1

3 A comparative study of continuous beams prestressed with bonded FRP4 and steel tendons

5

6

7 Tiejiong Lou a,⇑, Sergio M.R. Lopes a, Adelino V. Lopes b8 a CEMUC, Department of Civil Engineering, University of Coimbra, Coimbra 3030-788, Portugal9 b Department of Civil Engineering, University of Coimbra, Coimbra 3030-788, Portugal

10

1 2a r t i c l e i n f o

13 Article history:14 Available online xxxx

15 Keywords:16 Prestressed concrete17 Continuous beams18 FRP tendons19 Finite element analysis20

2 1a b s t r a c t

22This paper presents a numerical investigation of the performance of continuous prestressed concrete23beams with bonded fiber reinforced polymer (FRP) and steel tendons. A finite element model has been24developed and is validated against the available test data. A numerical test is carried out on two-span25continuous bonded prestressed concrete beams. Three types of tendons are considered for a comparative26study: aramid FRP (AFRP), carbon FRP (CFRP) and prestressing steel. Various levels of xp (reinforcement27index of prestressing tendons) are used. The results indicate that, with increasing xp, the failure mode of28FRP prestressed concrete beams would transit from tendon rupture to concrete crushing while the crush-29ing failure always takes place in steel prestressed concrete beams. Moreover, FRP tendons mobilize sig-30nificantly different behavior regarding the crack pattern, deformation characteristics and neutral axis31depth compared to steel tendons. The moment redistribution at ultimate in AFRP prestressed concrete32beams is comparable to that in steel prestressed concrete beams while, at a low xp level, CFRP tendons33register obviously lower redistribution than steel tendons.34� 2015 Published by Elsevier Ltd.35

36

3738 1. Introduction

39 Fiber reinforced polymer (FRP) composites have been widely40 employed for reinforcing concrete structures instead of traditional41 steel reinforcement [1–3]. FRP materials offer attractive benefits,42 including high tensile strength, noncorrosive and nonmagnetic43 properties, favorable fatigue resistance and low weight [4]. In the44 field of prestressing systems, FRP tendons are a promising alterna-45 tive to traditional steel tendons which are susceptible to corrosive46 damage. One of the primary concerns for practical applications of47 FRP tendons is related to anchorages [5]. Over past years, many48 studies have been undertaken regarding the anchorage systems49 for FRP tendons [6–10]. There are three basic composite materials50 that may be used for prestressing tendons: glass FRP (GFRP), ara-51 mid FRP (AFRP) and carbon FRP (CFRP). GFRP composites are not52 recommended for bonded tendons because of the poor resistance53 to alkaline environment and also to creep under sustained loads54 [11]. AFRP and CFRP composites are both desired for composite55 tendons and have been widely used for prestressing applications.56 FRP tendons possess different material properties compared to57 steel tendons. FRP composites are brittle in nature, with linear58 elastic behavior up to rupture. AFRP tendons have usually a much

59lower modulus of elasticity than steel tendons, while the elastic60modulus of CFRP tendons covers a wide range from 80 to61500 MPa [12]. Therefore, the common knowledge established from62conventional steel prestressed concrete beams may not be applica-63ble to FRP prestressed concrete beams. Many efforts have been64made to understand the behavior of concrete beams prestressed65with FRP tendons. Pisani [13] described a numerical study on the66flexural behavior of simply supported bonded and unbonded pre-67stressed concrete beams with steel and FRP tendons. Park and Naa-68man [14] reported the test results regarding the shear behavior of69CFRP prestressed concrete simply supported beams. Toutanji and70Saafi [15] tested a series of four simply supported concrete beams71prestressed AFRP tendons to study the flexural performance of72these beams. The tests indicated that the use of combined bonded73and unbonded AFRP tendons or of additional nontensioned rebars74can improve notably the ductile behavior. Stoll et al. [16] presented75the results of a test program performed on two full-scale simply76supported high-strength concrete bridge girders prestressed with77CFRP tendons. Dolan and Swanson [17] conducted a theoretical78and experimental study on the flexural capacity of simply sup-79ported beams prestressed with vertically distributed FRP tendons.80Kim [18] examined the effect of prestress level on the behavior of81simply supported AFRP prestressed concrete beams and also82checked the applicability of several design codes and existing pre-83dictive equations.

http://dx.doi.org/10.1016/j.compstruct.2015.01.0090263-8223/� 2015 Published by Elsevier Ltd.

⇑ Corresponding author. Tel.: +351 239797253.E-mail address: loutiejiong@dec.uc.pt (T. Lou).

Q1

Q2

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84 Although extensive works have been carried out regarding the85 behavior of prestressed concrete beams with bonded FRP tendons,86 most of the past works dealt only with the simply support condi-87 tion. Few attempts have so far been made to investigate the behav-88 ior of continuous FRP prestressed beams. The authors [19–21] have89 recently conducted a set of theoretical studies to evaluate the90 response of continuous external FRP tendon systems, with particu-91 lar emphasis placed on the redistribution of moments. Because of92 the bond effects and nonexistence of second-order effects, internal93 bonded tendon systems would behave differently from external94 tendon systems.95 This paper describes a numerical study that is conducted to96 reveal the flexural behavior of two-span continuous concrete

97beams prestressed with internal bonded FRP and steel tendons. A98numerical model for geometric and material nonlinear analysis of99bonded prestressed concrete continuous beams has been devel-100oped. The proposed model is verified with the experimental results101available in the literature. A comparative study on the performance102of AFRP, CFRP and steel prestressed concrete continuous beams is103carried out by using the proposed model. Various amounts of pre-104stressing tendons are used for the numerical evaluation. Some105findings are concluded.

AFRP

CFRPSteel

2

1 ( 2)c

cm

kf kσ η η

η−=

+ −

1/

1{1 [ / ( )] }p p p R Rp p py

QE QE Kf

σ εε

⎡ ⎤−= +⎢ ⎥+⎢ ⎥⎣ ⎦

(a) (b)

(c) (d)

Strain Strain

Strain Strain

Stre

ss

Stre

ssSt

ress

Stre

ss

Fig. 1. Stress–strain curves of materials. (a) Concrete in compression; (b) concretein tension; (c) prestressing tendons; (d) nonprestressed steel.

1 2Centroidal axis

Original tendon segment Idealized tendon segment

x

1e2e

1 2( ) / 2e e+

l

y

Fig. 2. Beam element with idealized tendon segment.

P P

4876.8 mm 2621.3 mm 2621.3 mm 4876.8 mm

203.2 mm

374

mm

291.

1 m

m

7498.1 mm 7498.1 mm

15392.4 mm Center support section

406.

4 m

m Tendon

Bars

Fig. 3. Test beams by Lin [26].

0 50 100 150 200-250

-200

-150

-100

-50

0

50

100

150

200

Mom

ent (

kN m

)

Applied load (kN)

Experimental Computational

Center support

Load point

Beam A

0 50 100 150 200 250-300

-200

-100

0

100

200

Experimental Computational

Mom

ent (

kN m

)

Applied load (kN)

Load point

Center support

Beam B

Fig. 4. Comparison between experimental and computational results regarding theload-moment response for the test beams.

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106 2. Nonlinear model

107 2.1. Stress–strain curves of materials

108 A stress–strain equation for concrete in uniaxial compression is109 recommended in Eurocode 2 [22] for structural analysis of con-110 crete members. The stress–strain curve is shown in Fig. 1(a), where111 g ¼ ec=ec0; rc and ec are the concrete stress and strain, respec-112 tively; k is a coefficient depending on the concrete modulus of elas-113 ticity Ec, strain at peak stress ec0 and mean compressive strength114 fcm. The concrete is assumed to be crushed when its strain reaches115 the specified ultimate compressive strain eu.

116The concrete in tension is assumed to be linear elastic prior to117cracking and linear strain-softening after cracking, as shown in118Fig. 1(b). The concrete tensile strain at the end of strain-softening119is taken as 10 ecr , where ecr is the cracking strain.120The stress–strain equation for prestressing steel proposed by121Menegotto and Pinto [23] is used in this study. The stress–strain122curve is shown in Fig. 1(c), where rp and ep are the tendon stress123and strain, respectively; Ep is the tendon modulus of elasticity;124f py is the yield stress of prestressing steel; and K, Q and R are empir-125ical parameters which can be determined by experimental data.

P P6667 mm

4444 mm

5000 mm 5000 mm 5000 mm 5000 mm

10000 mm 10000 mm

s1A

As2

As1

Center support sectionMidspan section

As2

s1A 600

mm

550

mm

440

mm

pA

300 mm

600

mm

550

mm

440

mm

300 mm

AppA =variable

A =1000 mms1 22

s2A =500 mm

Fig. 5. Details of continuous prestressed concrete beam for numerical evaluation.

Table 1Material properties of FRP and steel tendons.

Tendons Tensile strength (MPa) Elastic modulus (GPa) Ultimatestrain (%)

AFRP 1500 68 2.2CFRP 1840 147 1.25Steel 1860 195 >3.5

Table 2Normalized ultimate tendon stress and concrete strains as well as failure mode of thebeams.

Tendons xp rp=f pu rp=f f ec1=eu ec2=eu Failure mode

AFRP 0.024 – 1.0 0.56 0.62 Rupture0.048 – 1.0 0.68 0.72 Rupture0.084 – 1.0 0.85 0.87 Rupture0.108 – 0.99 1.0 0.97 Crushing

Approx. rupture

CFRP 0.144 – 0.91 1.0 0.99 Crushing0.204 – 0.83 1.0 0.99 Crushing0.024 – 1.0 0.45 0.51 Rupture0.048 – 1.0 0.6 0.63 Rupture0.084 – 1.0 0.84 0.82 Rupture0.108 – 0.98 1.0 0.98 Crushing

Approx. rupture

Steel 0.144 – 0.89 1.0 0.98 Crushing0.204 – 0.77 1.0 1.0 Crushing0.024 0.97 – 1.0 0.89 Crushing0.048 0.96 – 1.0 0.89 Crushing0.084 0.95 – 1.0 0.92 Crushing0.108 0.93 – 1.0 0.97 Crushing0.144 0.89 – 1.0 0.98 Crushing0.204 0.78 – 1.0 1.0 Crushing

0.0 0.5 1.0 1.5 2.0-0.005

0.000

0.005

0.010

0.015

0.020

ωp=0.024 AFRP CFRP Steel

Con

cret

e st

rain

X/L

Top fiber Bottom fiber

0.0 0.5 1.0 1.5 2.0-0.005

0.000

0.005

0.010

0.015

ωp=0.084

Con

cret

e st

rain

X/L

AFRP CFRP Steel

Top fiber Bottom fiber

0.0 0.5 1.0 1.5 2.0-0.005

0.000

0.005

0.010

ωp=0.204

AFRP CFRP Steel

Con

cret

e st

rain

X/L

Top fiberBottom fiber

Fig. 6. Concrete strain distribution over the length for different tendon types andvarious xp levels.

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Original text:Inserted Textstress-strain

Original text:Inserted Textstress-strain

Original text:Inserted Textstress-strain

126 The prestressing steel used in the present numerical investigation127 is assumed to be Grade 270, 7-wire low relaxation strands and the128 values of K, Q and R are 1.0618, 0.01174 and 7.344, respectively.129 The prestressing FRP tendons are linear elastic up to rupture, as130 shown in Fig. 1(c).131 The nonprestressed steel is assumed to be linear elastic prior to132 yielding and perfectly plastic after yielding, as shown in Fig. 1(d).

133 2.2. Finite element model

134 A finite element model based on the Euler–Bernoulli beam the-135 ory has been developed. The model is an extension of a previously

136developed model for unbonded prestressed concrete continuous137beams [24], in which the contribution of unbonded tendons was138not included in the stiffness matrix but was made by transforming139the prestressing force into equivalent nodal loads. For analysis of140bonded prestressed concrete beams, however, the contribution of141bonded tendons to the stiffness matrix must be considered because142of the compatibility of strains between prestressing tendons and143the surrounding concrete.144Consider a two-node plane beam element to be described in the145local coordinate system (x, y), as shown in Fig. 2. The node points146are assumed to be at the centroid of the cross-sections. The pre-147stressing tendon segment in the beam element is idealized to be

(a) (b)

0 20 40 60 80 1000

50

100

150

200

250

Appl

ied

load

(kN

)

Midspan deflection (mm)

AFRP CFRP Steel

ωp=0.024

0

50

100

150

200

250

-40 -30 -20 -10 0 10 20 30 40

ωp=0.024 AFRP CFRP Steel

Curvature (10-6 rad/mm)

Appl

ied

load

(kN

)

MidspanCenter support

0

100

200

300

400

500

600

700

800

-20 -10 0 10 20

ωp=0.204 AFRP CFRP Steel

Curvature (10-6 rad/mm)

Appl

ied

load

(kN

)

Center support Midspan

0 20 40 60 80 1000

50

100

150

200

250

300

350

400

450

ωp=0.084 AFRP CFRP SteelAp

plie

d lo

ad (k

N)

Midspan deflection (mm)

0

50

100

150

200

250

300

350

400

450

-30 -20 -10 0 10 20 30

ωp=0.084 AFRP CFRP Steel

Curvature (10-6 rad/mm)

Appl

ied

load

(kN

)

MidspanCenter support

-10 0 10 20 30 40 50 60 700

100

200

300

400

500

600

700

800

ωp=0.204 AFRP CFRP SteelA

pplie

d lo

ad (k

N)

Midspan deflection (mm)

Fig. 7. Load-deformation response for different tendon types and various xp levels. (a) Load–deflection response; (b) load-curvature response.

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148 parallel with the centroidal axis, with eccentricity equal to the149 mean value of the eccentricities at the end nodes. Each node has150 three degrees of freedom: axial displacement u, transverse dis-151 placement v, and rotation h. Assuming that u and v are a linear152 function and a cubic polynomial of x, respectively, they can be153 expressed in terms of element nodal displacements as follows:154

u ¼ ð1� nÞu1 þ nu2 ð1aÞ156156157

v ¼ 1� 3n2 þ 2n3� �

v1 þ lnð1� nÞ2h1 þ 3n2 � 2n3� �

v2þ l �n2 þ n3� �

h2 ð1bÞ159159

160 where l = length of the beam element; and n ¼ x=l.161 The axial strain eO on any point of the beam element is given by:162

eO ¼@u@xþ 1

2@v@x

� �2ð2Þ

164164

165 The second term of the right side of the above equation represents166 the large displacement effects. Assuming that the shear deforma-167 tion of the element is negligible, which is a reasonable approxima-168 tion for slender (Euler–Bernoulli) beams commonly used for169 prestressing applications, the section curvature / is expressed by170

/ ¼ � @2v@x2

ð3Þ172172

173 Assuming that a plane section remains plane during bending and174 that perfect bond exists between the reinforcement and the sur-175 rounding concrete, the strain e at any fiber of the section is given176 by:177

e ¼ eO þ y/ ð4Þ179179

180 The element nodal displacement vector ue and element nodal181 force vector Pe may be written as182

ue ¼ fu1 v1 h1 u2 v2 h2 gT ð5Þ184184185

Pe ¼ fN1 V1 M1 N2 V2 M2 gT ð6Þ187187

188 According to the total Lagrangian description, the tangent stiffness189 equations for an element can be determined by applying the princi-190 ple of virtual work as follows:191

dPe ¼ Ket due ¼ ðKe1 þ K

e2 þ K

e3Þdue ð7Þ193193

194

Ke1 ¼Z

lBTl DtBldx ð8aÞ196196

197

Ke2 ¼Z

lBTl DtBndxþ

Zl

BTnDtBldxþZ

lBTnDtBndx ð8bÞ199199

200

Ke3 ¼Z

lNJT Jdx ð8cÞ

202202

203

Bl ¼� 1l 0 0 1l 0 00 6

l2� 12n

l24l �

6nl 0 � 6l2 þ

12nl2

2l �

6nl

" #ð9aÞ

205205

206

Bn ¼ ½1 0 �T ueT JT J ð9bÞ208208209

J ¼ 0 � 6nl þ6n2

l 1� 4nþ 3n2 0 6nl �

6n2

l �2nþ 3n2

h ið10Þ211211

212

Dt ¼d11 d12d21 d22

� �ð11Þ

214214

215

d11 ¼X

i

EtciAci þX

j

EtsjAsj þX

k

EtpkApk ð12aÞ217217

218

d12 ¼ d21 ¼X

i

EtciAciyci þX

j

EtsjAsjysj þX

k

EtpkApkypk ð12bÞ220220

221

d22 ¼X

i

EtciAciy2ci þX

j

EtsjAsjy2sj þX

k

EtpkApky2pk ð12cÞ223223

224

N ¼X

i

rciAci þX

j

rsjAsj þX

k

rpkApk ð13Þ226226

227where the summation symbol signifies that the cross section is228divided into layers to employ a layered approach; the subscripts229ci, sj and pk represent the ith concrete layer, jth nonprestressed steel230layer and kth tendon layer, respectively; Et is the tangent modulus231for materials; A corresponds to the area and r corresponds to the232stress.233After assembling the equilibrium equations for the structure in234the global coordinate system and imposing an appropriate bound-235ary condition, a load or displacement control incremental method236in combination with the Newton–Raphson iterative algorithm is237used for the solution of the nonlinear equilibrium equations. Dur-238ing the solution process, when any of the constituent materials239reaches its ultimate strain or strength capacity, failure is assumed240to take place and the analysis is therefore terminated. The pro-241posed model is capable of conducting the geometric and material242nonlinear analysis of continuous concrete beams prestressed with243bonded steel and FRP tendons over the entire loading range up to244the ultimate. Time-dependent effects such as tendon relaxation245and concrete creep are not covered in the present study, but the246modeling of these effects may be seen elsewhere [25].

2473. Verification of the proposed model

248In order to validate the proposed nonlinear model, two of the249continuous prestressed concrete beams tested by Lin [26] have250been analyzed. These beams were designated as Beams A and B,251and were tested under static loads up to failure. The structure252and section details of the specimens are shown in Fig. 3. The beams253were designed to be identical except for the content of nonpre-254stressed steel: Beam B contained two 14 mm tensile steel bars over255the midspan and center support regions while Beam A did not. The256beams had a rectangular section (203.2 � 406.4 mm), and were25715392.4 mm long with two equal spans to which two concentrated258loads were applied at 2621.3 mm from the center support. The ten-259don profile, which was designed to be concordant, was curved over260the center support region and straight over other regions. The ten-261don consisted of 32 parallel 5 mm high-strength steel wires. The

0.00 0.03 0.06 0.09 0.12 0.15 0.18 0.21 0.2440

50

60

70

80

90

100

AFRP CFRP Steel

Ulti

mat

e de

flect

ion

at m

idsp

an (m

m)

ωp

Fig. 8. Variation of ultimate deflection with xp level for different tendon types.

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Original text:Inserted TextNewton-Raphson

Original text:Inserted Textteminated.

Original text:Inserted Text(203.2×406.4

262 ultimate tensile strength, yield stress and elastic modulus of the263 prestressing tendon were 1765, 1572 MPa and 200 GPa, respec-264 tively. The yield strength and elastic modulus of nonprestressed265 steel were 314 MPa and 196 GPa, respectively. The cylinder com-266 pressive strengths of concrete at age of 28 days were 36.2 MPa267 for Beam A and 41.3 MPa for Beam B. The tendon was tensioned268 to have an initial prestress of 979 MPa when the beams were269 14 days old. About two weeks later the beams were loaded up to270 collapse.271 Fig. 4 shows a comparison between experimental and computa-272 tional results regarding the load versus moment responses at the273 center support and load point for the two test beams. The experi-

274mentally obtained moments were calculated according to the275experimental values of the reaction at the end support reported276in the literature. It can be seen in the graphs that proposed analysis277reproduces with remarkable accuracy the moment evolution in278both of the continuous prestressed concrete beams throughout279the loading history up to failure.

2804. Numerical investigation

281A bonded prestressed concrete rectangular beam continuous282over two equal spans to which two centre-point loads are symmet-283rically applied, as shown in Fig. 5, is used as a control beam for the

(a) (b)

0 50 100 150 200 2500

100

200

300

400

500

600

ωp=0.024

AFRP CFRP Steel

Neu

tral a

xis

dept

h (m

m)

Applied load (kN)

Midspan

Center support

50 100 150 200 250 300 350 400 4500

100

200

300

400

500

600

ωp=0.084

AFRP CFRP Steel

Neu

tral a

xis

dept

h (m

m)

Applied load (kN)

Center support

Midspan

200 300 400 500 600 700100

200

300

400

500

600

ωp=0.204

AFRP CFRP Steel

Midspan

Center support

Neu

tral a

xis

dept

h (m

m)

Applied load (kN)

Center support

Midspan

-40 -30 -20 -10 0 10 20 30 400

100

200

300

400

500

600

ωp=0.024 AFRP CFRP Steel

Neu

tral a

xis

dept

h (m

m)

Curvature (10-6 rad/mm)

MidspanCenter support

-30 -20 -10 0 10 20 300

100

200

300

400

500

600

Center support Midspan

ωp=0.084 AFRP CFRP Steel

Neu

tral a

xis

dept

h (m

m)

Curvature (10-6 rad/mm)

-20 -15 -10 -5 0 5 10 15 20100

200

300

400

500

600

Center support Midspan

ωp=0.204 AFRP CFRP Steel

Neu

tral a

xis

dept

h (m

m)

Curvature (10-6 rad/mm)

Fig. 9. Development of neutral axis depth for different tendon types and various xp levels. (a) Load versus neutral axis depth; (b) curvature versus neutral axis depth.

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284 numerical evaluation. The prestressing tendons are assumed to285 have an idealized parabolic profile with eccentricities at the end286 support, midspan and center support of 0, 140 and 140 mm,287 respectively. The beam contains nonprestressed tensile steel over288 the positive moment region (As1 = 1000 mm2) and negative289 moment region (As2 = 500 mm2). The yield strength and elastic290 modulus of nonprestressed steel are 450 MPa and 200 GPa, respec-291 tively. The concrete characteristic cylinder compressive strength fck292 (note: f ck ¼ f cm � 8) is taken as 60 MPa. Three types of prestressing293 tendons are used: AFRP, CFRP and steel tendons. The material294 properties (tensile strength, elastic modulus and ultimate strain)295 for each type of tendons are summarized in Table 1. The yield296 stress of prestressing steel is taken as 90% of the tensile strength.297 The initial prestress fp0 is taken as 950 MPa. Various tendon areas298 Ap are used so as to generate various levels of reinforcement index299 of prestressing tendons xp, which is defined as300

xp ¼Apf p0bdpf ck

ð14Þ302302

303 where b is the section width and dp is the effective depth of pre-304 stressing tendons at the center support or midspan. In this study,305 the xp level ranges from a minimum of 0.024 to a maximum of306 0.204. In the finite element idealization, the concrete beam is307 divided into 36 two-node beam elements (18 elements for each308 span) with length of 555.56 mm each. Each element is subdivided309 into 10 concrete layers, 1 tendon layer and 1 nonprestressed steel310 layer.

311 4.1. Failure mode and crack pattern

312 A summary of the normalized tendon stress (rp=f pu or rp=f f )313 and concrete strains (ec1=eu and ec2=eu) at failure as well as the fail-314 ure mode of the beams is given in Table 2, where rp is the ultimate315 stress in tendons; f pu and f f are the tensile strengths of steel and316 FRP tendons, respectively; ec1 and ec2 are the concrete strains at317 ultimate in extreme compressive fiber of the midspan and center318 support sections, respectively. A normalized ultimate tendon stress319 of 1.0 indicates a rupture failure while a normalized ultimate con-320 crete strain of 1.0 signifies a crushing failure. According to the pres-321 ent numerical tests, all the steel prestressed concrete beams have322 failed due to crushing of concrete at the critical section. At the low-323 est xp level of 0.024, failure of the steel prestressed beam takes324 place at midspan while at failure the center support section is still325 far below its ultimate strain capacity. As xp increases, the exploi-326 tation of the center support is enhanced. At the highest xp level327 of 0.204, the concrete is crushed both at midspan and at the center328 support. For FRP prestressed concrete beams, on the other hand,329 failure may take place either due to concrete crushing or tendon330 rupture, depending on the xp level. According to the present anal-331 ysis, the tendon rupture and concrete crushing take place simulta-332 neously when xp reaches around 0.108. A rupture failure appears333 for xp lower than 0.108 while a crushing failure occurs for xp334 higher than 0.108. At a low xp level (e.g., 0.024), the critical sec-335 tions of a FRP prestressed concrete beam are far below their ulti-336 mate capacities when the tendons are ruptured. At a high xp337 level (e.g., 0.204), on the other hand, the FRP tendons are far below338 their ultimate tensile strength when the concrete is crushed.339 At the failure loads, the concrete strain distribution over the340 length for different tendon types and various xp levels is displayed341 in Fig. 6, where X/L is the ratio of the distance from the end support342 to the span. From the graphs in this figure the crack pattern may be343 deduced. At xp of 0.024, the steel prestressed concrete beam devel-344 ops very large tensile strains in the critical positive and negative345 moment regions, while the tensile strains over other regions are346 small. This observation indicates that there are a few main cracks347 with large widths in the critical regions while the cracks in other

348regions are insignificant. In other words, there appears crack con-349centration in the beam containing a low amount of steel reinforce-350ment. As xp increases, the crack pattern of the steel prestressed351concrete beam appears to be improved, that is, the maximum crack352widths are reduced and the crack zones are extended. The crack353pattern for FRP prestressed concrete beams is related to the failure354mode. At a low xp level (e.g., 0.024), failure is caused by rupture of355FRP tendons and, therefore, crack concentration is not as important356as the case of the steel prestressed concrete beam. At a high xp357level (e.g., 0.204), crushing failure happens. In this case, the crack358pattern of the CFRP prestressed concrete beam is very similar to359that of the steel prestressed concrete beam, while the AFRP pre-360stressed concrete beam develops obviously bigger crack widths361at the critical regions than the CFRP or steel prestressed concrete362beam.

3634.2. Deformation characteristics

364Prior to the application of external loads, there are initial defor-365mations as a result of combined prestressing and self-weight. At366the lowest xp level of 0.024, the self-weigh effect prevails over367the prestressing effect, thereby resulting in a downward deflection368of the beam as well as a sagging curvature at midspan and a hog-369ging curvature at the center support. For xp greater than 0.048, the370prestressing effect overrides the self-weight effect, causing a cam-371ber of the beam as well as a hogging curvature at midspan and a372sagging curvature at the center support. The entire load–deflection373and load-curvature responses for different tendon types and vari-374ous xp levels are shown in Fig. 7(a) and (b), respectively. Before375cracking, the behavior is primarily controlled by concrete and,376therefore, the responses for different tendon types are almost iden-377tical. After cracking, the contribution of prestressing reinforcement378is increasingly important. Due to lower modulus of elasticity, AFRP379tendons mobilize smaller bending stiffness of a beam than CFRP or380steel tendons. At a high xp level (in the case of crushing failure),381the entire load-deformation response characteristics of the CFRP382prestressed beam are similar to those of the steel prestressed383beam. However, at a low xp level, the load-deformation behavior384of FRP prestressed beams may be undesirable because of prema-385ture failure caused by tendon rupture.386Fig. 8 shows the variation of ultimate deflection with the xp387level for different tendon types. At the lowest xp level of 0.024,388the ultimate deflections of AFRP and CFRP prestressed beams are389respectively 31.3% and 46.9% lower than that of the steel pre-390stressed beam. As xp increases, the ultimate deflection for steel391tendons gradually decreases while the ultimate deflections for

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Fig. 10. Variation of neutral axis depth with xp level for different tendon types.

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392 FRP tendons increase quickly. When xp reaches 0.108 (transition393 from a rupture failure to a crushing failure), the deflections for394 FRP tendons turn to be obviously higher than that for steel ten-395 dons. With the continuing increase of xp, the deflections for both396 steel and FRP tendons decrease, but the decrease in deflection for397 steel tendons tends to be slower than those for FRP tendons. At398 the highest xp level of 0.204, the discrepancy between the deflec-399 tions for steel and FRP tendons appears to be insignificant. At a low400 level of xp, the deflection for CFRP tendons is smaller than that for401 AFRP tendons, but the difference gradually diminishes with the402 increase of xp, and appears to be not noticeable as xp increases403 to 0.144 or above.

4044.3. Neutral axis depth

405The initial neutral axis depth for a prestressed concrete beam406under prestressing and self-weight may be positive or negative,407depending on the xp level. At xp of 0.024, the initial neutral axis408depths are about 900 mm at midspan and 680 mm at the center409support, that is, the neutral axis is initially located below the bot-410tom fiber of the midspan section and above the top fiber of the cen-411ter support section. At xp of 0.204, the initial neutral axis depths412are about �50 mm at midspan and 90 mm at the center support,413that is, the neutral axis is initially located outside the midspan sec-414tion (above the top fiber) while inside the center support section.

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Fig. 11. Development of support reaction and bending moment for different tendon types and various xp levels. (a) Load-reaction response; (b) load-moment response.

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415 The neutral axis moves rapidly as the external loads are applied416 and gradually increased.417 The evolution of neutral axis depth with the applied load and418 curvature, after the neutral axis reaches the extreme tension fiber419 of the section, for different tendon types and various xp levels is420 shown in Fig. 9(a) and (b), respectively. It is seen in Fig. 9(a) that421 at a low xp level, the development of neutral axis depth exhibits422 a significantly nonlinear manner and the neutral axes at the center423 support and midspan move at almost the same rate. As xp424 increases, the nonlinear manner is obviously reduced and the425 movement of neutral axis at the center support may become faster426 than that at midspan. After the evolution of cracks stabilizes, the427 shift of neutral axis in the AFRP prestressed concrete beam appears428 to be faster than that in the CFRP or steel prestressed concrete429 beam. From Fig. 9(b) it is seen that the neutral axis moves rapidly430 at first but the movement tends to slow down as the hogging or431 sagging curvature increases. After stabilizing of the crack develop-432 ment, at a given curvature, AFRP tendons mobilizes lower neutral433 axis depth than CFRP or steel tendons, particularly obvious at a434 high xp level.435 Fig. 10 shows the variation of neutral axis depth at ultimate436 with the xp level for different tendon types. It is seen that the neu-437 tral axis depth increases as xp increases. The rate of increase in438 neutral axis depth for steel tendons tends to be faster than that439 for FPP tendons. At a given xp level, the midspan section develops440 higher neutral axis depth than the center support section. AFRP441 tendons mobilize lower neutral axis depth at the midspan or center442 support section than CFRP tendons. At a low xp level, the neutral443 axis depth for steel tendons is close to that for AFRP tendons, while444 at a high xp level the neutral axis depths for steel and CFRP ten-445 dons are comparable.

446 4.4. Development of support reaction and bending moment

447 The development of reactions at the end and center supports for448 different tendon types and various xp levels is illustrated in449 Fig. 11(a), while the evolution of bending moments at the midspan450 and center support is shown in Fig. 11(b). The elastic values, calcu-451 lated based on the linear-elastic theory, are also displayed in the452 graphs. The secondary reactions and moments due to prestressing453 are included in the values shown in these graphs. The magnitude454 and direction of secondary reactions and moments are dependent455 on the layout of tendons. For the beams analyzed in this study,456 the tendon line is below its linearly transformed concordant line.457 As a result, the secondary reaction at the end support is positive458 but negative at the center support, causing positive secondary459 moments in the beams.460 It can be seen in Fig. 11 that at initial loading, the actual reac-461 tion and moment develop linearly with the applied load, indicating462 there is no redistribution of moments in this elastic stage. After463 cracking, the actual values appear to deviate from the elastic ones464 due to redistribution of moments. Because cracking appears firstly465 at the center support, upon cracking the moments tend to be redis-466 tributed from the center support zone towards the midspan zone.467 As a consequence, there is a diminution of the rate of increase in468 reaction and moment at the center support and, correspondingly,469 a growth of the rate of increase in end support reaction and mid-470 span moment. In the inelastic ranges, the deviation of the actual471 reaction and moment from the elastic values is notable at a low

472 xp level but tends to diminish as xp increases, as can be observed473 in Fig. 11.

474 4.5. Degree of moment redistribution

475 Fig. 12 shows the development of the degree of moment redis-476 tribution for different tendon types and various xp levels. The

477degree of moment redistribution b is defined by: b ¼ 1� ðM=MeÞ,478where M and Me are the actual and elastic moments corresponding479to a load level, respectively. The degree of moment redistribution is480equal to zero until cracking. After cracking, the redistribution of481moments takes place. The redistribution is positive at the center482support while negative at midspan. At low and medium xp levels,483the evolution of redistribution after cracking is obviously affected484by several typical phases, namely, the stabilization of crack devel-

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Fig. 12. Development of the degree of moment redistribution for different tendontypes and various xp levels.

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485 opment, yielding of nonprestressed steel at the center support and486 yielding of nonprestressed steel at midspan. At a high xp level, on487 the other hand, the influence of these phases on the redistribution488 development appears to be not so noticeable. For steel tendons, the489 maximum redistribution occurs at the ultimate limit state irre-490 spective of the xp level. For FRP tendons, on the other hand, the491 maximum redistribution may not take place at failure at a low

492 xp level.493 Fig. 13 shows the variation of bu (degree of moment redistribu-494 tion at ultimate) with the xp level for different tendon types. It is495 seen that the bu value decreases as xp increases. At a given xp496 level, the bu values for AFRP and steel tendons are almost identical.497 The redistribution for CFRP tendons is obviously lower than that498 for steel tendons at a low xp level, but the difference between499 the bu values for CFRP and steel tendons becomes negligible when

500 xp is greater than 0.108.

501 5. Summary and conclusions

502 A numerical investigation has been carried out to reveal the503 flexural behavior of continuous concrete beams prestressed with504 bonded FRP and steel tendons. A finite element model based on505 the layered Euler–Bernoulli beam theory for full-range nonlinear506 analysis of continuous FRP and steel prestressed concrete beams507 has been developed. The model is validated against the experimen-508 tal results. A comparative study is conducted on two-span contin-509 uous prestressed concrete beams with bonded AFRP, CFRP and510 steel tendons using the proposed model. A wide range of xp is511 used. Typical aspects of behavior of the beams are examined,512 including the failure mode and crack pattern, deformation charac-513 teristics, neutral axis depth and moment redistribution. Based on514 the results obtained from the analysis, the follow conclusions515 may be drawn:

516 � Failure of steel prestressed concrete beams is always attributed517 to crushing of concrete. On the other hand, FRP prestressed con-518 crete beams may fail due to crushing of concrete or rupture of519 FRP tendons, depending on the xp level: when xp reaches520 around 0.108, concrete crushing and tendon rupture take place521 simultaneously; a rupture failure happens for xp lower than522 0.108 while a crushing failure occurs for xp greater than 0.108.523 � At a low reinforcement index, crack concentration appears in524 the steel prestressed concrete beam while this phenomenon is525 not so important in FRP prestressed concrete beams. At a high526 reinforcement index, AFRP tendons mobilize larger crack width527 than CFRP and steel tendons.

528� At a low xp level, the ultimate deflection for FRP tendons is sig-529nificantly lower than that for steel tendons. As xp increases, the530ultimate deflection for steel tendons consistently decreases531while the deflection for FRP tendons quickly increases up to532xp of 0.108 and then turns to decrease. The deflection difference533between FRP and steel tendons appears to be insignificant at a534high xp level.535� Due to a lower modulus of elasticity, AFRP tendons register536lower neutral axis depth at ultimate than CFRP tendons. At a537low xp level, the neutral axis depth mobilized by steel tendons538is close to that mobilized by AFRP tendons while, at a high xp539level, it is close to that mobilized by CFRP tendons.540� The maximum redistribution of moments in steel prestressed541concrete beams takes place at ultimate, but this may not true542for FRP prestressed concrete beams. At a given xp level, the543redistribution at ultimate for AFRP tendons is almost identical544to that for steel tendons. At a low xp level, CFRP tendons mobi-545lize obviously lower moment redistribution than steel tendons546but the redistribution difference between CFRP and steel ten-547dons is negligible at a high xp level.548

549

550Acknowledgments

551This research is sponsored by FEDER funds through the program552COMPETE (Programa Operacional Factores de Competitividade)553and by national funds through FCT (Fundação para a Ciência e a554Tecnologia) under the project PEst-C/EME/UI0285/2013. The work555presented in this paper has also been supported by FCT under556Grant No. SFRH/BPD/66453/2009.

557References

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0.00 0.03 0.06 0.09 0.12 0.15 0.18 0.21 0.240.00

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-0.06

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