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Computer-Aided Civil and Infrastructure Engineering 20 (2005) 354–368

Numerical Analysis of Debonding Mechanismsin FRP-Strengthened RC Beams

Hedong Niu & Zhishen Wu∗

Department of Urban and Civil Engineering, Ibaraki University, 4-12-1 Nakanarusawa-cho,Hitachi 316-8511, Japan

Abstract: Fiber-reinforced polymer (FRP) compositeshave been increasingly used as externally bonded rein-forcement in lieu of their steel counterpart in the rehabili-tation and retrofit of existing concrete structures. Withoutproper understanding of interfacial fracture behavior andfailure mechanisms, it is impossible to efficiently developan effective and rational FRP bonding technique. Thisarticle is mainly focused on clarifying the debonding be-havior and failure mechanisms caused by different typesof flexural crack distributions in FRP-strengthened R/Cbeams, which has not been solved so far. Using a dis-crete crack model for concrete crack propagation and abilinear bond–slip relationship with softening behavior torepresent FRP–concrete interfacial behavior, a nonlinearfracture mechanics-based finite-element analysis is per-formed to investigate the effects of crack spacing and in-terfacial parameters such as stiffness, local bond strength,and fracture energy on the initiation and propagation ofthe debonding and the structural performance. It is shownthat the debonding behavior and load-carrying capacityare significantly influenced by two important factors: in-terfacial fracture energy and crack spacing in relation tothe effective transfer length of FRP sheets. Based on thenumerical results, some suggestions concerning the effectof interfacial properties are made as practical design aids.

1 INTRODUCTION

Fiber-reinforced polymer (FRP) composites have beenincreasingly used as externally epoxy-bonded reinforce-ment to enhance structural stiffness and strength due totheir superior characteristics such as high strength- and

∗To whom correspondence should be addressed. E-mail: [email protected].

stiffness-to-weight ratio, high corrosion resistance, elec-tromagnetic neutrality, inherent tailorability, and easeof application in the field. Central to FRP bonding tech-nique is to ensure composite action between concreteand FRP composites, loss of which would cause unde-sirable premature failure prior to the theoretical or pre-dicted ultimate load. For an FRP-strengthened RC beamwith sufficient shear capacity, typical interfacial failuremodes may include the debonding from the cut-off pointto the midspan (Figure 1a) and the debonding fromthe end of flexural crack to the end of FRP composites(Figure 1b). Unlike the failure modes of concrete crush-ing, shear failure and FRP rupture, such debonding fail-ures cannot be predicted by conventional RC theory. Sofar, considerable research work has been directed to thedebonding at the cut-off point of FRP/steel plates causedby shear and normal stress concentrations. For this typeof debonding, criteria have been developed for pre-dicting the premature failure load (e.g., Roberts, 1989;Ziraba et al., 1994; Malek et al., 1998; Mukhopadhyayaand Swamy, 1999). In practice, FRP debonding from theend of an intermediate crack sometimes is unavoidableand more dominant despite careful surface preparationand good bond between FRP composites and concrete(in particular for concrete structures strengthened withthin FRP sheets instead of plates). However, very lim-ited literature can be found concerning such debondingfailures, and the associated debonding mechanisms stillremain unknown.

As far as the bond behavior between FRP sheetsand concrete is concerned, considerable investigationshave been done with simple shear (Nishida et al., 1999;Kamiharako et al., 1999; Sato et al., 2000; Yoshizawaet al., 2000; Nakaba et al., 2001) or flexural (Lorenziset al., 2001) test specimens. Local bond–slip curveswere identified from the experiments and schematically

C© 2005 Computer-Aided Civil and Infrastructure Engineering. Published by Blackwell Publishing, 350 Main Street, Malden, MA 02148, USA,and 9600 Garsington Road, Oxford OX4 2DQ, UK.

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Numerical analysis of debonding mechanisms 355

Fig. 1. Observed interface-related failure modes in FRP-strengthened RC beams. (a) Debonding at the cut-off point of FRPplate. (b) Debonding from flexural cracks.

shown in Figure 2. These curves are characterized byan ascending branch before reaching the local bondstrength, followed by plastic (debonding within the adhe-sive from flexural test) or softening behavior (debondingwithin concrete from shear test) up to an ultimate slip.By using the concept of fracture mechanics, the ultimateload-carrying capacity of FRP-bonded joints can be pre-dicted by a same simple equation for different kinds of in-terfacial constitutive relationships (Taljsten, 1996; Yuanet al., 2001; Wu et al., 2002b; Wu and Yin, 2002), which isonly related to the equivalent stiffness of FRP compos-ites (defined by multiplication of elastic modulus andthickness) and interfacial fracture energy. Yoshizawaet al. (2000) found that local shear stress distribution,effective transfer length beyond which nearly no stressis transferred, initiation and propagation of debondingcould be well described by a simplified model as shownin Figure 2. They also identified an average bond–sliprelationship with a local bond strength of 8 MPa andfracture energy, defined as the area under the bond–slipcurve, of 1.2 N/mm.

Little literature can be found concerning the effect ofconcrete cracking on the bond behavior between FRPcomposites and concrete. Bizindavyi and Neale (1999)used a single-lap shear test to investigate the bond behav-ior and observed that a crack in the concrete block at the

Slip,

Bon

d st

ress

,

Shear test Flexural test Simplified model

Fig. 2. Schematic bond–slip relationships from differentbond test methods.

loaded end may increase the stress transfer length. Uedaet al. (2002) confirmed the strengthening effect of CFRPsheets in RC tension members and found that cracksmay deteriorate the nearby bond properties. Accordingto the authors, more data are needed to verify this phe-nomenon. Wu et al. (1997) investigated the debonding inconcrete beams strengthened with FRP sheets in flexureby using the compliance method. They found that thenumber of flexural cracks and the interfacial fractureenergy may have considerable effects on the debondingload, although this needs further investigation. Wu andNiu (2000a,b) analytically investigated the stress concen-trations near cracks and concluded that interfacial sheartransfer and debonding propagation at the end of a singlelocalized crack may be similar to those of simple sheartests. This implies that the debonding failure caused bya single localized crack in an FRP-strengthened beammay be predicted by the ultimate load-carrying capacityobtained by simple shear tests.

Despite the volume of research to date, the debondingbehavior and failure mechanisms caused by multiple ordistributed flexural cracks are still unknown, and a de-sign criterion for debonding failure is still unavailable.Therefore, the main objective of this study is to presenta clear understanding of how the initiation and propa-gation of debonding is influenced by the distribution offlexural cracks and the interfacial properties, thus, pro-viding some implications for future design codes.

2 EXPERIMENTAL OBSERVATIONS

A series of experiments were conducted on beamsretrofitted with FRP sheets (Wu et al., 1998, 1999;Ichikawa and Wu, 1998; Wu and Kurokawa, 2002) to in-vestigate the strengthening performance, failure modes,bond mechanism, and effect of some important designparameters such as concrete strength, surface prepara-tion, number of plies, types of FRP sheets, prestress level,

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356 Niu & Wu

FRP sheets

Concrete

(a)

Lc FRP sheets

Reinforcing steel bars

Concrete

(b)

Fig. 3. Typical debonding failure modes. (a) Single localizedcrack pattern. (b) Multiple or distributed crack pattern.

anchorage treatment, and diameter of reinforcing steelbar. There were two typical flexural crack patterns thataccompanied the debonding failure mode, as shown inFigure 3: the single localized crack near the maximummoment region in the strengthened plain concrete beamsand multiple or distributed cracks in the strengthenedRC beams.

In what follows, a CFRP-strengthened RC beam sub-jected to three-point bending (Wu and Kurokawa, 2002)was analyzed using a commercial finite-element programDIANA (1998). The beam was 150 mm wide, 200 mmdeep, and 2,100 mm long with 1,700-mm-long axiallyoriented unidirectional CFRP sheets bonded to its fullwidth. Deformed reinforcing bars were used in tensionand compression. The geometric and reinforcement de-tails of the beam are shown in Figure 4. The mechanicalproperties of the materials are given in Table 1. The finalfailure mode of this beam was observed to be debondingof FRP from the flexural cracks near the midspan.

2,100 1,700 150 150 50 50

150 70 40 40

120

40

40

(Unit: mm)

2-D16

2-D13

FRP Sheets

24 10

Fig. 4. Details of the investigated RC beam strengthened with CFRP sheets.

Table 1Summary of material properties

Materials Mechanical properties Values

Concrete Young’s modulus (GPa) 35.1Compressive strength (MPa) 49.3Tensile strength (MPa) 3.0Poisson’s ratio 0.13

Reinforcing bars Young’s modulus (GPa) 210Poisson’s ratio 0.3

D16 Yield strength (MPa) 364D13 Yield strength (MPa) 358

CFRP sheets Young’s modulus (GPa) 230Tensile strength (GPa) 4.1Thickness (mm) 0.111Poisson’s ratio 0.3

3 FINITE-ELEMENT MODELING

For simplicity, stirrups, which were used to ensure thebeams would not fail in shear, are not considered in thefollowing simulations. To simulate the real response ofthe composite beam and to investigate the failure mech-anisms caused by flexural cracks, it is important to ac-curately model the crack propagation behavior in con-crete, the bond–slip relationships between reinforcingsteel bar and concrete, and interfacial bond behavior ofFRP–concrete interface in addition to constitutive be-haviors of concrete, steel, and FRP sheets.

3.1 Concrete cracking and interfacial models

3.1.1 Discrete crack model. Generally, two different ap-proaches for modeling the cracking discontinuity canbe used to describe the concrete cracking behavior:the smeared crack approach and the discrete crack ap-proach. The smeared crack approach is based on the as-sumption that the fracture is distributed over a domainand thus may be described by continuum models with



P

Crack opening, w

Cohesive crack

Traction free crack

ft

Crack opening, w

Gfc

Stress, σ

Unloading/reloading

Initiation of cohesive crack

Macro-crack

Fig. 5. Linear tension softening behavior of concrete.

the adoption of appropriate stress–strain relationships.Because it is difficult to trace the exact crack locationwith this approach, it may be inappropriate to capturethe effect of cracks on the initiation and propagation ofthe debonding between FRP composites and concrete.Therefore, in the present study, the discrete crack modelwas adopted to simulate the cracking behavior of con-crete. Possible flexural crack locations were predefinedto be vertical along the depth of the beam with inter-face elements, which describe the interface behavior interms of a relation between the normal and shear trac-tions and the normal and shear relative displacementsacross the interface. A very large value was assigned tothe initial stiffness of the interface elements to ensureinner continuity of the concrete. Under loading, a linearsoftening curve (Figure 5) was employed to model modeI cracking behavior of concrete, where f t is concrete ten-sile strength and Gc

f is the mode I fracture energy ofconcrete. Unloading and reloading behaviors are mod-eled by a secant path, which means following a straightline back to the origin upon unloading the stress. Aftercracking, no shear stress is assumed to be transferredalong the crack surface.

3.1.2 Steel–concrete interface. In an early study, thebond mechanism between concrete and the deformedreinforcing bars was investigated theoretically and ex-perimentally by Morita et al. (1967). A bond–slip modeldeveloped from their experimental data was used to sim-ulate the interfacial behavior between concrete and thedeformed steel bar (Figure 6).

3.1.3 FRP–concrete interface. As far as the bond be-havior of FRP composites in the strengthened concretebeams is concerned, it is generally accepted that debond-ing propagation resembles mode II fracture behavior inthat FRP composites are primarily loaded in tension andthe adhesive layer transfers stresses from concrete toFRP composites by shear. Even for the case of flexural-shear or shear cracks, the induced debonding propaga-tion is still mainly governed by the mode II fracture be-havior due to a small peel angle as shown by Niu and Wu(2001). However, in a strict sense (microscopic), only the

debonding within the adhesive layer may be like mode IIfracture behavior, whereas the general debonding withinthe adjacent concrete layer may be associated with a con-crete mode I fracture and the shearing fracture behav-iors (Figure 7a). By using a displacement discontinuitymodel, Wu et al. (2002a) found that there exists a linearcorrelation between mode I concrete and a mode II in-terfacial fracture energy values for a given shearing frac-ture energy introduced on the crack surface. This meansthat the overall debonding behavior can be regarded asa mode II fracture no matter where it occurs.

A simplified bilinear bond curve with softening be-havior was employed to simulate the real FRP–concretebond behavior. As shown in Figure 7b, when local bondstress attains local bond strength τ f, micro-debonding(softening) initiates, after that the capacity of stresstransfer decreases linearly to zero until the occurrenceof macro-debonding. The area under the curve is definedas the interfacial fracture energy Gb

f . The scheme of un-loading and reloading is represented by the secant path.This model is capable of simulating the bond behaviorregardless of whether debonding occurs within the con-crete substrate or within the adhesive layer. The differ-ence only lies in the choice of the parameters: local bondstrength τ f, initial stiffness ks, and the fracture energyGb

f . As reviewed previously, concrete flexural crackingnear the extreme tensile fibers of the beam can damagethe bond to FRP composites. Due to the scarcity of data,

0

2

4

6

8

0 0.2 0.4 0.6 0.8 1

Local slip (mm)

Loc

al b

ond

stre

ss (

MP

a)

Unloading/reloading

Fig. 6. Bond behavior of steel–concrete interface.


358 Niu & Wu

(b)

(a)

Local bond stress,

f

Slip, δ

Unloading/reloading

Gfb

ks

1

δ0

Initiation of micro-debonding

Maro-debonding

Concrete

Macro-debonding Micro-debonding Continuum

FRP sheet

Adhesive Mode II fracture

Mode I fracture

Stress transfer length

Macro bond stress transfer

Fig. 7. Bond behavior of FRP–concrete interface. (a) Insight into interfacial fracture process (Wu et al., 2002). (b) Simplifiedbilinear bond model.

the deterioration of bond properties caused by crackingin concrete was not considered here.

3.2 Material models

3.2.1 Concrete model. In the locations other than wherediscrete cracks are prescribed in advance, the nonlin-ear behavior of concrete is modeled by Drucker–Pragerplasticity. The yield surface F is defined in the (I1,

√J2)

space as follows:

F(I1, J2) =√

3J2 + αI1 − k (1)

where I1 = the first invariant of the stress tensor σ, J2 =the second invariant of the deviatoric stress tensor s, α =

2 sin φ

3 − sin φ, k = 6c cos φ

3 − sin φ, φ is the angle of internal friction,

c is the cohesion and can be expressed by the uniaxialcompressive strength f c: c = fc

1 − sin φ

2 cos φ.

In the present study, the associated flow rule was as-sumed for modeling the compressive behavior of con-crete. To avoid the computational problems and to focuson the interfacial behavior, the hardening in compression

was assumed for simplicity to follow the curve (Figure 8)specified by JSCE (1996) but without the limitation ofstrain capacity.

3.2.2 Reinforcing steel. Reinforcing steel was treated asa linear elastic–perfectly plastic material, as shown inFigure 9.

Compressive stress, c(MPa)

fc

Strain, c

0.002

−=

002.0002.0

2 2cc

cc f

Fig. 8. Strain hardening behavior of concrete in compression.



Strain, ε

Stress, σ

fy

1

Es

Fig. 9. Elastic–perfectly plastic model for reinforcing bars.

3.2.3 FRP sheets. Unidirectional FRP sheets generallybehave in a linear elastic fashion until rupture. In prac-tice, FRP sheets are mainly used to carry the tensionstress along their longitudinal direction in the strength-ened beam. The rupture of FRP sheets can be analyzedafterwards and is not the focus of this study. So a lin-ear elasticity was assumed for FRP sheets in the presentanalysis.

3.3 Structural model

Due to symmetry of the structure, only half of thebeam was analyzed with the appropriate boundary con-ditions and the applied load, as shown in Figure 10. Theretrofitted beam was loaded by displacement control inthe vertical direction at the top of the midspan. Thehorizontal thick lines represent FRP sheets and rein-forcing bars, vertical thick lines represent the flexuralcrack interface, and diagonal hatching represents steel–concrete and FRP–concrete interfaces. Concrete was

Lc

Steel-concrete interface

FRP-concrete interface

FRP sheets

Rebar Flexural crack interface

Concrete

RebarFlexural cracks

FRP sheets

Fig. 10. Structural model for FRP-strengthened RC beam specimens.

modeled by four-node plane stress elements, whereasreinforcing bars and the FRP sheets were modeled bytwo-node linear truss elements connected to concreteby zero-thickness line interface elements. The flexuralcracks were modeled by zero-thickness line interface el-ements at a spacing of Lc from the midspan to the sup-port of the beam. The aforementioned bond–slip curvesand concrete mode I fracture behavior can be easily in-troduced in the interface elements. As mentioned previ-ously, a very large value was assigned to initial interfacialstiffness in both directions normal and parallel to the in-terface in order to model the initial continuity.

4 NUMERICAL SIMULATIONSAND DISCUSSIONS

In practice, many cracks will have already formed withsome spacing before the application of FRP compos-ites. How do those cracks affect the failure behavior ofthe strengthened structure? And how do the bond prop-erties affect the strengthened performance of crackedstructures? A clear explanation of these influencing fac-tors is very essential to an effective application of FRPcomposites in practice. As mentioned previously, quan-titative evaluation of the detrimental effect of crackingon the bond is still unavailable. So no deterioration ofthe bond properties is considered due to concrete crack-ing. In the present study, no attempt was made to per-form the phased structural analysis of the applicationof FRP to predamaged (preloaded) structures. Instead,a virgin beam strengthened with FRP sheets was usedto investigate the effect of relevant parameters such as


360 Niu & Wu

crack spacing, local bond strength, interfacial stiffness,and interfacial fracture energy on the debonding behav-ior and the structural performance of the beam. Herein,the tensile fracture energy of concrete, Gc

f , was taken asa normal value of 0.12 N/mm due to lack of experimentaldata. The friction angle of concrete, φ, was taken as 10◦

to avoid overestimation of the biaxial strength accordingto DIANA (1998).

Determination of crack spacing Lc is very complicatedand may be related to many factors such as diameter ofrebar, depth of concrete cover, FRP reinforcing stiffness,and concrete properties. According to Yoshizawa andWu (1999), crack spacing and crack width were signifi-cantly smaller for both tensile and flexural RC memberswhen CFRP sheets were bonded. Average crack spacingwas only slightly affected by these factors and remained

Table 2The main parameters investigated in FE analysis

at about 100 mm for RC tension specimens and 70 mmfor flexural RC specimens.

According to the values identified by experimentaldata (Yoshizawa et al., 2000), the values such as ks =160 MPa/mm, τ f = 8 MPa, and Gb

f = 2.0 N/mm werechosen as reference values for the bond–slip relation-ship with linearly ascending and descending branchesdescribing the local interface behavior. As shown inTable 2, the value of interfacial stiffness, ks was variedfrom 80 to 3,200 MPa/mm to simulate different transfer-ring capacity exhibited by a highly flexible layer existingbetween concrete and FRP sheets (Maeda et al., 2001;Gao et al., 2001). The local bond strength, τ f was var-ied from 4 to 16 MPa to simulate the bond conditionaffected by surface preparation and interfacial fractureenergy, Gb

f was varied from 0.5 to 2.0 N/mm to simulate



0102030405060708090

0 2 4 6 8 10 12 14 16 18 20Deflection (mm)

Loa

d (k

N)

Coarse mesh (50mm*40mm)

Moderate mesh (10mm*20mm)

Fine mesh (10mm*10mm)

0102030405060708090

100

0 5 10 15 20 25 30

Deflection (mm)

Loa

d (k

N)

Coarse mesh (37.5mm*40mm)

Moderate mesh (12.5mm*20mm)

Fine mesh (5mm*5mm)

(a) (b)

Fig. 11. Effect of different FE meshes on load–deflection curves (ks = 160 MPa/mm, τ f = 8 MPa, Gbf = 2.0 N/mm). (a) Case of

single localized crack. (b) Case of crack spacing Lc = 75 mm.

debonding within the adhesive layer through the adja-cent concrete substrate.

To accurately simulate the complicated fracturebehavior involving concrete cracking and interfacialdebonding, a proper mesh needs to be used. Too coarseof a mesh may lead to inaccurate results, and too fine of amesh may require more computation time. As shown inFigure 11, a coarse mesh often overestimates the struc-tural performance, and this can be improved by refiningthe mesh. It shows that the moderate mesh used in thisarticle was adequate to be used for further analysis.

4.1 Effect of crack spacing Lc

Figure 12 demonstrates the effect of crack spacing, Lc onthe load–deflection behavior of the FRP-strengthenedbeam. It can be seen that the stiffness of the strength-ened beam decreases with the decrease of crack spacing.Single localized crack and large crack spacing give almostthe same ultimate load, which is lower than that of smallcrack spacing. Figure 12b also shows that cases of smallcrack spacing present a behavior reasonably close to theexperimental one. The difference may be attributed toincorrect values taken for the FRP–concrete interface. In

0102030405060708090

0 3 6 9 12 15 18 21 24 27

Deflection (mm)

Loa

d (k

N)

Single localized crack450mm300mm225mm

0102030405060708090

0 3 6 9 12 15 18 21 24 27

Deflection (mm)

Loa

d (k

N)

Single localized crack112.5mm75mm56.25mmExp.

(a) (b)

Fig. 12. Load versus deflection for various crack spacings. (a) Large crack spacings. (b) Small crack spacings.

Figure 13a, FRP exhibited linear behavior for four seg-ments corresponding to pre- and post-cracking, yieldingof the rebar and interface debonding. Once yielding oc-curs in the reinforcing bars, FRP stress increases at amuch higher rate until it becomes constant when inter-facial debonding occurs. For the case of large crack spac-ing, debonding propagation along the FRP–concrete in-terface occurs easily, and thus FRP stress no longerincreases. However, with regard to the case of small crackspacing, FRP stress continues to increase, contributingto a sustained increase in load-carrying capacity. Fromthese results, the decrease in crack spacing may be help-ful to utilize the full strengthening effect of both internalrebar (Figure 13b) and external FRP sheets.

The above results can be clearly explained by consider-ing three cases: single localized crack and two crack spac-ings at 300 and 75 mm. For the case of a single localizedcrack (Figure 14), once macro-debonding occurs at themidspan, FRP stress attains the maximum value and thenremains constant during the debonding propagationfrom the midspan to the end of FRP sheets. As taken ei-ther from the interfacial shear stress distribution or theFRP stress distribution shown in Figure 14, the effec-tive shear transfer length required to attain the ultimate


362 Niu & Wu

0

500

1000

1500

2000

2500

3000

3500

4000

0 3 6 9 12 15 18 21 24 27Deflection (mm)

Str

ess

(MP

a)Rebar (localized crack) FRP (localized crack)Rebar (300mm) FRP (300mm)Rebar (75mm) FRP (75mm)

0

50

100

150

200

250

300

350

400

0 150 300 450 600 750 900Distance from the midspan (mm)

Reb

ar S

tres

s (M

Pa)

Localized crack300mm75mm

(a) (b)

Fig. 13. Reinforcement stresses for various crack spacings. (a) Reinforcement stresses vs. deflection (midspan). (b) Rebar stressdistributions (deflection of 20 mm).

load-carrying capacity may be regarded as about 80 mm.For the case of Lc = 300 mm (Figure 15), similar debond-ing behavior is observed until the debonding propagatesto the adjacent crack. Herein, it should be noted that theexistence of the adjacent crack may be helpful to increasethe shear transfer length across the crack and the FRPstresses. This can be attributed to the fact that additionalwork is required for the debonding to propagate beyondthe crack (Figure 15b). This yields a slight increase in theultimate load, as shown in Figure 12a. In practice, suchphenomenon may not occur due to the fact that the effectof a certain length of debonded FRP sheets may quickenthe debonding propagation and fail the structures with-out further increase in load-carrying capacity. Unlikethe aforementioned debonding behavior, the debondingpropagation appears very complicated and more difficultdue to the existence of many cracks in the case of Lc =75 mm. Though macro-debonding occurs at some loca-tions, the FRP sheets can continue to carry additionalload, thus allowing the beam to carry an increasing load.As shown in Figure 16, macro-debonding is first formedat the midspan, but due to the fact that the crack spacing

(a) (b)

0

2

4

6

8

0 50 100 150 200 250 300 350 400 450 500Distance from the midspan (mm)

Shea

r st

ress

(M

Pa)

Debonding propagation

Effective transfer length=80mm 0

500

1000

1500

2000

2500

3000

0 100 200 300 400 500 600 700 800 900Distance from the midspan (mm)

FR

P s

tres

s (M

Pa)

Micro-debondingYielding of rebarMacro-debondingDeflection of 10mmDeflection of 20mm


Effective transfer length=80mm

Fig. 14. Stress distributions for the case of a single localized crack. (a) Interfacial shear stresses. (b) FRP stresses.

is smaller than the observed effective transfer length ofabout 80 mm, the debonding propagation encounters re-sistance from the opposite direction near the location ofthe adjacent crack, which leads to an increased equiva-lent transfer length as shown in Figure 16b. The debond-ing propagation in this case does not appear very smoothlike that observed for the cases of a single localized crackand the large crack spacing. It appears that more en-ergy is required for the debonding to propagate throughthe cracks, which contributes to the increase in externalload.

4.2 Effect of interfacial parameters

In this section, an effort is made to investigate the ef-fect of interfacial properties on the debonding behaviorand the strengthening effect of FRP in the strengthenedRC beam with multiple or distributed flexural cracks.The studied interfacial parameters included initial stiff-ness, ks, local bond strength, τ f, and interfacial fractureenergy, Gb

f . Details of the parameters can be found inTable 2.



(a) (b)

-8

-6

-4

-2

0

2

4

6

8

0 50 100 150 200 250 300 350 400

Distance from the midspan (mm)

Shea

r st

ress

(M

Pa)


Effective transfer length

Increased transfer length 0

5001000150020002500300035004000

0 100 200 300 400 500 600 700 800 900


FR

P s

tres

s (M

Pa)

Micro-debondingYielding of rebarMacro-debondingDeflection of 10mmDeflection of 20mmDeflection of 25mm


Fig. 15. Stress distributions for the case of crack spacing Lc = 300 mm. (a) Interfacial shear stresses. (b) FRP stresses.

(a) (b)

0500

1000150020002500300035004000

0 100 200 300 400 500 600 700 800 900Distance from the midspan (mm)

FR

P s

tres

s (M

Pa)

Micro-debondingYielding of rebarMacro-debondingDeflection of 10mmDefelction of 15mm

Equivalent transfer length-8-6-4-202468

0 50 100 150 200 250 300 350 400


Shea

r st

ress

(M

Pa)

Fig. 16. Stress distributions for the case of crack spacing Lc = 75 mm. (a) Interfacial shear stresses. (b) FRP stresses.

4.2.1 Effect of initial stiffness ks. Through varying theinitial stiffness of the FRP–concrete interface and fix-ing the other two parameters as τ f = 8 MPa andGb

f = 2.0 N/mm, the corresponding effect can be ob-tained in terms of the load-carrying capacity, yielding

(a) (b)

0

2

4

6

8

0 5 10 15 20

Deflection (mm)

She

ar S

tres

s (M

Pa) ks=80MPa/mm

ks=160MPa/mm

ks=320MPa/mm

ks=3200MPa/mm

Micro-debonding

Yielding of rebar

Macro-debonding 0

15

30

45

60

75

90

0 5 10 15 20 25

Deflection (mm)

Loa

d (k

N) ks=80MPa/mm

ks=160MPa/mm

ks=320MPa/mm

ks=3200MPa/mm

Fig. 17. Effect of interfacial stiffness. (a) Load vs. deflection. (b) Interfacial shear stress vs. deflection (midspan).

of the rebar, and interfacial shear stress distributions.Figure 17a shows that nearly no difference is observedin the load–deflection curves for the various values ofthe initial stiffness. As shown in Figure 17b, higher stiff-ness facilitates micro-debonding. This parameter may


364 Niu & Wu

(a) (b)

-8

-6

-4

-2

0

2

4

6

8

0 200 400 600 800


Shea

r st

ress

(M

Pa)

050

100150200250300350400


Reb

ar s

tres

s (M

Pa)

Micro-debonding(ks=80MPa/mm)

Yielding of Rebar(ks=80MPa/mm)

Deflection of 14mm(ks=80MPa/mm)

Micro-debonding(ks=3200MPa/mm)

Yielding of Rebar(ks=3200MPa/mm)

Deflection of 14mm(ks=3200MPa/mm)

Fig. 18. Stress distributions due to different initial interfacial stiffnesses. (a) Rebar stresses. (b) Interfacial Shear Stresses.

have little effect on the yield load due to the occur-rence of softening behavior along the FRP–concrete in-terface prior to the yielding of the rebar. The interfacialshear stress increases with the deflection until micro-debonding is initiated and thereafter decreases slightlydue to the softening behavior of the interface. Upon theyielding of the rebar, the axial stress in FRP sheets in-creases rapidly, thus quickly decreasing the correspond-ing interfacial shear stress until the initiation of macro-debonding. Low interfacial stiffness indicates low stresstransfer rate between concrete and FRP sheets and mayyield a high stress distribution in the internal rebar ascompared to that of high interfacial stiffness, as shownin Figure 18a. However, no significant difference can beobserved in both rebar and FRP sheets after yielding ofthe steel rebar. The difference in interfacial shear stressdistribution is found to be significant between stiffnessvalues of 80 and 3,200 MPa/mm (Figure 18b), but thishas no effect on the load-carrying capacity. This may beinterpreted as the interfacial stiffness not influencing theFRP stress distribution once the rebar yields, providedthat transferring of stresses from concrete to FRP sheetis ensured.

In fact, initial interfacial stiffness affects the rate of in-crease of FRP stress and thus may have an effect on theyield load. However, as softening of FRP–concrete in-terface occurs before yielding of the rebar (Figure 17b),this phenomenon could not be captured clearly in theabove simulation. To clearly demonstrate the effect ofinterfacial stiffness on the yield load, a linear bond–slip model without softening behavior was employed.Figure 19 illustrated that the higher the interfacial stiff-ness, the higher the yield load. But after the yielding ofthe rebar, FRP stress increases to the same degree andthe interfacial stiffness makes no positive contributionto the ultimate load-carrying capacity.

4.2.2 Effect of local bond strength τ f . By fixing ks =160 MPa/mm and Gb

f = 2.0 N/mm, the effect of localbond strength on structural performance was investi-gated. As shown in Figure 20, higher bond strength yieldshigher load before the formation of macro-debonding,but the ultimate load remains the same for different bondstrengths due to the debonding propagation. Figure 20balso shows that high bond strength may slow the initi-ation of micro-debonding, thus transferring more stressto the FRP sheets, which in turn may increase the yieldload. For these cases, it can be seen that yielding of therebar is preceded by the initiation of micro-debonding,thus insignificantly affecting the yield load. It shouldalso be noted that high bond strength induces earlymacro-debonding, which in turn may reduce the struc-tural ductility.

4.2.3 Effect of interfacial fracture energy Gbf . As dis-

cussed previously, the load-carrying capacity of theretrofitted beam is significantly affected by whether ornot crack spacing exceeds the effective transfer length of

0

20

40

60

80

100

120

140

0 5 10 15 20 25Deflection (mm)

Loa

d (k

N)

Linear softening model(ks=160MPa/mm)

Linear model(ks=160MPa/mm)

Linear model(ks=3200MPa/mm)

Yielding of rebar

Yielding of concrete

Fig. 19. Effect of interfacial stiffness on the yield load.



(a) (b)

0

15

30

45

60

75

90

0 5 10 15 20 25Deflection (mm)

Loa

d (k

N)

Bond strength=4MPaBond strength=8MPaBond strength=16MPa

024

68

1012

1416

0 5 10 15 20Deflection (mm)

Shea

r st

ress

(M

Pa)

Micro-debonding

Yielding of rebar

Macro-debonding

Fig. 20. Effect of local bond strength. (a) Load vs. deflection. (b) Interfacial shear stress vs. deflection (midspan).

FRP sheets. Therefore, the effect of interfacial fractureenergy on the structural performance was investigatedfor two kinds of crack spacings: Lc = 300 mm and Lc =112.5 mm. Herein, the parameters are ks = 160 MPa/mm,τ f = 8 MPa, Gb

f = 0.5, 1.0, and 2.0 MPa/mm.

0

15

30

45

60

75

90

0 5 10 15 20

Deflection (mm)

Loa

d (k

N)

Gf=0.5N/mm

Gf=1.0N/mmGf=2.0N/mm

0

500

1000

1500

2000

2500

3000

3500

0 150 300 450 600 750 900


FR

P s

tres

s (M

Pa) Gf=0.5N/mm

Gf=1.0N/mm

Gf=2.0N/mm

2980MPa

2200MPa

1520MPa

(a) (b)

Fig. 21. Effect of interfacial fracture energy for the case of crack spacing Lc = 300 mm. (a) Load vs. deflection. (b) FRP stressdistributions (final step).

0

15

30

45

60

75

90

0 5 10 15 20 25

Deflection (mm)

Loa

d (k

N)

Gf=0.5N/mm

Gf=1.0N/mm

Gf=2.0N/mm

0500

10001500200025003000350040004500


FR

P s

tres

s (M

Pa) Gf=0.5N/mm

Gf=1.0N/mmGf=2.0N/mm

4020MPa

3040MPa

2170MPa

(a) (b)

Fig. 22. Effect of interfacial fracture energy for the case of crack spacing Lc = 112.5 mm. (a) Load vs. deflection. (b) FRP stressdistributions (same debonded length).

As shown in Figures 21 and 22, the load-carrying ca-pacity increases with the interfacial fracture energy. Thiscan be attributed to the fact that large interfacial frac-ture energy yields a large shear transfer length, thusmore external work is required to create the interfacial


366 Niu & Wu

debonding. As for the case of Lc = 300 mm, the loaddoes not increase any more due to the propagation ofinterfacial debonding once macro-debonding is initiated(Figure 21). But for the case of Lc = 112.5 mm, the loadmay still increase even after the formation of macro-debonding, thus yielding a higher load. This is due tothe fact that small crack spacing leads to a complicatedinteraction of cracks and the propagation of interfacialdebonding, thus increasing the load-carrying capacity.

Yuan et al. (2001) and Wu et al. (2002a) derived asimilar formula for predicting the ultimate load-carryingcapacity (Pmax) of FRP-bonded joints irrespective of theconstitutive relationship of the interface:

Pmax = b1

√2Gb

f E1t1 (2)

where E1, t1, and b1 are Young’s modulus, thickness, andwidth of FRP, respectively.

Although this equation was developed from simpleFRP-bonded joints, to some degree, it can be used toclarify how large fracture energy contributes to increas-ing the load-carrying capacity of FRP-strengthened RCbeams. In Figure 21b, the ratio of the maximum FRPstress from 1,520 to 2,200 is about 1.45 and the one from1,520 to 2,980 is 1.96, which is very close to the squareroot of the ratio of two fracture energy values,

√2 and

2.0. The same results can be found in the case of Lc =112.5 mm (Figure 22b). In addition, almost the same ra-tio (about 1.4) of the maximum FRP stress from Lc =300 mm to Lc = 112.5 mm can be found for the same in-terfacial fracture energy, which accounts for the effect ofcrack spacing on the load-carrying capacity. As discussedpreviously, with the decrease of crack spacing, the distri-bution of FRP stresses (e.g., for the case of Lc = 75 mm)may present a monotonic and continuous curve shape,analogous to the case of a single localized crack withthe difference lying in the shear transfer length, whichis clearly shown in Figure 23. This fact may be used for

0500

1000150020002500300035004000

0 100 200 300 400 500 600 700 800 900


FR

P s

tres

s (M

Pa) Single localized crack

Crack spacing=75mm

Equivalent transfer length

Effective transfer length

Approximate curve

Fig. 23. Effect of crack spacing on stress transfer length.

establishing a unified analytical methodology for pre-dicting the debonding failure caused by cracks.

5 CONCLUSIONS

In the present study, a specially designed finite-elementmodel is used to characterize the debonding behaviorand failure mechanisms by multiple or distributed flexu-ral cracks in FRP-strengthened R/C beams, which are yetunknown and very important in the development of anefficient strengthening method with externally bondedFRP sheets. To clarify how the debonding mechanism isaffected by different types of crack distributions in con-crete, a series of parameters such as crack spacing, localbond strength, initial interfacial stiffness, and interfacialfracture energy are varied to investigate the correspond-ing effects on the interfacial debonding behavior and theload-carrying capacity. The following conclusions can bedrawn from the numerical analyses and may be used asknowledge for practical design:

1. The flexural crack spacing has a significant effecton the interfacial debonding mechanism and theultimate load-carrying capacity. For the case ofthe crack spacing larger than the effective trans-fer length of FRP sheets, the debonding mecha-nism and the structural performance are similar tothat of a case with a single localized crack. Oncemacro-debonding is initiated, the debonding wouldpropagate toward the end of FRP sheets and theload would remain constant until the final debond-ing failure. But for the case of crack spacing lessthan or close to the effective transfer length of FRPsheets, the initiation of macro-debonding wouldnot lead to the complete debonding failure of theretrofitted structure. Because more external work isneeded to redistribute the stress between adjacentcracks, this contributes to increase the load-carryingcapacity.

2. The stiffness of FRP–concrete interface may mainlyaffect the yield load, and may have no significant ef-fect on the ultimate load-carrying capacity. A higherinterfacial stiffness will result in a higher yield loadof the FRP-strengthened structure.

3. The local bond strength may only affect the struc-tural behavior prior to the initiation of macro-debonding and have no effect on the ultimateload-carrying capacity. Low bond strength facil-itates mirco-debonding and thus may be of norole in increasing the yield load. However, highbond strength may quicken the initiation of macro-debonding and be unfavorable to the increasing ofthe ductility of the structure. There should be a



balanced design criterion for choosing an adhesivewith appropriate bond strength.

4. The interfacial fracture energy has a decisive effecton the load-carrying capacity. High interfacial frac-ture energy requires more external work to producethe interfacial debonding and yields a large sheartransfer length. Therefore, this will significantly in-crease the load-carrying capacity.

5. With the decrease of crack spacing, the distributionof FRP stresses becomes gradually continuous andfor the case of very close crack spacing the distri-bution of FRP stresses may have a monotonicallycontinuous curve shape analogous to the case of asingle localized crack with the difference lying inthe shear transfer length. This fact may be used forestablishing the analytical methodology for predict-ing the debonding failure load caused by multipleor distributed flexural cracks in RC structures.

ACKNOWLEDGMENTS

The writers wish to acknowledge the partial financialsupports from the National High Technology Researchand Development Program of China (863 Program) un-der grant no. 2001AA336010 and the Joint ResearchFund for Overseas Chinese Young Scholars of the Na-tional Natural Science Foundation of China (grant no.50228808). The insightful and constructive commentsfrom the anonymous reviewers are also greatly appre-ciated in the final manuscript preparation.

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