nrica,

C. Numerical analysis

icale inl noer,stun tingter

in theter undpects ohe strecase

, the s

sented below. These equations are dependent on the type of thelocal bondslip law. Local laws dened by a linear or bilinear func-tion are known to be the simplest approximations and have beenproposed to represent the debonding phenomenon, but nonlinearbondslip approximations are more representative of the FRP-to-concrete interface behaviour [18] as shown in Fig. 1 becausethey can modulate the concrete nonlinearities involved in the deb-onding phenomenon more accurately. Other bondslip laws can be

along the FRP-to-concrete interface.The advantage of the proposed model is that other formulae be-

sides that of Popovics can be used to simulate different bondsliplaws. The authors have shown elsewhere [19] that several bondslip laws can successfully predict the performance of linear andnonlinear FRP-to-concrete systems. For example, Ferracuti [4,5]has proposed a model that solves a rst order differential equationbut it only takes into account Popovics formula. Martinelli et al.[23] proposed a model based on the FDM also and on a bi-linearbondslip law. Their model solves a 4th order differential equationand considers the interface normal and shear stresses and the

Corresponding author. Tel.: +351 962309126.

Composites: Part B 50 (2013) 210223

Contents lists available at

te

evE-mail address: hb@fct.unl.pt (H.C. Biscaia).FRP and concrete requires further knowledge. The stress transferand the performance of the interface between materials are crucialissues and must be well understood if an efcient design of thestrengthening of reinforced structural elements is to be provided.The failure of the beams at the anchorage region of the FRP rein-forcement illustrates the need for additional data.

According to Neubauer and Rostsy [1] such failure can be stud-ied by double shear tests, as shown in Fig. 1, and simple or doubleshear tests are widely described in the literature [217]. The gov-erning equations that model the debonding phenomenon are pre-

too much time may be being spent on such 3D analysis and othermethods could be excellent alternatives.

This paper reports a simple but fairly accurate solution for thenonlinear debonding process using the nite difference method(FDM) in association with a local nonlinear bondslip law similarto that adopted by Popovics for the stressstrain relation of con-crete [20]. When the nonlinear behaviour of the interface is intro-duced, the numerical procedures provide more accurateinformation about the debonding phenomenon so that suitablechoices can be made with respect to the size of the steps usedConcrete

1. Introduction

The emergence of new materialsled to new problems that require betstudy. This is the case with some asmers (FRPs) that have been used for tment of concrete structures. In thebeams with external strips of FRP1359-8368/$ - see front matter Crown Copyright 2http://dx.doi.org/10.1016/j.compositesb.2013.02.013building industry haserstanding and furtherf bre reinforced poly-ngthening or reinforce-of reinforced concretetress transfer between

found in [19] and the authors studied some of them in depth, e.g.Popovics formula [20], Ueda and Dais formula [2] and modiedPopovics formula. Some authors, e.g. [4,5,21], have been employ-ing Popovics formula in recent years to model FRP-to-concreteinterfaces, and the constant nP in the formula has been adjustedfor several FRP-to-concrete interfaces.

Some efforts have been made recently to modulate the FRP-to-concrete debonding with 3D nite elements, e.g. [22]. However,B. DebondingB. Interface the inuence of the local bondslip law on the debonding process is discussed.

Crown Copyright 2013 Published by Elsevier Ltd. All rights reserved.Nonlinear numerical analysis of the deboof FRP-to-concrete interfaces

Hugo C. Biscaia , Carlos Chastre, Manuel A.G. SilvaUNIC, Faculdade de Cincias e Tecnologia, Universidade Nova de Lisboa, 2829-516 Capa

a r t i c l e i n f o

Article history:Received 21 September 2012Received in revised form 4 February 2013Accepted 11 February 2013Available online 26 February 2013

Keywords:A. Laminates

a b s t r a c t

The paper analyses numerpolymers (FRP)-to-concretone end. Any realistic locathe present study. Howevdue to its use in differentan expression depending oin pure Mode II, the debondin the FRP plate along the in

Composi

journal homepage: www.els013 Published by Elsevier Ltd. Allding failure process

Portugal

solutions for the process leading to debonding failure of bre reinforcedterfaces in shear tests with the FRP plate subjected to a tensile load atnlinear bondslip law can be used in the numerical analysis proposed inonly a Popovics type expression is employed in the numerical processdies found in the literature. Effective bond length (Leff) is discussed andhe Popovics constant (nP) is proposed to calculate it. Assuming a fractureprocess is analysed in detail and distributions of bond stresses and strainsface are presented. The loaddisplacement behaviour is also presented and

SciVerse ScienceDirect

s: Part B

ier .com/locate /composi tesbrights reserved.

crete, respectively.

rela

H.C. Biscaia et al. / Composites: Part B 50 (2013) 210223 211treatment, environmental conditions, etc.) are centralized in thebondslip law adopted, i.e., they change the shape of the bondsliplaw.

Based on nonlinear bondslip laws, particularly Popovics for-mula, the current numerical model also enables the full denitionof the FRP-to-concrete debonding process, i.e., up to the completedebonding of the FRP plate from the concrete surface. The com-plete identication of the debonding process helped to dene theeffective bond length of the FRP-to-concrete interfaces, and whenthe bonded length is shorter than the effective bond length it alsoenabled a parameter to be dened that affects the maximum loadtransmitted to the FRP plate. This method and the analysis of thenumerical results are presented and discussed in the sections thatfollow.

2. Governing equations

The equilibrium of the horizontal forces acting on an innitesi-mal length dx of the FRP-to-concrete interface, Fig. 2, is given bybending moment of the FRP plate. However, the effectiveness ofthese models are unknown if another bondslip law is used. An-other important advantage of the numerical model herein pre-sented is that any aspects that might affect the performance ofthe FRP-to-concrete interfaces (e.g. concrete strength, surface

Fig. 1. Scheme of the FRP anchorage region(e.g. [19]):

drf xdx

sstf

0 1

where rf is the longitudinal stress acting on the FRP plate; s is thebond stress associated with the slip, s, between the FRP plate andconcrete; and tf is the FRP plate thickness.

Fig. 2. Equilibrium of the forces inThe interfacial slip (s) between the FRP plate and concrete is de-ned as the relative displacement between them:

s uf uc 3where uf and uc are the displacements in the FRP plate and concrete,respectively.

Assuming that the concrete displacements are very small whencompared with the displacements in the FRP plate and introducingEqs. (2) and (3) into Eq. (1), a differential equation is obtained:

d2s

dx2 ssEf tf 0 4

3. Local nonlinear bondslip law

The solution for Eq. (4) is highly dependent on the local bondslip law s(s), and exponential and other nonlinear bondslip lawscan accurately t data on the interface behaviour, for the FRP-to-Linear stressstrain relationships are assumed for FRP:

rf x Ef dufdx 2

where Ef and Ec are the Young modulus of the FRP plate and con-

ted to the double shear test. Based on [1].concrete interface. However, the corresponding analytical solu-tions for Eq. (4) are unavailable except for linear and bilinear laws.

In general, the ruptures of the FRP plates from the concrete sur-faces are cohesive in concrete, i.e., by detaching small amounts ofconcrete, the bondslip laws that characterise the FRP-to-concreteinterfaces are similar (in their shape) to the stressstrain relationof concrete in tension. Therefore, we opted for a realistic nonlinear

the FRP-to-concrete interface.

model for s(s) at the cost of using numerical treatment for theboundary value problem. The bondslip law adopted is similar tothat adopted by Popovics [20] for the concrete stressstrainrelation:

ssmax

ssmax

nPnP 1 ssmax

nP 5where smax is the maximum bond stress at the interface; smax is theslip for the maximum bond stress; and nP is a constant which,according to Ferracuti [4], must be higher than 2. This numbercan be adjusted according to the experimental data available, andNakaba [21] proposed nP = 3.0. In the shear tests performed by Fer-racuti and co-workers [4,5,24] to evaluate the bond behaviour be-tween CFRP and concrete, the authors determined experimentallythat nP varies between 2.86 and 4.44 depending on the shear testconditions used and the consequent stress eld.

: ; > >

212 H.C. Biscaia et al. / Composites:The constant nP helps with the tting of the post-peak behav-iour of the bondslip law: when it increases the post-peak curvedecays very quickly. Increasing nP corresponds to a reduction ofthe energy per unit area of the interface required for the rupture.This fracture energy of Mode II (GF) can be calculated by the areaunder the bondslip curve. Fig. 3 shows the inuence of the con-stant on the shape of the bondslip law and the hashed area is de-ned by the limits found in literature [4,21,2527], where2.39 6 nP 6 4.44 is for the materials under consideration.

4. Numerical integration

The FDM was used to determine the slip distribution along thebonded length. The method is based on the substitution of Eq. (5)by nite differences and then solving the resulting system of non-linear equations.

The procedure was set up as follows. Introducing (5) into (4) weobtain:

d2s

dx2 Ls smax

Ef tf s

smax nPnP 1 ssmax

np Ls0B@

1CA 6

where L(s) is a linear function of swhich allows the use of the xed-point iteration method explained below. For simplicity, let:

Ls c sGs smaxEf tf

ssmax

nPnP1 ssmax np

(7

withFig. 3. Bondslip curves based on Popovics formula for several nP.c smaxEf tf smax

npnp 1 8

and the equation to be integrated can be written as:

d2s

dx2 Ls Gs Ls 9

valid in the interval (0, Lb) where Lb is the bonded length. The dis-cretization step (h) was:

h Lbn

10

The distance between consecutive points xi, i = 0,1,2, . . . ,n, xi =i h and n was selected as the largest integer above which no gainin accuracy could be detected. During the solution procedure, aswitch of the boundary conditions is implemented to obtain thefull solution of the FRP-to-concrete debonding. First, the numericalintegration adopted to control the debonding process is the load(or strain) and in this case the boundary conditions are the sim-plest case for a Neumann type problem:

dsdx

ef x0j F

Af Ef 11:a

dsdx

ef xLbj 0 11:b

where Af and ef are the section and the strain of the FRP plate,respectively and F is the load transmitted to the FRP plate. The nitedifferences for Eqs. (11.a) and (11.b) are:

dsdx

0 FAf Ef

si1 si12h

for i 0 : FAf Ef

s1 s12h

() s1 s1 2 FAf Ef h 12:a

dsdx

Lb 0 si1 si12h for i n : 0

sn1 sn12h

() sn1 sn1 12:b

When the maximum load is nally reached, the control param-eter is changed and the displacement or slip at x = Lb is the newcontrol parameter. From this stage on, the boundary condition ex-pressed in (11.a) is modied to:

s xLbj sLb 13and at the point x = Lb

sLb sn 14The problem was then formulated based on the known para-

bolic formulae:

dsdx

xi si1 si12h ;d2s

dx2xi si1 2si si1

h2for i

0;1; . . . ;n 15leading, during the load increment phase, to a system of nonlinearequations of the type:

1

h2

k2 1 0 0 0 01 k 1 0 0 0... ..

. . .. . .

. . .. ..

. ...

0 0 0 1 k 1

26666666664

37777777775

s0

s1

..

.

sn1

8>>>>>>>>>>>>>>>>>

9>>>>>>>>>=>>>>>>>>>

Gs0Ls02 ah

Gs1Ls1...

Gsn1Lsn1

8>>>>>>>>>>>>>>>>>>

9>>>>>>>>>>=>>>>>>>>>

Part B 50 (2013) 2102230 0 0 0 1 k2 sn GsnLsn2: ;

16

6 7 >>>> >>>> Gs2Ls2>>>> >>>>

These systems can be rewritten as:

ticat(seintlowterco

Soth

zeen

len De

tes:A Sk1 FSk 20

and were solved by an iterative algorithm. The starting values foreach increasing magnitude of the external force leading to a weregiven by a unit S vector, a procedure that was possible given the lin-earised problem being solved. After the maximum load is reachedand the control parameter has also been changed, the starting val-ues for each increasing magnitude of slip at x = Lb are implemented,leading to b, given by an initial S vector equal to that obtained fromthe previous iteration.

The criterion assumed for convergence was the usual relativeerror of the modulus of S calculated between two consecutive iter-ations, k and k + 1 according to:

kSk1k kSkkkSkk

100 < 0:1% 21

where ||Sk|| and ||Sk+1|| are the vectors corresponding to the slipsalong the interface obtained from interactions k and k + 1,respectively.

5. Debonding process

The process that leads to the complete debonding of the FRP-to-concrete interface is dependent on the initially bonded length (Lb)[1,1618,2830]. There are, essentially, two different situationsthat lead to two different interface performances: (1) the bondedlength is the same as or longer than the effective bond length (Lb -P Leff) or (2) the bonded length is shorter than the effective bondlength (Lb < Leff). The resulting debonding failure processes arebriey described next.

5.1. Debonding process (1): with LbP Leff

Initially, before any load is applied the bonded area is unde-formed. When the loads applied to the FRP plate are increasedthe interface is subdivided according to the strains and displace-ments it undergoes: (i) quasi-undeformed qU; (ii) nonlinear elas-tic nlE; (iii) softened S; and (iv) debonded D. Depending onseveral parameters, such as the load applied at the end of the1

h2

0 1 k 1 0 0 00 0 1 k 0 0 0... ..

. ... . .

. . .. . .

. ... ..

.

0 0 0 0 0 1 k0 0 0 0 0 0 2

666666664

777777775

s1s2

..

.

sn2sn1

>>>>>>>>>>>>:

>>>>=>>>>>>>>>;

Gs3Ls3...

Gsn1Lsn1 bh2GsnLsn kbh2

>>>>>>>>>>>>>:

>>>>=>>>>>>>>>>;18

where

b sLb and k 2 c h2 19where

a FAf Ef and k 2 c h

2 17

and during the next stage (after maximum load is reached), the sys-tem of nonlinear equations is dened as:

1 k 1 0 0 0 02 3 s08> 9> Gs1Ls18>> 9>>

H.C. Biscaia et al. / ComposiFRP plate, shape of the bondslip law or FRP mechanical properties,during the course of loading the interface may exhibit the follow-ing states:begins to decrease.

Fig. 4 illustrates the debonding failure process 1 for a bondedlength the same as or higher than the effective bond length, show-ing the correspondence of all interface stages with the respectiveloaddisplacement behaviour.

Debonding failure for LbP Leff follows these stages:

(i) Stage I (nlE-qU) The maximum bond stress of the shear testis not yet reached, the load is smaller than Fmax, and themaximum longitudinal stress in the FRP plate (rmax) is notreached either. In Fig. 4 this state is represented by I.

(ii) Stage II (nlE-qU) As the load increases a higher bondedlength is mobilised, but a quasi-undeformed bonded lengthis still present. In Fig. 4, the maximum bond stress is reachedat point A and from there on smaller bond stresses will bepresent at A. The maximum longitudinal stress is not yetreached.

(iii) Stage III (S-nlE-qU) The maximum bond stress travels tothe right and part of the interface is in the post-peak soften-ing side of the local bondslip law. The longitudinal stressdistribution now has two different concavities. The transi-tion point of the concavities can be found differentiatingEq. (1) in order to x.

tf d2rf xdx2

dssds

dsdx

0 22

When the maximum bond stress occurs on the bondedlength, (point II in Fig 4), the derivative of the bond stressin order to the slip is zero:

dssds

0 23

leading, according to Eq. (22), to the following equation:

d2rf xdx2

0 24

that allows the inection point of the longitudinal stresses inthe FRP plate to be found (see bold point at stages IIIV, Fig. 4).At this stage, the maximum longitudinal stress has not yetbeen reached, as shown in the loadslip curve of Fig. 4.

(iv) Stage IV (S-nlE-qU) The maximum load capacity has beenreached (see loadslip curve) and, therefore, so has the max-imum longitudinal stress. The interface debonding begins atgth.bonding-Softening (D-S) when the bond stress diagram Debonding-Softening-Nonlinear Elastic (D-S-nlE), which onlyoccurs if the bonded length is greater than the effective bondro at point A of the bonded length, i.e., maximum fractureergy is attained.of the applied load. This state will end when the bond stress isnvergence.ftened-Nonlinear Elastic-quasiUndeformed (S-nlE-qU) wheree maximum bond stress begins to migrate to the opposite sideains or stresses (0) and the other part has a nonlinear elas-response. This state will end when the maximum bond stresspoint A, i.e., at the rst point of the bonded length, is reachede Fig. 4). In the numerical analysis, the quasi-undeformederface region can be dened as the region with bond stresseser than 0.1% of the maximum bond stress which was the cri-ion established previously in Eq. (21) in the numerical Nonlinear Elastic-quasiUndeformed (nlE-qU) where a part of theinterface remains quasi-undeformed, i.e., with insignicantstr

Part B 50 (2013) 210223 213this stage. The bonded length mobilised (excluding thequasi-undeformed region) is the effective bond length,

ress

214 tes:Fig. 4. Interfacial bond stress and FRP longitudinal st(v)

(vi)H.C. Biscaia et al. / Composimeaning that after this stage no more load increments canbe applied to the FRP plate. The bond stress diagram travelsto the right, keeping the same effective bond length, Leff.Stage V (D-S-nlE) The bond stress diagram is totally mobi-lised up to the unloaded end of the FRP plate and a debond-ing region develops at the end of the loaded plate. In theloadslip diagram, Fig. 4, this stage is represented by the lineparallel to the slip axis (line segments IVV). According toequilibrium.

F Z Lb0

sx bf dx 25

the distance between the initial points IV and V increases withthe bonded length of the interface. At this stage, it can be alsoseen that the FRP longitudinal stress maintains its maximumvalue across the debonding region, and starts to decreasewhen bond stress begins.Stage VI (D-S) The length of the bond stress diagram startsto decrease and from then on the maximum load alsodecreases. In terms of the slip at the loaded plate end andneglecting the concrete deformations, differentiating Eq.(3) and introducing into Eq. (2).dsdx

rf xEf

26

and the slip along the interface at this stage is

sx Z LD0

rmaxEf

dxZ LSLD

rf xEf

dx sx Lb 27

where s(x = Lb) is the slip at point x = Lb; the upper limit of therst integral (LD) is the debonding length of the interface andthe limits of the second integral (LS) correspond to the bondedlength of the interface that remains bonded, i.e.,distributions at different loading stages for L P L .5.2. D

Whlengthare alis sligcan n

Nodifslippounforin

A NshobeFRin

A Sres(nlenrigPart B 50 (2013) 210223Lb LS LD 28The slip at point A of the FRP plate (see Fig. 4) will de-

crease as the load decreases. When the load applied to theFRP plate decreases, according to Eq. (27) the slip at pointA will be equal to the slip at x = Lb, which is the ultimate slipof the bondslip law (represented by point IV in Fig. 4).

ebonding process (2): Lb < Leff

en the bonded length is shorter than the effective bond, Lb < Leff, all four states identied in the previous processso observed. However, in each state the interface behaviourhtly different from that in previous process. Those statesow be classied as:

nlinear Elastic-quasiUndeformed (nlE-qU). There may be twoferent paths, depending on the bonded length and the bondlaw. Assuming that the maximum bond stress is reached at

int A of the FRP plate and an interface region remains quasi-deformed, the state is still nonlinear Elastic-quasiUnde-med (nlE-qU) (state IIa in Fig. 5), while the interface is alla nonlinear Elastic (nlE) state (state IIb in Fig. 5).onlinear Elastic (nlE) state may occur when the interface isrt and the right FRP plate end starts to develop bond stressesfore the maximum bond stress is reached at point A of theP plate. After that, a Softening-nonlinear Elastic (S-nlE) occursthe interface.oftening-nonlinear Elastic-quasiUndeformed (S-nlE-qU) stateults from a previous nonlinear Elastic-quasiUndeformedE-qU) state, according to Stage IIa shown in Fig. 5, i.e., a soft-ing state appears in the interface before bond stresses at theht end of the plate.

b eff

tes:H.C. Biscaia et al. / Composi Softening-nonlinear Elastic (S-nlE) state: if the previous statewas Nonlinear Elastic (Stage IIb in Fig. 5) then the maximumbond stress migrates from point A to the right end of the FRPplate. On other hand, if the previous state was Softening-nonlin-ear Elastic-quasiUndeformed (Stage III in Fig. 5), the quasi-undeformed interface region on the right side of the interfacedisappears. The Softening-nonlinear Elastic (S-nlE) state van-ishes when the bond stress at point A is zero. The interface deb-onding starts from this point forward.

Debonding-Softening (D-S) state.

Fig. 5 illustrates the debonding failure at the interface for abonded length shorter than Leff, showing the correspondence ofall interface stages with the respective loaddisplacement behav-iour. Note here that at the right end of the FRP plate in the nonlin-ear elastic state, the derivative of the bond stress in order to x iszero. This effect can be explained with Eq. (22) and the boundarycondition (11.b).

6. Numerical examples

The two debonding failure processes identied above are ana-lysed below. Two numerical examples are presented and dis-

Fig. 5. Interfacial bond stress and FRP longitudinal stressPart B 50 (2013) 210223 215cussed. In these numerical examples certain referenceparameters which were used in several GFRP-to-concrete bondtests [9,25] are assumed unless otherwise stated. The thicknessof the FRP plate is tf = 2.54 mm (1.27 mm per layer), the width isbf = 80 mm, Youngs modulus is Ef = 20.39 GPa and the mean con-crete tensile strength is fctm = 2.69 MPa. The local bondslip lawwas determined from the experimental tests. The bond stress dis-tribution along the interface was calculated from expression:

s De Ef tfDL

29

where De is the difference in strain between two consecutive straingauges attached at a distance of DL from one another.

The local bondslip law obtained from the experimental datawas then adjusted to t Popovics formula (5) where smax = 4.24MPa, smax = 0.101 mm, nP = 3.01 [25]. With these values the frac-ture energy of Mode II (GF) can be calculated according to (e.g.[25]):

GF Z sult0

ssds 30

and then obtaining GF = 1.206 N/mm by solving the integral in (27)numerically.

distributions at different loading stages for Lb < Leff.

6.1. Debonding process 1: with LbP Leff

Once the nonlinear bondslip law is dened, Eq. (5) can besolved by the FDM as explained above. The bonded length assumedwas longer than the effective bond length calculated according to[28]:

Leff Ef tfc2 fctm

s31

where fctm is the average tensile strength of concrete and c2 = 0.8 inthe case of GFRP composites [25].

The value found was Leff = 155 mm and the assumed bondedlength Lb = 250 mm.

The numerical process begins, assuming a low strain in FRP. Thebonded length was divided into 50 equal intervals. The solution ofEq. (20) gives the slip distribution along the interface, leading tothe bond stresses according to the local bondslip law. The longi-tudinal stress distribution is then calculated according to Eq.(26). Unbalanced forces (errors from the numerical method)are also calculated according to equation:

ef Ef Af Z Lb0

sx bf dx R 32

where ef is the strain initially assumed for the FRP plate; Af is thetransversal section of the FRP plate; and R gives the unbalancedforces or residual of the numerical approximation.

The method is controlled by the initial boundary condition at

method are zero. The numerical procedure revealed that the max-imum residual was approximately 1.0 kN.

The results obtained with the FDM are shown in Fig. 6 wherethe stresses at the interface are shown for the four different stagesthat represent the four initial states previously discussed and out-lined in Fig. 4.

Tljsten [26] used nonlinear fracture mechanics to develop aformula for the maximum load transmittable to FRP plate:

Fmax bf 2GF Ef tf

q33

where bf is the width of the FRP plate. The authors proved [19] thatEq. (33) can be used regardless of the shape (linear or nonlinear) ofthe bondslip law. In this example, comparison with the maximumtransmittable load obtained experimentally in [9,25] showed verygood agreement, proving the consistency of Eq. (33). Experimen-tally, 25.4 kN was obtained, whilst Eq. (33) gives 28.3 kN and thepresent numerical procedure gave 27.7 kN.

Fig. 7 shows a comparison between the numerical model andtest results obtained by Biscaia [25]. From the point of view ofloadslip behaviour, the numerical model shows good accuracyand the snap-back branch observed is in accordance with, e.g.[17,29,30]. The failures observed in the GFRP-to-concrete interfacewere very brittle and the descending segment of the loadslipcurve was not recorded. The bond and longitudinal stresses alongthe GFRP-to-concrete interface for maximum load showed fair de-gree of accuracy.

6.2. Debonding process 2: with Lb < Leff

216 H.C. Biscaia et al. / Composites: Part B 50 (2013) 210223x = 0 of the FRP plate and nishes when the FDM provides a bondstress distribution, s(x), equal to zero. At that moment, the load ap-plied to the FRP plate is zero due to equilibrium conditions (see Eq.(25)), and according to Eq. (32) the residuals of the numericalFig. 6. Behaviour of the interface atThe debonding failure process 2 (bonded length is shorter thanthe effective bond length) starts in the same way as the debondingfailure process 1 (bonded length is the same as or longer than thedifferent stages with LbP Leff.

tes:H.C. Biscaia et al. / Composieffective bond length). For low loads, an undeformed region of theinterface remains undeformed and another is in elastic stage. Incontrast to debonding process 1, the distribution of the stressesreaches the right end of the FRP plate rapidly because of the shortbonded length. Therefore, as the maximum bond stress migratesupward, the maximum load transmittable to the FRP plate isreached faster and for loads lower than that of debonding process1. As a consequence, the strains on the FRP at maximum load arelower than those obtained during the debonding process 1. Themaximum slips are also smaller in debonding process 2. Fig. 8shows the stresses within the interface for three stages: stages I,IIb and IV (see Fig. 5). Note here that stage I is dened in Fig. 8by point I of the bondslip law and represents an elastic state (E)of the interface, which is only different from that of stage IIb shownin Fig. 4 by the bond stress level achieved at the rst point A of theinterface.

The numerical calculus ends, as explained above, when thebond stress distribution, s(x), is zero and the load applied tothe FRP plate is zero as well. In this debonding process example,the maximum residual reached was approximately 1.0 kN. Fromthe point of view of the loadslip curve, a softening branch is ob-served which is in accordance with the work of, e.g., Cornetti andCarpinteri [29] or Caggiano et al. [30].

In these examples with Lb < Leff, the maximum load obtainedfrom Eq. (33) must be multiplied by a parameter which, accordingto Neubauer and Rostsy [1], in the case of CFRP plates, is given byequation:

bL j 2 j 34where j is the ratio between the bonded length and the effectivebond length:

Fig. 7. Comparison between the numerical and test results: bPart B 50 (2013) 210223 217j LbLeff

35

Teng et al. [31], proposed an alternative formula for bL denedaccording to:

bL sinp2 j

36

Fig. 9 shows a comparison between the numerical results andthe formulae proposed by [1,31]. When the formulae are comparedwith the numerical method the results of this one are higher. Theseresults will be discussed in more detail in Section 7.3 below.

7. Interface behaviour and choice of Popovics constant

As discussed before, the representation of the local bondsliplaw is a factor of great importance. Some authors [4,21,2527]have chosen Popovics formula in order to approximate the localbondslip law and others [4,27] have shown that Popovics con-stant is not greater than 4.5 for CFRP-to-concrete interfaces. Ferra-cuti, for instance, demonstrated that nP must be higher than 2.0 [4]and Nakaba recommends nP = 3.0 [21]. Table 1 shows the values formaximum bond stress (smax), slip at maximum bond stress (smax)and Popovics constant (nP) determined from the tests carried outby different authors [4,21,2527]. All the authors worked onCFRP-to-concrete interfaces, except for Biscaia et al. [28] whoworked on GFRP-to-concrete interfaces.

The next sections show the inuence of Popovics constant onthe interface behaviour and its consequences for the slip, stresses,maximum loads and effective bond length.

ehaviour of the interface at different stages with Lb > Leff.

218 H.C. Biscaia et al. / Composites: Part B 50 (2013) 2102237.1. Slip behaviour

The slip distribution along the interface was dened by thesolution of the system of equations in (13). Several values of nPwere tested in order to nd the best value of nP to represent theexperimental slip distribution found in [25] at failure. Some results

Fig. 8. Behaviour of the interface a

Fig. 9. Parameter bL used in the examples with Lb < Leff.

Table 1Parameters of Popovics formula determined by different authors.

Parameter Ferracuti [4] Nakaba [21]

smax (MPa) 5.506.63 5.609.10smax (mm) 0.0260.044 0.0520.087nP 2.864.44 2.503.70t different stages with Lb < Leff.are shown in Fig. 10 for nP = 2.40, 3.01 and 3.14; when nP increasesslightly from 3.01 to 3.14, the slip distribution along the interfaceis overestimated, revealing the high sensitivity of slip distributionto nP. The curves in Fig. 10 show that for lower nP values the slipdistribution underestimates the experimental results while forhigher values the numerical results overestimate the experimentalresults. The best t selected for nP was 3.01.

The increment of the number of intervals used in the FDM hasthe advantage of approximating the numerical solution to the ex-act solution. This was also analysed by implementing 50 equalintervals with 4.8 mm each and the solution was compared withthe rst numerical solution where only 10 intervals were used.The results of the slip distribution for nP = 3.01 using 10 and 50intervals were not signicantly affected, meaning that the numer-ical method presented in this paper does not have much inuenceon the slip distribution.

7.2. Maximum load

As shown in Section 3, for high values of the Popovics constant(nP), the initial ascending branch is almost linear until it reachesthe maximum bond stress, while the softening branch decreasesabruptly. As a consequence, the fracture energy decreases accord-ing to Eq. (30), which is why the load capacity in Fig. 11 is highestfor nP = 2.40. This is consistent with Tljsten [26], Eq. (33), i.e., with

Biscaia [25] Tljsten [26] Chajes [27]

4.24 5.29 6.640.101 0.043 0.0393.01 2.64 2.39

tes:H.C. Biscaia et al. / Composiincreasing fracture energy the maximum load transmittable to theFRP plate increases as well. Table 2 presents the errors between Eq.(33) and the numerical method with a maximum of 5.06%.

Fig. 11 also shows that the maximum load is reached in a lessstiff manner, for lower values of nP. This can again be explainedby the local bondslip law, i.e., for a local bondslip law with a verysignicant post-peak regime (nP very low) the bond stress distribu-tion takes place over a greater bonded length and the effectivebond length is also greater than for higher nP values. This issue isdiscussed in more detail in the next section.

7.3. Effective bond length

The effective bond length based on the nonlinear bondslip lawwas studied for several values of bond length and nP, assuming all

Fig. 10. Inuence of nP on the slip d

Fig. 11. Inuence of nP in the loadslip behaviour of the interface.Part B 50 (2013) 210223 219the others parameters remained constant. The results are shown inFig. 12. Maximum load decreases for higher nP values as discussed

istribution along the interface.

Table 2Comparison between the maximum loads transmittable to the FRP plate using thenumerical method and using Eq. (33).

nP GF (N/mm) Fmax (kN) Fmax (kN) Error (%)Numerical modela Tljsten [26]

2.40 2.015 34.70 36.55 5.063.01 1.206 27.66 28.28 2.183.14 1.116 26.68 27.20 2.025.00 0.619 20.08 20.26 0.90

10.00 0.397 16.21 16.22 0.06

a With 50 equal intervals of 4.8 mm each.

Fig. 12. Inuence of nP on the maximum load for different bonded lengths of theinterface.

earlier, while effective bond length appears to increase for lower nPvalues. The effective bond length was dened based on thecriterion:

FmaxjLb2 FmaxjLb1FmaxjLb1

100 < 0:1% 37

where Fmax|Lb1 and Fmax|Lb2 are the maximum loads transmitted tothe FRP plate with a bonded length of Lb1 and Lb2, respectively.Eq. (37) represents the increment of the load transmittable to theFRP plate of the curve maximum load versus bonded length be-tween two consecutive bonded lengths Lb1 and Lb2. Note thatLb1 < Lb2.

For nP = 3.01 the effective bond length is 225 mm. This value ishigher than that obtained in previous bond tests [25], where theeffective bond length was determined experimentally and set toapproximately 160 mm. This can be explained by the fact that noslip limits were introduced in the bondslip law and so a bondstress distribution was promoted along the bonded length thatwas greater than that reported in [25]. Therefore, the maximumload transmitted to the FRP plate according to Eq. (25) is higherthan that reported in [25], as mentioned above.

The concept of effective bond length can be also dened by thestrain or bond stress distributions at maximum load capacity. Forbond stress, the effective bond length is taken to be the length overwhich the bond stresses are different from zero, whilst for straindistribution the effective bond length can be taken as the distancebetween the maximum strain (in the debonding state) and zero

as concrete strength are used by several authors or codes[3,12,35,36]. In this work we assumed that the inuence of con-crete strength or other parameters has been included in the deni-tion of the shape of the local bond slip law and not in the denitionof an expression for the effective bond length of the FRP-to-con-crete interface. Therefore, Popovics constant (nP) and the FRP stiff-ness were the parameters assumed to have a direct inuence onthe expression for calculating the effective bond length. The resultsare based on the study of the inuence of six different thicknessesof the FRP plate, and the inuence of Popovics constant was alsotaken into account. Results are shown in Fig. 14. The numerical re-sults made it possible to make the following proposal to calculatethe effective bond length:

Leff CnP Ef tf 0:3 38

where units are in N and mm and CnP is a factor dependent on thePopovics constant, (nP) dened according to the black trend lineshown in Fig. 14 where it was assumed that Popovics constant isalways greater than 2.

CnP 10 nP 20:2 with nP > 2 39Results obtained from the numerical method are compared

with some expressions reported in literature where the authorsconsider that effective bond length is only dependent on the stiff-ness of the FRP [2,3234] (see Table 3). However, none of themconsider the Popovics constant because they do not approximatethe bondslip law to the Popovics formula dened in (5). The re-

220 H.C. Biscaia et al. / Composites: Part B 50 (2013) 210223strains. Fig. 13 shows the effective bond length according to thestrain or bond stress distributions when nP is equal to 2.40, 3.01,5.00 and 10.00.

The literature [2,3,12,3236] reports some proposals for the cal-culation of effective bond length. All of them have a commonparameter, which is the FRP stiffness, Ef tf. Other parameters suchFig. 13. Inuence of nP on thsults show some divergences for nP = 3.01 but are consistent withthe experiments carried out by Biscaia [25]. However, in terms ofsimulation, the shape of the local bondslip law has great inuenceon the effective bond length of the FRP-to-concrete interface. Onother hand, by limiting the ultimate slip (sult) in the nonlinearbondslip law, the effective bond length can be dened withe effective bond length.

ess

tes:Fig. 14. Inuence of FRP stiffnH.C. Biscaia et al. / Composigreater accuracy, for instance, by means of the length of the bondstress distribution at maximum load capacity. As shown above, theexact denition of the effective bond length from the bond stressor strain distribution was only obtained for higher nP values forwhich the post-peak curves of the bondslip laws decay rapidlyto zero. However, in FRP-to-concrete interfaces the Popovics con-stant is not greater than 5 [4,21,2527] and therefore limitationof the ultimate slip was not considered.

7.4. Strength of the FRP-to-concrete interface

With the effective bond length determined according to Eq.(38), the ratio between the maximum load reached in an interfacewith Lb < Leff and the maximum load transmittable to the FRP withLbP Leff can be dened. This ratio is established by the bL parame-ter that depends on the ratio between the bonded length and theeffective bond length (j), as shown in Fig. 15. The numerical re-sults are higher than the proposals of Neubauer and Rostsy [1]and Teng et al. [31]. The curve that was the best t for the numer-

Table 3Comparison between the effective bond length using the model and using the equations f

nP Numerical modela Ueda and Dai [2] (mm) Maeda

2.40 450 162 473.01 2253.14 2355.00 200

10.00 120

a With 50 equal intervals with 4.8 mm each.

Fig. 15. Inuence of Popovics conson the effective bond length.

Part B 50 (2013) 210223 221ical method was determined and is shown in Fig. 15, according tothe formula:

bL arctan2p jarctan2p 40

Eq. (33) can be rewritten, leading to:

Fmax bf

2GF Ef tf

pif jP 1

bL bf 2GF Ef tf

pif j < 1

(41

where bL is calculated according to Eq. (40).Fig. 15 also shows that Popovics constant (nP) has very little

inuence on the bL parameter. Eq. (40) assumes that bL only de-pends on the ratio between the bonded length and the effectivebond length (j). The clear differences from the Neubauer and Ros-tsy [1] and Teng et al. [31] proposals (see Fig. 15) may be due tothe fact that these authors limited the ultimate slip whereas nolimits were assumed for slip in the numerical method presentedin this paper.

ound in the literature [2,3234].

et al. [32] (mm) Miller and Nanni [33] (mm) JCI [34] (mm)

151 61

tant (nP) on the bL parameter.

tes:8. Conclusions

Analysis of the FRP-to-concrete interface using the FDM is agood alternative solution for studying the behaviour of the bondbetween FRP and concrete. The method allows the accurate calcu-lation of some important parameters of the FRP-to-concrete inter-face, such as the maximum load transmittable to the FRP oreffective bond length.

The slip distribution is also reproduced well, even for largesteps. However, the step used in the numerical method must besmall in order to approximate the bond and longitudinal stressesdistribution along the interface to the exact solution. A more accu-rate solution of maximum load can thus be expected since theloads are calculated according to Eq. (25), i.e., based on the bondstress distribution. In previous work [37], the maximum load wasdetermined by the numerical method with a very good approxima-tion, using a step of 4.8 mm with a relative error of 4.84% whencompared to the experimental data.

The bond and longitudinal stresses distribution along the FRP-to-concrete interface were adequately reproduced. Therefore, thenumerical method allowed the simulation of various scenariosfor different states of the interface. The debonding processes (withLbP Leff and Lb < Leff) could be estimated satisfactorily.

Using a nonlinear bondslip law based on the Popovics formula,the value of nP was found to have a great inuence on the simula-tion of the interface performance. The values recommended for nPmust be set close to 3.0. For higher values of nP, the results tend toshow greater clarity in the denition of the effective bond length.However, this scenario is very unlikely to occur in reality and add-ing a criterion in the numerical method, by limiting the ultimateslip, for instance, will allow a clear denition of the effective bondlength.

As mentioned above, the type of the local bondslip laws has aninuence on the effective bond length. The concept of the effectivebond length has been discussed in depth and an expression to cal-culate it that depends on the Popovics constant (nP) has been pre-sented. Equations reported in the literature show a consistentdependency on the FRP stiffness and some include the concretestrength as well. Thus, as shown above, in future work the effectivebond length should probably include coefcients that can intro-duce the inuence of the local bondslip laws.

The strength of the FRP-to-concrete interface can be determinedusing an alternative bL parameter that is less than one and greaterthan zero, calculated from Lb/Leff. This parameter provided a rea-sonable prediction of the maximum load transmittable to the FRPplate when the bonded length was smaller than the effective bondlength. Assuming, in the numerical method, the Popovics formulafor the local bondslip law, the parameter bL had greater accuracythan that found in literature [1,31]. This parameter showed also avery low sensitivity to Popovics constant (nP).

Acknowledgments

The authors are grateful to the Fundao para a Cincia e Tecn-ologia for partial nancing of the work under Project PTDC/ECM/100538/2008. The authors would also like to thank Professor NadirArada from the Mathematics Department of FCT/UNL for hisimportant help with the nonlinear numerical model approach.

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H.C. Biscaia et al. / Composites: Part B 50 (2013) 210223 223

Nonlinear numerical analysis of the debonding failure process of FRP-to-concrete interfaces1 Introduction2 Governing equations3 Local nonlinear bondslip law4 Numerical integration5 Debonding process5.1 Debonding process (1): with LbLeff5.2 Debonding process (2): Lb