er

lblo

Reinforced concrete beamFinite element analysis (FEA)

ttede ectoryunThed w

The analyses results show good agreement with the experimental data regarding loaddisplacement

structue mayeteriorck of much as

installation time compared to conventional materials.The application of CFRP as external reinforcement to strengthen

concrete beams has received much attention from researchers [15],but only very few studies have focused on structural membersstrengthened after preloading [6,7]. The behaviour of structureswhich

structures have, however, not considered the effect of the interfa-cial behaviour at all [1113].

In this paper, we use the nite element method to model thebehaviour of beams strengthened with CFRP. For validation, thestudy was carried out using a series of beams that had beenexperimentally tested for exural behaviour and reported byObaidat [14]. Two different models for the CFRP and two differ-ent models for the concreteCFRP interface are investigated. Themodels are used for analysing beams with different lengths ofCFRP applied.

* Corresponding author.

Composite Structures 92 (2010) 13911398

Contents lists availab

Composite S

sevE-mail address: Yasmeen.Obaidat@byggmek.lth.se (Y.T. Obaidat).years, the development of strong epoxy glue has led to a techniquewhich has great potential in the eld of upgrading structures. Basi-cally the technique involves gluing steel or FRP plates to the sur-face of the concrete. The plates then act compositely with theconcrete and help to carry the loads.

The use of FRP to repair and rehabilitate damaged steel and con-crete structures has become increasingly attractive due to the well-known good mechanical properties of this material, with particularreference to its very high strength to density ratio. Other advanta-ges are corrosion resistance, reduced maintenance costs and faster

laminate ends, due to high concentration of shear stresses at dis-continuities, where shear cracks in the concrete are likely to de-velop [9]. Thus, it is necessary to study and understand thebehaviour of CFRP strengthened reinforced concrete members,including those failures.

Several researchers have simulated the behaviour of the con-creteCFRP interface through using a very ne mesh to simulatethe adhesive layer dened as a linear elastic material [10]. How-ever, they have not used any failure criterion for the adhesive layer.Most researchers who have studied the behaviour of retrotted1. Introduction

Upgrading of reinforced concretemany different reasons. The concretally inadequate for example, due to dinitial design and/or construction, laof design loads or accident events s0263-8223/$ - see front matter 2009 Elsevier Ltd. Adoi:10.1016/j.compstruct.2009.11.008response, crack pattern and debonding failure mode when the cohesive bond model is used. The perfectbond model failed to capture the softening behaviour of the beams. There is no signicant differencebetween the elastic isotropic and orthotropic models for the CFRP.

2009 Elsevier Ltd. All rights reserved.

res may be required forhave become structur-ation of materials, pooraintenance, upgradingearthquakes. In recent

have been preloaded until cracking initiates deserves more attention,since this corresponds to the real-life use of CFRP retrotting.

Researchers have observed new types of failures that can re-duce the performance of CFRP when used in retrotting struc-tures [8]. These failures are often brittle, and includedebonding of concrete layers, delamination of CFRP and shearcollapse. Brittle debonding has particularly been observed atLaminateCohesive model

elastic isotropic and orthotropic models were used for the CFRP and a perfect bond model and a cohesivebond model was used for the concreteCFRP interface. A plastic damage model was used for the concrete.The effect of CFRP and CFRP/concrete intretrotted RC beams with FEM

Yasmeen Taleb Obaidat *, Susanne Heyden, Ola DahDivision of Structural Mechanics, Lund University, Lund, Sweden

a r t i c l e i n f o

Article history:Available online 14 November 2009

Keywords:Carbon bre reinforced plastic (CFRP)Strengthening

a b s t r a c t

Concrete structures retroin the last decade due to this validated against laborageometry and were loadedforced plastic (CFRP) plate.rial models were evaluate

journal homepage: www.elll rights reserved.face models when modelling

m

with bre reinforced plastic (FRP) applications have become widespreadonomic benet from it. This paper presents a nite element analysis whichtests of eight beams. All beams had the same rectangular cross-section

der four point bending, but differed in the length of the carbon bre rein-commercial numerical analysis tool Abaqus was used, and different mate-ith respect to their ability to describe the behaviour of the beams. Linear

le at ScienceDirect

tructures

ier .com/locate /compstruct

2. Experimental work

Experimental data was obtained from previous work by Obaidat[14]. Eight identical RC beams were loaded with a four point bend-ing conguration with a span of 1560 mm, and distance betweenloads of 520 mm. All beams were 300-mm high, 150-mm wide,and 1960-mm long. The longitudinal reinforcement consisted oftwo / 12 for tension and two / 10 for compression. Shear rein-forcement was sufciently provided and consisted of / 8 c/c100 mm, as seen in Fig. 1.

mode I, Gf, is the area under the softening curve and was assumedequal to 90 J/m2, see Fig. 4b.

The stressstrain relationship proposed by Saenz [17] was usedto construct the uni-axial compressive stressstrain curve forconcrete:

rc Ecec1 R RE 2 ece0

2R 1 ece0

2 R ece0

3 3where

R RERr 1Re 12

1Re

; RE EcE0 ; E0 f 0ce0

4

and, e0 = 0.0025, Re = 4, Rr = 4 as reported in [18]. The stressstrainrelationship in compression for concrete is represented in Fig. 5.

Poissons ratio for concrete was assumed to be 0.2.

(c) Retrofitted beam RB3

P/2P/2

520 mm

Fig. 2. Length of CFRP laminates in test series RB1, RB2 and RB3.

0 2 4 6 8 10 120

50

100

150

200

Deflection (mm)

Loa

d (k

N)

RB1RB2RB3Control Beam

Fig. 3. Load versus mid-span deection for un-strengthened and strengthenedbeams.

1392 Y.T. Obaidat et al. / Composite StruTwo control beams were loaded to failure and the other beamswere loaded until cracks appeared, then retrotted using differentlengths of CFRP, see Fig. 2. The CFRP was adhered to the bottomsurface of the beams with their bre direction oriented in the axialdirection of the beam. Each CFRP plate was 1.2 mm thick and50 mm wide. Finally the beams were retested, while the deectionand load were monitored.

A comparison of loaddeection curves of retrotted beamsand control beams is presented in Fig. 3. The experimental re-sults showed that the retrotting using CFRP increased thestrength of the beam and the effect increased with the lengthof the CFRP plate. All retrotted beams failed due to debondingof the CFRP.

3. Finite element analysis

Finite element failure analysis was performed to model thenonlinear behaviour of the beams. The FEM package Abaqus/stan-dard [15] was used for the analysis.

3.1. Material properties and constitutive models

3.1.1. ConcreteA plastic damage model was used to model the concrete behav-

iour. This model assumes that the main two failure modes are ten-sile cracking and compressive crushing [15]. Under uni-axialtension the stressstrain response follows a linear elastic relation-ship until the value of the failure stress is reached. The failurestress corresponds to the onset of micro-cracking in the concretematerial. Beyond the failure stress the formation of micro-cracksis represented with a softening stressstrain response. Hence, theelastic parameters required to establish the rst part of the relationare elastic modulus, Ec, and tensile strength, fct, Fig. 4a. The com-pressive strength, f 0c , was in the experimental work measured tobe 30 MPa. Ec and fct were then calculated by [16]:

Ec 4700f 0c

q 26;000 MPa 1

fct 0:33f 0c

q 1:81 MPa 2

where f 0c , is given in MPa.To specify the post-peak tension failure behaviour of concrete

the fracture energy method was used. The fracture energy for

P/2P/2

300

mm

150 mm

210

2128/100 mm

520 mm 520 mm 520 mm

1960 mm

Fig. 1. Geometry, reinforcement and load of the tested beams.(a) Retrofitted beam RB1

(b) Retrofitted beam RB2

1560 mm

P/2P/2

1040 mm

P/2P/2

ctures 92 (2010) 139113983.1.2. Steel reinforcementThe steel was assumed to be an elasticperfectly plastic mate-

rial and identical in tension and compression as shown in Fig. 6.

d. (b) Post-peak stress deformation relationship.

2

1.5

1

0.5

0 0.00010.00005

t, MPa

1.81

0.44

t, MPa

, mm0.03 0.1

Gf

der

Y.T. Obaidat et al. / Composite Structures 92 (2010) 13911398 1393(a) Stress-strain relationship up to ultimate loaFig. 4. Concrete un

35

30

25

20

15

10

5

c, MPaThe elastic modulus, Es, and yield stress, fy, were measured in theexperimental study and the values obtained were Es = 209 GPaand fy = 507 MPa. These values were used in the FEM model. APoissons ratio of 0.3 was used for the steel reinforcement. Thebond between steel reinforcement and concrete was assumed asa perfect bond.

3.1.3. CFRPTwo different models for the CFRP were used in this study. In

the rst model, the CFRP material was considered as linear elasticisotropic until failure. In the second model, the CFRP was mod-elled as a linear elastic orthotropic material. Since the compositeis unidirectional it is obvious that the behaviour is essentiallyorthotropic. In a case like this however, where the composite isprimarily stressed in the bre direction, it is probable that themodulus in the bre direction is the more important parameter.This is why an isotropic model is considered suitable. The elasticmodulus in the bre direction of the unidirectional CFRP materialused in the experimental study was specied by the manufac-

Es

fy

y

Fig. 6. Stress strain behaviour of steel.

00

0.0025 0.005 0.01 0.015

Fig. 5. Stressstrain relationship for conuni-axial tension.turer as 165 GPa. This value for E and m = 0.3 was used for the iso-tropic model. For the orthotropic material model E11 was set to165 GPa. Using Rule of Mixture [19], Eepoxy = 2.5 GPa and thatthe bre volume fraction was 75%, Ebre was found to be219 GPa and m12 = m13 = 0.3. By use of Inverse Rule of Mixture[19], E22 = E33 = 9.65 GPa and G12 = G13 = 5.2 GPa. m23 and G23 wereset to 0.45 and 3.4, respectively.

3.1.4. CFRPconcrete interfaceTwo different models were used to represent the interface be-

tween concrete and CFRP. In the rst model the interface was mod-elled as a perfect bond while in the second it was modelled using acohesive zone model. Fig. 7 shows a graphic interpretation of asimple bilinear tractionseparation law written in terms of theeffective traction s and effective opening displacement d. The inter-face is modelled as a rich zone of small thickness and the initialstiffness K0 is dened as [20]:

Gcr

max

o f

Ko

Fig. 7. Bilinear tractionseparation constitutive law.

0.02 0.025 0.03

crete under uni-axial compression.

K0 1tiGi tcGc

5

where ti is the resin thickness, tc is the concrete thickness, and Giand Gc are the shear modulus of resin and concrete respectively.The values used for this study were ti = 1 mm, tc = 5 mm, Gi= 0.665 GPa, and Gc = 10.8 GPa.

From Fig. 7, it is obvious that the relationship between the trac-tion stress and effective opening displacement is dened by thestiffness, K0, the local strength of the material, smax, a characteristicopening displacement at fracture, df, and the energy needed foropening the crack, Gcr, which is equal to the area under the trac-tiondisplacement curve. Eq. (6), [21], provides an upper limit forthe maximum shear stress, smax, giving smax = 3 MPa in this case:

smax 1:5bwft 6where

ponent, see Fig. 9b. The values used for this study were r0n fct 1:81 MPa, and s0s s0t 1:5 MPa.

Interface damage evolution was expressed in terms of energyrelease. The description of this model is available in the Abaqusmaterial library [15]. The dependence of the fracture energy onthe mode mix was dened based on the BenzaggahKenane frac-ture criterion [15]. BenzaggahKenane fracture criterion is particu-larly useful when the critical fracture energies during deformationpurely along the rst and the second shear directions are the same;i.e., GCs GCt . It is given by:

GCn GCs GCn GS

GT

g GC 8

where GS GS Gt , GT Gn Gs; and g are the material parame-ter. Gn, Gs and Gt refer to the work done by the traction and its con-

1394 Y.T. Obaidat et al. / Composite Structures 92 (2010) 13911398bw 2:25 bf

bc

1:25 bf

bc

s

and bf is CFRP plate width, bc is concrete width and fct is concretetensile strength.

Numerical simulations showed that this value is too high; sinceCFRP rupture or concrete crushing induced the failure, instead ofthe CFRP debonding that occurred in the experimental study, seeFig. 8. The two curves representing smax = 3 MPa show increasingload up to failure, and the simulations ended with CFRP ruptureor concrete crushing. Hence, smax was reduced to 1.5 MPa.

For fracture energy, Gcr, previous researches have indicated val-ues from 300 J/m2 up to 1500 J/m2 [22,23]. To investigate to whatextent Gcr affects the results, numerical simulations were per-formed for Gcr = 500 J/m2 and 900 J/m2. The simulations showedthat Gcr has in this case only a moderate inuence on the loaddeformation behaviour, as seen in Fig. 8. For this study the value900 J/m2, in the middle of the interval proposed by previous stud-ies, was used.

The initiation of damage was assumed to occur when a qua-dratic traction function involving the nominal stress ratios reachedthe value one. This criterion can be represented by [15]:

rnr0n

2 sn

s0s

2 st

s0t

2 1 7

where rn is the cohesive tensile and ss and st are shear stresses ofthe interface, and n, s, and t refer to the direction of the stress com-

0 2 4 6 8 10 120

50

100

150

200

Deflection (mm)

Load

(kN

)

Interfacial shear= 3 MPa, Fracture energy= 900 J/m2Interfacial shear= 3 MPa, Fracture energy= 500 J/m2Interfacial shear= 1.5 MPa, Fracture energy= 900 J/m2Interfacial shear= 1.5 MPa, Fracture energy= 500 J/m2RB1 (Exp)Fig. 8. Comparison between the experimental and the FE analysis results fordifferent model of interfacial behaviour and isotropic behaviour for CFRP for beamRB1.jugate separation in the normal, the rst and the second sheardirections, respectively. The values used for this study wereGCn 90 J=m2, GCt GCs 900 J/m2, and g = 1.45.

3.2. Numerical analysis

4-Node linear tetrahedral elements were used for the rein-forced concrete, reinforcement steel, steel plates at supportsand under the load, and CFRP in this model. The element cong-uration is shown in Fig. 9a. 8-Node 3-D cohesive elements wereused to model the interface layer. The cohesive interface elementsare composed of two surfaces separated by a thickness, Fig. 9b.The relative motion of the bottom and top parts of the cohesiveelement measured along the thickness direction representsopening or closing of the interface. The relative motion of theseparts represents the transverse shear behaviour of the cohesiveelement.

To show the effect of the bondmodel and the behaviour of CFRP,four combinations of bond model and CFRP model were analysed;Perfect bond with isotropic CFRP, perfect bond with orthotropicCFRP, cohesive bond model with isotropic CFRP, and cohesive bondmodel with orthotropic CFRP.

One quarter of the specimen was modelled, as shown in Fig. 10,by taking advantage of the double symmetry of the beam. Theboundary conditions are illustrated in Fig.11. A ne mesh is neededto obtain results of sufcient accuracy. The pre-crack was modelledby making a gap of 0.1 mm width and 10 mm depth between thecontinuum elements, 20 mm from the centre of the beam. Table1 shows the number of elements, number of degrees of freedomand CPU time. The processor type used for this study was 2 Xeon5160 (3.0 GHz, dual core).

87

3

2

1

5 4

n

s

6t

(a) 4-node linear tetrahedral element.(b) 8-node 3-D cohesive element.Fig. 9. Elements used in the numerical analysis.

tropic and orthotropic cohesive models.There are several possible causes for the differences between

Fig. 10. Geometry and elements used in the numerical analysis.

Fig. 11. Boundary conditions used in numerical work.

Table 1Model size and computational time.

Model Number ofelements

Number of degrees offreedom (DOF)

CPU time (h)

Control beam 150,813 79,428 2:54

Isotropic CFRP/perfect bondRB1 168,630 89,595 3:41RB2 169,019 89,583 4:25RB3 168,917 89,406 4:05

Orthotropic CFRP/perfect bondRB1 168,630 89,595 6:36RB2 169,019 89,583 2:57RB3 168,917 89,406 2:40

Isotropic CFRP/cohesive modelRB1 168,669 90,075 4:06RB2 169,656 90,189 3:08RB3 170,307 90,240 2:30

Orthotropic CFRP/cohesive modelRB1 168,669 90,075 3:41RB2 169,656 90,189 2:50RB3 170,307 90,240 2:52

Y.T. Obaidat et al. / Composite Struthe experimental data and the nite element analysis. One is, asfor the control beam, the assumed perfect bond between concreteand steel reinforcement. In addition, the location and dimensionsof the pre-crack were not represented exactly as it appeared inthe experimental work; another reason is due to the estimationof the behaviour of the interface between CFRP and concrete. Thismay lead to the overestimation of the stiffness and capacity of thereinforced concrete structural element.

The results show that a cohesive model gives good agreementwith experimental results, but the perfect bond model does not,3.3. Nonlinear solution

In this study the total deection applied was divided into a ser-ies of deection increments. Newton method iterations provideconvergence, within tolerance limits, at the end of each deectionincrement. During concrete cracking, steel yielding and the ulti-mate stage where a large number of cracks occur, the deectionsare applied with gradually smaller increments. Automatic stabil-ization and small time increment were also used to avoid a di-verged solution.

4. Results

4.1. Loaddeection curves

The loaddeection curves obtained for control beam and retro-tted beams from experiments and FEM analysis are shown inFig. 12. Four different combinations of models for CFRP and con-crete/CFRP bond were used.

There is good agreement between FEM and experimental resultsfor the control beam, Fig. 12a. The FEM analysis predicts the beamto be slightly stiffer and stronger, probably because of the assumedperfect bond between concrete and reinforcement. The good agree-ment indicates that the constitutive models used for concrete andreinforcement can capture the fracture behaviour well.

When comparing Fig. 12ad, it can be seen that the length ofthe CFRP signicantly inuences the behaviour of the beam. Thelonger CFRP, the higher is the maximum load.

For the retrotted beams, the results from the four differentFEM models are close to identical during the rst part of the curve,all slightly stiffer than the experimental results, Fig. 12bd.

After cracks start appearing, the perfect bond models increas-ingly overestimate the stiffness of the beam. This is due to thefact that the perfect bond does not take the shear strain betweenthe concrete and CFRP into consideration. This shear strain in-creases when cracks appear and causes the beam to become lessstiff.

The perfect bond models also fail to capture the softening of thebeam, a fact that is most obvious for RB1. Debonding failure, whichoccurred in the experiments, is not possible with the perfect bondmodel. Thus, it is possible to increase the load further until anothermode of failure occurs. In this case shear exural crack failure orCFRP rupture. The curves for isotropic and orthotropic perfect bondmodels are close to coincident, but the orthotropic perfect bondmodel gives a maximum load value that is slightly smaller thanthe isotropic perfect bond model. This is possibly because the unre-alistically high stiffness in the transverse direction and shear of theisotropic CFRP provides a strengthening connement.

The cohesive models show good agreement with the experi-mental results. There are only small differences between the iso-

ctures 92 (2010) 13911398 1395at least not for high load levels. The results also show that it isnot necessary to take into account the orthotropic properties ofthe unidirectional CFRP.

Stru50

100

150

200Lo

ad (k

N)

Control Beam (FEM)

1396 Y.T. Obaidat et al. / Composite4.2. Effect of retrotting on the stress

Fig. 13 shows the differences between the axial stress for thecontrol specimens and the retrotted beam RB2 at load equal to10 kN. All models which were used in this study gave the sameindication for this point. Also the parts of the strengthened beamsa long distance from the CFRP have a different stress distributioncompared to those of the un-strengthened specimens at the corre-sponding location. This indicates that the effect of the strengthen-ing is not local but it affects the stress distribution of the beam as awhole.

(a) Control beam

(b) RB1

(c) RB2

(d) RB3

0 2 4 6 8 10 120Deflection (mm)

Control Beam (Exp)

0 2 4 6 8 10 120

50

100

150

200

Deflection (mm)

Load

(kN

)

1- Isotropic CFRP/ Perfect Bond2- Orthotropic CFRP/ Perfect Bond3- Isotropic CFRP/ Cohesive Model4- Orthotropic CFRP/ Cohesive Model5- Experimental

1

43

2

5

0 2 4 6 8 10 120

50

100

150

200

Deflection (mm)

Loa

d (k

N)

1- Isotropic CFRP/ Perfect Bond2- Orthotropic CFRP/ Perfect Bond3- Isotropic CFRP/ Cohesive Model4- Orthotropic CFRP/ Cohesive Model5- Experimental

5

1

2

43

0 2 4 6 8 10 120

50

100

150

200

Deflection (mm)

Loa

d (k

N)

1- Isotropic CFRP/ Perfect Bond2- Orthotropic CFRP/ Perfect Bond4- Isotropic CFRP/ Cohesive Model4- Orthotropic CFRP/ Cohesive Model5- Experimental

5

3

4

2 1

Fig. 12. Loaddeection curves of beams, obtained by experiments and differentmodels.Fig. 13. Comparison of axial stress distribution between un-strengthened beamand strengthened beam, RB2.

ctures 92 (2010) 139113984.3. Evolution of cracks

As the concrete damage plasticity model does not have a nota-tion of cracks developing at the material integration point, it wasassumed that cracking initiates at the points where the maximumprincipal plastic strain is positive, following Lubliner et al. [24].Fig. 14 shows a comparison between plastic strain distributionsobtained from the nite element analysis and crack patterns ob-tained from the experiments for the control beam and strength-ened beams. The cracks obtained in the experiments and in thesimulations are similar, which indicates that the model can capturethe mechanisms of fracture in the beams.

4.4. Failure mode

The perfect bond model does not include fracture of the bond,and is thus unable to model the debonding fracture mode whichthe experiments showed. The cohesive model, on the other hand,can represent debonding. When the cohesive bond model was useddebonding fracture occurred, just like in the experiments. This isillustrated in Fig. 15.

4.5. Stress in bond layer

Debonding of CFRP is likely to initiate at the stress concentra-tion in the bond layer, which occur in the plate end region andaround cracks. A simplied illustration of the axial stress in thecomposite and the corresponding shear stress in the bond layerfor a beamwith a constant moment and a mid-span crack is shownin Fig. 16. In the anchorage zone the axial stress in the composite isincreasing and the axial force is transmitted to the compositethrough shear stress in the bond layer. In the crack zone axial forcecannot be sustained by the beam itself and axial force is thus trans-mitted to the composite, resulting in shear stress in the bond layer.

StruY.T. Obaidat et al. / CompositeThe stress state in the bond layer in the analysed beams is morecomplicated due to a complex crack pattern and 3-D effects. Still,it is possible to see the phenomena illustrated in Fig. 16.

Fig. 17 illustrates the shear stress in the cohesive layer for RB2at different load levels. Note that due to symmetry only one half ofthe beam is represented. From Fig. 17, it can be seen that when theload is equal to 8 kN (i.e., before cracking) there are shear stressconcentrations at the pre-cracked region and at the plate end. Byincreasing the load up to 100 kN (i.e., after the cracks are initiated)the interfacial shear stress increases, and has a maximum value atthe plate end. The shear stress also reaches maximum valuearound the pre-cracked zone due to rapidly transmitted force be-tween the concrete and CFRP. At the ultimate load, 140 kN, deb-onding has occurred at the plate end, and the maximum shearstress shifts to the mid-span which becomes a new anchorage zoneand where the exural cracks propagate.

Fig. 18 shows the shear stress in the cohesive layer for differentCFRP lengths at a load of 100 kN. At this load cracking has initiated.It is clear that when the CFRP length is short, the entire plate is ananchorage zone and the shear stress is high and almost constant,see RB3. For RB1, the anchorage length needed is provided outsidethe cracking region which leads to an improved performance. Since

Fig. 14. Comparison between plastic strain distribution from FEM analysis andcrack patterns from experiments.ctures 92 (2010) 13911398 1397the moment is decreasing towards the end of the beam, the shearstresses do not reach the same level in the anchorage zone as forRB2 and RB3.

Fig. 15. Comparison of failure mode from FEM analysis and experiment for beamRB2.

Fig. 16. Axial stress in composite material and shear stress in bond layer.

The cohesive model proved able to represent the bondbehaviour between CFRP and concrete. The predicted ulti-mate loads and the debonding failure mode were in excel-lent correlation with the experimental work.

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[2] Valivonis J, Skuturna T. Cracking and strength of reinforced concrete structuresin exure strengthened with carbon bre laminates. Civ Eng Manage J2006;13(4):31733.0100200300400500

-1.5

-1

-0.5

0

0.5

1

1.5

Inte

rfaci

al s

hear

st

ress

(MPa

)

140 kN100 kN8 kN

Precrack

1398 Y.T. Obaidat et al. / Composite Structures 92 (2010) 13911398Distance from midspan (mm)Fig. 17. Shear stress of the interface layer at different loads for beam RB2.

0200400600800-1.5

-1

-0.5

0

0.5

1

1.5

Distance from midspan (mm)

Inte

rfa

cia

l she

ar

st

ress

(MPa

)

RB3RB2RB1

Precrack

Fig. 18. Shear stress of the interface layer for different beams at 100 kN.5. Conclusions

A nite element model was developed to analyse beams retro-tted with CFRP. The nite element results show good agreementwith the experimental results. Elastic orthotropic and isotropicbehaviours were used to represent the CFRP behaviour; also acohesive model was used to address the interfacial behaviour be-tween CFRP and concrete. The following conclusions can be drawnfrom this study:

The behaviour of the retrotted beams is signicantly inu-enced by the length of CFRP. This is clear in experimentalresults as well as in numerical analysis. The ultimate loadincreases with the length of the CFRP.

The perfect bond model failed to capture the softening ofthe beams.

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[9] Teng JG, Smith ST, Yao J, Chen JF. Intermediate crack-introduced debonding inbeams and slabs. Construct Build Mater J 2003;17(67):44762.

[10] Ebead U, Marzouk H. Tensionstiffening model for FRP strengthened RCconcrete two-way slab. Mater Struct 2004:193200.

[11] Hu H-T, Lin F-M, Jan Y-Y. Nonlinear nite element analysis of reinforcedconcrete beams strengthened by bre-reinforced plastic. Compos Struct J2004;63:27181.

[12] Lundquist J, Nordin H, Tljsten B, Olafsson T. Numerical analysis of concretebeams strengthened with CFRP a study of anchorage lengths. In: FRP inconstruction, Proceeding of the international symposium of bond behaviour ofFRP in structures; 2005. p. 24754.

[13] Santhakumar R, Chandrasekaran E. Analysis of retrotted reinforced concreteshear beams using carbon bre composite. Electron J Struct Eng 2004;4:6674.

[14] Obaidat Y. Retrotting of reinforced concrete beams using compositelaminates. Master Thesis, Jordan University of Science and Technology; 2007.

[15] Hibbitt, Karlsson, and Sorensen, Inc. ABAQUS Theory manual, User manual andExample Manual, Version 6.7. Providence, RI; 2000.

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The effect of CFRP and CFRP/concrete interface models when modelling retrofitted RC beams with FEMIntroductionExperimental workFinite element analysisMaterial properties and constitutive modelsConcreteSteel reinforcementCFRPCFRPconcrete interface

Numerical analysisNonlinear solution

ResultsLoaddeflection curvesEffect of retrofitting on the stressEvolution of cracksFailure modeStress in bond layer

ConclusionsReferences