Statistical Prediction Of Fracture Parameters Of Concrete .
Cement and Concrete Research 32 (2002) 529 – 556Statistical prediction of fracture parameters of concreteand implications for choice of testing standardZdeněk P. Bažanta,*, Emilie Becq-GiraudonbaDepartment of Civil Engineering and Materials Science, Northwestern University, Evanston, IL 60208, USAbNorthwestern University, USAReceived 4 December 2000; accepted 26 October 2001AbstractThis article shows how the fracture energy of concrete, as well as other fracture parameters such as the effective length of the fractureprocess zone, critical crack-tip opening displacement and the fracture toughness, can be approximately predicted from the standardcompression strength, maximum aggregate size, water – cement ratio, and aggregate type (river or crushed). A database, consisting of 238test data, is extracted from the literature and tabulated, and approximate mean prediction formulae calibrated by this very large data set aredeveloped. A distinction is made between (a) the fracture energy, Gf, corresponding to the area under the initial tangent of the softeningstress – separation curve of cohesive crack model, which governs the maximum loads of structures and is obtained by the size effect method(SEM) or related methods (Jenq – Shah two-parameter method and Karihaloo’s effective crack model, ECM) and (b) the fracture energy, GF,corresponding to the area under the complete stress – separation curve, which governs large postpeak deflections of structures and isobtained by the work-of-fracture method (WFM) proposed for concrete by Hillerborg. The coefficients of variation of the errors in theprediction formulae compared to the test data are calculated; they are 17.8% for Gf and 29.9% for GF, the latter being 1.67 times higher thanthe former. Although the errors of the prediction formulae taking into account the differences among different concretes doubtless contributesignificantly to the high values of these coefficients of variation, there is no reason for a bias of the statistics in favor of Gf or GF. Thus, thestatistics indicate that the fracture energy based on the measurements in the maximum load region is much less uncertain than that based onthe measurement of the tail of the postpeak load – deflection curve. While both Gf and GF are needed for accurate structural analysis, itfollows that if the testing standard should measure, for the sake of simplicity, only one of these two fracture energies, then Gf is preferable.D 2002 Elsevier Science Ltd. All rights reserved.Keywords: Concrete; Fracture; Fracture energy; Testing; Statistics; Scatter; Randomness; Uncertainty; Prediction; Standards; Size effect1. Problem faced and its historyAfter a quarter century of intense research, it has nowbecome clear that the use of fracture mechanics can yieldsafer and more efficient design against all kinds of brittlefailures of concrete structures. This conclusion is nowsupported by extensive analytical studies and numerousscaled-down laboratory tests, as well as a few full-scalefield tests. The importance of adopting fracture mechanics isunderscored by recent studies of some well-known structural catastrophes. A few examples deserve to be noted:1. The toppling of the Hanshin Viaduct in Kobe duringthe Hyogo-Ken Nambu earthquake in 1995 was* Corresponding author. Tel.: 1-847-491-4025; fax: 1-847-491-3741.E-mail address: z-bazant@northwestern.edu (Z.P. Bažant).0008-8846/02/ – see front matter D 2002 Elsevier Science Ltd. All rights reserved.PII: S 0 0 0 8 - 8 8 4 6 ( 0 1 ) 0 0 7 2 3 - 2caused primarily by softening fracture at the compression side due to bending of massive columns.Although the primary cause of collapse was insufficient confining reinforcement, the size effect wasa major contributing factor, causing the nominalstrength in compression fracture of the columnssubjected to bending to be about 38% less than thestandard compression strength determined on thebasis of standard size test cylinders. Similar observations can be made about various bridge columns thatfailed in the Northridge earthquake in Los Angelesin 1994.2. The failure of the Cypress Viaduct in Oakland, CA,during the 1989 Loma Prieta earthquake. Although theprimary cause of failure was again insufficient confining reinforcement, the size effect must have reduced
530Z.P. Bažant, E. Becq-Giraudon / Cement and Concrete Research 32 (2002) 529–556the nominal strength of concrete failing in compression to 30% less than the value assumed in design.3. The failure of the Malpasset Arch Dam in the FrenchMaritime Alps in 1959. Although the main cause wasexcessive movement of rock in left abutment, the sizeeffect on bending fracture of plain concrete caused themaximum tolerable movement to be about 55% lessthan the prediction on the basis of laboratory strengthtests. A similar observation can be made about thefailure of San Francis dam near Los Angeles in 1928.4. The failure of Schoharie Creek Bridge on New YorkThruway, which was caused by fracture of a foundationplinth during a flood in 1987. Although the main causewas an unexpected scouring of the river bottom, thesize effect on bending fracture of plain concrete causedthe nominal tensile strength of the unreinforced plinthto be about 46% less than the tensile strength determined in standard laboratory tests.5. The sudden explosive failure of Sleipner A oil platform in 1991 during submergence test in a Norwegian fjord — a disaster with no loss of life, but a costof about US 300 million, which was caused bydiagonal shear failure of a very thick tri-cell wall.Although the layout of the reinforcement was incorrect and the elastic finite element analysis had anerror of about 40%, it has been estimated that the sizeeffect decreased the nominal shear strength byabout 34%.Although fracture mechanics was originated by Griffithalready in 1921, for half a century, it was consideredinapplicable to concrete. This is not surprising because avery different fracture mechanics theory is needed forconcrete than for homogeneous structural materials suchas fatigued steel or ceramics. The study of fracture mechanics of concrete originated in 1961 with Kaplan [53]. Kesleret al. [57] concluded in 1971 that the classical linear elasticfracture mechanics (LEFM), with only one fracture parameter, the fracture energy Gf (or the fracture toughness Kc),was inapplicable to concrete. At least two fracture parameters are required. This transpired in 1976 from the studiesof smeared-cracking finite element models [6] and notchedfracture tests [46] (also Ref. [43]), and was recognized in thedevelopment of the crack band model [6,21], in which thefracture properties are characterized by the average slope ofthe postpeak strain softening tied to a certain characteristicwidth of the crack band front (which together imply acertain fixed value of fracture energy Gf).A major step was made by Hillerborg et al. [46] whointroduced to concrete the cohesive (or fictitious) crackmodel, in which the initial slope of the softening stress –separation curve or the area under the curve, together with thetensile strength, implies two independent basic fractureparameters of the material. Two parameters were subsequently used in Karihaloo and Nallathambi’s [54 – 56,67]effective crack model (ECM) and in Jenq and Shah’s ‘‘twoparameter model’’ (TPM) [51]Practically, the most important attribute of fracturemechanics of concrete is the size effect. Although in thesolid mechanics and structural design communities it waswidely believed until the mid-1980s that all the size effectswere of statistical origin, and should therefore be relegatedto statisticians, Leicester in 1969 [58] suggested the ideathat the size effect in concrete may originate from fracturemechanics. Walsh in 1972 [89] demonstrated experimentally that similar notched concrete specimens with similarlarge cracks or notches exhibited a strong size effect thatwas transitional between the case of no size effect for smallsizes and the case of LEFM size effect for large sizes.The source of the size effect was recognized to be thesoftening damage of the material in a large fracture processzone. Failures caused by fracture were numerically simulated with the crack band model [6,21], and were describedin 1984 [7] by a simple size effect formula, justified byasymptotic arguments. Subsequently it was shown that theknowledge of the size effect law implies the fracturecharacteristics. If the classical size effect law [7] is fit tothe maximum load data (which is adequate for a size rangeonly up to about 1:30), two fracture characteristics can bedetermined—the fracture energy and the effective length ofthe fracture process zone [22]. The latter was later shown tobe related to the critical crack-tip opening displacement usedin TPM [16]. Asymptotic analysis further showed that thefracture model based on the size effect law and the Jenq –Shah’s TPM give about the same size effect and, therefore,are approximately equivalent [9,23,70]. Likewise, ECM wasshown to be approximately equivalent to TPM, and thus tothe size effect model. The reason that these three modelsyield similar results is that all the measurements are conducted at or near the maximum load.The fracture energy normally associated with the cohesivecrack model, GF, is different from Gf. It is determined as thearea under the entire measured load – deflection curve,divided by the ligament area. The GF values obtained bythis method, called the work-of-fracture method (WFM), arequite sensitive to the specimen size and shape, although thismight perhaps be avoided by very careful evaluation of thetail of the load – displacement curve and of all the energydissipation sources in the test (see Guinea et al. [40]). On theother hand, fracture energy Gf determined by the size effectmethod (SEM) is, by definition, independent of the structuresize as well as geometry [22,23]. Significant changes in thetesting size range will nevertheless affect the Gf value obtained (this could in principle be avoided by using the broadrange size effect law, with a series of fracture energies [13].While GF corresponds to the area under the completesoftening stress – separation curve of the cohesive crackmodel, the fracture energy of the size effect model denotedas Gf, corresponds to the area under the initial tangent of thestress –separation curve (Fig. 1). Planas and Elices [74]estimated GF/Gf 2.0 –2.5 (see also Ref. [23]). It was alsorealized that these two fracture energy definitions areappropriate for different objectives—Gf is suitable and
Z.P. Bažant, E. Becq-Giraudon / Cement and Concrete Research 32 (2002) 529–556Fig. 1. Softening stress – separation curve of cohesive crack model and areasrepresenting Gf and GF.entirely sufficient for predicting the maximum loads ofstructures (as well as the load – deflection curves up to thepeak), while GF is suitable and necessary for calculating theenergy dissipation in total failure of structures and fordetermining the tail of the postpeak softening load – displacement diagram of a structure [14].Since the test results from different laboratories, based ondifferent fracture specimens and different methods, exhibitvery large scatter, it has widely been considered futile toattempt predicting the fracture energy of concrete and otherfracture characteristics from the mix parameters and thestrength, bypassing the tests of notched fracture specimens.Several previous attempts to set up such formulae, admittedly rather crude, have nevertheless been made, asdescribed in Appendix B. However, these older formulaehave been based on a much more limited set of test data thanthose that can today be collected from the literature.The database acquired in concrete fracture testing worldwide has by now become enormous. Thus, the time seemsripe for the goal of this study — evaluate all these datastatistically in order to determine whether an improvedformula for fracture energy could be formulated, andwhether an additional formula (apparently not yet attempted) to predict the second parameter of quasibrittle fracture,such as the fracture process zone size, could be developed.While it is clear that estimation of fracture parameters fromnonfracture tests can never replace fracture testing ofnotched specimens, an improved formula with a knownstandard deviation can nevertheless serve a useful purposefor preliminary design and for approximate analysis ofstructures with not too high fracture sensitivity.2. Choice of prediction formulaeThe fracture energy Gf (or GF) and the effective length offracture process zone, cf, may be expected to be related tothe basic simple characteristics of concrete by equations ofthe type:fðGf ; fc0 ; da ; w c; E; rÞ ¼ 0 andyðcf ; fc0 ; da ; w c; E; rÞ ¼ 0ð1Þ531where f and y are certain functions, da is the maximumaggregate size, w/c is the water –cement ratio (by weight),E is the Young’s modulus, r is the unit weight of concrete,and fc0 is the mean value of standard 28-day cylindricalcompression strength, based on cylinders 6 in. (15 cm) indiameter and 12 in. (30 cm) in length (for some of the testsincluded in the statistical evaluation discussed later, thecubic compression strength was reported; its value wasconverted to the cylindrical strength using the standardapproximate formula).The parameters in Eq. (1) may be grouped into thefollowing dimensionless parameters (Eq. (2)):rda fc0 ; E fc0 ; EGf fc0 2 da ; fc0 rda ; GF Eda ;GF fc0 da ; GF rda2 ; cf da ; cf r fc0 ; cf r E:ð2ÞAccording to Buckingham’s theorem of dimensionalanalysis, any physical phenomenon must, in theory, bereducible to an equation in terms of dimensionlessparameters whose number is equal to the total number ofparameters, which is six in each of these equations, minusthe number of parameters with independent dimensions,which is two (length and stress). Thus, only four of theaforementioned dimensionless ratios are, in theory, allowedto appear in the prediction equations for Gf and cf. Anumber of simple relations of this kind have beenformulated and compared to the existing test data. However,clear statistical trends could not be detected.It must, therefore, be concluded that the six parameterslisted in functions f and y are insufficient to characterizethe relationship fully. This means that the Buckinghamtheorem cannot be applied and some of the parametersmay thus have physical dimensions. As we will see, asidefrom three dimensionless parameters a1, a2, and a3, we willneed two additional parameters with physical dimensions.We will also drop parameter E, because it is strongly relatedto fc0, which in turn is strongly related to w/c. Further, wewill drop parameter r—not because it would be insignificantbut because the vast majority of the existing data pertains tonormal-weight concretes. Thus, we will seek Gf and cf asempirical functions of fc0, da, and w/c, to which we will addparameter a0 depending on the type of aggregate — crushedor river aggregate (see Appendix A).It might seem that the tensile strength ft0, the modulus ofrupture fr, or the Brazilian split-cylinder strength fsc wouldbe better parameters than fc0. However, measurement of thedirect tensile strength is difficult and more sensitive tostatistical size effect than fc0, while the modulus of ruptureis subject to a strong deterministic and statistical size effects[19]. As for fsc, one might suspect it to be unduly influencedby large compressive stress parallel to the splitting plane andby frictional plastic deformation near the contact with theloading strips, although this has not been proven. Besides, agreater obstacle is that only very few of the reports onfracture tests in the literature give information on the values
532Z.P. Bažant, E. Becq-Giraudon / Cement and Concrete Research 32 (2002) 529–556of these parameters, rendering a systematic calibration of aformula with these parameters impossible. Similarly, manyof the fracture data unfortunately miss the measurement ofE, which is another reason for omitting that parameter.In the case of cf, the main influencing parameters mightof course be different than in the case of Gf. Otherparameters such as the air content, aggregate – sand ratio,grain size distribution, porosity, relative strength and stiffness of the aggregates, etc., might seem more appropriate topredict cf. Nevertheless, these parameters could not beconsidered, not only for cf but also for Gf, due to lack ofinformation in the published reports on the tests.The formula should preferably be such that, among thechosen parameters, as many as possible should be identifiablefrom the test data by linear regression. The simplest relationship of this type would be Gf a1fc0 a2da a3(w/c) a4, inwhich ai (i 1, 2, . . .) are the parameters of the predictionformula. However, this form is unsuitable because an extension of the range could give negative values of Gf.Therefore, it is preferable to seek a linear formula inlogarithmic coordinates, having the general form logGf a1 log fc0 a2 log da a3 log (w/c) log a4 or Gf a4fc0a1daa2(w/c)a3, which is also identifiable by linear regression. One advantage is that such a formula automaticallyprecludes negative values of Gf. Further refinements,though, are needed. When da ! 0, which is the case of purehardened cement paste, the formula should give Gf 0.Therefore, da is replaced by 1 da/a5. Thus, we are led tothe following prediction formula: G f ¼ a0 0 a1fca4 da a2 w a31þ:ca5N hXi2Gtestf i Gf i ¼ minSimilar to parameter ai, parameters g0, g1, g2, g3, and g4 areobtainable by linear regression, while parameter g5 isinvolved nonlinearly. For cf, we take the objective function:c2 ¼NX2logðctestf i Þ logðcf i Þ ¼ min;ð6Þi¼1ð3ÞParameters a4, a5, and a0 are introduced here so as to have thedimensions of stress (MPa), length (mm), and fracture energy(N/m or J/m2), respectively. Parameter a0 is also used todistinguish between crushed and river aggregates. Parametera3 is expected to be negative because a decrease of w/cincreases strength. As for GF, a similar formula is assumed.By analogous arguments, a similar formula is alsoassumed for the effective length of the fracture process zone.Parameters a0 and a4 in Eq. (3) could of course becombined into a single parameter, assuming different valuesfor different aggregate types. It is solely for convenience ofdimensions to keep these two parameters separate. Parameters a0, a1, a2, a3, and a4 are obtainable by linear regressionof log Gf, but inevitably parameter a5 is involved nonlinearly.The objective function (merit function) to be minimizedby fitting the test data is chosen asc2 ¼PAs an alternative, theobjective functions c2 ¼ i ½ðGtestf P2Gf Þi 1 2 and c2 ¼ i ½logðGtest GÞwerealsoconsidfifered. These functions give greater weights to smaller Gfithan does Eq. (4). Similar objective functions were assumedfor GF and cf. However, the results obtained with thesealternative objective functions were worse than thoseobtained with Eq. (4).Compared to Gf, the values of cf are much more uncertain.Since the objective function in the form of Eq. (4) is logicallyassociated with the assumption of a normal distribution, thefact that the standard deviation of cf is particularly largewould mean that negative values of cf would have a nonnegligible probability if a normal distribution of cf wereassumed. This is documented in Fig. 2, which shows thenormal distributions calibrated in the sequel by the availabledata sets for fracture energy ( Gf) (see Sets I, II, and IIIdefined later) and the data set for effective length of thecritical crack extension (cf). Indeed, the values of thedistribution on the negative side are seen to be negligiblefor Gf, but not for cf. Therefore, we better assume cf to followa lognormal distribution and introduce formula (5): 0 g1 fd a g2 w g31þ:ð5Þlogcf ¼ g0 ccg4g5which is logically associated with the assumption of alognormal distribution of cf (the ‘‘log’’ here is considered asthe decadic logarithm).As will be seen, the coefficient of variation of cf can bevery large. But that does not cause the prediction of cf to beuseless because what matters in practical calculations ismainly the order of magnitude of cf, and not so much theprecise value. This corresponds to the fact that the size effectplot descends with an increasing slope in the logarithmicscale but with a rapidly diminishing slope in the linear scale.Linear regression for parameters a1, a2, a3, a4, and a0may seem to imply the objective function to be the sum ofsquared differences in the logarithms. But this turns out notto be the best assumption, as already mentioned. For thisreason, and because parameter a5 cannot be identified bylinear regression, the objective functions (4) and (6) are usedin nonlinear optimization of all the available test data. Thestandard library subroutine for the Levenberg–Marquardtoptimization algorithm has been employed for this purpose.ð4Þi¼13. Statistical analysis of fracture test datain which subscript i labels the individual data valuesmeasured in tests in various laboratories by variousinvestigators.The prediction formulae need to be evaluated by statistical comparison to essentially all the relevant test data that
Z.P. Bažant, E. Becq-Giraudon / Cement and Concrete Research 32 (2002) 529–556533Fig. 2. Normal distributions resulting from fitting of the test data acquired, for fracture energy, from: (a) SEM, TPM, and ECM (Set I); (b) work-of-fracture(Set II); and (c) SEM, TPM, ECM, and work-of-fracture combined (Set III); and for critical effective crack extension cf, from (d) SEM and TPM.exist [1– 93] and statistics of their errors determined. Oneproblem with the database existing in the literature is thatdifferent investigators used different methods to determinethe fracture parameters (the fact that specimens of differentsizes and shapes were used is ignored because the results ofa good testing procedure, presumably, should not depend onthese factors, and if they do there is anyway insufficientinformation for compensating).The older tests (from 1961 until about 1980) wereintended for evaluation by the LEFM, which is inadequate,and therefore the results of these tests could not be used.The following four methods, representing the main testingmethods, will be considered:1. The WFM, which was proposed for concrete byHillerborg [44,45].2. The SEM [24,79], based on size effect law [7].3. The Jenq – Shah method based on their TPM [50,78].4. Karihaloo and Nallathambi’s [54 – 56,66] ECM(which is not a general model but is formulated onlyfor notched beam specimens).Because, as already mentioned, SEM, TPM, and ECM giveessentially equivalent results, and because the fractureparameters of SEM can be easily transformed to the fractureparameters of TPM and vice versa [16,23], all the test dataobtained by these three methods are grouped into the firstset, which comprises 77 data (Set I). The fracture toughness(Kc) measured by TPM or ECM has been transformed toSEM fracture energy ( Gf) (Fig. 1) according to the wellknown relation:Gf ¼ Kc2 E 0 :ð7ÞBecause the critical crack-tip opening displacementdCTOD (32Gfcf/E0p)1/2 (see Eq. (13)) [16,9], in which, forplane strain, E0 E/(1 n2), n Poisson ratio. The measureddCTOD values obtained by TPM were transformed to theSEM parameter cf by relation (8):cf ¼pE 0 2d:32Gf CTODð8ÞAs recently established, SEM, as well as TPM and ECM,gives the fracture energy value representing the area underthe initial tangent of the softening stress – separationdiagram of the cohesive crack model (or the area underthe analogous stress –strain diagram of the crack bandmodel multiplied by the crack bandwidth) (see Fig. 1). Thestatistical analysis is first conducted using only the datafrom SEM, TPM, and ECM. The value of E0 needed forthese calculations was reported by only a few experimenters. In the other cases, this value was estimated from the
534Z.P. Bažant, E. Becq-Giraudon / Cement and Concrete Research 32 (2002) 529–556reported compressive strength ( fc0) using the approximateACI equation. Since this equation uses the cylindricalstrength and some experimenters reported the cubicalstrength, the well-known approximate relation between thecylindrical and cubical strengths had to be used in thesecases. The Poisson ratio, which slightly affects E0, was takenas 0.18 unless reported. The approximation errors of theseequations contribute to the coefficient of variation of theerrors of Gf measured by these methods.At first thought, one might wonder whether the parameters of TPM (Jenq – Shah) and ECM are properly considered if the fitting is done in terms of Gf and cf. However, aslong as the relations (7) and (8) are accepted, fitting thevalues of Kc2 in terms of the parameters of TPM and ECMmust give the same results. If the fitted variable is Kc (whichis a parameter of TPM), rather than Kc2, the results woulddiffer because a different weighting of the data is implied,but they will not differ by much.The fracture energy GF determined by WFM corresponds to the area under the entire stress – separation curve,including its long tail [31] (Fig. 1). This area is much largerthan Gf (see the explanation in Appendix C). Therefore,the test results obtained by WFM have been considered inthe second statistical analysis as a separate set (Set II), inwhich GF was transformed into Gf according to the relationship [75]:Gf 0:4GF :ð9ÞThis is, admittedly, only a crude estimate. In reality, the ratioGf/GF of course depends on the shape of the softening curve[23] and is influenced by the dependence of the measuredGF values on the size and shape of the specimen, reportedby many experimenters. However, no formulae and no cleartrends for such influences are known, and so the constancyof the ratio GF/Gf is an inevitable simplifying hypothesis.An important point to note is that the assumption of a fixedratio in Eq. (9) does not bias the scatter statistics in favor ofone or another method. If one would do the symmetricalopposite—convert all the data on Gf to data on GF and thenfit the aggregate of all data in terms of GF, the resultingcoefficient of variation would be exactly the same. Similarcomments can be made in regard to replacing cf by d2CTODas the variable used in data fitting.The third statistical analysis deals with all the data fromSEM, TPM, ECM, and WFM combined into one set(Set III), with GF being transformed to Gf according toEq. (9). Since 77 usable data have been found in theliterature for Set I and 161 data for Set II, one has a totalof 238 data for Set III. Since the GF measurement as well asthe relationship in Eq. (9) is rather uncertain, such acombined set must be expected to exhibit a large scatter,which proved to be the case.In preliminary optimizations of the overall fit of Set III,the ratio Gf/GF was considered as an additional unknown.The statistical studies reported here were run for differentassumed values of this ratio and the standard deviations oferrors compared. It was found that indeed the value 0.4proposed in [75] and used in Eq. (9) is approximately theoptimum value.The dependence of parameter a0 on the type of aggregate(crushed or river aggregate) has been determined by considering a sequence of values a0 1, 1.01, 1.02, . . . 1.50 N/m.The values that provided the best statistics were identified as:for river aggregates:a0 1 N/m;for crushed aggregates:Set I: a0 1.44 N/m,Set II: a0 1.12 N/m,Set III: a0 1.11 N/m.The shape of the aggregate, unfortunately, is not reported formany of the data in the literature. When it is not, theaggregates have been automatically assumed to be smoothriver aggregates, for which, by choice, a0 1 N/m.For the remaining parameters, the following optimalvalues have been identified by Levenberg – Marquardtoptimization algorithm:Set I: a1 0.46, a2 0.22, a3 0.30, a4 0.051,a5 11.27.Set II: a 1 0.40,a2 0.43, a3 0.18,a4 0.058,a5 1.94.Set III:a1 0.43,a2 0.47, a3 0.20, a4 0.062,a5 3.95.After obtaining the optimal values of all the parameters,the values of the predicted fracture energies ( Gf pred) werecomputed for each data set from Eq. (3) and were comparedto the corresponding values Gftest measured in the test. Thenthe diagram of Gftest versus Gf pred could be plotted. Suchdiagrams are shown for Sets I, II, and III in Fig. 3.Furthermore, the diagrams of Gftest/Gf pred versus Gf predare shown in Fig. 3, showing that the scatter in Sets IIand III is more significant. In order to facilitate comparisons, all the statistical plots for GF presented here areplotted in terms of the scaled values Gf 0.4GF. This ispossible since the scaling by the factor 0.4 has no effect onthe coefficient of variation.Although the Levenberg– Marquardt algorithm suppliesstatistical characteristics, it is preferable, and clearer for theuser, to determine the statistical characteristics on the basisof the plots in Fig. 3. To this end, a linear regression isperformed in the plot of Gftest versus Gf pred (Fig. 3), and thecoefficient of correlation r between Gftest and Gf pred iscalculated. Also calculated is the coefficient of variationwG of the ratio Gftest/Gf pred, which indicates the relativevariation of the measured fracture energy with respect to thepredicted means Gf pred. Calculations of this type can berepeated many times until the parameter values that giveeither the lowest coefficient of variation of the ratio Gftest/Gf pred or the highest correlation coefficient r are obtained.
Z.P. Bažant, E. Becq-Giraudon / Cement and Concrete Research 32 (2002) 529–556535Fig. 3. Plots of measured versus predicted values of Gf or 0.4GF, obtained for (a) SEM, TPM, and ECM (Set I, 77 data); (b) work-of-fracture (Set II, 161 data);(c) SEM, TPM, ECM, and work-of-fracture combined (Set III, 238 data). Note: syjx: standard deviation of vertical differences of data from line of slope 1; syj1:standard deviation of the diffe
This article shows how the fracture energy of concrete, as well as other fracture parameters such as the effective length of the fracture process zone, critical crack-tip opening displacement and the fracture toughness, can be approximately predicted from the standard . Asymptotic analysis further showed that the fracture model based on the .
A.2 ASTM fracture toughness values 76 A.3 HDPE fracture toughness results by razor cut depth 77 A.4 PC fracture toughness results by razor cut depth 78 A.5 Fracture toughness values, with 4-point bend fixture and toughness tool. . 79 A.6 Fracture toughness values by fracture surface, .020" RC 80 A.7 Fracture toughness values by fracture surface .
Fracture Liaison/ investigation, treatment and follow-up- prevents further fracture Glasgow FLS 2000-2010 Patients with fragility fracture assessed 50,000 Hip fracture rates -7.3% England hip fracture rates 17% Effective Secondary Prevention of Fragility Fractures: Clinical Standards for Fracture Liaison Services: National Osteoporosis .
hand, extra-articular fracture along metaphyseal region, fracture can be immobilized in plaster of Paris cast after closed reduction [6, 7]. Pin and plaster technique wherein, the K-wire provides additional stability after closed reduction of fracture while treating this fracture involving distal radius fracture.
6.4 Fracture of zinc 166 6.5 River lines on calcite 171 6.6 Interpretation of interference patterns on fracture surfaces 175 6.6.1 Interference at blisters and wedges 176 6.6.2 Interference at fracture surfaces of polymers that have crazed 178 6.6.3 Transient fracture surface features 180 6.7 Block fracture of gallium arsenide 180
Fracture is defined as the separation of a material into pieces due to an applied stress. Based on the ability of materials to undergo plastic deformation before the fracture, two types of fracture can be observed: ductile and brittle fracture.1,2 In ductile fracture, materials have extensive plastic
Fracture control/Fracture Propagation in Pipelines . Fracture control is an integral part of the design of a pipeline, and is required to minimise both the likelihood of failures occurring (fracture initiation control) and to prevent or arrest long running brittle or ductile fractures (fracture propagation control).
the Brittle Fracture Problem Fracture is the separation of a solid body into two or more pieces under the action of stress. Fracture can be classified into two broad categories: ductile fracture and brittle fracture. As shown in the Fig. 2 comparison, ductile fractures are characterized by extensive plastic deformation prior to and during crack
1) DNA is made up of proteins that are synthesized in the cell. 2) Protein is composed of DNA that is stored in the cell. 3) DNA controls the production of protein in the cell. 4) The cell is composed only of DNA and protein. 14) The diagram below represents a portion of an organic molecule. This molecule controls cellular activity by directing the
