Articulo a Design Model for Reinforced WITH Bars Concrete Beams

8/13/2019 Articulo a Design Model for Reinforced WITH Bars Concrete Beams

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A design model for fibre reinforced concrete beams pre-stressed with

steel and FRP bars

Joaquim A.O. Barros, Mahsa Taheri, Hamidreza Salehian, Pedro J.D. Mendes

ISISE, Dept. of Civil Engineering, Univ. of Minho, Campus de Azurm, 4810-058 Guimares, Portugal

a r t i c l e i n f o

Article history:

Available online 21 March 2012

Keywords:

Fibre reinforced concreteStrain hardeningStrain softeningFlexural reinforcementPre-stressed steel barsPre-stressed FRP bars

a b s t r a c t

This paper presents a design oriented model to determine the momentcurvature relationship of ele-ments of rectangular cross section failing in bending, made by strain softening or strain hardening fibrereinforced concrete (FRC) and reinforced with perfectly bonded pre-stressed steel and fibre reinforcedpolymeric (FRP) bars. Since FRP bars are not affected by corrosion, they have the minimum FRC coverthickness that guaranty proper bond conditions, while steel bars are positioned with a thicker FRC coverto increase their protection against corrosion. Using the momentcurvature relationship predicted by themodel in an algorithm based on the virtual work method, a numerical strategy is adopted to evaluate theloaddeflection response of statically determinate beams. The predictive performance of the proposedformulation is assessed by simulating the response of available experimental results. By using this model,a parametric study is carried out in order to evaluate the influence of the main parameters that charac-terize the post cracking behaviour of FRC, and the pre-stress level applied to FRP and steel bars, on themomentcurvature and loaddeflection responses of this type of structural elements. Finally the shearresistance of this structural system is predicted.

2012 Elsevier Ltd. All rights reserved.

1. Introduction

The corrosion of steel bars reinforcement is the major cause ofpathologies observed in reinforced concrete (RC) structures. Therelatively small concrete cover of the steel reinforcement contrib-utes for the initiation and development of this phenomenon, lead-ing to a significant decrease of the load carrying capacity of themember. The costs for the rehabilitation of corroded RC structuresare, in certain cases, so high that a decision for demolition is rela-tively frequent, with the consequent economic, social and environ-mental adverse impacts.

The knowledge acquired at the level of the behaviour of fibrereinforced polymer (FRP) materials applied in the industry of CivilConstruction has increased significantly in the last two decades. Infact, the possible substitution of conventional steel reinforcementby FRPbars has been investigated[13] to avoidcorrosion problemsandto improve the durability of concrete structures in adverse envi-ronmental conditions (marine, under-ground, and chemical indus-trial plants), and in thin structural elements. When compared tosteel, the FRPs have higher resistance to corrosion, and higherstrength-to-weight ratio. Furthermore, they are non-conductivefor electricity and non-magnetic. However, the major obstacles of

the application of FRP bars as a reinforcing material for concretestructures are the high initial costs, low modulus of elasticity, lackof ductility (linear stressstrain diagram up to rupture with nodiscernible yield point) and the small number of reliable design for-mulations to predict the behaviour of concrete elements reinforcedinternally with FRP bars[2,48]. Concrete membersreinforced withFRP subject to bending behave linearly up to cracking, and almostlinearly after cracking with a lower flexural stiffness when com-pared with homologous beams reinforced with steel bars. Deflec-tions and strains of concrete members reinforced with FRP barsare generally larger than of homologous members reinforced withsteelbars. This is due to the low modulus ofelasticity and the differ-ent bond characteristics of the FRP reinforcements[3,9,10]. In addi-tion, as a result of larger crack width and smaller compressive stressblock, the shear capacity of concrete beams is lower than whenusing high bond steel bars[7].

In an attempt of overcoming these drawbacks, some researchers[1013] proposed a combination of FRPand steel reinforcements forconcrete beams.Combining these reinforcement materials and con-sideringthe minor concrete cover required forFRP, an effectiverein-forcement solution in terms of durability is obtained by placing theFRP bars near the outer surface of the tensile zone, and steel bars ataninnerlevelofthetensilezone(Fig.1).Thepresenceofsteelbarsinthe above mentioned hybrid reinforcement system provides asignificant contribution in terms of ductility and stiffness. Theexperimental tests where this hybrid reinforcement concept wasused, in spite of being scarce, have confirmed the potentialities of

0263-8223/$ - see front matter 2012 Elsevier Ltd. All rights reserved.http://dx.doi.org/10.1016/j.compstruct.2012.03.007

Corresponding author. Tel.: +351 253510210; fax: +351 253510217.E-mail addresses:[email protected](J.A.O. Barros),[email protected]

(M. Taheri), [email protected] (H. Salehian), [email protected](P.J.D. Mendes).

Composite Structures 94 (2012) 24942512

Contents lists available at SciVerse ScienceDirect

Composite Structures

j o u r n a l h o m e p a g e : w w w . e l s e v i e r . c o m / l o c a t e / c o m p s t r u c t


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Notation

Ac the concrete cross sectionAf cross sectional area of FRP barAs cross sectional area of steel barAsl cross sectional area of the reinforcementb beam width

bw smallest width of the cross-section in the tensile areaCs central distance of steel bars from tensile face of sectionCf central distance of FRP bars from tensile face of sectiond effective depth of beamds central distance of steel bars from top face of sectionds,eq equivalent steel effective depthdf central distance of FRP bars from top face of sectionD1 steel bar diameterD2 steel bar diameterE tensile modulus of elasticity of FRCEc compressive modulus of elasticity of concreteEcr tensile post cracking modulus of FRCEs modulus of elasticity of steel barsEf modulus of elasticity of FRP barsFprf

pre-stressing load of FRP bars

FFtuk characteristic value of the ultimate residual tensilestrength of FRC

Fprs pre-stressing load of steel barsFpr total pre-stressing loadfcd the design value of the concrete compressive strengthfctk the characteristic value of the tensile strength for the

concrete matrixfck the characteristic value of cylindrical compressive

strength for the concrete matrixEI flexural stiffnessk the neutral axis depth ratiokd factor to take into account the size effectL beam total lengthlbd design anchorage lengthLs beam span lengthM bending momentM0i normalized bending moment (M/Mcr) in stage iMi bending moment in stageiMcr bending moment at FRC crack initiationMmid bending moment at beam mid-spanMprs bending moment corresponding to pre-stress load of

steel barsMpr

f bending moment corresponding to pre-stress load of

FRP barsDM bending moment increment with respect to non pre-

stressed beamMpr=25/50 the moment for a pre-stress level of 25% or 50%Mpr=0.0 the moment for non pre-stressed beamNEd axial force due to loading or pre-stressing actionsP total applied load on beamPfl ultimate applied load corresponding to flexural resis-

tancePsh ultimate applied load corresponding to shear resistanceDP total applied load increment with respect to nonpre-

stressed beamPpr=25/50 the load for a pre-stress level of 25% or 50%Ppr=0.0 the load for non pre-stressed beamDPd=10mm total applied load increment at d = 10 mm for a equal

to 10, 50, or 150Pa1:01d10mm the load atd = 10 mm for a = 1.01

VRd design value of shear resistanceVRd,F design shear resistance attributed to the FRCVRd,max maximum design value of shear resistanceVRd,s design shear resistance provided by shear reinforce-

ment

w width of CFRP laminatea normalized transition strainb normalized tensile strain at bottom fibrebtu normalized ultimate tensile strainc normalized compressive modulus of elasticity of FRCcc partial safety factor for the concrete material propertiescf normalized modulus of elasticity of FRP barscF partial safety factor for FRCcs normalized modulus of elasticity of steel barsdmid flexural beam deflection at mid-spanecy compressive yield strain of FRCecu ultimate compressive strain of FRCecr tensile strain at crack initiation of FRCesy tensile yield strain of steel barsesu ultimate tensile strain of steel bars

etrn tensile strain at transition point of FRCetbot tensile strain at the bottom of FRCectop compressive strain at the top of FRCetu ultimate tensile strain of FRCefu ultimate tensile strain of FRP barseprs pre-stressing strain of steel barseprf

pre-stressing strain of FRP barsf normalized transition tensile strain of steel barsg normalized post-crack modulus of FRCk normalized compressive strain at the FRC top fibrekcu normalized ultimate compressive strain of FRCl normalized post-crack residual strength of FRCm normalized tensile strain of FRP barsmfu normalized ultimate tensile strain of FRP barsqs reinforcement ratio of longitudinal steel barsqs,eq equivalent steel reinforcement ratioqf reinforcement ratio of longitudinal FRP barsql reinforcement ratio of the longitudinal reinforcementrc compressive stress of FRCrcp average stress acting on the concrete cross sectionrcy compressive yield stress of FRCrt tensile stress of FRCrs tensile stress of the steel barsrf tensile stress of the FRP barsrsy tensile yield stress of steel barsrsr the maximum steel stress in a crack in the crack forma-

tion stagercr tensile strength of FRCrcst residual tensile stress of FRCv curvaturevcr curvature at crack initiation of FRCv0i normalized curvaturev/vcrvi curvature in stageiw normalized tensile strain of steel barswsu normalized ultimate steel tensile strain of steel barsx normalized compressive yield strain of FRCDs normalized cover thickness of steel barsDf normalized cover thickness of FRP bars

J.A.O. Barros et al. / Composite Structures 94 (2012) 24942512 2495


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this reinforcement system. For example, Tian and Yuan [10]con-cluded that the deflection of concrete beams reinforced simulta-neously with GFRP (glass fibres) and steel bars was smaller thanthat of beams only reinforced with GFRP bars. Aiello and Ombres[11]verified that, in comparison with beams reinforced only withFRP bars, the participation of steel bars as part of the reinforcementsystem has reduced the crack width and crack spacing.

Pre-stressing the FRP bars can mobilize more effectively thestrengthening potentialities of these reinforcing elements. Further-

more, by applying a certain pre-stress to the bars, a significantincrease in terms of load carrying capacity can be obtained fordeflection levels corresponding to the serviceability limit states.Applying steel and FRP bars with a certain pre-stress level can alsocontribute for the shear resistance of the element[11].

The research conducted in this paper is part of a research pro-ject aimed at developing high durable precast beams reinforcedwith a hybrid reinforcing system (pre-stressed steel and FRP bars),and adopting a high performance fibre reinforced concrete (HPFRC)to suppress the use of steel stirrups. According to this concept(Fig. 1), the steel reinforcement ratio should be designed in orderto assure the safety of the structure in case of a fire occurrenceand the consequent loss of FRP reinforcing capacities. The FRPand steel bars are applied with a certain pre-stress for the optimi-

zation of their reinforcing capabilities, to overcome the drawbacksderived from the relatively low elasticity modulus of FRP bars, andto increase the shear capacity of the beams. These beams can beused in multi-storey car parking, shopping centres, and residentialand commercial buildings based on a precast constructive system.These beams can have a span between 6 and 11 m.

Available research[1519]evidences that steel fibres can sub-stitute steel stirrups, especially when a high strength concrete isused and when beams are relatively shallow [17,18]. Steel fibresalso reduce the width of shear cracks, thus also improving concretedurability[19]. In the study carried out by Meda et al. [19] thecrack spacing in FRC beams was reduced in about 20% when com-pared to reference beams of conventional concrete with and with-out stirrups. Available research shows that up to a maximum crackwidth of 0.25 mm, steel fibres are not affected by corrosion[20,21].Furthermore, advances in the manufacture technology of synthetic

fibres show that these non-corrodible fibres have high possibilitiesfor the shear resistance of RC elements[22].

According to the structural concept proposed in the presentwork, the total and relative reinforcement ratio of steel and FRPbars, as well as the pre-stress level, should be selected in orderto assure that at the beams failure the steel has already yielded.Furthermore, the increase of load carrying capacity, mainly at ser-viceability limit states, the reduction of costs maintenance and theincrease of life cycle should justify the relatively higher initial costs

of the materials used in this structural system. The pre-stress levelto be applied to the flexural reinforcements, and the performanceof the FRC, mainly its post-cracking residual strength, should bedesigned in order to avoid the formation of cracks in the topsurface of the beam during the application of the pre-stress inthe production process, and to prevent the occurrence of shear fail-ure mode.

This paper proposes a design approach for rectangular crosssection FRC beams reinforced with pre-stressed FRP and steel bars.After the model description, its predictive performance is appraisedand a parametric study is carried out in order to evidence the influ-ence of relevant parameters of the model on the load carryingcapacity and ductility performance of FRC-hybrid reinforced simplesupported pre-stressed beams. The proposed formulation is pre-

pared to work with FRC that has tensile strain softening (SS-FRC)or tensile strain hardening (SH-FRC) behaviour [23]. Finally, theshear capacity of this structural system is predicted by using anadapted version of the formulation proposed by CEB-FIP ModelCode 2010[24].

2. Numerical strategy for the evaluation of the moment

curvature and force deflection of FRC-hybrid pre-stressed

beams

2.1. Constitutive laws of materials under consideration

The stressstrain (r e) response in compression consideredfor the FRC is based on the model proposed by Soranakom and

Mobasher, represented inFig. 2 [21]. As shown inFig. 2a, the linearportion of an elasticperfectly plastic compressive stressstrain re-

Fig. 1. (a) Concept of FRC-hybrid reinforcing system and (b) variables involved in the analytical model.

2496 J.A.O. Barros et al. / Composite Structures 94 (2012) 24942512


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Table 3

Equations for the evaluation of the normalized moment, M0, and normalized curvature, v 0, for each stage[22].

Stage (i) M0

1M01

2bc 1k313BsBf 1k216Bs1 Ds Bf1 Df 0:5k13Bs1 Ds2 Bf1 Df2 11k1 M

0pr

1

M0pr

16Fprs1k1Ds 6Fprf1k1Df

2.1.1.1

M021112bc

C2111

k32111

3

bBs

bBf

0:5C2111

k22111

12

0:25C2111

b

Bs

1

Ds

Bf

1

Df

k2111

6b

Bs

1

Ds

2

Bf

1

Df

2

C2111

1k2111 M0pr

211

C21112gb3 3b2g 1 g 1

b2 ; M0pr

2111


2.1.2.1M02121

2bcC2121k321216fBsbBf 0:5C2121k22121120:25C2121 0:5fBsDs2 bBf1 Dfk21216fBs1 Ds bBf1 Df2 C21211k2121

M0pr

2

C21212gb3 3b2g 1 g 1

b2

; M0pr

21216Fprf1k2121Df

2.2.1.1M02211

3xcC2211k322113xcC22112bBs2bBfk22211120:25C2211bBs1 Ds Bf1 Dfk22116bBs1 Ds2 bBf1 Df2 C22111k2211

C22112gb3 3b21 g x3c g 1

b2 ; M0pr

2211


2.2.2.1M02221

3xcC2221k3222160:5xc 3C2221 fBsbBfk22221120:25C2221fBs1 0:5Ds bBf1 Dfk22216fBs1 DS bBf1 Df2 C22

1

k2221

C22212gb3 3b21 g x3c g 1

b2 ; M0pr

2221

6Fprf1k2221Df

3.1.1.1M03111

C31112bck3311132bBs2bBfC3111k23111120:25C3111bBs1 Ds Bf1 Dfk31116bBs1 Ds2 Bf1 Df2 C3111k31111

M0pr

3111

C31113lb2 a21 l g g2a3 1 1

b2 ; M0pr

3111


3.1.2.1M03121

C31212bck331216fBsbBf 0:5C3121k23121120:25C3121 0:5fBs2 Ds bBf1 Dfk31216fBs1 Ds bBf1 Df2 C3121k31211

M0pr

3

C31113lb2 a21 l g g2a3 1 1

b2 ; M0pr

3121

6Fprf1k3121Df

3.2.1.1M03211

3xcC3211k3321160:5xcC3211 bBsBfk23211120:25C3211bBs1 Ds Bf1 Dfk32116bBs1 Ds2 Bf1 Df2 C32111k3211

C32113lb2

a2

1 l g g2a3

1 x3

c 1b2

; M0pr


3.2.2.1M03221

3xcC3221k3322160:5xcC3221 fBsbBfk23221120:25C32210:5fBs2 Ds bBfDf 1k32216fBs1 Ds bBf1 Df2 C321k3221

C32113lb2 a21 l g g2a3 1 x3c 1

b2 ; M0pr

3221

6Fprf1k3221Df

Bs csqs1 Ds; Bf cfqf1 Df; Fpr Fprs F

prf .


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3. Model appraisal

To evaluate the accuracy of the proposed model, the results ofthe software developed according to the described algorithm werecompared to the results obtained from DOCROS software[27]. The

model implemented in DOCROS assumes that a plane section re-mains plane after deformation and bond between materials is per-fect. The section is divided in layers, and the thickness and width ofeach layer is user-defined and depend on the cross-section geom-etry. DOCROS can analyze sections of irregular shape and size,composed of different types of materials subjected to an axial forceand variable curvature. DOCROS has a wide database of constitu-tive laws for the simulation of monotonic and cyclic behaviour ofcement based materials, polymer based materials and steel bars.

The predictive performance of the model was assessed by eval-uating the momentcurvature relationship for a rectangular crosssection of 250 mm width and 500 mm depth, of a beam reinforcedlongitudinally with a percentage of steel bars of 0.2 and a percent-age of FRP bars of 0.1. FRP bars have a concrete cover of 30 mm,

while steel bars are positioned deeper, at a distance of 80 mm from

tensile face of the section. Furthermore, a pre-stress percentage of50% was applied to both, steel and FRP bars. The values of themodel parameters are included inTable 4(seeTable 5).

Momentcurvature relationships predicted by the proposedmodel and those obtained from DOCROS software are compared

inFig. 6a and b for cross section of beams made by strain softeningand strain hardening FRC, respectively, revealing the high accuracyof the developed model.

The predictive performance of the model was also evaluated bysimulatingexperimentaltestswithFRPstrengthenedRCbeams,car-ried out by Badawi and Soudki[28], and by Xue et al. [29]. AsFig. 7shows, two different strengthening techniques were adopted: thefirst one applying a pre-stressed longitudinal GFRP bar (glass fibresreinforced polymer) placedinto a grooveopen on the concrete coverof the beam, in agreement with the procedures of the near surfacemounted (NSM) technique [28]; and the second one applying apre-stressed CFRP laminate (carbon fibre reinforced polymer)according to the externally bonded reinforcement (EBR) technique[29]. The data to define the geometry, the reinforcement and the

strengthening arrangements is included inTable 6, whileTable 7

Fig. 5. Numerical approach to simulate the forcedeflection response of simple supported beams failing in bending.

Table 4

Data for the model parameters used in the examples for the assessment of the predictive performance of the developed model.

Geometric parameters (mm) Mechanical parameters

Concrete (tension) Concrete (compression) Steel (tension) FRP (tension)

b 250 ecr E a btu l x c kcu f cs wsu mfu cfd 500 () (GPa)Cs 80 0.1 30 10 150 1.1 (SH-FRC) 10 1 40 20 6.67 120 300 1Cf 30 0.33 (SS-FRC)

Table 5

Data to define the geometry, the reinforcement and the strengthening systems of the beams represented in Fig. 7.

Specimens designation Ref. L(mm) l1 (mm) l2 (mm) Sec. type W(mm) D1 (mm) D2 (mm) Pre-stressing (%)

S1 [24] 3500 1100 1100 T1 40.0S2 [24] 3500 1100 1100 T1 60.0S3 [25] 2700 950 600 T2 50 12 14 42.1S4 [25] 2700 950 600 T2 20 12 12 50.6



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presents the relevant values of the parameters that define theconstitutive laws of the intervening materials. Since non-fibrousconcrete was used in these RC beams, the contribution of the post-cracking residual strength of this material for the flexural resistanceof these beams was neglected (=0). Fig. 8 compares the loaddeflec-tion responses predicted by the proposed formulation and those

recorded in the tests, which evidences the capability of the modelto predict with good accuracy the deflection response of this typeof structural elements.

4. Parametric studies

To assess the influence of the relevant mechanical properties ofFRC, and the pre-stress level applied to FRP and steel bars, on themomentcurvature relationship and on the forcedeflection of hy-brid reinforced FRC beams, a parametric study was carried outadopting a simply supported beam with the geometry, the rein-forcement arrangement and the loading conditions representedin Fig. 9. Three distinct pre-stress levels were considered, 0%(non pre-stressed), 25%, and 50%, which is a percentage of the yieldstress of the steel bars and a percentage of the tensile strength of

Fig. 7. Geometry of the beams, reinforcement and strengthening configurations (dimensions in mm).

Table 7

Values considered for the constitutive parameters for the simulation of the series of beams.

Specimens designation ecr() a l btu c x kcu qs(%) cs f wsu qf(%) cf vfu eprs (%) eprf (%)

S1 0.125 2 1e8 150 1 13.98 28 1.04 6.29 18.53 120 0.2082 4.50 116 0.0 0.56S2 0.125 2 1e8 150 1 13.98 28 1.04 6.29 18.53 120 0.183 4.50 116 0.0 0.84S3 0.111 2 1e8 150 1 13.97 32 1.25 4.36 24.41 120 0.187 4.62 150 0.0 0.70S4 0.111 2 1e8 150 1 13.97 32 1.01 4.46 26.71 120 0.075 4.62 150 0.0 0.84

Table 6

Data to define the constitutive laws of the intervenient materials in the beams of

Fig. 7.

Specimensdesignation

Ref. rcy(MPa)

rcr(MPa)

Ec(GPa)

rsy(MPa)

Es(GPa)

rfu(MPa)

Ef(GPa)

S1 [24] 53 3.79 30.20 440 190 1970 136S2 [24] 53 3.79 30.20 440 190 1970 136S3 [25] 50.3 3.60 32.50 383 142 2500 150S4 [25] 50.3 3.60 32.50 429 145 2500 150

Fig. 6. Momentcurvature responses predicted by the model and DOCROS software for the cross section of a beam made by: (a) strain softening FRC and (b) strain hardeningFRC (the dimensions of the cross section are in mm).



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the FRP bars. However, due to the susceptibility to creep rupture ofsome types of FRP bars (mainly those made by glass fibres, GFRP),the limits recommended by some standards[58]for the stresslimits in these reinforcements under sustained stresses should beconsidered. If FRP bars are subjected to cyclic or fatigue loading,the stress limits proposed by these standards should be also taken

into account.

For the influence of the FRC post-cracking performance, the val-ues of 0.0, 0.4, 0.8, and 1.2 for the normalized residual strength (l)were adopted, maintaining constant the normalized transitionstrain ofa = 10.0. In this context, the influence of the a parameterwas also assessed by adopting values of 1.01, 10, 50, and 150,keeping constant the normalized residual strength (l = 0.4). For

the parametric study, the values of the parameters that define the

Fig. 9. Geometry and reinforcement data for the beam of the parametric study (dimensions in mm).

Table 8

Values for the parameters of the materials constitutive laws adopted in the parametric study.

Geometric parameters (mm) Mechanical parameters

Concrete (tension) Concrete (compression) Steel (tension) GFRP (tension)

b 100 ecr E a l btu x c kcu f cs wsu mfu cfd 220 (GPa)Cs 50 0.1 35 [1.01, 10, 50, 150] 0.4 150 20 1 35 75 5.71 150 166.7 1.71Cf 25 10 [0.0, 0.4, 0.8, 1.2]

Fig. 8. Force versus deflection relationship determined from the developed model and registered in the experimental tests for: (a) S1[24], (b) S2[24], and (c) S3[25], (d) S4[25].



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constitutive laws of the intervening materials are indicated in Table8. For this parametric study GFRP bars were considered.

The momentcurvature and the loaddeflection curves corre-sponding to this parametric study are presented in Figs. 1015.As expected, for the considered statically determinate beam thevariation of loaddeflection follows the variation of the corre-sponding momentcurvature.

4.1. Influence ofa parameter and pre-stress level on the momentcurvature and loaddeflection responses of hybrid reinforced FRC

beams

For each adopted pre-stress level of FRP and steel bars, theinfluence ofa FRC-related parameter in terms of momentcurva-ture and loaddeflection responses is represented in Fig. 10acand df, respectively. The points corresponding to the concrete

crack initiation and the steel yield initiation are also signalized inthe curves ofFig. 10. Since a is as a post cracking parameter of

FRC, it has no effect in the responses before crack initiation. How-ever, after crack initiation the flexural capacity of the cross section

and the load carrying capacity of the beam are significantly in-creased with the increase of a parameter. In fact, the momentand the load at yield initiation of steel bars increase with a, andthis tendency is also observed for the corresponding curvaturesand deflections. Therefore, the residual strength of FRC betweenecrand etrn= aecr(seeFig. 2a) has a significant favourable impacton the flexural and load carrying capacities corresponding to thelevel of curvatures and deflections installed in this type of struc-tural elements at serviceability limit states.

According toFig. 10, the momentcurvature and loaddeflec-tion diagrams corresponding to the lowest adopted values of nor-malized transition strain (a = 1.01 and a = 10) are only differentin a relatively small amplitude of curvature and deflection justafter crack initiation. This difference, which is more significant in

terms of loaddeflection, is a consequence of the post-crackingresidual strength of the concrete between ecrand etrn= aecr= 10ecr

Fig. 10. The effect ofa parameter on the momentcurvature and loaddeflection responses for l = 0.4, and steel and FRP bars pre-stressed at 0.0, 25, 50%.



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Fig. 11. Effect of the pre-stress level on the: (ad) momentcurvature response; (eh) increase of the resisting bending moment; for l = 0.4 and a equal to 1.01, 10.0, 50.0 and150.0.



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whena = 10. Furthermore, the increase ofaparameter from 10 to150 provides significant improvement of those responses. For thedeflection corresponding to the serviceability limit state conditions(d =Ls/250 = 2500/250 = 10 mm [14]), the increase percentage inthe load carrying capacity (DPd10mm=P

a1:01d10mm, whereP

a1:01d10mmis the

load at d = 10 mm for a = 1.01) by adopting the a values of 10, 50and 150 is 8%, 35% and 47% for pr = 0%; 4%, 20% and 27% forpr = 25%; and 3%, 14% and 19% for pr = 50%. Due to the linearbehaviour of FRP bars, the momentcurvature and the loaddeflec-tion diagrams vary almost linearly between steel yield point andultimate condition (all the analysis were interrupted when the ten-sile strength of FRP was attained).

The influence of the pre-stress percentage on the momentcur-vature and loaddeflection responses is illustrated in Figs. 11 and12ad, respectively, for the different values of a considered. Asexpected, for a given a value, the moment and the load at crack ini-tiation has increased with the applied pre-stress, but the momentand the load at yield initiation of the steel bars was not signifi-

cantly affected by the pre-stress level. However, due to the initialtensile strain introduced in the steel bars when pre-stress isapplied, the curvature and the deflection at yield initiation de-crease with the increase of the pre-stress level, and this is morepronounced with the increase of the pre-stress level. Due to similarreason, the curvature and the deflection at the rupture of the FRPbars decrease with the increase of the pre-stress level applied tothese bars.Fig. 11eh shows that the DM=Mpr=25/50 Mpr=0.0 in-creases with the pre-stress level, being Mpr=25/50 the moment fora pre-stress level of 25% or 50%, and Mpr=0.0 the moment for nonpre-stressed beam. However, the maximum increase of DM isalmost the same regardless the value ofaconsidered. Similar ten-dency is observed for the increase of DP=Ppr=25/50 Ppr=0.0 withthe pre-stress level (Fig. 12eh).

Figs. 11 and 12ad also show that the curvature and thedeflection at steel yield initiation decrease with the increase ofthe pre-stress level applied to steel and FRP bars, while thedeflection at crack initiation is not affected significantly. There-

Fig. 13. Effect of the l parameter on the momentcurvature and loaddeflection responses for a = 10, and steel and FRP bars pre-stressed at 0.0%, 25%, 50%.



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Fig. 14. Effect of the pre-stress level on the: (ad) momentcurvature response; (eh) increase of the resisting bending moment; for a = 10 andlequal to 0.0, 0.4, 0.8, 1.2.



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fore, the deflection amplitude between crack initiation and steelyield initiation decreases with the increase of the pre-stress

level, reducing the ductility of the response of the beams. How-ever, a hybrid reinforced FRC beam can be designed in order that

the maximum DP occurs at a deflection level larger than thedeflection at serviceability limit states (with an amplitude

decided by the designer), as is the case of the presentanalysis.

Fig. 15. Effect of the pre-stress level on the: (ad) Loaddeflection response; (eh) increase of the load carrying capacity; for a = 10 and l equal to 0.0, 0.4, 0.8, 1.2.



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4.2. Influence ofl parameter and pre-stress level on momentcurvature and loaddeflection responses of hybrid reinforced FRC

beams

Fig. 13ac and df represent the influence of normalized resid-ual strength, l, in terms of momentcurvature and load deflec-tion responses, respectively. The increase of this parameter

provides a significant increase of the flexural strength and loadcarrying capacity. In fact, for the deflection corresponding to theserviceability limit states conditions (d = 10 mm) the DPd10mm=Pl0:0

d10mm (where Pl0:0d10mm is the load at d = 10 mm for l = 0.0) for

l values of 0.4, 0.8 and 1.2 is 31%, 68% and 1037% for pr= 0%;20%, 41% and 61% for pr= 25%; and 14%, 29% and 42% for pr=50%. The increase level in terms of flexural strength and load car-rying capacity provided by the increase ofl remains almost con-stant up to the rupture of the FRP (the occurred failure condition).The moment at yield initiation of steel bars and its correspondingcurvature increase with l. The same tendency occurs in the loadcarrying capacity at yield initiation of steel bars. The increase oflhas also a favourable effect on increasing the deflection corre-sponding to the load at yield initiation of steel bars when thepre-stress level increases.

The influence of the pre-stress percentage on the momentcur-vature and loaddeflection responses is illustrated in Figs. 14 and15ad, respectively, for the different values of l considered. Asexpected, for a given l value, the moment and the load at crack ini-tiation increase with the applied pre-stress, but the moment andthe load at yield initiation of the steel bars were not significantlyaffected by the pre-stress level. The difference between the curva-tures at yield and crack initiation decreases with the increase of thepre-stress level, which also occurs in the loaddeflection response,indicating a decrease of the ductility performance of the beam. Asexpected, the curvature and the deflection at failure of the FRP alsodecrease with the increase of the pre-stress level.

Fig. 14eh shows that the DM=Mpr=25/50 Mpr=0.0 increaseswith the pre-stress level. However, the maximum increase ofDM

is almost the same regardless the value oflconsidered. Similartendency is observed for the increase of D P=Ppr=25/50 Ppr=0.0

with the pre-stress level (Fig. 15eh).

5. Shear resistance

The load carrying capacity of a FRC beam flexurally reinforcedwith pre-stressed steel and FRP bars can be limited by its shearresistance. To predict the shear resistance of this new structuralsystem, the recommendations of the CEB-FIP Model Code 2010[24]are adopted. According to this document, the shear resistanceof a FRC beam that has longitudinal reinforcement can be deter-mined from the following equation:

VRdVRd;sVRd;F6 VRd;max 21where the equations for the evaluation of the contribution of thesteel stirrups and to avoid crushing of the compression struts areindicated in the prEN 1992-1-1[30]. The term VRd,Frepresents thecontribution of the FRC for the shear resistance, and is obtainedfrom equation:

VRd;F 0:18

cckd 100 q1 1 7:5

fFtukfctk

fck

1=3 0:15 rcp

( )

bWd22

where[24]:

cc is the partial safety factor for the concrete (1.5);kd is a factor that takes into account the size effect and

is equal to: 1 ffiffiffiffiffiffi

200d

q 6 2:0 withdbeing the effective

depth of the cross section in mm;ql is the reinforcement ratio of the longitudinal

reinforcement, equal to ql=Asl/bwd, being the Asl

[mm2

] the cross sectional area of the reinforcementwhich extendslbd+dbeyond the considered section(lbdis the design anchorage length [mm]), and bw[mm] is the smallest width of the cross-section inthe tensile area;

fctk [MPa]is the characteristic value of the tensile strength for the

concrete matrix;fck [MPa]is the characteristic value of cylindrical compressive

strength for the concrete matrix;rcp [MPa]=NEd/ Ac< 0.2fcd[MPa] is the average stress acting on

the concrete cross section, Ac[mm2], for an axial

forceNEd[N], due to loading or pre-stressingactions (NEd> 0 for compression), and fcdis thedesign value of the concrete compressivestrength;

fFtuk [MPa]is the characteristic value of the ultimate residual

tensile strength of FRC, that can be determinedfollowing the recommendations of[24].

To adapt Eq.(22)for the case of a hybrid reinforced beam, qlis

replaced by the equivalent steel reinforcement ratio:

qs;eq Asbds

EfEs

Afbdf

23

andd is substituted by the equivalent steel effective depth

ds;eqAsds Ef=EsAfdfAs Ef=EsAf 24

where the meaning of the symbols were already introduced. Sincein the parametric studies, design values were assumed for theparameters that define the constitutive laws of the materials, inthe present approach it is considered that fFtuk= cF(lrcr), wherecF= 1.5 is the partial safety factor recommended by the Model Code

[24]for FRC. Considering the beam ofFig. 9adopted in the paramet-ric studies, and the properties ofTable 8, by fixinga = 10 and vary-

Table 9

Flexural versus shear resistance of hybrid reinforced FRC beams.

Pre-stress level (%) l Psh(kN) Pfl(kN) Failure mode

0 0 27.58 33.22 Shear0.4 43.78 38.97 Flexure0.8 52.76 44.50 Flexure1.2 59.42 49.82 Flexure

25 0 33.84 35.49 Shear0.4 50.05 41.20 Flexure0.8 59.02 46.70 Flexure1.2 65.69 51.99 Flexure

50 0 40.10 35.18 Flexure0.4 56.31 40.80 Flexure0.8 65.29 46.25 Flexure1.2 71.95 51.51 Flexure



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ing the pre-stress level applied to the steel and FRP bars, and rang-ing the l according to the values indicated in this table, the loadcarrying capacity of the beams limited by the shear (Psh= 2 VRd)and flexural resistance (Pfl) are compared inTable 9. In this study,Pflis the load when the minimum strain between ecu=kcuecr(con-crete crushing)esu= wsu ecr(steel rupture) and efu= mfuecr(FRP rup-ture) is attained. From the obtained results it can be concludedthat shear failure only occur in non-fibrous concrete beams whenthe pre-stress level is lower than 50%. When fFtuk is higher than40% of the characteristic value of the stress at crack initiation,0.4fctk, flexural failure mode is always guaranteed for the analyzedbeams. However, since the CEB-FIP Model Code formulation wasdeveloped by considering, mainly, the data available for steel fibrereinforced concrete beams flexurally reinforced with passive steelbars, the use of Eq.(22)for FRC hybrid pre-stressed beams shouldbe used with caution. In fact, according to the knowledge of theauthors the predictive performance of Eq. (22) for this type ofbeams was not yet assessed because experimental data is notavailable.

6. Conclusions

In this work a design oriented model was proposed for deter-mining the momentcurvature response of rectangular cross sec-tion of FRC members reinforced by longitudinal pre-stressed steeland FRP bars that fail in bending. By using a trilinear stressstraindiagram for the tensile behaviour of FRC, the proposed model iscapable of simulating both strain softening and strain hardeningFRC materials. A relatively small number of parameters is neces-sary to characterize the FRC behaviour in tension and in compres-sion, as well as the behaviour of steel and FRP bars in tension.Using the momentcurvature relationship predicted by the modeland implementing an algorithm based on the virtual work method,a numerical strategy was developed for the prediction of the forcedeflection response of statically determinate beams. The good pre-dictive performance of the model was assessed by simulating the

forcedeflection responses registered in experimental programs.The model is capable of simulating the behaviour of beams inter-nally reinforced with steel and FRP bars, and can also be used topredict the forcedeflection relationship of RC beams flexurallystrengthened with pre-stressed FRP systems applied according tothe near surface mounted (NSM) and externally bonded reinforce-ment (EBR) techniques.

The proposed methodology was used to execute a parametricstudy to evaluate the influence of the following parameters onthe momentcurvature and forcedeflection responses:a = etrn/ecrand l (normalized residual strength) FRC-related parameters andpre-stress level. From this parametric study the following mainobservations can be pointed out:

The flexural capacity of the cross section and the load carryingcapacity of the beam increase significantly with the increase ofland aparameters;

The moment at yield initiation of steel bars and its correspond-ing curvature increase withlanda. The same tendency occursfor the load carrying capacity at yield initiation of steel bars;

The increase ofl anda has also the favourable effect of increas-ing the deflection corresponding to the load at yield initiation ofsteel bars when the pre-stress level increases;

For the deflection corresponding to the serviceability limitstates conditions, the increase ofl and a leads to a significantincrease of the load carrying capacity;

By increasing the pre-stress level in the steel and FRP bars, thecurvature and the deflection at steel yield initiation, as well the

curvature and the deflection at failure decrease. Therefore, since

the deflection at crack initiation is not affected significantly bythe applied pre-stress level, the deflection amplitude betweencrack initiation and steel yield initiation decreases with theincrease of the pre-stress level, reducing the ductility of theresponse of the beams. However, the FRC can be optimized inorder to provide values for thel anda parameters that guaran-tee the aimed degree of ductility when applying a certain pre-

stress level in a hybrid reinforced beam.- For the beams considered in the parametric studies, shear fail-ure never occur if FRC with lP 0.4 is adopted, regardless thepre-stress level applied to the longitudinal bars.

Acknowledgements

The study reported in this paper is part of the research programDURCOST Innovation in reinforcing systems for sustainable pre-fabricated structures of higher durability and enhanced structuralperformance supported by FCT, PTDC/ECM/105700/2008. The sec-ond and forth authors acknowledge the research grant under theproject QREN number 3456 PONTALUMIS, while the third authoracknowledges the support provided by FCT Grant SFRH/BD/71934/2010.

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