1

Prediction of Concrete Cover Separation Failure for RC Beams Strengthened with CFRP Strips

Bo GAO1, Christopher K. Y. LEUNG2* and Jang-Kyo KIM1

1Department of Mechanical Engineering and 2Department of Civil Engineering,

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

*Tel: 852-23588183; Fax: 852-23581534; Email: ckleung@ust.hk.

Abstract

External bonding of fibre reinforced plastic (FRP) strips to reinforced concrete (RC) beams has been

widely accepted as an effective method for strengthening. The ultimate flexural strength of

strengthened RC beams can be improved significantly, but it is often impaired by premature failure

modes, such as concrete cover separation. The objective of this paper is to establish a simple and

accurate design methodology to predict the load carrying capacity of a strengthened RC beam when

concrete cover separation takes place. An analytical expression is developed taking into account the

stress concentrations in concrete near the tension rebar closest to the cut off point of FRP strip. The

derivation of the expression involves two major steps: i) determination of the tensile stress in the

FRP strips assuming a full composite action; and ii) obtaining the local stresses and comparison with

the concrete strength. The predictions based on the present analytical model are compared to 58

experimental data from the literature and good agreement has been obtained. The expressions

derived in this paper therefore have potential for application in the design of FRP-strengthened

beams.

Keywords: Prediction; Concrete cover separation; RC beam; Strengthening; Fibre reinforced plastic

This is the Pre-Published Version

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1. Introduction

The need for increased maintenance is inevitable, with the increase in the number of aging

structures in the world. Complete replacement is likely to become an increasing financial burden

and is certainly a waste of natural resources if upgrading is a viable alternative. Therefore,

strengthening and rehabilitation of these structures are considered to be the most practical approach.

In response to the growing needs for strengthening and rehabilitation of concrete beams, many

techniques have been developed [1,2], in which external bonding of fibre reinforced plastic (FRP)

strips to the beam has been widely accepted as an efficient and effective method. Generally, the

ultimate flexural strength of strengthened RC beams can be improved significantly.

In strengthening reinforced concrete beams with FRP strips, different failure modes have

been investigated and studied [3-7]. Some failure modes including concrete crushing in

compression before or after yielding of steel and rupture of FRP strips are similar to those in

conventional RC beams in flexure. For these failure modes, the FRP plate can be assumed to have

perfect bonding with the original beam. The perceptions on failure mechanism and analytical

methods for these failure modes have already been successfully established [8-15]. Also, shear

failure is not totally different from that in conventional RC beams, which is caused by low shear

capacity. For shear strengthening, several parameters have been studied, such as FRP U strips or U

jackets, FRP fibre orientation, mechanical type anchors, concrete strength, steel shear reinforcement

and shear span to depth ratio [16-21].

Besides the ‘conventional’ failure modes described above, failure of FRP strengthened

beams may also occur by interfacial debonding or concrete cover separation. The focus of the

present study is on concrete cover separation. To clearly define the specific mode of failure we are

studying, both interfacial debonding failure and concrete cover separation failure are described

below. Considering the interfacial debonding failures, Teng et al. [6] reported three separated

debonding modes, consisting of plate end interfacial debonding, interfacial debonding induced by

intermediate flexural crack and interfacial debonding induced by intermediate flexural shear cracks.

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Moreover, Oehlers et al. [7] identified three major debonding mechanisms: plate end debonding,

intermediate crack debonding and critical diagonal crack debonding. Notice that in this paper

“delamination of FRP strips” is referred to elsewhere as various “debonding” failures, which is the

combination of debonding mechanisms described by Teng et al. [6] or Oehlers et al. [7]. The

significant characteristic is that there is only a very thin layer of concrete attached on the debonded

FRP strips and the concrete cover stays essentially intact in the vicinity of FRP end. Photographs

showing the delamination of FRP strips are given in Fig. 1. Fig. 1(a) shows the case with no cover

failure along the debonded FRP strips and Fig. 1(b) shows the situation with some concrete cover

debonding inside the span but the cover concrete stays intact in the vicinity of FRP end.

Another failure mode, concrete cover separation, which is shown in Fig. 2, is also frequently

observed in the experiments. In many investigations [22], it is suggested that failure of the concrete

cover is initiated by the formation of a crack at the end of FRP end, due to high stress concentration

caused by the abrupt termination of FRP plate. After the formation of a crack, the crack propagates

to the level of the tension reinforcement and then progresses horizontally along the level of the steel

reinforcement, and thus resulting in the separation of concrete cover [22]. This failure mode has

other names, such as concrete rip-off, local shear failure, and so on. The main characteristic of this

mode is that the concrete cover is damaged in the vicinity of the end of FRP and afterwards the

separated concrete covers debond along the steel rebar. In an effort to identify the load capacity of a

strengthened RC beam with this failure mode, many studies have bean carried out and a number of

models have been proposed [22-30]. Most of them are based on the derivation of elastic stress

concentrations at the FRP strip curtailments [25-29]. However, it is found in experimental

investigations [31,32] that inclined concrete cracks always appear at the FRP end before the

ultimate load reached. This means that ultimate failure is not associated with elastic stress

concentrations at the plate end. Other models have been developed based on the shear capacity of

the beam [23,24]. Based on the physical observation that the damage of the cover may lead to the

formation of concrete blocks that resembles ‘teeth’ along the bottom of the beam, concrete tooth

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models have also been proposed [30]. A survey of existing models in Teng et al [6] and Smith and

Teng [22] shows that no existing models can accurately predict the failure load when cover failure

occurs. A new model is hence proposed [6]. However, while this new model provides a lower

bound to test results, it may underestimate the load capacity significantly, thus compromising the

effectiveness of the FRP strengthening technique. For practical design, a better model is hence

necessary.

The objective of this paper is to develop a simple and accurate design methodology for

predicting load carrying capacity of the RC beams strengthened with FRP strips with concrete cover

separation failure. In order to verify this method, the results obtained in this study are compared

with experimental results in the literature that show concrete cover separation as the dominant

failure mode. Also, other different types of analytical models are considered to compare.

2. Analytical Model

This analytical expression is developed for predicting the stress concentrations in concrete near the

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

a failure criterion. The following assumptions are made: linear elastic and isotropic behaviour for

concrete, FRP, epoxy, and steel reinforcement; perfect bonding between concrete and FRP strips;

and linear strain distribution through the full depth of the section with cracked concrete in tension.

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 a

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) with linear elastic assumption is employed to obtain accurate stress profiles in the model.

The finite element results are then fitted with simple empirical equations, which can be used in

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practical design. A modification factor is then applied to the theoretical results to obtain good

agreement with test data.

2.1 Stage I

The cross section of a strengthened RC beam is shown in Fig. 3. Notice that d’, d, and df denote the

depths of compressive steel, tensile steel and FRP strips, respectively; As and As’ are the cross-

sectional areas of tensile and compressive steel reinforcements; bc and bf are the widths of concrete

and FRP strips; and x, h, and h’ are the depths of the neutral axis, concrete beam, and concrete

cover, respectively. When the beams are subjected to the applied load, and assuming elastic

behaviour, the tensile stress in the FRP strips, ff , can be obtained from conventional beam theory

as

( )xhI

Mf f −= . (1)

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

FRP plate, and M is the bending moment. Therefore, one can get 0ff , the tensile stress of FRP

strips at the curtailment location,

( )xhI

Mf f −= 00 (2)

where M0 is the bending moment at the plate curtailment location.

When calculating the shear stress in concrete at the location of steel tension bar, only the

tensile stress in the FRP strips is considered, because of the very small shear force and bending

moment in the thin FRP strips as well as negligible influence of concrete cover in tension. Thus,

Iτ , the shear stress in concrete at the location of steel tension bar, can be determined from

conventional beam theory as

( ) ffc

I tbxhIbV

−=τ . (3)

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where tf is the thickness of FRP plate. Specifically, I0τ , the shear stress in concrete near the tension

rebar closest to the cut off point of FRP strip is

( ) ffc

I tbxhIbV

−= 00τ . (4)

V and V0 are the shear force in the beam and the shear force at the plate curtailment location,

respectively.

2.2 Stage II

In reality, the axial stress 0ff at the end of FRP does not exist. In the solution stage II, an opposite

force, - fff tbf 0 is applied, to the end of FRP plate as shown in Fig. 4. Notice that the opposite

moment and transverse shear force at the end of FRP are not applied because the FRP sheet is very

thin.

One can expect that many cracks appear in the tension side of the beam. An extension of the

classical theory of cracking can be used for calculating the minimum and maximum stabilized crack

distances, flmin and flmax , respectively, for RC beams with externally bonded FRP plate. It is

suggested that one adopt flmin for design purposes, which could give a safe solution [30]. It is

shown that

ffbars

tef

buOufA

l∑ +

=min . (5)

In eqn. (5), us and uf are the average bond strengths for steel/concrete and FRP/concrete interfaces,

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

tension. Also, one can take cus fu 280.= and cuf fu 280.= [30]. Indeed, it is found that there is

the largely insignificant influence of exact value chosen for the parameter uf on the prediction of

load capacity. ft and fcu are the concrete tensile strength and cube compressive strength,

respectively. Furthermore, it is assumed that complete shear stress transfer (between FRP and

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concrete due to the opposite axial force at FRP strip end) takes place over the length of flmin .

Consequently, only the nearest concrete cover block to the end of FRP strips is considered to

calculate stress distribution, as shown in Fig. 4. Many researchers may doubt the accuracy of

calculated flmin . In fact, the application of flmin is to calculate the stress concentrations at failure

initiation point. However, our results show that the parameter flmin only has minor effect on the

concentrated stress (Note: this aspect will be further discussed below). Therefore, some inaccuracy

in the determination of flmin seems to be acceptable, provided that the appropriate forms are used

for predicting the stresses.

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

( fff tbf 0 ), one may attempt to calculate II0σ and II

0τ (the vertical normal and shear stresses in

concrete near the tension rebar closest to the cut off point of FRP in stage II), based on the

conventional cantilever beam theory. It is noted, however, that the cantilever beam length is too

short compared with the dimensions in cross section for the conventional cantilever beam theory to

be valid. Therefore, the finite element method (FEM) may be applied to obtain II0σ and II

0τ . The

rectangular cover region 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.

Four possible simulated FEM models are shown in Fig 5. The actual physical problem can

be completely solved by a 3-D model with FRP as well as adhesive, shown in Fig. 5a. If the FRP

strip is not considered in the FEM model, a 3-D model without FRP is obtained as Fig. 5b. In

comparison, a 2-D model with FRP and adhesive (see Fig. 5c) is more common for the practical

problems. With further simplification, the simplest one is introduced in Fig. 5d, as a 2-D model

without FRP and adhesive.

In order to investigate the performance of individual model, four models mentioned above

have been performed to analyse the stress distributions. In the ANSYS 5.7 program used, the

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element of “Plane 183” and “Solid 185” is applied in 2-D and 3-D models, respectively. And the

mesh size is 1mm x 1mm in 2-D models and 1mm x 1mm x 1mm in 3-D models, respectively. The

linear elastic material property is assumed for concrete, adhesive and FRP. The material properties

are determined by the standard test or obtained from the information in the manufacture report. For

example, for two samples by Gao et al. [32], the Young’s modulus for concrete, adhesive and FRP

is 25GPa, 1GPa and 235GPa, respectively. Of note that if a FRP plate does not cover the full width

of the concrete beam, the Young’s modulus for adhesive and FRP is to be correspondingly linearly

reduced in terms of the ratio of the widths of FRP over concrete.

The comparisons among various FEM models are discussed in the following. Firstly, one

may think a 3-D FEM model could give rise to better results than that for a 2-D FEM model. Our

preliminary study indicates, however, that a 3-D model may not give more accurate prediction than

a 2-D model in general regardless of including FRP in the model. In fact, they give similar results,

in which the difference is mostly lower than 10%. Then, considering the influence of FRP, it is

found that a FEM model with FRP layers can always have better predictions than that without FRP,

when the modification factor is not contained. A 2-D FEM model with FRP could increase about

10% in the ratio of predicted/experimental value than that for a 2-D FEM model without FRP.

However, once the modification factor is incorporated, the results do not differ significantly from

one another, as shown in Fig. 6. This figure shows the predicted/experimental load capacity ratios

for many real cases, while using various FEM models. One can see that with respect to the

modification factor, the 2-D FEM model without FRP can give good predictions, compared to other

models. In summary, the simplest 2-D FEM model without FRP is adopted in further analysis, in

view of convenience and less time consuming.

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

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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

(6)

unitII ,0τ =

'

)'

( min2

hbh

lF

c

f

(7)

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 general, most probable ratios of flmin / h’ exist between 1 and 15. Fig. 7 presents the

predicted values of F1 and F2, based on the FEM simulations, with setting h’ to be 1 and varying the

flmin from 1 to 15. It is found that when the ratio flmin / h’ exceeds 3, the F1 and F2 values almost

remain constant values of 3.7Pa and 0.66Pa, respectively. In practical design, with the known

values of flmin , bc and h’, the F1 and F2 values can be measured through Fig. 7, or be calculated

from the following empirical equations:

34324972923605402

1 ≤+⎟⎟⎠

⎞⎜⎜⎝

⎛×−⎟

⎟⎠

⎞⎜⎜⎝

⎛×= '

min'

min'

min ,...h

lh

lh

lFfff

(8a)

7.31 =F , 3>'min

hl f

(8b)

37982173870119702

2 ≤+⎟⎟⎠

⎞⎜⎜⎝

⎛×−⎟

⎟⎠

⎞⎜⎜⎝

⎛×= '

min'

min'

min ,...h

lh

lh

lFfff

(9a)

6602 .=F , 3>'min

hl f

(9b)

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As a result, the unitII ,0σ and unitII ,

0τ can be obtained from eqns. (6) and (7).

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:

unitIIfff

II tbf ,0000 σσσ == (10)

( ) unitIIfffff

c

III tbftbxhIbV ,

000

000 ττττ +−=+= . (11)

2.3 Failure criterion

The failure criterion used in this study is such that when the maximum principal tensile stress 10,σ

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 , concrete cover separation failure occurs. 10,σ can be

obtained by the classical stress transformation equations for a 2-D plane stress condition,

( )20

200

10 22τ

σσσ +⎟

⎠

⎞⎜⎝

⎛+=, . (12)

Of note is that tf was defined in ACI code 318-95 (1999) as follows,

'. ct ff 530= , (13)

where 'cf is the concrete cylinder compressive strength. Therefore, concrete cover separation failure

takes place when

( )20

200

22τ

σσ+⎟

⎠

⎞⎜⎝

⎛+ = '. cf530 . (14)

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

load, 2P, are given

sfPLM −=0 ; PV =0 . (15)

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Considering eqns. (10) and (11), eqn. (14) can be transferred to

( )( )

( ) ( )2

0

2

0

0 2

2

⎟⎟⎠

⎞⎜⎜⎝

⎛−+−+

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛ −

+−

−

−

−

unitIIff

sfff

c

unitIIffsf

unitIIffsf

tbxhI

PLtbxh

IbP

ItbxhPL

ItbxhPL

,

,

,

τ

σ

σ= '. cf530 . (16)

Consequently, P can be determined from eqn. (16) as

( )( )

( ) ( )⎟⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜⎜

⎝

⎛

⎟⎟⎠

⎞⎜⎜⎝

⎛−+−+

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛ −

+−

=

−

−

−

2

0

2

0

0

1

2

2

530

unitIIff

sfff

c

unitIIffsf

unitIIffsf

c

tbxhI

Ltbxh

Ib

ItbxhL

ItbxhL

fP

,

,

,

'.

τ

σ

σ

. (17)

In order to obtain good prediction of test results, a modification factor has been considered

in many analytical models [5,27]. This is because final failure is preceded by the propagation of

cracks near the end of the plate and along the level of the concrete cover. A modification factor is

hence necessary to account for the nonlinear effect. The predicted load is then modified empirically

by:

PP Ψ=* (18)

Note that Ψ is an empirical function involving a number of fitting parameters. In this present

investigation, Ψ is assumed to be related to the ratios LL sf /− and ccff AEAE / . Herein, Lf-s and L

represent the distance from the end of FRP to the support and total span length, respectively; Ef and

Ec represent the elastic modulus of FRP strips and concrete beam, respectively; and Af and Ac

represent the cross section area of FRP strips and concrete beam, respectively. The parameter

LL sf /− is to account for the effect of relative FRP plate length on the prediction. And ccff AEAE /

is to show the influence of relative rigidity of FRP strips. Fig. 8 shows the effect of LL sf /− and

ccff AEAE / on the ratio of experimental/predicted results (i.e. the modification factor required for

exact prediction) without use of modification factor. From the Fig. 8, most data approximately

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locate in a flat plane, and thus assuming that Ψ varies linearly with LL sf /− and ccff AEAE / .

Finally, the full function obtained from empirical fitting is given by:

03503

1713

34

1719834

.,.

.

≥+−⎟⎟⎠

⎞⎜⎜⎝

⎛+×=

−⎟⎟⎠

⎞⎜⎜⎝

⎛×+⎟⎟

⎠

⎞⎜⎜⎝

⎛×=Ψ

−−

−

cc

ffsf

cc

ffsf

cc

ffsf

AEAE

LL

AEAE

LL

AEAE

LL

(19)

3. Verification of the model

This analytical model can be applied to predict the load carrying capacity for an RC beam

strengthened with FRP strips, which fails with concrete cover separation. In order to verify this

analytical expression, published experimental results [32-47] pertaining to strengthened RC beams

that fail due to concrete cover separation are analysed. Totally, 58 samples are selected, in which

the former 23 samples (up to P5) in the references are also used to obtain the expression for the

modification factor Ψ (eqn (19)), and the latter 35 samples are only employed in the verification

part. The geometries and material properties of the collected specimens are summarized in

Appendix A. The RC beams were strengthened using CFRP (Carbon Fibre Reinforced Plastics) or

GFRP (Glass Fibre Reinforced Plastics) plates. Note that these samples generally had different

material properties and dimensions, such as concrete compressive strength, FRP modulus, thickness

and width of FRP, steel reinforcement, FRP strip curtailment distance, and concrete beam

geometry. However, all specimens were subjected to four point bending test. The 23 samples

chosen for obtaining the modification factor, cover an extensive range of various parameters, such

as beam size, FRP material and so on. For example, the beam length varied from 1m to 3m; the

elastic modulus of FRP varied from 37.2GPa to 235GPa; the concrete strength was from 35MPa to

52.3MPa.

Many analytical models have been developed, which can be used to predict the load

carrying capacity of FRP strengthened RC beam with concrete cover separation. Generally, these

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existing models can be classified into three categories based on their approaches, namely interfacial

stress based models, concrete tooth models, and shear capacity based models [22]. In order to

compare the present model with these existing models, four previous analytical models are

employed. These include an interfacial stress based model by El-Mihilmy and Tedesco [5], a

concrete tooth model by Raoof and Hassanen [30], and two shear capacity based models by Oehlers

[23] and Smith and Teng [6].

The failure loads predicted based on the proposed model and four existing representative

models and the ratios of predicted/experimental failure load for all samples are presented in

Appendix A and Fig. 9. Also, Table 1 summarizes the average ratios of predicted/experimental

failure load and the standard deviations of the prediction for these models, as well as the coefficient

of variation. Good agreement of the proposed model is achieved with the average ratio of

predicted/experimental failure load being 0.94 and majority of data ranging between 0.8 and 1.2.

And the new model shows the least scatter, because the standard deviation and coefficient of

variation are 0.19 and 0.20, respectively. From Table 1, it can be seen that El-Mihilmy and

Tedesco’s model (1.42 in average) and Raoof and Hassanen’s upper bound model (1.94 in average)

are very un-conservative. The relatively conservative models are that given by Oehlers (0.44 in

average) and Smith and Teng (0.73 in average). In comparison, the proposed model and the

prediction of Raoof and Hassanen’s lower bound model show more accurate predictions, namely

0.94 and 1.04 in average, respectively. Of note is that the proposed model produces more

conservative predictions than Raoof and Hassanen’s lower bound model. Also, the standard

deviation is much lower (0.20 vs 0.54). Considering both the accuracy of load prediction, and the

variability of predicted results, the proposed model is clearly an improvement over existing models.

The present model could correctly predict the effect of FRP thickness on load capacity of

FRP strengthened beam. It is noted that when concrete cover separation is the dominant failure

mode, the experimental results show that the effect of FRP thickness is not fixed. It means that with

the increase of FRP thickness, the load capacity of FRP strengthened beams could decrease [33], or

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remain the same [44,45], even increase significantly [39,47]. This observation can be explained by

the ameliorating effect of reducing the beam rotation when the thicker FRP is used, thus improving

the crack restraint effect of the beam [36]. Unfortunately, most models cannot show the right trend

of effect of FRP thickness, because they always predict a significant decrease of load carrying

capacity with the increase of FRP thickness. Notice that with the use of modification factor in the

proposed model, much closer prediction can be obtained, although it is still not possible to have

perfect prediction of all the experimental trends. For example, in the tests by Beber et al. [47], the

prediction on the trend of effect of FRP thickness agrees well with the experimental trend, namely

the increase of ultimate load capacity with FRP thickness. This correct trend comes from the proper

consideration of ccff AEAE / in the modification factor. Eqn. (19) shows that the modification

factor increases with the increase of ccff AEAE / . Therefore, although the original model produces

lower maximum load with increasing FRP thickness, the final prediction gives the right trend after

taking into account the modification factor.

Moreover, the effect of FRP length (or the distance from the FRP end to the support) on the

load carrying capacity of strengthened beam can be accurately predicted by the proposed model and

Oehlers’s model [23]. The concrete tooth model by Raoof and Hassanen [30] cannot properly

consider the influence of this parameter. This point becomes clear when one try to apply the model

to the results in references [35, 43]. The interfacial stress based model by El-Mihilmy and Tedesco

[5] takes the FRP length into consideration, but sometimes predicts the wrong trend (e.g., when

employed to analyse the beams tested by Nguyen et al. [43]). The reason is that the applied

modification factor ( )2512351 LL sf /.. −×− is too strongly affected by the distance from the FRP

end to the support. The excessive correction may produce wrong results.

Fig. 10 shows the ratios of predicted/experimental failure load vs two parameters of LL sf /−

and ccff AEAE / . It is seen that there is NO clear relationship between the ratio of

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predicted/experimental failure load and the two parameters, indicating that the proposed equation

has similar accuracy within different ranges of the two parameters.

In summary, a simple analytical model for concrete cover failure is proposed in this work.

The model is quite simple to use (with similar degree of simplicity to many existing models), but

can produce relatively accurate prediction of measured load carrying capacity, with low scatter in

the ratio of predicted/measured results. Furthermore, the trend of load capacity with FRP thickness

and FRP length can be accurately measured. It is found to be an improvement over existing models,

and has good potential for application in practical design.

4. Conclusions

External bonding of fibre reinforced plastic (FRP) strips can significantly improve the ultimate

flexural strength and stiffness of strengthened reinforced concrete (RC) beam. However, the high

ultimate loading capacity is often impaired by premature failure modes, such as concrete cover

separation.

In this paper, for failure due to concrete cover separation, an analytical expression is

developed based on the consideration of stress concentrations in concrete near the tension rebar

closest to the cut off point of FRP strip. The analysis is performed in two stages, which includes (i)

a simple stress analysis assuming full composite action and (ii) the determination of local stresses

right below the steel reinforcement by analysing the concrete cover between two adjacent cracks.

An empirical modification factor is included to account for the effect of relative length and rigidity

of FRP strips on the nonlinear failure process. Compared to existing models, the present model can

more accurately predict the experimental load carrying capacity of strengthened RC beams, and

showing less scatter in the ratio of predicted/measured load. The expression proposed in the paper is

an improvement over existing models, and has good potential for application in practical design.

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Acknowledgements

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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Appendix A. Experimental Database for Concrete Cover separation Failure

Reference 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)

Ga1 150 200 2000 1200 75 0.44 2Φ 10 2Φ 8 162 27 150 Gb1 150 200 2000 1200 150 0.44 2Φ 10 2Φ 8 162 27 150 32 Gb2 150 200 2000 1200 150 0.66 2Φ 10 2Φ 8 162 27 150 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 33 MB5 115 150 1500 1200 115 0.444 3Φ 10 2Φ 10 125 25 75

RHB5 200 150 2300 1930 150 1.2 2Φ 10 2Φ 8 120 30 85 34

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

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 35

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

B2 100 100 1000 860 80 1.2 3Φ 6 2Φ 6 85 15 20 36

B4 100 100 1000 860 60 1.6 3Φ 6 2Φ 6 85 15 20 A1c 100 100 1000 860 80 1.2 3Φ 6 2Φ 6 85 15 20

A2b 100 100 1000 860 80 1.2 3Φ 6 2Φ 6 85 15 20 37 A2c 100 100 1000 860 80 1.2 3Φ 6 2Φ 6 85 15 20

1U,1.0m 100 100 1000 860 67 0.82 3Φ 6 2Φ 6 84 16 20 38

2U,1.0m 100 100 1000 860 67 0.82 3Φ 6 2Φ 6 84 16 20

P2 150 300 2800 2400 100 1.2 2Φ 14 — 257 — 200

P3 150 300 2800 2400 100 1.2 2Φ 14 — 257 — 200

P4 150 300 2800 2400 100 2.4 2Φ 14 — 257 — 200 39

P5 150 300 2800 2400 100 2.4 2Φ 14 — 257 — 200

1Au 100 100 1000 860 90 0.5 3Φ 6 2Φ 6 84 16 20

1Bu 100 100 1000 860 65 0.7 3Φ 6 2Φ 6 84 16 20

1Cu 100 100 1000 860 45 1.0 3Φ 6 2Φ 6 84 16 20

2Au 100 100 1000 860 90 0.5 3Φ 6 2Φ 6 84 16 20

2Bu 100 100 1000 860 65 0.7 3Φ 6 2Φ 6 84 16 20

2Cu 100 100 1000 860 45 1.0 3Φ 6 2Φ 6 84 16 20

3Au 100 100 1000 860 90 0.5 3Φ 6 2Φ 6 84 16 20

3Bu 100 100 1000 860 65 0.7 3Φ 6 2Φ 6 84 16 20

40

3Cu 100 100 1000 860 45 1.0 3Φ 6 2Φ 6 84 16 20

B 205 455 4880 4260 152 6 2Φ 25 2Φ 13 400 55 155 41 C 205 455 4880 4260 152 6 2Φ 13 2Φ 13 400 55 155

18

Reference 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)

A3 200 200 2000 1700 150 1.3 2Φ 14 2Φ 14 163 37 150 42 A4 200 200 2000 1700 150 1.3 2Φ 14 2Φ 14 163 37 150

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 43

NB2 120 150 1500 1100 80 1.2 2Φ 20 2Φ 6 120 23 115

1T6LN 150 200 2000 1460 150 0.66 2Φ 10 2Φ 8 162 27 20

2T6LN 150 200 2000 1460 150 0.66 2Φ 10 2Φ 8 162 27 20

2T6L1a 150 200 2000 1460 150 0.66 2Φ 10 2Φ 8 162 27 20

2T4LN 150 200 2000 1460 150 0.44 2Φ 10 2Φ 8 162 27 20

44

2T4L1a 150 200 2000 1460 150 0.44 2Φ 10 2Φ 8 162 27 20

DF2 125 225 1500 1400 75 0.334 3Φ 8 2Φ 6 193 32 50

DF3 125 225 1500 1400 75 0.501 3Φ 8 2Φ 6 193 32 50 45 DF4 125 225 1500 1400 75 0.668 3Φ 8 2Φ 6 193 32 50

AF3 125 225 1500 1300 75 0.334 2Φ 8 2Φ 6 193 32 100

CF2-1 125 225 1500 1300 75 0.334 2Φ 8

1Φ 62Φ 6 193 32 100

CF3-1 125 225 1500 1300 75 0.334 3Φ 8 2Φ 6 193 32 100 46

CF4-1 125 225 1500 1300 75 0.334 2Φ 10

1Φ 82Φ 6 193 32 100

VR5 120 250 2500 2200 120 0.44 2Φ 10 2Φ 6 214 34 75

VR6 120 250 2500 2200 120 0.44 2Φ 10 2Φ 6 214 34 75

VR7 120 250 2500 2200 120 0.77 2Φ 10 2Φ 6 214 34 75

VR8 120 250 2500 2200 120 0.77 2Φ 10 2Φ 6 214 34 75

VR9 120 250 2500 2200 120 1.1 2Φ 10 2Φ 6 214 34 75

47

VR10 120 250 2500 2200 120 1.1 2Φ 10 2Φ 6 214 34 75

19

Appendix A. (Continued)

Reference Beam fc’

(MPa) ft

(MPa) Ec

(GPa) fy

(MPa) Es

(GPa) Ef

(GPa) Shear span

(m) Ga1 43.1 3.5 31 531 200 235 0.5

Gb1 30 2.9 25 531 200 235 0.5 32 Gb2 30 2.9 25 531 200 235 0.5

MB3 30.3 2.9 26 534 183.6 230 0.5

MB4 30.3 2.9 26 534 183.6 230 0.5 33 MB5 30.3 2.9 26 534 183.6 230 0.5

RHB5 52.3 3.83 34.2 575 210 127 0.75 34

RHB6 52.3 3.83 34.2 575 210 127 0.75

FKF5 80 5 39.2 532 204 155 1.1

FKF6 80 5 39.2 532 204 155 1.1

FKF7 80 5 39.2 532 204 155 1.1 35

FKF10 80 5 39.2 532 204 155 1.1

B2 45.1 3.56 32 350 215 49 0.3 36

B4 45.1 3.56 32 350 215 49 0.3

A1c 59.5 4.1 36.5 350 210 49 0.3

A2b 35.7 3.2 28.3 350 210 49 0.3 37 A2c 35.7 3.2 28.3 350 210 49 0.3

1U,1.0m 45.9 3.6 32 350 215 111 0.3 38

2U,1.0m 45.9 3.6 32 350 215 111 0.3

P2 40 3.4 30 500 200 150 0.933

P3 40 3.4 30 500 200 150 0.933

P4 40 3.4 30 500 200 150 0.933 39

P5 40 3.4 30 500 200 150 0.933

1Au 50.2 3.8 33.5 350 215 111 0.3

1Bu 50.2 3.8 33.5 350 215 111 0.3

1Cu 50.2 3.8 33.5 350 215 111 0.3

2Au 50.2 3.8 33.5 350 215 111 0.34

2Bu 50.2 3.8 33.5 350 215 111 0.34

2Cu 50.2 3.8 33.5 350 215 111 0.34

3Au 50.2 3.8 33.5 350 215 111 0.4

3Bu 50.2 3.8 33.5 350 215 111 0.4

40

3Cu 50.2 3.8 33.5 350 215 111 0.4

B 35 3.14 28 456 200 37.2 1.983 41

C 35 3.14 28 456 200 37.2 1.983

20

Reference Beam fc’

(MPa) ft

(MPa) Ec

(GPa) fy

(MPa) Es

(GPa) Ef

(GPa) Shear span

(m) A3 33 2.6 25 540 200 167 0.7

42 A4 33 2.6 25 540 200 167 0.7

A950 27.3 2.8 25 384 200 181 0.44

A1100 27.3 2.8 25 384 200 181 0.44

A1150 27.3 2.8 25 384 200 181 0.44 43

NB2 37.9 3.23 29.1 384 200 181 0.44

1T6LN 47.8 3.7 32.5 531 200 235 0.5

2T6LN 62.1 4.2 37.1 531 200 235 0.5

2T6L1a 62.1 4.2 37.1 531 200 235 0.5

2T4LN 62.1 4.2 37.1 531 200 235 0.5

44

2T4L1a 62.1 4.2 37.1 531 200 235 0.5

DF2 46 3.6 30 568 185 240 0.5

DF3 46 3.6 30 568 185 240 0.5 45 DF4 46 3.6 30 568 185 240 0.5

AF3 46 3.6 30 568 185 240 0.5

CF2-1 46 3.6 30 568 185 240 0.5

CF3-1 46 3.6 30 568 185 240 0.5 46

CF4-1 46 3.6 30 586 185 240 0.5

VR5 33.6 3.1 27.4 565 200 230 0.783

VR6 33.6 3.1 27.4 565 200 230 0.783

VR7 33.6 3.1 27.4 565 200 230 0.783

VR8 33.6 3.1 27.4 565 200 230 0.783

VR9 33.6 3.1 27.4 565 200 230 0.783

47

VR10 33.6 3.1 27.4 565 200 230 0.783

21

Appendix A. (Continued)

Reference Beam Pexp (kN)

Pmodel (kN)a

Pmodel/ Pexp

PET

(kN)b PET/ Pexp

PRH l

(kN)c PRH l/ Pexp

PRH u

(kN)d PRH u/ Pexp

PO

(kN)e PO/ Pexp

PST

(kN)f PST/ Pexp

Ga1 92 114.8 1.25 277.3 3.01 95.1 1.03 190.3 2.07 28.6 0.31 66.5 0.72

Gb1 76 83.2 1.10 285.2 3.75 50.6 0.67 101.2 1.33 24.0 0.32 59.0 0.78 32 Gb2 75 83.3 1.11 231.0 3.08 50.2 0.67 100.4 1.34 21.1 0.28 59.0 0.79

MB3 86 73.5 0.86 152.8 1.78 67.3 0.78 134.7 1.57 27.8 0.32 47.7 0.55

MB4 82 70.4 0.86 124.3 1.52 67.1 0.82 134.3 1.64 24.9 0.30 47.7 0.58 33 MB5 79 70.9 0.90 108.6 1.37 66.8 0.85 133.7 1.69 22.8 0.29 47.7 0.60

RHB5 69.7 62.0 0.89 49.3 0.71 74.3 1.07 148.5 2.13 41.4 0.59 72.3 1.04 34

RHB6 69.6 62.0 0.89 49.3 0.71 74.3 1.07 148.5 2.13 41.4 0.59 72.3 1.04

FKF5 100 100.9 1.01 517.4 5.17 103.6 1.04 207.2 2.07 54.4 0.54 120.0 1.20

FKF6 103 100.9 0.98 517.4 5.02 103.6 1.01 207.2 2.01 54.4 0.53 120.0 1.17

FKF7 97.5 99.9 1.02 498.7 5.11 103.6 1.06 207.2 2.13 49.9 0.51 120.0 1.23 35

FKF10 82 98.9 1.21 478.9 5.84 103.6 1.26 207.2 2.53 45.5 0.55 120.0 1.46

B2 34 34.5 1.02 43.4 1.28 36.9 1.09 49.2 1.45 22.1 0.65 27.4 0.81 36

B4 35 37.6 1.07 38.5 1.10 37.5 1.07 49.3 1.41 21.8 0.62 27.4 0.78

A1c 44 32.6 0.74 50.0 1.14 40.1 0.91 54.0 1.23 24.4 0.55 30.1 0.69

A2b 36.7 35.5 0.97 37.8 1.03 34.3 0.93 44.9 1.22 20.2 0.55 25.4 0.69 37 A2c 37.3 35.5 0.95 37.8 1.01 34.5 0.92 45.3 1.21 20.2 0.54 25.4 0.68

1U,1.0m 36.5 43.2 1.18 33.8 0.93 35.6 0.98 71.1 1.95 20.3 0.56 27.6 0.76 38

2U,1.0m 32 43.2 1.35 33.8 1.06 35.6 1.11 71.1 2.22 20.3 0.63 27.6 0.86

P2 136.0 109.4 0.80 137.7 1.01 149.4 1.10 298.8 2.20 57.5 0.42 102.3 0.75

P3 142.2 109.4 0.77 137.7 0.97 149.4 1.05 298.8 2.10 57.5 0.40 102.3 0.72

P4 156.0 105.4 0.68 108.4 0.69 147.1 0.94 294.1 1.89 47.7 0.31 102.3 0.66 39

P5 159.0 105.4 0.66 108.4 0.68 147.1 0.93 294.1 1.85 47.7 0.30 102.3 0.64

1Au 39.6 37.8 0.95 42.2 1.07 36.2 0.91 51.0 1.29 22.0 0.56 28.5 0.72

1Bu 36.5 38.1 1.04 33.8 0.93 36.7 1.01 51.3 1.41 21.4 0.59 28.5 0.78

1Cu 31.9 37.8 1.19 26.3 0.82 33.6 1.05 47.0 1.47 20.5 0.64 28.5 0.89

2Au 38.5 37.8 0.98 42.2 1.10 31.9 0.83 45.0 1.17 22.0 0.57 28.5 0.74

2Bu 34.0 38.1 1.12 33.8 0.99 32.4 0.95 45.2 1.33 21.4 0.63 28.5 0.84

2Cu 35.5 37.8 1.07 26.3 0.74 29.6 0.83 41.5 1.17 20.5 0.58 28.5 0.80

3Au 39.0 37.8 0.97 42.2 1.08 27.2 0.70 38.3 0.98 22.0 0.56 28.5 0.73

3Bu 34.5 38.1 1.10 33.8 0.98 27.6 0.80 38.5 1.12 21.4 0.62 28.5 0.83

40

3Cu 30.7 37.8 1.23 26.3 0.86 25.2 0.82 35.3 1.15 20.5 0.67 28.5 0.93

B 250 238.4 0.95 160.6 0.64 155.2 0.62 218.9 0.88 175.3 0.70 219.5 0.88 41

C 190 117.6 0.62 69.7 0.37 99.0 0.52 136.6 0.72 111.9 0.59 142.1 0.75 a —representing the predicted load capacity using the present model; b—using El-Mihilmy and Tedesco’s model [5]; c—using Raoof and Hassanen’s lower bound model [30]; d— using Raoof and Hassanen’s upper bound model [30]; e—using Oehlers’s model [23]; f—using Smith and Teng’s model [6].

22

Reference Beam Pexp (kN)

Pmodel (kN)

Pmodel/ Pexp

PET

(kN) PET/ Pexp

PRH l

(kN) PRH l/ Pexp

PRH u

(kN) PRH u/ Pexp

PO

(kN) PO/ Pexp

PST

(kN)f PST/ Pexp

A3 106 107.2 1.01 87.7 0.83 130.3 1.23 260.7 2.46 31.2 0.29 92.2 0.87 42

A4 104 107.2 1.03 87.7 0.84 130.3 1.25 260.7 2.51 31.2 0.30 92.2 0.89

A950 56.2 43.6 0.78 98.4 1.75 48.4 0.86 96.8 1.72 9.7 0.17 47.4 0.84

A1100 57.3 53.2 0.93 70.9 1.24 55.8 0.97 111.6 1.95 13.8 0.24 47.4 0.83

A1150 58.9 59.8 1.02 43.5 0.74 55.8 0.95 111.6 1.89 16.1 0.27 47.4 0.80 43

NB2 130.1 103.1 0.79 150.6 1.16 103.6 0.80 207.2 1.59 26.2 0.20 73.3 0.56

1T6LN 116.2 162.4 1.40 193.0 1.66 116.2 1.00 232.4 2.00 51.4 0.44 68.9 0.59

2T6LN 135.9 153.5 1.13 222.6 1.64 126.9 0.93 253.8 1.87 57.2 0.42 75.1 0.55

2T6L1a 139.6 153.5 1.10 222.6 1.59 126.9 0.91 253.8 1.82 57.2 0.41 75.1 0.54

2T4LN 133.3 92.9 0.70 254.9 1.91 128.0 0.96 255.9 1.92 58.3 0.44 75.1 0.56 44

2T4L1a 137.7 92.9 0.67 254.9 1.85 128.0 0.93 255.9 1.86 58.3 0.42 75.1 0.55

DF2 120.6 82.0 0.68 96.2 0.80 132.8 1.10 265.6 2.20 45.2 0.37 63.6 0.53

DF3 120.0 96.0 0.80 85.8 0.72 132.3 1.10 264.6 2.21 43.2 0.36 63.6 0.53 45 DF4 125.6 109.5 0.87 80.4 0.64 131.7 1.05 263.5 2.10 41.7 0.33 63.6 0.51

AF3 96.6 91.1 0.94 109.9 1.14 324.4 3.36 648.8 6.72 33.8 0.35 55.5 0.57

CF2-1 104.8 104.8 1.00 126.4 1.21 322.8 3.08 645.6 6.16 37.0 0.35 60.4 0.58

CF3-1 118.2 115.2 0.97 139.0 1.18 331.3 2.80 662.5 5.60 39.3 0.33 63.6 0.54 46

CF4-1 140.2 141.7 1.01 170.9 1.22 356.6 2.54 713.1 5.09 44.5 0.32 70.7 0.50

VR5 102.2 66.2 0.65 59.4 0.58 72.8 0.71 145.7 1.43 39.2 0.38 60.0 0.59

VR6 100.6 66.2 0.66 59.4 0.59 72.8 0.72 145.7 1.45 39.2 0.39 60.0 0.60

VR7 124.2 85.0 0.68 52.8 0.43 71.9 0.58 143.8 1.16 35.8 0.29 60.0 0.48

VR8 124.0 85.0 0.69 52.8 0.43 71.9 0.58 143.8 1.16 35.8 0.29 60.0 0.48

VR9 129.6 103.0 0.79 50.6 0.39 51.1 0.39 102.2 0.79 33.6 0.26 60.0 0.46

47

VR10 137.0 103.0 0.75 50.6 0.37 51.1 0.37 102.2 0.75 33.6 0.25 60.0 0.44

23

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29

Figure Captions

Fig. 1: Delamination of FRP strips. (a) without cover failure along the debonded FRP strip; (b) with

some concrete cover debonding inside but NO cover failure at the vicinity of FRP end

Fig. 2: Concrete cover separation

Fig. 3: Cross section dimensions of a strengthened RC beam

Fig. 4: Analysis in stage II with opposite axial force in FRP strips

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

0τ in stage II: (a) 3-D with FRP; (b) 3-D without

FRP; (c) 2-D with FRP; and (d) 2-D without FRP

Fig. 6: The comparison of FEM model with/without FRP

Fig. 7: The normal and shear stresses concentrations for varying flmin / h’ ratios

Fig. 8: The effect of two parameters on modification factor

Fig. 9: The ratios of predicted/experimental failure load

Fig. 10: The ratios of predicted/experimental failure load vs two parameters of (a) LL sf /− and (b)

ccff AEAE /

30

(a)

(b)

Fig. 1. Delamination of FRP strips. (a) without cover failure along the debonded FRP strip; (b) with

some concrete cover debonding inside but NO cover failure at the vicinity of FRP end.

31

Fig. 2. Concrete cover separation.

32

Fig. 3. Cross section dimensions of a strengthened RC beam.

d df

d’ x

h’

h

As

As’

bf

bc

33

IIσ , IIτ

fff tbf 0 flmin

Fig. 4. Analysis in stage II with opposite axial force in FRP strips.

h’

34

unitII ,0σ and unitII ,

0τ at the critical point

a unit force lmin

Fig. 5. FEM models for predicting unitII ,

0σ and unitII ,0τ in stage II: (a) 3-D with FRP; (b) 3-D without

FRP; (c) 2-D with FRP; and (d) 2-D without FRP.

h’

(a)

(b)

(c)

(d)

35

Fig. 6: The comparison of FEM model with/without FRP.

0.0

0.5

1.0

1.5

2.0

Ppred

icte

d /Pre

al

FEM model (with FRP, without factor)FEM model (with FRP, with factor)FEM model (without FRP, without factor)FEM model (without FRP, with factor)

Ga1

Gb1

Gb2

MB

3M

B4

MB

5R

HB

5R

HB

6FK

F5FK

F6FK

F7FK

F10

B2

B4

A1c

A2b

A2c

1 U,1

.0m

2 U,1

.0m

1Au

1Bu

1Cu

36

Fig. 7: The normal and shear stresses concentrations for varying flmin / h’ ratios.

Normal stress Shear stress

0

2

4

6

8

10

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

Lminf/h'

Stre

ss C

once

ntra

tions

(Pa)

Normal stress function F1 Shear stress function F2

37

Fig. 8: The effect of two parameters on modification factor.

0.0000.0050.0100.0150.0200.0250.0300.0350.040

0

2

4

6

8

0.000.05

0.100.15

0.20

Mod

ifica

tion

fact

or

L f-s/LE

f Af /Ec A

c

38

Specimens collected

Fig. 9. The ratios of predicted/experimental failure load.

0

1

2

3

4

5

6

7

Ppred

icte

d /Pre

al

Ga1

Gb1

Gb2

MB

3M

B4

MB

5R

HB

5R

HB

6FK

F5FK

F6FK

F7FK

F10

B2

B4

A1c

A2b

A2c

1 U,1

.0m

2 U,1

.0m P2 P3 P4 P5 1Au

1Bu

1Cu

2Au

2Bu

2Cu

3Au

3Bu

3Cu B C A3

A4

A95

0A

1100

A11

50N

B2

1T6L

N2T

6LN

2T6L

1 a2T

4LN

2T4L

1 aD

F2D

F3D

F4A

F3C

F2-1

CF3

-1C

F4-1

VR

5V

R6

VR

7V

R8

VR

9V

R10

proposed model El-Mihilmy and Tedesco [5] Oehlers [23]

Raoof and Hassanen (lower bound) [30] Raoof and Hassanen (upper bound) [30] Smith and Teng [6]

39

Fig. 10. The ratios of predicted/experimental failure load vs two parameters of (a) LL sf /− and (b)

ccff AEAE / .

0.0

0.5

1.0

1.5

0 0.05 0.1 0.15 0.2Lf-s/L

Ppred

icte

d /Pre

al

0.0

0.5

1.0

1.5

0 0.01 0.02 0.03 0.04 0.05EfAf/EcAc

Ppred

icte

d /Pre

al

(a)

(b)

40

Table 1

Summary of prediction of the proposed model and four existing representative models

Model Average of predicted/experimental failure load

Standard deviation

Coefficient of variation

The proposed model 0.94 0.19 0.20

El-Mihilmy and Tedesco [5] 1.42 1.24 0.87

The lower bound in Raoof and Hassanen [30] 1.04 0.56 0.54

The upper bound in Raoof and Hassanen [30] 1.94 1.19 0.61

Oehlers [23] 0.44 0.14 0.32

Smith and Teng [6] 0.73 0.21 0.29