. L

y of

Sci

30

habac

research on FRP strengthened RC beams has shown ve most common modes, including (i) rupture of FRP strips; (ii) compression fail-ure after yielding of steel; (iii) compression failure before yielding of steel; (iv) delamination of FRP strips due to crack; and (v) concrete

in recent years. Upgrading structural load capacity is a sub-stantial part of the rehabilitation market, and seismic retro-

as an eective and convenient method. The main advantages

aration. Moreover, the strengthening system can be easilymaintained.

ing, there exist six distinct failure modes (see in Fig. 1),as described in the following:

(i) Compression failure before yielding of steel: the con-crete crushes in compression (i.e., the strain in theconcrete exceeds the ultimate value of 0.0035) before

* Corresponding author. Tel.: +852 2358 7207; fax: +852 2358 1543.E-mail address: mejkkim@ust.hk (J.-K. Kim).

Composite Structures 77 (2t of concrete components in earthquake regions is nowbecoming a mainstream. As a combined result of structuralrehabilitation needs, strengthening and rehabilitation ofconcrete structures have become the industrys majorgrowth area. Amongst various methods developed forstrengthening and rehabilitation of reinforced concrete(RC) beam structures, external bonding of bre reinforcedplastic (FRP) strips to the beam has been widely accepted

Signicant progress has been made based on experi-ments, theoretical analysis, and numerical simulation todemonstrate that the bonding of FRP strips to the tensionsot of reinforced concrete beams can improve much theultimate exural strength and stiness, although somereduction in ductility of the beam is caused. In strengthen-ing reinforced concrete beams with FRP strips, dierentfailure modes have been observed [13]. Generally speak-cover separation. In this paper, a failure diagram is established to show the relationship and the transfer tendency among dierent failuremodes for RC beams strengthened with FRP strips, and how failure modes change with FRP thickness and the distance from the end ofFRP strips to the support. The idea behind the failure diagram is that the failure mode associated with the lowest strain in FRP or con-crete by comparison is mostly likely to occur. The predictions based on the present failure diagram are compared to 33 experimental datafrom the literature and good agreement on failure mode and ultimate load has been obtained. Some discussion and recommendation forpractical design are given. 2005 Elsevier Ltd. All rights reserved.

Keywords: Failure mode; RC beam; Strengthen; Diagram; Fibre reinforced plastic

1. Introduction

Infrastructure repair and rehabilitation has become anincreasingly important challenge to the concrete industry

of FRP include high strength and stiness, high resistanceto corrosion and chemicals, as well as light weight due tolow density. The retrotting can be applied economically,as there is no need for mechanical xing and surface prep-Failure diagrams of FRP

Bo Gao a, Christopher K.Ya Department of Mechanical Engineering, Hong Kong Universit

b Department of Civil Engineering, Hong Kong University of

Available online

Abstract

Amongst various methods developed for strengthening and rebre reinforced plastic (FRP) strips to the beam has been widely0263-8223/$ - see front matter 2005 Elsevier Ltd. All rights reserved.doi:10.1016/j.compstruct.2005.08.003strengthened RC beams

eung b, Jang-Kyo Kim a,*

Science and Technology, Clear Water Bay, Hong Kong, China

ence and Technology, Clear Water Bay, Hong Kong, China

September 2005

ilitation of reinforced concrete (RC) beams, external bonding ofcepted as an eective and convenient method. The experimental

www.elsevier.com/locate/compstruct

007) 493508

tru494 B. Gao et al. / Composite Syielding of reinforcing steel and fracture of FRPstrips;

(ii) Compression failure after yielding of steel: the reinforc-ing steel yields due to tensile exure. This is followedby crushing of the concrete in the compression zone,before the tensile rupture of the FRP strips;

(iii) Rupture of FRP strips: the FRP strips rupture at theultimate strain following the yielding of reinforcingsteel rebar in tension;

Fig. 1. Failure modes of FRP strengthened RC beams: (a) compression failure;and (e) concrete cover separation.ctures 77 (2007) 493508(iv) Shear failure: the shear cracks extend from the vicin-ity of the support to the loading point, when the shearcapacity of the beam is exceeded;

(v) Delamination of FRP strips: delamination of CFRPstrip occurs rather catastrophically in an unstablemanner, with a thin layer of concrete residue attachedto the delaminated FRP sheets. The crack initiatesfrom the end of FRP strips or the bottom of a exuralor shear/exural crack in the concrete member;

(b) rupture of FRP strips; (c) shear failure; (d) delamination of FRP strips;

Stru(vi) Concrete cover separation: after crack initiation at theCFRP strip end, the CFRP strip is gradually peeledo with lumps of concrete detached from the longitu-dinal steel rebar.

These modes can be divided into two general categories,namely exural failures and local failures. The exural fail-ures include compression failure before yielding of steel,compression failure after yielding of steel and rupture ofFRP strips; shear failure, delamination of FRP strips andconcrete cover separation belong to local failures. Flexuralfailure modes are a typical of those encountered in conven-tional concrete beams, and, therefore, the perception onfailure mechanism and analytical methods for these failuremodes have already been successfully established.Although FRP rupture without yielding of steel reinforce-ment is sometimes regarded as a kind of exural failuremode, it is unlikely to occur unless the steel in tension islocated very near the centre of beam. In most exural equa-tions in the literatures for design recommendations, themost preferred failure mode to be designed for is compres-sion failure following yielding of steel reinforcement. Rup-ture of FRP strips following yielding of steel reinforcementis also acceptable. In comparison, compression failurebefore yielding of steel should be avoided as far as possible.In the above, the steel reinforcement mostly refers to steelrebar in tension. The yielding of tension steel rebar canensure the formation of large exural cracks, which pro-vides warning before ultimate failure.

Shear failure is caused generally by low shear reinforce-ment due to relatively large stirrup spacing. It may also oc-cur when only exural strengthening is applied, because theFRP strips along the bottom of reinforced concrete beamsdoes not improve the shear strength of beam remarkably. Itis found out that, however, restoring or upgrading beamshear strength using side FRP strips can result in increasedshear strength and stiness by substantially reducing shearcracking [46]. Many parameters including reinforcementconguration (U strip, side strip, full wrap), FRP orienta-tion, the use of mechanical type anchors, concrete strength,steel shear reinforcement and shear span to depth ratio [79], have been studied. Generally speaking, shear failure canbe eliminated by the appropriate shear strengthening of thebeam as mentioned above, and it is not to be discussed inthe following sections.

In the delamination of FRP strips, the bond between theFRP strip and the concrete fails in a sudden manner as aresult of the catastrophic propagation of a crack alongthe FRP concrete interface. In general, several reasonsmay cause this failure, such as: (a) technical aws includingimperfections in the spreading of the adhesive and signi-cantly uneven concrete tensile faces; (b) exural and ex-ural/shear cracks in the concrete that result in horizontalinterface cracks developed from the bottom tip of the ex-ural cracks; and (c) high shear and normal stress concentra-

B. Gao et al. / Compositetion at the end of FRP due to discontinuity [10]. Correctpreparation and operation can avoid aforementioned tech-nical aws. To analyse the initiation of failure at the end ofFRP strips, a number of models are available. These in-clude closed-form high order analytical models to solvefor stress distributions [11,12], shear-capacity-based models[13,14], and interfacial stress-based models [1517]. How-ever, experimental results show that, delamination alongthe concrete/FRP interface is most likely to occur fromexural and exural/shear cracks. High stress concentra-tion at the end of FRP strips may induce concrete coverseparation instead of delamination. Therefore, only delam-ination resulting from the exural and exural/shear crackson the tensile side is considered in the following, and theexisting analytical models will be discussed.

Concrete cover separation is a very common failuremechanism observed in experimental work. For this failuremode, a crack initiates in the vicinity of one of the FRPplate ends, then develops to the level of the tension steelreinforcement, and propagates horizontally towards themid span along the steel rebar. It is noticed that in the pro-cess many shear/exural cracks are developed in the con-crete cover forming tooths between the cracks. Basedon this mechanism, many theoretical models have beenbuilt.

From the design point of view, the relationship and thetransition guideline among the various failure modes haveto be understood. Currently there are very few papers thatstudy the varying trend of failure mode in terms of thechange of strengthening parameters (e.g., FRP thickness,FRP length, etc.), and identify which failure mechanismis dominant for the beam design. The objective of this pa-per is to build a diagram showing the relationship and thetransition among dierent failure modes for RC beamsstrengthened with FRP strips, and how failure modes varywith FRP thickness and the distance from the end of FRPstrips to the support. The failure mode prediction diagramis useful in establishing an FRP material selection proce-dure for external strengthening of RC beams. A review ofprevious theoretical models for these failure modes is givenrst, and appropriate expressions are chosen for failuremode prediction. A step-by-step procedure to establishthe failure mode diagram is also presented. Furthermore,a design example is provided to demonstrate the applicabil-ity of this approach. The applicability of the approach willthen be veried with a signicant number of experimentalresults. Finally, some discussions and recommendationsfor practical design are given.

2. Theoretical expressions for various failure modes

2.1. Flexural failure modes

To date, numerous exural design equations have beenproduced, and also existing research suggests that the ulti-mate exural strength of FRP strengthened RC beams canbe predicted using existing RC beam design approaches

ctures 77 (2007) 493508 495with appropriate modications to account for the brittlenature of FRPs [2,10,16,1823]. Some similar assumptions

in exural strength design equations are (a) plane sectionremaining plane after bending; (b) zero tensile strength inconcrete; (c) adhesive being omitted; and (d) the perfectbonding between the concrete and FRP plate.

Fig. 2 shows the cross section of a rectangular beam sub-jected to bending and the resultant strain distribution alongthe depth of the beam as well as a simplied equivalentrectangular stress block. Notice that d 0, d, and df denotethe depths of compressive steel, tensile steel and FRPstrips, respectively; As and A

0s are the area of tensile and

compressive steel reinforcement; bc and bf are the width

taneously. As the failure mode transitions from FRP rup-ture to compression failure, dierent expressions forAf,min can be obtained for dierent compressive steelconditions,

x hecuecu efu ; 5

Af;min a1f0cbcxb1 Ese0sA0s fyAs

Efefu; e0s 0:0035

x d 0x

< esy;

6a1f 0cbcxb1 f 0yA0s fyAs 0 x d 0

496 B. Gao et al. / Composite Structures 77 (2007) 493508of concrete and FRP strips; and x, h, and h 0 are the depthof the neutral axis, concrete beam, and concrete cover,respectively. In addition, ec, es, e0s, and ef are the strains ofconcrete, tensile steel rebar, compressive steel rebar andFRP strips, respectively. With the reference to Fig. 2, theinternal force components related to concrete and FRPstrips are:

Cc a1f 0cbcxb1; 1T f EfefAf ; 2where a1 (the ratio of the uniform stress in the rectangularcompression block to the maximum compressive strength)and b1 (the ratio of the depth of the rectangular compres-sion block to the depth to the neutral axis). Dierent valuesof a1 and b1 are dened as follows. El-Mihilmy and Tede-sco [2] set a1 and b1 to be 0.85 and 1:09 0:008f 0c , respec-tively. In Ng and Lee [23], the adopted values are 0.67 and0.9 for a1 and b1. Considering the eect of compressiveconcrete strength on these two factors, Chaallal et al. [19]dened a1 and b1 as follows:

a1 0:85 0:0015f 0c P 0:6; 3b1 0:97 0:0015f 0c P 0:6; 4which are also recommended in this paper.

Since there are three main exure failures, two balancedlimited values of cross section area of FRP are employed,Af,min and Af,max. If Af < Af,min, the rupture of FRP stripsmode can dominate. If Af,min < Af < Af,max compressionfailure after yielding of steel must take place. If Af > Af,max,compression failure before yielding of steel is to occur.

In the calculation of Af,min, ec = ecu(0.0035) and ef = efu(the fracture strain of FRP) are assumed to happen simul-Fig. 2. Cross section dimensions with stAf;min Efefu ; es 0:0035 x P esy.7

In the calculation of Af,max, ec = ecu (0.0035) and es = esy(the yielding strain of tension steel) are assumed to occursimultaneously. As the failure mode transitions fromcompression failure after yielding of steel to compressionfailure before yielding of steel, Af,max for dierent compres-sive steel conditions, can be obtained as follows:

x decuecu esy ; ef 0:0035

d f xx

; 8

Af;max a1f0cbcxb1 Ese0sA0s fyAs

Efef; e0s 0:0035

x d 0x

< esy;

9

Af;max a1f 0cbcxb1 f 0yA0s fyAs

Efef; e0s 0:0035

x d 0xP esy.

10Although the main objective of this failure diagram is toshow the relationship and the transition among dierentfailure modes, it can also predict the ultimate exuralstrength of FRP strengthened RC beams. Only briefdescriptions for expression are presented in Appendix A.

2.2. Delamination of FRP strips

Besides the end of FRP strips, exural and exural/shear cracks are also possible locations for delaminationto occur. While the beam is loaded, these cracks tend toopen and may induce high interfacial shear stress, thusresulting in crack propagation along the interface. Com-rain distribution and stress diagram.

Strupared to the existing stress analysis for delamination fromthe end of FRP, not much research has been carried out fordelamination initiating from cracks.

Triantallou and Plevris [10] suggested that the failurewas due to vertical (v) and horizontal (w) concrete crackopenings, which were resulted from the dowel action andaggregate interlock mechanisms. Also, it was assumed thatthe dowel deformation in the longitudinal steel and theFRP at the crack location were primarily due to shear.Therefore, when the shear force reached a critical value,the failure occurred as follows:

V cr vw

crGsAs Gfbf tf. 11

With the equation, the corresponding load capacity couldbe obtained. Nevertheless, (v/w)cr that was a characteristicproperty of the FRP-concrete bond, was not supported bynecessary experimental results.

In the study by Buyukozturk and Hearing [1], it wasshown that exural cracks in large moment region couldinitiate interfacial fracture in shear mode, and exural/shear cracks in mixed shear and moment region could in-duce mixed mode fracture. With the concept of fracturemechanics, when the strain energy release rate reaches theinterfacial fracture resistance, failure takes place. The crit-ical strain energy release rate can be measured with the sin-gle lap test.

Normal and shear stress distributions along the interfacebetween concrete and FRP have been studied in many pa-pers. Of note is that the normal stress perpendicular to theplate under exural cracks is compression due to bending.Since compressive normal stress cannot lead to delamina-tion, only shear stress under the cracks was responsiblefor delamination [24]. At the two sides of a crack, the max-imum shear stress smax at the adhesive/concrete interfacecan be calculated if the longitudinal stress in FRP plate isknown [17]. An approximate equation for smax is givenby [20]

smax GatfEf ta

rff ; 12

where the ff was the axial stress in the FRP plate.A theoretical framework was developed to analyse the

delamination at the location of a exural crack in the beam[24]. A fracture mechanics analysis was applied to get therelationship among M (moment), a (crack length), and w(crack mouth). The iterative calculation gave rise to Mfor a given crack size. Then, the maximum shear stress con-centration at the crack could be obtained from

smax wG2ta

. 13

By repeating the computation for various crack sizes, therelationship between smax and M could be established.

It was shown that crack induced delamination of FRP

B. Gao et al. / Compositehad much in common with debonding failures observedin the simple shear test [25]. In the literature, several bondstrength models based on the fracture mechanics have beendeveloped [26,27]. By modifying these models and with theempirical tting of experimental data, the following expres-sion was obtained by Teng et al. [25]:

rf;max 1:1bf

Ef

f 0c

ptf

s; 14

bf 2 bf=bc1 bf=bc

s15

in which rf,max was the maximum tensile stress permitted inFRP plate. 1.1 is a factor that will provide the best t toexperimental results. When the tensile stress in FRP stripsreaches rf,max in a strengthened RC beam subjected tobending, FRP debonding occurs. Obviously, with knownrf,max, the maximum moment or load capacity of the beamcan be calculated. Since this model can provide reasonableprediction while being simple for practical use, it is em-ployed in this paper.

2.3. Concrete cover separation

In an eort to identify the strength of a strengthened RCbeam failed by concrete cover separation, many studieshave bean carried out and several analytical models wereformulated. In general, two categories of analytical theo-retical solutions exist, including interfacial stress modeland tooth model. For interfacial stress model, most papersattempt to predict the stress distribution along the interfacebetween FRP and concrete, especially stress concentrationat the end of FRP. A simplied and approximate analyticalmodel to produce the shear and normal stress concentra-tions at the cut o point of FRP strips was developed byRoberts [15]. Actually, this model has been widely acceptedby many researchers, and equations based on its modica-tion were given in many studies [3,28]. Also, the papers byMalek et al. [17] and Saadatmanesh and Malek [20] devel-oped a methodology based on the linear elastic behaviourof the material and compatibility of deformation to predictthe interfacial stresses. Moreover, other analytical modelsconsidering more information, such as orthotropic materialproperties, have been developed [29,30]. Elastic models areusually not accurate in predicting the failure load [25].Also, some elastic models are cumbersome and not suitablefor hand calculation. In fact, for concrete cover separation,an inclined concrete crack is always observed to form at theplate end before the ultimate loading is reached in theexperiments. This means that the elastic analysis is no longervalid when failure is approached.

On the other hand, using the concept of concrete tooth,tooth-based models have been developed [31,32]. A con-crete tooth is a part of the concrete cover between twoadjacent cracks. It deforms like a cantilever under the ac-tion of horizontal shear stresses at the bottom of the con-

ctures 77 (2007) 493508 497crete beam. Concrete cover separation was deemed tooccur when the tensile stress at the root of the tooth

(tf represents the thickness of FRP strips) is applied, to theend of FRP plate as shown in Fig. 3. As shown above, onecan expect that many cracks appear in the tension side ofthe beam. The crack spacing model for conventional rein-forced concrete is extended for calculating the minimumstabilised crack spacing, lfmin in the case of RC beams with

tructures 77 (2007) 493508exceeded the tensile strength of concrete. Knowing the min-imum crack spacing, the critical shear stress can be deter-mined by using conventional cantilever beam theory,based on the above failure criterion. Herein, the criticalshear stress is assumed to act over an eective length deter-mined from empirical tting of experimental data. Then,from stress equilibrium of the FRP plate over the eectivelength, the limited maximum tension stress in FRP can becalculated, and thus the ultimate load or moment of thestrengthened beam can be obtained. A major limitationof the approach is that the cantilever length (i.e., the con-crete cover depth) is very short compared with its height(which is the minimum crack spacing). As a result, the con-ventional cantilever beam theory employed to obtain therelation between the tensile stress at the root of the toothand the applied shear stress is not valid.

In the following, a new model is proposed to predict thefailure of the concrete tooth. This analytical expression wasdeveloped for predicting the stress concentrations in con-crete near the tension rebar closest to the cut o point ofthe FRP strip, and then obtaining the load capacity basedon a specic failure criterion. The following assumptionswere made: (i) linear elastic and isotropic behaviour for con-crete, FRP, epoxy, and steel reinforcement, (ii) perfectbonding between concrete and FRP strips, and (iii) linearstrain distribution through the full depth of the section withcracked concrete. The methodology is implemented in twostages: (I) prediction of the tensile stresses in the FRP stripsat the curtailments and corresponding shear stress at thelocation of steel bar in tension assuming full composite ac-tion; and (II) solving the stress concentrations caused by re-verse tensile force of FRP strips at the curtailment locationdue to the cut o of FRP strips, and comparing the super-posed stresses with the concrete strength. In the secondstage, the nite element method (FEM) is employed to ob-tain accurate stress proles in the model, and a statisticalanalysis of experimental results gives rise to a modicationfactor that will lead to accurate predictions.

In the rst stage, if considering the full composite actionand elastic behaviour, the tensile stress of FRP strips at thecurtailment location, ff0, can be obtained from conven-tional beam theory as

ff0 M0I h x. 16

Herein, I is the cracked transformed moment of inertia ofbeam cross section in terms of the FRP plate, and M0 isthe bending moment at the plate curtailment location.The shear stress in concrete near the tension rebar closestto the cut o point of FRP strip, sI0, is

sI0 V 0Ibc

h xbf tf ; 17

where V0 is the shear force at the plate curtailmentlocation.

498 B. Gao et al. / Composite SIn the second stage, since the axial stress ff0 at the endof FRP does not actually exist, an opposite force, Wff0bftfexternally bonded FRP plate, as presented below:

lfmin Aeft

usP

Obar ufbf . 18

In this equation, us and uf is the average bond strength forsteel/concrete and FRP/concrete, respectively.

PObar is

the total perimeter of the tension bars, and Ae is the areaof concrete in tension. Also, one can take us 0:28

fcu

pand uf 0:28

fcu

p. Indeed, the results are found not too

sensitive to the exact value chosen for the parameter uf.In this model, W is an empirical function obtained fromempirical tting of experimental results. It is found that acomplete quadratic equation of W in terms of Lfs/L andbf/bc, as given in Eq. (19) below, will give the best agree-ment with test results.

W 3:527 35:827 Lfs=L 4:972 bf=bc 240:124 Lfs=L2 3:080 bf=bc2 1:635 Lfs=L bf=bc; Lfs=L 6 0:1 19

in which, Lfs and L represent the distance from the end ofFRP to the support and total span length, respectively. Thecomparison between predicted and experimental values for39 strengthened beams, with and without the modicationfactor W, are shown in Fig. 4. The details of selected sam-ples are presented in Gao et al. [33]. From the gure, it isclear the modication factor is necessary to obtain goodagreement between predicted and experimental results.

Under the applied force in Fig. 3, we assume that com-plete shear stress transfer between FRP and concrete takesplace over lfmin, the concrete cover block nearest to the endof FRP strips. When the individual concrete block at theend of FRP strips is subjected to a force (Wff0bftf), the ver-tical normal stress and shear stress in concrete near the ten-sion rebar closest to the cut o point of FRP strips in stageII, rII0 and s

II0 , can be calculated. As mentioned, the cantile-

ver beam length is too short compared to its depth for theconventional cantilever beam theory to be valid. Therefore,the nite element method (FEM) is applied to obtain rII0and sII0 . The rectangular cover region (one piece of tooth)between two cracks is modelled, and a unit force is appliedat the end of FRP strips for convenience. rII;unit0 and s

II;unit0 ,

the vertical normal and shear stresses for a unit force inFig. 3. Analysis in stage II with opposite axial force in FRP strips.

stage II, can be obtained. We have attempted to solve theproblem with three dierent models: (i) a 3D model withthe FRP and adhesive considered (Fig. 5(a)), (ii) a 2D modelwith the FRP and adhesive considered (Fig. 5b), and (iii) a2D model neglecting the presence of adhesive and FRP,

Specimens co

0.0

0.5

1.0

1.5

2.0

Predi

cted

/ ex

perim

enta

l fai

lure

lo

ad

ratio

With m

Withou

Ga1

Gb1

Gb2

MB

3M

B4

MB

5R

HB

5R

HB

6FK

F5FK

F6FK

F7FK

F10

B2

B4

B6

A1c

A2b A2c

Fig. 4. The comparison of predicted and experime

B. Gao et al. / Composite Struwith loading applied directly onto the concrete (Fig. 5c).The results indicate that as long as an appropriate modi-cation factor (obtained from empirical tting) is used withthe nite element results, each of the models can predictfailure loads in good agreement with experimental data.For convenience, we have decided to adopt the simplestmodel (Fig. 5c) for further analysis. More details on themodels and comparisons with test results can be found [33].Fig. 5. FEM models for predicting rII;unit0 and sII;unit0 : (a) 3D with FRP;

(b) 2D with FRP; and (c) 2D without FRP.In practical design, it is inconvenient to run nite ele-ment analysis every time. A better alternative is to provideequations for, rII;unit0 and s

II;unit0 , the stresses resulted from a

unit load applied on the plate end, based on a series of -nite element analysis. From the geometry of the problem,it is clear that the stresses are a function of lfmin=h

0, wherelfmin is the minimum stabilised crack spacing and h

0 is thedepth of concrete cover. Moreover, the stress for a unit ap-plied load must be inversely proportional to the width ofthe beam (bc) as well as the cover depth h

0. For a largercover depth, if lfmin=h

0 is xed, the same loads is appliedto a larger member, so the stress will decrease proportion-ally. Summarizing the above, one can write the stresses perunit load in the following form:

rII;unit0 F 1l

fmin

h0 bch

0 ; 20

sII;unit F 2lfmin

h0 ; 21

llected

odification factor

t modification factor

1 U,1

.0m

2 U,1

.0m

1Au

1Bu

1Cu

2Au

2Bu

2Cu

3Au

3Bu

3Cu B C

AA

3A

A4

AA

5SM

6A

950

A11

00A

1150

NB

2

ntal results, with/without modication factor.

ctures 77 (2007) 493508 4990 bch0

where bc and h0 are dimensionless that are the relative ra-

tios to 1 m. Through a systematic nite element analysis,the functions F1 and F2 can be numerically obtained. Inpractical design, with the known values of lfmin, bc and h

0,the F1 and F2 values can be calculated from the followingstatistical equations:

F 1 0:6054 lfmin

h0

23:7292 l

fmin

h0

9:4324; l

fmin

h06 3;

22a

F 1 3:7; lfmin

h0> 3; 22b

F 2 0:1197 lfmin

h0

20:7387 l

fmin

h0

1:7982; l

fmin

h06 3;

23a

F 2 0:66; lfmin

h0> 3. 23b

The complete solutions for the vertical normal and shearstresses in concrete near the tension rebar closest to the cuto point of FRP strips (r0 and s0), can be determined bysuperposition

r0 rII0 WM0Ih xbf tfrII;unit0 ; 24

s0 sI0 sII0 V 0Ibc

h xbf tf WM0I h xbf tfsII;unit0 . 25

The failure criterion for concrete cover separation failure is

ft 0:53f 0c

p. 27

If a strengthened RC beam is subjected to four point bend-ing, M0 and V0 in terms of the totally applied load, 2P, aregiven

M0 PLfs; V 0 P . 28

view, the thickness of FRP is a sensitive and important fac-tor that will aect the ultimate failure mode. With gradu-ally increasing FRP thickness to strengthen a RC beam,the probable order for failure occurrence is rupture ofFRP, delamination of FRP, concrete cover separationand then compression failure. As a result, it is reasonableto set thickness of FRP (tf) as a variable, which inuencesthe ultimate failure mode. Another important variable isthe distance from support to cut o point of FRP strips(Lfs), although only concrete cover separation failure is

facturer or measurement. For compression failure whetherit occurs before or after steel yielding, the failure strain isthe concrete ultimate strain (ecu), which is taken to be0.0035 in general. Considering the delamination of FRPstrips due to crack, the maximum corresponding strain inFRP (edf ) and strain in concrete (e

dc ) at failure are obtained

from Eqs. (14) and (15) as

0c

I

d 0

x

0

500 B. Gao et al. / Composite Structures 77 (2007) 493508Consequently, P can be determined as

3. Procedure for constructing the failure diagram

In this paper, the authors attempt to draw a failure dia-gram to predict the failure mode for a given strengthenedRC beam. There are ve possible failure modes include:(a) rupture of FRP strips; (b) compression failure afteryielding of steel; (c) compression failure before yieldingof steel; (d) delamination of FRP strips due to crack; and(e) concrete cover separation. From the practical point of

P 0:53f

pWLfsh xbf tfrII;unit0

2I

WLfsh xbf tfrII;unit0

2I

!2vuut0@

X WLfsh xbf tfrII;unit0

2I

WLfsh xbf tfrII;unit0

2I

!2

vuut

epf 0:53

f 0c

pLLs

X Efbf tfh 0:5bx d xh x

EsAsd 0:5bx xh

epc 0:53

f 0c

pLLs that when the maximum principle tensile stress r0,1 in con-crete near the tension rebar closest to the cut o point ofFRP strips is greater than the ultimate tensile strength ofconcrete ft, failure occurs. r0,1 can be obtained by the clas-sical stress transformation equations for a plane stresscondition

r0;1 r02

r02

2 s02

r. 26

And ft was dened in ACI code 318-95 (1999) as follows:X Efbf tfh 0:5bx d xh x EsAsd 0:5bx x dh xedf 1:12 bf=bc1 bf=bc

f 0c

pEf tf

s; 30

edc 1:12 bf=bc1 bf=bc

f 0c

pEf tf

sx

h x . 31

For concrete cover separation failure, in terms of Eq.(29), we can get the maximum corresponding strain inFRP (epf ) and strain in concrete (e

pc ) at failure below:

1

Ibch xbf tf W LfsI h xbf tfs

II;unit0

21A.29

1

bch xbf tf W LfsI h xbf tfs

II;unit0

2; 32

E0sA0s0:5bx d 0

; 33

x ; 34associated with this parameter. For a particular beam tobe strengthened and a given FRP material, tf and Lfsare the only parameters governing the failure diagram.

To identify the failure mode of a strengthened RC beam,the maximum strain in concrete or FRP at failure is calcu-lated for each individual failure mode. The actual failuremode is the one that gives rise to the lowest failure strain.

When rupture of FRP strips occurs, the failure strain isthe ultimate axial strain in FRP (efu) obtained from manu-E0sA0s0:5bx d 0

h x

where LLs is the distance from the support to the loadingpoint, and x and Itr according to tf can be obtained asshown in Eqs. (35) and (36).

The procedure for failure diagram construction is de-scribed in detail as follows:

(1) Firstly, the critical FRP thickness, trcf separatingFRP rupture and compression failure after yielding of steeland tcacbf separating compression failure after yielding of

can be obtained from Eq. (31), with the assumption ofedc ecu. When tf reaches tdrf , delamination of FRP stripsstarts to occur in place of compression failure.

(3) Lastly, concrete cover separation failure is consid-ered. As mentioned above, thickness of FRP (tf) and thedistance from support to cut o point of FRP strips (Lfs)are set as variables, with tf as horizontal axis and Lfs asvertical axis. Considering four point bending test, most

x Es

EfAs EsEf A

0s bf tf

EsEf

As EsEf A0s bf tf

2 2Ecbc

Ef

EsEf

Asd EsEf A0sd0 bf tfd f

s

EcbcEf

; 35

I tr EcEf bcx3=3 Es

EfAsd x2 EsEf A

0sd 0 x2 bf tfd f x2. 36

B. Gao et al. / Composite Structures 77 (2007) 493508 501steel and compression failure before yielding of steel, are gi-ven by

trcf Af;minbf

; 37

tcacbf Af ;maxbf

; 38

where Af,min and Af,max are given by Eqs. (5)(10). WhenFRP thickness (tf) exceeds trcf , the failure mode changesfrom rupture of FRP to compression failure after yieldingof steel. While tf continues to increase to tcacbf , compres-sion failure before yielding of steel may take the place ofcompression failure after yielding of steel.

(2) Secondly, the occurrence of crack-induced delamina-tion of FRP strips is analysed. Setting edf efu, we can gettdlf using Eq. (30). If t

dlf 6 trcf , it means that when tf in-

creases to tdlf the failure mode changes from rupture ofFRP to delamination of FRP. If trcf < 0 or t

dlf > t

rcf , t

drfFig. 6. A typical failure diagram in the third stecases show that Lfs is not allowed to be longer than LLs,the distance from the support to the loading point, whichmeans that the cut o point of FRP must be outside theconstant moment region. Two situations should be consid-ered. Fig. 6(a) and (b) shows a typical failure diagram, fortdlf 6 trcf and tdlf > trcf , respectively.

When tdlf 6 trcf in the second stage, the occurrence ofconcrete cover separation is divided into two parts, namelythe left part and the right part relative to tdlf . Settingepf efu, the relationship of Lfs and tf is obtained fromEq. (33), and the upper region of the curve on the left sideof tdlf is the left part. In order to predict the right part, thecomparison between edf and e

pf have to be done. By assum-

ing edf epf , one can get the transfer curve of Lfs and tffrom delamination of FRP to concrete cover separation.Consequently, the upper region of the curve of Lfs vs tfon the right side of tdlf is the right part, referring to Eqs.(30) and (33).p: (a) tdlf 6 trcf and (b) tdlf > trcf (using tdrf ).

fs fand the upper region of the curve on the left side of trcf is

Therefore,

As a result, one can get

below:

trcf a1f 0cbcxb1Ese0sA0s fsAs

b E e 0:77252:30:20:0340:8920:0004210;0000:00014600:000157

0:1 0:79mm.

tcacbf a1f 0cbcxb1 f 0yA0s fyAs 0:772 52:3 0:2 0:076 0:892 460 0:0001 460 0:000157 127; 000 0:0034 8:05 mm.

tructures 77 (2007) 493508Next, in terms of Eqs. (8)(10) and (38), one can obtain

x decuecu esy

0:12 0:00350:0035 0:002 0:076 m;

f f futhe left part. In comparison, the upper region of the curveof Lfs vs tf between trcf and t

drf is the middle part, referring

to Eq. (34) on account of epc ecu. Furthermore, withedc epc , one can obtain the transfer curve of Lfs and tffrom delamination of FRP to concrete cover separationreferring to Eqs. (31) and (34), and thus the right part isdetermined as the upper part of the curve of Lfs vs tf onthe right side of tdrf .

4. Derivation of the failure diagrama specic example

Several simply supported beams under four pointbending [34] are employed as examples to demonstratethe establishment of the failure diagram. Appendix Bpresents the beam dimensions and material properties, aswell as the failure mode and ultimate load. The estab-lishment of failure diagram for this particular case isshown in the following, and the results are shown inFig. 7(g).

(1) Firstly, determine a1 and b1 from Eqs. (3) and (4):

a1 0:85 0:0015 52:3 0:772 > 0:6;b1 0:97 0:0015 52:3 0:892 > 0:6.Then, using Eqs. (5)(7) and (37), we can get

x hecuecu efu

0:15 0:00350:012 0:0035 0:034m;

e0s 0:0035x d 0x

0:00350:034 0:030:034

0:0004< 0:002 esy;

es 0:0035d xx 0:00350:12 0:034

0:034 0:009> 0:002 esy.

bfEfef 0:15When tdlf > trcf in the second stage, the occurrence of

concrete cover separation is divided into three parts,namely the left part (left of trcf ), the middle part (betweentrcf and t

drf ) and the right part (right of t

drf ). Setting e

pf efu,

the transfer curve of L and t is obtained from Eq. (33),

502 B. Gao et al. / Composite Sef 0:0035 h xx 0:00350:15 0:076

0:076 0:00341:1

2bf=bc1bf=bc

f 0c

pEf tf

s

0:53f 0c

pLLs

5127;0000:012(2) Considering the occurrence of delamination ofFRP strips due to crack, we can get tdlf using Eq. (30)below:

0:012 1:12 0:15=0:21 0:15=0:2

52:3p

127;000 tdlf

s) tdlf 0:34mm.

Since tdlf 0:34 mm 6 trcf 0:79 mm, it means that withincreasing tf to tdlf the failure mode changes from ruptureof FRP to delamination of FRP, without chance to failwith compression failure.

(3) Since tdlf 0:34 mm 6 trcf 0:79 mm in the secondstage, the occurrence of concrete cover separation is di-vided into two parts, namely the left part and the right partrelative to tdlf . Setting e

pf efu, Eq. (33) is changed to as

follows:

0:012 0:5352:3

p 0:75=

X

127; 000 tf 0:15

0:15 0:5 0:892x 0:12 x0:15 x

210; 000

0:000157 0:12 0:5 0:892x x 0:030:15 x

210; 000 0:0001 0:5 0:892x 0:03

.

The upper region of the curve on the left side of tdlf(0.34 mm) is the left part. In order to predict the right part,with the assumption of edf epf , one can get the transfercurve of Lfs and tf from delamination of FRP to concretecover separation referring to Eqs. (30) and (33), as givenand

e0s 0:0035x d 0x

0:0035 0:076 0:030:076

0:0021> 0:002 esy.X Efbf tf h0:5bx dxhx EsAsd0:5bxxd 0hx E

0sA

0s0:5bxd 0

B. Gao et al. / Composite Structures 77 (2007) 493508 503and

Fig. 7. Demonstration of established failure diagram compared to experiments done by: (a) Alagusundaramoorthy et al. [37]; (b) Arduini et al. [38];(c) Fanning and Kelly [36]; (d) Gao et al. [39]; (e) Maalej and Bian [40]; (f) Nguyen et al. [35]; (g) Rahimi and Hutchinson [34]; and (h) Triantallou andPlevris [10].

1:1

20:15=0:210:15=0:2

52:3p

127;000 tdlf

s

0:5352:3

p 0:75X 127;000 tf 0:150:150:50:892x 0:12x0:15x

210;0000:0001570:120:50:892x x0:030:15x 210;0000:00010:50:892x0:03 .

truConsequently, the upper region of the curve of Lfs vs tfon the right side of tdlf (0.34 mm) is the right part.

5. Verication and discussions

In order to verify the applicability of the failure dia-gram, published experimental results pertaining tostrengthened RC beams are analysed. Totally, 33 samplesare selected from eight references, showing results that cov-er various failure modes. Two series of tests carried out byNguyen et al. [35] and Fanning and Kelly [36] focused onthe eect of the Lfs on the strengthening performance.The other papers investigated the inuence of the thicknessof FRP strips. The various values of Lfs and tf, the ulti-mate loads and the failure modes from experiments andtheoretical models in failure diagram as well as detail infor-mation for all samples collected are summarised in Appen-dix B. The corresponding failure diagrams for the eightgroups of tests are shown in Fig. 7.

The comparison between experiments and prediction byfailure diagram shows that this method could predict thefailure mode for a strengthened RC beam. Also, the ulti-mate load capacity can be calculated by individual theoret-ical expression, after the particular failure mode isobtained. The failure diagram shows that failure modefor a strengthened RC beam may vary from rupture ofFRP strips, to delamination of FRP strips, and then toconcrete cover separation, with increasing FRP thickness.With the correct design, compression failure after yieldingof steel may take place before local failure. Reducing thedistance from support to cut o of FRP may decreasethe likelihood of concrete cover separation.

From the practical application point of view, com-pression failure after yielding of steel is most preferablein design. However, the occurrence of local failuressuch as delamination of FRP strips and concrete coverseparation precludes the chance of compression failureafter yielding of steel. In order to have compression fail-ure after steel yielding, besides reducing the distancefrom support to cut o of FRP as far as possible, appropri-ate selection of FRP properties is very important. FRPwith good performance, such as high strength, highelongation at failure and high modulus, may not be eec-tive in practical applications, because failure may occurby delamination of FRP strips early with low axial strainin FRP.

Before closing, a few remarks should be made on the useof the failure diagram in practice. To perform strengthen-ing of a given RC beam, the beam dimensions and rein-forcement ratio are xed. Also, the selection of FRPproperties is perhaps limited by the availability of commer-cial products. Consequently, the variables to be chosen areonly the FRP dimensions, including the FRP thickness,length and width. For a particular concrete beam, theFRP width can be selected as a certain percentage of the

504 B. Gao et al. / Composite Sbeam width. The two parameters represented on the failurediagram are then sucient to determine the failure mode.After knowing the possible failure mode, the load capacityand deection of the beam can be accurately predicted.Since the failure diagram summarizes all possible failuremodes, a clear picture of all possibilities are provided toguide the designer in choosing the best combination ofplate thickness and length. The plotting of failure diagramswill also facilitate the selection of the best material. More-over, with the failure mode predicted, the critical failureinitiation location can be known. That information is veryuseful for the continuous monitoring of strengthenedbeams, as well as the determination of appropriate posi-tions for the application of anchors.

The idea behind the failure diagram is that the failuremode associated with the lowest strain in FRP or concreteby comparison is most likely to occur. In this paper, a gen-eral concept is proposed. With future development leadingto better methods for local failure, the equations in this pa-per can be further rened.

6. Conclusions

Numerous studies including experimental research, the-oretical analysis, and numerical simulation have demon-strated that epoxy bonding of bre reinforced plastic(FRP) strips to the tension sot of reinforced concrete(RC) beams can signicantly improve the ultimate exuralstrength and stiness. Several important failure modeshave been studied, such as compression failure before orafter yielding of steel, rupture of FRP strips, delaminationof FRP strips and concrete cover separation.

This paper attempts to build a failure diagram to showthe relationship and the transition among dierent failuremodes for RC beams strengthened with FRP strips, andhow failure modes vary with FRP thickness and the dis-tance from the end of FRP strips to the support. The ideabehind this failure diagram is that the failure mode associ-ated with the lowest strain in FRP or concrete by compar-ison is most likely to occur. By comparison betweenpredictions based on failure diagram and experimental re-sults, we show that this method could predict the failuremode for a strengthened RC beam. Knowing the failuremode, the ultimate load capacity can be calculated. Thefailure diagram provides guidelines to practical design,and is useful in establishing a procedure for selecting thetype and size of FRP for the external strengthening ofRC beam.

Acknowledgements

The Research Grants Council of the Hong Kong SAR(Project No. HKUST 6050/99E), provided the nancialsupport of this work. The authors wish to thank the Con-struction Materials Laboratory, Advanced EngineeringMaterial Facilities, and Design and Manufacturing Ser-

ctures 77 (2007) 493508vices Facility in HKUST for their technical supports.

ef 0:0035 x . 46

occur after steel yielding. A can be obtained from the

increased to a value within the range of Af,min and Af,max,

ing of steel. The ultimate moment resistance Mu, which can

Mu, can be calculated by the equations below for dierent

f f,min

h x

Ese0A0 h d 0; e0 0:0035 x d < esy; 53

Struf

Eq. (9) or (10), as well as Mu from Eq. (39). If es < esyand ef < efu, the yielding of steel in tension can not be ob-tained before failure. Therefore, in order to have enoughductility and warning before failure, the area of FRP mustbe reduced to Af,max. The ultimate moment resistance, M

0u

is then lower than Mu. The calculation of Af,max has beenintroduced in Eqs. (8)(10). Then, we can get M 0u, asfollows:

M 0u a1f 0cbcb1xh 0:5b1x fyAsh d0If es > esy and ef < efu, the FRP area is within the rangeAf,min 6 Af 6 Af,max. In this case, compressive failure willAppendix A. Prediction of the ultimate exural strength of

strengthened RC Beams

Generally speaking, two situations should be consid-ered, including (a) obtaining the quantity of FRP to satisfythe requirement of moment capacity for a RC beam and(b) calculating the moment capacity of a strengthenedRC beam. Both cases are the reverse processes.

From the practical design point of view, the former isthe more common case. To archive the targeted moment,Mu, the following formula is employed:

Mu a1f 0cbcb1xh 0:5b1x fyAsh d f 0yA0sh d 0.39

Only one unknown variable, x, exists in the equation,which can be obtained as the solution of a quadraticequation

x b1

b21 4a1c1

q2a1

; 40

a1 0:5a1b21f 0cbc; 41

b1 a1b1f 0cbch; 42

c1 Mu fyAsh d f 0yA0sh d 0. 43

Knowing x, we can get, es, e0s, and ef by linear strain distri-bution with ec = 0.0035.

es 0:0035 d xx ; 44

e0s 0:0035x d 0x

; 45

d f x

B. Gao et al. / Composite Ese0sA0sh d 0; e0s 0:0035x dx

< esy; 47s s s x

M 0u a1f 0cbcb1xh 0:5b1x EsesAsh d0When Af,min 6 Af 6 Af,max,

M 0u a1f 0cbcb1xh 0:5b1x fyAsh d

Ese0sA0sh d 0; e0s 0:0035x d 0x

< esy; 51

M 0u a1f 0cbcb1xh 0:5b1x fyAsh d

f 0yA0sh d 0; e0s 0:0035x d 0xP esy. 52

When Af > Af,max,

M 0u a1f 0cbcb1xh 0:5b1x EsesAsh d0M 0u AfEfefuh 0:5b1x fyAsd 0:5b1x

Ese0sA0s0:5b1x d 0; e0s efux d 0h x < esy. 49

M 0u AfEfefuh 0:5b1x fyAsd 0:5b1x

f 0yA0s0:5b1x d 0; e0s efux d 0

P esy. 50situations.When A < A ,be obtained using Eqs. (51) and (52), will then be higherthan Mu.

To calculate the moment capacity of a given strength-ened RC beam, two balanced limited values of cross sec-tion area of FRP, Af,min and Af,max are calculated rst, todetermine the failure mode in terms of Af. For each individ-ual failure mode, x can be obtained from force equilibriumof the cross section and the relationship among strain com-ponents (Eqs. (44)(46)). The ultimate moment resistance

0to assure the occurrence of compression failure after yield-0M 0u a1f 0cbcb1xh 0:5b1x fyAsh d

f 0yA0sh d 0; e0s 0:0035x d 0xP esy. 48

When ef > efu, rupture of FRP strips is to occur instead ofcompression failure. In this case, the area of FRP can be

ctures 77 (2007) 493508 505 f 0yA0sh d 0; e0s 0:0035x dxP esy. 54

Appendix B. Details of experiments

Experiments Beam Beamwidth(mm)

Beamdepth(mm)

Beamlength(mm)

FRPlength(mm)

FRPwidth(mm)

FRPthickness(mm)

As A0s d

(mm)d 0

(mm)Lfs(mm)

Alagusundaramoorthyet al. [37]

CB11-1F 230 380 4576 4370 203 0.18 225 29 342 25 103CB11-1F 230 380 4576 4370 203 0.18 225 29 342 25 103CB11-2F 230 380 4576 4370 203 0.36 225 29 342 25 103CB11-2F 230 380 4576 4370 203 0.36 225 29 342 25 103

Arduini et al. [38] A3 200 200 2000 1700 150 1.3 214 214 163 37 150A4 200 200 2000 1700 150 1.3 214 214 163 37 150A5 200 200 2000 1700 150 2.6 214 214 163 37 150

Fanning andKelly [36]

FKF5 155 240 3000 2030 120 1.2 312 212 203 37 385FKF6 155 240 3000 2030 120 1.2 312 212 203 37 385FKF7 155 240 3000 1876 120 1.2 312 212 203 37 462FKF10 155 240 3000 1700 120 1.2 312 212 203 37 550

Gao et al. [39] T1 150 200 2000 1200 75 0.11 210 28 162 27 150T2 150 200 2000 1200 75 0.22 210 28 162 27 150T4 150 200 2000 1200 75 0.44 210 28 162 27 150T6 150 200 2000 1200 75 0.66 210 28 162 27 150

Maalej andBian [40]

MB2 115 150 1500 1200 115 0.111 310 210 125 25 75MB3 115 150 1500 1200 115 0.222 310 210 125 25 75MB4 115 150 1500 1200 115 0.333 310 210 125 25 75MB5 115 150 1500 1200 115 0.444 310 210 125 25 75

Nguyen et al. [35] A950 120 150 1500 950 80 1.2 310 26 120 28 190A1100 120 150 1500 1100 80 1.2 310 26 120 28 115A1150 120 150 1500 1150 80 1.2 310 26 120 28 90A1500 120 150 1500 1500 80 1.2 310 26 120 28 0

Rahimi andHutchinson [34]

RHB3 200 150 2300 1930 150 0.44 210 28 120 30 85RHB4 200 150 2300 1930 150 0.44 210 28 120 30 85RHB5 200 150 2300 1930 150 1.2 210 28 120 30 85RHB6 200 150 2300 1930 150 1.2 210 28 120 30 85

Triantallou andPlevris [10]

3 76 127 1220 1070 60.5 0.2 24.6 111 754 76 127 1220 1070 63.2 0.65 24.6 111 755 76 127 1220 1070 63.2 0.65 24.6 111 756 76 127 1220 1070 63.3 0.9 24.6 111 757 76 127 1220 1070 63.3 0.9 24.6 111 758 76 127 1220 1070 63.9 1.9 24.6 111 75

Experiments Beam f 0c(MPa)

ft(MPa)

Ec(GPa)

Es(GPa)

Ef(GPa)

Pmodel

(kN)Pexp

(kN)

aFailuremodemodel

aFailuremodeexp

Alagusundaramoorthyet al. [37]

CB11-1F 31 3.0 26.3 200 228 229 219 RF RFCB11-1F 31 3.0 26.3 200 228 229 223 RF RFCB11-2F 31 3.0 26.3 200 228 233 263 DF DFCB11-2F 31 3.0 26.3 200 228 233 270 DF DF

Arduini et al. [38] A3 33 2.6 25 200 167 94.1 106 CS CSA4 33 2.6 25 200 167 94.1 104 CS CSA5 33 2.6 25 200 167 67.8 84 CS CS

Fanning and Kelly [36] FKF5 80 5 39.2 204 155 112.75 100 CS CSFKF6 80 5 39.2 204 155 112.75 103 CS CSFKF7 80 5 39.2 204 155 94.83 97.5 CS CSFKF10 80 5 39.2 204 155 80.23 82 CS CS

506 B. Gao et al. / Composite Structures 77 (2007) 493508

cPa

StruAppendix B (continued)

Experiments Beam f 0c(MPa)

ft(MPa)

E

(G

Gao et al. [39] T1 43.1 3.5 25T2 43.1 3.5 25T4 43.1 3.5 25T6 43.1 3.5 25

Maalej and Bian [40] MB2 30.3 2.9 26MB3 30.3 2.9 26MB4 30.3 2.9 26MB5 30.3 2.9 26

Nguyen et al. [35] A950 27.3 2.8 25A1100 27.3 2.8 25A1150 27.3 2.8 25A1500 27.3 2.8 25

Rahimi and Hutchinson [34] RHB3 52.3 3 25RHB4 52.3 3 25RHB5 52.3 3 25RHB6 52.3 3 25

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200 235 71.4 73.2 RF RF200 235 80.9 80.7 DF DF200 235 94.7 86.4 DF CS200 235 73.5 86.3 CS CS

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508 B. Gao et al. / Composite Structures 77 (2007) 493508

Failure diagrams of FRP strengthened RC beamsIntroductionTheoretical expressions for various failure modesFlexural failure modesDelamination of FRP stripsConcrete cover separation

Procedure for constructing the failure diagramDerivation of the failure diagram mdash a specific exampleVerification and discussionsConclusionsAcknowledgementsPrediction of the ultimate flexural strength of strengthened RC BeamsDetails of experimentsReferences