table3-2:Mechanicalpropertiesoftheconcretemixtures

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3.2 Three-pointbendtest

Three-point bend tests (3PBT) are often used to determine the fracture properties ofstructuralmaterials, suchas cementbased composites (Seitl,Veselý,&Řoutil, 2011). Thesingleedgenotchedbeam is auseful configuration for fracture toughness testing since itcanbeeasilyshapedandtested. Itsgeometry is includedinall internationalstandardsforfracturetoughnesstesting(Guinea,Pastor,Planas,&Elices,1998).Thisparagraphdescribesthetestspecimensandtheset-upusedinthelaboratory.

3.2.1 Testspecimen

Thestaticandcyclic3PBT’swereperformedoncastconcretespecimenswithasingleedgenotchedbeam-shapedgeometryasshowninFigure3-1.ThebeamsdimensionsaremainlybasedonRILEMrecommendations(RILEMTC89-FMT:Fracturemechanicsofconcrete-Testmethods,1991).Multiplexmouldswereusedforcasting.Afterasealedcuringperiodof24hours,thespecimensweredemoulded.Approximately48hourspriortotesting,thenotchwasprovidedbywetdiamondsawingthehardenedspecimens(Korte,2014).

Figure3-1:Three-pointbendtestspecimen(dimensionsinmm)

3.2.2 Statictests

Prior to the cyclic fatigue tests, static 3PBT’s were performed, in order to obtain themaximal static load of the samples. The maximal static load plays an important role indetermining the parameters of applied load cycle in the cyclic tests. The values for theaveragestaticloadofVC,SCC1anSCC2aregiveninthefirstrowofTable3-3.Forthestatictests,a25kNcapacitycompressiontestdevicewasused.Withtheuseofarollerbearingdevice, the vertical force from the test device converted to a linear load, applied in themiddle of the beam’s top surface. Under the applied vertical load, the line supportedsamplestartscrackingatthenotchtipuntilfailureoccurs(Figure3-2).Duringthetests,thevertical displacementwas constantly increased by 0.2mm/min up to the peak load, andsubsequentlyloweredtoaspeedof0.02mm/min.AclipgaugewasplacedattheedgesofthecrackopeningtomeasuretheCMODduringtheentiretest.Theappliedforcewasalsoregisteredduringtheentiretest.

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Figure3-2:Three-pointbendtestset-up(Korte,2014)

3.2.3 Cyclictests

Thecyclic3PBT’swereperformedina15kNcapacitycompressiontestdevice.Theloadwasappliedinthemiddleofthebeam,accordingtoasinusoidalloadfunction.Aconstantspeedof 15 kN/s was applied, resulting in a frequency of approximately 0.33 Hz. The exertedverticalloadandtheCMODevolution(atthenotchend)werecontinuouslyregistered.Thelowerloadlimitofthisfunctionwaschosentobe10%oftheaverageultimateloadofthestatic3PBT’s. For theupper limitvariouspercentageswere selected:70%,75%,80%,and90%.(Korte,2014).ToevaluatethefourdifferentstressratiosR,themodelisloadedaccordingtothemaximalandminimalstressvaluesfromtheseratios(σmaxandσmin).Herein,Risdefinedastheratiobetweentheminimalandmaximalstress:

eq.3-1: P = QRSTQRUV

.

Usingthisequation,Rcanbecalculatedforthefourtestedstressratios:• 10-70% R=10/70=0.1429• 10-75% R=10/75=0.1333• 10-80% R=10/80=0.1250• 10-90% R=10/90=0.1111

Theappliedminimalandmaximalloads,usedinthecyclic3PBT’saregiveninTable3-3.

Load VC(kN) SCC1(kN) SCC2(kN)

Ultimatestaticload(100%) 6.1091 6.9398 6.120010% 0.6109 0.6940 0.612070% 4.2764 4.8579 4.284075% 4.5818 5.2048 4.590080% 4.8873 5.5518 4.896090% 5.4982 6.2458 5.5080

Table3-3:loadvaluesforthetestedconcretetypes

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3.3 Wedgesplittingtest

Thewedge splitting test (hereafter noted asWST)was first introduced by Linsbauer andTschegg (Linsbauer& Tschegg, 1986) and further developedbyBrühwiler andWittmann(Brühwiler&Wittmann,1990). It’san interesting test set-up; just like the3PBT, it canbeperformedusing an ordinary electromechanical testingmachinewith a constant actuatordisplacementrate(Seitl,Veselý,&Řoutil,2011).

3.3.1 Testspecimen

The static and cyclicWST’s were performed on cast concrete cubes with a geometry asdisplayed in Figure 3-3. The main notch was created by placing a wooden bar withrectangularsection(30mm×22mm×150mm)atthesideofastandardcubemould.Aswiththe3PBTspecimens,thenotchwasprovidedbywetdiamondsawingofthehardenedspecimens,approximately48hourspriortotesting.(Korte,2014).

Figure3-3:Wedgesplittingtestspecimen(dimensionsinmm)

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3.3.2 Statictests

Theprincipleof theWST is to apply a vertical loadontoa transferbeamwith twometalwedgeswithanangleof30°,seeFigure3-4(Pease,Skocek,Geiker,Weiss,&Stang,2007).Thesewedgesmove between two roller bearing loading devices,mounted on twometalcaps,whichrestontheedgesofthespecimensguidinggroove.Theverticalforceappliedbythe compression test device is thus transformed into two horizontal splitting forces Fsp,causingthecubestostartcrackingat thenotchtip (Korte,Boel,DeCorte,&DeSchutter,2014).Nexttothesplittingforce,theloadappliedbythewedgeontothetworollerbearingloadingdevicesalsoresults inaverticalforceFvontheWSTspecimen,asshowninFigure3-4.

Figure3-4:SchematicofWSTgeometry:Specimenisplacedonlinesupport,tworoller

bearingloadingdevicesaremounted,andwedgeappliessplittingload

Thetestswereperformedina25kNcapacitycompressiontestdevice,withthespecimensrestingontwolinearsupports,spaced75mmapart(seeFigure3-3).Aconstantincrementrate of the vertical displacement of 0.2 mm/min was applied until the peak load wasreached,andsubsequently,thisspeedwasloweredtoavalueof0.02mm/min.Duringthetests, the exerted load was recorded continuously with a computer-controlled dataacquisition systemand the CMODwas registered by a clip gauge, fixed at the notch end(Korte,2014).

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3.3.3 Cyclictests

ThecyclicWSTprocedurewasverysimilartotheoneusedforthecyclic3PBT’s.Thetestswerealsoperformedbyapplyingasinusoidalloadwitha15kNtestdevice.Equalminimumand maximum load levels were applied (10-70%, 10-75%, 10-80% and 10-90%) for thesinusoidal function with a frequency of about 0.33 Hz (corresponding with a speed of15kN/s). During the entire test duration, the load as well as the CMOD evolution wereregistered(Korte,2014).Likethecyclic3PBT,thesetestswereperformedforthedifferentmaximalloadvaluesusedintheexperimentaltestset-up.Morespecific:70%,75%80%and90%oftheultimatestaticload.Table3-4containsthedifferentloadvaluesforthetestedconcretetypes.

Load VC(kN) SCC1(kN) SCC2(kN)

Ultimatestaticload(100%) 10.450 10.369 9.99010% 1.045 1.037 0.99970% 7.315 7.258 6.99375% 7.837 7.777 7.49380% 8.360 8.295 7.99290% 9.405 9.332 8.991

Table3-4:loadvaluesforthetestedconcretetypes

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Conclusionsofpreviousresearch

ThischaptergivesaverybriefoverviewoftheconclusionsfromtheresearchofSaraKorte(Korte,2014).TheseincludethedifferencesbetweenVCandSCCinthetestdatafromboththe3PBTandWST,aswellasacomparisonbetweenbothtestmethods. Inchapter8,thehereafter stated conclusions will be compared to the results and conclusions from theresearchinthisthesis.

4.1 Experimentalresearch

4.1.1 Wedgesplittingtest

FromtheWST’s, itcanbeconcludedthatVC is thetoughestconcretetype.Thismightbeattributed to the largeramountof coarseaggregates inVCand thus themoreprominentinterlockingmechanismduringfracture.SCC1andSCC2lackalargeamountofbridgingandtougheningelements,resultinginamorebrittlebehaviour.Asaresult,thecracksurfacesofthefracturedspecimensshowthatlessaggregatesexperiencepull-outincaseofSCC1andSCC2(Korte,2014).

4.1.2 Three-pointbendtest

LiketheresultsoftheWST,adifferentcrackingbehaviourisnoticedbetweenVCandSCC.Inthe case of the 3PBT, SCC is consistently more brittle than VC. This smaller crackingresistanceofSCCmightbeattributedtoacombinationoftheabsenceoflargeamountsofcoarseaggregateparticles,whichinduceinterlockandallowstressestobetransferredalongtheFPZ,andaweakercementpasteduetothehigherw/c-ratio(Korte,2014).

4.1.3 3PBTversusWST

TheresearchofS.Korterevealssomesignificantdifferencesbetweenthe3PBTandtheWSTregardingthemutualrelationshipofthestudiedconcretetypes.Incaseofthe3PBT,amoreimportant influence of the cement paste strength is noticed, resulting in the smallestfractureparametersforSCC1(whichhasthehighestw/cratio).VCandSCC2showasimilarcrackingresistanceandtougher.FromtheresultsoftheWST’s,itappearsthattheaggregatesplayamorecrucialrole.Themaximumsize,aswellastheamountandconcentrationareimportantforthetougheningmechanismandthebridgingabilityduringfracture.Forexample,VC(whichhasthelargestamountofcoarseaggregates)istougherthantheSCCtypes(Korte,2014).

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4.2 Generalconclusions

Sincethefatiguefailuremechanismofconcreteisstronglyrelatedtocrackpropagationthecrackingresistanceofconcreteiscrucialforitsfatigueperformance.Basedontheoutcomeofthefracturemechanicstests,fromwhichitwasdemonstratedthatSCCismorebrittle,ashorterfatiguelifecouldbeexpectedforthisconcretetype,opposedtoVC.Whencomparingthedifferentstressratios,VCshowedthelargestfatigueresistanceincaseofthe lower loading levels (upto70%ofthestaticultimate load),whereasforthehigherloading intervals, SCC performed best. In addition, the vertical displacements during theexperimentswere larger forbothSCC1andSCC2subjectedtocyclic loadingwithahigherupper load limitandtheconcretestrain increasewasgenerally larger,withrespecttoVC.Basedonthesefindings,cautionisrecommendedwhenSCCisappliedinlow-cyclefatiguesituation,giventhefasterdeteriorationprocess,asopposedtoVC(Korte,2014).

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Dataevaluation3PBT

5.1 Introduction

Inorder tocompareandevaluate thecycle fatiguebehaviourofVCand the twotypesofSCCwiththeuseoftheParis-Erdoganlawparameters,theCMODmeasurementsfromthe3PBT’smustbemanipulated.Figure5-1, shows theCMODmeasurements for the10-70%stress ratio, performed by S. Korte. This chapter provides an overview of the calculationmethods used to correlate the CMODdata to the Paris-Erdogan law. First, the necessarycalculationstepsarepresented,whicharethendiscussedinfurtherdetail.AfterwardstheresultsfromthedataevaluationareusedtocomparethefatiguefracturepropertiesofVC,SCC1andSCC2underdifferentstressratios.

Figure5-1:CMODdatafromthree-pointbendtests(Korte,2014)

Asmentioned in the literatureoverview, theParis-Erdogan lawdescribes the relationshipbetween the crack propagation rate da/dN and the stress intensity factor interval ∆K.Neitherda/dNnor∆Kcanbedirectlymeasuredduringa3PBToraWST.TheycanonlybeobtainedusingacombinationofFiniteElementAnalysisandseveralcalculationprocedures.ToobtaintheParis-ErdoganlawparametersCandminequation2-1fromthetestdata,thefollowingcalculationstepsarerequired:

- Creatingadetailedfiniteelementanalysismodel- Calculating the relationship between the measured CMOD and the crack length

throughcurvefitting- Calculatingtherelationshipbetween∆Kandda/dNthroughcurvefitting- CalculatingtheParis-Erdoganlawparametersfromtheobtained(da/dN–∆K)-curves

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

1 10 100 1000

measuredCM

OD[m

m]

NumberofloadcyclesN [-]

VCAVCBSCC1ASCC1BSCC2ASCC2B

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5.2 MeasuredCMODandcracklengthrelationship

Duringthestaticanddynamic3PBT’s,onlytheappliedforceFvandtheCMODatthecrackopening were measured. Therefore, the crack length a has to be calculated using finiteelementanalysissoftware.ThisparagraphdescribesthenumericalmodelusedtocalculatetheCMODandcracklengthrelationship,aswellastheresultsobtainedfromthismodel.

5.2.1 FEM-analysis:ANSYSmodel

The finite element analysis software ANSYS was used to create and evaluate variousnumerical 3PBTmodels. Thesemodelswere built usingmacro’s in the ANSYS ParametricDesign Language (APDL) (ANSYS Inc., 2011). TheAPDL codeused in the3PBT calculationswasadded inAppendixA. InANSYS,onlyonehalf of the testpiece ismodelled, since itsshapeissymmetrical,asshowninFigure5-2.Whenacrackpropagatesthroughaconcretespecimen, a typical stress field around the crack tip canbeobserved, as shown in Figure5-3.c. The stresses near the crack tip increase quickly. In order to obtain an accuratenumericalmodel,intheregionclosetothecracktipadensermeshwasapplied.Infurtheroptimizationof themesh, thespecialANSYScommand ‘KSCON’wasused.Asaresult, thetwelvemeshelementsnearesttothecracktiparetriangularandareagoodwaytomodelthestressfieldnearthecracktip.

Figure5-2:Lefthalfofthe3PBT-modelinANSYS

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Figure5-3:Detailedviewof(a)thestressfieldnearthecracktip

and(b)(c)themeshdensitynearthecracktip

All numerical calculations were executed as a simplified 2D model, using 8-nodeisoparametricelements.Sincethedeformationsarerathersmall, linearstaticanalysiswasused.Fromliterature, it isknownthatthedifferences intheresultsforthedeflectionandthe stress fields for both 2D and a 3D TBPT-models are very small (Ostergaard, 2003),(Korte, 2014). Therefore, using a 2Dmodel is preferred, since it requires little computingpower compared to complex 3Dmodels. Thematerial input parameters for the Young’smodulus E and the Poisson ratio v are given in Table 3-2: Mechanical properties of theconcretemixtures.

23

5.2.2 Cracklengthcalculation

ToobtaintheCMODvscracklengthrelationshipfromthe3PBTgeometry,reverseanalysiswasused.Herein,theCMODwascalculatedfordifferentvaluesoftherelativecracklengtha/W, hereafterwrittenasα, thedimensionless ratiobetween the crack lengtha and theheightWofthe3PBTspecimen.Figure5-4depictstheANSYS3PBTmodelcalculationresultsfor three different α values (0.3, 0.5 and 0.7). The colours represent the amount ofdeformation inthehorizontaldirection.Theshapedeformation isexaggeratedbyafactor100.

Figure5-4:Horizontaldeformationfrom3PBTcalculationsfor⍺ =0.3,0.5and0.7

Fromthesecalculations,aCMODvsαgraphisobtained,asshowninFigure5-5.ThegraphontheleftdepictstheCMODforseveralαvaluesfromcalculationsonVC,under70%oftheaverage ultimate load. As shown in Figure 5-4, the CMOD increases as the relative cracklength α increases. As shown in Figure 5-5.b, an exponential relationship between theCMODandtherelativecracklengthcanbefound.Thisrelationship,intheformof'WXY =Z ∗ [\∗],isdepictedbytheexponentialfittingcurve.ThecurvehasahighvalueofR2,whichindicatesanaccuratecurvefitting.

⍺ = 0.3

⍺ = 0.5

⍺ = 0.7

24

Figure5-5:CMODvsa/WgraphforVCunder70%load,(a)linearverticalaxis,(b)logaritmicverticalaxisandexponentialfittingcurve

The values of the CMOD were calculated for values of α between 0.1 and 0.9, withincrementsof0.1.Sincethe3PBTspecimenaregivenanotchtoadepthof⍺ =0.33, theCMOD for this specific relative crack length is also calculated. In order to obtain moreaccuracy,itcanbeinterestingtouseonlythevaluesintherangebetween0.3and0.7,sinceit results in curve fittingwith a corresponding higher R2 value. Especially forα = 0.9 thefittingcurvedoesnotcorrelatewell,asshowninFigure5-5.b.Sincethetestspecimenhaveaninitialnotchdepthof⍺=0.33,smallervaluesfor⍺cannotbeused.Figure 5-6 shows that the exponential curve fitting applies best to the range between0.3and0.7;theR2valueinthisregionisveryhigh.Moreoverthemathematicalequationsofthefittingcurvesdiffersignificantly;theparameterAintherelationship'WXY = Z. [\∗] istwice as big in the 0.3 – 0.7 range compared to the 0.1 – 0.9 range. In all furthercalculations,the0.3–0.7rangewasused.

Figure5-6:ExponentialfittingcurvecomparisonforVCunder70%load

0

0.2

0.4

0.6

0.8

1

1.2

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

calculatedCMOD[m

m]

Relativecracklengthα [-]

y=0.0012e6.8974xR²=0.97226

0.001

0.01

0.1

1

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

calculatedCMOD[m

m]


y=0.0012e6.8974xR²=0.97226

0.001

0.01

0.1

1

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

calculatedCMOD[m

m]


y=0.0025e5.7326xR²=0.99634

0.001

0.01

0.1

1

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

calculatedCMOD[m

m]


25

5.2.3 Fittingcurves

UsingthemathematicalanalysissoftwareMAPLE(WaterlooMaple Inc,2008),exponentialfittingcurvesinthe0.3–0.7intervalwereobtainedforallloads(70%,75%,80%and90%)andallconcretetypes(VC,SCC1andSCC2).ThemathematicalfunctionsobtainedthroughthecurvefittingareshowninTable5-1andwillbeusedinfurthercalculations.

Concrete Load Equation

VC 70% 'WXY = 0.0017516 ∗ [cd(5.732587349 ∗ j) 75% 'WXY = 0.0018767 ∗ [cd(5.732586733 ∗ j) 80% 'WXY = 0.0020019 ∗ [cd(5.732579488 ∗ j) 90% 'WXY = 0.0022521 ∗ [cd(5.732576014 ∗ j)

SCC1 70% 'WXY = 0.0020071 ∗ [cd(5.732582331 ∗ j) 75% 'WXY = 0.0021440 ∗ [cd(5.725856669 ∗ j) 80% 'WXY = 0.0022869 ∗ [cd(5.725851013 ∗ j) 90% 'WXY = 0.0025728 ∗ [cd(5.725856079 ∗ j)


Table5-1:MathematicalrelationshipbetweenCMODandα,obtainedthroughexponentialcurvefittinginMAPLE.

Basedonthesecalculations,thefollowingconclusionscanbedrawn:- TheCMODvalue for fixed values ofα increaseswith increasing load, as shown in

Figure5-7.a;theCMODunder90%oftheaverageultimateloadisthegreatestforallvaluesofα.

- TheCMODvalueforVCisthesmallestofalltestedconcretetypeswhiletheCMODvalueforSCC1isthegreatest,forallvaluesofα,asshowninFigure5-7.b.

26

Figure5-7:(a)comparisonofstressratiosonVC,(b)comparisonofconcretetypesundera90%load

5.2.4 Meshsize

Inordertofindasuitablemeshsize,whichprovidesaccurateresultsanddoesn’trequireanimmense amount of calculation power, four models with different mesh sizes werecalculatedandcompared(4mm,2mm,1mmand0.5mm).ThesecondcolumnofTable5-2showstheresultingCMOD-valuesfora1mmmesh,fromcalculationsonSCC1undera70%load. The next columns show the calculated CMOD values for the othermesh sizes, andtheir relative difference compared to the CMOD from the 1 mmmesh calculations. Therelativedifferenceswerecalculatedusingeq.5-1andareplottedinFigure5-8.

eq.5-1: ∆'WXY% = 1 − lmnopRRRqrslmnmtusqvRqrsrSwq

∗ 100%

Based on the values in Table 5-2, the following can be concluded: the calculated CMODvalues increasewith a smallermesh size, for all valuesofα, especially forα ≥ 0.7. For⍺between0.2and0.5,thedifferencesaresmallest.Forhighervalues,forexampleforα=0.8,theCMODvaluefroma4mmmeshcalculation(0.29128)is3.5%smallerthenfroma1mmmesh(0.30138).Thesedifferencescannotbeneglected;asmallermeshsizeisnecessaryinordertoobtaingoodprecisionintheresults.Whencomparingameshsizeof1mmagainsta0.5mmmesh,thedifferencesarelessthan1%forallvaluesofα.Moreover,forvaluesofαbetween0.3and 0.7, the difference in result is even smaller than 0.40%. For the remaining finiteelementanalysis 3PBT calculations, amesh sizeof 1mmwas chosen.Calculationswithasmaller0.5mmmeshsizerequirea largeramountofcomputingpowerandareverytimeconsumingwhiletheydon’tdeliversignificantlybetterresults.

0.00

0.02

0.04

0.06

0.08

0.10

0.12

0.14

0.2 0.3 0.4 0.5 0.6 0.7 0.8

calculatedCMOD[m

m]


VC70% VC75% VC80% VC90%

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.2 0.3 0.4 0.5 0.6 0.7 0.8

calculatedCMOD[m

m]


VC90% SCC190% SCC290%

27

α 1mm 4mm 2mm 0.5mm

0.1 0.00306 0.00297 -2.9% 0.00303 -1.1% 0.00308 0.52%

0.2 0.00665 0.00653 -1.8% 0.00661 -0.7% 0.00667 0.31%

0.3 0.01165 0.01148 -1.6% 0.01159 -0.6% 0.01168 0.26%

0.33 0.01362 0.01341 -1.5% 0.01355 -0.5% 0.01365 0.26%

0.4 0.01949 0.01919 -1.6% 0.01938 -0.5% 0.01954 0.26%

0.5 0.03305 0.03252 -1.6% 0.03286 -0.6% 0.03314 0.27%

0.6 0.05939 0.05826 -2.0% 0.05900 -0.7% 0.05958 0.31%

0.7 0.11961 0.11689 -2.3% 0.11862 -0.8% 0.12008 0.39%

0.8 0.30138 0.29128 -3.5% 0.29778 -1.2% 0.30306 0.55%

0.9 1.33318 1.25451 -6.3% 1.30283 -2.3% 1.34579 0.94%

Table5-2:3PBTmeshcomparison:CMODvaluesfordifferentmeshsizes(inmm).

Figure5-8:3PBTmeshcomparison:relativedifferenceinCMODcomparedto1mmmesh.

-7.0%

-6.0%

-5.0%

-4.0%

-3.0%

-2.0%

-1.0%

0.0%

1.0%

2.0%

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

Relativ

eCM

ODdiffe

rence[%]


4mm

2mm

0.5mm

28

5.3 Crackpropagationrate

5.3.1 Crackpropagationcurve

The crack propagation rate for a dynamic test on SCC2 under a 10-70% stress ratio wasobtainedfromthemeasuredCMODvalues.Byapplyingtheinversemathematicalfunctionobtained in the inverseanalysis (Table5-1) to theCMODmeasurements, thecrack lengthforeachloadcyclecanbecalculated.Figure5-9showsthepropagationofthecracklengthforSCC2under70%oftheaverageultimateload.

Figure5-9:CrackpropagationinSCC2TPBsample,under70%cyclicload.

The detailed view in the figure above shows there is a certain amount of scatter in theresults.Themainreasonforthescatterisbecauseconcreteisanon-homogeneousmaterial.Intheory,acrackpropagateswitheveryloadcycle,andtherefore,aftereveryloadcyclethecrack length increases a finite amount. In reality however, when the crack reaches anobstacle,forexamplea largeaggregateparticle,thecrackpropagationcanbetemporarilystopped or slowed down. In that case, it might take a certain number of load cycles toovercomethisobstaclebeforethecrackcancontinuetopropagate.

40

50

60

70

80

0 100 200 300 400 500 600 700

Cracklengtha[m

m]


29

5.3.2 Crackpropagationratethroughcurvefitting

Afterafirstcomparisonitcanbeassumedthatthecrackpropagationrateisconstantduringacertainperiodof time.Thisassumption isaprofoundsimplification, since in reality, thecrackpropagationrateisnotconstant.However,asshowninFigure5-10,thecrackgrowthandthenumberofloadcycleshaveamoreorlesslinearcorrelationbetweenN=100andN=600.Therefore,aconstantvalueforthecrackpropagationrateda/dNcanbecalculatedfrom the analytical equation obtained from linear fitting curve in the aforementionedinterval.

Figure5-10:Linearcurvefittingoncrackpropagationcurve

fromaSCC2sampleunder10-70%stressratio

Ingeneral,themathematicalequationforalinearfittingcurvecanbewrittenas:

eq.5-2: x = W ∗ c + y.

In the case of the fitting curve in Figure 5-10,M is the crack propagation rate; da/dN =0.0138. Inotherwords:onaverage,thecrack lengthincreases0.0138mmaftereachloadcycle(betweenN=100andN=575).Since each cyclic test resulted in a different number of load cycles prior to failure, astandardizedmethodwasused to compare the crackpropagation rates.More specific, inorder to take into account only the region of the crack propagationwhere the growth isrelativelyconstant,thefirstandlast15%oftheloadcycleswereignoredinthelinearfittingcalculation.Forexample:ifaspecimenfailsafterNtot=1000cycles,linearfittingisappliedtothedatapointsintheregionbetweenN=150andN=850.

y=0.0138x+48.605R²=0.99093

45

50

55

60

65

70

0 100 200 300 400 500 600 700

Cracklenghta[m

m]


30

ThecalculatedcrackpropagationratesforalltestsaregiveninTable5-3.Themissingvaluesin this tablearedueto the fact thatsomeof the test samples failedafter the firstor thesecondloadcycle.

Concrete Stressratio N Crackpropagationrate R2

VC 10-70% 25 y=0.6228x+50.611 0.99672 51 y=0.2852x+52.719 0.99669 10-75% 30 y=0.4203x+50.012 0.98981 10-80% 1 - - 2 - - 10-90% 1 - -

SCC1 10-70% 18 y=0.6218x+43.627 0.95373 3 - - 10-75% 6 y=2.1383x+40.424 0.99447 10-80% 2 - - 3 - - 10-90% 1 - -

SCC2 10-70% 406 y=0.0198x+45.163 0.98725 678 y=0.0138x+48.605 0.99093 10-75% 65 y=0.2131x+41.622 0.97283 10-80% 3 - - 38 y=0.2917x+39.159 0.98673 10-90% 8 y=1.1197x+38.574 0.99168

Table5-3:3PBT:Crackpropagationratesobtainedthroughlinealcurvefitting

5.3.3 Conclusions

Basedon thesecalculations it is verydifficult todrawanyconclusions,due to the limitedamountofdata.Thereisabigdifferenceinthenumberofcyclestofailure,evenfortestset-upswiththesameconcretetypeandstressratio.Moreover,mostofthesamplesfromVCandSCC1whichweretestedunderthe10-80%and10-90%stressratiofailedafterveryfewloadcyclesanddon’tresultinusefuldata.However,somegeneraltrendscanbeobserved:

- Thecrackpropagationrateincreaseswithincreasingstressratio- SCC2canwithstandsignificantlymore loadcyclesunderacertain stress ratio than

VCorSCC1undersimilarloadconditions.Therefore,lowervaluesforda/dNcanbeobserved.

31

5.4 Stressintensityfactor

5.4.1 ModellinginANSYS

Thestressintensityfactor(SIF)KforthedifferentloadswascalculatedinANSYS.Inordertoobtainmoreaccuratevalues,theANSYScommands‘KSCON’and‘KCALC’wereused(ANSYSInc.,2011).First,whilebuildingthemesh,the‘KSCON’commandisusedtocreateadensemeshregionnearthecracktip,asshowninFigure5-11.Then,inthepost-processingphaseofthecalculation,the‘KCALC’commandwasusedtocalculatetheSIFatthecracktip.

Figure5-11:Increasedmeshdensitynearthecracktip,using‘KSCON’command

TheSIFKwascalculatedfordifferentvaluesofα,rangingfrom0.1to0.9,andfordifferentloads(10%,70%,75%,80%and90%).IncontrasttotheCMOD-calculations,the10%isalsoincluded,sincetheminimalstressintheappliedcyclicloadingfunctionsisnot0.Fromthesevalues,thestressintensityfactordifference∆Kwascalculatedforthedifferentstressratios(10-70%,10-75%,10-80%and10-90%)withtheuseofequation2-1.ThevaluesoftheSIFwerecalculatedforαvaryingbetween0.1and0.9,withincrementsof0.1.Sincethe3PBTspecimenaregivenanotchuptoadepthof⍺=0.33,thestressintensityfactor for this specific relative crack length is also calculated. Figure 5-12 shows thecalculatedK-valuesforVCunder70%oftheaverageultimateload.

32

Figure5-12:3PBT:CalculatedSIFKforVCunder70%load

5.4.2 Fittingcurves

As with the CMOD calculations, the linear fitting curves used in further calculations arebasedon thevalues in the rangebetween0.3and0.7 since this results inmoreaccuratecurvefittingandcorrespondinghigherR2values.UsingthemathematicalanalysissoftwareMAPLE,exponential fittingcurveswereobtainedforall stressratios (10-70%,10-75%,10-80%and10-90%)andallconcretetypes(VC,SCC1andSCC2).Themathematicalfunctionsobtained through the curve fitting are shown in Table 5-4 and will be used in furthercalculations.ForVCandSCC2,thesamefunctionsareusedsincethetestspecimenshavethesamegeometryandhavenearlyequalvaluesforthemaximalload(0.18%difference).

Concrete Stressratio Equation

VC 10-70% ∆" = 1.51911 ∗ 10z ∗ [cd(3.790759592 ∗ j) 10-75% ∆" = 1.64572 ∗ 10z ∗ [cd(3.790714665 ∗ j) 10-80% ∆" = 1.77230 ∗ 10z ∗ [cd(3.790745346 ∗ j) 10-90% ∆" = 2.02546 ∗ 10z ∗ [cd(3.790738594 ∗ j)

SCC1 10-70% ∆" = 1.72571 ∗ 10z ∗ [cd(3.790745012 ∗ j) 10-75% ∆" = 1.86950 ∗ 10z ∗ [cd(3.790769380 ∗ j) 10-80% ∆" = 2.01328 ∗ 10z ∗ [cd(3.790761954 ∗ j) 10-90% ∆" = 2.30095 ∗ 10z ∗ [cd(3.790740202 ∗ j)


Table5-4:3PBT:MathematicalrelationshipbetweenKandα,obtainedthroughexponentialcurvefittinginMAPLE

0

5

10

15

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Stressintensity

factorK[M

Pa.√m]

Relativecracklenght⍺ [-]

y=0.1671e4.2133xR²=0.94935

0.1

1

10

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Stressintensity

factorK[M

Pa.√m]


33

5.5 Paris-Erdoganlawparameters

5.5.1 Plottingdatapoints

Inafinalstep,theda/dNvs∆Kcurveisobtained.Cementbasedcomposites likeconcreteare complex heterogeneous mixtures with inherent flaws like pores, water inclusions,microscopiccracksduetoshrinkage,etc.(Jenq&Shah,1991).Consequently,thetestdatashow a certain amount of scatter. As a result, the crack propagation shows someirregularities, as shown in detailed view in Figure 5-9. To reduce the influenceof scatter,largeintervalswereusedintheprocessofobtainingdatapoints.Toachievethis,thetotalamountofloadcyclesNtotwasdividedintonequalsections.Dependingonthetotalamountofloadcyclesofeachtestspecimen,calculationswithn=10,20and/or50wereexecuted.Forexample,foraspecimenthatfailsafter500,threecalculationswereexecuted:

- n=10 sectionsof50loadcycles- n=20 sectionsof25loadcycles- n=50 sectionsof10loadcycles

Hereafter,thesuffix“x”willbeusedtoexpressacertainsection.Forexample:CMODxistheCMOD-value for section x. For each of these sections, the average CMOD-value wascalculated. From these values, the crack length ax and their corresponding SIFx werecalculated.Then,foreachconsecutivepairofcracklengthvalues,anaveragevalueof∆aavgwascalculated,withtheuseofeq.5-3.

eq.5-3: ∆0${| = $V}$V~p

�

Inafinalstep,these∆aavgvaluesareplottedagainsttheircorresponding∆Kvalues,resultinginplotsshowninFigure5-13.

34

Figure5-13:Paris-Erdoganlawdataplotsobtainedfrom

(a)dN=40(15datapoints),(b)dN=20(32datapoints),(c)dN=10(65datapoints),(d)dN=5(134datapoints)

0.001

0.01

0.1

1

800000 1300000 1800000

Crackprop

agationrateda/dN

[-]

Stressintensityfactor∆K [Pa.√m]

0.001

0.01

0.1

1

800000 1300000 1800000

Crackprop

agationrateda/dN

[-]


0.001

0.01

0.1

1

800000 1300000 1800000

Crackprop

agationrateda/dN

[-]


0.001

0.01

0.1

1

800000 1300000 1800000

Crackprop

agationrateda/dN

[-]


35

5.5.2 Curvefitting

Figure5-14.adepictsthelog-loggraphforSCC2undera70%load.HereinthedatapointsdonotresembleatypicalParis-ErdogancurveasshowninFigure2-4.The linearfittingcurvehasaverylowvalueR2=0.00074.However,whenplottingthesepointsonalog-loggraphaccording to the Paris-Erdogan law, a linear relationship is expected. This mathematicalrelationshipistypicallyoftheform:

eq.5-4: x = W ∗ c + y,

whereM>0.However,forpointswith∆K≤1MPa.√m,thereisadownwardtrendinsteadofa lineargrowth.Forthisreason, linearcurvefitting is inaccurate,whichresults ina lowvalueofR2.The aforementioned phenomenon exists due to the fact that in concrete, two stages ofcrack growth can be observed: deceleration and acceleration (Kruzic, Cannon, Ager, &Ritchie,2005).ConcretefatiguefractureintheaccelerationstagefollowstheParis-Erdoganlaw (Kolluru,O’Neil, Popovics,&Shah,2000), (Bazant&Xu,1991). Therefore, inorder toobtaina fitting curvewitha reasonableR2 value,only thedatapoints in theaccelerationstageareusedwhilethegreydatapointswere ignored.TheresultinggraphisdepictedinFigure 5-14.b, which shows a more accurate linear fitting curve, with a correspondingsteepercurveandhighervalueofR2=0.91912.

Figure5-14:(a)originaland(b)manipulatedParis-Erdogangraphdatapoints

y=0.1734x- 1.5271R²=0.00074

-2.5

-2

-1.5

-1

-0.5

-0.15 -0.05 0.05 0.15

Crackprop

agationratelog(da

/dN)[-]

Stressintensityfactorlog(∆K)[MPa.√m]

y=10.686x- 1.9034R²=0.91912

-2.5

-2

-1.5

-1

-0.5

-0.15 -0.1 -0.05 0 0.05 0.1 0.15

Crackprop

agationratelog(da

/dN)[-]


36

5.5.3 Overviewofresults

ThecalculatedfittingcurvesonthemanipulatedParis-ErdogangraphsforalltestsaregiveninTable5-5.Foreachdataset,multiplefittingcurveswereobtained.Thefittingcurvesinthetable below are the ones with the best value for R2. They were mainly obtained fromdatasetswith a small number of data points, since these greatly reduce the influence ofscatter,asstatedinparagraph5.5.1.Themissingvaluesinthetableareduetothefactthatsomeofthetestsamplesfailedafterthefirstorsecondloadcycle.Becauseofthemissingvaluesforthe10-80%and10-90%stressratios,comparisonbetweenVCandSCCbasedonthe given 3PBT data is not possible for these stress ratios. These results are furtherdiscussedinChapter7.

Concrete Stressratio N Paris’slawcurvefitting R2

VC 10-70% 25 y=2.3094x-0.5433 0.85936 51 y=5.4506x-1.7672 0.9073 10-75% 30 y=4.5047x-1.0792 0.96086 10-80% 1 - - 2 - - 10-90% 1 -

SCC1 10-70% 18 y=4.5187x-0.2661 0.84298 3 - - 10-75% 6 y=1.8504x-10.705 0.80821 10-80% 2 - - 3 - - 10-90% 1 -

SCC2 10-70% 406 y=9.7762x-1.8863 0.96082 678 y=4.7330x-2.1646 0.82632 10-75% 65 y=2.6385x-0.6689 0.482 10-80% 3 - - 38 y=5.5326x-0.5146 0.7098 10-90% 8 y=2.3999x+0.1180 0.78862

Table5-5:Mathematicalrelationshipbetweenda/dNand∆K,obtainedthroughlinearcurvefittingfrommanipulatedParis-Erdogangraphdatapoints.

37

DataevaluationWST

6.1 Introduction

InordertocompareandevaluatethefatiguebehaviourofVCandthetwotypesofSCCwiththeuseoftheParis-Erdoganlawparameters,theCMODmeasurementsfromtheWSTmustbemanipulated.Figure6-1shows theCMODmeasurements for theWSTundera10-80%stress ratio on VC, SCC1 and SCC2, performed by S. Korte. The methods used to obtainda/dN and∆K and theParis-Erdogan lawcorrelationareverysimilar to thoseused in thedataevaluationofthe3PBT’s.Inthefollowingsection,theusedtestmethodsareexplained,whilefocusingonthedifferenceswiththe3PBTdataevaluation.ToobtaintheParis-Erdoganparameters,similarcalculationstepswereused:

- Creatingadetailedfiniteelementanalysismodel- Calculating the relationship between the measured CMOD and the crack length

throughcurvefitting- Calculatingtherelationshipbetween∆Kandda/dNthroughcurvefitting- CalculatingtheParis-Erdoganlawparametersfromtheobtained(da/dN–∆K)-curves

Figure6-1:WST:CMODdatafrom10-80%stressratio(Korte,2014)

0

0.02

0.04

0.06

0.08

0.1

0.12

1 10 100 1000 10000 100000

CMOD[m

m]

NumberofcyclesN [-]

VC

SCC1

SCC2

38

6.2 MeasuredCMODandcracklengthrelationship

6.2.1 FEM-analysis:ANSYSmodel

Like the 3PBT, finite element analysis software ANSYS was used to create and evaluatevariousnumericalWSTmodels.Figure6-2.bdepictsaWSTmodel.Aswiththe3PBTmodel,onlyonehalfofthetestpieceismodelled,sincetheshapeissymmetrical.Instead of modelling the rather complex role bearing system with moving steel parts, asimplifiedmodelconsistingofasetoftwoequivalentforcesFvandFsp isused.Thevectorsumoftheseforcesrepresentstheforceappliedbythewedgeontherollerbearingloadingdevices. Figure 6-2 depicts the experimental test set-up (Löfgren, 2004) next to thesimplifiednumericalmodelwiththeequivalentforcesFvandFsp(redcolour).Eventhoughonlythesplittingforceisaccountableforthecrackpropagation,theverticalforceFvcannotbeneglected.In2011,Seitletal.publishedapapershowingthatneglectingtheinfluenceofthe vertical force can causemajor errors, especiallywhen theWST is performed using asinglesupport(Seitl,Veselý,&Řoutil,2011).Therefore,theverticalforceisalsoincludedinthe numericalmodel. All numerical calculationswere executed as a simplified 2Dmodel,using8-nodeisoparametricelements.Sincethedeformationsarerathersmall,linearstaticanalysiswasused.TheAPDLcodeusedintheWSTcalculationswasaddedinAppendixB.

Figure6-2:(a)WSTlaboratoryset-up(Korte,2014),(b)simplifiedFEAmodelinANSYS

39

6.2.2 Modelcomparison

InordertoobtainanumericalmodelwhichrepresentsWSTset-upinarealisticway,threedifferent numerical models were built and compared. In the first model (model1), theequivalentforcesFvandFspwereapplieddirectlyontotheconcretesurface,asalinearforce(Figure 6-3, left). As expected, this results in very high stress values in the region of theloadingpoint.Therefore,inthesecondmodel(model2),theforcesFvandFspwereappliedon2.5mmthicksteelplates(Figure6-3,centre).Themodelledsteelhasstandardvaluesforthe Young’smodulus Es=210GPa and Poisson ratio v = 0.3. The steel plates spread theforcemoreevenlyalongtheconcretesurface.Consequently,thestresslevelsintheregionof the forces are significantly lower compared to the first model. In the last model(model3), an L-shaped steel supportwasmodelled,with the samegeometry as the steelrollerbearingloadingdevicesusedintheexperimentalWSTset-up(Figure6-3,right).Thismodel,howevermorecomplexthantheprevious,isafarbetterrepresentationfortherealtestset-up.

Figure6-3:ANSYSWSTmodels1,2and3

Inordertoevaluateandcomparethesethreenumericalmodels,CMODcalculationswereperformedonVCunder70%oftheaverageultimateload.Inallcalculations,a1.5mmmeshsizewasused,inordertoeliminatetheinfluenceofthisparameter.TheCMODvaluesfromthese calculations are given in Table 6-1, aswell as their relativedifference compared tomodel3.Thesevalueswerecalculatedusingeq.6-1.TherelativedifferencesarealsoshowninFigure6-4.

eq.6-1: ∆'WXY% = 1 − lmnoRtÄqÅÇlmnoRtÄqÅV

∗ 100%

40

Fromtheseresults,itisclearthattherearesomedifferencesbetweenthethreemodels.Asshown inTable6-1, thedifferences inCMODaregreatest forsmallvaluesof⍺.For thesevalues of⍺, the crack tip region is closest to the regionwhere the forces Fv and Fsp areapplied. Therefore, changes in the way these forces are applied have a larger effect onCMOD calculations for these values of ⍺, rather than on larger values. Since model 3resemblestherealtestset-upthemost,andthedifferencesinCMODarerelativelysmallfor⍺≥0.33,thismodelwasusedinallfurthercalculations.

α model3 model1 model2

0.2 0.00591 0.00569 -3.9% 0.00569 -3.9%

0.3 0.01362 0.01335 -2.1% 0.01335 -2.0%

0.33 0.01641 0.01614 -1.7% 0.01613 -1.7%

0.4 0.02440 0.02410 -1.2% 0.02410 -1.2%

0.5 0.04146 0.04114 -0.8% 0.04112 -0.8%

0.6 0.07202 0.07168 -0.5% 0.07166 -0.5%

0.7 0.13866 0.13838 -0.2% 0.13824 -0.3%

0.8 0.33920 0.33880 -0.1% 0.33840 -0.2%

0.9 1.50440 1.50340 -0.1% 1.49620 -0.5%

Table6-1:CMODvaluesandrelativedifferencecomparedtomodel3.

Figure6-4:RelativedifferenceinCMODresultscomparedmodel3.

-4.0%

-3.5%

-3.0%

-2.5%

-2.0%

-1.5%

-1.0%

-0.5%

0.0%

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Relativ

eCM

ODdiffe

rence[%]

Relativecracklength⍺ [-]

model1model2

41

6.2.3 Cracklengthcalculation

DuringthestaticanddynamicWST’s,onlytheappliedverticalforceonthewedgeandtheCMODatthecrackopeningwereregistered.Thehorizontalsplittingforcedependsonthewedge angle θ and is calculated using eq. 6-2. In theWST from the test data thewedgeangleθ=30°,thereforeFspisobtainedineq.6-3.

eq.6-2: MÉÑ =ÖÜ

<∗á$�(à/<)

eq.6-3: MÉÑ =ÖÜ

<∗á$�(Hâ/<)= M{ ∗ 1.866025

The CMOD vs crack length relationship was obtained using reverse analysis. Figure 6-5depictsthenumericalWST-modelcalculationresultsforfourdifferentαvalues(0.2,0.4,0.6and0.8).Thecoloursrepresenttheamountofdeformationinthehorizontaldirection.Inallfigures,theshapedeformationisexaggeratedbyafactor100.

Figure6-5:HorizontaldeformationfromWSTcalculationsfor⍺ =0.2,0.4,0.6and0.8

42

As shown in Figure6-5, thewidthof the crack at the crackmouth increasesquickly asαincreases.Similargraphsas thoseobtained in the3PBTevaluationcanbeexpected.Fromthenumericalcalculations,CMODvsαgraphsareobtained, liketheonesshowninFigure6-6.Herein,thegraphontheleftdepictstheCMODforseveralαvalues,fromcalculationsonSCC2,undera70%oftheaverageultimatestaticload.SimilartotheTPBT,inthegraphontheright,anexponentialrelationshipbetweentheCMODandtherelativecracklength⍺can be found. This relationship, in the form of 'WXY = Z ∗ [\∗], is depicted by theexponentialfittingcurve,withanaccordinglyhighR2value.

Figure6-6:CMODvs⍺ graphforSCC2under70%load,(a)linearverticalaxis,(b)logaritmicverticalaxisandexponentialfittingcurve

ThevaluesoftheCMODwerecalculatedforrelativecracklengthsαbetween0.2and0.9,withincrementsof0.1.For⍺=0.1,nocalculationswerepossible,sincethecracktipcannotexistforthisvalue,duetothepresenceofthemain22mmdeepnotch,asshowninFigure3-3.Liketheevaluationofthe3PBTdata,onlythevaluesintherangebetween0.3and0.7are used, since this results in more accurate curve fitting, and a different fitting curveequation.UsingthemathematicalanalysissoftwareMAPLE,exponentialfittingcurvesinthe0.3 – 0.7 intervalwere obtained for all loads (70%, 75%, 80% and 90%) and all concretetypes(VC,SCC1andSCC2).ThemathematicalfunctionsobtainedthroughthecurvefittingareshowninTable6-2andwillbeusedinfurthercalculations.Basedonthesecalculations,thefollowingconclusionscanbedrawn:

- TheCMODvalue for fixed values ofα increaseswith increasing load, as shown inFigure6-7.a;theCMODfora90%loadisthegreatestforallvaluesofα.

- The CMOD value for VC is the smallest of all tested concrete types, although itsCMODvaluesarealmostidenticaltothoseofSCC1.TheCMODvalueforSCC2isthegreatest,forallvaluesofα,asshowninFigure6-7.b.

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

calculatedCMOD[m

m]


y=0.0013e7.2261xR²=0.97008

0.001

0.01

0.1

1

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

calculatedCMOD[m

m]


43

Concrete Load Equation

VC 70% 'WXY = 0.0024555 ∗ exp(5.726692532 ∗ j) 75% 'WXY = 0.0026332 ∗ [cd(5.725455765 ∗ j) 80% 'WXY = 0.0028088 ∗ [cd(5.725452545 ∗ j) 90% 'WXY = 0.0031599 ∗ [cd(5.725449393 ∗ j)



Table6-2:MathematicalrelationshipbetweenCMODandα,obtainedthroughexponentialcurvefittinginMAPLE.

Figure6-7:(a)comparisonoftestedloadsonSCC1,(b)comparisonofthetestedconcretetypesundera90%load

6.2.4 Meshcomparison

Comparable to the numerical model in the 3PBT, in order to find a suitable mesh size,modelswiththreedifferentmesheswerecalculatedandcompared.Thesethreemodelsallhavethegeometryof‘model3’,whichwaschosenafterthemodelcomparisoninparagraph6.2.2.Inthefirstmodel,a1.5mmmeshwasusedfortheentirespecimen;itwasusedinboththeconcreteandthesteelparts.Inthesecondandthirdmodel,fortheconcretepart,adensermeshwasusednearthecracktipregion.Inthesecondmodel,anoverall1.5mmmeshwasused,withadenser0.375mmmeshnearthecrack.Furthermore,intheregionofthecrack

0.00

0.02

0.04

0.06

0.08

0.10

0.12

0.14

0.16

0.18

0.20

0.2 0.3 0.4 0.5 0.6 0.7 0.8

calculatedCMOD[m

m]


SCC170% SCC175% SCC180% SCC190%

0.00

0.02

0.04

0.06

0.08

0.10

0.12

0.14

0.16

0.18

0.20

0.2 0.3 0.4 0.5 0.6 0.7 0.8

calculatedCMOD[m

m]


VC90% SCC190% SCC290%

44

tip, the ANSYS command ‘KSCON’ was used to further refine the mesh. The last modelconsistsofapproximately four timesmoreelements thanthesecondmodel; ingeneral,a0.75mmmeshisused,witha0.188mmmeshnearthecrack.Againthe‘KSCON’commandisusedtofurtherrefinethemeshnearthecracktip.ThesecondcolumnofTable6-3showstheresultingCMOD-valuesfora1.5mmmeshwithadenser0.375mmmeshnearthecrack,from calculations onVCunder a 70% load. Thenext columns show the calculatedCMODvaluesforthemeshes,andtherelativedifference.

α1.5mm,densenearcrack

1.5mmoverall0.75mm,dense

nearcrack

0.2 0.00603 0.00591 -2.06% 0.00603 -0.13%

0.3 0.01382 0.01362 -1.47% 0.01381 -0.06%

0.4 0.02472 0.02440 -1.31% 0.02472 0.00%

0.5 0.04200 0.04146 -1.30% 0.04200 0.00%

0.6 0.07302 0.07202 -1.39% 0.07300 -0.03%

0.7 0.14080 0.13866 -1.54% 0.14080 0.00%

0.8 0.34560 0.33920 -1.89% 0.34560 0.00%

0.9 1.54920 1.50440 -2.98% 1.54920 0.00%

Table6-3:WST:meshcomparisonformodel3

Figure6-8:WST:model3meshcomparison

-3.00%

-2.50%

-2.00%

-1.50%

-1.00%

-0.50%

0.00%0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Relar-tiv

eCMODdiffe

rence[%]

Relativecracklength[-]

1.5mmoverall 0.75mm,densenearcrack

45

6.3 Crackpropagationrate

6.3.1 Crackpropagationratethroughcurvefitting

In this paragraph the samemethods from the 3PBT data evaluation are used for a firstcomparisonofthecrackpropagationrate.Byapplyingthe inversemathematical functionsobtained in the inverseanalysis (Table6-2) to theCMODmeasurements, thecrack lengthforeachloadcyclecanbecalculated.ThisresultsinagraphliketheoneinFigure5-9.Oncethevalueofthecracklengthforeachloadcycleisknown,da/dNcaneasilybederived.Asshown inParagraph5.3.2, foracyclic testwithatotalnumberofcyclesNtot, thecrackgrowthandthenumberof loadcycleshaveamoreor less linearcorrelation intheregionbetween 0.15Ntot and 0.85Ntot. The crack propagation rate can be calculated from theanalyticalequationfromthelinearfittingcurvefortheaforementionedinterval.ThecalculatedcrackpropagationratesforalltestsaregiveninTable6-4.Themissingvaluesinthistableareduetothefactthatsomeofthetestdatashowedstrangecrackpropagationbehaviour,sometimesimplyinganegativecrackpropagation.ThesedataplotsarediscussedindetailinParagraph6.5.2.

Concrete Stressratio N Crackpropagationrate R2

VC 10-70% - - - 10-75% - - - 10-80% 13088 y=0.0012x+57.608 0.97207 10-90% 93* y=0.1820x+54.337 0.99523 271 y=0.0498x+61.459 0.99693

SCC1 10-70% 1168 y=0.0095x+62.308 0.76466 10-75% 383 y=0.0234x+59.034 0.98169 7061 y=0.0017x+58.621 0.93384 10-80% 2066 y=0.0017x+60.424 0.84132 10-90% 334 y=0.0186x+58.948 0.9815

SCC2 10-70% 3099 y=0.0071x+55.522 0.90696 10-75% 11002 y=0.0012x+62.500 0.95777 10-80% 545 y=0.0304x+51.596 0.87768 10-90% 739 y=0.0062x+60.028 0.41023 724 y=0.0161x+54.543 0.98386

Table6-4:WST:crackpropagationrates

46

6.3.2 3PBT:Discussionofresults

Basedonthesecalculations,itisdifficulttodrawconclusions,duetothelimitedamountofdataforthe10-70%and10-75%stressratioinVC.Thereisabigdifferenceinthenumberofcyclestofailure,evenfortestset-upswiththesameconcretetypeandstressratio.Despitetheselimitations,somegeneraltrendscanbeobserved:

- ForVC,thevaluesofR2arehigherthanforSCC1andSCC2.ThecrackgrowthinVCintheregionbetween0.15Ntotand0.85Ntotresemblesverywellaregimeofacrackpropagationrate.

- SCC2canwithstandsignificantlymoreloadcyclesunderacertainstressratio,thenVCorSCC1undersimilarloadconditions.Therefore,lowervaluesforda/dNcanbeobserved.

FromthetestdataofthefirstcyclictestonVCforthe10-90%stressratio,aratherstrangecrackpropagationcurvewasobtained,as shown inFigure6-9.a.During the first275 loadcycles,thecrackbarelygrows(exceptforthefirst25cycles).Then,after275loadcycles,thecrackstarts topropagateratherquicklyandthetestspecimenbreaksafter the368th loadcycle.The reason for thisbehaviour isuncertain. It couldbecausedby thepresenceofalargeaggregatepiece in the regionof thecrackprocessingzone,whichcould temporarilyslowdownthecrackpropagation.Inordertoobtainthefittingcurveforthisdataset,onlythedatapointsafterthe275thcyclewereused,sincethesepointsshowindicateanormalcrackpropagationbehaviourpattern,asshowninFigure6-9.b.Thenumberof loadcycleswas thereforeadjusted to368 -275=93,andwasmarkedwithanasterisk symbol (*) inTable6-4.

Figure6-9:(a)crackpropagationcurvefromWSTonVCunder10-90%stressratio,(b)detailedviewbetweenN=275andN=375

40

50

60

70

80

90

0 100 200 300 400

Cracklengtha[m

m]


50

60

70

80

90

275 300 325 350 375

Cracklengtha[m

m]


47

6.4 Stressintensityfactor

6.4.1 ModellinginANSYS

Thestressintensityfactor(SIF)KforthedifferentloadswascalculatedinANSYS.Similartothe data analysis for the TPBT, the ANSYS commands ‘KSCON’ and ‘KCALC’ were used(ANSYSInc.,2011).ThevaluesoftheSIFKwerecalculatedforvaluesofαbetween0.2and0.9,withincrementsof0.1.Figure5-12depictsthecalculatedK-valuesforVCunder70%oftheaverageultimateload.

Figure6-10:CalculatedSIFKforVCin70%load

6.4.2 Fittingcurves

As with the CMOD calculations, the linear fitting curves used in further calculations arebasedonthevalues in therangebetween0.3and0.7,sincetheyresult inmoreaccurateexponentialcurvefitting,andcorrespondinghigherR2values.Figure6-10depictsthefittingcurvesforboththerangeof⍺between0.2and0.9andtherangeof⍺between0.3and0.7.Theexponentialfittingcurvefortherangeof⍺between0.2and0.9 issteeperandis lessaccurateduetoalowervalueofR2.UsingthemathematicalanalysissoftwareMAPLE,exponentialfittingcurveswereobtainedforallstressratios(10-70%,10-75%,10-80%and10-90%)andallconcretetypes(VC,SCC1and SCC2). The mathematical functions obtained through the curve fitting are shown inTable6-6andwillbeusedinfurthercalculations.

0

5

10

15

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Stressintensity

factorK[M

Pa.√m]


K =0.1776e4.1543 ⍺R²=0.90988

y=0.2681e3.0088xR²=0.97694

0.1

1

10

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Stressintensity

factorlog(K)[M

Pa.√m]


48

Concrete Stressratio Equation

VC 10-70% ∆" = 2.70247 ∗ 10z ∗ [cd(3.00876470 ∗ j) 10-75% ∆" = 2.92768 ∗ 10z ∗ [cd(3.00876468 ∗ j) 10-80% ∆" = 3.15289 ∗ 10z ∗ [cd(3.00876470 ∗ j) 10-90% ∆" = 3.60330 ∗ 10z ∗ [cd(3.00876469 ∗ j)



Table6-5:WST:MathematicalrelationshipbetweenKandα,obtainedthroughexponentialcurvefittinginMAPLE

49

6.5 Paris-Erdoganlawparameters

6.5.1 Plottingdatapoints

Thesameprocessfromthe3PBTevaluationwasusedtoobtainthelog-logplotswhichshowtherelationbetweenda/dNand∆KfortheWSTdata.6.5.2 WST:Discussionofresults

ThecalculatedfittingcurvesonthemanipulatedParis-Erdogangraphsforalltestsaregivenin Table 6-6, along with their corresponding R2 values. For each dataset, multiple fittingcurveswereobtained.Thefittingcurvesinthetablebelowaretheoneswiththebestvaluefor R2. Like the fitting curves used in thedata evaluation for the 3PBT results, theyweremainly obtained from datasets with a small number of data points, since these largelyreducetheinfluenceofscatter.

Concrete Stressratio N Paris’slawcurvefitting R2

VC 10-70% 110941 - - 10-75% 77409 - - 10-80% 13088 y=3.6086x-24.913 0.5729 10-90% 93* y=2.6103x-16.398 0.82739 271 y=5.0523x-31.907 0.76516

SCC1 10-70% 1168 y=7.6693x-48.390 0.99226 10-75% 383 y=9.0568x-56.368 0.83846 7061 y=12.056x-75.999 0.90169 10-80% 2066 y=35.651x-218.91 0.99961 10-90% 334 y=20.393x-126.55 0.97867 6 - -

SCC2 10-70% 3099 y=12.873x-80.308 0.89778 10-75% 11002 y=8.3470x-53.874 0.45461 10-80% 545 y=6.0156x-37.862 0.77218 10-90% 739 y=19.633x-122.04 0.98338 724 y=9.6688x-60.700 0.97021

Table6-6:Mathematicalrelationshipbetweenda/dNand∆K,obtainedthroughlinearcurve

fittingfrommanipulatedParis-Erdogangraphdatapoints.

Themissingvaluesinthetableareduetothefactthatsomeofthetestsamplesfailedafterveryfewloadcycles.Forexample,theSCC1WSTspecimenfailedafteronly6cyclesundera10-90%stressratio.Forthe10-70%and10-75%stressratiotestsonVC,itwasimpossibletoobtain useful results. The crack propagation curves for these test show large errors, asshown inFigure6-11 (VC10-70%onthe left,VC10-75%ontheright). Inbothcurves thecracklengthstartstodecreaseafterapproximately40%ofthetotalnumberofloadcycles.Thisresultsinnegativevaluesforda/dN,whichcannotbeplottedinalog-loggraph,sinceit

50

isnotpossibletoobtainthelogarithmicofanegativenumber.Asaresult,nousefulfittingcurves were obtained for VC for the aforementioned stress ratios. Due to thesemissingvalues,comparisonbetweenVCandSCCbasedonthegivenWSTdataisnotpossibleforthe10-70%and10-75%stressratio.

Figure6-11:Crackpropagationcurvesfor(a)VC10-70%and(b)VC10-75%

Asmentionedbefore,fromthetestdataofthefirstcyclictestonVCforthe10-90%stressratio,a ratherstrangecrackpropagationcurvewasobtained,asshown inFigure6-9.a. Inorder toobtain theParis-Erdogancurve from thisdata set,only thedatapointsafter the275th cyclewere used, since these points indicate a normal crack propagation behaviourpattern. The number of load cycles was therefore adjusted to 93, and marked with anasterisksymbol(*)inTable6-6.

50

60

70

80

90

0 30000 60000 90000 120000

Cracklengtha[m

m]


45

50

55

60

65

0 20000 40000 60000 80000

Cracklengtha[m

m]


51

Discussionofresults

In this chapter, the calculation results obtained in Chapter 5 and 6 are compared andevaluated. In the first two paragraphs, the curve fitting results from the 3PBT will bediscussed. First, a comparisonwill bemade for the different stress ratios, followed by acomparison for the different concrete types, in a subsequent paragraph. Similar toparagraph 1 and 2, in paragraphs 3 and 4, the curve fitting results from theWSTwill bediscussedinasimilarway.The final conclusions and a general comparisonbetween the results of the 3PBT and theWST will be presented in Chapter 8. Herein, the results will also be compared with theresultsfromtheresearchofSaraKorte.

7.1 3PBT:stressratiocomparison

First, thefittingcurves fromtheParis-Erdogandataplots forthethreeconcretetypesarecompared separately for the four stress ratios. Since the difference in stress range Rbetweentheseratiosisrathersmall,onlythelowestandhigheststressratioareplotted.ForVCandSCC1thismeansplottingthefittingcurvesfor10-70%and10-75%,sincenoresultswere obtained for the 10-80% and 10-90% stress ratios (Figure 7-1 and Figure 7-2). ForSCC2,thefittingcurvesforthe10-70%and10-90%stressratioareplotted(Figure7-3).Themathematicalequationsofthefittingcurvesfromalldatasetsaregiven inTable7-1,alongwiththeircorrespondingnumberofloadcyclesNtot.Dependingonthetotalnumberof cycles and thequalityof the curve fitting, twoor threeequationswereobtained fromeachcyclic3PBT.

52

7.1.1 Overviewtablecyclic3PBT

Concrete Stressratio Test Equationm.x+log(C) m C Ntot

VC 10-70% 1 1.7301x-0.4349 1.7301 0.6473 25 2.3094x-0.5433 2.3094 0.5808 25 2 3.2763x-1.2861 3.2763 0.2763 51 3.8338x-1.4086 3.8338 0.2445 51 4.7943x-1.6396 4.7943 0.1941 51

10-75% 1 4.2383x-1.065 4.2383 0.3447 30 3.9467x-1.0177 3.9467 0.3614 30 4.5047x-1.0792 4.5047 0.3613 30

10-80% 1 - - - 1 2 - - - 2

10-90% 1 - - - 1

SCC1 10-70% 1 - - 3 2 4.1851x-0.2647 4.1851 0.7674 18 4.5187x-0.2661 4.5187 0.7664 18

10-75% 1 2.6576x+0.3541 2.6576 1.4249 6 1.8504x+0.3972 1.8504 1.4877 6

10-80% 1 - - - 2 2 - - - 3

10-90% 1 - - - 1

SCC2 10-70% 1 8.9098x-1.9121 8.9098 0.1478 406 9.1797x-1.8743 9.1797 0.1535 406 8.2086x-1.8755 8.2086 0.1533 406 2 8.7199x-2.6571 8.7199 0.0702 678 9.6656x-2.7707 9.6656 0.0626 678 8.1233x-2.6207 8.1233 0.0728 678

10-75% 1 2.5099x-0.7097 2.5099 0.4918 65 2.1634x-0.679 2.1634 0.5071 65 2.6385x-0.6689 2.6385 0.5123 65

10-80% 1 - - - 3 2 5.6393x-0.5134 5.6393 0.5985 39 5.5326x-0.5146 5.5326 0.5977 39 3.2637x-0.5198 3.2637 0.5946 39

10-90% 1 2.5419x+0.0334 2.5419 1.0340 9 2.3999x+0.1180 2.3999 1.1252 9

Table7-1:3PBT:Paris-Erdoganlawfittingcurves

53

7.1.2 Vibratedconcrete

FromthefittingcurvesforVC,giveninthegraphandParis-Erdoganlawparametersinthetablebelow,thefollowingconclusionscanbedrawn:

• The average value mavg is greater for the 10-75% stress ratio compared to the10-70%stressratio.Asaresult,when∆Kincreases,thecrackpropagationofthe10-75%stressratioincreasesfastercomparedtothe10-70%stressratio.

• BasedonthevaluesofCavg,noconclusionscanbedrawn.

Figure7-1:Paris-Erdoganlawfittingcurvesfor3PBTVC

Stressratio Test Equationm.x+log(C) m mavg C Cavg Ntot

10-70% 1 1.7301x-0.4349 1.7301 2.0198 0.6473 0.6141 25

2.3094x-0.5433 2.3094 0.5808

2 3.2763x-1.2861 3.2763 3.9681 0.2763 0.2383 51

3.8338x-1.4086 3.8338 0.2445

4.7943x-1.6396 4.7943 0.1941

10-75% 1 4.2383x-1.065 4.2383 4.2299 0.3447 0.3558 30

3.9467x-1.0177 3.9467 0.3614

4.5047x-1.0792 4.5047 0.3613

Table7-2:Paris-Erdoganlawfittingcurvesfor3PBTVC

-1.0

-0.8

-0.6

-0.4

-0.2

0.0

0.2

0.4

0.6

0.8

1.0

0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

Crackprop

agationratelog(da

/dN)[mm]


Linear(10-701.1)

Linear(10-701.2)

Linear(10-702.1)

Linear(10-702.2)

Linear(10-702.3)

Linear(10-751.1)

Linear(10-751.3)

Linear(10-751.2)

54

7.1.3 Self-compactingconcrete1

FromthefittingcurvesforSCC1andthetablebelow,thefollowingcanbeconcluded:• Theaveragevaluemavg is smaller for the10-75%stress ratio compared to the10-

70%stressratio. ItshouldbementionedthattheSCC1specimenfromthe10-75%stressratiotestfailedafteronly6cycles.Thevalidityofthisconclusioncanthereforebequestioned.

• TheaveragevalueCavgisgreaterforthe10-75%stressratiocomparedtothe10-70%stressratio.Forafixedvalueof∆K,thecrackpropagationrateundera10-75%stressratioisgreatercomparedtothe10-70%ratio.

Figure7-2:Paris-Erdoganlawfittingcurvesfor3PBTSCC1


10-70% 1 4.1851x-0.2647 4.1851 4.3519 0.7674 0.7669 18 4.5187x-0.2661 4.5187 0.7664

10-75% 1 2.6576x+0.3541 2.6576 2.2540 1.4249 1.4563 6 1.8504x+0.3972 1.8504 1.4877

Table7-3:Paris-Erdoganlawfittingcurvesfor3PBTSCC1

-0.5

-0.25

0

0.25

0.5

0.75

1

1.25

-0.05 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45

Crackprop

agationratelog(da

/dN)[mm]


Linear(10-701.1)

Linear(10-701.2)

Linear(10-751.2)

Linear(10-751.1)

55


FromthefittingcurvesforSCC2thefollowingconclusionscanbedrawn:• The average value mavg is smaller for the 10-90% stress ratio compared to the

10-70% stress ratio.When ∆K increases, the crack propagation of for the 10-70%stressratioincreasesfastercomparedtothe10-90%stressratio.

• TheaveragevalueCavgisconsiderablygreaterforthe10-90%stressratiocomparedto the 10-70% stress ratio. For a fixed value of ∆K, the crack propagation for the10-90%stressratioissignificantlygreater.

Figure7-3:Paris-Erdoganlawfittingcurvesfor3PBTSCC2


10-70% 1 8.9098x-1.9121 8.9098 8.7660 0.1478 0.1515 406 9.1797x-1.8743 9.1797 0.1535 8.2086x-1.8755 8.2086 0.1533 2 8.7199x-2.6571 8.7199 8.8363 0.0702 0.0685 678 9.6656x-2.7707 9.6656 0.0626 8.1233x-2.6207 8.1233 0.0728

10-90% 1 2.5419x+0.0334 2.5419 2.4709 1.0340 1.0796 9 2.3999x+0.1180 2.3999 1.1252

Table7-4:Paris-Erdoganlawfittingcurvesfor3PBTSCC2

-2

-1.5

-1

-0.5

0

0.5

1

1.5

-0.05 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

Crackprop

agationratelog(da

/dN)[mm]


Linear(10-901.1)

Linear(10-901.2)

Linear(10-701.1)

Linear(10-701.2)

Linear(10-701.3)

Linear(10-702.2)

Linear(10-702.3)

Linear(10-702.1)

56

7.2 3PBT:concretetypecomparison

7.2.1 Stressratio10-70%

Fromthefittingcurvesforthe10-70%stressratio,thefollowingconclusionscanbedrawn:• TheaveragevaluemavgissmallestforVC,andgreatestforSCC2.Asaconsequence,

when∆Kincreases,thecrackpropagationrateforSCC2increasesfastercomparedtoVCandSCC1.

• In these tests, theaveragevalueCavg isgreatest forSCC1.Asaconsequence, forafixed value of ∆K, the crack propagation for SCC1 is the greatest. However, thevalidityof this conclusioncouldbequestionedsince theSCC1 test specimen failedafteronly18loadcycles.

Figure7-4:Paris-Erdoganlawfittingcurvesfor3PBT10-70%stressratio

Concrete Test Equationm.x+log(C) m mavg C Cavg Ntot

VC 1 1.7301x-0.4349 1.7301 2.0198 0.6473 0.6141 25 2.3094x-0.5433 2.3094 0.5808 2 3.2763x-1.2861 3.2763 3.9681 0.2763 0.2383 51 3.8338x-1.4086 3.8338 0.2445 4.7943x-1.6396 4.7943 0.1941

SCC1 1 4.1851x-0.2647 4.1851 4.3519 0.7674 0.7669 18 4.5187x-0.2661 4.5187 0.7664

SCC2 1 8.9098x-1.9121 8.9098 8.7660 0.1478 0.1515 406 9.1797x-1.8743 9.1797 0.1535 8.2086x-1.8755 8.2086 0.1533 2 8.7199x-2.6571 8.7199 8.8363 0.0702 0.0685 678 9.6656x-2.7707 9.6656 0.0626 8.1233x-2.6207 8.1233 0.0728

Table7-5:Paris-Erdoganlawfittingcurvesfor3PBT10-70%stressratio

-2.5

-2

-1.5

-1

-0.5

0

0.5

1

-0.05 0.05 0.15 0.25 0.35 0.45

Crackprop

agationratelog(da

/dN)[mm]


Linear(SCC11.1)Linear(SCC11.2)Linear(SCC11.3)Linear(SCC21.2)Linear(SCC21.1)Linear(SCC21.3)Linear(SCC22.2)Linear(SCC22.1)Linear(SCC22.3)Linear(VC2.1)Linear(VC2.2)Linear(VC1.2)Linear(VC1.3)Linear(VC1.1)

57


Fromthefittingcurvesforthe10-75%stressratio,thefollowingconclusionscanbedrawn:• The average value mavg is greatest for VC and smallest for SCC1, although the

difference between SCC1 and SCC2 is rather small. As a consequence, when ∆Kincreases,thecrackpropagationforVCincreasesfastercomparedtoSCC.

• The average value Cavg is greatest for SCC1. For a fixed value of ∆K, the crackpropagationSCC1 is thegreatest.Similar tothe3PBT’s for the10-70%stressratio,thevalidityofthisconclusioncanbequestionedsincetheSCC1testspecimenfailedafteronly6loadcycles.

Figure7-5:Paris-Erdoganlawfittingcurvesfor3PBT10-75%stressratio


VC 1 4.2383x-1.0650 4.2383 4.2299 0.3447 0.3558 30 3.9467x-1.0177 3.9467 0.3614 4.5047x-1.0792 4.5047 0.3613

SCC1 1 2.6576x+0.3541 2.6576 2.2540 1.4249 1.4563 6 1.8504x+0.3972 1.8504 1.4877

SCC2 1 2.5099x-0.7097 2.5099 2.4373 0.4918 0.5037 65 2.1634x-0.6790 2.1634 0.5071 2.6385x-0.6689 2.6385 0.5123

Table7-6:Paris-Erdoganlawfittingcurvesfor3PBT10-75%stressratio

Forboththe10-80%and10-90%stressratio,nocomparativegraphsweremade,sincetestdata from VC and SCC1 for these stress ratios is not available, due to failure of the testspecimensafterveryfewloadcycles.

-1

-0.5

0

0.5

1

1.5

-0.1 0 0.1 0.2 0.3 0.4 0.5

Crackprop

agationratelog(da

/dN)[mm]


Linear(VC1)

Linear(VC2)

Linear(VC3)

Linear(SCC11)

Linear(SCC12)

Linear(SCC21)

Linear(SCC22)

Linear(SCC23)

58

7.3 WST:stressratiocomparison

Like the firstparagraph, the fittingcurves fromtheParis-Erdogandataplots for the threeconcretetypearecomparedseparatelyforthefourstressratios.Sincethestressrangeofthese ratios isquite similar,only the results fromthe lowestandhighest stress ratiosareplotted.ForVCthismeansplottingthefittingcurvesfor10-80%and10-90%,sincenousefulresultswereobtained for the10-70%and10-75%stress ratios (Figure7-6). ForSCC1andSCC2,thefittingcurvesforthe10-70%and10-90%stressratioareplotted(Figure7-7andFigure7-8).ThemathematicalequationsofthefittingcurvesfromallthedatasetsaregiveninTable7-7,alongwiththeircorrespondingnumberofloadcyclesNtot.Dependingonthenumberofloadcycles and quality of the curve fitting, two or three equations were obtained from eachcyclicWST.AsmentionedinParagraph6.5.2,noresultswereobtainedfromtheWSTonVCforthe10-70%and10-75%stressratios.

59

7.3.1 OverviewtablecyclicWST

Concrete Stressratio Test Equationm.x+log(C) m C Ntot

VC 10-70% - - - 110941

10-75% - - - 77409

10-80% 1 3.6086x-3.2615 3.6086 0.0383 13088

4.2947x-3.3270 4.2947 0.0359

5.3274x-3.5448 5.3274 0.0289

10-90% 1 2.2933x-0.7533 2.2933 0.4708 93*

2.6103x-0.7364 2.6103 0.4788 2 5.2891x-1.6165 5.2891 0.1986 271

5.0523x-1.5937 5.0523 0.2032

5.0132x-1.6181 5.0132 0.1983

SCC1 10-70% 1 7.6693x-2.3741 7.6693 0.0931 1168

7.9848x-2.7067 7.9848 0.0668

8.6825x-2.3679 8.6825 0.0937

10-75% 1 8.8130x-2.0132 8.813 0.1336 383

9.0568x-2.0271 9.0568 0.1317

9.8247x-2.0533 9.8247 0.1283

2 12.056x-3.6606 12.056 0.0257 7061

12.437x-3.6476 12.437 0.0261

10.418x-3.5515 10.418 0.0287

10-80% 1 35.651x-4.9991 35.651 0.0067 2066

35.964x-5.0091 35.964 0.0067

37.539x-5.1235 37.539 0.0060

10-90% 1 20.721x-4.2487 20.721 0.0143 334

20.393x-4.1926 20.393 0.0151 17.617x-3.8780 17.617 0.0207

2 - - - 6

SCC2 10-70% 1 12.873x-3.0718 12.873 0.0463 3099

12.060x-3.0050 12.06 0.0495

10-75% 1 6.7485x-3.5836 6.7485 0.0278 11002

8.3470x-3.7921 8.347 0.0225

8.1971x-3.6990 8.1971 0.0247

10-80% 1 6.0156x-1.7681 6.0156 0.1707 545

5.6734x-1.6908 5.6734 0.1844

6.0127x-1.6912 6.0127 0.1843

10-90% 1 19.633x-4.2489 19.633 0.0143 739

19.495x-4.1261 19.495 0.0161

18.918x-4.0160 18.918 0.0180

2 8.3252x-2.5857 8.3252 0.0753 724

10.282x-2.7674 10.282 0.0628

9.6688x-2.6870 9.6688 0.0681

Table7-7:WST:Paris-Erdoganlawfittingcurves

60

7.3.2 Vibratedconcrete

FromthefittingcurvesforVCthefollowingconclusionscanbedrawn:• Based on the values ofmavg, no certain conclusions can be drawn from the given

fittingcurves.Thismightbeaconsequenceofthefactthatthedifferencebetween10-80%and10-90%thestressratiosisverysmall.

• TheaveragevalueCavg issignificantlygreaterforthe10-90%stressratiocomparedtothe10-80%stressratio.Therefore,forafixedvalueof∆K,thecrackpropagationrateforthe10-90%isconsiderablyfaster.

Figure7-6:Paris-ErdoganlawfittingcurvesforWSTVC


10-80% 1 3.6086x-3.2615 3.6086 4.4102 0.0383 0.0344 13088

4.2947x-3.3270 4.2947 0.0359

5.3274x-3.5448 5.3274 0.0289

10-90% 1 2.2933x-0.7533 2.2933 2.4518 0.4708 0.4748 93*

2.6103x-0.7364 2.6103 0.4788 2 5.2891x-1.6165 5.2891 5.1182 0.1986 0.2000 271

5.0523x-1.5937 5.0523 0.2032

5.0132x-1.6181 5.0132 0.1983

Table7-8:Paris-ErdoganlawfittingcurvesforWSTVC

-3.5

-3

-2.5

-2

-1.5

-1

-0.5

-0.05 0 0.05 0.1 0.15 0.2

Crackprop

agationratelog(da

/dN)[mm]


Linear(10-901.1)

Linear(10-901.2)

Linear(10-902.1)

Linear(10-902.2)

Linear(10-902.3)

Linear(10-801)

Linear(10-802)

Linear(10-803)

61


FromthefittingcurvesforSCC1thefollowingconclusionscanbedrawn:• Theaveragevaluemavg is smaller for the10-70%stress ratio compared to the10-

90%stressratio.Asaconsequence,thecrackpropagationratefora10-90%stressratio increases faster compared to a 10-70% stress ratiowhen the stress intensityratio∆Kincreases.

• TheaveragevalueCavgissmallerforthe10-90%stressratiocomparedtothe10-70%stress ratio.However, the fitting curves intersect in the regionof log(∆K)between0.1 and 0.16 (1.105 and 1.174MPa.√m). Therefore, for smaller values of ∆K, thecrackpropagation for the10-70%ratiooccurs faster,while for largervaluesof∆K,thecrackpropagationrateforthe10-90%stressratioisfaster.

Figure7-7:Paris-ErdoganlawfittingcurvesforWSTSCC1


10-70% 1 7.6693x-2.3741 7.6693 8.1122 0.0931 0.0845 1168

7.9848x-2.7067 7.9848 0.0668

8.6825x-2.3679 8.6825 0.0937

10-90% 1 20.721x-4.2487 20.721 20.557 0.0143 0.0147 334

20.393x-4.1926 20.393 0.0151

Table7-9:Paris-ErdoganlawfittingcurvesforWSTSCC1

-2.5

-2

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-1

-0.5

0

0 0.05 0.1 0.15 0.2 0.25

Crackprop

agationratelog(da

/dN)[mm]


Linear(10-701)Linear(10-702)Linear(10-703)Linear(10-901)Linear(10-902)Linear(10-903)

62


FromthefittingcurvesforSCC2thefollowingconclusionscanbedrawn:• Basedonthevaluesofmavg,noconclusionscanbedrawn.Thedifferencesinresult

between the two testswith a 10-90% stress ratio are greater than the differencewiththetestresultfromthe10-70%stressratiotest.

• In addition, based on the values ofCavg, no conclusions can be drawn either. Thedifferencesbetween the two testswitha10-90%stress ratioaregreater than thedifferencewiththetestresultfromthe10-70%stressratiotest.

Figure7-8:Paris-ErdoganlawfittingcurvesforWSTSCC2


10-70% 1 12.873x-3.0718 12.873 12.467 0.0463 0.0479 3099

12.060x-3.0050 12.06 0.0495

10-90% 1 19.633x-4.2489 19.633 19.349 0.0143 0.0161 739

19.495x-4.1261 19.495 0.0161

18.918x-4.0160 18.918 0.0180

2 8.3252x-2.5857 8.3252 9.4253 0.0753 0.0687 724

10.282x-2.7674 10.282 0.0628

9.6688x-2.6870 9.6688 0.0681

Table7-10:Paris-ErdoganlawfittingcurvesforWSTSCC2

-3

-2.5

-2

-1.5

-1

-0.5

0 0.05 0.1 0.15 0.2 0.25

Crackprop

agationratelog(da

/dN)[mm]


Linear(10-901.1)

Linear(10-901.2)

Linear(10-901.3)

Linear(10-902.1)

Linear(10-902.2)

Linear(10-902.3)

Linear(10-701)

Linear(10-702)

63

7.4 WST:concretetypecomparison


Fromthefittingcurvesforthe10-70%stressratio,thefollowingconclusionscanbedrawn:• IncomparisonofthetwoSCCspecimens,theaveragevaluemavgissmallerforSCC1

compared to SCC2. Therefore, when ∆K increases, the crack propagation rate forSCC2isgreater.

• TheaveragevalueCavgisgreatestforSCC1.Asaconsequence,forafixedvalueof∆K,thecrackpropagationforSCC1isthegreatest.

• NoconclusionscanbedrawnforacomparisonwithVC,sincenodataisavailable.

Figure7-9:Paris-ErdoganlawfittingcurvesforWST10-70%stressratio

Concrete Test Equationm.x+log(C) m mavg C Cavg N

SCC1 1 7.6693x-2.3741 7.6693 8.1122 0.0931 0.0845 1168

7.9848x-2.7067 7.9848 0.0668

8.6825x-2.3679 8.6825 0.0937

SCC2 1 12.873x-3.0718 12.873 12.4665 0.0463 0.0479 3099

12.060x-3.0050 12.06 0.0495

Table7-11:Paris-ErdoganlawfittingcurvesforWST10-70%stressratio

-3

-2.5

-2

-1.5

-1

-0.5

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2

Crackprop

agationratelog(da

/dN)[mm]


Linear(SCC21)

Linear(SCC22)

Linear(SCC11)

Linear(SCC12)

Linear(SCC13)

64


Fromthefittingcurvesforthe10-75%stressratio,thefollowingconclusionscanbedrawn:• TheaveragevaluemavgisgreatestforSCC1,althoughthereisabigdifferencewithin

the results of the two cyclic WST specimens of this concrete type. When ∆Kincreases,thecrackpropagationforSCC1increasesfastercomparedtoSCC2.

• BasedonthevaluesofCavg,noconclusionscanbedrawn,althoughwithintheregionofthefittingcurves,log(∆K)between0.07and0.1(1.073and1.105MPa.√m),acleardifferencecanbeseen,asshowninthegraphbelow.Withinthisregion,forafixedvalueof∆K, thecrackpropagation forbothSCC1specimens is faster compared toSCC2.

• NoconclusionscanbedrawnforacomparisonwithVC,sincenodataisavailable.



SCC1 1 8.8130x-2.0132 8.813 9.2315 0.1336 0.1312 383

9.0568x-2.0271 9.0568 0.1317

9.8247x-2.0533 9.8247 0.1283

2 12.056x-3.6606 12.056 11.6370 0.0257 0.0268 7061

12.437x-3.6476 12.437 0.0261

10.418x-3.5515 10.418 0.0287

SCC2 1 6.7485x-3.5836 6.7485 7.7642 0.0278 0.0250 11002

8.3470x-3.7921 8.347 0.0225

8.1971x-3.6990 8.1971 0.0247


-3.5

-3.0

-2.5

-2.0

-1.5

-1.0

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2

Crackprop

agationratelog(da

/dN)[mm]


Linear(SCC11.2)

Linear(SCC11.3)

Linear(SCC12.1)

Linear(SCC12.2)

Linear(SCC12.3)

Linear(SCC21)

Linear(SCC22)

Linear(SCC23)

65


Fromthefittingcurvesforthe10-80%stressratio,thefollowingconclusionscanbedrawn:• The average valuemavg is by far the greatest for SCC1. The average valuemavg of

SCC2isslightlygreaterthenVC.Therefore,as∆Kincreases,thecrackpropagationforSCC1increasesalotfastercomparedtoSCC2andVC.

• AlthoughSCC1andVChavesimilaravaluesofmavg,thereisabigdifferenceinthevalueofCavg: inall fitting curves, theparameterC is significantlygreater forSCC2,comparedtoVC.Therefore,forafixedvalueof∆K,thecrackpropagationforSCC2stressratioisconsiderablyfastercomparedtoVC.

• For0.06≤log(∆K)≤0.1(1.062≤∆K≤1.105MPa.√m),thevalueofCavgisgreaterthanVCyetsmallerthanSCC2.Forlog(∆K)<0.06(1.062MPa.√m),thevalueofCavgisthesmallest.Forlog(∆K)>0.1(1.105MPa.√m),thevalueofCavgisthegreatest.


Concrete Test Equationm.x+log(C) m mavg C Cavg N

VC 1 4.2947x-3.3270 4.2947 4.8111 0.0359 0.0324 13088

5.3274x-3.5448 5.3274 0.0289

SCC1 1 35.651x-4.9991 35.651 36.385 0.0067 0.0065 2066

35.964x-5.0091 35.964 0.0067 37.539x-5.1235 37.539 0.0060

SCC2 1 6.0156x-1.7681 6.0156 5.9006 0.1707 0.1798 545

5.6734x-1.6908 5.6734 0.1844 6.0127x-1.6912 6.0127 0.1843


-3.5

-3.0

-2.5

-2.0

-1.5

-1.0

-0.5

0.0

0 0.05 0.1 0.15 0.2 0.25

Crackprop

agationratelog(da

/dN)[mm]


Linear(VC2)

Linear(VC3)

Linear(SCC11)

Linear(SCC12)

Linear(SCC13)

Linear(SCC21)

Linear(SCC22)

Linear(SCC23)

66


Fromthefittingcurvesforthe10-80%stressratio,thefollowingconclusionscanbedrawn:• TheaveragevaluemavgisclearlythesmallestforVC.TheaveragevaluemavgofSCC1

isslightlygreaterthenSCC2forthefirsttestspecimen,butalotbiggercomparedtothe second specimen. Therefore, as ∆K increases, the crack propagation for SCC1increasesslightlyfastercomparedtoSCC2andalotfastercomparedtoVC.

• TheaveragevalueCavgisgreatestforthetwotestedVCspecimens,comparedtotheSCCsamples.Asaconsequence,forafixedvalueof∆K,thecrackpropagationforVCis the greatest. Based on the results for Cavg no conclusion can be drawn incomparisonofSCC1andSCC2.



VC 1 2.2933x-0.7533 2.2933 2.4518 0.4708 0.4748 93*

2.6103x-0.7364 2.6103 0.4788 2 5.2891x-1.6165 5.2891 5.1182 0.1986 0.2000 271

5.0523x-1.5937 5.0523 0.2032

5.0132x-1.6181 5.0132 0.1983

SCC1 1 20.721x-4.2487 20.721 19.577 0.0143 0.0167 334

20.393x-4.1926 20.393 0.0151 17.617x-3.8780 17.617 0.0207

SCC2 1 19.633x-4.2489 19.633 19.349 0.0143 0.0161 739

19.495x-4.1261 19.495 0.0161

18.918x-4.0160 18.918 0.0180

2 8.3252x-2.5857 8.3252 9.4253 0.0753 0.0687 724

10.282x-2.7674 10.282 0.0628

9.6688x-2.6870 9.6688 0.0681


-2.5

-2

-1.5

-1

-0.5

0

-0.05 0 0.05 0.1 0.15 0.2

Crackprop

agationratelog(da

/dN)[mm]


Linear(VC1.1)Linear(VC1.2)Linear(VC2.1)Linear(VC2.2)Linear(VC2.3)Linear(SCC11.1)Linear(SCC11.2)Linear(SCC11.3)Linear(SCC21.2)Linear(SCC21.3)Linear(SCC22.1)Linear(SCC22.2)Linear(SCC22.3)

67

Conclusions

In this final chapter, the overall conclusions from the Paris-Erdogan law data correlationfromboththe3PBTandWSTresultsaregiven.Afterwards,thedatafrombothtestmethodsarecompared.Thegeneralconclusionfromthisresearchisgiveninthefinalparagraph.

8.1 Three-pointbendtest

The 3PBT shows that SCC with similar strength (SCC1) has the highest overall value ofparameterCofalltestedconcretes,forboththe10-70and10-75%stressratio.Thisimpliesthatforagivenstressintensityrange∆K,thecrackpropagationrateinSCC1willbelarger,comparedtoVCandSCC2.Furthermore,theresultsforthe10-70and10-75%stressratioshowthatSCCwithequalwater-cementratio(SCC2)hasthelowestoverallvalueofC.Foragivenstressintensityrange∆K,thecrackpropagationrateinSCC2willbesmallercomparedtoVC.Forthe10-70%stressratio,SCC2hasthehighestvalueofmwhileVChasthelowestvalue,thus implying that when ∆K increases, the crack propagation in SCC2 rate increases thequickest.However, in the10-75%stress ratio,VChas thegreatest valueofparameterm,whilethemparametersofSCC1andSCC2aremerelyidentical.Ingeneral,itcanbeconcludedthatSCCwithcomparablestrengthhasthehighestvalueofparameterC,whileSCCwithacomparablew/c-ratiohasthelowestvalue.Furthermore,forthelowest10-70%stressratio,SCC2hasahighervalueofparameterC,whileinthe10-75%stress ratio VC has the highest value. The results don’t always allow conclusions to bedrawn.Thismightbecausedbythefactthatconcreteisaheterogeneousmaterialknowntobepronetoscatterintheresults.

68

8.2 Wedgesplittingtest

Thetestresultsdon’talwaysallowdrawingconclusionswithhighcertitude.This iscausedbythefactthatconcreteisaheterogeneousmaterialknowntobepronetoscatter intheresults. These variations and exceptions can be caused by amultitude of factors. One ofthembeingthefactthatthecracksdon’talwaysfollowthetheoreticalpathdown,asshowninFigure8-1.Despitethesevariations,certaintrendscanbeobserved.

Figure8-1:WST:crackpatternsinVCspecimens(Korte,2014)

Whenobservingtheresultsfromthefourstressratios,acleartrendcanbeobservedintheresultsofSCCwithequalstrength(SCC1)andSCCwithcomparablew/c-ratio(SCC2).Forthelowest 10-70% stress ratio, the fitting curves for SCC1 are less steep compared to thosefromSCC2;mavg,SCC1<mavg,SCC2.Thesameresultcanbefoundintheresultsfromthe3PBT.Forthe10-75%stressratioshowever, liketheresultsfromthe3PBT,thefittingcurvesforSCC1andSCC2haveacomparablesteepness;mavg,SCC1≈mavg,SCC2.Finally,forthehighest10-80% and 10-90% stress ratios, it is the other way round. The fitting curves for SCC1 aresteepercomparedtothosefromSCC2;mavg,SCC1>mavg,SCC2.Thisimpliesthatforanincreasingstressratio,SCC2performsbetterthanSCC1.Inotherwords,theinfluenceoftheincreasingstressratioisgreaterinSCC1comparedtoSCC2.Moreover,whencomparingVCtotheSCCspecimensforthe10-80%and10-90%stressratios,VChasthelowestvalueofminalltests.Asaresult,anincreasein∆KwillhavethesmallesteffectonthecrackpropagationrateofVC.WhencomparingtheparameterCforthetwotypesofSCC,asimilarshiftcanbeobserved;for the lower 10-70% and 10-75% stress ratios, SCC2 has a higher value. The differencebetweenSCC1andSCC2for theparameterC ishoweversmaller than in the3PBTresults.Forthe10-80%and10-90%stressratioontheotherhand,SCC2clearlyhasthehighestC-value. From the comparison of the C-parameter of VC to SCC, it is difficult to drawconclusions;inthe10-80%stressratiotests,VCclearlyhasalowervalueofC,whileinthe10-90%stressratio,thingsarethetheopposite.

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8.3 3PBTandWSTcomparison

Comparedtothe3PBT,especiallyforsmallerstressratioslike10-70%and10-75%,theWSTrequiresahighernumberofloadcyclesbeforefailureofthetestspecimen.Oftenupto100timesmore.Whenperformingmanyofthesetests,timecanbecomeamajorfactor.Whenaspecimenrequires100000cyclesbeforefailure,theexperimentisratheruneconomical.However,forhigherstressratios,the3PBTsamplesoftenfailafteronlyasingleloadcycle.In this case, no useful information about the fatigue crack propagation can be obtained.Therefore, the3PBT ismoreuseful forsmallerstress ratios,while theWST ismoreusefulwhentestingspecimensunderhigherstressratios.

Stressratio ConcreteThree-pointbendtest Wedgesplittingtest

mavg Cavg N mavg Cavg N

10-70% VC 2.0198 0.6141 25 - - 110941

3.9681 0.2383 51

SCC1 4.3519 0.7669 18 8.1122 0.0845 1168 SCC2 8.7660 0.1515 406 12.4665 0.0479 3099

8.8363 0.0685 678

10-75% VC 4.2299 0.3558 30 - - 77409 SCC1 2.2540 1.4563 6 9.2315 0.1312 383 11.6370 0.0268 7061 SCC2 2.4373 0.5037 65 7.7642 0.0250 11002

10-80% VC - - 2 4.8111 0.0324 13088 SCC1 - - 1 36.385 0.0065 2066 SCC2 4.8119 0.5969 39 5.9006 0.1798 545

10-90% VC - - 3 2.4518 0.4748 93* 5.1182 0.2000 271 SCC1 - - 1 19.577 0.0167 334 SCC2 2.4709 1.0796 9 19.349 0.0161 739 9.4253 0.0687 724

Table8-1:3PBTandWSTcomparison

InTable8-1,theaveragevaluesformandCfromallthetestswhichresultedinusefuldataareshown.DuetothemissingvaluesfromVCandSCC1forthe10-80%and10-90%stressratios,afullcomparisonforallstressratiosisimpossible.However,asstatedinthepreviousparagraph,forthe10-70%and10-75%stressratios,comparabletrendscanbeobservedinthedatafromthe3PBT’sandtheWST’s.Therefore,itcanbeconcludedthatboththe3PBTandtheWSTcanbeusedintheanalysisofconcretefatiguefracture.

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8.4 Generalconclusion

Asageneralconclusionfromthe3PBT’sandtheWST’s,itcanbestatedthatbothtestscanbeused toobtain valuable informationabout the fatigue crackpropagationpropertiesofbothvibratedconcreteandself-compactingconcrete.Forsmallstressratioslike10-70%the3PBT ismoreusefulsince itusuallydoesnotrequiremorethan1000 loadcyclesuntil thetestspecimenfails.Forhigherstressratiosontheotherhand,theWSTismoreuseful,sinceforthesehigherstressratios,the3PBTspecimensfailafterveryfewloadcycles.From the data correlation it can be stated that VC performs better under cyclic loadingsituationscomparedtoSCCwithcomparablestrength.Inalltests,exceptthe10-90%WST,VCperformsbetter.FromthetestsonSCCwithacomparablew/c-ratiothedifferenceswithVCarelesspronounced.Overall,itcanbestatedthatSCCismorebrittlethanVC.Thisconclusioncorrespondswiththe conclusions stated in thework of Sara Korte. It can therefore be concluded that theParis-Erdogan law isavaluable tool for theevaluationof the fatiguecrackpropagation incement based composites. Due to its alternate behaviour, precaution is required whenusingSCCinsteadofVCincyclicloadingconditions.

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9.1.1 Authorreferences

1. Seitl,S.&Thienpont,T. (2016).Numericalsupport forstudyofstressratioeffectonfatigue crack behaviour in three-point bend specimen made from vibratedconcrete.Proceedingsofscientificconference"Structuralreliability&Modelling in

Mechanics 2016. VŠB-TU Ostrava, Faculty of Civil Engineering, May 2016, 6 pp.ISBN:978-80-248-3917-2

2. Seitl,S.&Thienpont,T.(2016).Fatiguecrackbehaviour:Comparingthree-pointbendtestandwedgesplittingtestdataonvibratedconcreteusingParis'law.Transactionof VŠB, Technical University of Ostrava, Civil Engineering series. VŠB-TU Ostrava,FacultyofCivilEngineering.ISSN:1804-0993(inpress).

3. Seitl,S.,Thienpont,T.&DeCorte,W. (2016)Evaluationof fatiguecrackbehaviorofself-compacting through correlation of three-point bend test data to Paris’ law.ProcediaEngineering.ProceedingsofXVIIIInternationalColloquiumonMechanical

FatigueofMetals(ICMFMXVIII)(inpress)4. Seitl, S., Thienpont, T. & De Corte, W. Fatigue crack behaviour: Comparing Self-

compacting concrete and vibrated concrete using Paris' law. Construction andbuildingmaterials.(tosend)

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9.1.2 AuthorCV

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Appendices

10.1 AppendixA:3PBTmodel-ANSYSAPDLcode/TITLE, Tree point bend test: CMOD vs a FINISH /CLEAR /PREP7 ! all Units are SI-units!!! !--- Element type choice ----! ET,1,PLANE183 ! 8-node element with 4 edges !---- Material Properties ----! properties > doc. thesis Sara Korte E = 38.423e9 ! Vibrated Concrete !E = 38.093e9 ! SCC1 !E = 35.290e9 ! SCC2 nu=0.2 MP,EX ,1, E MP,PRXY,1, nu !---- General Parameters ------! W =0.100 ! in meters R1 =0.1*W ! 10 mm R2 =0.75*R1 ! 7,5 mm R =0.1*R1 ! 1 mm a =0.5*W ! relative crack length (between 0.1 and 0.9) M1 =0.02*W ! coarse mesh = 5 mm M2 =0.01*W ! fine mesh = 1 mm MC =0.1*R ! mesh in inner circle = 0.1mm !----- Load Parameters ----! L = 6109 ! VC !L = 6939.8 ! SCC1 !L = 6120 ! SCC2 !load = 0.10*L load = 0.70*L !load = 0.75*L !load = 0.80*L !load = 0.90*L !---- Creating Key-points ----! k,1, 0,0 k,2, -R,0 k,3, -R2,0 k,4, -R1,0 k,5, -a,0 k,6, -a,.15 k,7, -a,.20 k,8, W-a,.20 k,9, W-a,0

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k,10, R1,0 k,11, R2,0 k,12, R,0 k,13, 0,R k,14, 0,R2 k,15, 0,R1 k,16, -a,1.5*R1 k,17, 0,.20 k,18, W-a,1.5*R1 k,19, 0,1.5*R1 !---- Creating lines ----! L,1,2 L,2,3 L,3,4 L,4,5 L,5,16 ! line 5 L,16,6 L,6,7 L,7,17 L,17,8 L,8,18 ! line 10 L,18,9 L,9,10 L,10,11 L,11,12 L,12,1 ! line 15 L,19,16 L,19,17 L,19,18 ARC,12,13,2,R ARC,2,13,12,R !line 20 ARC,11,14,3,R2 ARC,3,14,11,R2 ARC,10,15,4,R1 ARC,4,15,10,R1 L,15,19 ! line 25 !---- Creating Area ----! AL,1,20,19,15 AL,2,22,21,14,19,20 AL,3,24,23,13,21,22 AL,4,5,16,24,25 AL,12,23,25,18,11 AL,6,7,8,17,16 AL,17,9,10,18 !---- Boundary conditions and Loads ----! DK,6,UX,0,,,,,,, !support 3PBT DL,12,,SYMM !symmetric crack zone DL,13,,SYMM DL,14,,SYMM DL,15,,SYMM FK,9,FX,-load*0.5*10, !half load in middle of the span !———Visual control ---! /PNUM,KP,1 ! turn on key-point numbering /PNUM,LINE,1 ! turn on line numbering LPLOT ! does a multi-plot

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!---- Mesh-controls and Meshing Area 4 en 5 ----! LESIZE,4,M2,,, LESIZE,12,M2,,, LESIZE,5,,,15,2 LESIZE,11,,,15,.5 LESIZE,23,M2,,, LESIZE,24,M2,,, LESIZE,25,,,5,.5 LESIZE,16,M1,,, LESIZE,18,M1,,, AMESH,4 AMESH,5 !---- Mesh-controls and Meshing Area 6 en 7 ----! LESIZE,6,M1,,, LESIZE,7,M1,,, LESIZE,8,M1,,, LESIZE,9,M1,,, LESIZE,10,M1,,, LESIZE,17,M1,,, AMESH,6 AMESH,7 !---- Mesh-controls and Meshing Area 3 ----! LESIZE,21,.75*M2,,, LESIZE,22,.75*M2,,, LESIZE,3,M2,,, LESIZE,13,M2,,, AMESH,3 !---- Mesh-controls and Meshing Area 1 ----! KSCON,1,0.05*R,1,12,0, LESIZE,19,,,20, LESIZE,20,,,20, LESIZE,1,,,10, LESIZE,15,,,10, AMESH,1 !---- Mesh-controls and Meshing Area 2 ----! LESIZE,2,,,20,10 LESIZE,14,,,20,.1 AMESH,2 !---- Solution ----! /SOL SOLVE !---- KCALC ----! /POST1 FLST,2,3,1 FITEM,2,17406 FITEM,2,17408 FITEM,2,17409 PATH,pad1,3,30,20, PPATH,P51X,1 !PATH,STAT KCALC,1,1,0,0

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10.2 AppendixB:WSTmodel-ANSYSAPDLcode/TITLE, Wedge splitting test: CMOD ifv a FINISH /CLEAR /PREP7 ! all Units are SI-units!!! !--- Element type choice ----! ET,1,PLANE183 ! 8-node element with 4 edges !---- Material Properties concrete -----! properties > doc. thesis Sara Korte !in N/m^2 !E = 38.423e9 ! Vibrated Concrete !E = 38.093e9 ! SCC1 E = 35.290e9 ! SCC2 nu=0.2 MP,EX ,1, E MP,PRXY,1, nu !---- Material Properties steel ——! E2 = 210e9 nu2 = 0.3 MP,EX,2,E2 MP,PRXY,2,nu2 !---- General Parameters ------! W=0.150 ! edge cube DR=0.022 ! depth ridge = 22 mm BR=0.015 ! half width ridge = 15 mm S=0.075 ! distance between supports t=0.012 g=0.002 ! gap 2 mm WE=W-.005 ! total crack length R=0.01 M=0.01*W ! mesh size = 1.5 mm a=0.9*WE+.005 ! relative crack length (between 0.2 and 0.9) !----- Load Parameters ----! !L = 10450*20/3 ! VC !L = 10368.9*20/3 ! SCC1 L = 9990*20/3 ! SCC2 !load = 0.10*L !load = 0.70*L !load = 0.75*L !load = 0.80*L load = 0.90*L !load = L !---- Creating Concrete part ----! k,1, 0, 0 k,2, DR-a, 0 k,3, DR-a, BR

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k,4, -a, BR k,5, -a, 0.5*W k,6, W-a, 0.5*W k,7, W-a, 0.5*S k,8, W-a, 0 k,9, DR-a-g, BR k,10, .005-a, BR-t k,11, -a-t, BR-t k,12, -a-t, BR+.025 k,13, -a-t, BR+.025+t k,14, -a, BR+.025+t k,15, DR-a-g, BR-t k,16, W-a, BR k,17, 0, R k,18, R, 0 k,19, -R, 0 L,1,19 L,19,2 L,2,3 L,3,9 L,9,4 ! line 5 L,4,14 L,14,5 L,5,6 L,6,7 L,7,16 ! lijn10 L,16,8 L,8,18 L,18,1 L,9,15 L,15,10 ! line 15 L,10,11 L,11,12 L,12,13 L,13,14 AL,14,15,16,17,18,19,6,5 ! area 1 steel ARC,18,17,19,R ! line 20 ARC,19,17,18,R AL,1,21,20,13 ! area 2 beton semi circle L,3,16 ! line 22 AL,22,4,5,6,7,8,9,10 ! area 3 concrete big area AL,21,2,3,22,11,12,20 ! area 4 concrete rest !---- Boundary conditions and Loads ----! DK,7,UX,0,,,,,,, !support WST DL,12,,SYMM !symmetric crack zone DL,13,,SYMM !symmetric crack zone FK,10,FY,load, FK,12,FX,load/1.8660254, !--- Visual control ---! /PNUM,KP,1 ! turn on key-point numbering /PNUM,LINE,1 ! turn on line numbering LPLOT ! does a multi-plot !---- Mesh-controls and Meshing Area 2 ----!

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KSCON,1,0.02*R,1,12,0, LESIZE,20,M/2,,, LESIZE,21,M/2,,, LESIZE,1 ,,,30,2 LESIZE,13,,,30,.5 MSHKEY,0 MAT,1 AMESH,2 ! mesh area 2 !---- Mesh-controls and Meshing Area 1 ----! LESIZE, 5,M,,, LESIZE, 6,M,,, LESIZE,14,M,,, LESIZE,15,M,,, LESIZE,16,M,,, LESIZE,17,M,,, LESIZE,18,M,,, LESIZE,19,M,,, MSHKEY,0 MAT,2 AMESH,1 ! mesh area 1 !---- Mesh-controls and Meshing Area 3 ----! LESIZE,22,M,,, LESIZE,4,M,,, LESIZE,7,M,,, LESIZE,8,M,,, LESIZE,9,M,,, LESIZE,10,M,,, MSHKEY,0 MAT,1 AMESH,3 ! mesh area 3 !---- Mesh-controls and Meshing Area 4 ----! LESIZE,2,M/2,,, LESIZE,3,M,,, LESIZE,11,M,,, LESIZE,12,M/2,,, MSHKEY,0 MAT,1 AMESH,4 ! mesh area 4 !---- Solution ----! /SOL SOLVE /POST1 /SHOW,WIN32C !---- KCALC ----! /POST1 !FLST,2,3,1 !FITEM,2,1 !FITEM,2,3 !FITEM,2,4 !PATH,pad1,3,30,20, !PPATH,P51X,1 !PATH,STAT !KCALC,1,1,0,0

Thomas Thienpont

self-compacting concreteNumerical analysis of Paris-Erdogan law parameters of

Academic year 2015-2016Faculty of Engineering and ArchitectureChair: Prof. dr. ir. Luc TaerweDepartment of Structural Engineering

Master of Science in de industriële wetenschappen: bouwkundeMaster's dissertation submitted in order to obtain the academic degree of

of Materials of the Academy of Sciences of the Czech Republic, Brno, Tsjechië)Supervisors: Prof. dr. ir. Wouter De Corte, Prof. Stanislav Seitl (Institute of Physics

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