Correlation of Compressive Strength and OtherEngineering Properties of High-Performance

Steel Fiber–Reinforced ConcreteRamadoss Perumal, Ph.D.1

Abstract: In this paper, correlations among the compressive, flexural, and splitting tensile strengths of high-performance steel fiber-reinforced concrete (HPSFRC) were analyzed. For the investigation, a large amount of data was collected from the published papers withwater/binder ratio in the range of 0.25–0.48, steel fiber volume fraction of 0.5–2.0% with aspect ratio of 40–80, and specimens of 150 mm Øcylinders and 100 × 100 × 500 mm size prisms. Results on the evaluation of the published empirical relations using the collected data in-dicate a large variation and inapplicability to steel fiber reinforced concrete, which necessitates the determination of empirical relationsbetween the mechanical properties of SFRC. Through a statistical analysis on the data collected, power relations for flexural and splittingtensile strengths as a function of compressive strength, and the relation between flexural and splitting tensile strengths of HPSFRC withr ¼ 0.94, were obtained. The validity of the proposed models was examined with the experimental data of the present study and earlierresearcher, and the integral absolute error obtained is 5.1. DOI: 10.1061/(ASCE)MT.1943-5533.0001050.© 2014 American Society of CivilEngineers.

Author keywords: High-performance steel fiber reinforced concrete; Engineering properties; Correlation equation; Prediction; Validation.

Introduction

Concrete fiber composites have been found more economical foruse in various applications in the construction industry. Earlierinvestigators [ACI 544 (ACI 2006); Ahmed and Shah 1985; Wafaand Ashour 1992; Banthia and Trottier 1995a, b; Bindiganavile andBanthia 2001; Balendran et al. 2002] have reported that theaddition of steel fibers to the concrete matrix improves the engi-neering properties of concrete, especially tensile strength, impactstrength, fracture toughness, energy absorption capacity, anddurability. Numerical works pertaining to experimental andanalytical methods for determining strength properties of steelfiber-reinforced concrete (SFRC) have been reported, with consid-eration of various influencing parameters, and it is well acceptedthat steel fibers can improve the mechanical properties andbehavior of concrete (Wafa and Ashour 1992; Banthia and Trottier1995a, b; Bindiganavile and Banthia 2001; Balendran et al.2002; Ezeldin and Balaguru 1989; Banthia 1990; Sing and Kaushik2001; Mohammadi and Kaushik 2003; Song and Hwang 2004;Banthia and Sol-Imani 2005; Dhonde et al. 2007; Thomas andRamasamy 2007).

The two important indices used for evaluating the mechanicalproperties of concrete are compressive strength and indirect tensilestrength (flexural and splitting tensile). Flexural and splitting tensilestrengths can be estimated from the compressive strength of concretethrough various empirical relations as proposed by the differentconcrete institutes and researchers [Ahmed and Shah 1985; Wafaand Ashour 1992; ACI 363 (ACI 2004); ACI 318 (ACI 2004);CEB-FIP 1990; Nilson 1987; Raphael 1984; Oluokun 1991;

Rashid et al. 2002; Zam and Mahmud 2002; Atis and Taurikulu2003; Hueste et al. 2004; Choi and Yuan 2005; Bhanja and Sengupta2005; Arioglu et al. 2006; Thomas and Ramasamy 2007; Ramadoss2008; Ramadoss et al. 2009; Xu and Shi 2009] and those empiricalrelations can be summarized by the following general equation:

f ¼ Aðf 0cÞB ð1Þ

where f = flexural tensile strength/ splitting tensile strength (MPa);f 0c = cylinder compressive strength (MPa); and A and B = regression

coefficients.It was observed from the literature that most of the published

empirical expressions were for normal strength/high-strength con-cretes. Only a few researchers by using their own experimental testdata have expressed empirical power relations for evaluating theflexural strength (modulus of rupture)/splitting tensile strengthas a function of compressive strength of SFRC, and the relationbetween flexural strength and splitting tensile strength of SFRC.Ramadoss (2008) reported that flexural strength of HPFRC canbe nonlinearly related to splitting tensile strength using a powerrelation. Xu and Shi (2009) have suggested that flexural strengthof SFRC is nonlinearly correlated to splitting tensile strength(frf ¼ 1.63f0.85spf ). Nataraja et al. (2001) have suggested the linearrelation between the flexural and splitting tensile strengths(frf ¼ 1.49fspf ), and between the splitting tensile and compressivestrengths (fspf ¼ 0.09f 0

cf) for normal strength SFRC. Ramadossand Nagamani (2006) have suggested the linear relation betweenthe flexural and splitting tensile strengths (frf ¼ 1.321fspf ) ofHPFRC. Since correlation equations among the mechanical proper-ties of SFRC and high-performance steel fiber concrete (HPSFRC)using a database containing a large number of data sets are notreported and unclear, investigation into the correlations of compres-sive strength, flexural strength, and splitting tensile strength ofHPSFRC/SFRC by collecting a large number of data sets isneeded. An evaluation of the published empirical relations of theseproperties indicates a large variation in predictions and their

1Associate Professor, Civil Engineering, Pondicherry EngineeringCollege, Puducherry 605014, India. E-mail: dosspr@pec.edu

Note. This manuscript was submitted on September 17, 2013; approvedon February 5, 2014; published online on July 10, 2014. Discussion periodopen until December 10, 2014; separate discussions must be submitted forindividual papers. This paper is part of the Journal of Materials in CivilEngineering, © ASCE, ISSN 0899-1561/04014114(8)/$25.00.

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inapplicability to SFRC, which confirms the necessity of develop-ing the empirical relations among the mechanical properties.

Research Significance

An evaluation of the published relations on mechanical propertiesof SFRC reveals a large variation in predictions and their inappli-cability to SFRC. Correlation relations (equations) between themechanical properties of normal strength SFRC/high strengthSFRC/HPSFRC using a database are not reported by earlierresearchers, meaning these aspects are unclear and informationis lacking. Therefore, large experimental data sets (920 data points)were statistically analyzed in order to develop the empiricalrelations among the mechanical properties of SFRC/HPSFRC.The models were also tested and validated for their performanceand suitability with the test results obtained from the presentinvestigation and the experimental values of previous researcher.

Database on Engineering Properties of SFRC

Earlier researchers (Soroushian and Bayasi 1991; Mansur et al.1994; Day 1994; Zhou et al. 1998; Trottier and Banthia 1994;Shaaban and Gesund 1993; Gao et al. 1997; Yan et al. 1999;Khaloo and Kim 1996; Yao et al. 2003; Balaguru and Najim2004; Yazici and Sezer 2007; Sivakumar and Santhanam 2007;Koksal et al. 2008; Ismail and Riza 2011) have reported that themechanical properties of fiber-reinforced concrete are affected bymany factors, including specimen geometry, water-binder (w/b)ratio, curing method and time, type and grade of cement,cement replacement materials, fiber geometry, aspect ratio, fiberdosage, super plasticizer, etc. Since limited information of theseaffecting factors is available in the literature, Table 1 presents generalinformation for those available. Test results for the engineeringproperties are based on 28-day strengths of concrete since28-day strengths are only considered in the design of structuralcomponents. The design procedure contained in various standardsand codes of practice [Canadian Standard Association 1994;ACI 318 (ACI 2004); CEB-FIP 1990] for structural design are basedon 150Ø × 300 mm cylinder concrete strength, and hence strengthin specifications related to that of 150 mm Ø cylinders. Crimped,hooked end, and twin cone steel fibers were mostly used by theresearchers in the investigation, which may be due to the significanteffect on fiber-matrix bond/fiber pullout effect in concrete (Ezeldinand Balaguru 1989; Banthia 1990; Banthia and Trottier 1995a, b).Most researchers chose fiber volume fraction and aspect ratio in therange of 0.5–1.5% and 40–80, respectively, and therefore, the samerange was chosen in this investigation. Test data used in this studywas obtained considering the rest results of 150 mm (5.91 in.) diam-eter cylinder and 100 × 100 × 500 mm (3.94 × 3.94 × 19.69 in:)size prism specimens, since the empirical expressions available

for normal strength or HSC to convert the test results (mechanicalproperties) of different geometry of specimens to the required size ofspecimen. But the available empirical relations for conversion cannotbe applicable for SFRC. Therefore, to avoid large errors in the con-verted equivalent values, test data (data points) are confined to thetest results of standard specimens.

For the purpose of investigation, database containing a largenumber of data sets (380þ 249þ 291 ¼ 920 data points) has beencreated from the published literature on SFRC/HSSFRC/HPSFRChaving two or three mechanical properties reported at a time. Thew/b ratio was taken in the range of 0.25–0.48, the steel fiberdosage in the range of 0–2%, with a fiber aspect ratio of 40–80.A significant volume of test data on tensile strengths (both flexuraland splitting) of concrete has been collected from the existingliterature, which includes the experimental test data of presentstudy. Experimental data collected in this investigation are basedon the tests on 150 mm Ø cylinders and 100 × 100 × 500 mm sizeprism specimens.

Assessment on the Applicability of ExistingEmpirical Relations to SFRC

Already-published empirical expressions for flexural strength(modulus of rupture) as a function of compressive strength of nor-mal strength concrete (NSC)/HSC, and splitting tensile strength asa function of compressive strength of NSC/HSC are presented inTables 2 and 3, respectively. The empirical relations presented inTables 2 and 3 predict the strengths (flexural and splitting tensile)more accurately to the test data of compressive strength within25 MPa. Above this value (range: 25–120 MPa), the predictiondeviates more significantly from the experimental data, whichindicates a large variation in strength assessments. This largevariation is due to significant improvements in tensile strengths(flexural and splitting tensile) in the SFRC. This improvementin tensile strengths is attributed to the addition of discrete steelfibers in the concrete matrix, in which the reinforcement mecha-nism is mainly composed of fiber-matrix bond/fiber pullout effect.Therefore, with the increase of the concrete’s compressive strength,fibers that increase the tensile strength will become more effectivedue to the fiber-matrix bond/fiber pullout effect, but fibers haslittle effect in increasing compressive strength. This finding wasreported by earlier researchers (Banthia and Trottier 1995a, b;Balendran et al. 2002; Ezeldin and Balaguru 1989; Banthia1990; Mohammadi and Kaushik 2003; Song and Hwang 2004;Banthia and Sol-Imani 2005; Ramadoss 2008; Shaaban andGesund 1993; Yan et al. 1999) and also evidenced in the presentinvestigation. In order to assess the performance of various empiri-cal expressions of the earlier researchers/institutes, the deviationbetween the test data points and the predictions are evaluated byemploying the statistical parameters of integral absolute error(IAE) and absolute percent variation.

Table 1. Database of Components for Steel Fiber-Reinforced Concrete

Componentw=cm (or)w/b ratio

SF/PFA(%) Steel fiber

Volumefraction (%)

Aspectratio (l=d) 28 day-strength (MPa) Specimen

Range 0.25–0.48 0–20 crimped (corrugated),hooked end, straight,

twin cone

0–2 40–80 Compressive: 30-120Flexural: 3-20 Split

tensile: 3-16.5

For compressiveand splitting tensile: 150Ø

cylinder. for flexure:100 × 100 × 500 mm

size prism

Note: SF = condensed silica fume; PFA = fly ash.

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Table 2 presents the empirical relations between flexuralstrength and compressive strength of normal strength/high-strengthconcretes. The comparison of evaluation of predictions of the em-pirical relations reported by the earlier researchers/ institutes isgiven in Table 2. The IAE values of empirical relations are all above20% (average above 25%) except that for SFRC found by Xuand Shi (2009), which indicates the inapplicability of theseempirical relations to SFRC. Similarly, Table 3 presents the empir-ical relations between splitting tensile strength and compressivestrength of normal strength/high-strength concretes. The compari-son of evaluation of predictions of the empirical relations reportedby the earlier researchers is given in Table 3. The IAE values ofempirical relations are all above 25% except that for SFRC foundby Xu and Shi (2009), which again indicates the inapplicability ofthese empirical relations to SFRC. It was observed from thepredicted values of the previous empirical relations that as the com-pressive strength of SFRC increased, the more the experimentaldata points deviate from the prediction values. The IAE valuesof these empirical expressions given in Tables 2 and 3, respectively,

are significantly large, all being above 25%. Obviously, this may beattributed to the different development trends of the ratios of themechanical properties of normal strength/medium strength/high-strength concretes, and steel fiber reinforced concretes.

Experimental Details

Materials and Mix Proportions

Ordinary portland cement-53 grade having 28-day compressivestrength of 54.5 MPa, and silica fume containing 88.7% ofSiO2, having a fineness by surface area of 23000 m2=kg, and aspecific gravity of 2.25 were used. Fine aggregate (river sand)conforming to grading zone-II, having a specific gravity of 2.63,and coarse aggregate (crushed granite stone) with maximum sizeof 12.5 mm (0.492 in.) having a specific gravity of 2.70 wereused. Superplasticizer of sulphonated naphthalene formaldehydecondensate conforming to ASTM Type F (ASTM C494) was used.

Table 2. Empirical Relation between Flexural Strength and Compressive Strength of High-Strength Concrete and SFRC and Corresponding IAE andAbsolute Variation (%)

Code of practice/ researcher Empirical relation

Predicted strength

Absolute variation Standard deviation IAE

ACI-363R-92 (2004) fr ¼ 0.94ðf 0cÞ0.5 for f 0

c < 85 MPa 18.68 1.70 20.80ACI-318-95 (2004) fr ¼ 0.62ðf 0

cÞ0.5 for f 0c < 83 MPa 38.45 2.26 43.08

Ahmed and Shah (1985) fr ¼ 0.44ðf 0cÞ0.67 for f 0

c < 84 MPa 19.20 1.59 21.52Nilson (1987) fr ¼ 0.90

ffiffiffiffiffif 0c

p19.30 1.71 21.75

Oluokun et al. (1991) fr ¼ 0.79ffiffiffiffiffif 0c

p24.58 2.03 28.78

Wafa and Ashour (1992) fr ¼ 1.03ffiffiffiffiffif 0c

p17.49 1.50 18.23

Rashid et al. (2002) fr ¼ 0.42f 00.68c for 5 < f 0

c < 120 MPa 19.44 1.60 21.77Atis and Taurikulu (2003) fr ¼ 0.405f 00.69

c 19.07 1.61 21.11Hueste et al. (2004) fr ¼ 0.83

ffiffiffiffiffif 0c

pfor 40 < f 0

c < 90 MPa 22.53 1.90 25.93Bhanja and Sengupta (2005) fr ¼ 0.275f 00.81

c 16.24 1.34 17.39Ramadoss (2008) fr ¼ 0.45f 00.67

c 18.33 1.53 20.36Bakhsh et al. (1990) fr ¼ 0.80f 00.5

c 23.92 2.00 27.16Thomas and Ramasamy (2007) fr ¼ 0.79

ffiffiffiffiffif 0c

pfor 30 < f 0

c < 75 MPa 24.58 2.03 27.87Xu and Shi (2009) frf ¼ 0.39ðf 0

cfÞ0.75 15.96 1.06 14.75

Table 3. Empirical Relation between Splitting Tensile Strength and Compressive Strength of High-Strength Concrete and SFRC and Corresponding IAE andAbsolute Variation (%)

Code of practice/researcher Empirical relation

Predicted strength

Absolute variation Standard deviation IAE

ACI-363R-92 (2004) fsp ¼ 0.59ðf 0cÞ0.5 for f 0

c < 85 MPa 23.91 1.24 27.05ACI-318-95 (2004) fsp ¼ 0.56ðf 0

cÞ0.5 for f 0c < 83 MPa 26.49 1.33 30.04

CEB-FIP (1990) fr ¼ 1.4ðf 0c=10Þ0.67 21.66 1.10 24.17

Ahmed and Shah (1985) fsp ¼ 0.46ðf 0cÞ0.55 for f 0

c < 84 MPa 25.81 1.28 29.16Raphael (1984) fsp ¼ 0.31f 00.67

c 19.99 1.04 22.19Nilson (1987) fspf ¼ 0.62ðf 0

cfÞ0.5 19.37 1.03 21.06Oluokun et al. (1991) fspf ¼ 0.21ðf 0

cfÞ0.55 48.06 1.45 50.41Wafa and Ashour (1992) fsp ¼ 0.58

ffiffiffiffiffif 0c

p24.72 1.27 28.02

Rashid et al. (2002) fsp ¼ 0.47ðf 0cÞ0.56 for 5 < f 0

c < 120 MPa 22.58 1.17 25.40Hueste et al. (2004) fsp ¼ 0.55

ffiffiffiffiffif 0c

pfor 40 < f 0

c < 90 MPa 27.52 1.35 31.14Bhanja and Sengupta (2005) fsp ¼ 0.248f 00.717

c 21.35 1.06 23.57Choi and Yuan (2005) fsp ¼ 0.60

ffiffiffiffiffif 0c

p23.13 1.21 26.10

Arioglu et al. (2006) fsp ¼ 0.39f 00.55c 35.78 1.43 39.28

Thomas and Ramasamy (2007) fsp ¼ 0.57ffiffiffiffiffif 0c

pfor 30 < f 0

c < 75 MPa 25.58 1.30 29.01Ramadoss (2008) fsp ¼ 0.485f 00.56

c 21.08 1.11 23.58Bakhsh et al. (1990) fr ¼ 0.62f 00.5

c 21.70 1.15 24.30Xu and Shi (2009) fspf ¼ 0.21ðf 0

cfÞ0.83 15.10 0.60 13.21

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Steel fibers (crimped type) conforming to ASTM A820-2001were used. They had the following physical properties:length ¼ 36 mm (1.417 in.); diameter ¼ 0.45 mm (0.018 in.); as-pect ratio ðl=dÞ ¼ 80; ultimate tensile strength ðfuÞ ¼ 910 MPa;and Esf ¼ 210 GPa.

Mixtures were proportioned in accordance with the recom-mended guidelines of the American Concrete Institute. Mixtureproportions used in this test program are summarized in Table 4.For each water-binder ratio (w/b), two silica fume concrete [high-performance concrete (HPC)] mixes and six fibrous concrete mixeswith fiber volume fractions, Vf ¼ 0.5, 1.0 and 1.5% by volume ofconcrete [39, 78 and 117.5 kg=m3, (1 kg=m3 ¼ 1.684 lb=yd3) re-spectively] were prepared. Specimens were cast and water cureduntil the age of testing at 28 days.

Compressive Strength Test

Compressive strength tests were carried out according toASTM C 39-1992 standards using 150 diameter × 300 mm(5.91 × 11.82 in:) cylinders loaded uniaxially. The tests were donein a servo-controlled compression testing machine. A minimum ofthree specimens were tested to assess average strength. A databaseconsisting of test results on 96 concrete specimens was used fortesting the prediction models developed.

Tensile Strength Test

Flexural strength (modulus of rupture) tests were conducted as perthe specification of ASTM C78 (ASTM 1994) using 100 × 100 ×500 mm (3.94 × 3.94 × 19.69 in:) prisms under third-point load-ing on a simply supported span of 400 mm (15.748 in.). The testswere conducted in a 100 kN (22.48 kpi) hydraulically operated uni-versal testing machine (UTM). Samples were tested at a deforma-tion rate of 0.1 mm=min. Three samples were used for computingthe average strength.

Splitting tensile strength tests were conducted according to thespecification of ASTM C 496-1990 using 150Ø × 300 mm(5.91 × 11.82 in:) cylindrical specimens. The tests were conductedin a 1000 kN (22.48 kpi) hydraulically operated UTM. Three sampleswere used for computing the average strength.

Analysis Results and Discussion

The 28-day compressive strength (fcf) obtained for concrete cyl-inder specimens on the effect of fiber content in terms of the fiber-reinforcing index is presented in Table 4. Test results show that theaddition of steel fiber (vf ¼ 0.5 to 1.5%) to the HPC matrix (silicafume concrete) increases the compressive strength to a maximum of12.4% at 1.5% volume fraction (RI ¼ 3.88) of concrete. Table 4presents the variations of flexural strength (frf) (modulus of rup-ture) and splitting tensile strength (fspf ) on the effect of fiber con-tent in terms of fiber reinforcing index (RI). It is observed from thetest results (refer Table 4) that there is a significant improvement inmodulus of rupture (varying in the range of 16–38% compared toHPC) due to increase in fiber content from 0 to 1.5% (RI ¼ 0 to3.88) in the concrete matrix with a maximum increase in strength of38% at 1.5% volume fraction of concrete. It is observed from thetest results (Table 4) that there is a significant improvement in split-ting tensile strength (varying in the range of 24–56% compared toHPC) due to increase in fiber content from 0 to 1.5% (RI ¼ 0 to3.88) in the concrete matrix with a maximum increase in strength of56% at 1.5% volume fraction of concrete.

Relation between Flexural Strength andCompressive Strength

Using the experimental data sets collected, the relation betweenflexural strength (modulus of rupture) and compressive strengthof HPSFRC has been obtained by performing the statisticalanalysis, is shown in Fig. 1. The empirical (correlation) equationobtained for flexural strength of HPSFRC with correlation coeffi-cient, r ¼ 0.90 is given as

frf ¼ 0.259f0.843cf ð2Þ

The average absolute variation, standard error of the estimate(s), and standard deviation (σ) for the estimated flexural strengthwere obtained as 15.18%, 1.501 and 1.04, respectively, which in-dicate higher accuracy in the relationship obtained. In order to fur-ther evaluate the deviation between experimental data points andpredicted values, integral absolute error (IAE) is assessed, whichis written as

IAE ¼ ΣðQ − PÞΣQ

× 100% ð3Þ

where Q = actual flexural strength/ split tensile strength and P =predicted value of flexural strength/split tensile strength.

Table 4. Mix Proportions for HPSFRC (Data for 1 m3)

Mixdesignation w=cm

C(kg)

FA(kg)

CA(kg)

SF(kg)

W(kg)

SP(kg)

Steel fiber

Vf (%)

FC1-0 0.4 416 691 1088 22 175 7.66 0FC1-0.5 0.4 416 687 1079 22 175 7.66 0.5FC1-1 0.4 416 682 1071 22 175 7.66 1FC1-1.5 0.4 416 678 1062 22 175 7.66 1.5FC1*-0 0.4 394.2 691 1088 43.8 175 7.66 0FC1*-0.5 0.4 394.2 687 1079 43.8 175 7.66 0.5FC1*-1 0.4 394.2 682 1071 43.8 175 7.66 1FC1*-1.5 0.4 394.2 678 1062 43.8 175 7.66 1.5FC2-0 0.35 461.7 664 1088 24.3 170 9.72 0FC2*-0 0.35 437.4 664 1088 48.6 170 9.72 0FC3-0 0.3 522.5 624 1088 27.5 165 13.75 0FC3*-0 0.3 495 624 1088 55 165 13.75 0FC4-0 0.25 608 562 1088 32 160 17.60 0FC4*-0 0.25 576 562 1088 64 160 17.60 0

Note: In mix designation FC1 to FC4, and FC1* to FC4*, silica fumereplacement is 5 and 10%, respectively by weight of cementitiousmaterials (Cm) or binder material. Cm ¼ Cþ SF = cement + silicafume; Vf denotes steel fiber volume fraction in percent in volume ofconcrete; in each w=cm ratio of mix, four fiber volume fractions(Vf ¼ 0, 0.5, 1.0 and 1.5%) are used; SP (%)—Super plasticizer inpercent by weight of binder material.

proposed model

y = 0.252x 0.851

R = 0.90

2

4

6

8

10

12

14

16

18

20

22

20 40 60 80 100 120Compressive strength (MPa)

Fle

xura

l str

eng

th (M

Pa)

Fig. 1. Correlation between flexural strength and compressive strength(MPa) of HPSFRC

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The integral absolute error (IAE) for this model-I [Eq. (2)] isobtained as 14.32. The validity of correlation Eq. (2) was investi-gated by examining relevant statistical coefficients. A comparisonof the IAE values shown in Table 2 with the IAE value of 14.3 forthis proposed model-I suggests higher accuracy and reliability ofthe proposed relation.

Relation between Splitting Tensile Strength andCompressive Strength

The statistical analysis carried out on the data sets collected for therelation between split tensile strength and compressive strength ofHPSFRC is shown in Fig. 2. The empirical equation obtained forsplitting tensile strength of HPSFRC with correlation coefficient,r ¼ 0.92 is given as

fspf ¼ 0.188f0.84cf ð4Þ

The average absolute variation, standard error of the estimate (s)and standard deviation for the estimated splitting tensile strengthwere obtained as 12.62%, 0.96 and 0.552, respectively, which in-dicate higher accuracy in the relationship obtained. The integralabsolute error (IAE) for this model-II [Eq. (4)] is obtained as11.8. The validity of correlation Eq. (4) was investigated by exam-ining relevant statistical coefficients. A comparison of the IAE val-ues shown in Table 3 with the IAE value of 11.8 for this proposedmodel-II suggests higher the accuracy and reliability of the pro-posed relation.

Relation between Flexural Strength and SplittingTensile Strength

The statistical analysis carried out on the data sets collected forthe relation between flexural strength and split tensile strengthHPSFRC is shown in Fig. 3. The empirical equation obtainedfor flexural strength of HPSFRC with correlation coefficient, r ¼0.94 is given as

frf ¼ 1.741f0.88spf ð5Þ

The average absolute variation, standard error of the estimate (s)and standard deviation for the estimated flexural strength were ob-tained as 10.64%, 1.14 and 0.748, respectively, which indicatehigher accuracy in the relationship obtained. The validity of corre-lation Eq. (5) was investigated by examining relevant statistical co-efficients. The integral absolute error (IAE) value of 9.85 obtainedfor this proposed model-III [Eq. (5)], is suggesting higher the ac-curacy and reliability of the proposed relation. This result is in good

agreement with the values obtained by the expression suggestedby Xu and Shi (2009); however, the linear relation reported byNataraja et al. (2001) is not in agreement, and the linear relationreported by Ramadoss and Nagamani (2006) is comparable.

Discussion

Researchers have reported that the mechanical properties of SFRCare affected by many factors, such as specimen geometry, w/b ratio(w=cm ratio), mixing procedure, curing type/time, type/grade ofcement, cement replacement material, size and shape of aggregate,geometry of fiber, volume fraction, aspect ratio, etc. This investi-gation reports on the results of a statistical analysis on collecteddata points regarding the correlation between flexural strength,splitting tensile strength, and compressive strength of HPSFRC/SFRC without considering these affecting factors. However, thestatistical analysis indicates that strong correlations exist betweenthe mechanical properties of SFRC. Figs. 1–3 show the scatter ofthe data sets show the influence of affecting factors as mentionedabove. Further, highly influencing factors as variables can be con-sidered in the nonlinear regression analysis on correlation ofmechanical properties.

Validation of the Proposed Models with theExperimental Data

The proposed prediction models are testified for validation againstthe experimental results of the present investigation and the datafrom earlier researchers. The model’s performance was assessedusing experimental data points of 150 mm diameter cylinder com-pressive and splitting tensile strengths, and flexural strengths ob-tained using 100 × 100 × 500 mm size beams. The absolutepercent variations determined from the predicted values of flexuralstrength, splitting tensile strength, and flexural strength using themodels I, II and III, respectively, are presented in Table 5. The pre-diction models are found to give good correlations with the testresults obtained from the present investigation. A comparison ofexperimental values of flexural strength of the present investigationand the values obtained by Thomas and Ramasamy (2007) to thepredictions of the proposed model-I [Eq. (2)] is shown in Fig. 4.The IAE values and average absolute variation obtained are 7.29and 7.5%, respectively. The standard error of the estimate is0.822. A comparison of experimental values of splitting tensilestrength of the present investigation and the values obtained by

Proposed model

y = 0.235x 0.79

R = 0.92

2

4

6

8

10

12

14

20 40 60 80 100 120Compressive strength (MPa)

Sp

lit te

nsi

le s

tren

gth

(MP

a)

Fig. 2. Correlation between splitting tensile strength and compressivestrength (MPa) of HPSFRC

Proposed model

y = 1.741x0.879

R = 0.94

2

4

6

8

10

12

14

16

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2 4 6 8 10 12 14Splitting tensile strength (MPa)

Fle

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eng

th (

MP

a)

Fig. 3. Correlation between flexural strength and splitting tensilestrength (MPa) of HPSFRC

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Thomas and Ramasamy (2007) to the predictions of the proposedmodel-II [Eq. (4)] is shown in Fig. 5. The average absolutevariation and IAE values obtained are 10.3 and 10.98%, respec-tively. The standard error of the estimate (s) is 0.806. A comparisonof experimental values of flexural strength of the presentinvestigation and the values obtained by Thomas and Ramasamy(2007) to the predictions of the proposed model-III [Eq. (5)] isshown in Fig. 6. The IAE value and average absolute variationobtained are 5.12 and 5.29%, respectively. The standard error ofthe estimate is 0.542. It is observed that the proposed models I,II and III (empirical relations) perform very well with theexperimental data of previous researchers and the experimentalvalues obtained from the the present investigation, and predictthe engineering properties quite accurately. For further testamentto the proposed model for validation, statistical parameters ofaverage absolute variation, standard Deviation, and confidenceinterval at 95% significance value determined with other models,respectively, are given as follows:• Proposed model: 0.705, 0.574, and 0.169;• ACI 318: 1.090, 0.803, and 0.237;• ACI 363: 3.451, 1.078, and 0.318;

• Ahmad and Shah: 1.413, 0.863, and 0.255;• Oluokun: 2.136, 0.998, and 0.295.

In Fig. 7, the different curves drawn for models [ACI committee363 (ACI 2004); ACI committee 318 (ACI 2004); Ahmed and Shah1985; Oluokun 1991] based on the test data of the present study and

Table 5. 28-Day Compressive, Flexural and Split Tensile Strengths ofHigh-Performance Steel Fiber-Reinforced Concrete and AbsoluteVariation in Percent

Mixdesignation w=cm RI

Compressivestrength

Predicted by the models

Eq. (2),Model-1

Eq. (4),Model-2

Eq. (5),Model-3

f 0cf(MPa)

Absoluteerror (%)

Absoluteerror (%)

Absoluteerror(%)

FC1-0 0.4 0 46.85 16.24 18.26 7.36FC1-0.5 0.4 1.29 48.94 2.18 4.40 6.80FC1-1 0.4 2.58 52.00 0.24 2.80 10.52FC1-1.5 0.4 3.88 52.68 2.59 8.95 11.53FC1*-0 0.4 0 52.56 15.79 19.33 2.79FC1*-0.5 0.4 1.29 54.77 5.05 3.49 8.74FC1*-1 0.4 2.58 56.01 0.97 9.38 10.82FC1*-1.5 0.4 3.88 57.40 4.56 13.83 9.23FC2-0 0.35 0 52.69 15.73 17.60 3.07FC2-0.5 0.35 1.29 55.64 3.99 0.91 9.78FC2-1 0.35 2.58 57.85 0.14 6.16 10.70FC2-1.5 0.35 3.88 58.23 6.25 10.76 9.51FC2*-0 0.35 0 55.85 13.14 19.25 1.56FC2*-0.5 0.35 1.29 59.65 0.20 2.13 3.55FC2*-1 0.35 2.58 61.05 3.55 7.09 7.94FC2*-1.5 0.35 3.88 61.44 9.49 14.46 8.83FC3-0 0.3 0 60.10 11.19 17.57 4.31FC3-0.5 0.3 1.29 62.81 0.49 0.49 4.37FC3-1 0.3 2.58 64.01 5.25 4.69 2.93FC3-1.5 0.3 3.88 64.56 9.83 9.68 2.56FC3*-0 0.3 0 63.86 15.65 18.05 1.04FC3*-0.5 0.3 1.29 67.12 1.92 4.87 1.04FC3*-1 0.3 2.58 68.91 2.04 5.30 5.81FC3*-1.5 0.3 3.88 69.67 9.03 11.05 3.64FC4-0 0.25 0 71.64 16..98 18.33 0.79FC4-0.5 0.25 1.29 74.15 6.66 9.49 0.25FC4-1 0.25 2.58 75.65 2.74 2.54 6.64FC4-1.5 0.25 3.88 76.09 2.24 7.51 6.16FC4*-0 0.25 0 74.87 15.26 18.17 6.48FC4*-0.5 0.25 1.29 77.42 5.22 7.26 0.03FC4*-1 0.25 2.58 79.96 0.01 5.01 5.27FC4*-1.5 0.25 3.88 80.41 5.45 9.42 3.22

Note: f 0cf -represents cylinder compressive strength of HPSFRC; fiber

reinforcing index ðRIÞ ¼ wf � ðl=dÞ; weight fraction ðwfÞ ¼ ðdensityof fiber=density of fibrous concreteÞ � Vf .

Fig. 4. Comparison of experimental values of flexural strength (MPa)with the predictions by the model [Eq. (2)]

Fig. 5. Comparison of experimental values of splitting tensile strength(MPa) with the predictions by the model [Eq. (4)]

2

4

6

8

10

12

2 4 6 8 10 12Experimental flexural strength, MPa

Pre

dic

ted

fle

xura

l str

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Pa

Author

Thomas andRamasamy 2007

Zero variation

Predicted by Model- III

Fig. 6. Comparison of experimental values of flexural strength (MPa)with the predictions by the model [Eq. (5)]

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earlier researchers fall well below the data points distributed in theplane except that of the proposed model, indicating the underesti-mation of tensile strength (MOR) and that therefore the IAE valuesare too high, meaning those prediction equations (models) areinvalid for SFRC (HPSFRC). On the other hand, the predictioncurve of the proposed model falls well within the regression planeat a 95% confidence interval as shown in Fig. 7, indicating the goodcorrelation of data points, the best estimation of MOR, and theaccuracy of the model developed. A similar trend has also beenobserved for the proposed model in predicting the splitting tensilestrength of HPSFRC. Overall, the proposed relations (models) werevalidated with the experimental data of both the present investiga-tion and those of earlier researchers, in which the IAE valuesobtained are indicating that the proposed correlation equationsperform very well with the test data.

Conclusions

Based on the investigation of the applicability of availableempirical relations to SFRC and the statistical analysis on exper-imental data of engineering properties, the following conclusioncan be made:1. Empirical relations already proposed by the institutes and other

researchers for normal strength concrete/HSC are not applicableto SFRC/HPSFRC and hence, the development of potentialcorrelation relations among the mechanical (engineering)properties of SFRC/ HPSFRC should be emphasized;

2. Correlation relations between flexural and compressivestrengths, splitting tensile, and compressive strengths, andflexural and splitting tensile strengths are found with IAEvalues of 14.32, 11.86 and 9.84, respectively.

3. The models I [Eq. (2)], II [Eq. (4)] and III [Eq. (5)] are found togive good correlations with experimental data, with obtainedIAE values obtained of 7.29, 10.29 and 5.12, respectively.

4. Incorporation of steel fibers up to the volume fraction Vf ¼1.5% in HPC results in significant improvement on indirecttensile strengths.

5. The proposed power relations (equations) are tested for valida-tion against the test data of present study and earlier researchers.

6. It was observed that the performances of the proposed modelsare quite accurate in estimating 28-day flexural and split ten-sile strengths of HPSFRC, where 90% of the estimated valuesare within �5% of the actual values.

Notation

The following symbols are used in this paper:f 0cf = cylinder compressive strength of HPSFRC, MPa;

frf = flexural strength (modulus of rupture) of HPSFRC,MPa;

fspf = splitting tensile strength of HPSFRC, MPa;HPSFRC = high-performance steel fiber reinforced

concrete;IAE = integral absolute error; andRI = fiber reinforcing index;

SFRC = steel fiber reinforced concrete;s = standard error of the estimate.

Vf = fiber volume fraction in percent;

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ACI 318

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Oluokun

Data points

flex

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rf (

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Compressive strength, f'cf (MPa)

Fig. 7. Comparison of flexural strength prediction capabilities ofdifferent models with the proposed model

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