A Suggested Model Macr63-197

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Magazine of Concrete Research, 2011, 63(3), 197214

doi: 10.1680/macr.9.00085

Paper 900085

Received 04/06/2009; last revised 14/04/2010; accepted 26/04/2010

Published online ahead of print 14/02/2011

Thomas Telford Ltd & 2011

Magazine of Concrete Research

Volume 63 Issue 3

A suggested model for European code to

calculate deflection of FRP reinforced

concrete beams

Rafi and Nadjai

A suggested model forEuropean code to calculatedeflection of FRP reinforcedconcrete beamsM. M. RafiDepartment of Civil Engineering, NED University of Engineering andTechnology, Karachi, Pakistan

A. NadjaiFireSERT, University of Ulster at Jordanstown, Shore Road,Newtownabbey, UK

The theoretical deflection behaviours of concrete beams reinforced with fibre-reinforced polymer bars were

investigated and compared with the experimental data. The Eurocode 2 Part 1-1 deflection model, which is used for

conventional steel-reinforced structures, was tried for theoretical predictions. Experimentally recorded deflections of

75 simply supported specimens (beams and slabs), including the beams tested by the authors, were compared with

the Eurocode 2 method of deflection calculation. This method was found to be inaccurate for beams/slabs with

different fibre-reinforced polymer bar elastic moduli and reinforcement ratios. An appropriate modification for

theoretical beam deflection is proposed. The suggested expression includes effects of reinforcement amount relative

to the balanced condition and ratio of modulus of elasticity of fibre-reinforced polymer/steel bar. The results of the

proposed equation compared well with the recorded deflection for every fibre-reinforced polymer bar type.

Notationas shear span

b width of section

Ec modulus of elasticity of concrete

Ef modulus of elasticity of FRP bar

Es modulus of elasticity of steel bar

ff ultimate strength of FRP bar

fy yield strength of steel bar

h height of section

I moment of inertia

Icr cracking moment of inertiaIg gross moment of inertia neglecting the reinforcement

Iuncr uncracked moment of inertia of a transformed section

M applied moment

Mcr cracking moment

Mu ultimate moment

P applied load

crack deflection in fully cracked condition

uncrack deflection in uncracked condition

c concrete strain

c concrete stress

f FRP stress

IntroductionThe strength and compatibility with concrete are those qualities

which make steel a very effective reinforcing material for

reinforced concrete (RC) structures. However, steel is highly

susceptible to oxidation when exposed to chlorides. To arrest

rusting of steel, remedial work often has to be carried out in order

to achieve the full potential of the structure. These structural

repairs incur exorbitant costs to owners and stakeholders. For

example, countries in the UK and European Union spend around

20 billion annually on repair and maintenance of infrastructure

because of the problems associated with corrosion of steel

(ConFibreCrete, 2000). Recently non-metallic fibre-reinforced

polymer (FRP) materials have been introduced in the construction

industry to deal with unreliable durability problems of steel RC

structures.

A significant amount of research work has been carried out in

order to investigate the behaviour of FRP RC. As a result of these

efforts, worldwide interest in the use of non-metallic bars has

significantly increased over the last 25 years. FRP has thus

emerged as a potential alternative material to traditional steel. It

is expected that the use of FRP rebars in structural elements will

reduce maintenance cost to a great extent.

Fibrous bars have been successfully used in commercial applica-

tions in Japan, USA and Canada. These countries have estab-

lished design procedures specifically for the use of FRP rods in

concrete structures (ACI, 2006; CSA, 2002; JSCE, 1997).

Commercial application of FRP bars is still not fully exploited in

Europe compared to North America and Japan. The major

obstacle to its wider acceptance in the construction industry is the

absence of proper design guidelines. As FRP bars have low

modulus of elasticity compared to steel, the design of FRP

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reinforced structures is often based on the serviceability limitstate (SLS) as opposed to the ultimate limit state (ULS), which is

used for conventional steel RC. The control of both crack width

and deflection () becomes important in the SLS design. The use

of low-modulus FRP bars makes cracked FRP RC less stiff

compared to similar steel-reinforced concrete elements and

results in wider cracks and larger deflections. It is not hard to

understand that deflection calculations for FRP reinforced ele-

ments could lead to inaccurate predictions if these are based on

the design provisions intended for steel-reinforced concrete. This

fact has been realised in some international codes and appropriate

modifications have been made in the deflection calculation meth-

ods for FRP reinforced elements. This paper evaluates suitability

of the present Eurocode 2 Part 1-1 (CEN, 2004) deflection

prediction method for FRP RC flexural members and focuses on

the appropriateness of using its modified form. Specimens, which

were tested by the authors and other researchers, have been

included in this study. Equation1 has been employed to calculate

theoretical deflections of these specimens.

uncrack crack uncrack 1a:

where is given as

1 Mcr

M

2

1b:

where is coefficient related to duration of loading and is taken

as 1.0 for short-term loading. All other terms are defined in

Figure1 and the list of notation.

This equation is recommended by Eurocode 2 for deflection

calculation of steel-reinforced structures. The variation in the

stiffness of a cracked member is dealt with in Equation 1a

whereas Equation 1b accounts for tension stiffening as a func-

tion of the level of Mcr/M. Information on the accuracy ofEquation 1 for the deflection predictions of FRP reinforced

beams is scarce in the available literature. Although Pecce et al.

(2000) reported a good correlation between the recorded and

predicted deflections, Al-Sunna (2006) has indicated 20% under-

predicted deflection results for FRP RC with this equation. It is

important to note that Pecce et al. (2000) compared only a few

glass FRP (GFRP) reinforced beams whereas Al-Sunna (2006)

analysed 28 RC beams and slabs reinforced with both carbon

FRP (CFRP) and GFRP bars. One of the major shortcomings of

the investigations on FRP bars, which is evident in the published

research, is its concentration on GFRP bars. This has been

recognised by other researchers (Abdalla, 2002; Al-Sunna, 2006;

Mota et al., 2006). Consequently, deflection prediction models

of the international codes have been compared and/or validated

against the results of mainly GFRP reinforced concrete, whereas

it is imperative for the accuracy of a deflection prediction

method that elements reinforced with every type of FRP bar be

investigated and verified. Al-Sunna (2006) presented the idea of

using a reduced effective modulus and a 10% reduction ofcrack

in Equation 1 for the design of, respectively, CFRP and GFRP

reinforced beams. This approach requires different deflection

calculation procedures for these bars and, in the authors

opinion, it is an unrealistic approach to have more than one

method for different types of FRP bars. The challenge of using

Equation 1 for FRP RC is to verify this equation for beamsreinforced with FRP bars of different moduli and reinforcement

ratios.

CFRP bars are mostly considered in prestressing applications

owing to their high tensile strength (Rafi et al., 2008). The

authors carried out experimental testing of RC beams reinforced

with CFRP or steel rods. Complete details of the experimental

testing work and results can be found in Rafi et al. (2007a,2008).

This experimental testing was augmented by a strain compat-

ibility analysis (SCA) to predict theoretical behaviours of the

beams, which were compared with the experimentally recorded

data. The Eurocode 2 Part 1-1 (CEN, 2004) model (Equation 1)

400

L 1750

200

120

2 T8 bars6 mmstirrups

as 675 675

100 mm c/c2 T10 steel/2 95 mmCFRP bars

2P

600 600

Strain gauge

All dimensions in mm

125 125

Figure 1.Details of a typical beam

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was used for deflection predictions and a stiffer response of FRPreinforced beams tested by the authors was found. Finite-element

modelling (FEM) was also undertaken to help to understand the

beam behaviour. A sizeable amount of data for beams and slabs,

which has been reported in the literature by various researchers,

was analysed. This paper identifies the limitations of Equation 1

in relation to FRP reinforced structures. A modified expression

has been suggested and the results of both the existing and

modified equations have been compared with recorded data and

discussed. This comparison showed the effectiveness of the

suggested expression over Equation1. First, a short description of

the beams tested by the authors is given in the following section;

this is followed by a brief review of the experimental deflection

behaviour of these beams so as not to distract from the main

emphasis of the present paper.

Test specimensThe experimental programme comprised duplicate steel and

CFRP reinforced beams. The overall length of the beam was

2000 mm and the cross-section was 120 3 200 mm. Each beam

was reinforced with two longitudinal bars on the tension face

(9.5 mm diameter CFRP bars for FRP reinforced beams and

10 mm diameter steel bars for steel-reinforced beams). The CFRP

(ff 1676 MPa and Ef 135.9 GPa) and steel (fy 530 MPa

and Es 201 GPa) rods are shown in Figure 2. A 20 mm

concrete cover was used all around the beam. The area andnominal yield strength of the compression steel (8 mm diameter,

fy 566 MPa) and nominal concrete strength were kept constant

for all beams. The shear reinforcement consisted of smooth 6 mm

diameter (fy 421 MPa) closed rectangular stirrups spaced at

100 mm centre to centre. The beams were cast separately using

identical concrete mixes and were tested as simply supported

beams over a span (L) of 1750 mm under four-point static load,

as shown in Figure 1. These beams were part of a programme

that was designed to study the behaviour of FRP RC beams both

at normal and elevated temperatures.

Each tested beam is defined by letters comparing its reinforcingmaterial and temperature conditions. The notation of the beam as

is follows: the first letter (B) stands for beam; the second letter

indicates testing temperature as R for room temperature; the third

letter represents the type of tension reinforcing bar material such

as S for steel and C for CFRP bars. Table 1 shows equivalent

cylindrical strength (using strength of three cubes for each beam)

of the concrete (fc) and age of beams on the day of testing. It

was stated earlier that design guidelines are unavailable in the

Eurocode for FRP RC. Therefore, the design of these beams was

based on the ACI code approach (ACI, 2002; ACI, 2003). BRC

beams were designed as over-reinforced whereas BRS beams

were under-reinforced beams. Balanced (rb) and actual (r) rein-

forcement ratios for both types of beams are given in Table1.

Loaddeflection responseThe ultimate load (Pu) and corresponding deflection of the beams

is presented in Table 1. The ultimate load here is considered as

the maximum load carried by the beam. Figure 3 shows the

recorded loaddeflection responses of both types of beams. The

initial linear parts of the curves correspond to the uncracked

conditions of these beams. As can be seen in Figure 3, the

behaviour of both types of beams is similar before cracking when

the beams are stiff. The end point of this linear part is an

indication of the initiation of cracking in the beam.

The next segment that immediately follows this initial linear

part provides information about the bond quality and tension

stiffening effects due to crack spacing. The slope of this part is

smaller than the slope of the initial linear segment. This shows

that the amount of deflection per unit load is higher after the

beam has cracked, which is an indication of a reduction in the

stiffness of the cracked beam. Stiffness here is defined as load

per unit deflection. It can be seen in Figure 3 that the gap

between BRS and BRC beam curves widened as load increased.

This indicates that reduction in the stiffness of BRC beams was

higher compared to BRS beams with increase in load (Rafi et

al., 2008).

The last part of the deflection curve provides an indication of

a possible failure mechanism of a structure. As observed in

Figure 3, both BRS beams showed a very ductile behaviour

and both beams failed at nearly the same load after under-

going considerable deformation with very small increase inFigure 2.CFRP and tension steel bar

Beam fc: MPa Age: days r: % rb: % Pu: kN at Pu: mm Modes of failure

BRS1 46.52 61 0.77 2.84 41.9 29.16 Steel yielding

BRS2 44.64 85 0.77 2.78 40.1 27.78 Steel yielding

BRC1 42.55 78 0.70 0.37 88.9 35.26 Shear compressionBRC2 41.71 77 0.70 0.35 86.5 35.50 Compression

Table 1.Properties, ultimate load, deflection and failure modes

of beams

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load once steel yielded. On the other hand BRC beams

exhibited a linear elastic behaviour up to failure. The ultimate

load of the BRS beams was around 53% lower than the BRC

beams, while the deflection of BRC beams at ultimate state

(u) was 25% greater than the BRS beams (Table 1). The

observed modes of failure of the beams are mentioned in

Table 1. The behaviour of both BRS beams was similar as

they both failed by the crushing of concrete after the tension

reinforcement yielded, whereas BRC beams failed in compres-

sion (Rafi et al., 2008).

Strain compatibility analysisRafi (2010) implemented a numerical model to carry out strain

compatibility analysis of an RC section. This model is based on

the layer-by-layer approach of calculating section forces, compat-

ibility of strain, equilibrium of forces and a perfect barconcrete

bond. A maximum of 100 layers was used for a section. A non-

linear constitutive relation for uniaxial concrete compressive

strength was employed in order to calculate rebar strain and depth

of neutral axis (NA) with respect to concrete strain at the extreme

fibres. The concrete contribution below the NA was taken into

account before cracking and the tensile strength of concrete wasneglected for the cracked section. A linear stressstrain relation-

ship was used for the FRP bars up to the ultimate strength. The

actual steel stressstrain curve, which was obtained during the

tensile test of steel bars, was employed to calculate stress in steel.

The correlation of the experimental and analytical load capacity

was found to be remarkably good for both under- and over-

reinforced members.

The design of a conventional steel RC structure is based on ULS

and its deflection is checked at service load level. However,

service load for FRP RC structures has yet to be defined by the

international codes (Mota et al., 2006). Therefore, a comparison

of full experimental and analytical loaddeflection histories has

been made to ensure accuracy of the method at all stages of

applied load. The deflection before and after cracking of beam

was calculated with Equation 2 by using uncracked and cracked

moment of inertia, respectively.

PL3

6EcI

as

4L3

3 3L2 4a2s

2:

Analyticalexperimental deflectioncomparisonA comparison of deflection predictions for the BRS1 beam tested

by the authors has been made in Figure 4with the experimental

record. Theoretical deflection was calculated with the help of

Equation 1, which employed Equation 2 to compute deflections

both at uncracked and cracked states. Note that Equation 1 was

based on linear elastic behaviour of a steel-reinforced section and

may not provide accurate results beyond yielding of steel.

However, the analysis was not interrupted and a good agreement

between the theoretical and recorded deflection was found in this

case up to the failure of the beam, as can be seen in Figure 4.

The results for beam BRS2 are similar to BRS1.

Figure 5 compares the experimental and predicted load deflec-

tion curves for BRC beams tested by the authors. Equation 1has

been used for the theoretical deflection calculation of these beams

in the absence of an existing method for FRP RC structures in

the Eurocode, as mentioned previously. It can be seen in Figure 5

that the predictions in the initial stages of loading up to 35 kN

are quite close to the experimental results. However, Equation 1overestimated the stiffness of BRC beams with increase in load

and as a result deflection was underpredicted, as can be seen in

Figure 5. Note that the recorded and predicted cracking loads

(Pcr) in Figure 5 are very close to each other for BRC beams,

which minimises the influence of this factor on the theoretical

deflection. Theoretical uncr and cr, which are based, respec-

tively, on Iuncr and Icr and have been calculated with the help of

Equation2, have also been plotted in Figure5. The line foruncr

is the stiffest curve which is based on the uncracked beam state

whereas cr represents the least stiff behaviour, neglecting the

entire concrete in tension. It can be seen in Figure 5 that the

measured response crosses over the cracked deflection line at alow level of load (35 kN approximately). From a theoretical

403020100

20

40

60

80

100

0

Deflection: mm

Load:kN

50

BRC1

BRC2

BRS1

BRS2

Figure 3.Load deflection response of beams

40200

10

20

30

40

50

0

Deflection: mm

Load:kN

60

BRS1 (Exp.)

Eurocode 2

Figure 4.Load deflection curves for beam BRS1

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standpoint it is impossible for the member response to cross over

the Icrresponse and this behaviour is atypical of steel RC flexural

members. Note that except for heavily reinforced sections Iuncr is

generally replaced byIg to calculate uncr.

The stiffness of a partially cracked RC beam is not homogeneous

and an effective moment of inertia (Ie) is considered in the design

of a flexural member to account for the contribution of uncrackedconcrete between cracks in resisting tensile stresses. This is

termed the tension stiffening effect of concrete, which reduces

rebar strain between the consecutive cracks compared to the

strain at the crack location. It is also an alternate way of

modelling the barconcrete bond and plays a significant role in

the overall response of flexural elements. Tension stiffening is

important at loads close to cracking and its effects reduce at

higher loads. Ie provides a transition between Ig and Icr as a

function ofMcr/M. The deflection of a flexural member is derived

from its curvature (k) profile, as given by Equation 3. Shear

induced deflection may cause an increase in the curvature of a

beam owing to a shear flexure interaction. This results in

additional bar strain and a consequent increase in the total

deflection along the span of a beam. Therefore, it is imperative to

investigate the amount of shear-induced deflection in BRC beams.

Various approaches that were employed in this regard are

explained below.

k d2

dx2

MEcIe3:

Approach 1 analysis of rebar strain

First of all, recorded data of rebar strain were analysed. This

strain both at the mid-span and in the shear-span of BRC beams

is traced in Figure 6. The positions of strain gauges on the bars

are shown in Figure 1. Although this method is not very exact

owing to the dependence on the data of only one strain gauge in

the shear-span, it clearly indicates that bar strain in the shear-span

is considerably less than that at mid-span. Therefore, shear-

induced deflections can be assumed negligible. Note that in

Figure 1 the strain gauges in the shear span were fixed at an

equal distance from the beam centre. The data of only one strain

gauge are plotted in Figure 6 for clarity as both the gauges

recorded similar strain.

Approach 2 deflection comparison of CFRP RC beams

In another attempt to investigate further the possibility of shear

deformation in BRC beams, deflection characteristics of a num-

ber of CFRP reinforced beams and slabs, which were tested by

other researchers, were compared with BRC beams. The results

0

20

40

60

80

100

0

Deflection: mm

(b)

Load:kN

50

Uncrackeddeflection

BRC2 (Exp.)

Eurocode 2

FEM

Equation 5

Crack

40302010

40302010

0

20

40

60

80

100

0

Deflection: mm

(a)

Load:kN

50

Uncrackeddeflection

BRC1 (Exp.)

Eurocode 2

FEM

Equation 5

Crack

Figure 5.(a) Loaddeflection curves for beam BRC1; (b) load

deflection curves for beam BRC2

00150010005

00150010005

0

20

40

60

80

100

0Strain: m/m

(a)

Load:kN

Exp shear-span

FEM shear-span

Exp mid-span

FEM mid-span

0

20

40

60

80

100

0

Strain: m/m(b)

Load:kN

Exp shear-span

FEM shear-span

Exp mid-span

FEM mid-span

Figure 6. (a) Rebar strain in BRC1 in shear-span and at mid-span;

(b) rebar strain in BRC2 in shear-span and at mid-span

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of a few beams have been presented in Figure 7 for clarity. Thedeflection responses of slab LL-200-C (Abdalla, 2002), and

beams BC2a (Al-Sunna, 2006), F-29f (Orozco and Maji, 2004)

and B1 (Wilson et al., 2003) have been compared with BRC

beams in Figure7. Since all these specimens vary in strength and

stiffness from each other, ultimate load and ultimate deflection

have been normalised by using ratios ofM/Mu and/u, respec-

tively, in order to provide a unified basis of comparison. It can be

seen in Figure 7 that the normalised stiffness of all these

specimens is nearly the same. This takes out the effects of

specimen size, shear and effective spans, and effects of CFRP

bars produced by different manufacturers. A close correlation in

the stiffness of all these specimens is an indication that shear-

induced deflections are insignificant in BRC beams since it is

unlikely that all beams can have the same amount of shear

deformations.

Approach 3 finite-element modelling

As a next step towards understanding the beam behaviour in

relation to Equation 1, non-linear FEM of BRC beams was

carried out using the computer code Diana (TNO, 2005). The

concrete stiffness was based on secant moduli, which were taken

perpendicular and parallel to the direction of crack. The analy-

tical model, which is based on the total strain, was used to

idealise the response of cracked concrete. The cracks were

considered as smeared cracks and a rotating crack approach wasemployed to simulate the formation and propagation of cracks.

The behaviour of concrete in compression, effects of tension

softening and stiffening, and the behaviour of tension reinforce-

ment were considered in the analytical model. An incremental

iterative non-linear solution procedure was used for the analysis.

Complete information of this analytical work is available in Rafi

et al. (2007b). The analytical deflection behaviours of both BRC

beams have been plotted in Figure 5. It can be seen that the

predictions of ultimate capacity and stiffness of the beams are

fairly good. It is noted in Figure 5 that the theoretical analysis

slightly overestimated post-cracking stiffness of the beams, which

may be due to the use of a too stiff tension stiffening model.Nevertheless, this overestimation is typical of this type of analysis

(Zhao et al., 1997) as the effects of other factors such as local-

bond slip and shrinkage stresses are unaccounted for in the

analysis (Rafi et al., 2007b). The correlation for the initial

stiffness and for the overall non-linear behaviour is very exact.

Since shear-induced deflections are not included in the analytical

models, the results correlate closely with the observation of

negligible shear deformations in BRC beams.

A comparison of the analytical strain of the CFRP bar at the mid-

span and in the shear-span of BRC beams is presented in Figure

6. A stiffer analytical response of the beam in the post-cracking

stage can be seen in Figure 6 compared to the observed response,

especially for BRC2 beam. However, considering the influence of

cracking on recorded strain, the predicted results are fairly close

to the experimental plot and a good correlation exists between

the two results. These results also confirm that there is no

additional bar strain other than that induced by the flexural

deflection.

Further, it was noticed during the experimental testing of beams

that cracking in both the BRS and BRC beams stabilised after an

applied load of 30 kN and both types of beams developed almost

the same number of cracks up to their failure with similar average

spacing (Rafi et al., 2007a). Since the additional deflection inBRC beams is induced after a load of 35 kN (Figure5) it cannot

be associated with shear deflection, as the development of shear

deflection must coincide with the formation and spread of cracks

within the shear span. Based on all these results it can be

concluded that the additional deflection in BRC beams is not

caused by shear deformation. This is in agreement with the

conclusions made byAl-Sunna (2006).

Approach 4 stress comparison

Theoretical concrete compressive and FRP tensile stress have

been traced in Figure 8 against the applied loads only for beam

BRC1, owing to the similarity of results for both BRC beams.Both the compressive and tensile stresses have been normalised

using ratios ofc/fc andf/ff. It can be seen in Figure 8 that the

slope of the initial part of the concrete curve is reasonably

constant. If a tangent is drawn to the curve at the origin (Figure

8), the slope of this tangent and the initial part of the curve

remains the same up to 35 kN, which represents the linear part of

the curve. This load corresponds to approximately 40%Pu and

can be regarded as low load level. As the load is further increased

the concrete behaviour becomes significantly non-linear. This is

due to the use of low modulus FRP tension bars which require

large concrete compressive force to maintain equilibrium. It is

noted in Figure8 that the ultimate strength of concrete is reached

at 60% of FRP stress. The corresponding load comes out to be

65 kN, which is nearly 75% ofPu. This indicates that service load

levels could be higher for CFRP RC compared to the suggested

range of 3550% in the published literature. Note that severe

microcracking usually results close to the ultimate strength of

/ u

1210080604020

02

04

06

08

10

12

0

M

M/

u BRC1

BRC2

LL-200-C

BC2a

F-2 f

B1

Figure 7.Comparison of stiffness of CFRP RC beams tested by

other researchers

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concrete, which may create creep effect in the concrete. As can

be seen in Figure8, these effects in BRC beams would be present

only after a load level of 50 kN and may not affect the deflection

behaviour prior to this load level.

Approach 5 concrete constitutive models

As a last attempt, different concrete constitutive relations, as

suggested in the available technical literature, were varied to

study the effects of variation in the behaviour of compression

concrete. The relations which were tried include those suggestedby Hognestad (1951), Kent and Park (1971), Popovics (1973),

Sheikh and Uzumeri (1982), Mander et al. (1988), Hillerborg

(1989) and Almusallam and Alsayed (1995). This attempt also

failed to produce any significant improvement in the predicted

deflection response of BRC beams. Therefore, it is hard with the

present level of knowledge to explain the reasons for the stiffer

crresponse compared to the recorded deflection. The behaviour

of the beam is as if the tension stiffening effects are negative.

The above analysis provides sufficient evidence that the measured

deflection in BRC beams is a result of only flexural curvature.

This can have significant implications upon the theoretical back-ground and formulation of conventional RC design. A stiffer

flexural cr compared with the measured response would imply

that concrete compressive stress is a non-linear function of strain

(c f(c)) instead of being proportional to strain (c c).

Therefore, Hookes law cannot be used to determine concrete

compressive stress. This invalidates linear elastic theory, which is

the backbone of RC design. Consequently, the forcedeformation

relationship, such as given by Equation 3, does not apply to BRC

beams as it has been obtained from the elastic deflection theory

of beams. This possibly explains underestimation of deflection of

BRC beams by Equation 1. Non-linear concrete behaviour as

traced in Figure 8 provides evidence to support this type of

concrete response. However, the above presented work in this

study is by no means conclusive and specialised investigative

research is suggested to confirm the findings of this study. The

analysis, which is carried out in the subsequent sections of this

paper, provides firm ground for a future study.

It was mentioned earlier that reduction in the recorded stiffnessof BRC beams was higher compared to BRS beams during their

testing (Figure3). The average difference in the stiffness of both

types of beams at the yielding of steel bars was about 38% (Rafi

et al., 2008). Yost et al. (2003) reported a higher loss of stiffness

and a rapid change from gross to fully cracked section properties

in GFRP RC beams compared to similar steel-reinforced beams.

The stiffness of cracked RC is primarily dependent on the correct

estimate of its tension stiffening characteristics, which in turn are

related to elastic modulus, bond and reinforcement ratio of rebars.

Therefore a review of these for BRC beams tested by the authors

seems appropriate at this stage. Bond characteristics of the CFRP

bars were found satisfactory in BRC beams. The bars carried a

stress between 80 and 90% of their tensile strength (Rafi et al.,

2007a). As noted earlier, cracking in both the BRS and BRC

beams stabilised after an applied load of 30 kN and both types of

beams developed almost the same number of cracks up to their

failure with similar average spacing. Since bond properties influ-

ence the spacing of cracks, similar crack spacing in BRS and

BRC beams indicates comparable bond of both the steel and

CFRP bars and strengthens the observation of satisfactory

CFRPbarconcrete bond. This was also confirmed by the

aforementioned FEM results and discussion (Figure 6). The

deflection beyond this load (30 kN) mainly resulted in increased

width of existing cracks. Therefore, it can be inferred that

Equation 1 provided higher tension stiffening estimates for BRCbeams. Note that Al-Sunna (2006) has also found lesser tension

stiffening effects in CFRP RC beams compared to steel-rein-

forced beams and indicated higher tension stiffening represent-

ation by Equation1 for FRP RC under certain conditions.

In order to evaluate the effects of the remaining two variables

(i.e. bar modulus and reinforcement ratio) on the predictions from

Equation 1, a more detailed analysis of tested specimens was

carried out with the help of available test results in the existing

technical literature. The beams were selected according to the

reinforcing amount and modulus of elasticity of FRP bars.Yost et

al. (2003) and Razaqpuret al. (2000) have reported the influenceof theoretical Mcr on the stiffness results of cracked beams. The

subsequent discussion included beam data based on two criteria

in order to simplify comparison the beams have closely

matched theoretical and experimental Mcrand either r or r/rb of

the beam is similar to BRC beams tested by the authors.

Effects of reinforcement ratio

Figure 9 shows the effects of reinforcement ratio on the

analytical results of two beams, which were tested by Yost et al.

(2003). GFRP reinforcing bars with the same Ef were used in

both beams. Beam 1a-NL had a low r and r/rb compared to

beam 4b-HL which was designed with a high r. Details of these

beams have been summarised in Table 2 where it can be seen

that r for beam 1a-NL is the same as for BRC beams. The

theoretical deflection is slightly overestimated for beam 1a-NL,

whereas the predictions are reasonably good for beam 4b-HL,

which has a high r and r/rb. Cracked deflections have also

c c t t/ or /f f

105070350035070

20

40

60

80

100

105

Load:kN

CFRP bar

Tangent

Concrete

35 kN

Figure 8.Theoretical concrete and CFRP stress in BRC1 beam

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been plotted in Figure9 which indicate a large tension stiffening

effect for beam 1a-NL when compared with the measured

response. The effect of tension stiffening is more significant in a

lightly reinforced beam (1a-NL) as the depth of NA is small. As

a result, overall the beam response in tension is dictated by the

tensile response of both the concrete and the rebars. These

effects were underestimated by Equation 1, which resulted inhigher deflections compared to the recorded deflection. It is

noted in Figure 9that tension stiffening effects for the beam 4b-HL (high r value) are lower. This indicates that the beam

stiffness changes very quickly from Ig to a level close to Icr.

Since the tension stiffening effects are insignificant in this case,

theoretical deflections (Equation 1) are very close to both the

cracked and recorded deflection.

Note that these are not isolated results and have been further

verified for beams RC-A1 and RC-A5 which were tested by

Nakano et al. (1993). These beams were reinforced with 8 mm

and 16 mm diameter aramid FRP (AFRP) bars, respectively,

which had nearly the same modulus as can be seen in Table 2.

Both the experimental and theoretical behaviours are plotted in

Figure 10 for beams RC-A1 and RC-A5. Reinforcement ratios

for both beams are provided in Table 2, where it can be

noticed that beams RC-A1 and 1a-NL had nearly the same r/

rb. As beam RC-A1 was lightly reinforced compared to beam

RC-A5 the theoretical deflection was overestimated (similar to

beam 1a-NL) whereas for beam RC-A5 the predicted deflection

matches well with the recorded deflection. A comparison

between cracked and measured beam behaviour in Figure 10

indicates that tension stiffening effects are higher in beam RC-

A1 (similar to 1a-NL). However, contrary to beam 1a-NL, the

crcurve gets closer to the theoretical curve of Equation 1 for

beam RC-A1, which strengthens the observation of under-

predicted tension stiffening effects from Equation 1. Tensionstiffening is less in beam RC-A5 (similar to 4b-HL) and

measured beam response, in this case, crosses over the cracked

response at low load level (82 kN). Beyond this load level,

cracked response is similar to the theoretical deflection from

Equation 1. Although the possibility of shear deflection was not

investigated for beam RC-A5 it is clear that the shear deflec-

tion in beam RC-A5 cannot be more than RC-A1 as the former

was a heavily reinforced beam and shear deflection reduces

with an increase in either the modulus or amount of reinfor-

cing. Since beams 4b-HL and RC-A5 had similar amount of

reinforcing, a stiff cracked response in beam RC-A5 is thought

to be due to higher modulus AFRP bars and is furtherinvestigated in the following sections.

10080604020

80604020

crack

0

10

20

30

40

0

Deflection: mm

(a)

Load:kN

120

crack

1a-NL (Exp.)

Eurocode 2

Equation 5

0

10

20

30

40

50

60

0

Deflection: mm

(b)

Load:kN

100

4b-HL (Exp.)

Eurocode 2

Equation 5

Figure 9.(a) Loaddeflection curves for beam 1a-NL (Yost et al.,

2003); (b) loaddeflection curves for beam 4b-HL (Yost et al.,

2003)

Beam b: mm h: mm r: % r=rb Ef: GPa Bar type

1a-NL (Yost et al., 2003) 254 184 0.71 1.27 40.30 GFRP

4b-HL (Yost et al., 2003) 178 184 2.32 2.43 40.30 GFRP

RC-A1 (Nakano et al., 1993) 200 300 0.28 1.38 65.00 AFRP

RC-A5 (Nakano et al., 1993) 200 300 3.03 13.60 57.00 AFRP

Coated FRP (Nanni, 1993) 100 150 0.70 2.61 63.80 AFRP

CB2B (Benmokrane and Masmoudi, 1996) 200 300 0.70 1.24 37.65 GFRP

BG3b (Al-Sunna, 2006) 150 250 3.93 5.42 42.75 GFRPBC2a (Al-Sunna, 2006) 150 250 0.65 1.13 131.80 CFRP

F1 (Saadatmanesh and Ehsani, 1991) 200 460 1.53 6.03 53.60 GFRP

Table 2.Summary of the properties of the beams

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Effects of bar modulus

Figure 11 presents experimental and theoretical loaddeflection

curves for the beams tested by Nanni (1993) and Benmokrane

and Masmoudi (1996). The beams were, respectively, reinforced

with AFRP and GFRP bars. Both beams have the same r as beam

1a-NL and BRC beams whereas the coated FRP beam had a highr/rb compared to beam CB2B. Details of the beams have been

provided in Table 2. It is evident in Figure 11 that deflection is

overestimated for beam CB2B and a good correlation between

the recorded and predicted behaviour exists for the beam coated

FRP. Note that this beam had the same r as beams 1a-NL and

CB2B, and the Ef of FRP rebars was higher (Table 2). Cracked

responses of both beams have been traced in Figure 11 and it is

noted that tension stiffening effects of beam BC2B are largely

similar to beam 1a-NL. As can be expected, Equation 1 under-

estimated these effects which resulted in overestimation of beam

deflection. Measured deflection for coated FRP beam crosses over

the cracking response similar to beam RC-A5 and, subsequently,

theoretical creventually surpasses the beam predicted deflection

(Equation 1). This confirms that bar modulus is the main factor

to cause this type of beam behaviour.

Figure12 further evaluates the accuracy of Equation1in relation

to Ef of FRP rods. The recorded deflections of beam BG3b (Al-

Sunna, 2006) reinforced with GFRP bars (Ef 41.95 GPa) and

beam BC2a (Al-Sunna, 2006) reinforced with CFRP bars

(Ef 131.8 GPa) have been compared in Figure 12 with the

predictions made by Equation1. Properties of the test specimens

can be reviewed in Table2, where it can be seen that beam BC2a

had low r value, which was also nearly the same as BRC beams

and r/rb of this beam is very close to beam 1a-NL ( Yost et al.,

2003). On the other hand beam BG3b has high randr/rb values.

As can be expected from the above, Equation 1 gave very

accurate results for beam BG3b. The theoretical curve, on the

other hand, significantly deviates from the measured deflection of

beam BC2a, which was reinforced with higher modulus rebars. A

comparison of the coated FRP beam (Figure 11(a)) with BC2a

reveals that this deviation is proportional to the bar modulus as

both these beams have the same reinforcing amount. The cracked

deflection responses are also plotted for beams BG3b and BC2a

in Figure 12 which confirms the underestimation of tension

stiffening from Equation1b as was noted in Figures 911.

The results of Figures 912 are telling in several respects. First,

2010

crack

0

20

40

60

80

100

120

0Deflection: mm

(a)

Load:kN

60

crack

Nakano (RC-A1)

Eurocode 2

Equation 5

0

50

100

150

200

250

0Deflection: mm

(b)

Load:kN

30

Nakano (RC-A5)

Eurocode 2

Equation 5

4020

Figure 10.(a) Loaddeflection curves for beam RC-A1 (Nakano et

al., 1993); (b) loaddeflection curves for beam RC-A5 (Nakano et

al., 1993)

0

15

30

45

60

75

0

Deflection: mm

(a)

Load:kN

12

crack

Coated FRP (Exp.)

Eurocode 2

Equation 5

crack

0

20

40

60

80

100

0

Deflection: mm(b)

Load:kN

CB2B (Exp.)

Eurocode 2

Equation 5

963

80604020

Figure 11.(a) Loaddeflection curves for the coated FRP beam

(Nanni, 1993); (b) loaddeflection curves for beam CB2B

(Benmokrane and Masmoudi, 1996)

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it is clear that tension stiffening of the beams is not correctly

represented by Equation 1. This is particularly true for lightly

reinforced sections. Although additional deflection, which is

observed in beams BC2a (Al-Sunna, 2006), BRC (Rafi et al.,

2008), RC-A5 (Nakano et al., 1993) and coated FRP (Nanni,

1993), cannot be explained satisfactorily it was observed that,

for lightly reinforced sections, this deflection is independent ofeither FRP bar type or amount and results for FRP bars with a

modulus greater than 50 GPa. Since all the above tested beams

with Ef> 50 GPa were reinforced with either AFRP or CFRP,

beam F1 (Saadatmanesh and Ehsani, 1991) was analysed

additionally to verify this observation. This beam is moderately

reinforced with GFRP bars (Ef 53.60 MPa). Other details of

the beam F1 are presented in Table 2. Recorded and predicted

deflection responses of the beam are traced in Figure 13. It can

be seen in Figure 13 that the measured deflection crosses over

the cr curve at low load level and the theoretical beam

deflection (Equation 1) is the same as cr. This type of response

was not noted in the GFRP RC beams in Figures 9, 11 and 12

as the GFRP bar moduli were less than 50 GPa for these beams.

Modified form of Eurocode equationIt becomes clear in the above discussion that tension stiffening

effects of FRP RC are different from steel-reinforced concrete

and the results of predicted deflection (Equation 1) vary with

both Ef and, to a certain extent, r/rb. Contrary to what can be

expected in steel RC elements, reduction in a cracked FRP

reinforced beam stiffness increases with an increase in both

parameters. This may appear counter-intuitive from a structural

engineering point of view that higher modulus reinforcing bars

reduce stiffness of RC. It is imperative that the tension stiffening

of FRP RC as represented by Equation1b is brought to a realisticlevel. This is possible by softening the crresponse as suggested

byAl-Sunna (2006).

An attempt has been made here to modify Equation 1 empiri-

cally to develop a more accurate estimation of beam stiffness.

Efforts have been made to introduce such changes that will allow

the basic form of this equation to remain close to the original

Eurocode 2 expression (Equation 1). It is worth mentioning here

that beams 1a-NL (Yost et al., 2003) and 4b-HL (Yost et al.,

2003) did not have any shear reinforcement. All other beams

contain adequate stirrups to keep diagonal tension cracks tight.

The presented discussion did not provide any evidence of shear-induced deflection in BRC beams. Furthermore, Al-Sunna (2006)

indicated the possibility of higher shear deformation with low

modulus bars (typically GFRP) compared to higher modulus bars

(CFRP). The results in Figures 912 do not indicate any such

possibility in any of the GFRP reinforced beams. Therefore,

shear deformations were not considered for simplicity in the

analytical work described in the next section. Similarly, any

local-bond slip can be reflected by the concrete tension stiffening

relation and can be accounted for by appropriate modification in

Equation 1. Al-Sunna (2006) has also pointed out a dependency

of deflection more on Ef and r of FRP bars than its bond with

concrete.

Based on the above theoretical analysis and presented discussion

a new factor of the form given in Equation 4is suggested here

in order to take into account differences in the stiffness of FRP

and steel RC.

0

20

40

60

80

100

120

140

0

Deflection: mm

(a)

Load:kN

crack

BG3b

Eurocode 2

Equation 5

0

20

40

60

80

100

120

0

Deflection: mm

(b)

Load:kN

40

crack

BC2a (Exp.)

Eurocode 2

Equation 5

30252015105

302010

Figure 12.(a) Loaddeflection curves for beam BG3b (Al-Sunna,

2006); (b) loaddeflection curves for beam BC2a (Al-Sunna, 2006)

0

100

200

300

400

0

Deflection: mm

Load:kN

40

crack

F1 (Exp.)

Eurocode 2

302010

Figure 13.Loaddeflection curves for beam F1 (Saadatmaneshand Ehsani, 1991)

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a1 1 a2 EfEs

4:

This factor can be included in Equation 1 for short-term

deflection calculation which can be rewritten as Equation5

uncrack crack uncrack 5a:

1 Mcr

M

2

" #5b:

The value ofa2 as 0.5 was selected based on some trial and error

calculations anda1 is considered a function of r/rb. Substitution

of a typical value ofEs 200 GPa in Equation4 yields

a1 1 Ef

400

6:

A statistical approach has been followed in order to obtain a

relation between a1 and r/rb. The reported test results in the

literature are included in order to increase the population size. Atotal population of 73 beams, including BRC beams tested by the

authors, and two slabs was selected with a range of r/rb and Ef

values. These will collectively be referred to as specimens here.

These specimens were tested in either a three-point or four-point

load. FRP bars consisted of GFRP, AFRP and CFRP, which were

placed in either one or two layers. However, GFRP rods were

used in the majority of the specimens because they attracted more

attention in the researchers community owing to their lower cost,

as mentioned earlier. The ratio of r/rb varied between 0.27 and

13.59 where the concrete consisted of normal-, high- and very

high-strength concrete. Details of the specimens are given in

Table 3. Specimens with a wide variety of bars were used inTable 3 in order to minimise the effects of FRP manufacturing

processes which are employed by various manufacturers across

the globe. The researchers for the designated specimens are given

in Table4.

For each specimen, values of (which were calculated from

Equation1 at different load levels) were substituted in Equation5

to determine corresponding, which was found to be the same at

all the load steps. A unique value of a1 was then determined

using Equation 6 for that specimen. This was then changed in

close intervals of 0.01 and a value of a1 for the best fitting

experimental curve was obtained. Particular attention was paid to

ensure closest correlation of the experimental and theoretical

curves in the range of 35% and 90% Pu. The same method was

followed for all the specimens in Table 3. A typical example of

the method has been illustrated in Figure 14 for beam BRC2

tested by the authors. The initial value of a1 for this beam,

corresponding to Equation1, came out to be 0.75. It can be seenin Figure 14 that the deflection curves from Equation 1 and

Equation 5 using a1 0.75 are a perfect match. The predicted

curves at a few more a1 values have also been included in Figure

14 and it becomes clear that the theoretical predictions at

a1 0.90 provided the closest correlation with the experimental

curve for beam BRC2.

The values of a1 were then plotted with corresponding r/rb of

the specimens. The results are graphically represented in Figure

15 and a simplified relationship of a1 was obtained by linear

regression, which is given in Equation7.

a1 0:0121 r

rb

0:85817:

The correlation coefficient for Equation 7 comes out to be 0.21.

A low coefficient is owing to a few higher r/rb values, as can be

seen in Figure 15. The correlation coefficient increases if these

higher values are excluded from the data. However, this was not

considered necessary as Equation 5 provided satisfactory results,

after substitution of and a1 from Equation 6 and Equation 7,

respectively. The obtained results have been plotted in Figure 5

and Figures912. These figures show a good correlation between

the modified equation and the experimental results. The value of can be taken as 1 for steel-reinforced beams.

To assess the effectiveness and repeatability of Equation 5 (in

combination with Equations 6 and 7), all the beams in Table 3

were analysed and the deflections predicted by both the original

(Equation1) and modified (Equation 5) equations were compared

with the experimental data. This comparison at three load stages

(35% Pu, 50% Pu and 90% Pu) has been illustrated in Figure 16

for a few of the beams for clarity. For the beams in Table 3 the

maximum coefficient of variation of the ratio of experimental and

theoretical deflection (using Equation 5) at the above-mentioned

three load levels comes out to be approximately 21.5% as

opposed to 32.3% for a similar ratio with the Eurocode 2 method

(Equation 1). The 95% confidence interval is approximately in

the range 0.971.13 for the former method and 1.101.35 for the

latter.

Full analytical loaddeflection histories of six beams from Table

3 are traced in Figure 17. A comparison of the experimental

curve is made with the original equation (Equation 1) and

proposed equation (Equation 5). It is evident in Figure 17 that

Equation 5 predicts deflection more accurately. These results are

typical for almost all the beams in Table 1.

It can be seen in Figure 15 that Equation 7 improves the

correlation of theoretical deflection with the measured deflection.

The use of Equation 7 is, therefore, recommended for more

accurate deflection calculation of FRP reinforced structures.

Alternatively, an average value ofa1 can be obtained from Figure

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Set Beam b: mm h: mm L: mm as: mm fc: MPa r: % r=rb Ef: GPa ff: MPa Bar type

1 BF6 127 305 3048 1067 32.43 1.39 3.49 26.22 696.6 GFRP

BF7 127 305 3048 1067 29.67 2.09 5.92 26.22 696.6 GFRP

BF9 127 305 3048 1067 29.67 1.81 3.66 26.22 696.9 GFRP

2 D 152 305 2750 917 51.75 1.00 0.96 44.82 591 GFRP

3 F1 200 460 3050 1295 31.00 1.53 6.03 53.60 1180 GFRP

4 VH2 152 305 2750 917 44.81 0.38 0.74 44.82 591 GFRP

H 152 305 2750 917 44.81 0.38 0.38 44.82 591 GFRP

E 152 304.8 2750 917 51.75 0.94 0.89 44.82 591 GFRP

5 RC-A1 200 300 2400 900 29.43 0.28 1.38 65.00 1413 AFRP

RC-A3 200 300 2400 900 29.43 0.21 0.27 65.00 1413 AFRP

RC-A4 200 300 2400 900 29.43 1

.71 7

.70 56

.00 1265 AFRPRC-A5 200 300 2400 900 29.43 3.03 13.60 57.00 1265 AFRP

6 Coated 100 150 800 350 43.60 0.70 2.61 63.80 1400 AFRP

7 G II 200 210 2700 1250 31.30 3.60 7.58 35.63 700 GFRP

G III 200 260 2700 1250 31.30 1.20 3.31 43.37 886 GFRP

GIV 200 300 2700 1250 40.70 1.16 2.02 35.63 700 GFRP

G V 200 250 2700 1250 41.00 2.87 6.11 35.63 700 GFRP

8 CB2B 200 300 3000 1300 52.00 0.70 1.24 37.65 773 GFRP

CB3B 200 300 3000 1300 52.00 1.05 1.86 37.65 773 GFRP

CB4B 200 300 3000 1300 45.00 1.40 2.68 37.65 773 GFRP

CB6B 200 300 3000 1300 45.00 2.10 4.00 37.65 773 GFRP

9 ISO1 200 550 3000 1000 43.00 1.13 1.54 45.00 690 GFRP

ISO3 200 550 3000 1000 43.00 0.57 0.78 45.00 690 GFRP

10 M1 150 300 2750 917 31.00 1.08 1.38 44.82 590 GFRPM2 150 300 2750 917 31.00 2.15 2.77 44.82 590 GFRP

11 F-1-GF 154 254 2100 700 35.70 1.55 2.23 34.00 586 GFRP

12 GB10 150 250 2300 767 33.70 1.36 4.33 45.00 1000 GFRP

13 B1 200 400 2300 750 25.10 0.07 0.63 52.97 1775 AFRP

A1 175 350 2300 750 29.76 0.13 0.79 52.97 1775 AFRP

14 GB5 150 250 2300 767 28.14 1.30 4.71 45.00 1000 GFRP

15 cb-st 152 292 2743 1372 48.26 0.26 1.07 147.00 2250 CFRP

16 BC2NA 130 180 1500 500 53.10 1.24 2.15 38.00 773 GFRP

BC2HA 130 180 1500 500 57.20 1.24 2.06 38.00 773 GFRP

BC4VA 130 180 1500 500 93.50 2.47 2.23 38.00 773 GFRP

BC2VA 130 180 1500 500 97.40 1.24 1.21 38.00 773 GFRP

17 F1 500 185 3400 1200 30.00 1.22 3.57 42.00 886 GFRPF2 500 185 3400 1200 30.00 0.70 1.58 42.00 886 GFRP

18 L.4 500 250 2300 700 30.00 0.47 2.24 147.00 1970 CFRP

L.2 500 250 2300 700 30.00 0.20 0.95 147.00 1970 CFRP

I.4 500 250 2300 700 30.00 0.38 0.78 42.00 692 GFRP

LL-200-C 1000 200 3000 700 30.00 0.30 0.87 147.00 1970 CFRP

19 CB-4 200 300 2750 875 39.90 0.52 2.24 122.00 1988 CFRP

CB-6 200 300 2750 875 44.80 0.78 3.61 122.00 1988 CFRP

CB-8 200 300 2750 875 44.80 1.04 4.84 122.00 1988 CFRP

20 GB1 180 300 2800 1200 35.00 0.53 0.92 40.00 695 GFRP

GB2 180 300 2800 1200 35.00 0.79 1.44 40.00 695 GFRP

GB3 180 300 2800 1200 35.00 1.05 5.74 40.00 695 GFRP

21 B1 180 300 2000 850 71.70 0.49 1.87 147.00 2550 CFRP

B2 180 300 2000 850 71.70 0.32 1.24 147.00 2550 CFRP

B3 180 300 2000 850 71.70 0.49 0.87 147.00 2550 CFRP

B4 180 300 2000 850 71.70 0.49 0.87 147.00 2550 CFRP

( continued)

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15.For this average value r/rb is taken in the range 1.202.70 as

suggested by Yost et al. (2003). As can be seen in Figure15 the

average value of a1 in this range of r/rb comes out to be 0.88,

which can be used as a simplification to Equation 7. It is

important to note here that Al-Sunna (2006) suggested a bond

factor of 0.5 and 10% reduction in the cracked stiffness in order

to calculate deflection of GFRP RC beams with Equation 1.

Owing to variations in FRP bar properties this method will

require different deflection calculation methods depending on bar

type used. In fact with FRP bar types differing from that used by

Set Beam b: mm h: mm L: mm as: mm fc: MPa r: % r=rb Ef: GPa ff: MPa Bar type

22 1a-NL 254 184 2896 1372 40.37 0.71 1.27 40.30 690 GFRP

2b-NL 305 184 2896 1372 40.37 0.94 1.67 40.30 690 GFRP

3a-NS 254 286 2134 991 36.36 2.05 3.89 40.30 690 GFRP

3a-HS 165 286 2134 991 79.70 2.10 2.20 40.30 690 GFRP

3a-HL 152 184 2896 1372 79.56 1.88 2.00 40.30 690 GFRP

4b-NL 203 184 2896 1372 40.37 1.41 2.51 40.30 690 GFRP

4b-HL 178 184 2896 1372 79.56 2.32 2.43 40.30 690 GFRP

4a-NS 229 286 2134 991 36.36 2.28 4.32 40.30 690 GFRP

23 F-29f 102 102 1016 339 46.54 0.81 4.18 144.80 2490 CFRP

F-29g 102 102 1016 339 46.54 0.81 4.18 144.80 2490 CFRP

24 B2 150 200 2700 850 45.70 0

.34 2

.16 49

.00 1674 AFRP25 F-3 102 102 1016 432 46.54 1.21 6.37 144.80 2490 CFRP

F-6 102 102 1016 432 46.54 2.41 9.68 144.80 2490 CFRP

26 DF2T1 150 300 2400 800 84.50 0.40 2.63 53.00 1760 AFRP

DF3T2 150 300 2400 800 84.50 0.59 2.66 53.00 1760 AFRP

DF3T3 150 300 2400 800 84.50 0.59 2.33 53.00 1760 AFRP

CF3T1 150 300 2400 800 85.60 0.59 3.21 53.00 1760 AFRP

DF4T1 150 300 2400 800 84.50 0.85 3.36 53.00 1760 AFRP

27 SG2a 500 120 2100 750 32.96 0.79 1.06 42.75 665 GFRP

BG2a 150 250 2300 767 38.61 0.77 0.92 41.60 620 GFRP

BC2a 150 250 2300 766 50.30 0.65 1.13 131.80 1325 CFRP

BG3b 150 250 2300 767 34.20 3.93 5.42 41.95 670 GFRP

28 BRC1 120 200 1750 675 42.55 0.70 1.94 135.90 1676 CFRP

BRC2 120 200 1750 675 41.71 0.70 1.99 135.90 1676 CFRP

Table 3.Material and geometrical properties of beams analysed

Set Researcher Set Researcher

1 Nawy and Neuwerth (1977) 2 Faza and GangaRao (1990)

3 Saadatmanesh and Ehsani (1991) 4 Faza and GangaRao (1992)

5 Nakano et al. (1993) 6 Nanni (1993)

7 Al-Salloum et al. (1996) 8 Benmokrane and Masmoudi (1996)

9 Benmokrane et al. (1996) 10 Vijay and GangaRao (1996)

11 Swamy and Aburawi (1997) 12 Duranovic et al. (1997)13 Tan (1997) 14 Zhao et al. (1997)

15 Grace et al. (1998) 16 Theriault and Benmokrane (1998)

17 Pecce et al.(2000) 18 Abdalla (2002)

19 Kassem et al.(2003) 20 Toutanji and Deng (2003)

21 Wilson et al. (2003) 22 Yost et al.(2003)

23 Orozco and Maji (2004) 24 Aiello and Ombres (2005)

25 Maji and Orozco (2005) 26 Rashid et al. (2005)

27 Al-Sunna (2006) 28 Rafi et al. (2008)

Table 4.List of researchers for designated specimens

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Al-Sunna (2006) this method and/or factors (0.5 and 10%) may

not work at all with sufficient accuracy. On the other hand, the

single equation proposed in this study (Equation 5) is free from

this dependency on the bar type.

In summary, current knowledge of RC flexural design is largely

derived from the behaviour of an under-reinforced steel RC

section whose response is predominantly controlled by steelbehaviour. The behaviour of a steel RC beam to applied load is

linearly elastic up to steel yielding. Therefore, elastic deflection

theory is able to determine deflection behaviour satisfactorily. On

the other hand, FRP RC beams are designed as over-reinforced to

avoid brittle bar failure. In this case concrete behaviour in

compression is largely non-linear and controls beam behaviour.

This may require a re-examination of some of the concepts of

conventional steel RC flexure design before they are applied to

FRP RC. Most importantly a review of the linear stressstrain

relationship for concrete compressive stress is appropriate. This

may become helpful in explaining the overestimation of tension

stiffening in the present Eurocode 2 formulation. A simplistic

approach has been followed in this study to modify tension

stiffening estimation from Equation 1b, which is included in

Eurocode 2 for steel RC. This would leave the present familiar

form of the Eurocode 2 equation the same for use by academics

and practising engineers. A factor has been suggested in order

to soften the cr response of an FRP RC beam. This factor can

be calculated from Equation6. a1 in Equation6may be taken as

an average value of 0.88 or may be obtained from Equation 7.

These modifications improve deflection behaviour appropriately

compared to predictions for FRP RC beams with the existing

Eurocode 2 method.

ConclusionsThe results of a theoretical investigation of FRP RC beam

behaviour are presented in this paper. The analytical study was

403020100

20

40

60

80

100

0

Deflection: mm

Load:kN

50

Eurocode 2

075a1

080a1

084a1

090a1

Figure 14.Loaddeflection curves for beam BRC2 with differentvalues of a1

/ b

161412108642

y x00121 08581

0

04

08

12

16

0

a1

Averagea1

Figure 15.Results of regression analysis

10080604020

0

10

20

30

40

0

Exp: mm

(a)

Theo:mm

40

Eurocode 2 Equation 5

0

20

40

60

80Eurocode 2 Equation 5

0

20

40

60

80

100

120

140

0 120

Eurocode 2 Equation 5

302010

0 20 40 60

Exp: mm

(b)

Theo:mm

Exp: mm

(c)

Theo:mm

Figure 16.(a) Measured and predicted deflection of beams at

35%Pu; (b) measured and predicted deflection of beams at 50%

Pu; (c) measured and predicted deflection of beams at 90%Pu

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15/18

based on strain compatibility analysis and employed the Eurocode

2 Part 1-1 deflection model. The beams tested by the authors

(BRS and BRC beams) and various other investigators were

analysed. Non-linear FEM was also carried out for BRC beams.

The main findings of this investigation are listed below.

(a) An over-reinforced (r . rb) design is generally

recommended for FRP reinforced concrete beams. Results

show that recorded tension stiffening effects are higher in

FRP RC beams with low r/rb compared to heavily

reinforced beams. The current Eurocode 2 method of tension

stiffening estimation underestimates this parameter for FRP

beams. The degree of underestimation in the tension

stiffening is correlated with the relative amount of FRP

reinforcing. The error of underestimation decreases as the

ratio r/rb increases.

2010

40302010 100755025

2010 8642

0

10

20

30

40

50

0

Deflection: mm

(a)

Load:kN

100

2b-NL (Exp.)

Eurocode 2

Equation 5

0

50

100

150

200

0 30

RC-A4 (Exp.)

Eurocode 2

Equation 5

0

15

30

45

60

75

90

0 50

B1 (Exp.)

Eurocode 2

Equation 5

0

50

100

150

200

250

300

0 125

LL-200-C (Exp.)

Eurocode 2

Equation 5

0

50

100

150

200

250

0 30

B3 (Exp.)

Eurocode 2

Equation 5

0

4

8

12

16

0 10

F-2 g (Exp.)

Eurocode 2

Equation 5

755025

Load:kN

Deflection: mm

(b)

Deflection: mm

(c)

Load:kN

Load:kN

Deflection: mm

(d)

Deflection: mm

(e)

Load:kN

Load:kN

Deflection: mm

(f)

Figure 17.(a) Loaddeflection curves for beam 2b-NL (Yost et

al., 2003); (b) loaddeflection curves for beam RC-A4 (Nakano et

al., 1993); (c) loaddeflection curves for beam B1 (Tan, 1997);

(d) load deflection curves for slab LL-200-C (Abdalla, 2002);

(e) loaddeflection curves for B3 (Wilson et al., 2003);

(f) loaddeflection curves for F-29g (Orozco and Maji 2004)

211


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(b) In the uncracked state of a beam, deflection is calculatedusing Iuncrwith the slope of the uncracked deflection line

equal to EcIuncr. For a cracked section, deflection calculation

is based on Icrand the resulting line has a slope ofEcIcr. The

former line represents the stiffest response whereas the latter

is the representation of least stiff behaviour. Actual beam

behaviour lies somewhere in between the two owing to

concrete tension stiffening effects. Conversely, peculiar beam

behaviour was identified in FRP reinforced beams as the

recorded beam response crossed over the theoretical

deflection curve based on Icr. This additional deflection

beyondcroccurred after a critical bar modulus (Ef 50

GPa) for beams with higher relative amount of reinforcement

and was found to be proportional to barEf. It was noted that

beam deflection was not influenced by shear-induced

deformations as these decrease with an increase in

reinforcing ratio. Shear-induced deflections were separately

investigated for BRC beams, which were tested by the

authors, using various approaches including FE modelling.

These deflections and creep effects were found to be

insignificant and it was noted that the beam deflection was

based on flexural curvature.

(c) The behaviour of concrete in compression becomes important

in over-reinforced RC design. For FRP beams, the behaviour

of concrete in compression is non-linear at an early stage of

load application. This may require a re-examination of someof the fundamental concepts applied to the design of steel RC

as these are based on linear elastic material behaviour.

(d) A simplistic approach has been used in this study and a

modified expression has been suggested for the deflection

calculation of FRP reinforced structures. The factor

proposed in this study is given in Equation6 and includes

effects of ratio of modulus of FRP/steel bar andr/rb. The

relation forr/rb with Equation6 has been derived with the

help of the recorded deflection of 75 beams and slabs and is

given in Equation7. Alternatively, an average value of 0.88

can be used.

(e) A wide range of experimental data was theoretically analysedusing the original and modified expressions. The suggested

equation provided satisfactory correlation with the measured

deflection for the majority of the specimens. The maximum

coefficient of variation was found to be 21.5% with the

modified method in comparison to a value of 32.3% with the

existing equation.

It is understood that the modifications suggested in this study are

empirical and by no means is it an alternative to the proper

understanding of the beam behaviour. However unless a different

approach of designing FRP reinforced concrete beams is required

by Eurocode 2, the suggested modification can provide satisfac-

tory results.

AcknowledgementsThe authors wish to acknowledge the support provided for this

research by the School of Built Environment, University of

Ulster. The first author also acknowledges the support fromProfessor Sarosh H. Lodi, Chairman, Department of Civil

Engineering at the NED University of Engineering and Technol-

ogy, for discussing some of the pertinent issues which arose

during this study.

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