Research Article Compressive Behavior and Mechanical ...downloads.hindawi.com/journals/amse/2016/6465218.pdfFiber-Reinforced Reactive Powder Concrete. Figure shows typical stress-strain

Research ArticleCompressive Behavior and Mechanical Characteristicsand Their Application to Stress-Strain Relationship of SteelFiber-Reinforced Reactive Powder Concrete

Baek-Il Bae1 Hyun-Ki Choi2 Bong-Seop Lee2 and Chang-Hoon Bang2

1Research Institute of Industrial Science Hanyang University 17 Haengdang-Dong Seongdong-Gu Seoul 04763 Republic of Korea2Department of Fire and Disaster Prevention Engineering Kyungnam University Gyeongsangnam-do 51767 Republic of Korea

Correspondence should be addressed to Hyun-Ki Choi chk7796kyungnamackr

Received 21 April 2016 Revised 26 May 2016 Accepted 7 June 2016

Academic Editor Juan J Del Coz Dıaz

Copyright copy 2016 Baek-Il Bae et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

Althoughmechanical properties of concrete under uniaxial compression are important to design concrete structure current designcodes or other empirical equations have clear limitation on the prediction ofmechanical properties Various types of fiber-reinforcedreactive powder concrete matrix were tested for making more usable and accurate estimation equations for mechanical propertiesfor ultra high strength concrete Investigated matrix has compressive strength ranged from 30MPa to 200MPa Ultra high strengthconcrete was made by means of reactive powder concrete Preventing brittle failure of this type of matrix steel fibers were usedThe volume fraction of steel fiber ranged from 0 to 2 From the test results steel fibers significantly increase the ductility strengthand stiffness of ultra high strength matrix They are quantified with previously conducted researches about material properties ofconcrete under uniaxial loading Applicability of estimation equations for mechanical properties of concrete was evaluated withtest results of this study From the evaluation regression analysis was carried out and new estimation equations were proposedAnd these proposed equations were applied into stress-strain relation which was developed by previous research Ascending partwhich was affected by proposed equations of this study directly well fitted into experimental results

1 Introduction

The compressive strength of concrete is an important param-eter in the design of reinforced concrete structures accordingto current design criteria [1] In recent years performance-based designs have been increasingly used with accompa-nying increases in the diversity of types and strengths ofconcrete and reinforcement used As a result prediction ofthe compressive strength and other mechanical propertiesof concrete have become important to use various types ofmaterials because these parameters control the stress-strainbehavior of concreteThe concrete stress-strainmodels devel-oped in previous studies [2ndash11] and used extensively are basedon coefficients determined from experiments These stress-strain relations cannot be used without additional investi-gation because many coefficients for stress-strain relation ofconcrete are determined by limited number of experimentsThe reason for this is that there are limits to the strength

ranges to which such models are applicable depending onthe conditions of the tests conducted to develop thesemodelsTherefore an equation for use in estimating the mechanicalproperties of ultrahigh strength fiber-reinforced concrete wasderived These equations can serve as a basis for describingthe stress-strain relationships for suchmaterials even beyondthe limits of the currently used empirical formulas or codeprovisions

Usually normal strength concrete members are designedusing rectangular stress block parameters under flexureCurrent design codes provide the rectangular stress blockparameters for simplified design methodology Howeverthese stress blocks are semiempirical parameters They aredetermined by tests of reinforced concrete columns and theyhave apparent limitations Rectangular stress block can beused because the shape of stress-strain relation of concreteis similar to the trapezoid However shape of stress-strainrelationship of concrete changed into triangle as increase of

Hindawi Publishing CorporationAdvances in Materials Science and EngineeringVolume 2016 Article ID 6465218 11 pageshttpdxdoiorg10115520166465218

2 Advances in Materials Science and Engineering

Table 1 Mix proportions

ID wb Weight (kgm3) 119891119888119896

Cement Water Silica fume Sand Filler Steel fiberlowastlowast Super-plasticizer [MPa]30-0 and f serieslowast 043dagger 344 172 mdash 635 1180daggerdagger 0 37 74 147 00 3080-0 and f serieslowast 030 780 255 60 1097 114 0 37 74 147 05 80100-0 and f series 025 809 222 80 1052 162 0 37 74 147 1 100150-0 and f series 020 820 190 112 918 186 0 37 74 147 104 150200-0 and f series 017 830 176 207 912 246 0 37 74 147 108 200lowastf series means fiber-reinforced mixlowastlowast0 means no fiber 37 for 05 volume fraction of fiber contents 74 means 1 volume fraction of fiber contents and 147 means 2 volume fraction of fibercontentsdaggerWater-cement ratio was used for normal strength concretedaggerdaggerCoarse aggregates were used for normal strength concrete

compressive strength of concrete For this reason rectangularstress block parameters depend on the compressive strengthof concrete For the compressive strength of concrete higherthan 76MPa rectangular stress block parameters 120572

1and

1205731of ACI318 [1] are limited by 085 and 065 respectively

However in this case relation between ultimate strain ofconcrete and peak stress should be checked again Becausehigh strength concrete failed with brittle manner usuallythey cannot experience the slight and gradual decrease ofcompressive stress It may cause the unexpected failure underflexure especially for ultrahigh strength concrete member

Making brittle and unexpected failure of ultrahighstrength concrete matrix under compression more ductilesteel fiber can be included in the matrix Inclusion of steelfiber can change explosive failure of ultrahigh strengthconcrete and providemore tensile strength and deformability[12] So steel fiber can usually be used for ultrahigh strengthconcrete matrix

Ultrahigh performance concrete usually has much highercompressive strength and tensile strength than normalstrength concrete [13 14] Shape of stress distribution undercompression and effect of tensile strength of concrete shallbe considered in design of section Many related designguidelines were suggested the design methodologies for ultrahigh performance concrete flexural member but their safetyshould be investigated and more easy way to design thesection shall be found Therefore in this study various typesof stress block and distribution combinations were evaluatedwith experimental result and previous research results

2 Compression Testing of Ultrahigh StrengthSteel Fiber-Reinforced Concrete

In this study compression tests of cylindrical concrete speci-mens with compressive strengths in the range of 30ndash200MPawere carried out to derive equations that can be safely used toestimate the mechanical properties of concrete even beyondthe strength limits given in the design criteriaThe test resultswere analyzed to determine the stress-strain relationshipof concrete over a range of design concrete strengths andcompare the relationship to others described by existingequations The important mechanical properties related tothe stress-strain relationship of concrete under compressive

loading that were considered in this study were the elasticmodulus the stress-to-matrix strength ratio and the strainat the maximum stress

21 Experiment Design According to the previous researcheson mechanical properties of concrete and fiber-reinforcedconcrete [2ndash11] themechanical properties that determine thestress-strain relationship of fiber-reinforced concrete dependon the proportion of fibers contents and matrix strength ofconcrete Therefore the compressive strength of the concretematrix and the contents of steel fiber are important parame-ters in this study The compressive strength range consideredwas 30ndash200MPa Because it is hard to make compressivestrength exceeding 100MPa we use reactive powder concrete(RPC) as steel fiber-reinforced concrete matrix

Steel fibers were mixed into concrete batches at volu-metric ratios of 05 to 2 to ensure both improvement ofthe structural performance of the concrete and workabilityof the concrete after addition of the fibers Five cylindricalspecimens (120593100 times 200mm in size) were produced for eachmix Loading was performed using KS F 2405 [15]The strainrate of the specimenwasmeasured using a compressormeterTable 1 lists the mixes used in the testing

22 Experimental Results for theMechanical Properties of SteelFiber-Reinforced Reactive Powder Concrete Figure 1 showstypical stress-strain relation of steel fiber-reinforced powderconcrete cylinder specimens According to Figure 1 steelfiber can increase ductility and strength of brittle matrixTherefore the most important investigation shall be theincrease rate of ductility and strength of matrix according tothe inclusion of steel fiber So we listed the test results forthe elastic modulus the maximum stress and the strain atthe maximum stress which show the ductility and strength ofmaterial in Table 2

The values shown are the average values for the fivespecimens from each mix The test results indicate that steelfiber significantly affects the properties related to the ductilityof material Figure 2 shows the trends in the parameters ofelastic modulus and strain at peak strength with respect tothe steel fiber content

The mixes with design strengths of 30 80 100 150and 200MPa that were reinforced with fibers at a volume

Advances in Materials Science and Engineering 3

Table 2 Test results (mean value of test specimens)

ID 119881119891

119864119888

119891119888119900

120576119888119900

[] [MPa] [MPa] [permil]C30-0 0 26756 3254 265C30-f05 05 27659 3332 298C30-f10 10 28751 3472 332C30-f20 20 31221 3498 343C80-0 0 32970 8079 316C80-f05 05 33597 8260 355C80-f10 10 34097 8517 364C80-f20 20 34768 8901 404C100-f0 0 36233 10486 339C100-f05 05 37376 10739 371C100-f10 10 38732 11193 395C100-f20 20 38099 11692 409C150-0 0 42023 14940 397C150-f05 05 41203 15496 447C150-f10 10 42365 15960 477C150-f20 20 43222 16240 479C200-0 0 45512 19821 487C200-f05 05 45019 20270 497C200-f10 10 46734 21040 526C200-f20 20 47515 21652 539119881119891 volume fraction of steel fiber () 119864119888 elastic modulus (MPa) 119891119888119900 com-pressive strength of concrete (tested value MPa) 120576119888119900 strain correspondingpeak stress

Com

pres

sive s

tress

(MPa

)

Com

pres

sive s

tress

(ksi)

Strain

No fiber2 volume fraction

Strength of matrix

Strength of matrix

35

30

25

20

15

10

5

0

250

200

150

100

50

00000 0002 0004 0006 0008 0010 0012 0014

fck = 200MPa

fck = 80MPa

Figure 1 Typical uniaxial compressive stress-strain relation of steelfiber-reinforced powder reactive concrete

of 2 exhibited compressive strength increases of 75102 115 87 and 92 respectively The strains at themaximum stress increased by 294 278 235 206and 106 respectively with 2 fiber volume fraction Theincreases of strain at peak stress are significant in comparisonto the increases in strength Significant increase of strain atpeak stress would be caused by the confinement effect of

steel fiber to the formation of cracks along the specimen axisas shown in Figure 3 the failure aspects of fiber-reinforcedspecimen

This reinforcing effect on the strain at peak stress wasshown in all the cases of compressive strength of matrixHowever the ductility increase of the concrete specimenswith strengths greater than 100MPa was smaller than theincrease of the ductility of the 30 and 80MPa concrete Thatis at higher compressive strengths the effect of reinforcingeffect of steel fibers on improving the ductility decreases

3 Estimation of the MechanicalProperties of Steel Fiber-ReinforcedReactive Powder Concrete

We collected concrete material test results from a numberof studies [30ndash44] and analyzed them together with thetest results obtained in this study to estimate the importantparameters needed to derive a stress-strain relationshipthat can be easily applied over a wider range of concretestrengths than the range encompassed by previous studiesThe data collected included 295 results for compressivestrength increase 134 results for the strain at the maximumstress and 1486 results for the elastic modulus In this studythe reinforcing effect of the fiber was characterized using areinforcing index (RI) associated with the aspect ratio andother physical properties of the fiber The value of RI for aparticular type of fiber is determined using (1) The aspectratio of the fibers the fiber end shape and the amount of fiberreinforcement are important parameters in this equationConsider

RI =119881119891119871119891119889119891

119863119891

(1)

where RI is reinforcing index 119881119891is volume fraction of steel

fiber 119871119891is length of steel fiber and 119863

119891is diameter of steel

fiberThe term 119889

119891in (1) is a correction factor for the end shape

of the fiber In this study we used correction factors of 120 and 15 for straight fibers hooked fibers and crimpedfibers respectively to correct for the effect of fiber shape assuggested in the literature [45]

In order to compare the proposed estimation methodsand find the important parameters on mechanical character-istics on steel fiber-reinforced reactive powder concrete wealso collect the empirical equations about mechanical char-acteristic estimation equations for concrete under uniaxialcompression

Table 3 shows an equation for estimating the strain atthe maximum stress as a function of the reinforcing effect ofthe steel fibers and the compressive strength of the concreteTable 4 shows an equation for estimating the elastic modulusWhen examining equations about compressive mechanicalcharacteristics on concrete and fiber-reinforced concretethe most important parameter is compressive strength ofconcrete and (1) reinforcing index Most of the equationsconsider the effect of steel fiber as additional value inde-pendent of compressive strength of concrete However as

4 Advances in Materials Science and EngineeringSt

reng

th (M

Pa)

Stre

ngth

(ksi)

Volume fraction of steel fiber ()

300

250

200

150

100

50

000 05 10 15 20

40

35

30

25

20

15

10

5

0

Matrix type 30MPaMatrix type 80MPaMatrix type 100MPa

Matrix type 150MPaMatrix type 200MPa

(a) Compressive strength

Volume fraction of steel fiber ()00 05 10 15 20



50000

45000

40000

35000

30000

25000

20000

15000

Elas

tic m

odul

us (M

Pa)

Elas

tic m

odul

us (k

si)

7000

6000

5000

4000

3000

(b) Elastic modulus




6

4

2

0

6

5

4

3

2

1

0

Stra

in at

pea

k str

ess(10minus3)

Stra

in at

pea

k str

ess(10minus3)

(c) Strain at peak stress

Figure 2 Mechanical characteristics of fiber-reinforced reactive powder concrete under uniaxial compression

Figure 3 Failure of specimens under uniaxial compression

shown in test results increase rate ofmechanical properties ofsteel fiber-reinforced reactive powder concrete is affected bycompressive strength of concrete In this section we quantify

the increase rate of mechanical properties of concrete con-sidering combination of compressive strength of matrix andsteel fiber contents

31Mechanical Properties of Reactive Powder ConcreteMatrixFor fiber-reinforced concrete the elastic modulus and thestrain at the maximum stress are often determined fromequations that are based on test results for nonreinforcedmixesThe accuracy of such estimates is significantly affectedby the strength ranges of the specimens used to derive theequations Elastic modulus and 120576

119888119900values derived from the

results of material testing conducted in previous studies [30ndash44] and in the present study are shown in Figures 4 and 5respectively The associated equations are also shown Tables5 and 6 list the statistical parameters of the equations

As shown in Table 5 and Figure 4 the current codeprovisions show the highest accuracy in the estimation of the


Table 3 Prediction equations for strain at peak stress (previous researches)

Researcher Equation Limitation [MPa]

Collins et al [6] 120576119888119900=119891119888119896

119864119888

119899

119899 minus 1 119899 = 08 +

119891119888119896

17 119864119888= 332011989105

119888119896+ 6900 119891

119888119896le 100

Wee et al [7] 120576119888119900=180 (119891

119888119896)025

106119891119888119896le 120

Ros [16] 120576119888119900=(00546 + 0003713119891

119888119896)

100119891119888119896le 43

Fafitis and Shah [17] 120576119888119900= 1491 times 10minus5119891

119888119896+ 000195 119891

119888119896le 66

De Nicolo et al [18] 120576119888119900= 000076 + [(0626119891

119888119896minus 433) times 10minus7]

05

119891119888119896le 90

CEB-fip 228 [9] 120576119888119900=07 (119891

119888119896)031

1000119891119888119896le 100

EC2 [19] 120576119888119900=(20 + 0085 (119891

119888119896minus 50)

053

)

1000 for 119891

119888119896ge 50MPa otherwise 120576

119888119900= 0002 119891

119888119896le 90

Soroushian and Lee [20] 120576119888119900119891= 0007RI + 00021 RI =

119881119891119871119891

119863119891

mdash

Nataraja et al [21] 120576119888119900119891= 0002 + 0006(RI) RI =

119882119891119871119891

119863119891

119891119888119896le 50

Dhakal et al [22] 120576119888119900119891= 0002 + 2000120572

1198881198812119891 120572119891= 10 (flat) 120572

119891= 24 (hooked) 119891

119888119896le 40

119864119888 secant elastic modulus of concrete at 045119891119888119896 (MPa) 120576119888119900 strain corresponding to peak stress (normal concrete)119891119888119896 compressive strength of concrete (MPa)RI reinforcing index119863119891 diameter of steel fiber (mm) 119871119891 length of steel fiber (mm) 119881119891 volume fraction of steel fiber119882119891 weight fraction of steel fiber

Table 4 Prediction equations for modulus of elasticity (previous researches)

Researcher Equation LimitationKCI2007 [23] 119864

119888= 8500 3radic119891

119888119906 119864119888= 007711990815

119888

3radic119891119888119906 119891119888119906= 119891119888119896+ 8

Did not specify but radic119891119888119896

cannot exceed 84MPaKCI2012 [24]119864119888= 8500 3radic119891

119888119906 119864119888= 007711990815

119888

3radic119891119888119906 119891119888119906= 119891119888119896+ Δ119891 where

Δ119891 = 4 when 119891119888119896le 40MPa and Δ119891 = 6 when 119891

119888119896gt 60MPa

Interpolate between 40MPa and 60MPaACI318-11 [1] 119864

119888= 4700radic119891

119888119896(normal weight concrete) 119864

119888= 119908151198880043radic119891

119888119896

CEB-fip 228 [9] 119864119888119894= 119864119888119900[(119891119888119896+ Δ119891)

119891119888119898119900

]

13

Δ119891 = 8 119891119888119898119900= 10 119864

119888119900= 22GPa 119891

119888119896le 90

Martinez et al [25] 119864119888= 3320radic119891

119888119896+ 6900 21MPa lt 119891

119888119896lt 83MPa

Cook [26] 119864119888= 3385 times 10minus5119908255

119888(119891119888119896)0315 mdash

Ahmad and Shah [27] 119864119888= 3385 times 10minus5119908255

119888(119891119888119896)0325

119891119888119896lt 84MPa

Graybeal [11] 119864119888= 3840radic119891

119888119896119891119888119896lt 200MPa

Gao et al [28] 119864119888119891= 119864119888(1 + 0173(RI)) 70MPa lt 119891

119888119896lt 85MPa

Padmarajaiah [29] 119864119888119891= 119864119888+ 2440(RI) 119891

119888119896lt 69MPa

119864119888 secant elastic modulus of concrete at 045119891119888119896 (MPa) 119908119888 unit weight of concrete (kgm3) 119891119888119896 compressive strength of concrete (MPa) RI reinforcingindex RI = 119881119891119871119891119863119891

elastic modulus As shown in statistical values for KCI2012current Korean design code provision a standard deviationis not the lowest among other estimation methods

However mean value coefficient of variation and IAE(integrated absolute error) values for KCI2012 have low-est value among other prediction methods In generalthe estimation equation for the elastic modulus tends tounderestimate the modulus of elasticity for normal strengthconcrete and overestimate the modulus of elasticity for highstrength concrete However highest accuracy of KCI2012occurs because the estimation equation was developed toprovide relatively safe estimates for both normal strength andhigh strength concrete using different value of Δ119891 accordingto the compressive strength of concrete Therefore even foran ultrahigh strength concrete there should be no significant

problems associated with using the estimation equationpresented in the current design standards to estimate theelastic modulus of the concrete

Because 120576119888119900 the strain at the maximum stress is an

important boundary condition in defining the stress-strainrelationship and important design parameter we also com-pared the results of previous studies and the results of thisstudy with respect to this parameter In previous studies onthe stress-strain relationship of concrete the typical valueused for the strain at the maximum stress is 0002 HoweverFigure 5 confirms that the strain at the maximum stress tendsto increase with the compressive strength of the concreteIt was confirmed that equations from previous studies forestimation of 120576

119888119900 which take the form of an exponential

function of the compressive strength underestimate the


2 4 6 8 10 12 14 160

10000

20000

30000

40000

50000

60000

70000

80000

Elas

tic m

odul

us (M

Pa)

Square root of compressive strength of concrete (MPa)

0 500 1000 1500 2000

0

2000

4000

6000

8000

10000

Elas

tic m

odul

us (k

si)

Square root of compressive strength of concrete (psi)

KCI2007KCI2012ACI318-11CEB-fip MC90CEB-fip 228Martinez et al

CookAhmad and ShahRadainGraybealExperimental data

Figure 4 Elastic modulus test results and prediction equations

0 50 100 150 200 2500000

0001

0002

0003

0004

0005

Compressive strength of concrete (MPa)

0 5 10 15 20 25 30 35

0

2

4

6

8

10

Stra

in at

pea

k str

ess

Stra

in at

pea

k str

ess

Compressive strength of concrete (ksi)

Collins et alWee et alRosShah-Fafits

De Nicolo et alCEB-fip 228EC2C29

Figure 5 Strain at peak stress test results and prediction equations

increase in 120576119888119900with increasing compressive strength Table 6

confirms that the accuracy of these equations increases asthe compressive strength range considered increases Giventhat the equation proposed by Collins et al [6] has the lowestIAE and average value Collins et al [6] equation for 120576

119888119900was

concluded to be suitable for use with high strength mixesCollins et al [6] equation shown in Table 1 reflects the effectof the elastic modulus Because our analysis confirmed thatthe equation presented in the current design standards yieldsthe highest accuracy in the estimation of the elastic modulusthis equation was used with Collins et al [6] equation to

Table 5 Statistics on the prediction methods for modulus ofelasticity

Researcher Mean SD CV IAEKCI2007 [23] 099 012 012 852KCI2012 [24] 101 012 012 842ACI318-11 [1] 093 013 014 1372CEB-fip 228 [9] 083 010 012 2065Martinez et al [25] 105 013 012 945Cook [26] 075 009 012 3244Ahmad and Shah [27] 107 012 012 983Graybeal [11] 114 016 014 1354SD standard deviation CV coefficient of variation IAE integrated absoluteerror ()

Table 6 Statistics on the predictionmethods for strain at peak stressunder uniaxial compression

Researcher Mean SD CV IAE ()Collins et al [6] 099 012 012 874Wee et al [7] 116 017 015 1637Ros [16] 078 019 025 4204Fafitis and Shah [17] 084 009 011 1949De Nicolo et al [18] 093 011 012 1158CEB-fip 228 [9] 099 013 013 1056EC2 [19] 109 015 014 1252SD standard deviation CV coefficient of variation IAE integrated absoluteerror ()

estimate 120576119888119900 As a result the IAE decreased by 867 and

the average of the ratios of experimental values to estimatedvalues was found to be 102 indicating that the estimatedvalues were on the safe side

32 Effect of Steel Fiber Contents on Compressive StrengthIncrease Figure 6 shows test results from previous studies[30ndash44] and from this study on the effect of the steel fiber con-tents on the increase in concrete compressiveThe increase ofmaximum stress of test results was expressed as the ratio ofthe compressive strength of fiber-reinforced specimens to thecompressive strength of non-fiber-reinforced specimensTheaverage compressive strength increase due to fiber reinforce-ment was 97 However for specimens of the same designcompressive strength the compressive strength tended toincrease linearly with increasing proportions of fiber Exam-ination of the results classified by the design compressivestrength of the concrete indicated that the increase achievedin compressive strength with increasing fiber content tendedto decrease with increasing design compressive strengthHowever it is difficult to confirm that a clear correlation existsbecause of the high degree of variation in the test resultsIn particular the correlation between compressive strengthincrease and fiber content tended to decrease with increasingdesign compressive strengthThe results of our study suggestthat the increases achieved in compressive strength withincreasing fiber content were comparable over the range of


00 05 10 15 20 25 30 3509

10

11

12

13

14

15

Incr

ease

rate

of c

ompr

essiv

e stre

ngth

Reinforcing index

Test results from previous researchesThis study C30This study C80This study C100This study C150This study C200

Figure 6 Compressive strength increase rate according to reinforc-ing index

compressive strengths considered A maximum compressivestrength increase of 10 was verified

33 Effect of Steel Fiber Contents on Strain at Peak StressFigure 7 illustrates the effect of the steel fiber contents on thechange in the strain at the maximum stress as indicated bythe results of previous studies and by the results of this studyThe change in the strain at the maximum stress exhibited ahigher correlation to the contents of steel fibers than did theincrease with compressive strength of matrix The results ofthe experiments conducted in our study are shown in Figure 7along with the results from previous studies The test resultsindicate that for mixes with design compressive strengths of80MPa 120576

119888119900increases with increasing steel fiber content but

for mixes with strengths of more than 100MPa 120576119888119900decreases

with increasing fiber content This phenomenon was partic-ularly significant for the specimens with design compressivestrengths of 200MPa Normal strength concrete has shownsimilar change according to the steel fiber inclusion

34 Effect of Steel Fiber Contents on Elastic Modulus of Con-crete Higher compressive strength of concrete is achievedby homogenization of the constituent materials and of theirstrengths Accordingly the elastic modulus of concrete tendsto increase with increasing concrete strength The elasticmodulus derived from test results obtained in previousstudies and in our study was examined to assess the trendin the elastic modulus of fiber-reinforced reactive powderconcrete with respect to the previously defined reinforcingindex Figure 8 shows the distribution of elastic modulusof concrete test results according to the reinforcing indexBecause the elastic modulus is affected by the strengthand strain simultaneously the increases in strength and

00 05 10 15 20 25 30 3509

10

11

12

13

14

15

16

17

18

Incr

ease

rate

of s

trai

n at

pea

k str

ess

Reinforcing index


Figure 7 Strain at peak stress increase rate according to reinforcingindex

stress exhibit similar trends The degree to which the elasticmodulus increased with the fiber reinforcing index decreasedwith increasing concrete compressive strength However theaverage increase was 65 which is lesser than that for thestrength or strain Our test results revealed similar trendsSpecimens in the C80 test group (with design compressivestrengths of 80MPa) exhibited the greatest increases amonghigh strength concrete mixes Specimens in test groups withdesign compressive strength greater than 100MPa exhibitedsmaller increases and trends similar to those observed in theresults from previous studies

4 Equation for Estimating the Effect ofthe Steel Fiber Contents on the MechanicalProperties of Steel Fiber-ReinforcedReactive Powder Concrete underUniaxial Compression

The effects of steel fiber on concrete strength improvementobserved in the test results obtained in this study and inprevious studies [30ndash44] were quantified for compressivestrength ranges in increments of 20MPa as shown in Fig-ure 9 The magnitude of the improvement within each rangewas determined from linear regression analysis of the data forthe specimens within that range The regression analysis wasperformed using the model form shown as

119896119888119891

119896119888119900

= 1 + RI times 120572 (2)

where 119896119888119891is mechanical properties for steel fiber-reinforced

concrete 119896119888119900is mechanical properties on nonreinforced con-

crete RI is reinforcing index and 120572 is regression coefficient


00 05 10 15 20 25 30 3509

10

11

12

13

14

Incr

ease

rate

of e

lasti

c mod

ulus

Reinforcing index


Figure 8 Elastic modulus increase rate according to reinforcingindex

20 40 60 80 100 120 140 160 180 20000

01

02

03

04

05

Incr

ease

rate


Compressive strengthStrain at peak stressElastic modulus

0 5000 10000 15000 20000 25000Compressive strength of concrete (ksi)

Figure 9 Change of statistical values and trend lines onmechanicalcharacteristics of steel fiber-reinforced reactive powder concreteunder uniaxial compression

The increase in compressive strength with inclusion ofsteel fiber exhibited a clear decreasing trend with increasingconcrete compressive strength The results of the regressionanalysis suggest that themagnitude of the improvement in thecompressive strength which depends on the elastic modulusthe strain at the maximum stress and the compressivestrength of concrete can be expressed by

119896119888119891

119896119888119900

= 1 + RI (1198861minus 1198871119891119888119896) (3)

In the above equation 1198861and 119887

1are regression coeffi-

cientsThe values of 1198861and 1198871were determined to be 029 and

00013 respectively for change in the compressive strength052 and 00026 respectively for change in 120576

119888119900 and 020 and

000092 respectively for change in elastic modulusIn order to verify the applicability of proposed equations

predicting mechanical properties of steel fiber-reinforcedreactive powder concrete under uniaxial compression weapply these equations to the stress-strain relation of concreteunder uniaxial compression

Most of the stress-strain relation of concrete are deter-mined by the important mechanical properties elastic mod-ulus of concrete and strain at peak stress Stress-strainrelations proposed by previous researches were investigatedThere are several types of stress-strain relation of concreteBut in this study we investigated the way to use the elasticmodulus for predicting stress-strain relation of concretebecause stress-strain relations of high strength or ultrahighstrength concrete depend on the elastic modulus as weinvestigated in this studyThe first one of stress-strain relationof concrete we investigated is suggested by Collins et al[6] and the other is suggested by Attard and Setunge [10]Collins et al use the differential of plasticity as base modeland Attard and Setunge [10] use the mathematical model topredict stress-strain relationship Both of them use the elasticmodulus of concrete as main variable for prediction of stress-strain relation However mathematical model which wassuggested by Attard and Setunge [10] needs more boundaryconditions such as inflection points after experiencing peakstress Inflection points of descending curve of stress-strainrelation are highly dependent on the experimental equip-mentTherefore in this study stress-strain relation suggestedby Collins et al [6] is used for prediction of stress-strainrelation by using proposed equation of main variables suchas elastic modulus and strain at peak stress The stress-strainrelation suggested by Collins et al [6] can be described using

119891119888

1198911015840119888

=119899 (1205761198881198911205761015840119888)

119899 minus 1 + (1205761198881198911205761015840119888)119899119896 (4)

where1198911015840119888is peak stress obtained from cylinder test 1205761015840

119888is strain

when 119891119888reaches 1198911015840

119888 119899 is curve fitting factor equal to 119864

119888(119864119888minus

1198641015840119888) 119864119888is tangent stiffness when 120576

119888119891is zero 1198641015840

119888is secant

stiffness when 120576119888119891

is 1205761015840119888 and 119896 model the strain decay before

and after experiencing peak stress 119899 and 119896 are suggested byusing data of high strength concrete and we decided to usethese two variables without change 119899 and 119896 can be calculatedby using

119899 = 08 +1198911015840119888

17

119896 = 067 +1198911015840119888

62

(5)

For the verification of the applicability of suggestedequations stress-strain relationships of concrete were con-structed and compared with experimental results As canbe seen in Figure 10 predicted stress-strain relationship


0 0002 0004 0006Strain

0

50

100

150

200

Stre

ss (M

Pa)

0

10

20

30

Stre

ss (k

si)Stress-strain model

Test resultsSymbols(+ 0 ) middot middot middot )

(a) 119881119891 = 0

0 0002 0004 0006Strain

0

50

100

150

200

0

10

20

30

Stre

ss (M

Pa)

Stre

ss (k

si)

Stress-strain model


(b) 119881119891 = 20

Figure 10 Applicability for stress-strain relationship prediction and experimental results

of ascending curve was well fitted into the experimentalresults However descending curve of prediction shows lessaccuracy in descending part of the total stress-strain relationThe inaccuracy of descending curve of stress-strain relationprediction might be caused by the difference of stiffness ofexperiencing equipmentMakingmore exact solution energyabsorption capacity under compression shall be investigated

5 Conclusion

In this study we conducted material testing to evaluate theproperties of ultrahigh strength concrete which was madefrom reactive powder concrete reinforced with steel fibersunder compressive loading and we evaluated the suitabilityof equations developed in previous studies for use in estimat-ing material properties of ultrahigh strength concrete Theresults of this study are summarized as follows

(1) The compression test results for non-fiber-reinforcedandfiber-reinforced ultrahigh strength concrete spec-imens indicated that non-fiber-reinforced concretespecimens exhibited brittle fractures whereas fiber-reinforced concrete specimens did not The testresults suggest that reinforcing fibers resisted the hor-izontal tensile forces induced by vertical compressiveloading

(2) The test results confirmed that the effects of steelfiber on the concrete compressive strength strain atmaximum stress and elasticmodulus exhibited lineartrends regardless of the compressive strength of themix Compressive strength and elastic modulus werenot significantly affected by the amount of reinforcing

fiber in the mix However a relatively strong effect onthe strain at the maximum stress was confirmed

(3) The mechanical properties of the concrete such asthe elastic modulus and the strain at the peak stresswere estimated with a high degree of accuracy usingthe secant modulus estimation equation presented inthe current design standards and by Collins et alrsquosequation respectively Thus the applicability of theseequations to concrete in the ultrahigh strength rangewas confirmed

(4) Regression analyses were performed on the testresults from this study and previous studies todescribe the effects of fiber reinforcement on thematerial properties of interest The analysis resultsshowed that the reinforcing effects of steel fiber canbe expressed in terms of the fiber reinforcing index

(5) Analysis of the test results confirmed that the rel-evant mechanical properties of concrete mix at agiven strength level improve in proportion to thefiber reinforcement index Linear regression analyseswere performed for the mechanical properties usingequations of the same form for all strength levelsThe equation formwas chosen so that different valuesof the regression coefficients were obtained for eachstrength level

(6) Proposed equations for mechanical properties underuniaxial compression on concrete can be used asmain variables of stress-strain relation Accordingto comparison ascending curve of stress-strain rela-tion can be well fitted into experimental resultsHowever descending part of stress-strain relation


cannot be well fitted because prediction method didnot consider the energy dissipation capacity afterexperiencing peak stress In order to construct fullrange of stress-strain relation of fiber-reinforced con-crete boundary conditions after peak stress shall beconsidered

Competing Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This research was supported by Basic Science ResearchProgram through theNational Research Foundation of Korea(NRF) funded by theMinistry of Science ICTamp Future Plan-ning (15CTAP-C097470-01 and NRF-2014R1A1A1005444)

References

[1] ACI Committee 318 Building Code Requirements for StructuralConcrete (ACI 318-11) and Commentary American ConcreteInstitute Farmington Hills Mich USA 2011

[2] S Popovics ldquoAnumerical approach to the complete stress-straincurve of concreterdquo Cement and Concrete Research vol 3 no 5pp 583ndash599 1973

[3] M Sargin Stress-Strain Relationship for Concrete and the Anal-ysis of Structural Concrete Sections Study 4 Solid MechanicsDivision University of Waterloo Waterloo Canada 1971

[4] A Tomaszewicz Betongens Arbeidsdiagram SINTEF STF65A84065 Trondheim 1984

[5] D J Carreira and K-H Chu ldquoStress-strain relationship forplain concrete in compressionrdquo Journal of the American Con-crete Institute vol 82 no 6 pp 797ndash804 1985

[6] M P Collins D Mitchell and J G MacGregor ldquoStructuraldesign considerations for high-strength concreterdquo ConcreteInternational vol 15 no 5 pp 27ndash34 1993

[7] T H Wee M S Chin and M A Mansur ldquoStress-strainrelationship of high-strength concrete in compressionrdquo Journalof Materials in Civil Engineering vol 8 no 2 pp 70ndash76 1996

[8] P TWang S P Shah andA ENaaman ldquoStress-strain curves ofnormal and lightweight concrete in compressionrdquo ACI JournalProceedings vol 75 pp 603ndash611 1978

[9] Comite Euro-International du Beton-Federation Internationalede la Precontrainte ldquoHigh performance concretemdashrecommended extensions to the model code 90 researchneedsrdquo CEB Bulletin no 228 p 60 1995

[10] M M Attard and S Setunge ldquoStress-strain relationship ofconfined and unconfined concreterdquo ACI Materials Journal vol93 no 5 pp 432ndash442 1996

[11] B A Graybeal ldquoCompressive behavior of ultra-high-performance fiber-reinforced concreterdquo ACI Materials Journalvol 104 no 2 pp 146ndash152 2007

[12] A K H Kwan and L G Li ldquoCombined effects of water filmthickness and paste film thickness on rheology of mortarrdquoMaterials and StructuresMateriaux et Constructions vol 45 no9 pp 1359ndash1374 2012

[13] F de Larrard and T Sedran ldquoOptimization of ultra-high-performance concrete by the use of a packing modelrdquo Cementand Concrete Research vol 24 no 6 pp 997ndash1009 1994

[14] L G Li and A K H Kwan ldquoPacking density of concrete mixunder dry and wet conditionsrdquo Powder Technology vol 253 pp514ndash521 2014

[15] KS F 2405 Standard Test Method for Compressive Strength ofConcrete Korean Agency for Technology and Standards 2010

[16] M Ros Material-Technological Foundation and Problems ofReinforced Concrete Bericht no 162 Eidgenossische Materi-alprufungs- und Versuchsanstalt fur Industrie Bauwesen undGewerbe Zurich Switzerland 1950

[17] A Fafitis and S P Shah ldquoPredictions of ultimate behavior ofconfined columns subjected to large deformationsrdquo Journal ofthe American Concrete Institute vol 82 no 4 pp 423ndash433 1985

[18] B De Nicolo L Pani and E Pozzo ldquoStrain of concrete at peakcompressive stress for a wide range of compressive strengthsrdquoMaterials and Structures vol 27 no 4 pp 206ndash210 1994

[19] EuropeanCommitee for Standardization (CEN)Design of Con-crete StructuresmdashPart 1-1 General Rules and Rules for BuildingsEurocode 2 Brussels Belgium 2004

[20] P Soroushian and C H Lee ldquoConstitutive modeling of steelfiber reinforced concrete under direct tension and compressionfibre reinforced cements and concretes recent developmentsrdquoin Proceedings of the International Conference held at theUniversity of Wales Collige of Cardiff School of EngineeringUnited Kingdom London UK September 1989

[21] M C Nataraja N Dhang and A P Gupta ldquoStress-strain curvesfor steel-fiber reinforced concrete under compressionrdquo Cementand Concrete Composites vol 21 no 5-6 pp 383ndash390 1999

[22] R P Dhakal C Wang and J B Mander ldquoBehavior of steel fibrereinforced concrete in compressionrdquo in Proceedings of the Nan-jing International Symposium on Innovation amp Sustainability ofStructures in Civil Engineering 2005

[23] Korea Concrete Institute Concrete Design Code and Commen-tary Kimoondang Publishing Company Seoul Republic ofKorea 2007

[24] Korea Concrete Institute Concrete Design Code and Commen-tary Kimoondang Publishing Company Seoul Korea 2012

[25] SMartinez A HNilson and F Slate ldquoSpirally reinforced high-strength concrete columnsrdquo ACI Journal vol 81 no 5 pp 431ndash442 1984

[26] J E Cook ldquo10000 PSI concreterdquo Concrete International Designand Construction vol 11 no 10 pp 67ndash75 1989

[27] S H Ahmad and S P Shah ldquoComplete triaxial stress-straincurves for concreterdquo Journal of the Structural Division vol 108no 4 pp 728ndash742 1982

[28] J Gao W Sun and K Morino ldquoMechanical properties of steelfiber-reinforced high-strength lightweight concreterdquo Cementand Concrete Composites vol 19 no 4 pp 307ndash313 1997

[29] S K Padmarajaiah Influence of fibers on the behavior ofhigh strength concrete in fullypartially prestressed beams anexperimental and analytical study [PhD thesis] Indian Instituteof Science Bangalore India 1999

[30] H-K Choi B-I Bae and C-S Choi ldquoMechanical character-istics of ultra high strength concrete with steel fiber underuniaxial compressive stressrdquo Journal of the Korea ConcreteInstitute vol 27 no 5 pp 521ndash530 2015

[31] S V T J Perera H Mutsuyoshi and S Asamoto ldquoProperties ofhigh-strength concreterdquo in Proceedings of the 12th InternationalSummer Symposium of Japan Society of Civil Engineers (JSCErsquo10) Funabashi Japan 2010


[32] K F Sarsam I A S Al-Shaarbaf and M M S RidhaldquoExperimental investigation of shear-critical reactive powderconcrete beams without web reinforcementrdquo Engineering ampTechnology Journal vol 30 no 17 pp 2999ndash3022 2012

[33] D A Pandor Behavior of high strength fiber reinforced concretebeams in shear [MS thesis] Massachusetts Institute of Technol-ogy 1994

[34] J Thomas and A Ramaswamy ldquoMechanical properties ofsteel fiber-reinforced concreterdquo Journal of Materials in CivilEngineering vol 19 no 5 pp 385ndash392 2007

[35] S Kang and G Ryu ldquoThe effect of steel-fiber contents on thecompressive stress-strain relation of Ultra High PerformanceCementitious Composites (UHPCC)rdquo Journal of the KoreaConcrete Institute vol 23 no 1 pp 67ndash75 2011

[36] P Bhargava U K Sharma and S K Kaushik ldquoCompressivestress-strain behavior of small scale steel fibre reinforced highstrength concrete cylindersrdquo Journal of Advanced ConcreteTechnology vol 4 no 1 pp 109ndash121 2006

[37] Y-C Ou M-S Tsai K-Y Liu and K-C Chang ldquoCompressivebehavior of steel-fiber-reinforced concrete with a high reinforc-ing indexrdquo Journal of Materials in Civil Engineering vol 24 no2 pp 207ndash215 2012

[38] A Samer Ezeldin and P N Balaguru ldquoNormal- and high-strength fiber reinforced concrete under compressionrdquo Journalof Materials in Civil Engineering vol 4 no 4 pp 415ndash429 1992

[39] B-W Jo Y-H Shon and Y-J Kim ldquoThe evalution of elasticmodulus for steel fiber reinforced concreterdquo Russian Journal ofNondestructive Testing vol 37 no 2 pp 152ndash161 2001

[40] O Kazuhiro S Shunsuke A Hideo et al An ExperimentalStudy on the Compressive Properties of the Super-High StrengthConcrete vol 25 Architectural Institute of Japan China BranchResearch Report Collection 2002

[41] A R Murthy N R Iyer and B K R Prasad ldquoEvaluation ofmechanical properties for high strength and ultrahigh strengthconcretesrdquo Advances in Concrete Construction vol 1 no 4 pp341ndash358 2013

[42] N N Meleka A A Bashandy and M A Arab ldquoUltra highstrength concrete using economical materialsrdquo InternationalJournal of Current Engineering and Technology vol 3 no 2 pp393ndash402 2013

[43] D Moldovan and C Magureanu ldquoStress-strain diagram forhigh strength concrete elements in flexurerdquo in Proceedings ofthe 3rd International Conference Advanced Composite MaterialsEngineering (COMAT rsquo10) pp 137ndash142 Transilvania UniversityPress of Brasov Brasov Romania 2010

[44] W I Khalil and Y R Tayfur ldquoFlexural strength of fibrous ultrahigh performance reinforced concrete beamsrdquoARPN Journal ofEngineering andApplied Sciences vol 8 no 3 pp 200ndash214 2013

[45] R Narayanan and I Y S Darwish ldquoUse of steel fibers as shearreinforcementrdquo ACI Structural Journal vol 84 no 3 pp 216ndash227 1987

Submit your manuscripts athttpwwwhindawicom

ScientificaHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CorrosionInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Polymer ScienceInternational Journal of



CeramicsJournal of


CompositesJournal of

NanoparticlesJournal of



International Journal of

Biomaterials


NanoscienceJournal of

TextilesHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Journal of

NanotechnologyHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

CrystallographyJournal of


The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


CoatingsJournal of

Advances in

Materials Science and EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Smart Materials Research



MetallurgyJournal of


BioMed Research International

MaterialsJournal of


Nano

materials


Journal ofNanomaterials


Table 1 Mix proportions

ID wb Weight (kgm3) 119891119888119896

Cement Water Silica fume Sand Filler Steel fiberlowastlowast Super-plasticizer [MPa]30-0 and f serieslowast 043dagger 344 172 mdash 635 1180daggerdagger 0 37 74 147 00 3080-0 and f serieslowast 030 780 255 60 1097 114 0 37 74 147 05 80100-0 and f series 025 809 222 80 1052 162 0 37 74 147 1 100150-0 and f series 020 820 190 112 918 186 0 37 74 147 104 150200-0 and f series 017 830 176 207 912 246 0 37 74 147 108 200lowastf series means fiber-reinforced mixlowastlowast0 means no fiber 37 for 05 volume fraction of fiber contents 74 means 1 volume fraction of fiber contents and 147 means 2 volume fraction of fibercontentsdaggerWater-cement ratio was used for normal strength concretedaggerdaggerCoarse aggregates were used for normal strength concrete

compressive strength of concrete For this reason rectangularstress block parameters depend on the compressive strengthof concrete For the compressive strength of concrete higherthan 76MPa rectangular stress block parameters 120572

1and

1205731of ACI318 [1] are limited by 085 and 065 respectively

However in this case relation between ultimate strain ofconcrete and peak stress should be checked again Becausehigh strength concrete failed with brittle manner usuallythey cannot experience the slight and gradual decrease ofcompressive stress It may cause the unexpected failure underflexure especially for ultrahigh strength concrete member

Making brittle and unexpected failure of ultrahighstrength concrete matrix under compression more ductilesteel fiber can be included in the matrix Inclusion of steelfiber can change explosive failure of ultrahigh strengthconcrete and providemore tensile strength and deformability[12] So steel fiber can usually be used for ultrahigh strengthconcrete matrix

Ultrahigh performance concrete usually has much highercompressive strength and tensile strength than normalstrength concrete [13 14] Shape of stress distribution undercompression and effect of tensile strength of concrete shallbe considered in design of section Many related designguidelines were suggested the design methodologies for ultrahigh performance concrete flexural member but their safetyshould be investigated and more easy way to design thesection shall be found Therefore in this study various typesof stress block and distribution combinations were evaluatedwith experimental result and previous research results

2 Compression Testing of Ultrahigh StrengthSteel Fiber-Reinforced Concrete

In this study compression tests of cylindrical concrete speci-mens with compressive strengths in the range of 30ndash200MPawere carried out to derive equations that can be safely used toestimate the mechanical properties of concrete even beyondthe strength limits given in the design criteriaThe test resultswere analyzed to determine the stress-strain relationshipof concrete over a range of design concrete strengths andcompare the relationship to others described by existingequations The important mechanical properties related tothe stress-strain relationship of concrete under compressive

loading that were considered in this study were the elasticmodulus the stress-to-matrix strength ratio and the strainat the maximum stress

21 Experiment Design According to the previous researcheson mechanical properties of concrete and fiber-reinforcedconcrete [2ndash11] themechanical properties that determine thestress-strain relationship of fiber-reinforced concrete dependon the proportion of fibers contents and matrix strength ofconcrete Therefore the compressive strength of the concretematrix and the contents of steel fiber are important parame-ters in this study The compressive strength range consideredwas 30ndash200MPa Because it is hard to make compressivestrength exceeding 100MPa we use reactive powder concrete(RPC) as steel fiber-reinforced concrete matrix

Steel fibers were mixed into concrete batches at volu-metric ratios of 05 to 2 to ensure both improvement ofthe structural performance of the concrete and workabilityof the concrete after addition of the fibers Five cylindricalspecimens (120593100 times 200mm in size) were produced for eachmix Loading was performed using KS F 2405 [15]The strainrate of the specimenwasmeasured using a compressormeterTable 1 lists the mixes used in the testing

22 Experimental Results for theMechanical Properties of SteelFiber-Reinforced Reactive Powder Concrete Figure 1 showstypical stress-strain relation of steel fiber-reinforced powderconcrete cylinder specimens According to Figure 1 steelfiber can increase ductility and strength of brittle matrixTherefore the most important investigation shall be theincrease rate of ductility and strength of matrix according tothe inclusion of steel fiber So we listed the test results forthe elastic modulus the maximum stress and the strain atthe maximum stress which show the ductility and strength ofmaterial in Table 2

The values shown are the average values for the fivespecimens from each mix The test results indicate that steelfiber significantly affects the properties related to the ductilityof material Figure 2 shows the trends in the parameters ofelastic modulus and strain at peak strength with respect tothe steel fiber content

The mixes with design strengths of 30 80 100 150and 200MPa that were reinforced with fibers at a volume



ID 119881119891

119864119888

119891119888119900

120576119888119900


Com

pres

sive s

tress

(MPa

)

Com

pres

sive s

tress

(ksi)

Strain


Strength of matrix

Strength of matrix

35

30

25

20

15

10

5

0

250

200

150

100

50

00000 0002 0004 0006 0008 0010 0012 0014

fck = 200MPa

fck = 80MPa







RI =119881119891119871119891119889119891

119863119891

(1)










reng

th (M

Pa)

Stre

ngth

(ksi)


300

250

200

150

100

50

000 05 10 15 20

40

35

30

25

20

15

10

5

0







50000

45000

40000

35000

30000

25000

20000

15000

Elas

tic m

odul

us (M

Pa)

Elas

tic m

odul

us (k

si)

7000

6000

5000

4000

3000

(b) Elastic modulus




6

4

2

0

6

5

4

3

2

1

0

Stra

in at

pea

k str

ess(10minus3)

Stra

in at

pea

k str

ess(10minus3)













Collins et al [6] 120576119888119900=119891119888119896

119864119888

119899

119899 minus 1 119899 = 08 +

119891119888119896

17 119864119888= 332011989105

119888119896+ 6900 119891

119888119896le 100

Wee et al [7] 120576119888119900=180 (119891

119888119896)025

106119891119888119896le 120

Ros [16] 120576119888119900=(00546 + 0003713119891

119888119896)

100119891119888119896le 43


119888119896+ 000195 119891

119888119896le 66

De Nicolo et al [18] 120576119888119900= 000076 + [(0626119891


05

119891119888119896le 90

CEB-fip 228 [9] 120576119888119900=07 (119891

119888119896)031

1000119891119888119896le 100

EC2 [19] 120576119888119900=(20 + 0085 (119891

119888119896minus 50)

053

)

1000 for 119891

119888119896ge 50MPa otherwise 120576

119888119900= 0002 119891

119888119896le 90


119881119891119871119891

119863119891

mdash


119882119891119871119891

119863119891

119891119888119896le 50

Dhakal et al [22] 120576119888119900119891= 0002 + 2000120572

1198881198812119891 120572119891= 10 (flat) 120572

119891= 24 (hooked) 119891

119888119896le 40




119888= 8500 3radic119891

119888119906 119864119888= 007711990815

119888

3radic119891119888119906 119891119888119906= 119891119888119896+ 8



119888119906 119864119888= 007711990815

119888

3radic119891119888119906 119891119888119906= 119891119888119896+ Δ119891 where


119888119896gt 60MPa


119888= 4700radic119891


119888= 119908151198880043radic119891

119888119896

CEB-fip 228 [9] 119864119888119894= 119864119888119900[(119891119888119896+ Δ119891)

119891119888119898119900

]

13

Δ119891 = 8 119891119888119898119900= 10 119864

119888119900= 22GPa 119891

119888119896le 90


119888119896+ 6900 21MPa lt 119891

119888119896lt 83MPa

Cook [26] 119864119888= 3385 times 10minus5119908255

119888(119891119888119896)0315 mdash


119888(119891119888119896)0325

119891119888119896lt 84MPa

Graybeal [11] 119864119888= 3840radic119891

119888119896119891119888119896lt 200MPa

Gao et al [28] 119864119888119891= 119864119888(1 + 0173(RI)) 70MPa lt 119891

119888119896lt 85MPa

Padmarajaiah [29] 119864119888119891= 119864119888+ 2440(RI) 119891

119888119896lt 69MPa










2 4 6 8 10 12 14 160

10000

20000

30000

40000

50000

60000

70000

80000

Elas

tic m

odul

us (M

Pa)


0 500 1000 1500 2000

0

2000

4000

6000

8000

10000

Elas

tic m

odul

us (k

si)





0 50 100 150 200 2500000

0001

0002

0003

0004

0005


0 5 10 15 20 25 30 35

0

2

4

6

8

10

Stra

in at

pea

k str

ess

Stra

in at

pea

k str

ess







119888119900was










00 05 10 15 20 25 30 3509

10

11

12

13

14

15

Incr

ease

rate

of c

ompr

essiv

e stre

ngth

Reinforcing index









00 05 10 15 20 25 30 3509

10

11

12

13

14

15

16

17

18

Incr

ease

rate

of s

trai

n at

pea

k str

ess

Reinforcing index






119896119888119891

119896119888119900

= 1 + RI times 120572 (2)





00 05 10 15 20 25 30 3509

10

11

12

13

14

Incr

ease

rate

of e

lasti

c mod

ulus

Reinforcing index



20 40 60 80 100 120 140 160 180 20000

01

02

03

04

05

Incr

ease

rate






119896119888119891

119896119888119900

= 1 + RI (1198861minus 1198871119891119888119896) (3)





119888119900 and 020 and




119891119888

1198911015840119888

=119899 (1205761198881198911205761015840119888)

119899 minus 1 + (1205761198881198911205761015840119888)119899119896 (4)


119888is strain

when 119891119888reaches 1198911015840


119888(119864119888minus


119888119891is zero 1198641015840

119888is secant

stiffness when 120576119888119891



119899 = 08 +1198911015840119888

17

119896 = 067 +1198911015840119888

62

(5)



0 0002 0004 0006Strain

0

50

100

150

200

Stre

ss (M

Pa)

0

10

20

30

Stre

ss (k



(a) 119881119891 = 0

0 0002 0004 0006Strain

0

50

100

150

200

0

10

20

30

Stre

ss (M

Pa)

Stre

ss (k

si)

Stress-strain model


(b) 119881119891 = 20



5 Conclusion











Competing Interests


Acknowledgments


References






















































CeramicsJournal of







Biomaterials




Journal of


Journal of





CoatingsJournal of

Advances in








MaterialsJournal of


Nano

materials





ID 119881119891

119864119888

119891119888119900

120576119888119900


Com

pres

sive s

tress

(MPa

)

Com

pres

sive s

tress

(ksi)

Strain


Strength of matrix

Strength of matrix

35

30

25

20

15

10

5

0

250

200

150

100

50

00000 0002 0004 0006 0008 0010 0012 0014

fck = 200MPa

fck = 80MPa







RI =119881119891119871119891119889119891

119863119891

(1)










reng

th (M

Pa)

Stre

ngth

(ksi)


300

250

200

150

100

50

000 05 10 15 20

40

35

30

25

20

15

10

5

0







50000

45000

40000

35000

30000

25000

20000

15000

Elas

tic m

odul

us (M

Pa)

Elas

tic m

odul

us (k

si)

7000

6000

5000

4000

3000

(b) Elastic modulus




6

4

2

0

6

5

4

3

2

1

0

Stra

in at

pea

k str

ess(10minus3)

Stra

in at

pea

k str

ess(10minus3)













Collins et al [6] 120576119888119900=119891119888119896

119864119888

119899

119899 minus 1 119899 = 08 +

119891119888119896

17 119864119888= 332011989105

119888119896+ 6900 119891

119888119896le 100

Wee et al [7] 120576119888119900=180 (119891

119888119896)025

106119891119888119896le 120

Ros [16] 120576119888119900=(00546 + 0003713119891

119888119896)

100119891119888119896le 43


119888119896+ 000195 119891

119888119896le 66

De Nicolo et al [18] 120576119888119900= 000076 + [(0626119891


05

119891119888119896le 90

CEB-fip 228 [9] 120576119888119900=07 (119891

119888119896)031

1000119891119888119896le 100

EC2 [19] 120576119888119900=(20 + 0085 (119891

119888119896minus 50)

053

)

1000 for 119891

119888119896ge 50MPa otherwise 120576

119888119900= 0002 119891

119888119896le 90


119881119891119871119891

119863119891

mdash


119882119891119871119891

119863119891

119891119888119896le 50

Dhakal et al [22] 120576119888119900119891= 0002 + 2000120572

1198881198812119891 120572119891= 10 (flat) 120572

119891= 24 (hooked) 119891

119888119896le 40




119888= 8500 3radic119891

119888119906 119864119888= 007711990815

119888

3radic119891119888119906 119891119888119906= 119891119888119896+ 8



119888119906 119864119888= 007711990815

119888

3radic119891119888119906 119891119888119906= 119891119888119896+ Δ119891 where


119888119896gt 60MPa


119888= 4700radic119891


119888= 119908151198880043radic119891

119888119896

CEB-fip 228 [9] 119864119888119894= 119864119888119900[(119891119888119896+ Δ119891)

119891119888119898119900

]

13

Δ119891 = 8 119891119888119898119900= 10 119864

119888119900= 22GPa 119891

119888119896le 90


119888119896+ 6900 21MPa lt 119891

119888119896lt 83MPa

Cook [26] 119864119888= 3385 times 10minus5119908255

119888(119891119888119896)0315 mdash


119888(119891119888119896)0325

119891119888119896lt 84MPa

Graybeal [11] 119864119888= 3840radic119891

119888119896119891119888119896lt 200MPa

Gao et al [28] 119864119888119891= 119864119888(1 + 0173(RI)) 70MPa lt 119891

119888119896lt 85MPa

Padmarajaiah [29] 119864119888119891= 119864119888+ 2440(RI) 119891

119888119896lt 69MPa










2 4 6 8 10 12 14 160

10000

20000

30000

40000

50000

60000

70000

80000

Elas

tic m

odul

us (M

Pa)


0 500 1000 1500 2000

0

2000

4000

6000

8000

10000

Elas

tic m

odul

us (k

si)





0 50 100 150 200 2500000

0001

0002

0003

0004

0005


0 5 10 15 20 25 30 35

0

2

4

6

8

10

Stra

in at

pea

k str

ess

Stra

in at

pea

k str

ess







119888119900was










00 05 10 15 20 25 30 3509

10

11

12

13

14

15

Incr

ease

rate

of c

ompr

essiv

e stre

ngth

Reinforcing index









00 05 10 15 20 25 30 3509

10

11

12

13

14

15

16

17

18

Incr

ease

rate

of s

trai

n at

pea

k str

ess

Reinforcing index






119896119888119891

119896119888119900

= 1 + RI times 120572 (2)





00 05 10 15 20 25 30 3509

10

11

12

13

14

Incr

ease

rate

of e

lasti

c mod

ulus

Reinforcing index



20 40 60 80 100 120 140 160 180 20000

01

02

03

04

05

Incr

ease

rate






119896119888119891

119896119888119900

= 1 + RI (1198861minus 1198871119891119888119896) (3)





119888119900 and 020 and




119891119888

1198911015840119888

=119899 (1205761198881198911205761015840119888)

119899 minus 1 + (1205761198881198911205761015840119888)119899119896 (4)


119888is strain

when 119891119888reaches 1198911015840


119888(119864119888minus


119888119891is zero 1198641015840

119888is secant

stiffness when 120576119888119891



119899 = 08 +1198911015840119888

17

119896 = 067 +1198911015840119888

62

(5)



0 0002 0004 0006Strain

0

50

100

150

200

Stre

ss (M

Pa)

0

10

20

30

Stre

ss (k



(a) 119881119891 = 0

0 0002 0004 0006Strain

0

50

100

150

200

0

10

20

30

Stre

ss (M

Pa)

Stre

ss (k

si)

Stress-strain model


(b) 119881119891 = 20



5 Conclusion











Competing Interests


Acknowledgments


References






















































CeramicsJournal of







Biomaterials




Journal of


Journal of





CoatingsJournal of

Advances in








MaterialsJournal of


Nano

materials




reng

th (M

Pa)

Stre

ngth

(ksi)


300

250

200

150

100

50

000 05 10 15 20

40

35

30

25

20

15

10

5

0







50000

45000

40000

35000

30000

25000

20000

15000

Elas

tic m

odul

us (M

Pa)

Elas

tic m

odul

us (k

si)

7000

6000

5000

4000

3000

(b) Elastic modulus




6

4

2

0

6

5

4

3

2

1

0

Stra

in at

pea

k str

ess(10minus3)

Stra

in at

pea

k str

ess(10minus3)













Collins et al [6] 120576119888119900=119891119888119896

119864119888

119899

119899 minus 1 119899 = 08 +

119891119888119896

17 119864119888= 332011989105

119888119896+ 6900 119891

119888119896le 100

Wee et al [7] 120576119888119900=180 (119891

119888119896)025

106119891119888119896le 120

Ros [16] 120576119888119900=(00546 + 0003713119891

119888119896)

100119891119888119896le 43


119888119896+ 000195 119891

119888119896le 66

De Nicolo et al [18] 120576119888119900= 000076 + [(0626119891


05

119891119888119896le 90

CEB-fip 228 [9] 120576119888119900=07 (119891

119888119896)031

1000119891119888119896le 100

EC2 [19] 120576119888119900=(20 + 0085 (119891

119888119896minus 50)

053

)

1000 for 119891

119888119896ge 50MPa otherwise 120576

119888119900= 0002 119891

119888119896le 90


119881119891119871119891

119863119891

mdash


119882119891119871119891

119863119891

119891119888119896le 50

Dhakal et al [22] 120576119888119900119891= 0002 + 2000120572

1198881198812119891 120572119891= 10 (flat) 120572

119891= 24 (hooked) 119891

119888119896le 40




119888= 8500 3radic119891

119888119906 119864119888= 007711990815

119888

3radic119891119888119906 119891119888119906= 119891119888119896+ 8



119888119906 119864119888= 007711990815

119888

3radic119891119888119906 119891119888119906= 119891119888119896+ Δ119891 where


119888119896gt 60MPa


119888= 4700radic119891


119888= 119908151198880043radic119891

119888119896

CEB-fip 228 [9] 119864119888119894= 119864119888119900[(119891119888119896+ Δ119891)

119891119888119898119900

]

13

Δ119891 = 8 119891119888119898119900= 10 119864

119888119900= 22GPa 119891

119888119896le 90


119888119896+ 6900 21MPa lt 119891

119888119896lt 83MPa

Cook [26] 119864119888= 3385 times 10minus5119908255

119888(119891119888119896)0315 mdash


119888(119891119888119896)0325

119891119888119896lt 84MPa

Graybeal [11] 119864119888= 3840radic119891

119888119896119891119888119896lt 200MPa

Gao et al [28] 119864119888119891= 119864119888(1 + 0173(RI)) 70MPa lt 119891

119888119896lt 85MPa

Padmarajaiah [29] 119864119888119891= 119864119888+ 2440(RI) 119891

119888119896lt 69MPa










2 4 6 8 10 12 14 160

10000

20000

30000

40000

50000

60000

70000

80000

Elas

tic m

odul

us (M

Pa)


0 500 1000 1500 2000

0

2000

4000

6000

8000

10000

Elas

tic m

odul

us (k

si)





0 50 100 150 200 2500000

0001

0002

0003

0004

0005


0 5 10 15 20 25 30 35

0

2

4

6

8

10

Stra

in at

pea

k str

ess

Stra

in at

pea

k str

ess







119888119900was










00 05 10 15 20 25 30 3509

10

11

12

13

14

15

Incr

ease

rate

of c

ompr

essiv

e stre

ngth

Reinforcing index









00 05 10 15 20 25 30 3509

10

11

12

13

14

15

16

17

18

Incr

ease

rate

of s

trai

n at

pea

k str

ess

Reinforcing index






119896119888119891

119896119888119900

= 1 + RI times 120572 (2)





00 05 10 15 20 25 30 3509

10

11

12

13

14

Incr

ease

rate

of e

lasti

c mod

ulus

Reinforcing index



20 40 60 80 100 120 140 160 180 20000

01

02

03

04

05

Incr

ease

rate






119896119888119891

119896119888119900

= 1 + RI (1198861minus 1198871119891119888119896) (3)





119888119900 and 020 and




119891119888

1198911015840119888

=119899 (1205761198881198911205761015840119888)

119899 minus 1 + (1205761198881198911205761015840119888)119899119896 (4)


119888is strain

when 119891119888reaches 1198911015840


119888(119864119888minus


119888119891is zero 1198641015840

119888is secant

stiffness when 120576119888119891



119899 = 08 +1198911015840119888

17

119896 = 067 +1198911015840119888

62

(5)



0 0002 0004 0006Strain

0

50

100

150

200

Stre

ss (M

Pa)

0

10

20

30

Stre

ss (k



(a) 119881119891 = 0

0 0002 0004 0006Strain

0

50

100

150

200

0

10

20

30

Stre

ss (M

Pa)

Stre

ss (k

si)

Stress-strain model


(b) 119881119891 = 20



5 Conclusion











Competing Interests


Acknowledgments


References






















































CeramicsJournal of







Biomaterials




Journal of


Journal of





CoatingsJournal of

Advances in








MaterialsJournal of


Nano

materials






Collins et al [6] 120576119888119900=119891119888119896

119864119888

119899

119899 minus 1 119899 = 08 +

119891119888119896

17 119864119888= 332011989105

119888119896+ 6900 119891

119888119896le 100

Wee et al [7] 120576119888119900=180 (119891

119888119896)025

106119891119888119896le 120

Ros [16] 120576119888119900=(00546 + 0003713119891

119888119896)

100119891119888119896le 43


119888119896+ 000195 119891

119888119896le 66

De Nicolo et al [18] 120576119888119900= 000076 + [(0626119891


05

119891119888119896le 90

CEB-fip 228 [9] 120576119888119900=07 (119891

119888119896)031

1000119891119888119896le 100

EC2 [19] 120576119888119900=(20 + 0085 (119891

119888119896minus 50)

053

)

1000 for 119891

119888119896ge 50MPa otherwise 120576

119888119900= 0002 119891

119888119896le 90


119881119891119871119891

119863119891

mdash


119882119891119871119891

119863119891

119891119888119896le 50

Dhakal et al [22] 120576119888119900119891= 0002 + 2000120572

1198881198812119891 120572119891= 10 (flat) 120572

119891= 24 (hooked) 119891

119888119896le 40




119888= 8500 3radic119891

119888119906 119864119888= 007711990815

119888

3radic119891119888119906 119891119888119906= 119891119888119896+ 8



119888119906 119864119888= 007711990815

119888

3radic119891119888119906 119891119888119906= 119891119888119896+ Δ119891 where


119888119896gt 60MPa


119888= 4700radic119891


119888= 119908151198880043radic119891

119888119896

CEB-fip 228 [9] 119864119888119894= 119864119888119900[(119891119888119896+ Δ119891)

119891119888119898119900

]

13

Δ119891 = 8 119891119888119898119900= 10 119864

119888119900= 22GPa 119891

119888119896le 90


119888119896+ 6900 21MPa lt 119891

119888119896lt 83MPa

Cook [26] 119864119888= 3385 times 10minus5119908255

119888(119891119888119896)0315 mdash


119888(119891119888119896)0325

119891119888119896lt 84MPa

Graybeal [11] 119864119888= 3840radic119891

119888119896119891119888119896lt 200MPa

Gao et al [28] 119864119888119891= 119864119888(1 + 0173(RI)) 70MPa lt 119891

119888119896lt 85MPa

Padmarajaiah [29] 119864119888119891= 119864119888+ 2440(RI) 119891

119888119896lt 69MPa










2 4 6 8 10 12 14 160

10000

20000

30000

40000

50000

60000

70000

80000

Elas

tic m

odul

us (M

Pa)


0 500 1000 1500 2000

0

2000

4000

6000

8000

10000

Elas

tic m

odul

us (k

si)





0 50 100 150 200 2500000

0001

0002

0003

0004

0005


0 5 10 15 20 25 30 35

0

2

4

6

8

10

Stra

in at

pea

k str

ess

Stra

in at

pea

k str

ess







119888119900was










00 05 10 15 20 25 30 3509

10

11

12

13

14

15

Incr

ease

rate

of c

ompr

essiv

e stre

ngth

Reinforcing index









00 05 10 15 20 25 30 3509

10

11

12

13

14

15

16

17

18

Incr

ease

rate

of s

trai

n at

pea

k str

ess

Reinforcing index






119896119888119891

119896119888119900

= 1 + RI times 120572 (2)





00 05 10 15 20 25 30 3509

10

11

12

13

14

Incr

ease

rate

of e

lasti

c mod

ulus

Reinforcing index



20 40 60 80 100 120 140 160 180 20000

01

02

03

04

05

Incr

ease

rate






119896119888119891

119896119888119900

= 1 + RI (1198861minus 1198871119891119888119896) (3)





119888119900 and 020 and




119891119888

1198911015840119888

=119899 (1205761198881198911205761015840119888)

119899 minus 1 + (1205761198881198911205761015840119888)119899119896 (4)


119888is strain

when 119891119888reaches 1198911015840


119888(119864119888minus


119888119891is zero 1198641015840

119888is secant

stiffness when 120576119888119891



119899 = 08 +1198911015840119888

17

119896 = 067 +1198911015840119888

62

(5)



0 0002 0004 0006Strain

0

50

100

150

200

Stre

ss (M

Pa)

0

10

20

30

Stre

ss (k



(a) 119881119891 = 0

0 0002 0004 0006Strain

0

50

100

150

200

0

10

20

30

Stre

ss (M

Pa)

Stre

ss (k

si)

Stress-strain model


(b) 119881119891 = 20



5 Conclusion











Competing Interests


Acknowledgments


References






















































CeramicsJournal of







Biomaterials




Journal of


Journal of





CoatingsJournal of

Advances in








MaterialsJournal of


Nano

materials




2 4 6 8 10 12 14 160

10000

20000

30000

40000

50000

60000

70000

80000

Elas

tic m

odul

us (M

Pa)


0 500 1000 1500 2000

0

2000

4000

6000

8000

10000

Elas

tic m

odul

us (k

si)





0 50 100 150 200 2500000

0001

0002

0003

0004

0005


0 5 10 15 20 25 30 35

0

2

4

6

8

10

Stra

in at

pea

k str

ess

Stra

in at

pea

k str

ess







119888119900was










00 05 10 15 20 25 30 3509

10

11

12

13

14

15

Incr

ease

rate

of c

ompr

essiv

e stre

ngth

Reinforcing index









00 05 10 15 20 25 30 3509

10

11

12

13

14

15

16

17

18

Incr

ease

rate

of s

trai

n at

pea

k str

ess

Reinforcing index






119896119888119891

119896119888119900

= 1 + RI times 120572 (2)





00 05 10 15 20 25 30 3509

10

11

12

13

14

Incr

ease

rate

of e

lasti

c mod

ulus

Reinforcing index



20 40 60 80 100 120 140 160 180 20000

01

02

03

04

05

Incr

ease

rate






119896119888119891

119896119888119900

= 1 + RI (1198861minus 1198871119891119888119896) (3)





119888119900 and 020 and




119891119888

1198911015840119888

=119899 (1205761198881198911205761015840119888)

119899 minus 1 + (1205761198881198911205761015840119888)119899119896 (4)


119888is strain

when 119891119888reaches 1198911015840


119888(119864119888minus


119888119891is zero 1198641015840

119888is secant

stiffness when 120576119888119891



119899 = 08 +1198911015840119888

17

119896 = 067 +1198911015840119888

62

(5)



0 0002 0004 0006Strain

0

50

100

150

200

Stre

ss (M

Pa)

0

10

20

30

Stre

ss (k



(a) 119881119891 = 0

0 0002 0004 0006Strain

0

50

100

150

200

0

10

20

30

Stre

ss (M

Pa)

Stre

ss (k

si)

Stress-strain model


(b) 119881119891 = 20



5 Conclusion











Competing Interests


Acknowledgments


References






















































CeramicsJournal of







Biomaterials




Journal of


Journal of





CoatingsJournal of

Advances in








MaterialsJournal of


Nano

materials




00 05 10 15 20 25 30 3509

10

11

12

13

14

15

Incr

ease

rate

of c

ompr

essiv

e stre

ngth

Reinforcing index









00 05 10 15 20 25 30 3509

10

11

12

13

14

15

16

17

18

Incr

ease

rate

of s

trai

n at

pea

k str

ess

Reinforcing index






119896119888119891

119896119888119900

= 1 + RI times 120572 (2)





00 05 10 15 20 25 30 3509

10

11

12

13

14

Incr

ease

rate

of e

lasti

c mod

ulus

Reinforcing index



20 40 60 80 100 120 140 160 180 20000

01

02

03

04

05

Incr

ease

rate






119896119888119891

119896119888119900

= 1 + RI (1198861minus 1198871119891119888119896) (3)





119888119900 and 020 and




119891119888

1198911015840119888

=119899 (1205761198881198911205761015840119888)

119899 minus 1 + (1205761198881198911205761015840119888)119899119896 (4)


119888is strain

when 119891119888reaches 1198911015840


119888(119864119888minus


119888119891is zero 1198641015840

119888is secant

stiffness when 120576119888119891



119899 = 08 +1198911015840119888

17

119896 = 067 +1198911015840119888

62

(5)



0 0002 0004 0006Strain

0

50

100

150

200

Stre

ss (M

Pa)

0

10

20

30

Stre

ss (k



(a) 119881119891 = 0

0 0002 0004 0006Strain

0

50

100

150

200

0

10

20

30

Stre

ss (M

Pa)

Stre

ss (k

si)

Stress-strain model


(b) 119881119891 = 20



5 Conclusion











Competing Interests


Acknowledgments


References






















































CeramicsJournal of







Biomaterials




Journal of


Journal of





CoatingsJournal of

Advances in








MaterialsJournal of


Nano

materials




00 05 10 15 20 25 30 3509

10

11

12

13

14

Incr

ease

rate

of e

lasti

c mod

ulus

Reinforcing index



20 40 60 80 100 120 140 160 180 20000

01

02

03

04

05

Incr

ease

rate






119896119888119891

119896119888119900

= 1 + RI (1198861minus 1198871119891119888119896) (3)





119888119900 and 020 and




119891119888

1198911015840119888

=119899 (1205761198881198911205761015840119888)

119899 minus 1 + (1205761198881198911205761015840119888)119899119896 (4)


119888is strain

when 119891119888reaches 1198911015840


119888(119864119888minus


119888119891is zero 1198641015840

119888is secant

stiffness when 120576119888119891



119899 = 08 +1198911015840119888

17

119896 = 067 +1198911015840119888

62

(5)



0 0002 0004 0006Strain

0

50

100

150

200

Stre

ss (M

Pa)

0

10

20

30

Stre

ss (k



(a) 119881119891 = 0

0 0002 0004 0006Strain

0

50

100

150

200

0

10

20

30

Stre

ss (M

Pa)

Stre

ss (k

si)

Stress-strain model


(b) 119881119891 = 20



5 Conclusion











Competing Interests


Acknowledgments


References






















































CeramicsJournal of







Biomaterials




Journal of


Journal of





CoatingsJournal of

Advances in








MaterialsJournal of


Nano

materials




0 0002 0004 0006Strain

0

50

100

150

200

Stre

ss (M

Pa)

0

10

20

30

Stre

ss (k



(a) 119881119891 = 0

0 0002 0004 0006Strain

0

50

100

150

200

0

10

20

30

Stre

ss (M

Pa)

Stre

ss (k

si)

Stress-strain model


(b) 119881119891 = 20



5 Conclusion











Competing Interests


Acknowledgments


References






















































CeramicsJournal of







Biomaterials




Journal of


Journal of





CoatingsJournal of

Advances in








MaterialsJournal of


Nano

materials





Competing Interests


Acknowledgments


References






















































CeramicsJournal of







Biomaterials




Journal of


Journal of





CoatingsJournal of

Advances in








MaterialsJournal of


Nano

materials

























CeramicsJournal of







Biomaterials




Journal of


Journal of





CoatingsJournal of

Advances in








MaterialsJournal of


Nano

materials










CeramicsJournal of







Biomaterials




Journal of


Journal of





CoatingsJournal of

Advances in








MaterialsJournal of


Nano

materials



Documents

Research Article Compressive Behavior and Mechanical ...downloads.hindawi.com/journals/amse/2016/6465218.pdfFiber-Reinforced Reactive Powder Concrete. Figure shows typical stress-strain