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PROCEEDINGS OF THE 9T11 INTERNATIONAL CONFERENCE ONAPPLICATIONS OF STATISTICS AND PROBABILITY IN CIVIL ENGINEERINGSAN FRANCISCO, CALIFORNIA, USA, JULY 6-9,2003ApPLICATIONS OF STATISTICSAND PROBABILITYIN CIVIL ENGINEERINGEdited byArmen Der Kiureghian, Samer Madanat & Juan M. PestanaDepartment of Civil and Environmental EngineeringUniversity of California. Berkeley, USAVOLUME 1Millp essMILLPRESS ROTIERDAM NETHERLANDS2003

Applications of Statistics and Probability in Civil Engineering, Der Kiureghian, Madanat & Pestana (eds) 2003 Millpress, Rotterdam, ISBN 90 5966 004 8Computational modeling of statistical size effect in quasibrittle structuresD. NovakInstitute o/Structural Mechanics, Brna University of Technolagy, Brna, Czech RepublicZ. P. BazantDepartment afCivil Engineering and Materials Science, NorthwesternUniversit} EVallJ'toll,lllinois, USAM. VorechovskyInstitute0/ Structural Mechanics,Bma Universit}' a/Technology, Brno, Czech RepublicKeywords: fracture mechanics, statistical size effect, theory of extreme values, failure probability, Weibulltheory, random fields, stochastic finite element method, nonlinear fracture mechanics, cohesive crackABSTRACT: The paper begins by discussing some fundamental features of the statistics of extremes importantfor the computational modeling of the statistical size effect, whose asymptotic behavior is not correctly reproduced by the existing stochastic finite element methods. A simple strategy for capturing of statistical size effectusing stochastic finite element methods in the sense of extreme value statistics is suggested. Such probabilistic treatment of complex fracture mechanics problems using the combination of feasible type of Monte Carlosimulation and nonlinear fracture mechanics computational modeling are presented using numerical exampleof crack initiation problem - size effect due to bending span of four-point bending tests.INTRODUCTIONLarge concrete structures usually fracture under alower nominal stress than geometrically similar smallstructures (the nominal stress being defined as theload divided by the characteristic cross section area).This phenomenon, called the size effect, has in general two physical sources - deterministic and statistical. The deterministic source consists of the stressredisrribution and the associated energy release described by nonlinear fracture mechanics (in finiteelement setting, the crack band model or cohesivecrack model). The deterministic size effect representsa transition from ductile failure with no size effect,asymptotically approached for very small structures,to brittle failure with the strongest possible size effect, asymptotically approached for very large structures (Bafant & Planas 1998).The classical explanation of size effect used to bepurely statistical - simply the fact that the minimumrandom local strength of the material encountered ina structure decreases with an increasing volume ofthe structure. This idea was qualitatively proposedalready in the middle of the 17th century by Mariotte. Although what became known as the Weibulldistribution was in mathematics discovered alreadyin 1928 by Fisher & Tippett (1928) (in connectionwith Tippett's studies of the length effect on thestrength of long fibers), the need for this extremevalue distribution in describing fatigue fracture ofmetals and the size effect in structural engineeringwas first developed, independently of Fisher & Tippett's cardinal contribution, by Weibull (1939). Hispioneering work was subsequently refined by manyother researchers, mainly mathematicians; e.g. Epstein (1948) and Saibel (1969).The classical (but erroneous) view that any observed size effect should be described by extremevalue statistics prevailed in structural engineering until about 1990. However, beginning with the studiesat Northwestern University initiated in the mid 1970s,it gradually emerged that there exists a purely deterministic size effect, caused by energy release associated with stress redistribution prior to failure, and thatthis energetic size effect usually dominates in the 50called quasibrittle structures (i.e., structures in whichfracture propagation is preceded by a relatively largefracture process zone which, in contrast to brittleductile fracture of metals, exhibits almost no plasticdeformations but undergoes progressive softening dueto microcracking). Beginning with the 1990s, manystudies focused on the deterministic size effect; seethe reviews in Bazant (1986), Bazant & Chen (1997),Bazant & Planas (1998), Bazant (l999a). The recent621

development of nonlocal Wei bull theory by Bazant &Novak (2000ab) in connection with statistical studies of the modulus of rupture (or flexural strength)of plain concrete beams showed the that, for largequasibrittle structures failing at crack initiation, thedeterministic energetic size effect needs to be combined with the Weibull probabilistic size effect. Inthis connection, some fundamental questions aroseregarding the applicability of various statistical approaches to the statistical size effect. As shown byBazant & Novak (2000a) and Bazant (2002), the existing stochastic finite element method (SFEM) doesnot have the correct large size asymptotic behaviorand fails to capture the statistical size effect on nominal strength.The decisive parameter in SFEM is the correlation length which governs spatial correlation over thestructure. The correlation length modifies the sizeeffect curve in the region where this parameter issmaller than the element size. There is a clear relationship - the larger the correlation length, thestronger is the spatial correlation of strength alongthe structure and, consequently, the weaker is the decrease (due to local strength randomness) of the nominal strength with increasing structure size. Computational problems, however, develop in trying to simulate the extreme value asymptotic size effect using therandom field approach. Approximately, the requirement is that the ratio of the correlation length to theelement size should not drop bellow one. This poses amajor obstacle to using SFEM for describing the sizeeffect, especially for large structure sizes.Some advances in this problem were achieved byseveral authors, e.g. Gutierrez & de Borst (200 I)who, however, confined their studies to the sizerange of real structures. The ratio of the correlationlength to the element size implies, unfortunately, asevere limitation. To actually compute the extremevalue asymptote using the random field approach, thenumber of discretization points (e.g. nodes in a finiteelement mesh) would have to increase proportionallyto the structure size, which is in practice impossiblesince an extremely large structure size would have tobe considered to approach the asymptotic behaviorclosely. To make computations feasible, it is necessary to devise a way to increase the element size inproportion to the structure size, keeping the numberof elements constant. Therefore, the aims of thispaper are:1. To introduce the problem by summarizing the vitalfeatures of the statistics of extrcmcs established bymathematical statisticians in a form meaningful toengineers, pUlling emphasis on the philosophy ofderivation of the probability distribution of extremevalues in a set of independent stochastic variableshaving an arbitrary elemental probability distribution.6222. To draw the consequence for capturing the statistical size effect with the help of SFEM.3. To propose a method for computer simulation ofthe statistical size effect based directly on the basicconcept of extreme value statistics in combinationwith nonlinear fracture mechanics, and verify it by anexample.2 WEAKEST LINK CONCEPT AND THEORYOF EXTREME VALUESThe weakest link concept for the strength of a chainlike structure with N elements is equivalent to the distribution of the smallest values in samples of size N.If one element, the weakest element, fails, the wholestructure fails, i.e., the failure is governed solely bythe element of the smallest strength. To clarify theproblem, it will be useful to recall some basic formulae of the statistical theory of extremely small values.The strength distribution of an element of a chainlike (or statically determinate) structure, i.e., the distribution of the failure probability of an element as afunction of the applied stress 0', may be characterizedby continuous probability density function PI (0') withthe associated cumulative distribution function PI (0')(in statistical literature called the elemental. underlying, basic or primary distribution). Then the cumulative distribution of the failure probability of a structure of N elements (or the distribution of the smalleststrength value in samples of size N) is given byand the failure probability density isThese basic equations provide an overall representation of the failure distribution PN(O') (or PN(O') corresponding to a given elemental distribution Pl (0').Different elemental distributions can give differentfailure distributions PN (a), however, it is remarkablethat the asymptotic forms PN(O') can be only three.Before discussing this fact, let us illustrate the influence of the type of elemental distribution on the failure distribution graphically.Figure I shows the plots of the failure probabilitydensity functions PN(O') and the cumulative distribution functions PN(O') calculated for N 1,10,100and 1000 according to (1)-(2) for the various elemental distributions, in particular the (a) normal, (b)Weibull and (c) rectangular distributions (the last oneis included merely for comparison purposes). All theelemental distributions are chosen to have the samemean value, I, and the same standard deviation, 0.2.Novak, D., Bazant, l.P. & Vorechovsky, M.

0.8 zQ.0.80.6b0.4Q.Zb0.4Q.0.20.200a)6 ---'.54321000.5 Coo f---.,.IIIC. -,.------101.50.40.276543210bO.60.5C) 60 . , - - - - - - - , - - - - - - ,5040.E.z 30Co20I\.10O ----- -- 1.5o0.51.5Figure I: Failure probability density and cumulative distribution function for different elemental distributions: a) normal;b) Weibull; c) rectangular.-. -.'-.-------Weibull. pf O.5--Weibull. pf O.OS. - - Welbull.pf 1 E-6- - - normal, pf 0.5- - - normal. pf O.OS. normal,pf 1 E-6--lognormal. pf O.S--lognormal. pf OOS- -. lognormal. pf 1 E-60.1I.E OO- - - , - - -----r--I.E OlI.E 02 --.- -.-.I.E-02I.E-Ol- ----rI.E 03N- - . -,------1 E 04I.E 05I.E 06I.E 02I.E 03I.E 04I.E OlI.E 02I.E 03----,------- -----,-I.E OOI.E OlSize 20 (m)--,----I.E-03lE-02I.E-OlI.E OOSize 3D (m)Figure 2: Dependence of strength on N for elemental distributions.A general trend may be noticed: Both the mean valueand the variance decrease with an increasing samplesize (i.e., number N of elements). Cases (a) and (b)are very similar in these overall plots, having a bellshaped form. But, as discussed later. for large N, thedifferences are becoming very significant especiallyfor very small probabilities normally required in design. When the elemental distribution is rectangular(case c), the extreme value is seen to converge veryquickly to the threshold of the rectangular distribution. This distribution exhibits no size effect, whichmakes it unacceptable (aside from physical reasons)for strength modeling. But the elemental normal andlognormal distributions give also a physically unacceptable distribution of structural strength, since forsmall enough probability they give a negative strengthvalue. Thus Figure I provides a qualitative insightinto the statistics of extremes.Differences in structural strength for various elemental distribution are particularly pronounced forlarge N and small probabilities (i.e., in the tail). Thisphenomenon is illustrated in Figure 2, in which thebasic equation (I) is used in the inverse: For a chosenfailure probability PN (a), the strength a is solved.Naturally, even for the elemental distributions, themain differences lie in their tails (case N 1). Butas N increases, the differences in strength get largerand larger, not only for the tails but also for the medians. The dependence of strength on N is plottedin Figure 2 for selected failure probabilities PN 0.5,0.05,10 -6. Three elemental distributions, normal.Weibull and lognormal. with the same mean, 1, andthe same standard deviation, 0.1, are considered, andenormous differences among them are found. For theelemental normal distribution, the fact that the sizeeffect on the mean is stronger than on the tail is unrealistic. A more realistic, and much stronger, size effect is observed for the Weibull elemental distribution. For the elemental mean I and standard deviation0.1, the statistical parameters for the Weibull (twoparametric) distribution are: m 12.15 (WeibuJl modulus) and ao 1.043 (scale parameter). In the doublelogarithmic plot of Figure 2, the Weibull size effect is,for any specified failure probability, a straight line ofslope -11m.Computational modeling of statistical size effect in quasibrittle structures623

To show the differences among structures that arescaled in one-, two- and three-dimensions (10, 20,3D), Figure 2 includes three horizontal scales. For thevalidity of (1) and (2) in muti-dimenensional situations, it is required that the whole structure fails whena single element fails. This is a property of a chain aswell as all statically determinate structural systems,and is also a good approximation for fracture of unnotched structures of positive geometry (e.g., unreinforced concrete beams in flexure). In that case, N represents the ratio of the structure volume to the characteristic volume v;, of the micro-heterogeneous material. Vc is here understood as the volume havingthe size of the autocorrelation length of the randomfield of the local material strength, in which case thestrength limits of various characteristic volumes canbe considered as statistically independent (uncorrelated) random variables, a basic hypothesis in the statistical theory of extremes (note that Vc is in generaldifferent (and larger) than the representative volumeVr of the material, which is the smallest volume forwhich the continuum concepts of stress and strengthmake sense, or a volume for which the mean strengthis unaffected by randomness of microstructure as thisvolume is shifted through the material). With respectto the situation in concrete structures, Vr may be considered to be approximately 0.01 m 2 (for 20) and0.001 m3 (for 3D).The foregoing illustrations bring to light a salientpoint (which will be discussed in detail later)namely, the selection of the elemental probabilitydistribution is of fundamental importance for thestatistical size effect, and must therefore be realistic.3IMPLICATIONSMETHODFORFINITEELEMENTSince the failure probabilities acceptable for designare of the order of 10- 7 , at least 1 billion materialtests of identical specimens would be needed to verify the elemental statistical distribution purely experimentally. This is obviously impossible. However, averification is made possible by scaling up the structure to a very large size, a size that would comprise10003 characteristic volumes. Thus a verification ofthe strength distribution of such a structure is equivalent to conducting 1 billion material tests, providedthat the structure is of a type for which the failure ofone element causes the whole structure to fail. Thestrength distribution of such a structure is known,based on a mathematical argument. Therefore, oneneeds to consider the large size asymptotic behaviorand verify that it conforms to this distribution.The asymptotic behavior rests on the so-called stability postulate of extreme value statistics, generally624accepted beginning with Frechet (1927). In this postulate, the extreme value of a set of v Nn identical independent random variables x (the strengths) isregarded as the extreme of the set of n extremes ofthe subsets of N variables. When both n - 00 andN - 00, it is perfectly reasonable to postulate thatthe distribution of the extreme of set N n must be similar to the distribution of the extreme of each subset N(i.e., related to it by a linear transformation). In otherwords, the asymptotic form of the distribution mustbe stable. From this property it can be shown that thesurvival probability f N of a structural system with avery large size N as a function of applied strength amust asymptotically satisfy the functional equation(3)where a.N and bN are functions of size N. In themost important paper of extreme value statistics motivated by the strength of textile fibers, Fisher & Tippett (1928) showed that this recursive functional relation for function f( a) can be satisfied by three andonly three distributions. One of them had already beenfound by Frechet (1927) and the other two have laterbecome known as the Gumbel and Wei bull distributions (curiously, not the Fischer and Tippett distributions). The first two distributions have no thresholdand admit negative values of the argument, and soare unsuitable for strength. Hence, the Wei bull distribution is the only realistic distribution for structuralstrength.Consequently, the only way to ensure the correctness of SFEM for failure analysis is to make it matchthe large size asymptotic behavior, in particular, theWeibull power law size effect, typical of structuresfailing at crack initiation. But how to overcome theobstacle of a forbiddingly large number of randomvariables associated with all the finite elements?The basic idea proposed here is to exploit directlythe fundamental stability postulate from which Fisher& Tippett derived the asymptotic forms of the extreme value distributions. In regard to SFEM, thispostulate may be literally implemented as follows: Instead of subdividing a very large structure into the impracticably large number v of finite elements havingthe fixed size of the characteristic volume, we mustuse a mesh with only n macroelements (finite elements) associated with n random strength variables,keeping n fixed and increasing the macroelement sizewith the structure size, while the subdivision N ofeach macroelement is defined as the ratio of its volume to the characteristic volume of the material. Theneach of these n subsets of N variables is simulatedstatistically, and for each subset the extreme is selected to be the representative statistical property ofthe finite clement (macroelement). These n extremesof the subsets of N variables are then used in FEMNovak, D., Bazant, Z.P. & Vorechovsky, M.

analysis of the whole structure. This procedure ensures that the extreme value statistics is correctly approached, with one crucial advantage-the number nof finite elements (macroelements) remains reasonable from the computational point of view. AlthoughN increases with the structure size, the determinationof the extreme from the subdivision of each macroelement does not add to the computational burden sinceit is carried out outside FEM analysis.One basil: hypothesis of the classical Weibull theory of structural strength is the statistical independence of the strengths of the individual characteristic volumes Lo 2 (in 2D) or 10 3 (in 3D), where lois the characteristic length. The strength of each ofthese volumes can be described by Weibull distribution with Weibull modulus m and scale parameter ao(the threshold being taken as zero, as usual). Each ofthe aforementioned macroelements, whose characteristic size is Lo and characteristic volume Lo 2 or Lo 3 ,may be imagined of being discretized into N charac. .tIC vo Iumes·o1 2 or·o1 3·3tens,I.e. N L 0 2/L0 2 or L (I :I/L o·This consideration provides, according to (1) or (2),the statistical properties of the macroelement. Sincewe are interested only in very small tail probabilities,we may substitute in these equations the tail approximation of the elemental (generic) Weibull distributionwith a ccrtain modulus and scale parameter. The tailapproximation is the power function am (times a constant), and

APPLICATIONS OF STATISTICS AND PROBABILITY IN CIVIL ENGINEERING SAN FRANCISCO, CALIFORNIA, USA, JULY 6-9,2003 ApPLICATIONS OF STATISTICS AND PROBABILITY IN CIVIL ENGINEERING Edited by Armen Der Kiureghian, Samer Madanat & Juan M. Pestana Department of Civil and Environmental Engineering University of California. Berkeley, USA VOLUME 1 Millp ess

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