a fracture-based model for frp debonding in strengthened beams

$: A fracture-based model for FRP debonding in strengthened beams$
Engineering Fracture Mechanics 76 (2009) 1897–1909

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Engineering Fracture Mechanics

journal homepage: www.elsevier .com/locate /engfracmech

A fracture-based model for FRP debonding in strengthened beams

Oguz Gunes a, Oral Buyukozturk b,*, Erdem Karaca c

a Atilim University, Department of Civil Engineering, Kizilcasar Mahallesi, Incek Golbasi, 06836 Ankara, Turkeyb Massachusetts Institute of Technology, 33 Massachusetts Ave., Room 1-280, Cambridge, MA 02139, USAc Swiss Reinsurance America Corporation, 175 King St., Armonk, NY 10583, USA

a r t i c l e i n f o

Article history:Received 12 December 2008Received in revised form 16 April 2009Accepted 27 April 2009Available online 5 May 2009

Keywords:FRPStrengtheningBeamsDebondingFracture

0013-7944/$ - see front matter � 2009 Elsevier Ltddoi:10.1016/j.engfracmech.2009.04.011

* Corresponding author. Tel.: +1 617 2537186; faE-mail addresses: [email protected] (O. Gun

a b s t r a c t

This paper presents an experimental and analytical research study aimed at understandingand modeling of debonding failures in fiber reinforced polymer (FRP) strengthened rein-forced concrete (RC) beams. The experimental program investigated debonding failuremodes and mechanisms in beams strengthened in shear and/or flexure and tested undermonotonic loading. A newly developed fracture mechanics based model considers the glo-bal energy balance of the system and predicts the FRP debonding failure load by character-izing the dominant mechanisms of energy dissipation during debonding. Validation of themodel is performed using experimental data from several independent research studiesand a design procedure is outlined.

� 2009 Elsevier Ltd. All rights reserved.

1. Introduction

FRP composites are becoming a material of choice in an increasing number of rehabilitation and retrofitting projectsaround the world. Depending on the design objectives, these materials can be used to improve one or more of the structuralmember characteristics such as the load capacity, ductility, and durability. A multi-national effort is underway to developproper codes and guidelines to set the standards for material selection, design, installation, inspection, maintenance, and re-pair of FRP applications. Design of structural strengthening applications using externally bonded FRP composites is usuallybased on conventional design approaches with improvements to account for the presence and characteristics of the FRPmaterial. Nonconventional design issues that are specific to the type of application require special considerations for theirproper inclusion in the design process. One such design issue is the debonding problems in externally bonded FRP strength-ening applications, the incorporation of which in the design has been a research challenge since the initial developmentstages of the method [1–3]. Due to the typical premature and brittle nature of debonding failures, inadequately designedstrengthening applications may not only become ineffective, but may also reduce the level of safety of the member bydecreasing its ductility. Design procedures that properly consider debonding problems are needed to ensure the safetyand reliability of flexural members strengthened using FRP composites. This paper presents an experimental and analyticalresearch study aimed at the understanding of debonding failures in FRP strengthened beams, and proposes a fracture-basedmodel for the prediction of such failures as a basis for design of these systems.

2. Failure modes of FRP strengthened beams

Failure of FRP strengthened beams may take place through several mechanisms depending on the beam and strengthen-ing parameters. In the recent publications of the ACI 440 Committee on Fiber Reinforced Polymer Reinforcement, these

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1898 O. Gunes et al. / Engineering Fracture Mechanics 76 (2009) 1897–1909

failure modes are classified as: (1) concrete crushing before reinforcing steel yielding, (2) steel yielding followed by FRP rup-ture, (3) steel yielding followed by concrete crushing, (4) cover delamination, (5) FRP debonding [4,5]. Oehlers [6] furtherclassified the debonding failure modes according to the type of crack causing debonding. In addition to these, shear failureoccurs if the shear capacity of the beam cannot accommodate the increase in the flexural capacity. An investigation of each ofthese failure modes is required in the design process to ensure that the strengthened beam will perform satisfactorily.

2.1. Debonding failure mechanisms

The term debonding failure is often associated with a significant decrease in member capacity due to initiation and prop-agation of debonding. Debonding initiation in beams strengthened with FRP composites generally take place in regions ofhigh stress concentration at the FRP–concrete interface. These regions include the ends of the FRP reinforcement, and thosearound the shear and flexural cracks. Fig. 1 shows examples of debonding failures from laboratory tests. The cover debondingmechanism shown in Fig. 1a is usually associated with high interfacial stresses, low concrete strength, and/or with extensivecracking in the shear span. If the concrete strength and the shear capacity of the beam are sufficiently high, potential deb-onding failure is most likely to take place through FRP debonding, as shown in Fig. 1b and c. Depending on the beam param-eters and the strengthening configuration, such failures may initiate at the areas of high stress concentration at laminateends and propagate towards the center of the beam, or may initiate at flexure-shear cracks and propagate towards the endsof the beam. Depending on the material properties, debonding may occur within the FRP laminate, at the FRP–concrete inter-face, or a few millimeters within the concrete.

A noteworthy issue regarding the failure behavior of FRP strengthened beams is the interaction between shear and deb-onding failures, which may have a causal relationship and sequential occurrence. It is often the case that the debonding fail-ures and debonding + shear failures are not properly differentiated and reported. This is partly justified considering that themember is considered as failed in both cases. However, a fundamentally important difference between debonding and shearfailures is the ductility behavior. Debonding failures significantly reduce the beam flexural capacity, however, provided thatthe beam has adequate shear capacity, it can still display the ductile failure behavior of a regular reinforced concrete beam.This is not the case for shear failures where total beam failure takes place in a brittle fashion. Thus, it is important to makesure that the beam has a shear capacity that is sufficiently higher than its flexural capacity and debonding failure load.

3. Previous research on debonding problems

Characterization and modeling of debonding in structural members strengthened with externally bonded reinforcementshas long been a popular area of interdisciplinary research due to critical importance of debonding failures in bonded joints.In the last decade, there has been a concentration of research efforts in this area with respect to FRP strengthened flexuralmembers, and considerable progress has been achieved in understanding the causes and mechanisms of debonding failuresthrough numerous experimental, analytical, and numerical investigations [7,8]. Modeling research in this area can be clas-sified in general terms by their approach to the problem as strength or fracture mechanics approaches. In addition to these, anumber of researchers have proposed relatively simple semi-empirical and empirical models that avoid the complexities ofstress and fracture analyses and can be relatively easily implemented in design calculations [7].

Strength approach involves prediction of debonding failures through calculation of the interfacial or bond stress distribu-tion in FRP strengthened members based on elastic material properties. Calculated stresses are compared with those corre-sponding to the strength of the materials to predict the debonding failure load and mechanism. The fact that debonding isessentially a crack propagation promoted by local stress intensities has raised interest among some researchers to take afracture mechanics approach to the problem and develop predictive models that utilize elastic and fracture material prop-erties [7]. Several recent studies have conducted detailed investigations of opening mode [9–11], shear mode [9,10,12–17]and mixed mode [18–21] fracture processes during debonding. Despite the demonstrated success of various fracture modelsfor specific bond test configurations, there is a need for models that can satisfactorily predict the debonding failure loads forthe general case of FRP strengthened beams in which multiple mechanisms of debonding processes are simultaneously at

Fig. 1. Debonding failure modes.

O. Gunes et al. / Engineering Fracture Mechanics 76 (2009) 1897–1909 1899

work due to multiple cracks at unknown locations along the beam. Gunes [22] and Achintha and Burgoyne [23] used theglobal energy balance in a strengthened beam, with fundamental differences in characterization of energy componentsand debonding fracture, to predict the debonding failure loads based on a fracture mechanics approach. The general objec-tive of empirical models is to provide a simple methodology to predict debonding failures without going into complex stressor fracture analyses. Several such models were proposed for FRP strengthened beams based on certain parameters that influ-ence their debonding behavior [7]. The reader is referred to [6–8,12,16,20,24] for a comprehensive review of debonding re-search and modeling.

Guidelines by the ACI Committee 440 enforce a limit on the strain level developed in the FRP reinforcement to preventdebonding failures [4,5]. This limit is given by the following expression:

Table 1Propert

Materia

Concret#3 andD4 defoCFRP plEpoxy a

efe ¼ ecuh� c

c

� �� ebi 6 jmefu ð1Þ

where efu and efe are the ultimate strain and the maximum allowed effective strain in the FRP reinforcement, respectively, ecu

is the ultimate strain of concrete, h is the beam height, c is the neutral axis depth, ebi is the concrete substrate strain at thetime of the FRP installation, and the limiting strain coefficient jm is given by:

jm ¼1

60efu1� nEf tf

360;000

� �6 0:90 for nEf tf 6 180;000

160efu

90;000nEf tf

� �6 0:90 for nEf tf > 180; 000

8><>: ðSIÞ ð2Þ

where n is the number of FRP reinforcement layers, tf is the thickness of each layer, and Ef is the elastic modulus of the FRPreinforcement. From Eq. (2), it is apparent that the limiting strain in the FRP reinforcement is defined by the geometric andmaterial properties of the FRP reinforcement only.

4. Experimental study

The experimental study presented herein is part of a comprehensive experimental program carried out to investigate themonotonic and cyclic load performance of precracked reinforced concrete beams strengthened in flexure and/or shear usingFRP composite plates and sheets [22]. The focus of the study was characterization and prevention of debonding failures asaffected by the shear strengthening and anchorage conditions. In this paper, primary test results from this experimental pro-gram are presented as a basis for the model presented in Section 5.

Laboratory size reinforced concrete beams were strengthened using carbon FRP composite plates in shear and/or flexurewith and without anchoring of the flexural FRP reinforcement, and were loaded in four-point bending until failure. Propertiesof the materials used in the experimental program are given in Table 1. All beams were precracked prior to strengthening.The geometry and reinforcement details of the control specimen (CM1) are shown in Fig. 2a and the strengthening config-urations of the tested beams are shown in Fig. 2b. All specimens shown in Fig. 2b were strengthened in flexure using1270 mm (50 in) long, 38.1 mm (1.5 in) wide, and 1.2 mm (0.047 in) thick unidirectional FRP plates. For shear strengthening,40-mm wide straight (beams S3PS1M and S3PS2M) and L-shaped (beams S4PS1M and S4PS2M) unidirectional FRP plateswere used, the latter of which also served as anchorage for the flexural FRP reinforcement. In order to compare the influenceof external shear strengthening vs. higher internal shear capacity on the debonding behavior, the shear capacity of a beamwas increased through use of larger internal shear reinforcement (see beam S2PF7M in Fig. 2b). Using section analysis, thecalculated flexural load capacity was 118.6 kN for the control beam (CM1) and 158.6 kN for all strengthened beams, with allexpected to fail through concrete crushing. Calculated shear capacities were 202 kN for the control beam (D4 shear rein-forcement), 339 kN for S2PF7M (#3 rebar shear reinforcement), and 300 kN in the sections of beams with FRP shear rein-forcement [22].

The load–deflection curves obtained from the tests are shown in Fig. 3a and the corresponding load vs. mid-span FRPstrain curves are shown in Fig. 3b. Table 2 summarizes the experimental results and indicates the observed failure modefor each specimen. All beams shown in Fig. 2b failed through FRP debonding except for beam S1PF1M which failed throughcover debonding followed by shear failure. Comparing the load–deflection curves for beam S1PF1M and S2PF7M, the influ-ence of the shear capacity of a beam on its debonding failure behavior is immediately apparent. Both beams were strength-ened in the same configuration and essentially both failed through debonding, however, the failure load of S2PF7M, which

ies of materials used in the experimental program.

l Compressive strength (MPa) Yield strength (MPa) Tensile strength (MPa) Elastic modulus (MPa) Ult. tensile strain (%)

e 41.4 – – 25,000 –#5 rebars – 440 – 200,000 –rmed bars – 620 – 200,000 –ate – – 2800.0 165,000 1.69dhesive – – 24.8 4500 1.00

75450 450 450

75

P/2 P/2

1500

75

150

30

120

30

2#3

2#5

D4 @ 75

CM1

(a) control specimen

S2PF7MP/2

#3 @ 75

S1PF1M

D4@75

P/2

100 P/240S3PS2M

100100S4PS2M

P/2

S3PS1M100 P/2

40

100100S4PS1M

P/2

40D4@75

D4@75

D4@75

D4@75

(b) beams strengthened in shear and/or flexure

Fig. 2. Geometry and strengthening configurations of beam test specimens (in mm).

0 5 10 15 200

20

40

60

80

100

120

140

160

180

200

Mid−span deflection (mm)

Load

(kN

)

CM1S1PF1M

S2PF7MS3PS1M

S3PS2MS4PS1M

S4PS2M

0 0.2 0.4 0.6 0.8 10

20

40

60

80

100

120

140

160

180

200

FRP strain at mid−span (%)

Load

(kN

) S1PF1M

S2PF7M

S3PS1MS3PS2M

S4PS1M

S4PS2M

ACI 440.2R

strain limit

Manufacturer’s

strain limit

(a) load-deflection curves (b) load vs. mid-span FRP strain curves

Fig. 3. Experimental results.

Table 2Summary of experimental results.

Beam Py (kN) dy (cm) efm,y (%) Pu (kN) du (cm) efm,u (%) Failure mode

CM1 110.0 5.2 – 117.4 17.9 – –S1PF1M 123.0 5.8 0.31 131.9 7.7 0.40 Cover debonding + shear failureS2PF7M 127.5 5.5 0.32 148.3 9.3 0.59 FRP debondingS3PS1M 124.0 5.4 0.31 143.1 9.2 0.55 FRP debondingS3PS2M 129.1 5.6 0.36 145.6 9.0 0.61 FRP debondingS4PS1M 128.2 5.4 0.33 153.8 11.5 0.68 FRP debondingS4PS2M 127.9 4.9 0.31 168.2 14.7 0.87 FRP debonding



had sufficiently high shear capacity, was approximately 12% higher than that of beam S1PF1M. The beams strengthened inshear with side bonded plates along the half and full shear span, S3PS1M and S3PS2M, respectively, displayed essentially thesame performance as S2PF7M. This suggests that the shear capacity of a strengthened beam is especially critical in the plate-end region, where the flexure-shear cracks initiated at plate ends propagate towards the beam center. The influence of shearstrengthening combined with anchorage of the flexural reinforcement, which was achieved by L-shaped plates, was signif-icant as shown in Fig. 3. Unlike the case for side bonded plates, increasing L-shaped plate bonding from half shear span to fullshear span resulted in a large performance increase due to increased bond area and fracture surface.

For the particular FRP reinforcement used, the limiting effective strain was calculated using Eqs. (1) and (2) as efe = 0.0076,which is shown with a dashed line as the ACI 440.2R strain limit in Fig. 3b. Comparison of this strain limit with the exper-imental results indicates that the ACI 440 strain limit is unconservative in most cases. The practical FRP strain limit recom-mended by the manufacturer, efe = 0.006 provides a better estimation of the FRP strain at debonding since this limit is basedon targeted experimental studies using the particular reinforcement. However, Fig. 3b shows that this limit cannot be con-sidered as generally applicable since it is unconservative for the case of beam S1PF1M which failed through cover debondingat an FRP strain level of 0.004. It should also be noted that the manufacturer-recommended strain limit is over-conservativefor beams with bond anchorage (S4PS1M and S4PS2M), which has economic significance.

The above discussion shows that development of an accurate and reliable debonding model is needed from both safetyand economy viewpoints. The following sections present a fracture model developed to predict FRP debonding failures.Interaction between shear and debonding behavior of FRP strengthened beams and characterization of the required shearcapacity to prevent cover debonding failures is the subject of ongoing research and is not included in this discussion.

5. A fracture model for FRP debonding

The experimental results and the discussions presented in the previous sections illustrate the significance and importanceof debonding failures in performance of FRP strengthened concrete beams. Accurate prediction of these premature type offailures and their consideration in the design process are essential to ensure the safety of strengthened members. In this sec-tion, a newly developed engineering model based on fracture mechanics approach is described to predict FRP debonding fail-ures by means of a global failure criterion [22].

Debonding and associated fracture processes result in global energy transformations in FRP strengthened members. In theearly stages of loading, these fracture processes may be gradual and stable, whereas upon reaching a critical energy state, asudden brittle failure may take place. The global energy dissipation, dD, in the system can be described in terms of thechanges in the amount of externally supplied work, Wext , and the energy stored in the system as free energy, W, by the fol-lowing expression:

dD ¼ dWext � dW � 0 ð3Þ

Introducing the potential energy of the system, P, in the following form:

P ¼W �U ð4Þ

where U is the external work done by prescribed surface forces, the expression for total dissipation in Eq. (3) can be rewrit-ten for constant prescribed surface forces and displacements as follows:

dD ¼ �dP � 0 ð5Þ

Thus, the amount of energy dissipated in the system during debonding can be determined by calculating the change in thepotential energy of the system.

5.1. Energy dissipation during debonding

The mechanisms of energy dissipation in FRP strengthened RC beams under loading include cracking and crushing ofconcrete, reinforcement yielding and pullout, and FRP debonding. These mechanisms are shown in Fig. 4. Debonding fail-ure may take place before or after steel reinforcement yielding depending on the RC beam geometry and reinforcement,and FRP strengthening configuration. The potential energy difference in strengthened beams upon debonding failure is de-picted in Fig. 5 for the cases before and after reinforcement yielding. The difference between Fig. 5a and b in terms of en-ergy dissipation is that the latter involves an added plastic energy dissipation term due to reinforcement yielding. Thus,the energy dissipation, DD, given by the change in potential energy during debonding failure, can be written in generalterms as:

DD ’ dD ¼Z�dXþ

Zr � depdXþ

ZGf dAf � 0; ep ¼ es � ey � 0 ð6Þ

whereRr � depdX is the plastic energy dissipation due to steel yielding when the strain in the reinforcing steel, es, is greater

than its yield strain, ey, and is equal to zero otherwise. The termR

Gf dAf represents dissipation due to debonding processevaluated over the crack surface defined by the energy per unit area necessary for the crack formation called the interfacefracture energy, Gf, and the interfacial bond area Af , and the term

R!dX represents the bulk energy dissipation within the

P/2P/2

Reinforcement yieldingand pullout

FRP debonding

Concrete cracking and crushing

P/2P/2

Reinforcement yieldingand pullout

FRP debonding

Concrete cracking and crushing

Fig. 4. Energy dissipation mechanisms in FRP strengthened beams.

Fig. 5. Energy dissipation during debonding failure.


system due to remaining mechanisms shown in Fig. 4 which consist mainly of concrete cracking under bending and sheareffects.

Examination of Eq. (6) shows that debonding failures, before or after steel yielding, are not pure fracture processes. Thus,formulation of a debonding failure criteria based on fracture mechanics requires quantification of different energy dissipa-tion mechanisms that are of significance. Experimental evidence shows that the bulk energy dissipation

R�dX included in

Eq. (6) is less significant compared to the remaining dissipation terms since much of the concrete cracking takes place beforedebonding, and only limited cracking occurs during debonding due to constant curvature and small change in the location ofthe neutral axis. As a first approximation, the bulk energy dissipation during debonding failure can be assumed to be insig-nificant and that the dominant modes of energy dissipation are the debonding fracture process and the plastic energy dis-sipation at the rebar. Thus, the total energy dissipation can be approximated as:

DD �Z

r � depdXþZ

Gf dAf � 0; ep ¼ es � ey � 0 ð7Þ

Eq. (7) assumes that debonding failure before reinforcement yielding is a pure debonding fracture process, and that the onlyadditional dissipation term in case of debonding after reinforcement yielding is the plastic energy dissipation due to rebaryielding. Quantification of these two mechanisms is sufficient for debonding failure modeling.

5.2. Plastic energy dissipation due to reinforcement yielding

In order to define a debonding criterion, an essential step is to characterize the plastic energy dissipation term in Eq. (7). Itis apparent from Fig. 5 that in a displacement-controlled experiment the beam deflection and thus the curvature essentiallystays constant upon debonding. Fig. 6 shows the strain profile in the beam cross-section before and after debonding failure.From the definition of curvature, u, one can write:

u ¼ ec

c¼ e0c

c0ð8Þ

where ec and e0c are the maximum concrete strain, and c and c0 are the neutral axis depth before and after debonding, respec-tively. Strain at rebars before and after debonding can be expressed using strain compatibility as:

Fig. 6. Strain profile in beam cross-section before and after debonding.


es ¼ uðd� cÞ; e0s ¼ uðd� c0Þ ð9Þ

Thus, the change in rebar strain upon debonding is given by:

Des ¼ uDc ¼ ec 1� c0

c

� �ð10Þ

Using Eq. (10), the plastic energy dissipation at the rebars during debonding failure can be determined as:

Wps ¼

Zr � depdX ¼ rsDesAsls ¼ fyec 1� c0

c

� �Aslc ð11Þ

where As and fy are the total cross-sectional area and yield strength of the steel reinforcement, and lc is the length of the con-stant moment region.

5.3. Fracture energy dissipation due to FRP debonding

The energy dissipated at the FRP concrete interface region during debonding goes to creating new surfaces along the bondarea. Depending on the fracture properties of the materials that form the strengthened system, debonding fracture may takeplace within or at the interfaces of the materials, taking the path that requires the least amount of energy. The interface frac-ture energy Gf in Eq. (7) can be expressed as:

Gf ¼ CðhÞ ð12Þ

where the toughness of the interface C(h) can be regarded as an effective surface energy that depends on the mode of loadinggiven by the phase angle h:

h ¼ tan�1ðK II=K IÞ ð13Þ

which is a measure of Mode II to Mode I loading acting on the interface crack [25]. The case in which h = 0� corresponds topure Mode I fracture and h = 90� corresponds to pure Mode II fracture. In the case of brittle, isotropic and homogenous mate-rials, fracture propagation follows a trajectory for which KII = 0, i.e. cracks for which the Mode II stress intensity is nonzerodeflect out of their plane until the crack propagation is in pure Mode I. Interface cracks, however, may deviate from thisbehavior due to the inhomogeneous fracture energy of the bimaterial system. If the interface fracture energy is relativelysmall, the crack can propagate along the interface even when h – 0� [25,26]. Depending on the fracture properties of thematerials and interfaces, kinking of interface cracks into materials take place according to the following expression:

GGt

max

<CðhÞCc

ð14Þ

where C(h) and Cc are the interface fracture energy and Mode I fracture toughness of the substrate material, G is the energyrelease rate for continued interface cracking, and Gt

max is the maximum energy release rate at the kinked crack tip.Experimental observations during laboratory tests have shown that debonding at the FRP–concrete interface generally

takes place within concrete, although limited cases of kinking into the adhesive and the FRP composite were encountered.For a system of bonded dissimilar isotropic and elastic materials, crack propagation within the substrate paralleling theinterface means that the Mode II loading conditions vanish at a small depth below the surface of the substrate [27,28]. Inthe case of FRP–concrete interface that is predominantly under shear loading, a possible mechanism for debonding propa-gation in the concrete substrate paralleling the FRP–concrete interface is the initial formation of inclined tensile microcracksin concrete under shear and their subsequent coalescence through various energy dissipating micromechanisms such ascrushing of the concrete struts between the microcracks, and asperity contact and plasticity[26,29,30,16]. Such mechanismsresult in an apparent shear mode (Mode II) fracture energy that is not a material property but depends on other material


properties as well as the geometry and loading. This shear mode fracture energy typically has a magnitude that is consid-erably higher than the opening mode (Mode I) fracture energy [26,30].

In bonded layers of dissimilar materials, mode mixity is a natural result of elastic mismatch even for symmetric loadingor geometry [25]. Hence, all interface debonding processes are essentially mixed-mode fracture processes. There is awealth of literature on mixed-mode cracking in layered dissimilar materials mainly developed for peeling of thin filmsand delamination of composites [25,26,31,32]. In the case of FRP–concrete interfaces, fracture under shear loading (suchas those in simple bond shear tests or at mid-span regions of FRP-bonded beams) is generally referred to as shear mode(Mode II) fracture [33,13,16,17] whereas fracture due to combined shear and tensile loading, such as that takes place undera flexure-shear crack (also called intermediate crack debonding) [9,10,18–20], or edge debonding [21] is described asmixed-mode fracture.

Extensive mixed-mode cracking around flexure-shear cracks can be detrimental to the integrity of FRP strengthenedbeams [10,19,20]. Experimental observations show that this is especially the case if extensive transverse shear cracking takesplace in the strengthened beam, which result in intermediate crack (IC) debonding or even critical intermediate crack (CIC)debonding failure [6]. Vertical component of the flexure-shear crack mouth displacement, which results in debondingthrough Mode I fracture is of critical importance considering that the Mode I fracture energy is an order of magnitude lessthan the Mode II fracture energy. However, if the strengthened beam is sufficiently strong in shear, as is the case in this re-search, not only the flexure-shear crack mouth displacements will be limited, but also the mixed-mode nature of the deb-onding fracture will quickly merge to near Mode II conditions upon debonding propagation. The study by Wang [20] verifiesthis argument as his numerical case study showed that the phase angle h around a flexure-shear crack neared Mode II con-ditions within millimeters of debonding propagation while the typical length of debonding that causes failure is an order ofmagnitude longer. Additionally, Neubauer and Rostasy [34] investigated the effect of mixed-mode fracture on the carbonFRP–concrete bond strength in beams using a truss model with shear crack friction and concluded that the reduction in bondstrength due to mixed-mode fracture around flexure-shear cracks, including the combined effects of multiple flexure-shearcracks, is within 10% in most cases. Hence, the effect of mixed-mode debonding on the performance of FRP strengthenedbeams with sufficient capacity in shear can be neglected without significant error in the analysis.

Based on the above discussion, it is assumed in this research that all debonding at the FRP–concrete interface takes placesufficiently close to Mode II conditions, i.e.

h ¼ tan�1ðK II=K IÞ � 90� ð15Þ

Furthermore, examination of the debonding surfaces during laboratory tests have revealed that the debonding at FRP–con-crete interface generally takes place within the concrete substrate, while debonding of the bond anchorage takes place at theFRP composite interfaces. Thus, considering the assumption in Eq. (15), the associated fracture energies can be taken as:

Gf ¼GFðhÞ � GFIIðFRP—concrete interfaceÞCFðhÞ � CFIIðFRP—FRP interface� transverse anchorageÞ

�ð16Þ

where GF(h) and GFII are the mixed mode and Mode II fracture energies of concrete, and CF(h) and CFII are the mixed modeand Mode II fracture energy of the FRP–FRP bond anchorage interface. Now, the debonding energy dissipation term in Eq. (7)can be rewritten as:
Z
Gf dAf �Z

GFIIdAfb þZ

CFIIdAfa ð17Þ

where Afb = lfbf is the bond area at the FRP–concrete interface and Afa = laba is the bond area between the FRP reinforcementand the bond anchorage reinforcement in the transverse direction.

5.4. Change in potential energy during debonding failure

The total change in the potential energy of the system after debonding failure is the difference between the recoverableenergy stored in the beam before and after debonding. Eq. (5) gives the total dissipation as the negative change in the po-tential energy of the system. This change in potential energy can be calculated by means of the load–deflection curves at loadpoints as shown in Fig. 5 based on the idealizations that the load–deflection relationship is bilinear, and that the unloadingstiffnesses are equal to the pre-yield loading stiffnesses. Considering that the strain energy density is equal to the comple-mentary strain energy density (w = w*) by the linearity assumption, the change in potential energy is equal to the change inglobal free energy or strain energy, W of the system, shown by the shaded areas in Fig. 5. Based on the homogeneous andlinearly elastic materials assumption, the global free energy in beam elements is given by:

W ¼Z

L

M2

2EIþ V2

2GA

( )dx ð18Þ

where L is the beam length, M is moment, V is shear, E and G are the elastic and shear moduli, and A and I are the area andmoment of inertia of the beam cross-section, respectively. Neglecting the shear component and assuming that the FRP


reinforcement length is close to the span length, which is mostly the case in field applications, the strain energy in the beamunder a total load P can be expressed by:

W ¼ P2

4EIl2s L2� 2l3s

3

" #ð19Þ

where ls is the shear span. Before debonding, W = W2 and I = I2 and after debonding, W = W1 and I = I1. For ls = L/3 as was thecase in this research, the strain energy simplifies to W = �P2L3/(216EI). Assuming a constant elastic modulus for concrete inthe strain energy calculation is consistent with the aforementioned assumption that the unloading stiffness of the beam be-fore and after strengthening is equal to the respective pre-yield loading stiffness.

The load values in Fig. 5 for before and after debonding can only be associated through the displacement; hence, the load–deflection (or moment–curvature) curve for the loading points must be constructed. This can be performed either through aniterative approach to construct an accurate nonlinear curve, or a bilinear curve defined by the calculated yield point and thepoint of ultimate failure. The former approach was used in this research [22], but the latter approach is more suitable forpractical design since it significantly simplifies the problem for debonding after reinforcement yielding as shown in Fig. 5.Since the load capacity of an unstrengthened beam stays constant after reinforcement yielding, so does the strain energyof the strengthened beam after debonding, i.e. W2 = Wy = const. where Wy is the strain energy of the system at steel yielding.From elasticity, the deflection at loading points is given by the following expression:

dL ¼u24ð3L2 � 4l2s Þ ð20Þ

where the curvature, u is given by:

u ¼ ec

c¼ M

EIe¼ Pls

2EIeð21Þ

Once the load–deflection curves are constructed, for the deflection at which debonding takes place, dLd, the total dissipationin the system is given by:

DD ¼ �DP ¼ P22d

2K2� P2

1d

2K1ð22Þ

where P2d = P(dL = dLd) and P1d = P(dL = dLd) are the load values before and after debonding that takes place at deflection dLd

under the load application points. From Fig. 5 and Eqs. (19) and (22), the stiffness values for the strengthened andunstrengthened beams, K2 and K1 respectively, are given by:

K2 ¼ 2EI2l2s L2� 2l3s

3

", #; K1 ¼ 2EI1

l2s L2� 2l3

s

3

", #ð23Þ

where I1 and I2 are the moments of inertia of the transformed beam sections in cracked condition. With the total potentialenergy difference at hand, use of Eq. (7) now allows development of a debonding failure criterion.

5.5. Debonding failure criterion

Using Eqs. (7), (11), (17), and (22) a global debonding criterion can be developed based on the assumption that debondingtakes place along the entire bond surface along the FRP reinforcement with concrete and if present with the transverseanchorage reinforcement:

DD ¼ P22d

2K2� P2

1d

2K1¼ ðGFIIlf bf þ CFIIlabaÞ þWp

s P 0 ð24Þ

Eq. (24) indicates that for increasing beam curvature/deflection under loading, the portion of the energy stored in thestrengthened beam in excess of that stored in the unstrengthened beam reaches a critical value that causes debonding failureand its dissipation through reinforcement yielding and debonding fracture. Using Eq. (24), the debonding failure load can bedetermined through iteration, trial and error, or through explicit solutions based on bilinear assumption for the beam load–deflection curves. If debonding takes places after reinforcement yielding, a simplification in Eq. (24) can be made by assum-ing P1d � Py as illustrated in Fig. 5, in which case the expression becomes:

DD ¼ P22d

2K2�

P2y

2K1¼ ðGFIIlf bf þ CFIIlabaÞ þWp

s P 0ðP2d > PyÞ ð25Þ

By Eq. (25) the strain energy of the unstrengthened beam becomes a constant after yielding, which greatly simplifies theanalysis and design problem by enabling an explicit solution. A critical issue in use of Eq. (25) is the estimation of the fracture


energies GFII and CFII which are not well known. A possible approximation for GFII is the simple relation developed by Neu-bauer and Rostasy [35,36] based on the study by Holzenkämpfer [37] who developed a bond strength model for steel platesbonded to concrete using nonlinear fracture mechanics [38], in which the shear mode fracture energy is given by the follow-ing expression:

GF ¼ GFII ¼Z sf

0sðsÞds � cf k2

bfctm ð26Þ

where s is the shear stress, s is the shear slip, fctm is the pull-off tensile strength of concrete measured according to DIN-1048[39], kb is a geometric factor that considers the influence of the plate width, bp, relative to the width of the concrete member,bc, according to the following expression:

kb ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1:125

2� bp=bc

1þ bp=400

sðSIÞ ð27Þ

and cf is an experimentally determined constant that contains all secondary effects. Neubauer and Rostasy [35,36] deter-mined this constant for carbon fiber reinforced polymer (CFRP) bonded concrete as cf = 0.202 from 70 double shear bondtests and concluded that shear fracture in CFRP–concrete bond can be modeled using a triangular shear-slip model [38].It is important to note that the model by Neubauer and Rostasy [35,36] is valid for normal strength concrete only since CFRPdelamination failures were observed for high strength concrete with a compressive strength of 55 MPa. Lu et al. [12] testedthe performance of this model along with 11 other bond strength models using 253 test results compiled from the literatureand found that the model by Neubauer and Rostasy [35,36] is one of the better performing models with a high correlationcoefficient (0.885) and a low coefficient of variation (0.168). Noting that the bond strength is directly proportional to thesquare root of the interface fracture energy (

ffiffiffiffiffiffiGFp

) regardless of the shape of the bond–slip curve, performance of the modelin estimating the bond strength also shows its performance in estimating the interface fracture energy [11]. Simplicity of themodel by Neubauer and Rostasy [35,36] combined with its demonstrated accuracy justifies its use as a basis for estimatingthe shear fracture energy of the FRP–concrete bond in this research.

The expression proposed by Neubauer and Rostasy [35,36] gives a fracture energy that is at least an order of magnitudehigher than the Mode I fracture toughness of concrete that can be calculated using the CEB–FIP Model Code expression[40,41]:

GF ¼ GFI ¼ aff 0c10

� �0:7

ðSIÞ

af ¼ ð0:0469d2a � 0:5da þ 26Þ � 10�3

ð28Þ

where GF is in N/mm and af, a function of the maximum aggregate size da (in mm), was calculated as 0.026 for the concreteused in this research.

Additional discussions and experimental studies [30,42–47] suggest that the Mode II fracture toughness of concrete mayrange from 10 to 25 times its Mode I fracture toughness, i.e. GFII � (10–25)GFI [22,30,16]. Although not a material property,Mode II fracture of concrete involves several effects such as friction, asperity contact and plasticity which considerably in-crease its fracture resistance [26]. The expression for the fracture energy given by Eq. (26) was developed from double shearbond tests where the bonded FRP reinforcement was subjected to uniaxial tension [36]. It is conceptually clear and exper-imentally evident [48] that the Mode II fracture energy of the FRP–concrete interface under flexural loading is likely to behigher for interface crack (IC) debonding than that of the bond test configuration due to the curvature effect that exerts addi-tional compression on the interface. Considering the results obtained from laboratory tests and the reported range of Mode IIfracture energy GFII in relation to the Mode I fracture energy [22,30,16], the experimental constant in Eq. (26) was modifiedas cf = 0.23 mm in this research, which produced a Mode II fracture energy for the FRP–concrete interface that is approxi-mately 20 times the Mode I fracture energy of concrete calculated using the CEB–FIP Model Code [40] expression. Also, inorder to make the expression simpler and more generally applicable, the pull-off tensile strength in Eq. (26) was approxi-mated as:

fctm � fct ¼ 0:53ffiffiffiffif 0c

qðSIÞ ð29Þ

where fct is the split cylinder tensile strength. With these modifications, Eq. (26) becomes:

GFII ¼ 0:122k2b

ffiffiffiffif 0c

qðSIÞ ð30Þ

Knowledge of interface fracture energy, CFII, in Mode II between the longitudinal FRP reinforcement and the transverseanchorage reinforcement is very limited at this time. However, experimental observations indicate that the interface orinterlaminar fracture energy of FRP, CFII, is higher than GFII but is in the same order since limited kinking of crack propagationinto the composite was observed during the laboratory tests. Therefore, based on existing experimental observations and


data, CFII was assumed to be twice the fracture energy of that between FRP–concrete interface, GFII, for the purposes of thisresearch [22].

6. Model implementation to experimental data

Implementation of the developed model was performed for the beam tests presented in Section 4 to compare the modelprediction with the experimental results. Fig. 7a shows the experimental results obtained from representative beam tests,excluding the cover debonding failure case. This figure also shows the nonlinear load–deflection curves for the controland strengthened beams. These curves, shown with the dashed lines in the figure, were iteratively constructed using Hog-nestad’s nonlinear concrete model [49], and were used as a basis for determining the debonding failure loads according to Eq.(25) to increase the accuracy of the model prediction. The reader is referred to Gunes [22] for the details of constructingthese curves. For practical design, a simpler approach can be taken by using bilinear load–deflection curves as shown inFig. 5.

A comparison of the model predictions with the experimental results is shown in Fig. 7b. As can be seen from the figure,the developed fracture model yields a satisfactory prediction of the FRP debonding failure loads and performs better thanthe ACI 440 [4,5] provision given by Eqs. (1) and (2) which yield a constant debonding load for all three cases shown inFig. 7b.

In order to perform further validation, the developed model was tested on a number of FRP debonding failure data setsproduced by various independent research studies [29,50,51]. Fig. 8 compares the experimental results of these studiesincluding this research with the model predictions. The overall success of the model in predicting FRP debonding failureloads for various sizes of beams and strengthening configurations shows the potential of fracture mechanics modeling ap-proach for design against debonding failures.

7. Design of beams against FRP debonding failures

The developed FRP debonding failure model can be easily integrated into design of FRP strengthened beams to achievesafety against FRP debonding failures. The design approach is described in the following steps starting from the design ofFRP strengthened beams for flexure and shear effects:

1. Perform the strengthened beam design using conventional ultimate strength analysis for design flexural loads [4,5]. Theoutcome of this step is the cross-sectional area of the bonded FRP reinforcement Af = bftf needed for strengthening.

2. Perform design for shear strengthening of the beam using side bonded or wrapped FRP composites if the design shearload exceeds the beam shear capacity.

3. In order to ensure ductile behavior and failure of the beam, equate the debonding failure design load to the flexural capac-ity of the beam calculated in step 1, and calculate the total bond fracture resistance needed to resist the debonding failureload using Eq. (25):

Dd ¼ ðGFIIlf bf þ CFIIlabaÞ ¼P2

2d

2K2�

P2y

2K1�Wp

s P 0 ð31Þ

0 5 10 15 200

20

40

60

80

100

120

140

160

180

200

Mid−span deflection (mm)

Load

(kN

)

CM1S3PS2M

S4PS1MS4PS2M

Calculated response after strengthening

Calculated response before strengthening

130 140 150 160 170 180130

140

150

160

170

180

Experimental debonding load (kN)

Estim

ated

deb

ondi

ng lo

ad (k

N)

Gunes (2004)ACI 440.2R (2002)

(a) experimental results and predictions (b) experimental vs. estimated debonding loads

Fig. 7. Comparison of debonding model predictions with experimental results.

50 100 150 200 250 300

50

100

150

200

250

300

Experimental Debonding Load (kN)

Estim

ated

Deb

ondi

ng L

oad

(kN

)

Gunes (2004)Hearing (2000)Taljsten (1999)Leung (2004)

Fig. 8. Model implementation to multiple sets of experimental data.


4. Once the required total debonding energy is determined, one has to make sure that the total fracture energy of the FRP–concrete bond and possible anchorage is sufficient to meet this energy demand. Since the required FRP reinforcementarea Af is known from step 1, the first try would be to check if the bond area of the FRP (plate) reinforcement designedin step 1 is sufficient (or to arrange the width and thickness of the FRP (sheet) reinforcement to provide sufficient bondarea) without any anchorage:

bf ¼Dd

GFIIlf6 b; tf ¼

Af

bfð32Þ

Special attention must be paid not to design the FRP reinforcement too thin to avoid FRP rupture due to stress concen-trations at crack locations. If the bond area without any anchorage is not enough to meet the energy demand, thenanchorage requirement needs to be calculated to provide additional fracture energy:

laba ¼Dd � GFIIlf bf

CFIIð33Þ

so that the integrity of the bond is ensured under the design load. The calculated anchorage reinforcement should beplaced close to the FRP reinforcement end regions.

5. It should be noted that the developed FRP debonding model does not address cover debonding failures since this failuretype appears to be mainly influenced by the shear behavior and capacity of the beam. Until an accurate model is devel-oped to address cover debonding failures, specification of a minimum bond anchorage in the FRP reinforcement endregions with a length approximately equal to the beam height is recommended as a safety assurance.

8. Summary and conclusions

Through experimental research and analytical modeling studies, a global fracture model was developed to predict FRPdebonding failures in strengthened beams. The model includes the beam geometry, strengthening configuration, and addi-tional bond anchorage effects considering energy balance in the system and energy dissipation through steel reinforcementyielding and FRP debonding. Implementation of the model to several sets of independently reported experimental datashows that the model can satisfactorily predict the FRP debonding failure loads for various sizes of beams strengthenedin various configurations, with or without bond anchorage. The model can be further improved through better characteriza-tion of its components such as bulk energy dissipation in the concrete beam during debonding, and mixed-mode fractureenergy values at the FRP–concrete and FRP–FRP interfaces.

The developed model can easily be integrated into the design of FRP strengthened beams to ensure that the debondingfailure load is higher than the flexural capacity of the beam. A possible design approach is outlined to determine the bondarea or additional bond anchorage area required to prevent brittle debonding failures. With further improvements and val-idation, the model may be used as a code provision for preventing FRP debonding failures.

Acknowledgements

The research work presented in this paper was funded by NSF under the project title ‘‘Failure Behavior of FRP BondedConcrete Affected by Interface Fracture” through the research grant CMS 0010126 to Massachusetts Institute of Technology.The authors thank Dr. Christopher Leung at Hong Kong University of Science and Technology for generously making available


the unpublished experimental data of his laboratory. The FRP materials used in the experimental program were donated bySika Corporation.

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2004.

a fracture-based model for frp debonding in strengthened beams

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