Construction and Building Materials 70 (2014) 243–253

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Construction and Building Materials

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Backcalculation of residual tensile strength of regular and highperformance fiber reinforced concrete from flexural tests

http://dx.doi.org/10.1016/j.conbuildmat.2014.07.0370950-0618/� 2014 Elsevier Ltd. All rights reserved.

⇑ Corresponding author. Tel.: +1 (480)965 0141; fax: +1 (480)965 0557.E-mail address: barzin@asu.edu (B. Mobasher).

Barzin Mobasher a,⇑, Mehdi Bakhshi b, Christopher Barsby c

a School of Sustainable Engineering and Built Environment, Arizona State University, Tempe, AZ 85287-5306, United Statesb AECOM, New York, NY 10005, United Statesc PK Associates Structural Engineers, Scottsdale, AZ 85250, United States

h i g h l i g h t s

� Closed form equations for measuring tensile constitutive response from flexural tests.� Parameters obtained from routine experimental data can be used for design of FRC elements.� Correlation of backcalculated tensile data from flexural and direct tension tests.� Comparison of nature of the stress distribution under the two tension and flexural tests.� Residual tensile strength, and post crack stiffness correlated with the fiber type and content.

a r t i c l e i n f o

Article history:Received 4 October 2013Received in revised form 4 July 2014Accepted 16 July 2014

Keywords:ConcreteFiberFlexural behaviorMoment–curvature responsePostcracking tensile strengthStress–strain response

a b s t r a c t

The tensile stress–strain response of a fiber reinforced concrete dominates the performance under manyloading conditions and applications. To represent this property as an average equivalent response, aback-calculation process from flexural testing is employed. The procedure is performed by model fittingof the three-point and four-point bending load deflection data on two types of macro synthetic polymericfibers, one type of steel fiber and one type of Alkali Resistant (AR) glass fiber. A strain softening tensilemodel is used to simulate the behavior of different FRC types and obtain the experimental flexuralresponse. The stress–strain model for each age, fiber type and dosage rate is simulated by means ofthe inverse analysis procedure, using closed-form moment–curvature relationship and load–deflectionresponse of the piecewise-linear material. The method of approach is further applied to one external dataset for High Performance Fiber Reinforced Concrete (HPFRC) with two different types of steel fibers andvalidated by tensile test results reported. Results of back-calculation of stress–strain responses bytri-linear tensile model for all mixtures are compared and correlated with the corresponding standardmethod parameters used for post crack behavior characterization and a regression analysis for compar-ative evaluation of test data is presented.

� 2014 Elsevier Ltd. All rights reserved.

1. Introduction

Fiber reinforced concrete is widely used in infrastructureapplications because of improved mechanical properties such asfracture toughness, ductility, durability, and crack-width control[1–5]. Steel, glass, natural, and synthetic fibers have been used over40 years in industrial slabs, floors, and pavements to primarilyreduce shrinkage and thermal cracking [6–11], reduce the requiredslab thickness, and increase the allowable joint spacing [12–18].Experimental tests show that fibers increase the flexural and

ultimate load carrying capacity in proportion to their volume andaspect ratio [19–24]. Fiber reinforced concrete is used in elevatedslabs and water distribution infrastructure. Structural applicationsof fibers include but are not limited to precast structural elements[25], tunnel linings [26,27], shotcrete [28–32], offshore structures,seismic applications, thin and thick repairs [33], crash barriers,footings, and hydraulic structures [34,35]. The fibers are also addedto concrete to enhance spalling resistance during exposure to hightemperature [36].

The mechanical properties depend on the characteristics of theconcrete matrix but also on the type and geometry of the fibersthat governs their bond mechanism with the matrix [37,38]. Fibersoffer increased abrasion and impact resistance as well [39,40]. The

244 B. Mobasher et al. / Construction and Building Materials 70 (2014) 243–253

effectiveness of short, randomly distributed fibers may be superiorto other forms of reinforcement such as welded wire mesh, orrebars since the small diameter of the individual fibers ensures amore uniform dispersion, along with a far superior bond strength.Moreover, due to the reduced specific spacing, fibers strengthenthe composite at the micro level by bridging the microcracksbefore they reach the critical flaw size [41]. Among all mechanicalparameters, residual tensile strength and toughness are the mostimproved parameters which are a direct consequence of macrofiber bridging mechanisms across the crack surfaces [42,43].Hybrid fiber reinforced concrete combining micro- with macro-fibers with an improved resistance against both types of cracks isalso useful for a variety of applications, including thin repairsand patching [44,45].

Flexural tests are routinely done as a means of quality controland limited material properties are extracted from their results.Furthermore, the scatter and variations in these tests due tonotched or un-notched samples, or the choice of control variableused in experiments, are compounded by the methods used toreport the results especially in the post-peak region. For example,scatter is much smaller for synthetic fibers than steel fibers due tothe higher number and more homogeneous distribution across thefracture surface [42]. Scatter is also lower for samples tested asround panel specimens tested under ASTM C1550 than ASTMC1609 beam specimens [46]. Scatter in the case of ASTM C1609may also be attributed to the degree of rigidity of the support reac-tions, or frictional sliding at the supports. There is a need to betterutilize the flexural test data for realistic materials property.

This paper validates a back-calculation procedure for flexuraltest results and obtains tension stress–strain response from avariety of tests conducted on notched and un-notched beams ofdifferent sizes, fiber types, shapes, lengths, and dosage rates. Theobjective is to correlate the residual strength results with empiricalresidual strength methods of ASTM C1609 [46], RILEM TC 162-TDF[47], and JCI-SF4 [48] which propose calculation of residualstrength using simple engineering bending theory for linear elasticmaterials and uncracked section properties. A database used foranalysis containing three internal data sets for tests conductedon polymeric, AR Glass and steel fibers at the Structural Engineer-ing Laboratory at the Arizona State University, and one externaldata set for reported test results of Kim et al. [49] on HighPerformance Fiber Reinforced Concretes (HPFRCs). A correlationis studied between backcalculated residual strengths and various

Table 1Properties of fibers used in study.

Fiber type P-type ABase Monofilament polypropylene/polyethylene blend

Length (mm) 50Density (g/cm3) 0.92Tensile strength (MPa) 600–650Elastic modulus (GPa) 5

Table 2Mixture proportions and compressive strength of all mixtures.

Set Mix ID Portlandcement (kg/m3)

Fly ash(kg/m3)

Silica fume(kg/m3)

Fine aggregate(kg/m3)

Coa(kg/

1 P3-A 475 60 15 1100 45P3-B 475 60 15 1100 45

2 G6-6 796 80 0 578 76G6-12 796 80 0 578 76G6-24 796 80 0 578 76

3 S13-HL 380 125 0 1343 181S26-HL 380 125 0 1343 181S39-HL 380 125 0 1343 181

standard flexural parameters. In lieu of empirical correlation val-ues between these parameters that are currently in use in theFRC industry, this paper provides a theoretical approach to obtainsuch correlation factor.

2. Materials and methods

2.1. Flexural tests

Set one of internal database consisted of two polymeric fibers of modifiedpolypropylene, polyethylene and olefin blends, both at a dosage rate of 3 kg/m3

(5 lb/yd3). Set two consisted of AR Glass fibers at three different fiber lengths, andSet 3 consisted of one type of steel fibers at three different dosage rates. All sampleswere tested under flexural testing configuration and the load-deformation responsein the post-peak region was measured. Physical and mechanical properties of thefibers used in the test program are presented in Table 1. The analysis section alsodiscusses results from published work on four different mixtures of HPFRC byKim et al. [49]. This was designated as Set 4 and included both tensile and flexuraltest results.

2.2. Testing program

Proportions of eight different mixtures prepared and tested under three-pointbending configuration are shown in Table 2. The first letter on the samples’ labelsrefers to the general type of fiber used, i.e. ‘‘P’’ in case of polymeric, ‘‘G’’ in case ofglass fiber and ‘‘S’’ in case of steel fiber. The following number is the dosage ofthe fiber presented in kg/m3. For polymeric and steel fibers, the letter following thisnumber refers to the type of fibers shown in Table 1, while for glass fibers; the num-ber following this number is the length of fiber. In the results section, a final num-ber added at the end of the labels designates the age at testing. In addition to thesamples tested, one set of published HPFRC data by Kim et al. [49] was used withemployed two different types of steel fibers, ‘‘H’’ for hooked fibers and designation‘‘T’’ was introduced to refer to longitudinally twisted fibers. Subsequently, parame-ter ‘‘L’’ refers to large size of specimen with depth, width and span of 150, 150 and450 mm, respectively, to differentiate the results from results of medium size spec-imens reported by Kim et al. [49].

Closed loop control flexural tests were conducted on pre-notched FRC samplesof polymeric and AR glass fibers in accordance with RILEM TC 162-TDF recommen-dation in order to monitor post-peak response [47]. Dimensions of Set 1 Polymeric-FRC sample and Set 2 AR glass-FRC samples were 450 mm � 100 mm � 100 mmwith an initial notch length of 12 mm and test span of 400 mm. Un-notchedsteel-FRC samples in Set 3 were tested in accordance with ASTM C1609 underfour-point bending loading configuration using 510 mm � 150 mm � 150 mmspecimens with a test span of 450 mm. The diameter of steel fibers used was0.3 mm. Test setup, specimen dimensions and instrumentation are shown in Fig. 1.

Tests were performed under closed loop control with Crack Mouth OpeningDeformation (CMOD) as the controlled variable for testing sets one and two, andload point deflection as the controlled variable for testing set three. Both the CMODand deflection were measured using a Linear Variable Differential Transformer(LVDT) with a working range of 2.5 mm. In notched specimens, cracks initiatedfrom the notch and extended up along the depth of the beam. The crack opening

P-type B Glass (G) Steel (S)Modified olefin Alkali resistant glass Hooked (H)

50 6, 12, 24 500.92 2.7 7.9552 1724 230010 69 200

rse aggregatem3)

Water(kg/m3)

Fiber type/dosage(kg/m3)

w/c s/c Compressivestrength (MPa)

0 230 P-A/3 0.42 2 290 230 P-B/3 0.42 2 34

0 350 G-6 mm/10 0.4 0.66 410 350 G-12 mm/10 0.4 0.66 410 350 G-24 mm/10 0.4 0.66 41

6 242 S-HL/13 0.48 2.66 286 242 S-HL/26 0.48 2.66 286 242 S-HL/39 0.48 2.66 28

Fig. 1. Test setup for three-point bend notched flexural test.

B. Mobasher et al. / Construction and Building Materials 70 (2014) 243–253 245

was resisted by bridging fibers which pulled out under this loading. The presence offiber significantly increases the ductility and resulted in a stable crack opening up tohigh range of deflections. The load–deflection curve is characterized by the maxi-mum load and its associated deflection, elastic stiffness, maximum flexural strengthand flexural toughness. The post-peak behavior of the samples was also reported aselastically equivalent residual strengths as measured by three alternative methodsof ASTM C1609 (f D

150), JCI-SF4 (rb) and RILEM TC 162-TDF (feq,3).

3. Test results

Results of experimental analysis on three- and four-pointbending tests on different macro synthetic, glass and steel fibersare summarized in Table 3. A wide range of responses in the datasuch as apparent flexural strength and toughness correlate withthe fiber type, fiber content, loading rate, and age at testing. Thetoughness measure is obtained as the area under the load deflec-tion curve and as an age-dependent property correlates withstrength gain. Therefore, toughness after 28 days of curing wasused as the control and results obtained after 14 and 56 days forsteel fibers were correlated with the 28 day results.

Effect of curing duration on flexural response of polymeric fibertypes A and B in Set 1 are shown in Fig. 2a. Average elastic flexuralstiffness of P3-A samples increased by 20%. The maximum loadincreased by 22%, and the deflection associated with the maximumload level did not change significantly, while the toughness dou-bled from 14 to 28 days. The increase in apparent flexural strengthfrom 14 to 28 days was from 1.57 to 1.92 MPa (+22%). In P3-Bsamples with 3 kg/m3 of type B polymeric fibers, elastic flexural

Table 3Summary of average experimental analysis for all FRC samples.

Set Sample ID Age (days) Elastic flexuralstiffness (kN/mm)

Deflection at mflexural load (m

1 P3-A-14d 14 149 0.067P3-A-28d 28 180 0.069P3-B-14d 14 149 0.068P3-B-28d 28 189 0.059

2 G6-6-28d 28 249 0.050G6-12-28d 28 249 0.048G6-25-28d 28 249 0.050

3 S13-HL-28d 28 822 0.043S13-HL-56d 56 822 0.042S26-HL-28d 28 822 0.054S26-HL-56d 56 731 0.052S39-HL-28d 28 548 0.082S39-HL-56d 56 548 0.080

stiffness increased by 27%. The maximum load increased by 18%and its associated deflection decreased slightly by 13%, while thetoughness increased from 2.2 to 4.3 kN mm (+95%) from 14 to28 days.

It is clear that the primary parameter that differentiates amongthe age of these systems is the toughness which is affected by thepost cracking response. When overall toughness is specified as adesign parameter, standard procedures can be utilized to selectfiber type, length, and volume content. Such procedures, however,are costly and vary for each fiber type. It would be ideal to developa procedure to back-calculate the tensile response from each flex-ural test so that the design procedures can utilize these results.

Effect of fiber length at 28 days on flexural response of AR-glassfibers in Set 2 are shown in Fig. 2b for three different fiber lengthsof 6, 12 and 24 mm. Results show that the glass fiber length doesnot affect the elastic flexural stiffness and deflections at maximumflexural load; however, flexural strength increased by 12% from5.79 to 6.50 MPa at 28 days as the fiber length changes from 12to 25 mm. The flexural toughness is showing a marginal decreasewith increasing fiber length.

Comparative evaluation of the mixtures shows that there is nodiscerning of the effect of fiber type at these loading levels, and asshown in Fig. 2, minimal changes are observed for all mixes of Set1, namely P3-A, P3-B, and Set 2 AR-Glass fiber mixtures. As far asflexural strength is concerned, little or no effect on the effect ofage or fiber length is observed. However, the flexural toughnessis clearly affected in the case of polymeric fibers. Therefore, it isconcluded that the general increase in the post peak response from14 to 28 days is the main parameter affected by the curingduration.

Effect of steel fiber dosage rate on samples of testing Set 3 isshown in Fig. 3. Note that for the low fiber contents in the rangeof 13–26 kg/m3, the effect of steel fibers is observed in the postcrack response while as the fiber content increases, behaviorchanges from strain softening to strain hardening. The transitionfrom strain softening to hardening is best shown by the increasein the ultimate strength and post crack resistance in terms oftoughness.

4. Analysis

4.1. Strain softening and hardening models

A formulation is presented to back-calculate material propertiesby fitting experimental data with a closed form relationship of theload deflection using a nonlinear material model [50–52]. Theadaptation of this tri-linear model provides a precise correlationof the flexural response to back-calculate material parameters

axm)

Maximum flexuralload (kN)

Bending strength,fp (MPa)

Flexural toughness,T (kN mm)

5.63 1.57 2.066.89 1.92 4.355.17 1.44 2.196.08 1.70 4.28

7.73 5.87 1.337.63 5.79 1.058.56 6.50 0.95

23.34 3.01 32.2721.18 2.74 31.3626.07 3.10 46.2024.93 3.22 77.9825.07 3.24 122.6022.23 2.87 109.37

Fig. 2. (a) Effect of curing time on load deflection response for polymeric fiber typeA and B with fiber content of 3 kg/m3 (Set 1), (b) Effect of fiber type on loaddeflection response for glass with three different lengths at 6 kg/m3 at age of28 days (Set 2).

Fig. 3. Effect of fiber dosage on load deflection response for hooked steel fibers withlarge sample size at 28 days (Set 3).

Fig. 4. Material models for FRC materials: (a) tension for strain softeningcomposites, (b) compression.

246 B. Mobasher et al. / Construction and Building Materials 70 (2014) 243–253

and could explain the differences between the tensile and flexuralstrengths of strain hardening and strain softening composites[53,54].

Fig. 4 presents the constitutive model for homogenized strainsoftening/hardening fiber reinforced concrete. The tension model

in Fig. 4a is described by a tri-linear response with an elastic rangeE, first cracking tensile strain (ecr) and post cracking modulusEcr = gE , which g is assigned a negative or positive scalar valuein order to simulate either strain softening or hardening materials.The third region in the tensile response is a constant stress rangedefined with stress rcst in the post crack region. Two strain mea-sures define the first cracking and transition strains (ecr, etrn). Thetensile response terminates at the ultimate tensile strain level ofetu. The linear portion of an elastic-perfectly-plastic compressivestress–strain response with compressive modulus of Ec terminatesat yield point (ecy, rcy). The response remains constant at acompressive yield stress level of rcy until reaching the ultimatecompressive strain ecu shown in Fig. 4b. To convert this approachinto a closed-form solution of moment–curvature response andload deflection calculation, parameters must be expressed in nor-malized terms. Two intrinsic material parameters of first crackingtensile strain ecr and tensile modulus E are used to define sevennormalized parameters as shown in Fig. 4a and b and Eq. (1):

x ¼ ecy

ecr; a ¼ etrn

ecr; btu ¼

etu

ecr; kcu ¼

ecu

ecr; c ¼ Ec

E; g ¼ Ecr

E;

l ¼ rcst

Eecrð1Þ

In a flexural test, the moment curvature diagram for a rectangu-lar cross section with a width b and depth d is derived using theassumption of plane sections remaining plane. The maximum ten-sile strain b and maximum compressive strain k are linearly relatedto the neutral axis depth ratio, k as presented in Eq. (2). ectop andetbot are the compressive strain at the top fiber and the tensilestrain at the bottom fiber, respectively.

b ¼ etbot

ecr; k ¼ ectop

ecr;

kecr

kd¼ becr

d� kdor k ¼ k

1� kb ð2Þ

Using the normalized parameters defined in Eqs. (1) and (2), thecompressive stress (rc), the tensile stress (rt) and the toughness(Gf) are expressed as:

rc kð ÞEecr

¼ck 0� k�xcx x< k� kcu

0 kcu < k

8><>:

rt bð ÞEecr

¼

b 0� b�11þg b�1ð Þ 1< b�al a< b� btu

0 btu� b

8>>><>>>:

ð3Þ

Gf ¼12

Ee2cr a� lð1þ a� 2btuÞ½ � ð4Þ

By assuming linear strain distribution across the depth andignoring shear deformations, stress distribution across the crosssection at three stages of imposed tensile strain: 0 6 b 6 1,1 < b 6 a and a < b 6 btu are obtained in closed form [53]. Internalmoment is obtained using the force components and their distancefrom the neutral axis and the curvature is determined as the ratioof compressive strain at top fiber (ectop = kecr) to the depth ofneutral axis kd. The moment Mi and curvature /i at each stage i

ε λεctop cr=

ε βεtbot cr=

1

1hc1

ht1

kh

h1

1yc1

yt1

Fc1

yt2ft1

fc1

2 ht2

εcrFt22

ft2

Ft1

2.1

(a)

ε λεctop cr=

ε βεtbot cr=

1

1hc1

ht1

kh

h 1

1yc1

yt1

Fc1

yt2ft1

fc1

2 ht2

εcrFt22ft2

Ft1

3.1

3 ht3 3 Ft3

yt3

ft3

εtrn

(b)

Fig. 5. Strain and stress diagrams at the post crack stage, (a) Stage 2.1 in Table 4; (b)Stage 3.1 in Table 4.

B. Mobasher et al. / Construction and Building Materials 70 (2014) 243–253 247

are then normalized with respect to the values at cracking Mcr and/cr and presented in Eqs. (5) and (6). The transition from deflectionsoftening to deflection hardening is defined by critical normalizedpost-peak tensile strength (lcrit) as defined in Eq. (7).

Mi ¼ M0Mcr ; Mcr ¼16

bd2Eecr ð5Þ

/i ¼ /0i/cr ; /cr ¼2ecr

dð6Þ

lcrit ¼x

3x� 1ð7Þ

Calculation of k, M0 and /0 for the five stages of governing strainis presented in Table 4. During stage 1, the tensile and compressivezones are both elastic with a linear moment–curvature plot andthe neutral axis at the centroid of the sample. This case continuesuntil reaching the point of first cracking. There are two potentialregions when the elastic Stage 1 ends and the tensile cracking asdefined in Stage 2 starts. The compression side may or maynot enter the plastic zone. Elastic compression shown in Fig. 5ais denoted as Stage 2.1, while tensile cracking, with thecompression in plastic range is defined as Stage 2.2 (tension-plasticcompression).

Two potential regions at the end of Stage 2 depending onwhether the transition takes place form region 2.1 or 2.2 exist.Stage 3.1 shown in Fig. 5b is an elastic response in compressionwhile plastic compression is defined as Stage 3.2. It is importantto note that depending on the relationship among material param-eters, any of the stages 2.1, and 2.2, or 3.1, and 3.2 are potentiallypossible in succession.

By applying the moment-area method to the bilinear momentcurvature response, mid-span deflection of three-point bendingtests can be derived explicitly [50]. After cracking, the curvaturedistribution depends on the normalized post-peak tensile strain.The maximum deflection during the elastic stage of loading isdetermined from the curvature at cracking (/cr) and Eq. (8). Ifl > lcrit, as the post-crack curvature increases, the moment contin-ues to increase with the deflection determined by Eq. (9). On theother hand, if l < lcrit, as the post-crack curvature increases, themoment either increases or decreases at the levels below the bilin-ear cracking moment Mcr, the deflection during this stage is deter-mined by Eq. (10), and the term Lp represents the length oflocalization zone.

dcr ¼1

12L2/cr ð8Þ

Table 4Governing equations for the calculation of k, M0 and /0 for each stage specified by strains

Stage Parameters k

1 0 < b < 1k1 ¼

12 for c ¼ 1

�1þ ffifficp�1þc for c–1

(

2.1 1 < b < a k21 ¼ D21�ffiffiffiffiffiffiffiffiffiffiffiD21cb2p

D21�cb2

0 < k < x D21 ¼ g b2 � 2bþ 1� �

þ 2b� 1

2.2 1 < b < a k22 ¼ D22D22þ2xcb,

x < k < kcu D22 ¼ D21 þ cx2

3.1 a < b < btu k31 ¼ D31�ffiffiffiffiffiffiffiffiffiffiffiD31cb2p

D31�cb2

0 < k < x D31 ¼ g a2 � 2aþ 1� �

þ 2l b� að Þ þ 2a� 1

3.2 a < b < btu k32 ¼ D32D32þ2xcb,

x < k < kcu D32 ¼ D31 þ cx2

du ¼L2

24M2u

2M2u �MuMcr �M2

cr

� �/u þ M2

u þMuMcr

� �/cr

h il > lcrit ð9Þ

du ¼/uLp

82L� Lp� �

þMu/crL12Mcr

L� 2Lp� �

l < lcrit ð10Þ

Similarly, a set of equations for the four-point bending can bewritten as

dcr ¼23

216L2/cr ð11Þ

du¼L2

216M2u

23M2u�4MuMcr�M2

cr

� �/uþ 4M2

uþ4MuMcr

� �/cr

h il>lcrit ð12Þ

du ¼5/uL2

72þMuL2/cr

27Mcrl < lcrit ð13Þ

at top and bottom fibers (c = 1 in the present study).

M0 = M/Mcr /0 = ///cr

M01 ¼2b c�1ð Þk3

1þ3k21�3k1þ1½ �

1�k1

/01 ¼b

2 1�k1ð Þ

M021 ¼2cb3�C21ð Þk3

21þ3C21k221�3C21 k21þC21

1�k21

/021 ¼b

2 1�k21ð Þ

C21 ¼ ð2b3�3b2þ1Þgþ3b2�1b2

M022 ¼ 3cxb2 þ C22

� �k2

22 � 2C22k22 þ C22, /022 ¼b

2 1�k22ð Þ

C22 ¼ C21 � cx3

b2

M031 ¼2cb3�C31ð Þk3

31þ3C31k231�3C31 k31þC31

1�k31

/031 ¼b

2 1�k31ð Þ

C31 ¼ð2a3�3a2þ1Þg�3l a2�b2ð Þþ3a2�1

b2

M032 ¼ 3cxb2 þ C32

� �k2

32 � 2C32k32 þ C32, /032 ¼b

2 1�k32ð Þ

C32 ¼ C31 � cx3

b2

248 B. Mobasher et al. / Construction and Building Materials 70 (2014) 243–253

From the approximate bilinear moment–curvature diagram, thetotal load Pi at a given stage of loading i can be calculated by Eq.(14) for /i through /u, where S = L/2 for three point bending tests,respectively.

Pi ¼2Mi

Sð14Þ

When a flexural specimen is loaded well into the post peakregion, two distinct zones develop and the deformation localizesin the cracking region, while the remainder of the specimen under-goes general unloading. To correlate the stress-crack width rela-tionship into the stress–strain approach, localization is treated asan average response within the cracking region. Results are usedas a smeared crack in conjunction with the moment–curvaturediagram to obtain load deformation behavior as presented bySoranakom and Mobasher, and Bakhshi et al. [53,55].

Fig. 6. (a) Effect of curing time on back calculated tensile stress strain response. (b)Effect of curing time on experimental and simulated load deflection response forpolymeric fibers. (Set 1).

4.2. Prediction of load–deflection response of FRC

The back-calculation procedure computes the tensile materialproperties from experimental three- and four-point bendingload–deflection data. Results of back-calculation of stress–strainresponses by trilinear tensile model for all mixtures are shown inTable 5. Fig. 6 represent the effect of curing time on the back cal-culated tensile stress–stain response and flexural load–deflectionresponse of type A and B macro synthetic fibers. The initialresponse is linear elastic up to the first crack stage at about2 MPa for 14 day and increased to 2.3–2.6 MPa for 28 day samples.After cracking, load is transferred to the fibers bridging the cracksresulting in the significant drop in the sample stiffness and increas-ing the crack width. Back-calculated tensile stress–strain responsesshow that after an average strain level of about 0.003–0.004 mm/mm, the residual strength of the macro synthetic fiber compositesreaches a constant value and that strain is maintained until 3–4%level. The post-crack residual strength at this plateau zoneincreased from about 0.4 to 0.7 MPa between 14 to 28 days.

The load versus deflection response based on the simulated fitof the data matches the experimental response as shown inFig. 6b. The overall predictions are well established. Representativeproperties for the simulation of upper and lower bound valuesobtained from these samples indicate E = 18–21 GPa, a = 30–40,l = 0.21–0.3, g = 0.02–0.026 and ecr = 107–125 lstr. In all these fits,the parameters for the ratio of compressive to tensile stiffness andstrength were held constants at c = 1, and x = 10. The limits of themodeling were set at btu = 267–406 and kcu = 70.

Back-calculated tensile stress–strain response and experimen-tal and simulated load–deflection response for AR-glass fibers are

Table 5Average back calculated tensile parameters.

Sample ID Young’smodulus

First cracktensile strain

First cracktensile strength

Post crackmodulus

Post cracktensile stre

E, GPa ecr, lstr rcr MPa g l

P3-A-14d 19 107 2.07 �0.020 0.21P3-A-28d 21 125 2.63 �0.026 0.26P3-B-14d 18 107 1.88 �0.020 0.23P3-B-28d 20 115 2.3 �0.024 0.3G6-6-28d 32 90 2.92 �0.026 0.11G6-12-28d 32 98 3.18 �0.023 0.05G6-24-28d 33 110 3.64 �0.049 0.06S13-HL-28d 31 61 1.89 �0.126 0.12S13-HL-56d 31 54 1.68 �0.106 0.15S26-HL-28d 31 63 1.95 �0.074 0.33S26-HL-56d 28 72 1.99 �0.111 0.22S39-HL-28d 21 89 1.84 �0.064 0.42S39-HL-56d 21 76 2.09 �0.049 0.41

shown in Fig. 7a and b, respectively. The tensile strength of theglass fiber campsites are affected only marginally by the fiberlength as the tensile strength increased from about 2.92 to3.6 MPa by increasing the fiber length from 6 to 24 mm. Theback-calculated tensile strength for parameter l in this case is0.11, 0.05 and 0.06, representing the effect of fiber length from 6to 12 and 24 mm and corresponds to residual tensile strength atthe plateau zone for glass fiber reinforced samples in the rangeof 0.2–0.3 MPa. Representative properties for the simulation of

ngthTransitional tensile strain Ultimate tensile strain Tensile toughness

a etrn, lstr etu, mm/mm btu Gf, MPa

40 4280 0.029 267 0.01630 3750 0.045 359 0.03440 4280 0.031 292 0.01630 3450 0.047 406 0.03535 3150 0.025 286 0.01242 4116 0.033 340 0.01120 2200 0.018 160 0.008

8 488 0.033 545 0.0089 486 0.029 543 0.008

10 630 0.032 509 0.0218 576 0.043 597 0.019

10 890 0.038 431 0.03013 988 0.041 542 0.036

B. Mobasher et al. / Construction and Building Materials 70 (2014) 243–253 249

upper and lower bound values obtained from these samples indi-cate E = 32–33 GPa, a = 20–42, l = 0.06–0.11, g = 0.023–0.049,ecr = 90–110 lstr. The simulated load–deflection responses showgood agreements with experimental data and the descending partof load–deflection response is fitted quite well.

Effect of steel fiber was evaluated using different dosages of 13,26 and 39 kg/m3 using data from Set 3. Hooked-end steel fibersdesignated as type H fiber were used in the concrete mixes pouredinto samples specified as type L specimens (450 mm � 150 mm �150 mm). At 28 days, the steel fiber reinforced samples showedincreases in flexural toughness as fiber dosage increased. This isevidenced by the calculation of the area under load–deflection dia-gram shown in Fig. 8b. Flexural toughness increased by 43% and165% by increasing steel fiber dosages from 13 kg/m3 to 26 kg/m3

and 39 kg/m3, respectively. The residual flexural loads increasedproportionally with the fiber dosage. While this improvement isclearly evident in the measured toughness (l = 0.12–0.15 to0.22–0.33 and to 0.42 for the 13, 26 and 39 kg/m3 dosages, respec-tively), first crack tensile strength is not largely affected by thefiber dosage and is stable at around 1.89, 1.95 and 1.84 MPa withincreasing fiber content. Simulations presented in Fig. 8b arereasonable fits for the 13 and 26 kg/m3 dosage curves, but fail tocapture the almost linear unloading in the post cracking regionof the 39 kg/m3. This may be attributed to uneven distribution offibers in the mix or larger concentrations of steel in the tensileregion. Representative properties for the simulation of upper andlower bound values obtained from these samples indicate E = 21–31 GPa , a = 8–13, g = 0.049–0.126, ecr = 54–89 lstr. The choice of

Fig. 7. a) Effect of fiber length on back calculated tensile stress strain response, b)Effect of fiber length on experimental and simulated load deflection response forglass fibers. (Set 2).

the model used in the back calculation procedure may be alteredusing parameters, a and, g to change from a strain softening tostrain hardening model to properly capture both the peak andresidual strength values. It is noted that there is a clear and consis-tent post crack residual strength measure that is similar to metalplasticity as the yielding behavior extends to deflections in excessof 4 mm.

4.3. Model extension to HPFRC

Kim et al. [49] performed an experimental study on the effect ofHooked (H) and twisted (T) steel fibers on flexural and tensileresponses of high strength cementitious matrix (84 MPa) with afiber content of 79 kg/m3. Flexural tests were performed on threedifferent geometries of specimens, S (small) for 50 mm �25 mm � 300 mm specimens, M (medium) for 100 mm � 100 mm� 300 mm and L (large) for 150 mm � 150 mm � 450 mm. Proper-ties of hooked fibers in this study are very similar to the ones pre-sented in the previous section, with the exception of length offibers and diameter of the hooked fibers which are 30 mm, and0.38 mm in Kim’s study, respectively. The ratio of water to cemen-titious materials was 0.26, and other details of mix design can befound in the reference paper [49].

Present method of approach is validated by comparing resultsof back calculated stress–strain responses with experimentaltensile results. As shown in Fig. 9, the present approach predictsthe experimental results quite well. The results of flexural testson HPFRC are also shown in Fig. 10b which represents the compar-ison of two steel fiber types of hooked (H) and twisted (T) at two

Fig. 8. (a) Effect of steel fiber dosage on back calculated stress strain response. (b)Effect of steel fiber dosage on experimental and simulated load deflection response.(Set 3).

Fig. 9. (a) Simulation of flexural responses of HPFRC materials with hooked-endfibers, (b) comparing back calculated tensile stress–strain responses with experi-mental tensile stress–strain response for data set of Kim et al. [45]. (Set 4). Fig. 10. (a) Effect of sample size and steel fiber deformation on back calculated

stress strain response (H or T represents hooked or twisted and M or L representsMedium or Large), (b) Effect of sample size and steel fiber content on experimentaland simulated load deflection responses for data set of Kim et al. [45]. (Set 4).

250 B. Mobasher et al. / Construction and Building Materials 70 (2014) 243–253

different specimen sizes. The fiber content in all mixtures is 79 kg/m3. The HPFRC showed very clear delineations between samplesize (M or L) and fiber deformation type (H or T). The twisted fibersin both the M and L sample sizes showed increases in flexuraltoughness, (+116%) and hooked fibers (+190%).

Peak tensile strength of about 5 MPa and peak flexural strengthin the range of 12–14 MPa are observed in these samples and donot seem to be influenced by sample size and fiber distributions.The maximum loads are 40, 57, 87 and 98 kN for the HM, TM, HLand TL samples respectively. The larger (L) samples show a slightlyhigher deflection capacity with 6 mm total deflection compared tothe medium (M) samples at 4 mm deflection. This additional duc-tility could be from the combined effect of length and high dosagerate of steel fibers which deform and yield as the load increases.

Back-calculated tensile stress–strain responses resulted in sim-ulated load–deflection responses for HPFRC mixtures with steelfibers are shown in Fig. 10a and b for the Twisted fibers with theflexural simulation which compare the twisted and hooked fibersand show an excellent fit for the experimental data through samplefailure. Parameters related to this simulation are summarized inTable 6. As shown in this table and Fig. 10a, back calculatedstress–strain responses for hooked fibers for both medium andlarge size specimens are very similar. Back calculated stress–strainresponses for twisted fibers however differ from medium to largesize samples. The first crack tensile strength of the twisted fibersin medium specimens are 30% higher than large specimens whichmay be attributed to a more uniform fiber distribution in largesamples. The back-calculated tensile strength parameter l in casesof both fibers are also as much as 35% higher for medium size

specimens. Similar to residual stress parameter, transitional tensilestrains are 20 + 25% more in favor of medium size samples, butultimate tensile strains are almost identical for all samples. None-theless, the difference between flexural test results of differentsizes are much more significant than predicted stress–strainresponses using this method of approach.

4.4. Residual strength comparison with ASTM C1609 (f D150), RILEM, and

JCI-SF4 (rb)

Flexural FRC beams results are also analyzed using the datareduction approach according to ASTM C1609 [46]. Load and netdeflection are recorded up to an end-point deflection of L/150.Residual strength (f D

150) is calculated using an elastically equivalentapproach:

f D150 ¼

PD150L

bd2 ð15Þ

where L is the span length, PD150 is the residual load at net deflection

of L/150, b and d are the average width and depth. ASTM C 1609method uses an elastically equivalent elastic measure and overesti-mates the residual uniaxial tensile strength lEecr obtained based onthe present approach by almost three times. Therefore, it is imper-ative to note that the f D

150parameter is not even an equivalently elas-tic stress and can not to be associated with the post crack tensilestrength parameter rcst in Fig. 4a.

Table 6Average 28-day back calculated tensile parameters of HPFRC (Kim et al. [45]).

Sample ID Young’smodulus

First cracktensile strain

First cracktensile strength

Post crackmodulus

Post cracktensile strength

Transitional tensile strain Ultimate tensile strain Tensile toughness

E, GPa ecr, lstr rcr MPa g l a etrn, lstr etu, mm/mm btu Gf, MPa

S79-HM-28d 20 260 5.2 �0.006 0.2 140 0.0364 0.061 235 0.0532S79-TM-28d 20 380 7.6 �0.007 0.2 110 0.0418 0.062 163 0.0916S79-HL-28d 20 260 5.2 �0.008 0.13 105 0.0273 0.061 235 0.0378S79-TL-28d 20 290 5.8 �0.008 0.13 115 0.0334 0.062 214 0.0439

cr

cr

eq, 3 cr2

b2

cr2

Fig. 11. Comparison of residual strength (lrcr) with JCI-SF4, RILEM and ASTMC1609 residual parameters (Sets 1,2, 3, 4).

B. Mobasher et al. / Construction and Building Materials 70 (2014) 243–253 251

Similar to ASTM C 1609, JCI-SF recommends testing fiber rein-forced concrete by third-point loading and measuring the netdeflection by Linear Variable Differential Transformers (LVDTs).Equivalent flexural strength (rb) is calculated by Eq. (16) [48].

rb ¼Tb

dtb:

L

bd2 ð16Þ

where rb is the equivalent flexural strength (N/mm2), Tb is the flex-ural toughness (N mm), L is the span length (mm), dtb is the deflec-tion of 1/150 of span (mm), b is the width of failed cross-section(mm) and d is the height of failed cross-section (mm).

According to RILEM TC 162-TDF [47] bending test method canbe used for the determination of residual flexural tensile strengthas well. The tensile behavior is obtained by the load–deflectioncurve of a simply supported notched beam of 150 � 150 mm crosssection and 500 mm loaded under three-point bending arrange-ment tested using CMOD (Crack Mouth Opening Displacement)control. The residual flexural tensile strength (feq,3) is defined withrespect to d3, defined as:

d3 ¼ dL þ 2:65 mmðmmÞ ð17Þ

where dL is the deflection at the limit of proportionality (mm).The energy absorption capacity, DBz,3 is measured as the area

under the load–deflection curve up to a deflection d3 and consistsof two parts. The part that includes the influence of steel fibers (Df-BZ,3) is used for calculation of the equivalent flexural tensilestrength, feq,3, by means of the following equation.

f eq;3 ¼32

DfBZ;3

2:5

!:

L

bh2sp

ð18Þ

where L is the span length (mm), b is the width of the specimen(mm), and hsp is the distance between tip of the notch and top ofcross section (mm).

As shown in Fig. 11, direct correlation of JCI residual strengthand the present method indicate JCI-SF4 method overestimatesthe residual uniaxial tensile strength lEecr by as much as 3.22times. The exact correction factor for the JCI method is 1/(3.22).A plot of corresponding values from two tests reflects the relation-ship between the two residual strength measures. It is imperativeto note that the f D

150 parameter can be used as a tensile stress mea-sure associated with the post crack tensile strength parameter rcst

in Fig. 4a, so long as this parameter is corrected by a scale factor of1/(2.94). Correction factors for presented standard parameters areas follows:

f D150 ¼ 2:94lrcr

f eq;3 ¼ 3:10lrcr

rb ¼ 3:22lrcr

ð19Þ

Similar to other test methods, direct correlation of RILEM resid-ual strength and the present method indicates that RILEM methodoverestimates the residual uniaxial tensile strength lEecr by asmuch as three times. Alternatively, standard residual flexuralstrength parameters can be correlated to the tensile strength by

a coefficient factor of 1/3. This value is in accordance with the draftof ACI 544.3R report based on the stress coefficients values adoptedby Barros 2004 [56] who presented a linear relationship betweentensile stress at large strains and flexural strength using a coeffi-cient factor of 0.27.

r3 ¼ 0:27f R 4 ð20Þ

It is noted that in the proposed methods for design by FIB [57], acorrection factor of 1/3 is used for scaling the parameter fR3 fromflexural tests to obtain fFtu as the ultimate residual strength. Thiscorrection factor can be justified by calibration of various specimensizes, and various fiber types and dosages. The proposed value andthe present calculation therefore correlate quite well. The presentapproach can be used as theoretical justification for the empiricalvalues obtained and used in the FIB model code.

Since the inherent assumption of the available standard methodassumes that the neutral axis is still at the centroid of the speci-men, and the stress distribution is linear throughout. This leadsto very high nominal flexural stress levels in tension fiber whichare far more than tensile strength. Extreme caution must beexercised in application of the ASTM 1609, JCI-SF4 and RILEM TC162-TDF methods in design and analysis of fiber reinforced con-crete sections, as the results show overestimation of the residualparameter by as much as 2.94–3.22 times. These results are verysimilar to the results of previous study on toughness parametersof early age fiber reinforced concrete materials [58].

5. Conclusions

Characterization of tensile-flexural strain softening of fibercement composites with alternative fiber types, fiber content andlengths shows that the presence of fiber significantly increases

252 B. Mobasher et al. / Construction and Building Materials 70 (2014) 243–253

the ductility of the material. Based on the results of experimentsand analyses in this study, following conclusions may be drawn:

1. By applying the load deflection back-calculation technique onecan generate tensile constitutive data with a higher degree ofaccuracy than the current standard methods.

2. Using a closed form set of governing parameters and variablesapplied through each stage of material response, the stress dis-tribution that considers a shifting neutral axis also provide amore accurate representation of the residual strength andtoughness of FRC.

3. The inherent assumption of the available standard methodassumes that the neutral axis is still at the centroid of thecracked specimen, and the stress distribution is linear through-out. This leads to very high nominal flexural stress levels in ten-sion fiber which are far more than residual tensile strength.

4. Caution must be exercised in application of the ASTM C1609,JCI-SF4 and RILEM TC 162-TDF methods in design and analysisof fiber reinforced concrete sections, as the results show overes-timation of the residual parameter by as much as 2.94–3.22times. Strength parameters provided by these standards mustbe scaled by as much as 0.31–0.34 before designing FRC ele-ments for bending loadings.

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