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VIBRATIONFATIGUEANALYSISOFSTRUCTURESUNDERBROADBANDEXCITATION

ATHESISSUBMITTEDTOTHEGRADUATESCHOOLOFNATURALANDAPPLIEDSCIENCES

OFMIDDLEEASTTECHNICALUNIVERSITY

BY

BLGEKOER

INPARTIALFULFILLMENTOFTHEREQUIREMENTSFOR

THEDEGREEOFMASTEROFSCIENCE

INMECHANICALENGINEERING

JUNE2010


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Approvalofthethesis:

VIBRATIONFATIGUEANALYSISOFSTRUCTURESUNDER

BROADBANDEXCITATION

submittedbyBLGEKOERinpartialfulfillmentoftherequirementsforthedegreeofMasterofScience inMechanicalEngineeringDepartment,MiddleEastTechnicalUniversityby,

Prof.Dr.Cananzgen ________________ Dean,GraduateSchoolofNaturalandAppliedSciences

Prof.Dr.SuhaOral ________________ HeadofDepartment,MechanicalEngineering

Assoc.Prof.Dr.SerkanDa ________________ Supervisor,MechanicalEngineeringDept.,METU

Asst.Prof.Dr.YiitYazcolu ________________ CoSupervisor,MechanicalEngineeringDept.,METU

ExaminingCommitteeMembers:

Asst.Prof.Dr.ErginTnk _____________________ MechanicalEngineeringDept.,METU

Assoc.Prof.Dr.SerkanDa _____________________ MechanicalEngineeringDept.,METU

Asst.Prof.Dr.YiitYazcolu _____________________ MechanicalEngineeringDept.,METU

Asst.Prof.Dr.EnderCierolu _____________________ MechanicalEngineeringDept.,METU

Dr.VolkanParlakta _____________________ MechanicalEngineeringDept.,HacettepeUniversity

Date: 24062010


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Iherebydeclarethatallinformationinthisdocumenthasbeenobtainedandpresented inaccordancewithacademic rulesandethicalconduct. Ialso declare that, as required by these rules and conduct, I have fullycitedandreferencedallmaterialandresults thatarenotoriginal to thiswork.

Name,Lastname : Bilge,Koer

Signature :


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ABSTRACT

VIBRATIONFATIGUEANALYSISOFSTRUCTURESUNDERBROADBANDEXCITATION

Koer,BilgeM.S.,DepartmentofMechanicalEngineeringSupervisor:Assoc.Prof.Dr.SerkanDaCoSupervisor:Asst.Prof.Dr.YiitYazcolu

June2010,117pages

Thebehavior of structures is totally differentwhen they are exposed to

fluctuating loading rather than static one which is a well known

phenomenon inengineeringcalled fatigue.When the loading isnotstatic

butdynamic, thedynamicsof the structure shouldbe taken intoaccount

since there isahighpossibility to excite the resonance frequenciesof the

structureespeciallyiftheloadingfrequencyhasawidebandwidth.Inthese

cases,thestructuresresponsetotheloadingwillnotbelinear.Therefore,in

the analysis of such situations, frequency domain fatigue analysis

techniquesareusedwhichtakethedynamicpropertiesofthestructureinto

consideration.Vibrationfatiguemethodisalsofast,functionalandeasyto

implement.

Inthisthesis,vibrationfatiguetheoryisexamined.Throughouttheresearch

conductedforthisstudy,theultimateaim istofindsolutionstoproblems

arising from test application for the loadingswith nonzeromean value

bringing a newperspective tomean stress correction techniques.A new


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method isdeveloped togenerateamodified input loadinghistorywitha

zeromeanvaluewhich leads infatiguedamageapproximatelyequivalent

to damage induced by input loading with a nonzero mean value. A

mathematicalprocedure isproposed to implementmean stresscorrection

to the output stress power spectral density data and a modified input

loadingpowerspectraldensitydataisobtained.Furthermore,thismethod

is improved formultiaxial loadingapplications.A loadinghistorypower

spectraldensitysetwithzeromeanbutmodifiedalternatingstress,which

leadsinfatiguedamageapproximatelyequivalenttothedamagecausedby

the unprocessed loading setwith nonzeromean, is extracted taking all

stress components into account using full matrixes. The proposed

techniques efficiency is discussed throughout several case studies and

fatiguetests.

Keywords: Vibration Fatigue, Mean Stress Correction, Power Spectral

Density,FiniteElementMethod.


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Z

YAPILARINGEN BANTLITAHRKALTINDATTREMKAYNAKLIYORULMANCELEMELER

Koer,BilgeYksekLisans,MakineMhendisliiBlmTezyneticisi:Do.Dr.SerkanDaYardmctezyneticisi:Yrd.Do.Dr.YiitYazcolu

Haziran2010,117sayfa

Yaplar statik ykleme yerine dalgalanan bir yklemeye maruz

kaldklarnda, davranlar tamamen deiiklik gsterir. Bu husus,

mhendislikte sklkla karlalan bir durum olup yorulma olarak

adlandrlr.Yklemestatikolmaktanziyadedinamikiseyapdinamiide

gz nnde bulundurulmaldr nk zellikle ykleme frekans band

genise yapnn rezonans frekanslarn tahrik etme olasl yksektir.Bu

trdurumlarda,yapnnyklemeyekartepkisidorusalolmayacaktr.Bu

yzden,butarzproblemlerinincelenmesinde,yapnndinamikzelliklerini

dedikkatealanfrekanstemelliyorulmaanaliztekniklerikullanlr.Ayrca,

titreimkaynaklyorulmayntemihzl,kullanlveuygulamaskolaydr.

Butezde,titreimkaynaklyorulmateorisiincelenmektedir.Bualmaiin

gerekletirilen aratrma srecinde esas ama, ortalama deeri sfrdan

farkl yklemeler iin test uygulamalarndan kaynaklanan problemleri,

ortalama gerilme dzeltme tekniklerine yeni bir bak as getirerek

zmektir. Ortlamas sfr olmayan bir girdi yknn sebep olduu


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yorulma hasarna yaknbir hasarmeydana getirecek ve ortalamas sfr

olacak ekildemodifiye edilmi bir girdi yk oluturmak iin yenibir

yntemgelitirilmitir.Ortalamagerilmedzeltmeformlnktgerilme

g spektrum younluk datasna uygulamak iin matematiksel bir

prosedr nerilmitir ve modifiye edilmi girdi ykleme g spektrum

younluk datas elde edilmitir. Ayrcabu yntem, ok ynl ykleme

uygulamalar iin de kullanlacak ekilde gelitirilmitir.Ortalamas sfr

olmayanilenmemi girdiyksetininyarattyorulmahasarnayaknbir

hasar oluturacak ve ortalamas sfr olacak ekilde dalga deerleri

modifiye edilmi biryklemeg spektrumyounluk setibtngerilme

elemanlarn dikkate alarak ve tmmatrisleri kullanarak elde edilmitir.

nerilen tekniklerin etkinlii eitli rnek almalarla ve yorulma

testleriyledeerlendirilmitir.

Anahtar kelimeler: Titreim Kaynakl Yorulma, Ortalama Gerilme

Dzeltmesi,GSpektrumYounluk,SonluElemanMetodu


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Tomydearfamily,withloveandgratitude...


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ACKNOWLEDGEMENTS

Iamvery thankful tomy thesissupervisors,Assoc.Prof.Dr.SerkanDa,

andAsst. Prof.Dr. Yiit Yazcolu,whose valuable guidance, technical

support,andcontributions throughout thisstudyenabledme tocomplete

this thesiswork successfullyandgavemeenthusiasm indevelopingnew

techniquesinscientificstudies.

Iampleased toexpressmy special thanks tomy family for theirendless

support and help, especially tomymotherwho supportedmemorally

throughout my life. I also heartily thank to my fianc, and colleague,

MehmetErsinYmer,withmydeepestappreciationforhisunderstanding,

encouragementandwillingassistance.

I am alsograteful toTBTAKSAGE, especially to StructuralMechanics

Division, for their technicalguidance inacademicstudies. Iwould like to

thanktoDr.A.SerkanGzbykandDr.zge enwho inspiredmeto

achievenewsolutionstotheproblemsarisingfromtestapplications.Also,I

sincerely thank to my colleagues Ahmet Akbulut and Harun Davut

Kendzlerfortheirhelpinmanufacturingtestequipments.

Lastly,Ioffermyregardstomycolleaguesandfriendswhosupportedme

inanyrespectduringthecompletionofthisthesis.


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TABLEOFCONTENTS

ABSTRACT.............................................................................................................iv

Z............................................................................................................................vi

ACKNOWLEDGEMENTS...................................................................................ixTABLEOFCONTENTS.........................................................................................xLISTOFFIGURES................................................................................................xiiLISTOFTABLES.................................................................................................xviLISTOFSYMBOLS............................................................................................xviiCHAPTERS

1.INTRODUCTION...............................................................................................11.1 MetalFatigueDamage............................................................................11.1.1 PropertiesofFatigueFailure...........................................................21.1.2 MajorFatigueDamageAccidentsDuringInService..................51.2 ScopeoftheThesis..................................................................................91.3 OutlineoftheThesis.............................................................................102.LITERATURESURVEY...................................................................................132.1 HistoricalOverviewofFatigue...........................................................132.2 BackgroundofVibrationFatigue........................................................163.FATIGUETHEORY.........................................................................................243.1 TimeDomainFatigueApproaches.....................................................253.1.1 StressLife(SN)Approach............................................................253.1.2 StrainLifeApproach......................................................................373.1.3 CrackPropagationApproach.......................................................373.2 FrequencyDomainApproach.............................................................373.2.1 NarrowBandApproach................................................................48


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3.2.2 EmpiricalCorrectionFactors........................................................503.2.3 SteinbergsSolution........................................................................513.2.4 DirliksEmpiricalFormulation....................................................52

4.ANEWMETHODFORGENERATIONOFALOADINGHISTORY

WITHZEROMEAN............................................................................................534.1 Theory.....................................................................................................554.2 CaseStudy..............................................................................................574.2.1 FiniteElementAnalyses................................................................59

5.GENERATIONOFAZEROMEANEXCITATIONFORMULTIAXIAL

LOADING ....................................................................................................705.1 Theory.....................................................................................................715.2 CaseStudies...........................................................................................745.2.1 CaseStudyITestComparison..................................................745.2.2 CaseStudyIIMultiaxialLoading............................................90

6.CONCLUSIONSANDDISCUSSION.........................................................102REFERENCES.....................................................................................................106

APPENDICES

A.TESTFIXTURE...............................................................................................113

B.COMPARISONOFSOLIDANDSHELLFEM..........................................116


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LISTOFFIGURES

FIGURES

Figure1.1Schematicsectionthroughafatiguefractureshowingthethree

stagesofcrackpropagation[4].............................................................................3Figure1.2StageI:CrackInitiation....................................................................3Figure1.3Afatiguefailuresurface[52]............................................................4Figure1.4Failureofarailwaytrackcomponent[52].....................................5Figure1.5LusakaAccident,1977[8]................................................................6Figure1.6ChicagoAccident,1979:a)Anamateurphoto,as theaircraft

rollingpasta90bankangle,b)explosionastheairplaneimpacts[9]..........7Figure1.7ACrackedFlangeofAmericanAirplane[8]................................8Figure1.8AlohaAccident,1988[10]................................................................9Figure1.9OutlineoftheThesis........................................................................12Figure3.1TimeDomainvs.FrequencyDomainSchematicRepresentation

.................................................................................................................................24Figure3.2StressCycles;(a)fullyreversed,(b)offset...................................26Figure3.3StandardformofthematerialSNcurve[52]............................27Figure3.4SNcurvesforferrousandnonferrousmetals[52]..................28Figure3.5MeanStressModificationMethods[1].........................................31Figure3.6BlockLoadingSequence................................................................33Figure3.7BroadbandRandomLoading........................................................34Figure3.8RainflowCycles[52].......................................................................35Figure 3.9 Rainflow Counting Example: a) TimeHistory b) Reduced

Historyc)RainflowCount[52]..........................................................................36Figure3.10FourierTransformation................................................................40


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Figure3.11SchematicRepresentationofPowerSpectralDensity.............41Figure3.12TimeHistoriesandPSDs.............................................................42Figure3.13ProbabilityDensityFunction......................................................44Figure3.14PSDMomentsCalculation...........................................................47Figure3.15ExpectedZerosandExpectedPeaksintheTimeHistory......47Figure3.16NarrowBandSolution.................................................................49Figure4.1PlateSubjectedtoBaseExcitation................................................55Figure4.2InputAccelerationLoadingTimeHistory..................................58Figure4.3InputAccelerationLoadingPSD..................................................59Figure4.4FirstModeShape(1stBending)oftheCantileverBeam............60Figure4.5SecondModeShape(1stTorsion)oftheCantileverBeam........60Figure4.6ThirdModeShape(2ndBending)oftheCantileverBeam.........61Figure4.7FourthModeShape(2ndTorsion)oftheCantileverBeam........61Figure4.8FifthModeShape(3rdBending)oftheCantileverBeam...........62Figure4.9FrequencyResponseAnalysisResultsof theCantileverPlate

forFirstFiveNaturalFrequenciesfornode511(LogarithmicScale)...........63Figure4.10LoadandBoundaryConditionsofStaticAnalysis..................64Figure4.11StaticvonMisesStressontheCantileverPlate........................64Figure 4.12 Modified Input Loading and Original Input Loading for

CantileverPlate.....................................................................................................66Figure4.13FatigueLifeforthePlate(inseconds): a)withoutMeanStress

Correction..............................................................................................................66Figure4.14FiniteElementModelofDisplayingthePlateNodeNumbers

.................................................................................................................................68Figure5.1Dimensionsofthetestspecimen(Dimensionsin[mm])..........74Figure5.2Verticaltestconfiguration.............................................................76Figure5.3Horizontaltestconfiguration........................................................76


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Figure5.4AccelerationInputPSD..................................................................77Figure5.5OriginalandcorrectedaccelerationinputPSDs........................78Figure 5.6 Differencebetween corrected andoriginal acceleration input

PSDs(Corrected Original)................................................................................78Figure5.7Finiteelementmodel......................................................................79Figure5.8Fatiguelifefortheplate:withoutmeanstresscorrection.........80Figure5.9Fatiguelifefortheplate:withoutmeanstresscorrection(close

up)...........................................................................................................................80Figure5.10Fatiguelifefortheplate:withmeanstresscorrection.............81Figure5.11Fatigue lifefortheplate:withmeanstresscorrection (close

up)...........................................................................................................................81Figure 5.12 Fatigue life for the plate:withmean stress correction via

proposedmethod.................................................................................................82Figure 5.13 Fatigue life for the plate:withmean stress correction via

proposedmethod(closeup)...............................................................................82Figure 5.14 Fatigue life for the plate: without mean stress correction

(closeup,logarithmicscale)...............................................................................84Figure5.15Fatigue lifefortheplate:withmeanstresscorrection (close

up,logarithmicscale)...........................................................................................84Figure5.16Fatigue life for theplate:withproposedmethod (closeup,

logarithmicscale)..................................................................................................85Figure5.17Damagedspecimen1:withoutmeanstress..............................87Figure5.18Damagedspecimen1:withoutmeanstress (closeup)..........87Figure5.19Damagedspecimen2:withmeanstress....................................88Figure5.20Damagedspecimen2:withmeanstress(closeup).................88Figure5.21Damagedspecimen3:withproposedmethod.........................89Figure5.22Damagedspecimen3:withproposedmethod(closeup)......89


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Figure5.23Fintypeaerodynamicsurface.....................................................91Figure5.24FEMofthefintypeaerodynamicsurface.................................92Figure5.25Inputtimehistoryinxaxis.........................................................92Figure5.26InputPSDinxaxis.......................................................................93Figure5.27Inputtimehistoryinyaxis.........................................................93Figure5.28InputPSDinyaxis.......................................................................94Figure5.29Inputtimehistoryinzaxis.........................................................94Figure5.30InputPSDinzaxis.......................................................................95Figure5.31CorrectedandOriginalInputPSDsinxaxis...........................96Figure5.32CorrectedandOriginalInputPSDsinyaxis...........................96Figure5.33CorrectedandOriginalInputPSDsinzaxis...........................97Figure5.34Fatiguelifeforthefin:withoutmeanstresscorrection...........98Figure5.35Fatiguelifeforthefin:withmeanstresscorrection.................98Figure5.36Fatiguelifeforthefin:correctedwiththeproposedmethod 99Figure5.37Fatiguelifeforthefin:withoutmeanstresscorrection (close

up)...........................................................................................................................99Figure5.38Fatiguelifeforthefin:withmeanstresscorrection (closeup)

...............................................................................................................................100Figure5.39Fatigue life for the fin:with theproposedmethod (closeup)

...............................................................................................................................100FigureA.1TestFixture...................................................................................113FigureA.2TestFixtureSpecimenAssemblyTypes................................114FigureA.3FirstModeShapeoftheTestFixture........................................114FigureA.4SecondModeShapeoftheTestFixture...................................115FigureA.5ThirdModeShapeoftheTestFixture......................................115FigureB.1PlateDimensions...........................................................................116


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LISTOFTABLES

TABLES

Table4.1DimensionsofthePlate...................................................................55Table4.2ComparisonsofFatigueLifeValues..............................................68Table5.1InputPSDData.................................................................................77Table5.2Lifevaluesfromanalysis.................................................................83Table5.3Lifevaluesfromtests.......................................................................86Table5.4Meanvalueoflifefromtests...........................................................86Table5.5Comparisonoflifevaluesfromtestandanalysis........................90TableB.1ConstantAfora/b=5.......................................................................116

TableB.2NaturalFrequenciesErrors............................................................117


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LISTOFSYMBOLS

S :Stress

nN, :Numberofcycles

a :Cracksize

aS :Alternatingstressamplitude

rS :Stressrange

mS :Meanstress

maxS :Maximumstressamplitude

minS :Minimumstressamplitude

R :Stressratio

b :Basquinexponent

C :Materialconstant

oS :Fatiguelimit

eS :Endurancelimit

iK :Stressconcentrationfactor

fK :Fatiguestrengthconcentrationfactor

q :NotchSensitivity

ak

:Surfaceconditionmodificationfactorbk :Sizemodificationfactor

ck :Loadmodificationfactor

dk :Temperaturemodificationfactor

ek :Reliabilityfactor

fk :Miscellaneouseffectsmodificationfactor


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'

eS :Laboratorytestspecimenendurancelimit

aS :Alternatingstress

'

aS :Equivalentalternatingstress

mS :Meanstress

uS :Ultimatetensilestrength

yS :Tensileyieldstrength

[ ]DE :Expecteddamage

PSD :Powerspectraldensity

pT :Period

)(ty :Timehistory

nno BAA ,, :Fouriercoefficients

PSD :Powerspectraldensity

)(fy :Frequencyspectrumoftimehistory

FFT :FastFourierTransform

IFFT :InverseFourierTransform

)(fH :Transferfunction

)(* fH :Complexconjugateoftransferfunction

)(fGr

:ResponsestressPSD

)(fGi :InputaccelerationPSD

)(Sp :Probabilitydensityfunction(PDF)ofrainflowstressranges

nm :ThenthspectralmomentofthePSDofstress

)(PE :Expectednumberofpeakspersecond

)0(E :Expectednumberofupwardzerocrossingspersecond

:Irregularityfactor

)(fGrm :ModifiedoutputstressPSD


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CHAPTER1

INTRODUCTION

The aerospace industrys increasing demands for durability, safety,

reliability, long life and low cost lead in high interest in studies for

improvingstrength,quality,productivity,and longevityofcomponents in

engineeringscience.Consequently,theseproductsshouldbedesignedand

tested for sufficient fatigue resistance over a desired range of product

populationstosatisfytheneeds.

1.1 MetalFatigueDamage

Thebehavior ofmachine partsbecomes totally differentwhen they aresubjected to fluctuating loading rather than static one. Often, machine

membersfailundertheexcitationofrepeatedoralternatingstresses;yetthe

most careful analysis shows that the actualmaximum stresses are well

below the ultimate strength of the material, and quite frequently even

below the yield strength. Themost distinguishing characteristic of these

failuresisthatthestressesrepeat inacyclicmannervery largenumberoftimes. Such a failure is called a fatigue failure [1]. In otherwords,metal

fatigueistheprogressiveandlocalizedstructuraldamagethatoccurswhen

amaterialissubjectedtoalternating,cyclic,varyingloading.


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1.1.1 PropertiesofFatigueFailure

Afatiguefailurelookslikeabrittlefracture,sincethefracturesurfacesare

flat and perpendicular to the stress axis without necking occurring.

However,fracturecharacteristicsoffatiguefailurearequitedifferentfrom

the staticbrittle fracture in the aspect that fatigue failure comes out in

stages (Figure1.1). InForsythsnotation [2], [3],Stage I is the initiationof

oneormoremicrocracksandthisstagescrackpropagationisanextension

of the initiatedmicrocrackwithout change ofdirection.Hence, a Stage I

crackpropagateswithinaslipbandthat isonaplaneofhighshearstress

(Figure 1.2).The term fatigue crack initiation sometimes involves Stage I

fatigue crack growth. A Stage I crack turns into a Stage II crack as it

achieves a critical length. It changes itsdirection and starts topropagate

normaltothemaximumprincipaltensilestress.Afterthetransition,Stage

II,thecrackpropagatesthroughoutthemajorityofthecrosssection.More

descriptive terms, microcrack and macrocrack, are sometimes used, for

Stage I and II, respectively. Similarly, unqualified references to fatigue

crack propagation refer tomacrocrack propagation.After two preceding

stages, finally, the cross section so shrinks that even one load cycle is

sufficienttoconstitutetheconditionsforfailure.Thisprocessissometimes

called Stage III and canbe resulted from crack propagationbybrittle

fracture,ductilecollapseorboth.


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Figure1.1Schematicsectionthroughafatiguefractureshowingthe

threestages

of

crack

propagation

[4].

Figure1.2StageI:CrackInitiation


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Most of the static failures give a visible warning in advance, such as

developinga largedeflectionduetotheexceedingofyieldstrengthofthe

material,before the total fracture occurs. However, fatigue failures are

suddenand totalusuallywithoutawarning, thereforedangerous.Figure

1.3 showsapost fatigue failure surface at thepointof failure.Figure 1.4

showsafailurecrackwhichoccursforarailwaycomponent.

Figure1.3Afatiguefailuresurface[52]


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Figure1.4Failureofarailwaytrackcomponent[52]

1.1.2 MajorFatigueDamageAccidentsDuringInService

The first known disastrous fatigue failure accident is the France railway

accidenttakesplacein1842[5].Thebodyoftheleadingenginefallsdown

since the front axle of the front, fourwheeled engine collapses suddenly

because of metal fatigue. Then, the second enginebreaks up. Follower

carriagespassoverruinwhichstartsafireandresultsinmajorlossoflife.

Thesekindsoffailuresoccurringonrailwaytransportationpushscientists

startextensiveexaminationsonthenatureofmetalfatigue[5],[6],[7].

Comet I aircraft GALYP disintegrates after 3680 flight hours and 1286

flightcyclesat30,000feetaltitudein1954sincefatiguecrackinitiatesatthe

cornerofautomaticdirectionfindingwindow[8].


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ABoeing707300runbyDANAirlosesthecontrolofpitch,then,theright

handhorizontalstabilizerandelevatorseparatesinflightafter16,723flight

cyclesand47,621hoursin1977(Figure1.5).Therearspartopchordandis

exposed to longterm fatigue damage. Besides, the redundant fail safe

structurecannotsupporttheflightloadsforaperiodlongenoughtoenable

thefatiguecracktobedetectedduringaroutineinspection[8].

Figure1.5LusakaAccident,1977[8]

Left engine and pylon of Douglas DC10No.22, operatedbyAmerican

Airlines, separates from thewingas theairplane liftsoff the runwayon

takeoff and the airplane crashes, in Chicago, in 1979. The accident


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investigation [9] shows that captains control panel isdisabledwhen the

engineseparates.Theseveredhydraulic linesallowed theslatson the left

wing to gradually retract. The stall speed on the left wing increases

noticeably.Astheaircraftsspeeddecreases,theleftwingaerodynamically

stallsduetoitscleanconfiguration,whiletherightwingcontinuestogain

liftwhileitsslatsarestillintakeoffposition.Sinceonewingstallsandthe

otherwing generates full lift continuously, the airplane eventually rolls

past a 90bank, and fallsdown to ground (Figure 1.6 Figure 1.7).The

reason of the crash is stated as the crackwhich occurs due to improper

maintenance.

a) b)Figure1.6ChicagoAccident,1979:a)Anamateurphoto,astheaircraft

rollingpasta90bankangle,b)explosionastheairplaneimpacts[9]


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Figure1.7ACrackedFlangeofAmericanAirplane[8]

Boeing737200aircraftMSN152belongingtoAlohaAirlinesisexposedto

explosivedecompressionat24,000feetaltitudeafter89,681flightcycles in

1988 (Figure 1.8). The accident is causedby fatigue failurewhich takes

placeinlapspliceatoneofthestringersthatisacoldbondedandriveted

joint. Fatigue failure occurs because of knife edge effect due to deep

countersunkandthereforesubsequentmultiplesitedamage[8].


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Figure1.8AlohaAccident,1988[10]

1.2 ScopeoftheThesis

Lifetimepredictionassessmentforcomponentsunderrandomloadingisan

importantconcerninengineering.Thestudycarriedoutinthisthesisaims

to investigate vibration fatigue approach. Throughout the research, the

ultimate goal is to bring new perspectives to mean stress correction

techniquesandfindsolutionstoproblemsarisingfromtestapplicationsfor

theloadingswithnonzeromeanvalues.

Infatiguecalculations,meanstresseffectresultingfromthemeanvalueof

the loading can be taken into account rather easily but experimental

verification is troublesome because any mean value of an acceleration

excitationother than 1g in thedirectionofgravity isdifficult to simulate

withtraditionaltestingequipmentslikevibrationshakers.Toovercomethis

situation,amethodisdevelopedtocreateamodifiedinputloadinghistory


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withazeromeanwhichcauses fatiguedamageapproximatelyequivalent

to that createdby input loadingwith a nonzeromean.Amathematical

procedure isproposed to implementmean stresscorrection to theoutput

vonMisesstresspowerspectraldensitydata.Furthermore,thismethod is

extendedtogeneratealoadinghistorysetinmultiaxiswithzeromeanbut

modifiedalternatingstress,which leads in fatiguedamageapproximately

equivalent to the damage causedby the unprocessed loading set with

nonzero mean, taking all stress components into account using full

matrices. The efficiency of proposed solutions is discussed throughout

severalcasestudiesandfatiguetests.

1.3 OutlineoftheThesis

Chapter1beginswithpreliminaryinformationonfatiguedefinition,metal

fatigue mechanism, fatigue failure and the importance of design

considering fatigue criterion illustrating major fatigue accidents duringservice.

InChapter2, theemphasis ison literaturesurvey.Historicaloverviewof

fatiguetheorywiththemilestonesofthefatiguephenomenonispresented

in this chapter in a chronological order. Furthermore, a subchapter is

completely separated to investigate the progress in vibration fatigueapproachmentioningtheresearchandstudiescarriedoutinthisfield.

Chapter 3 is devoted to fatigue theory. Time domain methods and

frequencydomainmethodsarediscussed.Stresslifeapproachisexamined

indetailwheremeanstresseffectsonfatiguelifeandcountingmethodsare


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presented.Besides,stresslifeapproachandcrackpropagationapproachare

mentionedbriefly.Moreover,frequencydomainvibrationfatiguetheoryis

examined indetailwhere narrowband solutions andwideband solutions

arediscussed.

InChapter4,anewmethod forgenerationofa loadinghistorywithzero

mean isproposed,whoseefficiency isdiscussed throughoutacasestudy.

The method aims to obtain zero mean loading for a structure,

approximately equivalent in fatigue damage that is subjected to random

loadingwithnonzeromeanmakinguseof the frequencydomain fatigue

calculationtechniquesandoutputstresspowerspectraldensity.

Chapter 5 is dedicated to the extension of the technique proposed in

Chapter 4.The goal is to generate a loading historywith zeromean for

multiaxial loading. A multiaxial input loading with zero mean but

modified alternating stress,which creates fatigue damage approximately

equivalent to the damage causedby the unprocessed loading set with

nonzero mean, is extractedby means of the frequency domain fatigue

calculation techniques and output stress power spectral density. The

efficiencyofproposedapproach isdiscussed throughoutcasestudiesand

fatiguetests.

Finally,Chapter6summarizestheworkdonethroughouttheresearch,and

evaluatestheoutcomesandcontributionstotheliterature.

SchematicrepresentationofthethesisoutlineisgiveninFigure1.9.


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Figure1.9OutlineoftheThesis


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CHAPTER2

LITERATURESURVEY

2.1 HistoricalOverviewofFatigue

The first fatigue examinations seems to havebeen reportedbyAlbert, a

miningengineer,whocarriedoutsomerepeatedloadingtestsonironchain

in1829[11].

As railway service started to improve rapidly throughout the nineteenth

century, fatigue failures of railway axlesbecome awidespread problem.

This situation requires that the cyclic loading effect shouldbe taken into

consideration. The firstmajor impact of failures resulting from repeatedstressesappears in therailway industry in the1840s. It isrecognized that

railroadaxles failsregularlyatshoulders [12].Then, theremovalofsharp

cornersisrecommended.

The term fatigue is introduced to explain failures happening due to

alternating stresses in the 1840s and 1850s. The first usage of thewordfatigue in print comes into viewby Braithwaite, although Braithwaite

statesinhispaperthatitiscoinedbyMr.Field[13].Then,ageneralopinion

starts todevelop insuchaway that thematerialgets tiredofbearing the

load or repeating application of a load exhausts the capability of the

materialtocarryloadwhichsurvivestothisday[4].


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14

AugustWhler, aGerman railway engineer, sets and performs the first

systematic fatigue examination from 1852 to 1870. He carries out

experiments on fullscale railway axles and also on small scale torsion,

bending, and axial cyclic loading test specimens for differentmaterials.

WhlersdataforKruppaxlesteelareplottedasnominalstressamplitude

versus cycles to failure.Thispresentation of fatigue life leads in the SN

diagram. Furthermore,Whler indicates that the range of stress ismore

importantthanthemaximumstressforfatiguefailure[11],[14].

Gerber, Goodman and some other researchers examine the influence of

meanstressinloadingthroughout1870sand1890s.

Bauschinger [15] points out that the yield strength in compression or

tensiondecreasesafterapplyinga loadof theoppositesign thatresults in

inelastic deformation. It is the first indication that a single exchange of

inelasticstraincouldalterthestressstrainbehaviorofmetals.

EwingandHumfrey[16]studyonfatiguemechanismsinmicroscopicscale

observing microcracks in the early 1900s. Basquin [17] represents

alternatingstressversusnumberofcycles to failure (SN) in the finite life

regionasalogloglinearcorrelationin1910.

Grififth [18], an important contributor to fracture mechanics, presents

theoretical calculations and experiments onbrittle fracturebymeans of

glassinthe1920s.Hestatesthattherelation aS =constant,whereSisthe

nominalstressatfractureand a isthecracksizeatfracture.


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Palmgren [19] introduces a linear cumulative damagemodel for loading

withvaryingamplitude in1924.Neuber [20]demonstratesstressgradient

effects at notches in the 1930s.Miner [21] formulates linear cumulative

fatigue damage criterion proposedby Palmgren in 1945 which is now

knownasPalmgrenMinerlineardamagerule.

Irwin [22] introduces the stress intensity factorKI,which isknownas the

basis of linear elastic fracturemechanics and the origin of fatigue crack

growth life calculations. The Weibull distribution [23] grants a two

parameter and a threeparameter statistical distribution for probabilistic

fatiguelifeanalysisandtesting.

MansonCoffin relationship [24][25] investigates the relationshipbetween

plastic strain amplitude at the crack tip and fatigue life. The idea is

developed by Morrow [26]. Matsuishi and Endo [27] formulate the

rainflowcounting algorithm to determine stress ranges for variable

amplitudeloading.

Elber [28] develops a quantitativemodelwhich figures crack closure on

fatigue crack growth in 1970. Paris [29] shows that a threshold stress

intensityfactorcouldbeobtainedforwhichfatiguecrackgrowthwouldnot

occurin1970.

Throughout the 1980s and the 1990s, many researchers study on the

complexproblemof inphaseandoutofphasemultiaxial fatigue,critical

planemodelsareproposed.


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2.2 BackgroundofVibrationFatigue

Rice [30] develops the very important relationships for the number of

upwardmean crossings per second and peaks per second in a random

signal expressed solely in terms of their spectralmoments of the power

spectraldensityof thesignal.This is the firstseriouseffortatprovidinga

solutionforestimatingfatiguedamagefrompowerspectraldensity.

Bendat [31] presents the theoreticalbasis for the socalledNarrow Band

solution. The expression used for the estimation of expected value of

damage is defined only in terms of the spectralmoments of the power

spectraldensityuptothefourthdegree.

Manyexpressionsareproposedtoimprovenarrowbandsolution[31]and

adapt it tobroadband type.Mostof themaredevelopedwith regards to

offshoreplatformdesignwhere interest in the techniqueshas existed for

manyyears.Themethodsareproducedbygeneratingsampletimehistories

frompowerspectraldensitiesusingInverseFourierTransformtechniques.

Thenprobabilitydensityfunctionofstressrangesisestimatedempirically.

The solutionsofWirsching etal. [32],ChaudhuryandDover [33],Tunna

[34],Hancock[35],andKamandDover[35]formulasareallderivedusing

mentioned technique. They are all expressed in terms of the spectralmomentsofpowerspectraldensityuptothefourthdegree.

Steinberg [36] presents three band technique, based on Gaussian

distribution, for fatigue failure under random vibration for electronic

components. It is a simplified approachwhich does not require a large


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17

computationaleffort.Thereforeitisvaluableasatimesavingmethod.The

proposedmethodisbasedonalargevolumeoftestdatathatwasarranged

andrearranged toverify thatasimplification indeterminingstressranges

distribution is usable to analyze random vibration fatigue failureswith

reasonableaccuracy.

Dirlik[37]developsanempiricalclosedformexpressionfordetermination

oftheprobabilitydensityfunctionofrainflowrangesdirectlyfrompower

spectral density. The solution is obtained using extensive computer

simulationstomodelthesignalsusingtheMonteCarlotechnique.

Wuetal.[38]workonagroupofspecimensmadeof7075T651aluminum

alloytoexaminetheapplicabilityofmethodsproposedintheestimationof

fatigue damage and fatigue life of components under random loading.

Fatigue test results illustrate that the fatiguedamage estimatedbasedon

thePalmgrenMiner rule canbe improvedby applyingMorrowsplastic

work interactiondamage rule.Furthermore, randomvibration theory for

narrow band process is utilized to estimate fatigue life where the

probability density function of the stress amplitudes is determined by

means of Rayleigh distribution for a zeromean narrowband Gaussian

randomprocess.

Bishopetal.[39]presentastateoftheartperspectiveofrandomvibration

fatigue technology. Several design applications are presented. Time and

frequency domain methods are investigated and compared performing

finiteelementanalysisusingamodelofabracket,whichisfullyfixedatthe

position of a round hole inside. Time domain and frequency domain


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calculationsshowagreement ifa transientdynamicanalysis iscarriedout

with time domain fatigue life calculations for the excitations whose

frequencyrangeincludesthenaturalfrequenciesofthespecimen.Also,itis

stated that the time and frequency domain processes are actually very

similar.Theonlydifferencesarethestructuralanalysisapproachused(time

orfrequencydomain)andthefactthatafatiguemodelerisneededasatool

toextracttherainflowcyclehistogramfromthePSDofstress.

Liouetal.[40]studythetheoryofrandomvibrationwhichisincorporated

into the calculationof the fatiguedamageof the component subjected to

variableamplitudeloadingmakinguseofthefundamentalstresslifecycle

relationship and different accumulation rules. The purpose is to derive

readytouseformulasforthepredictionoffatiguedamageandfatiguelife

when a component is subjected to statistically defined random stresses.

Morrowsplasticworkinteractiondamageruleisconsidered,inparticular,

in thederivation inwhich themaximumstresspeakshouldbe taken into

considerationanditgivesratheraccuratefatiguelifepredictioninthelong

liferegionandslightlyconservativelifeintheshortliferegionwiththeuse

ofMinersrule.Moreover,aseriesoffatiguetestsareconductedandsome

oftheanalyticalinadditiontoexperimentalresultsarepresentedtoverify

theapplicabilityofthederivedformulas.

Pitoisetetal.[41]makeuseofthefrequencydomainmethod,directlyfrom

a spectralanalysis for the estimationofhighcycle fatiguedamageunder

randomvibrationleadingmultiaxialstresses.Thismethodisderivedfrom

anewdesignationof thevonMises stress as a randomprocesswhich is

usedasanequivalentuniaxialcountingvariable.Also,thisapproachcanbe


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19

generalized to include a frequencydomain formulation of themultiaxial

rainflowmethodforbiaxialstressstates.Furthermore,aprocedureforthe

generation ofmultiaxial stress tensor histories of a given PSDmatrix is

described.

Petrucci [42] studies the fatigue life prediction of components and

structuresunderrandomloadingmainlyconcerningthegeneralproblemof

directlyrelatingfatiguecycledistribution to thepowerspectraldensityof

thestressrangesbymeansofclosedformexpressionsthatavoidexpensive

digital simulations of the stress process. Themethod illustrates that the

statisticaldistributionof fatiguecyclesdependson fourparametersof the

PSD calculated from spectral moments making use of numerical

simulations and theoretical considerations while the present methods,

proposedtoobtainstresscycledistribution,arebasedontheuseofasingle

parameterofthePSDwhich is irregularityfactor.Theproposedapproach

givesreliableestimatesofthefatiguecycledistribution,withtheuseofthe

rangemean countingmethod,when the stress process has narrowband

derivatives.

Petrucci et al. [43] develop Petruccis approximatedmethod [42] for the

high cycle fatigue life prediction of structures subjected to Gaussian,

stationary,wideband random loading.Fatigue lifeof componentsunderuniaxialstressstatecandirectlybeestimatedfromthestresspowerspectral

densityofanyshape.Inparticular,approximatedclosedformralationships

between the spectralmomentsof the stresspower spectraldensityof the

fatiguecyclesoftheGoodmanequivalentstress,andtheirregularityfactor

alsowithabandwidthparameteraredetermined.Theaccuracyof fatigue


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20

lifepredictionsobtainedwith thismethod is comparedwith someof the

frequencybased techniques in literature and found tobemore accurate

thansomeofthem.

Pitoiset etal. [44]dealwith frequencydomainmethods todetermine the

highcycle fatigue life of metallic structures under random multiaxial

loading. The equivalent vonMises stressmethod is reviewed and it is

assumedthatthefatiguedamageundermultiaxialloadingcanbepredicted

bycalculatinganequivalentuniaxialstressonwhich theclassicaluniaxial

randomfatiguetheoryisapplied.Themultiaxialrainflowmethod,initially

formulated in the time domain, canbe implemented in the frequency

domaininaformallysimilarway.Theconsistencyoftheresultsisverified

by means of a time domain method based on the critical plane.

Furthermore, it is showed that frequency domain methods are

computationally efficient and correlate fairlywellwith the time domain

methodintermsoflocalizingthecriticalareasinthestructure.Afrequency

domainimplementationofCrosslandsfailurecriterionisalsoproposed;it

isfoundingoodagreementandfasterthanitstimedomaincounterpart.

Tovo [45] studies fatiguedamageunderbroadband loadingby reviewing

analytical solutions in the literature states and verifies the relationships

betweenvariouscyclecountingmethodsandavailableanalyticalsolutionsofexpected fatiguedamage in the frequencydomain.Furthermore,anew

method isdeveloped starting from knowledge ofpower spectraldensity

distributions for rainflow damage estimation which gives accurate

approximationsoffatiguedamageunderbothbroadbandandnarrowband

Gaussian loading. The proposed technique is based on the theoretical


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21

examination of possible combinations of peaks and valleys in Gaussian

loadingandfitsnumericalsimulationsoffatiguedamagewell.

Siddiqui et al. [46] states that a tension leg platform, a deep water oil

exploration offshore compliant system, is subjected to fatigue damage

significantlybecauseof thedynamic excitations causedby the oscillating

waves and wind over its design life period. The reliability estimation

against fatigue and fracture failuredue to random loading considers the

uncertainties associatedwith the parameters and procedures utilized for

the fatigue damage assessment. Calculations for fatigue life estimation

require a dynamic response analysis under various environmental

loadings.Anonlineardynamicanalysisoftheplatformisimplementedfor

responsecalculations.Theresponsehistoriesareemployedforthestudyof

fatigue reliability analysis of the tension legplatform tethersunder long

crestedrandomseaandassociatedwind.Fatiguedamageoftetherjointsis

estimatedusingPalmgrenMinersruleandfracturemechanicstheory.The

stressrangesaredescribedbyRayleighdistribution.Firstorderreliability

method andMonteCarlo simulationmethod are employed for reliability

estimation. The influence of various random variables on overall

probabilityoffailureisstudiedthroughsensitivityanalysis.

Tuetal.[47]conductBGAsolderjointsvibrationfatiguefailureanalysisthatarereflowedwithdifferent temperatureprofiles,andagingat120 oC

for1,4,9,16,25,36days.Also,Ni3Sn4andCuSnintermetalliccompound

(IMC) effects are consideredon the fatigue lifetimedetermination in this

work.Duringthevibrationfatiguetests,electrical interruption isfollowed

continuously to see the failure of BGA solderjoint. The results of the


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experiments indicate that the fatigue lifetime of the solderjoint firstly

increasesandthendecreaseswithincreasingheatingfactor,integralofthe

measured temperatureover thedwell timeabove liquidus (183 oC) in the

reflowprofile.Furthermore,solderjoint lifetimedecreasesalmost linearly

asfourthrootoftheagingtimeincreases.Accordingtothetestresults,the

intermetallic compounds contributemainly to the fatigue failure ofBGA

solderjoints.Also,thefatiguelifetimeofsolderjointisshorterforathicker

IMClayer.

Benasciutti et al. [48] work on fatigue damage caused by wideband

stationary and Gaussian random loading. Expected rainflow damage is

estimatedapproximatelyforagivenstationarystochasticprocessdescribed

infrequencydomainbyitspowerspectraldensity.Numericalsimulations

onpowerspectraldensitieswithdifferentshapesareperformedinorderto

establish proper dependence between rainflow cycle counting, linear

cumulative fatiguedamageandspectralbandwidthparameters.Expected

rainflow fatigue damage is taken as weighted linear combination of

expected damage intensity givenby the narrowband approximation of

Rychlik[49]andexpectedrangecountingdamageintensityapproximately

obtainedbyMadsen et al. [50]with theweighting factor approximated

using Tovos approach [45] dependent on PSD through bandwidth

parameters. Comparison is made between numerical results and someanalyticalpredictionformulaeavailableforfatiguedamageevaluation.

Aykan [51] analyzes helicopters selfdefensive systems chaff/flare

dispenserbracketby applying vibration fatiguemethod as a part of an

ASELSAN project. Operational flight tests are performed to obtain the


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acceleration loadingasboundaryconditions.Theaccelerationversus time

data is gathered and converted into power spectral density to obtain

frequency components of the signal. Bracket is modeled in the finite

elementenvironmentinordertoobtainthestressesforfatigueanalysis.The

validityofdynamiccharacteristicsofthefiniteelementmodelisverifiedby

conducting modal tests on a prototype. As the reliability of the finite

elementmodel is confirmed, stress transfer functions that are frequency

response functions are found and combined with the loading power

spectral density to acquire the response stress power spectral density.

Narrowbandandbroadbandvibration fatiguecalculation techniquesare

applied and compared to each other for fatigue life.The fatigue analysis

resultsareverifiedbyacceleratedlifetestsontheprototype.Furthermore,

the effect of single axis shaker testing for fatigue on the specimen is

obtained.


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CHAPTER3

FATIGUETHEORY

Fatigueisexaminedintwodomainsbasically.Firstoneisthetimedomain

methodsand thesecondone is the frequencydomainmethodswhichare

similarinapproachFigure3.1andexplainedindetailinthischapter.

Figure3.1TimeDomainvs.FrequencyDomainSchematic

Representation


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26

Figure3.2StressCycles;(a)fullyreversed,(b)offset

aS :Alternatingstressamplitude

rS :Stressrange

mS :Meanstress

maxS :Maximumstressamplitude

minS :MinimumstressamplitudeR :Stressratio, maxmin/ SS

Themostcommontestspecimeninthepastisacylinderwithlittlechanges

of crosssection, andwith a polished surface in the regionwhere crack

formation tends to start. It is loaded inbending and calledWhler test

which has limitations. Now, a cylinder loaded in axial tension is the

preferredtestspecimenwithfreeofsuddenchangesofgeometryandalso

with a polished surface at the section where cracks are likely to start.

Severalidenticalspecimensareexaminedandthenumberofcyclesneeded

fortotalseparationisacceptedasN.Load,notstress,iskeptconstantinthe

test. For each specimen a nominal stress, S, is calculated from elastic

formulaeandtheresultsareplottedastheunnotched NS curvewhichis

abasicmaterialproperty[52]. N isplottedonthexaxisinthelogarithmic


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scalewhile Sconstitutestheyaxisplottedinlinearorlogarithmicscale,but

logarithmic isbecoming thenorm.Themean line in the finitelife region

(10000 to 10million cycles) is usually straight Figure 3.3, and presented

with the convenient equation where the inverse slope of the line is b,

namelyBasquinexponent,andCisrelatedtotheinterceptontheyaxis.

Figure3.3StandardformofthematerialSNcurve[52]

(3.1)

NS plotofsomemetals,mostlylowalloysteels,isconstitutedfromtwo

lines.The linemaybecomehorizontal and thematerial is said tohave afatigue limit, 0S , which is important when the aim is infinite life. For

materials which do not show a clear fatigue limit, tests are generally

terminatedatbetween 710 and 810 cycles.Thecorresponding S isnamed

asanendurance limit,eS ,at thespecifiedN,Figure3.4.There isnostrict

convention about theuse of the terms fatigue limit and endurance limit,

bSCN = .


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andtheoriginaltestdocumentationshouldbeobservedifthedifferenceis

acriticalissue[52].

Figure3.4SNcurvesforferrousandnonferrousmetals[52]

3.1.1.1 StressConcentrationandNotchSensitivity

NS curvepropertiesofamaterialareextractedforthespecimensthatare

free from geometrical factors that cause high stress gradients and create

localhigh stress regions.However, componentsdesigned for real life canhaveholes,grooves,changesofsectionetc.whichleadinlocalhotspotsof

highstress.Then,fatiguestartsatthesepoints.Therefore,theireffectmust

betakenintoconsiderationinlifecalculations.Geometricalpropertiesthat

causehighlocalstressesarenamedasnotchesand,thestressconcentration

factor, tK ,isusedtohandlethiseffect[52].


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(3.2)

fK is a reduced value of tK and generally called fatigue strength

concentrationfactoranddescribedby(3.3)[1].

(3.3)

Then,notchsensitivity, q,isdefinedas(3.4)[1].

(3.4)

For simple loading, it is convenient to reduce the endurance limitby

dividing theunnotched specimen endurance limitbyfK ormultiplying

thereversingstressbyfK [1].

3.1.1.2 EnduranceLimitModifyingFactors

Itisnotrealistictoexpecttheendurancelimitofastructuralparttomatch

the values obtained in the laboratory [1].All effects of endurance limit

modifyingfactorsareinvestigatedintheequationgivenbelow(3.5).

(3.5)

where

ak :surfaceconditionmodificationfactor

notchthefromremotestressNominal

notchtheofregionin thestressMaximum=tK

1

1

=

t

f

K

Kq

specimenfreenotchinStress

specimennotchedinstressMaximum=fK

'

efedcbae SkkkkkkS =


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bk :sizemodificationfactor

ck :loadmodificationfactor

dk :temperaturemodificationfactor

ek :reliabilityfactor

fk :miscellaneouseffectsmodificationfactor

'

eS :laboratorytestspecimenendurancelimit

eS : endurance limit at the critical location of amachine part in the

geometryandconditionofuse.

3.1.1.3 MeanStressEffect

Fatigue lifedependsessentiallyon theamplitudeof stress existing in the

component,but amodification is neededwhen amean value of stress

exists.Componentscancarrysome formofdead loadbefore theworking

stresses are applied or the input loading can have a mean value. Thegeneraltrendintraditionalmethodsissuchthattheallowableamplitudeof

fatigue stress gets smaller as themean stressbecomesmore tensile for a

givenlife[52].

Thebasic idea istoassumethatmeanstress lessens theallowableapplied

amplitude of stress in a linearway. It is expected that any fatigue load

cannot be carried when the mean stress reaches the ultimate tensile

strengthofthematerial.Iffatiguestrengthatanymeanisknown,theline

representing fatigue life canbedefined.This is a commonly known and

mostly used method and called Goodmans Rule. [52]. There are other

meanstresscorrectionapproaches, likeGerberandSoderbergthatmodify


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alternatingstressaccordingtomeanstress.Meanstresscorrectionmethods

graphicalrepresentationsareillustratedinFigure3.5.

Figure3.5 MeanStressModificationMethods[1]

GoodmansModel:

(3.6)

GerbersModel:

(3.7)

1'

=+u

m

a

a

S

S

S

S

1

2

' =

+

u

m

a

a

S

S

S

S


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SoderbergsModel:

(3.8)

aS :Alternatingstress

'

aS :Equivalentalternatingstress

mS :Meanstress

uS :Ultimatetensilestrength

yS :Tensileyieldstrength

3.1.1.4 VariableAmplitudeLoadingandFatigueDamage

The investigation of metal fatigue under variable amplitude loading is

based on the study of cumulative damage. The variable amplitude time

history consists of 1n cycles of amplitude 1S , 2n of amplitude 2S , 3n of

amplitude 3S and so on.Usually the pattern repeats itself after a small

number of stress cycles,nS (Figure 3.6). The sequence up to nS is then

calledblock,andtheaimistodeterminethenumberoftheblocksthatcan

beapplieduntil failureoccurs.Themethod commonlyused isknown as

Miner or PalmgrenMiner rule. According toMiners hypothesis, lineardamage accumulation is assumed. First,when the 1S level of stress that

repeats 1n cycles isconsidered, thenumberof lifecyclesatstress level 1S

thatwould cause fatigue failure ifno other stresseswerepresent canbe

obtained from NS curve . Calling this number of cycles as 1N , it is

assumedthat 1n cyclesof 1S useupafraction 11/Nn ofthetotalfatiguelife.

1' =+

y

m

a

a

SS

SS


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This shows the damage fraction that 1n cycles cause. The total damage

fractionforoneblockisobtainedbydoingasimilarcalculationforallother

stressesandsummingall thedamage fractionresults [52].Then,expected

damage, [ ]DE , is equated to this summation and set to one to calculate

fatigue life (3.9). The order of stresses occurring in history is ignored in

Minersrule.

Figure3.6BlockLoadingSequence

(3.9)

3.1.1.5 CountingMethods

Most of the engineering components in real life are exposed to stress

responses that are more complex than shown in Figure 3.6. When

individual stress cycles causing fatigue damage cannotbe distinguished

[ ] 1...1 2

2

1

1 =+++=== n

nn

i i

i

N

n

N

n

N

n

N

nDE


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easily, as inbroadband random loading (Figure 3.7), a cycle counting

method is needed to determine discrete cycles, and hence allow the

applicationofMinersrule.

Figure3.7BroadbandRandomLoading

Rainflow counting, peak counting, level crossing counting and range

countingprocedures [53]aremethodscommonlyused to identify loading

cycle occurrences, amplitudes and mean values in a time history.

Developmentoftheseempiricalcyclecountingmethodsismainlybasedon

a trialanderror,andallhaveshortcomings [4].Themost reasonableand

widelyusedcyclecountingmethodistherainflowcounting[54].Itderives

its name from the first practical algorithm, developedbyMatsuishi and

Endo [27], inwhich it is imagined thatwater flowsdown the load time


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history,with the axes reversed.An algorithm is developed for counting

Rainflowcycles.Themostcommonprocedureisasfollows[52]:

Peaksand troughs fromthe timesignalareextractedsothatallpointsbetweenadjacentpeaksandtroughsarediscarded.

Thebeginningandendofthesequencearearranged tohavethesamelevel.Thesimplestway is toaddanadditionalpointat theendof the

signaltomatchthebeginning.

The highest peak is found and the signal is reordered so that thisbecomes thebeginning and the end. Thebeginning and end of the

originalsignalhavetobejoinedtogether.

Procedure is startedat thebeginningof the sequenceand consecutivesetsoffourpeaksandtroughsarepicked.Aruleisappliedthatstates,

If thesecondsegment isshorter (vertically) than the first,and the third is

longerthanthesecond,themiddlesegmentcanbeextractedandrecorded

asaRainflowcycle.Inthiscase,BandCarecompletelyenclosedbyAand

D(Figure3.8).

Figure3.8RainflowCycles[52]


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Ifnocycleiscountedthenacheckismadeonthenextsetoffourpeaks,i.e.peaks2to5,andsoonuntilaRainflowcycleiscounted.Everytime

aRainflowcycleiscountedtheprocedureisstartedfromthebeginning

ofthesequenceagain.

AnexampleofRainflowcyclesextractionispresentedin Figure3.9.

a)

b)

c)

Figure3.9RainflowCountingExample:a)TimeHistory

b)ReducedHistoryc)RainflowCount[52]


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3.1.2 StrainLifeApproach

Whenloadingcyclesaresevere,anothertypeoffatiguebehavioremerges.

In this regime, the cyclic loadsare relatively largeand lead in significant

amountsofplasticdeformationresultinginrelativelyshortlives.

In this approach, StrainCycle curve shouldbe used rather than Stress

Cycle curve ( NS ) to obtain fatigue life. StrainCycle curve can be

obtainedby cyclic testswhere the strain isheld constantvia closed loop

controltestequipment.

3.1.3 CrackPropagationApproach

When an initial crack exists in the structure, then crack propagation is

followed. The crack growth analytical calculation is doneby means of

LinearElasticFractureMechanicstheory.

3.2 FrequencyDomainApproach

Whenthe loading isnotstaticbutdynamic, thedynamicsof thestructure

should be taken into account. There is high possibility to excite the

resonance frequenciesof thestructure if the loadingfrequencyhasawidebandwidth.When this situation occurs, it cannotbe assumed that the

structures response to the loading will remain linear in the frequency

domain.Therefore,toovercomesuchsituations,frequencydomainfatigue

analysismethodsareappliedwhichdonotneglectthevibrantpropertiesof

thestructure.


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Ageneralwayofrepresentingtherandomdatainthefrequencydomainis

using Power Spectral Density (PSD) which is an alternative way of

specifyingthetimesignal.ThePSDillustratesthefrequencycontentofthe

timesignal.ItisobtainedbyutilizingFourierTransformation.

Aperiodic timehistory, )(ty , canbe representedby the summation of a

seriesofsineandcosinewavesofdifferentamplitude,frequencyandphase

whichisthebasisofFourierseriesexpansionexpressedby(3.10)[55].

(3.10)

wherepT denotesforperiodand

(3.11)

(3.12)

(3.13)

0A , nA and nB are Fourier coefficientswhich yield information about the

frequency content of the time history. 0A denotes themeanvalue of the

timehistoryasnA and nB show theamplitudesof thevarious sinesand

cosineswhichconstitutetimehistorywhenaddedtogether.

=

+

+=

1

0

2sin

2cos)(

n p

n

p

n tT

nBt

T

nAAty

=2

2

0 )(1

p

p

T

Tp

dttyT

A

=

2

2

2cos)(

2p

p

T

T pp

n dttT

nty

TA

=

2

2

2sin)(

2p

p

T

T pp

n dttT

nty

TB


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39

To express the time series with its frequency content, a transformation

between the timeand frequencydomain isachievedbymeansofFourier

transformation.Inthissense, )(fy entailsadescriptionofthetimehistory

)(ty inthefrequencydomain f .TheFouriertransformpairgivenin(3.14)

and(3.15)enablestransformationsbetweenthetwodomainseffectively.

(3.14)

(3.15)

As well as the integral form, the Fourier transformation can also be

depicted in adiscrete form.Thisusage ismostly favorablebecause time

historiesaregenerallymeasured inadiscrete,digitized formwithequally

spacedintervalsintime.Intheseconditions,theintegralformisdifficultto

process. Therefore, some numerical calculations are carried out on the

measuredtimehistoryto transform it intothefrequencydomainwhich is

called Discrete Fourier Transform. The result obtained from a discrete

transformationisliabletoapproximatetheresultobtainedfromanintegral

transformationasthesamplelengthandsamplingfrequencyincrease[11].

Cooley andTukey [56]develops a very rapiddiscrete Fourier transform

algorithm called Fast FourierTransform (FFT) and has a reverseprocess

namedtheInverseFourierTransform(IFFT). ThediscreteformofFourier

transformationisgivenby

(3.16)

(3.17)

= dtetyfy tfi )2()()(

= dfefyty tfi )2()()(

kN

ni

k

k

p

n etyN

Tfy

=2

)()(

nN

ki

n

n

p

k efyT

ty

=2

)(1

)(


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wherepT istheperiodofthefunction )( kty and N isthenumberofdata

pointsforFouriertransform.

Random time histories canbe expressed in the frequencydomain under

certain circumstances. A random time history cannot be periodic by

definition,but it canbe expressed in the frequency domain if the time

historyistakenfromastationaryrandomprocess.Astationarytimehistory

isdescribedasatimeserieswhosestatisticsarenotinfluencedbyashiftin

thetimeorigin. Inotherwords, thestatisticswhichbelongto timehistory

y(t) are the same as those of time historyy(t+) for all values of . The

statistics obtained from a sampled time history are representative of the

statistics of the whole random process. Figure 3.10 illustrates the

applicationoftheFouriertransformationappliedtoarandomtimehistory

takenfromastationaryrandomprocess.

Figure3.10FourierTransformation


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41

PowerSpectralDensity(PSD)isawayofillustratingtheamplitudecontent

ofatimesignal inthefrequencydomainasaspectrum.It iscalculatedby

takingsquareofthemodulusoftheFFTanddividingby2timestheperiod,

pT asshownin(3.18).

(3.18)

Only the amplitudes are retained in PSD but phase information is

discarded.

TheareaundereachspikeinFigure3.11representsthemeansquareofthe

sinewaveat that frequencyand the totalureaunder thePSDcurvegives

themeansquareofthetimehistory.

Figure3.11SchematicRepresentationofPowerSpectralDensity

PSD shows the frequency content and the type of the time history. For

example,a sinusoidal time signal comesoutasa single spike centeredat

2)(

2

1n

p

def fyT

PSD =


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42

the frequencyof the sinewaveon thePSDplotand the spike shouldbe

infinitelytallandinfinitynarrowforapuresinefunctiononthePSDgraph

intheorywhileitalwayshasafiniteheightandfinitewidthbecauseasine

wave has a finite length.However, the areaunder thePSD graph is the

interest.Broadband timehistoryappearsas awidebandpeakor several

peaks covering a wide range of frequencies in the PSD plot while

narrowbandprocesscomesoutascentered,narrowbandplotwhenPSDis

calculated.Whitenoiseincludessinewavesofthesameamplitudeateach

frequency(Figure3.12).

Figure3.12TimeHistoriesandPSDs


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43

In linear systems, the output is related to the inputby a linear transfer

function. The response of a linear system to a single random process is

obtainedbymeans of transfer functionsobtainedvia frequency response

analysis of the structure where for each frequency a different transfer

functioniscalculated:

(3.19)

where )(fFFTr

and )(fFFTi standsforFFToftheresponse,stress in this

thesis,andFFToftheinputloading,accelerationinthisthesis,respectively.

)(fH representsthetransferfunctionofthesysteminfrequencydomain.It

isconvenient toobtain the response, stress,asaPSD, )(fGr (3.20). If the

input,acceleration,isalsoexpressedasPSD, )(fGi ,thenthePSDofoutput

isgivenbymeansofalineartransferfunction, )(fH ,itsconjugate, )(* fH

and input PSD in the frequency domain where )(* fFFTi is complex

conjugateoftheinputFFT(3.21).

(3.20)

(3.21)

Then,(3.21)canberewritten intheformof(3.22),where )(fGr isequalto

inputPSDtimesmodulussquaredoftransferfunction.

(3.22)

)()()( fFFTfHfFFT ir =

( ))()()()(2

1)(

**fFFTfHfFFTfH

TfG ii

p

r

=

)()()()( * fGfHfHfG ir =

)()()( 2

fGfHfG ir =


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44

StressPSDisdirectlyusedforvibrationfatiguelifecalculationswhichwill

beexplainedintheproceedingpartsofthisthesisindetail.

In vibration fatigue calculations, the frequency content of the stresses is

accounted forby a probability density function (PDF) of rainflow stress

ranges, )(Sp .Themost convenientway,mathematically,of storing stress

rangehistograminformationisintheformofaprobabilitydensityfunction

of stress rangesFigure 3.13.Thebinwidth, dS, and the totalnumberof

cyclesrecordedinthehistogram,tS ,arerequiredtotransformfromastress

rangehistogramtoaPDF,orback.

Figure3.13

Probability

Density

Function

Theprobabilityofthestressrangeoccurringbetween2

dSSi and

2

dSSi+

iscalculatedasthemultiplicationof ( )iSp and dS while togetPDFfrom


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45

rainflowhistogram,eachbinheight isdividedby dSSt .Then,numberof

cyclesattheparticularstresslevel S, )(Sn isstatedasin(3.23).

(3.23)

Thetotalnumberofcyclesatthestresslevel S thatcausesfailureaccording

toWoehlercurve, )(SN ,isexpressedasin(3.24)whereC and b standfor

materialconstantandBasquinexponentrespectively.

(3.24)

Then, fatigue life calculation isperformedbyutilizing (3.25) that equates

(3.26)inturnwheretotalnumberofcyclesinrequiredtime,tS ,isequalto

[ ]TPE . T is the fatigue life in seconds and [ ]DE is expected value of

damage.Fatigue life is found setting [ ]DE tounity.Theorderof stresses

occurring in history is ignored inMiners rule.However, this does not

createaproblemforvibrationfatiguesolutionsincetheloadingsstatistical

properties are important and the loading is random, stationary and

Gaussian.

(3.25)

(3.26)

( )dSStSpSni =)(

bi S

CSN =)(

[ ] [ ] ( )

==0

)(

)(dSSpS

C

TPE

SN

SnDE b

i i

i

[ ] ( )

==0

)()( dSSpS

CS

SNSnDE bt

i i

i


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46

To calculate the probability density function of rainflow stress ranges,

severaldifferentempiricalsolutionsareproposedbyresearcherstoobtain

the )(Sp fromoutputstressPSDandcalculatefatiguelifewhichconstitutes

thebasis of vibration fatigue approach. Statistical properties of random

historyshouldbeexaminedtoobtain )(Sp .

Randomstress timehistoriescanbedescribedusingstatisticalparameters

properly.Anysampletimehistorycanberegardedasonesamplefroman

infinite number of possible samples which can occur for the random

process.Eachtimesamplewillbedifferentbutthestatisticsofeachsample

shouldbeconstantaslongasthesamplesarereasonablylong[39],[57].

Twoof themost important statisticalparameters areundertakenbyRice

[30],whicharerelationshipsforthenumberofupwardmeancrossingsper

secondandpeakspersecondinarandomsignalusingspectralmomentsof

thePSD.Thisisthefirstseriouseffortatprovidingasolutionforestimating

fatiguedamagefromPSDs.

ThenthspectralmomentofthePSDofstress,whichisnm ,isdefinedasthe

summationofeachstripsarea times the frequency raised to thepowern

(3.27). It isnecessary todivide thePSD curve into small stripes tomake

thesecalculationsasshowninFigure3.14.

(3.27)

=

==0 1

).(..).(m

k

kk

n

k

n

n ffGfdfffGm


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47

Figure3.14PSDMomentsCalculation

Expected number of peaks per second )(PE and expected number of

upward zero crossings per second, )0(E , in the time history, whose

schematic representation are shown in Figure 3.15, are calculated using

spectralmomentsup to 4thdegree, 0m , 1m , 2m , 3m , 4m as formulated in

(3.28)and(3.29).

Figure3.15ExpectedZerosandExpectedPeaksintheTimeHistory


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48

(3.28)

(3.29)

Another important statisticalparameter is irregularity factor, ,which is

theratioofexpectednumberofzerospersecondoverexpectednumberof

peaksper second. It isameasureof frequencyband thehistory coveron

PSDgraphinthesensethatastheirregularityfactorapproachestoone,the

processgets closer tonarrowbandprocess,andas the irregularity factor

approachestozero,theprocessgetsclosertobroadbandprocess. Inother

words,avalueofone corresponds toanarrowband signalwhichmeans

that thesignalcontainsonlyonepredominant frequencywhileavalueof

zero implies that the signal contains an equal amount of energy at all

frequencies.

(3.30)

All these statistical information gathered from the stressPSD isused for

vibrationfatiguecalculations.

3.2.1 NarrowBandApproachThe first frequency domainmethod on calculating fatigue damage from

PSDsiscallednarrowbandapproach.Bendat[31]assumesthatallpositive

peaks in the timehistory is followedbycorresponding troughsofsimilar

magnitude regardless of whether they actually formed stress cycles, in

0

2

2

4

]0[

][

mmE

m

mPE

=

=

40

2

2

][

]0[

mm

m

PE

E==


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49

otherwords,itassumesthatthePDFofpeaksisequaltothePDFofstress

amplitudes.Negativepeaks andpositive troughs in the time history are

ignored. For wide band response data, the method overvalues the

probability of large stress ranges. Therefore, calculated fatigue damage

becomesconservative.InFigure3.16, thetimeseriesshown isacceptedas

the black history instead of blue one. Probability density function

calculated according to narrowband solution comes out as (3.31) and

expecteddamageequation(3.26)turnsinto(3.32).

(3.31)

(3.32)

Figure3.16NarrowBandSolution

( ) 02

8

04

m

S

em

SSp

=

[ ] [ ]

==

0

8

0

0

2

4)(

)(dSe

m

SS

C

TPE

SN

SnDE

m

S

b

i i

i


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50

3.2.2 EmpiricalCorrectionFactors

Manyexpressionsareproposedtocorrecttheconservatismofnarrowband

solution. Generally, they are constituted by generating sample time

histories from PSDs by means of inverse Fourier transformation. A

conventionalrainflowcyclecountisobtainedfromthesedatasets.

The solutionsofWirsching et al [32],Chaudhury andDover [33],Tunna

[34]andHancock [35]arealldevelopedmakinguseof thisapproachand

areallexpressedintermsofthespectralmomentsuptotheforthdegree.

WirschingsEquation:

(3.33)

where

(3.34)

(3.35)

(3.36)

HancocksEquation:

(3.37)

This solution is proposed in the form of an equivalent stress range

parameter,hS ,where

(3.38)

[ ] [ ] ( ) ( )( )( )( ))(11 bcNB babaDEDE +=

( ) bba 033.0926.0 =

( ) 323.2587.1 = bbc

21 =

+

= 1

222 0

bmS

bbb

h

( )

= dSSpSS bbh


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51

ChaudhuryandDoverEquation:

(3.39)

TunnaEquation:

(3.40)

3.2.3 SteinbergsSolution

Steinberg [36]proposesa threeband techniqueas a rough estimation for

the fatigue damage calculation. This method isbased on the Gaussian

distribution. The instantaneous stress values in the time history are

assumedforthe3bandswhere standsforstandarddeviation:

1 valuesoccur68.3%ofthetime



Therefore, the fatigue damage is estimatedbased on these three stresslevels.This approach is a very simple solutionbased on the assumption

that no stress cycles occur with ranges greater than 6 rms values. The

distribution of stress ranges is then specified to follow a Gaussian

distribution.Thismethod ispreferred intheelectronics industry,asatool

forcomparison.

[ ] [ ] bhSCTPEDE =

( ) 02

8

04

me

m

SSp

=


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52

3.2.4 DirliksEmpiricalFormulation

Abetter approach is to approximatePDFdirectly from thePSDwithout

using the narrow band approach as a starting point because PDF of

rainflow stress ranges is the issue controlling fatigue life. Dirlik [37]

develops an empirical closed form expression for the PDF of rainflow

ranges,which isobtainedusingextensivecomputersimulations tomodel

thesignalsusingtheMonteCarlotechnique.Dirlikssolutionisillustrated

intheequation[11].

(3.41)

where

2

11

2

1

1

23

213

2

1122

2

1

4

2

0

1

2

4

0

2

0

1

)(25.1

11

1

1

)(2

][]0[

][

]0[

2

DD

DxR

D

RDDQ

DDDR

DDD

xD

m

m

m

mx

m

mPE

m

mE

PE

E

m

SZ

m

m

m

i

i

+

=

=

=

+=

+

=

=====

DirliksempiricalformulaforthePDFofrainflowstressrangesisshownto

be far superior, in terms of accuracy, then the previously available

correctionfactorsfornarrowbandsolutionconservatism[59].

0

23

22

21

2)(

2

2

2

m

eZDeR

ZDe

Q

D

Sp

iii Z

iR

Z

iQ

Z

i

++=


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53

CHAPTER4

ANEWMETHODFORGENERATIONOFALOADINGHISTORY

WITHZEROMEAN

Whenacomponent issubjected toanexcitation thathasanonzeromean,

thismeanvaluecouldhaveanoteworthyeffectonthestressesandthuson

thefatiguelifeofthecomponent.Infatiguecalculations,meanstresseffect

canbetakenintoconsiderationrathereasilybutexperimentalverificationis

troublesomesincewithtraditionaltestingequipmentlikevibrationshakers,

anymeanvalueofanaccelerationexcitationotherthan1ginthedirection

of gravity is difficult to simulate. In the content of this thesis study, a

methodisdevelopedtocreateamodifiedinputloadinghistorywithazero

mean which causes fatigue damage approximately equivalent to thatcreated by input loading with a nonzero mean. For this purpose, a

mathematicalprocedureisdevelopedtoapplymeanstresscorrectiontothe

output vonMises stress power spectral density data.Amodified input

acceleration power spectral density is generated by means of transfer

functions calculated via frequency response analysis. Thiswill enable to

performfatiguetests.Themathematicalapproachdevelopedtomodifytheoutput von Mises stress power spectral density takes the multiaxial

stressesintoaccountatthepointofconsiderationandprovidestheneeded

input loadingpowerspectraldensity.Thecorrected inputwithzeromean

value enables experimentalverificationwhile themeanvalue, other than

1g,oftheinputcannotbeappliedinstandardvibrationtests.


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54

Timedomainfatiguelifecalculationtechniquesdirectlyutilizemeanstress

correctionmethods to amplify the stress ranges to take themean stress

effectintoaccount.Infrequencydomainfatiguelifecalculationapproaches,

when the power spectral density of the stress is determined, only the

variablestresspart isconsidered,however, theconstantmeancomponent

isignored[60].Inthisstudy,themeanstresscorrectionisaccomplishedon

vonMisesstresspowerspectraldensitytobeabletoincludetheloadings

biaseffect.GoodmancorrectiontechniqueispreferredtoGerbersinceitis

reportedin[61]thatGerberrelationshipgenerallygiveshigherfatiguelives

compared to the experimental results. Dirlik method is chosen for

application of vibration fatigue approach. According to [59], Dirlik

approach leads tobetter results in comparison to the corresponding time

domainandfrequencyresponsemethods.

Analuminum (2024T4)plate fixedandexcitedatoneend isused for the

implementation which is shown schematically in Figure 4.1, whose

dimensionsaregiveninTable4.1.Thebaseexcitationleadstoafullythree

dimensionalmultiaxialstateofstressintheplate.

It is shown that the developedmethod is useful in creating amodified

inputaccelerationdatawithazeromean that can simulatedamage foraselected point in the structure under consideration. Equivalent input

loading obtainedbymeans of the aforementionedmethod is convenient

andconservativeforexperimentalapplications.Besidesbeingatimesaving

method,itissuitabletobeimplementedforanytypeofloading.


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55

Figure4.1PlateSubjectedtoBaseExcitation

Table4.1DimensionsofthePlateLength 750mm

Width 100mm

Thickness 5mm

4.1 Theory

The developed method to calculate zero mean loading for a structure,

approximatelyequivalentinfatiguedamage,whichissubjectedtorandom

loading with nonzeromean, isbased on the frequency domain fatigue

calculationtechniques.

TocomputeazeromeaninputaccelerationPSD,whichismentionedasthe

modifiedaccelerationinputPSD,Gim(f),firstofall,meanstresscorrectionis

performed on the vonMises stress output PSD, Gr(f).Goodmansmean

stress correction shown in Equation (3.6) is used to obtain themodified

outputstressPSD,Grm(f).NotethatGoodmansformulaiswidelyusedfor

Fixedendoftheplate

Acceleration(BaseExcitation)


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56

mean stress correction for stress amplitudemodification [62],but in this

studyitisutilizedtocorrectPSDofvonMisesstressoutputdata.IfthePSD

issplit intoequalstrips,theareaofeachstripcanbeutilized toobtainan

equivalentsinewave.Theamplitudeofeachequivalentsinewaveisequal

tothesquarerootoftheareatimes 2.Thisisbecauseofthefactthatthe

rootmeansquareofasinewave isequal to itsamplitudedividedby 2,

andtherootmeansquareofeachstripinthePSDisequaltothesquareroot

ofitsarea[39].Inthisstudy,thestressPSDvalue,Gr(f),ismultipliedbythe

corresponding frequencyresolutionand thesquarerootof it is takenand

multipliedwith 2 toobtain thealternatingstressamplitude,Sa, foreach

frequency.Then,meanstresscorrectionisperformedwith(3.6)tocalculate

themodifiedalternatingstressamplitude,Sa.Afterwards, thecalculations

arecarriedoutinthereversedirectiontoreachmodifiedoutputstressPSD,

Grm(f).

Theoretically,thereshouldbeaninputaccelerationPSDthatresultsinthe

modifiedoutputstressPSD,Grm(f),whenappliedtothestructuresincethe

transferfunctionofthestructureremainsthesameregardlessofinputtype.

Thisphysicalrelationcanberepresentedbythefollowingequationwhich

isrewrittenfrom(3.21)foradifferentinputoutputcouple.

(4.1)

Accordingto(3.21);

(4.2)

)()()()( * fGfHfHfG imrm =

1* )()()()( = fxGfGfHfH ir


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57

andfrom(4.1);

(4.3)

To obtainGim(f), (4.2) and (4.3) are equated to each other and reordered

whichresultsin:

(4.4)

The variables in equation (4.4) are vectors,which in turnmakes regular

matrixinversionimpossible.Therefore,theinversecalculationsinequation

(4.4) are carried outby using pseudoinverse.A corrected input loading

with zero mean but modified alternating stress, which creates fatigue

damage approximately equivalent to the damage caused by the

unprocessed loading with nonzeromean, is extractedbymeans of this

technique. It shouldbe noted that, this approach is focusing to find a

modified loading that simulates damage for a selected point on the

structure,whichinturnwillbeacceptableinthevicinityofthatpointonly.

4.2 CaseStudy

The proposed correction method is demonstrated on an applicationanalyzedaccordingtothemethodpresentedinSection4.1.

The plate shown in Figure 4.1 is excited from its cantileveredbasewith

stationary,Gaussian, random, 10 second acceleration timehistorywith a

nonzeromeanvalueof3g,asshowninFigure4.2.Powerspectraldensity

1* )()()()( = fxGfGfHfH imrm

[ ]111 )()()()( = fxGfxGfGfG irrmim


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58

of the input is also shown in Figure 4.3. The input loading consists of

harmonics that possess different amplitudes withbroadband frequency

range,upto2000Hz,involvingthefirst20ofthenaturalfrequenciesofthe

specimen.Thedef

vibration infomation

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