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Failure Diagrams of FRP Strengthened RC Beams

Bo GAOa, Christopher K. Y. LEUNGb and Jang-Kyo KIMa*

aDepartment of Mechanical Engineering and bDepartment of Civil Engineering

Hong Kong University of Science & Technology Clear Water Bay, Hong Kong, China

Abstract

Amongst various methods developed for strengthening and rehabilitation of reinforced

concrete (RC) beams, external bonding of fibre reinforced plastic (FRP) strips to the beam has

been widely accepted as an effective and convenient method. The experimental research on

FRP strengthened RC beams has shown five most common modes, including (i) rupture of

FRP strips; (ii) compression failure after yielding of steel; (iii) compression failure before

yielding of steel; (iv) delamination of FRP strips due to crack; and (v) concrete cover

separation. In this paper, a failure diagram is established to show the relationship and the

transfer tendency among different failure modes for RC beams strengthened with FRP strips,

and how failure modes change with FRP thickness and the distance from the end of FRP

strips to the support. The idea behind the failure diagram is that the failure mode associated

with the lowest strain in FRP or concrete by comparison is mostly likely to occur. The

predictions based on the present failure diagram are compared to 33 experimental data from

the literature and good agreement on failure mode and ultimate load has been obtained. Some

discussion and recommendation for practical design are given.

Keywords: Failure mode; RC beam; Strengthen; Diagram; Fibre reinforced plastic *Corresponding author. Tel: 852-2358 7207; Fax: 852-2358 1543; Email: mejkkim@ust.hk

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1. Introduction

Infrastructure repair and rehabilitation has become an increasingly important challenge to the

concrete industry in recent years. Upgrading structural load capacity is a substantial part of

the rehabilitation market, and seismic retrofit of concrete components in earthquake regions is

now becoming a mainstream. As a combined result of structural rehabilitation needs,

strengthening and rehabilitation of concrete structures have become the industry’s major

growth area. Amongst various methods developed for strengthening and rehabilitation of

reinforced concrete (RC) beam structures, external bonding of fibre reinforced plastic (FRP)

strips to the beam has been widely accepted as an effective and convenient method. The main

advantages of FRP include high strength and stiffness, high resistance to corrosion and

chemicals, as well as light weight due to low density. The retrofitting can be applied

economically, as there is no need for mechanical fixing and surface preparation. Moreover,

the strengthening system can be easily maintained.

Significant progress has been made based on experiments, theoretical analysis and

numerical simulation to demonstrate that the bonding of FRP strips to the tension soffit of

reinforced concrete beams can improve much the ultimate flexural strength and stiffness,

although some reduction in ductility of the beam is caused. In strengthening reinforced

concrete beams with FRP strips, different failure modes have been observed [1-3]. Generally

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

(i) Compression failure before yielding of steel: the concrete crushes in compression (i.e. the

strain in the concrete exceeds the ultimate value of 0.0035) before yielding of reinforcing steel

and fracture of FRP strips;

(ii) Compression failure after yielding of steel: the reinforcing steel yields due to tensile

flexure. This is followed by crushing of the concrete in the compression zone, before the

tensile rupture of the FRP strips;

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(iii) Rupture of FRP strips: the FRP strips rupture at the ultimate strain following the yielding

of reinforcing steel rebar in tension;

(iv) Shear failure: the shear cracks extend from the vicinity of the support to the loading point,

when the shear capacity of the beam is exceeded;

(v) Delamination of FRP strips: delamination of CFRP strip occurs rather catastrophically in

an unstable manner, with a thin layer of concrete residue attached to the delaminated FRP

sheets. The crack initiates from the end of FRP strips or the bottom of a flexural or

shear/flexural crack in the concrete member;

(vi) Concrete cover separation: after crack initiation at the CFRP strip end, the CFRP strip is

gradually peeled off with lumps of concrete detached from the longitudinal steel rebar.

These modes can be divided into two general categories, namely flexural failures and

local failures. The flexural failures include compression failure before yielding of steel,

compression failure after yielding of steel and rupture of FRP strips; Shear failure,

delamination of FRP strips and concrete cover separation belong to local failures. Flexural

failure modes are a typical of those encountered in conventional concrete beams, and,

therefore, the perception on failure mechanism and analytical methods for these failure modes

have already been successfully established. Although FRP rupture without yielding of steel

reinforcement is sometimes regarded as a kind of flexural failure mode, it is unlikely to occur

unless the steel in tension is located very near the centre of beam. In most flexural equations

in the literatures for design recommendations, the most preferred failure mode to be designed

for is compression failure following yielding of steel reinforcement. Rupture of FRP strips

following yielding of steel reinforcement is also acceptable. In comparison, compression

failure before yielding of steel should be avoided as far as possible. In the above, the steel

reinforcement mostly refers to steel rebar in tension. The yielding of tension steel rebar can

ensure the formation of large flexural cracks, which provides warning before ultimate failure.

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Shear failure is caused generally by low shear reinforcement due to relatively large stirrup

spacing. It may also occur when only flexural strengthening is applied, because the FRP strips

along the bottom of reinforced concrete beams does not improve the shear strength of beam

remarkably. It is found out that, however, restoring or upgrading beam shear strength using

side FRP strips can result in increased shear strength and stiffness by substantially reducing

shear cracking [4-6]. Many parameters including reinforcement configuration (U strip, side

strip, full wrap), FRP orientation, the use of mechanical type anchors, concrete strength, steel

shear reinforcement and shear span to depth ratio [7-9], have been studied. Generally

speaking, shear failure can be eliminated by the appropriate shear strengthening of the beam

as mentioned above, and it is not to be discussed in the following sections.

In the delamination of FRP strips, the bond between the FRP strip and the concrete fails

in a sudden manner as a result of the catastrophic propagation of a crack along the FRP

concrete interface. In general, several reasons may cause this failure, such as: (a) technical

flaws including imperfections in the spreading of the adhesive and significantly uneven

concrete tensile faces; (b) flexural and flexural/shear cracks in the concrete that result in

horizontal interface cracks developed from the bottom tip of the flexural cracks; and (c) high

shear and normal stress concentration at the end of FRP due to discontinuity [10]. Correct

preparation and operation can avoid aforementioned technical flaws. To analyse the initiation

of failure at the end of FRP strips, a number of models are available. These include closed-

form high order analytical models to solve for stress distributions [11,12], shear-capacity-

based models [13,14], and interfacial stress-based models [15-17]. However, experimental

results show that, delamination along the concrete/FRP interface is most likely to occur from

flexural and flexural/shear cracks. High stress concentration at the end of FRP strips may

induce concrete cover separation instead of delamination. Therefore, only delamination

resulting from the flexural and flexural/shear cracks on the tensile side is considered in the

following, and the existing analytical models will be discussed.

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Concrete cover separation is a very common failure mechanism observed in experimental

work. For this failure mode, a crack initiates in the vicinity of one of the FRP plate ends, then

develops to the level of the tension steel reinforcement, and propagates horizontally towards

the mid span along the steel rebar. It is noticed that in the process many shear/flexural cracks

are developed in the concrete cover forming “tooths” between the cracks. Based on this

mechanism, many theoretical models have been built.

From the design point of view, the relationship and the transition guideline among the various

failure modes have to be understood. Currently there are very few papers that study the

varying trend of failure mode in terms of the change of strengthening parameters (e.g. FRP

thickness, FRP length, etc.), and identify which failure mechanism is dominant for the beam

design. The objective of this paper is to build a diagram showing the relationship and the

transition among different failure modes for RC beams strengthened with FRP strips, and how

failure modes vary with FRP thickness and the distance from the end of FRP strips to the

support. The failure mode prediction diagram is useful in establishing an FRP material

selection procedure for external strengthening of RC beams. A review of previous theoretical

models for these failure modes is given first, and appropriate expressions are chosen for

failure mode prediction. A step-by-step procedure to establish the failure mode diagram is

also presented. Furthermore, a design example is provided to demonstrate the applicability of

this approach. The applicability of the approach will then be verified with a significant

number of experimental results. Finally, some discussions and recommendations for practical

design are given.

2. Theoretical expressions for various failure modes

2.1 Flexural failure modes

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To date, numerous flexural design equations have been produced, and also existing research

suggests that the ultimate flexural strength of FRP strengthened RC beams can be predicted

using existing RC beam design approaches with appropriate modifications to account for the

brittle nature of FRPs [2,10,16,18-23]. Some similar assumptions in flexural strength design

equations are (a) plane section remaining plane after bending; (b) zero tensile strength in

concrete; (c) adhesive being omitted; and (d) the perfect bonding between the concrete and

FRP plate.

Fig. 2. shows the cross section of a rectangular beam subjected to bending and the

resultant strain distribution along the depth of the beam as well as a simplified equivalent

rectangular stress block. Notice that d’, d, and df denote the depths of compressive steel,

tensile steel and FRP strips, respectively; As and As’ are the area of tensile and compressive

steel reinforcement; bc and bf are the width of concrete and FRP strips; and x, h, and h’ are the

depth of the neutral axis, concrete beam, and concrete cover, respectively. In addition,

cε , sε , 'sε , and fε are the strains of concrete, tensile steel rebar, compressive steel rebar and

FRP strips, respectively. With the reference to Fig. 2., the internal force components related to

concrete and FRP strips are,

1'

1 βα xcbcfcC = (1)

fAffEfT ε= (2)

where 1α (the ratio of the uniform stress in the rectangular compression block to the

maximum compressive strength) and 1β (the ratio of the depth of the rectangular

compression block to the depth to the neutral axis). Different values of 1α and 1β are defined

as follows. El-Mihilmy and Tedesco [2] set 1α and 1β to be 0.85 and '008.009.1 cf− ,

respectively. In Ng and Lee [23], the adopted values are 0.67 and 0.9 for 1α and 1β .

Considering the effect of compressive concrete strength on these two factors, Chaallal et al.

[19] defined 1α and 1β as follows,

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6.0'0015.085.01 ≥−= cfα (3)

6.0'0015.097.01 ≥−= cfβ (4)

which are also recommended in this paper.

Since there are three main flexure failures, two balanced limited values of cross

section area of FRP are employed, Af,min and Af,max . If Af < Af,min, the rupture of FRP strips

mode can dominate. If Af,min < Af < Af,max compression failure 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, cε = cuε (0.0035) and fε = fuε (the fracture strain of FRP)

are assumed to happen simultaneously. As the failure mode transitions from FRP rupture to

compression failure, different expressions for Af,min can be obtained for different compressive

steel conditions,

fucu

cuhx

εεε+

= (5)

syxdx

sEAfAExbf

fAfuf

syssscc εεεεβα

<−=−+

='''

1'

1 0035.0',min (6)

syxdx

sEAfAfxbf

fAfuf

sysycc εεε

βα≥−=

−+=

'''1

'1 0035.0',min (7)

In the calculation of Af,max, cε = cuε (0.0035) and sε = syε (the yielding strain of

tension steel) are assumed to occur simultaneously. As the failure mode transitions from

compression failure after yielding of steel to compression failure before yielding of steel,

Af,max for different compressive steel conditions, can be obtained as follows.

xxd

fd

x f

sycu

cu −=

+= 0035.0, ε

εεε

(8)

syxdx

sEAfAExbf

fAff

syssscc εεεεβα

<−=−+

='''

1'

1 0035.0',max (9)

8

syxdx

sEAfAfxbf

fAff

sysycc εεε

βα≥−=

−+=

'''1

'1 0035.0',max (10)

Although the main objective of this failure diagram is to show the relationship and the

transition among different failure modes, it can also predict the ultimate flexural strength of

FRP strengthened RC beams. Only brief descriptions for expression are presented in

Appendix A.

2.2 Delamination of FRP strips

Besides the end of FRP strips, flexural and flexural /shear cracks are also possible locations

for delamination to occur. While the beam is loaded, these cracks tend to open and may

induce high interfacial shear stress, thus resulting in crack propagation along the interface.

Compared to the existing stress analysis for delamination from the end of FRP, not much

research has been carried out for delamination initiating from cracks.

Triantafillou and Plevris [10] suggested that the failure was due to vertical (v) and

horizontal (w) concrete crack openings, which were resulted from the dowel action and

aggregate interlock mechanisms. Also, it was assumed that the dowel deformation in the

longitudinal steel and the FRP at the crack location were primarily due to shear. Therefore,

when the shear force reached a critical value, the failure occurred as follows,

( )ftfbfGsAsGcrw

vcrV +⎟

⎠⎞⎜

⎝⎛= (11)

With the equation, the corresponding load capacity could be obtained. Nevertheless, (v/w)cr

that was a characteristic property of the FRP–concrete bond, was not supported by necessary

experimental results.

In the study by Buyukozturk and Hearing [1], it was shown that flexural cracks in

large moment region could initiate interfacial fracture in shear mode, and flexural/shear

cracks in mixed shear and moment region could induce mixed mode fracture. With the

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concept of fracture mechanics, when the strain energy release rate reaches the interfacial

fracture resistance, failure takes place. The critical strain energy release rate can be measured

with the single lap test.

Normal and shear stress distributions along the interface between concrete and FRP have

been studied in many papers. Of note is that the normal stress perpendicular to the plate under

flexural cracks is compression due to bending. Since compressive normal stress cannot lead to

delamination, only shear stress under the cracks was responsible for delamination [24]. At the

two sides of a crack, the maximum shear stress maxτ at the adhesive/concrete interface can be

calculated if the longitudinal stress in FRP plate is known [17]. An approximate equation for

maxτ is given by [20]

ftEtG

faf

fa=maxτ (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

flexural crack in the beam [24]. A fracture mechanics analysis was applied to get the

relationship among M (moment), a (crack length), and w (crack mouth). The iterative

calculation gave rise to M for a given crack size. Then, the maximum shear stress

concentration at the crack could be obtained from

atwG2max =τ . (13)

By repeating the computation for various crack sizes, the relationship between maxτ and M

could be established.

It was shown that crack induced delamination of FRP had much in common with

debonding failures observed in the simple shear test [25]. In the literature, several bond

strength models based on the fracture mechanics have been developed [26,27]. By modifying

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these models and with the empirical fitting of experimental data, the following expression was

obtained by Teng et al [25]:

f

cf

tfE

ff'

1.1max, βσ = (14)

cf

cfbbbb

f /1/2

+−

=β (15)

in which max,fσ was the maximum tensile stress permitted in FRP plate. 1.1 is a factor that

will provide the best fit to experimental results. When the tensile stress in FRP strips reaches

max,fσ in a strengthened RC beam subjected to bending, FRP debonding occurs. Obviously,

with known max,fσ , the maximum moment or load capacity of the beam can be calculated.

Since this model can provide reasonable prediction while being simple for practical use, it is

employed in this paper.

2.3 Concrete cover separation

In an effort to identify the strength of a strengthened RC beam failed by concrete cover

separation, many studies have bean carried out and several analytical models were

formulated. In general, two categories of analytical theoretical solutions exist, including

interfacial stress model and tooth model. For interfacial stress model, most papers attempt to

predict the stress distribution along the interface between FRP and concrete, especially stress

concentration at the end of FRP. A simplified and approximate analytical model to produce

the shear and normal stress concentrations at the cut off point of FRP strips was developed by

Roberts [15]. Actually, this model has been widely accepted by many researchers, and

equations based on its modification were given in many studies [3, 28]. Also, the papers by

Malek et al. [17] and Saadatmanesh and Malek [20] developed a methodology based on the

linear elastic behaviour of the material and compatibility of deformation to predict the

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interfacial stresses. Moreover, other analytical models considering more information, such as

orthotropic material properties, have been developed [29,30]. Elastic models are usually not

accurate in predicting the failure load [25]. Also, some elastic models are cumbersome and

not suitable for hand calculation. In fact, for concrete cover separation, an inclined concrete

crack is always observed to form at the plate end before the ultimate loading is reached in the

experiments. This means that the elastic analysis is no longer valid when failure is

approached.

On the other hand, using the concept of concrete tooth, tooth-based models have been

developed [31,32]. A concrete tooth is a part of the concrete cover between two adjacent

cracks. It deforms like a cantilever under the action of horizontal shear stresses at the bottom

of the concrete beam. Concrete cover separation was deemed to occur when the tensile stress

at the root of the tooth exceeded the tensile strength of concrete. Knowing the minimum crack

spacing, the critical shear stress can be determined by using conventional cantilever beam

theory, based on the above failure criterion. Herein, the critical shear stress is assumed to act

over an effective length determined from empirical fitting of experimental data. Then, from

stress equilibrium of the FRP plate over the effective length, the limited maximum tension

stress in FRP can be calculated, and thus the ultimate load or moment of the strengthened

beam can be obtained. A major limitation of the approach is that the cantilever length (i.e., the

concrete cover depth) is very short compared with its height (which is the minimum crack

spacing). As a result, the conventional cantilever beam theory employed to obtain the relation

between the tensile stress at the root of the tooth and the applied shear stress is not valid.

In the following, a new model is proposed to predict the failure of the concrete tooth.

This analytical expression was developed for predicting the stress concentrations in concrete

near the tension rebar closest to the cut off point of the FRP strip, and then obtaining the load

capacity based on a specific failure criterion. The following assumptions were made: (i) linear

elastic and isotropic behaviour for concrete, FRP, epoxy, and steel reinforcement, (ii) perfect

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bonding between concrete and FRP strips, and (iii) linear strain distribution through the full

depth of the section with cracked concrete. The methodology is implemented in two stages: I)

prediction of the tensile stresses in the FRP strips at the curtailments and corresponding shear

stress at the location of steel bar in tension assuming full composite action; and II) solving the

stress concentrations caused by reverse tensile force of FRP strips at the curtailment location

due to the cut off of FRP strips, and comparing the superposed stresses with the concrete

strength. In the second stage, the finite element method (FEM) is employed to obtain accurate

stress profiles in the model, and a statistical analysis of experimental results gives rise to a

modification factor that will lead to accurate predictions.

In the first stage, if considering the full composite action and elastic behaviour, the

tensile stress of FRP strips at the curtailment location, 0ff , can be obtained from

conventional beam theory as

( )xhI

Mff −= 0

0 . (16)

Herein, I is the cracked transformed moment of inertia of beam cross section in terms of the

FRP plate, and M0 is the bending moment at the plate curtailment location. The shear stress in

concrete near the tension rebar closest to the cut off point of FRP strip, I0τ , is

( ) ftfbxhcIb

VI −= 00τ . (17)

where V0 is the shear force at the plate curtailment location.

In the second stage, since the axial stress 0ff at the end of FRP does not actually

exist, an opposite force, - ftfbff 0Ψ ( ft represents the thickness of FRP strips) is applied, to

the end of FRP plate as shown in Fig. 3. As shown above, one can expect that many cracks

appear in the tension side of the beam. The crack spacing model for conventional reinforced

concrete is extended for calculating the minimum stabilized crack spacing, flmin in the case

of RC beams with externally bonded FRP plate, as presented below,

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( )ffbars

tebuOu

fAfl∑ +

=min . (18)

In this equation, us and uf is the average bond strength for steel/concrete and FRP/concrete,

respectively. ∑ barO is the total perimeter of the tension bars, and Ae is the area of concrete in

tension. Also, one can take cufsu 28.0= and cuffu 28.0= . Indeed, the results are

found not too sensitive to the exact value chosen for the parameter fu . In this model, Ψ is an

empirical function obtained from empirical fitting of experimental results. It is found that a

complete quadratic equation of Ψ in terms of LsfL /− and cbfb / , as given in Eqn (19)

below, will give the best agreement with test results.

( ) ( ) ( )( ) ( ) ( ) 1.0/,//635.12/080.3

2/124.240/972.4/827.35527.3

≤−×−×−×+

−×+×−−×−=Ψ

LsfLcbfbLsfLcbfb

LsfLcbfbLsfL (19)

in which, Lf-s and L represent the distance from the end of FRP to the support and total span

length, respectively. The comparison between predicted and experimental values for 39

strengthened beams, with and without the modification factor Ψ , are shown in Fig. 4. The

details of selected samples are presented in Gao et al. [33]. From the figure, it is clear the

modification factor is necessary to obtain good agreement between predicted and

experimental results.

Under the applied force in Fig. 3, we assume that complete shear stress transfer

between FRP and concrete takes place over flmin , the concrete cover block nearest to the

end of FRP strips. When the individual concrete block at the end of FRP strips is subjected to

a force ( ftfbff 0Ψ ), the vertical normal stress and shear stress in concrete near the tension

rebar closest to the cut off point of FRP strips in stage II, II0σ and II

0τ , can be calculated.

As mentioned, the cantilever beam length is too short compared to its depth for the

conventional cantilever beam theory to be valid. Therefore, the finite element method (FEM)

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is applied to obtain II0σ and II

0τ . The rectangular cover region (one piece of tooth) between

two cracks is modelled, and a unit force is applied at the end of FRP strips for convenience.

unitII ,0σ and unitII ,

0τ , the vertical normal and shear stresses for a unit force in stage II, can

be obtained. We have attempted to solve the problem with three different models: (i) a 3D

model with the FRP and adhesive considered (Fig. 5a), (ii) a 2D model with the FRP and

adhesive considered (Fig. 5b), and (iii) a 2D model neglecting the presence of adhesive and

FRP, with loading applied directly onto the concrete (Fig. 5c). The results indicate that as

long as an appropriate modification factor (obtained from empirical fitting) is used with the

finite element results, each of the models can predict failure loads in good agreement with

experimental data. For convenience, we have decided to adopt the simplest model (Fig. 5c)

for further analysis. More details on the models and comparisons with test results can be

found [33].

In practical design, it is inconvenient to run finite element analysis every time. A

better alternative is to provide equations for, unitII ,0σ and unitII ,

0τ , the stresses resulted

from a unit load applied on the plate end, based on a series of finite element analysis. From

the geometry of the problem, it is clear that the stresses are a function of flmin / h’, where

flmin is the minimum stabilized crack spacing and h’ is the depth of concrete cover.

Moreover, the stress for a unit applied load must be inversely proportional to the width of the

beam (bc) as well as the cover depth h’. For a larger cover depth, if flmin / h’ is fixed, the

same loads is applied to a larger member, so the stress will decrease proportionally.

Summarizing the above, one can write the stresses per unit load in the following form:

unitII ,0σ = '

min1 )

'(

hbh

lF

c

f

(20)

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unitII ,0τ = '

min2 )

'(

hbh

lF

c

f

(21)

where bc and h’ are dimensionless that are the relative ratios to 1m. Through a systematic

finite element analysis, the functions F1 and F2 can be numerically obtained. In practical

design, with the known values of flmin , bc and h’, the F1 and F2 values can be calculated

from the following statistical equations,

3'

min,4324.9'

min7292.32

'min6054.01 ≤+⎟

⎟

⎠

⎞

⎜⎜

⎝

⎛×−⎟

⎟

⎠

⎞

⎜⎜

⎝

⎛×=

h

fl

h

fl

h

flF (22a)

,7.31 =F 3'

min >h

fl (22b)

3'

min,7982.1'

min7387.02

'min1197.02 ≤+⎟

⎟

⎠

⎞

⎜⎜

⎝

⎛×−⎟

⎟

⎠

⎞

⎜⎜

⎝

⎛×=

h

fl

h

fl

h

flF (23a)

66.02 =F . 3'

min >h

fl (23b)

The complete solutions for the vertical normal and shear stresses in concrete near the

tension rebar closest to the cut off point of FRP strips ( 0σ and 0τ ), can be determined by

superposition:

( ) unitIIftfbxh

IMII ,

00

00 σσσ −Ψ== (24)

( ) ( ) unitIIftfbxh

IM

ftfbxhcIb

VIII ,0

00000 ττττ −Ψ+−=+= . (25)

The failure criterion for concrete cover separation failure is that when the maximum principle

tensile stress 1,0σ in concrete near the tension rebar closest to the cut off point of FRP strips

is greater than the ultimate tensile strength of concrete tf , failure occurs. 1,0σ can be

obtained by the classical stress transformation equations for a plane stress condition,

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( )202

221,000 τ

σσσ +⎟

⎠⎞

⎜⎝⎛+= . (26)

And tf was defined in ACI code 318-95 (1999) as follows,

'53.0 cftf = , (27)

If a strengthened RC beam is subjected to four point bending, M0 and V0 in terms of the

totally applied load, 2P, are given

sfPLM −=0 ; PV =0 . (28)

Consequently, P can be determined as

( )( )

( ) ( )⎟⎟⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜⎜⎜

⎝

⎛

⎟⎟⎠

⎞⎜⎜⎝

⎛−

−Ψ+−+

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛ −−Ψ

+−−Ψ

=

2,

01

2

2

,0

2

,0

'53.0

unitIIftfbxh

IsfL

ftfbxhcIb

I

unitIIftfbxhsfL

I

unitIIftfbxhsfL

cfP

τ

σ

σ

(29)

3. Procedure for constructing the failure diagram

In this paper, the authors attempt to draw a failure diagram to predict the failure mode for a

given strengthened RC beam. There are five possible failure modes include: (a) rupture of

FRP strips; (b) compression failure after yielding of steel; (c) compression failure before

yielding of steel; (d) delamination of FRP strips due to crack; and (e) concrete cover

separation. From the practical point of view, the thickness of FRP is a sensitive and important

factor that will affect the ultimate failure mode. With gradually increasing FRP thickness to

strengthen a RC beam, the probable order for failure occurrence is rupture of FRP,

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delamination of FRP, concrete cover separation and then compression failure. As a result, it is

reasonable to set thickness of FRP (tf) as a variable, which influences the ultimate failure

mode. Another important variable is the distance from support to cut off point of FRP strips

(Lf-s), although only concrete cover separation failure is associated with this parameter. For a

particular beam to be strengthened and a given FRP material, tf and Lf-s are 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 calculated for each individual failure mode. The actual failure mode is the

one that gives rise to the lowest failure strain.

When rupture of FRP strips occurs, the failure strain is the ultimate axial strain in FRP

( fuε ) obtained from manufacturer or measurement. For compression failure whether it

occurs before or after steel yielding, the failure strain is the concrete ultimate strain ( cuε ),

which is taken to be 0.0035 in general. Considering the delamination of FRP strips due to

crack, the maximum corresponding strain in FRP ( dfε ) and strain in concrete ( d

cε ) at

failure are obtained from Eqns. (14) and (15) as

ff

c

cf

cf

tEf

bbbbd

f'

/1/2

1.1 ⎟⎟⎠

⎞⎜⎜⎝

⎛+−

=ε (30)

xhx

tEf

bbbbd

cff

c

cf

cf

−⎟⎟⎠

⎞⎜⎜⎝

⎛+−

='

/1/2

1.1ε . (31)

For concrete cover separation failure, in terms of Eqn. (29), we can get the maximum

corresponding strain in FRP ( pfε ) and strain in concrete ( p

cε ) at failure below,

18

( )( )

( ) ( )2

,0

1

2

2

,0

2

,0

⎟⎟⎠

⎞⎜⎜⎝

⎛−

−Ψ+−

+⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛ −−Ψ

+−−Ψ

=Ω

unitIIftfbxh

IsfL

ftfbxhcIb

I

unitIIftfbxhsfL

I

unitIIftfbxhsfL

τ

σ

σ

(32)

( ) ( ) ⎟⎟

⎠

⎞

⎜⎜

⎝

⎛⎟⎠⎞⎜

⎝⎛ −⎟

⎟

⎠

⎞

⎜⎜

⎝

⎛

−−

+−⎟⎠⎞

⎜⎝⎛

−−

+−Ω

−='5.0''

'5.05.0

'53.0

dxsAsExh

dxxdsAsExhxdxhftfbfE

sLLcfpf

βββ

ε

(33)

( ) ( )( )xh

x

dxsAsExh

dxxdsAsExhxdxhftfbfE

sLLcfpc −

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛⎟⎠⎞⎜

⎝⎛ −⎟

⎟

⎠

⎞

⎜⎜

⎝

⎛

−−

+−⎟⎠⎞

⎜⎝⎛

−−

+−Ω

−='5.0''

'5.05.0

'53.0

βββ

ε

(34)

where LL-s is the distance from the support to the loading point, and x and Itr according to tf

can be obtained as shown in Eqns. (35) and (36).

fEcbcE

fdftfbdsAfEsE

dsAfEsE

fEcbcE

ftfbsAfEsE

sAfEsE

ftfbsAfEsE

sAfEsE

x⎟⎟⎠

⎞⎜⎜⎝

⎛+++

⎟⎟⎠

⎞⎜⎜⎝

⎛+++

⎟⎟⎠

⎞⎜⎜⎝

⎛++−

=

''22

'

'

(35)

( ) ( )22''23/3 xfdftfbxdsAfEsExdsA

fEsExcb

fEcE

trI −+⎟⎠⎞⎜

⎝⎛ −+−+⎟

⎠⎞⎜

⎝⎛= (36)

The procedure for failure diagram construction is described in detail as follows:

19

1) Firstly, the critical FRP thickness, tf r-c separating FRP rupture and compression failure

after yielding of steel and tf ca-cb separating compression failure after yielding of steel and

compression failure before yielding of steel, are given by:

fb

fAcrft min

=− (37)

fbfAcbca

ft max=− (38)

where Af,min and Af,max are given by Eqns. (5)-(10). When FRP thickness (tf) exceeds tfr-c, the

failure mode changes from rupture of FRP to compression failure after yielding of steel.

While tf continues to increase to tf ca-cb, compression failure before yielding of steel may take

the place of compression failure after yielding of steel.

2) Secondly, the occurrence of crack-induced delamination of FRP strips is analysed. Setting

dfε = fuε , we can get tf

dl using Eqn. (30). If tfdl ≤ tf

r-c, it means that when tf increases to tfdl

the failure mode changes from rupture of FRP to delamination of FRP. If tfr-c<0 or tf

dl > tfr-c,

tfdr can be obtained from Eqn. (31), with the assumption of d

cε = cuε . When tf reaches tfdr,

delamination of FRP strips starts to occur in place of compression failure.

3) Lastly, concrete cover separation failure is considered. As mentioned above, thickness of

FRP (tf) and the distance from support to cut off point of FRP strips (Lf-s) are set as variables,

with tf as horizontal axis and Lf-s as vertical axis. Considering four point bending test, most

cases show that Lf-s is not allowed to be longer than LL-s, the distance from the support to the

loading point, which means that the cut off point of FRP must be outside the constant moment

region. Two situations should be considered. Fig. 6 (a) and (b) show a typical failure diagram,

for tfdl ≤ tf

r-c and tfdl > tf

r-c, respectively.

20

When tfdl ≤ tf

r-c in the second stage, the occurrence of concrete cover separation is

divided into two parts, namely the left part and the right part relative to tfdl. Setting

pfε = fuε , the relationship of Lf-s and tf is obtained from Eqn. (33), and the upper region of

the curve on the left side of tfdl is the left part. In order to predict the right part, the

comparison between dfε and p

fε have to be done. By assuming dfε = p

fε , one can get

the transfer curve of Lf-s and tf from delamination of FRP to concrete cover separation.

Consequently, the upper region of the curve of Lf-s vs tf on the right side of tfdl is the right part,

referring to Eqns. (30) and (33).

When tfdl > tf

r-c in the second stage, the occurrence of concrete cover separation is

divided into three parts, namely the left part (left of tfr-c), the middle part (between tf

r-c and

tfdr) and the right part (right of tf

dr). Setting pfε = fuε , the transfer curve of Lf-s and tf is

obtained from Eqn. (33), and the upper region of the curve on the left side of tfr-c is the left

part. In comparison, the upper region of the curve of Lf-s vs tf between tfr-c and tf

dr is the

middle part, referring to Eqn. (34) on account of pcε = cuε . Furthermore, with d

cε = pcε ,

one can obtain the transfer curve of Lf-s and tf from delamination of FRP to concrete cover

separation referring to Eqns. (31) and (34), and thus the right part is determined as the upper

part of the curve of Lf-s vs tf on the right side of tfdr.

4. Derivation of the failure diagram—a specific example

Several simply supported beams under four point bending [34] are employed as examples to

demonstrate the establishment of the failure diagram. Appendix B presents the beam

dimensions and material properties, as well as the failure mode and ultimate load. The

21

establishment of failure diagram for this particular case is shown in the following, and the

results are shown in Fig. 7(g).

1) Firstly, determine 1α and 1β from Eqns. (3) and (4):

6.0892.03.520015.097.01

6.0772.03.520015.085.01>=×−=>=×−=

βα

.

Then, using Eqns. (5)-(7) and (37), we can get

mh

xfucu

cu 034.00035.0012.0

0035.015.0 =+×=

+=

εεε

syxdx

s εε =<=−=−= 002.00004.0034.0

03.0034.00035.00035.0' '

syxxd

s εε =>=−=−= 002.0009.0034.0

034.012.00035.00035.0 .

Therefore,

mm

EbAfAExbfcr

ftfuff

ssssscc

79.0012.012700015.0

000157.04600001.02100000004.0892.0034.02.03.52772.0

''1

'1

=××

×−××+××××=

−+=−

εεβα

Next, in terms of Eqns. (8)-(10) and (38), one can obtain

0034.0076.0

076.015.00035.00035.0,076.0002.00035.0

0035.012.0 =−×=−==+

×=+

=x

xhfm

dx

sycu

cu εεε

ε

and

syxdx

s εε =>=−

×=−= 002.00021.0076.0

03.0076.00035.00035.0' '.

As a result, one can get

mm

EbAfAfxbfcbca

ftfff

sysycc

05.80034.012700015.0

000157.04600001.0460892.0076.02.03.52772.0

''1

'1

=××

×−×+××××=

−+=−

εβα

.

2) Considering the occurrence of delamination of FRP strips due to crack, we can get tfdl

using Eqn. (30) below,

22

mmdlft

t dlf

34.0127000

3.522.0/15.012.0/15.021.1012.0 =⇒

×⎟⎠⎞⎜

⎝⎛+−=

Since tfdl =0.34mm≤ tf

r-c=0.79mm, it means that with increasing tf to tfdl the failure mode

changes from rupture of FRP to delamination of FRP, without chance to fail with

compression failure.

3) Since tfdl =0.34mm≤ tf

r-c =0.79mm in the second stage, the occurrence of concrete cover

separation is divided into two parts, namely the left part and the right part relative to tfdl.

Setting pfε = fuε , Eqn. (33) is changed to as follows,

( )

( ) ( )⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

−××××⎟⎠⎞

⎜⎝⎛

−−

+×−×

××⎟⎠⎞

⎜⎝⎛

−−

+×−×××Ω

××=

03.0892.05.00001.021000015.0

03.0892.05.012.0

000157.021000015.012.0892.05.015.015.0127000

75.03.5253.0012.0

xx

xx

xxxft

.

The upper region of the curve on the left side of tfdl (0.34mm) is the left part. In order to

predict the right part, with the assumption of dfε = p

fε , one can get the transfer curve of Lf-s

and tf from delamination of FRP to concrete cover separation referring to Eqns. (30) and (33),

as given below,

( ) ( )

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛

⎟⎠⎞⎜

⎝⎛ −⎟

⎟

⎠

⎞

⎜⎜

⎝

⎛

−−

+

−⎟⎠⎞

⎜⎝⎛

−−

+−

Ω

−=⎟⎟⎠

⎞⎜⎜⎝

⎛+−

'5.0'''

5.05.0

'53.0/1/2

1.1'

dxsAsExh

dx

xdsAsExhxdxhftfbfE

sLLcftEf

bbbb

ff

c

cf

cf

β

ββ

and

23

( )

( )

( ) ⎟⎟⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜⎜⎜

⎝

⎛

−××××⎟⎠⎞

⎜⎝⎛

−−

+

×−×××⎟⎠⎞

⎜⎝⎛

−−

+×−×××

Ω

××=

×⎟⎠⎞⎜

⎝⎛+−

03.0892.05.00001.021000015.0

03.0

892.05.012.0000157.021000015.012.0

892.05.015.015.0127000

75.03.5253.0127000

3.522.0/15.012.0/15.021.1

xx

x

xxx

xftt dl

f

.

Consequently, the upper region of the curve of Lf-s vs tf on the right side of tfdl (0.34mm) is the

right part.

5. Verification and Discussions

In order to verify the applicability of the failure diagram, published experimental results

pertaining to strengthened RC beams are analysed. Totally, 33 samples are selected from eight

references, showing results that cover various failure modes. Two series of tests carried out by

Nguyen et al. [35] and Fanning and Kelly [36] focused on the effect of the Lf-s on the

strengthening performance. The other papers investigated the influence of the thickness of

FRP strips. The various values of Lf-s and tf, the ultimate loads and the failure modes from

experiments and theoretical models in failure diagram as well as detail information for all

samples collected are summarised in Appendix B. The corresponding failure diagrams for the

eight groups of tests are shown in Fig. 7.

The comparison between experiments and prediction by failure diagram shows that

this method could predict the failure mode for a strengthened RC beam. Also, the ultimate

load capacity can be calculated by individual theoretical expression, after the particular failure

mode is obtained. The failure diagram shows that failure mode for a strengthened RC beam

may vary from rupture of FRP strips, to delamination of FRP strips, and then to concrete

cover separation, with increasing FRP thickness. With the correct design, compression failure

24

after yielding of steel may take place before local failure. Reducing the distance from support

to cut off of FRP may decrease the likelihood of concrete cover separation.

From the practical application point of view, compression failure after yielding of steel

is most preferable in design. However, the occurrence of local failures such as delamination

of FRP strips and concrete cover separation precludes the chance of compression failure after

yielding of steel. In order to have compression failure after steel yielding, besides reducing

the distance from support to cut off of FRP as far as possible, appropriate selection of FRP

properties is very important. FRP with good performance, such as high strength, high

elongation at failure and high modulus, may not be effective in practical applications, because

failure may occur by delamination of FRP strips early with low axial strain in FRP.

Before closing, a few remarks should be made on the use of the failure diagram in

practice. To perform strengthening of a given RC beam, the beam dimensions and

reinforcement ratio are fixed. Also, the selection of FRP properties is perhaps limited by the

availability of commercial products. Consequently, the variables to be chosen are only the

FRP dimensions, including the FRP thickness, length and width. For a particular concrete

beam, the FRP width can be selected as a certain percentage of the beam width. The two

parameters represented on the failure diagram are then sufficient to determine the failure

mode. After knowing the possible failure mode, the load capacity and deflection of the beam

can be accurately predicted. Since the failure diagram summarizes all possible failure modes,

a clear picture of all possibilities are provided to guide the designer in choosing the best

combination of plate thickness and length. The plotting of failure diagrams will also facilitate

the selection of the best material. Moreover, with the failure mode predicted, the critical

failure initiation location can be known. That information is very useful for the continuous

monitoring of strengthened beams, as well as the determination of appropriate positions for

the application of anchors.

25

The idea behind the failure diagram is that the failure mode associated with the lowest

strain in FRP or concrete by comparison is most likely to occur. In this paper, a general

concept is proposed. With future development leading to better methods for local failure, the

equations in this paper can be further refined.

6. Conclusions

Numerous studies including experimental research, theoretical analysis and numerical

simulation have demonstrated that epoxy bonding of fibre reinforced plastic (FRP) strips to

the tension soffit of reinforced concrete (RC) beams can significantly improve the ultimate

flexural strength and stiffness. Several important failure modes have been studied, such as

compression failure before or after yielding of steel, rupture of FRP strips, delamination of

FRP strips and concrete cover separation.

This paper attempts to build a failure diagram to show the relationship and the

transition among different failure modes for RC beams strengthened with FRP strips, and how

failure modes vary with FRP thickness and the distance from the end of FRP strips to the

support. The idea behind this failure diagram is that the failure mode associated with the

lowest strain in FRP or concrete by comparison is most likely to occur. By comparison

between predictions based on failure diagram and experimental results, we show that this

method could predict the failure mode for a strengthened RC beam. Knowing the failure

mode, the ultimate load capacity can be calculated. The failure diagram provides guidelines to

practical design, and is useful in establishing a procedure for selecting the type and size of

FRP for the external strengthening of RC beam.

Acknowledgements

26

The Research Grants Council of the Hong Kong SAR (Project No. HKUST 6050/99E),

provided the financial support of this work. The authors wish to thank the Construction

Materials Laboratory, Advanced Engineering Material Facilities, and Design and

Manufacturing Services Facility in HKUST for their technical supports.

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30

Appendix A. Prediction of the Ultimate Flexural Strength of Strengthened RC Beams

Generally speaking, two situations should be considered, including a) obtaining the quantity

of FRP to satisfy the requirement of moment capacity for a RC beam, and b) calculating the

moment capacity of a strengthened RC beam. Both cases are the reverse processes.

From the practical design point of view, the former is the more common case. To

archive the targeted moment, Mu, the following formula is employed,

( ) ( ) ⎟⎠⎞⎜

⎝⎛ −+−−−= '''

15.01'

1 dhsAyfdhsAyfxhxcbcfuM ββα (39)

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

a quadratic equation,

1

112

112

4a

cabbx

−−−= (40)

cbcfa '2115.01 βα= (41)

hcbcfb '111 βα−= (42)

( ) ⎟⎠⎞⎜

⎝⎛ −−−+= '''

1 dhsAyfdhsAyfuMc . (43)

Knowing x, we can get, sε , 'sε , and fε by linear strain distribution with cε =0.0035,

xxd

s−= 0035.0ε (44)

xdx

s'

0035.0' −=ε (45)

xxd

ff −= 0035.0ε (46)

If sε > syε and fε < fuε , the FRP area is within the range Af,min ≤Af ≤Af,max. In this case,

compressive failure will occur after steel yielding. Af can be obtained from the Eqns. (9) or

(10), as well as Mu from Eqn. (39). If sε < syε and fε < fuε , the yielding of steel in tension

31

can not be obtained before failure. Therefore, in order to have enough ductility and warning

before failure, the area of FRP must be reduced to Af,max. The ultimate moment resistance, Mu’

is then lower than Mu. The calculation of Af,max has been introduced in Eqns. (8), (9) and (10).

Then, we can get Mu’, as follows,

( ) ( ) syxdx

sdhsAssEdhsAyfxhxcbcfuM εεεββα <−=⎟⎠⎞⎜

⎝⎛ −+−−−=

'0035.0','''

15.01'

1' (47)

( ) ( ) syxdx

sdhsAyfdhsAyfxhxcbcfuM εεββα ≥−=⎟⎠⎞⎜

⎝⎛ −+−−−=

'0035.0','''

15.01'

1' (48)

When fε > fuε , rupture of FRP strips is to occur instead of compression failure. In this case,

the area of FRP can be increased to a value within the range of Af,min and Af,max, to assure the

occurrence of compression failure after yielding of steel. The ultimate moment resistance Mu’,

which can be obtained using Eqns. (51) and (52), will then be higher than Mu.

To calculate the moment capacity of a given strengthened RC beam, two balanced

limited values of cross section area of FRP, Af,min and Af,max are calculated first, to determine

the failure mode in terms of Af. For each individual failure mode, x can be obtained from force

equilibrium of the cross section and the relationship among strain components (Eqns. (44)-

(46)). The ultimate moment resistance Mu’, can be calculated by the equations below for

different situations.

When Af < Af,min,

( ) ( ) syxhdx

fusdxsAssExdsAyfxhfufEfAuM εεεβεββε <−−=⎟

⎠⎞⎜

⎝⎛ −+−+−=

'','15.0''

15.015.0'

(49)

( ) ( ) syxhdx

fusdxsAyfxdsAyfxhfufEfAuM εεεβββε ≥−−=⎟

⎠⎞⎜

⎝⎛ −+−+−=

'','15.0''

15.015.0'

(50)

When Af,min ≤Af ≤Af,max,

( ) ( ) syxdx

sdhsAssEdhsAyfxhxcbcfuM εεεββα <−=⎟⎠⎞⎜

⎝⎛ −+−−−=

'0035.0','''

15.01'

1'

(51)

32

( ) ( ) syxdx

sdhsAyfdhsAyfxhxcbcfuM εεββα ≥−=⎟⎠⎞⎜

⎝⎛ −+−−−=

'0035.0','''

15.01'

1'

(52)

When Af >Af,max,

( ) ( ) syxdx

sdhsAssEdhsAssExhxcbcfuM εεεεββα <−=⎟⎠⎞⎜

⎝⎛ −+−−−=

'0035.0','''

15.01'

1'

(53)

( ) ( ) syxdx

sdhsAyfdhsAssExhxcbcfuM εεεββα ≥−=⎟⎠⎞⎜

⎝⎛ −+−−−=

'0035.0','''

15.01'

1'

(54)

33

Appendix B. Details of Experiments

Experiments Beam

Beam

width

(mm)

Beam

depth

(mm)

Beam

length

(mm)

FRP

length

(mm)

FRP

width

(mm)

FRP

thickness

(mm)

As

As’

d

(mm)

d’

(mm)

Lf-s

(mm)

Alagusundaramo

orthy et al.

[37]

CB11-1F 230 380 4576 4370 203 0.18 2Φ 25 2Φ 9 342 25 103

CB11-1F 230 380 4576 4370 203 0.18 2Φ 25 2Φ 9 342 25 103

CB11-2F 230 380 4576 4370 203 0.36 2Φ 25 2Φ 9 342 25 103

CB11-2F 230 380 4576 4370 203 0.36 2Φ 25 2Φ 9 342 25 103

Arduini et al.

[38]

A3 200 200 2000 1700 150 1.3 2Φ 14 2Φ 14 163 37 150 A4 200 200 2000 1700 150 1.3 2Φ 14 2Φ 14 163 37 150

A5 200 200 2000 1700 150 2.6 2Φ 14 2Φ 14 163 37 150

Fanning and

Kelly

[36]

FKF5 155 240 3000 2030 120 1.2 3Φ 12 2Φ 12 203 37 385 FKF6 155 240 3000 2030 120 1.2 3Φ 12 2Φ 12 203 37 385

FKF7 155 240 3000 1876 120 1.2 3Φ 12 2Φ 12 203 37 462

FKF10 155 240 3000 1700 120 1.2 3Φ 12 2Φ 12 203 37 550

Gao et al. [39]

T1 150 200 2000 1200 75 0.11 2Φ 10 2Φ 8 162 27 150

T2 150 200 2000 1200 75 0.22 2Φ 10 2Φ 8 162 27 150

T4 150 200 2000 1200 75 0.44 2Φ 10 2Φ 8 162 27 150

T6 150 200 2000 1200 75 0.66 2Φ 10 2Φ 8 162 27 150

Maalej and Bian

[40]

MB2 115 150 1500 1200 115 0.111 3Φ 10 2Φ 10 125 25 75

MB3 115 150 1500 1200 115 0.222 3Φ 10 2Φ 10 125 25 75

MB4 115 150 1500 1200 115 0.333 3Φ 10 2Φ 10 125 25 75

MB5 115 150 1500 1200 115 0.444 3Φ 10 2Φ 10 125 25 75

Nguyen et al.

[35]

A950 120 150 1500 950 80 1.2 3Φ 10 2Φ 6 120 28 190

A1100 120 150 1500 1100 80 1.2 3Φ 10 2Φ 6 120 28 115

A1150 120 150 1500 1150 80 1.2 3Φ 10 2Φ 6 120 28 90

A1500 120 150 1500 1500 80 1.2 3Φ 10 2Φ 6 120 28 0

Rahimi and

Hutchinson

[34]

RHB3 200 150 2300 1930 150 0.44 2Φ 10 2Φ 8 120 30 85

RHB4 200 150 2300 1930 150 0.44 2Φ 10 2Φ 8 120 30 85

RHB5 200 150 2300 1930 150 1.2 2Φ 10 2Φ 8 120 30 85

RHB6 200 150 2300 1930 150 1.2 2Φ 10 2Φ 8 120 30 85

Triantafillou

and Plevris

[10]

3 76 127 1220 1070 60.5 0.2 2Φ 4.6 - 111 - 75

4 76 127 1220 1070 63.2 0.65 2Φ 4.6 - 111 - 75

5 76 127 1220 1070 63.2 0.65 2Φ 4.6 - 111 - 75

6 76 127 1220 1070 63.3 0.9 2Φ 4.6 - 111 - 75

7 76 127 1220 1070 63.3 0.9 2Φ 4.6 - 111 - 75

8 76 127 1220 1070 63.9 1.9 2Φ 4.6 - 111 - 75

34

Appendix B. (Continued)

Experiments Beam fc

’

(MPa)

ft

(MPa)

Ec

(GPa)

Es

(GPa)

Ef

(GPa)

Pmodel

(kN)

Pexp

(kN)

aFailure

modemodel

aFailure

modeexp

Alagusundaramoorthy

et al.

[37]

CB11-1F 31 3.0 26.3 200 228 229 219 RF RF

CB11-1F 31 3.0 26.3 200 228 229 223 RF RF

CB11-2F 31 3.0 26.3 200 228 233 263 DF DF

CB11-2F 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 CS

A4 33 2.6 25 200 167 94.1 104 CS CS

A5 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 CS

FKF6 80 5 39.2 204 155 112.75 103 CS CS

FKF7 80 5 39.2 204 155 94.83 97.5 CS CS

FKF10 80 5 39.2 204 155 80.23 82 CS CS

Gao et al. [39]

T1 43.1 3.5 25 200 235 71.4 73.2 RF RF

T2 43.1 3.5 25 200 235 80.9 80.7 DF DF

T4 43.1 3.5 25 200 235 94.7 86.4 DF CS

T6 43.1 3.5 25 200 235 73.5 86.3 CS CS

Maalej and Bian [40]

MB2 30.3 2.9 26 183.6 230 71.7 72 CC RF

MB3 30.3 2.9 26 183.6 230 73.3 86 CC CS

MB4 30.3 2.9 26 183.6 230 90.5 82 CS CS

MB5 30.3 2.9 26 183.6 230 73.5 79 CS CS

Nguyen et al.

[35]

A950 27.3 2.8 25 200 181 35.4 56.2 CS CS

A1100 27.3 2.8 25 200 181 55.0 57.3 CS CS

A1150 27.3 2.8 25 200 181 67.3 58.9 CS CS

A1500 27.3 2.8 25 200 181 74.6 118.0 DF CC

Rahimi and

Hutchinson

[34]

RHB3 52.3 3 25 210 127 48.5 55.2 DF DF

RHB4 52.3 3 25 210 127 48.5 52.5 DF DF

RHB5 52.3 3 25 210 127 52.6 69.7 CS CS

RHB6 52.3 3 25 210 127 52.6 69.6 CS CS

Triantafillou

and Plevris

[10]

3 44.7 3.5 31.6 200 186 16.3 17.27 RF RF

4 44.7 3.5 31.6 200 186 31.7 29.56 DF DF

5 44.7 3.5 31.6 200 186 31.7 25.59 DF DF

6 44.7 3.5 31.6 200 186 35.6 30.50 DF DF

7 44.7 3.5 31.6 200 186 35.6 27.90 DF DF

8 44.7 3.5 31.6 200 186 46.7 37.33 CS DF aCC = Compression failure; RF = Rupture of FRP strips; DF = Delamination of FRP strips; CS = Concrete cover separation

35

Figure Captions

Fig. 1: Failure modes of FRP strengthened RC beams: (a) Compression failure; (b) Rupture of

FRP strips; (c) Shear failure; (d) Delamination of FRP strips; and (e) Concrete cover separation

Fig. 2: Cross section dimensions with strain distribution and stress diagram

Fig. 3: Analysis in stage II with opposite axial force in FRP strips

Fig. 4: The comparison of prediction with/without modification factor

Fig. 5: FEM models for predicting unitII ,0σ and unitII ,

0τ : (a) 3-D with FRP; (b) 2-D with

FRP; and (c) 2-D without FRP

Fig. 6: A typical failure diagram in the third step: (a) tfdl ≤ tf

r-c and (b) tfdl > tf

r-c (using tfdr)

Fig. 7: Demonstration of established failure diagram compared to experiments done by: (a)

Alagusundaramoorthy et al., 2003; (b) Arduini et al. 1997; (c) Fanning and Kelly, 2001; (d) Gao

et al., 2003; (e) Maalej and Bian, 2001; (f) Nguyen et al., 2001; (g) Rahimi and Hutchinson,

2001; and (h) Triantafillou and Plevris, 1992.

36

(a)

(b)

(c)

(d)

(e)

Fig. 1. Failure modes of FRP strengthened RC beams: (a) Compression failure; (b) Rupture of FRP

strips; (c) Shear failure; (d) Delamination of FRP strips; and (e) Concrete cover separation

37

Fig. 2. Cross section dimensions with strain distribution and stress diagram

d df

d’ x

h’

h

As

As’

bf

bc

ε s’

ε s

ε f

ε c

Tf

Ts

Cc

Cs

1βx

38

IIσ , IIτ

ftfbff 0Ψ lmin

Fig. 3. Analysis in stage II with opposite axial force in FRP strips

h’

39

Specimens collected

Fig. 4: The comparison of predicted and experimental results, with/without modification factor

p p

0.0

0.5

1.0

1.5

2.0P

redi

cted

/ ex

perim

enta

l fai

lure

load

ratio

With modification factor

Without modification factor

Ga1

Gb1

Gb2

MB

3M

B4

MB

5R

HB

5R

HB

6FK

F5FK

F6FK

F7FK

F10

B2

B4

B6

A1c

A2b

A2c

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

40

unitII ,0σ and unitII ,

0τ at the critical point

a unit force lmin

Fig. 5. FEM models for predicting unitII ,0σ and unitII ,

0τ : (a) 3-D with FRP; (b) 2-D with FRP;

and (c) 2-D without FRP

h’

(a)

(b)

(c)

41

Fig. 6: A typical failure diagram in the third step: (a) tf

dl ≤ tfr-c and (b) tf

dl > tfr-c (using tf

dr)

tf dl tf r-c

Left part Middle part Right part of concrete cover separation

Compression failure Rupture of FRP strips Delamination of FRP strips

Lf-s

tf

(a)

tf r-c tf dr

Lf-s

tf

(b)

42

(a)

0

0.2

0.4

0.6

0.8

1

0 0.5 1 1.5 2tf (mm)

L f-s (m

)

0

0.1

0.2

0.3

0.4

0.5

0 0.5 1 1.5 2 2.5 3tf (mm)

L f-s (m

)

(b)

0

0.2

0.4

0.6

0.8

1

0 1 2 3 4 5tf (mm)

L f-s (m

)

(c)

0.000.050.100.150.200.250.300.350.400.450.50

0 0.2 0.4 0.6 0.8 1tf (mm)

L f-s

(m)

(d)

43

0.00

0.10

0.20

0.30

0.40

0.50

0 0.2 0.4 0.6 0.8 1tf (mm)

L f-s

(m)

(e)

0.0

0.1

0.2

0.3

0.4

0 0.5 1 1.5 2 2.5 3tf (mm)

L f-s (m

)

(f)

0.00

0.25

0.50

0.75

0 0.5 1 1.5 2 2.5 3tf (mm)

L f-s

(m)

(g)

0

0.1

0.2

0.3

0.4

0 0.5 1 1.5 2tf (mm)

L f-s (m

)

(h)

44

Fig. 7. Demonstration of established failure diagram compared to experiments done by: (a)

Alagusundaramoorthy et al., 2003; (b) Arduini et al. 1997; (c) Fanning and Kelly, 2001; (d) Gao et

al., 2003; (e) Maalej and Bian, 2001; (f) Nguyen et al., 2001; (g) Rahimi and Hutchinson, 2001; and

(h) Triantafillou and Plevris, 1992.

Delamination of FRP strips Concrete cover separation

Experiment (Correctly predicted) Experiment (wrongly predicted)

Compression failure Rupture of FRP strips