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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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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
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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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Volume 63 Issue 3
A suggested model for European code to
calculate deflection of FRP reinforced
concrete beams
Rafi and Nadjai
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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
200
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
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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
201
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
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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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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
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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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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)
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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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