Magazine of Concrete Research, 1998, 50, No.3, Sept., 237-246

Scale dependence of tensile strength of concretespecimens: a multifractal approach

A. Carpinteri", B. Chiaia" and G. Ferro"

Politecnico di Torino

The tensile strength is a parameter that is usually very difficult to determine. Nevertheless, knowledge of itseffective value is of strategie importance, as materials are often used at the limit of their performance. In all tensileexperiments done in the last few decades (direct tensile test, splitting test, pull-out test, double punch test) it isevident that the tensile strength varies with the dimension of the specimen. Increasing the specimen size results in adecrease in the nominai tensile strength. We propose a new size-effect law, the multifractal scaling law, whichdescribes satisfactorily the relationship between the strength parameter and the structural size for disorderedmaterials. The law considers the geometrical multifractality of the resisting section at the peak load, as aconsequence of the decreasing injluence of the disorder with increasing size. Extensive application of this law tovarious experimental results reported in the literature, and concerning direct and indirect tensile tests, is presented.A simpler version of this law is introduced and a comparison between the linear and nonlinear best-fits isdiscussed.

NotationaA,BbC,Dddrnaxftf~u.r;RX*y*XY

notch depthMFSL fitting parameterssecondary structural sizeSEL fitting parameterscharacteristic structural sizemaximum aggregate sizetensile strength for infinitely large structurescompressive strengthcharacteristic lengthmaximum tensile loadcorrelation coefficientabscissa of linearized MFSLordinate of linearized MFSLabscissa of linearized SELordinate of linearized SELnominai tensile strength

* Department of Structural Engineering, Politecnico di Torino, CorsoDuca degli Abruzzi 24, 10129 Torino, Italy.

(MCR 614) Paper received 4 November 1996; last revised 29 Ju1y1997; accepted 23 September 1997

IntroductionThe problem of determining the tensile failure prop-

erties of materials is an old one. Nevertheless, only inthe last few decades have researchers tackled it in amore systematic way, focusing on the peculiar aspectsof the phenomenon. The classical approach of limitanalysis and in generai of all the theories based onclassical mechanics, that in the past permitted us tobuild very reliable and durable structures, is now inade-quate to meet the requirements of very small (electro-nic structures) and very large structures (e.g. dams andatomi c power plants), where safety is a fundamentalaspect of designo In the same way, the new very highperformance composite materials, initially developedfor use in the aerospace industry, are now used ingenerai engineering practice (laminate and fibre-rein-forced composites in the manufacture industry, high-resistance concrete in the building industry, and metalmatrix composites such as aluminium-lithium, etc.).

In ali these cases, the inadequacy of classical mech-anics is evident: material faiIure is not completely de-scribed, and consequentIy only the mean strength valuesare used. Such mean values constitute only partial infor-mation about the complex phenomena. In particular, theclassical approach does not describe the strength scale

237

0024-9831 © 1998 Thomas Te/ford Ltd

Carpinteri et al.

effects, while it is clear from experimental observationsthat strength decreases with increasing structural size.

Fracture mechanics can be considered to originatefrom the observation of rnaterial-strength scale effects.The unexpected failure of the Liberty ships (1940-1945)under tensile stresses lower than the standard safetyvalue determined in the laboratory, and in a very brittleway compared with the experimental responses ob-served in specimens of the same material, represents oneof the first indicators of the importance of structure sizeon the global structural response (ductility and strength).Analogously, the very high ductility and strength ofmicroscopi c glass filaments (whiskers) observed byGriffith l cannot be explained if the structure size is notconsidered as an essential variable ofthe problem.

In this paper attention is focused on the variation inthe material tensile strength, which experimentally al-ways increases with decreasing structure size, regard-less of the kind of test carri ed out. This variation isinterpreted satisfactorily by the multifractal scaling law(MFSL),2 which can be considered as an extension ofthe monofractal scaling theory, proposed by Carpin-teri.3,4 In the assumed hypotheses, the resisting sectionof a specimen at the peak load is modelled as a lacunarfractal set, with a Hausdorff dimension less than 2. Onincreasing the specimen size, the progressive homo-genization of the random fields comes into play. Thus,for the largest sizes, fractality disappears (geometricalmultifractality) at the structural level, and the classicalEuclidean approach can he adopted. The main problemlies in the evaluation of the asymptotic value for thestrength, which is lower than the value obtained fromlaboratory specimens. The MFSL is based on the as-sumption that the resisting section shows geometricalmultifractality, and allows the strength of large struc-tures to be determined by extrapolating a sufficientrange of experimental data obtained in the laboratory.

In the present work, the MFSL is applied to a wideseries of relevant experimental results reported in theliterature. The results were obtained using direct tensiletests, splitting (or Brazilian) tests, double-punch testsand pull-out tests. The experimental values confirm thevalidity of the MFSL, which can be very useful indesign, by extrapolating the strength variation over anunbounded scale range.

In addition, a linear version of the law is intro-duced. With this version it is possible to determine in asimple way the values ofthe two free parameters using asimple linear regression, and it could represent a usefultool for the building codes. A comparison between thenon-linear version and the linear approach is also given.

Experimental tests for the tensile strengthIn this section the relevant literature on the British

and American standard tests for evaluating experimen-tally the tensile properties of concrete is reviewed. It is

238

very difficult to determine the mechanical parameter'tensile strength' far concrete-like materials. It is easierto test steel-wire ropes and cables because it is simpleto grasp the ends and roll them up on the cylinder of awinch or windlass. It is more difficult to grasp lessdeformable solids in order to perform a tensile test. Farthis reason, for a long time such tests have been limitedto compression and bending.

Normally, indirect tensile tests are used to evaluatethe tensile strength. We use the term 'indirect tensiletest' to indicate a test in which the tensile strength isdetermined by loading the specimen with a stress fielddifferent from the uniform tensile one. The most com-monly used indirect tensile tests for evaluating thetensile strength of concrete are the splitting test (BS1881: Part 1175 and ASTM C 4966) (Fig. l(a)) and thethree-point (ASTM C 787

) flexural loading and four-point flexural loading tests (BS 1881: Part 1188

)

(Fig. 1(b)). Other indirect tensile tests are: the wedge-splitting test (Fig. l(d)) (which is the equivalent of thecompact test used for metals), the double-punch tese(Fig. l(c)) and the pull-out test (Fig. l(e)). These testsare easy to perform, although the results obtained arenot always very significant, particularly with regard tothe complete experimental load versus relative defor-mation curve. This curve is fundamental in the descrip-tion and definition of the material behaviour undertensile load. For this reason, in the last thirty yearsnumerous researchers have proposed different experi-mental procedures.

The first set-up for performing direct mechanicaltests was designed with jaw-like clamps (cali ed frictionjaws), and was designed for doing tensile tests on ironbars. In practice, this solution was unsatisfactory, be-cause the jaws damage the extremities of the specimen,and thus the results obtained are unreliable. For thisreason, dog-bone specimens were used, the rationalebeing that the failure was localized in the middle zonewhere the specimen is thinner. However, the design andconstruction of satisfactory specimens requires an ad-vanced level of knowledge and experience, because theoptimum shape depends on the material to be tested.Some researchers tried to design specimens in whichthe compressive load was converted into a tensile one,by means of two co-axial perforations'" (Fig. 2(d)).

The problem is even more complex when we want todetermine the complete stress -deformation curve. Firstof all, it is necessary that the test is controlled in astable way, in order to obtain the post-linear part of thediagram. For quasi-brittle materials, such as concrete,the post-linear part of the curve is characterized by asoftening response, with a negative slope, due to thelocalization of failure. It is then necessary that the fail-ure occurs in a prefixed zone, where the extensometersare placed. Moreover, and this is the most delicaterequirement, it is necessary that the resultant of theload is maintained perfectly centred, in order to avoidbending moments, which can complicate the interpreta-

Magazine of Concrete Research, 1998, 50, No.3

(a)

(c)

I

////////

I

Tensile strength of concrete

I

Fig. l. Indirect tensile tests; (a) Brazilian splitting test; (b) Jour-pointflexural test; (c) double-punch test;9 (d) wedge-splittingtest; (e) pull-out test

(a) (b)

I""""",(b)

t

i::.i~·:I ..:",i' " ,I.' ::.'i. . ~~I I

I· ..",'. rJ!i. ".~~.I I

, "

r

ti ZI

(d) (e)

t t

(d)

Fig. 2. Direct tensile schemes: (a) Evans and Marathe; Il (b) Reinhardt et al.; 12 (c) Petersson; 13 (d) LuonglO

tion. Even if we centre the resultant at the beginning ofthe test, the growth of the cracks in the loading processleads to a cross-sectional asymmetry and hence pro-duces flexural stresses.

It thus appears logical to use servo-controlled ma-chines to keep the flexural forces under control alongtwo mutually orthogonal planes. For this purpose, acompletely new testing apparatus made up of threemutually orthogonal actuators was proposed by Carpin-teri and Ferro14 (Fig. 3). Two different series of tests

Magazine of Concrete Research, 1998, 50, NO.3

(c)

were performed on concrete specimens in the scalerange 1:16. Only when the experimental results areobtained over a scale range of at least one order ofmagnitude is it possible to evaluate the relationshipbetween tensile strength and structure dimension.

In other experimental studies reported in the litera-11-131516 .ture "the complete stress-stram curve was ob-

tained from direct traction (see Fig. 2). Nevertheless, themachines were not suitable for testing specimens ofdifferent sizes. Only Bazant and Pfeiffer17 and Nooru-

239

Carpinteri et al.

d= 20 cm

~ d

d

d

dd= 10cm

d=2122Sd= 5 cm

d d d

(a)

Fixed cross-head

(b)

Fig. 3. Testing scheme with a three-jacks solution: (a) scaling of specimens tested; (b) experimental set-up proposed byCarpinteri and Ferro 14

Mohamed and van Mierl8 obtained results in a scalerange of 1:4. The tests performed by Nooru-Mohamedand van Mier, on the other hand, were conducted bycontrolling the load eccentricity in the specimen pIaneonIy, using two actuators, whereas in the tests done byBazant and Pfeiffer no eccentricity control was per-formed.

Linearization of the MFSLThe MFSL proposed by the authors is based on the

hypothesis of geometricai multifractality of the resistingsection of disordered materials at the peak Ioad. In fact,the Iigament can be considered as a lacunar fractal set,with a fractal dimension l'l = 2 - da, where da is thedimensionai decrement due to non-homogeneous da-mage in the material (Fig. 4). The fractal dimension playsa fundamental role at small structural sizes, whereas atlarger scales the heterogeneities of the material becomeless dominant, being limited by the maximum size oftheaggregates. At the largest scales, the resistant cross-sec-tion couId be considered as a Euclidean set with theclassical physicai dimension l'l = 2. More detaiis on theMFSL are reported in Carpinteri et al.19 Moreover, theeffects of microstructurai disorder become progressivelyless important at larger scales, whereas they represent afundamentai effect at smaller scales.

The statistical analysis performed using the MFSLaims, first, at the validation of the multifractal modelby means of the goodness of fit of the experimentaldata and, secondly, to the extrapolation of reliable val-ues to very Iarge structures. The analytical expressionof the MFSL is

( l) 1/2ON = Il 1 + ~h

where the non-dimensionai term in the brackets repre-sents the positive deviation, due to disorder, from a

240

limit nominaI strength Il> valid for infinitely Iargesizes. The parameter leh represents a characteristiclength related to the internai material length, and de-pends on the microstructure. Unfortunately, leh is de-pendent on the test geometry, and is slightly influencedalso by the presence of a notch. The parameters leh andIl represent the two constants to be determined fromthe best-fit of the experimental data. The non-linearbest-fit algorithm proposed by Levenberg" and Mar-quardt'" was used, this representing the optimal proce-dure for the application of the MFSL. Unfortunately,this method is not well suited for engineering purposes,due to the relatively complex algorithms that are in-volved in the computation. An alternative, easier wayof determining the MFSL parameters from experimen-tal data is represented by the Iinearization of the fune-tion (l) by means of changing the coordinates.

lt is possible to rewrite the MFSL in the followingform?

(1)

Fig. 4. Self-similar array of cracks, complementary to thelacunar resisting section at the peak load. PII' maximumtensile load

Magazine of Concrete Research, 1998, 50, NO.3

where A = I; and B = I; Zch. Taking the square of thetwo members of equation (2), we obtain

2 B°N = A =:

Instead of fitting the data in the 0N, d piane, one mayconsider transforming the data to the y* = o~ versusX* = 1/ d piane, where the datapoints tend to be dis-posed along a narrow strip. The MFSL is described bya linear function in the X*, y* piane:

y* = A + BX*

where A and Bare the same parameters considered inthe non-linear expression (equation (2». The best-fitparameters can therefore be determined by a simplelinear regression of the experimental data in theX*, Y* plane. Note that Bazant's size effect law(SEL)22 can also be linearized, 17 although in a differentX, Y piane. The expression of the SEL can be conven-iently written as

AON = -(B-+---=-d)--:-I/=2

Taking the square of equation (5), and consideringits reciprocal, one obtains

l l-=-(B+d)o~ A2

Assuming Y = I / o~ and X = d, the following \inearexpression is obtained for the SEL:

where e = B/A2 and D = 1/A2. Once e and D havebeen determined by linear regression in the X, Y piane,the SEL parameters A and Bare given by: A = 1/ VDand B = e/D.

In the following section comparison is made betweenthe MFSL and SEL linearized fitting of experimentaldata. Even a rough comparison shows that the MFSLparameters are rather insensitive to linearization, whilethe SEL parameters determined by \inear regressiondiffer substantially from the corresponding non-linearones.23

Application of the MFSL to tensileexperimental data

The MFSL was applied to nearly 60 sets of relevantexperimental data reported in the literature.v' The ex-periments concerned both direct and indirect tensiletests. In the present section, the most significant experi-mental data are presented. The data analysed in thiswork relate to the largest scale ranges. In any case,fitting of the experimental data appears to be consistent

Magazine oJ Concrete Research, 1998, 50, No.3

Tensile strength 01 concrete

(2)only if (at least) one order of magnitude is consideredin the scale range. In addition, we present a comparisonof the non-linear version of the MFSL and the linearversion described in the previous section.

(3)Indirect tensile tests

The first geometry investigated is represented by theBrazi\ian test (splitting cylinder tests) carried out byHasegawa et aZ?4 Concrete cylinders, geometricallysimilar in two dimensions, were tested over a widescale range (l :30), the smallest dimension being 100mm and the largest 3000 mm. Ali the specimens were500 mm thick and the maximum aggregate size ofconcrete drnax was 25 mm. The average compressivestrength of the 100 mm diameter and 200 mm highcylinders was 23-4 MPa. Note that the norninal tensilestrength was defined according to the maximum normalstress given by the theory of elasticity:

2PuON = :rcbd (8)

(4)

(5)

where Pu is the failure load, and b and dare the speci-men thickness and diameter, respectively. The values ofthe MFSL constants, obtained by using the non-linearfitting method, are reported in Table l. The asymptoticstrength Il is equal to 80% of the mean strength(l.80 MPa), but to only 56% of the strength obtainedfrom the smallest specimens. This result is very signifi-cant in appreciating the importance of the scale effecton the real structures. The MFSL and its \inearizationin the modified piane are shown in Fig. 5, and theMFSL and the SEL in the double logarithmic strengthversus dimension piane are shown in Fig. 6. It ispossible to evaluate the large difference between thetwo laws. The same diagram shows the values obtainedwith the linear versions of the two laws. The lineariza-tion of the MFSL seems to be closer to its non-\inearversion than is the linear version of the SEL to its non-linear formo The percentage difference between linearand non-linear versions of the MFSL and the SEL,respectively, are reported in Table l. The good approx-imation of the MFSL obtained with linear fitting is avery interesting result, and implies that the method issuitable for implementation in building codes.

The same conc1usion was obtained using the resultsof the sp\it-cylinder tests performed at the Northwes-tern University (Fig. 7)?S The tests were conducted onthe scale of 1:27 on cylinders of diameters d = 19, 38,76, 152, 254 and 508 rom. The thickness of all thespecimens (i.e. the cylinder length) was b = 51 rom.The maximum aggregate size was drnax = 5 rom. Thecompressive strength I~was 51A MPa.

The nominai stress values from measured maximumload are plotted as a log ON versus log d diagram inFig. 8. The non-linear MFSL fitting provides the valuesreported in Table l; Il = 4·80 MPa represents nearlyone-tenth of the compressive strength I~, and 82% ofthe mean strength. In this case, as in the previous one,

(6)

(7)

241

.,c,V) o N '<t N \O r-:'8 '<l: 00 V) <'! 00 e-- ~

~-.i ci N ci <'l

:3' '" o r- \O o-; 00 oU.l \O r- r- o. e-- 00 N

~ \O V) e-- o. e-- o. \Oci ci ci ci ci ci ci

C>:;

~ :3'Vj C/l \O N \O <'l N r- N"<:! •... \O r- e-- O V) '<t

6 e-- 00 o. o. e-- o; 00><: ci ci ci ci ci ci ci<::!ç;3 C>:;

~"-s 00

.~ § o r- \O \O.. ~ <:'l ~ ";I: ";I:"<:! " ..

'" (;j~ o'- C/l

<R'-'"t:>..~ ~'" ...Q "<::! !:: ~~><: ~<::! ;1; '" Jj ",'"ed o<l -.;'" <; > t::

"Q) " <; ~ "O ~''::: o " "'o- '"'".~ e ~ '" .€ c... ., "O., '" 1A "Cl "O §~ ~ 1:i

.,~ '" :'8 '" "'" ~ ~)~

., es !il '':..c: 1:i "'" ~ IIIQ) ..c: 1:i';S .~ ., o °B:r: >N

~ra ee.s III ee

U"<:! o

'" o.::; Z~

-<:>Cl

J:l;::s~ ..c:

~ 2<.>2O:::; 1;; '" .- .;;;" V>2 0.'" '" '"-; '- 00 00_ ~ :: B 2o :€ .5 " " "~ " .~ 9 J::> o

-<:> $ '3 '" " Q) "P., P., o . == .== ..~ C/l C/l c... Cl Cl Cl Ci

242

Carpinteri et al.

3·00

o

Id2·55Scale range 1:30

ctlo..~'",'"

o1·65 o

8 o

1·20O 500 1000 1500 2000 2500 3000 3500d:mm

(a)

Fig. 5. The MFSL for the splitting test results reported byHasegawa et al.: 24 (a) non-linear MFSL; (b) linearization ofthe MFSL in the modified plane

r-oO\<o::to\O\'<:Too<o::tO\r-\O("") •......•v),....;o\~o~.....:

......•o V') \O N --IM N-<o::t\OooMC'ir---:""';O.....:tri~~'<t

Il<:j

\Ooo<o::toor-O\O\O\N('f')~Nr-: -q: 0\ ,....; N N .

~ - N

88

NNlIì\Ot-NOOMo\~ociNo\~OO\N("t') N<o::tN

'<t -N 5t5

Il>- 4

0·8

"'zCl 0·6.Q

8,-----------------------------------,o

7 -- y= 1·841 + 462·12 x

6

3

2

0·002 0·004 0·006 0·008X= 1/d

0·010 0·012

(b)

1-4 r------------------p

1·2- Non-linear---Linear1·0

0·0 Scale range 1:30

-0.~L1~--~~0--~--~--~~2~--~~3------4~----~5

log d

Fig. 6. Application of the MFSL and SEL, as linear and non-linear regressions, to the splitting test results reported by

24Hasegawa et al.

the MFSL describes the phenomenon of the scale effectin a better way than does the SEL. Noting the weakadherence of the SEL to the Brazilian splitting testresults, Bazant et al.25 proposed the introduction of an

Magazine of Concrete Research, 1998, 50, NO.3

Tensile strength 01 concrete

60

Id 50

o'" 40

8 ~Ilo ).. 30

7·5

7·0

6·5

6·0'"o,~ 5·5~

5·0o

4·5 o

4·0o

3·5o 100

o

oo

Scalerange:1:27

300d:mm

500 600200 400

(a)

08o - y = 23·513+ 543·29x

O R= 0·810

O

O

}tp

0·05 0·06

20

10~--~--~~---~---~---~---~0·00 0·01 0·02 0·03 0·04

X= l/d

(b)

Fig. 7. The MFSLJor the splitting test results reported by Baiant et al.:25 (a) non-linear MFSL; (b) linearization oJthe MFSLin the modified piane

1-4,.---....,.-------

1·2

Scalerange1:27

__ Non·linear......... Linear

1·0

SEL O0·8 O

~ R = 0·570······················ MFSLCl pi Qg R = 0·872.Q

0·6 O

0-4 E0·2

Pi0·0

-1 O 2 3 4logd

Fig. 8. Application of the MFSL and SEL, as linear and non-linear regressions, to the splitting test results reported byBaiant et al. 25

asymptote for d .....•00. They affirmed that the damageextension at failure is constant for specimens largerthan dlim, and then is not proportional to the specimendiameter. The linearized MFSL gives an excellent ap-proximation of the non-linear version (Fig. 8).

The linearization of MFSL seems to approximate themechanical behaviour in, a better way than the corre-sponding linear version of the SEL for the pull-out testperformed by Eligehausen et al?6 (Fig. 9). Specimensand headed anchor bolts of three different sizes but thesame geometrical shape were used. To obtain the corre-sponding dimensions the scaling ratio 1:3:9 was ap-plied, the actual embedment lengths being 50, 150 and450 mm. AH the specimens were made of concrete ofnominally identical quality with a specified cubestrength I~ of 30 MPa, while the tensile strength li,measured using the splitting test, was 3·0 MPa. Themean value of the fracture energy was measured as

Magazine of Concrete Research, 1998, 50, NO.3

1·5,.--------------------,

1·0

0·5~ SEL~ R = 0·977 .

0·0

-- Non-Iinear- - - - Linear

MFSLR= 0996

-0,5

_1·0'-- .L. ~ '_____ ..L.. __ ....J.... __ .._...J

-15 o 2logd

53 4

Fig. 9. Application of the MFSL and SEL, as linear and non-linear regressions, to the pull-out test results reported byEligehausen et al. 26

155 N/m. The water/cement ratio was 0·58 and themaximum aggregate size dmax was 22 mm. In a11thetests failure was caused by pulling out of a concretecone, the shape of which was approximately similar foralI embedment depths. Due to this kind of failure, thenominaI collapse stress used in the statistical analysiswas calculated as

(9)

where Pu is the ultimate load, normalized with re-spect to the cube compressive strength, and Acone =

n(3d)2/4 is the nominaI surface area of the failurecone, d being the embedrnent depth. The results of thestatistical analysis are reported in Table l.

The double-punch tests performed by Marti9 show abetter adherence of the results to the SEL. Specimenrepresentative size d (which is the diameter of the bases

243

Carpinteri et al.

of the cylinders) ranges from 76 to 1219 mm (scalerange l: 16). The maximum aggregate size drnax wasIO mm. The average compressive strength determinedfrom cylinder tests was 23·6 MPa. The nominaistrength used in the statistical analysis was given by:

PuaN= OA d2 (lO)

where P; is the failure load and d is the specimendiameter. The nominai stress values at the measuredmaximum load are plotted in Fig. lO as a log aNversuslog d diagram.

Direct tensile testsOnly three investigations using the direct tensile test

have been reported in the literature. In the study reportedby Bazant and Pfeiffer,17 the cross-section of the speci-mens was rectangular, and the length/depth ratio was 8:3in all cases. The cross-sectional heights d of the speci-mens were 38·1, 76·2 and 152-4 mm, the scale rangebeing 1:4. The thickness of the tension specimens b was19 mm. Notches of depth d/6 and thickness 2·5 mm(same thickness for all specimen sizes) were cut into thehardened samples with a diamond saw. The maximumgrave I size drnax was 12·7 mm, with a maximum grainsize of 4·83 mm. The mean compressive strength I~was33·5 MPa, with a standard deviation of 3·79 MPa. Thetensile strength was estimated as Il= 6..f1'c andYoung's modulus as E, = 57000 v7'c (f~in psi).

A special Ioading grip was produced for the tensilespecimens. It consisted of two tests of aluminium platescompressed together by bolts. The nominai tensilestrength can be defined as

(Il)

where Pu is the maximum load, b is the specimenthickness and d is the width of the ligament. As can beseen from Fig. Il, the two linear versions give very

1·2

1·0

o-a0·6

ztl 0-4O>

.Q

0·2

-- Non-linear- - - - - Linear

L~I d ISEL

R~ 0·996._.._...•.MFSL

R~ 0·9030·0

-0·2 - Scale range1:16-0.4 '-----~~'--~~.L........~~'---~~.L........~_.L........~____..J

-1 o 3 42log d

Fig. lO. Application of the MFSL and SEL. as linear andnon-linear regressions, to the results of the double-punch testsreported by Marti9

244

1·2,------.------------------,

1·0Non-IinearLinear

Scale range 1:4

o-eSEL

0·6 R ~ 0·999~---~

cf 0-4O>.Q

MFSL~:------R ~0·952

0·2

0·0

~-0,2

-0·4-1 O 2

log d3 4 5

Fig. Il. Application o[ the MFSL and SEL. as linear andnon-linear regressions. to the results of the direct tensile testsreported by Baiant and Pfeiffer+'

good approximations in both cases, due to the smallscale range. It can be noted how the limited scale rangeof these tests does not permit one to appreciate thecomplete trend of the change in the strength on varyingthe dimension. In Fig. Il the two laws have the sametangent in the specimen range, and a linear law couldbe used to locally interpolate the data.

The same result can be obtained from the directtensile test described by Nooru-Mohamed and vanMier.18 In this study double-edge-notched concreteplate specimens of three different sizes (200 X 200 X

50, 100 X 100 X 50 and 50 X 50 X 50 mm) were testedin traction under displacement control. The a/ d ratioof 0·125 was adopted for all specimen sizes (where a isthe notch depth and d is the specimen depth). The scalerange was 1:4. The maximum aggregate size drnax was2 mm. The concrete was composed of Portland cementtype B (500 kg/m '), with a water/cement ratio of 0·5.The average 28-day compressive strength I~ and theaverage splitting tensile strength I; were 49·66 and3·76 MPa, respectively. The tests were controlled byusing an average of four vertical LVDTs mounted nearthe notches.

The nominaI tensile strength can be defined as

PuaN=----

b(d - 2a)(12)

5

where Pu is the maximum Ioad, b is the specimenthickness and (d - 2a) is the ligament width. TheMFSL is plotted in Fig. 12, and the correspondingparameters are reported in Table l.

The last tests are those reported by Carpinteri andF 27 ?8 h' h f d . .erro, ,- w ic were per orme usmg a new testingapparatus made up of three orthogonally disposed ac-tuators. The specimens were flared in the centre andtheir thickness (lO cm) was maintained constant for theentire set, while the transverse dimension was 0'25,0'5, l, 2 and 4 times that of the thickness (see Fig.

Magazine of Concrete Research, 1998, 50, NO.3

1·2

1·0

0·8

0·6 SEL

;s- R = 0·988O> 0·4 q.Q

0·2

0·0

-0·2 _d ___

-0-4-1 O

Scale range 1:4

Non-linear____ - - Linear

~~ MFSL

R = 0·917

3 42log d

Fig. 12. Application of the MFSL and SEL, as linear andnon-linear regressions, to the results of the direct tensile testsreported by Nooru-Mohamed and van Mier 18

3(a)). The scale range was l: 16, which is the largestrange for the direct tensile tests found in the literature.The test specimen was glued to steel supporting platesso that it could be attached to the load-bearing system.The dimensions of the minimum cross-section of thespecimens were 2·5 X lO, 5 X lO, lO X lO, 20 X lOand 40 X lO cm. Eight specimens of each size werecast. The water/cement ratio of the concrete mix was0·5. The maximum gravel size drnax was 16 mm, andthe mean compressive strength I~ was 36·9 MPa. Thenominal tensile strength can be defined as in equation(11).

The parameters Iland leh of the MFSL in this caseare equal to 3·67 MPa and 20·32 mm, respectively. Thevalue of 1(, which represents the tensile strength ford ......,00, is nearly one-tenth of the compressive strengthI~,while the quantity lehl drnax is equal to 1·27.

The fitting was performed, for both the MFSL andthe SEL, using the average values. In this case, theMFSL provides a correlation coefficient R equal to0·842, whereas fitting data using the SEL givesR = 0·620 (Fig. 13). The linearization of the MFSLgives an approximation of 1·14% with respect to thenon-linear prediction, while the linear version of theSEL presents an approximation of 5·81%. This result,obtained using data from unnotched specimens and overa scale range larger than one order of magnitude, is verysignificant in confirming the validity ofthe MFSL.

ConclusionsThe relevant literature on the nominal tensile

strength indicates that this mechanical parameter is notconstant across all the (direct and indirect) tensile testsreported. The nominal tensile strength decreases withan increase in the structure size. The MFSL, proposedby the authors, satisfactorily approximates the trend ofthe strength over an unbounded size range, whereas the

Magazine of Concrete Research, 1998, 50, No. 3

Tensile strength 01 concrete

1·5 ~-------------------,Scale range 1:16-- Non-linear- - - - - Linear

1·0

~~==~~~~~ ~MFSLR= 0·842

;s-O> 0·5.Q

5 2 3log d

4 5 6O

Fig. 13. Application of the MFSL and SEL, as linear andnon-linear regressions, to the results of the direct tensile testsreported by Carpinteri and Ferro 27.28

SEL assumes a zero strength for structure sizes tendingto infinity. The MFSL aims at extrapolation to obtainreliable values of the nominal tensile strength for verylarge structures. From the obtained results, we canaffirm that it is only possible to obtain correct predic-tions when tests are done over a scale range greaterthan one order of magnitude. When the scale range issmall, only the tangenti al slope of the non-linear trendcan be captured. In the latter case, a linear (monofrac-tal) scaling law may be used, and only limited predic-tions can be made.

Finally, we wish to underline how the MFSL linearregression provides an excellent approximation of thenominal tensile strength. It is of practical use and wecan suggest its implementation in building codes.

AcknowledgementsThe present research was carried out with the finan-

cial support of the Ministry of University and ScientificResearch (MURST) and the National Research Council(CNR). Financial support from the European Commu-nity, THR contract No. ERBFMRXCT 960062, is alsogratefully acknowledged.

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Discussion contributions Oll this paper should reach the editor by27 March 1999

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