Size and Shape Effects on the Compressive Strength of High StrengthConcreteJ.R. del Viso, J.R. Carmona & G. Ruiz,E.T.S. de Ingenieros de Caminos, Canales y Puertos, Universidad de Castilla-La ManchaABSTRACT: In this paper we investigate the inuence of the shape and of the size of the specimens on thecompressive strength of high strength concrete. We use cylinders and cubes of different sizes for performingstable stress-strain tests. The tests were performed at a single axial strain rate, 106s1. This value was keptconstant throughout the experimental program. Our results show that the post-peak behavior of the cubes ismilder than that of the cylinders, which results in a strong energy consumption after the peak. This is consistentwith the observation of the crack pattern: the extent of micro-cracking throughout the specimen is denser in thecubes than in the cylinders. Indeed, a main inclined fracture surface is nucleated in cylinders, whereas in cubeswe nd that lateral sides get spalled and that there is a dense columnar cracking in the bulk of the specimen.Finally, we investigate the relationship between the compressive strength given by both types of specimen forseveral specimen sizes.1 INTRODUCTIONByfar themost commontest carriedout oncon-crete is the compressive strength test. The main rea-son to understand this fact is that this kind of test iseasy and relatively inexpensive to carry out (Mindesset al., 2003). Testing Standard requirements use dif-ferent geometries of specimens to determine the com-pressive concrete strength, fc. The most used geome-triesarecylinderswithaslendernessequal totwoand cubes. Shape effect on compression strength hasbeenwidelystudiedanddifferent relationshipsbe-tweencompressionstrengthobtainedfor thesege-ometries have been proposed, mainly from a techno-logical standpoint. Such approach eludes the fact thatthere is a direct relation between the nucleation andpropagationoffractureprocessesandthefailureofthe specimen. Indeed, experimental observations con-rm that a localized micro-cracked area develops atpeak stress (Shah and Sankar, 1987) or just prior to thepeak stress (Torreti et al., 1993). For this reason com-pressive failure is suitable to be analyzed by means ofFracture Mechanics (Van Mier, 1984).Somerecent experimental worksbasedonFrac-ture Mechanics to study the compressive behavior ofconcrete are especially noteworthy. Jansen and Shah(Jansen and Shah, 1997) planned an experimental pro-gram aimed at analyzing the effect of the specimenslenderness on the compressive strength. Also notableis the work by Choi et al. (Choi et al., 1996), on thestrainsofteningincompressionunderdifferent endconstraints. Borges et al. (Borges et al., 2004) studiedthe ductility of concrete in uniaxial and exural com-pression. The results of these experimental programssuggest that the compressive test may be consideredas an structural test, because the results depend notonly on the actual mechanical properties but on thegeometry of the specimen and on the boundary con-ditions, like end constraints, feedback signal or spec-imen capping (Shah et al., 1995).In the last decades concrete technology has madeiteasiertoreachhigherstrengthsandtheso-calledHigh-StrengthConcrete (HSC) has appearedas anew construction material. This amount of the com-pressive strength provokes that the normalized speci-mens for normal strength concrete, e.g. 150 300mmcylinder (diameter height), may surpass the capac-ityofstandardlaboratoryequipment. Toovercomethis drawback HSC mixtures are often evaluated us-ing 100 200mm cylinders, which also meet the re-quirements of ASTMC39. The strength obtained withthis specimen is higher in average than that obtainedwith 150 300mm cylinders. This size effect on thecompressive strength manifests that the specimen isnot behaving as simply as it may look. Indeed, sizeeffect is understood as the dependence of the nominalstrength of a structure on its size (dimension) whencompared to another geometrically similar structure(Ba zant and Planas, 1998). For quasi-brittle materialsTrue crack FPZ Intact materialftFigure 1: Fracture process zone (FPZ).as concrete, the presence of a non-negligible fractureprocess zone at the front of the crack, which is com-mensurate with an internal characteristic length andwith the typical size of the material inhomogeneities,provokes a deterministic size effect (please, see Fig-ure 1).In this paper we investigate the mechanical behav-ior of HSC (around 100 MPa) tested in a compressivesetup in axial strain control. We are particularly inter-ested in the inuence of the shape and of the size ofthe specimens on the compressive strength, fc, of thematerial. According to our results we propose a newrelationship between cylinder and cube specimens.The article is organized as follows. An outline ofthe experimental program is given in Section 2. Con-crete production and specimens are described in Sec-tion3. InSection4wedescribethetestingproce-dures. The experimental results are presented and dis-cussed in Section 5. Section6 includes a simple anal-ysis of size effect and establishes a relationship be-tween the compressive strength measured from stan-dard cylinders and the one obtained from cubes of dif-ferentsizes.Finally,inSection7someconclusionsare extracted.2 OVERVIEWOFTHEEXPERIMENTALPRO-GRAMThe experimental programwas designedtostudythesizeandshapeeffect oncompressivetestsper-formed on High-Strength Concrete (HSC) specimens.Specically,wewantedtodisclosetheinuenceofthe size in cubes and the relationship between com-pressivestrengthobtainedfromstandardcylinders,ASTMC39 (100 200mm) and the strength got fromcubes. We also intended to analyze the variations inthe crack pattern and in the mechanical behavior dueto the size and shape of the specimens.With these intentions in mind, we chose the spec-imenssketchedinFigure2. Weusecylindersandcubes of different sizes: the dimensions of the cylin-ders are 75150 and 100200mm(diame-ter height); the edges of the cubes are 33, 50, 67 100

75100

67

50 33

D2

D3 C1

C2C3 C4Cylinders specimenCubes specimenDimensions in mm.200150Figure 2: Specimen geometry.and 100 mm long. The dimensions of the cylinderswere scaled to the diameter D, while the dimensionsof the cylinders were scaled to the edge L, please seeFigure 2. Each specimen was named by a letter indi-cating the shape, D for cylinders and C for cubes, andone number which indicates the size. For example, D2names a cylinder of 100 200mm. We performed atleast four tests for each type of specimen.An important feature of our experimental programunlike most of the tests available inthe scien-tic literature consists of providing complete mate-rial characterization obtained by independent tests, inmany cases performed according to Standard recom-mendations. Specically, we measured the compres-sive strength, the tensile strength, the elastic modulusand the fracture energy of the concrete.3 MATERIAL AND SPECIMENSA single high-strength concrete mix was usedthroughout theexperimental program. It wasmadewith a andesite aggregate of 12mmmaximumsize andASTM type I cement. Microsilica fume slurry and su-perplasticizer (B-255, BASF) were used in the con-cretecomposition. Thewatertocement ratio(w/c)was xed to 0.28.There was a strict control of the specimen-makingprocess, tominimizescatter intest results. Inthecase of cylinders the concrete was poured into steelmolds in three layers and compacted on a vibrationtable. Inthecaseofthecubesthespecimensweretaken by wet-sawing from 100 100 420mm con-crete prisms. The prisms were also made using steelmoldsandavibrationtable. Cylindersandprismswere wrap-cured for 24 hours, demolded and storedinamoist chamber at 20oCand96%relativehu-midity. Right after making the cubes from the prismstheir sides, as well as the bases of the cylinders,were surface-ground with a wet diamond wheel to en-Table 1: High strength concrete mechanical properties.fc(a)fts(b)EcGF

ch(c)MPa MPa GPa N/m mmmean 89.6 5.4 36.1 119.0 147.3std. dev. 7.13 0.6 1.1 13.5 -(a)Cylinder, compression tests, D2.(b)Cylinder, splitting tests.(c)

ch=EcGFf2tsure perfectly-plane and parallel surfaces. Of course,the specimens were immediately stored again in thechamber until testing time (six months from concretemaking). Just before testing the length, diameter andweight of the cylinders and the height and weight ofcubes were measured.Table 1 shows the characteristic mechanical param-eters of the micro-concrete determined in the variouscharacterization tests. In the table it is also shown thecharacteristic length of the concrete,ch, (Petersson,1981), which is dened as ch = EcGF/f2t , being Ectheelasticmodulus, GFthefractureenergyandftthe tensile strength. Please, notice that the character-istic length, ch, of this concrete is roughly 150 mm,closely half the ch of normal concrete (ch=300mm).Thismeansthat, fromaFractureMechanicsstand-point, weexpect that high-strengthconcrete(HSC)may present a more brittle behavior than NSC for thesame specimen size. As mentioned above, the char-acteristic length is related with the extension of thefracture processes zone (FPZ), see Figure 1.4 EXPERIMENTAL PROCEDURES4.1 Characterization testsCompressive tests were carried out on 4 cylin-drical specimens according to ASTMC-39 on100 200mm (diameter height), except for a re-duction of the axial displacement rate of the machineactuator, which was 0.018mm/min.The Elastic modulus was obtained accordingto ASTM C-469 on cylindrical specimens of150 300mm. The strain was measured over a 50mmgagelengthbymeansoftwoinductiveextensome-ter placed symmetrically, on a similar setup that it isshown in Figure 3. The tests were run under displace-ment control, at a rate of 0.3mm/min.Brazilian tests were also carried out on cylindricalspecimens following the procedures recommended byASTMC496on150 300mmcylinders (diame-ter height). The specimen was loaded through ply-wood strips with a width of 1/6 of the specimen di-ameter. The velocity of displacement of the machineactuator was 0.18mm/min.Stable three-point bend tests onFigure 3: Testing setup for D3 specimen.100100420mmnotched beams were car-ried out to obtain the fracture properties of concretefollowing the procedures devised by Elices, Guineaand Planas (Guinea et al., 1992; Elices et al., 1992;Planas et al., 1992). During the tests the beams restedon two rigid-steel semi-cylinders laid on two supportspermitting rotation out of the plane of the beamandrollingalongthebeamslongitudinalaxiswithnegligible friction. These supports roll on the upperface of a very stiff steel beam fastened to the machineactuator.Table 1 shows the characteristic mechanical param-eters of the micro-concrete determined in the variouscharacterization tests.4.2 Testing procedure to obtain stable - curvesAll the specimens were tested on a compressive setupas illustrated in Figure 3. The experiments were per-formedonanINSTRON8800closed-loopservo-hydraulicdynamictestingmachine,withacapacityof1000kN. Theloadingplatensaremadeofhard-ened steel and have polished surfaces. Two linear vari-able differential transformers (LVDTs), having a cali-brated range of 2.5mm, were placed opposed to thespecimen in a symmetric fashion. These LVDTs wereused to measure axial platen-to-platen displacement,. The axial deformation referred to in this paper isthe average value of the two LVDTs.For controlling the tests, the signal of the averageaxial displacement from the LVDTs was chosen. Thisaverage signal represents the axial displacement ve-locityatthecenteroftheloadingplaten.Thistypeof control leads to stable tests for our particular ma-terial and specimens geometry. The axial strain rateis dened as =/L, where is the axial platen-to-platen displacement rate and L is the initial distancebetween the top and bottom plane surfaces over theheight of the specimen. This axial strain rate was keptconstant for all tests having a value of106s1. Theaxial displacement rates selected for each specimenTable2: Axialdisplacementrateforeachofthespeci-mens.Denomination Axial displacement rate,mm min1D2 1.8102D3 1.2102C1 0.9102C2 0.6102C3 0.4102C4 1.98103are showed in Table 2.An elastic band was stuck around the specimen, seeFigure 3, (1) to preserve the fragments together for aposterior analysis, and (2) to avoid a interchange ofhumiditywiththeenvironmentduringthetest.Thetransversal force provoked by this elastic band is neg-ligible.Theload, P, andtheaxialdisplacement, ,werecontinually monitored and recorded. We used a pairof clip extensometer centered on the specimen in D2and D3 to measure the displacement in the central partof the specimen in order to get the elastic modulus. Tocomplete the experimental information, we also tookpictures of the crack pattern resulting from each test.5 RESULTS AND DISCUSSIONThe discussion of the results is organized as follows.Firstwepresentanddescribethe curvesob-tained from the tests. Then we proceed to discuss sizeeffect in cubes and cylinders. Finally the crack pat-terns are analyzed. Let us emphasize again that cylin-ders and cubes were tested in strain control keepingTable 3: Strength and strain at peak and ultimate load.Specimen c

cuuMPa % MPa %D2 mean 89.6 0.37 31.4 0.71desv. std. 7.11 0.02 23.31 0.51D3 mean 89.9 0.34 41.6 0.22desv. std. 4.65 0.02 22.08 0.42C1 mean 96.1 0.57 12.7 1.53desv. std. 1.63 0.01 1.55 0.12C2 mean 102.4 0.61 20.9 1.59desv. std. 9.93 0.03 1.81 0.19C3 mean 104.2 0.66 38.5 1.46desv. std. 2.19 0.06 6.63 0.13C4 mean 110.0 0.85 33.8 1.94desv. std. 7.34 0.09 5.09 0.35 LDC1C2C3C4a)b)D2D3

0204060801001200 0,5 1 1,5 (MPa) (%)0204060801001200 0,5 1 1,5 (MPa) (%)Figure 4: Curves corresponding to: (a) Cubes; (b)Cylinders.constanttheaxialstrainratethroughouttheexperi-mental program.5.1 curvesTable 3 lists the average value and the standard devia-tion for the peak stresses, c, the strain at peak stress,c, the ultimate stress recorded,u, and the ultimatestrain recorded, u for all specimens. As expected, cincreases noticeably as the size of the specimen de-creases. The strain at peak load increases also as sizedecreases.Figure 4a and 4b compare curves for selectedsizes. Specically, Figure 4a shows the obtainedfor cubes and Figure 4b for cylinders. The x-axis cor-responds to the average strain, i.e. the displacementbetween the top and bottom plane surfaces over theheight of the specimen. The y-axis corresponds to theconcretestress(c=PArea). Atypical curvestarts with a linear ramp-up. The initial slope showsno signicant variations due to the size in the case ofcubes while in cylinders the shorter specimen showsa higher slope. In both cases there is a loss of linear-ity before reaching the peak load, which indicates theinitiation of fracture processes.The post peak behavior depends on the specimenshape. In the case of cylinders, Figure 4b, a suddendrop in the load takes place after the peak load andnosignicantchangesarereportedinthesofteningC1C2C3C47510012540 60 80 100a) (MPa)L (mm)D2 D3b) L75100125125 150 175 200 225 (MPa)D (mm) D

Figure 5: Size effect on the peak strength: (a) Cubes; (b)Cylinders.branch between the scaled tested cylinders. In the caseof cubes, Figure 4a, curves show a not so steepsoftening branch which is associated to a milder fail-ure localization and to a large amount of volumetricenergy dissipation, in contrast with the strong failurelocalization observed in cylinders.5.2 Size effectFigures5aandbplot thepeakloadsormaximumstrengthobtainedinthe curves for thedif-ferent testedspecimens, against acharacteristicdi-mension of the specimen. In Figure 4a is drawn thecase of cubes specimens. It is clearly observed thatlarge specimens resist less in terms of stress than thesmaller ones. A interpolation line has been plotted at-tached to the tests results to facilitate the comparison.Size effect is less noticeable as size increases.Bycontrast, theaveragestress, , obtainedfromcylinders isroughlyconstantwithinthe experimen-tal size interval planned for this research and approx-imately 10% lower than the horizontal asymptote forthe cubes as it can be observed in Figure 4b.5.3 Crack patternThe crack pattern observed after the test is sensitiveto the shape of the specimen as Figure 6 shows. Theextent of micro-cracking throughout the specimen isdenserinthecubesthaninthecylinders.Indeed,aa)b)C1C2C4C3D1D2D3Figure 6: Crack pattern for: (a) Cubes; (b) Cylinders.main inclined fracture surface is nucleated in cylin-ders, whereas in cubes we nd that the lateral sidesget spalled and that there is a dense columnar crack-inginthebulkofthespecimen. ThisisconsistentwiththeobservationmadeinSection5.1aboutthedifferencesinthe curvesbetweencubesandcylinders. The steeper response observed in the curves for cylinders may be associated to the inclinedfracture surface or cone type failure, see Figure 6b,with a initial localization failure in the central part ofthe specimen. In the case of cubes the fracture pro-cessisprovokedbyastressconcentrationnearthecubecorners. Inclinedmicro-cracksappearandco-alescenearthecornersandprovokethecrackpat-tern observed. The stress state inside the cubes pro-vokes vertical cracks that en-up forming column-likefragments. Theextentofthemicro-crackinginthisfracture process leads to a higher energy consumptionthan in the case of cylinders.Another interesting point is that the failure patternobservedisalmostindependentforscaledelementsin the size interval planned for the experimental pro-gram. Cylinders broke in all cases by a diagonal frac-ture plane and cubes present a bursting rupture com-bined with a dense columnar cracking. We have alsoincluded the crack pattern obtained for a cylinder of150 300mm, called D1, in Figure 6b.6 RELATIONSHIP BETWEEN CYLINDER ANDCUBESTRENGTHBASEDONFRACTUREMECHANICS THEORIESIn this section we analyze the test results showed insection5withinaFractureMechanicsframe. Theaim of this analysis is to get a relationship betweencylinder and cube strength. According to Bazant andPlanas (Ba zant and Planas, 1998) the maximum loadPc is a function of the specimen geometry (shape andsize), boundaryconditionsandconcreteproperties.In this investigation only the size and shape are var-ied, keeping constant the boundary conditions. For thesake of simplicity, the nominal strength at peak load,N, for a concrete specimen is represented by the gen-eralized deterministic energy-based formula for sizeeffect at crack initiation proposed by Ba zant (Ba zant,1998). Its applicability is based on the fact the failureis provoked by crack initiation, which is our case asshown in the previous section. So, we can writeNas:N=

1 +BH1r(1) = 0whereis the theoretical strength for a specimenof a innite size and 0 is a reference strength, whichinourcasewetakeasthestrengththatisobtainedfrom the standard test (ASTM C-39 on 100 200mmcylinders). Bandareempirical constantsdepen-dentonthespecimenshapeandfracturepropertiesofthematerial, butnotonthestructuralsize.Bothconstants can be obtained from test results. H is thedenominated Hillerborgs brittleness number (Ba zantand Planas, 1998). It is dened as the ratio betweenthe representative size of the specimen representedby the depth D in the case of cylinders, and L in thecase of cubes and the characteristic length of theconcrete. The values of the exponent r using in con-crete ranges from 1 to 2 (Ba zant, 1998). The formula-tion when the exponent r is equal to 2, is identical tothe formula proposed on the basis of strictly geomet-ric arguments proposed by Carpinteri and coworkers(Carpinteri et al., 1998). The size effect law can berewritten as:N=

1 +BH(2)Applicability of Eqs. 1 and 2 requires not only thatthe specimens are scaled to each other, but also thatthe shape of cracks patterns must be similar. In ourcase the crack patterns for cubes are very similar, aswe showed in Section 5.3.Figure 7a shows the linear regression made with thecubes to get the constants andB. Please note thatN/Cylinder D2HC2C1C4C3C1 C2C4C32B28500075001000012500150000 1 2 3 4 5= 8273 + 887 1/HN2a) L 882 82H1.01.21.40.1 1b) L DHN8= 1+

0.107Figure 7: Size effect in the peak load: (a) Results for theregressiontocalibrateandBcoefcientsforcubesspecimens; (b) Size effect law.the Pearsons correlation coefcient, R in Figure 7ais close to 1. Figure 7b show the test results comparedto the obtained size effect law in a non-dimensionalfashion. It may be pointed out that size effect tends todisappear whenD . The compressive strengthconverges to a value that in our case is quite similarto 1, that is, for big sizes the compressive strength incubes converges to the compressive strength obtainedwith the standard procedure. Further analysis wouldbe necessary to evaluate the inuence of the displace-mentcontrolplaten-to-platenvelocity. Thestrengthforaspecimenofinnitesizecanbeevaluatedas = 0. In our case is equal to 88.7 MPa.Basedontheobtainedlawacorrelationbetweencompressive strength obtained for different cubesspecimens, cub, and for Standard tests, fc, can be de-rived. In a general way for concrete which compres-sion standard is around 100MPa and have a character-istic length of 150 mm, the next simplied relation-ship arises. For HSC like the one used in this researchfc = 0 100MPa, ch 150mm and cub = N, theexpression is:fc = cub

LL+L0(3)where L is the side of the cube andL0is a empiri-cal constant equal to 20mm in this case. The fact thatcubes are somewhat easier to test, since cubes providedirectlytwocouplesofperfectly-planeparallelsur-faces and they not require capping, together with thenecessity of using specimens whose strength do notsurpass the capacity of standard equipment, suggestthat cubes would be a solution for compressive testcharacterization. Thetendencytoprefercylindertocubes, in our opinion does not have a solid foundationdue to the structural character of the compressive test.There is not a reason to think that cylinders catch con-cretecompressivebehaviorbetterthancubes, sincecompressivetestsdependstronglyonmanyfactorsother than the actual material properties. The standardcompressive strength of HSC could be obtained fromcubes using the simplied expression showed in Eq.3.With this example we want to show how that us-ingFracture Mechanics theories provide solutionsfor concrete technological problems. These solutionsmay help to understand concrete behavior and struc-tural failures better.7 CONCLUSIONSIn this paper we investigate the mechanical behaviorof high-strength concrete (around 100 MPa) in com-pression and tested in strain-control. We are particu-larly interested in the inuence of the shape and thesize of the specimens on the compressive strength, fc,ofthematerial. Cylindersandcubesweretestedinstrain control at a rate that was kept constant through-out the experimental program. Concrete making andtesting procedures were closely controlled to reduceexperimental scatter. The following conclusions canbe drawn from the study:1. The prepeak and post-peak behavior in the curves are dependent on the specimen size andshape. Cubes show a mild failure localization incontrast with the strong failure localization ob-served in cylinders.2. Test results showsize effect. Large specimens re-sist less in terms of stress than the small ones.Thesizeeffect inthecubes is quitestrongerthan in cylinders, where the average strength ob-tained is roughly constant within the size intervalplanned for this research.3. The crack pattern observed after the test is sen-sitive to the shape of the specimen. A main in-clined fracture surface is nucleated in cylinders,whereas in cubes we nd that the lateral sides getspalled and that there is a dense columnar crack-ing in the bulk of the specimen. This is consistentwith the differences observed in the curvesbetweencubesandcylinders. Interestingly, thefailure pattern does not change with the size.4. Thesizeeffect inthecompressivestrengthofcubes is described by a simple model based onFracture Mechanics concepts. A relationship be-tweenstandardcylinder strengthandstrengthobtained from cubes of any size is derived. It dis-closes the inuence of the cube dimension andmay help to deduce relations between the com-pressive strength determined from cylinders andcubes, insideatheoretical FractureMechanicsframe.ACKNOWLEDGEMENTSThe authors gratefullyacknowledge nancial sup-port for this research provided by the MinisteriodeFomento, Spain, undergrantBOE305/2003, andthe Junta de Comunidades de Castilla -La Mancha,Spain, under grant PAI05-028. Javier R. del Visoand Jacinto R. Carmona also thank the Junta de Co-munidadesdeCastilla-LaMancha, Spain, andtheFondo Social Europeo for the fellowships that supporttheir research activity.REFERENCESBa zant, Z. P. (1998). Sizeeffect intensileandcom-pression fracture of concrete structures: Computa-tional modeling and desing. In Fracture Mechanicsof Concrete Structures, pages 19051922, Freiburg,Germany. F.H. Wittmann, Ed., Aedicatio Publish-ers.Ba zant, Z. P. and Planas, J. (1998). Fracture Size Effectin Concrete and Other Quasibrittle Materials. CRCPress, Boca Raton.Borges, J. U. A., Subramaniam, K. V., Weiss, W. J.,Shah, S. P., and Bittencourt, T. (2004). Length ef-fect on ductility of concrete in uniaxial and exuralcompression. ACI Structural Journal, 101(6):765772.Carpinteri, A., Chiaia, B., andFerro, G. (1998). Sizeeffect on nominal tensile strength of concrete struc-tures: Multifractality of material ligaments and di-mensional transition from order to disorder. Mate-rials and Structures, 28:311317.Choi, S., Thienel, K. C., and Shah, S. P. (1996). Strainsoftening of concrete in compression under differ-ent end constraints. Magazine of Concrete Reseach,48(175):103115.Elices, M., Guinea, G. V., and Planas, J. (1992). Mea-surement ofthefractureenergyusingthree-pointbend tests. 3. Inuence of the cutting the P tail.Materials and Structures, 25:327334.Guinea, G. V., Planas, J., and Elices, M. (1992). Mea-surement ofthefractureenergyusingthree-pointbend tests. 1. Inuence of experimental procedures.Materials and Structures, 25:121128.Jansen, D. C. and Shah, S. P. (1997). Effect on length oncompressive strain softening of concrete. Journal ofEngineering Mechanics-ASCE, 123(1):2535.Mindess, S., Young, J. F., and Darwin, D. (2003). Con-crete. Prentice Hall, Pearson Education, Inc. UnitedStates of America.Petersson, P. E. (1981). CrackGrowthandDevelop-ment of Fracture Zones in Plain Concrete and Simi-lar Materials. Report No. TVBM-1006, Division ofBuilding Materials, Lund Institute of Technology,Lund, Sweden.Planas, J., Elices, M., and Guinea, G. V. (1992). Mea-surement ofthefractureenergyusingthree-pointbend tests. 2. Inuence of bulk energy dissipation.Materials and Structures, 25:305312.Shah, S. P. and Sankar, R. (1987). Internal cracking andstrain-softening response of concrete under uniax-ial compression. ACI Materials Journal, 84(3):200212.Shah, S. P., Swartz, S. E., and Ouyang, C. (1995). Frac-ture Mechanics of Concrete. Wiley, New York.Torreti, J. M., Benaija, E. H., and Boulay, C. (1993). In-uence of boundary conditions on strain softeningin concrete compression tests. Journal of Engineer-ing Mechanics-ASCE, 119(12):23692384.Van Mier, J. G. M. (1984). Strain-softening of concreteunder multiaxial loadingconditions. PhDthesis,EindhovenUniversityof Technology, Eindhoven,The Netherlands.