am am te

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D ynamic Charact eristics and M o dal Parameter s Of a Pla t e With a Surfac e Crack

by

@YinChen,D.Eng.,M.Eng.

A lhesiJsubmit ted to theSchool of GraduateStudies in partial fulfillmentof the requirements(or thedegree of

Masterof Engineering

FacultyorEngineering and AppliedScience MemorialUniversity or Newfoundland

~mber.l992

St.John's Newfoundla.nd Canada

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Abstract

Asystematic study of the behaviour of " platewit hi'surfa cecrackis Ci\r rjl'd out usingthefiniteelemen tmethod.Static, stea dystate, resonantandtran sit'll l responses ofthe cracked plate are examined.Strain ,displa cementand al'O:clcralio ll responses without / with the creekarc computed .Modalanalysisis performed to determine the natural freq uencies,mode shape>!andlllr a;n/tlisp iaccllwnl (r(,II'"'lley responsefunctions. Frequencyrespo nsefuudionswitholll/wi~htill'crackaT(' cern- puted.Strain mod e shapes, as well as displaceme ntmodeshapca, arc olJl a ill(xl.TIll' differenceof the strainmode shapes betweentheuncrackedplate1\111\thoI'T;wkl'(l plateis calculated.Each of the parametersmentionedaboveisexa minedtodeter- minethe sensitivityofthisparamete rtothecracktha t occursilltill! !Ilrudurc'.

By using allthe methods ofanalysismentionedabove, andcomparingtheS'~Il' sitlvity ofdifferent parameters to cracking,itis foundtha tthesur facecrack in the structurewillaffect mostof the dynamiccharacteristicssuchas stfllin/stressfield aro un d the crack,naturalfrequ enciesof tile structure,amplitude sor the r,!sl mlls,!

and mode sha pes.Someof themost sensitiveparam ete rs aretiledjfferenccoftill!

strai nmode shapesandthe loca l strain frequency responsefuncti on s.By monitoring thecha ngesin the local strain freq uencyresponsefuncti ons anddifferenceof stra in

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modeshapes, thelocation andseverityoC thecrackthat occursinthestructurecan hedete rmined.Based on tneseresulh ,suitableprocedures Cordetectingthecrack in reAl structuresarc suuested.

iii

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Acknowledgements

Firstof all.Iwould!Ike tothankOr. A.S.J.Swamidu,mysupervisor.Ili~

supervisionandIinanclel assistancearesraciously acknowledged.The linam'ialsup' portfromFacul ty of Engineering andAppliedandtheSchoolofGtaduate Slu,l;,~

isalsogreatlyappreciate d.

rwouldalsoliketoextendmy grati tudetomyfriendsMr.JiugXiaoilndMr.

Bin Zou, fortheir helpin usingtheABAQUSandtheLatex,andforthe many usefuldiscussions withthem.

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sr

7·(

.'i.:1 ImpulseExcitation .. .').:1.1

.').:1.2

6 Conclusion

Refe re n ce s Time lIist ory PowerSpectrum.

vii

so

80 81 87

91

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List of Figures

3.1 Geometric dimen sions ofthe plate. ::!{)

3.2 Finite clementmesh ofthe plate.. 20

3.3 Line springmodelling. 26

3.4 Line spring compliancecalibrationmodel. :'W

3.5 Geometri c profileof thesurfacecrack... '27

3.6 Contou rs of the:normal stress(1rwithoutasurfacecrack :!!I 3.7 Contou rsof the normalstressI7rwitha surfacecrack. '2!1 3.8 Contoursofthe surfacest rainlrwithouta surfaceCrack. :11 3.9 Contours ofthe surfacest rain!rwith a surface crack ... :II 3.10Thenormalized strainlevel decrease as the depthof thesurfacecrack

increases.

3.11Contour s of displacementwof theuncrackedplate.

3.12 Contoursof displac ementwofthecrackedplale.. 3.13Contoursof therot ationd1~withoutthecrack

viii

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3.1-1 Contoursoftherotation 9, withthe crack ... .... ..• 36 .1.15Contours of displacementu atthe referencesurface for a crackedplate 37

1.1 Mode shapesof the uncrackedplate .. 48

·1.2 Accelerat ionFRFsofuncrackedand cracked plet es ...• ... 49 ,1.3 DisplacementFRFs ofuncracked lind cracked plates. 49

4.·1 StrainFRFs,1.6 mm away fromthe crack 55

4.5 StrainFRFs,7.8mm away fromthe crack.. 55

·1.6 StrainFRFs,15.8mm awayfrom the crack... . . 56 4.7 Waterfallplot ofstrainFRFs,7.8mm away fromthe crack.... 58

5.1 Displacement and strain responsesfor the firstresonant excitation frequency

5,2 Displacement andstrain responses forthe secondresonant excitation 63

frequency ... .. .. . . .. . . . 64

5.3 Displacementand strainresponsesfor thethird resonantexcitation

frequency .. . . .. . . 65

5.4 Strai nresponsesoftheuncrackedandcracked plate,at a point7.8 mmaway from thecrack,forthe Restbendingmodalfrequency 66 5.5 Powerspectrumof theresponseswhen excitedatWI •• 69 5.6 Powerspectrumorthe responseswhen excited at W2.... 70

ix

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5.7 Powerspectru mof the responses when excitedatwJ • 71 5.8 Powerspectrum of the strain response,with/without thecrack.. 73 5.9 Powerspectrumof thedisplacementICSpOMC,with/withoutthe creek7~

5.10Firstthreebending strainmode shapes .•. ....

5.11Strainmodeshapechanges due to increaseinthe depthofthe crack. 7i 5.12Differencesof thestrain mode shapeIorthe firstthree bendingmcrlcs78 5.13Time historyoftheexcitation force., . 5.14Displacementr~ ponseunder the impulseexcitat ion. , 5.15Strainresponse under theimpulse excitation.

5.16 Powerspectrumof the displacementresponse 5.17Powerspectrum ofthe strain response

85 85

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List of Tables

3.1 Strain valuest:~for the untrackedand crackedplate around thecrack 32

4.1 Frequ ency changes duetothe growth of thedepthofthe surfacecrack46 1.2 Changesofthe amplitudeof displacementand accelerationF;~.',5at

resonantfrequenciesat the cent re ofthe frccend. 1.3 Changes orthe amplitu de ofstrainFRFs..

50 57

·1.4 Changes in am plitudesofstrain FRFs at nonresonantpoints.. . .. 58

5.1 Changes inthe-Iisplace rnent andstrain responses . . .•.. .. . 82 5.2 Changesin amplitudes of the powerspectrumof thestr ain response

atnonresonantpoints ,7.8awayfrom the crack , ..,• .. 86

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Nomenclature

["II mesamatr ix

IKI

^stiffness^matrix

Ie] dampingmatrix

(F) force vector

IU] ^modal^matrix

[mlmoda/ diagonalmedalmass matrix

[k]•• ,., diagonal modal stiffnessmatrix

(e]m...1 diagonalmodal dampingmatrix

W; ith natur alfrequen cy

.;

ithdisplace mentmodeshape

<N Nthmembe r deformation

strain displacement

<- modal strai nforrthmode

xii

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(GI.(I/] (/1'1 [AI [. ') E

st rain frequencyresponsefunction system matri x

force-strainttanderfunction fOfce-displa.ccn.cnttran !(crfunct ion strain modeshape matrix You ng'smodulus poisson'sratio massdensity

N shapefunction s

[e"] nodaldisplacement stress

o

elasticity mat rix

exte rnalconcen tratednodalforces rotationof theplate T H thickness oftheplate

depthofthe surfa ce crack Laplace transform operator Laplacevariable Fouriervariable

xiii

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H

"

transferfunction ith eigenvalue

xiv

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Chapter 1 Introduction

Structurceunderrepeated regular/irregular loadings inevitably developcracks.For inst ance, offsho restructuresarc subjectedtorepe atedloadingsdue toocean waves whichim pose a.largenumber of cyclicstresses;under these varyingcyclicstresses fatigue cracksdevelopat criticalwelded junction sDCoffshorestructures.Therefore the prediction, detect ion andmonitoring of cracks instr uctur eshave beenthe sub-

jed of intensiveinvestigationsduringrecentyears.Us import anceis shownin the strict guidelinesenforcedonthedesignof struct u resand structuralcompo nents,as wellas onthe maintenanceandrepaircarried out on in-situstructuresfor maxi- mumeffectivenessatminimum cost.Many theoreticaland experim entalstudiesof thebehaviour of struct ureswithcrackshavebeen carried outtodevelop afeasible methodo logy.Thousandsoftheoreticaland experimenta lstudies have been carried

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outinthealliedare a offract uremechanics.r..Ieanwhil e,manytechniqu es have been developedto quantifycra.cki ngin st.ruetures, such as ecoustjcemissio n,mag n clic particleinspection, alt ernati ng / direc t curren tfieldme asurement,edd y carrcnt,ul- trescnlcs an dmodaltesting.

Inrecent years,Non-Dest.ruetlveEvaluation(NOE) hasga i neda greatera.llen·

tion for mon itoringand det ect ingcrack!developed in opera ti ngstruct ures,and sev e raltech niques ha ve beendeveloped,as mentionedbefore,Each ofthesemoth- ode hasitsownadva ntagesanddisadvantages .Newte chniques arebei ng develop ed tomeetthe newrequ irementofstruct uresin more an dmoresevereenvironm e n ts.

Amo ngthese newlyemergingtechniques, the procedur efor mod alana lysisiLll t.!tesl·

ingisbeingexplore dbymanyreeeae c hers todelermine whet heri~can he uscd1\.1 an efficientte chniqu e ,especiallyif the modal strainsaround thecritica lzonesco uld be monitore d.Earlierstudie s onmodal analysisprovedto beineffec ti ve sincethe researchers considered only the Irequen ey shiftsandthes e shiftscould be causedby a num berorfac torssuc hIUcra ck ingand damage,erosion andst rength deg radat ionof the foundatio n,addition orextr aneou s mMSto tht!sub/s upt!rst r ucture,etc.Rece nt st ud iesin modal ana lysishasbeenfoc ussed onthe totaldynam ic behaviour.

Thephys i cal pro pertieaof any str uctureca.nbe expresse-t intermof itsmNl ~, stiffness and damping.Anydamagethat occu rsin thestructu r ewillca.usechan gcs in the ses phy alcalproperties; and thesechangeswillbereflectedinthe chanKcof the

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dy namic characteristicsofthestructure,suc has natural frequencies,mode shapes, displacements,straiJls/slresscs,and damping.This study invest igates thesensitiv- itiesofthedynamic characteristicsof any structu reto crackdevelopmentsoas to developanew tech niqueto monitorandde te ct crac ksorda magesin struct ures, es pecially inolrshore/onshorestructures.As an initial studyin this area of crack llct ectionandpred ictionforsmallcracksusi ng modalanalysis, the scope ofthis stu dyisto determ inethe changein the dynamiccharacterist icsand modal parem- etcrs of aplate witha surfacecrack.lIow dothese parameterschangeas the de pth ofthecrackincreases ?Differentmodes of response he....e been exami nedto relate theglobaland localchangesshownI\Saconsequenceof crack ing.Sta tic,steady stat e,resouantnndtransient(dyna m ic)TeSpOIlSC\lhavebeen consideredas asmall cruckincreas es in depth due to stress cycling.Grea te r emphasishas been laid on 1I11!examin a tionofstrainfrequencyresponsefunct ion s (FR Fs)andstrainmodal sha pes,which arcindependent to the amplitu d eof ap pliedforces,andgiveabetter andmore sensitiveindication ofthe locationand severityorthecrack.

Tile sub j ectma tter ofthe above study is develo ped fur t her inthis thes isas mentionedbelow..viz.•

l,Litera t urereview ofearlie rstudies on damage det ection,andthe latestmethod- ology of exam i ningthemodal straindevelopme ntare givenin Chapte r2.

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2. Thestatic analysis of thestr uctureusing finiteclement analysis is deta iled ill Chapter 3.Strain, displacementandrotationdevelopedaroundalocation, whereinthe possibility{orcracking exists, is investigatedin detailfor the case ofa structurewithout/withthe crack t.o deter minethe limits of changes that could occurin these parameters.

3.The fourthchapterdealswiththe steadystate analysisof astructure without/witha crack. Frequency response functi o ns of surfacest rains ncar the crack,and the accelerationanddisplacementfrequencyresponse functions away fromthecrackarecomputed andcompa redas thecrackgrowl!through the thickness.

4.The resonantand the transientresponse analysesaredetailed in Chapter5to outlinethemodal information thatcouldbe obtainedfro mthese investigations to identifyandlocali ze crackingzones andextentof cracking. Power spectra l responseis alsoanalyzedto determine the changesthat occurduetocracking.

5.The salient conclusionsfromtheabove studiesare summa rizedin Chapter6 and possibleareas forfurtherdetailed inveatlg atien arcalsoout lined.

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Chapter 2 Literature Review

2.1 Nondestruct ive Eval uation and Damage Detection by M odal Analys is

The detectionofcrack!incomponentsof struc tures inserviceisvery essential and importa ntto cn,uretheinte~rityof struct uressinceth~tTackstendto propagateandcaul!e ecdden{ailureswhichareusuallyveryecetly interm s of huma.nlifeandpropertydamage,Therefore nondestructiveevaluation (NDE) techniques arc widelyusedfordamagedetection.Scme cfthe wen-knowntech- niquesareacoustic;emission,magneticparticleinspection,alternating/direct currentfieldme asurement,eddycurrent and ultrasonics,Each of these tech-

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niques has its own advantages and disadvantages.As pointedout hyGome~cl al. (1990)the useof mostof the wellknown NDEtechniques may beincOll\'c, nicntin many situationsdue to the need forthe invcetiget oe to have access to the componentunder operating conditions.Therefore a new NOEtechnique usingthe vibrationalcharacter isticsof structu resvia modalanalysishas been recently considered by manyresearchers.

Shahrivar et al. (1986) determined the natural frequ ency sh ifts ill anoffshore structuredueto the presence

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damages instructu ralelements. Modal testing of an offshoreframed plat formmodel,withgrossly cracked structuralmembers (severedmembers),was performedanda reductioninna!.uralfrequencies and increase in structuralresponseswerereported.

Springer et al.(1990) reportedthat a boxbeamstructure was builtand tested atthe MarshallSpace FlightCenter (MSFC)(or thepurposeof evaluating the abilityof the modal test to detect the presenceofstructuralfaults inspace flight hardware. A finiteelement model of thisstructu re was also developed and usedto simu latethe modal testresults. The results o( finiteclement analysiscomparedfavourably with the test data. Nofurth er resultswere presented in the paper.

Gomesand Silva(1990,1991and1992) conducte da series of studies using

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modalanalysis for crack identificati oninsimplestruc tures. Modal tests of cantileverbeamswith fatiguccracksat diffcrent locat ions for differentcrack depthswert'perfor med.Theintluc n ceof a lumpedmas s,which was attached tothe struct ure (tosimulat etheendloads in struct u res),was alsostudied.

The main modalparametersthat we re monitoredwer e the nat uralfreq uencies ofthe vibratingbeam.A seriesofnondimensional curves and tablesdepicting the changes in nat u ralfrequencies forvarious dept hsof thecrack,location of the crack,andthelumped masswerepresented. Use ofthesetablesand figureswould lead to the identificationof the probabledepthand locatio nof the crackin any can t ileverbeam.

RichardsonandMannen(1991)developedmodalsens it ivity functionsforthe locat ion ofstructu ra l faults due toachange ofmass, st iffnessand/ordamping of structuralcompo nents. Equations ofmotion consideri ngmodes of vibration andstructu ralresp onseweregivenby:

IM ]{ XI'))

+

ICI{.t( .))

+

IK ]{ X(t ))=(Fl' )) (2.1)

where1M]is the massmatrix;[K]isthe stiffness matrix; and[e]isthedamping matrix.

Fora lightlydamp edstructure,thefollowingorthogonality condit ionsare

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valid:

WJ'[·I/Jll'j=(mJ•• ••

IWICJ W! = ['1_.,

(U)' IKJlUJ=(k)•• •,

(2.:1)

12..1)

where(m]",ri.'isthediagonalmed al mans matrix ; IkJ...1i~thediAgollal modal stiffnessmatrix;[eJ....,..'isthediagona lmodalIIAmpingmatrix,:UIIIIUI isthemodaldisplacementmatrix.

Twootherrelat ion shipswhich resul t fromthe orthogonalitycondition,Are:

(2.5)

(2.6 )

and:

lo1llI.do: =

fre quencyand dampinsoftheuncrackedst ruct ure.

Ifonlystiffness chances occurin the structu reduetocrackin&.then F"clns.

(2.4).(2.5)and(2. 6)become:

[U

+

dU)' [K+dKJ{lI

+

dU)

=

[k

+

dkl••••, (2.7)

2du = ~m, ^(2.9)

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where:nl ~=""l ~+d1t

and:""I~' dl~=frequency and dampingof the crackedstructu re.

Scalingthe mode shapesto getunity modalmasses so thatmt ::1{orall modesIk}, andsub t.eaetlngEqn. (2.4) {rom Eqn . (2.7),the s tiffncsssensitivity equationcanbeobtainedas:

(U.

+

dU.)'[dKJ{UddU,)+2{dU.)'[K!{U,)+{dU,jT[KJ{dU.)

= w"'-,,,,'

(2.10) For small srillness changes,where thefault is small en oughsothat themode shapesdon'tchangesubstantially{i.e.,dU.=0), the stiffness sensitivity equa- tiongiveninEqn.(2.10)issimplifiedas:

(2.11)

UsingEqn. (2.10)or(2.11),whenthecha ngein natu ralfre quency ofkth mod eisknown, th ecorres pondin gchangeinstiffnes scan beobtai ned,and vice-versa.Normallycracks inthe structure tend tore ducethestiffness of the structure.FromEqn.(2.10),a natur al frequ ency red uct ion ispredicted .For messanddamping changes.similarsensitivity equati on s were derived intheir paper.

Stubbe etal,(1990 )alsoderivedexpressions for changes in modal sti ffness in termsofmodalmasses,modal damping, eigenfrequencies,eigenvecto rs, and

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10 theirrespectivechanges.

Most researchers have statedthat damagesinstructures lendto reduce the natural frequ encies;consequentlyknowingthe changesinnature lIrcqucucics it is possibleto determine theextent of cracksordama gesin a structure .BuL it is hardto determ inethelocationof thecrackby this procedure since cracks at two differentlocat io ns,associated withcertaincrack lengths ordepths,may cause the same amou ntof frequencyshiftatcertainmodes. HoweverHearn et al.(1991)havepoi nted outintheir paperth atthema gnitud es ofchengcs for differentnaturalfrequenciesis a functionof the severit yandthelocatlon offlawsin thestruct u re.The reforeratios of changesin natllral frequencies, normalizedwit h respectto thelargestfrequency change,are independent

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severityforsmall (Jaws and canserveto indicatethe locatio n ofnaw~di- reed y. The theorywas based onthe conceptthat asinglecrackwillalTccL eachvibration modedifferentl y, havinga strongaffectoncerta in modesand a weak effectonothe rs.Thisdissim ilarity ofeffect on various modes,since itcanbe pre dicted, is thebasi8for tile identification of dama ged members.

The procedurewasstated asfollows: Natu ralvibration frequencies arctohe measured per iodically. When changesin natur alfrequenci esarc observed,the setofratios ofchangesarecom putedandcomparedwit htheva rious member

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11 characte rist ic ralio ensem blesobtaine dfromthe equation:

(2.12)

whereLlw;andLlwJarethe changes of the naturalfrequencyof itbmodeand jth mode.r()~pcclivc1y,due tocracking. fNaretheNthmemberdeform atio ns andhave tile followingrelationshipwithdisplacementmode sha peand5tj fFn~s oftilewhole structure:

(2.13)

whererPiis theith displacemen tmode shape;Mis the mass ofthe structur e;

Kisthe stillnessof thewholestructure;andkNisthestiffnessof Nthmember.

Thelocat ion of the crackisdeterminedby selecting the memberchara cteristic ensem ble that most closelymatch the observedratiosof frequency changes.

This met hod canidentify thelocationof crackedmembers ofthe structure.

Pandyclal. (1991)usedchanges in curvaturemode shap estodetermine thelocationofthedamage. They examined a cantil ever bea m and asimply supported beam.Instead of displacementmode shapes,curvaturemode shapes werecalculated. Absolute differences betweenthe curvaturemodesh apesfor the intactand the damaged beamwere calculated. Thelar gest differences occurredat the damaged point .Thereforethey statedthat absolutedifferences

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12 ofthecurva t uremodeshapes betwee ntheint actand thedamaged stru ctures couldservetoindicatethelocationof damage inthestr ucture.

Recer••lya studyconcerning boththeseveri ty &lid locationof the(ru kin offshorestructureswucarried outbySwam idu and Chen(1992). They usedstrain gauges inthemod altesting, monitoringboththe gl.)balandlocal changesin frequenc y and amplitudeof strai nfrequency response functi ons.

Alsotheycarr ied outafinite elementanalysis (orthe modalresponseofthe str uc tureanddetermi ned the freque ncy respo nse[u ncti ons fordisplaceme nts and accelerations.Theresultsshowed thatlocalsurface strain frequencyre- spon sefunct ions wereverysensit ivetothepresence of crackinginthestructure and wasstronglyrecommended forthe detection ofcrac.kin~inslruclu res.

2.2 Strain Modal Tes ting

As pointedoutintheprev iousparagraph,modalstrains areverysensit ive tolocaldamage.ChenandSwamid.a..s(1992) conductedanexperiment which showed thattheamplitude ofthe strainfrequencyresponsefunctionncarthe crack zonedecreased considerablywhen the crack size increased.They reportedthat in themain columnof a tripodtowerplaHormmodel,witha diameterof 300 mm and athickness of 3mm,the amplitu de of the local

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13 strainFRF atthefirstresonantfrequencydecreasedbymorethan 60%when thecrackbecame a through crack,with a length of90 mm(45⁰inclina tion ).

Theyusedacceleromete rs,linearvariable displacementtransducers(LVDT ), andstra ingauges in their exper iment.Theyproposeda proceduretodete ct cracksinoffshorest ruct ures:fromtheglobal sensors such as accelerometers determinethenaturalfrequency changes that occur in the structure .When there is a changein thenaturalfrequenciesofthestructure,there is a pes- sibilitythat a crack hasst artedto growinthestructure.Then monitor the local sensors (straingauges applied in the critica lareas of thest ruct ur e) to find outthe locationand magnitud eofthecrack,

O.Bernasconi and D.J.Ewins (1989) have examined in detailthebehaviour of modal strain/stressfields.Theydefined"modal strain"from thegeneral theory ofsolid elastodynamics.Itisassumedthatanydisplacementfieldfor thebody may be expandedas a convergentseriesof modeshapes:

(2.14)

whereC.=JvMIJ~.dVande,isthe rthmode shape.

Forsmall displacements , thestrain field can be definedas:

(2. 15) whereD isII.lineardifferentialoperator.

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\.\

Themodalstrain can then be defined as

<,=D(.,) (2.\6)

Usingth is definition, th e strainfrequencyresponsefunction isderived as:

(2.17)

where k istheexcita tionpointandjisthe response point .

Later on,they used this theoryand appliedthe strain modal testingonreal structureswhichincluded:(i)a discontinuous cantileverbeam.(ii) a curved plate,and(ii i)a steamturb ineblad e.The conclusiontheyreachedin their study was that the mass-normalizedmodal strainscouldbe eithercompute d usingfiniteelementmethod,umeasured usingstrain gauges.

Tsang(1990)proposeda governingequationlor strainresponsefunction which was obtainedina mannersimilartothatused fordeterminingaMetofstruc- turalsystemmatricesfro m experimental inertancemeasurementsona.vibrat- ingstru ct ure. Usinga formsimilarto thedisplacementresponse functio n, viz"

(-w'IMI+ [KII {u ("' ))=(I(w))

heproposedusing the following for m:

(-w'IGI

+

IHlH«w))=(I(w))

(2.\8)

(2.\9)

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I .

where[MJand \KJarc mass and stiffness matrices, respectiv ely, jG} and[HI arc definedallsyste mmatricesfor strainwhich did not haveanyphysical inter pr etationyet,«(w)isthe strain response function andf(w)is the forcing funct ion used intheinvestigation. The proposedmethodology was verified usingfiniteclement analysis theory andan algorithm to formul atethe system matrices(01and [HIfrom experimentalstrai ndata was derived.

Lictal. (lU8!)) alsoderived an expressionforthe strain transfer funct ion.

Theirderiva t ionwas also from the vibration equat ion which was the same asEqu.(2.1).Then based onthe relationshipsbetweenstrainand displacement,atheoret ical expressionforthe strainresponse wasder ived.Forone dimensional problems,ithas the form asfollows:

(,I

~

WIIAr'['J(F) =

[H'J(F ) (2.20)

whereIH']istheforce-straintransfermatrixor straintransferfunction;(9]

the displacement modeshape matr ix.[¢<]thestrainmodeshape matrix, and [A]the transfer function matrixfor displacements. Usingthistheory, they obtainedthestraintransferfunctionofa plateand abeam fromthestrain gauge readings. Consecuti vely,they identified the modal parametersand the displacementand the strainmode shapes.Thedisplacement mode shapecb- tninedbythestrai ntransferfunction was compared withthe displacement

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16 modeshape obtainedbyusing the accelerometers, and a good matchWiLS found.

In the studycited earlier Pandy et al. (1991)had used changesin curvature mode shapes to determinethe location ofthedamage;sincethecurvaturei!

proportionalto the surfacestrain, the strainmode shapewouldas wellgivea similarinformationon thedamage that occurs inthestructu re.

2.3 Miscellaneous

Besides the literaturereviewedabove,manyotherresearcher!have carried outstudies onthe behaviour ofstructurewith cracks. Collinsctal.(1992) studied the free andforced longitudinalvibrationsof11.cantilever barwith a crack, and the effect of thecrack locationandcomplianceon the fundamental nlltura lfrequencieswasdetermined . Rajah etaI.(1991)examined the vibra- tiona lcharacteristics of crackedshafts,derivedrelationships bet ween the crack dept hand location on theshaft to changesintheliulfewnaturl\.1 frequencies.

Collinsetal.(1991) report ed theresponse ofII.rotating 'I'imoehcnkoeheft with asingletransverse crack under axial im pulses . Therelationships betweenthe maximum displacementand the crack depth was presented.Chondroset al.

(1989)studiedthe changein thenaturalfrequenciesand modes of vibrationor

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17

the structure wit h .. given geometryof acrack.EarlierChondrosetal.(1980) hi dinvestigatedtherelationshipsbetweenthechangeinnatu ralfrequency orvibrationof.. cant ilever beam and the cruk depththatoccu rredat the

weldedjunc tions.Besidesthesemany otherresearchershavecarriedoutst udies on frequencychangesthat occur instruc turesusingconventionalvibration studies.Mostofthesestudiesconsideredonlythefrequency shifts that occur due to theexistenceandgrowthin crackdepth andwidth ;and mostofthese studies wereon grosscracking in members suchallsevering ofmemb ers, etc.

2.4 Summary

Earlierstudieswhichhave beenreviewedabovehave investi ga tedthebe-

haviourofcrackedstructures in a globalsenile.Moetofthese studies have examinedthe changein natural frequencies of the crecked structure.Some have studied thechange inthe curvaturemodeshapesand the maximumdis- placement. The .ize or thecrack consideredinthese.tudiesis relatively large.

Theuseofstrainmoda l analysisand tes ting, for the det ection ofcrack, in the structure ,is.... relativelynew conceptandveryfewst udies have been re- porledso farusingthistechnique.Thestudy present ed in tbis thesis is focused ona sma llcrackand theconsequent changesthat occur inthe structuralre-

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18 spouse.Besidesthenaturalfrequencies,diaplaccmcnta.modal displacemen t

&ITlp litudesand displace mentmode shapeswhich have been examined in the

earlier studies.otherparameterssuchallstr&inamplitudes,modal strains And strai nmodesh a pesArcexamined.Local para meters...well uglohalparame- ters,have beenexami ned to monito rthechan ges that occurduring thellfowth incrackdepth.

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Chapter 3 Static Analysis of a Plate with a Surf ace Crack

3.1 Finite Element Modelling

Inorder tohave a better unden t an ding:of the behaviour of structures with surfacecracks,a finite element model

o r

a simplecentileverplate ,witha surface crack.Wallinvestigated inthisthesis; the plateWa.!Ifixed at oneend, and{teeat allother three edges.Fig.3.1shewsthe geometricdimensionof theplateandthe location ofthe sur face crack. Thechoiceof thisboundary conditionwas to salisrytherequirementsforan experiment alinvestigation

19

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:.!o

hit:lCII

6SO.00

1 _g _- _6Z.50~

L ^", r"."",

'~2.==============

Figure 3.1:Geometricdimension, orthe plAte

p o,

L -1(..) ("

Figure 3.2:Finiteelementmesh orthe plate

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21

whichwillbecarried out inthencarfutureonasimilar plate.Themat erial ofthe plateis steel witha thickness of 9.525 mm,Young', modulusE= 20h: IO' N/m¹,musdensityp ::::;7860kg/m³,And Poisson's rat ioIl

=

0.29.

The coordinatesystemis showninFig.3.2,with the x-axisalongthelong sideof the plate,and the y-axisalong theshortside.The finite elementmesh isalso shown in Fig.3.2.In order to getbetterresulllAround thecrack.

a filler mesh was int roducedaroundthe crack area. Thegeneral purpose finiteclementcomputer softwareABAQUSwasusedto carryoutthe finite clement analysis.Quadrilatera l shell elements with 8nodesand a 2x2reduced integration.and6node sline springelementswere used. The su rface crackwu simulatedby theline springelements. Inthis part o( thestudy,thecantilever plate is analyzedCoritsstalic responsetoident ifypAtternsfor nondestructive evaluation.

3.1.1 BasicFiniteElem entEquations

Infinite clementanalysis,thedisplacement a.t anypointLJgivenby:

u~u

=

ENieJ;=Nat (3.1)

whereN,the shape(unctionmatr ix,isdependent ontheposition ofthe node inthe xy-plane and0-isa listofnodal displacements.

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Wit hdisplacementsknown at allpointsthe strain at any point canhe,!dl'r·

minedas:

c=SU'::::.'L B;a;=Ba' B;=SNi

also,

B=SN

whereS isasuitable linear operator.

Thenforthe generallinear clasticbehaviour,the stressescan be determined

(:1·' 1 where D is an elasticitymatrixcontain ingthe appropriatematerial properties;

£0isthe initialstrain and (To isthe initialstress.

Fromtheprinciple ofvirtualdisplacements,thefollowingapproximateequi- libriumequations can he obtained:

Ka+1

=

^T

Kij=

J

^{vB? DBjdV}

(:1.6) p.7) where K is thestiffnessmatrix ;r is the externalconcentratednodal forcesi andfrepresen tsforces due to body forces, surface forces ,initialstrain,and initia lstress(Zienkiewicz and Taylor,1989).

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23 3.1.2 ShellElem ent

The clement typeused in this stu dyisS8R,with 8nodes;itisa doubly curvedshell element . lt hasa3)(3 middlesurfaceintegration formass, body forcesandsur face pressurecalc ulation and a 2x2 integration{or constitu tive calculation and out put. Five integrationpointsarechosen throughthe shell thickness.They arc locatedfromthe bottomto the top surface

or

theshell, wit hequaldistancesbetween twoadj acent points (Hibbittet al.'sABAQUS manuals,1!.l89).Thestrain and stresscouldbeoutputat anyofthesefive pointsthroughthe shellthickness.Inthisst udy,surface strainsarefocu sed ontocomparethemeasurementsfrom straingauges. Therefore, onlythe surfacestrainvaluesarcchosenfor the outpu t .

Fordlaplaccmcnt out put ,S8Rtype shel1elemen tclABAQUScanonlyout- putnodal displacementsatthereference surface, which isthemiddle ofthe shellsect ion(ifthethicknessisequalthroughoutthewhole model,which isthe situation Icr theproblemconsidered in this study ).Butin prac t ical measur ements,it isimpossible tomeasure thedisplacementsat themiddle surface.Sincethisst udyistrying todevelopa techniquetodetect cracking instructures,andunder pract ical situatio ns, onlysurface displacements could be measured , only the surfacedisplacementsshouldbe calculatedandoutput.

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.n

TheABAQUS shell element formulationis based onthe Kirchhoff constraint assumpt ionsof the"thin"shelltheory.The constraints require a materialline, thatis originally normal to theshell'sreferencesurface,to remain normalto tha t surfacethroughoutthe deformation state. Sothe surface {lisplal'cmClllis equal tothedisplacement at referencesurfaceplusthecorrespondingrotat ion multiplied by the thicknessofthe shellsection.

(:1.81

where^U."risthe displacement IIIthe surface of the plate,^{Ur . j}isthe displace...

ment atthe referencesurface,4Jis therotationat the referencesurface,and THisthethickness oftheplate.

Whenthe displacementat referencesurfaceis verysmall (it is equaltoaero in pure bending)in comparison to the rotation times thethicknessof the plate , then thefirstterm onthe righthandside ofthe Eqn.(3.8)can he ignored.

Thesurface displacementcanbecalculated by:

(:l.9)

3.1.3 Line Spring Ele me nt

Line spring elementsprovideacomputationally inexpensivetool for the modelling of surfacecracksin platesand shells. The basicconceptwas firstpro-

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25 posedbyRice(1972)and h4S beenfurthe rdiscussedbyParksandWhite (1982). The "llnesprins" isiI.series of one-dimensional finite elementsplaced iILlons:thepart-t hrous:h flaw,whichallows localflexibilit y ofone sideofthe fla.wwithrespecttothe other(pointA andBinfig_3.3.).This localfiexibil- ilyis calculatedfromexisting solutions for singleedge notch specimensunder plane strainconditions(bymean s offracturemechanics) (Fi g.3.4 ). The wholeapproach is com puta ti onally inexpen sivein comparisonto fullthree- dimensionalmodelsofthe vicinityofthe flaw.It is also pointedoutin the AIlAQUSmanu althatthepracticalexperie nce withthis method on typical geometrieshas shown that,for several important geometries,the methodpro- videsacceptable accuracy.Themajorapprox imAtion occursinthevicinityof the flawas itpenet ratesItosurface,if theresulte are compar edwith the results ofa fullthree-dimensional model.Since thisstudywu to developapractical method to deledcr.ckins in structures,endin the actualapplication, it is impossibleloapplysensors(strain gauges) veryclosetothe crack,therefore the output valuesw~lichare notin the vicinityof the crack,would bebetter represented byresults fromthemodel using the line spring element s.The geomet ricprofile ofthesurfacecrack istakentobeasemi-elliptical one,U

shown in Fig.3.~.

In this study,thedepth ofthecrack meansthemaximumcrack depth in the

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Figure3.3:Line springmodelling

Figure3.4: Line springcompliancecalibrationmodel

ze

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27

d•maximumflowdlpttl TI/- w U l tliclu'lest

Figure3.5; Geometricprofileof the surfacecrack

middleofthecrack.Thesurfacelengthof thecrack to be sensedistaken as 40mm .ThedepthoCthe crackvaries from zero to 6.667mm (70%of the thicknessoftheplate).The incrementofthechange of the crack dept h is taken as 5%ofthethickness.

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2S

3.2 Ana lysis and Res ults

3.2 .1 Sur faceStrain/S tres s

A concentratedvertical load was applied at the centreofthe free(."1111of the plate.The locationofthe crack,as couldbe noted fromFigs.3.1and3.2,is close tothe fixedend. Therefore, the concentratedloadwillnot have a local effectonthe strain/s tress field aroundthecrack. In the followingdiscussions the crackedplat emeansthattheplatehasamaximum crackdepthof6.667 mmand a surfacecracklengt h of 10 mm.The appliedload was 500N.

Figs.3.6and3.7showthe contours of the normalstresst1zof theplate witll·

out/withthe surface crack.Aheavyst ress concentrationis observedaround the two endsof thecrack; while away from thecrack, thereis almostn:>dil- Ierence forthe stressofthe plate withand withoutthe crack.Itshows that theuse of linespring elementsfor simu latinga surface crack is successfulin showingthe actual stress variationaround thecrackzona.

In this study,a major concernisthe nature of surfacestrainswhich couldhe measuredby straingaugesinthe real structure. Sincetheconcentratedload is appliedverticallyat the freeend,thebending stresses/strainsrlz/!zare dominant.The largestvaluesorthe stresses/strainsOCCU falong thex-e xis.

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Unit:Njm^J 29

Figure3.6:Contours01the normalstress^dr without asurfacecrack

.' 1 " ::'::.. ..

1 ""n.n

J 'l.'lE'OJ

• . ,_'D~.OJ

, , •.'If·oJ

• o , I U · O I

J .5.SI.OJ

• ·" )f·a'

!~ 1mm

Unit:Nlm^J

Figu re 3.7: Conto urlofthenorm alstre81CTrwitha surface crack

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30

Conseque ntlye..,at the surfaceofthepla te. iscalcu lat edandcompared, Fig. 3.8showsthe contoursof the surface strain£...wit houtthe surfacecrack, andFig. 3.9 showsthecontO UTSof th esamestraincomponen twiththesurface

crack. The crack depthis 6.667 mm, wh ichis;0%

o r

the thicknes s of the plate.It can be observedthatatthe tipof thecrack there are heavystrain concentrationareas; whileat the middleofthe cracktherearc somestrain

rel ease areas .Itis alsonot icedtha t with this crac ksize anddepth ,a large portionofarea aroundthe crackareaffected; hence a straingauge, locatedat asuitabledistance(around16 mm) awayfrom the crackwillbe able tosense thepresen ceofth ecrackshowing a.change of8.8%instrain(500Table3.1).

Fig.3.10 shows the decreeseclthe normal izedstrain level asthedepth ultho surface cr ac kincreases.The measur ementpointsare1.6mm,7.8mm and15.8 mmaway from the crack.Itisobse rvedtha t whcnthecrackdepthisequalto 70%of thethicknessoftheplate, thestrain leveldecre asesarc Around 37.4% , 20.9% an d8.8%fe.rthemcasurementpoints 1.6mm,7.8mm and1.~.8mrn away from the crack,respect ively. This showsthatevenwhen a crackissmall, it ispossible10ident ify thecrack in thest r uct ure usingstat ic measurement s.

Itmustbe emphas izedhere thatthestrain gaugemustbe loc ated aro undthe shadow reg ion of th ecrack to sense an app reciable lev elof strain cha ngesdu e to crack growt h.Itshould alsobepointedout lhat thest a ticmees u rcmenta

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:1 " ::'~:,-"

a •B~~E·1l5

J .1.2 11:_0 4

~ .~,13E·1' 4 5₁ •• 2_.1:_,$~E·O._~BE·O_•

• •J .'f - O·

~ •l,B~ £·O . 10 ••21E·0 . 11 ''5 13. - 0.

Unit:m/m 31

Figure 3.8:Co ntoursofthe sur facestrainfrwitho ut asurface Crack

Unit:m(m

Figure3.9:Contours

o r

the surface straine..witha surfacecrack

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32

Table3.1:Str a invalues(.,forthe untracked and crackedplalearoundthe cr<lCk x coord(mm ) distothecrack(mm) (.,(/H)uncreckcd) (.,(u)(c racked) chauges t%)

15.4 47.1 .1."'1 5.20 ·1..11

32.8 29.7 ·1.94 ,1.79 :1.11-1

46.7 15.8 4.66 ,1.25

s.s u

5·1.7 7.80 ·1.'16 3.53 2D.!.!

60.9 1.60 ,1,41 2.76 :17.·1

67.2 4.70 -1.35 3.07 2!J..I

73A 10.9 4.31 3.71 13.!)

84.8 22.3 ·1.22 ·U 7 1.[8

100. 37.5 4.13 0\.18 .t.z t

120. 57.5 4,02 ·\.08 -I..I!I

156. 93,5 3.80 3.83 ·D.7!!

note: theorigin ofx-exia is at the fixed end; x coordin at e forthe crackXc ::fi2.5m lll.

Nmm"li7.cdllr~jnle~1

~----

i , ^'.:~~~ ^~_. ^_,----

-g ..•...~.

j n: ~ .

:;~0.7S ..,Iidline:1.6 mmawayf,oml be crnk

~ 0.7 d~<lH:dline:7.8mm awayfromthe crack

~0.6.1 d~.hdolline: U,Smmaway from lhecruek

~ 0.6,;'.- - ; - - ; - - - ,- . .--;--.-~

Cr~ckdeplh

Figure3.10:The normalized strainleveldecrease as thedepth of thesurface crack increases

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33 are dependenton the amplitudeof applied forces.In order to get a good and reliableresults from experimental measurements, the appliedforces must be large enough;also theapplied forces must bethesame forthe uncrac kedand cracked structu re.

3.2.2 Out-of-planeandSurfaceDisplacem ents

The largest displacementcomponent of the plate withthe vertical load atthe endis the out-of-planedisplacementWihence displacementwis studiedfirst.

Figs.3.11and 3.12show the displacementcontours ofwof the untracked and crackedplates.Itis observed that thereis verylittl e differencebetween these two sets ofcontours; thisindicates that the out-of- plane displacement is not sensitive to the presence ofa small surfacecrack.

Correspondingto the surfacestrainsconsideredearly, thein-plane surface displacementsu, atthe surfaceof the plate. are determined. The in-pla ne surfacedisplacements u, withand withoutthe surface crack,are calculated and output.As mentionedbefore, ABAQUS cannot compute thesurface displacementsof shellelementdirectly.So Eqn.(3.8)or (3.9) isused.In this part icularcase,the displacementatreference surface (which is theneutr al axis) is almostzero; consequentlyEqn. (3.9) is used. Sincethethickness

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Unit:m 31

Figure3.11:Contours

o r

displacementwofthe uncrackcdplate

Unit:m

Figure 3.12:Contoursof displacementw of thecracked plate

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35 of the plate is a constant. the surface displacements arcproportionaltothe correspondingrotationsat thereferencesurface; hence the contours ofro tat ion , atthe referencesurface,tend to show the same pattern as the corresponding surfacedisplacement .

Figs.3,13 and :1.14 givethe contours of therotation about the y-exla at the referencesurface wit hout/withthesurface crack (which are proportionalto the surfacedisplacem ent u andshowthe same pattern).The crackdepth is takento be 70%of t.hethickness cithe plate.It is observ edthat therearesome increasesinthe x-axissurface displacements(givenby therotationaldegree of freedomof theelemen t)at thecenterofthe crack. In comparisonto the surface strai ns(shownin Figs. 3.8 and 3.9),the change in the surfacedisplacement fieldis small.Thereforeforpract ical measurements,strainmeasurements give a better methodfor crack det ection.

Fig.3.15showsthe contoursofthein-plane displacement u at thereference surfaceduetothe presenceof a crack. Itis observed thattherearemany displacementcontou rsaround thecrack; this is dueto thechangeofin-plane disp lacements, afterthesurfacecrackwasintrod uced.Itshould be pointed out thatthese valuesare very small compared to the correspondi ng rotatio ns at thereference surfaceand the in-planesurface displacements.Moreover, even thoughthemiddle planedisplacementsushow quite afew interesting details

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Unit:rad

Figure3.13:Contoursor therotation_<$~without the crack

Unit :red

Figure 3.14:Contoursor therotationr/Jv withthecrack

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37

Unit:m

Figure3.15: Contours ofdisplacement uat therefere ncesurface for acrackedplate

in displacementvariation,thi.~cannot be sensedbyany sensingdevice;also thesevalues are verysmall.

3.3 Summary

Cracksoccurring in astructurechangethe stress/st r ain and displace ment fieldsaroundthecracks. Among th esechangesalarg edecre aseof surface strain, withina cert ainareaaro und the crack,has beenobservedinthis study.

Thereforeitis possibleto apply strain gauges torealstructuresto detect cracksoccurringinthestructure. Furtherinve~ligationinvolvingdyn amic

t I

j

I

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measurementsand mod al~l1l\lysisarc ca rriedout inthefo llowingchaptersto findoutthe bestmetho d foecrackdetectionin realstruct ures.

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Chapter 4 Steady State Analysis of a Plate with a Surface Crack

4. 1 Modalanalysisan d Fre quencyRe spo n s e Function

4.1.1 Theor et icalBasis

The modalanalysisdeals withthemethodof determiningthe basicvibration properties of ageneral linear struct ure.Itisbased00.thecon cept thatthe structure 'sbehaviourcanhe describe dbya set ofvibrati on modes:themodal

39

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·10 model.Thismodelis defined byaset of naturalfrequencies with ccrrcspomling vibrationmode shapesand modaldampingfactors.

Forthe typicaltheoreticalrouteto vibrationanal)'sis(bymodal analysts],cue beginswiththe description of the structure',physical characteristics,usuo\lIy in termsofitsmass,stiffness anddampingproperties.Then anal\a.t~·tic/ll modalanalysisis performed, and themodalmodelis dete rmi ned. FiMll y,the response of the structureto a given excitationis obtained.

Thegoverningequat ionsof vibra tion for thestruct ure are expressedas:

jM]{X }

+

[C]{X}

+

(KJ(X)=(Fl')) (U)

where[M).

I CJ

^and^[K]are the mass,dampin gandstiffness matrices.rcspec- tively,{X}isthe displacementvector andF(t) isthe appliedtime-dependent force.

Applying Laplace transformtothe equations:

L(IMIlX) +ICJ{.t}+ IKIlX))=L({Fl 'll} ('.2) oneobtains:

([M !. '+[C/.+ K){Xl.ll=(F(. ))+( [MI.+ICIHX(O))+IMIlXIOll('.3)

IBI.)IlX(,))={F(. ))+ (IMI.+[CJ){X(O))+[M Il X(O)) I' "I

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41

Re arr a ngementoftheaboveequati o n leads to

where (11(5)1

=

[8-¹("llis called the transferfunction matrix.

For thehomo geneoussolut io n:

[B(,1I1X (,)} =0 The characte risticpolynomia l equa t ion isobtainedfrom:

(4.6)

(4.7)

He re theLaplacevariable s

=

tj

+

^jw

The rootsofthe characteristicequation(8i) are calledeigenvalues.Substi- tuting aneigenvalueinto theequation ofmotion and solving forU yieldsthe associatedeigenvecto r:

[B(,;)){U ;}=(OJ (4.8)

whereSiis th e eigenvalueandUiis the associated eigenvector.Theeigen value .!Ii(or sometimescalled complex naturalfrequency) bas two parts: (i)the imaginarypartwhich gives the damped natura lfrequency;(ii)therealpart whichgives thedamping factor.

Itcan beshown that modalvectorsareorthogonalwithrespect to one an- otheriftheyareweighted withrespectto the stiffnessmatrix(KJ and mass

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12 matrix[1\I}.Itcan alsobeshownthat mod al vectorsareorthogonaltonile anotherifthey are weightedwith respect to the dampingmatrix

ICJ.

when the damping matrix is proportionalto themassmatrix andstilfness matrix (theproportionallydamped systcm).

Eqn. (4.1)canhewritt enin the modal space. Thetransformationfrom physicalspace tomodal space is givenby

{X) ..,=1U1..,{p}

where:

{X}:displacementof the physical degreesoffreedom.

{p}:displacementin modalspace.

n: numberofphysical degrees offreedom.

m:number of modesevaluated .

Using Eqn.(4.9), Eqn.(4.1)canbe writt enll.!I:

(4.91

[MIIUHp)

+

^[CIIUHP}

+

^[KIIUJ{p)^.,^{F(t)) ^(1.101

Pre-multi plyingby[VIT ,we get:

[UJ'IMJIUJ{p)

+

IUIT[CIIUJ{p}

+

IUIT[K Ilu]{p)

=

IUIT{F(I)} (1.111

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43

for theproportionallydamped system, usinS the orthogonality propert iesof the modal vectors'-'eightcdwithrespectto the mass,stiffnessandda.mpin g matrices,onecan define the diagonal modalmassmatrix as:

IMJ....,.,

=

IUJT(M IIUI

di&!on almodaldamping matrixas:

IC).M., =

(UITICIIUI

diagonal modalstiffness matrixas:

WI••,..

⁼

IUITIKIIUI

andmodalforce vectoras:

(J'I....,⁼IUI^T{ F) Rewrit ingEqn,(4.1)u:

(M!•••(p} ••,+~••• (Pl.+[7i'J•••(p}...

=

{Ji'). (' .12)

O!lCcan noticethateachequationisuncoupledCromtheotherequationand representsanindividua lmodalresponse

o r

^{tbe system}^[Ewins,^1984).

4.1.2 Frequency ResponseFunctio n

FromEqn.(.1.5),ifone assumesa zeroinitialcond ition,oneobtains:

(1l( )J

,

=~

(F(,)) ^(4.13)

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---_._--- - - - _.

II Let the Fouriervariables= j"",then wehave:

(H( '

)J =

{U(jw)}

JW {F(jw)}

where[H (jw )]is calledtheFrequency Response Function(FRF) (·1.1.1)

Thefrequ encyresponsefunctionhasmany forms in lermsofi nput (excitati on) andoutpu t (response).Thetraditional formsinclude:

(a)recept ance,intheformof(displacement)j( forcc), wheredisplacement is theoutput.

(b)mobility,intheformof (vclocity){(force), wherevelocit y isthe output . (e) inerta nce, intheform of(acce[crati on)!( (orce ).wherethe accelera tionis

the output .

Inallcas es,theinputistheexcita tionforce.

Recently, thecons ideration ofsurface strainshas been introducedinto moda l analysis. Thestrainfrequencyresponsefunction meansthe transform func.

ticnbetweeninput (the excitation force) and output (the strainlevel ata certa inpoint)in frequency domain.The strictly theoreticaldefinit ioncanbe found inthe articl es ofBernasconiandEwina(1989),andLi(1989).IIIlh i~

study, thestrain FRFs areanalyzed inthe same manneras thedispleccmcnt or accelerationFRFs.An unitforceis appliedto theplate,then steadystate

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lineardynamicanalysisisperformed by a.!signinga specificran geoffrequen- cies.The excitingforcewillbevaryingwithinthe range ofthefrequencies consideredin thestudy.The strainresponse,as well&Sthe dlsptacement and acceleratio nrespenseawillhedetermined.inthefrequencydomaiu. Thusthe frequencyresponse Iuncrions ofstrain, displacement,andacceleration arecb- tained.The mod alst rainsare definedasthestrainsmeasuredatthedifferent response pointsofthe structu re,when thestructu reisexcited byaunitforce ataresonantfrequency.

4.2 Change of Modal Parameters Due to the Growth in Depth of t he Surface Crack

4.2.1 Global Chang esofthe Moda l Parameters

It isknownthat anycrackordamageoccurringinthestructurewillaffect the physicalproperti~of thestructure.Thechangesinphysicalpropertieswill beshownbythechanges in modalparameters.Thesemodalparamete rs are independent ofthe amplitude of applied forces.Earlier studies on cracking of str uctureshad focused onlyonthenaturalfrequencychanges.Butthese cha ngesof nat uralfrequencies areverysmall,alt houghnotimpossibletomen-

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·16

Table4.1:Frequenc ychanges due to the growth of the depth ofthe surfacecrack Crack depth D(mm) wl(Hz) W](Hz) w3(Hz) w4(Hz) ws(Hzl(",'o(lIz) w7{lIz)

0 18.991 118.62 125.57 332.65 367.75 391.99 653.02 0.00048 18.989 118.62 125.57 332.64 367.75 3!H.97 653.01 0.00095 18.985 tlS.51 125.56 332.63 367.75 391.96 653.01 0.00140 18.979 118.59 125.56 332.62 367.74 391.95 65:1.01 0.00191

is.sn

118.57 125.55 332.61 367.74 391.9:1 653.0 1 0.00238 18.961 118.55 125.54 332.59 367.73 391.91 653.01 0.00286 18.951 118.52 125.54 332.58 367.73 391.89 653.01 0.00333 18.939 118.49 125.53 332.56 361. 72 391.88 653.01 0.00381 18.923 118.45 125.53 332.53 367.71 391.86 65:1.00 0.00429 18.915 118.43 125.53 332.52 367.70 391.85 653.00 0.00476 18.903 118.40 125.52 332.50 367.69 391.84 653.00 0.00523 18.890 118.37 125.52 332.48 367.68 391.83 65:1.00 0.00·571 18.880 118.35 125.52 3:12.46 367.67 391.83 653.00 0.00619 18.870 118.32 125.51 332.44 367.65 39l.S2 653.00 0.00667 18.862 lIS.3D 125.51 332.43 367.4 3 391.82 6.')3. 0 0

iter.Acertainamount of frequencyshift wouldoccuronlywhen a largecrack hasfor med in the structure.

In this study,naturalfrequenciesof the plate arcdetermined throughthe eigenvalueextractionproceduregivenby ABAQUS.The nat uralfrequency shifts dueto the growt hofthedepth of the surfacecrack(each crac kdepth increaseconstit utes a5%changeincrackdept h)arecalcul ated andshown in Table 4.1. It is observed that even forthelargest crack depth change, which occurs for the first mode,the frequencycha ngesbyless than 1%.The corres pondingsur facecrackis6.667 mm deep,40mm long: thepla te is6.~O

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47 mmlong, 200 mm wide.and 9..52.5mm thick.

Themodeshapes correspond ing to these natural frequ encies are shownin Fig.

4.1 (uncreckcdplate).Itis observedthatthe first,second, fourthandsevent h modes arebendingmodes; and the thirdand sixth modes aretorsionmodes;

thefift h mode is thetransverse mode.

Theanalytical modal analysisisalso carriedouttodetermine theglobal changesof modalparametersdue to increasein crackdepth . The frequency rangeis chosento vary from ItoiOOHz, whichinclu desthefirst sevenresonant frequencies.The accelerationsand displacementsobtai nedatthefreeend ,Iar away from the surfacecrack.are chosento represent the global par ameters.

The accelerationFRFsand displace mentFRFsatthecent reofthefree end are calculated and plotted.Inthemodalanalysis,II.unit vertical excitationforce i~appliedto the cornerofthefree endofthe plate. Theresponse measurements for theacceleration anddisplacementareobtai nedatthe centreof the freeend.Therefore, inthefrequ encyresponse functio ns,the bendin gmodes will be dominantwhiletheresponse correspondingto thetc esicnmodeswill besmal l,and theresponsecorrespondingto thetransversemodewill be almostzero.Figs. 4.2 and4.3 show theacceleration anddisplacement FRFsof untr ack ed plate andthe plate withthe larges tcrackdepth.In the calculat ion orthe FRFs,adamping ratioof1% was introduced into the system .

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" -_.=:....::.:=---~

I ,":~~I

'. I ^~~ - ~I ^1i~! """

.1-.

101,-

·M _.·. .. .. .'...

^~,

...

^~

_ "'M.. ..

l.~1

Figure4.1:Mode.bapesoftheuncracked plate

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49

Ampliludeof ll!cacc.clclalionfRFa\ll!eu nlrcDfl hcfrcccnd

I"

Frequency(Hz)

Figure -1.2:Accelerat ionFRFs of uncrecked andcracked plate s

Amplilildeofthcdlsplacemcn'FRF81!heentreollhe free end

IOI.-_~ ~

Solid~nc:\IIl-cnckcd plalc;

Dn hcd 1illc'Qackcdp llllc.

700

Figure4.3:Displacement fRFsofuncreck..dand cracked plates

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50

Table 4.2:Changes oftheamplitudeof displacementandaccclcratiollFRFs at resonant frequenciesat the centre ofthe Irccend

lstmode 2nd mode 3rd mode nilmode Ace.Ampl,(m/~IN )(uncracked) 19.229 15.617 l2.459 9.01495

Ace.Amp!.m/3ljN )(cracked) 19.1S7 15.638 12.479 9.'148

Change%) 0.22 -0.11 -0.16 0,02

Dis.Ampl.{mmN)uncracked 1.3506 0.0281 0.0029 0.0006 Dis.AmpJ.(mm/ N)cracked) 1.3660 0.0283 0.0029 0.0006

Chane(%) ·1.11 -0.11 0.00 0.00

As could be seenfromFigs.4.2and 4.3,it isobserved that thereisI\hno~t nodifference between thetwo curves forcases of withand withoutthe crack.

Table4.2gives changesofthe amplitudeof displacementand acceleration FRFsat each resonant frequency.Thelargestamplitude change occursat the first resonantfrequencyofthe displacementFRF, andthechangeleIc.'ill than1.2%.Also as seen from Table4.1, the largestfrequencyshift is less than 1%.However from an earlierstudyona more complex structu re(Chen and Swamidas, 1992)witha.larger creek,it was observedthatit was possible to monitortheglobalchangesfromglobalsensorssuch as aceelcromcter eor linear variabledisplacement transducers.Inthat experiment,a. 2%ofnatural frequency shift was observed.Fromthatexperimental investigat ion and other similar studiesconsidered earlier (Chondros ctaI.1980 and1989,Collinsct al. 1991and 1992,Hanko et al.1991,Gomes ct al. 1990 and 1991,and

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51 Guignect al.1992), one can concludethatit ispossible to sense the cracki ng in any st ructur eby observingthechangesin natural freq uencies thatoccur in theglobal sensors.Inthe present study sincethe cracking is verysmall ,a shield ingof its presenceoccursintheoverall vib ra tionresponse;the struc t ure vibratesin a global manner asifno crackispresentinit.Hence the chan ges inthe FRF'arevery marginal.

Ilhas beenstaledbyearlier resea rc hersthat dampingrat ios of the st ructu re,at differentreson ant freque ncies,change consider ably due to cracking.Sanli t urk cta!'(1991)reporteda theoreticalstudy of the dampingproduced by fat igue cracks.The damping increasedue to a fatiguecrackin beam -like struct ures

Wa9predicted.Arubbing damping modelofenergydissipatio n was considered in their study. However,from theearlierexperiment conductedintheStr engt h Labora to ryat MemorialUniversity(ChenandSwamidas, 1992), thechange ofdampin g ratios didnotalwaysincrease, but sometimesit also decreased.

It was concludedfromthatexperiment thatdampingchangesdue to cra ck

growthneededmore st udiesowingto the complexmechani smof damping ill real struct ures. Itwas also shown thatthe other modal par a mete rssuch as amp lit udes and mode shape undergochangesdue to crac kingin struct ures.

Based onthefactmentione dabove, a cons tantdamping rati o011%was chose n for allresonantfrequenciesinthisstudy.

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52 Once thefrequency shiftsare observed from the globalsensors,the next st ep will he to determine where thecrackislocatedand howsevere it 19.But for thedetermin at ionofthelocat ionandseverity ofthe crackthatoccur sintile st ructu re, monitoring oftheglobal changes in themodalpar am eterssuch;\.5

naturalfrequen ciesalone isnot enough.Becau-othesame amount offrequency shiftmaybecaused by cracks atdifferentlocation shavingdifferentdepths or lengt hs,one needs tohavea procedureto determinethelocationand severity of cracking. Hearn etal.(1991)proposeda meth odby comparing theratioof changesin naturalfrequencies to the membercharacteri sticratioensembles, However,thismethod islimite dtoskeletalstruct uressuchas weldedCraml's or bridges,andonlythe damaged member

or

theskelet alatructure canhe identified.From the literatu res reviewed andtheexperiment alstudycarried outearlier atMemorialUnivers ity,it app earsthatthemostsensitive.~etof modal parameterswill hetheset oflocalmodal paramet ers; and oneofthese localparameterswillbe thestrain levelaroundthe crack andthenearbyarea.

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53 4.2 .2 Loc alChanges ofthe ModalParamet er s

The crack in the structurewillchange thedynamic propert iesofthe structure.

Thiseffectwill be more significantillthelocal area around the crack than in any otherarea ofthe structure. Since modalparameters,whether at the localor global level,represent the structural properties,monitoring of the localchanges of the modal parameterswill giveAmoredirectand significant indicat ion of the creekoccurring in the st ructure.Inthe classical modaltheory, modalparameters norma llymean naturalfrequencies, damping factors and

displacement mode shapes.Normally,these parametershaveonly a global meaning. for example, the natural freque nciesusually mea n the resonant frequencies forthe wholestructure.

Since strain modal testing has been used inthis study, thelocal strainvalues canalso be designatedas an important local parameter.Whenoneobserves thestrainFRFs,thelocations ofthe peaksindicatetheresonant frequencies of the structure.Theseresonantfrequencieswillbe the same throughoutthe structu re.Thelocal effects will affect onlythe amplitudesofthe peaks of the strainFRFfunct ions.Theamplitudeofthe strail1FRFlI measuredfroma local area c105e to thecrackis different from theFRFsfromany other area in the structure. This local strainFRFs contain informationofthepart icular

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crack andcan beutilizedtoindicatethe locationandseverityof the crnrk, Thereforeby performing st raiu modalanalysis or testing, record ing thestrain FRFs in thelocalareaatdifferenttimes, or at different crack depthsin an analytical investigationandcompar ingthenew FRFs withpreviousones. ;t ispossibleto findoutthe location andseverityof the crackinthe struct ure.

Based onthe methoddiscussed above the strain responseIunctio ns of the plate.

w;.hand withoutcrack,arccalculated .Steady stateanalysisisperformed with a harmonicexcitation whosefrequenciesrangefromI to 700II;.:,Thercapous c is calculated in the frequency dcme ln, with aunitforceinput,andthusthe strainfrequencyresponsefunctions arc obtained.

Three measurement pointsfor strainresponseoutputs arechosen.Thedis- tancefromthe measurementpointsto thecrackare 1.6mm,7.80111IiUI.1 15.8 mm,respectively.Figs.H,4.5and 4.6show thestrain FltFsforthe uncracked plateand forthe platewiththelargestcrackdept h (70%oftill!

thicknessof the plate).Figs.4.4,01.5, and 4.6 aretheFRFsdeterminednt distancesof 1.6mm,7.8mm and15.8 mm away fromthe crack,respect ively.

Significant changes of the amplitudeof the strainFRFs arcobserved.This indicates that thelocal effect hasa strong influence on the amplitude of tile st rain FRFs. Table4.3givesthechanges of thed.lllplitu deofst rainFRFsat eachresonantfrequency.Itis observedthat forthepoint7.8mmawayfrom

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,,'

".

10'°__

o

Amplilud<:oftilemainfRF,1.6 !!1m....

a '

^from^the^Clack

So", ,,~ ~<".,"

^P"O,

1 ~

¹⁰⁰ ^~ ^•flequen<y( lb)⁰⁰ ^~ ^~ ^m

Figure4.4: StrainFRFs, 1.6 roma.wa.yfrom the crack

Amplitudeof ehe ltla in FRF,1.8mmawa,fromthetratk

\0' _

10'

i

_-: _10"

, J

10'

Solidlinc: un-crlckedplltc;

O..hcdlinc;c'lcke d plotc.

FrcqllCn<y{fu)

l

Figure 4.5:StrainFRFs, 7.8 romawayfrom thecrack

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Solidline:un-tT..:k.dplal.:

D.llhc<lli"":.raclu:dpL11e.

"'~~

1 I

\ I \'!

.~ ^.. ~ / \

----_._,~- - - -

..

_...

~ ~ ~ ~ ~ ~

AmpJilud.of 1110lIf~infRF,15.8mm a"'ayfT~m1110«lICk

IO'~__ '._'__ "' •.•._~_~_.• ._ .

".

a

Froquonqo( Hz)

Figure 4.6:Strain FRFs,15.8mm awayfromthecrack the surface crack, the changesof the amplitu desof thestrainrHFsare21.0¹/(1, 20.9%and 18.6%COfthefirst,second andthirdbending modes,respectively.

Fromthe earlierstatic analysis,the same crackdepth at the same measurementpoint causedastatic straindecreaseof20.9%.lieneeCrom theresults ofthe firstthreemodalresponses,theredosenol seem10beanyApprcciilhll!

differencebetweenthe resultsof static anddynamic straingaugetcstiJl~.R,,- suitsfrom the measurementpoint1-5.8 mm awayfrom thecrack.showchanges of9.25%,14.8%,22.1 %and 36.3% forthefirst fourbending modes, from the results of modal analysisgivenin Table4.3. Thestatic measurements given earliershow a 8.8%changeatthis point.Therefore,strainmodaltestinggives