ShearStrengthPredictionModelofFRPBar-ReinforcedConcrete

Research ArticleShear Strength PredictionModel of FRP Bar-Reinforced ConcreteBeams without Stirrups

Danying Gao12 and Changhui Zhang 13

1School of Water Conservancy Engineering Zhengzhou University Zhengzhou Henan 450001 China2Henan University of Engineering Zhengzhou Henan 451191 China3North China University of Water Resources and Electric Power Zhengzhou Henan 450046 China

Correspondence should be addressed to Changhui Zhang changhuizhangfoxmailcom

Received 25 January 2020 Revised 29 February 2020 Accepted 10 March 2020 Published 9 April 2020

Academic Editor A M Bastos Pereira

Copyright copy 2020 Danying Gao and Changhui Zhang -is is an open access article distributed under the Creative CommonsAttribution License which permits unrestricted use distribution and reproduction in anymedium provided the original work isproperly cited

-e shear strength prediction model for fiber-reinforced polymer (FRP) bar-reinforced concrete beams without stirrups inACI4401R-2015 does not consider the ldquosize effectrdquo and the effect of shear span-to-depth ratio and predicts the zero-shear strengthfor concrete members without longitudinal reinforcement A modified shear strength prediction model for FRP bar-reinforcedconcrete beams without stirrups was presented in this paper -e proposed model takes into account the effect of concretestrength size of the beam shear span-to-depth ratio reinforcement ratio and modulus of elasticity of the longitudinal rein-forcement and the ldquosize effectrdquo -e superiority of the proposed model has been evaluated by comparing the calculated shearstrength of FRP bar-reinforced concrete beams without stirrups by the proposed model with the experimental results andcalculated values by the models in design codes respectively It confirmed that the shear strength of FRP bar-reinforced concretebeams without stirrups by the proposed model was in better agreement with the experimental results

1 Introduction

Fiber-reinforced polymer (FRP) bars have gained the ac-ceptance as an alternative to conventional steel bars forconcrete structures due to their corrosion resistance highstrength-to-weight ratio and magnetic neutrality It hasbeen recognized that the flexural capacity of FRP bar-reinforced concrete beams can be predicted by the tradi-tional assumptions used in steel bar-reinforced concretebeams However while FRP bars have two drawbacks in-cluding brittle failure and low modulus of elasticity com-pared with the steel bars the shear behavior (including shearstrength deformation and crack width) of concrete beamsreinforced with the FRP bar is different from those rein-forced with similar amount of steel reinforcement [1ndash4] Forexample the stiffness [3] and dowel action [4] of FRP bar-reinforced beams are smaller compared to the concretebeams with steel bars subsequently resulting in the largerdeformation and lower shear strength -erefore the

existing shear strength prediction models for steel bar-reinforced beams cannot be directly applied to FRP bar-reinforced beams

-e shear strength prediction model of FRP bar-rein-forced concrete beams in ACI4401R-2015 [5] was based onthe research by Tureyen and Frosch [6] -e ACI4401R-2015 model was a function of the width (b) and effectivedepth (d) of the beam concrete compressive strength (fc

prime)reinforcement ratio (ρf) and modulus of elasticity (Ef) ofFRP bars and the shear span-to-depth ratio (ad) was notincluded [7 8] Several investigations confirmed that shearstrength Vc of FRP bar-reinforced concrete beams decreasesas the ad increases [7 9ndash13] Some investigators [7]revealed that Vc decreases almost linearly with (ad)23while the others [10] reported that it decreases linearly with(ad)13 In addition the ACI4401R-2015 model did nottake into account the effect of concrete in the tensile zone ofthe beam on the shear strength for FRP bar-reinforcedconcrete beams without longitudinal reinforcement and

HindawiMathematical Problems in EngineeringVolume 2020 Article ID 7516502 11 pageshttpsdoiorg10115520207516502

predicted the zero-shear strength for concrete beamswithout longitudinal reinforcement [8] -irdly theACI4401R-15 model did not take into account the ldquosizeeffectrdquo [14ndash17] which explains the phenomenon that thenormalized shear strength of FRP bar-reinforced concretebeams decreases with the increasing of beam depth

It has been recognized that fcprime b d ad ρf and Ef are

the important parameters affecting the shear strength of FRPbar-reinforced concrete beams without stirrups An equa-tion that cannot predict the effects of known parameterswould lack generality and its applicability to general designsituations would be uncertain [18] -us there is a need todevelop a modified shear strength prediction model forproperly reflecting the effects of important parameterswhich are known to affect the shear strength of FRP bar-reinforced beams without stirrups

In this paper a database of published test results on shearstrength of 369 beams reinforced with FRP bars withoutstirrups was compiled Considering the effects of ad on theshear strength of FRP bar-reinforced concrete beamswithout stirrups and the ldquosize effectrdquo and the contribution ofthe concrete to the shear strength for concrete beamswithout longitudinal reinforcement a more accurate andrational-modified prediction model was proposed based onthe ACI4401R-2015 model -e efficiencies of the proposedmodel and ACI4401R-2015 CANCSA-S806-2012 [19]JSCE-1997 [20] AASHTO LRFD-2017 [21] and CNR-DT203-2006 [22] models were evaluated by comparing thecalculated values with the experimental values in the da-tabase and the test data which are not contained in thedatabase

2 Experimental Database

To present a new shear strength prediction model of FRPbar-reinforced concrete beams without stirrups a relativelylarge database including 369 beams reinforced with FRP barswithout stirrups from 42 different investigations wasestablished -e criteria for collecting the data of beamspecimens were as follows (1) rectangular cross sections (2)simply supported (3) tested under one- or two-pointloading (4) statically loaded and (5) failed in shear -isdatabase included various parameters which are known toaffect the shear strength of FRP bar-reinforced concretebeams without stirrups such as ad fc

prime b d ρf and Ef -evariation range of each parameter and the correspondingshear strength of the beam used in this study are given inTable 1 which is collected from the original source

3 Review of Shear Strength PredictionModels in Design Codes

-e shear strength prediction models of FRP bar-reinforcedconcrete beams without stirrups were highly valued bymanycountries such as America Japan Canada and Italy -eldquosize effectrdquo and effects of ad ρf and Ef on the shearstrength of FRP bar-reinforced concrete beams withoutstirrups were mainly considered in the shear strength pre-diction models of different design codes

31 ACI4401R-2015 Guidelines [5] -e shear strengthcalculating model of FRP bar-reinforced concrete beamswithout stirrups recommended by ACI Committee 440 is asfollows

Vacic

25

fcprime

1113969

b(kd) (1)

where k 2ρfnf + (ρfnf)2

1113969minus ρfnf nf EfEc

It can be clearly seen that equation (1) of the ACI4401R-2015 model predicts zero-shear strength for concrete beamswithout longitudinal reinforcement and does not considerthe effect of ad on the shear strength of FRP bar-reinforcedconcrete beams without stirrups and the ldquosize effectrdquo

32 JSCE-1997 Design Recommendations [20] -e shearstrength calculating model of FRP bar-reinforced concretebeams without stirrups recommended by Japan Society ofCivil Engineering (JSCE-1997) is as follows

Vjscec βdβρβnfvcdb d (2)

where βd (1000d)14 le 15 βρ (100ρEfEs)13 le 15 and

fvcd 02

fcprime3

1113969

le 072MPaIt is the same as equation (1) of the ACI4401R-15 model

that equation (2) of the JSCE-1997 model predicts zero-shear strength for concrete beams without longitudinalreinforcement and does not consider the effect of ad on theshear strength of FRP bar-reinforced concrete beamswithout stirrups although it takes into account the ldquosizeeffectrdquo through βd

33 CANCSA-S806-2012 Design Provisions [19] -e shearstrength calculating model of FRP bar-reinforced concretebeams without stirrups recommended by the TechnicalCommittee on Design and Construction of BuildingStructures with Fibre-Reinforced Polymers of CanadianStandards Association (CANCSA-S806) is as follows

Vs806c 005kmkrkaks

fcprime3

1113969b d (3)

011

fcprime

1113969b dleV

s806c le 022

fcprime

1113969

b d (4)

where km da

radicle 10 kr 1 +

Efρf

31113969

10le ka

25 dale 25 and ks 750(450 + d)le 10Equation (3) of the CANCSA-S806-2012 model con-

siders the effects of nearly all parameters on the shearstrength of FRP bar-reinforced concrete beams withoutstirrups

34 AASHTO LRFD-2017 Design Guide Specifications [21]-e shear strength calculating model of FRP bar-reinforcedconcrete beams without stirrups recommended by theAmerican Association of State Highway and TransportationOfficials is as follows

2 Mathematical Problems in Engineering

Vlrfdc 00676

fcprime

1113969

+ 46ρf

d

a1113888 1113889bdle 0126

fcprime

1113969

bd (5)

Equation (5) of the AASHTO LRFD-2017 model con-siders the effects of ad and ρf on the shear strength of FRPbar-reinforced concrete beams without stirrups while theeffect of Ef and the ldquosize effectrdquo are not included

35 CNR-DT203-2006 [22] -e shear strength of FRP bar-reinforced concrete beams without stirrups recommended byAdvisory Committee Technical Recommendations Con-struction of Italian National Research Council is as follows

Vcnrc 13 EfEs1113872 1113873

12τrkd 12 + 40ρf1113872 1113873bd (6)

where 13(EfEs)12 le 1 τr 025ft kd 16 minus (d1000)ge

1 and ρf le 002

Equation (6) of the CNR-DT203-2006 model considersthe ldquosize effectrdquo through kd and does not consider the effectof ad on the shear strength of FRP bar-reinforced concretebeams without stirrups

4 Proposed Shear Strength Prediction Model

According to the experimental database mentioned abovethe normalized shear strength V

expc

fcprime

1113969

bd was plottedagainst ρfnf (nf EfEc) as shown in Figure 1(a) It can beseen that the relationship between the normalized shearstrength V

expc

fcprime

1113969

bd and ρfnf fits into the trendline inFigure 1(a) except some plots To express this trendline witha reasonable equation the normalized shear strengthV

expc

fcprime

1113969

bd was also plotted against k (k

Table 1 Specimen details

Investigator No of specimens ad fcprime (MPa) b (mm) d (mm) Ef (GPa) ρf () Vc (kN)

Ashour and Kara [15 23] 18 25sim59 23sim502 150sim200 163sim371 32sim142 012sim139 9sim361Gross [4 24 25] 42 41sim65 363sim814 65sim279 141sim225 41sim139 033sim256 88sim51El-Sayedand El-Salakawy [26ndash28] 18 31sim65 40sim63 250sim1000 155sim326 39sim135 039sim263 60sim190Razaqpur et al [10] 7 18sim42 405sim49 200 225 145 025sim088 361sim962Benmokrane [29] 12 28sim37 341sim432 130sim160 310sim346 42sim120 072sim154 427sim637Tureyen and Frosch [30] 6 34 397sim426 457 360 38sim47 096sim192 947sim177Nanni [31] 3 26sim27 241 178 279sim287 40 077sim23 361sim534Deitz and Harik [32] 5 45sim58 27sim308 305 158 40 073 268sim292Mizukawa [33] 1 27 347 200 260 130 13 622Duranovic [34] 3 37 329sim381 150 210 45sim130 131sim136 262sim622Swamy [35] 2 32sim41 38sim39 154sim305 192sim222 34sim42 036sim155 195sim267Suzuki [36] 3 3 343 150 250 105 151sim302 405sim46Alam and Hussein [7 17] 37 15sim35 345sim883 250sim300 291sim744 47sim144 018sim147 437sim1558Nakamura and Higai [37] 2 3 227sim278 300 150 29 13sim18 33sim36Bentz et al [14] 6 33sim41 35sim46 450 188sim937 37 051sim254 545sim232Bazant and Yu [38] 1 31 40 450 970 40 046 136Wakui and Tottori [39] 4 32 446sim469 200 325 58sim192 07sim09 87sim118Nagasaka and Fukuyama [40] 2 31 229sim341 250 265 56 19 83sim113Issa et al [41] 6 57sim7 359 300 165sim170 48sim53 08sim412 293sim515Tomlinson and Fam [42] 3 41sim45 565sim60 150 245sim270 70 039sim085 209sim292Abed et al [3] 9 1sim15 43sim65 200 230sim330 51 092sim184 1166sim3739Guadagnini et al [43] 3 11sim33 428sim477 150 223 45 128 272sim81Kim and Jang[44] 40 15sim45 30sim403 150sim200 214sim216 40sim148 033sim079 166sim851Olivito and Zuccarello [45] 20 56 204sim272 150 180 115 087sim145 166sim299Matta et al [46] 12 31 295sim597 114sim457 146sim883 41sim49 012sim028 179sim2207-omas and Ramadass [9] 8 05sim18 406sim653 100sim170 270sim416 40 116sim175 30sim300Wegian and Abdalla [47] 6 65sim95 325 1000 105sim155 42sim147 023sim096 235sim127Andermatt and Lubell [11] 12 11sim21 399sim685 300sim310 257sim891 38sim43 147sim213 96sim11345Huaxin and Genjin [13] 13 11sim2 349sim546 200 260sim360 52sim210 076sim116 838sim3097Xiaoliang and Wenjun [48] 3 25sim27 473sim504 300 300sim315 45 214sim562 853sim1227El Refai and Abed [8] 8 25sim33 49 152 195sim215 50 031sim153 169sim316Chang and Seo [49] 14 58sim8 30 1200 130sim182 44sim50 024sim122 263sim159Abdul-Salam et al [50] 16 57sim63 413sim862 1000 134sim150 41sim148 051sim378 94sim213Farghaly and Benmokrane [51] 4 11 387sim493 300 1088sim1111 48sim144 026sim124 5955sim953Ali et al [12] 12 23sim3 13sim335 130 196sim200 52 03sim091 127sim394Omeman et al [1] 8 14sim23 347sim631 150 387sim662 134 113sim226 458sim2341Total 369 05sim95 13sim883 65sim1200 105sim1111 29sim210 012sim562 88sim11345

Note if the concrete cylinder compressive strength fcprime and modulus of elasticity of concrete Ec were not provided by the investigator while the concrete cube

compressive strength fcu was only measured it is assumed that fcprime 08fcu and Ec 4733

fcprime

1113969

If the unit of data was BS provided by the investigator it is

converted by the following factors 1 ksi 6895MPa 1 in 254mm and 1 kip 4448KN

Mathematical Problems in Engineering 3

2ρfnf + (ρfnf)2

1113969minus ρfnf) as shown in Figure 1(b) where k

can be defined as the relative depth of the compressive zonebecause the depth of compressive zone c can be calculated bymultiplying d by k (c k d) From the regression of theresults of FRP bar-reinforced concrete beams withoutstirrups there is a linear equation between V

expc

fcprime

1113969

bd andk which can be written as

Vexpc

fcprime

1113969

bd 04(k + 01) (7)

Equation (7) can take into account the effects of ρf andEf on the shear strength for FRP bar-reinforced concretebeams without stirrups and the contribution of concrete onshear strength for FRP bar-reinforced concrete beamswithout longitudinal reinforcement

41 Effect of Shear Span-to-Depth Ratio (ad) on the ShearStrength In order to eliminate the influence of ρf and Ef onshear strength of FRP bar-reinforced concrete beamswithout stirrups the normalized shear strength was taken asV

expc 04

fcprime

1113969

b(k + 01)d on the basis of equation (7) whichwas plotted against ad according to the experimental da-tabase mentioned above as shown in Figure 2 EvidentlyV

expc 04

fcprime

1113969

b(k + 01)d decreases as ad increases and

there is a trendline between Vexpc 04

fcprime

1113969

b(k + 01)d with

ad -e relationships between Vexpc 04

fcprime

1113969

b(k + 01)d andad can be obtained by the regressions based on the fol-lowing three criteria

(1) -e regressions can be described in four formsdepending upon the value of ad or determined bythe types of failure referred to those for steel bar-reinforced beams [52]Diagonal compression failure (when 0le adlt 1)

Vexpc

04

fcprime

1113969

b(k + 01)d f1(ad) (8)

Shear compression failure (when 1le adlt 3)

Vexpc

04

fcprime

1113969

b(k + 01)d f2

a

d1113874 1113875 (9)

Diagonal tension failure (when 3le adlt 6)

Vexpc

04

fcprime

1113969

b(k + 01)d f3

a

d1113874 1113875 (10)

Bending failure (when adge 6)

Vexpc

04

fcprime

1113969

b(k + 01)d f4

a

d1113874 1113875 (11)

(2) -e regressions remain coordinated on boundaryconditions

f1(1) f2(1)

f2(3) f3(3)

f3(6) f4(6)

(12)

(3) According to Reinforced Concrete Design toEurocode 2 [53] when ρf 0 and ad 0 the shearstrength can be calculated by the following equation

Vρf0ad0c τbd (13)

-e shear strength of concrete τ can be expressed by thetension strength of concrete ft as follows [52]

τ αft (14)

where α is a constant generally α 15 minus 25 Conserva-tively here α 16

-e concrete tension strength ft can be evaluated withthe equation in ACI318-14 [54] as follows

ft 58

fcprime

1113969

(15)

0 002 004 006 008 010

02

04

06

08

1

12

V cex

p radicf prime c

bd

ρfnf = ρfEfEc

Trendline

(a)

Vcexpradicf primecbd = 04(k + 01)

0 01 02 03 040

02

04

06

08

1

12

V cex

p radicf prime c

bd

k = radic2ρf nf + (ρf nf)2 ndash ρf nf

(b)

Figure 1 Effect of ρfnf and k on Vexpc

fcprime

1113969

b d


Substituting the values of ft in equation (15) intoequation (14) the equation for shear strength of concrete canbe rewritten as

τ

fcprime

1113969 (16)

Substituting the values of τ in equation (16) intoequation (13) when ρf 0 and ad 0 the equation for theshear strength of the FRP bar-reinforced concrete beamswithout stirrups can be rewritten as

Vρf0ad0c

fcprime

1113969bd (17)

Based on the regressions of the database aforemen-tioned as shown in Figure 2 the functions betweenV

expc 04

fcprime

1113969

b(k + 01)d and ad can be obtained as follows

f2a

d1113874 1113875

25(ad) minus 05

f3a

d1113874 1113875 1

f4a

d1113874 1113875

6ad

(18)

As there are only two specimens in the database whoseshear span-to-depth ratio is below 10 no more data can beemployed for regression therefore the function of f1(ad)

could be set up according to the criterions (2) and (3) as follows

f1a

d1113874 1113875

(1 minus (ad))(1 +(ad))

04(k + 01)+ 5

a

d1113874 1113875 (19)

-en the equation for shear strength of FRP bar-rein-forced concrete beams without stirrups which considers theeffect of shear span-to-depth ratio (ad) can be modified asfollows

Vc kfad

fcprime

1113969bd (20)

where

kfa d

(1 minus (ad))

(1 + (ad))+ 2(k + 01)

a

d1113874 1113875 while

a

dle 1

(k + 01)

((ad) minus 05) 1lt

a

dle 3

04(k + 01) while 3lta

dle 6

24(k + 01)

(ad) while adgt 6

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(21)

42 ldquoSize Effectrdquo on the Shear Strength According to theexperimental database mentioned above and equation (21)the normalized shear strength V

expc kfad

fcprime

1113969

bd was plottedagainst ad fc

prime b d ρf and Ef as shown in Figure 3 re-

spectively It can be seen that Vexpc kfad

fcprime

1113969

bd does not havethe obvious variation with the increase of b dfc

prime ρfEf andad respectively but there is a relation betweenV

expc kfad

fcprime

1113969

bd and d It indicates that the influence of b dfcprime ρf Ef and ad upon the shear strength of FRP bar-

reinforced concrete beams without stirrups has been em-bodied reasonably well by equation (20) except the influenceof d -is phenomenon demonstrates the ldquosize effectrdquo existson the shear strength of FRP bar-reinforced concrete beamswithout stirrups mentioned in the literature review

-en referring to Eurocode 2 for the shear capacity ofconcrete reinforced with the steel bar the equation for shearstrength of FRP bar-reinforced concrete beams withoutstirrups which has taken into account the ldquosize effectrdquo by alinear reduction form [15] can be established as follows

Vc kfadkd

fcprime

1113969bd (22)

where

0 1 2 3 4 5 6 7 8 9 100

2

4

6

8

10

1lt ad le 3kυ = 25(ad ndash 05)

3lt ad le 6

ad gt 6kυ = 1

k υ

ad

kυ = 6(ad)

kυ = Vcexp04radicf primecb(k + 01)d

Figure 2 Effect of ad on Vexpc 04

fcprime

1113969

b(k + 01)d


0 300 600 900 1200 15000

1

2

3

4

b (mm)

V cexp

k fadradic

f prime cbd

(a)

0

1

2

3

4

V cex

p kfa

dradicf prime c

bd

0 20 40 60 80 100f primec (MPa)

(b)

0

1

2

3

4

V cex

p kfa

dradicf prime c

bd

0 1 2 3 4 5 6ρf ()

(c)

0 300 600 900 12000

1

2

3

4

Trendline

d (mm)

V cexp

k fadradic

f prime cbd

(d)

0

1

2

3

4

V cex

p kfa

dradicf prime c

bd

0 30 60 90 120 150 180 210Ef (GPa)

(e)

0

1

2

3

4

V cexp

k fadradic

f prime cbd

0 2 4 6 8 10ad

(f )

Figure 3 Effect of parameters on Vexpc kfad

fcprime

1113969

bd

0

300

600

900

1200

V cexp (k

N)

ACI 4401Rndash2015

0 300 600 900 1200Vc

calc (kN)

DatabaseTest dataVc

exp = Vcclac

(a)

JSCEndash1997

0

300

600

900

1200

V cexp (k

N)

0 300 600 900 1200Vc

calc (kN)

DatabaseTest dataVc

exp = Vcclac

(b)

CANCSA-S806-2012

0

300

600

900

1200V cex

p (kN

)

0 300 600 900 1200Vc

calc (kN)

DatabaseTest dataVc

exp = Vcclac

(c)

V cexp (k

N)

0 300 600 900 12000

300

600

900

1200

Vccalc (kN)

AASHTO LRFDndash2017

DatabaseTest dataVc

exp = Vcclac

(d)

V cexp (k

N)

0

300

600

900

1200

0 300 600 900 1200Vc

calc (kN)

CNRndashDT203ndash2006

DatabaseTest dataVc

exp = Vcclac

(e)

V cexp (k

N)

0

300

600

900

1200

0 300 600 900 1200Vc

calc (kN)

Proposed model

DatabaseTest dataVc

exp = Vcclac

(f )

Figure 4 Comparison of Vexpc and Vcalc

c


16

12

8

4

0

12

9

6

3

06543210

15

12

9

6

3

065432104

3

2

1

0

V cexp V

cprop

V cexp V

ccnr

V cexp V

clr fd

V cexp V

cs806

V cexp V

cjsec

V cexp V

caci

0 2 4 6 8 10ad

0 01 02 03 04k = radic2ρf n f + (ρf nf)2 ndash ρf n f

0 300 600 900 1200d (mm)

DatabaseTest data

Data (adlt1)

Figure 5 Influence of ad k and d on shear strength predictions


kd 135 minusd

1000ge 075 (23)

5 Comparison of Predicted Shear Strength andExperimental Results

Figure 4 presents the correlations of the experimental shearstrength V

expc of all 369 specimens in the database mentioned

above and another 18 test data of specimens from Jumaa andYousif [55] and Ovitigala [56] with the calculated shearstrengths Vaci

c Vjscec Vs806

c Vlrfdc Vcnr

c and Vpropc respectively

A line with tolerance of 0 has been represented in thegraph which indicates that the exact prediction V

expc Vcalc

c

1 of the shear strength It can be seen that the ACI4401R-2015 JSCE-1997 and AASHTO LRFD-2017 models provideconservative predictions of the shear strengths of the mostspecimens (Vexp

c Vcalcc gt 1) the predictions of the CNR-

DT203-2006 model are highly unconservative for manyspecimens the CANCSA-S806-2012 model shows betteragreement with the experiment results than other models inthe design codes aforementioned and the predictions of theproposed model by equation (22) fit better with the ex-periment results than all the models in the design codesaforementioned

Figure 5 illustrates the relationships among the shearstrength ratio of the experimental shear strength to thecalculated shear strength V

expc Vcalc

c and ad k and d re-spectively As shown in Figure 5 the ratio of V

expc Vcalc

c

decreases as ad increases while adle 25 for all the modelsin the design codes aforementioned both ratios of V

expc Vs806

c

and Vexpc Vlrfd

c increase as k increases for the CANCSA-S806-2012 model does not well consider the effects of ρf norEf and the AASHTO LRFD-2017 model does not considerthe effect of Ef and the ratio of V

expc Vprop

c does not haveobvious variation with the increase of ad k or d Moreoverthe proposed model has the least scatter range of V

expc Vprop

c

from 039 to 263 whilst the scatter range of Vexpc Vcalc

c forACI4401R-2015 JSCE-1997 CANCSA-S806-2012 AASHTOLRFD-2017 and CNR-DT203-2006 models is from 082 to1738 044 to 1288 024 to 458 039 to 1391 and 018 to 487respectively Hence the effect of ad ρf Ef and d on shearstrength normalized by the proposed model of equation (22) iscaptured reasonably well It should be mentioned that thepredicted values by the proposed model for the two specimenswith shear span-to-effective depth less than 1 are in goodagreement with the experimental values (Vexp

c Vpropc 1)

though the function of the proposed model (adle 1) was notobtained by the regression method

-e ratios of Vexpc Vprop

c are presented graphically using ahistogram as shown in Figure 6 -e horizontal axis of thefigure shows the ratio of V

expc Vprop

c and the vertical axisrepresents the frequency of the specimens for a certainV

expc Vprop

c ratio It can be seen that the values of Vexpc Vprop

c

follow a normal distribution and 60 of the values is in anarrow range 08 to 12

To further investigate the superiority of the proposedshear strength prediction model of FRP bar-reinforcedconcrete beams without stirrups a total of three

performance checks were adopted -e mean standarddeviation (SD) and coefficient of variation (COV) of theratio V

expc Vcalc

c are given in Table 2 It can be observed thatthe proposed model as a whole predicts the shear strengthof FRP bar-reinforced concrete beams without stirrups withsmaller SD and COV values than ACI4401R-2015 JSCE-1997 AASHTO LRFD-2017 CNR-DT203-2006 and CANCSA-S806-2012 models

6 Conclusions

-e impacts of shear span-to-depth ratio upon the shearstrength of FRP bar-reinforced concrete beams withoutstirrups and the ldquosize effectrdquo were investigated by analyzingthe collected experimental results A newmodel for the shearstrength prediction of FRP bar-reinforced concrete beamswithout stirrups was proposed by using the regressionmethod based on the experimental database -e mainconclusions of this research are summarized as follows

(1) -e proposed model considers the ldquosize effectrdquo andthe effect of shear span-to-depth ratio on the shearstrength for FRP bar-reinforced concrete beamswithout stirrups which decrease as the shear span-to-depth ratio and effective depth increaserespectively

(2) -e values of experimental results for the shearstrength of FRP bar-reinforced concrete beamswithout stirrups to calculated values by the proposedmodel follow a normal distribution and 60 of thevalues distribute in a narrow range 08 to 12

0 05 1 15 2 250

5

10

15

20

25

Freq

uenc

y (

)

VcexpVc

prop

Figure 6 Histogram of the predictions of the proposed model

Table 2 Mean SD and COV of the ratio Vexpc Vcalc

c

Performance checksRegression database Test dataMean SD COV Mean SD COV

Vexpc Vaci

c 261 184 070 195 055 028Vexp

c Vlrfdc 255 173 068 209 064 030

Vexpc Vjsce

c 192 145 076 163 050 030Vexp

c Vs806c 115 052 045 112 019 017

Vexpc Vcnr

c 090 069 077 069 012 018Vexp

c Vpropc 109 031 028 114 019 017


(3) -e proposed model has more reasonable and reli-able predictions for the shear strength of FRP bar-reinforced concrete beams without stirrups incomparison with the models in design codesmentioned

Notations

fcprime Cylinder compressive strength of concrete MPa

fcu Cube compressive strength of concrete MPaft Tension strength of concrete MPaa Shear span mmb Beam width mmd Effective depth mmad Shear span-to-depth ratiok Relative depth of the compressive zonec Depth of the compressive zone mmρf Reinforcement rationf Modular ratioEf Modulus of elasticity of FRP bars MPaEc Modulus of elasticity of concrete MPaVc Shear strength of FRP bar-reinforced concrete

beams NV

expc Experimental shear strength N

Vcalcc Calculated shear strength N

Vpropc Calculated shear strength using the proposed model

NVaci

c Calculated shear strength using the ACI4401R-15model N

Vjscec Calculated shear strength using the JSCE-1997

model NVs806

c Calculated shear strength using the CANCSA-S806-2012 model N

Vlrfdc Calculated shear strength using the AASHTO LRFD-

2017 model NVcnr

c Calculated shear strength using the CNR-DT203-2006 model N

Data Availability

All data included in this study are available upon request bycontact with the corresponding author

Conflicts of Interest

-e authors declare that they have no conflicts of interest

Acknowledgments

-is research was financially supported by the NationalNatural Science Foundation of China (Grant no U1704254)

References

[1] Z Omeman M Nehdi and H El-Chabib ldquoExperimentalstudy on shear behavior of carbon-fiber-reinforced polymerreinforced concrete short beams without web reinforcementrdquoCanadian Journal of Civil Engineering vol 35 no 1 pp 1ndash102008

[2] M Abed Z Omeman and H El-Chabib ldquoOptimal efficiencyfactor in strut-and-tie model for FRP-reinforced concreteshort beams with (15 lt ad lt 25)rdquo Materials and Structuresvol 41 no 10 pp 1713ndash1727 2008

[3] F Abed H El-Chabib and M AlHamaydeh ldquoShear char-acteristics of GFRP-reinforced concrete deep beams withoutweb reinforcementrdquo Journal of Reinforced Plastics andComposites vol 31 no 16 pp 1063ndash1073 2012

[4] J R Yost S P Gross and D W Dinehart ldquoShear strength ofnormal strength concrete beams reinforced with deformedGFRP barsrdquo Journal of Composites for Construction vol 5no 4 pp 268ndash275 2001

[5] ACI Committee 440 Guide for the Design and Construction ofConcrete Reinforced with Fiber Rienforced Polymers (FRP)Bars (ACI 4401R-15) ACI Committee Farmington Hills MIUSA 2015

[6] A K Tureyen and R J Frosch ldquoConcrete shear strengthanother perspectiverdquo ACI Structural Journal vol 100 no 5pp 609ndash615 2003

[7] M S Alam and A Hussein ldquoUnified shear design equationfor concrete members reinforced with Fiber-Reinforcedpolymer without stirrupsrdquo Journal of Composites for Con-struction vol 17 no 5 pp 575ndash583 2013

[8] A El Refai and F Abed ldquoConcrete contribution to shearstrength of beams reinforced with basalt Fiber-Reinforcedbarsrdquo Journal of Composites for Construction vol 20 no 4pp 1ndash13 2016

[9] J -omas and S Ramadass ldquoParametric study of shearstrength of concrete beams reinforced with FRP barsrdquo Journalof Fe Institution of Engineers (India) Series A vol 97 no 3pp 273ndash284 2016

[10] A G Razaqpur B O Isgor S Greenaway and A SelleyldquoConcrete contribution to the shear resistance of fiber rein-forced polymer reinforced concrete membersrdquo Journal ofComposites for Construction vol 8 no 5 pp 452ndash460 2004

[11] M F Andermatt and A S Lubell ldquoBehavior of concrete deepbeams reinforced with internal Fiber-Reinforced Polymer--Experimental studyrdquo ACI Structural Journal vol 110 no 4pp 585ndash594 2013

[12] I Ali R -amrin A A S Abdul and M Noridah ldquoDiagonalshear cracks and size effect in concrete beams reinforced withglass fiber reinforced polymer (GFRP) barsrdquo Applied Me-chanics and Materials vol 621 pp 113ndash119 2014

[13] L Huaxin and L Genjin ldquoShear capacity of basalt fiberreinforced polymer reinforced recycled concrete deep beamwithout web reinforcementrdquo Journal of Sichuan University(Engineering Science Edition) vol 47 no 5 pp 17ndash22 2015

[14] E C Bentz L Massam and M P Collins ldquoShear strength oflarge concrete members with FRP reinforcementrdquo Journal ofComposites for Construction vol 14 no 6 pp 637ndash646 2010

[15] A F Ashour and I F Kara ldquoSize effect on shear strength ofFRP reinforced concrete beamsrdquo Composites Part B Engi-neering vol 60 pp 612ndash620 2014

[16] M S Alam and A Hussein ldquoEffect of member depth on shearstrength of high-strength fiber-reinforced polymer-reinforcedconcrete beamsrdquo Journal of Composites for Constructionvol 16 no 2 pp 119ndash126 2012

[17] M S Alam and A Hussein ldquoSize effect on shear strength ofFRP reinforced concrete beams without stirrupsrdquo Journal ofComposites for Construction vol 17 no 4 pp 507ndash516 2013

[18] A G Razaqpur ldquoProposed shear design method for FRP-Reinforced concrete members without stirrupsrdquo ACI Struc-tural Journal vol 103 no 1 2006


[19] Canadian Standards Association Design and Construction ofBuilding Structures with Fibre-Reinforced Polymers (CSA-S806-12) Canadian Standards Association MississaugaCanada 2012

[20] Japan Society of Civil Engineers Recommendation for Designand Construction of Concrete Structures Using ContinuousFiber Reinforcing materials (JSCE-1997) Japan Society of CivilEngineers Tokyo Japan 1997

[21] AASHTO Bridge Design Guide Specifications for GFRP-Reinforced Concrete Bridge Decks and Traffic Railings(AASHTO LRFD-17) AASHTO Washington DC USA2017

[22] Advisory Committee Technical Recommendations Con-struction Guide for the Design and Construction of ConcreteStructures Reinforced with Fiber-Reinforced Polymer Bars(CNR-DT203-06) Advisory Committee Technical Recom-mendations Construction Rome Italy 2006

[23] A F Ashour ldquoFlexural and shear capacities of concrete beamsreinforced with GFRP barsrdquo Construction and BuildingMaterials vol 20 no 10 pp 1005ndash1015 2006

[24] JR Yost and S P Gross ldquoEffective moment of inertia for glassFiber-Reinforced Polymer-Reinforced concrete beamsrdquo ACIStructural Journal vol 100 no 6 pp 732ndash739 2003

[25] S P Gross J R Yost DW Dinehart E Svensen and N LiuldquoShear strength of normal and high strength concrete beamsreinforced with GFRP barsrdquo in Proceedings of the Interna-tional Conference on High Performance Materials in Bridge-sKona HI USA 2001

[26] A K El-Sayed and E F El-Salakawy ldquoShear strength of FRP-reinforced concrete beams without transverse reinforcementrdquoACI Structural Journal vol 103 no 2 pp 235ndash243 2006

[27] A K El-Sayed and E F El-Salakawy ldquoShear capacity of high-strength concrete beams reinforced with FRP barsrdquo ACIStructural Journal vol 103 no 3 pp 383ndash389 2006

[28] A El-Sayed E El-Salakawy and B Benmokrane ldquoShearstrength of one-way concrete slabs reinforced with Fiber-Reinforced polymer composite barsrdquo Journal of Compositesfor Construction vol 9 no 2 pp 147ndash157 2005

[29] M Benmokrane and J P Newhook ldquoShear testing of frpreinforced concrete without transverse reinforcementrdquo inProceedings of the Annual Conference of the Canadian Societyfor Civil Engineering Moncton Canada 2003

[30] A K Tureyen and R J Frosch ldquoShear tests of FRP-reinforcedconcrete beams without stirrupsrdquo ACI Structural Journalvol 99 no 4 pp 427ndash434 2002

[31] T A M W Nanni ldquoShear strength of GFRP RC beams andslabsrdquo in Proceedings of the 4th International SymposiumFiber Reinforced Polymer Reinforcement for Reinforced Con-crete Structures Porto Portugal 2001

[32] D H Deitz and I E Harik ldquoOne-way slabs reinforced withglass fiber reinforced polymer reinforcing barsrdquo in Proceed-ings of the 4th International Symposium Fiber ReinforcedPolymer Reinforcement for Reinforced Concrete StructuresPorto Portugal 1999

[33] Y S Mizukawa ldquoA study on shear fatigue behavior of con-crete beams with FRP rodsrdquo in Proceedings of the Fird In-ternational Symposium on Non-metallic (FRP) Reinforcementfor Concrete structures (FRPRCS-3) Sapporo Japan 1997

[34] N Duranovic ldquoTests on concrete beams reinforced with glassfiber reinforced plastic barsrdquo in Proceedings of the FirdInternational Symposium on Non-metallic (FRP) Reinforce-ment for Concrete structures (FRPRCS-3) Sapporo Japan1997

[35] N A M Swamy ldquoStructural implications of using GFRP barsas concrete reinforcementrdquo in Proceedings of the Fird In-ternational Symposium on Non-metallic (FRP) Reinforcementfor Concrete structures (FRPRCS-3) Sapporo Japan 1997

[36] H Suzuki W Zhao and K Maruyama ldquoShear behavior ofconcrete beams reinforced by FRP rods as longitudinal andshear reinforcementrdquo in Proceedings of the Second Interna-tional Rilem Symposium on Non-metallic (FRP) Reinforcementfor Concrete Structures (FRPRCS-2) Ghent Belgium 1995

[37] H Nakamura and T Higai ldquoEvaluation of shear strength ofconcrete beams reinforced with FRPrdquo in Proceedings of theJapan Society of Civil Engineers pp 111ndash123 Tokyo Japan1995

[38] Z P Bazant and Q Yu ldquoSafe shear design of large widebeamsrdquo Concrete International vol 26 no 8 pp 14ndash17 2004

[39] H Wakui and S Tottori ldquoShear capacity of RC and PC beamsusing FRP reinforcementrdquo ACI Special Publication vol 138no 27 pp 615ndash632 1993

[40] T Nagasaka and H Fukuyama ldquoShear performance ofconcrete beams reinforced with FRP stirrupsrdquo ACI SpecialPublication vol 138 pp 789ndash811 1993

[41] M AM U Issa T Ovitigala and T Ibrahim ldquoShear behaviorof basalt fiber reinforced concrete beams with and withoutbasalt FRP stirrupsrdquo Journal of Composites for Constructionvol 20 no 4 2016

[42] D Tomlinson and A Fam ldquoPerformance of concrete beamsreinforced with basalt FRP for flexure and shearrdquo Journal ofComposites for Construction vol 19 no 2 2015

[43] M Guadagnini K Pilakoutas and P Waldron ldquoShear re-sistance of FRP RC beams experimental studyrdquo Journal ofComposites for Construction vol 10 no 6 pp 464ndash473 2006

[44] C H Kim and H S Jang ldquoConcrete shear strength of normaland lightweight concrete beams reinforced with FRP barsrdquoJournal of Composites for Construction vol 18 no 2pp 1090ndash1268 2014

[45] R S Olivito and F A Zuccarello ldquoOn the shear behaviour ofconcrete beams reinforced by carbon fibre-reinforced poly-mer bars an experimental investigation by means of acousticemission techniquerdquo Strain vol 46 no 5 pp 470ndash481 2010

[46] F Matta A K El-Sayed A Nanni and B Benmokrane ldquoSizeeffect on concrete shear strength in beams reinforced withFiber-Reinforced polymer barsrdquo ACI Structural Journalvol 110 no 4 pp 617ndash628 2013

[47] F M Wegian and H A Abdalla ldquoShear capacity of concretebeams reinforced with fiber reinforced polymersrdquo CompositeStructures vol 71 no 1 pp 130ndash138 2005

[48] Z Xiaoliang and Q Wenjun ldquoShear behavior test of GFRP-reinforced concrete beams without stirrupsrdquo China Journal ofHighway and Transport vol 23 no 5 pp 51ndash57 2010

[49] K Chang and D Seo ldquoBehavior of One-Way concrete slabsreinforced with GFRP barsrdquo Journal of Asian Architecture andBuilding Engineering vol 11 no 2 pp 351ndash358 2012

[50] B Abdul-Salam A S Farghaly and B BenmokraneldquoMechanisms of shear resistance of one-way concrete slabsreinforced with FRP barsrdquo Construction and Building Ma-terials vol 127 pp 959ndash970 2016

[51] A S Farghaly and B Benmokrane ldquoShear behavior of FRP-Reinforced concrete deep beams without web reinforcementrdquoJournal of Composites for Construction vol 17 no 6 ArticleID 04013015 2013

[52] Z GUO Principles of Reinforced Concrete TsinghuaUniversity Press Peiking China 2014


[53] G Toniolo and M di Prisco Erratum to Reinforced ConcreteDesign to Eurocode 2 Springer Tracts in Civil EngineeringMilan Italy 2017

[54] ACI Commiting 318 Building Code Requirements for Struc-tural Concrete and Commentary (ACI318-14) ACI Com-miting Farmington Hills MI USA 2014

[55] G B Jumaa and A R Yousif ldquoSize effect in shear failure ofhigh strength concrete beams without stirrup reinforced withbasalt FRP barsrdquo KSCE Journal of Civil Engineering vol 23no 4 pp 1636ndash1650 2019

[56] T Ovitigala Structural Behavior of Concrete Beams Reinforcedwith Basalt Fiber Reinforced Polymer (BFRP) Bars Universityof Illinois at Chicago Chicago IL USA 2012


predicted the zero-shear strength for concrete beamswithout longitudinal reinforcement [8] -irdly theACI4401R-15 model did not take into account the ldquosizeeffectrdquo [14ndash17] which explains the phenomenon that thenormalized shear strength of FRP bar-reinforced concretebeams decreases with the increasing of beam depth

It has been recognized that fcprime b d ad ρf and Ef are

the important parameters affecting the shear strength of FRPbar-reinforced concrete beams without stirrups An equa-tion that cannot predict the effects of known parameterswould lack generality and its applicability to general designsituations would be uncertain [18] -us there is a need todevelop a modified shear strength prediction model forproperly reflecting the effects of important parameterswhich are known to affect the shear strength of FRP bar-reinforced beams without stirrups

In this paper a database of published test results on shearstrength of 369 beams reinforced with FRP bars withoutstirrups was compiled Considering the effects of ad on theshear strength of FRP bar-reinforced concrete beamswithout stirrups and the ldquosize effectrdquo and the contribution ofthe concrete to the shear strength for concrete beamswithout longitudinal reinforcement a more accurate andrational-modified prediction model was proposed based onthe ACI4401R-2015 model -e efficiencies of the proposedmodel and ACI4401R-2015 CANCSA-S806-2012 [19]JSCE-1997 [20] AASHTO LRFD-2017 [21] and CNR-DT203-2006 [22] models were evaluated by comparing thecalculated values with the experimental values in the da-tabase and the test data which are not contained in thedatabase

2 Experimental Database

To present a new shear strength prediction model of FRPbar-reinforced concrete beams without stirrups a relativelylarge database including 369 beams reinforced with FRP barswithout stirrups from 42 different investigations wasestablished -e criteria for collecting the data of beamspecimens were as follows (1) rectangular cross sections (2)simply supported (3) tested under one- or two-pointloading (4) statically loaded and (5) failed in shear -isdatabase included various parameters which are known toaffect the shear strength of FRP bar-reinforced concretebeams without stirrups such as ad fc

prime b d ρf and Ef -evariation range of each parameter and the correspondingshear strength of the beam used in this study are given inTable 1 which is collected from the original source

3 Review of Shear Strength PredictionModels in Design Codes

-e shear strength prediction models of FRP bar-reinforcedconcrete beams without stirrups were highly valued bymanycountries such as America Japan Canada and Italy -eldquosize effectrdquo and effects of ad ρf and Ef on the shearstrength of FRP bar-reinforced concrete beams withoutstirrups were mainly considered in the shear strength pre-diction models of different design codes

31 ACI4401R-2015 Guidelines [5] -e shear strengthcalculating model of FRP bar-reinforced concrete beamswithout stirrups recommended by ACI Committee 440 is asfollows

Vacic

25

fcprime

1113969

b(kd) (1)

where k 2ρfnf + (ρfnf)2

1113969minus ρfnf nf EfEc

It can be clearly seen that equation (1) of the ACI4401R-2015 model predicts zero-shear strength for concrete beamswithout longitudinal reinforcement and does not considerthe effect of ad on the shear strength of FRP bar-reinforcedconcrete beams without stirrups and the ldquosize effectrdquo

32 JSCE-1997 Design Recommendations [20] -e shearstrength calculating model of FRP bar-reinforced concretebeams without stirrups recommended by Japan Society ofCivil Engineering (JSCE-1997) is as follows

Vjscec βdβρβnfvcdb d (2)

where βd (1000d)14 le 15 βρ (100ρEfEs)13 le 15 and

fvcd 02

fcprime3

1113969

le 072MPaIt is the same as equation (1) of the ACI4401R-15 model

that equation (2) of the JSCE-1997 model predicts zero-shear strength for concrete beams without longitudinalreinforcement and does not consider the effect of ad on theshear strength of FRP bar-reinforced concrete beamswithout stirrups although it takes into account the ldquosizeeffectrdquo through βd

33 CANCSA-S806-2012 Design Provisions [19] -e shearstrength calculating model of FRP bar-reinforced concretebeams without stirrups recommended by the TechnicalCommittee on Design and Construction of BuildingStructures with Fibre-Reinforced Polymers of CanadianStandards Association (CANCSA-S806) is as follows

Vs806c 005kmkrkaks

fcprime3

1113969b d (3)

011

fcprime

1113969b dleV

s806c le 022

fcprime

1113969

b d (4)

where km da

radicle 10 kr 1 +

Efρf

31113969

10le ka

25 dale 25 and ks 750(450 + d)le 10Equation (3) of the CANCSA-S806-2012 model con-

siders the effects of nearly all parameters on the shearstrength of FRP bar-reinforced concrete beams withoutstirrups

34 AASHTO LRFD-2017 Design Guide Specifications [21]-e shear strength calculating model of FRP bar-reinforcedconcrete beams without stirrups recommended by theAmerican Association of State Highway and TransportationOfficials is as follows


Vlrfdc 00676

fcprime

1113969

+ 46ρf

d

a1113888 1113889bdle 0126

fcprime

1113969

bd (5)



Vcnrc 13 EfEs1113872 1113873

12τrkd 12 + 40ρf1113872 1113873bd (6)


1 and ρf le 002




expc

fcprime

1113969


expc

fcprime

1113969


expc

fcprime

1113969







fcprime

1113969




2ρfnf + (ρfnf)2



expc

fcprime

1113969


Vexpc

fcprime

1113969

bd 04(k + 01) (7)



expc 04

fcprime

1113969


expc 04

fcprime

1113969



fcprime

1113969

b(k + 01)d with


fcprime

1113969



Vexpc

04

fcprime

1113969

b(k + 01)d f1(ad) (8)


Vexpc

04

fcprime

1113969

b(k + 01)d f2

a

d1113874 1113875 (9)


Vexpc

04

fcprime

1113969

b(k + 01)d f3

a

d1113874 1113875 (10)


Vexpc

04

fcprime

1113969

b(k + 01)d f4

a

d1113874 1113875 (11)


f1(1) f2(1)

f2(3) f3(3)

f3(6) f4(6)

(12)


Vρf0ad0c τbd (13)


τ αft (14)



ft 58

fcprime

1113969

(15)

0 002 004 006 008 010

02

04

06

08

1

12

V cex

p radicf prime c

bd

ρfnf = ρfEfEc

Trendline

(a)


0 01 02 03 040

02

04

06

08

1

12

V cex

p radicf prime c

bd


(b)


fcprime

1113969

b d



τ

fcprime

1113969 (16)


Vρf0ad0c

fcprime

1113969bd (17)


expc 04

fcprime

1113969


f2a

d1113874 1113875

25(ad) minus 05

f3a

d1113874 1113875 1

f4a

d1113874 1113875

6ad

(18)



f1a

d1113874 1113875

(1 minus (ad))(1 +(ad))

04(k + 01)+ 5

a

d1113874 1113875 (19)


Vc kfad

fcprime

1113969bd (20)

where

kfa d

(1 minus (ad))

(1 + (ad))+ 2(k + 01)

a

d1113874 1113875 while

a

dle 1

(k + 01)

((ad) minus 05) 1lt

a

dle 3


dle 6

24(k + 01)

(ad) while adgt 6

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(21)


expc kfad

fcprime

1113969




fcprime

1113969



expc kfad

fcprime

1113969




Vc kfadkd

fcprime

1113969bd (22)

where

0 1 2 3 4 5 6 7 8 9 100

2

4

6

8

10


3lt ad le 6

ad gt 6kυ = 1

k υ

ad

kυ = 6(ad)



fcprime

1113969

b(k + 01)d


0 300 600 900 1200 15000

1

2

3

4

b (mm)

V cexp

k fadradic

f prime cbd

(a)

0

1

2

3

4

V cex

p kfa

dradicf prime c

bd

0 20 40 60 80 100f primec (MPa)

(b)

0

1

2

3

4

V cex

p kfa

dradicf prime c

bd

0 1 2 3 4 5 6ρf ()

(c)

0 300 600 900 12000

1

2

3

4

Trendline

d (mm)

V cexp

k fadradic

f prime cbd

(d)

0

1

2

3

4

V cex

p kfa

dradicf prime c

bd

0 30 60 90 120 150 180 210Ef (GPa)

(e)

0

1

2

3

4

V cexp

k fadradic

f prime cbd

0 2 4 6 8 10ad

(f )


fcprime

1113969

bd

0

300

600

900

1200

V cexp (k

N)

ACI 4401Rndash2015

0 300 600 900 1200Vc

calc (kN)

DatabaseTest dataVc

exp = Vcclac

(a)

JSCEndash1997

0

300

600

900

1200

V cexp (k

N)

0 300 600 900 1200Vc

calc (kN)

DatabaseTest dataVc

exp = Vcclac

(b)

CANCSA-S806-2012

0

300

600

900

1200V cex

p (kN

)

0 300 600 900 1200Vc

calc (kN)

DatabaseTest dataVc

exp = Vcclac

(c)

V cexp (k

N)

0 300 600 900 12000

300

600

900

1200

Vccalc (kN)


DatabaseTest dataVc

exp = Vcclac

(d)

V cexp (k

N)

0

300

600

900

1200

0 300 600 900 1200Vc

calc (kN)


DatabaseTest dataVc

exp = Vcclac

(e)

V cexp (k

N)

0

300

600

900

1200

0 300 600 900 1200Vc

calc (kN)

Proposed model

DatabaseTest dataVc

exp = Vcclac

(f )


c


16

12

8

4

0

12

9

6

3

06543210

15

12

9

6

3

065432104

3

2

1

0

V cexp V

cprop

V cexp V

ccnr

V cexp V

clr fd

V cexp V

cs806

V cexp V

cjsec

V cexp V

caci

0 2 4 6 8 10ad


0 300 600 900 1200d (mm)

DatabaseTest data

Data (adlt1)



kd 135 minusd

1000ge 075 (23)





c Vjscec Vs806

c Vlrfdc Vcnr



expc Vcalc

c





expc Vcalc


expc Vcalc

c


expc Vs806

c

and Vexpc Vlrfd


expc Vprop


expc Vprop

c



c Vpropc 1)




expc Vprop


expc Vprop


c




expc Vcalc


6 Conclusions




0 05 1 15 2 250

5

10

15

20

25

Freq

uenc

y (

)

VcexpVc

prop



c


Vexpc Vaci

c 261 184 070 195 055 028Vexp

c Vlrfdc 255 173 068 209 064 030

Vexpc Vjsce

c 192 145 076 163 050 030Vexp

c Vs806c 115 052 045 112 019 017

Vexpc Vcnr

c 090 069 077 069 012 018Vexp

c Vpropc 109 031 028 114 019 017



Notations



beams NV




NVaci



model NVs806



2017 model NVcnr


Data Availability




Acknowledgments


References




























































Vlrfdc 00676

fcprime

1113969

+ 46ρf

d

a1113888 1113889bdle 0126

fcprime

1113969

bd (5)



Vcnrc 13 EfEs1113872 1113873

12τrkd 12 + 40ρf1113872 1113873bd (6)


1 and ρf le 002




expc

fcprime

1113969


expc

fcprime

1113969


expc

fcprime

1113969







fcprime

1113969




2ρfnf + (ρfnf)2



expc

fcprime

1113969


Vexpc

fcprime

1113969

bd 04(k + 01) (7)



expc 04

fcprime

1113969


expc 04

fcprime

1113969



fcprime

1113969

b(k + 01)d with


fcprime

1113969



Vexpc

04

fcprime

1113969

b(k + 01)d f1(ad) (8)


Vexpc

04

fcprime

1113969

b(k + 01)d f2

a

d1113874 1113875 (9)


Vexpc

04

fcprime

1113969

b(k + 01)d f3

a

d1113874 1113875 (10)


Vexpc

04

fcprime

1113969

b(k + 01)d f4

a

d1113874 1113875 (11)


f1(1) f2(1)

f2(3) f3(3)

f3(6) f4(6)

(12)


Vρf0ad0c τbd (13)


τ αft (14)



ft 58

fcprime

1113969

(15)

0 002 004 006 008 010

02

04

06

08

1

12

V cex

p radicf prime c

bd

ρfnf = ρfEfEc

Trendline

(a)


0 01 02 03 040

02

04

06

08

1

12

V cex

p radicf prime c

bd


(b)


fcprime

1113969

b d



τ

fcprime

1113969 (16)


Vρf0ad0c

fcprime

1113969bd (17)


expc 04

fcprime

1113969


f2a

d1113874 1113875

25(ad) minus 05

f3a

d1113874 1113875 1

f4a

d1113874 1113875

6ad

(18)



f1a

d1113874 1113875

(1 minus (ad))(1 +(ad))

04(k + 01)+ 5

a

d1113874 1113875 (19)


Vc kfad

fcprime

1113969bd (20)

where

kfa d

(1 minus (ad))

(1 + (ad))+ 2(k + 01)

a

d1113874 1113875 while

a

dle 1

(k + 01)

((ad) minus 05) 1lt

a

dle 3


dle 6

24(k + 01)

(ad) while adgt 6

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(21)


expc kfad

fcprime

1113969




fcprime

1113969



expc kfad

fcprime

1113969




Vc kfadkd

fcprime

1113969bd (22)

where

0 1 2 3 4 5 6 7 8 9 100

2

4

6

8

10


3lt ad le 6

ad gt 6kυ = 1

k υ

ad

kυ = 6(ad)



fcprime

1113969

b(k + 01)d


0 300 600 900 1200 15000

1

2

3

4

b (mm)

V cexp

k fadradic

f prime cbd

(a)

0

1

2

3

4

V cex

p kfa

dradicf prime c

bd

0 20 40 60 80 100f primec (MPa)

(b)

0

1

2

3

4

V cex

p kfa

dradicf prime c

bd

0 1 2 3 4 5 6ρf ()

(c)

0 300 600 900 12000

1

2

3

4

Trendline

d (mm)

V cexp

k fadradic

f prime cbd

(d)

0

1

2

3

4

V cex

p kfa

dradicf prime c

bd

0 30 60 90 120 150 180 210Ef (GPa)

(e)

0

1

2

3

4

V cexp

k fadradic

f prime cbd

0 2 4 6 8 10ad

(f )


fcprime

1113969

bd

0

300

600

900

1200

V cexp (k

N)

ACI 4401Rndash2015

0 300 600 900 1200Vc

calc (kN)

DatabaseTest dataVc

exp = Vcclac

(a)

JSCEndash1997

0

300

600

900

1200

V cexp (k

N)

0 300 600 900 1200Vc

calc (kN)

DatabaseTest dataVc

exp = Vcclac

(b)

CANCSA-S806-2012

0

300

600

900

1200V cex

p (kN

)

0 300 600 900 1200Vc

calc (kN)

DatabaseTest dataVc

exp = Vcclac

(c)

V cexp (k

N)

0 300 600 900 12000

300

600

900

1200

Vccalc (kN)


DatabaseTest dataVc

exp = Vcclac

(d)

V cexp (k

N)

0

300

600

900

1200

0 300 600 900 1200Vc

calc (kN)


DatabaseTest dataVc

exp = Vcclac

(e)

V cexp (k

N)

0

300

600

900

1200

0 300 600 900 1200Vc

calc (kN)

Proposed model

DatabaseTest dataVc

exp = Vcclac

(f )


c


16

12

8

4

0

12

9

6

3

06543210

15

12

9

6

3

065432104

3

2

1

0

V cexp V

cprop

V cexp V

ccnr

V cexp V

clr fd

V cexp V

cs806

V cexp V

cjsec

V cexp V

caci

0 2 4 6 8 10ad


0 300 600 900 1200d (mm)

DatabaseTest data

Data (adlt1)



kd 135 minusd

1000ge 075 (23)





c Vjscec Vs806

c Vlrfdc Vcnr



expc Vcalc

c





expc Vcalc


expc Vcalc

c


expc Vs806

c

and Vexpc Vlrfd


expc Vprop


expc Vprop

c



c Vpropc 1)




expc Vprop


expc Vprop


c




expc Vcalc


6 Conclusions




0 05 1 15 2 250

5

10

15

20

25

Freq

uenc

y (

)

VcexpVc

prop



c


Vexpc Vaci

c 261 184 070 195 055 028Vexp

c Vlrfdc 255 173 068 209 064 030

Vexpc Vjsce

c 192 145 076 163 050 030Vexp

c Vs806c 115 052 045 112 019 017

Vexpc Vcnr

c 090 069 077 069 012 018Vexp

c Vpropc 109 031 028 114 019 017



Notations



beams NV




NVaci



model NVs806



2017 model NVcnr


Data Availability




Acknowledgments


References




























































2ρfnf + (ρfnf)2



expc

fcprime

1113969


Vexpc

fcprime

1113969

bd 04(k + 01) (7)



expc 04

fcprime

1113969


expc 04

fcprime

1113969



fcprime

1113969

b(k + 01)d with


fcprime

1113969



Vexpc

04

fcprime

1113969

b(k + 01)d f1(ad) (8)


Vexpc

04

fcprime

1113969

b(k + 01)d f2

a

d1113874 1113875 (9)


Vexpc

04

fcprime

1113969

b(k + 01)d f3

a

d1113874 1113875 (10)


Vexpc

04

fcprime

1113969

b(k + 01)d f4

a

d1113874 1113875 (11)


f1(1) f2(1)

f2(3) f3(3)

f3(6) f4(6)

(12)


Vρf0ad0c τbd (13)


τ αft (14)



ft 58

fcprime

1113969

(15)

0 002 004 006 008 010

02

04

06

08

1

12

V cex

p radicf prime c

bd

ρfnf = ρfEfEc

Trendline

(a)


0 01 02 03 040

02

04

06

08

1

12

V cex

p radicf prime c

bd


(b)


fcprime

1113969

b d



τ

fcprime

1113969 (16)


Vρf0ad0c

fcprime

1113969bd (17)


expc 04

fcprime

1113969


f2a

d1113874 1113875

25(ad) minus 05

f3a

d1113874 1113875 1

f4a

d1113874 1113875

6ad

(18)



f1a

d1113874 1113875

(1 minus (ad))(1 +(ad))

04(k + 01)+ 5

a

d1113874 1113875 (19)


Vc kfad

fcprime

1113969bd (20)

where

kfa d

(1 minus (ad))

(1 + (ad))+ 2(k + 01)

a

d1113874 1113875 while

a

dle 1

(k + 01)

((ad) minus 05) 1lt

a

dle 3


dle 6

24(k + 01)

(ad) while adgt 6

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(21)


expc kfad

fcprime

1113969




fcprime

1113969



expc kfad

fcprime

1113969




Vc kfadkd

fcprime

1113969bd (22)

where

0 1 2 3 4 5 6 7 8 9 100

2

4

6

8

10


3lt ad le 6

ad gt 6kυ = 1

k υ

ad

kυ = 6(ad)



fcprime

1113969

b(k + 01)d


0 300 600 900 1200 15000

1

2

3

4

b (mm)

V cexp

k fadradic

f prime cbd

(a)

0

1

2

3

4

V cex

p kfa

dradicf prime c

bd

0 20 40 60 80 100f primec (MPa)

(b)

0

1

2

3

4

V cex

p kfa

dradicf prime c

bd

0 1 2 3 4 5 6ρf ()

(c)

0 300 600 900 12000

1

2

3

4

Trendline

d (mm)

V cexp

k fadradic

f prime cbd

(d)

0

1

2

3

4

V cex

p kfa

dradicf prime c

bd

0 30 60 90 120 150 180 210Ef (GPa)

(e)

0

1

2

3

4

V cexp

k fadradic

f prime cbd

0 2 4 6 8 10ad

(f )


fcprime

1113969

bd

0

300

600

900

1200

V cexp (k

N)

ACI 4401Rndash2015

0 300 600 900 1200Vc

calc (kN)

DatabaseTest dataVc

exp = Vcclac

(a)

JSCEndash1997

0

300

600

900

1200

V cexp (k

N)

0 300 600 900 1200Vc

calc (kN)

DatabaseTest dataVc

exp = Vcclac

(b)

CANCSA-S806-2012

0

300

600

900

1200V cex

p (kN

)

0 300 600 900 1200Vc

calc (kN)

DatabaseTest dataVc

exp = Vcclac

(c)

V cexp (k

N)

0 300 600 900 12000

300

600

900

1200

Vccalc (kN)


DatabaseTest dataVc

exp = Vcclac

(d)

V cexp (k

N)

0

300

600

900

1200

0 300 600 900 1200Vc

calc (kN)


DatabaseTest dataVc

exp = Vcclac

(e)

V cexp (k

N)

0

300

600

900

1200

0 300 600 900 1200Vc

calc (kN)

Proposed model

DatabaseTest dataVc

exp = Vcclac

(f )


c


16

12

8

4

0

12

9

6

3

06543210

15

12

9

6

3

065432104

3

2

1

0

V cexp V

cprop

V cexp V

ccnr

V cexp V

clr fd

V cexp V

cs806

V cexp V

cjsec

V cexp V

caci

0 2 4 6 8 10ad


0 300 600 900 1200d (mm)

DatabaseTest data

Data (adlt1)



kd 135 minusd

1000ge 075 (23)





c Vjscec Vs806

c Vlrfdc Vcnr



expc Vcalc

c





expc Vcalc


expc Vcalc

c


expc Vs806

c

and Vexpc Vlrfd


expc Vprop


expc Vprop

c



c Vpropc 1)




expc Vprop


expc Vprop


c




expc Vcalc


6 Conclusions




0 05 1 15 2 250

5

10

15

20

25

Freq

uenc

y (

)

VcexpVc

prop



c


Vexpc Vaci

c 261 184 070 195 055 028Vexp

c Vlrfdc 255 173 068 209 064 030

Vexpc Vjsce

c 192 145 076 163 050 030Vexp

c Vs806c 115 052 045 112 019 017

Vexpc Vcnr

c 090 069 077 069 012 018Vexp

c Vpropc 109 031 028 114 019 017



Notations



beams NV




NVaci



model NVs806



2017 model NVcnr


Data Availability




Acknowledgments


References





























































τ

fcprime

1113969 (16)


Vρf0ad0c

fcprime

1113969bd (17)


expc 04

fcprime

1113969


f2a

d1113874 1113875

25(ad) minus 05

f3a

d1113874 1113875 1

f4a

d1113874 1113875

6ad

(18)



f1a

d1113874 1113875

(1 minus (ad))(1 +(ad))

04(k + 01)+ 5

a

d1113874 1113875 (19)


Vc kfad

fcprime

1113969bd (20)

where

kfa d

(1 minus (ad))

(1 + (ad))+ 2(k + 01)

a

d1113874 1113875 while

a

dle 1

(k + 01)

((ad) minus 05) 1lt

a

dle 3


dle 6

24(k + 01)

(ad) while adgt 6

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(21)


expc kfad

fcprime

1113969




fcprime

1113969



expc kfad

fcprime

1113969




Vc kfadkd

fcprime

1113969bd (22)

where

0 1 2 3 4 5 6 7 8 9 100

2

4

6

8

10


3lt ad le 6

ad gt 6kυ = 1

k υ

ad

kυ = 6(ad)



fcprime

1113969

b(k + 01)d


0 300 600 900 1200 15000

1

2

3

4

b (mm)

V cexp

k fadradic

f prime cbd

(a)

0

1

2

3

4

V cex

p kfa

dradicf prime c

bd

0 20 40 60 80 100f primec (MPa)

(b)

0

1

2

3

4

V cex

p kfa

dradicf prime c

bd

0 1 2 3 4 5 6ρf ()

(c)

0 300 600 900 12000

1

2

3

4

Trendline

d (mm)

V cexp

k fadradic

f prime cbd

(d)

0

1

2

3

4

V cex

p kfa

dradicf prime c

bd

0 30 60 90 120 150 180 210Ef (GPa)

(e)

0

1

2

3

4

V cexp

k fadradic

f prime cbd

0 2 4 6 8 10ad

(f )


fcprime

1113969

bd

0

300

600

900

1200

V cexp (k

N)

ACI 4401Rndash2015

0 300 600 900 1200Vc

calc (kN)

DatabaseTest dataVc

exp = Vcclac

(a)

JSCEndash1997

0

300

600

900

1200

V cexp (k

N)

0 300 600 900 1200Vc

calc (kN)

DatabaseTest dataVc

exp = Vcclac

(b)

CANCSA-S806-2012

0

300

600

900

1200V cex

p (kN

)

0 300 600 900 1200Vc

calc (kN)

DatabaseTest dataVc

exp = Vcclac

(c)

V cexp (k

N)

0 300 600 900 12000

300

600

900

1200

Vccalc (kN)


DatabaseTest dataVc

exp = Vcclac

(d)

V cexp (k

N)

0

300

600

900

1200

0 300 600 900 1200Vc

calc (kN)


DatabaseTest dataVc

exp = Vcclac

(e)

V cexp (k

N)

0

300

600

900

1200

0 300 600 900 1200Vc

calc (kN)

Proposed model

DatabaseTest dataVc

exp = Vcclac

(f )


c


16

12

8

4

0

12

9

6

3

06543210

15

12

9

6

3

065432104

3

2

1

0

V cexp V

cprop

V cexp V

ccnr

V cexp V

clr fd

V cexp V

cs806

V cexp V

cjsec

V cexp V

caci

0 2 4 6 8 10ad


0 300 600 900 1200d (mm)

DatabaseTest data

Data (adlt1)



kd 135 minusd

1000ge 075 (23)





c Vjscec Vs806

c Vlrfdc Vcnr



expc Vcalc

c





expc Vcalc


expc Vcalc

c


expc Vs806

c

and Vexpc Vlrfd


expc Vprop


expc Vprop

c



c Vpropc 1)




expc Vprop


expc Vprop


c




expc Vcalc


6 Conclusions




0 05 1 15 2 250

5

10

15

20

25

Freq

uenc

y (

)

VcexpVc

prop



c


Vexpc Vaci

c 261 184 070 195 055 028Vexp

c Vlrfdc 255 173 068 209 064 030

Vexpc Vjsce

c 192 145 076 163 050 030Vexp

c Vs806c 115 052 045 112 019 017

Vexpc Vcnr

c 090 069 077 069 012 018Vexp

c Vpropc 109 031 028 114 019 017



Notations



beams NV




NVaci



model NVs806



2017 model NVcnr


Data Availability




Acknowledgments


References




























































0 300 600 900 1200 15000

1

2

3

4

b (mm)

V cexp

k fadradic

f prime cbd

(a)

0

1

2

3

4

V cex

p kfa

dradicf prime c

bd

0 20 40 60 80 100f primec (MPa)

(b)

0

1

2

3

4

V cex

p kfa

dradicf prime c

bd

0 1 2 3 4 5 6ρf ()

(c)

0 300 600 900 12000

1

2

3

4

Trendline

d (mm)

V cexp

k fadradic

f prime cbd

(d)

0

1

2

3

4

V cex

p kfa

dradicf prime c

bd

0 30 60 90 120 150 180 210Ef (GPa)

(e)

0

1

2

3

4

V cexp

k fadradic

f prime cbd

0 2 4 6 8 10ad

(f )


fcprime

1113969

bd

0

300

600

900

1200

V cexp (k

N)

ACI 4401Rndash2015

0 300 600 900 1200Vc

calc (kN)

DatabaseTest dataVc

exp = Vcclac

(a)

JSCEndash1997

0

300

600

900

1200

V cexp (k

N)

0 300 600 900 1200Vc

calc (kN)

DatabaseTest dataVc

exp = Vcclac

(b)

CANCSA-S806-2012

0

300

600

900

1200V cex

p (kN

)

0 300 600 900 1200Vc

calc (kN)

DatabaseTest dataVc

exp = Vcclac

(c)

V cexp (k

N)

0 300 600 900 12000

300

600

900

1200

Vccalc (kN)


DatabaseTest dataVc

exp = Vcclac

(d)

V cexp (k

N)

0

300

600

900

1200

0 300 600 900 1200Vc

calc (kN)


DatabaseTest dataVc

exp = Vcclac

(e)

V cexp (k

N)

0

300

600

900

1200

0 300 600 900 1200Vc

calc (kN)

Proposed model

DatabaseTest dataVc

exp = Vcclac

(f )


c


16

12

8

4

0

12

9

6

3

06543210

15

12

9

6

3

065432104

3

2

1

0

V cexp V

cprop

V cexp V

ccnr

V cexp V

clr fd

V cexp V

cs806

V cexp V

cjsec

V cexp V

caci

0 2 4 6 8 10ad


0 300 600 900 1200d (mm)

DatabaseTest data

Data (adlt1)



kd 135 minusd

1000ge 075 (23)





c Vjscec Vs806

c Vlrfdc Vcnr



expc Vcalc

c





expc Vcalc


expc Vcalc

c


expc Vs806

c

and Vexpc Vlrfd


expc Vprop


expc Vprop

c



c Vpropc 1)




expc Vprop


expc Vprop


c




expc Vcalc


6 Conclusions




0 05 1 15 2 250

5

10

15

20

25

Freq

uenc

y (

)

VcexpVc

prop



c


Vexpc Vaci

c 261 184 070 195 055 028Vexp

c Vlrfdc 255 173 068 209 064 030

Vexpc Vjsce

c 192 145 076 163 050 030Vexp

c Vs806c 115 052 045 112 019 017

Vexpc Vcnr

c 090 069 077 069 012 018Vexp

c Vpropc 109 031 028 114 019 017



Notations



beams NV




NVaci



model NVs806



2017 model NVcnr


Data Availability




Acknowledgments


References




























































16

12

8

4

0

12

9

6

3

06543210

15

12

9

6

3

065432104

3

2

1

0

V cexp V

cprop

V cexp V

ccnr

V cexp V

clr fd

V cexp V

cs806

V cexp V

cjsec

V cexp V

caci

0 2 4 6 8 10ad


0 300 600 900 1200d (mm)

DatabaseTest data

Data (adlt1)



kd 135 minusd

1000ge 075 (23)





c Vjscec Vs806

c Vlrfdc Vcnr



expc Vcalc

c





expc Vcalc


expc Vcalc

c


expc Vs806

c

and Vexpc Vlrfd


expc Vprop


expc Vprop

c



c Vpropc 1)




expc Vprop


expc Vprop


c




expc Vcalc


6 Conclusions




0 05 1 15 2 250

5

10

15

20

25

Freq

uenc

y (

)

VcexpVc

prop



c


Vexpc Vaci

c 261 184 070 195 055 028Vexp

c Vlrfdc 255 173 068 209 064 030

Vexpc Vjsce

c 192 145 076 163 050 030Vexp

c Vs806c 115 052 045 112 019 017

Vexpc Vcnr

c 090 069 077 069 012 018Vexp

c Vpropc 109 031 028 114 019 017



Notations



beams NV




NVaci



model NVs806



2017 model NVcnr


Data Availability




Acknowledgments


References




























































kd 135 minusd

1000ge 075 (23)





c Vjscec Vs806

c Vlrfdc Vcnr



expc Vcalc

c





expc Vcalc


expc Vcalc

c


expc Vs806

c

and Vexpc Vlrfd


expc Vprop


expc Vprop

c



c Vpropc 1)




expc Vprop


expc Vprop


c




expc Vcalc


6 Conclusions




0 05 1 15 2 250

5

10

15

20

25

Freq

uenc

y (

)

VcexpVc

prop



c


Vexpc Vaci

c 261 184 070 195 055 028Vexp

c Vlrfdc 255 173 068 209 064 030

Vexpc Vjsce

c 192 145 076 163 050 030Vexp

c Vs806c 115 052 045 112 019 017

Vexpc Vcnr

c 090 069 077 069 012 018Vexp

c Vpropc 109 031 028 114 019 017



Notations



beams NV




NVaci



model NVs806



2017 model NVcnr


Data Availability




Acknowledgments


References





























































Notations



beams NV




NVaci



model NVs806



2017 model NVcnr


Data Availability




Acknowledgments


References









































































































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