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f.t THE SOCIETYOF NAVALARCHITECTSAND MARINEENGINEERS

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Probabilistic Fatigue Crack GrowthAnalysis of Offshore Structures, withReliability Updating Through InspectionHenrik O. Madsen and Rolf K. Skiona, A. S VerNas Research, Oslo. NorwavAndrew G. Taliin, Polytechnic Unker;iy, Brooklyn, New York Finn Kirkemo, A. S Veritec, Oslo, NofwayAssfltACr

A stochastic mtiel for fatiguecrackgrowth isapplied.which accountsfor uncertaintiesin loading,initialand criticaldefectsizes,materialparametersincludingspatialvariation,and in the uncertaintyrelatedto computationof the stressintensityfactor.Failu~ probabilitiesare computed by first-andsecond-orderreliabilitymethods and sensitivityfac-torsare &termined. Model updating based on in-serviceinspectionresultsis formulated witfdn thefirstkrderreliabilitymethod. Updated failureproba-bilitiesarecomputed and thedistributionsofthebasicvariablesareupdated.Two typesof h-serviceinspec-tionresultsare used to update the computed failureprobabilities.Inspectionswhich do not detect a crackare used and the inspection uncertainty k included interms of the distribution of nondetwted crack sizes bythe specific tnspectlon methcd. inspections whichdetect a crack are also included and the inspectionuncertainty l.s Included through the uncertainty m themeasured crack size. The formulation are presentedfor u@ating based on one or more fnspectioms. Asimilar formulation for reliability updating afterrepair 1sprovided within the same framework.

INTRODUSXION

In offshore steel structures flaws are inherent dueto, e.g., notches, weldiag defects and voids, Macrocracks can originate from thew flaws and under timevarying 100ding grow to a critical size causing mtas-trophic failure. The conditions governing the fatiguecrack growth are the geometry of the structure andcrack initiation site, the material characteristics, theenvironmental conditions and the loading. In general,these conditions are of random nature. The appropri-ate analysis and design methodologies should there-fore be based m probabilistic rnethcds.

In recent years considerable research efforts havebeen reported on probabilistic modellng of fatiguecm.ck growth based m a fracture mechanim approach,see, e.g., [1-81. In particular, stable crack growth dueto cyclic kmding has been studied. This paper presentsa stochastic model for this crack growth phase forwhich linear elastic fracture mechanics 1s applicable.

A common mcdel is formulated for constant and vari-able amplitude loadfng. The model is &veloped for acracked panel and has hem shown to be in goodagreement with experimental test restits. A generafi-zatkm to a semi-elliptical surface crack is straightfor-ward and has been successfully implemented. Uncef-taintks fn the loading conditions, in the computationof the stress intensity factor, in the initial crackgeometry, and in tie material properties are included.fn particulm the material resistance against crackgrowth ts mcdeled as a spatial random process thusaccounting for material variations within each spec-men.

The probability that the crack size exceeds a criU-cal size during some time Prlcd fs of interest. It f.sdemonstrated how thfs event is formulated in termsof a limit state function with a corresponding safetymargin and how the probability of falfure can ke cal-culated by a first- or second-mder reliability method.The critical crack size may refer to growth throughthe ttdcknm m to a size where a brittle fracture orplastic collapse occur. The critical crack size can bemcdeled as a determirdstlc or as a random quantity.

impactions are frequently made for structures inservice. some inspections rt.mdt in the detection of acrack while others give no detection. The size of adetected crack is measumd either directly mindirectly through a non destructive inspectionmethcxi, where the meammd signal k interpreted as acrack size. Neither the measurement nor the interpr~tation are possible In an exact way and the resultingInspection result is consequently of random nature.When the ins~tkm dc+s not reveal a crack this doesnot necessarily mean that no crack fs present. Adetectable crack is only detected by a certain probabil-ity depsnding on the size of the crack and on theinspection method. Whether or not a crack kdetected, the inspection provides additional informat-ion which can be used to update the reliability and/orthe distribution of the basic variables. This can leadto, e.g., modifications of inspection plans, change ininspection methcd, or a decision on repair or replace-ment. The pap?r describes ins~ction results in termsof event margins and formulates the updating in

45L

.terms of these event margins and the safety margin.The use of first-order reliability methods to performthe calculations h demonstrated.

When a repair of a detected crack is made and anew refinability analysk is performed, it is importantthat the new analysis accounts for the informationthat a repair was necessary. Often it is not pmsible todetermine If the unexpected large crack size has keencaused by a large initial size, by material propertiespoorer than anticipated, or by a loading of the crackUP area larger than anticipated. The paper demon-strates how information obtained III connection with arepair is introduced.

For welded structures a crack is generallyassumed to te present after fabrkation. The analysismethcd can, however, in a simple manner include arandom crack initiation pericd for which a separatemcdel can be formulated.

FATfGUE CRACK GROWTH MODELIn a linear elastic fracture mechanics approach

the increment in crack tie, Aa, durfng a load cycle ISrelated to the range of the stress intensity factor, AK,for the load cycle. A simple relation which ksuflcient for most pm-poses was proposed hy Paris andErdcgan, [9]

Aa=c(AKY, AK>O (1)The crack growth equation IS used without a positivelower threshold on AK below which no crack growthoccurs. The equation was proposed based on experi-mental results, but is also the result of variousmechanical and energy based modefs, see, e.g., [9,101.C and m are material constants. The crack incrementin one cycle is genemdly very small compared to thecrack size and (1) is consequently written in a kineticfmm as

.@. c(AKr, AK>odN (2)

where N is the number of stress cycles. The stressintensity factor K IS computed by ltnear elastic frac-ture mechani~ and is expressed as

K=u Y(a)& (3)where o is the far-field strm and Y(a) IS thegeometry furmtlon. To explicitly account for uncer-tainties in the calculation of K, the geometry functionis written as Y(a) = Y (a ,Y), where Y ts a vettor ofrandom parameters. Inserting (3) tn (2) and separatingthe variables leads to the differential equation

daY(a,YF(&Y

= C (AaTdN, a [O)=ao [4)

where Q~ fs the hdtial crack size. The equation ISappli~ hth for constant and for variable amplitu&leading, thus ignoring possible sequence effects. Also apxmlble etfect of the mean stress or R -ratio is ignored.

Eqs.( 1L4) describe the crack size as a scalar a,which for a cracked panel IS the crack length. For a

surface m embedded crack a description of the crackdepth, crack length and crack shape is necessary. It !scommon practice to assume a semi+ lliptical initialshape for a surface crock and to assume that the shaperemains semi-elliptical during the crack growth. Inthat WI.%the crack depth a and the length 2C describethe crack. TIM differential equation (2) is replacd bya pair of coupled equations,seee.g.[11].

8olutions to (4) are smcmtb curves which do notintermingle. TMs is in contrast to experimentalresults as reported in, e.g., [12]. As a consequence thecrack growth model is randomized as, [7]

(5)

where C, b a random variable nmdeung variations inC from specimen to specimen, while C*(a) is a sta-tionary 1~-nomml process mcdeling variations ti Cwithin each specimen. The expected value of C ~(a ) tstaken as one. The random model in (5) has th~ pm-pertie.s, which me ex~rfrnentally observed in the testresults reported in [121

sample curves of a versus N me irregular andnot very snmotb,sample curves of a versus N become moresmooth for larger values of a,sample curves of a versus N intermingle, tn pm.titular for smaller values of o.

To estimate the correlation proyrties of the randomprocess C 2(u ) a statistical analysis of the @t datafrom [12] has been carried out, [7]. The correlationfunction p2( Aa ) for C ~(a ) IS shown to decrease tozero very rapidly with Aa. The correlation radius rcISdefined as

.

r= = jp2(x ) dx (6)

and has been estimated as 0.12 mm for the alwninumalloy in the experiments of [12]. The variance of C ~has been estimated as 0.062 for the same data. Thevariance fs expected to be signtfcantly larger for crackgrowth in material in the heat affwted zone or In theweld material. Non-proprietary &td are, however,not available for e&fmatfon of the variance in theseCircumstances.

A damage function V(. ) is introduced from (4)as

ac *(x)Y(=)=/ Y(x,YY(&Y dx (7)

The stress ranges are denoted St= Ao ~ and solution of(4) gives

N C , S N comtani ~plitieY(a) =C, fSmdN= (8)

o c I is, .~~le amPlicude,=,

The d~erence between the two cases of constant andvariable mnplltnde Ioadfng therefore only concerns

the loading statistics. fn what follows, um.stantamplitude loading 1s considenxi, while variable ampli-tude loading is considered again at the end of thepaper.

fn th? pm.sentation it has so far been a,s.sunmdthat a crack k present at the time the loading isapplied. With an Initial crack initiation Periwi beforethe crack reaches a size a ~, for which fracturemechanics can be applied with some confidence todescribe the fatigue crack growth, the solution to (4)k

/y(x>y;:(;&)mdx = C,Sm (NNO) (9)where No IS the (random) crack Initiation pericd forwhich a separate mcdel can Ix formulated.

The samnd moment statistics for the damagefunction conditioned upon (a ~,Y,m ) are

E[v(4 )lao,Y,ml = Jy(x,y;,=y ~(o)

C, v~[czldxVclr [w(. )lr30,Y,nI] = J., Y (x ,YY (7X F (11)

p[W(a ~),Y(a J Iao,Y,m 1= (12)~im[a,,nJ

] dx.0 Y (x ,YP (mx Y

The approxlmatkms for the variamx and the cmreL-tkm function are justh%d by the short correlationlength of C J.a ) compared b crack size increments ofinterest. The random variable W(U) Ia ~,Y,m IS essen-tially the sum of many Independent random variablesof approximately the same variance. The distributionIS therefore well approximatai by a normal distribu-tion.

The failure criterion is taken as exceedence of acritical crack size a= In a time period with N stresscycles,

q+N

The crack size present after repair and a ~hleinspectkm is a random variable o_ and the materialproperties after repair are ~ and C,,-. Thesafety margin aftir repair is Mm

cl,- s- (NNrq )and the failure probability after repair is

PF = P(MW

?(.i.f

EXAMPLE.Consider a panel with a center crack as in the

experiments of [12]. The loading is a constant ampli-tude loading leading to a far-field stress range S. Thegeometry function is mtiled as

Y(a ,Y) = exp(Yl(~)y9 (45)

The geometry function takes the value one for a=O.Lengths are measured in mm and stresses in N /mm 2.The distribution of the basic variabks k taken as

S 6 N (60, 102)Y1 c LN(l, O.29Y*e LN(2,0.12)cloc?x(l) (46)

ac 6 N (50, l@)(lnC,, m ) c N,(33.CXJ, 0.472,3.5, 0.32; -0.9)

N (g, 02) denotes a normal distribution with meanvalue # and variance 02. Similarly LN (#,oz) denotesa 1~-normal distribution with mean value p andvariance 02. N JLLI,o ~,wz,ul;p) denotes a blvariatenormal distribution with mean values PI and p2, vari-ances o ~ and o; and correlation cmf?icient p. EX (p)denotes an exponential dktrlbution with mean valuep. The negative correlation between lnC, and m knot reflecting a physical depmdence, but IS introducedby the form of the crack growth equation (2). Statis-tics for C *(a ) are taken as those reported in [7], seesection 2 of this paper. The example has eight basicvariables and the transformation into standardizedand Independent normal variables has been describedin [13,19,20].

~

,03 ,!0 5!0 10 ,10

Figure 1. First- and second-order relia-bility index from design calculation.

The firs-order and improved second-orderaPProximauons to the reliability index are shown InFig. 1 for variom life times expressed III terms of thenumber of stress cycles N. The two approximationsare close implying that the curvatures of the limitstate surface are mcderate at the design point. Stat&tics for the distribution of life time T CaIIbe directlyapprO~mati frorntheresultsof Fig. 1. For the nleanlife time the approximation Is

EIT]=f(l P(T

_-..

Next, the situation where a crack h found in thefirst insped.ion is considered. It is envisaged that theinspection is carried out after N,= I@ stress cyclesand a crack length of 3.9 mm is measured. The mem-urement error ls assumed to be normally distributedwith standard deviation UA. Figure 2 shows theu@t~ mUability index as a function of OA, when(43) has been applied with (rs )==(0,1). The result isalmost independent of OA tn this example as theuncertainty in the initial crack size fs dominating theuncertdlnty in A,. When the crack Is detected, a tii-sion has to be ma& and two options are present. Itmay be decided to repair the crack now or to leave thecrack as it is and base a decision on repair on moreinspection results. With just one inspection it k notpossible to determine if the crack was initially fmgebut grows slowly enough that repair is not needed, orthe crack was inlttally fairly small but is growing fastand must be repaired. ff a requirement on the reliabil-ity index in a period without lnsFctiom is fornm-Iated, e.g., 51>2, the latest time of the next lnspe-s-tion ts determined from Fig.2.

4

3

~ :\:

CA. ..5

2%.?0.02om

0

.,

N,05 2,.5 5,0, ,Os m. ,1.6 10,

Figure 2. U@atmf Jimt-ofder reliabilitytn&x after first inspection with crackmeasurement 3.9 mm.

Assume that the crack is not repalre?f but asecond inspection at N =2.l@ stress cycles I.$required.Let the InspectIon methcd be the same as in the firstimpection and let the measured crack size h 4.0 nun.The measurement error is again assumed to be nor-mally distributed Ith standard &viation OA and thetwo measurement errors are assumed to be stmsu-callyindependent. Figure 3 shows the updated relfa-billt.y index after tbk. second Inspection. Differentinspection qualities now lead to very different results.With CA=0 the negative slope of the reiiabilit y indexcurve becomes very large demonstrating that thecrack growth behavior is basically determined by two

B r ., 05 02 0? 002 Orm 74

3

2

0

::\-1

N

,0, 2d 51$ 10s 210 5.10 10

Figure 3. Updated first-order reflabilltyti&x after second inspecthm with crackmeasurements 3.9 mm and 4.0 mm.

combinations of the basic varkibks. With a iargemeasurement uncertainty there is an hnmedlate andlarge increase in reliability, but after some time thecurve become almost identical to the curve resultingafter the first InspectIon. Due to large uncertainty inboth Inspdions only ifttle foformation Is gained onthe crack growth rate. If the Inspection quallty isvery high It may be possible to state that the crackdoes not grow to a critical size wtti the design Metime. Repair and further inspections are then unneces-sary. For a poorer inspection quality a time perioduntil the next inspection can be determined and thedecision on repair be further delayed.

4

t

3

2I01.,

t:: ;;\:,0, 2,0, 5)06 ,0. 2,. * ,m. ,.:

Figure 4. U@ded first~rder reliabilityin&x after second inspection with crackmeasurements 3.9 mm and 4.0 mm,importance of inhomogeneit y.

N

51

Figure 4 shows the results of Flg.3 together withsimilar results for a homogeneous material. It isobserved that only for very small ins~ion uncer-tainty dces the material inhomogeneity sigticantlyzdfect results. The estimates for material inhomu.genelty used in this example are for base material andthe conclusion may be somewhat different for crackgrowth In weld material or in base material in a heata5ected zone.

Figure 5 presents results similar to those in Fig.3,but for the case where a crack size of 5 mm fs reportedin the second Inspection. Tc.gether, the two inspectionresults now indicate fiat a large and fast growingcrack IS present. Repair is therefore necessary witbfna short period.

B

4

3

2

0

-,

~ \.o

d. o.. ..~-.).2

Nm= m 510 ,0 >,0 ,,0 ,.

Figure 5. Updated 6mt-onier reflabllityindex after second in,s~cm with crackmeasurements 3.9 mm and 5.0 mm.

Consider now difTwent situations where the~-OIIS do nOt result in crack detecticm. hattempt 1s ma& to lflustrate possible means to achievea rff@~ re~bflity. Let the reliablflty requirementbe BR>3.0 and let the design fife time correspond to1.5. I@ tires cycles. Figure 6 shows the refinabilityindex as a funcflon of numbw of stress cycles for twopfate thicknesses. With a plate thickness t the refia-blfity requirement is fuffilled for the design life tfmeand no tnspectlons are needed. With a plate thtcfcmsof only 60?3 of t the refinability requirement isfuffdkd for the period until N =2.l@ stress wcks,where an inspection is neded. The quallty of theinspection IS reflected fn the distribution of non-detected cracks. An exponential distribution isassumed with a mean value k. Cracks initiallypresent are cracks which have passed the inspwtion atthe production We either because they were notdetected or bemuse they were below the acceptancelevel. ~ no cracks were accepted in fabrication, thefabrication inspection therefore corresponds to k= 1.

I I \lo

.,

05

,5)0I I I N

# 2)0 510 m 2Q 5m m

Figure 6. First-.mder r-efhbflity Index fortwo plati thickne53es.

Figure 7 shows the initial reliability index andu~t~ mliabiEty ~&ces for three inspection quali-ties. The best inspection quality A=O.3 is betkr thanthe fabrication imspecUon quality and if no crack ISfound with this method the increase in reflabillty issw?tcient to make further inspections unnecessary. Forthe two other inspection qualities, Prbds are deter-mined until the next In5pwtion.

@ PF5

, 10-,..s

4to.I

30001

.Oo1

2

01

Figure 7. Updated first-order reliabilityindex after first inspection with no crackdetection.

52

Figure8 shows the totalinspd.ionrequirementfork= 1 when no crackisdetectedin anY inspection.For thiscase two tn.s~ctionsare needed. Finally,Fig.9 shows the total inspection requirement for A=3when no crack is detected fn any InspectIon, and forthis case five inspections are needed. It is thus demon-stratti that different strategies on design and inspec-tion planning can be used to achieve a required relia-bility.6 p,I I 1

0

-,

0,

,,10sL I I

w m 5703 ?. ,m 3w ,. N

Figure 8. Updatkd fimt-order refiabifftyfndex after inspections with no crack&tectfOn, mean sfze of non-detecticracks 1 mm.

., IL

,0 ,!0 510 10

q

00.7

Oa

5>0

2m 510 10

Figure 9. Updated first-order reliabilityindex after inscectlons with no crackdetection,rnea~ size of nondetectdcracks3 mm.

0>

05

N

The results of a reliability analysis following arepair of a detected crack is lllustmted in Fig.10. It isassumed that a crack size of arep=8 mm is repairedafter N*P =2.105 stress cycles. The distrlbt uon of theinitial crack size after repair a- is taken as anexponential distribution with a mean value of 1 mm,I.e., as the same initial dktributlon as after fabrim-tion. Two situations are considered with either idenU-cal or independent nmterlal properties before and afterrepair. When independent pmpwties are assumed thesame distribution k used for the properties kefore andafter repair. If follows from the resuft.s that there isan immediate increase in refiabifity after repair, butthe rellabtllty quickly drops to a level below the levelobtained for the calculations before repair. Thisreflects the possibility @t the cause for the largerepaired crack sfze fs a larger than anuci@ed loadingof the crack tip, which is afso acting after the repair.

-1.

INDEPENDENT

NO REPAIR

~+@ 2m5 5W5 @ ~.@ 5.w6 107

Figure 10. Updated first-order refiabifityinckx after repair of an 8 mm crack atN =2.l& stress cycles.

The results presented fn this example have keenfor a constant amp fftude loading. For offshore struc-tures a long tsrm stress range distribution fs generallyappfied fn fatigue analyses. Due to uncertainty fn theenvironmentzd statistics, load models, global stmc-tural analysis and local stress analysis, the parametersof the long term distribution should be modeled asrandom variabks. A Weibull distribution is oftenused

FS[S ) = lexP(-(sl A )B), s >0 (49)where A and B are random variables. A calibrationof the statistics for A and B, based on an uncertaintymodeling for the above mentioned sources, can be per-formed by a modification of the probabilistic fatigueanal ysis presented fn [21]. The factor ~N= ,Spmin (8)1s replaced by the expected value, which for Weibulldistributed stress ranges becomes

53

E[$S:]=EW]E[S ]= E[N]A I(1+~) (50),==1

The expected vdUi? is random due to the random dl.-tributicm parameters, but the uncertainty in the sumfor 6xd distribution parameters is neglected. flIIS isreasonable due to the large number of random vari-ables with little correlation in the summation.

For variable ampUtude leading it is also ofInterest to study the eflect of a non-zero thresholdvalue for AK in (2), which becomes

~= C(AKY, AK> AKttidN

(51)

Eq.(4) is then replaced byda

= c (Amy 1Y(a,Yp(GY K* dN (52)

i ~where 1 denotes the indicator function. Replaclng asan approximation

(AoF 1 MmAu> Y(n.Yh!m

by its expected value, [221, yieldsda

Y(a,YY(~P G(a) = C E[(AuY ]dN (53)

The reduction factor G(a), O

9.

10.

11.

12.

13.

14.

15.

16.

P. Paris,and F.Erdogan, A Critical &MlysIS ofCrack Propagation faws, Journal of BasicEngineering, Trans. ASME, Vol. 85, 1%3, pp.528-534.P.E. Irving, and L. N. McCartney, Prediction ofFatQue Crack Growth Rat-es: nw~, M&h~.i.sms and Experimental Result-s, Fatigue 77Clmferenw, University of Cambridge, in A&alScience, Aug./Sept. 1977, pp. 351-36L

W. Sfmng-Mam, Shape change of Surface CrackDuring Fatigue Growth, .%gineering FractureMechanics, Vol. 22, 1985, pp. 897-913.D. A. Vlrkler, B. M. Hllberry, and P. K. God,The StaUStical Nature of Fatigue Crack Propaga-tion, JOWM3 of Mu!eria& and Tedndogy, Vol.101,1979, pp. 148-153.H. O. Madsen, S. Krenk, and N. C. f,ind, Methcdsof .Wucturai .Szfery, PrentirnHall Inc., Engle-wood Cliffs, New Jersey,1986.M. Hohenbichler, and R. Rackwltz, NonnornmlDependent Vectors in Strwtural Refhblfity,Journal of the .Ezgineering Mechanics Division,ASCE, Vol. 107, 1981, pp. 1227-1238.M. Hohenbichler, An Asymptotic Formula forthe ProbabilNy of fn&I.SWUom,- &rfiMe ~rZuverfassigtiitstbrie de? Euuwerke, Heft 69,LKJ, Techrdsche Universftat Munchm, 1984, pp.21-48,

fL Breitung, Asymptotic Approxirnauon.s forMuluMmud fntegrafs: JwmaI of the figineer-ing A&hanks Divfs&m, ASCB, Vol. 110, 1984,pp.377-386.

17.

18.

19.

20.

21.

22.

M. Hohenbichler, S. Gollwitzer, W. Kruse, and R.Rackwltz, New Light on Firm- and Second-OrderReliability Methods, Manuscript, TeCbmWIUniversity of Munich, 1986.M. Hohenbidder, Mathematische Grundlagen derZuverlassigkeitsmethode Erste Ordnung undEinlge Erweiterun.gen; Dcctoral Thesis at theTecfmfcal University of Munich, Murdch, WestGermany, 1984.H. O. Madsen, Random Fatigue Crack Growthand inspection, in fiructural Safety and Ref.ia6u-ity, Fmceedings of ICOSSAR85, Kobe, JapWIAS.SAR, 1985, Vol.1,pp.475484.H. O. Madsen, Model U@aUng in First-OrderRefinabilityTheory with Applicationto FaUgueCrack Growth, in Pxcwdings, Second lntema-Uonal Workshop on Stcda.stlcMethcds InStruc-tural Mechanics, Universityof Pavia, Pav[a,Italy,1985.

H. O. Madsen, R. Skjong,and M. Moghtaderi-Zadeh, Experlencx on PmbabllisUc FatigueAnalysfsof OEshom Structures,in Proceedings,OMAFJ86,Tokyo, JaPan,1986,VOLK, pp.14.

P.H. Wimddng, K. Ortiz,and Y. N. Chen,Fmc-ture Mechanics FatigueMcdel in a ReliabilityFormat,. ti p~ 0Mm87, H~uston,Texas,1987,Vol.fff,~, 331.337.

55