research article a modified model for deflection...

Research ArticleA Modified Model for Deflection Calculation ofReinforced Concrete Beam with Deformed GFRP Rebar

Minkwan Ju1 Hongseob Oh2 Junhyun Lim3 and Jongsung Sim4

1Department of Civil Engineering Kangwon National University Gangwon-do 25913 Republic of Korea2Department of Civil Engineering Gyeongnam National University of Science and Technology 150 Chilam-dongJinju Gyeongsangnam-do 52725 Republic of Korea3Sambo Engineering 30 Bang i-dong Wiryeseong-daero 16-gil Songpa-gu Seoul 05640 Republic of Korea4Department of Civil and Environmental Engineering Hanyang University Ansan 15588 Republic of Korea

Correspondence should be addressed to Hongseob Oh opera69cholcom

Received 27 March 2016 Revised 9 July 2016 Accepted 19 July 2016

Academic Editor Ulrich Maschke

Copyright copy 2016 Minkwan Ju 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

The authors carried out experimental and analytical research to evaluate the flexural capacity and the moment-deflectionrelationship of concrete beams reinforced with GFRP bars The proposed model to predict the effective moment of inertia for RCbeam with GFRP bars was developed empirically based on Bransonrsquos equation to have better accuracy and a familiar approach to astructural engineer For better prediction of the moment-deflection relationship until the ultimate strength is reached a nonlinearparameter (119896) was also considered This parameter was introduced to reduce the effect of the cracked moment of inertia for thereinforced concrete member including a lower reinforcement ratio and modulus of elasticity of the GFRP bar In a comparativestudy using six equations suggested by others the proposed model showed better agreement with the experimental test results Itwas confirmed that the empirical modification based on Bransonrsquos equation was valid for predicting the effective moment of inertiaof RC beams with GFRP bar in this study To evaluate the generality of the proposed model a comparative study using previoustest results from the literature and the results from this study was carried out It was found that the proposed model had betteraccuracy and was a familiar approach to structural engineers to predict and evaluate the deflection behavior

1 Introduction

Service life of reinforced concrete (RC) structures can bedecreased by a number of factors including harsh environ-mental conditions and unexpected excessive external loadsOne of the main factors contributing to degradation ofstructural condition is the corrosion of steel reinforcementHence using a noncorrosive reinforcement can be an effec-tive solution to increase the service life of RC structures Inmany regions fiber-reinforced polymer (FRP) bars have beenof considerable interest to civil (structures) and structuralengineers for strengthening and reinforcing concrete as asubstitute for steel bars Their high strength-to-weight ratiocorrosion-free properties and ease of handling during con-struction are considered advantages for application in civilstructures Much structural research using FRP bars has beenperformed in field applications and there are now guidelines

for the design and construction of concrete reinforced usingFRP bars such as the AASHTO LRFD Bridge Design Speci-fication ACI 440 guidelines and the Canadian Design Code[1ndash3]

The flexural capacity of reinforced concrete memberswith GFRP bar has been an issue in structural design due tothe relatively low modulus of elasticity which causes a largerdeflection and crack width Thus the flexural behavior ofreinforced concretememberswithGFRPbar should be inves-tigated further with respect to serviceability The predictionof deflection behavior is one of the most important criteria inevaluating and ensuring the serviceability of concrete mem-ber ACI 318-14 [4] uses an equation for themoment of inertiabased on Bransonrsquos equation [5] to calculate the deflectionof reinforced concrete beams Recently ACI 4401R-15 [2]recommended a new model for the moment of inertia forreinforced concrete members with GFRP bar that was not

Hindawi Publishing CorporationInternational Journal of Polymer ScienceVolume 2016 Article ID 2485825 10 pageshttpdxdoiorg10115520162485825

2 International Journal of Polymer Science

based on Bransonrsquos equation in contrast with ACI 4401R-06 [6] The Branson-based equation has long been familiarto most structural engineers in designing flexural concretemembers For Reinforced concrete members with GFRP barBransonrsquos equation had been modified to predict the deflec-tion as accurately as possible The significant modificationswere to correct the power of119898 and to add a parameter

In this study we suggest a modified effective momentof inertia and carried out a comparative study regardingthe deflection behavior of RC beams with GFRP bar withexperimental tests For the comparative study six equationsincluding some from individual research were consideredThe proposed model was developed based on Bransonrsquosequation to provide a familiar approach to calculate themoment of inertia for RC beams with GFRP barThis modelwas empirically modified according to the test results of thesix test specimens with variables of the reinforcement ratioFor better prediction of deflection until ultimate strength wasreached an empirical nonlinear parameter was introduced toreduce the effect of the cracked moment of inertia Amongthe equations the degree of accuracy in the prediction ofdeflection behavior for the new moment of inertia suggestedin this study was analyzed and predictability was discussed

2 Existing and Proposed Equations for theMoment of Inertia for FRP Bar-ReinforcedConcrete Flexural Members

Bransonrsquos equation generally underestimates the deflectionof FRP-reinforced concrete beams Benmokrane et al [7]modified the equation to make it more suitable for evaluatingthe deflection of FRP-reinforced concrete beams based onexperimental data The equation is as follows

119868119890= (

119872cr119872119886

)

3 119868119892

120573+ 120572[1 minus (

119872cr119872119886

)

3

] 119868cr le 119868119892 (1)

where 119868119892is the gross moment of inertia (mm4) 119868cr is the

moment of inertia of transformed cracked section (mm4)119872cr is the cracking moment (Nsdotm) and119872

119886is the maximum

service load moment in member (Nsdotm)The noticeable difference lies in the modification of 120572

and 120573 120572 reflects the reduced composite action betweenthe concrete and FRP bars However 120573 has no physicalsignificance because there was no justification for reducing119868119892 120572 and 120573 were 084 and 7 respectivelyACI 4401R-06 [6] recommended an equation for the

effective moment of inertia based on Bransonrsquos model Therewas an additional factor for considering the reduced tensionstiffening of FRP-reinforced concrete members This modelhas been commonly used to calculate the moment of inertiaof FRP-reinforced concrete members so that the deflectionof the cracked section can be calculated

119868119890= (

119872cr119872119886

)

3

120573119889119868119892+ [1 minus (

119872cr119872119886

)

3

] 119868cr le 119868119892 (2)

where 120573119889is the reduction factor related to the reduced

tension stiffening exhibited by RC member with FRP bar

(= (15)(120588119891120588119891119887) le 10) 120588

119891is the reinforcement ratio of

GFRP bar and 120588119891119887

is the balanced reinforcement ratio ofGFRP bar

Toutanji and Saafi [9] empirically suggested an equationfor the effective moment of inertia for reinforced concretebeam with GFRP bar Their equation focused on the mod-ification factor of the power of 119898 in (1) The factor wasbased on applying the modulus ratio to the reinforcementratio of the FRP bar By only modifying the power of 119898the conventional form of the equation which is familiar tostructural engineers wasmaintainedThe equation predictedthe deflection of the tested RC beams with GFRP wellConsider

119868119890= (

119872cr119872119886

)

119898

119868119892+ [1 minus (

119872cr119872119886

)

119898

] 119868cr le 119868119892 (3)

where119898 = 6 minus 10120588F119864F119864119904For the Canadian Code for reinforced concrete mem-

bers with FRP bar CANCSA S806-12 [3] suggested thefollowing equation (see (4)) for calculating the deflectionThe equation was based on the conventional equation forcalculating deflection under four-point loading It uses thecracked moment of inertia while ACI 4401R-15 [2] uses theeffective moment of inertia However additional equationterms referring to shear span span length and uncrackedlength in half of the beam were included This equationrequires a calculation-intensive process subject to humanerror thus the code also provides closed-form equations forcommon loading and support conditions Hence

Δ =119875119886

48119864119888119868cr

(3(119886

119871) minus 4 (

119886

119871)

3

minus 8120578(119871119892

119871)

3

) (4)

where 119886 is the shear span (mm) 119875 is the total applied load(N) 119871 is the span length (mm) 119871

119892is the uncracked length in

half of the beam (mm) (= 119886(119872cr119872119886)) and119864119888is themodulus

of elasticity of concrete (MPa) 120578 = (1 minus 119868cr119868119892)Recently the other semiempirical model was suggested

by modifying Bransonrsquos equation according to experimentalresults and a genetic algorithm approach [10] For betterprediction some of the factors were developed empiricallyThe model which has two multiplying factors and an expo-nential factor 119898 was analyzed using experimental data for55 FRP-reinforced concrete beams for the load-deflectionrelationship The effects of the elastic modulus of FRP barsreinforcement ratio and level of loading on the power of 119898in Bransonrsquos equation are taken into account in (5) as follows

119868119890= 013 (

119872cr119872119886

)

119898

119868119892+ 089 [1 minus (

119872cr119872119886

)

119898

] 119868cr le 119868119892 (5)

where119898 = minus024(120588119891120588119891119887) + 535(119872cr119872119886) + 228(119864

119891119864119904) 119864119891

is the modulus of elasticity of FRP bar (MPa) and 119864119904is the

modulus of elasticity of steel bar (MPa)ACI 4401R-15 [2] suggested an equation for calculating

the effective moment of inertia for reinforced concrete beamswith FRP bar This equation is based on Bischoff rsquos proposedapproach which represents a weighted average of flexibility

International Journal of Polymer Science 3

Milled glass fiber ribs

(a) Detailing of GFRP reinforcing bar

Lug GFRP core

Polymer rib with milled glass fiber

P = 063 120572014

014

w = 023

P = Lug pitch120572 = 80∘

d

d

d

dd

d

998400

t = 021d

(b) Surface pattern of GFRP reinforcing bar

Figure 1 GFRP reinforcing bar used in this study [8]

(1119864119888119868) as shown in (6) It was reported that the equation

works equally well for both steel- and GFRP-reinforced con-crete members with no empirical parameter [11] Therefore

119868119890=

119868cr

1 minus 120574 (119872cr119872119886)2

[1 minus 119868cr119868119892]le 119868119892 (6)

where 120574 is the parameter to account for the variation instiffness along the length of the member for four-pointbending Hence

120574 =3 (119886119871) minus 16 (119872cr119872119886) (119886119871)

2

+ 12 (119886119871)3

3 (119886119871) minus 4 (119886119871)3

(7)

Intelligent sensing for innovative structures [12] recom-mended following (8) of the effective moment of inertia forFRP bar-reinforced concrete beams This equation addedadditional corrective terms in a modified Branson equationwith more experimental data The notations are introducedabove in equations from (1) to (6) Here we have

119868119890=

119868119892119868cr

119868cr + (1 minus 05 (119872cr119872119886))2

(119868119892minus 119868cr)

(8)

3 Experimental Tests

31 Description of the GFRP Bar The GFRP reinforcing barused in this study is developed reinforcement having a similarouter shape to conventional steel bars It consists of contin-uous longitudinal glass fibers with a 67 volume fractionin a thermosetting epoxy A typical pultrusion process wasadopted To enhance the shear resistance under bondingmilled glass fiber ribs were formulated on the GFRP coreTo form the fiber ribs a steel mold was used in the curingprocess The rib section of the GFRP bar was manufacturedby mixing milled glass fiber and epoxy at a ratio of 1 1 byweight and it was cured for 15min at a temperature above160∘CDetails of the external shape of theGFRPbarwith fiberribs were provided by Ju and Oh [8] (Figure 1(b))

The GFRP bar used for the tensile area had a nominaldiameter of 953mm Tensile tests were conducted with eighttest specimens according to ACI 4403R-04 [13] The tensilespecimens were loaded through thick plates at the anchoredends A universal testing machine (UTM) with a capacityof 2000 kN was used and the loading rate was 178 kNminAmong the test specimens the maximum tensile strengthwas found to be 8714MPa Table 1 summarizes the tensilestrength of the GFRP bar The guaranteed tensile strengthwith standard deviation was calculated to be 6160MPa Thedesigned tensile strength was calculated by multiplying the


Table 1 Mechanical properties of the GFRP bar used in this study

Average tensilestrength (MPa)

Guaranteed tensilestrength (MPa)

Guaranteedultimate strain ()

Design tensilestrength (MPa)

Design rupturestrain ()

Modulus ofelasticity (GPa)

8410 plusmn 236 7702lowast 165lowast 5391lowastlowast 116lowastlowast 421lowastACI Committee 440 [2] average ndash 3 times standard deviationlowastlowastEnvironmental reduction factor (119862

119864) is applied with 07 exposed to earth and weather

200200 1600

25

180

186

0

230

18017

35

230

180

186

0

230

685 685

ldquoA-A sectionrdquo

230

LVDT

Strain gauge

FB-1 FB-2 FB-3

P2P2

A

A2D10

Figure 2 Test setup and measurement detail (in mm)

environmental reduction factor (07 for external exposure)in compliance with ACI 4401R-15 [2] resulting in 5391MPaThe modulus of elasticity was found to be 429GPa withinthe general range of the modulus of elasticity for GFRP bars

32 Test Setup For a GFRP bar the conventional ductilitydesign used for steel bars is not appropriate because ofthe absence of a yield point Three types of RC beamswith GFRP bar were designed according to ACI 4401R-15 [2] FRP rupture (FB-1) balanced (FB-2) and concretecrushing (FB-3) failure Three different amounts of longi-tudinal reinforcement GFRP bar were used 2D10 for FB-23D10 for FB-3 and 4D10 for FB-4 D indicates the nominaldiameter of the GFRP bar For the balanced FB-2 it maybe regarded as FRP rupture failure with concrete crushingEach specimen consisted of two identical beamsThe flexuraltest was conducted by four-point bending Figure 2 showsthe test setup and measurement details The dimensionsof the test beams were as follows 180mm wide 230mmdeep and 2000mm long The pure span was 1600mmThe shear span of 119886119889 which can determine the governedbehavior of the flexural beam was calculated to be between37 and 39 thus the beam was regarded as being subjectedto flexural behavior To monitor the structural behavior ofthe beam linear variable differential transformers (LVDTs)were installed at the bottom surface of the concrete inthe midsection Two electric resistance strain gauges were

attached to the surface of the centered GFRP bar for FB-2and the outer GFRP bar for FB-1 and FB-3 at the midsectionThe two loads were automatically applied to the beam at arate of 2 kN per minute using MTS loading machine Alldata (forces strains and deflections) were collected by anautomated data acquisition system For crack width a crackmeasure was used and crack width was investigated visuallyat the individual loading step For the concrete the average28-day compressive strength was 270MPa and the flexuraltensile strength of the concrete was approximately 24MPa

33 Flexural Test Results and Discussion Flexural failure andcrack patterns are shown in Figure 3 The specimens failedin a typical flexural failure manner FB-1 showed a crackingload of 120 kN The failure was governed by rupture of theGFRP bars The maximum crack width at the midsectionwas investigated visually and found to be 09mm The FB-2 specimen initiatally failed by crushing of the concreteand then finally collapsed due to rupture of the GFRPbar Consequently the failures showed compression tensionfailure with rupture of the GFRP bar and the maximumcrack width was 07mm In the case of the FB-3 specimenit showed conventional concrete crushing failure withoutrupture of the GFRP bar The maximum crack width wasmeasured as 04mm at the midsection For flexural capacityit was found that the designed failure modes representedthe experimental modes of failure and the crack width was


(a) FB-1 (b) FB-2 (c) FB-3

Figure 3 Flexural failure and crack patterns of FB-1 FB-2 and FB-3

Table 2 Result of experimental test of GFRP bar reinforced concrete beams

Specimen Reinforcementratio (balanced)

Average load atinitial cracking

(kN)

Average load atultimate state

(kN)

Averagedeflection atultimate state

(mm)

Nominalmoment (exp)

(kNsdotm)

Nominalmoment (cal)

(kNsdotm)Mode of failure

FB-1 000427(00069) 120 544 344 186 136 FRP rupture

FB-2 00064(00069) 100 648 391 222 199 Compression

tension

FB-3 000903(00069) 120 740 266 254 210 Concrete

crushing

reduced as the reinforcement ratiowas increased Cracking inthe flexural zone predominantly consisted of vertical cracksperpendicular to the direction of maximum principle stressinduced by the pure flexural moment Cracking was initiatedat the middle of the span and then propagated toward thesupports

Eventually shear stress became more important andinduced inclined cracks When reaching ultimate strengthflexural cracks propagated towards the vicinity of the loadpoints on the compressive face of the beams All test beamsshowed significant flexural cracking before inclined cracksjoined flexural cracks For an analytical approach regardingthe nominal flexuralmoment the equation fromACI 4401R-15 [2] was used with a varying reinforcement ratio When120588119891

lt 120588119891119887 the controlling limit state is rupture of the FRP

bar and the nominal flexural strength can be computedBased on the equilibrium of forces and strain compatibility(10) can be derived Otherwise when 120588

119891gt 120588119891119887 the design

tensile strength (119891119891119906) in (10) is changed to the stress in FRP

(119891119891) in tension (see (9)) Table 2 shows the experimentally

and analytically obtained flexural moment strengths of RCbeams with GFRP bar The theoretical moment strength wasevaluated and was about 20 lower than that of the momentstrength in the experimental testThismay be due to variation

resulting from the small number of test specimens Howeverthe calculated moment strength could well represent thestructural capacity as a conservative prediction For struc-tural stiffness defined by dividing the average load by theaverage deflection at ultimate strength FB-1 FB-2 and FB-3 showed values of 158 165 and 278 respectively It wasfound that the structural stiffness increased according to theincrease in the reinforcement ratio of the GFRP bar

119891119891= (radic

(119864119891120576119888119906)2

4+

08512057311198911015840

119888

120588119891

119864119891120576119888119906

minus 05119864119891120576119888119906)

le 119891119891119906

(9)

119872119899= 119860119891119891119891119906

(119889 minus119886

2) (10)

where 119872119899is the nominal flexural strength (kNsdotm) 119886 is the

depth of equivalent rectangular stress block (mm) 119891119891is the

stress in the FRPbar in tension (MPa)119891119891119906is the design tensile

strength of the FRP bar (MPa)119864119891is the design or guaranteed

modulus (MPa) 120576119888119906is the ultimate strain in concrete (0003)

1205731is an empirical factor 1198911015840

119888is the specified compressive

strength of concrete (MPa) 120588119891is the FRP reinforcement ratio


Deflection (mm)

Without k

With k

Ie reduced

Stiffness softened for lower120588f and Ef reinforced

concrete members

Mom

ent (

kNmiddotm

)

Figure 4 Basic concept of considering parameter 119896

and 120588119891119887

is the balanced FRP reinforcement ratio as given byACI Committee 440 [2]

4 Comparative Study for the Prediction ofDeflection Behavior Using the Proposed andExisting Models for Calculating the EffectiveMoment of Inertia

In this paper a semiempirical prediction model for theeffective moment of inertia is proposed The model is basedon Bransonrsquos equation and the modification methodologyfollowed the empirical approach of Toutanji and Saafi [9] Asshown in Figure 4 deflection of the RC beam with GFRPbars was affected by the reinforcement ratio of GFRP barsas well as the elastic modulus of the GFRP bar A notableparameter 119870 to reflect the nonlinear behavior of the RCbeams with GFRP bar is considered empirically to providegood agreement with the experimental tests in this study (see(11)) This factor was used to reduce the effect of the crackedmoment of inertia for the reinforced concrete memberby including a lower reinforcement ratio and modulus ofelasticity for GFRP bar This considering parameter is thecurve fitting factor Its concept was empirically derived byinvestigating the results of moment-deflection relationshipfrom the considering equation commented on above in thisstudy In Figure 6 the considering equations showed thestiff curve as a bilinear behavior up to failure of the testspecimen However the test result exhibited a nonlinearbehavior up to failure so that it can be estimated that areducing factor should be needed for the good curve fittingto the experimental results It resulted in a decrease in theeffective moment of inertia so that the calculated deflectionwas increased according to the increase in applied loadingFigure 4 illustrates the basic concept of considering 119870 inthe proposed model Using 119870 the stiffness of the deflectionbehavior can be softened slightlyThis analytical concept maybe more appropriate for a concrete member reinforced with

Experiment Proposed model

ACI 440 1R-06 [6]

Toutanji and Saafi [9]

ISIS Canada [15]ACI 440 1R-15 [2]Mousavi et al [10]Benmokrane et al [7]

5 10 15 200Applied moment (kNmiddotm)

0

05

1

15

2

Effec

tive m

omen

t of i

nert

ia(times

108

mm

4)

Ie [7]Ie [10]Ie [2]

Ie [9]Ie [15]Ie [6]

Figure 5 Effective moment of inertia of FB-1

a material with a lower reinforcement ratio and modulus ofelasticity

119868119890= (

119872cr119872119886

)

119898

119868119892+ [1 minus (

119872cr119872119886

)

119898

minus 119870] 119868cr le 119868119892 (11)

where 119898 = 6 minus 13120588F119864F119864119904 and 119870 is a nonlinear parameter(= 111(119872cr119872119886))

4In total six codes and developed equations were inves-

tigated for a comparative study of the moment of inertiaand load deflection according to the experimental tests andthe model proposed in this study For this a representativespecimen for each reinforcing group was considered for thecomparison study because of their similarity of the testedresults There are some studies showing that the evaluationof structural capacity of FRP bar-reinforced concrete beamusing only one representative specimen for each reinforcinggroup was successfully done [14 15] Figure 5 shows theresults of the comparative study on the effective momentof inertia There is a noticeable discrepancy between theexperiment and equation approaching an applied momentof 5 kNsdotm ACI 440 Committee [6] Toutanji and Saafi [9]and the proposed model showed better agreement with goodnonlinear prediction of the experimentally obtained effectivemoment of inertia after the cracking of the concrete Theother equations such as those of ACI Committee 440 [2]ISIS Canada [12] Mousavi et al [10] and Benmokrane etal [7] showed large drops in the gross moment of inertia(119868119892) after the cracking of the concrete They did not represent

the hardening behavior of the experimental results well Twoof the prediction models modified from Bransonrsquos equationshowed good agreement with the experimental results whilethe other two models with modified Branson equationsshowed relatively larger discrepancies This was caused by


Experiment

Proposed model

Benmokrane et al [7]

Mousavi et al [10]

ISIS Canada [15]Toutanji and Saafi [9]ACI 440 1R-15 [2]

ACI 440 1R-06 [6]

Δ [6]Δ [15]Δ [9]

Δ [2]Δ [3]Δ [10]Δ [7]

5 10 15 20 25 30 35 400Midspan deflection (mm)

0

5

10

15

20

25

30

Appl

ied

mom

ent (

kNmiddotm

)

Figure 6 Moment-midspan deflection of FB-1

application of the empirical parameters such as the power of119898 or the multiplying constants

Figure 6 shows the moment and midspan deflectioncurve for the FB-1 specimen which consisted of two RCbeams with GFRP bar Except for Toutanji and Saafi [9]ACI 440 Committee [6] and Benmokrane et al [7] equa-tions plasticity behavior was detected after cracking Theseequations underestimated the cracking behavior of the FB-1specimen while the experimental results showed hardeningbehavior with the applied moment The equation of Mousaviet al [10] showed the highest stiffness in predicting thedeflection They used almost-identical multiplying constantfor the gross and cracked moments of inertia howeverthe power of 119898 was different from that of Benmokraneet al [7] This difference might make the flexural stiffnessin the prediction of deflection more relaxed than that ofBenmokrane et al [7]

ACI 440 Committee [6] and Toutanji and Saafi [9]showed good accordance in deflection behavior until aroundhalf of the applied moment however after the loading stagethese models behaved as a linear-dependent prediction ofdeflectionThus the difference in the prediction of deflectionwas increased until the ultimate moment For the proposedequation with a nonlinear parameter 119870 it was found thatit best predicted the deflection behavior in the experimentaltest until failure In Figures 7ndash10 the analytical effectivemoment of inertia-applied moment strength and moment-deflection curves obtained from the six equations and theproposed model are compared with the experimental resultsfor FB-2 and FB-3 specimens The trends in the prediction ofdeflection were similar to that of the FB-1 specimen wherethe ACI 440 Committee [6] Toutanji and Saafi [9] andthe proposed models still showed better agreement withgood nonlinear predictions of the experimentally obtained

ACI 440 1R-06 [6]Toutanji and Saafi [9]


5 10 15 20 250Applied moment (kNmiddotm)

0

05

1

15

2

Effec

tive m

omen

t of i

nert

ia(times

108

mm

4)


Ie [7]Ie [10]

Ie [2]

Ie [9]Ie [15]Ie [6]



ISIS Canada [15]Toutanji and Saafi [9]ACI 440 1R-15 [2]Mousavi et al [10]

ACI 440 1R-06 [6]

Experiment

Proposed model

Δ [6]Δ [15]Δ [9]

Δ [2]Δ [3]

Δ [10]Δ [7]

10 20 30 400Midspan deflection (mm)

0

5

10

15

20

25

30

Appl

ied

mom

ent (

kNmiddotm

)


effective moment of inertia after cracking of the concreteThe experimental moment-deflection curves of FB-2 and FB-3 did not show good agreement with the analytical curvesderived from the six equations considered herein but are ingood agreement with the proposed model

The six equations evaluated the moment-deflectionresponse which was linear compared with the actualresponse of the test specimens after cracking until ultimatestrength is reached Unlike the ACI 4401R-06 [6] Toutanjiand Saafi [9] and the proposed model the other equations


ACI 440 1R-06 [6]


ISIS Canada [15]Mousavi et al [10]Benmokrane et al [7]

ACI 440 1R-15 [2]

5 10 15 20 25 300Applied moment (kNmiddotm)

0

05

1

15

2

Effec

tive m

omen

t of i

nert

ia(times

108

mm

4)


Ie [7]Ie [10]

Ie [2]

Ie [9]Ie [15]Ie [6]


Mousavi et al [10]

ACI 440 1R-06 [6] ISIS Canada [15]Toutanji and Saafi [9]ACI 440 1R-15 [2]Benmokrane et al [7]


0

5

10

15

20

25

30

Appl

ied

mom

ent (

kNmiddotm

)

Experiment

Proposed model

Δ [6]Δ [15]Δ [9]

Δ [2]Δ [3]

Δ [10]Δ [7]


did not represent the tension-stiffening effect of the testspecimens until around half of the ultimate strength Thereason may be that the equations evaluate the effectivemoment inertia to be much less than that of the testspecimen until the loading stage and then they respondwith a linearly hardening prediction until ultimate strengthThe proposed model however represented the moment-deflection response well even the nonlinear behavior untilultimate strength

5 Comparative Study forValidation of the Proposed Model

To evaluate the generality of the proposed model some ofthe previous test results were considered specimen A1 fromAiello and Ombres [16] specimen BC2HA from Theriaultand Benmokrane [17] specimen F1 from Pecce et al [18]specimen Series 1 from Benmokrane et al [7] and specimenGroup 2 from Al-Salloum et al [19] Furthermore thetested specimens FB-1 FB-2 and FB-3 from this studywere also compared according to the order of the calculatedvalue of equivalent reinforcement ratio with the modulusratio 120588F119864F119864119904 120588F119864F119864119904 normalized the reinforcement ratioof the FRP bar to steel bar properties and must be animportant index to investigate the validation of moment-deflection behavior with the proposed model The appli-cation criteria of 120588F119864F119864119904 can be determined to evaluatethe structural behavior of concrete beams reinforced withvarious reinforcement ratios Figure 11 shows the results ofthe comparative study using the proposed model The resultsshowed that the proposed model reasonably described themoment-deflection behavior of the considered test specimenswhen 120588F119864F119864119904 was varied from 000068 to 0006 Howeverthe proposed model showed overestimation as 120588F119864F119864119904 wasincreased for example to 0006 forGroup 2 so that the appli-cation boundary should be investigated furtherThe FB seriesshowed relatively good agreement with the experimental testsdue to the reference specimens used in this study

For the other specimens the ascending trend wasdescribed well with the experimental results and some dis-crepancies were detected after cracks occurred There aresome influencing parameters such as concrete property sizeeffect and bar type for bonding property In particular thebond performance of FRP bar in concrete beammay bemoreaffected by flexural loads than uniaxial tensile load due toits different surface treatment including chemical adhesionproperty Further accurate analysis about this should bediscussed by experimental and analytical study

6 Conclusions

In this study we carried out experimental and analyticalresearch to evaluate the flexural capacity and the moment-deflection relationship of concrete beams reinforced withGFRP bars The proposed model suggested for the effectivemoment of inertia of RC beams with GFRP bar couldreasonably describe the moment-deflection relationshipTheconclusions drawn are as follows

(i) This study suggested a new equation for the effectivemoment of inertia for concrete beams reinforcedby GFRP bars The new equation was modifiedfrom Bransonrsquos equation which has long been usedin this field by structural engineers The power of119898 was modified based on Toutanji and Saafirsquos [9]equation and the nonlinear parameter 119870 was alsointroduced This factor was used to reduce the effectof the cracked moment of inertia for the concretemember reinforced with a lower reinforcement ratio


FB-1 (test) FB-1 (proposed model)FB-2 (test) FB-2 (proposed model)A1 (test) A1 (proposed model)FB-3 (test) FB-3 (proposed model)BC2HA (test) BC2HA (proposed model)F1 (test) F1 (proposed model)Series 1 (test) Series 1 (proposed model)Group 2 (test) Group 2 (proposed model)

F1

Series 1

Group 2

50 100 150 200 250 300 350 400 450 500 550 6000Deflection (mm)

0

10

20

30

40

50

60

70

80

90

100

Appl

ied

mom

ent (

kNmiddotm

)

FB-1 FB-2

A1

FB-3

BC2HA

Over120588fEfEs = 0002

120588fEfEs = 0002Under Over 120588fEfEs = 0006

Figure 11 Comparison results between experimental tests and the proposed model

and a material with a lower modulus of elasticityFor comparison with experimental tests three typesof RC beams with GFRP bar were designed andtested The predictability of the proposed model wasevaluated

(ii) The comparative study used six equations and theproposed model to calculate the effective moment ofinertia and applied moment relationship and foundthat the equations of ACI 4401R-06 [6] Toutanjiand Saafi [9] and the proposed models showedbetter agreement with the experimental results Theother three equations considerably underestimatedthe moment of inertia immediately after concretecracking From this result it was confirmed that theempirical modification based on Bransonrsquos equationwas valid for predicting the effective moment ofinertia and applied moment of the RC beams withGFRP bar

(iii) For the prediction of deflection in the experimentaltests the proposed model showed the best pre-dictability among the equations considered The newmodel showed better agreement with the deflectionbehavior of the GFRP bar-reinforced concrete beamuntil ultimate strength even with respect to thenonlinear behavior To evaluate the generality of theproposed model a comparative study using previous

test results as well as the results from this studywas carried out regarding the moment-deflectionrelationship For further study with regard to thedifference in bonding properties of FRP bars the pro-posed model could reasonably describe the moment-deflection relationship for the test results consideredfrom previous research and the test results in thisstudy

(iv) This study confirmed the predictability of the pro-posed model for the effective moment of inertia Itwas found that the modification methodology withan empirical approach was applicable In terms offuture research it is important that a comparativestudy with varying reinforcement ratios bondingproperties of FRP bars and size effects of concretebeams is conducted

Competing Interests

The authors declare that they have no competing interests

Acknowledgments

Thisworkwas supported by a grant (2015R1A2A2A01005286)from the National Research Foundation of Korea (NRF) anda grant (16CTAP-C117247-01) by the RampD Program from the


Ministry of Land Infrastructure and Transport of the KoreanGovernment

References

[1] AASHTO LRFD Bridge Design Guide Specifications for GFRP-Reinforced Concrete Bridge Decks and Traffic Railings AmericanAssociation of State Highway and Transportation OfficialsWashington DC USA 2009

[2] ACI Committee 440 Guide for the Design and Construction ofConcrete Reinforced with FRP Bars (ACI 4401R-15) AmericanConcrete Institute Farmington Hills Mich USA 2015

[3] CANCSA S806-12 Design and Construction of Building Struc-tures with Fibre-reinforced Polymers Canadian Standards Asso-ciationNational Standard of Canada Ontario Canada 2012

[4] ACI 318 ldquoBuilding code requirements for structural concreteand commentaryrdquo ACI 318-14 American Concrete InstituteFarmington Hills Mich USA 2014

[5] D E Branson ldquoInstantaneous and time-dependent deflectionsof simple and continuous reinforced concrete beamsrdquo HPRReport no 7 part 1 Alabama Highway Department Bureau ofPublic Roads Alabama 1965


[7] B Benmokrane O Chaallal and R Masmoudi ldquoFlexuralresponse of concrete beams reinforced with FRP reinforcingbarsrdquo ACI Structural Journal vol 93 no 1 pp 46ndash55 1996

[8] M Ju and H Oh ldquoExperimental assessment on the flexuralbonding performance of concrete beam with GFRP reinforcingbar under repeated loadingrdquo International Journal of PolymerScience vol 2015 Article ID 367528 11 pages 2015

[9] H A Toutanji and M Saafi ldquoFlexural behavior of concretebeams reinforced with glass fiber-reinforced polymer (GFRP)barsrdquo ACI Structural Journal vol 97 no 5 pp 712ndash719 2000

[10] S R Mousavi M R Esfahani and M Arabi ldquoAn equationfor the effective moment of inertia for FRP-reinforced concretebeamsrdquo in Proceedings of the CICE Rome Italy 2012

[11] P H Bischoff ldquoReevaluation of deflection prediction for con-crete beams reinforced with steel and fiber reinforced polymerbarsrdquo Journal of Structural Engineering vol 131 no 5 pp 752ndash762 2005

[12] ISIS Canada Reinforced Concrete Structures with Fibre Rein-forced Polymers Design Manual No 3 vol 3 ISIS CanadaManitoba Canada 2007

[13] ACI Committee 440 ldquoGuide test methods for fiber-reinforcedpolymers (FRPs) for reinforcing or strengthening concretestructuresrdquo ACI 4403R-04 American Concrete InstituteFarmington Hills Mich USA 2004

[14] C Barris L I Torres A Turon M Baena and A Catalan ldquoAnexperimental study of the flexural behaviour ofGFRPRCbeamsand comparison with predictionmodelsrdquo Composite Structuresvol 91 no 3 pp 286ndash295 2009

[15] M Noel and K Soudki ldquoEstimation of the crack width anddeformation of FRP-reinforced concrete flexural members withandwithout transverse shear reinforcementrdquo Engineering Struc-tures vol 59 pp 393ndash398 2014

[16] M A Aiello and L Ombres ldquoLoad-deflection analysis of FRPreinforced concrete flexuralmembersrdquo Journal of Composites forConstruction vol 4 no 4 pp 164ndash170 2000

[17] MTheriault and B Benmokrane ldquoEffects of FRP reinforcementratio and concrete strength on flexural behavior of concretebeamsrdquo Journal of Composites for Construction vol 2 no 1 pp7ndash16 1998

[18] M Pecce GManfredi and E Cosenza ldquoExperimental responseand code models of GFRP RC beams in bendingrdquo Journal ofComposites for Construction vol 4 no 4 pp 182ndash190 2000

[19] Y A Al-Salloum S H Alsayed and T H AlmusallamldquoEvaluation of service load deflection for beam reinforced byGFRP barsrdquo in Proceedings of the 2nd International Conferenceon Advanced Composite Materials in Bridges and Structures(ACMBS-II rsquo96) pp 165ndash172 Montreal Canada 1996

Submit your manuscripts athttpwwwhindawicom

ScientificaHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CorrosionInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Polymer ScienceInternational Journal of



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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


based on Bransonrsquos equation in contrast with ACI 4401R-06 [6] The Branson-based equation has long been familiarto most structural engineers in designing flexural concretemembers For Reinforced concrete members with GFRP barBransonrsquos equation had been modified to predict the deflec-tion as accurately as possible The significant modificationswere to correct the power of119898 and to add a parameter

In this study we suggest a modified effective momentof inertia and carried out a comparative study regardingthe deflection behavior of RC beams with GFRP bar withexperimental tests For the comparative study six equationsincluding some from individual research were consideredThe proposed model was developed based on Bransonrsquosequation to provide a familiar approach to calculate themoment of inertia for RC beams with GFRP barThis modelwas empirically modified according to the test results of thesix test specimens with variables of the reinforcement ratioFor better prediction of deflection until ultimate strength wasreached an empirical nonlinear parameter was introduced toreduce the effect of the cracked moment of inertia Amongthe equations the degree of accuracy in the prediction ofdeflection behavior for the new moment of inertia suggestedin this study was analyzed and predictability was discussed

2 Existing and Proposed Equations for theMoment of Inertia for FRP Bar-ReinforcedConcrete Flexural Members

Bransonrsquos equation generally underestimates the deflectionof FRP-reinforced concrete beams Benmokrane et al [7]modified the equation to make it more suitable for evaluatingthe deflection of FRP-reinforced concrete beams based onexperimental data The equation is as follows

119868119890= (

119872cr119872119886

)

3 119868119892

120573+ 120572[1 minus (

119872cr119872119886

)

3

] 119868cr le 119868119892 (1)

where 119868119892is the gross moment of inertia (mm4) 119868cr is the

moment of inertia of transformed cracked section (mm4)119872cr is the cracking moment (Nsdotm) and119872

119886is the maximum

service load moment in member (Nsdotm)The noticeable difference lies in the modification of 120572

and 120573 120572 reflects the reduced composite action betweenthe concrete and FRP bars However 120573 has no physicalsignificance because there was no justification for reducing119868119892 120572 and 120573 were 084 and 7 respectivelyACI 4401R-06 [6] recommended an equation for the

effective moment of inertia based on Bransonrsquos model Therewas an additional factor for considering the reduced tensionstiffening of FRP-reinforced concrete members This modelhas been commonly used to calculate the moment of inertiaof FRP-reinforced concrete members so that the deflectionof the cracked section can be calculated

119868119890= (

119872cr119872119886

)

3

120573119889119868119892+ [1 minus (

119872cr119872119886

)

3

] 119868cr le 119868119892 (2)

where 120573119889is the reduction factor related to the reduced

tension stiffening exhibited by RC member with FRP bar

(= (15)(120588119891120588119891119887) le 10) 120588

119891is the reinforcement ratio of

GFRP bar and 120588119891119887

is the balanced reinforcement ratio ofGFRP bar

Toutanji and Saafi [9] empirically suggested an equationfor the effective moment of inertia for reinforced concretebeam with GFRP bar Their equation focused on the mod-ification factor of the power of 119898 in (1) The factor wasbased on applying the modulus ratio to the reinforcementratio of the FRP bar By only modifying the power of 119898the conventional form of the equation which is familiar tostructural engineers wasmaintainedThe equation predictedthe deflection of the tested RC beams with GFRP wellConsider

119868119890= (

119872cr119872119886

)

119898

119868119892+ [1 minus (

119872cr119872119886

)

119898

] 119868cr le 119868119892 (3)

where119898 = 6 minus 10120588F119864F119864119904For the Canadian Code for reinforced concrete mem-

bers with FRP bar CANCSA S806-12 [3] suggested thefollowing equation (see (4)) for calculating the deflectionThe equation was based on the conventional equation forcalculating deflection under four-point loading It uses thecracked moment of inertia while ACI 4401R-15 [2] uses theeffective moment of inertia However additional equationterms referring to shear span span length and uncrackedlength in half of the beam were included This equationrequires a calculation-intensive process subject to humanerror thus the code also provides closed-form equations forcommon loading and support conditions Hence

Δ =119875119886

48119864119888119868cr

(3(119886

119871) minus 4 (

119886

119871)

3

minus 8120578(119871119892

119871)

3

) (4)

where 119886 is the shear span (mm) 119875 is the total applied load(N) 119871 is the span length (mm) 119871

119892is the uncracked length in

half of the beam (mm) (= 119886(119872cr119872119886)) and119864119888is themodulus

of elasticity of concrete (MPa) 120578 = (1 minus 119868cr119868119892)Recently the other semiempirical model was suggested

by modifying Bransonrsquos equation according to experimentalresults and a genetic algorithm approach [10] For betterprediction some of the factors were developed empiricallyThe model which has two multiplying factors and an expo-nential factor 119898 was analyzed using experimental data for55 FRP-reinforced concrete beams for the load-deflectionrelationship The effects of the elastic modulus of FRP barsreinforcement ratio and level of loading on the power of 119898in Bransonrsquos equation are taken into account in (5) as follows

119868119890= 013 (

119872cr119872119886

)

119898

119868119892+ 089 [1 minus (

119872cr119872119886

)

119898

] 119868cr le 119868119892 (5)

where119898 = minus024(120588119891120588119891119887) + 535(119872cr119872119886) + 228(119864

119891119864119904) 119864119891

is the modulus of elasticity of FRP bar (MPa) and 119864119904is the

modulus of elasticity of steel bar (MPa)ACI 4401R-15 [2] suggested an equation for calculating

the effective moment of inertia for reinforced concrete beamswith FRP bar This equation is based on Bischoff rsquos proposedapproach which represents a weighted average of flexibility




Lug GFRP core


P = 063 120572014

014

w = 023

P = Lug pitch120572 = 80∘

d

d

d

dd

d

998400

t = 021d





119868119890=

119868cr

1 minus 120574 (119872cr119872119886)2

[1 minus 119868cr119868119892]le 119868119892 (6)


120574 =3 (119886119871) minus 16 (119872cr119872119886) (119886119871)

2

+ 12 (119886119871)3

3 (119886119871) minus 4 (119886119871)3

(7)


119868119890=

119868119892119868cr

119868cr + (1 minus 05 (119872cr119872119886))2

(119868119892minus 119868cr)

(8)














200200 1600

25

180

186

0

230

18017

35

230

180

186

0

230

685 685


230

LVDT

Strain gauge

FB-1 FB-2 FB-3

P2P2

A

A2D10







(a) FB-1 (b) FB-2 (c) FB-3





(kN)


(kN)


(mm)

Nominalmoment (exp)

(kNsdotm)

Nominalmoment (cal)


FB-1 000427(00069) 120 544 344 186 136 FRP rupture

FB-2 00064(00069) 100 648 391 222 199 Compression

tension

FB-3 000903(00069) 120 740 266 254 210 Concrete

crushing





119891gt 120588119891119887 the design





119891119891= (radic

(119864119891120576119888119906)2

4+

08512057311198911015840

119888

120588119891

119864119891120576119888119906

minus 05119864119891120576119888119906)

le 119891119891119906

(9)

119872119899= 119860119891119891119891119906

(119889 minus119886

2) (10)










Deflection (mm)

Without k

With k

Ie reduced


concrete members

Mom

ent (

kNmiddotm

)


and 120588119891119887





ACI 440 1R-06 [6]




0

05

1

15

2

Effec

tive m

omen

t of i

nert

ia(times

108

mm

4)

Ie [7]Ie [10]Ie [2]

Ie [9]Ie [15]Ie [6]



119868119890= (

119872cr119872119886

)

119898

119868119892+ [1 minus (

119872cr119872119886

)

119898

minus 119870] 119868cr le 119868119892 (11)






Experiment

Proposed model


Mousavi et al [10]


ACI 440 1R-06 [6]

Δ [6]Δ [15]Δ [9]

Δ [2]Δ [3]Δ [10]Δ [7]


0

5

10

15

20

25

30

Appl

ied

mom

ent (

kNmiddotm

)








0

05

1

15

2

Effec

tive m

omen

t of i

nert

ia(times

108

mm

4)


Ie [7]Ie [10]

Ie [2]

Ie [9]Ie [15]Ie [6]




ACI 440 1R-06 [6]

Experiment

Proposed model

Δ [6]Δ [15]Δ [9]

Δ [2]Δ [3]

Δ [10]Δ [7]


0

5

10

15

20

25

30

Appl

ied

mom

ent (

kNmiddotm

)





ACI 440 1R-06 [6]



ACI 440 1R-15 [2]


0

05

1

15

2

Effec

tive m

omen

t of i

nert

ia(times

108

mm

4)


Ie [7]Ie [10]

Ie [2]

Ie [9]Ie [15]Ie [6]


Mousavi et al [10]



0

5

10

15

20

25

30

Appl

ied

mom

ent (

kNmiddotm

)

Experiment

Proposed model

Δ [6]Δ [15]Δ [9]

Δ [2]Δ [3]

Δ [10]Δ [7]






6 Conclusions





F1

Series 1

Group 2

50 100 150 200 250 300 350 400 450 500 550 6000Deflection (mm)

0

10

20

30

40

50

60

70

80

90

100

Appl

ied

mom

ent (

kNmiddotm

)

FB-1 FB-2

A1

FB-3

BC2HA









Competing Interests


Acknowledgments




References



























CeramicsJournal of







Biomaterials




Journal of


Journal of





CoatingsJournal of

Advances in








MaterialsJournal of


Nano

materials






Lug GFRP core


P = 063 120572014

014

w = 023

P = Lug pitch120572 = 80∘

d

d

d

dd

d

998400

t = 021d





119868119890=

119868cr

1 minus 120574 (119872cr119872119886)2

[1 minus 119868cr119868119892]le 119868119892 (6)


120574 =3 (119886119871) minus 16 (119872cr119872119886) (119886119871)

2

+ 12 (119886119871)3

3 (119886119871) minus 4 (119886119871)3

(7)


119868119890=

119868119892119868cr

119868cr + (1 minus 05 (119872cr119872119886))2

(119868119892minus 119868cr)

(8)














200200 1600

25

180

186

0

230

18017

35

230

180

186

0

230

685 685


230

LVDT

Strain gauge

FB-1 FB-2 FB-3

P2P2

A

A2D10







(a) FB-1 (b) FB-2 (c) FB-3





(kN)


(kN)


(mm)

Nominalmoment (exp)

(kNsdotm)

Nominalmoment (cal)


FB-1 000427(00069) 120 544 344 186 136 FRP rupture

FB-2 00064(00069) 100 648 391 222 199 Compression

tension

FB-3 000903(00069) 120 740 266 254 210 Concrete

crushing





119891gt 120588119891119887 the design





119891119891= (radic

(119864119891120576119888119906)2

4+

08512057311198911015840

119888

120588119891

119864119891120576119888119906

minus 05119864119891120576119888119906)

le 119891119891119906

(9)

119872119899= 119860119891119891119891119906

(119889 minus119886

2) (10)










Deflection (mm)

Without k

With k

Ie reduced


concrete members

Mom

ent (

kNmiddotm

)


and 120588119891119887





ACI 440 1R-06 [6]




0

05

1

15

2

Effec

tive m

omen

t of i

nert

ia(times

108

mm

4)

Ie [7]Ie [10]Ie [2]

Ie [9]Ie [15]Ie [6]



119868119890= (

119872cr119872119886

)

119898

119868119892+ [1 minus (

119872cr119872119886

)

119898

minus 119870] 119868cr le 119868119892 (11)






Experiment

Proposed model


Mousavi et al [10]


ACI 440 1R-06 [6]

Δ [6]Δ [15]Δ [9]

Δ [2]Δ [3]Δ [10]Δ [7]


0

5

10

15

20

25

30

Appl

ied

mom

ent (

kNmiddotm

)








0

05

1

15

2

Effec

tive m

omen

t of i

nert

ia(times

108

mm

4)


Ie [7]Ie [10]

Ie [2]

Ie [9]Ie [15]Ie [6]




ACI 440 1R-06 [6]

Experiment

Proposed model

Δ [6]Δ [15]Δ [9]

Δ [2]Δ [3]

Δ [10]Δ [7]


0

5

10

15

20

25

30

Appl

ied

mom

ent (

kNmiddotm

)





ACI 440 1R-06 [6]



ACI 440 1R-15 [2]


0

05

1

15

2

Effec

tive m

omen

t of i

nert

ia(times

108

mm

4)


Ie [7]Ie [10]

Ie [2]

Ie [9]Ie [15]Ie [6]


Mousavi et al [10]



0

5

10

15

20

25

30

Appl

ied

mom

ent (

kNmiddotm

)

Experiment

Proposed model

Δ [6]Δ [15]Δ [9]

Δ [2]Δ [3]

Δ [10]Δ [7]






6 Conclusions





F1

Series 1

Group 2

50 100 150 200 250 300 350 400 450 500 550 6000Deflection (mm)

0

10

20

30

40

50

60

70

80

90

100

Appl

ied

mom

ent (

kNmiddotm

)

FB-1 FB-2

A1

FB-3

BC2HA









Competing Interests


Acknowledgments




References



























CeramicsJournal of







Biomaterials




Journal of


Journal of





CoatingsJournal of

Advances in








MaterialsJournal of


Nano

materials













200200 1600

25

180

186

0

230

18017

35

230

180

186

0

230

685 685


230

LVDT

Strain gauge

FB-1 FB-2 FB-3

P2P2

A

A2D10







(a) FB-1 (b) FB-2 (c) FB-3





(kN)


(kN)


(mm)

Nominalmoment (exp)

(kNsdotm)

Nominalmoment (cal)


FB-1 000427(00069) 120 544 344 186 136 FRP rupture

FB-2 00064(00069) 100 648 391 222 199 Compression

tension

FB-3 000903(00069) 120 740 266 254 210 Concrete

crushing





119891gt 120588119891119887 the design





119891119891= (radic

(119864119891120576119888119906)2

4+

08512057311198911015840

119888

120588119891

119864119891120576119888119906

minus 05119864119891120576119888119906)

le 119891119891119906

(9)

119872119899= 119860119891119891119891119906

(119889 minus119886

2) (10)










Deflection (mm)

Without k

With k

Ie reduced


concrete members

Mom

ent (

kNmiddotm

)


and 120588119891119887





ACI 440 1R-06 [6]




0

05

1

15

2

Effec

tive m

omen

t of i

nert

ia(times

108

mm

4)

Ie [7]Ie [10]Ie [2]

Ie [9]Ie [15]Ie [6]



119868119890= (

119872cr119872119886

)

119898

119868119892+ [1 minus (

119872cr119872119886

)

119898

minus 119870] 119868cr le 119868119892 (11)






Experiment

Proposed model


Mousavi et al [10]


ACI 440 1R-06 [6]

Δ [6]Δ [15]Δ [9]

Δ [2]Δ [3]Δ [10]Δ [7]


0

5

10

15

20

25

30

Appl

ied

mom

ent (

kNmiddotm

)








0

05

1

15

2

Effec

tive m

omen

t of i

nert

ia(times

108

mm

4)


Ie [7]Ie [10]

Ie [2]

Ie [9]Ie [15]Ie [6]




ACI 440 1R-06 [6]

Experiment

Proposed model

Δ [6]Δ [15]Δ [9]

Δ [2]Δ [3]

Δ [10]Δ [7]


0

5

10

15

20

25

30

Appl

ied

mom

ent (

kNmiddotm

)





ACI 440 1R-06 [6]



ACI 440 1R-15 [2]


0

05

1

15

2

Effec

tive m

omen

t of i

nert

ia(times

108

mm

4)


Ie [7]Ie [10]

Ie [2]

Ie [9]Ie [15]Ie [6]


Mousavi et al [10]



0

5

10

15

20

25

30

Appl

ied

mom

ent (

kNmiddotm

)

Experiment

Proposed model

Δ [6]Δ [15]Δ [9]

Δ [2]Δ [3]

Δ [10]Δ [7]






6 Conclusions





F1

Series 1

Group 2

50 100 150 200 250 300 350 400 450 500 550 6000Deflection (mm)

0

10

20

30

40

50

60

70

80

90

100

Appl

ied

mom

ent (

kNmiddotm

)

FB-1 FB-2

A1

FB-3

BC2HA









Competing Interests


Acknowledgments




References



























CeramicsJournal of







Biomaterials




Journal of


Journal of





CoatingsJournal of

Advances in








MaterialsJournal of


Nano

materials




(a) FB-1 (b) FB-2 (c) FB-3





(kN)


(kN)


(mm)

Nominalmoment (exp)

(kNsdotm)

Nominalmoment (cal)


FB-1 000427(00069) 120 544 344 186 136 FRP rupture

FB-2 00064(00069) 100 648 391 222 199 Compression

tension

FB-3 000903(00069) 120 740 266 254 210 Concrete

crushing





119891gt 120588119891119887 the design





119891119891= (radic

(119864119891120576119888119906)2

4+

08512057311198911015840

119888

120588119891

119864119891120576119888119906

minus 05119864119891120576119888119906)

le 119891119891119906

(9)

119872119899= 119860119891119891119891119906

(119889 minus119886

2) (10)










Deflection (mm)

Without k

With k

Ie reduced


concrete members

Mom

ent (

kNmiddotm

)


and 120588119891119887





ACI 440 1R-06 [6]




0

05

1

15

2

Effec

tive m

omen

t of i

nert

ia(times

108

mm

4)

Ie [7]Ie [10]Ie [2]

Ie [9]Ie [15]Ie [6]



119868119890= (

119872cr119872119886

)

119898

119868119892+ [1 minus (

119872cr119872119886

)

119898

minus 119870] 119868cr le 119868119892 (11)






Experiment

Proposed model


Mousavi et al [10]


ACI 440 1R-06 [6]

Δ [6]Δ [15]Δ [9]

Δ [2]Δ [3]Δ [10]Δ [7]


0

5

10

15

20

25

30

Appl

ied

mom

ent (

kNmiddotm

)








0

05

1

15

2

Effec

tive m

omen

t of i

nert

ia(times

108

mm

4)


Ie [7]Ie [10]

Ie [2]

Ie [9]Ie [15]Ie [6]




ACI 440 1R-06 [6]

Experiment

Proposed model

Δ [6]Δ [15]Δ [9]

Δ [2]Δ [3]

Δ [10]Δ [7]


0

5

10

15

20

25

30

Appl

ied

mom

ent (

kNmiddotm

)





ACI 440 1R-06 [6]



ACI 440 1R-15 [2]


0

05

1

15

2

Effec

tive m

omen

t of i

nert

ia(times

108

mm

4)


Ie [7]Ie [10]

Ie [2]

Ie [9]Ie [15]Ie [6]


Mousavi et al [10]



0

5

10

15

20

25

30

Appl

ied

mom

ent (

kNmiddotm

)

Experiment

Proposed model

Δ [6]Δ [15]Δ [9]

Δ [2]Δ [3]

Δ [10]Δ [7]






6 Conclusions





F1

Series 1

Group 2

50 100 150 200 250 300 350 400 450 500 550 6000Deflection (mm)

0

10

20

30

40

50

60

70

80

90

100

Appl

ied

mom

ent (

kNmiddotm

)

FB-1 FB-2

A1

FB-3

BC2HA









Competing Interests


Acknowledgments




References



























CeramicsJournal of







Biomaterials




Journal of


Journal of





CoatingsJournal of

Advances in








MaterialsJournal of


Nano

materials




Deflection (mm)

Without k

With k

Ie reduced


concrete members

Mom

ent (

kNmiddotm

)


and 120588119891119887





ACI 440 1R-06 [6]




0

05

1

15

2

Effec

tive m

omen

t of i

nert

ia(times

108

mm

4)

Ie [7]Ie [10]Ie [2]

Ie [9]Ie [15]Ie [6]



119868119890= (

119872cr119872119886

)

119898

119868119892+ [1 minus (

119872cr119872119886

)

119898

minus 119870] 119868cr le 119868119892 (11)






Experiment

Proposed model


Mousavi et al [10]


ACI 440 1R-06 [6]

Δ [6]Δ [15]Δ [9]

Δ [2]Δ [3]Δ [10]Δ [7]


0

5

10

15

20

25

30

Appl

ied

mom

ent (

kNmiddotm

)








0

05

1

15

2

Effec

tive m

omen

t of i

nert

ia(times

108

mm

4)


Ie [7]Ie [10]

Ie [2]

Ie [9]Ie [15]Ie [6]




ACI 440 1R-06 [6]

Experiment

Proposed model

Δ [6]Δ [15]Δ [9]

Δ [2]Δ [3]

Δ [10]Δ [7]


0

5

10

15

20

25

30

Appl

ied

mom

ent (

kNmiddotm

)





ACI 440 1R-06 [6]



ACI 440 1R-15 [2]


0

05

1

15

2

Effec

tive m

omen

t of i

nert

ia(times

108

mm

4)


Ie [7]Ie [10]

Ie [2]

Ie [9]Ie [15]Ie [6]


Mousavi et al [10]



0

5

10

15

20

25

30

Appl

ied

mom

ent (

kNmiddotm

)

Experiment

Proposed model

Δ [6]Δ [15]Δ [9]

Δ [2]Δ [3]

Δ [10]Δ [7]






6 Conclusions





F1

Series 1

Group 2

50 100 150 200 250 300 350 400 450 500 550 6000Deflection (mm)

0

10

20

30

40

50

60

70

80

90

100

Appl

ied

mom

ent (

kNmiddotm

)

FB-1 FB-2

A1

FB-3

BC2HA









Competing Interests


Acknowledgments




References



























CeramicsJournal of







Biomaterials




Journal of


Journal of





CoatingsJournal of

Advances in








MaterialsJournal of


Nano

materials




Experiment

Proposed model


Mousavi et al [10]


ACI 440 1R-06 [6]

Δ [6]Δ [15]Δ [9]

Δ [2]Δ [3]Δ [10]Δ [7]


0

5

10

15

20

25

30

Appl

ied

mom

ent (

kNmiddotm

)








0

05

1

15

2

Effec

tive m

omen

t of i

nert

ia(times

108

mm

4)


Ie [7]Ie [10]

Ie [2]

Ie [9]Ie [15]Ie [6]




ACI 440 1R-06 [6]

Experiment

Proposed model

Δ [6]Δ [15]Δ [9]

Δ [2]Δ [3]

Δ [10]Δ [7]


0

5

10

15

20

25

30

Appl

ied

mom

ent (

kNmiddotm

)





ACI 440 1R-06 [6]



ACI 440 1R-15 [2]


0

05

1

15

2

Effec

tive m

omen

t of i

nert

ia(times

108

mm

4)


Ie [7]Ie [10]

Ie [2]

Ie [9]Ie [15]Ie [6]


Mousavi et al [10]



0

5

10

15

20

25

30

Appl

ied

mom

ent (

kNmiddotm

)

Experiment

Proposed model

Δ [6]Δ [15]Δ [9]

Δ [2]Δ [3]

Δ [10]Δ [7]






6 Conclusions





F1

Series 1

Group 2

50 100 150 200 250 300 350 400 450 500 550 6000Deflection (mm)

0

10

20

30

40

50

60

70

80

90

100

Appl

ied

mom

ent (

kNmiddotm

)

FB-1 FB-2

A1

FB-3

BC2HA









Competing Interests


Acknowledgments




References



























CeramicsJournal of







Biomaterials




Journal of


Journal of





CoatingsJournal of

Advances in








MaterialsJournal of


Nano

materials




ACI 440 1R-06 [6]



ACI 440 1R-15 [2]


0

05

1

15

2

Effec

tive m

omen

t of i

nert

ia(times

108

mm

4)


Ie [7]Ie [10]

Ie [2]

Ie [9]Ie [15]Ie [6]


Mousavi et al [10]



0

5

10

15

20

25

30

Appl

ied

mom

ent (

kNmiddotm

)

Experiment

Proposed model

Δ [6]Δ [15]Δ [9]

Δ [2]Δ [3]

Δ [10]Δ [7]






6 Conclusions





F1

Series 1

Group 2

50 100 150 200 250 300 350 400 450 500 550 6000Deflection (mm)

0

10

20

30

40

50

60

70

80

90

100

Appl

ied

mom

ent (

kNmiddotm

)

FB-1 FB-2

A1

FB-3

BC2HA









Competing Interests


Acknowledgments




References



























CeramicsJournal of







Biomaterials




Journal of


Journal of





CoatingsJournal of

Advances in








MaterialsJournal of


Nano

materials





F1

Series 1

Group 2

50 100 150 200 250 300 350 400 450 500 550 6000Deflection (mm)

0

10

20

30

40

50

60

70

80

90

100

Appl

ied

mom

ent (

kNmiddotm

)

FB-1 FB-2

A1

FB-3

BC2HA









Competing Interests


Acknowledgments




References



























CeramicsJournal of







Biomaterials




Journal of


Journal of





CoatingsJournal of

Advances in








MaterialsJournal of


Nano

materials





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



research article a modified model for deflection...

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