Numerical Study of Using FRP and Steel Rebars in Simply Supported Prestressed Concrete Beams with External FRP Tendons

polymers

Article Numerical Study of Using FRP and Steel Rebars in Simply Supported Prestressed Concrete Beams with External FRP Tendons

Miao Pang 1, Zhangxiang Li 2 and Tiejiong Lou 2,*

1 Department of Civil Engineering, Zhejiang University, Hangzhou 310058, China; [email protected] 2 Hubei Key Laboratory of Roadway Bridge & Structure Engineering, Wuhan University of Technology, Wuhan 430070, China; [email protected] * Correspondence: [email protected]

Received: 2 November 2020; Accepted: 18 November 2020; Published: 24 November 2020

Abstract: This study aimed at examining the feasibility of using fiber-reinforced polymer (FRP) rebars instead of steel ones in prestressed concrete beams (PCBs) with external FRP tendons. By applying an experimentally validated program, numerical tests were performed on simply supported PCBs, with investigated variables including rebars’ type and area. Three types of rebars were considered, i.e., carbon, glass FRPs (CFRP, GFRP), and reinforcing steel. The ratio of tensile rebars ranged from 0.22% to 2.16%. The results indicated that the beams with CFRP rebars exhibited better crack mode and higher ultimate load than the beams with GFRP or steel rebars. GFRP rebars led to considerably higher ultimate deflection and tendon stress increment than steel rebars. In addition, several models for calculating the ultimate stress in unbonded tendons were assessed. An analytical model was also proposed to predict the tendon stress at ultimate and flexural strength in externally PCBs with steel and FRP rebars. The model predictions agreed well with the numerical results.

Keywords: ﬁber-reinforced polymer; beams; rebars; ﬂexural strength; structural analysis

1. Introduction Steel corrosion would lead to the deterioration of reinforced or prestressed concrete beams (RCBs or PCBs) [1]. An effective solution to this problem is to replace steel reinforcement with fiber-reinforced polymer (FRP). In addition to their non-corrosive property, FRP composites are high-strength, light-weight, and nonmagnetic. These composite materials are increasingly used for strengthening structural elements [2–8]. Different types of FRP composites are available, such as aramid, basalt, carbon, and glass FRPs (AFRP, BFRP, CFRP, and GFRP). Unlike steel reinforcement with ductile characteristics, FRP composites are linear-elastic materials without yielding [9,10]. In addition, the FRP modulus of elasticity is usually lower than that of steel reinforcement [9,10]. Hence, some concerns on the use of FRP reinforcement instead of steel reinforcement may arise, e.g., ductility and deflection issues due to the lack of yielding and low modulus of elasticity for FRP composites. The bond performance of FRP reinforcement in concrete under environmental exposure is also a concern. It was generally demonstrated that harsh environments have adverse effects on FRP reinforcement’s bond durability [11–13]. External prestressing has been widely used in the retrofit or construction of various concrete structures. Extensive research has been conducted concerning PCBs with external FRP tendons. Grace and Abdel-Sayed [14] carried out tests on four specimens to examine the behavior of PCBs with combined externally unbonded and internally bonded CFRP tendons. Their test results showed that the ductility could be improved by strengthening using draped external CFRP tendons.

Polymers 2020, 12, 2773; doi:10.3390/polym12122773 www.mdpi.com/journal/polymers Polymers 2020, 12, x FOR PEER REVIEW 2 of 16

Beeby [15] presented the test results of 16 PCBs with external AFRP tendons. They studied several Polymers 2020, 12, 2773 2 of 16 factors influencing the ultimate stress in external FRP or steel tendons and proposed modifying the BS8110 equation. Abdel Aziz et al. [16] developed an analytical method to predict the load-deflection Ghallabbehavior and of PCBs Beeby with [15] presentedexternal CFRP the test tendons. results ofWang 16 PCBs et al. with [17 external] tested AFRPfour specimens tendons. They, including studied 3 severalPCBs with factors external influencing BFRP the tendons ultimate and stress one incontrol external reinforced FRP or steel concrete tendons beam and (RCBs).proposed Their modifying study theshowed BS8110 that equation. using BFRP Abdel compositesAziz et al. [16 as] developed external tendons an analytical for strengthening method to predict RCBs the load-deflectioncan effectively behaviorimprove ofstructural PCBs with perform externalance. CFRP Zhu tendons. et al. [18 Wang] experimentally et al. [17] tested investigated four specimens, the influence including of bending 3 PCBs withangle external and radius BFRP on tendons external and FRP one tendonscontrol reinforced’ performance concrete. They beam concluded (RCBs). Theirthat FRP study tendons showed’ load that usingcapacitie BFRPs are composites reduced as as the external bending tendons angle increases for strengthening or the bending RCBs canradius effectively decreases. improve Both numerical structural performance.and experimental Zhu investigations et al. [18] experimentally [19,20] showed investigated that behaviors the influence of PCBs of with bending external angle CFRP and radiusand steel on externaltendons FRPare similar. tendons’ performance. They concluded that FRP tendons’ load capacities are reduced as theIn bending PCBs with angle external increases tendons, or the some bending bonded radius rebars decreases. need to be Both provided numerical to avoid and behaving experimental as a investigationstied arch and limit [19,20 the] showed crack width that behaviors and spacing of PCBs [21]. with Previous external studies CFRP demonstrated and steel tendons that areexternally similar. PCBsIn without PCBs with bonded external rebars tendons, exhibit somesignificant bonded crack rebars concentration, need to be which provided can to be avoid effectively behaving relieved as a tiedby providing arch and limitbonded the rebars crack width[19]. In and existing spacing works [21]. on Previous PCBs with studies external demonstrated FRP tendons, that steel externally rebars PCBswere usually without provided. bonded rebars The performance exhibit significant of conventional crack concentration, RCBs with which FRP rebars can be instead effectively of steel relieved ones byhas providing been extensively bonded studied rebars [ 19[22].– In30]. existing However, works the onfindings PCBs withobtained external from FRP the tendons, study of steelRCB rebarss with wereFRP/steel usually rebars provided. may not The be performance valid for PCBs of conventional with external RCBs FRP with tendons FRP rebars as the insteadlatter is of a steel different ones hasstructural been extensively system from studied the former [22–30 due]. However, to external the prestressing. findings obtained The influence from the of study using of FRP RCBs rebars with FRPinstead/steel of rebarssteel ones may on not the be behavior valid for of PCBsPCBs with external external FRP FRP tendons tendons has as yet the to latter be inv isestigated. a different structuralThis paper system presents from the a numerical former due and to analytical external prestressing.study to evaluate The the influence feasibility of usingof providing FRP rebars FRP insteadrebars instead of steel ones of steel on the ones behavior in PCBs of PCBs with with external external FRP FRP tendons. tendons A has comprehensive yet to be investigated. numerical evaluationThis paper was conducted presents a on numerical simply supported and analytical PCBs study using to an evaluate experimentally the feasibility validated of providing program, FRPwhere rebars investigated instead ofvariables steel ones includ in PCBsed the with typeexternal and amount FRP tendons.of bonded A rebars. comprehensive An analytical numerical model evaluationwas also proposed was conducted to predict on the simply tendon supported stress at PCBs ultimate using and an flexural experimentally strength validatedin externally program, PCBs wherewith steel investigated and FRP rebar variabless. included the type and amount of bonded rebars. An analytical model was also proposed to predict the tendon stress at ultimate and flexural strength in externally PCBs with2. Numerical steel and Program FRP rebars. and Verification

2.2.1. Numerical Numerical ProgramProgram and Verification A numerical program employing the Euler–Bernoulli beam element was developed [31]. The 2.1. Numerical Program finite element assumes negligible shear deformation, small strains with large displacements, and moderateA numerical rotations. program The p employinglane section the hypothesis Euler–Bernoulli was also beam adopted. element The was element developed has [six31]. degrees The finite of elementfreedomassumes as shown negligible in Figureshear 1, where deformation, u, v, and θ small are the strains axial, with transverse large displacements, displacements and, and moderate rotation, rotations.respectively The, and plane the section subscripts hypothesis 1 and 2 was refer also to adopted. the two end The elementnodes. A has linear six degreesvariation of for freedom u and as a showncubic variation in Figure for1, where v alongu ,thev, andx-axisθ are is assumed. the axial, transverseBy applying displacements, an updated Lagrangian and rotation, approach, respectively, the andgoverning the subscripts equations 1 and were 2 refer developed to the two, where end nodes. the stiffnessA linear variation matrix consists for u and of a cubic the material variation and for vgeometricalong the stiffx-axisness is assumed.matrices. The By applying effects of an external updated tendons Lagrangian are transformed approach, the into governing equivalent equations loads wereacting developed, on the finite where elements. the stiff nessA detailed matrix consistsformulation of the of material the numerical and geometric program sti canffness be matrices. seen in TheReference effects [31 of]. external tendons are transformed into equivalent loads acting on the finite elements. A detailed formulation of the numerical program can be seen in Reference [31].

Figure 1. Beam element, u, v, and θ are the axial, transverse displacements, and rotation, respectively.

Polymers 2020, 12, x FOR PEER REVIEW 3 of 16

Figure 1. Beam element, u, v, and θ are the axial, transverse displacements, and rotation, respectively.

The stress–strain relationships for materials are demonstrated schematically in Figure 2. The Polymersstress2020–strain, 12, 2773relationship for concrete in compression recommended in Eurocode 2 [32] was ad3opted. of 16 2 It is shown in Figure 2a and expressed by c/f cm  ( k    ) /[1  ( k  2)  ], where  cc/  0 The stress–strain relationships for materials are demonstrated schematically in Figure0.31 2. ;  and  are the concrete stress and strain, respectively;  (‰ ) 0.7f 2.8 ; The stress–strainc c relationship for concrete in compression recommended inc0 Eurocode 2 cm [32] was f = (k 2) [ + (k ) ] = adopted.ffcm It ck is shown8 in which in Figure fck2 ais and the expressed concreteby characteristicσc/ cm η cylinderη / 1 compressive2 η , where strength,η ε inc/ ε MPa;c0; − 0.31− σc and εc are the concrete stress and strain, respectively;0.3 εc0(%) = 0.7 fcm < 2.8; fcm = fck + 8 in k1.05 E / f ; and Ef 22( /10) , in GPa. As shown in Figure 2b, a bilinear elastic and which fck is thec c0 concrete cm characteristicc cylinder cm compressive strength, in MPa; k = 1.05Ecεc0/ fcm; 0.3 andstrainEc =-sof22tening( fcm/10 law) is, in adopted GPa. As for shown concrete in Figurein tension.2b, a FRP bilinear prestressing elastic and tendons strain-softening are linearly law elastic is , adoptedas shown for concrete in Figure in 2 tension.c. FRP rebars FRP prestressing are also linearly tendons elastic are, linearly while steel elastic, rebars as shown are elastic in Figure-perfectly2c. FRPplastic rebars, as are shown also linearlyin Figure elastic, 2d. while steel rebars are elastic-perfectly plastic, as shown in Figure2d.

Figure 2. Stress–strain curves of materials. (a) concrete in compression recommended in Eurocode 2 [32]; Figure 2. Stress–strain curves of materials. (a) concrete in compression recommended in Eurocode 2 (b) concrete in tension; (c) Fiber-reinforced polymer (FRP) tendons; (d) FRP and steel rebars. [32]; (b) concrete in tension; (c) Fiber-reinforced polymer (FRP) tendons; (d) FRP and steel rebars. A typical analysis consists of two steps: load-control analysis at prestress transfer and A typical analysis consists of two steps: load-control analysis at prestress transfer and displacement-control analysis at loading up to the ultimate. The above-mentioned numerical procedure displacement-control analysis at loading up to the ultimate. The above-mentioned numerical can simulate the nonlinear behavior of PCBs with combined external FRP tendons and internal steel/FRP procedure can simulate the nonlinear behavior of PCBs with combined external FRP tendons and rebars under short-term loading up to failure. internal steel/FRP rebars under short-term loading up to failure. 2.2. Verification with Experimental Results 2.2. Verification with Experimental Results Bennitz et al. [20] conducted a series of laboratory tests to investigate the flexural behavior of PCBs withBennitz external et al. CFRP[20] conducted tendons. a The series main of testlaboratory variables tests were to investigate the tendon the eccentricity, flexural behavior effective of prestress,PCBs with and external deviator CFRP arrangement. tendons. FourThe main of the test beams variables (B2, B3, were B6, the and tend B7)on were eccentricity, analyzed hereineffective toprestress verify the, and proposed deviator model. arrangement. The beams Four were of the of T-shapedbeams (B2, section, B3, B6, simplyand B7) supported were analyzed over herein a span to ofverify 3.0 m, the and proposed were under model. third-point The beams loading were of up T to-shaped failure, section, as shown simply in Figure supported3. The over beams a span were of 3.0 post-tensionedm, and were withunder two third horizontal-point loading straight up external to failure, CFRP as tendonsshown in (8 mmFigure in diameter3. The beams each) were with post an - initialtensioned effective with depth two ofhorizontal 200 mm. straight Beams B2external and B3 CFRP were tendons provided (8 mm with in one diameter deviator each) at the with midspan, an initial whileeffective Beams depth B6 and of 200 B7 hadmm. no Beams deviator. B2 and The B3 bottom were provided reinforcement with one consisted deviator of at two the deformed midspan steel, while rebars, each having 16 mm in diameter, while the top reinforcement consisted of four deformed steel rebars, each having 8 mm in diameter. The material parameters of the specimens are given in Table1. Polymers 2020, 12, x FOR PEER REVIEW 4 of 16 Polymers 2020, 12, x FOR PEER REVIEW 4 of 16 Beams B6 and B7 had no deviator. The bottom reinforcement consisted of two deformed steel rebars, Beamseach having B6 and 16 B7 mm had in no diameter, deviator. while The bottom the top reinforcementreinforcement consistedconsisted ofof twofour deformed deformed steel steel rebars rebars, , Polymerseacheach having having2020, 12 16 ,8 2773 mmmm inin diameter.diameter, Thewhile material the top parameters reinforcement of the consisted specimens of four are deformedgiven in Tabl steele 1.rebars4 of 16, each having 8 mm in diameter. The material parameters of the specimens are given in Table 1.

' Figure 3. Prestressed concrete beam (PCB) specimens with external CFRP tendons [20], As, As and ' Figure 3. Prestressed concrete beam (PCB) specimens with external CFRP tendons [20], As, A0 and Ap FigureAp are 3.the Prestressed areas of tensile, concrete compressive beam (PCB steel) specimens rebars and with prestressing external CFRP tendons, tendons respectively. [20], As, sAs and are the areas of tensile, compressive steel rebars and prestressing tendons, respectively. Ap are the areas of tensile, compressive steel rebars and prestressing tendons, respectively. According to the proposed analysis, crushing failure occurs at the midspan, after significant According to the proposed analysis, crushing failure occurs at the midspan, after signiﬁcant yielding of tensile rebars (formation of a plastic hinge with sufficient rotation). This failure mode is yieldingAccording of tensile to rebars the proposed (formation analysis, of a plastic crushing hinge failure with suoccursﬃcient at rotation). the midspan, This after failure significant mode is consistent with the experimental observation. The load-displacement curves and the tendon force’s consistentyielding of with tensile the rebars experimental (formation observation. of a plastic The hinge load-displacement with sufficient rotation). curves and This the failure tendon mode force’s is development generated by the proposed analysis are compared to the experimental data in Figure 4. developmentconsistent with generated the experimental by the proposed observation. analysis The are load compared-displacement to the experimentalcurves and the data tendon in Figure force4’.s It was observed that although the proposed analysis overestimated the tendon force at ultimate in Itdevelopment was observed generated that although by the theproposed proposed analysis analysis are compared overestimated to the the experimental tendon force data at ultimatein Figure in4. some specimens (B2, B3, and B7), the numerical results were generally in satisfactory agreement with someIt was specimens observed (B2,that B3,although and B7), the the proposed numerical analysis results overestimate were generallyd the in satisfactorytendon force agreement at ultimate with in the experimental ones over the entire ranges of loading. thesome experimental specimens (B2, ones B3 over, and the B7), entire the numerical ranges of loading.results were generally in satisfactory agreement with the experimental ones over the entire ranges of loading.

Figure 4. Veriﬁcation with experimental results. (a) load versus midspan deﬂection; (b) load versus tendon force. Polymers 2020, 12, x FOR PEER REVIEW 5 of 16

Figure 4. Verification with experimental results. (a) load versus midspan deflection; (b) load versus tendon force.

Table 1. Material parameters for specimens [20].

Steel Rebars FRP Tendons Concrete Polymers 2020, 12, 2773 5 of 16 ' ' ' Beam As fy Es A f y E Ap Ef ff σpe s s ck 2 2 f (MPa) (mm ) (MPa) (GPa) 2 (mm ) (GPa) (MPa) (MPa) Table(mm 1. Material) (MPa) parameters (GPa) for specimens [20]. B2 402 560 172 201 510 187 100.5 158 2790 895 35.2 B3 402 560 172Steel Rebars201 510 187 100.5 FRP158 Tendons2790 1382 Concrete33.4 Beam B6 402 560 172 2010 5100 1870 100.5 158 2790 917 40.6 As fy Es As fy Es Ap Ef ff σpe fck B7 402(mm 2) (MPa)560 (GPa)172 (mm201 2) (MPa)510 (GPa)187 (mm100.52 ) (GPa)158 (MPa)2790 (MPa)1407 (MPa)35.9

B2Note: A 402s, fy and E 560s = area, 172yield strength 201, and 510 elastic modulus 187 100.5 of tensile 158 steel rebars, 2790 respectively; 895 35.2 B3 402 560 172 201 510 187 100.5 158 2790 1382 33.4 B6, and 402 = 560 area, yield 172 strength 201, and elastic 510 modulus 187 of 100.5 compressive 158 steel 2790 rebars, 917 respectively; 40.6 B7 402 560 172 201 510 187 100.5 158 2790 1407 35.9 Ap = tendon area; Ef and ff = elastic modulus and tensile strength of FRP composites, respectively; σpe Note: A , f and E = area, yield strength, and elastic modulus of tensile steel rebars, respectively; A , f and E = = effectives y prestress.s s0 y0 s0 area, yield strength, and elastic modulus of compressive steel rebars, respectively; Ap = tendon area; Ef and ff = elastic modulus and tensile strength of FRP composites, respectively; σpe = eﬀective prestress. 3. Numerical Evaluation 3. Numerical Evaluation A simply supported PCB with external tendons draped at two deviators, as shown in Figure 5, is usedA simplyherein for supported the study. PCB The with depths external of external tendons tendons draped at at the two end deviators, supports as and shown deviators in Figure were5, 300is used and herein 500 mm, for the respectively. study. The depths The prestressing of external tendons atwe there end 1000 supports mm2 in and area, deviators made of were CFRP 300 compositesand 500 mm, having respectively. the tensile The prestressingstrength of 1840 tendons MPa were and 1000 the mmmodulus2 in area, of elasticity made of CFRPof 147 composites GPa. The initialhaving prestress the tensile prior strength to prestress of 1840 transfer MPa and wa thes 1104 modulus MPa, namely of elasticity 60% of 147the tensile GPa. The strength. initial The prestress area ofprior tensile to prestress rebars, A transferr, varied was from 1104 360 MPa, to 3560 namely mm2. 60% Thus, of the value tensile of strength. ρr (ratio Theof tensile area of rebars) tensile range rebars,d 2 fromAr, varied 0.22% from to 2.16%, 360 to where 3560 mmρr = A.r Thus,/(bdr) in the which value b ofis ρther (ratio cross of-sectional tensile rebars) width; rangedand dr is from the depth 0.22% of to 2 tensile2.16%, where rebars.ρ Ther = A arear/(bd r of) in compressive which b is the rebars cross-sectional was 360 mm width;. Three and d typesr is the of depth bonded of tensile rebars rebars.were considered,The area of compressivenamely, steel, rebars CFRP was, and 360 GFRP. mm2. ThreeThe mechanical types of bonded properties rebars of were the rebars considered, are given namely, in Tablesteel, 2. CFRP, The concrete and GFRP. strength The mechanical fck is 60 MPa. properties of the rebars are given in Table2. The concrete strength fck is 60 MPa.

Figure 5. Simply supported beam for numerical investigation, Ar and A'r0 are the areas of tensile and Figure 5. Simply supported beam for numerical investigation, Ar and A are the areas of tensile and compressive rebars, respectively. r compressive rebars, respectively. Table 2. Mechanical properties of rebars.

Rebars TensileT Strengthable 2. Mechanical (MPa) Yieldproperties Strength of rebars. (MPa) Elastic Modulus (GPa) Rebars SteelTensile Strength (MPa) 450 Yield Strength 450 (MPa) Elastic Modulus 200 (GPa) Steel CFRP450 1840450 - 147200 CFRP GFRP1840 750- - 40147 GFRP 750 - 40 3.1. Failure and Cracking Modes

Figure6 shows the ultimate strain distribution in the top and bottom fibers of the beams with different ρr levels. The failure and cracking modes can be seen in the graphs of this figure. Failure of all the investigated beams occurs due to concrete crushing at midspan when the specified ultimate compressive strain of 0.003 is reached. For a beam with no rebars (ρr = 0.0%), the noncritical sections were still far below their ultimate strain capacity at failure. When a minimum content of tensile rebars Polymers 2020, 12, x FOR PEER REVIEW 6 of 16

3.1. Failure and Cracking Modes Figure 6 shows the ultimate strain distribution in the top and bottom fibers of the beams with different ρr levels. The failure and cracking modes can be seen in the graphs of this figure. Failure of Polymersall the 2020investigated, 12, 2773 beams occurs due to concrete crushing at midspan when the specified ultimate6 of 16 compressive strain of 0.003 is reached. For a beam with no rebars (ρr = 0.0%), the noncritical sections were still far below their ultimate strain capacity at failure. When a minimum content of tensile rebars was provided (ρr = 0.22%), the exploitation of noncritical sections was improved. Such improvement was provided (ρr = 0.22%), the exploitation of noncritical sections was improved. Such improvement is pronounced for FRP (especially CFRP) rebars and relatively not so notable for steel rebars. However, is pronounced for FRP (especially CFRP) rebars and relatively not so notable for steel rebars. at a high ρr level of 2.16%, the exploitation of noncritical sections for steel rebars is nearly comparable However, at a high ρr level of 2.16%, the exploitation of noncritical sections for steel rebars is nearly to that for FRP rebars. At failure, both FRP tendons and rebars did not reach their rupture strength comparable to that for FRP rebars. At failure, both FRP tendons and rebars did not reach their rupture while the tensile steel rebars had yielded. strength while the tensile steel rebars had yielded.

FigureFigure 6. 6.Concrete Concrete strain strain distribution distribution at at ultimate. ultimate. (a ()aρ) rρ=r =0.0%; 0.0%; ( b(b) )ρ ρr r= = 0.22%;0.22%; ((cc))ρ ρrr == 2.16%.

CracksCracks of of concrete concrete occur occur once once the the tensile tensile strain strain reaches reaches its cracking its cracking strain. strain. Over Over the cracking the cracking zone, thezone, crack the width crack maywidth be may represented be represented by the ultimate by the ultimate tensile strain. tensile In strain. the case In fortheρ caser = 0%, for thereρr = 0%, appears there aappears huge tensile a huge strain tensile at the strain midspan at the against midspan marginal against ones marginal over the ones other over zones. the Thisother indicates zones. This an unfavorableindicates an crackunfavorable mode, crack i.e., the mode, beam i.e. has, the only beam one has large only crack one large at the crack midspan, at the andmidspan the concrete, and the overconcrete other over zones other is nearly zones uncracked. is nearly uncracked. By providing By providing a minimum a minimum content of content tensile rebarsof tensile (ρr =rebars0.22%), (ρr the= 0.22%), crack mode the wascrack substantially mode was improved, substantially i.e., improved,the crack width i.e., atthe the crack midspan width was at reduced, the midspan and more was cracksreduced occurred, and more at the cracks noncritical occurr zones.ed at the The noncritical use of CFRP zones. rebars The is use more of eCFRPﬀective rebars than theis more use ofeffective GFRP orthan steel the rebars use of to GFRP improve or steel the crackrebars mode. to improve At a high the crackρr level mode. of 2.16%, At a high the crack ρr level modes of 2.16%, for the the beams crack withmodes CFRP for andthe beams steel rebars with CFRP are similar, and steel while rebars GFRP are rebars similar, lead while to smaller GFRP crackrebars zone lead and to smaller larger crack crack widthzone and over larger the ﬂexural crack width span thanover steelthe flexural rebars. span than steel rebars.

3.2.3.2. TendonTendon Stress Stress Development Development Figure7a shows the stress increase in external tendons versus midspan deflection curves for the beams with different types of rebars (ρr = 1.19%). There was a roughly linear relationship between the tendon stress and the deflection. The slopes for the beams with steel, CFRP, and GFRP rebars were 2.12, 2.2, and 2.46 MPa/mm, respectively. Figure7b shows the stress increase in external tendons with the applied load. The beams with FRP rebars exhibited bilinear behavior with a turning point due Polymers 2020, 12, x FOR PEER REVIEW 7 of 16

Figure 7a shows the stress increase in external tendons versus midspan deflection curves for the beams with different types of rebars (ρr = 1.19%). There was a roughly linear relationship between the tendon stress and the deflection. The slopes for the beams with steel, CFRP, and GFRP rebars were 2.12, 2.2, and 2.46 MPa/mm, respectively. Figure 7b shows the stress increase in external tendons Polymers 2020, 12, 2773 7 of 16 with the applied load. The beams with FRP rebars exhibited bilinear behavior with a turning point due to concrete cracking, while the beam with steel rebars exhibit trilinear behavior with turning topoints concrete due cracking, to concrete while cracking the beam and with steel steel yielding, rebars exhibit respectively. trilinear Since behavior the with elastic turning behavior points is duedominated to concrete by concrete, cracking the and type steel of yielding, rebars appears respectively. to have Since practically the elastic no behavior influence is on dominated the tendon by concrete,stress evolution the type in of the rebars elastic appears range toof haveloading. practically Beyond no cracking, inﬂuence GFRP on the rebars tendon develop stress substantially evolution in thelower elastic rebar range stress of because loading. of Beyond the smaller cracking, elastic GFRP modulus rebars and, develop therefore, substantially a higher lower stress rebar increase stress in becauseexternal oftendons the smaller is required elastic at modulus a given and,load therefore,level to satisfy a higher the force stress equilibrium increase in externalwhen compared tendons to is requiredCFRP or atsteel a given rebars. load level to satisfy the force equilibrium when compared to CFRP or steel rebars.

FigureFigure 7. 7.Tendon Tendon stress stress development development for the for beams the bea withms di withﬀerent different types of types rebars. of (a rebars.) midspan (a) deﬂection midspan versusdeflection tendon versus stress tendon increment; stress ( b increment;) load versus (b tendon) load versus stress increase; tendon stress (c) variation increase; of ultimate (c) variation tendon of stressultimate increment tendon withstress varying incrementρr;( withd) variation varying of ρr ultimate; (d) variation load withof ultimate varying loadρr. with varying ρr.

TheThe variationsvariations ofof thethe ultimateultimate stressstress incrementincrement inin externalexternal tendonstendons ((Δ∆σσpp)) andand thethe ultimateultimate loadload ((PPuu) with with the the ρρrr levellevel are are illustratedillustrated in in FigureFigure7 c,d,7c,d respectively., respectively. As As ρρrr increases,increases, the the value value of of ∆ Δσσpp quicklyquickly decreasesdecreases forfor thethe beamsbeams withwith CFRPCFRP rebarsrebars whilewhile itit decreasesdecreases slightlyslightly forfor thethe beamsbeams withwith GFRPGFRP oror steelsteel rebars.rebars. AsAs expected,expected, GFRPGFRP rebarsrebars resultedresulted inin substantiallysubstantially higherhigher (around(around 70%70% higher)higher) ∆Δσσpp thanthan steelsteel rebars.rebars. At At ρρrr == 0.22%,0.22%, the the Δ∆σσp pratioratio between between the the beams beams with with CFRP CFRP and and steel steel rebars rebars was was as ashigh high as as1.73, 1.73, att attributedributed to to the the fact fact that that CFRP CFRP rebars rebars led led to to much better exploitation ofof noncriticalnoncritical sectionssections thanthan steelsteel rebars.rebars. ThisThis ratioratio waswas reducedreduced to to 1.12 1.12 at atρ ρrr= = 2.16%. This couldcould be explained byby thethe comparablecomparable exploitation exploitation of ofnoncritical noncritical sections sections of the of beams the beams with CFRP with and CFRP steel and rebars, steel as mentioned rebars, as inmentioned the previous in the section. previous The ultimate section. load The isultimate dependent load upon is dependent the ultimate upon stresses the ultimate in external stresses tendons in andexternal tensile tendons rebars. and Due tensile to higher rebars. reinforcement Due to higher stresses reinforcement at ultimate, stresses CFRP rebars at ultimate, led to substantiallyCFRP rebars higherled to substantially (37.1% higher) higher ultimate (37.1% load higher) than steelultimate rebars load at thanρr = 0.22%.steel rebars The diatﬀ ρerencer = 0.22%. tends The to difference decrease with increasing ρr due to the reduced diﬀerence between reinforcement stresses in the beams with

CFRP and steel rebars. At a low ρr level of 0.22%, GFRP rebars lead to a slightly higher ultimate load than steel rebars. As ρr increases, the increase in ultimate load for the beams with GFRP rebars is slower than that for the beams with steel rebars. As a result, when ρr increases to a level greater than Polymers 2020, 12, x FOR PEER REVIEW 8 of 16

tends to decrease with increasing ρr due to the reduced difference between reinforcement stresses in Polymersthe beams2020, 12with, 2773 CFRP and steel rebars. At a low ρr level of 0.22%, GFRP rebars lead to a slightly 8higher of 16 ultimate load than steel rebars. As ρr increases, the increase in ultimate load for the beams with GFRP rebars is slower than that for the beams with steel rebars. As a result, when ρr increases to a level 0.77%,greater the than ultimate 0.77%, load the for ultimate the beams load with for the GFRP beams rebars with turns GFRP to be rebars lower turns than to that be for lower the beamsthan that with for steelthe rebars.beams with steel rebars. 3.3. Deformation Behavior 3.3. Deformation Behavior The moment-curvature and load-deflection curves for the beams with different types of rebars The moment-curvature and load-deflection curves for the beams with different types of rebars (ρr = 1.19%) are shown in Figure8a,b, respectively. In the precracking stage, the e ffect of rebars on (ρr = 1.19%) are shown in Figure 8a,b, respectively. In the precracking stage, the effect of rebars on the response characteristics is negligible due to slight stress increments in rebars. The responses for the response characteristics is negligible due to slight stress increments in rebars. The responses for the beams with FRP and steel rebars differ after cracking because the rebar contribution becomes the beams with FRP and steel rebars differ after cracking because the rebar contribution becomes increasingly important. The reduction in flexural stiffness due to cracking is highly dependent on the increasingly important. The reduction in flexural stiffness due to cracking is highly dependent on the rebar modulus of elasticity, i.e., the higher the rebar modulus of elasticity, the less the reduction in rebar modulus of elasticity, i.e., the higher the rebar modulus of elasticity, the less the reduction in member stiffness. Therefore, at a given load level, GFRP rebars lead to higher post cracking deformation member stiffness. Therefore, at a given load level, GFRP rebars lead to higher post cracking than CFRP or steel rebars. The post cracking deformation for the beams with FRP rebars develops deformation than CFRP or steel rebars. The post cracking deformation for the beams with FRP rebars linearly until failure. For the beam with steel rebars, a significant further reduction in flexural stiffness develops linearly until failure. For the beam with steel rebars, a significant further reduction in occurs on steel yielding. flexural stiffness occurs on steel yielding.

FigureFigure 8. 8.Deformation Deformation behavior behavior for for the the beams beams with with di differentfferent types types of of rebars. rebars. (a ()a moment) moment versus versus curvaturecurvature at at midspan; midspan; (b ()b load) load versus versus midspan midspan deflection; deflection; (c )(c variation) variation of of ultimate ultimate curvature curvature with with varyingvaryingρr ρ;(r;d ()d variation) variation of of ultimate ultimate deflection deflection with with varying varyingρr. ρr.

FigureFigure8c,d 8c, illustratesd illustrate thes the variation variation of of ultimate ultimate midspan midspan curvature curvature ( κ u(κ)u and) and deflection deflection (∆ (uΔ)u with) with varyingvaryingρr ,ρ respectively.r, respectively. It is It seen is seen in Figure in Figure8c that 8c the that value the ofvalueκu decreases of κu decreases as ρr increases. as ρr increases. GFRP rebars GFRP mobilizerebars mobilize a smaller a κsmalleru at low κuρ atr levels low ρ whereasr levels whereas a larger aκ largeru at high κu atρr highlevels ρr in levels comparison in comparison with steel with rebars.steel rebars. CFRP rebarsCFRP mobilizerebars mobilize lower κ loweru than κ steelu than rebars; steel rebars; the diff erencethe difference is substantial is substantial at low ρ atr levels low ρr but reduced with increasing ρr. It was noted that the above observation was similar to the effect of rebars on the crack width (represented by the ultimate concrete tensile strain), as discussed previously. Polymers 2020, 12, x FOR PEER REVIEW 9 of 16

Polymers 2020, 12, 2773 9 of 16 levels but reduced with increasing ρr. It was noted that the above observation was similar to the effect of rebars on the crack width (represented by the ultimate concrete tensile strain), as discussed Thispreviously. was because This the was ultimate because curvature the ultimate was directly curvature proportional was directly to the ultimate proportional concrete to tensile the ul straintimate inconcrete terms of tensile the plane strain section in terms hypothesis. of the plane As section shown hypothesis. in Figure8d, As GFRP shown rebars in Figure register 8d substantially, GFRP rebars higherregister∆ substantiallyu than steel rebarshigher Δ becauseu than steel of a rebars significantly because lowerof a significantly modulus oflower elasticity. modulus CFRP of elasticity. rebars registerCFRP rebars considerably register higherconsiderably∆u than higher steel rebars Δu than at steel low ρ rebarsr levels, at whilelow ρ ther levels, difference while isthe reduced difference with is increasingreduced withρr. This increasing could also ρr. beThis attributed could also to the be rebarattributed effect onto the exploitationrebar effect ofon noncritical the exploitation sections, of asnoncritical explained sections, previously. as explained previously.

3.4.3.4. NeutralNeutral AxisAxis DepthDepth andand RebarRebar StrainStrain SinceSince the the neutral neutral axis axis depth depth and and rebar rebar strain strain are are key key parameters parameters describing describing flexural flexural ductility ductility [33], it[33 is], important it is important to understand to understand their their behavior behavior well. well The. movementThe movement of the of neutralthe neutral axis, axis, after after it rises it rises to theto the midspan midspan section’s section bottom’s bottom fiber, fiber with, with the the moment moment for for the the beams beams with with di differentfferent types types of of rebars rebars (ρ(ρr r= = 1.19%1.19%)) isis illustratedillustrated inin FigureFigure9 a.9a. For For the the beams beams with with FRP FRP rebars, rebars, the the neutral neutral axis axis shifts shifts rapidly rapidly withwith the the increasing increasing moment, moment and, and then then the the shift shift gradually gradually slows slows down. down. SimilarSimilar behaviorbehavior was was observed observed forfor the the beam beam with with steel steel rebars rebars until until yielding. yielding. After After that, that, a a fastfast movementmovement of of the the neutralneutral axis axis is is resumedresumed duedue to to steel steel yielding. yielding. Figure Figure9bshows 9b shows the strain the strain development development in FRP in and FRP steel and rebars steel with rebars the bending with the momentbending ( ρmomentr = 1.19%). (ρr The= 1.19 behavior%). The forbehavior the beam for withthe be steelam rebarswith st exhibitedeel rebars three exhibit stagesed three with transitionsstages with causedtransitions by cracking caused by and cracking yielding, and respectively. yielding, respectively. Two-stage behavior Two-stage was behavior observed was for observed the beams for with the FRPbeams rebars with because FRP rebars of the because lack of yielding. of the lack Due of toyielding. a lower modulusDue to a oflower elasticity, modulus GFRP of rebarselasticity, exhibited GFRP arebars significantly exhibit fastered a significantly increase in strainfaster afterincrease cracking in strain but after a slower cracking stress but development a slower stress when development compared towhen steel compared or CFRP rebars. to steel or CFRP rebars.

FigureFigure 9. 9.Neutral Neutral axis axis depth depth and and bar bar strain strain for for diﬀ differenterent types types of rebars. of rebars. (a) moment (a) moment versus versus neutral neutral axis depthaxis depth at midspan; at mids (pab)n moment; (b) moment versus versus rebar rebar strain strain at midspan; at midspan; (c) variation (c) variation of ultimate of ultimate neutral neutral axis depthaxis depth with varyingwith varyingρr;(d )ρ variationr; (d) variation of ultimate of ultimate tensile tensile rebar rebar strain strain with varyingwith varyingρr. ρr.

Polymers 2020, 12, 2773 10 of 16

Figure9c,d shows the variation of neutral axis depth ( cu) and tensile bar strain at ultimate (εr) with varying ρr, respectively. Comparing these graphs to the graph of Figure8c, it is seen that the e ﬀect of rebars on κu is opposite to that on cu while coincident to that on εr. Their theoretical relationships can explain this observation. According to the plane section hypothesis, cu and εr are related to κu by: cu = εu/κu; εr = κu(dr cu) = κudr εu, where dr is the depth of tensile rebars, equal to 550 mm; εu is − − the ultimate concrete compressive strain, equal to 0.003. Because of a smaller value of κu, CFRP rebars mobilize a higher cu value and a lower εr value than steel rebars. Likewise, the values of cu and εr mobilized by GFRP rebars could be higher or lower than those by steel rebars, depending on the ρr level.

4. Analytical Modeling

4.1. Existing Models Using Combined Reinforcing Index for Prediction of Ultimate Stress in Unbonded Tendons Owing to strain incompatibility, the stress in external or unbonded tendons is member-dependent. The quantification of the tendon stress is a key task in design practice. The combined reinforcing index is considered one of the best parameters used for calculating the tendon stress, as this parameter involves several important factors, including the tendon area and depth, effective prestress, rebar area, and concrete strength. The combined reinforcing index for the beams with steel rebars is defined by

Apσpe + Ar fy ω0 = (1) bdp fck where dp is the tendon depth. At ultimate, FRP rebars often reach a stress level far below their rupture strength. Therefore, the combined reinforcing index for the beams with FRP rebars could be expressed by Apσpe + Arσr ω0 = (2) bdp fck where σr is the stress in tensile rebars at ultimate. Figure 10a shows the numerical results regarding the relationship between ∆σp and ω0 for the beams with FRP and steel rebars. It is interesting to note that the variations of ∆σp with varying ω0 for the beams with CFRP and GFRP rebars appear to be consistent. Since CFRP represents the highest modulus of elasticity and GFRP the lowest one amongst FRP groups, it can be concluded that the beams with diﬀerent types of FRP rebars follow approximately the same ∆σp-ω0 response. At a given ω0 value, the beams with FRP rebars led to substantially higher tendon stress than the beams with steel rebars. The tendon stress tends to decrease with increasing ω0, while the decrease for the beams with FRP rebars is much quicker than that for the beams with steel rebars. Polymers 2020, 12, 2773 11 of 16 Polymers 2020, 12, x FOR PEER REVIEW 11 of 16

FigureFigure 10. 10.Relationship Relationship between between tendon tendon stress stress increment increment and combined and combined reinforcing reinforcing index. (a) numerical index. (a) results;numerical (b) comparisonresults; (b) comparison between simpliﬁed between models simplified and numericalmodels and data numerical for the beamsdata for with the steelbeams rebars; with (csteel) comparison rebars; (c) between comparison simpliﬁed between models simplified and numerical models and data numerical for the beams data withfor the FRP beams rebars. with FRP rebars. By performing laboratory tests of 22 PCBs with unbonded tendons, Du and Tao [34] found an approximatelyBy performing linear laboratory relationship tests between of 22 PCBs the tendonwith unbonded stress and tendons, the combined Du and reinforcingTao [34] found index. an Theyapproximately recommended linear the relationship following expression between the for tendon calculating stress∆σ andp: the combined reinforcing index. They recommended the following expression for calculating Δσp: ∆σp = 786 1920ω (3) − 0 p 786  1920 0 (3) JGJ/T 92-93 [35] suggested the following Equation for the computation of ∆σp: JGJ/T 92-93 [35] suggested the following Equation for the computation of Δσp:

∆σp = 500 770ω0 (4) p 500−  770 0 (4) forforL L/d/dpp ≤ 35; and ≤ ∆σp = 250 380ω0 (5) p 250−  380 0 (5) for L/dp > 35. for L/dp > 35. JGJ 92-2016 [36] proposed a modiﬁcation of the above Equation, which is expressed as follows: JGJ 92-2016 [36] proposed a modification of the above Equation, which is expressed as follows: L2 L ∆σp = (240 335ω0)(0.45 + 5.5h/L0) 2 (6) p (240−  33500 )(0.45  5.5hL /L1 ) (6) L1 where ω0 is not greater than 0.4; h is the cross-sectional depth; L0 is the span length; L1 is the tendon where ω0 is not greater than 0.4; h is the cross-sectional depth; L0 is the span length; L1 is the tendon length between end anchorages for continuous beams, and L2 is the total length of loaded spans. length between end anchorages for continuous beams, and L2 is the total length of loaded spans. It should be noted that the above models were developed for the beams with steel rebars. To evaluate the applicability of the models, the predictions by the simplified equations against the numerical data for the beams with steel rebars are presented in Figure 10b, while those for the beams

Polymers 2020, 12, 2773 12 of 16

Polymers 2020, 12, x FOR PEER REVIEW 12 of 16 It should be noted that the above models were developed for the beams with steel rebars. Towith evaluate FRP rebars the applicabilityare presented of in the Figure models, 10c. theIt wa predictionss observedby that, the in simplified general, the equations Du and againstTao model the andnumerical JGJ/T 92 data-93 forare theunsafe beams, while with JGJ steel 92- rebars2016 is areconservative presented when in Figure predicting 10b, while the thosetendon for stress the beamsin the withbeams FRP with rebars steel arerebars. presented In addition, in Figure the Du10c. and It was Tao observed model substantially that, in general, overestimates the Du and the Tao influence model ofand ω0 JGJ. For/T the 92-93 beams are unsafe, with FRP while rebars, JGJ 92-2016the effect is conservativeof ω0 on the tendon when predictingstress is overestimated the tendon stress by the in Du the andbeams Tao with model steel while rebars. underestimated In addition, the by DuJGJ/T and 92 Tao-93 and model JGJ substantially 92-2016. Moreover, overestimates JGJ 92-2016 the influence appears toof beω0 .overly For the conservative. beams with FRP rebars, the effect of ω0 on the tendon stress is overestimated by the Du and Tao model while underestimated by JGJ/T 92-93 and JGJ 92-2016. Moreover, JGJ 92-2016 appears 4.2.to be Proposed overly conservative.Model

4.2. ProposedAccording Model to the linear fit to the numerical data of the beams with steel and FRP rebars as illustrated in Figure 11, the following Equation is proposed for predicting the stress increment in externalAccording tendons to at theultimate: linear ﬁt to the numerical data of the beams with steel and FRP rebars as illustrated in Figure 11, the following Equation is proposed for predicting the stress increment in external tendons at ultimate: p 303  220 0 (7) ∆σp = 303 220ω (7) for the beams with steel rebars; and − 0 for the beams with steel rebars; and p 626  1032 0 (8) ∆σp = 626 1032ω0 (8) for the beams with FRP rebars. − for the beams with FRP rebars.

Figure 11. Linear ﬁt to the numerical data. (a) beams with steel rebars; (b) beams with FRP rebars. Figure 11. Linear fit to the numerical data. (a) beams with steel rebars; (b) beams with FRP rebars.

It is worth mentioning that at given cross-sectional and material properties, the value of ω0 in the beamsIt withis worth steel mentioning rebars is known that at (see given Equation cross (1)),-sectional whereas and that material in the beamsproperties, FRP rebarsthe value is unknown of ω0 in (seethe beams Equation with (2)). steel Consequently, rebars is known the value(see Equation of ∆σp in (1)) the, beamswhereas with that steel in the rebars beams can FRP be calculated rebars is unknowndirectly by (see using Eq Equationuation (2)). (7), Consequently, while the computation the value ofof ∆Δσσpp inin thethe beamsbeams with with FRP steel rebars rebars needs can be to calculatedcombine Equation directly (8)by using with the Equation section ( equilibrium7), while the equation.computation of Δσp in the beams with FRP rebars needsThe to combine axial equilibrium Equation equation (8) with the of the section critical equilibrium section of equation. the beams with steel rebars is The axial equilibrium equation of the critical section of the beams with steel rebars is

0.85 fckbβ1cu = Ap(σpe + ∆σp) + Ar fy Ar0 fy''0 (9) 0.85fck b1 c u A ppe (     p )  A ry f−  A ry f (9) where β1 is the stress-block factor for concrete, taken equal to 0.85; Ar and Ar0 are' the area of tensile and where β1 is the stress-block factor for concrete, taken equal to 0.85; Ar and Ar are the area of tensile compressive rebars, respectively. Substituting Equation (7) into Equation (9) results in and compressive rebars, respectively. Substituting Equation (7) into Equation (9) results in '' Ap(σpe + 303 220ω0) + Ar fy Ar0 fy0 Ap( pe 303 − 2200 )  A r f y−  A r f y ccu= (10) u 0.85 fckbβ1 (10) 0.85 fbck 1 Hence, the flexural ﬂexural strength for the beams with steel rebars is calculatedcalculated fromfrom

' ' '2 2 MAuMu = ppe(Ap(σ pe + ∆ peσ )p dAfdAfd)de + A ryrr fydr A0 ryrf 0d0 0.85 0.85f fbcb ck(β (c u)1 u/ )2 / 2 (11 (11)) − r y r − ck 1 ' where dr and dr are the effective depth of tensile and compressive rebars, respectively; de is the effective depth at ultimate of external tendons. The value of de can be obtained by

Polymers 2020, 12, 2773 13 of 16

where dr and dr0 are the eﬀective depth of tensile and compressive rebars, respectively; de is the eﬀective depth at ultimate of external tendons. The value of de can be obtained by

de = Rddp (12) where Rd is the depth reduction factor as a result of second-order eﬀects of externally PCBs, which may be calculated from R = 1.14 0.005(L/dp) 0.19(S /L) 1.0 (13) d − − d ≤ for center-point loading; and

R = 1.25 0.01(L/dp) 0.38(S /L) 1.0 (14) d − − d ≤ for third-point loading. On the other hand, the axial equilibrium of the critical section of the beams with FRP rebars is given by 0.85 f bβ cu = Ap(σpe + ∆σp) + Arσr A0σ0 (15) ck 1 − r r where ! dr σr = E f εu 1 (16) cu − ! dr0 σr0 = E0f εu 1 (17) − cu where Ef and E0f are the elastic modulus of tensile and compressive FRP rebars, respectively; εu is taken equal to 0.003. Combining Equations (2), (8), (16), and (17) with Equation (15) leads to

B + √B2 4AC c = − − (18) u 2A where A = 0.85 fckbβ1 B = ArE εu(1 1032ρp/ f ) + A E εu Ap(σpe + 626 1032ωp) f − ck r0 0f − − C = ArE εudr(1 1032ρp/ f ) A E εud − f − ck − r0 0f r0 where Ap ρp = (19) bdp

Apσpe ωp = (20) bdp fck Hence, the ﬂexural strength for the beams with FRP rebars is obtained by

2 Mu = Ap(σpe + ∆σp)de + Arσrdr A0σ0d0 0.85 f b(β cu) /2 (21) − r r r − ck 1 A comparison of the ultimate tendon stress increment and ﬂexural strength predicted by the proposed analytical model with the numerical results for the beams with steel and FRP rebars is presented in Table3 and Figure 12. It is seen that there is a good agreement between the analytical predictions and numerical data. The mean discrepancy for the ultimate tendon stress increment is 1.03% with a standard deviation of 4.08%, while that for the ultimate moment is 4.33% with a standard − deviation of 2.32%. Polymers 2020, 12, 2773 14 of 16 Polymers 2020, 12, x FOR PEER REVIEW 14 of 16

FigureFigure 12. 12. ComparisonComparison between between analytical analytical predictions predictions and and numerical numerical results. results. (a ()a ) tendon tendon stress stress incrementincrement at at ultimate; ultimate; (b (b) )ultimate ultimate moment. moment.

TableTable 3. 3. ComparisonComparison of of ultimate ultimate tendon tendon stress stress increment increment and and moment moment predicted predicted by by the the analytical analytical modelmodel with with numerical numerical results. results.

Δσp (MPa) Mu (kN m) Rebars ρr (%) ∆σp (MPa) Mu (kN m) Rebars ρr (%)Analytical Numerical Error (%) Analytical Numerical Error (%) Analytical Numerical Error (%) Analytical Numerical Error (%) 0.22 272.05 267.60 1.66 654.40 653.12 0.20 0.22 272.05 267.60 1.66 654.40 653.12 0.20 0.70 263.25 269.05 −2.15 812.72 831.63 −2.27 0.70 263.25 269.05 2.15 812.72 831.63 2.27 − − SteelSteel 1.19 1.19254.45 254.45 252.44 252.44 0.80 0.80 962.98 962.98 997.08 997.08 −33.42.42 − 1.67 1.67245.65 245.65 246.93 246.93 −0.520.52 1105.17 1105.17 1154.67 1154.67 −44.29.29 − − 2.16 2.16236.85 236.85 234.63 234.63 0.95 0.95 1239.31 1239.31 1315.42 1315.42 −55.79.79 − 0.22 448.74 463.02 3.09 837.89 874.32 4.17 0.22 448.74 463.02 −3.09− 837.89 874.32 −4.17 0.70 382.53 400.19 4.41 1054.40 1129.98 6.69 0.70 382.53 400.19 −4.41− 1054.40 1129.98 −6.69 CFRP 1.19 338.41 324.23 4.37 1189.31 1256.83 5.37 − CFRP 1.19 1.67338.41 304.47 324.23 287.09 4.37 6.05 1189.31 1288.01 1256.83 1366.46 −55.74.37 − 1.67 2.16304.47 276.65 287.09 261.86 6.05 5.65 1288.01 1365.57 1366.46 1466.04 −56.85.74 − 2.16 0.22276.65 483.51 261.86 435.02 5.65 11.15 1365.57 713.43 1466.04 706.31 − 1.016.85 0.70 455.67 451.08 1.02 808.85 840.51 3.77 0.22 483.51 435.02 11.15 713.43 706.31 −1.01 GFRP 1.19 433.69 437.72 0.92 882.11 928.20 4.97 0.70 455.67 451.08 1.02− 808.85 840.51 −3.77 1.67 415.19 428.42 3.09 942.27 1004.81 6.22 − − GFRP 1.19 2.16433.69 399.09 437.72 407.03 −0.921.95 882.11 993.58 928.20 1063.79 −46.60.97 1.67 415.19 428.42 −3.09− 942.27 1004.81 −6.22 2.16 399.09 407.03 −1.95 993.58 1063.79 −6.60 5. Conclusions

5. ConclusionsA numerical and analytical study was conducted on simply supported PCBs with external CFRP tendons, aimed at identifying the effect of providing FRP bonded rebars instead of steel ones. Based on A numerical and analytical study was conducted on simply supported PCBs with external CFRP the results of the study, the following conclusions can be drawn: tendons, aimed at identifying the effect of providing FRP bonded rebars instead of steel ones. Based on theFRP results (especially of the study, CFRP) the rebars following lead toconclusions better exploitation can be drawn: of noncritical sections than steel rebars, • particularly notable at a low ρ level. The crack mode is improved by providing a minimum  FRP (especially CFRP) rebars leadr to better exploitation of noncritical sections than steel rebars, amount of rebars, while the improvement is more effective using CFRP rebars than using GFRP or particularly notable at a low ρr level. The crack mode is improved by providing a minimum steel rebars. amount of rebars, while the improvement is more effective using CFRP rebars than using GFRP CFRP rebars lead to larger ultimate load and neutral axis depth but smaller ultimate curvature • or steel rebars.  CFRPand tensile rebars rebar lead strainto larger than ultimate steel rebars. load Suchand neutral values registeredaxis depth by but GFRP smaller rebars ultimate could becurvature larger or andsmaller tensile than rebar those strain by steelthan rebars,steel rebars. depending Such values on the ρregisteredr level. by GFRP rebars could be larger GFRP rebars mobilize substantially higher ultimate deflection and tendon stress increment than • or smaller than those by steel rebars, depending on the ρr level.  GFRPsteel rebars.rebars CFRPmobilize rebars substantially lead to similar higher observation ultimate deflection at a low ρr andlevel, tendon while stress the di ffincrementerence between than the values for the beams with CFRP and steel rebars diminishes as ρ increases. steel rebars. CFRP rebars lead to similar observation at a low ρr level,r while the difference between the values for the beams with CFRP and steel rebars diminishes as ρr increases.

Polymers 2020, 12, 2773 15 of 16

Both JGJ/T 92-93 and JGJ 92-2016 underestimated the inﬂuence of combined reinforcing index on • external tendons’ stress at ultimate in the beams with FRP rebars. Moreover, JGJ 92-2016 appears to be overly conservative for predicting the ultimate tendon stress. An analytical model was proposed to predict the tendon stress at ultimate and ﬂexural strength in • externally PCBs with steel and FRP rebars. The model predictions are in good agreement with the numerical results.

The present study is limited to simply supported conditions. Further studies on the use of FRP rebars instead of steel ones in continuous PCBs with external tendons shall be performed in the future.

Author Contributions: Conceptualization, M.P. and T.L.; Methodology, M.P.; Software, T.L.; Validation, Z.L.; Formal Analysis, Z.L.; Investigation, Z.L. and M.P.; Resources, T.L.; Data Curation, Z.L.; Writing—Original Draft Preparation, M.P.; Writing—Review & Editing, Z.L.; Visualization, Z.L.; Supervision, T.L. All authors have read and agreed to the published version of the manuscript. Funding: This research received no external funding. Conﬂicts of Interest: The authors declare no conﬂict of interest.

References

1. Ahmad, S. Reinforcement corrosion in concrete structures, its monitoring and service life prediction—A review. Cem. Concr. Compos. 2003, 25, 459–471. 2. Lou, T.; Karavasilis, T.L. Numerical evaluation of prestressed steel-concrete composite girders with external FRP or steel tendons. J. Constr. Steel Res. 2019, 162, 105698. 3. Lou, T.; Peng, C.; Karavasilis, T.L.; Min, D.; Sun, W. Moment redistribution versus neutral axis depth in continuous PSC beams with external CFRP tendons. Eng. Struct. 2020, 209, 109927. 4. Sun, W.; Jirsa, J.O.; Ghannoum, W.M. Behavior of anchored carbon fiber-reinforced polymer strips used for strengthening concrete structures. ACI Mater. J. 2016, 113, 163–172. 5. Sun, W.; He, T.; Wang, X.; Zhang, J.; Lou, T. Developing an anchored near-surface mounted (NSM ) FRP system for fuller use of FRP material with less epoxy filler. Compos. Struct. 2019, 226, 111251. 6. Mostofinejad, D.; Hosseini, S.M.; Tehrani, B.N.; Eftekhar, M.R.; Dyari, M. Innovative warp and woof strap (WWS) method to anchor the FRP sheets in strengthened concrete beams. Constr. Build. Mater. 2019, 218, 351–364. 7. Wang, L.; Kawaguchi, K.; Xu, J.; Han, Q. Effects of transverse constraints on the longitudinal compressive strength of unidirectional CFRP pultruded plates and rods. Compos. Struct. 2019, 207, 740–751. 8. Xu, J.; Wang, W.; Han, Q.; Liu, X. Damage pattern recognition and damage evolution analysis of unidirectional CFRP tendons under tensile loading using acoustic emission technology. Compos. Struct. 2020, 238, 111948. 9. ACI committee 440. Guide for the Design and Construction of Structural Concrete Reinforced with FRP Bars; ACI 440.1R-06: Farmington Hills, MI, USA, 2006. 10. ACI committee 440. Prestressing Concrete Structures with FRP Tendons; ACI 440.4R-04: Farmington Hills, MI, USA, 2004. 11. El Refai, A.; Abed, F.; Altalmas, A. Bond durability of basalt fiber–reinforced polymer bars embedded in concrete under direct pullout conditions. ASCE J. Compos. Constr. 2015, 19, 04014078. 12. Altalmas, A.; El Refai, A.; Abed, F. Bond degradation of basalt fiber-reinforced polymer (BFRP) bars exposed to accelerated aging conditions. Constr. Build. Mater. 2015, 81, 162–171. 13. Al-Tamimi, A.; Abed, F.H.; Al-Rahmani, A. Effects of harsh environmental exposures on the bond capacity between concrete and GFRP reinforcing bars. Adv. Concr. Constr. 2014, 2, 1–11. 14. Grace, N.F.; Abdel-Sayed, G. Behavior of externally draped CFRP tendons in prestressed concrete bridges. PCI J. 1998, 43, 88–101. 15. Ghallab, A.; Beeby, A.W. Factors affecting the external prestressing stress in externally strengthened prestressed concrete beams. Cem. Concr. Compos. 2005, 27, 945–957. 16. Abdel Aziz, M.; Abdel-Sayed, G.; Ghrib, F.; Grace, N.F.; Madugula, M.K.S. Analysis of concrete beams prestressed and post-tensioned with externally unbonded carbon fiber reinforced polymer tendons. Can. J. Civ. Eng. 2005, 31, 1138–1151. Polymers 2020, 12, 2773 16 of 16

17. Wang, X.; Shi, J.; Wu, G.; Yang, L.; Wu, Z. Effectiveness of basalt FRP tendons for strengthening of RC beams through the external prestressing technique. Eng. Struct. 2015, 101, 34–44. 18. Zhu, H.; Dong, Z.; Wu, G.; Chen, H.; Li, J.; Liu, Y. Experimental Evaluation of Bent FRP Tendons for Strengthening by External Prestressing. ASCE J. Compos. Constr. 2017, 21, 04017032. 19. Lou, T.; Lopes, S.M.R.; Lopes, A.V. Numerical analysis of behaviour of concrete beams with external FRP tendons. Constr. Build. Mater. 2012, 35, 970–978. 20. Bennitz, A.; Schmidt, J.W.; Nilimaa, J.; Taljsten, B.; Goltermann, P.; Ravn, D.L. Reinforced concrete T-beams externally prestressed with unbonded carbon fiber-reinforced polymer tendons. ACI Struct. J. 2012, 109, 521–530. 21. ACI Committee 318. Building Code Requirements for Structural Concrete (ACI 318-14) and Commentary (ACI 318R-14); American Concrete Institute: Farmington Hills, MI, USA, 2014. 22. Abed, F.; Oucif, C.; Awera, Y.; Mhanna, H.H.; Alkhraisha, H. FE modeling of concrete beams and columns reinforced with FRP composites. Def. Technol. 2020.[CrossRef] 23. Abed, F.; El Refai, A.; Abdalla, S. Experimental and finite element investigation of the shear performance of BFRP-RC short beams. Structures 2019, 20, 689–701. 24. Al-Rahmani, A.; Abed, F.H. Numerical investigation of hybrid FRP reinforced beams. In Proceedings of the 5th International Conference on Modeling, Simulation and Applied Optimization (ICMSAO), Hammamet, Tunisia, 28–30 April 2013; pp. 1–6. [CrossRef] 25. Ju, M.; Park, Y.; Park, C. Cracking control comparison in the specifications of serviceability in cracking for FRP reinforced concrete beams. Compos. Struct. 2018, 182, 674–684. 26. Alam, M.S.; Hussein, A. Relationship between the shear capacity and the flexural cracking load of FRP reinforced concrete beams. Constr. Build. Mater. 2017, 154, 819–828. 27. Barris, C.; Torres, L.; Vilanova, I.; Mias, C.; Llorens, M. Experimental study on crack width and crack spacing for Glass-FRP reinforced concrete beams. Eng. Struct. 2017, 131, 231–242. 28. Dundar, C.; Tanrikulu, A.K.; Frosch, R.J. Prediction of load-deflection behavior of multi-span FRP and steel reinforced concrete beams. Compos. Struct. 2015, 132, 680–693. 29. Lou, T.; Lopes, S.M.R.; Lopes, A.V. Neutral axis depth and moment redistribution in FRP and steel reinforced concrete continuous beams. Compos. Part B Eng. 2015, 70, 44–52. 30. Mahroug, M.E.M.; Ashour, A.F.; Lam, D. Experimental response and code modelling of continuous concrete slabs reinforced with BFRP bars. Compos. Struct. 2014, 107, 664–674. 31. Lou, T.; Xiang, Y. Finite element modeling of concrete beams prestressed with external tendons. Eng. Struct. 2006, 28, 1919–1926. 32. CEN. Eurocode 2: Design of Concrete Structures—Part 1-1: General Rules and Rules for Buildings. EN 1992-1-1; European Committee for Standardization: Brussels, Belgium, 2004. 33. Bernardo, L.; Nepomuceno, M.; Pinto, H. Neutral axis depth versus ductility and plastic rotation capacity on bending in lightweight-aggregate concrete beams. Materials 2019, 12, 3479. 34. Du, G.C.; Tao, X.K. Ultimate stress of unbonded tendons in partially prestressed concrete beams. PCI J. 1985, 30, 72–91. 35. JGJ/T 92-93. Technical Specification for Concrete Structures Prestressed with Unbonded Tendons; China Planning Press: Beijing, China, 1993. 36. JGJ 92-2016. Technical Specification for Concrete Structures Prestressed with Unbonded Tendons; China Architecture & Building Press: Beijing, China, 2016.

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