Article

Advances in Structural Engineering1–14� The Author(s) 2017Reprints and permissions:sagepub.co.uk/journalsPermissions.navDOI: 10.1177/1369433217729518journals.sagepub.com/home/ase

Fatigue behaviour of cracked steelbeams retrofitted with carbonfibre–reinforced polymer laminates

Qian-Qian Yu1,2 and Yu-Fei Wu3

AbstractIn recent years, externally bonded carbon fibre–reinforced polymer has been considered an innovative way to strengthen steel struc-tures attributed to its high strength-to-weight ratio, excellent corrosion resistance and fatigue performance. This article presents anexperimental and numerical study on the fatigue behaviour of defected steel beams strengthened with carbon fibre–reinforced poly-mer laminates, with a special focus on the effect of interfacial debonding. Analytical modelling and numerical simulation confirmed thatthe interfacial debonding had a pronounced effect on carbon fibre–reinforced polymer strain and stress intensity factor at the crackfront. After introducing interfacial debonding from experimental findings into the numerical analysis, the fatigue life and crack propaga-tion versus cycle numbers of the specimens compared well with the test results. Based on the current experimental program, speci-mens with Sikadur 30 were more prone to debonding failure; therefore, Araldite 420 is suggested for strengthening schemes.

Keywordscarbon fibre–reinforced polymer, debonding, fatigue, steel beam, strengthen, stress intensity factor

Introduction

Assessment and rehabilitation of fatigue behaviour ofmetallic structures is of great concern in the civil engi-neering community. Carbon fibre–reinforced polymer(CFRP) materials, attributed to the high strength-to-weight ratio, excellent corrosion resistance and fatigueperformance, have been widely used in retrofitting ofconcrete structures (Kezmane et al., 2016; Wu andLiu, 2013). In recent years, researchers have exploredCFRP application for fatigue strengthening of steelstructures (Teng et al., 2012; Zhao and Zhang, 2007).

Extensive studies have demonstrated that externallybonded CFRP could effectively improve the fatiguebehaviour of defected steel plates (Colombi et al., 2014;Feng et al., 2014; Jones and Civjan, 2003; Liu et al.,2009; Yu et al., 2013, 2014a). The overlay is helpful insharing the far-field load and restraining the crack open-ing. Parametric analysis was performed on the mechani-cal properties of overlays and bond configurations tooptimize the strengthening scheme (Brighenti, 2007;Taljsten et al., 2009; Wang et al., 2013; Yu et al., 2014b).

Research on strengthening of steel beams is com-paratively less reported. Jiao et al. (2012) tested 21 ini-tially cracked steel beams to investigate strengtheningefficiency by sole welding and welding together with

CFRP attachments. One layer of CFRP plate extendedthe fatigue life of the steel beams by approximatelyseven times. Four different retrofitting materials,including high-modulus carbon fibre–reinforced poly-mer (HM-CFRP) plates, high-strength carbon fibre–reinforced polymer (HS-CFRP) plates, steel-wirebasalt-fibre–reinforced polymer (SW-BFRP) platesand welded steel plates explored by Wu et al. (2012),showed that HM-CFRP exhibited the best fatigue per-formance while SW-BFRP was the most cost-effectivechoice. Kim and Harries (2011) performed an experi-mental and numerical study on the flexural behaviourof CFRP repaired steel beams. An empirical modelwas proposed to estimate the fatigue response ofthe CFRP–steel interface. An unbonded CFRP

1Department of Structural Engineering, Tongji University, Shanghai,

China2Department of Architecture and Civil Engineering, City University of

Hong Kong, Hong Kong, China3School of Engineering, RMIT University, Melbourne, VIC, Australia

Corresponding author:

Yu-Fei Wu, School of Engineering, RMIT University, Melbourne, VIC

3001, Australia.

Email: yufei.wu@rmit.edu.au

prestressing technique was developed by Ghafooriet al. (2012).

Most previous research on fatigue strengtheningwith CFRP materials is relied on adhesive bonding.Experimental and analytical work has been carried outon the interfacial behaviour between CFRP and steel(Liu et al., 2010; Nozaka et al., 2005; Xia and Teng,2005; Yu et al., 2012).

Hmidan et al. (2015) developed 1240 finite elementmodels of wide-flange steel girders and extracted thestress intensity factor (SIF) at the crack front. A broadrange of variables were taken into consideration toderive the correction factors for SIF. Strain compatibil-ity between steel and CFRP was adopted in the numeri-cal modelling. Fatigue tests on cracked steel beamsrepaired with CFRP plates by Colombi and Fava (2015)revealed the presence of a debonded area around thecrack front, and consequently, an elliptical debondedregion was introduced into the finite element model.The strain on the CFRP plates showed a good agree-ment with the test results. More recently, Zheng andDawood (2016) presented an experimental and numeri-cal analysis on CFRP strengthened steel plates andreported that the shape and size of the debonded zonegreatly influenced the crack growth rate up to 54 times.

This study aims to investigate the fatigue behaviourof defected steel beams retrofitted by CFRP laminates,and the effect of interfacial debonding is emphasized.Analytical modelling and numerical simulation con-firmed that interfacial debonding had a pronouncedeffect on CFRP strain and SIF at the crack front.Fatigue life and crack propagation with cycle numbersof the specimens compared well with the experimentalfindings with interfacial debonding taken into consid-eration. Based on current test results, the specimenwith Sikadur 30 was more prone to debonding failure,and therefore, Araldite 420 became the preferred prod-uct for the strengthening scheme.

Experimental program

Four cracked steel beams were tested under fatigueloading in the Heavy Structures Testing Lab at City

University of Hong Kong, China, among which twowere unstrengthened and the other two were retrofittedwith CFRP laminates.

Configuration of specimens

Figure 1 shows the specimen configuration and geome-try. I-shaped beams with a cross section of 150 3 753 53 7 mm were selected in the test program. Inorder to simulate an initial damage, a notch of 15 mmlong and 0.3 mm wide was artificially cut at the mid-span of the beam (Figure 1(b)). When strengthened,CFRP laminates, with a nominal cross section of50 3 1.4 mm, were attached to the tension flange asdepicted in Figure 1(a).

Material properties

The steel beams used in the test were Grade Q345b.Tensile coupon specimens were first extracted fromboth the flange and web and then tested according toAustralia Standard AS 1391-1991 (1991), as listed inTable 1. Normal modulus CFRP laminates, whichwere provided in rolls of 50 mm wide, were selectedfor the patch system. The mechanical properties werealso determined through tensile coupon tests accordingto ASTM D3039/D3039M-08 (2008). Two types ofstructural adhesives, Araldite 420 and Sikadur 30, wereadopted to bond the composite overlays. Their ulti-mate strength and tensile Young’s modulus based onthe manufacturers’ data sheet are given in Table 1.Araldite 420 is a nonlinear adhesive with a smallerelastic modulus while Sikadur 30 is a linear one with alarger elastic modulus (Yu et al., 2012).

Specimen preparation

The bond area on the tensile flange was sandblasted(coarse corundum power with a size from 1.5 to2.0 mm at a pressure of 8 bar) to remove the rust andcreate a rough surface. Afterwards, the substrate wascleaned with high-pressure air and acetone. In addi-tion, the surface of the CFRP laminate was wiped with

(a) (b)75

1365

77

150

15

Notch

1200800

205 205

1400

Figure 1. Specimen geometry (dimension in millimetre, not to scale): (a) strengthened specimen and (b) notch details.

2 Advances in Structural Engineering 00(0)

acetone before attachment. The two components ofthe adhesive were carefully mixed according to theoperation manual and used to glue the CFRP to steel.Measures were taken to ensure a uniform and desir-able bond thickness around 0.5 mm (Yu and Wu,2017). All specimens were cured at room temperaturefor 2 weeks before testing.

Fatigue loading and data acquisition

A four-point bending rig was selected to conduct theexperiment. The distance between the two supportswas 1200 mm, and the span between the two loadingpoints was 410 mm. Compressive cyclic loading with afrequency of 6 Hz and a stress ratio R (minimum load/maximum load) of 0.1 with the waveform being sinu-soidal was applied to the specimen through a spread

beam. Three loading levels were selected in the testprogram, that is, 2–20, 3.5–35 and 7–70 kN (Figure 2).The load P here refers to the force applied by theactuator to the spread beam.

Strain gauges were attached to the potential crackpath on the beam web at a certain interval to detectthe crack propagation against fatigue cycle numbers.When the crack grew to the position where a straingauge attached, there would be sharp change in thestrain reading.

Besides fatigue cycles, static loads were applied atthe beginning of the test and after a certain number offatigue cycles. The specimens were loaded from zero tothe maximum force of the fatigue load and thenunloaded to zero. During these loops, the digital imagecorrelation (DIC) system was used to monitor thestrain distribution along the CFRP patch (Wu andJiang, 2013).

Test results

Table 2 presents a summary of the test program andresults. Beams 1 and 2 were bare specimens withoutstrengthening. As the cycle number increased, the ini-tial slot began to propagate towards the compressionflange. When the crack reached a certain length, thereduced cross section of the beam could not bear theapplied loading, and the specimen ruptured to failure.

Beams 3 and 4, which were strengthened withCFRP materials, showed a significantly retarded crackpropagation in comparison with the non-strengthenedspecimens. As a result of stress concentration, interfa-cial debonding within the adhesive layer was initiatedbeneath the notch and expanded to the two ends of the

Table 1. Measured material properties of steel, CFRP and adhesive.

Steel CFRP laminate Araldite 420 Sikadur 30

Tensile strength (MPa) 519.0 3022.4 29 30Yield strength (MPa) 378.2 – – –Tensile modulus (GPa) 192.8 200.4 1.495 11.2

CFRP: carbon fibre–reinforced polymer.

Figure 2. Fatigue load spectrum.

Table 2. Fatigue life of test specimens.

Specimen Adhesive type Bond thickness (mm) DF (kN) Fatigue life

Beam 1 – – 2–20 71,259Beam 2 – – 3.5–35 16,700Beam 3 Araldite 420 0.45 7–70 71,736Beam 4 Sikadur 30 0.58 7–70 8776

Yu and Wu 3

CFRP laminate. Compared to Beam 4, Beam 3(repaired with Araldite 420) had a slower debondingprocess. Even when the crack reached the intersectionof the compressive flange and web, the debondinglength was still limited and extended slowly. The speci-men finally fractured when the debonded zone reacheda certain value. However, in Beam 4 which was retro-fitted with Sikadur 30, the debonded zone grew muchmore rapidly with the fatigue cycles. Before the crackdeveloped to the web tip, the CFRP patch was sud-denly detached from the steel and the crack spread tothe beam web. Cohesive failure (adhesive layer failure)was observed for both specimens, and Araldite 420was found to have a higher interfacial fracture energyand rougher bond than Sikadur 30 (Yu et al., 2012).

Figure 3 depicts the crack propagation with fatiguecycles of all the specimens. The horizontal axis andvertical axis represent the fatigue cycle numbers andthe crack length measured from the tension soffit tothe crack front, respectively. The difference betweenBeams 1 and 2 as shown in Figure 3(a) indicates that alarger fatigue loading can lead to a significantlyincreased fatigue crack growth (FCG) rate and conse-quently a considerably reduced fatigue life. Figure 3(b)shows that Beam 3 with CFRP retrofitting enduredmany more cycles under a doubled fatigue loading incomparison with Beam 2, which demonstrates thatstrengthening by CFRP can effectively improve thefatigue behaviour of defected steel beams. An apparentdifference was observed between specimens repairedwith different structural adhesives. Araldite 420yielded a much slower debonding extension as well asslower crack development compared to Sikadur 30.

During the static loading processes in between fati-gue cycles, the strain fields on the CFRP patch ofBeams 3 and 4 were captured by the DIC system and

are shown in Figures 4 and 5. High strain concentra-tion occurred at the centre notch. As the crack lengthincreased, the high strain zone spread towards the freeends of the laminate.

The variations in the strain plateau depicted by thesame colour in Figures 4 and 5 indicate the develop-ment of interfacial debonding. The debonding lengthversus crack length and fatigue cycles are furtherextracted and shown in Figures 6(a) and (b), respec-tively. It was found that the specimen repaired bySikadur 30 was more prone to debonding and theexpansion of debonded zone with crack length wasmuch faster than that in the specimen strengthened byAraldite 420. This indicated that Araldite 420 is pre-ferred for CFRP external bonding applied in steelstrengthening.

Analytical modelling

With the presence of retrofitting materials, the stressfield in a steel beam is reduced, leading to a retardedcrack propagation of the specimen. Therefore, it is ofgreat importance to accurately evaluate the stress dis-tribution between the steel component and attachedoverlay. With respect to the fatigue load applied in thecurrent program, the materials in this work (steel,CFRP and adhesive) were all considered to be underthe linear elastic stage. A simple analytical modellingproposed by Colombi and Fava (2015) is used in thissection for analytical study. The model is based on thefollowing assumptions:

1. All materials including steel, CFRP and adhe-sive are elastic.

2. A plane section remains plane during bending.

Fatigue cycle numbers (million)0.00 0.02 0.04 0.06 0.08

Cra

ck le

ngth

(mm

)

0

30

60

90

120

150

Beam 1Beam 2

Fatigue cycle number (million)0.00 0.02 0.04 0.06 0.08

Cra

ck le

ngth

(mm

)

0

30

60

90

120

150

Beam 3Beam 4

(a) (b)

Figure 3. Fatigue crack propagation: (a) steel beams without strengthening and (b) steel beams strengthened with CFRP laminates.

4 Advances in Structural Engineering 00(0)

3. Perfect bonding occurs between steel andCFRP, that is, no slippage, is considered.

Consequently, the strain distribution along the crosssection is displayed in Figure 7.

Analysis of the cracked section

For such a composite cross section, the neutral axis isfirst derived as expressed by equation (1)

y ¼bftf hw + 1:5tfð Þ+ 0:5 hw + tfð Þ2 � a2

h itw � a1bata 0:5tað Þ � a2bctc ta + 0:5tcð Þ

bftf + hw + tf � að Þtw +a1bata +a2bctcð1Þ

Postition along CFRP length (mm)-300 -200 -100 0 100 200 300

Posi

tion

alon

gC

FRP

wid

th (m

m)

-10

-5

0

5

100 1000 2000 3000 4000

Position along CFRP length (mm)-300 -200 -100 0 100 200 300

Posi

tion

alon

gC

FRP

wid

th (m

m)

-10

-5

0

5

100 1000 2000 3000 4000

(a) (b)

Position along CFRP length (mm)-300 -200 -100 0 100 200 300

Posi

tion

alon

gC

FRP

wid

th (m

m)

-10

-5

0

5

100 1000 2000 3000 4000 5000 6000

Position along CFRP length (mm)-300 -200 -100 0 100 200 300

Pos

ition

alo

ngC

FRP

wid

th (m

m)

-10

-5

0

5

100 1000 2000 3000 4000 5000 6000

(c) (d)

Position along CFRP length (mm)-300 -200 -100 0 100 200 300

Posi

tion

alon

gC

FRP

wid

th (m

m)

-10

-5

0

5

100 1000 2000 3000 4000 5000 6000

Position along CFRP length (mm)-300 -200 -100 0 100 200 300

Posi

tion

alon

gC

FRP

wid

th (m

m)

-10

-5

0

5

100 1000 2000 3000 4000 5000 6000

(e) (f)

Position along CFRP length (mm)-300 -200 -100 0 100 200 300

Posi

tion

alon

gC

FRP

wid

th (m

m)

-10

-5

0

5

100 1000 2000 3000 4000 5000 6000

Position along CFRP length (mm)-300 -200 -100 0 100 200 300

Posi

tion

alon

gC

FRP

wid

th (m

m)

-10

-5

0

5

100 1000 2000 3000 4000 5000 6000

(g) (h)

Position along CFRP length (mm)-300 -200 -100 0 100 200 300

Posi

tion

alon

gC

FRP

wid

th (m

m)

-10

-5

0

5

100 1000 2000 3000 4000 5000 6000

Position along CFRP length (mm)-300 -200 -100 0 100 200 300

Posi

tion

alon

gC

FRP

wid

th (m

m)

-10

-5

0

5

100 1000 2000 3000 4000 5000 6000

(i) (j)

Figure 4. Strain distribution on the CFRP laminate of Beam 3: (a) a = 15 mm, (b) a = 23 mm, (c) a = 33 mm, (d) a = 43 mm,(e) a = 53 mm, (f) a = 63 mm, (g) a = 75 mm, (h) a = 93 mm, (i) a = 113 mm and (j) a = 143 mm.

Yu and Wu 5

Position along CFRP length (mm)-300 -200 -100 0 100 200 300

Pos

ition

alo

ngC

FRP

wid

th (m

m)

-10

-5

0

5

100 1000 2000 3000 4000

Position along CFRP length (mm)-300 -200 -100 0 100 200 300

Pos

ition

alo

ngC

FRP

wid

th (m

m)

-10

-5

0

5

100 1000 2000 3000 4000

(a) (b)

Position along CFRP length (mm)-300 -200 -100 0 100 200 300

posi

tion

alon

gC

FRP

wid

th (m

m)

-10

-5

0

5

100 1000 2000 3000 4000

Position along CFRP length (mm)-300 -200 -100 0 100 200 300

Pos

ition

alo

ngC

FRP

wid

th (m

m)

-10

-5

0

5

100 1000 2000 3000 4000

(c) (d)

Position along CFRP length (mm)-300 -200 -100 0 100 200 300

Posi

tion

alon

gC

FRP

wid

th (m

m)

-10

-5

0

5

100 1000 2000 3000 4000

Position along CFRP length (mm)-300 -200 -100 0 100 200 300

Posi

tion

alon

gC

FRP

wid

th (m

m)

-10

-5

0

5

100 1000 2000 3000 4000

(e) (f)

Position along CFRP length (mm)-300 -200 -100 0 100 200 300

Posi

tion

alon

gC

FRP

wid

th (m

m)

-10

-5

0

5

100 1000 2000 3000 4000

(g)

Figure 5. Strain distribution on the CFRP laminate of Beam 4: (a) a = 15 mm, (b) a = 23 mm, (c) a = 33 mm, (d) a = 43 mm, (e)a = 53 mm, (f) a = 63 mm and (g) a = 75 mm.

Crack length (mm)0 30 60 90 120 150

Deb

ondi

ng le

ngth

(mm

)

0

200

400

600

800Beam 3Beam 4

Fatigue cycle numbers (million)0.00 0.02 0.04 0.06 0.08

Deb

ondi

ng le

ngth

(mm

)

0

200

400

600

800Beam 3Beam 4

(a) (b)

Figure 6. Comparison of debonding process between Beams 3 and 4: (a) debonding length versus crack length and (b) debondinglength versus fatigue cycle numbers.

6 Advances in Structural Engineering 00(0)

The second moment of area, I, of the composite sec-tion is composed of three parts, that is, Is, Ic and Ia,which are expressed by

I ¼ Is + Ia + Ic ð2Þ

Is ¼bft

3f

12+ bftf hw + 1:5tf � yð Þ2 + tw

12hw + tf � að Þ3

+ tw hw + tf � að Þ y� 1

2hw + tf + að Þ

� �2

ð3aÞ

Ia ¼ a1

bat3a

12+ bata y+ 0:5tað Þ2

� �ð3bÞ

Ic ¼ a2

bct3c

12+ bctc y+ 0:5tc + tað Þ2

� �ð3cÞ

Substitution of equations (3a) to (3c) into equation(2) results in equation (4). Finally, the strain on CFRPlaminate beneath the initial slot is calculated by equa-tion (5)

I ¼ bft3f

12+ bftf hw + 1:5tf � yð Þ2 + tw

12hw + tf � að Þ3

+ tw hw + tf � að Þ y� 1

2hw + tf + að Þ

� �2

+a1

bat3a

12+ bata y+ 0:5tað Þ2

� �

+a2

bct3c

12+ bctc y+ 0:5tc + tað Þ2

� �

ð4Þ

ec ¼M

EsIy+ ta + 0:5tcð Þ ð5Þ

where y represents the distance from the neutral axisto the bottom flange of the steel section; a is the cracklength measured from the tensile soffit to the crack tip;b and t indicate the width and thickness, respectively; Iis the second moment of area; E is the elastic modulus;e is the strain; the subscript s, f, w, a and c denote thesteel beam, beam flange, beam web, adhesive andCFRP, respectively; a1 and a2 are elastic modulusratios, which are defined as Ea/Es and Ec/Es,respectively.

Comparison between analytical results and test data

Figure 8 plots the comparison of the strain on theCFRP laminate against the crack length between theanalytical results and experimental findings for Beams3 and 4. Figure 8(a) shows that when the crack length aincreases from 15 to 63 mm, the strain results obtainedfrom the analytical solution compare well with the testdata of Beam 3. However, a significant differenceoccurs thereafter, and the analytical calculation overes-timates the strain on the CFRP by as much as 16%.

(a) (b)bc,bf

y

twt f

a

h w

tctf

ε′s

εs

εaεf

Figure 7. Strain distribution on the cross section: (a) crosssection and (b) strain distribution.

Crack length (mm)0 30 60 90 120 150

Stra

in a

t mid

span

of C

FRP

(με )

0

2000

4000

6000

8000

Analytical resultsTest results

Crack length (mm)0 30 60 90 120 150

Stra

in a

t mid

span

of C

FRP

(με )

0

2000

4000

6000

8000

Analytical resultsTest results

(a) (b)

Figure 8. Comparison between analytical results and test data: (a) Beam 3 and (b) Beam 4.

Yu and Wu 7

The experimental axial strain at the mid-span of theCFRP laminate of Beam 4 is significantly lower thanthe analytical one from the beginning to the finalfailure.

This phenomenon is mainly due to the developmentof the debonded zone. Considerable interfacialdebonding is observed after certain loading cycles inFigures 4 and 5, which violates the plane section’sremaining plane assumption. For Beam 3, the highstrain region keeps stable for the crack length rangingfrom 15 to 63 mm and shows remarkable extension toboth ends of CFRP from a = 75 mm. For Beam 4,the strain concentration quickly progresses from theinitial beginning. These trends are consistent with thevariation of difference between analytical and testresults shown in Figure 8.

After the occurrence of significant interfacialdebonding, slip between CFRP and steel was observedfrom experimental tests, as reported in Yu and Wu (inpreparation). Therefore, the assumption of strain com-patibility (Figure 7) becomes invalid. The analyticalsolution without considering the slip between CFRPand steel substrate would overestimate the strain onthe CFRP laminate, and the difference becomes morepronounced with crack propagation, leading to anoverestimation of the strengthening efficiency. Thisanalysis also suggests that debonding has an unfavour-able effect on repair efficiency. In Colombi and Fava(2016), analytical solution to evaluate the stress on theCFRP with the slip effect taken into consideration wasproposed.

Numerical modelling

Finite element modelling

Three-dimensional finite element modelling was devel-oped to assess the fatigue behaviour of the test speci-mens using the commercial software ABAQUS (2010).The contour integral method was used to define theonset of cracking and to calculate the output

parameter required for computation of SIF. With con-sideration of the symmetric specimen configurationand boundary condition, a quarter-scale model withsymmetrical constraints on corresponding boundaries,as shown in Figure 9(a), was adopted for the analysis.The geometry and dimensions of the models were thesame as those in the experimental program. A loadwas applied at the position of the spread beam (Figure9(b)).

The fatigue loading applied in the test program wasrelatively small so that the steel and CFRP were undera linear elastic stage. Consequently, linear elastic con-stitutive laws were adopted for the materials. The steelhad a Young’s modulus of 192.8 GPa and a Poisson’sratio of 0.3, and the CFRP had a Young’s modulus of200.4 GPa and a Poisson’s ratio of 0.3. Araldite 420and Sikadur 30 had a Young’s modulus of 1.495 and11.2 GPa, respectively. A Poisson’s ratio of 0.36 wasadopted for the structural adhesives (Fawzia, 2007).

Three-dimensional quadratic brick elements(C3D20R) were selected to mesh the steel plate, adhe-sive layer and CFRP laminate. The model of thestrengthened beam consisted of 29,952 elements and147,413 nodes. The stress and strain singularities at thecrack front were simulated using the 15-node wedgeelements collapsed from the 20-node brick elements, asdisplayed in Figure 10(a). Sixteen elements (11.25� perelement) were assigned around the semi-circumferenceto create a sharp crack front. The detailed mesh aroundthe crack front is depicted in Figures 10(b) and (c).

Mesh dependency analysis

A mesh dependency analysis was carried out to obtaina reasonable size of the element around the crack front.Models with ratios of the crack front element size l tothe crack length a ranging from 0.025 to 0.25 werecomputed. The case of a bare steel beam withoutstrengthening loaded under 20 kN was adopted andtwo crack lengths, that is, 30 mm (a/h = 0.2, h is the

Figure 9. Finite element modelling: (a) one quarter model and (b) boundary conditions.

8 Advances in Structural Engineering 00(0)

beam height) and 75 mm (a/h = 0.5), were selected. Atotal of 16 cases were calculated and the correspondingresults are presented in Figure 11. It was found thatbased on the current numerical results, the SIF had lit-tle dependency on the element size ratio. Consequently,the ratio value of 0.20 was used for the present study.

Validation of the numerical simulation

The theoretical solution of a steel plate under bending(Figure 12) was used to validate the numerical analysis.The SIF value of a steel plate with a mid-span crackwas calculated by equation (6) (Boresi and Schmidt,2003)

K ¼ sffiffiffiffiffiffipap

f lð Þ ð6aÞ

where l is the ratio of the crack length to the plateheight and s is the far-field stress, which are expressedby equations (6b) and (6c), respectively. f(l) is a cor-rection factor, and its value versus l is given in Table 3

l ¼ a

hð6bÞ

s ¼ 3M

0:5th2ð6cÞ

A finite element model similar to that of the steelbeam was adopted here, and the only difference wasthat the beam flanges were excluded. Consequently,the steel plate was 700 mm long, 150 mm high and2.5 mm thick. It was loaded under 20 kN and the com-parison between the theoretical solution and numericalresults is depicted in Figure 13. It is indicated that thetwo agreed well with the ratio of the crack length toplate height increasing from 0.1 to 0.6.

Effect of interfacial debonding

It was observed from the experimental study that inter-facial debonding developed under fatigue cycling, andthe analytical study demonstrated that the debondinghad a detrimental effect on the strengthening scheme.The length of the debonded region was considered todominate the performance of the repaired steel beamand, therefore, was focused in this section. Finite ele-ment models with different lengths of the debondedzone were studied and the SIF at the crack front wereextracted. The debonded region was modelled withoutconnection between the steel and CFRP while perfectbond was assumed for the remaining area. It wasfound that the strain distribution was approximatelyuniform in the width direction of the patch; therefore,a rectangle shape of the debonded zone was assumed

Crack(b)

midplane

edge plane

C3D20 (RH)

2to the same location

crack line3 nodes collapsed

to the same locationmidside nodes

moved to 1/4 pts

(a) (c)

l

Figure 10. Collapsed three-dimensional element and crack front mesh: (a) 15-node wedge elements, (b) crack front mesh size and(c) crack front mesh.

Ratio of crack front element size to crack length 'l/a'0.00 0.05 0.10 0.15 0.20 0.25 0.30

Stre

ss in

tens

ity fa

ctor

K (M

Pa·

mm

1/2 )

0

1000

2000

3000

4000

a = 30 mm (a/h = 0.2)a = 75 mm (a/h = 0.5)

Figure 11. Stress intensity factor versus crack front elementsize to crack length ratio.

Table 3. f(l) versus l.

l 0.1 0.2 0.3 0.4 0.5 0.6

f(l) 1.02 1.06 1.16 1.32 1.62 2.10

Yu and Wu 9

for the sake of simplicity. A uniform bond thickness of0.5 mm was adopted in all cases. The SIF results ver-sus the length of debonded region of the beams loadedunder 70 kN are displayed in Figure 14.

Similar to Figure 11, two crack lengths of 30 and75 mm as well as two structural adhesives of Araldite420 and Sikadur 30 were adopted. Six lengths ofdebonded region, that is, 0 (no debonding), 20, 40, 60,80 and 100 mm were analysed. The debonding lengthhere is for a quarter-scale beam and hence is half ofthe total debonding length of the entire beam.

It was found that the SIF notably increased with theincrease of the debonding length, especially in the casewith a larger crack length. When the crack length was30 mm, the SIF was increased from 1595 MPa mm1/2

(Sikadur 30, debonding length of 0 mm) to4434 MPa mm1/2 (Sikadur 30, debonding length of100 mm) and the corresponding variation percentagewas 178%. When the crack length reached 75 mm, amore significant effect of the debonding length on theSIF was demonstrated as the SIF was improved by360% for the specimen repaired by Sikadur 30.

Figure 14 shows that at a certain crack length, pro-vided the debonding lengths of the specimens withAraldite 420 and Sikadur 30 are the same, the one with

Sikadur 30 would lead to a slightly smaller SIF at thecrack front in comparison with that with Araldite 420attributed to the larger stiffness of Sikadur 30.However, in reality, specimens bonded with Sikadur30 were more prone to debonding with a largerdebonded zone under same cycles of loading due tothe smaller interfacial fracture energy, leading to apoorer performance in comparison with Araldite 420.

Numerical analyses of the test specimens

For the unstrengthened specimens, symmetric constraintswere applied to the corresponding boundaries, and thecrack face was left free. For the strengthened specimens,both the scenarios of perfect bonding and partial debond-ing were analysed. The length of debonded zone observedfrom the experimental results was introduced into thenumerical simulation for the scenario of partial debond-ing. The corresponding SIF at the maximum force of thefatigue load is plotted in Figure 15.

It was interesting to see that the SIF of the bare steelbeams increased quickly with the crack length whilethe strengthened specimens showed different variationtrends. In the perfect bonding cases, the SIF decreasedwith the crack length. This was because SIF dependson both far-field loading and crack length as expressedby equation (6a). As the crack propagated to the com-pressive flange, the CFRP shared more of the far-fieldloading, resulting in a decreased stress field in the steelbeam. Without considering the interfacial debonding,the stress field on the CFRP was significantly overesti-mated and the reduction of the stress field in the steelbeam governed the variation trend of SIF.

After the debonding was considered in the numeri-cal simulation, the SIF again increased with the cracklength. In Beam 3, when the crack length was less than75 mm, no apparent debonding was found based onthe experimental findings; therefore, no debonding wasset in the models; thus, the SIF results of the two sce-narios coincided.

With the SIF derived from the numerical simula-tion, the fatigue behaviour of the test specimens waspredicted based on the linear elastic fracture method(LEFM). Paris Law (JSSC, 1995) as expressed by equa-tion (7) was adopted and the fatigue life was integratedbased on equation (8)

da

dN¼ C DKð Þm ð7Þ

N ¼ðaf

ai

1

CDKmda ð8Þ

where a is the crack length, N is the number of fatiguecycles, C and m are material constants that are

h

a

t

MM

Figure 12. Schematic diagram of the classical solution withrespect to a steel plate under bending.

Ratio of crack length to plate height0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

Stre

ss in

tens

ity fa

ctor

K (M

Pa·

mm

1/2 )

0

2000

4000

6000

8000Numerical resultsClassical theoretical solution

Figure 13. Comparison between numerical results andclassical theoretical solution.

10 Advances in Structural Engineering 00(0)

Debonded length in the one-quarter model (mm)0 20 40 60 80 100

Stre

ss in

tens

ity fa

ctor

K (M

Pa·

mm

1/2 )

0

1000

2000

3000

4000

5000

6000

Araldite 420Sikadur 30

Debonded length of the one-quarter model (mm)0 20 40 60 80 100

Stre

ss in

tens

ity fa

ctor

K (M

Pa·

mm

1/2 )

0

1000

2000

3000

4000

5000

6000

Araldite 420Sikadur 30

(a) (b)

Figure 14. Stress intensity factor versus length of debonding region: (a) a = 30 mm (a/h = 0.2) and (b) a = 75 mm (a/h = 0.5).

Crack length (mm)0 30 60 90 120 150

Stre

ss in

tens

ity fa

ctor

K (M

Pa·

mm

1/2 )

0

2000

4000

6000

8000

10000

Beam 1Beam 2

(a)

Crack length (mm)0 30 60 90 120 150

Stre

ss in

tens

ity fa

ctor

K (M

Pa·

mm

1/2 )

0

500

1000

1500

2000

2500

3000

3500

Perfect bondingPartial debonding

Crack length (mm)0 30 60 90 120 150

Stre

ss in

tens

ity fa

ctor

K (M

Pa·

mm

1/2 )

0

1000

2000

3000

4000

5000

6000

7000

8000

Perfect bondingPartial debonding

(b) (c)Figure 15. Stress intensity factor versus crack length: (a) Beams 1 and 2, (b) Beam 3 and (c) Beam 4.

Yu and Wu 11

determined experimentally, DK is the SIF range at thecrack front and ai and af are the initial crack size andfinal crack size, respectively.

For the bare steel beams without strengthening, cand m were taken as 3.98 3 10–13 and 2.88, respec-tively (da/dN in mm/cycle and DK in MPa mm1/2),according to the recommendation by British StandardsInstitution, mean curve (BS 7910:2005); for the steelbeams strengthened by CFRP laminates, c and m weretaken as 6.77 3 10–13 and 2.88, respectively (da/dN inmm/cycle and DK in MPa mm1/2), according to therecommendation by British Standards Institution,mean + 2SD (BS 7910:2005).

Figure 16 shows the comparison of fatigue lifebetween prediction results and test data. It was foundthat the SIF derived from the numerical simulationcould be used to conservatively predict the fatigue life ofboth bare steel beams and strengthened components.All the deviations were less than 20% except for Beam 3(23.8%). Note that the cases displayed in Figure 16 tookthe interfacial debonding into consideration. The resultscorresponding to the scenario of perfect bonding wasdramatically different from the test data and was, there-fore, excluded from the figure. The test fatigue life ofBeams 3 and 4 was 71,736 cycles and 8,776 cycles,respectively, while the calculated results based on theperfect bonding were 915,453 and 467,236 cycles, respec-tively. It was, therefore, concluded that the interfacialdebonding with crack propagation should be taken intoconsideration; thus, future research is planned on theprogress of the deboning zone with fatigue cycles.

The fatigue crack propagation with cycle numbersof the four specimens is plotted in Figure 17, whichalso confirms that the numerical simulation with inter-facial debonding considered was feasible for estimatingthe fatigue behaviour of the steel beams.

Conclusion

In this article, an experimental and numerical studyon the fatigue behaviour of defected steel beamsstrengthened with CFRP laminates was presented.The effect of interfacial debonding was focused. Thefollowing observations can be made and conclusionsdrawn.

Specimens strengthened with CFRP laminates wit-nessed apparent interfacial debonding during the

Predicted fatigue life (million)

0.00 0.02 0.04 0.06 0.08

Test

fatig

ue li

fe (m

illio

n)

0.00

0.02

0.04

0.06

0.08

Beam 1Beam 2Beam 3Beam 4

Figure 16. Comparison of fatigue life between predictedresults and experiential data.

Fatigue cycle numbers (million)0.00 0.02 0.04 0.06 0.08

Cra

ck le

ngth

(mm

)

0

30

60

90

120

150Beam 1 (prediction)Beam 1 (test)Beam 2 (prediction)Beam 2 (test)

Fatigue cycle numbers (million)0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07

Cra

ck le

ngth

(mm

)

0

30

60

90

120

150

Beam 3 (prediction)Beam 3 (test)Beam 4 (prediction)Beam 4 (test)

(a) (b)

Figure 17. Comparison of fatigue crack propagation between predicted results and experiential data: (a) Beams 1 and 2 and(b) Beams 3 and 4.

12 Advances in Structural Engineering 00(0)

experimental study. Compared to Sikadur 30, Araldite420 with a larger interfacial fracture energy was recom-mended for the strengthening scheme. Specimensrepaired with Sikadur 30 were more prone to interfa-cial failure.

With the presence of interfacial debonding, theplane section assumption was no longer valid. Theanalytical modelling based on the perfect bondingoverestimated the field load shared by the CFRP mate-rials, which would lead to an overestimated strength-ening efficiency.

Parametric analysis of the debonding length showedthat the SIF increased considerably with the increaseof the debonding length, which implied that it is ofgreat importance to accurately assess the progress ofinterfacial debonding with fatigue cycles. Futureresearch is planned to study the development of inter-facial debonding under cyclic loading. With thedebonding length from experimental findings intro-duced into the numerical modelling, the fatigue lifeand crack propagation versus cycle numbers agreedwell with the test results.

Acknowledgements

The authors wish to express their heartfelt appreciation toProf. Xiang-Lin Gu from Tongji University and Prof. Xiao-Ling Zhao from Monash University for supervising the pres-ent work. Thanks are due to all the technical staff in theHeavy Structures Testing Lab at City University of HongKong, especially Mr Chan Wan Tong Allan and Mr ErnestCheung, for their assistance in carrying out the tests.

Declaration of Conflicting Interests

The author(s) declared no potential conflicts of interest withrespect to the research, authorship and/or publication of thisarticle.

Funding

The author(s) disclosed receipt of the following financial sup-port for the research, authorship and/or publication of thisarticle: The work described in this article was supported by the

National Natural Science Foundation of China (project no.51508406) and the Research Grant Council, Hong KongSpecial Administrative Zone, China (project no. CityU124113).

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