Designing Against Size Effect on Shear Strength of Reinforced Concrete Beams Without Stirrups

Transcription

1 Designing Against Size Effet on Shear Strength of Reinfore Conrete Beams Without Stirrups By Zeněk P. Bažant an Qiang Yu Abstrat: The shear failure of reinfore onrete beams is a very omplex frature phenomenon for whih a purely mathematial moeling approah is not possible at present. However, etaile moeling of the frature mehanism is not neessary for establishing the general form of the size effet. The mere knowlege of the fat that the failure is ause by ohesive (quasibrittle) frature propagation an that the maximum loa is attaine only after large frature growth (an not at frature initiation) is shown to suffie for etermining the general approximate mathematial form of the size effet law, while the numerial oeffiients of this law nee to be ientifie from experimental ata. Simple imensional analysis yiels the asymptoti properties of size effet, whih are haraterize by () a onstant beam shear strength v (i.e., absene of size effet) for suffiiently small beam epths, an () the LEFM size effet v / for very large beam epths. Together with the known small- an large-size seon-orer asymptoti properties of the ohesive (or fititious) rak moel, this suffies to unambiguously support a size effet formula of the general approximate form v = v (+/ ) / (where v, are onstants), whih was propose in 984 for shear failure of beams on the basis of less general an less funamental arguments. Calibration by least-square regression of the existing experimental atabase (onsisting of 398 ata) leas to several empirial expressions for v an with ifferent ompromises between auray an simpliity. Introution Although a provision for size effet in shear failure of reinfore onrete beams was inorporate into some esign oes more than a ozen years ago, a ompelling experimental eviene obtaine with properly sale large-size beams mae of normal onrete has beome available only uring the last few years. The ase has now beome lear: The esign formula must inlue the size effet. The problem is how to best interpret the test results an how to best esribe the size effet mathematially in a suffiiently simple an pratial manner without violating ertain restritions that have rystallize from theoretial researhes uring the last two eaes. To larify this timely question, is the objetive of this artile. The size effet is measure in terms of the nominal strength, generally efine as σ N = P/b where P is the maximum (or ultimate) loa (or loa parameter), b is the struture with an is the harateristi imension (or size) of the struture. The size effet is haraterize by omparing σ N for geometrially similar strutures of ifferent sizes. Aoring to the lassial allowable stress esign, as well as the theory of limit states (or plasti limit analysis) whih unerlies the urrent esign oes for reinfore onrete strutures, the nominal strength σ N is inepenent of the struture size. We say that in this ase there is no size effet. It has been generally proven that the size effet is absent from all strutural analysis methos in whih the material failure at a point of the struture is eie by the stress an strain tensors at that point. If the material failure riterion involves energy, a size effet inevitably arises. This MCormik Shool Professor an W.P. Murphy Professor of Civil Engineering an Materials Siene, Northwestern University, Teh-CEE, 45 Sherian R., Evanston, Illinois 68; z-bazant@northwestern.eu. Grauate Researh Assistant, Northwestern University.

2 is the ase of frature mehanis, provie that either the rak or the frature proess zone (miroraking zone) at failure (or both) is not negligible ompare to struture imensions. A size effet is exhibite by all the theories of failure whih involve some material harateristi length, l. This is always the ase when the failure riterion involves some quantity of the imensions of energy. An example is the ohesive rak moel, as well as the rak ban moel, whih is almost equivalent. The ohesive (or fititious) rak moel, pioneere for onrete by Hillerborg, is now regare as the best among the simple approximate moels for onrete frature. Metho of Interpretation of Existing Experimental Database Why an t a size effet formula be etermine purely empirially? We nee to aress this question to plae the problem in proper perspetive. Many formulas in onrete esign oes an of ourse be evelope purely empirially beause it is possible to obtain aequate test ata for the entire range of pratial interest, an to sample that range statistially uniformly, without bias. An example is the ratio of tensile an ompressive strengths, or the effet of reinforement ratio in various oe speifiations. Unfortunately, the size effet is not a problem of that kin. Fig. shows the histogram of the number of test ata in ACI-445 atabase as a funtion of beam epth. The size effet is of pratial onern mainly for beam epths ranging from m to m. Unfortunately, 86% of all the available test ata pertain to beam epths less than.5 m, an 99% to epths less than. m. The oeffiient of variation ω of the eviations of a size effet formula from the points of the atabase will therefore be totally ominate by small epths for whih the size effet is unimportant. Thus it an easily happen that some formula that gives the smallest ω for suh ata oul be ompletely wrong for very large sizes while another formula that might give a higher ω oul be muh more realisti for large sizes. This is not the way to justify a formula for the ACI oe. For an unbiase purely empirial valiation of a formula, the test ata woul have to be istribute uniformly over the entire range of interest. In view of the osts of large sale tests, we annot even hope to aquire suh a atabase. Therefore, we must extrapolate. Suh extrapolation, visualize by Fig., annot be aomplishe empirially. In keeping with the motto of ACI presient s inaugural message [4], ars sine sientia nihil est, a soun theoretial support is require. The theory, in turn shoul be verifie by properly sale size effet tests on one an the same onrete, an espeially by reue-sale moel tests in whih the imensionless sizes (haraterize in a shape-inepenent manner by their brittleness numbers β [, 4], efine later), an reah the highest possible values. A size effet formula for shear strength, motivate by frature mehanis, was propose in 984 [5]. Sine that time a number of other formulas have appeare; see Fig. whih inlues: the size effet law base on frature mehanis an energy release arguments [5,, 4, 4]; an extene form of that law for fratures in whih the ohesive stresses are never reue to zero but exhibit a finite resiual strength [6]; the CEB-FIP formula, introue empirially [34]; the formula of Japan Conrete Institute [43], motivate by Weibull theory, in whih the failure is assume to our right at the initiation of a marosopi rak an the size effet

3 is assume to be ause by ranomness of loal material strength; a formula resulting from an enhanement of the moifie ompression fiel theory (MCFT) base on a suppose effet of rak spaing [, 3, 33, 65]; an a power law orresponing to linear elasti frature mehanis (LEFM) an supplemente with an upper boun (small-size ut-off). For omparison, all the ata points of the existing ACI-445 atabase are also shown in Fig.. What is striking in this figure is that the very ifferent urves of the aforementione formulas look almost equally goo, or equally ba. The reason is that the size range overe by the ata is not broa enough an the satter is enormous. The size range annot be signifiantly extene without very large finanial outlays. The enormous satter is ause by mixing in one atabase test ata for ifferent onretes, ifferent shear spans, ifferent reinforement ratios, et. These influenes annot be eliminate beause they are highly ranom an poorly unerstoo. The problem is ompoune by the fat that most of the ata sets inlue in the atabase involve only a single beam size (epth) or a negligible size range (as their original purpose was to larify influenes other than the size effet). Some efforts are presently being mae to hoose among various formulas by omparing the oeffiients of variation of the errors of eah formula alulate for the existing ACI-445 atabase. But suh efforts are futile. The oeffiients of variation of the eviations of the formula from the ata points are almost the same for all the formulas, goo or ba. The arbitrariness of suh a omparison then inevitably leas a ommittee to a politial hoie. The most serious obstale to extrating size effet information from the ACI-445 atabase is the fat that the vast majority (more than 97%) of its 398 ata points ome from tests motivate by ifferent objetives (suh as the effet of onrete type, reinforement, shear span, et.), in whih the beam epth was not varie at all. To oument the problem, see Fig. 3, whih shows some suh test ata (marke by an oval) in omparison with the size effet law [5] an (at bottom right) with the ata points from two tests series of the broaest available size range. These ata ontaminate the atabase by unneessary satter ue to influenes that annot be eliminate beause they are poorly unerstoo. Suh ontamination greatly wiens the satter ban of the atabase an masks the size effet trens of the iniviual ata series with a signifiant size range. When this atabase is fitte with a power law (a straight line in the figure), the exponent (slope of the straight line) will epen on size istribution of the ata; see Fig. 5 whih illustrates how the shifting of a hypothetial ata lou to smaller or larger sizes an lea any power law exponent between an /. Obviously, suh statistial inferenes are not objetive; they epen on the frequeny of test ata in various size intervals, whih is a subjetive hoie of the experimenters influene by the funs available. So is the existing ACI-445 atabase useless? Not at all. But it shoul be use only for alibrating the size effet formula after the basi form of that formula has alreay been selete, an nothing else. The seletion of the best form of the formula must be base on a soun theory. The theory shoul of ourse be experimentally valiate. This an be one only by omparisons with iniviual test series with proper saling an a broa enough size range, mae on one an the same onrete. The theory must be suh that it oul also apture similar size effets in other types of failure with the same physial soure, ourring in onrete as well as other quasibrittle materials. Beause of the typial high ranom satter in beam shear tests, the size range shoul be at least :8. Geometrial similarity of the beams of various sizes shoul be maintaine as losely as possible, so as to prevent pollution of the ata set by unertain influenes other than those of the size. Currently there exist only about ata series (from 8 investigator teams [, 3, 4, 3, 4, 44, 5, 63]) satisfying these riteria at least to some extent 3

4 (a few more have a signifiant size range but grossly violate geometrial similarity). Only two of them, namely the 99 Northwestern tests [4] an the reent Toronto tests [, 3], satisfy these riteria quite losely. Of these two, the Northwestern ones [4] are reue-sale moel tests, whih have the avantage that they ahieve (thanks to the reue sale) the largest imensionless size so far (as measure by the brittleness number [4]). Reently it was attempte to eflet the foregoing ritiism by onsiering only the aforementione test series in whih the beam epth was signifiantly varie. However, the ombine group of suh test series was evaluate jointly, without paying attention to the trens of the iniviual test series. This is again misleaing. To illustrate it, onsier Fig. 4 showing bilogarithmi plots of log v versus log (on the left) for two sets of four hypothetial ata series with the same range of beam epth (in log-sale), generate so as to math perfetly the urves of the theoretial size effet law v = v ( + / ) / (isusse later), in whih v an are empirial parameters epening on the type of onrete. The set on top is obtaine by a frugal investigator, who has moest funing an must therefore test smaller (less expensive) beams, an the set at the bottom is obtaine by a wealthy investigator, who has greater funing an an thus affor to test larger beams. Eah investigator onuts the size effet tests for four ifferent onretes eah of whih is the same for both investigators (an all the other influening parameters, inluing the steel ratio ρ an shear span a/, are also the same for both). The urve of the size effet law for eah onrete is ifferent, haraterize by ifferent values v, v,, of the size effet law parameters v an. Assuming that both of these investigators o not know the theoretial size effet law an regar these perfet ata as one ombine atabase, they see only the ata pitures on the right of Fig. 4. Beause of the high satter of the ombine atabase on the right, eah investigator, looking at his ombine atabase, an at best infer a straight line tren in the bi-logarithmi plot, whih orrespons to a power-law size effet. By statistial regression, the frugal investigator thus fins the mean size effet v /4 (whih happens to oinie with the JSCE oe speifiation [43]), while the wealthy investigator fins the mean size effet v /3 (whih happens to oinie with a reent reommenation by one oe-proposing subommittee). Thus, beause they have not heke the trens of the iniviual ata series, both investigators are le to erroneous onlusions. Their onlusions epen on subjetive fators, suh as the hoie of beam sizes whih, in turn, epens on the funing of their sponsors. Making the tests for variously shifte size ranges, these investigators oul have obtaine as optimum any exponent between an /; see Fig. 5. Obviously, knowlege of a soun theory an of the trens of the iniviual test series is neee to obtain the orret onlusion. Variables in the Problem of Failure an Size Effet Problems of quasibrittle frature in the normal range of interest are very iffiult to solve. However, the asymptoti properties are simple. For very small struture sizes, the asymptoti solution an be obtaine by plasti limit analysis, an for very large sizes, by linear elasti frature mehanis (LEFM a theory in whih the frature proess zone is a point an the struture, whih may be inelasti, unloas uring frature propagation elastially). These asymptoti situations are often outsie the size range of pratial interest (being approahe losely, for example, only for onrete beams mm eep an m eep, respetively). It is nevertheless very helpful to know these asymptoti situations beause a goo approximate solution for the intermeiate pratial size range an be obtaine by some sort of interpolation between these asymptoti ases, alle asymptoti mathing. 4

5 Although the etaile mehanism of shear failure of reinfore onrete beams is ompliate, it learly onsists of some omplex form of ohesive softening frature. The material softening ue to istribute fraturing may also be taking plae but beause it must loalize into narrow bans, it may be approximate as ohesive softening frature. This fat alone suffies to establish the basi asymptoti properties of size effet. The problem of shear apaity of the ross setion involves the following variables: The shear strength v whih we want to preit, efine as v = V/b w (v plays here the role of nominal strength of struture, normally enote as σ N ); V = shear fore, whih is assume to be istribute uniformly, epth from the top fae of beam to the entroi of longituinal reinforement, an b w with of the web, whih equals the total beam with b if the ross setion is retangular. The harateristi size of the struture, hosen as. Parameters G f an σ of the ohesive rak moel, whih automatially exhibits size effet [, 6]. This moel was evelope by Barenblatt [4], Leonov an Panasyuk [5], Rie [6], Palmer an Rie [56] an others (Knauss, Smith, Wnuk an Kfouri an Rie[47, 48, 64, 67, 46]), an was pioneere for onrete by Hillerborg et al. [39] uner the name fititious rak moel [4, 58]. Frature energy G f represents the area uner the initial tangent of the softening urve of ohesive stress versus rak-fae separation (Fig. 6). The shape of the softening urve is assume to be fixe, whih means that all the other parameters of the softening urve, suh as the total frature energy G F, representing the area uner the entire urve, are relate to σ an G f. Geometry parameters ρ = A s /b w, L/, L /, L /,..., whih represent the reinforement ratio (A s = ross setion area of steel) an the ratios to of all imensions L, L, L,... efining the span L, the beam length, the over thikness, an the istanes efining the loations of loaing points. The beam with b w nee not be inlue sine its effet on v is known to be negligible. Some of the parameters L i haraterize the rak shape an rak tip loation at maximum loa. Normally the ohesive rak moel is onsiere to esribe the tensile frature, in whih ase σ = f t = tensile strength of onrete. However, the shear failure of beam is oubtless triggere by shear-ompression frature of onrete in the region above the tip of the main rak an elamination frature of onrete over (as a result of owel ation) an bon frature (propagation of a rak along the steel onrete interfae, with subsiiary raks emanating from bar eformations) may also play some role. 3 In the ase of rak ban or ohesive rak moels for ompression frature, the strength limit is the ompressive strength σ = f, an in the ase of bon frature it is the bon strength. For our general imensional analysis, we o not nee to istinguish among these iverse strength limits (an that is why the strength is generally enote as σ ). The frature energy together with σ imply a material harateristi length, l = EG f / σ () 3 Note that ompression shear-frature (or ompression rushing), whih is an essential riterion for brittle failure in the strut-an-tie moel, exhibits a pronoune size effet an is also esribe by the rak-ban moel (in the form of triaxial softening amage) or by the ohesive rak moel (aapte for ompression). Likewise, the over elamination an propagation of bon frature also require ohesive rak moel. 5

6 introue by Irwin s [4] for metals an by Hillerborg for onrete; l haraterizes the length of the frature proess zone at the tip of a rak or rak ban. The solution represents a funtional relation whih may generally be written as Φ(v, σ, EG f,, ρ, L, L, L,..., L m ) = () Here we multiply G f with Young s elasti moulus E beause it is known that maximum loas in frature mehanis o not epen separately on E an G f but only on their prout representing the frature toughness, given by Irwin s relation K = EG f. In (), we have a total of N t = 5 + m parameters. 4 Small- an Large-Size Asymptotes Ditate by Dimensional Analysis The number of unknowns may be reue by introuing imensionless parameters. Aoring to the Bukingham s Π-theorem 5 [8], the number, n, of inepenent imensionless parameters is equal to the total number, n t, of all parameters minus the number of inepenent physial imensions. Combining this theorem with the known physial meaning of l, we an etermine the asymptoti behaviors. First, onsier that l. Sine l is known to haraterize the size of the frature proess zone, it follows that this zone beomes infinitely smaller than. This means that, in relation to the beam epth, all of frature is happening at only one point, whih propagates. Consequently, the material strength σ an have no effet an an be remove from the list of parameters in (). Funtion Φ of 5 + m parameters then beomes a funtion, ˆΦ, of 4 + m parameters, i.e. ˆΦ(v, EG f,, ρ, L, L, L,..., L m ) = (4) There are only two inepenent physial imensions, Pa an m (in SI units), beause the imensions of all the other parameters in () an be obtaine as prouts an ratios of Pa an m (for instane, the physial imension of EG f is, in SI units, Pa J/m or N /m 3 ). Therefore, aoring to the Π-theorem, the problem of failure an be reast as a funtional relation among + m imensionless parameters, whih may be hosen as follows: ˆφ(v /EG f, ρ, L/, L /, L /,..., L m /) = (5) If the strutures of ifferent sizes are geometrially similar (whih inlues the onition that the main raks in speimens of ifferent sizes must be similar), then ρ, L/, L /, L /,... are all onstant, an so the first parameter in the foregoing equation must also be onstant, i.e,, v /EG f = onstant. It follows that for l : v = C EGf = onstant (6) 4 Base on Eqs. 9 for the fraturing truss moel [9], an example of funtion Φ is Φ(v,...) = k e v ( + a / ) EG f /( + /κ e l ) = (3) where k e an κ e are empirial funtions of f an ρ. However, although this moel yiels a realisti form of size effet, it turne out to be insuffiient to apture realistially the effets of other influening parameters, inluing a/, ρ, f an a. This is why we must justify the size effet by general imensional analysis whih is vali regarless of the etails of the moel an influenes other than the size. 5 This theorem woul be more properly alle Riabouhinsky-Bukingham s theorem sine the iea was introue earlier by Riabouhinski [6]. 6

7 where C is a onstant epening on the geometry parameters. Note that the foregoing argument requires that the frature length at the moment of failure must not be negligible. When the failure ours right at frature initiation from a smooth surfae (as in the moulus of rupture test of flexural strength of unreinfore beams), then the last equation oes not apply beause the energy release rate of an infinitely short rak is zero. Eq. (6) is a power saling law that is harateristi of linear elasti frature mehanis (LEFM), observe in the ase that the raks at maximum loa are large an geometrially similar, or that the strutures ontain geometrially similar nothes. 6 In a plot of log v versus log, this asymptoti saling is represente by a straight line of slope /. Seon, onsier that l. In this ase the frature proess zone oupies the entire ross setion, an so there an be no frature propagation. So, the failure loa must be inepenent of K or EG f. Aoring to the Π-theorem, the problem again reues to a funtional relation among + m imensionless parameters, whih may be hosen as follows: φ(v /σ, ρ, L/, L /, L /,..., L m /) = (7) Noting again that, for geometrially similar strutures, ρ, L/, L /, L /,... are onstant, we onlue that the first parameter in the foregoing equation must also be onstant, i.e,, v /σ = onstant, an so for l : v = C σ = onstant (8) where C = onstant. So, in this asymptoti ase the size effet is absent. This is harateristi of plasti limit analysis as well as any theory in whih the material failure onition is expresse in terms of stresses an strains. The foregoing asymptoti saling laws represent all that an be eue from imensional analysis alone. To learn more, one must take into aount some results eue from the ifferential equations an bounary onitions governing the mehanis of failure. Seon-Orer Asymptoti Properties It has been generally proven [] that the first two terms of the small-size an large-size asymptoti expansions of size effet base on the ohesive (or fititious) rak moel must have the following form: for : for : v =... (9) C σ v = ( ) C σ +... () ([], Eqs. 9.5, 9.6);,, C an C are onstants with respet to the size effet (i.e., they epen on struture geometry but not on size ). These asymptoti properties hol true uner the onition that the softening stress-separation urve of the ohesive rak moel begins its esent with a finite slope, whih is known to be true for onrete. The asymptoti properties in (8) an (6) apply to all types of failure ue to ohesive frature or loalization of istribute amage, provie that either there are large nothes (whih is 6 The power law of exponent /3, i.e. v D /3, whih has reently been propose on the basis of the ACI-445 atabase ontaminate by variation of highly sattere variables other than, woul be justifie by imensional analysis only if the frature energy ha the irrational imension of J/m 7/3 instea of J/m (where J = Nm = Joule). 7

8 not our ase) or large geometrially similar fratures evelop in a stable manner prior to the maximum loa. The fat that the frature patterns in small an large beams are approximately similar is oumente by many laboratory experiments as well as finite element simulations. If, for example, the epths of frature at maximum loa in small an large beams were % an 8% of ross setion epth, respetively, or if the fratures in small beams were almost vertial an in large beams almost horizontal, then this assumption woul not apply. But from experiene this is not the ase. The foregoing opposite asymptoti properties have been analytially erive by transformations of the ifferential equations an bounary onitions of ontinuum mehanis. The large-size asymptoti properties have further been erive by asymptoti expansion of (a) equivalent LEFM, or (b) the J-integral, or () the smeare-tip metho []. Knowing these properties, one an exten imensional analysis to obtain a simple expression for the transition between the asymptotes, whih we o next. Size Effet Transition Between the Asymptotes The fat that there are only two inepenent physial imensions, Pa an m (in SI units) means, that aoring to the Π-theorem, the number of inepenent imensionless variables is n = 3 + m. Although various sets of imensionless parameters an be introue, agreement with the asymptoti onitions (8) an (6) an be ahieve by the following hoie of 3 + m imensionless parameters: Π = v = v, EG f σ l Π = v σ, Π 3 = ρ, Π 4 = L,...Π n = L m () The equation governing the nominal strength of struture may then be written as F ( Π, Π,...Π n ) = () The fat that this hoie of parameters agrees with the first-orer asymptoti properties in (8) an (6) may be heke as follows. For, Eq. () takes the form: F (, Π,... ) = (3) Noting that all the parameters other than Π are onstant (an that funtion F must be, for physial reasons, uniquely invertible), we see that Π must be onstant, too, i.e. v /σ = onstant, whih agrees with (8). For, Eq. () takes the form: F ( Π,,... ) = (4) Again, noting that all the parameters other than Π are onstant, we may onlue that the first parameter, Π = (v /σ ) /l, must be onstant, too. This agrees with (6). Let us now try to obtain a simple transitional size effet law whih also agrees with the seon-orer terms in (9) an (). To this en, we may expan funtion F (Π, Π,...) into a Taylor series about some hosen state (D, v ) in the mile of the ranges of Π an Π ; F = F + F Π + F Π +... = (5) where Π = (v D v D )/σ l, Π = (v v )/σ 8 (6)

9 Here F = F/ Π an F = F/ Π are erivatives evaluate at that hosen state. Keeping only the linear terms as shown 7 an solving this equation for σ N, we obtain an expression of the following general form [5, 3]: v = v + / (7) in whih v an are onstants whih are expresse in terms of D, v, F, F, F, σ an l (an epen, in an unknown way, on struture shape). This formula, representing the firstorer asymptoti mathing, esribes a smooth transition between the aforementione two asymptotes (Fig. ); is the transitional struture size, separating sizes > for whih the failure is preominantly brittle from sizes < for whih the failure is preominantly utile (for this reason, the ratio β = / is alle the brittleness number [4, 6]). The use of this size effet law for shear failure of beams was propose in 984 [5], although on the basis of a muh more restritive an simplifie mathematial argument. The aeptability of the size effet law in (7) epens on satisfying both the first an seon-orer asymptoti terms in (9) an (). This is reaily heke by enoting β = / an ξ = /, an realizing that for β : ( + β) / β/ (8) for ξ : ( + /ξ) / = ξ / ( + ξ) / ξ( ξ/) (9) Note that the foregoing erivation of size effet law has not relie on ontinuum mehanis. Even if a speimen or struture is not large enough ompare to material inhomogeneities (aggregate size), the foregoing erivation remains vali beause, even for ranom isrete meia, the shear fore V, material strength σ an frature energy G f an be efine statistially. The Question of Uniqueness of Size Effet Law Let us now isuss the question of uniqueness of results. From the point of view of imensional analysis, it woul be perfetly legitimate to hoose other imensionless variables. If we hose Π = (v /σ ) /l an Π = v /l an proeee in the same manner as before, we woul get the result v = v /( + / ), whih is ifferent from (7). Although this expression has the same asymptotes as the size effet law in (7), it isagrees with the seon-orer asymptoti terms in both (9) an (), an so it must be rule out. As another example, one oul take Π = v p /σ p (/l ) q an Π = v r /σ r with any onstants p, q, r. But then it woul be foun that, unless p = r = an q =, the result woul isagree with even the first-orer asymptoti properties in asymptoti terms in (9) an (). On the other han, nothing woul, for example, prevent us from taking Π = v /σ (/l )(+ /l ). In that ase, the asymptoti terms in (9) an () woul be satisfie up to the seon orer, but the size effet expression woul beome onsierably more omplex. 7 Inluing higher-orer terms of Taylor series expansion, one oul obtain higher-orer asymptoti mathing approximations apable of greater auray over a very broa size range (whih have previously been obtaine in other ways [, ]). But their greater omplexity is, for onrete beams, unwarrante, in view of high satter of test results. 9

10 So. Eq. (7) appears to be the only simple size effet formula satisfying the require firstan seon-orer asymptoti properties in (9) an (). There are also physial arguments that lea to imensionless variables Π an Π as hosen. These variables haraterize failure in terms of energy. Π represents (exept for a geometryepenent fator) the ratio of the energy release rate of a rak to G f, whih is what eies the propagation of a rak when the struture is muh larger than the frature proess zone. Π represents (exept for a geometry-epenent fator) the ratio of the strain energy ensity to its value at the strength limit of the material, whih is what eies failure when the struture is smaller than a fully evelope frature proess zone. For ata fitting, the size effet law in (7) or () has the avantage that it an be algebraially onverte to a linear regression plot: Y = A + C () where Y = (v v r ), v =, C = C A () (Fig. 7 right). The regression is a onvenient way to ientify v an from size effet test ata. For less omplex problems in whih the frature path is simple an known, the size effet law in (7) has also been erive by several other methos, e.g., by asymptoti expansion of equivalent LEFM, asymptoti expansion of J-integral, an asymptoti expansion of smeare-tip metho []. The first of these methos further yiels expressions for the size effet parameters in terms of the energy release funtion of frature mehanis. Suh expressions, however, annot be obtaine for the beam shear problem beause the LEFM rak pattern orresponing to infinite size extrapolation is unknown. The size effet law in (7) has been verifie numerially for many problems by finite element simulations base on nonloal amage moel, ohesive rak moels, an lattie or partile moels of mirostruture. Experimental verifiations now inlue many types of failure of reinfore onrete strutures, as well as roks, toughene eramis, fiber omposites, brittle foams, snow slabs an sea ie (the last up to a reor size of 8m 8m.8m). In view of all this eviene, aumulate over the past two eaes, it woul be extremely surprising if the size effet law in (7) i not provie a goo approximation for the shear failure of reinfore onrete beams. 8 Comparison with Iniviual Size Effet Data The form of the size effet formula must be heke by omparison with ata sets that are not signifiantly ontaminate by variation of parameters other than size. Ieally, the beams 8 So far we have taitly assume that ohesive frature reues the stress all the way to zero. In ompression frature, though, if the softening zone is suffiiently onfine so as to evelop large fritional stresses, it is possible [] that ohesive frature terminates with some finite resiual ompressive stress. In that ase, the foregoing imensional analysis remains vali if v is replae by v v r, an the resulting formula, alle the extene size effet law, has the form: v v = + v r () + / However, in the ase of shear failure of normal reinfore onrete beams there is no way to evelop a triaxial onfinement strong enough to generate high fritional resistane in the ligament above the tip of the iagonal shear rak. Therefore (even though a finite value of v r makes it possible to obtain a slightly lower oeffiient of variation for the entire ACI-445 atabase), formula () is not evelope in etail.

11 teste must be geometrially sale, over a large size range, an use one an the same onrete (ientially ure an teste uner the same environmental onitions). Only two ata sets ame lose to this ieal situation: Reue-sale tests in 99 at Northwestern University [4], in whih beams having epths from. m to.33 m, maximum aggregate size 4.8 mm an reue-sale bars with stanar ASTM eformations (bought from PCA, Skokie) were teste (with 3 iential beams for eah size); an reent normal-sale tests at the University of Toronto [, 3], with beam epths from. m to.89 m an maximum aggregate size mm (only one speimen of eah size was teste, with an interval of up to two years elapsing between subsequent tests). Eq. (7) leas to a very goo agreement with both ata series, as an be seen from the optimum fits of these ata. This is oumente by Fig. 7, an valiates the orretness of the form of the formula. Three kins of omparisons are shown: those with the Northwestern ata alone (on top), those with the Toronto ata alone (in the mile), an those with both ata ombine (at the bottom). In the last ase, the optimum fit has been obtaine uner the onstraint that the minimum relative values of v /v woul be the same in both ases. The optimum fits in Fig. 7 (on both the left an right) have been obtaine by leastsquare fitting in the logarithmi sale using the Levenberg-Marquart nonlinear optimization algorithm. The optimum parameters of the size effet formula (7) are given in the figures, along with the oeffiient of variation whih is efine later. All the fits are shown both in the plots of log(v /v ) versus log(/ ) (on the left), an in linear regression plots of (v /v ) versus / (on the right). The top two rows show the iniviual fits of eah ata set, an the last row shows a ombine fit of both ata sets obtaine by onstraine optimization alreay esribe. Although linear regression in transforme variables has not been use to get the optimum fit, the linear regression plots are also shown in Fig. 7, on the right, along with their oeffiients of variation. The results of nonlinear optimization an linear regression are lose but not iential. The nonlinear optimization is slightly preferable beause it implies a better weighting of the ata [4]. What is partiularly noteworthy is that the test series at both Northwestern University an University of Toronto verify very well the asymptoti behavior of frature mehanis require by the foregoing asymptoti arguments ouple with imensional analysis (Eq. 6). This onfirms that the asymptoti slope of size effet in a oubly logarithmi plot must be /. In aition, it must be emphasize that the well-known Japanese tests [4, 63] o not at all ontrait this asymptoti slope, even though they were previously interprete by a power law of exponent /4; see Fig. 8. By impliation, the omparisons with all the relevant test ata thus onfirm that the orret explanation of the size effet lies in frature mehanis. This further verifies that the orretness of hoosing G f an σ as the governing material parameters in imensional analysis. The fat that the test ata are muh loser to the LEFM asymptote of slope / than to the horizontal asymptote means that the shear failure of beams is highly brittle. This further implies that the frature energy is a muh more important material parameter than the material strength, an that, if finite element programs are use, they must be base on frature mehanis (as another onsequene, the importane of introuing without elay a stanarize frature test of onrete [6] is highlighte). Aitional 7 ata sets with a muh narrower but non-negligible size range exist [44, 5, 3]. They are shown in Fig. 9. Unfortunately, geometrial similarity of saling was signifiantly vio-

12 late in these tests, ue to variation of other influening parameters suh as a/ an ρ. Beause of unertain influene of suh parameters, these ifferenes are simply neglete in making the omparisons in the figure, but the inevitable result is an inrease satter. Nevertheless, the size effet tren of these ata is still esribe reasonably well, an the asymptoti slope / is not ontraite. Expressions for Size Effet Law Parameters The question now is how to preit the values of v an. Expressions for these onstants were set up in 984 [5] on the basis of ertain simplifiations base on the beam theory an the iea of arh ation, whih however were supersee by later researh. Further stuy [9] le to an energeti fraturing generalization of the truss moel (also alle the strut-an-tie moel), whih was base on a simplifie estimation of the energy release rate from either the potential energy or the omplementary energy (uner the hypothesis that the failure at maximum loa is triggere by propagation of a ompression amage ban of a fixe with aross the ompression strut). Suh analysis le rigorously to the size effet law (7), whih provie aitional valuable support for the general form of this law, an it also gave an intuitive explanation as to why a size effet must arise. Reent omparisons with numerous test ata from the atabases ompile at Northwestern University an in ACI Committee 445 have nevertheless reveale that the etaile mehanism of failure assume in [9] for the fraturing truss moel was too simplifie for apturing the epenene of oeffiients v an on parameters other than size, partiularly the epenene on a/ an ρ. It transpire that the axis of the ompression strut annot be assume to pass through the points of appliation of the loa an the reation. A muh steeper ompression strut (skethe in Fig., top left) woul have to be onsiere, but the proper slope to onsier is unlear. Therefore, we nee to set up semi-empirial expressions for v an, whih nee to be alibrate by an experimental atabase. The following general expressions have been onsiere in optimization exerises: = f r ρ r (a/) r 3 a r 4 (3) v = k f r 5 ρ r 6 (a/) r 7, (4) where k,, r,...r 7 are empirial onstants; a/ = shear span (Fig. 6), ρ = steel ratio, f = ompressive strength of onrete, an a = maximum aggregate size. Why on t we use the Π- theorem to reue these expressions to a imensionless form? beause we o not know all the influening parameters. An why o we assume prouts of powers in these expressions? in orer to prelue the possibility of negative values an to have at the same time a linear form in terms of the logarithms. Calibration by Data Fitting Coeffiients, k, r, r,... of the formulas for v an have been alibrate by least-square optimum fitting of the ACI-445 atabase, onsisting of 398 ata points (see Fig. an the imensionless plots in Figs. 3). This atabase inlues only the tests mae uner threepoint loaing, an therefore exlues the Japanese tests for reor-size beams [4, 63] mae uner istribute loaing. 9 Although most tests in the atabase ha retangular ross setions, 9 The reue-sale three-point-loae tests at Northwestern University [4] were also exlue by ACI-445 from its atabase, base on three questionable arguments: () that the aggregate was supposely too small

13 4 ol ones ha a T-ross setion (these tests orrespon to the upper left points in the iagrams in Figs. 3, eviating from the general tren an ausing an inrease in the stanar error of regression). The ata fitting is a nonlinear statistial regression problem, an the hoie of approah alls for some isussion. Let us enote by ˆv i (i =,,...n, n = 398) the measure ata points, an by v i the orresponing values of v alulate from the formula. It appears that the best approah (Appenix II) is not to minimize sum of square errors (or resiuals) i(v i ˆv i ) beause the variane of ata (or the satter ban with) ereases with the inreasing size (i.e., the ata are heteroskeasti). To minimize statistial bias, one shoul transform the statistial variable v so as to make the variane approximately uniform (i.e., make the ata approximately homoskeasti, inepenent of ). This may be ahieve by least-square fitting in the logarithmi sale of v, i.e., by minimizing the square of the stanar error of regression, s L, the unbiase efinition of whih is s L = n p n i= ( ln v ) i (5) ˆv i where p is the number of free parameters in ata fitting; in our problem, p is at least 4 (parameters, k, r, r ) an better 5 (with r 3 ) (the reason why p > is that p ata points an be fitte perfetly). Beause ( ln v ) = (v ) /v, the sale transformation from linear to logarithmi has a similar effet as applying weights proportional to /v (exept for the fat that the implie error istribution is gaussian in the sale of ln v rather than v ). The minimization was aomplishe by a stanar library subroutine for the Levenberg-Marquart nonlinear optimization algorithm, whih reues the problem to a sequene of linear regressions. Sine s L, efine in the sale of natural (not eai) logarithm of v, is imensionless, it may at the same time be regare as the oeffiient of variation. The reverse transformation to the linear sale of v gives the following oeffiient of variation of regression, haraterizing the ratio of stanar error in the linear sale to the mean of all ata ˆv i ; ω = ( e s L e s L) / (6) ω an s L are almost equal beause usually s L <. (in whih ase their ifferene error is of the orer of s L 3 /3, i.e.,.8, as an be heke by Taylor series expansions). Note that, although the approximate equality of s L an ω requires fitting in the sale on natural logarithm, ln v, Figs. 3 are plotte, for onveniene, in terms of the eai logarithm, log v. The hoie of beam sizes for testing has been governe by funing limitations an other subjetive onsierations. Unfortunately, the ata points are rowe in the range of small sizes ( inhes, Fig. an ). This istorts the resulting fit, giving insuffiient weight to large beam ata whih are the most important for extrapolation to still larger sizes that oul be use in pratie. Ieally, the histogram of a atabase shoul be a horizontal line, but it is ebatable whether this shoul be the ase in the sale of or log. Therefore, two types of histogram-base ata weighting are onsiere. In the fist type, the range of was ivie into onstant intervals, of in. with, an the resulting histogram, plotte in Fig. (bottom right), was approximate by the smooth urve plotte. Eah point in the atabase was then assigne a weight inversely proportional to the smoothe histogram (beause this is (but this was avantageous for maximizing the brittleness number β); () that the beams were supposely too narrow; an (3) that the bon slip was supposely exessive (but so it was in many other tests, inluing those run in Toronto). 3

14 what is neee to make the weighte histogram a horizontal line in the sale of ). In the seon type, the range of log was ivie into onstant intervals, of with log. The resulting histogram, shown in Fig. (bottom right), was not approximate by a smooth urve but was use iretly to assign to eah point of the atabase a weight inversely proportional to this histogram. To provie aitional safety margin, the urve of esign formulas has generally been mae to pass not through the mile of the satter ban but near its lower margin. This provies an aitional safety margin, impose in aition to that provie by the loa fators, by the apaity reution fator, an by the fat that f is efine as a signifiantly smaller value than the mean ompression strength from testing. To avoi bias, this aitional safety margin must be obtaine aoring to the least-square metho by taking the preition formula obtaine by least-square regression (soli urves in Figs. an ) an reuing it by the stanar error of regression times a fator orresponing to the hosen probability ut-off, to be taken from the gaussian istribution table (whih yiels the ashe urves in Figs. ). In view of the large satter of the atabase (Fig. ), it is impossible to ientify all the 9 parameters,, k, r,...r 7 without high unertainty. Beause the fitting of the available atabase has shown extremely small sensitivity to exponents r 4, r 5, r 6 an r 7, it was simply assume that r 4 = r 6 = r 7 = an r 5 = / (the exponent of / was hosen simply to agree with the urrent ACI esign formula, v f, motivate by Pauw s [57] observation that the tensile strength is approximately proportional to f ). As for the shear span a/, although its effet shoul in priniple be signifiant, the sensitivity to exponent r 3 is low, beause of high satter of the atabase. Therefore, two smooth formulas, one without a/ an one with it, have been ientifie. The simpler formula not involving a/, whih is the number one proposal for the oe mae here an is plotte in Fig. (top left), reas: v = µ f + /, = ( ) /3 ρ (7) f where µ = 5 (mean), µ = 3.8 (esign) (s L = 6.%, ω = 6, %) (8) (v an f are in psi, an ρ is a number (not a perentage). The value µ = 5 gives the soli urve in Fig., representing the least-square fit (mean fit) of the atabase, while the value µ = 3.8 gives the ashe urve in Fig., whih is obtaine as the mean urve minus.65s L an orrespons to a 5% probability ut-off, i.e., the probability of v lying below the ashe urve is 5% (the optimize values of all parameters are roune off to the extent that ω woul not be appreiably affete). If the shear-span is inlue in the optimize formula, one gets a slightly more general formula, whih is plotte in Fig. (top right) an has the same form as (7) exept that the values of, µ, s L an ω are replae by: ( ρ = 35 f ) /3 ( ) /3 a with µ = 5.5 (mean), µ = 4. (esign) (s L = 6.%, ω = 6.%) (9) 4

15 However, as seen from the ω value, the improvement ahieve by onsiering a/ is negligible. Even though unsmooth formulas generally impair the onvergene of omputer algorithms (important, for example, for optimization of esign), there has been a preferene to use for oes straight-line formulas, possibly with inequality ut-off. In this spirit, the formula in Fig. (bottom left) (having, not surprisingly, an appreiably higher oeffiient of variation), has been obtaine (with D in inhes); v ( = min µ f D /, µ f ), D = 7 ρ /3 (3) with µ = µ = 3.5 (mean), µ =.5 (esign) (s L =.3%, ω =.5%) (3) Although µ shoul equal µ from the statistial viewpoint, one oul alternatively use for esign µ =, in whih ase the urrent shear strength f woul be retaine for small enough sizes. Fig. shows imensionless plots omparing the foregoing three formulas to the ACI-445 atabase, in whih the area of eah plotte ata irle is proportional to the weight assigne on the basis of the histogram in. Fig. gives analogous formulas obtaine when the weights are assigne aoring to the histogram in log. The ifferene in fits is insignifiant. For the sake of omparison, Fig. 3 gives analogous formulas obtaine when all the points of the atabase have equal weights. Here the ifferene in fits ompare to Fig. is more pronoune an, espeially, the ata points for the eepest beams, whih are the least numerous, are fitte poorly. Yet, a lose fit of these ata points is most important for extrapolation to larger sizes than teste. Amittely, the oeffiients of variation ω of all the formulas are quite large, but this is ue to the enormous ata satter. When the ontamination by non-size parameter is eliminate, the satter is far less; see Figs The fat that the optimize exponent of steel ratio is positive alls for a omment. One might take the viewpoint [45] that small enough ρ must lea to flexural failure ue to yieling of steel, whih is not brittle an orrespons to. This viewpoint suggests that an inrease of ρ shoul ause an inrease of brittleness, an thus shoul ause a erease of ; in other wors, the exponent of ρ may be expete to be negative. Yet fitting of the atabase learly iniates a positive exponent. The reason may be either that the atabase is spoile by mixing too many influenes, or that the viewpoint of transition to flexural failure is not germane. On the other han, sine high-strength onrete is more brittle than normal onrete an higher brittleness orrespons to smaller, it is logial that the optimize exponent of f in the expression for is negative [45]. Critial Examination of Other Formulas for Beam Shear The lose agreement with the broa-range Northwestern tests an Toronto tests (Fig. 7) an with the reor-size Japanese tests (Fig. 8), together with the lak of isagreement with the lassial size-effet tests (Fig. 9) of limite size ranges or small maximum sizes, provies strong experimental support for the frature mehanis size effet espouse in this stuy, as well as a strong argument against other size effet formulas in beam shear whih annot fit these tests losely. Two among the other formulas, namely the CEB-FIP formula, an the power law of the type v /3, are purely empirial an thus there is nothing to isuss beyon statistial omparisons with iniviual size-effet test series (regaring v /3 ; see the urves in Figs. 7 5

16 (left an right) an 8, an regaring CEB-FIP, see Fig. ). Theoretial justifiations, however, have been propose for three other formulas, whih will now be isusse. ) Formula of Japan Soiety of Civil Engineers (JSCE) As a onsequene of a pioneering proposal of Oakamura an Higai [55] mae in 98 (several years before the onset of the energeti frature-base theory), an of a later realibration in [53, 54], a power law of the type v /4 (propose on an empirial basis alreay by Kani [44]) was aopte for the onrete esign oe in Japan [43]. In 98, Okamura an Higai s proposal was a breakthrough an it was logial to unerpin it by Weibull s (939) statistial theory, the only size effet theory available at that time. That theory [4] iniates a power law size effet with exponent 3/m for three-imensional similarity an /m for two-imensional similarity, where m is the empirial Weibull moulus (shape parameter), the value of whih is normally etermine from the oeffiient of variation of tensile strength of many iential speimens. Base on the lassial work of Zeh an Wittmann [68], Okamura an Higai assume that m =. Further assuming three-imensional similarity to apply, they ame up with the exponent 3/m = /4. Reent in-epth stuies [8, 9, ], however, showe that the apparent value of m for onrete inreases markely with struture size an that, after separation of eterministi nonloal effets, the orret value of Weibull moulus for onrete is about m 4. This gives, for three-imensional similarity, the exponent 3/m = /8. Furthermore, it transpire that not only the exponent value but also the assumption of three-imensional similarity of beam shear failures nees to be revise. The reason is that material failure at one point in three imensions oes not suffie to ruin the beam. Rather, the beam must fail simultaneously over its whole with, whih means that the loation of the failure initiation point oul be ranom only in two imensions (in the length an epth oorinates, but not in the with oorinate). Therefore, if the Weibull theory were appliable, the size effet exponent woul have to be /m = /. This gives in the bi-logarithmi plot a straight line of slope / rather than /4. But suh a weak size effet blatantly isagrees with all the test ata. This onfirms that the Weibull theory annot be applie to the mean size effet in the beam shear problem (however, if extene to a nonloal form, it nevertheless appears appliable to the ranom satter [3]). Furthermore, as graually establishe, the assumption that the shear failure of reinfore onrete beams is governe by Weibull statistial theory is itself funamentally unaeptable. That theory is preiate on the hypothesis that the failure ours as soon as a marosopi rak initiates from one mirosopi flaw, before any signifiant stress reistribution in the struture is ause by the frature proess. This is true for fatigue frature of metals or finegraine eramis, but not onrete (exept perhaps on the sale of a large ams, the ross setion of whih is far larger than the harateristi length l of the frature proess zone in onrete, whih is typially.5 m). The beam oes not fail at the initiation of iagonal shear rak, as woul be require by Weibull theory, but only after this rak has propagate in a stable manner through most of the ross setion. The growth of the main iagonal shear rak is governe primarily by reistribution of the mean (eterministi) stress fiel. The ranomness of the loal strength of onrete at points loate far away from the rak path itate by frature mehanis annot signifiantly eflet that path. Therefore, the mean size effet observe in beam shear failure is ause preominantly by eterministi stress reistribution an the assoiate energy release prior to failure. The statistial ontribution esribe by Weibull s 6

17 statistial theory is negligible for the mean response [8, 9,,,, ], although it oubtless influenes the satter. Weibull s statistial size effet, whih is muh miler than the energeti size effet assoiate with stress reistribution, is appliable to the first raking loa. That this size effet exists, an that it is inee muh weaker than the size effet on nominal strength, is onfirme, e.g., by the test ata in Fig. 4 in [63]. ) Crak Spaing Theory an Role of Cohesive Stresses Aross the Crak The maximum rak opening with w is roughly proportional to effetive rak spaing s e, while s e is roughly proportional to. Hene, the eeper the beam, the larger is w at maximum loa. The larger the rak with, the smaller are the ohesive stresses σ transmitte aross the iagonal shear rak. These fats are inisputable. However, it was propose [65, 3] that the reution of the ohesive stresses with inreasing shoul be the physial soure of size effet. This proposition, whih they inorporate into their moifie ompression fiel theory (MCFT), is untenable. In this theory, it is assume that v ereases with w as v = onstant/( + w) where is a parameter epening on the maximum aggregate size. Assuming w to be approximately proportional to s e, one gets the relation v = v /( + s e ) where v, are ertain onstants. Consiering beams with no stirrups an no horizontal steel bars other than those at the bottom, an assuming s e to be proportional to [3], one gets for the MCFT rak spaing theory of size effet a formula of the type: v v = (3) + / where an v are parameters inepenent of struture size. For, this formula approahes the final asymptoti size effet v (whih is skethe by the ashe lines in Fig. ). Although the oeffiients of the formula are set up so that the test ata woul lie in the transitional range between the horizontal an inline asymptotes in the log-log plot, in whih the formula gives a muh miler slope than an an math some of the existing ata [3], the asymptoti size effet (propose by Leonaro a Vini [35] an ispute by Galileo [38]) is theoretially objetionable an in fat thermoynamially impossible (exponent / is the strongest size effet possible). This observation suffies to onlue that the rak spaing theory oes not have a soun theoretial basis. But there are more pratial objetions. Finite element simulation of propagation of a iagonal shear rak aoring to the ohesive (or Hillerborg s, fititious) rak moel [39] shows that, if the onrete is assume to have an unlimite ompression strength, the loa-efletion iagram is always rising, i.e., has no peak; see Fig. e. Therefore, the stresses transmitte aross the iagonal shear rak annot eie the maximum loa, ontrary to what is assume in the rak spaing theory. Rather, the maximum loa must be ontrolle by the shear-ompression frature of the ligament above the tip of the iagonal shear rak (Fig. e). This onlusion is supporte by the fat that the ompression stresses in the ligament attain the ompression strength of onrete (whih in turn implies that, in the strut-an-tie moel, the failure must be eie by the rushing of the ompression strut [9, ]). So the size effet in beam shear physially represents the size effet of shear-ompression frature. The fat that the ohesive stresses aross a iagonal shear rak annot have a signifiant influene on the maximum loa is further supporte by the experimental observations at the University of Toronto [, 3]. Fig. a,b shows the major raks observe near the maximum loa. The strain in the steel bar at the ross setion passing through the tip of the main 7

18 iagonal shear rak was also measure. From this, one an alulate the axial fore in the steel, an knowing this fore an the bening moment, one an fin the preise loation of the ompression fore resultant in this ross setion of the teste beam (Fig. b). Beause this resultant must also pass through the point of intersetion of steel bars an the vertial resultant at the support, one reaily ientifies the line of the ompression resultant, whih is rawn in Fig. b. Now it shoul be note that this resultant passes above the observe main iagonal rak an runs parallel to the top segment of this rak. Consequently, the shear an normal stresses transmitte aross that rak at maximum loa must be negligible. From the magnitue of this resultant one an also estimate the ompression stresses σ in the ligament above the rak tip. One fins them to be roughly equal to the ompression strength of onrete, f. Beause the tensile ohesive stresses σ in the iagonal rak at maximum loa must be muh less than the tensile strength f t (surely less than.5f t ), an beause f t.f, it follows that σ must be less than.5σ, probably muh less. So it must be onlue that the ontribution of the tensile ohesive stresses σ in the iagonal rak at maximum loa must be negligible ompare to the ontribution of the ompressive stresses σ parallel to the rak (Fig. ), an thus annot ontrol the maximum loa. This onlusion puts again in question the physial explanation of size effet by the rak spaing theory [3, 65]. The fat that the line of the ompression resultant as ientifie in Fig. is parallel to the en portion of the iagonal shear rak (or that it oes not interset this rak) further implies that shear stresses ue to aggregate interlok annot have an important effet on the maximum loa. Fratal Charateristis of Frature an Carpinteri s MFSL The role of the fratal harateristis of frature has been ebate for almost a eae, an a learer piture is graually emerging [6]. Although some questions remain unresolve [5], two salient points rystallize: () The so-alle multi-fratal saling law (MFSL) propose by Carpinteri [3] annot be applie to failures ourring after large stable rak growth, whih is the ase of iagonal shear raks in beams, the main reason being that the stress reistribution ue to large raks an the assoiate energy release are not taken into aount in the existing form of the fratal theories. () The MFSL is iential to a speial ase of a more general size effet formula for rak initiation erive from (non-fratal) frature mehanis [6, 7] an refine in [7] (see also [8, ]). So there is no longer any isagreement about the formula itself, but only about the physial justifiation of the formula [6, 5]. Beause this formula eals only with failures at rak initiation (e.g., moulus of rupture), there is no nee to well on the iffiult fratal questions any more. They are not relevant to our problem. Also note that, in the strut-an-tie (truss) moel, the straight line onneting the support an the applie loa (of slope a/) woul normally be assume as the axis of the imagine ompression strut. The fat that the Toronto measurements imply a muh steeper ompression resultant means that the simplisti version of the strut-an-tie moel (or truss moel) is invali. While the fraturing strut-an-tie moel [9] an explain the size effet in beam shear, unertainty about the orret slope of the ompression strut in beam shear remains to be one obstale to using the strut-an-tie moel as a preitive tool. 8

19 Conlusions. Beause the size range of main pratial interest lies outsie the range of the available test ata, the size effet law annot be set up purely empirially. A realisti theoretial founation is inevitable.. Sine onrete is just one of many quasibrittle materials failing ue to quasibrittle frature or softening amage, it woul be illogial to expet ifferent laws to govern failure. Laws that are ommon to all these materials are a more logial hoie, espeially when the experimental eviene is ambiguous ue to high ranom satter. 3. The hypothesis that the maximum loa in shear failure is ontrolle by propagation of ohesive frature or softening amage leas to the same size effet law as establishe for other quasibrittle materials. The experimental eviene an be mathe with this law as losely as one oul esire in view of the inevitable experimental satter. 4. Although a speialize frature-base moel, suh as the fraturing truss moel, yiels a realisti form of the size effet formula an intuitively explains the mehanism of size effet, this moel is insuffiient for preiting the epenene of size effet law oeffiients on the shear span, reinforement ratio, aggregate size, material strength, et. 5. In the asymptoti situations of infinitely small an infinitely large strutures, the analysis of failure an size effet beomes far simpler an learer than in the pratial size range. Therefore, it is appropriate to erive a theoretial formula by asymptoti mathing, a tehnique that interpolates between the known asymptoti behaviors at the opposite infinities. 6. Assuming the failure loa to be ontrolle by ohesive frature parameters (material strength σ an frature energy G f ), an exploiting the known first two asymptoti terms of the large- an small-size asymptoti expansions of the ohesive rak moel (or the nonloal amage moels), one an easily eue a simple transitional size effet law by means of imensional analysis. 7. Asymptoti mathing base on imensional analysis leas logially to the size effet law propose for beam shear by Bažant in 994. As a simplifiation, one an also use a power law of exponent /, supplemente with an upper boun for small sizes. 8. The size effet law establishe agrees with all of the existing test series in whih the beam epth was varie signifiantly, the number of whih is. Espeially, this size effet law agrees with the tests onute at Northwestern University an at University of Toronto, whih represent the only ata with a broa size range an almost perfet geometrial saling. 9. Most importantly, the Northwestern an Toronto test series, an the Japanese tests of Shioya et al. (whih are the only sale tests of a signifiant size range) onfirm that the large-size asymptoti slope of size effet is /. This in turn proves that the explanation of failure lies in frature mehanis.. The oeffiients of the size effet law are alibrate by optimum fitting of the ACI-445 atabase (whih inlues 398 ata) an of the Northwestern University atabase (664 ata) [9]. However, sine these large atabases are inevitably ontaminate by ranom 9

20 variation of fators other than the size, the ifferenes in the oeffiients of variation of errors in omparison to other propose formulas are relatively insignifiant.. The size effet law in (7) provies a goo an simple representation of the ACI-445 atabase. The oeffiient of variation of regression errors, whih is about 6%, allows etermining the 5% probability ut-off whih is suitable for a esign oe formula.. Comparisons with the require asymptoti behavior an with the test ata pertinent to the size effet in beam shear reveal serious ontraitions with the ol theories an formulas (inluing the JSCE formula base on Weibull statistial theory, the rak spaing enhanement of MCFT, the empirial CEB-FIP formula, an the appliation of MFSL to iagonal shear failure) an reveal signifiant isagreement with the empirial power law of exponent /3. Aknowlegment: Finanial support by the Infrastruture Tehnology Institute of Northwestern University is gratefully aknowlege. Professor M.T. Kazemi of Sharif University of Tehnology, Tehran, is thanke for valuable omments. Appenix I. The Question of Differene Between Nothe Speimens an Unnothe Strutures with Large Craks Appliation of the size effet law (7) to unnothe strutures with large raks rests on two hypotheses: Hypothesis I. The major raks at failure in small an large strutures are geometrially similar. Hypothesis II. The frature proess zones at the tips of a noth or a rak give approximately the same energy issipation rates. Aoring to the ohesive (or fititious) rak moel, both hypotheses are asymptotially exat for an thus are goo approximation for large enough beams. These hypotheses are justifie by the following theorem: As long as the frature problem has a unique solution, sharp (LEFM) raks in geometrially similar strutures have similar paths an, at maximum loa (as well as other orresponing stages of loaing), also geometrially similar lengths. This theorem an be rigorously proven by saling transformations of all the ifferential equations, bounary onitions an rak fae onitions of equivalent LEFM, whih is an approximation (wiely use in nonlinear frature mehanis) base on the assumption that a rak with a large frature proess zone is approximately equivalent to a sharp (LEFM) rak with a tip loate in the mile of the frature proess zone. The ohesive rak moel shows that the ohesive stress σ tail at the tail of the frature proess zone is zero for a naturally growing rak. But when the frature proess zone is attahe to the noth tip, σ tail is, for a finite beam size, non-zero (approahing zero only if the size tens to infinity). The onsequene is that the size effet with seon-orer asymptoti auray for large sizes is slightly ifferent from (7): v = v ( + + ) / (33) (see [], an in etail Eqs an 9.9 in []). Here an are onstants, an must be larger than. Obviously this law has the same asymptotes as (7) (whih is of ourse require by the imensional analysis presente here) an for beomes iential to (7).

21 For finite, the transition between the asymptotes is more abrupt (with a sharper urvature) than for (7), an the v values in the intermeiate range are always larger. But they are only slightly larger. Base on preliminary numerial ohesive rak simulations, 5 to, but then the ifferene from (7) is negligible ompare to the unertainty ue to the satter of test results. This is one justifiation of the use the simpler size effet law (7) for beam shear. Besies, (33) has one more unknown parameter than (7), an the ata satter makes it next to impossible to ientify it experimentally. Another justifiation stems from the fat that, before an overloa to failure, the beam may be subjete to yli loaing. Suh loaing reues the ohesive stresses in a naturally grown rak to almost zero, whih means that a preexisting fatigue rak is stress free an thus ats like a noth. Assuming that yli loaing may our is on the sie of safety. Finally, those who think that the ifferene between a noth an a natural rak is important shoul note that the growth of the iagonal shear rak is usually not what auses the loa to peak. Rather, it is the growth of shear-ompression frature aross the ligament above the tip of the iagonal shear rak. How the ifferene between a noth an a naturally grown rak might affet suh frature is not known. But the present asymptoti onsierations base on imensional analysis, whih le to (7), must apply to shear-ompression frature as well. Appenix II. Questions of Statistial Evaluation an Bias Although formula ientifiation from beam shear ata is a problem of statistial regression, in one urrent (unpublishe) investigation it has been trie to reue omparisons of various propose formulas for beam shear to elementary population (ensemble) statistis of the ratios γ i of the measure an alulate values. In this investigation, the oeffiient of variation has been efine as follows: COV = γ (γ i γ) n with γ = γ i (34) n i where γ i = ˆv i /v i (i =,,...n) (35) However, suh an approah is funamentally inorret, for four reasons. First, if the parameter ientifiation is not base on the metho of least squares, the results annot be unbiase, i.e., the resulting mean an variane of formula parameters annot be the mean an variane of the statistial istribution of these parameters. Seon, if the minimize expression is not a sum of squares of errors, the tangential linearizations mae in any ata fitting algorithm an lea to numerial instability. Thir, fitting of the ratio ˆv i /v i implies weighting of the ata ˆv i as a funtion of the unknown values of v i to be solve, whih is inamissible. Fourth, if all v i in (35) an (34) are replae by kv i, k being any onstant between an, then γ i an γ i are replae by γ i /k an γ i /k, an so it is foun that the value of COV oes not hange. Suh a efinition of the oeffiient of variation, whih is insensitive to multiplying the formula for v by any number, makes no sense at all. Obviously, minimization of (COV ) annot be use to alibrate formula parameters. As a workable alternative to the use of ratios γ i = ˆv i /v i, it has bee trie to ientify formula parameters by minimizing not (COV ) but the sum Φ = i(γ i ). This sum may be rewritten as: ( ) n ˆvi n Φ = = w i (ŷ i y i ) (36) v i i= i= i

22 where y i = /v i, ŷ i = /ˆv i, w i = ( v i ) (37) So, the minimization of Φ represents simply a weighte least-square regression of ata ȳ i, minimizing the sum of square errors in y i with weights w i proportional to the squares of the measure values ˆv i. Asie from a misguie esire to avoi regression statistis, the motivation for use of the ratios γ i = ˆv i /v i has been to inrease the weight of smaller v i values. But there is a problem with this motivation. As is well known [37,, 5, 36, 7, 59, 49], to minimize bias, the regression shoul be onute in suh variables in whih the variane (Var(v ), in our ase) is as uniform as possible, an nonuniform weights shoul not be use unless the variane varies by an orer of magnitue or more (i.e., if the ata are heteroskeasti). For beam shear size effet ata (as well as most size effet ata), the variane appears to be the least nonuniform in the plots of log v versus log, an so these are the preferable oorinates for statistial regression. The impliation of unjustifie shear strength epenene of the weights in (37) is that the minimization of (36) is not unbiase, i.e., the estimates of the mean an the stanar eviation (or COV) of the optimize parameters are not the atual mean an stanar eviation of the statistial istribution of these parameters (in other wors, are not the maximum likelihoo estimates). One onsequene is that the mean plus.65 times stanar eviation oes not give the orret (unbiase) value for the 5th perentile, neee for setting up the esign formula, an generally all the statistial estimates are not orret (note that ata weighting to ompensate for the rowing of ata points into the small-size range is an entirely ifferent matter). On the other han, the absene of shear-strength epenent weighting from the least-square fitting of ln v ensures that the optimize parameters are unbiase. Beause ((v ) = (v ) /(v ), the transformation of sale from linear to logarithmi has a similar effet as the weighting of the ata in proportion to /(v ), but without ausing any statistial bias. Referenes [] Angelakos, D, Bentz, E.C., an Collins, M.P. (). Effet of onrete strength an minimum stirrups on shear strength of large members. ACI Strutural J. 98 (3), 9 3. [] Ang, A. H.-S., an Tang, W.H. (976). Probability onepts in engineering planning an esign. Vol., Se.7. [3] Bahl, N.S. (968) Über en Einfluss er Balkenhöhe auf Shubtragfähighkeit von einfelrigen Stalbetonbalken mit un ohne Shubbewehrung. Dissertation, Universität Stuttgart. [4] Barenblatt, G.I. (96). The mathematial theory of equilibrium of raks in brittle frature. Av. Appl. Meh., 7, [5] Bažant, Z.P. (984). Size effet in blunt frature: Conrete, rok, metal. J. of Engrg. Mehanis, ASCE,, [6] Bažant, Z.P. (987). Frature energy of heterogeneous material an similitue. Preprints, SEM- RILEM Int. Conf. on Frature of Conrete an Rok (hel in Houston, Texas, June 987), e. by S. P. Shah an S. E. Swartz, publ. by SEM (So. for Exper. Meh.) [7] Bažant, Z.P. (998). Size effet in tensile an ompression frature of onrete strutures: omputational moeling an esign. Frature Mehanis of Conrete Strutures (Pro., 3r Int. Conf., FraMCoS-3, hel in Gifu, Japan), H. Mihashi an K. Rokugo, es., Aeifiatio Publishers, Freiburg, Germany, [8] Bažant, Z.P. (99). Why ontinuum amage is nonloal: Miromehanis arguments. Journal of Engineering Mehanis ASCE 7 (5), 7 87.

23 [9] Bažant, Z.P. (997). Fraturing truss moel: Size effet in shear failure of reinfore onrete. J. of Engrg. Mehanis ASCE 3 (), [] Bažant, Z.P. (999). Size effet in onrete strutures: nuisane or neessity? (plenary keynote leture), in Strutural Conrete: The Brige Between People, Pro., fib Symp. 999 (hel in Prague), Féération Internationale u Béton, publ. by Viaon Ageny, Prague, pp [] Bažant, Z.P. (). Probabilisti moeling of quasibrittle frature an size effet. (prinipal plenary leture, Pro., 8th Int. Conf. on Strutural Safety an Reliability (ICOSSAR), hel at Newport Beah, Cal., ), R.B. Corotis, e., Swets & Zeitinger (Balkema), 3. [] Bažant, Z.P. (). Saling of strutural strength. Hermes Penton, Lonon. [3] Bažant, Z.P. (3). Statistial istribution of size effet in quasibrittle frature. Report, in preparation. [4] Bažant, Z.P., an Kazemi, M.T. (99). Size effet on iagonal shear failure of beams without stirrups. ACI Strutural Journal 88 (3), [5] Bažant, Z.P., an Kim, Jenn-Keun (984). Size effet in shear failure of longituinally reinfore beams. Am. Conrete Institute Journal, 8, ; Dis. & Closure 8 (985), [6] Bažant, Z.P., an Li, Z. (995). Moulus of rupture: size effet ue to frature initiation in bounary layer. J. of Strut. Engrg. ASCE, (4), [7] Bažant, Z.P., an Li, Z. (996). Zero-brittleness size-effet metho for one-size frature test of onrete. J. of Engrg. Mehanis ASCE (5), [8] Bažant, Z.P., an Novák, D. (). Energeti probabilisti size effet, its asymptoti properties an numerial appliations. Pro., European Congress on Computational Methos in Applie Siene an Engineering (ECCOMAS ), Barelona, pp. 9. [9] Bažant, Z.P., an Novák, D. (a). Probabilisti nonloal theory for quasibrittle frature initiation an size effet. I. Theory. J. of Engrg. Meh. ASCE 6 (), [] Bažant, Z.P., an Novák, D. (b). Probabilisti nonloal theory for quasibrittle frature initiation an size effet. II. Appliation. J. of Engrg. Meh. ASCE 6 (), [] Bažant, Z.P., an Novák, D. (). Energeti-statistial size effet in quasibrittle failure at rak initiation. ACI Materials Journal 97 (3), [] Bažant, Z.P., an Novák, D. (). Proposal for stanar test of moulus of rupture of onrete with its size epenene. ACI Materials Journal 98 (), [3] Bažant, Z.P., an Oh, B.-H. (983). Crak ban theory for frature of onrete. Materials an Strutures (RILEM, Paris) 6, [4] Bažant, Z.P., an Planas, J. (998). Frature an Size Effet in Conrete an Other Quasibrittle Materials. CRC Press, Boa Raton an Lonon (Setions. 9., 9.3) [5] Bažant, Z.P., an Yavari, H. (3). Shoul the size effet in quasibrittle strutures be moele by multifratal or energeti saling law? Theor. an Appl. Meh. Report Northwestern University (in preparation). [6] Bažant, Z.P., Yu, Q., an Zi, G. (). Choie of stanar frature test for onrete an its statistial evaluation. Int. J. of Frature 8 (4), De., [7] Bek, J.V., an Arnol, K.J. (977). Parameter estimation in engineering siene. J. Wiley, New York. [8] Bukingham, E. (94). On Physially Similar Systems: Illustrations of the Use of Dimensional Equations. Phys. Rev. 4,

24 [9] Beq-Girauon, E. (). Size effet on frature an utility of onrete an fiber omposites Ph.D. Dissertation, Northwestern University. [3] Carpinteri, A. (994). Fratal nature of materials mirostruture an size effets on apparent material properties. Mehanis of Materials 8, 89. [3] Collins, M.P., an Kuhma, D. (999). How safe are our large, lightly reinfore onrete beams, slabs an footings. ACI Strutural J. 96 (4), [3] Collins, M.P., an Mithell, D. (99). Prestresse onrete strutures. Prentie Hall, Englewoo Cliffs, New Jersey 99 (setion 7.). [33] Collins, M.P., Mithell, S., Aebar, P. an Vehio, F.J. (996) General shear esign metho. ACI Strut. J., 93(), [34] Comité Euro-international u Béton (99) CEB-FIP Moel Coe 99. [35] a Vini, L. (5s) see The Notebooks of Leonaro a Vini (945), Ewar MCury, Lonon (p. 546); an Les Manusrits e Léonar e Vini, transl. in Frenh by C. Ravaisson-Mollien, Institut e Frane (88-9), Vol. 3. [36] Draper, N., an Smith, F. (98). Applie regression analysis. n e. J. Wiley, New York. [37] Fox, J. (997). Applie regression analysis, linear moels an relate methos. Sage Publiations (also: [38] Galileo Galilei Lineo (638) Disorsi i Demostrazioni Matematihe intorno à ue Nuove Sienze, Elsevirii, Leien. (English transl. by T. Weston, Lonon (73), pp. 78 8) [39] Hillerborg, A., Moée, M. an Petersson, P.E. (976). Analysis of rak formation an rak growth in onrete by means of frature mehanis an finite elements. Cement Conrete Res., 6, [4] Iguro, M., Shioya, T., Nojiri, Y. an Akiyama, H. (985) Experimental stuies on shear strength of large reinfore onrete beams uner uniformly istribute loa. Conrete Library International of JSCE, No.5 (translation from Proeeings of JSCE, No. 345/V-, August 984), [4] Irwin, G.R. (958). Frature. In Hanbuh er Physik, Vol. 6, Flügge, e., Springer-Verlag, Berlin, pp [4] Izquero-Enarnaión, J.M. (3). Ars sine sientia nihil est. Conrete International 5 (5), May, p. 7. [43] (99). Stanar speifiation for esign an onstrution of onrete strutures. Part I (Design), Japan So. of Civil Engrs., Tokyo. [44] Kani, G.N.J. (967). How safe are our large reinfore onrete beams? ACI J., 58(5), [45] Kazemi, M.T. (3). Private ommuniation to Z.P. Bažant. [46] Kfouri, A.P., an Rie, J.R. (977). Elasti-plasti separation energy rate for rak avane in finite growth steps. Frature 977 (Pro., 4th Int. Conf. on Frature, ICF4, Waterloo), D.M.R. Taplin, E., Univ. of Waterloo, Ontario, Canaa, Vol., [47] Knauss, W.C. (973). On the steay propagation of a rak in a visoelasti sheet; experiment an analysis. The Deformation in Frature of High Polymers, H.H. Kaush, E., Plenum, New York [48] Knauss, W.C. (974). On the steay propagation of a rak in a visoelasti plasti soli. J. of Appl. Meh. ASME 4 (), [49] Lehmann, E.L. (959). Testing statistial hypotheses. J. Wiley, New York. [5] Leonhart, F. an Walther, R. (96) Beiträge zur Behanlung er Shubprobleme in Stahlbetonbau. Beton-un Stahlbetonbau (Berlin), Marh, 54 64, an June,

25 [5] Leonov, M.Y., an Panasyuk, V.V. (959). Development of a Nanorak in a Soli. Priklnaya Mekhanika (transl. Soviet Applie Mehanis) 5 (4), 39 4; English transl. in Frature: A topial enylopeia of urent knowlege, G.P. Cherepanov, e., Krieger Publ. Co., Malabar, Floria 998. [5] Manel, J. (984). The statistial analysis of experimental ata. Dover Publiations. [53] Niwa, J., Yamaa, K., Yokozawa, K. an Okamura, M.P.S. (986). Reevaluation of the equation for shear strength of R.C. beams without web reinforement. Proeeings of the Japanese Soiety of Civil Engineers, Vol. 5, No.37, [54] Niwa, J., Yamaa, K., Yokozawa, K., an Okamura, H. (987). Reevaluation of the equation for shear strength of reinfore onrete beams without web reinforement. JSCE Conrete Library International 9, [55] Okamura, H. an Higai, T. (98). Propose esign equation for shear strength of reinfore onrete beams without web reinforement. Proeeings of Japanese Soiety of Civil Engineers, Vol. 3, 3 4 [56] Palmer, A.C. an Rie, J.R. (973). The growth of slip surfaes on the progressive failure of over-onsoliate lay Pro. Roy. So. Lon. A., 33, [57] Pauw, A. (96). Stati moulus of elastiity of onrete as affete by ensity Journal of the Amerian Conrete Institute., 3: [58] Petersson, P.E. (98). Crak growth an evelopment of frature zones in plain onrete an similar materials. Report TVBM-6, Div. of Builing Materials, Lun Inst. of Teh., Lun, Sween. [59] Plakett, R.L. (96). Priniples of regression analysis. Clarenon Press. [6] Riabouhinski D.P. (9-). Annual Report. British Avisory Committee for Aeronautis, Abstrat No. 34, p. 6 (also L Aerophile, Sept., 9). [6] Rie, J.R. (968). Mathematial analysis in the mehanis of frature, Frature An avane treatise, Vol., e. H. Liebowitz, Aaemi Press, New York, [6] RILEM (3) Committee QFS, Quasibrittle Frature Saling. State-of-Art Report; to be submitte to Materials an Strutures (RILEM). [63] Shioya, T. an Akiyama, H. (994). Appliation to esign of size effet in reinfore onrete strutures. Size Effet in Conrete Strutures (Pro., Japan Conrete Institute International Workshop, Senai), H. Mihashi, H. Okamura an Z.P. Bažant, es., E&FN Spon, Lonon, [64] Smith, E. (974). The struture in the viinity of a rak tip: A general theory base on the ohesive rak moel. Engineering Frature Mehanis 6, 3. [65] Vehio, F.J., an Collins, M.P. (986). The moifie ompression fiel theory for reinfore onrete elements subjete to shear. ACI J.. Pro. 83 (), 9 3. [66] Weibull, W. (939). A statistial theory of the strength of materials. Pro. Royal Sweish Aaemy of Eng. Si. 5, 45. [67] Wnuk, M.P. (974). Quasi-stati extension of a tensile rak ontaine in visoelasti plasti soli. J. Appl. Meh. ASME 4 (), [68] Zeh, B. an Wittmann, F.H. (977). A omplex stuy on the reliability assessment of the ontainment of a PWR, Part II. Probabilisti approah to esribe the behavior of materials. Trans. 4th Int. Conf. on Strutural Mehanis in Reator Tehnology, T.A. Jaeger an B.A. Boley, es., European Communities, Brussels, Belgium, Vol. H, J/, 4. 5

26 List of Figures Histogram of beam epths (number of test ata in eah beam epth interval of 5 in. versus the epth in inhes) ACI-445 atabase for beam shear an plots of various size effet formulas Example of ontamination of atabase ue to variation of unertain fators other than size, showing some typial test ata inlue in the ACI-445 atabase, ompare to the broa range ata from the Northwestern an Toronto tests an their ommon optimum fit Example of fallaious statistial analysis: (a,) Hypothetial perfet ata generate so as to math exatly the size effet law for four ifferent onretes; (b,) inorret inferene mae by regression of the ombine ata set. Note that shifting of the hosen size range of ata an yiel any esire slope of regression line Example of the effet of shifts in the size range of a highly sattere atabase on the slope of the regression line a) Softening stress-separation urve of ohesive (or fititious) rak moel; b) Geometry of reinfore onrete beam Two test series of geometrially similar beams with a signifiant size range, fitte by size effet law (Eq. 7). Top: optimum fit of ata of Bažant an Kazemi (99, Northwestern); mile: optimum fit of ata of Pogorniak-Stanik (998, Toronto) an Yoshia (, Toronto); bottom: onstraine optimization of ombine Northwestern an Toronto ata. Left: Plots use in nonlinear optimization; Right: Optimization results shown in linear regression plots (note the isagreement with power laws of exponents /3 an /4) Size effet of two lassial series of shear tests of very large beams onute in Japan [4, 63] Optimum fits by size effet law (Eq. 7) of remaining existing ata [44, 5, 3] that have a non-negligible size range but have gross eviations from geometrial similarity. 7 (a) () Beam shear failure pattern measure at University of Toronto an its interpretation; (e) loa-efletion iagram of a beam with growing iagonal shear rak, an imensionless loa-efletion iagrams with the peak ontrolle by shear-ompression failure (CCM = ohesive rak moel, LEFM = linear elasti frature mehanis).. 7 Three formulas optimize by least-square fitting of ACI-445 atabase, with weights inversely proportional to the histogram of ata (bottom right) as a funtion of (note that the orinates are the eai (not natural) logarithms)... 7 Same as Fig. but the histogram (at bottom right) is a funtion of log rather than Same as Fig. but the weights of all the ata points are taken as equal

27 a) Limite Size Range (up to 9 in.) 86% of ata Range of main pratial interest for size effet Number of Tests Extrapolation that must be anhore in a theory ACI 445 Data Bank Bažant an Yu, 3 6 Beam Depth h (inhes) 9 b) Full Size Range of Interest (up to 5 in.) 86% of ata Number of Tests Range of main pratial interest for size effet Extrapolation that must be anhore in a theory Fig. 5 5 Beam Depth h (inhes) 5

28 5, 3 4 log v C (MPa) 5 4 It is assume that all urves.5 3 are alibrate to pass through 4 the entroi of the atabase Leonaro size effet (a Vini) Fig. 7 Extrapolation Range of Conern for Pratie 6 3. Bažant's Size Effet Law Extene Size Effet Law 3 MCFT (rak spaing) 4 CEB-FIP 5 JSCE (Weibull) 6 MFSL 7 LEFM with boun 8 ACI-445 F (3/3) 5,7 8 log D (m)

29 v (MPa) a) b) ) ) Ahma et al. (986) ata Hanson (958) ata Islam et al. (998) ata Lambotte et al. (99) ata v (MPa) Toronto ata Northwestern ata. Depth (mm) Depth (mm) Depth (mm) Depth (mm) e) f) g) h) Mphone et al. (984) ata Niwa et al. (987) ata Remmel (99) ata Mooy et al. (99) ata. Depth (mm) Fig. 3 Depth (mm) Depth (mm) Depth (mm)

30 a) Perfet ata b) Wrong onlusion log v v v Data range log v Same? 4 log log ) Perfet ata ) Wrong onlusion log v Fig. 4 v v Data range log log v Same log 3?

31 v a) v log v Combine ata JSCE 4 log v b) Combine ata ACI- 445 F 3 v log v.46 log If we oul test > m only log ) ) log v /v Toronto tests Northwestern tests Fig. 5 log log /

32 σ σ G f G F σ w w f w w F F V=F a Fig. 6

33 log v /v v = 5.3 MPa = 8.89 mm ω =.% Bažant & Kazemi (99). v v 5 Bažant & Kazemi (99) 5 v = 5.3 MPa = 9.7 mm ω =.3% log v /v log v /v.. Toronto (998 - ) v =.46 MPa = 74 mm ω = 6.9% 4 Toronto (998 - ) Bažant & Kazemi (99) ω =.9%.. 3 v v v v 5 Toronto (998 - ) 8 Bažant & Kazemi (99) Toronto (998 - ) 5 v =.3 MPa = 76.7 mm ω =9.8% /4 /3 v ω =.8% log / Fig. 7 / v

34 Shear stress v (MPa) Iguro et al. (985) an Shioya & Akiyama (985) Aggr. Size = 5 mm COV = % 4 Atypial failure Iguro et al. (985) an Shioya & Akiyama (985) Aggr. Size = mm COV = % 3. Depth (m). Depth (m) Fig. 8

35 3 Kani (967) Kani (967) Kani (967) Kani (967) log v (MPa) a/=.5 v = 3. MPa = 76 mm log (mm).3 a/=3. v =.4 MPa = 46 mm log (mm).5 a/=4. v =.54 MPa = 66 mm log (mm).5 a/=5. v =.55 MPa = 68 mm log (mm) log v (MPa) Kani (967).5 a/=7. v =. MPa Fig. 9 = 33 mm log (mm) Leonhart & Walther (96) a/=3. v =.75 MPa Bahl (968) a/=3. v =.33 MPa = 386 mm = 388 mm.5 log (mm) log (mm)

36 a) b) ϕ R a P/= V b C C ϕ 7 5 V C V V ε s. ) Pogorniak-Stanik Measure steel strains ( 998), Toronto, Test BN V b e) Unrake beam ) CCM, Small LEFM, large CCM, large LEFM, small σ f ' Shear-ompression frature σ f ' t rak Fig. w/