Probabilistic analysis of linear elastic cracked structures

Akramin et al. 1J Zhejiang Unv SciA 2007 811):1795-17991795ISSN 1673-565X (Prit); ISSN 1862-1775 (Online)www. zu edu. cnjzus; www springerink. comE-mail: jzus@zju.edu.cnProbabilistic analysis of linear elastic cracked structuresAKRAMIN M.R.M., ALSHOAIBI Abdulnaser, HADI M.S.A., ARIFFIN A.K, MOHAMED N.A.N.(Department of Mechanical and Materials Engineering, Universiti Kebangsan Malaysia, Bangi 43600 Selangor Danu! Ehsan, Malaysia)E-mail: md. akramin@yahoo.comReceived Feb. 10, 2007; revision acepted June 23, 2007Abstract; This paper presents a probabilistic methodology for linear fracture mechanics analysis of cracked structures. The mainfocus is on probilistic aspect related to the nature of crack in matenial. The methodology involves finite element analysis; sta-tistical models for uncertainty in material properties, crack size, fracture toughness and loads; and standard reliablity methods forevaluating probabilistic characteristics of linear elastic fracture parameter. The uncertainty in the crack size can have a significanteffect on the probability of failure, particularly when the crack size has a large cofficient of variation. Numerial example ispresented to show that probabilistic methodology based on Monte Carlo simulation provides accurate estimates of failure prob-ability for use in linear elastic fracture mechanics.Key words: Probabilitic facture mechanics, Linear elastic fracture mechanics, Failure probability, First-order reiabilitymethodsdoi:10.1631/jzus.2007.A1795Document code: ACLC oumber: R683INTRODUCTIONA number of methods have been developed orimplemented for estimating statistics of variousProbabilistic fracture mechanics is becomingfracture response and reliability. Most of theseincreasingly popular for realistic evaluation of frac-methods are based on linear elastic fracture mechan-ture response and reliability of cracked structures.ics and finite element method (FEM) that employs theUsing probabilistic fracture mechanics, statisticalstress intensity factor as the primary crack drivinguncertainties can be incorporated in engineering de-force (Bestrfield et al, 1990). Although finite ele-sign and evaluation (Rahman, 2001). The theory ofment based methods are well developed, research infracture mechanics provides a mechanistic relation-probabilistic analysis has not been widespread and isship between the maximum permissible load actingonly currently gaining attention. Grigoriu et al.(1990)on a structural component to the size and location ofaapplied first- and second-order reliability methodscrack in that component. Cutrently, there are many(FORM/SORM) to predict the probability of fracturemethods and applications for probabilistic fractureinitiation and a confidence interval of the direction ofmechanics in oil and gas, nuclear, automotive, naval,crack extension, The methods can account for randomaerospace, and other industries, nearly all of whichloads, material properties, and crack geometry.have been developed based on linear elastic fractureHowever, the randomness in crack geometry wasmechanics models (Rahman and Kim, 2001). Prob-modelled by response surface approximations ofability theory determines bow the uncertainties instress intensity factor as explicit functions of crackcrack size, loads, and material properties, whengeometry. Furthemmore, the usefulness of responsemodelled accurately, affect the integrity of cracked surface based methods is limited, since they cannot bestructures. Probabilistice fracture mechanics providesapp中国煤化工ics analysis (Chena more rational means to describe the actual behav-et aMHE(2001) developediour and reliability of structures than traditional dePROc N M H Garlo with impor-terministic methods (Provan, 1987).tance sampling to calculate the probability of failure1796Akramin et el.1J Zhejiang Univ SciA 2007 8(1):1795-1799based on initiation of crack growth.shape became highly distorted due to large displace-This paper presents a computational methodol-ment. After a few deformnation increments, the wholeogy for probabilistic characterization of fracture ini-domain was remeshed based on a stress error normn.tiation in cracked structures. The methodology isThe strategy used to refine the mesh during analysisbased on linear FEM for deterministic stress analysis,process is adopted fom (Alshoaibi et al, 2007).statistical models for loads and materiai properties andMonte Carlo method for probabilistic analysis. E:ample is presented to ilustrate the proposed method-PROBABILISTIC ANALYSIS AND RELIABILITYology for 2D cracked structures. The results fromConsider a linear elastic cracked structure underthese examples show that the methodology is capableuncertain mechanical and geometric characteristicsof predicting deterministic and probabilistic charac-subject to random loads. Denote by X, anteristics for use in linear elastic fracture mechanics.N-dimensional random vector with components X,x....v characterizing uncertainties in the load,FINITE ELEMENT CALCULATIONcrack geometry and material properties. For example,if the crack size a, elastic modulus E, far field appliedIn order to perform linear elastic analysis, thestress magnitude o", and mode I fracture toughness atfinite element analysis needs to be well developedcrack intation Kie, are modelled as input random(Tada et al, 2000). In this study triangular meshvariables, then X={a, E, o", Klc}. Let stress intensitygeneration using the advancing front method wasfactor K be a relevant crack driving force that can beused (Zienkiewicz and Zhu, 1987). The mesh finallycalculated using standard finite element analysis.optimised by smoothing and associated boundarySuppose that the structure fails when K> Kie (Guineaconditions are found by interpolation from the initialet al, 2000). This requirement cannot be satisfiedgeometry conditions, then finally producing the out-with certainty, since K is dependent on the inputput files. The remeshing algorithms place a rosette ofvector X which is random, and K'e itself to be a ran-quarter point elements around the crack tip, and thendom variable.rebuild the mesh around the crack tip. A computerThe performance of the cracked structure withcode has been developed using FORTRAN pro-the above uncertainties consideration can be evalu-gramming language for finite element analysis cal-ated by the Monte Carlo failure analysis. In thisculation processes, which is based on displacement methodology, the random variables are generated bycontrol for linear elastic crack propagation modelling.some prescribed probability distribution functions,The stress intensity factors during crack propagation such as Lognormal and Gaussian. Then a satisticalsteps were calculated by using the displacement ex-analysis is carried out for each of the Monte Carlotrapolation method, which shown to be highly accu-samples to obtain some parameters such as mean andrate.coefficient of variation (COV) (Soong, 2004). TheThe mesh refinement is guided by a characteris-calculation of stress intensity factor is affected by thetic size of each element, and is predited according to randonness of crack size, elstic modulus and fara given error rate and the degree of the element in-field applied stress magnitude. Then the analysis forterpolation function. The error estimation for theeach of Monte Carlo samples is evaluated to see ifasimulation is based on stress smoothing. It was a pointfailure has occurred. Failure occurred when K calcu-wise error in stress indicator (ESI) to evaluate thelated from finite element analysis exceeds the value ofaccuracy of the finite element solution.Kix.In general, the smaller mesh sizes in a finiteConsider the limit state represented by X=g(a, E,element mesh give more accurate finite element ap-o , Kc) corresponding to a failure mode for a struc-proximate solution. However, reduction in the mesh ture. With all the random variables assumed to besize leads to greater computational effort. The errorstatis中国煤化工Carlo simulationestimator used in this paper was based on stress error approz:s of the variablesnom by Zienkiewicz et al.(2005). The adaptiveaccorYHc N M H Goution fnetioisremeshing technique was used when the element and then feeding them into the mathematical model.Akramin et ael. 1 J Zhejiang Univ SciA 2007 811):1795-17991797The samples thus otained gave the probabilitic RESULTS AND DISCUSSIONcharacteristics of the response random variableX. It isknown that if the value of K is less than Klc, it indi-Consider a 2D double edged notched tensioncates failure. Let Np be the number of simulation (DENT) specimen subjected to quasi-static far fieldcycles whenK is less than Kie and letN be the total tension stress o". The geometry of the DENT speci-number of simulation cycles. Therefore, an estimate men, shown in Fig.la, has width 2W, length 2L andof the probability of failure Pr can be expressed ascrack length a. The load, crack size and materialproperties were treated as statistically independentPp=N/N.(1) random variables. Table 1 presents the mean, COVand probability distribution for each of these pa-Then the probability of failure obtained from Monte rameters. The Poisson's ratio of v-0.3 was assumedCarlo failure analysis is compared with FORM to be deterministic. The mean of far field tensile stressmethodology.was arbitrarily varied from 140 MPa until 350 MPaThe FORM is based on linear approximation of and the COV of normalised crack length was arbi-the limit state surface g(x)=0 tangent to the closest trarily varied from 0 until 0.4. The probabilitic dis-point of the surface to the origin of the space. Thetribution is adopted from (Chen et al, 2001).FORM algorithm involves several steps. First, theFig.lb depicts a finite element mesh of DENTspace x of uncertain parameters X is transformed into specimen. A total of 1184 elements and 2451 nodesa new N-dimensional space u, consisting of inde- were used in the mesh. Both plane stress and planependent standard Gaussian variables U. The original strain conditions were studied. Focused elementslimit state g(x)=0 is then mapped into the new limit were used in the vicinity of crack tip. Using Montestate gu(u)=0 in the u space. Second, the point on theCarlo analyses, a number of probabilistic analyseslimit state gu(u)=0 having the shortest distance to thewere performned to calculate the probability of failureorigin of the u space is determined. This point is re-Pp of the DENT specimen, as a function of mean farferred to as the most probable point or the beta point, field tensile sress, [o"], where u[] is the expectationand has a distance BuL (known as rliability index) to (mean) operator. Fig.2 plots the Pp versus [o°] rethe origin of the u space. Third, the limit state gc(u)=0is approximated by a hyperplane 8gz(u)=0, tangent toitat the beta point. The probability of failure Pr is thusapproximated by Pr;= =Pr[gz(U)<0] in FORM and isgiven by (Madsen et al, 1986)CrackR=中( Bmn),(2)at+whereO(4)=二[°exp(-ξ2 12)d5，(3)↓↓↓↓↓↓↓↓(b)which is the cumulative probability distribution2Wfunction. A Hasofer and Lind algorithm (Madsen eta)al, 1986) was used to calculate analytically theFig.1 A DENT specimen under far-field uniform ten-probability of failure.sion; (a) Geometry and loads; {b) Finite element meshTable 1 Statisticat properties of random input for DENT specimnenRandom variableMeanCOVProbability distributionReferenceNormalised crack length a/wI nonormalElastic modulus E (GPa)75.2中国煤化工iInitiation fracture toughness KIe (MPa:m3)24.83Far field tensile stress。”Variable'CNMHG_5 COV-slandard deviation/mean; b: arbitrarily varied; a: arbirarily assumed1798Akramin et al. 1J Zhejiang Univ SciA 2007 81):1795-1799sults for Vaxw=0.2 and the plane stress condition, probability increases with Vonlr, and can be much lar-where Vaw is the COV of the normalised crack length ger than the probabilities calculated for a determinis-a/W. As can be seen in Fig.2, the probability of failuretic crack size, particularly when the uncertainty ofby (Chen et al, 2001) and FORM are in good a/W is large. The probability of failure in plane stressagreement with the present study results.is slightly larger than that in plane strain, regardless ofFigs.3a and 3b indicate the plots of Pr versus the load intensity, since K in plane stress is (1-以E[o门] using FORM and present study methodology times larger than K in plane strain. The predictedfor plane stress and plane strain conditions, for both finite element results from this study matched welldeterministic (vaw=0) and random (van=0.1, 0.2, 0.4) with the FORM resuts.crack sizes. The results indicate that the failure1.DE+00日CONCLUSIONVur=0.2E 1.0E-01The probabilistic method has been presented for要1.0E-02fracture mechanics analysis of linear elastic crackedstructures. The methodology involves development of言1.0E-03Chen et a.(2001)*10)finite element analysis codes, and statistical models& 1.0E-04. FORMfor uncertainty and probabilistic analyses usingPresent studyMonte Carlo simulation. The numerical example has1.0E-0580 120 160 200 240 280 320 360 400been presented to ilustrate the proposed methodologyMean of far field tensile stres u[o"](MPa)for 2D cracked structures. The results from this ex-Fig.2 Failure probability of DENT specimen by Chen etample indicate that the methodology is capable ofal.(2001)'s, FORM and present study for plane stressdetermining accurate probabilistic analyses in linearconditionelastic fracture mechanics.1.0E+0O[Referencesr 1.0E-01 iAlshoaibi, A.M., Hadi, M.S.A. Arffin, A.K., 2007. An adap-; 1.0E-02 Itive finite element procedure for crack propagationPreseatanalysis. Journal of Zhejiang University SCIENCE A,1.0E -04 IFORM study8(2):228-236. 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Intermational Journal of PressureInternational Journal for Numerical Methods In Engi-Vessels and Piping, 784):261-269. {doi:10. 1016/S0308-neering, 241:237-357.01610100006-0]Zienkiewicz, O.C, Taylor, R.L, Zhu, J.Z, 2005. The FiniteSoong, TT, 2004. Fundamentals of Probability and StatisticsElement Method: Its Basis and Fundamentals (6tb Ed).for Engineers. John Wiley & Sons, West Sussex.Elsevier Butterworth-Heinemann.Chief: Wei YANGISSN 1673-565X (Print); ISSN 1862-1775 (Online), monthlyJournal of Zhejiang UniversitySCIENCE AWww.zju.edu.cn/jzus; www.springerlink.comjzus@zju.edu.cnJZUS-A focuses on“Applied Physics & Engineering"JZUS-A has been covered by SCI-E since 2007➢Welcome your contributions to JZUS-AJournal of Zhejiang University SCIENCE A warmly and sincerely welcomes scientists all overthe world to contribute Reviews, Articles and Science Letters focused on Applied Physics & Eogi-neering. 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