Engineering Structures 86 (2015) 213–224

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Engineering Structures

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Finite element modelling of debonding failures in steel beams flexurallystrengthened with CFRP laminates

http://dx.doi.org/10.1016/j.engstruct.2015.01.0030141-0296/� 2015 Elsevier Ltd. All rights reserved.

⇑ Corresponding author. Tel.: +852 27666012.E-mail address: cejgteng@polyu.edu.hk (J.G. Teng).

J.G. Teng a,⇑, D. Fernando a,b, T. Yu a,c

a Department of Civil and Environmental Engineering, The Hong Kong Polytechnic University, Hong Kong, Chinab School of Civil Engineering, The University of Queensland, QLD 4072, Australiac School of Civil, Mining and Environmental Engineering, Faculty of Engineering and Information Sciences, The University of Wollongong, Northfields Avenue, Wollongong,NSW 2522, Australia

a r t i c l e i n f o a b s t r a c t

Article history:Received 25 March 2014Revised 2 January 2015Accepted 3 January 2015Available online 21 January 2015

Keywords:CFRPStrengtheningSteel beamsDebondingCohesive zone modellingFinite element analysis

A steel beam may be strengthened in flexure by bonding a carbon fibre-reinforced polymer (CFRP) plateto the tension face. Such a beam may fail by debonding of the CFRP plate that initiates at one of the plateends (i.e. plate end debonding) or by debonding that initiates at a local damage (e.g. a crack or concen-trated yielding) away from the plate ends (intermediate debonding). This paper presents the first finiteelement (FE) approach that is capable of accurate predictions of such debonding failures, with particularattention to plate-end debonding. In the proposed FE approach, a mixed-mode cohesive law is employedto depict interfacial behaviour under a combination of normal stresses (i.e. mode-I loading) and shearstresses (i.e. mode-II loading); the interfacial behaviour under pure mode-I loading or pure mode-II load-ing is represented by bi-linear traction–separation models. Damage initiation is defined using a quadraticstrength criterion, and damage evolution is defined using a linear fracture energy-based criterion.Detailed FE models of steel beams tested by previous researchers are presented, and their predictionsare shown to be in close agreement with the test results. Using the proposed FE approach, the behaviourof CFRP-strengthened steel beams is examined, indicating that: (1) if the failure is governed by plate enddebonding, the use of a CFRP plate with a higher elastic modulus and/or a larger thickness may lead to alower ultimate load because plate end debonding may then occur earlier; (2) plate end debonding is morelikely to occur when a short CFRP plate is used, as is commonly expected; and (3) the failure mode maychange to intermediate debonding or other failure modes such as compression flange buckling if a longerplate is used.

� 2015 Elsevier Ltd. All rights reserved.

1. Introduction

Similar to concrete beams, steel beams or steel–concrete com-posite beams (referred to collectively as ‘‘steel beams’’ hereafterfor simplicity) can be strengthened in flexure by bonding an FRP(generally CFRP) laminate to the tension face [1–11]. Such beamsare herein referred to as FRP-strengthened steel beams. Thebonded FRP laminate may be prefabricated (e.g. by pultrusion) orformed in-situ (e.g. by the wet-layup process), and is referred toas a plate for simplicity in this paper. CFRP is commonly preferredto other FRPs including glass FRP (GFRP) in the strengthening ofsteel structures due to the much higher stiffness of CFRP [7], so thispaper is focused on CFRP strengthening only. For simplicity of

discussion, only simply-supported beams are explicitly considered,so the CFRP plate is bonded to the soffit of the beam.

CFRP flexural strengthening can enhance both the stiffness andthe load-carrying capacity of a steel beam [7,12]. The load-carryingcapacity of such CFRP-strengthened beams may be governed byone or a combination of the many possible failure modes [7], whichinclude: (a) in-plane bending failure (i.e. CFRP failure, concretecrushing); (b) lateral buckling; (c) debonding at a plate end (i.e.plate-end debonding); and (d) debonding away from the plate endsinduced by cracking or concentrated yielding in the steel beam (i.e.intermediate debonding). Additional failure modes include: (e)local buckling of the compression flange; and (f) local buckling ofthe web. Among these failure modes, debonding of the CFRP plate[failure modes (c) and (d)] has been found to be common in labo-ratory tests on CFRP-strengthened steel beams [1,3,6,7,13].

In a CFRP-strengthened steel beam failing by debonding of theCFRP plate, the load-carrying capacity depends on the contribution

214 J.G. Teng et al. / Engineering Structures 86 (2015) 213–224

of the CFRP at the time of debonding, which in turn depends on theinterfacial stress transfer of the adhesive layer. Therefore, accuratesimulation of the bond behaviour of CFRP-to-steel interfaces is ofparticular importance in the theoretical modelling of debondingfailures.

As mentioned above, in a CFRP-strengthened steel beam, bothplate-end debonding and intermediate debonding are possible.Intermediate debonding initiates from the presence of a defect(e.g. a crack or concentrated yielding of the steel substrate) [3]where the FRP plate is highly stressed, and propagates towards aplate end where the stress in the FRP plate is lower. Very limitedresearch is available on intermediate debonding in CFRP-strength-ened steel beams and no theoretical modelling exists so far. Thiskind of debonding, similar to intermediate-crack induced debond-ing (IC debonding) in an FRP-strengthened concrete beam [14], isdominated by interfacial shear stresses. Therefore, accurate simu-lation of intermediate debonding requires an appropriate modelfor the damage behaviour of the interface in the shear direction(i.e. under mode-II loading). Such bond-slip models have previ-ously been developed by the authors’ group [15].

Compared to intermediate debonding, the modelling of plate-end debonding is more involved as it is governed by both interfa-cial shear stresses and interfacial normal stresses [12,16–18].Therefore, the effect of interaction between mode-I loading andmode-II loading on damage initiation and propagation within theadhesive layer needs to be appropriately addressed. A number oftheoretical studies (e.g. [12]) using strength-based approacheshave been conducted on plate-end debonding. These theoreticalanalyses have generally significantly underestimated the perfor-mance of the strengthened beam, as the bond strength dependson the interfacial fracture energy instead of the strength of theadhesive [19]. De Lorenzis et al. [18] have recently presented theonly existing reliable theoretical study based on the fracture-energy approach for plate end debonding of FRP-strengthenedsteel beams. However, De Lorenzis et al.’s study [18], which wasconducted after the present work [15] and based on the theoreticalconcepts presented in the present paper, employed a number ofassumptions to arrive at a simplified analytical solution, is onlyapplicable to cases where the steel beam is linear elastic and sub-jected to three-point bending. The analytical solution presented inRef. [18] was also verified using the FE method described in thepresent paper. No other theoretical (numerical or analytical) stud-ies on plate-end debonding in CFRP-strengthened steel beams havebeen found in which the mixed-mode damage/fracture behaviourof the adhesive layer is appropriately captured.

Against this background, this paper presents a finite element(FE) approach for CFRP-strengthened steel beams, with a particularemphasis on the accurate modelling of the bond behaviour anddebonding failures in such beams. It should be noted that the pres-ent work is based on the premise that debonding failures in FRP-strengthened steel beams occur by cohesion failure within theadhesive layer instead of adhesion failure at the steel/adhesivebi-material interface or the FRP/adhesive bi-material interface. Ofthese two adhesion failure modes, the latter one is much less likelyand the former one needs to be avoided in practice through theproper preparation of the steel surface [20].

2. Modelling of CFRP-to-steel interfaces

The successful prediction of debonding failures in CFRP-strengthened steel beams requires a model for the CFRP-to-steelinterface which has the following characteristics: (1) it accuratelypredicts the behaviour of the interface subjected to pure mode-Iloading and pure mode-II loading; (2) it appropriately accountsfor the effect of interaction between mode-I loading and mode-II

loading on damage propagation along the interface. For (1), anaccurate bond-slip model (e.g. such as those presented in Ref.[15]) and an accurate bond-separation model [21,22] can beemployed to predict the full-range interfacial behaviour underpure mode-II loading and pure mode-I loading respectively. For(2), the so-called mixed-mode cohesive law needs to be used.Among the existing modelling techniques, a coupled cohesive zonemodel appears to be the most suitable as it possesses the tworequired characteristics. Cohesive zone models have been usedfor simulating the fracture of ductile and brittle solids [23], thedelamination of composites [24,25] and the behaviour of adhe-sively bonded joints [26,27]. Bocciarelli et al. [28] presented anFE model for CFRP-to-steel bonded joints with a cohesive zonemodel and showed close predictions for results from double-shearlap tests, but the cohesive zone model used by them only consid-ered interfacial behaviour under pure mode-II loading. In the fol-lowing section, a coupled cohesive zone model is proposed forCFRP-to-steel interfaces, which consists of the following threekey components: a bond-slip model for mode-II loading, a bond-separation model for mode-I loading, and a mixed-mode cohesivelaw. It should be noted that the model presented in the followingsection is for linear adhesives (with a linear stress–strain curvebefore tensile rupture) only, but the general concepts of the modelare extendable to nonlinear adhesives.

2.1. Coupled cohesive zone model

2.1.1. Bond-slip modelThe bi-linear bond-slip model proposed by Fernando [15] for

linear adhesives has been shown to provide accurate predictionsfor the bond behaviour of CFRP-to-steel bonded joints subjectedto mode-II loading, and is thus adopted here. The bi-linear bond-slip model proposed by Fernando [15] can be written as:

s ¼

smaxdd1

if d 6 d1

smaxdf�d

df�d1if d1 < d 6 df

0 if d > df

8>><>>:

ð1Þ

where s is the bond shear stress, smax is the peak bond shear stress,d is the slip, d1 is the slip at peak bond shear stress, and df is the slipat complete failure. Based on the experimental results of CFRP-steelbonded joints with linear adhesives, Fernando [15] derived the fol-lowing expressions for the above bond-slip parameters:

smax ¼ 0:9rmax ð2Þ

d1 ¼ 0:3ta

Ga

� �0:65

rmax ðmmÞ ð3Þ

df ¼2Gf

smaxðmmÞ ð4Þ

where rmax is the tensile strength (in MPa in Eq. (3)) of the adhe-sive, ta and Ga are the thickness and shear modulus of the adhesivelayer respectively, and Gf is the interfacial fracture energy given by[15]:

Gf ¼ 628t0:5a R2 ðN=mm2 mmÞ ð5Þ

R is the tensile strain energy per unit volume of the adhesive whichis equal to the area under the uni-axial tensile stress (in MPa)–strain curve.

2.1.2. Bond-separation modelIt is common to obtain the bond-separation model and the

mode-I fracture energy using double cantilever beam tests (DCB)[21,29]. In the absence of these test data, the bi-linear bond-sepa-ration behaviour of a linear adhesive can be closely approximatedusing the tensile stress–strain data of the adhesive [30]. The peak

J.G. Teng et al. / Engineering Structures 86 (2015) 213–224 215

stress of the bi-linear bond-separation model can be assumed to bethe same as the tensile strength of the adhesive, the slope of theascending branch can be taken to be equal to the tensile elasticmodulus divided by the adhesive thickness (Eq. (8)), and the sepa-ration at complete failure, df, can be taken as the product of thetensile strain at complete failure and the adhesive thickness [30].In the present study, the tensile stress–strain data of the adhesivewere used to define the bi-linear bond-separation model, followingthe approach suggested by Campilho et al. [30].

2.1.3. Mixed-mode cohesive lawIn a coupled cohesive zone model, a mixed-mode cohesive law

is employed to account for the interaction between mode-I loadingand mode-II loading. As the fracture energy for mode-II loading isoften much larger than that for mode-I loading [31], a mixed-modelaw which properly accounts for this aspect is adopted, followingXu and Needleman [32] and Hogberg [31]. For ease of discussion,bond stresses are hereafter referred to collectively as tractionswhile interfacial deformations (i.e. displacements) are referred tocollectively as separations.

The mixed-mode cohesive law adopted in the present studyconsiders tractions and separations in all three directions: thosenormal to the interface and those parallel to the interface (i.e.the two shear components). The normal and the two shear trac-tions are denoted by tn, ts and tt respectively, while the correspond-ing separations are denoted by dn, ds and dt respectively. With thethickness of the cohesive element taken as the original thicknessof the adhesive layer T0, the strains in the normal (en) and thetwo shear directions (cs and ct) are given by:

en ¼dn

T0; cs ¼

ds

T0and ct ¼

dt

T0ð6Þ

2.1.3.1. Elastic behaviour. It is assumed that the interface behaveslinear-elastically until the initiation of damage [18,21,23,26]. Theinterfacial behaviour before damage initiation can thus be repre-sented by

tn

ts

tt

8><>:

9>=>; ¼

Knn 0 00 Kss 00 0 Ktt

264

375

dn

ds

dt

8><>:

9>=>; ð7Þ

where Knn, Kss, Ktt are the elastic stiffness values of the normal andthe two shear directions respectively. It is obvious that Knn shouldbe equal to the initial slope of the bond-separation model formode-I loading and is given by

Knn ¼Ea

T0ð8Þ

where Ea is the elastic modulus of the adhesive.Kss and Ktt are assumed to be the same, and should be equal to

the initial slope of the bond-slip model presented earlier for mode-II loading. From Eqs. (2) and (3):

Kss ¼ Ktt ¼ 3Ga

T0

� �0:65

ð9Þ

Eq. (9) suggests that Kss and Ktt depend on the shear modulus Ga ofthe adhesive. Therefore, the elastic stiffness in the two shear direc-tions and that in the normal direction are inter-related through thePoisson’s ratio.

2.1.3.2. Damage behaviour. Under pure mode-II loading, damage ofthe interface initiates when the shear stress reaches the peak bondshear stress [15]. Similarly, under pure mode-I loading, damageinitiates when the normal stress reaches the peak bond normalstress. Under mixed-mode loading, a strength criterion needs to

be adopted to define the initiation of damage, considering theinteraction between mode-I and mode-II loading. Following exist-ing studies [33,34], the following quadratic strength criterion isadopted in the present study:

htnirmax

� �2

þ ts

smax

� �2

þ tt

smax

� �2

¼ 1 ð10Þ

The symbol hi is the Macaulay bracket which is used to signify thatcompressive stresses do not lead to damage (i.e. when tn is negative,htni is equal to zero). Based on Eq. (10), the damage initiation pointcan be determined when the mode-mix ratio (i.e. the ratio betweenthe fracture energies of two different modes) is known.

After damage initiation, a scalar damage variable D is intro-duced. D is equal to zero at the initiation of damage and is equalto one at complete failure of the interface. The interfacial behav-iour can then be represented by the following equation:

tn

ts

tt

8><>:

9>=>; ¼

ð1� D�ÞKnn 0 00 ð1� DÞKss 00 0 ð1� DÞKtt

264

375

dn

ds

dt

8><>:

9>=>; ð11Þ

where ⁄ means that if tn is compressive, D⁄ is equal to zero.The complete failure of the interface, when D is equal to one, is

defined based on fracture energy considerations. While a few othercriteria for the definition of complete failure are available (e.g. thequadratic criterion or the BK criterion proposed by Benzeggagh andKenane [35]), the linear criterion is adopted in the present studydue to its simplicity and good performance for adhesive joints[34,36]. The linear criterion is expressed by:

G�nGIþ Gs

GIIþ Gt

GII¼ 1 ð12Þ

where Gn, Gs, Gt are the works done by the tractions and their con-jugate displacements in the normal and the two shear directionsrespectively (Fig. 1a). GI and GII represent the interfacial fractureenergies required to cause failure when subjected to pure mode-Iloading and pure mode-II loading respectively.

To describe the evolution of damage, the definition of an effec-tive displacement is introduced as follows:

dm ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffihdni2 þ d2

s þ d2t

qð13Þ

With this definition, the displacement at complete failure, dfm,

can be found using Eq. (12) for a certain mode-mix ratio. The dam-age variable D is then defined by the following equation assuminglinear softening of the interface [34]:

D ¼df

m dmaxm � d0

m

� �dmax

m dfm � d0

m

� � ð14Þ

where d0m is the effective displacement at the initiation of damage

and dmaxm is the maximum value of the effective displacement

attained in the loading process (Fig. 1b).

3. FE modelling of CFRP-strengthened steel I-beams

In this section, an FE approach for debonding failures in CFRP-strengthened steel I-beams is first presented, in which the coupledcohesive zone model presented above for CFRP-to-steel interfacesis employed. Numerical results obtained with the FE approachare then given to demonstrate the capability of the proposed FEapproach in predicting debonding failures as well as other possiblefailure modes (e.g. compression flange buckling), to clarify theeffect of approximating the mode-I fracture energy on damagepropagation in the adhesive layer, and to study the effect of plateaxial stiffness on plate end debonding failure. With these aims in

(a) Traction-separation curve

(b) Linear damage evolution under mixed-mode loading

, , , ,

Traction

Separation

σmax, τmax

Knn, Kss, Ktt (1-D)K

GI-Gn, GII-Gs,GII-Gf

Gn,Gs,Gt

, ,

Traction

Separation

n s t n s t n s t1

m

mc

m0

mf

1 1 f f f

max

G

Fig. 1. Traction–separation curves and linear damage evolution under mixed-modeloading. (a) Traction–separation curve. (b) Linear damage evolution under mixed-mode loading.

216 J.G. Teng et al. / Engineering Structures 86 (2015) 213–224

mind, FE models were developed for four beams tested by Dengand Lee [13], and were verified using the test results. Deng andLee’s tests [13] were selected for comparison among many otherexperimental studies (e.g. [1,3,7]) as Deng and Lee’s tests [13]had the following desirable characteristics: (1) debonding failurescontrolled by cohesion failure occurred; (2) experimental load–dis-placement curves were reported; (3) different failure modes (e.g.

LCFR

1100

127x76UB13 steel I beam

1 mm thick adhesive layer

127mm

76mm

76mm

Fig. 2. Details of test specime

plate-end debonding and compression flange buckling) wereobserved due to the use of CFRP plates with different lengths.These characteristics make Deng and Lee’s tests [13] the most suit-able for verifying the proposed FE approach, especially in terms ofthe behaviour of CFRP-to-steel interfaces.

3.1. Beam tests conducted by Deng and Lee [7]

Four of the beams tested by Deng and Lee [13] were selected forFE simulation, including one control beam without CFRP strength-ening and three beams strengthened with 3 mm thick CFRP platesof three different lengths (i.e. 300 mm, 400 mm, and 1000 mm)respectively. These beams were selected because they all had thesame loading configuration (i.e. three-point bending), the samesteel section, and a single continuous CFRP plate; the only variablewas the plate length to examine how the failure mode wouldchange with the plate length (from plate end debonding to buck-ling of compression flange) and whether this could be accuratelypredicted by the proposed FE approach. The four selected beamswere named by Deng and Lee [13] as specimens S300 (controlbeam), S303, S304 and S310 respectively, where the last two num-bers represent the length of the CFRP plate and the first number‘‘3’’ indicates that the beams were subjected to three-point bend-ing. These four steel beams all had a length of 1.2 m (with a clearspan between the supports being 1.1 m) and a cross-section of type127x76UB13; the dimensions of the steel beams are shown inFig. 2. Grade 275 steel was used, which means that the steel hada nominal yield strength of 275 MPa (with the actual yield strengthoften being larger than 275 MPa) and a tensile elastic modulus of205 GPa. The CFRP plates used all had a thickness of 3 mm, a widthof 76 mm, and an elastic modulus in the fibre direction of 212 GPa.To avoid premature flange buckling and web crushing, two 4 mmthick steel plate stiffeners were welded to each beam at the mid-span, one on each side of the web. For beams with a short CFRPplate (i.e. 300 mm or 400 mm), plate end debonding of the CFRPplate was observed. However, when a longer CFRP plate (i.e.1000 mm) was used, failure was controlled by the buckling ofthe compression flange of the steel section, which was the same

P

mm

3 mm thick CFRP plate

4mm

7.6mm

ns of Deng and Lee [13].

Table 1Details of the test beams and the FE models.

Specimen/modelname

Length of CFRP plate,mm

Elastic modulus of CFRP,GPa

Thickness of CFRP plate,mm

Mode Ibehaviour

Compression flangestrengthening

S300a N/A N/A N/A N/A NoS303a 300 212 3 N/A NoS304a 400 212 3 N/A NoS310a 1000 212 3 N/A NoS300-0-000b N/A N/A N/A N/A NoS303-1-212b 300 212 3 Model A NoS303-2-212b 300 212 3 Model B NoS303-1-330b 300 330 3 Model A NoS304-1-212b 400 212 3 Model A NoS310-1-212b 1000 212 3 Model A NoS310-1-212-Pb 1000 212 3 Model A 6 mm steel plate

a Test beams [13].b FE models.

J.G. Teng et al. / Engineering Structures 86 (2015) 213–224 217

as the failure mode of the control beam (i.e. specimen S300). Thedetails of the test beams are summarized in Table 1.

3.2. FE models

FE models were created using ABAQUS [37] for the four beams,with the exact dimensions and support conditions (i.e. simply-sup-ported boundary conditions) as given in Fig. 2 and Table 1. Thegeneral purpose shell element S4R with reduced integration wasadopted for both the steel section and the CFRP plate, while theadhesive layer was modelled using the cohesive element COHD8available in ABAQUS. The two full-depth stiffeners on the two sidesof the web in the mid-span region were included, and the threeedges of each stiffener in contact with the I beam were tied tothe top flange, the bottom flange and the web of the cross sectionrespectively. Similarly, the top surface and the bottom surface ofthe adhesive layer were tied to the bottom surface of the steelbeam and the top surface of the CFRP plate respectively. Basedon a mesh convergence study, 2.5 mm � 2.5 mm elements wereselected for the steel beam and the CFRP plate, while2.5 mm � 2.5 mm � 1 mm (with 1 mm being in the thicknessdirection) elements were selected for the adhesive layer. In allthe FE simulations, the analysis was terminated soon after the ulti-mate load had been reached.

The well-known J2 flow theory was employed to model thematerial behaviour of the steel. As the experimental stress–straincurve of the steel was not given by Deng and Lee [13], a tri-linear(i.e. elastic-perfectly plastic-hardening) stress–strain model [38]with a yield strength of 330 MPa (determined by a trial-and-errorprocess to match the linear portion of the experimental load–dis-placement curve of specimen S300) and a ultimate tensile stressof 430 MPa (as specified in BS EN 10025-1 [39]) was adopted.The use of such an idealized stress–strain curve for the steel isbelieved to be the best pragmatic solution possible in the absenceof the experimental stress–strain curve and has only minor effectson the predictions for the steel beam (see [15]).

The CFRP plate was treated as an orthotropic material in the FEmodels. In the fibre direction, the elastic modulus (i.e. E3) providedby Deng and Lee [13] was adopted (i.e. 212 GPa based on a nominalthickness of 3 mm). The elastic modulus in the other two directions

Table 2Key parameters for traction–separation models.

Loading mode Peak bond stress (MPa) Displacement a

Mode I (model A) 29.7 0.00371Mode I (model B) 29.7 0.00371Mode II 26.7 0.0526

(i.e. E1, E2), the Poisson’s ratios and the shear moduli were assumedthe following values respectively based on the values reported inDeng et al. [16]: E1 = E2 = 10 GPa, m12 = 0.3, m13 = m23 = 0.0058,G12 = 3.7 GPa and G13 = G23 = 26.5 GPa.

The coupled cohesive zone model presented earlier wasadopted to model the constitutive behaviour of the adhesive layer.Deng and Lee [13] provided only the tensile strength (29.7 MPa),the tensile elastic modulus (8 GPa) and the shear modulus(2.6 GPa) for the adhesive. Considering that the adhesive used byDeng and Lee [13] was a linear adhesive, the strain energy was cal-culated by assuming a bi-linear stress–strain curve with the slopeof the ascending branch being equal to the elastic modulus (i.e.8 GPa), the peak stress being equal to the tensile strength (i.e.29.7 MPa) and the ultimate strain being equal to 4% which is thevalue provided by the manufacturer [13]. With the above parame-ters, the bond-slip model for mode-II loading can be determinedusing Eqs. (1)–(4), and the key parameters of the so-obtainedbond-slip model are given in Table 2. The bond-separation modelfor mode-I loading can also be determined using the assumed bi-linear stress–strain curve; the key parameters of the so-obtainedbond-separation model are given in Table 2 as model A. Besidesmodel A whose mode-I fracture energy is 0.059 N/mm, anotherbond-separation model (i.e. model B, see Table 2) was also used,whose mode-I fracture energy (i.e. 0.11 N/mm) is twice the elasticenergy of model A. The use of two different bond-separation mod-els for mode-I loading was to explore the effect of mode-I fractureenergy on damage propagation in the adhesive layer.

As failure of specimens S300 and S310 was controlled by com-pression flange buckling, their behaviour may be affected by geo-metric imperfections such as those specified in Section 14.4.3 ofAS4100 [40]. As no measured geometric imperfections werereported by Deng and Lee [13] for the test beams, the out-of-square imperfection, which was found to be the most influentialfor flange compression buckling among the three types of imper-fections specified in AS4100 [40] (see [15]), was chosen for inclu-sion in the FE models; a magnitude of 1.3 mm was selected by atrial-and-error process to match the ultimate load of the controlbeam (i.e. specimen S300). Residual stresses, as described in Piand Trahair [41], were also included in the FE models [15]. Itshould be noted that although the geometric imperfection and

t peak bond stress, d1 (mm) Interfacial fracture energy, Gf (N/mm)

0.05940.1101.59

0

20

40

60

80

100

120

140

0 10 20 30

Load

(kN

)

Mid-span deflection (mm)

S300- Experimental(Deng and Lee 2007)

S300-0-000

Fig. 3. Load–displacement curves of a bare steel I beam.

218 J.G. Teng et al. / Engineering Structures 86 (2015) 213–224

the residual stresses adopted in the FE models may not be exactlythe same as those in the test beams, their effects on the predictionsare very limited: the geometric imperfection has little effect on theload–displacement curve before the ultimate load which is con-trolled by the buckling of the steel section, and the residual stres-ses only have some effects on the slope of the curve close to theyield load (i.e. the load at which the load–displacement becomesnonlinear) (see [15]).

In total six FE models were developed (Table 1). The controlbeam FE model is referred to as S300-0-000, where the first 4 digitsindicate the test specimen configuration, the middle digit indicatesthe model employed for the bond-separation behaviour undermode-I loading (‘‘0’’ for the control beam as no CFRP strengtheningwas provided), and the last three digits represent the elastic mod-ulus of the CFRP plate in the fibre direction given in GPa (‘‘000’’ asno CFRP strengthening was provided in the control beam). Thesame naming method is used in the paper for the other FE models.

Two FE models were developed for specimen S303 tested byDeng and Lee [13], and they differ only in the bond-separationmodel for mode-I loading. These two FE models are referred to asmodels S303-1-212 and S303-2-212 respectively, where the num-bers ‘‘1’’ and ‘‘2’’ in the middle indicate respectively the use of themodel A and the model B bond-separation laws for mode-I loading.Besides these two FE models, an additional FE model (model S303-1-330) was also created, with all the details being the same asmodel S303-1-212 except that the elastic modulus of the CFRPplate in the fibre direction was increased to 330 GPa in modelS303-1-330. This additional model was created to investigate theeffect of axial stiffness of CFRP plate.

FE models S304-1-212 and S310-1-212 were respectively cre-ated for beams S304 and S310 tested by Deng and Lee [13]. Toexamine the possibility of intermediate debonding, an additional

(a) Deformed shape of S303-1-212 at failure

Adhesive layer

Plate end debonding

Dark blue color: zero stress region Red color: high stress region

(c) Deformed shape of model S310-

Compression flange buckling

Fig. 4. Deformed shapes from FE model

FE model was built, where all the details are exactly the same asthose of model S310-1-212 except that an additional 6 mm thicksteel plate identical in material properties to the steel sectionwas added (using tied nodes on the plate edges in the FE model)to the top flange of the steel section, so that the buckling of thetop flange can be suppressed. This FE model is referred to asS310-1-212-P where ‘‘P’’ indicates the addition of a steel plate onthe top flange.

4. Results and discussions

4.1. Accuracy of assumed properties for the steel beam

The FE results are compared with the experimental load–dis-placement curve of the control beam (i.e. specimen S300) inFig. 3. As explained earlier, both the material stress–strain curve

(b) Deformed shape of model S304-1-212 at failure

Adhesive layer

Plate end debonding

Dark blue color: zero stress region Red color: high stress region

1-212 at failure

s for CFRP-strengthened specimens.

(a) Load-deflection curves for specimen S303

(b) Load-deflection curves for specimens S304 and S310

0102030405060708090

100110120130

0 2 4 6 8 10 12

Load

(kN

)

Mid-span deflection (mm)

S303-Experimental(Deng and Lee 2007)S303-1-212

S303-2-212

S303-1-330damage initiation of S303-1-330

damage initiation of S303-1-212 and S330-2-212

debonding initiation of S303-1-330

debonding initiation of S303-1-212

debonding initiation of S303-2-212

020406080

100120140160180200

0 10 20 30

Load

(kN

)

Mid-span deflection (mm)

S304- Experimental(Deng and Lee 2007)S310- Experimental(Deng and Lee 2007)S304-1-212

S310-1-212

S310-1-212-P

damage initiation of S310-1-212-P

debonding initiation of S310-1-212-P

damage initiation of S304-1-212

debonding initiation of S304-1-212

Fig. 5. Load–deflection curves. (a) Load–deflection curves for specimen S303. (b)Load–deflection curves for specimens S304 and S310.

(a) 92.6 kN (b) 118.3 kN

Z

Z Debonding initiation

Debonding initiation

76mm

300mm Mid-span

Red colour: high stress region; Green colour:intermediate stress region; Blue colour: low stress region

Fig. 6. Longitudinal shear stresses in the adhesive from S303-1-212. (a) 92.6 kN. (b)118.3 kN.

J.G. Teng et al. / Engineering Structures 86 (2015) 213–224 219

and the geometric imperfections of the steel beam employed in theFE model were obtained through a trial-and-error process toachieve a close prediction of the experimental load–displacementcurve of specimen S300. With these calibrated input data, the FEresults are seen to agree closely with the test results (Fig. 3). Thematerial and geometric properties adopted in the FE model forthe control beam are thus believed to approximate the experimen-tal values well, and any errors arising from these input data arebelieved to have negligible effects on the predicted response ofCFRP-strengthened steel beams.

Table 3Experimental and FE results.

FE model Experimental results FE results

Ultimate load,Pu (kN)

Deflection at ultimateload, Du (mm)

Ultimate load,Pu-FE (kN)

Deflectioload, Du-

S300-0-000a

123 20.7 120 21.0

S303-1-212b

120 5.12 125 7.05

S303-2-212b

120 5.12 125 7.07

S303-1-330b

N/A N/A 123 6.17

S304-1-212b

135 7.00 136 11.0

S310-1-212a

160 20.1 158 20.8

S310-1-212-Pc

N/A N/A 188 27.3

a Compression flange buckling.b Plate end debonding.c Intermediate debonding.

4.2. Plate end debonding failures

In Deng and Lee’s tests [13], specimens S303 and S304 werefound to fail by the plate end debonding of the CFRP plate. Thesame failure mode was also predicted by all the three FE modelsof S303 (i.e. S303-1-212, S303-2-212, and S303-1-330) and theFE model of S304 (i.e. S304-1-212). The failure mode (i.e. deformedshape at ultimate load) obtained from model S303-1-212 is shownin Fig. 4a while those for models S303-2-212 and S303-1-330 aresimilar; the failure mode from model S304-1-212 is shown inFig. 4b.

The load–deflection curves obtained from these FE models arecompared with the experimental curve in Fig. 5; other key resultsare summarized in Table 3. To further examine the FE results forbeams failing by debonding (i.e. S303, S304 and S310-1-212-P),two key points are marked on each of the predicted load–displace-ment curves in Fig. 5: (1) the point when damage initiates in the

n at ultimateFE (mm)

Load at debondinginitiation, Pd-FE (kN)

Deflection at debondinginitiation, Dp-FE (mm)

N/A N/A

118 5.08

122 5.78

115 4.60

132 8.00

N/A N/A

188 27.3

00.10.20.30.40.50.60.70.80.9

0 0.05 0.1 0.15

Nor

mal

ized

stre

ss

Strain

Normal stress-150mm from the mid-spanLongitudinal shear stress-150mm from the mid-spanNormal stress-147.5mm from the mid-spanLongituninal shear stress-147.5mm from the mid-span

Fig. 7. Interfacial stress–strain behaviour at the plate end from model S303-1-212.

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0 20 40 60 80 100 120 140

Nor

mal

ized

stre

ss

Load (kN)

Longitudinal shear-S303-1-212

Normal-S303-1-212

Longitudinal shear-S303-2-212

Normal-S303-2-212

Longitudinal shear-S303-1-330

Normal-S303-1-330

Fig. 8. Normalized interfacial stresses at the plate end for specimen S303.

220 J.G. Teng et al. / Engineering Structures 86 (2015) 213–224

adhesive layer (i.e. the interfacial stresses start to decrease withthe bond displacements); (2) the point when debonding initiates(i.e. when complete damage occurs at a certain location and theinterfacial stresses there reduce to zero). For the specimens failingby plate end debonding (i.e. S303 and S304), the predicted loads atthe damage initiation point are seen to be much lower than thecorresponding ultimate loads achieved in the tests (Fig. 5). Consid-ering that damage initiates when the strength of the adhesive isreached, this observation demonstrates that the strength-basedapproach (e.g. [2,12]) can substantially underestimate the load atplate-end debonding, as found by Colombi and Poggi [1]. It is alsointeresting to note that the predicted loads by FE models S303-1-212 and S304-1-212 at the debonding initiation point (i.e.118.3 kN and 132.0 kN) are both very close to their experimentalultimate loads respectively (i.e. 120.0 kN and 135.0 kN); the

(a) 102 kN (b) 140.5 k

Red colour: high stress region; Green colour: i

stress re

Z

1000mm

Mid-span

Z

76mm

Fig. 9. Longitudinal shear stresses in the adhesive from model S3

corresponding displacements predicted by the two FE models(i.e. 5.08 mm and 8.00 mm) are also close to their respective exper-imental displacements at the ultimate load (i.e. 5.12 mm and7.00 mm). This observation suggests that if failure of the strength-ened beam is assumed to occur at the debonding initiation point,these FE models can closely predict both the ultimate load andthe load–displacement curve up to the ultimate load. Such anassumption is regarded to be reasonable as after the initiation ofdebonding at a CFRP plate end, a sudden energy release can beexpected as the debonding propagation is a dynamic process dri-ven by both the interfacial normal stresses and the interfacial shearstresses. During this process, idealistic debonding propagation pre-dicted by a static analysis (i.e. the part after the debonding initia-tion point on the predicted load–displacement curve) cannot

N (c) 159.7 kN (peak load)

ntermediate stress region; Blue colour: low

gion

High shear stress

Softening near mid-span

10-1-212. (a) 102 kN. (b) 140.5 kN. (c) 159.7 kN (peak load).

J.G. Teng et al. / Engineering Structures 86 (2015) 213–224 221

occur; sudden debonding propagation is likely to happen insteadwhich means no further load increases can occur in a test.

The interfacial stress distributions over the adhesive layer inmodel S303-1-212 are shown in Fig. 6 for two different load levels.The interfacial stress distributions in the other two S303 FE modelsand the S304 FE model are similar and are thus not provided here.Damage is seen to initiate at the two plate ends and propagatestowards the mid-span (Fig. 6). Fig. 7 compares the normalized nor-mal stress (i.e. normalized by the tensile strength)–normal straincurves and the normalized shear stress (i.e. normalized by the peakbond shear stress)–shear strain curves for the adhesive at the endof the plate (i.e. 150 mm from the mid-span) and those for theadhesive at 2.5 mm away from the plate end (i.e. 147.5 mm fromthe mid-span). From Fig. 7 it can be seen that, after the initiationof damage at a plate end, both the interfacial normal stress andthe longitudinal shear stress at the very end of the plate decrease,but the shear stress in the region nearby starts to increase. Fig. 7also shows that the maximum normal stress and the maximumlongitudinal shear stress are equally high at the very end of theplate, but the former is significantly lower than the latter at a loca-tion 2.5 mm away from the plate end, suggesting that the signifi-cant effect of the normal stress on damage propagation of theinterface is limited only to a small region close to the plate end.

(a) 102 kN

(b) 140.5 kN

(c) 159.5 kN (peak load)

-0.2-0.15

-0.1-0.05

00.05

0.10.15

0.20.25

0 100 200 300 400 500

Nor

mal

ized

stre

ss (M

Pa)

Distance from the mid span (mm)

Normalstress

Longitudinalshear stress

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

0 100 200 300 400 500

Nor

mal

ized

stre

ss (M

Pa)

Distance from the mid span (mm)

Normal stress

Longitudinalshear stress

-1-0.8-0.6-0.4-0.2

00.20.40.60.8

1

0 100 200 300 400 500

Nor

mal

ized

stre

ss (M

Pa)

Distance from the mid span (mm)

Normal stress

Longitudinalshear stress

Fig. 10. Interfacial stress distributions along section Y–Y in Fig. 9a, from modelS310-1-212. (a) 102 kN. (b) 140.5 kN. (c) 159.5 kN (peak load).

4.3. Effect of mode-I fracture energy and CFRP plate stiffness

Fig. 5a shows that FE models S330-1-212 and S303-2303-2-212predict very similar load–deflection curves despite the use of dif-ferent bond-separation models. The load–deflection curve pre-dicted by model S303-1-330 is slightly higher than the other twoafter the initial linear portion because of the use of a stiffer CFRPplate, but ends at a lower ultimate load. It is also seen that thecurves predicted by FE models S303-1-212 and S303-2-212, forwhich the CFRP properties adopted are the same as those of thetest specimen, are very close to the experimental curve except thatthey both have a higher ultimate load (Fig. 5a).

The interfacial normal stresses and the interfacial longitudinalshear stresses at the plate end (i.e. 150 mm from the mid span)from the three models (i.e. S303-1-212, S303-2-212 and S303-1-330) are shown in Fig. 8. The magnitudes of the interfacial trans-verse shear stresses are very small, so they are not shown in thisfigure. It is clear that damage initiates at a load of 83 kN in FE mod-els S303-1-212 and S303-2-212, but initiates at a smaller load (i.e.71 kN) in model S303-1-330. Before damage initiation, the normal-ized normal stress and the normalized shear stress are similar in FEmodels S303-1-212 and S303-2-212, and are lower than those inmodel S303-1-330, indicating that a stiffer CFRP plate leads to lar-ger interfacial stresses. In addition, the load at the debonding initi-ation point is smaller from model S303-1-330 (i.e. 115 kN) thanthose from the other two models (i.e. around 120 kN), suggestingthat steel beams strengthened with a stiffer CFRP plate are likelyto fail at a smaller load by plate end debonding.

It is also interesting to note that although quite different bond-separation models for mode-I loading were employed, the predic-tions of FE models S303-1-212 and S303-2-212 are very similar(Fig. 5a). The predictions are exactly the same before the initiationof damage at the plate end (Fig. 8). After damage initiation, theinterfacial stresses in model S303-1-212 are seen to decreaseslightly more rapidly with the load than those in model S303-2-212, as the mode-I interfacial fracture energy adopted in the for-mer is smaller which leads to a smaller total interfacial fractureenergy at failure. However, as the mode-II fracture energy (i.e.1.59 N/mm) adopted by both FE models is much larger than themode-I fracture energy (i.e. 0.059 N/mm for model S303-1-212and 0.11 N/mm for model S303-2-212), the use of a larger mode-I fracture energy in model S303-2-212 has only a small effect onthe total fracture energy at failure for mixed-mode loading. Thisexplains the very similar predictions of the two models. It shouldbe noted that for linear adhesives commonly used in CFRP-to-steelbonded joints, the mode-II fracture energy is often much largerthan the mode-I fracture energy [31], so the debonding of suchjoints under mixed-mode loading is often governed by the mode-II fracture energy. This also suggests that the method adopted inthe present study for estimating the mode-I fracture energy (i.e.the method used for deriving bond-separation model A in Table 2)can work well for common linear adhesives.

4.4. Compression flange buckling failures

Deng and Lee [13] indicated that beam S310 failed by the buck-ling of the compression flange of the steel section. The same failuremode was also predicted by model S310-1-212. The deformedshape at failure obtained from model S310-1-212 is shown inFig. 4c. The load–deflection curve predicted by model S310-1-212is seen to compare very well with the experimental results(Fig. 5b). It should also be noted that, while S300-0-000 andS310-1-212 both failed due to compression flange buckling, thelatter achieved a higher load carrying capacity than the formerdue to the contribution of the bonded FRP plate.

(a) 33.7kN (b) 159.5kN

High longitudinal shear stresses

Mid-span

76mm

1000mm

Z

Z

(c) 188.05kN (peak load) (d) 186.7kN (post peak curve) Red colour: high stress region; Green colour: intermediate stress region; Blue colour: low

stress region

Debonding initiation

Debonding Mid-span

Fig. 11. Damage propagation in the adhesive layer in model S310-1-212-P. (a) 33.7 kN. (b) 159.5 kN. (c) 188.05 kN (peak load). (d) 186.7 kN (post peak curve).

222 J.G. Teng et al. / Engineering Structures 86 (2015) 213–224

The interfacial longitudinal shear stress patterns over the adhe-sive layer at different load levels are shown in Fig. 9, while the nor-malized interfacial stress distributions along section Z–Z atdifferent load levels are shown in Fig. 10. Before the load reaches102 kN, both the normal stress and the longitudinal shear stressare relatively low, and the maximum interfacial stresses occur atthe plate end (Fig. 10a). As the load increases, the longitudinalshear stress at the mid-span becomes higher than those at theplate ends (Fig. 10b). At the ultimate load, softening in the region

close to the mid-span has already begun (Fig. 10c), but no debond-ing occurs before the buckling of the compression flange.

4.5. Intermediate debonding failures

As expected, the failure mode predicted by model S310-1-212-Pis the intermediate debonding of the CFRP plate initiating fromnear the mid-span (Fig. 11c). The load–deflection curve predictedby model S310-1-212-P is shown in Fig. 5b. The interfacial

(a) 176.8 kN

(b) 184.8 kN

(c) 188.05 kN (peak load)

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

0 100 200 300 400 500

Nor

mal

ized

stre

ss (M

Pa)

Distance from the mid span (mm)

Normal stress

Longitudinalshear stress

-0.6-0.4-0.2

00.20.40.60.8

11.2

0 100 200 300 400 500

Nor

mal

ized

stre

ss (M

Pa)

Distance from the mid span (mm)

Normal stress

Longitudinalshear stress

-1-0.8-0.6-0.4-0.2

00.20.40.60.8

1

0 100 200 300 400 500

Nor

mal

ized

stre

ss (M

Pa)

Distance from the mid span (mm)

Normal stress

Longitudinalshear stress

(d) 186.7 kN (a peak-peak state)

-1-0.8-0.6-0.4-0.2

00.20.40.60.8

1

0 100 200 300 400 500

Nor

mal

ized

stre

ss (M

Pa)

Distance from the mid span (mm)

Normal stress

Longitudinalshear stress

Fig. 12. Interfacial stress distributions along section Z–Z in Fig. 11a, from modelS310-1-212-P. (a) 176.8 kN. (b) 184.8 kN. (c) 188.05 kN (peak load). (d) 186.7 kN (apeak–peak state).

J.G. Teng et al. / Engineering Structures 86 (2015) 213–224 223

longitudinal shear stress patterns over the adhesive layer at differ-ent load levels are shown in Fig. 11, while Fig. 12 shows thenormalized interfacial stress distributions along section Z–Z(Fig. 11a) at different load levels. The interfacial stress distributionsfrom model S310-1-212-P at low load levels are similar to thosefrom model S310-1-212. However, as the load increases, a largeincrease in the longitudinal shear stress near the mid-span is seen,which also means a higher contribution of the CFRP plate. At theultimate load, the longitudinal shear stress near the mid-span is

seen to have dropped significantly (Fig. 12c). The longitudinalshear stress distribution at a post-peak state (Fig. 12d) clearly indi-cates the existence of two debonded portions of the interface onthe two sides of the mid-span.

4.6. Possible failure modes of CFRP-strengthened steel beams

The discussions above indicate that the proposed FE approachcan provide accurate predictions for the response of CFRP-strengthened steel beams failing in different failure modes, interms of both the ultimate load and the load–displacement curve.

In CFRP-strengthened steel beams, the contribution of the CFRPplate relies on the stress transfer function of the adhesive layer, sothe ultimate load of such beams is often governed by the failure ofthe adhesive layer. When a short CFRP plate is used, the interfacialstresses (i.e. both the normal stress and the shear stress) at theplate end are large, and the failure is often governed by plate enddebonding. When a longer CFRP plate is used, the interfacial stres-ses at the plate end are smaller and the critical region of the adhe-sive layer may move to the mid-span region. In such cases, thefailure mode may change from plate end debonding to intermedi-ate debonding, buckling of the compression flange of the steel sec-tion, and tensile rupture of the FRP plate.

5. Conclusions

This paper has been concerned with the accurate prediction ofdebonding failures in simply-supported steel beams strengthenedin flexure with a bonded CFRP soffit plate using the FE method.An FE approach has been presented in the paper, in which bi-lin-ear traction–separation models are employed to represent puremode-I and pure mode-II responses of the interface for a linearadhesive; a mixed-mode cohesive law is employed to considerinteractions between mode-I loading and mode-II loading. Dam-age initiation is defined using a quadratic strength criterion,and damage evolution is defined using a linear fracture energy-based criterion, both of which take due account of mixed-modeloading. The proposed FE approach represents a significantadvancement in the modelling of debonding failures in CFRP-strengthened steel structures.

Predictions from the FE approach have been shown to comparewell with the test results reported by Deng and Lee [13] for CFRP-strengthened beams failing by either the plate-end debonding ofthe CFRP plate or the compression flange buckling of the steel sec-tion. It was also concluded from the study that when a static FEanalysis is conducted, the ultimate load of a beam failing by plateend debonding should be taken as the load at which debonding ini-tiates at the plate end.

Using the proposed FE approach, the behaviour of CFRP-strengthened steel beams was examined and it was found that:(1) a CFRP plate with a higher elastic modulus and/or a largerthickness leads to a lower ultimate load by plate end debonding;(2) plate end debonding is more likely to occur when a short CFRPplate is used, as is commonly expected; and (3) the failure modemay change to intermediate debonding or other modes such ascompression flange buckling if sufficiently long CFRP plate is used.

Acknowledgements

The authors are grateful for the financial support received fromThe Hong Kong Polytechnic University provided through an Inter-national Postgraduate Scholarship for PhD Studies to the secondauthor and through a Postdoctoral Fellowship to the third author.

224 J.G. Teng et al. / Engineering Structures 86 (2015) 213–224

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