4-Failure Criteria and Triaxial Behaviour of HPFRC

Construction and Building Materials 29 (2012) 215–229

Contents lists available at SciVerse ScienceDirect

Construction and Building Materials

journal homepage: www.elsevier .com/locate /conbui ldmat

Failure criteria and triaxial behaviour of HPFRC containing high reactivitymetakaolin and silica fume

Saeed Karim Babanajad, Yaghoob Farnam ⇑, Mohammad ShekarchiConstruction Materials Institute, School of Civil Engineering, College of Engineering, University of Tehran, Tehran, Iran

a r t i c l e i n f o

Article history:Received 5 October 2010Received in revised form 10 August 2011Accepted 16 August 2011Available online 24 November 2011

Keywords:Triaxial strengthHPFRCHigh reactivity metakaolinSilica fumeFailure criteria

0950-0618/$ - see front matter � 2011 Elsevier Ltd. Adoi:10.1016/j.conbuildmat.2011.08.094

⇑ Corresponding author. Tel.: +98 21 88973631.E-mail address: [email protected] (Y. Farnam).

a b s t r a c t

In this paper, the triaxial compressive behaviour of high performance fibre reinforced concrete (HPFRC)containing two types of pozzolans namely high reactivity metakaolin (MK) and silica fume (SF) is studied.Standard triaxial tests with four different confining pressure levels were performed on 75 � 150 mmcylindrical concrete specimens containing 2% steel fibres by volume. Companion uniaxial direct tensiletests are also performed on dog-bone shaped specimens to ascertain the strain-hardening behaviour ofthese composites. The results show that an increase in confining pressure leads to an increase in the peakadditional axial stress in both SF and MK specimens, and the increase is about the same for the two typesof composites. However, the peak additional axial stresses of MK specimens occur at higher axial strainsthus showing a more ductile behaviour with higher energy absorption capacity. To establish an empiricalrelation for prediction of axial strength as a function of confining pressure, three constitutive models i.e.Mohr–Coulomb two-parameter, Hsieh–Ting–Chen four-parameter and William–Warnke five-parametermodels are used and calibrated with the test results.

� 2011 Elsevier Ltd. All rights reserved.

1. Introduction

With the increasing demand for modern civil engineering mate-rials and their practical utilization in structural applications, thereis increasing need to develop reliable constitutive models for highperformance concretes. They include the effects of adding steel fi-bres as well as supplementary cementitious material to the con-crete. Enhancing the structural resistance and performance andalso reducing cracking and spalling phenomena is possible byincreasing the concrete’s toughness, ductility, tensile strengthand flexural strength under various kinds of loading. Fibre rein-forced concrete (FRC) is one of the commonplace engineeringmaterials, employed in several structural applications.

High and ultra high strength concrete, named as HSC and UHSCwith a very high compressive strength values remain principally abrittle material. Adding adequate additives such as appropriatepozzolans and fibres improves tensile strength, flexural strength,energy absorption, and impact resistance and also provides ductil-ity and intensifies their performance [1–13]. The fibres volumefraction of conventional FRC and high performance fibre-reinforcedconcrete (HPFRC) is generally in the range of 1–3%.

As shown in many research studies, the addition of pozzolansimproves the mechanical properties of concrete [1–13]. Therefore,by combining steel fibre and suitable pozzolans, the mechanical

ll rights reserved.

properties of concrete are expected to be improved. To the authors’knowledge, there are no published comprehensive studies investi-gating the properties of HPFRC containing pozzolans and subjectedto triaxial loading. Hence, in this research, the behaviours of HPFRCunder triaxial tests containing two different practical pozzolans i.e.high reactivity metakaolin and silica fume are investigated andcompared.

Silica fume is often introduced as a super-pozzolan that reactswith the lime produced from hydrated cement reducing the volumeof larger pores commonly found in cement paste. Likewise, highreactivity metakaolin is a highly active and effective pozzolan,which can be used as a cementitious material in partial replacementof cement in concrete. As known, energy absorption and toughnessof HPFRC increases in relation to adding the high-reactivity materi-als to mixture [1,6]. Generally, adding convenient pozzolans to theconcrete may enhance the mechanical properties i.e. compressivestrength, bending strength, and splitting tensile strength and alsoits performance, durability and toughness. Therefore, the use of var-ious mineral admixture is now becoming commonplace in concretefor structural applications especially in the place where both en-hanced durability and high toughness are required.

In relation to the recent published researches, Kim et al. [4] stud-ied HSC specimens using metakaolin from the aspects of strengthand durability properties. From this research, it is understood thatmetakaolin constitutes promising material as a substitute for silicafume. Gutierrez et al. [5] inspected the effect of pozzolans on theperformance of fibre-reinforced matrixes. Experimenting three

Table 1Chemical properties of ordinary Portland cement, silica fume and high reactivitymetakaolin.

Chemical analyses (%) Cement Silicafume

High reactivitymetakaolin

Silicon dioxide (SiO2) 21.03 93.16 51.85Aluminium oxide

(Al2O3)4.53 1.13 43.87

Ferric oxide (Fe2O3) 3.63 0.72 0.99Calcium oxide (CaO) 62.45 – 0.20Magnesium oxide

(MgO)3.42 1.6 0.18

Sodium oxide (Na2O) 0.21 – 0.01Potassium oxide (K2O) 0.55 – 0.12Sulfur trioxide (SO3) 2.23 0.05 –Loss on ignition (LOI) 1.37 1.58 0.57

216 S.K. Babanajad et al. / Construction and Building Materials 29 (2012) 215–229

different pozzolans i.e. fly ash, silica fume and metakaolin showsthat silica fume has an upper effective influence on the compressivestrength of concrete than metakaolin after a definite curing day.Shekarchi et al. [2] performed compressive, splitting tensile, waterabsorption, water penetration and gas permeability tests on highperformance concrete (HPC) containing high reactivity metakaolinand silica fume. The results showed the same role implemented bythese pozzolans in improving mechanical properties of the concrete.However, high reactivity metakaolin indicated better effects onworkability and durability of concrete. Dubey and Banthia [6] inves-tigated the influence of high reactivity metakaolin and silica fume onthe flexural toughness of high-performance steel fibre reinforcedconcrete (SFRC). It was concluded that high reactivity metakaolinis particularly effective in improving the post-peak energy absorp-tion. Poon et al. [7] investigated the compressive strength, chloridediffusivity and pore structure of high performance metakaolin andsilica fume concrete. In this investigation, an improvement in theinterfacial microstructure with the incorporation of the pozzolansis found. It is found that metakaolin has superior strength develop-ment, a very dense microstructure of the paste with a lower total ofporosity, finer pore size distribution, lower MIP (Mercury IntrusionPorosimetry) porosity and similar chloride resistance in comparisonto silica fume. Ding and Li [8] discussed on the effects of metakaolinand silica fume on concrete properties by casting concrete’s mixturewith the different mass of silica fume and metakaolin. Metakaolin-modified concrete showed a better workability than silica fume-modified concrete. As the replacement level was increased, thestrength of the metakaolin-modified concrete increased at all agessimilarly to that of the silica fume-modified concrete.

Bindiganavile and Banthia [9] studied the effect of pozzolans onthe improvement of shotcrete quality. Silica fume indicated supe-rior effects than high reactivity metakaolin in reducing rebounddue to its ultra-fine particle size. However, because of increasingthe brittleness and decreasing energy adsorption by silica fume, ablend of silica fume and high reactivity metakaolin was suggestedto be used. Banthia and Yan [10] found that high reactivity metaka-olin is more effective in improving the pull-out performance thansilica fume. In a comparative study between metakaolin and silicafume, Curcio et al. [11] showed that metakaolin has better pozol-anic activity in compare with silica fume. Based on Wong and Ra-zak research [12], metakaolin showed more efficiency factor (K) incompare with silica fume at long term ages. Also, Tafraoui et al.[13] stated that metakaolin is a very acceptable substitute for silicafume and can obtained almost the same mechanical properties ofsilica fume specimens and moreover in some cases such as avail-ability, colour and price is more suitable.

One of the major technical interests for different kinds of con-crete is the failure mechanism under uniaxial and triaxial loading.For designing of triaxially compressed structures such as the coreof mass concrete, and the columns bearing high loads, it is neces-sary to have some equations making relationship between the con-fining pressure, uniaxial compressive strength and the peak triaxialstrengths. Chern et al. [14] studied the response of FRC subjected tomultiaxial loading conditions and calibrated the William–Wrankemodel with the test results. Xie et al. [15] performed a series of tri-axial tests on HSC specimens and calibrated the modified Reimannfailure criteria model. Ansari and Li [16–18] performed a series ofunaxial and triaxial tests on HSC specimens involving concreteswith strengths as high as 103 MPa. They used Ottosen four-param-eter failure criteria for evaluating the test results. Lu and Hsu [19]studied on the behaviour of HSC and high strength SFRC undertriaxial compression. They also investigated on triaxial stress–strain relations and William–Warnke failure criteria. As the reviewof technical literature indicates, there is no comprehensiveresearch on the evaluation of most known failure criterions withuse of different pozzolans on the HSC and HPFRC specimens under

triaxial loading. Hence, the most applied failure criterions are usedfor calibration and comparing the literature and present paper’stest results.

In the current study, triaxial tests are carried out on cylindrical-shaped specimens (75 � 150 mm) containing 2% steel fibres byvolume with the objective to compare the effects of metakaolineand silica fume on triaxial compressive behaviour. Also, uniaxialcompression and tension tests are carried out. Furthermore, dueto developing failure surface procedures and prediction of HPFRCbehaviours containing high reactivity metakaolin and silica fume,the most common failure criteria i.e. Mohr–Coulomb, Hsieh–Ting–Chen and William–Warnke models are employed to calibratethese models with the test results. These models are well-known inusing for HSC and FRC [14–17,19–23] to establish an empiricalrelationship. The calibrated failure criterions are also comparedwith corresponding literature results.

2. Experimental program

To use HPFRC as a structural material, it is necessary to increase steel fibre vol-ume fraction more than 1%. In these dosages, fibres could play a considerable roleon mechanical and structural properties of HPFRC as a fractional replacement ofbar reinforcement. Therefore, in this research, 2% volume fraction (Vf) of steel fibrewas chosen to get considerably involved in improving structural properties ofHPFRC. Two types of HPFRC containing two different pozzolanic cementitious mate-rials were used in this study. Therefore, for the experimental program two types ofmatrix has been used. In the first matrix, silica fume as a pozzolan additive has beenreplaced by cement mass. In the second matrix, high reactivity metakaolin has beenused instead of silica fume. Cementitious materials were ordinary Portland cement(OPC), equivalent to ASTM Type II. The chemical analyses of materials are given inTable 1. The aggregate-sand was crushed limestone with a maximum size of 1 mm.It had a specific gravity and absorption values of 2.50% and 3.2%, respectively. Thewater-cementitious material ratio (w/cm) of the mixture is 0.33. Polycarboxylatechemical additive (1.3% of cementitious material mass) was used as a superplasti-sizer to make better mixtures flow and performance. Mix proportions of mortars arelisted in Table 2. Also the properties of smooth steel fibre used in the mortar areshown in Table 3.

In order to prepare specimens, at first cement, silica fume (or high reactivitymetakaolin) and aggregates were blended together in dry condition. Then watermixed with superplasticizer was gradually added to the dry mixture to provide ahomogeneous mortar. The prepared mortar was then poured into cylindrical anddog bone-shape moulds in several layers. After pouring of each layer, fibres wererandomly dispersed in a horizontal direction. These steps were repeated until themoulds were completely casted.

Freshly cast specimens were kept in the mould for 24 h; afterwards, they weredemoulded and stored in water which was saturated with calcium hydroxide at22 �C. Specimens were left there for 90 days. Then, the specimens were taken outof the curing room and were carefully capped at both ends to provide flat andsmooth bearing areas to be loaded.

3. Testing procedure

The experimental tests were conducted in two parts: (1) directtensile tests and (2) uniaxial and triaxial compression tests; whosedetails are separately presented and discussed.

Table 2Mix proportions of mortars.

SF MK kg/m3

Cement Portland (Type II) Portland (Type II) 1020Pozzolan Silica fume High reactivity metakaolin 175Aggregate Sand (0–1 mm) Sand (0–1 mm) 835Water – – 388w/cm – – 0.33Superplastisizer Polycarboxylate (glenium) Polycarboxylate (glenium) 15.3 (1.3% of cementitious material)

Table 3The properties of smooth steel fibre.

Fibre type Steel fibre

Section shape CircleLength (mm) 30Diameter (mm) 0.25l/d (aspect ratio) 120Tensile strength (MPa) 2200

Fig. 2. Cylindrical specimen under uniaxial compressive test.

S.K. Babanajad et al. / Construction and Building Materials 29 (2012) 215–229 217

All of the uniaxial and triaxial tests were performed by MTS ma-chine, with a rigid frame and capacity of 4500 KN [24]. The loadingprocedure according to ASTM C469 [25] was followed for all spec-imens. Each specimen was loaded to a stress of about 20% of itsultimate stress and unloaded to a stress level of about 5% of theultimate stress. The specimen was then reloaded to reach failure.The data taken from the last loading cycle were recorded as the fi-nal results.

In total, six direct tensile tests were carried out (three for eachMK and SF). The direct tensile specimens were prepared in dogbone-shape, as shown in Fig. 1a. Moreover, the details of tensilespecimens are indicated in Fig. 2b. The loading was applied witha strain rate of 0.00005 per second for the direct tensile tests.

Eight specimens of each high reactivity metakaolin and silicafume HPFRC specimens were tested in triaxial compression underconfining pressure up to 21.5 MPa. Two specimens were tested foreach parameter. Specimens tested herein and designations ofuniaxial and triaxial compression tests are listed in Table 4.

According to Li and Ansari research [17], there is a slight sizeeffect on the failure strength in triaxial compression. To facilitatetriaxial test procedure, therefore, cylindrical specimens with diam-eter of 75 mm and height of 150 mm were adopted. Considering

Fig. 1. (a) Dog-bone shape specimen under direct ten

specimens with minimum dimension of 2.5 times of fibre length,it could be assumed that there is no effect of specimen dimensiondue to fibre length. Accordingly, standard cylindrical specimenswith diameter and height of 75 � 150 mm were prepared in orderto carry out uniaxial and triaxial compressive tests (see Fig. 2). In

sile test; (b) details of direct tensile specimens.

Table 4Designation of uniaxial and triaxial compression tests.

Designation (fibre volume ratio) Specimendesignation

Confining pressure(MPa)

High reactivity metakaolin (MK) (Vf = 2%) MK0-1 0MK0-2 0MK5-1 5MK5-2 5MK15-1 15MK15-2 15MK21.5-1 21.5MK21.5-2 21.5

Silica fume (SF) (Vf = 2%) SF0-1 0SF0-2 0SF5-1 5SF5-2 5SF15-1 15SF15-2 15SF21.5-1 21.5SF21.5-2 21.5 Fig. 4. Triaxial test setup.


accordance with ASTM C801 [26], a triaxial Hoek cell [27] was usedto determine the behaviour of specimens under confined lateralconditions (see Fig. 3).

All triaxial tests were performed by the MTS machine, as shownin Fig. 4. The axial actuator and the confining pressure intensifierwere programmed together to provide the desired load paths. Dur-ing the tests, load and deformation data were recorded and storedby the data acquisition system and subsequently were processedfor analysis. A confining pressure intensifier was used to fill andpressurize the triaxial cell with confining fluid. The confining pres-sure (r3) is directly measured by a pressure transducer.

The load path for triaxial tests is given in Fig. 5. As shown, theaxial and confining pressures were simultaneously applied to thespecimens until the confining pressures reached the targeted va-lue. At this point, the confining pressure was kept constant andthe additional axial stress (Dr) was increased at a constant strainrate of 0.0001 per second through the platens located at its ends.It should be noted that r1 refers to axial stress and calculated by(r1 = r3 + Dr). A rubber bladder was utilized to isolate the speci-men from oil penetration.

Increasing the additional axial stress causes the specimen to di-late (or contract) in the triaxial cell. In order to keep the lateralconfining pressure constant, some oil should be drained out of(or inject into) the cell. To do this, an external hydraulic pressurecontroller equipped by a jack was used. In other words, in responseto the specimen dilation (or contraction) in the cell, the jack wasrelaxed (or compressed) to accommodate a volume of oil which

Fig. 3. Triaxial Hoek cell [27].

is equal to the specimen’s volume change. Therefore, oil transmit-ted from the cell to the jack (and vice versa) during the test to guar-antee constant load in the jack and consequently constant pressurein the cell. This arrangement kept confining pressure constant with0.5% accuracy. All data transition and process were performed bythe use of digital machine. Minicomputer was employed for thecontrol of experiments and acquisition of data.

4. Results and discussion

4.1. Direct tensile test results

The ultimate tensile strength (uniaxial tension strength f 0t ) forMK and SF specimens are reported in Table 5. Table 5 depicts thetensile yield stress ðrytÞ; tensile yield strain and tensile ultimatestress ðrutÞ of these specimens under direct tension loading condi-tions. The mean stress–strain responses of the HPFRC specimenssubjected to direct tension are given in Fig. 6. The direct tensiletests results will be later used only in calibration of ‘‘Hsieh–Ting–Chen four-parameter’’ failure criteria. Due to the lowamounts of tensile stresses some fluctuation was seen in this fig-ure. Therefore, for better scanning, the average of fluctuationswas taken and drawn by black highlight.

4.2. Uniaxial and triaxial compression test results

The results of uniaxial and triaxial compression tests are dis-cussed in two parts including effects of confining pressure andused pozzolans. It should be noted that maximum confining ratio

r3f 0c

� �among all tests were approximately 0.25.

4.2.1. Effects of confining pressureThe experimental responses of MK and SF specimens under dif-

ferent confining pressures are plotted in Figs. 7 and 8, respectively.It is obvious that additional axial stress (Dr) and triaxial compres-sive stress (r1 = r3 + Dr) were increased by increasing confiningpressure at a constant content of steel fibres. Concrete specimensbehave as a ductile and plastic material in the presence of confin-ing pressure.

Increasing confining pressure in triaxial tests raises the peakadditional axial stress. Also, the peak point moves towards right(higher axial strain) as the confining pressure increases. Elicitedfrom the observation, the dependency of triaxial mechanicalbehaviour especially peak additional axial stress to the value ofconfining pressure is decreased at higher confining pressure.However, the additional axial stress increases with the increase

Fig. 5. Loading path for triaxial tests.

Table 5Direct tensile results of HPFRC.

Specimen designation MK SF

MK1 MK2 MK3 Mean SF1 SF2 SF3 Mean

ryt tensile yield stress (MPa) 5.58 6.82 7.78 6.72 5.09 4.98 7.03 5.7eyt tensile yield strain (mm/mm � 10�5) 17.12 15.41 14.27 15.6 20.02 23.41 14.51 19.31rut tensile ultimate stress (MPa) 6.94 8.41 6.81 7.38 5.04 6.02 7.61 6.22

Fig. 6. Mean direct tensile stress–strain curves for MK and SF specimens.


of confining pressure. This effect is slight as the confining pressurereaches values over 15 MPa. These behaviours are seen in both MKand SF specimens. As a procedure occurred during applying axialload in triaxial tests, specimens contract in axial direction, whiledilation occurs in the lateral direction because of Poisson’s law.Three types of relationship can be observed between the evolutionof axial strain history and characteristics of compressive stress inthe plots due to the three types of material behaviour under differ-ent loading conditions. These behaviours are softening, plastic andhardening behaviours as shown in Fig. 9.

As another result observed in the Figs. 7 and 8, MK specimensshow softening behaviour up to 5 MPa confining pressure and forhigher confining pressure, the softening behaviour is changed toplastic behaviour. Albeit it is the same for SF specimens, but thetransferring from softening behaviour to plastic behaviour in SFspecimen occurs at higher confining pressure in comparison toMK specimens.

The increases of peak additional axial stresses are summarizedin Table 6. Also, Variation of peak additional axial stresses versus

confining pressure for MK and SF specimens are indicated inFig. 10.

4.2.2. Effects of high reactivity metakaolin and silica fume pozzolansAt point where the additional axial stress-axial strain curve

deviates from linearity, the initiation of some micro cracks occurs.Cracks extension within the concrete specimens will cause them tofail.

As known, when cylindrical plain concrete specimen undercompression load fails in a brittle mode, adding fibres in concretebridge matrix crack and apply a closing pressure at the crack front.This reduces the stress intensity at the crack tip and a higher en-ergy input is needed to further extend the crack. Additional energyneeded for crack propagation depends mainly to properties of fibreand matrix and also bond strength between them.

The fibre and matrix properties play an important role on inter-facial bond strength between the fibre and surrounding matrixwhich provides most of the pull out resistance. Also, the transmis-sion of forces between the fibres and the matrix is achieved

Fig. 7. Additional axial stress-axial strain curves for MK specimens under different confining pressure.

Fig. 8. Additional axial stress-axial strain curves for SF specimens under different confining pressure.

Fig. 9. The schematic additional axial stress versus axial strain responses incompression for different material behaviours.


through interfacial bond provided by the shearing stress at theinterface surface between the fibre and the surrounding matrix.Therefore, if this interfacial bond strength can be increased, theabsorbed energy increases. Matrix brittleness directly affects onthe interfacial bond strength and consequently on pull out resis-tance of fibre. In brittle matrix, the post-peak load carrying

capacity is expected to degenerate at a faster and steeper drop rate[6,28,29]. Therefore, it is clear that the brittleness has an adverseeffect on the post-peak toughness characteristics of concrete.Incorporation of pozzolans in matrix has different effect on theinterfacial bond strength and pull out resistance of fibre as wellas on matrix brittleness [6,28,29]. Some pozzolanic admixture suchas silica fume increases the brittleness of the matrix while someothers such as high reactivity metakaolin alleviates brittleness[6,9,10].

From the complete stress–strain responses of HPFRC specimens,shown in Figs. 7 and 8, it can be observed that the post-peakbehaviour of MK specimens is superior to SF specimens. SF speci-mens exhibit a relatively brittle behaviour and consequently a fall-ing trend is observed in stress–strain responses beyond the peakpoint. Moreover, it is obvious that MK specimens show more plas-tic behaviour especially in higher axial strains. With considering tothis fact, when a cylindrical specimen is exerted under compres-sion load, the height of specimen decreases and the diameter ofspecimen (lateral strain) increases (Poisson’s ratio law). Therefore,the steel fibres in horizontal direction carry the tensile load.According to Figs. 7 and 8, it seems that high reactivity metakolinpozzolan gives ductility to the mortar and creates effective bondstrength between fibre and matrix. In the case of MK specimensat the moment of crack extension, the fibre slowly develops fullanchorage and approaches ultimate conditions up to full pull outcapacity of fibre. In SF specimens, due to brittle properties ofmatrix provided by silica fume pozzolan, the connection between

Fig. 10. Variation of peak additional axial stress versus confining pressure forHPFRC.

Table 6Uniaxial and triaxial compression test results of HPFRC.

Specimen Confining pressure(MPa)

Peak additional axial stress(MPa)

Confining pressureratio

Average of peak additional axialstress (MPa)

Increase of peak additional axialstress (%)

MK0-1 0 94.1 0 95.10 –MK0-2 0 96.0MK5-1 5 106.5 0.05 109.9 15.6MK5-2 5 113.2MK15-1 15 147.4 0.16 148.6 35.2MK15-2 15 149.9MK21.5-1 21.5 157.1 0.23 161.0 8.30MK21.5-2 21.5 164.8

SF0-1 0 83.0 0 87.10 –SF0-2 0 91.2SF5-1 5 119.1 0.06 122.8 41.0SF5-2 5 126.4SF15-1 15 149.6 0.17 147.4 20.0SF15-2 15 145.2SF21.5-1 21.5 159.8 0.25 160.1 8.60SF21.5-2 21.5 160.4


fibre and matrix weakens and the complete pull out capacity of fi-bres cannot be provided.

In addition, based on Dubey and Banthia research [6] whoinvestigated the effect of silica fume and metakaolin pozzolanson the pull out bond capacity of hooked steel fibre reinforced con-crete, the photographs of fractured surfaces of beams with silicafume and high-reactivity metakaolin under flexural tests show thatin the beams with silica fume, the majority of the fibres were stillintact with no straightening of the hooks. These fibres then pulledout with minimal straightening of the hooks and absorbed muchless energies. For fibres in concrete with high reactivity metakao-lin, on the other hand, larger number of hooked ends of fibresstraightened implying an increased absorption of energy. It is con-cluded that high reactivity metakaolin is particularly effective inimproving the post-peak energy absorption capacity of concretewith fibres, and unlike in beams with silica fume, no particularpost-peak brittleness is seen to occur.

Accordingly, it seems that high reactivity metakaolin supplies abetter stress transfer between fibre and matrix than silica fume.

4.3. Energy absorption capacity

While the load–deflection curves can provide a qualitativeassessment of the performance of the various fibre-reinforcedcomposites, for a quantitative evaluation, energy absorption capac-ity is used to perform a suitable toughness analysis.

The area under the stress–strain curve is a measure of tough-ness of the material. Researchers have used different definitionsfor the toughness using the area under the curve. Fanella and Naa-man [30] have defined the toughness of FRC as the ratio of tough-ness of the fibre reinforced matrix to that of the unreinforcedcontrol matrix. Here, this ratio is greater than unity. It can alsobe defined as the ratio of area of the descending part, to the areaof the ascending part of the stress–strain curve. Nataraja et al.[31] have defined a toughness ratio (TR) by comparing the calcu-lated toughness to the toughness of a rigid plastic material. In thisform, the calculated ratio is less than unity for normal strengthFRC.

Most researchers [30–32] have obtained the energy absorptioncapacity or toughness of FRC in compression as the area under thestress–strain curve calculated up to a specified strain value nearthe strain of 0.015, though the specimens still had significant resis-tance left. According to the results obtained in current research,HPFRC under triaxial loading indicated totally different behaviourrather than normal FRC. Therefore, in present study in order to de-fine a convenient method for calculating toughness of HPFRC undertriaxial loading, a similar way corresponding to ASTM C1018 [33]for calculating flexural toughness and first-crack strength of FRCwas adopted. The suggested method was used for the determina-tion of energy absorption capability of material up to the selectedstrain criteria.

In present paper, three amounts of areas (representatives of en-ergy absorption capacity) were calculated for each triaxial testwhich are named with Areacr, Area3cr, and Area5cr where are theareas under the stress–strain curve up to the first-crack strainðecrÞ, three times of ecr, and five times of ecr, respectively. The sche-matic procedure of determining first crack strain and area calcula-tion is explained in Fig. 11. As shown in Fig. 11, first crack strain(ecr) is calculated from the additional axial stress-axial strain curvewhere the elastic behaviour ends and the linear axial stress-axialstrain relationship changes to nonlinearity. ecr corresponds to thepoint that cracks start and grow or where the curve deviates fromlinearity. By this approach, the area of the specimens can be easilyevaluated and the results are presented in Table 7. Here, Areacr un-der no confining pressure for MK and SF is equal to 0.24 and

Fig. 11. Schematic procedure of determining first crack strain and calculatingenergy absorption areas.


0.12 MPa.(mm/mm), respectively. The variation in the energyabsorption is between 0.24 and 5.06 MPa (mm/mm) for MK andalso is between 0.12 and 4.35 MPa (mm/mm) for SF. It is obviousfrom Table 7 that all of energy absorptions of MK specimens, ex-cept for two cases, are more than energy absorption of SF speci-mens. As explained in previous section, it might be due to higherinterfacial bond strength of fibres and cement paste with highreactivity metakaolin in compare with silica fume which leads toa higher ductility advantage. Furthermore as shown in Figs. 7and 8, in post-peak strains, SF specimens have steeper drops and

Table 7Energy absorption of HPFRC specimens under uniaxial and triaxial compression.

Specimen Confining pressure (MPa) ecr (%) Areacr (MPa

MK (Vf = 2%) MK0-1 0.51 0.24MK0-2 0.53 0.24MK5-1 0.49 0.25MK5-2 0.51 0.25MK15-1 0.61 0.37MK15-2 0.67 0.40MK21.5-1 0.70 0.46MK21.5-2 0.73 0.49

SF (Vf = 2%) SF0-1 0.39 0.14SF0-2 0.36 0.12SF5-1 0.48 0.24SF5-2 0.50 0.26SF15-1 0.61 0.34SF15-2 0.54 0.31SF21.5-1 0.64 0.40SF21.5-2 0.65 0.43

Fig. 12. (a) Linear superposition subjected to triaxial compression; (b

loose more strengths in comparison to MK specimens indicatinga greater rate of softening and damage in SF composites. Visually,one could notice that the SF matrix had cracked more extensively,and fibres as a result could not develop adequate mechanicalanchorage with the matrix. Hence, MK specimens show more en-ergy adsorption and plastic behaviour, especially in post-peakstrains.

The highest amounts of energy absorption is obtained by HPFRCwith MK under 21.5 MPa confining pressure. It is obvious that bothconfining pressure and pozzolan types have good effect on the en-ergy absorption.

4.4. Failure pattern

SFRC, especially HPFRC under triaxial condition behaves quitedifferently compared to what is usually observed in plain concrete.Besides the special mechanical behaviours of HPFRC like high mul-tiaxial strength and ductility, the failure modes of these specimensare considerable.

Based on the linear superposition of stress states as shown inFig. 12a, it is expected that specimens under triaxial compressiontest fail in a mode similar to those under uniaxial compression test(tensile failure), shown in Fig. 12b. However, it seems that lateralconfining pressures and adding steel fibres prevent the occurrenceof cracks parallel to the direction of force, and causes the specimenunder triaxial compression to fail in shear modes which areobserved in HPFRC specimens. Failure pattern of HPFRC specimensunder triaxial compression is shown in Fig. 13. Since the failuremodes were approximately similar in MK and SF specimens, onlythe pictures for MK specimens are depicted in this figure. Accord-ing to Fig. 13, increasing of confining pressure causes multiple

) (mm/mm) Area3cr (MPa) (mm/mm) Area5cr (MPa) (mm/mm)

1.12 –1.16 1.981.28 2.301.36 2.391.99 3.752.23 4.232.55 4.712.70 5.06

0.79 1.300.67 1.171.33 2.201.35 2.262.01 3.551.84 3.412.30 4.282.33 4.35

) mode of specimen’s failure subjected to uniaxial compression.

Fig. 13. Failure patterns of specimens under triaxial compression for (a) 0 MPa; (b) 5 MPa; (c) 15 MPa; (d) 21.5 MPa.

Fig. 14. (a) Deviatoric plane, (b) s� r coordinate system.


shear cracks during loading, which is typical of a semi ductilematerial behaviour. In the other hand, despite of shear failure inHPFRC under no confining pressure, they behave as a ductile mate-rial and multiple shear cracking increased gradually during the tri-axial loading.

4.5. Failure criteria

Three most applied failure criterions are used to calibrate themodels with triaxial compression test results of HPFRC, as describedin the following.

4.5.1. Mohr–Coulomb two-parameter failure criteriaThe Mohr–coulomb failure criterion model [34,35] is one of the

pressure-dependent failure criteria that the pure shear (or

octahedral shear (soct)) depends linearly on the hydrostatic stress(or octahedral normal stress (roct)). This model is the most appliedclassical model for explaining of failure envelope and is still usedextensively due to its simplicity and relatively adequate accuracy.This failure surface should passes through the point of uniaxialcompression exactly. The normalized form of Mohr–Coulomb canbe expressed as Eq. (1) and the deviatoric plane and s� r coordi-nate system illustrated in Fig. 14.

r1

f 0c

� �¼ aþ b � r3

f 0c

� �ð1Þ

a and b define as

a ¼ 2C � cos uð1þ sinuÞ � f 0c

; b ¼ 1� sinu1þ sinu

where u and C are the angle of internal friction and the cohesioncoefficient of concrete and f 0c denotes the uniaxial compressivestrength.

Calculated amounts of a and b for HPFRC triaxial data are com-pared with other researches in Table 8.

Xie et al. [15] obtained b values between 4.25 and 4.47 for dif-ferent kinds of HSC with uniaxial compressive strength upper than67 MPa. Lu and Hsu [19] obtained 3.65 and 3.55 as b value of HSCand high strength SFRC (containing 1 vol.% fraction of steel fibre)specimens respectively. Moreover, Candappa et al. [22] obtainedb value in range of 4.58 and 5.57 for HSC specimens. Accordingto Imran and Pantazapolu [21] results, the b value is calculated3.13 for HSC specimens and in range of 3.42 and 4.15 for normalstrength concrete (NSC). Also, from Nielsen [23] results, the b valueis calculated in range of 4.14 and 3.29 for HSC specimens. Based onthe data presented by Ansari and Li [16,17], the b values obtainedfor HSC specimens are in range of 3.25–3.80. Based on the resultsof present research as shown in Fig. 15, the b value for MK andSF specimens is 4.20 for both MK and SF, and the internal-frictionangle ðuÞ of specimens can be adopted as 38�. Also, values of a forMK and SF are obtained 1.01 and 1.10, respectively. The cohesioncoefficient (C) can be adopted as 23.4 MPa for both MK and SFspecimens. It can be concluded that the amount of f 0c has not anyconsiderable effects on the a and b in the range of tested confiningpressure. The values of a and b extracted from experimental resultsare in good agreement with those obtained from literature. It isobserved that the b values and the internal-friction angle vary

Table 8Empirical coefficients for Mohr–Coulomb model.

Specimen Max r3f 0c

� �Max r1

f 0c

� �f 0c a b uo C (MPa) R2

MK 0.2 2.0 95 1.01 4.20 38.0 23.4 0.986

SF 0.2 2.1 87 1.10 4.20 38.0 23.3 0.949

Nielsen [23] 0.4 3.0 164 1.26 4.14 37.7 50.8 0.9590.9 4.1 165 1.31 3.29 32.3 59.6 0.971

Xie et al. [15] 0.5 3.2 60 1.12 4.47 39.4 15.9 0.9850.5 3.2 92 1.19 4.45 39.3 25.9 0.9610.5 3.1 119 1.07 4.25 38.2 30.9 0.940

Ansari and Li [16] 0.9 3.5 47 1.17 3.80 35.7 14.1 0.9880.9 3.2 71 1.24 3.25 32.0 24.4 0.9780.8 2.9 107 1.16 3.43 33.3 33.5 0.983

Li and Ansari [17] 1.0 3.4 70 1.28 3.25 32.0 24.9 0.9820.8 3.2 103 1.25 3.56 34.2 34.1 0.972

Lu and Hsu [19] 0.8 4.1 67 1.16 3.65 34.7 20.3 0.9901.0 4.7 69 1.25 3.55 34.1 22.9 0.991

Imran and Pantazapolu [21]

HSC 0.7 3.3 73 1.14 3.13 31.0 23.5 0.9920.8 3.6 65 1.10 3.14 31.1 20.2 0.996

NSC-Type a 0.9 4.3 47 1.08 3.64 34.7 13.3 0.9971.0 4.3 43 0.98 3.42 33.2 11.4 0.994

NSC-Type b 0.7 4.0 29 1.02 4.15 37.7 7.3 0.9971.0 4.8 21 1.01 3.90 36.3 5.4 0.999

Candappa et al. [22] 0.3 2.5 42 1.04 5.13 42.4 9.6 0.9950.2 1.9 61 1.00 4.58 39.9 14.3 0.9990.2 1.9 73 1.05 5.35 43.2 16.6 0.9710.1 1.6 103 1.03 5.57 44.1 22.5 0.973

Average 1.12 3.98 36.3 22.8 0.981Standard deviation 0.1 0.7 3.8 12.6 0.016Minimum 0.98 3.13 31.1 5.37 0.94Maximum 1.31 5.57 44.1 59.6 0.999

Fig. 15. Triaxial strength ratio versus confining ratio (Mohr–Coulomb failurecriteria).


insignificantly with the change of pozzolan type and the model hasnot significant sensitivity to the types of pozzolans.

4.5.2. Hsieh–Ting–Chen four-parameter failure criteriaBesides the Mohr–Coulomb two-parameter model, another con-

stitutive model with four-parameter was suggested. This modelprepared the geometric requirements of the failure surface for con-crete materials and it can be an useful model with high accuracyfor triaxial compression test data, suggested by Hsieh–Ting–Chen

[34–36]. The following criterion is involving all three stress invari-ants ðn;q; hÞ and using Haigh–Westergaard coordinates system as afailure function with the following form of the following equation:

f ðn;q; hÞ ¼ aq2 þ kqþ dn� 1 ¼ 0 ð2Þ

where k is a function of cos h and defined as kðhÞ ¼ b � cos hþc ðfor jhj � 60�Þ, in which a, b, c and d are material constants.The hydrostatic and deviatoric planes can be represented by:n ¼ I1ffiffi

3p , q ¼

ffiffiffiffiffiffiffi2J2

p.

With defining of q � cos h ¼ffiffi32

qr1 �

ffiffi16

qI1

� �, the Eq. (2) can be

rewritten in terms of the new A, B, C and D material constants asmentioned in Eq. (3).

f ðI1; J2;r1Þ ¼ AJ2

ðf 0cÞ2 þ B

ffiffiffiffiJ2

pf 0cþ C

r1

f 0cþ D

I1

f 0c� 1 ¼ 0 ð3Þ

In which,

I1 ¼ r1 þ r2 þ r3 ¼X

rii;

J2 ¼12ðs2

1 þ s22 þ s2

3Þ ¼X1

2sij � sij

where I1 is the first invariant of the stress tensor, and J2 pertain tothe deviatoric principal stress invariant. rij denotes the stresstensor, and sij is the stress deviator tensor. Also, f 0c denotes the uni-axial compressive strength of concrete.

With compare of Eqs. (2) and (3), the relations between the twosets of material constants (a, b, c, d and A, B, C, D) are obtained as:a ¼ A

2 ; b ¼ffiffi23

qC; c ¼ Bffiffi

2p ; d ¼

ffiffiffi3p

Dþ bffiffi6p

� �.

Fig. 16. (a) Haigh–Westergaard coordinate system, (b) deviatoric plane.

Table 9Empirical coefficients for Hsieh–Ting–Chen model.a

Specimen Max r3f 0c

� �Max r1

f 0c

� �f 0c A B C D R2

MK 0.2 2.0 95 �1.2116 6.3681 �2.2562 0.0831 0.967

SF 0.2 2.1 87 �1.2272 5.7678 �1.9847 0.1187 0.960

Nielsen [23] 0.4 3.0 164 �0.7669 4.9291 �1.9175 0.2581 0.9690.9 4.1 165 �0.4665 4.6898 �1.9923 0.3171 0.953

Xie et al. [15] 0.5 3.2 60 �0.6659 5.3376 �2.1587 0.2443 0.9110.5 3.2 92 �0.7715 4.6800 �1.7852 0.2552 0.9050.5 3.1 119 �0.8565 4.7468 �1.6585 0.1773 0.876

Ansari and Li [16] 0.9 3.5 47 �0.3756 5.5237 �2.3289 0.2233 0.9360.9 3.2 71 �0.5279 4.2941 �1.6278 0.2337 0.9300.8 2.9 107 �0.5266 4.9376 �1.8459 0.1567 0.930

Li and Ansari [17] 1.0 3.4 70 �0.5123 4.0279 �1.5844 0.2888 0.9330.8 3.2 103 �0.6329 3.9822 �1.4336 0.2399 0.930

Lu and Hsu [19] 0.8 4.1 67 �0.3064 5.6037 �2.5629 0.3196 0.9381.0 4.7 69 �0.2134 5.5353 �2.6899 0.3942 0.924

Imran and Pantazapolu [21]HSC 0.7 3.3 73 �0.3902 5.3971 �2.2048 0.2001 0.963

0.8 3.6 65 �0.2562 5.6917 �2.3667 0.1727 0.965

NSC-Type a 0.9 4.3 47 �0.2491 6.0806 �2.6082 0.1847 0.9631.0 4.3 43 �0.1276 6.4589 �2.6639 0.0742 0.967

NSC-Type b 0.7 4.0 29 �0.3867 6.1726 �2.5720 0.1621 0.9641.0 4.8 21 �0.2085 6.5993 �2.8286 0.1379 0.965

Candappa et al. [22] 0.3 2.5 42 �1.1182 6.6078 �2.3470 0.0404 0.9650.2 1.9 61 �1.6143 7.0297 �2.3348 0.0032 0.9870.2 1.9 73 �1.8778 7.0558 �2.2762 0.0064 0.9700.1 1.6 103 �2.5133 7.4437 �2.2599 0.0021 0.984

Average �0.7418 5.6234 �2.1787 0.1789 0.948Standard deviation 0.5910 0.9712 0.3861 0.1064 0.027Minimum �2.5133 3.9822 �2.8286 0.0021 0.876Maximum �0.1276 7.4437 �1.4336 0.3942 0.987

a For the references in which there are no results for uniaxial tension and biaxial compression data, the uniaxial tension corresponding to f 0t ¼ 0:1 f 0c formula werecalculated and also the results taken from He and Song research [37] were employed for biaxial compression.


In this investigation, only the A, B, C and D are reported. Eqs. (2)and (3) define a failure surface with curved meridians and noncir-cular cross sections on the deviatoric planes. The Haigh–Westerg-aard coordinate system and the deviatoric plane are shown inFig. 16. The failure surface passes through the points of uniaxialcompression and uniaxial tension exactly. In Fig. 16.a, the pointqðr1;r2;r3Þ in the stress space is described by the coordinates(n, q, h), in which n is the projection on the hydrostatic axis and(q, h) are polar coordinates in the deviatoric plane, which is orthog-onal to the hydrostatic axis.

A, B, C and D are the four parameters in the failure criteria thatneed to be empirically evaluated through the least square fit of theexperimental data to Eq. (2) or Eq. (3). In Table 9, A, B, C and D coef-ficients of experimental data were compared with those takenfrom literature [15–17,19,21–23]. To evaluate the coefficients, fourstates of stress i.e. uniaxial compression, uniaxial tension, biaxialcompression and triaxial compression should be used.

The regression analysis results in terms of A, B, C and D coeffi-cients are calculated in Table. 9. The failure surface for MK and SFresults are illustrated in Fig. 17. It is obvious that the coefficients

Fig. 17. Failure surface for MK and SF (compressive meridian); Hsieh–Ting–Chenfour-parameter failure criteria.

Fig. 18. Mean shear stress ratio versus mean normal stress ratio (William–Warnkefive-parameter failure criteria).


values related to MK and SF are comparable with HSC specimen val-ues taken from other researches. Furthermore, in comparing withthe existence differences between literatures calibration, it can beconcluded that the difference between MK and SF specimens in thismodel is quite small. Therefore, it seems that the kinds of pozzolanhave a slight effect on determination of model coefficients and auniform compressive meridian can be adopted for both MK and SFdata.

4.5.3. William–Warnke five-parameter failure criteriaThe William–Warnke five-parameter model is widely used for

FRC [19,34,35]. This model proposes compressive meridiansexpressed by quadratic parabolas of the form of the followingequation

sm

f 0c¼ c0 þ c1

rm

f 0c

� �þ c2

rm

f 0c

� �2

ð4Þ

where rm ¼ I13 ¼ 1

3 ðr1 þ r2 þ r3Þ and sm ¼ 1ffiffiffiffi15p ½ðr1 � r2Þ2 þ ðr2�

r3Þ2 þ ðr3 � r2Þ2�1=2.In Eq. (4), rm and sm are mean normal stress and mean shear

stress, respectively. Also r1, r2 and r3 are the stresses in the threeprincipal directions. I1 is the first invariant of the stress tensor. f 0cdenotes the uniaxial compressive strength. C1, C2 and C3 are theWilliam–Warnke model coefficients.

Fig. 18 presents the variation of mean shear stress ratio versusmean normal stress ratio. Table 10 compares the model coeffi-cients of HPFRC with existing HSC results in the literatures. In Eq.(4), the absolute value of C2 coefficient represents the curve con-vexity and hence is the most important coefficient in this modelto be discussed. According to Eq. (4), if the value of C2 coefficientis obtained zero, the William–Warnke model will change toMohr–Coulomb model and, moreover, the parabolic relation be-tween sm and rm will become a linear relation. As shown in Table10, absolute values of 0.164 and 0.517 are obtained for MK and SFas C2 coefficient, respectively. Therefore, it is obvious that the spec-imens behave with a non-linear and parabolic relationship undertriaxial compression and sm is a parabolic function of rm. It shouldbe noted that the obtained value of C2 coefficient for MK is placednear the average value of literature data while for SF, it is placednear the minimum value of literature data. This difference mayexist due to the use of different pozzolans in HPFRC which greatlyaffects the behaviour of the concrete. It can be concluded that the

William–Warnke model can state these differences more signifi-cantly in compare to Mohr–Coulomb and Hsieh–Ting–Chenmodels.

4.5.4. Suitability analysis and comparison of failure criteriaIn statistics, the coefficient of determination (R2) is used in the

context of statistical models whose main purpose is the predictionof future outcomes on the basis of other related information. It isthe proportion of variability in a data set that is accounted forthe statistical model. It provides a measure of how well future out-comes are likely to be predicted by the model. There are severaldifferent definitions of R2. The most general definition of the coef-ficient of determination is:

R2 ¼ 1� ER ð5Þ

In which,

ER ¼ SSerr

SStot

where SStot and SSerr are total sum of squares and residual sum ofsquares, respectively. Moreover, ER is the proportional value ofresidual sum of squares divided by total sum of squares. ER, also,represents the amount of error between available data and modelledvalue (value predicted by the model). Higher value for ER shows lessaccuracy of proposed model and lower value proofs that the model ismore accurate. In other words, higher value of R2 shows suitable pro-posed model which can predict the behaviour with higher accuracy.Coefficient of determination (R2) and amount of error (ER) togetherwith minimum, maximum and average values for each failure crite-ria are respectively indicated in Table 11 and Fig. 19. Complete valuesof R2 have been calculated in Tables 8–10 for each data set corre-sponding to related failure criterion.

As shown in Table 11, highest values of R2 are corresponded toWilliam–Warnke model which shows that this model is more suit-able for predicting concrete behaviour. According to Fig. 19, more-over, smaller amounts of errors are obtained for William–Warnkemodel (less than 1%; including MK, SF and literature data set)which shows that William–Warnke model is fairly acceptable notonly for ordinary strength concrete, but also for high strengthconcrete and HPFRCs. Moreover, the differences between the

Table 10Empirical coefficients for William–Warnke model.

Specimen Max r3f 0c

� �Max r1

f 0c

� �f 0c C2 C1 C0 R2

MK 0.2 2.0 95 �0.164 0.754 0.129 0.995

SF 0.2 2.1 87 �0.517 1.193 0.025 0.997

Nielsen [23] 0.4 3.0 164 �0.280 1.032 0.054 0.9990.9 4.1 165 �0.106 0.721 0.152 0.993

Xie et al. [15] 0.5 3.2 60 �0.109 0.771 0.126 0.9970.5 3.2 92 �0.187 0.902 0.088 0.9980.5 3.1 119 �0.220 0.921 0.075 0.993

Ansari and Li [16] 0.9 3.5 47 �0.068 0.695 0.142 0.9980.9 3.2 71 �0.111 0.734 0.137 0.9990.8 2.9 107 �0.107 0.719 0.140 0.998

Li and Ansari [17] 1.0 3.4 70 �0.099 0.717 0.148 0.9990.8 3.2 103 �0.141 0.816 0.115 0.998

Lu and Hsu [19] 0.8 4.1 67 �0.054 0.638 0.164 0.9981.0 4.7 69 �0.055 0.653 0.160 0.999

Imran and Pantazapolu [21]HSC 0.7 3.3 73 �0.085 0.617 0.182 0.997

0.8 3.6 65 �0.043 0.544 0.200 0.998

NSC-Type a 0.9 4.3 47 �0.036 0.597 0.173 0.9991.0 4.3 43 �0.054 0.618 0.145 0.999

NSC-Type b 0.7 4.0 29 �0.035 0.636 0.153 0.9991.0 4.8 21 �0.013 0.572 0.172 0.999

Candappa et al. [22] 0.3 2.5 42 �0.099 0.769 0.120 0.9990.2 1.9 61 �0.013 0.610 0.162 0.9990.2 1.9 73 �0.384 1.062 0.053 0.9990.1 1.6 103 �0.519 1.162 0.035 0.999

Average �0.146 0.769 0.127 0.998Standard deviation 0.144 0.185 0.048 0.002Minimum �0.519 0.544 0.025 0.993Maximum �0.013 1.193 0.200 0.999

Table 11Coefficient of determination (R2) together with minimum, maximum and averagevalues for each failure criteria, calculated on HPFRCs results and literatures data.

Coefficient ofdetermination (%)

Mohr–Coulomb

Hsieh–Ting–Chen

William–Warnke

MK 98.6 96.7 99.5SF 94.9 96.0 99.7Minimum 94.0 87.6 99.3Average 98.1 94.8 99.8Maximum 99.9 98.7 99.9


maximum and minimum amount of error for William–Warnkemodel is in a very small range.

For Mohr–Coulomb model, obtained ER values are almost in alarge range starting from 0.1% amount of error to 6.0%. This showsthat in some cases, Mohr–Coulomb model could have good accu-racy to predict concrete behaviour. In these cases, the behaviourof concrete is similar to linear behaviour and this is the reason thatMohr–Coulomb model (linear model) can strongly adopted in suchsituation because of its simplicity. However, in some other cases,Mohr–Coulomb model could not accurately simulate materialbehaviour because of ignoring non-linear behaviour of materials.As an example, simulating behaviours of HPFRCs by Mohr–Cou-lomb model showed same internal-friction angle and almost samecohesion coefficient for both MK and SF, while the amount of errorfor SF is more than 5%.

Comparing calculated ER for MK and SF data in Mohr–Coulombmodel indicates that HPFRC with metakaolin pozzolan acts moresimilar to materials with linear behaviour than HPFRC with silicafume. This evidence also can be seen in William–Warnke model

(Fig. 18) where the value of C2 is more close to zero for MK datathan SF. However, the value of C2 is not zero and it shows thatHPFRC with metakaolin pozzolan does not behave purely linear.It should be noted that for MK, the amount of error in William–Warnke model (0.5%) is even less than the amount of error inMohr–Coulomb model (1.4%).

For Hsieh–Ting–Chen model, obtained ER values are in a verylarge range starting from 1.3% amount of error to 12.4%. This highamount of error could be attributed to consideration of tensilebehaviour of concrete which causes more errors to predict concretebehaviour.

5. Conclusions

In this paper, the stress–strain responses of HPFRC with 2% vol-ume fraction of steel fibres containing two types of pozzolans i.e.high reactivity metakaolin and silica fume subjected to triaxial com-pression and uniaxial direct tension are investigated via a compre-hensive experimental program. However, it seems that in order toinvestigate the effect of other parameters such as fibre content, poz-zolan content, and specimen size. on triaxial behaviour of HPFRC,more researches could be conducted. From the experimental resultsand related studies the following conclusion can be drawn:

� The triaxial mechanical properties of HPFRC directly depends onthree significant parameters i.e. the steel fibre volume, confin-ing pressure and pozzolan type. The increase of confining pres-sure leads to ductile behaviour in HPFRC and consequently itssoftening behaviour changes to plastic and hardening behav-iour. However, this effect is slight as the confining pressurereaches values over 15 MPa. For HPFRC, brittleness of the matrix

Fig. 19. Amount of errors (ER) together with minimum, maximum and average values for each failure criteria, calculated on HPFRCs results and literatures data.


itself can notably affect the toughness and ductility behaviour.Under multiaxial loading conditions, addition of silica fume tothe matrix increases the brittleness of the matrix and has anadverse effect on the ductility, energy absorption and toughnessperformance of the HPFRC. However, in the same case, additionof high reactivity metakaolin to the matrix increases the ductil-ity of the matrix. Therefore, high reactivity metakaolin due toproviding more ductile properties in matrix may supply a betterstress transfer between fibre and matrix and also can activatethe most pull out capacity of fibre. On the other hand, silicafume due to providing brittle behaviour in matrix decreasesactivation of whole pull out capacity of fibre. This also can bedrawn from uniaxial direct tensile test results.� Both confining pressure and pozzolan type have good effect on

the toughness indices and they can raise the toughness ofHPFRC up to 4 times.� Applying Mohr–Coulomb, Hsieh–Ting–Chen and William–

Warnke failure criteria for experimental data show that:1. Adding pozzolan has insignificant effect on related Mohr–

Coulomb coefficients and the internal friction angle isobtained 38� for both types of HPFRC. Moreover, calculatedvalues for a and b coefficients are in good agreement withthose obtained from literature for HSC specimens.

2. For Hsieh–Ting–Chen failure criteria, a uniform compressivemeridian can be acceptable for both types of HPFRC.

3. The results taken from William–Warnke failure criteriashow fairly appropriate differences between types ofHPFRCs. It seems that the William–Warnke model can statethe differences between pozzolans more significantly incomparison to Mohr–Coulomb and Hsieh–Ting–Chen mod-els. Therefore, it is proposed to calibrate HPFRCs with differ-ent types of materials. In addition, the calculated amount oferror for different types of concrete i.e. ordinary strengthconcrete, high strength concretes and HPFRCs shows suit-ability of William–Warnke failure criteria for modelling ofthese concretes.

Acknowledgments

The authors would like to thank Dr. M. Moosavi for his help inthe Rock Mechanic Lab of School of Mining Engineering, University

of Tehran, Iran, for supporting the mechanical tests and Prof. Rahi-mian, Prof. Mirgaderi, and Dr. Rahjoo, University of Tehran, Iran, foruseful scientific assistance. Also, the authors would like to expressspecial thanks to Prof. Naaman, University of Michigan, Ann Arbor,USA, and deeply appreciate his valuable review of the manuscript.

References

[1] Bonakdar A, Bakhshi M, Ghalibafian M. Properties of high-performanceconcrete containing high reactivity metakaolin. In: Proceedings of theseventh international symposium on the utilization of high-strength/high-performance concrete. ACI-SP-228, ACI, Washington, DC; 2005. p. 287–96.

[2] Shekarchi M, Bonakdar A, Bakhshi M, Mirdamadi A, Mobasher B. Transportproperties in metakaolin blended concrete. Constr Build Mater2010;24(11):2217–23.

[3] Farnam Y, Mohammadi S, Shekarchi M. Experimental and numericalinvestigations of low velocity impact behavior of high-performance fiber-reinforced cement based composite. Int J Impact Eng 2010;37(2):220–9.

[4] Kim HS, Lee SH, Moon HY. Strength properties and durability aspects of highstrength concrete using Korean metakaolin. Constr Build Mater2007;21:1229–37.

[5] Gutierrez RM, Diaz LN, Delvasto S. Effect of pozzolans on the performance offibre-reinforced mortars. Cem Concr Compos 2005;27:593–8.

[6] Dubey A, Banthia N. Influence of high-reactivity metakaolin and silica fume onthe flexural toughness of high-performence steel fibre-reinforced concrete. ACIMater J 1998;95(3):284–92.

[7] Poon CS, Kou SC, Lam L. Compressive strength, chloride diffusivity and porestructure on high performance metakaolin and silica fume concrete. ConstrBuild Mater 2006;20:858–65.

[8] Ding JT, Li Z. Effect of metakaolin and silica fume on properties of concrete. ACIMater J 2002;99(4):393–8.

[9] Bindiganavile V, Banthia N. Fibre reinforced dry-mix shotcrete withmetakaolin. Cem Concr Compos 2001;23:503–14.

[10] Banthia N, Yan C. Bond-slip characteristics of steel fibres in high reactivitymetakaolin modified cement-based matrices. Cem Concr Res1996;26(5):657–62.

[11] Curcio F, DeAngelis BA, Pagliolico S. Metakaolin as a pozzolanic microfiller forhigh-performance mortars. Cem Concr Res 1998;28(6):803–9.

[12] Wong HS, Razak AH. Efficiency of calcined kaolin and silica fume as cementreplacement material for strength performance. Cem Concr Res2005;35:696–702.

[13] Tafraoui A, Escadeillas G, Lebaili S, Vidal T. Metakaolin in the formulation ofUHPC. Constr Build Mater 2009;23(2):669–74.

[14] Chern JC, Yang HJ, Chen HW. Behaviour of steel fibre reinforced concrete inmultiaxial loading. ACI Mater J 1992;89(1):32–40.

[15] Xie J, Elwi AE, Mac Gregor JG. Mechanical properties of three high-strengthconcretes containing silica fume. ACI Mater J 1995;92(2):1–11.

[16] Ansari F, Li Q. High-strength concrete subjected to triaxial compression. ACIMater J 1998;95(6):747–55.

[17] Li Q, Ansari F. High-strength concrete in triaxial compression by different sizesof specimens. ACI Mater J 2000;97(6):684–9.

[18] Li Q, Ansari F. High-strength concrete in uniaxial tension. ACI Mater J2000;97(1):49–55.


[19] Lu X, Hsu CTT. Behaviour of high strength concrete with and without steelfibre reinforcement in triaxial compression. Cem Concr Res 2006;36:1679–85.

[20] Sfer D, Carol I, Gettu R, Etse G. Study of the behaviour of concrete under triaxialcompression. J Eng Mech 2002;128(2):156–63.

[21] Imran I, Pantazopoulou SJ. Experimental study of plain concrete under triaxialstress. ACI Mater J 1996;93(6):589–601.

[22] Candappa DC, Sanjayan JG, Setunge S. Complete triaxial stress strain curves ofhigh-strength concrete. J Mater Civ Eng 2001;13(3):209–15.

[23] Nielsen CV. Triaxial behaviour of high-strength concrete and mortar. ACIMater J 1998;95(2):144–51.

[24] Rock and Concrete Mechanics Testing System. Technical Description. MTSSystems Corp. Minneapolis: Minn; 1994.

[25] ASTM C469. Standard test method for static modulus of elasticity andpoisson’s ratio of concrete in compression. American Society for TestingMaterials. West Conshohocken: Annual book of ASTM Standards; 2002.

[26] ASTM C801. Standard test method for determining the mechanical propertiesof hardened concrete under triaxial loads. American Society for TestingMaterials. West Conshohocken: Annual book of ASTM Standards; 2004.

[27] Hoek E, Brown ET. Underground excavations in rock. England: Institution ofMining and Metallurgy; 1980.

[28] Banthia N. A study of some factors affecting the fibre–matrix bond in steelfibre reinforced concrete. Can J Civ Eng 1990;17:611–9.

[29] Balaguru P, Narahari R, Patel M. Flexural toughness of steel fibre reinforcedconcrete. ACI Mater J 1992;89(6):541–6.

[30] Fanella DA, Naaman AE. Stress-strain properties of fibre reinforced mortar incompression. ACI J 1985;82(4):475–83.

[31] Nataraja MC, Dhang N, Gupta AP. Stress-strain curves for steel fibre reinforcedconcrete under compression. Cem Concr Compos 1999;21:383–90.

[32] Poon CS, Shui ZH, Lam L. Compressive behaviour of fibre reinforced high-performance concrete subjected to elevated temperatures. Cem Concr Res2004;34:2215–22.

[33] ASTM C1018. Standard test method for flexural toughness and first-crackstrength of fibre-reinforced concrete (using beam with third-point loading).American Society for Testing Materials. West Conshohocken: Annual book ofASTM Standards; 2005.

[34] Chen WF, Han DJ. Plasticity for structural engineering. Springer-Verlag Pub.;1988.

[35] Chen WF. Plasticity in reinforced concrete. New York, USA: McGraw-Hill BookCompany; 1982.

[36] Hsieh SS, Ting EC, Chen WF. Plasticity-fracture model for concrete. Int J SolidsStruct 1982;18(3):577–93.

[37] He Z, Song Y. Failure mode and constitutive model of plain high-strength high-performance concrete under biaxial compression after exposure to hightemperatures. Acta Mech Solida Sinica 2008;21(2):149–59.

4-Failure Criteria and Triaxial Behaviour of HPFRC

Documents

Transcript of 4-Failure Criteria and Triaxial Behaviour of HPFRC